Turbulent Shear Flows 8: Selected Papers from the Eighth International Symposium on Turbulent Shear Flows, Munich, Germany, September 9-11, 1991

AD- A2

43

810

EIGHTH SYMPOSIUM ON

d'

LIII,

SEPTEMBER 9-11, 1991 TECHNICAL UNIVERSITY OF MUNICH MUNICH, GERMANY

VOLUME 2 SESSIONS 19-31, POSTER SESSIONS

91-17172

FTIG!III'l SYMPOSIUM ON 1 URBULENT SHEAR FLOWS TVechnical University of

September 9-1I, 1991

Munich

__9___

AN ALGEBRAIC PROPERTY OF THE TURBULENT STRESS AND ITS POSSIBLE USE IN SUBCRID MODELING

NT l

1

3

M. Germano

J......

Politecnico di Torino - Dip. di Ing. Aeronautica e Spaziale

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C.so Duca degli Abruzzi 24,10129 Torino,ltaly

A new approach to the study of turbulence,based on the general algebraic properties of the filtered representationsof a turbulent field at different filtering levelsis presented.New quantities, the gene-

must actively interact with the resolved scale.In this paper some possible ways to test the resolved scale in order to produce an interactive procedure that dynamically adapt the model to the calculations are presented.They are based on a new algebraic approach to the study of turbulence that in some sense

ralized central moments and the generalized turbu-

extends to a general filtering or averaging or split-

lent stresses are introduced,and an algebraic identity that relates these quantities at different levels is deduced.This new approach extends the statistical approach to a general implicit or explicit filtering operator and it represents in the opinion of the author a simple way to study the similarity of the turbulence at different levels. A particular application of this approach to the subgrid scale (SGS) modelling in the large eddy simulation (LES) of turbulent flows,based on the definition and the utilization of the resolved turbulence,is discussed.

ting operator the statistical approach.The need for a new theoretical approach intermediate between the statistical approach and the coherent structures approach seems largely requested at the present moment in the study of turbulence. New splitting procedures of the turbulent flow different from the Reynolds decomposition are actually proposed and in use not only in the LES computation of turbulent flows but also in some multi-level techniques and in alot of experimental procedures where the turbulent field is analyzed in terms of non statistical quantities. The main problem of such decompositions is that the usual algebraic properties of the statistical averages are no longer valid.In particular the mean value of the fluctuations is generally different from zero,so that its fundamental ro!e in the Reynolds decomposition is largely reduced when we consider a general filtering operator. As a consequence the formal analysis of the turbulence in terms of the fluctuations is not so simple as in the case of the statistical averageand a simple extension of the statistical approach can be founded on different quantities defined without recourse to the fluctations.ln the following such new quantities,the generalized central moments, and in particular the generalized central moments of the second order of the components of the velocity field, the generalized turbulent stresses,are defined.They do not involve explicitly the fluctuations,they reduce to the usual statistical expressions in the case of a statistical operator and they give a simple and unified description of the algebraic structure of the filtered turbulent field and the filtered turf ulent equations.This new approach is in particular well fitted to examine the similarity among the turbulence at different levels.Peculiar to

ABSTRACT

INTRODUCTION The idea that stands at the basis of the dynamic SGS models recently tested with success in the LES of the transitional and the turbulent channel flows, Germano et al. (1990),and in the LES of compressible flows and transport of a scalar, Moin et al. (1991),is very simple. In the LES of turbulent flows the large scale of the turbulent field can be captured,and a large amount of infor'nation is collected It is spontaneous to try to utilize th;s information in order to adapt the SGS model dynamically and in an interactive way to the real calculated turbulent field. Loosely speaking,if we interpret the SGS model mainly as a way of introducing an appropriate turbulent viscosity, we could guess on testing procedures that in some way or another check the real amount required by that particular turbulent flow.Such intelligent model,that in particular should recognize where the flow is laminar and should account for an eventual backscatter from small scales to larger scales,cannot have,in the opinion of the author,a universalform,but in some way or another

19-1-1A ' I= l

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a decomposition in terms of mean values and fluctuations is the vision of a turbulent field that is the sum of different contributions. Peculiar to this new approach is the idea of the comparison of different representations,or numerical realizations,at different filtering lkvels.ln particular it will be shown that it is quite spontaneous in a numerical realization to analyze the resolved turbulence in order to extract informations useful for an interactive modelling.Fundamental in order to do that is an algebraic identity that relates the generalized turbulent stresses at different levels.In the following all these points are exposed in detail. FILTERING ALGEBRA.THE GENERALIZED TURBULENT STRESSES. AVERAGING INVARIANCE OF THE FILTERED EQUATIONS. In the LES of the turbulent flows the large scale of the turbulent field is captured and the residual small scale is modeled The scale of the splitting between small and large scales is dictated by the numerical resolution,and in the physical space it is explicitly given by the grid length.As such the LES theory is intimately associated to the numerical computations,and,as Leonard (1974) pointed out,a LES splitting procedure can be formally represented by a filter generally expressed as a convolutional integral applied to the original turbulent field.More generally we will interpret an explicit or implicit filter as a linear operator F that when applied to a turbulent quantity a produces the filtered representation < a >f,and it is of some interest to examine the related algebraic properties.Pioneering works on the filtering algebra applied to turbulence are the papers of Kampi de Firiet (1957) and Rota (1960),but their studies were mainly directed to the analysis of the Reynolds operators, provided with the well known propetties

The main problem of turbulence modeling is to express the turbulent stress TY(ui, u,) =< uiu, >f - < ui >f< uj >f

(4)

in terms of the resolved filtered quantities.In the case of a statistical Reynolds operator tne turbulent stress is simply given in terms of the fluctuations by the well known expression rf(ui,uj) =
!

(5)

but when F is a general filtering operator the expression of the turbulent stress rf(uj, uj) in terms of the fluctuations is not so simple,Deardorff (1970), Leonard (1974),Schumann (1975).New terms,called the Leonard term and the cross term appear,and the separate modeling of each of them give some problems owing to their intrinsic non-Galileian invariance,Speziale (1985).A different algebraic approach to the problem has been followed by the present authorGermano (1986),(1987) and (1990).This approach does not require any decomposition in mean values and fluctuations,but is mainly based on the introduction of new quantities,the generalized central moments,Germano (1991),that in some sense extend to the non-Reynolds operators the statistical relations between the moments and the central moments. Given the turbulent fields a,/3,y ...... the moments are expressed by < /37 >f < ao >f a and the generalized central moments rf(a,,3) rf(a,fly). are defined by the statistical relations Tf(a,/ 3 )

= f

-

f< j3 >f

'/3t >f - < a >f Tf (P, 7) rf(a, /3, Y) = < - < 3 >! fj(-, a)- < -, >f rf(a,/3) - < a >f< )3 >< 7 >f (6) ....... .

(1)

The genera.:zed central moments of the second or-

(2) >f>f=< a >f operators,and The LES operators are not Reynolds in general the mean value of the mean value is not equal to the mean value.The analysis of the algebraic properties of the non-Reynolds operators has been conducted on the same basis of the statistical approach,and in particular the well known procedure of decomposing a turbulent quantity in a mean value and a fluctuation was adopted

der related to the Cartesian velocity components u,uj

f>f=
f< i >f

f +a'

'r"(u,, uj) =< Uiuj >f - < Ui >f< Uj >f will be in particular called generalized turbulent stresses,and we notice that in the case that F is a statistical Reynolds operator they reduce to the well known expressions ry (a,) = < a'/O' >f rf(a,/3,y) = f,
f; if, CS)

T(< it: >f, < Ili >f),so that it is able to recognize 9 when the flow is laminar at the resolved level. The optimal size of the ratio lfg/l is not so critical,Moin

the functional expression of the Smagorinsky model

at the F-level and with

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>fg; IA, CS)

,1I(< it, >f"' < I,

et al.,and a good choice is 2. This model was im-

, ) u"

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the corresponding expression at the FG-level~we can

filby applying explicit plemented h N sn cut-offs roib spectal tr n twstse

tln write on the basis of the algebraic re-

database of Kim,Moin & Moser (1987) for turbulent channel flow and that of Zang,Gilbert & Kleiser

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11.81 it is continued by a power-law description of the form u + = A(z+)B (with A=8.3 and B=1/7). The velocity components tangential to a wall at the grid point next to the wall (up, vp) can be related to the corresponding wall shear stress components by integrating the velocity distribution over the height of the first

INFLOW AND WALL BOUNDARY CONDITIONS At the inflow section, we used at each time step the instantancous flow field of a LES result of the corresponding (fully developed) channel flow (see figure 1). Boundary conditions at horizontal and vertical walls were specified by as-

components (rub, rTb) and (b) the instantaneous velocity dis-

19-4-1

gli(l element, and the resulting expression can be lesolved analytically foi tile wall shear stress component, i.e.

jr~mj

u-p,


for

"' AIA -z ] 1)

(-. -)- )uP[ I " A 2pL~

(Az is the vertical width of tie gfid volume next to the w m is the uppel boundaiyof validity of the linear-law-ofthe-wall). Though theoretically not fully satisfying, fiom an apphication point of view this procedure offers the advantage that the averages < 7,b > and < Utp > are not required (in paiticular in flow situations in which these variables may be slowly varying in time), and numerical problems are avoided in evaluating the commonly used relation rub = tip < Tub > / < tip > in reattachment regions. In consequence of the experimental results fiom Ruderich & Fernholz (1986) we have abandoned the use of the logarithmic-law-of-the-wall, DISCUSSION OF RESULTS AND CONCLUSIONS The results shown in this paper are made dimensionless using a reference height L,,f = H (H=cube height) and a reference velocity Un,, = Ub (Ub=mean bulk velocity, see fig. 1). Note, that in our nomenclatme, Z is the coordinate

normal to the walls of the plate channel. LES provides insight into the time-dependent large-scale structure of a turbulent flow field. An instantaneous view of

order statistics have been collected over 160 reference times (taking one sample at every 40th time step), samples for the second-order statistics have been collected over the latest 110 reference times. One reference time is equivalent to the time a tracer particle needs to travel with bulk velocity U6 (see fig. 1)over a distance of one obstacle height H. Figure 1 shows vertical profiles of the mean U-velocity component; the profiles are quite smooth, indicating that a sufficient number of samples has been taken for the first-order statistics. From distributions of the mean U-velocity component close to the bottom plate (Z=0.01=location of first vertical grid point) and close to the top face of the cube (Z=1.01), the different mean recirculation lenghts can be approximately determined to XF=0.9, XR1=1.75, XR2=0.2 and XT=0.75 (fig. 3a,b). In a vertical cut normal to the main flow direction at a dimensionless position X=0.88 behind the cube (fig. 3d), two pairs of counter-rotating mean vortices can be observed (one pair very close to the side walls of the cube, and the other one at Y=+1.5 and Y=-1.5 from the center of the cube). In that vertical plane, the lateral shape of the mean shear layer also becomes visible (fig 3d). As an example for the second order statistics, figure 6 shows vertical profiles of the Reynolds stress - < uw > and of the tumbulent energy (in a vertical plane through the center of the cube). The sm3othness of the profiles is not quite satisfying, indicating that still more samples are desirable for the second-order statistics.

The evaluation of the mean vorticity field in figure 7 gives an interesting view of the geometry of the mean shear layer behind the cube. In addition, figure 7a shows on the right

the velocity vectors projected onto a vertical plane through the center of the cube (fig 2a) exhibits an extremly complicated flow field. The intei action of diffei eat processes like the development of a three-dimensional shear layer, the reattachment of flow on the bottom plate behind the flow obstacle, the recirculatmon of highly turbilent flow and its reentrainment into the free shear layer takes place within a spatial regime which is significantly smaller compared to the case of flow over a square-rib, see Werner & Wengle (1988, 1989): the mean recirculation length is about 2.0 for the flow over a cube, and about 7.0 for the square-rib. Strong horizontal fluctuations in the lateral direction close to the walls can be observed in the small recirculation iegimes in front and on

hand side the mean vorticity vectors projected onto vertical planes normal to the main flow direction (the length of a vorticity vector is proportional to the strength of the local mean vorticity) On the left hand side, isolines of the mean total enstrophy are shown, indicating the same geometry of the 3D shear layer as exhibited by the arrows of the velocity vectors in fig. 3d An interesting perspective view of the shape of the 3D mean recirculation zone is given in fig. 7b by 2 an isosurface of the mean total enstrophy < w 2+wy2+W. > (for aii arbitrarily chosen value of 20.0). Finally, we evaluated the so-called helicity 6. 0 of the flow field: e.g. fig. 8a shows a perspective view of a mean helicity iso-suiface for a value of 0.25. If the instantaneous

top of the cube and in the large iecirculation iegime behind the flow obstacle (fig. 4) The creation of strong hoiizontal

ielicity density is normiahzed as cosO = . 0/1611017 it can be interpreted as tme cosine of the angle between velocity vector

fluctuations by splashing down o tongues of fluid material

and vorticity vector. For example, fig. 8b gives a perspective

can be seen in figure 5.

view of the instantaneous locations of cosO = 0.90(9 = 250).

The mean sti ucture of time flow field (e g. mean velocity, mean vorticity, second order statistucs, mean enstrophy, mean helicity) can be provided by time-averaging only. For time results shown here, we started -vcraging for the firstorder statistics after 60 -eference times, samples for the first-

Regions of small helicity values are equivalent to regions of large values of V x t (the non-linear term of the Navier-Stokes equation in rotation form), in regions of small angles between it is reasonable to expect that the local structure will 6 and LU be helical '(an ideal case would be cosO = 1.0, i.e. 6 and c are aligned).

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REFERENCES From the results presented hewe it can bc concluded that BAETKE, F., WERNER, H. & WENGLE, H.,. 1987 the case of turbulent flow over a cube in a plate channel xcComputation of turbulent flow around a cube on a vecpresents a flow prol~lein which is very well suited for testing tor computer, Proc.6th Symposium on Turbulent Shear and valhdating nnneiical sinnlation techniques and turbuFlows, Sept. 7-9, Toulouse, France. lence models. Coriesponding experimental data will be avaiKOBAYASHI,T., KANO,M. & ISHIHARA,T. 1985 lable in the near fulture from Larousse, Martinnzzi & TroPrediction of turbulent flow in two-dimensional channel pea (1991). For applying the solution concept "large-eddywith turbulence p~romoters, Bull. JSME,Vol. 28, No. simulation" about 300 CPU hours on a CRAY/Y-MP must b~espent to provide sufficiently stable second-order statistics24,94-5. For every mean variable to be evaluated a three-dimensionali4,24-93 LAROUSSE, A., MARITINUZZI, Rt.& TROPEA, C. 1991 array must be assigned in the computer programnn, and the Flow around surface-mounted three-dimensional obstacles, number of grid points used for the results presented here Proc 8th Symposium on Tuhrbulent Shear Flows, Sept. should be considcred to he a minimum to reach satisfying 9-11, Technical University of Munich, Munich. agreement with experimental data for the second-order staMURAKAMI, S., MOCHIDA, A. & HIBI, K. 1987 tistics and to provide sufficiently accurate results needed for Three-dimensional numerical simulation of air flow arothe validation of statistical turbulence models involved in solund a cubic model by means of large eddy simulation, vig the Reynolds-averaged Navir-Stokcs equations. J.Wind Eag. and Ind Aerodynamics, 25, 291-305. RUDERICH, R. & FERNHOLZ, H.H. 1986 An experimental investigation of a turbulent shear flow

with separation,

reverse flow, and reattachment,

J.Fhuid Mech. 163, 283-322.

Fo arab e ey eaetobee al atd a hredi e si na a m r u r at yI~ bmsi

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ACKNOWLEDGEMENT nu b e of gri p o nt use fo

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,

a n d imllg h el

pr s n e

h er

This work is currently supported by the German Research Society, Priority Research Program, Project No. (Rmer/Wengle). We also gratefully acknowledge o R 497/5 the support by the computing center of the Universitt der Bundeswehr Mia chen, by the Leibniz Computing Center of the Bavarian Academy of Sciences, and by Convex Computer GmbH, oS Bayern. sfit

•

SCHUMANN, U., 1975 AR Subgrid US Escale difference simulations A, model A TI forUZfinite I R9&-4-AC 1 9of turbulent flows in plane channels and annuli, 376-404. J.Comp.Phys.18, m lm r o un s u rf c e - m u nt e t r e d m ni o lnall l ob s t a c WERNER, H. & WENGLE, H. 1988, 1989 P

8t

Sy

p

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on

Tu b

l

n

Sh a

Fl

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Large-eddy simulation of turbulent flow over a square rib in a channel, Proc.2nd European Turbulence Conference, August 30 September 2,H1988, Berlin, in: Advances in Turbulence 2, Springer., Proc.7th Symposium on Turbulent Shear Flows, August 21-23, 1989, Stanford University, USA.

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TURBULENT S11EAR FLOWS

19"5

Technical University of Munich September 9-11, 1991

NUMERICAL SIMULATION OF TURBULENT CONVECTIVE SHEAR FLOWS OVER WAVY TERRAIN A. Dornbrack, K. Krettenauer and U. Schumann DLR, Institut fOr Physik der Atmosphdire W-8031 Oberpfaffenhofen, Germany ABSTRACT

Thermal convection of a Boussinesq fluid in a layer confined between two infinite horizontal walls is investigated by large eddy simulation (LES) for zero and finite horizontal mean motion. The lower surface height varies sinusoidally in one direction in order to investigate the effect of ccmplex terrain on the turbulent motion. Several cases are considered with amplitude 6 up to 0.15H and wavelength ,Abetween 1H and 8H In the windless cases, the gross features of the flow statistics such as profiles of turbulence variances and fluxes are found to be not very sensitive to the variations of wavelength, amplitude, domain-size and resolution whereas details of the flow structure are changed considerably. The mean flow causes a systematic change of the temporal development of the flow structure. INTRODUCTION Much is known about thermal convection over homogeneous horizontal surfaces. Land surfaces are, however, rarely homogeneous. They are often undulated and form hilly terrain. Even when the amplitude of such hilly terrain stays below the mean height of the atmospheric boundary layer, one might expect that topography has appreciable effect on the flow structure, From field observations on the structure of the atmospheric convective boundary layer, vatious authors (see for a review Krettenauer and Schumann, 1991) found that "gently rolling terrain" has generally little effect with respect to the intensity of turbulence but changes the flow structure as measured by the characteristic length-scales of the motions. E.g. Kaimal et al. (1982) found a 30 % increase in the length-scale of the spectrum of vertical velocity and a 60 % reduction of the length-scale of temperature. Most previous LES considered flows over plane surfaces (Schmidt and Schumann, 1989, Nieuwstadt, 1990). Krettenauer and Schumann (1989)

investigated the thermal convection for various finite Rayleigh numbers and zero mean flow oe iuodlwvsb waves by means en offdrc iu over sinusoidal direct simulation. In this paper, we extend the previous study to LES for infinite Rayleigh number including the case of nonzero mean wind speed. The effect of topography, turbulent convection and mean wind on the turbulence structure is investigated for a Boussinesq fluid confined in a layer between two infinite horizontal walls.

METHOD AND MODEL PARAMETERS The lower surface height h varies sinusoidally in the streamwise direction with an amplitude (5up to 0.15 H and a wavelength A of 1 OH to 40H (inclination up to 430) in different cases, where H Is the mean fluid layer height. We implemented a subgrid-scale (SGS) model which is similar to that used by Schmidt and Schumann (1889) into a numerical scheme which uses terrain following coordinates according to t(z,h) = H(z - H)/(H - h). The numerical method, which employs a second-order finite-difference scheme, follows proposals of Clark (1977). Constant heat flux (temperature flux Q, = W'9',) is prescribed at the lower surface of the fluid layer. The momentum fluxes at the lower rough surface (z/H= I 10 4) are computed using the Monin-Obuchow relationships. At the flat top an adiabatic frictionless boundary condition is applied which approximates a strong capping inversion of an atmospheric convective boundary layer. For cases UA40 we use the BKD boundary condition which reduces the reflection of gravity waves as described by Schmidt and Schumann (1989). In both horizontal directions, the model domain extends over the same length (from 4 to 8 H) with periodic lateral boundaty conditions. The horizontal mean velocity In the x-direction, U, Is prescribed and varies between 113 0 and 4 w., where w, = (figQz,) (z, = H for the runs with U = 0) is the convective velocity scale (fl = volumetric thermal expansion coefficient of the fluid, g = gravity); t. H/w is the convective time-scale.

19-5-1

(A) 6/H=0, U =0; (B) 61H = 0, U = 5 ms;

RESULTS

a) Convection for zero mean horizontal flow speed For flat surfaces the flow statistics have been avuemens re thery for veryflwitis well with measurements to scompare foundfato

(C)6H = 0.05, U= 0 mls. (D)61H = 0.05, U= 5 m/s. all simulations we assumed that the initial For FoalsiutonwesumdhtteIiil an uniwith we Is stably stratified temperature add 3 K/km. Further, a®/a3z form gradientfield

taken by Adrian et al. (1986) In a comparable laboratory study. Examples of such ccmparisons are shown in Fig. 1.

random perurbations rantto initiate turbulent motions:

For convection over undulating terrain, we have identified some rather surprising effects of.the unidirectional terrain-variation on three-dimensional motions. In particular we found that a two-dimensional model would be misleading since the results demonstrate that three-dimensional motions are enforced by terrain. So, we can observe circulations with axis parallel as well as perpendicular to the wave crests, which cannot be predicted with two-dimensional models. On the other hand, undulating terrain has rather small effects on the mean turbulence profiles as long as the wavelength Ais less than 4 H and the wave amplitude 6 less than 0.15 H.

® = 300K + --

In Fig. 2 we see convection patterns which are composed of large-scale components at scales of the order H plus small-scale random turbulent motions. The flow structure is little affected by the sinusoidal waves for A/H_< 2 and 5/H < 0.15. But, they persist over longer time intervalls (at least 5 convective time scales t.) in the presence of surface waves which fix the motion patterns (Fig. 3) Our results confirm field observations with respect to changes in the horizontal length scales (Kaimal et al., 1982). A complete documentation was published by Krettenauer (1991) b) Convection for positive mean horizontal flow speed As a next step in our investigations )wards realistic atmospheric cases, we began to simulate the flow over sinusoidal waves with prescribed mean horizontal flow velocity U. The aim of this part of our study is to investigate tle combined influence of the undulating surface, the convective motion and the mean advection of wind on the turbulence structure. We chose a special configuration which is relevant for applications in atmospheric physics. The domain size is typical for the meso-scale of atmospheric motions. The wavelength Is A/H = I and the prescribed temperature flux at the surface is Q,= 0.1 Km/s. We compare four cases:

19-5-2

z + T. rant

Depending on the height of the Inversion zi the ratio U/v' is about 3 to 4 for runs (B) and (D). Fig. 4a shows the temporal development of () (averaged over planes t = const.). These profiles look simlliar for all runs and mark typical mean profiles of a convective boundary layer (superadiabatic lapse rate in the surface layer, a well mixed layer with (0 / oz O,the entrainment zone, and above it the stable atmosphere). Fig. 4b depicts the temporal evolution of the volume averaged kinetic energy of turbulent motion (TKE) which first decreases to a minimum and then achieves an approximately steady state for times greater 100 min. This behaviour Is caused by the development of the transfer spectrum in wave number space. The following interval up to t = 200nin was chosen to average the vertical profiles of momentum flux Uw and heat flux w,9 which are displayed in Fig. 5. With regard to u'w', considerably differences between the runs can be seen whereas th' heat fluxes differ only little. T1"ie differences are caused by the effect of mean advection (compare (A) and (C) with (B) and (D) ) as well as the effect of orography (compare (A) with (C) and (B) with (D) ). As expected, the absolute values are larger for runs with U 0 0 because u'w Is proportional to the gradients of mean velocity. For (/H = 0 (case B) 1'w' I is nearly constant in the surface layer and decreases monotonously in the mixed layer. Above the Inversion height z, the momentum flux has a small negative value which is supposed to originate from the wave transport and can be denoted by wave drag. Consistently, I u'w' I has a larger nearly'conslant value for run (D) In this layer. Also the absolute value in the mixed layer shows the influence of the little mountain: a vertically more extended domain with a larger value of I jw I. The mean flow causes a systematic change of the flow structure (Fig. 6). Obviously, as can be

seen from contour plots of 0, the upward motion causes upwards propagating perturbations in the stable layer. These oscillations are of an irregular nature for U = 0 whereas for U:# 0 internal gravity waves grow in the stably stratified troposphe-9. In contrast to the horizontally homogeneous case, we observe for U = 0 in the layer up to z = zi that the thermals are fixed near the crest of the hill for all times (Krettenauer ,(1gg1). The situation in the well mixed layer is totally different for runs with mean wind. Due to the strong shear the updraughts are elongated and inclined. As argued by Mahrt (1991), these updraughts become asymmetric with sharp horizontal convergence and sharp horizontal changes of velocity and temperature at the upstream edge of the thermals. These so called microfront zones are generated by horizontal convergence between faster horizontal motion of descending air and slower horizontal motion of the updraughts Orography Included, we observe over the whole integration period upward flow at the upwind side and downward flow at the lee-side of the crest. Therefore, we conclude that the surface undulation triggers these flow patterns. Also the gravity waves above them are caused by the joint action of surface undulations and gradient of mean wind. The flow streamlines, as indicated by the isolines of E), oscillate at a comparable amplitude and have a wavelength of about A/2. Fig. Ta shows the contour plots of We vertical velocity field w at t =50 m. The runs A and C show randomly distributed thermal elements. These elements are deformed by the mean wind in the horizontally homogeneous case B. Due to this deformation in the streamwise direction, the convective cells grow together and form rolls with axis in the x-dire -ton. The vertical extension of these rolls is closely connected to the mean shear. When orographj is present, these flow patterns are overlayed by rolls with axis parallel to the crest and wave fronts. FIg. 7b doesn't show any systematic difference in the vertical velocity field between the runs. But, looking at the temporal development of these rolls we observe more or less stationary thermals for U = 0, which change their position only little. For U 0 (run B) the rolls walk through the computational domain and in layers with stiong mean sheat we also have rolls with axis in the streamwise direction. In contrast to run B, these structure are absent for run D and the thermal is fixed near the crest of the hill over

whole integration time. The mean statistics of turbulence variances in terms of horizontally averaged vertical profiles also show surprisingly little systematic variation between all cases. The rather good agreement of results from cases with U = 0 and U # 0 shows that finite difference errors from mean horizontal advection are small. However, for larger horizontal wind speed, we still have to make sure that the flow structure is only little affected by numerical approximation errors. As a next step, we plan a systematic study for various flow and surface parameters as well as a theoretical investigation of the complicated problem.

CONCLUSIONS This study presents results from two different numerical experiments by means of large eddy simulation. In the first part, we studied the effect of topography on the turbulence structure for thermal convection of a Boussinesq fluid without mean wind. In order to check the new subgridscale model we compared the LES results with laboratory measurements. Both results agree very well. The gross features of the flow statistics such as profiles of turbulence variances and fluxes are found to be not very sensitive to the variations of wavelength, amplitude, domainsize and resolution whereas details of the flow structure are chanced considerably. The orography has the largest effect for A= 4H. In the second part, first results have been presented of studies of combined action of wavy surface, thermal convection and mean advection. The mean flow causes a systematic change of the flow structure. Due to mean shear, roll pattern with axis along aE well as perpendicular to the streamwise direction develop. These were absent In the windless cases. REFERENCES Adrian, R. J., Ferreira, R. T. D. S., Boberg, T., 1986: Turbulent thermal convection in wide horizontal fluid layers. Experiments in Fluids 4 121-141. Clark, T., 1977: A small-scale dynamical model using a terrain-following coordinate transformation. J. Comput. Phys., 24, 186-215. Kaimal, J. C., Eversole, R. A., Lenschow, D. H., Stankow, B. B., Kahn, P. H., Businger, J. A., 1' 2: Spectral characteristics of the convecti.,e boundary layer over uneven terrain. J. Atmos. Sci. 39, 1098-1114. Krettenauer, K , 1991: Numerische Simulation Ober gewellten Konvektlon turbulenter Flachen. Dissertation, TU Munich.

19-5-3

Nieuwstadt, F. T. M., 1990: Direct and large-eddy simulation of free convection. Proc. 9th Intern. Heat Transfer Conference, Jerusalem, 19-24 August 1990, Vol. I, pp. 37-47. Coherent Schmidt, H., Schumann, U., 1989: structure of the convective boundary layer

Direct Krettenauer, K., Schumann, U., 1989: numerical simulation of thermal convection over a wavy surface. Meteorol. Atmos. Phys. 41, 165-179. Krettenauer, K., Schumann, U., 1991: Numericai simulation of turbulent convection over a wavy terrain. J. Fluid Mech., to appear. the sheared Mahrt, L., 1991: Eddy asymmetry iri heated boundary layer. J. Atmos. Sc. 48, 472-492.

1.0

deduced from large-eddy simulation. J. Fluid Mech., 200, 511-562.

° 0

Z

00 0

0

a)

b)

B o

00 0

00

00%

00 S0 00

0000\

0

0

00

0

o

0

00

0

0

00

00

00 0

0

00

0

0

0

d

0o0)

00

00

00 00

0.5-

0 .0°

0

0co C

!0

0 0

0 0

0.5

0

1.0 0

0t%

0

0

0

0

0

ga0

1.0 0

0.5

w'T'/w.T.

T'/T*

U'/W,,

W'/W.

.

1

5

1

000

Fig. I Root-mean-square fluctuations of a) vertical velocity variance w', b) horizontal velocity variance u', c) temperature variance T', versus height for the LES-case with flat surface. Full curves in a), b) and d) enclose the sum of resolved and SGS contributions, the curve c) depicts the resolved part only. The circles represent the experimental results of Adrian et al. (1986).

0

x/I

4

0

x/H

4

Fig. 2 Influence of wavelength of orography on velocity field in the LES in a vertical plane for various averages and wavelengths. Left: A = H. Right: A = 4 H. Top: Instantaneous result at t - 35 t.. Middle: Result averaged from t/t. = 30 to 35. Bottom: Result averaged over the same time interval and in addition over the y-coordinate. Maximum normalized velocity vectors, from top to bottom, left: 2.20, 1.50, 1.03; right: 2.59, 1.63, 1.34.

19-5-4

Y

4A

H

0 4

Yi

0

x/H

4

0

4

x/H

Fig. 3 Velocity field in a horizontal plane for various wavelength (time averaged from t/t. 30 to 35. Top-left: flat surface; top-right: 2 = H; bottom-left: ), = 2 H.; bottom-right: A= 4 H.. Maximum normalized velocity vectors: 1.69, 1.79, 1.93, 1.91, respectively.

1000

2000 . S-

300, 8t64000

-

'

f

1

6-00

N'Src

A

8'

500

300

301

302 0

304

305

0

308

100

50

/K

200

150

I /mi

Fig. 4 a) Temporal evolution of the profiles of mean potential temperature (averaged over planes q = const.) for run (C). b) Spatially averaged kinetic energy of turbulent motion versus integration time t for different runs.

--

500

-

1500

Ej I

00

I- 1.000

500

-01

5001)

01

03

05

07

J.4

09

-0 12

W'E'

k .... .0.

-009

-006

. .

-003

i

000

00,

006

M'l,,' s

Fig. 5 Turbulent temperature and momentum flux versus height q for runs (A) to (D). All curves are averaged over planes t/=const. and the time period 100min and < V 2> which is grossly overpredicted by an isotropic eddy-viscosity model and requires for a realistic simulation a Reynolds-stress-equation model (Taulbee and Tran, 1988). (ii)

Attached and Detached Shear Layers on Side Walls

In the vicinity of the side walls of a square cylinder, a complex flow field develops with partly attached and partly detached shear layers, possible reattachment on the side wall, as well as laminarturbulent transition. Here, inertial forces, viscous and turbulent stresses interplay, and the region is of considerable importance for the development of the vortex-shedding flow as here the origin of the vortex instability is located. The details of the flow mechanisms in this region are not well understood, and the phenomena are difficult to describe by a turbulence model, Because of the influence of viscous forces, wall functions are unlikely to be very suitable. (iii) Wake Region In the wake past the cylinder, convection and turbulent stresses dominate and the viscous forces are negligible. The vortices generated in the region discussed under (ii) are convected downstream forming a vortex street. The periodic vortex motion is damped by turbulent stresses. The modelling of turbulence in this region mainly affects the vortex frequency but not so much the amplitude of the forces acting on the body. History and transport effects are particularly important in this region so that an eddy-viscosity model is unlikely to do justice to this region (see Franke et al., 1989). Two different models for -imulating the turbulent stresses in the bulk of the flow away from walls were tested and are briefly introduced in the following. The two approaches used for treating the near-wall region are described thereafter. The actual turbulence-model equations are not given since only models have been used which are fairly standard by now and have been described elsewhere (e.g. Rodi, 1980). For all equations, the ensemble-average form has been employed,

k-E mod. The standard form of the k-c model was applied to many steady flows and had some success also for calculating separated flows, even though the accuracy of the predictions was not as high as for shear-layer flows. Here, the standard k-e model is simply adopted for ensemble-averaged quantities. The Reynolds stresses are calculated flor the eddy-viscosity relation ( < u;> . < uj > 2< Sk>, (4)

The eddy viscosity is related to the kinetic energy and its dissipation rate and the spatial and temporal distribution of these quantities is determined from model transport equations.The standard values were adopted for the empirical constants (see Rodi, 1980). Reynolds-Stress Equation (RSE) Model. In order to account better for the important transport and history effects in vortexshedding flows, a second-order closure model was tested which does not employ the eddy-viscosity hypothesis but determines the individual turbulent stresses from transport equations. The standard model of Launder, Reece and Rodi (1975) was adopted, with wall corrections to the pressure-strain terms due to Gibson and Launder (1978).The wall corrections effect damping of the normal fluctuations near the wall and enhance the fluctuations parallel to the wall. Again, standard values were taken for the empirical constants (see Franke, 1991). Near-Wall Treatment. For both turbulence models described above, two approaches were tested for handling the near-wall region. Wall functions arm still used in most practical calculations and were therefore tested also here. In this approach, the viscous sublayer is not resolved but the first grid point in a numerical calculation is located outside this layer. Based on the assumption of the logarithmic velocity distribution and of local equilibrium of turbulence (production = dissipation) the flow quantities at this grid point with1/wall distance yc are related to the friction velocity Ur = ('twalt/P) by t4 k '=-n(y+B) X (5) VA Here, y+ = UTy/v is a dimensionless wall distance. In connection with2 the RSE model, the shear stress at point c is set equal to U%, and invoking local equilibrium conditions on the stress equations yields the normal stresses at the first grid point. The assumptions of a logarithmic velocity distribution and of local equilibrium of turbulence are violated in separated flows (see e.g. Rodi, 1991), especially near separation and reattachment regions. One way around this problem would be to use lowReynolds-number versions of the models employed away from the wall. For the k-c model, a variety of such versions is available, but they have virtually all been tested only for attached boundary layers. For RSE models, low-Re versions are still under development and not yet ready for application to a complex flow. In both cases, such model versions require very fine numerical resolution near the wall which has an adverse effect on the computational effort required. For these reasons, a two-layer approach was adopted here in which the viscosity-affected nearwall region is resolved by a simpler one-equation model, which does not solve a dissipation-rate equation but uses a prescribed length-scale distribution and hence requires less numerical resolution near the wall. The one-equation model is then matched to the model used away from the wall. In calculations with both the k-c model and the RSE model away from the wall, the oneequation model due to Norris-Reynolds (1975) was adopted near the wall. The k-e-based two-layer model was already tested quite extensively for steady flows, including separated flows (see Rodi, 1991). The Norris-Reynolds model employs the eddyviscosity hypothesis and calculates the eddy viscosity from: < v' >= fc. vl k > ,

= I - ,t oi,

I,/)

(6)

where fp is a damping function similar to the van Driest function accounting for the reduction of turbulent momentum exchange very near the wall. The kinetic energy is obtained from the same -equation as used in the k-e model, while the length scale L is assumed to vary linearly with distance from the wall as L = Kyw/cg 3/4 in the very thin near-wall layer where the model is applied (K = 0.4187, c1 = 0.09). The dissipation rate appearing as source term in the -equation is not obtained from an equation but calculated from the following algebraic relation: 2g2 V (7) I. L 1 P (>Lj When used in combination with the RSE model, only the shear stress in the viscous sublayer is calculated with the aid of an eddy viscosity relation according to (6). The normal stresses appearing in the momentum equations are determined by assuming / = constant across the viscous sublayer. The ratio taken is the one resulting from the RSE model at the matchirg

20-1-2




2), the niodel represented by Eqs (,4)-(6) can be tested for prediction of two quantities lid and of the (normalized) pressure variance -7/p'.The pressure variance as a function of M is shown ii Fig 1, where the cross-hatched area represents the spread of DNS data reproduced froni Blaisdell (1990) (plottd only for t > 3S - 1).'Fhie values of p are very close to, 2but always smaller than, the equilibrium pressure variance p2(q , M) (not shown), the subtle difference between p2 and p2 is according to (5), responsible for the sign and magnitude of Hd. Comparison of the relative quantity Hi/(, is shown in Fig. 2. Both figures support the model mechanism of relaxation to equihbrium. We note that ,othier viable model for lldin shear turbulence has been proposed by Aupoix etal (1990). This model, howe ver, is not capable of dealing with the initial condition effects, RAPID HOMOGENEOUS COMPRESSION: 3D VS ID Recently, Coleman (11990) performed direct siniulations of compressible turbulence subjected to rapid spherical (3D) and ID compression for different initial values of the r.m.s. Mach number Mi. = 0.02 and 0.44. These DNS experiments have

-

I

5

,

I

5

,

I

I

75

10

b

I

125

,

I

15

,

I

175

,

I

20

225

5

dissipation; othervise same as in Fig. 1.

'(6)

with the acoustic time scale r, = 0.2Mlr. It is noted that in these flows. tie turbulence time scale is related to ortical turbulence. I e , 7 = q2/(,

,

St F:g. 2. Evolution of the pressu.e dilataton normalized by total

The equilibrium variance p2 was inferred from theory and DNS data as +m 2

A

02

and proposed a closure for Ildbased on an heuristic argument that in the absence of forcing, the pressure fluctuations will tend to relax to an equmlibrium value p,(M,) at a rate dictated by an acoustic time scale 7 (xL/a This has led to the closure

2

0

,)

p D .-'d l

T12 a

\

to-,

We xote that td Is ieghigble (w.r.t. %) when Mi < 0.3. Physically, it means that shocklets cannot occur below a certain threshold value of Mi In the simulations of Lee et al (1991), the shocklets were found only when Af exceeded values of about 0.3 In most of the computational examples presented in this paper, (d is unimportant, the solenoidal dissipation c isdetc-ninied from a standard modeling equation (independent of Mi).

2

, . "\' ...

(3)

00, if Mt < 025

F(At)

10-

is to elucidate the physics of rapidly compressed turbulence and describe the recent modeling effort in this Lrea. Perhaps the most important finding is that due to strong interactions between fluctuating pressure, dilatation (u 2 ,), and vorticity in the 11)rapid compression, the pressure dilatation term iHd cannot be discarded even for Mt > 1. In the Coleman simulations discussed here, Aor o -_ 47 and therefore, in principle, a rapid theory could be utilized to determine the evolution of q2 and, ii'need be, p. The 3D rapid compresson ofincompressible isotropic turbulence (with Mt. 1,certain rays that are originally directed away from the receiver plane get reflected from the uppei flow region and eventuall cross the receiver plane. Although not included on the figure, some rays that cross the receiver plane get reflected from the lower flow legion and cross the receiver plane once more.

EdO

I a A, 2

On die receiver plane, where M = 0, the wave fionts are perpendicular to the rays This gives A,.(x,O) = dx ccs00 , where dx is tile "footprint" of the ray tube oii the .r-axis and 00 the angle at which it crosses the x-axis (Fig 5). Also. we note that fl(x. 0) = w=constant, since M(x, 0)=0 For a = a=coistant, the intensity on the x-axis is

Knowing the geometry of the ray paths we can also construct the wave fronts, shown in Fig. 4. The time interval between wave fronts is 0.5, with a = I and yv = 1. As expected, the wave fronts also become distorted as the Mach number increases The wave-fiont geometries for the .M 1 =2 case are particularly inteiesting as they illustrate that. even though the velocity at the source is supersonic. the signal eventually piopagates upstream of the souice Thisis due to the lay reflections from the lowei flow region, as illustrated

doo a, dx cos0o

I(X)-

I this paper, we wl be compaing intensity distfibutois create byo a source with constant and w, hence the .first fraction on the right-hand side will remain constant. For the sake of simplicity, and without losing any information 2 on the Mach-number effect, we set w ',/ai = 1 and obtain dx

(

in Fig 7.

o s0o

I(X) =

-

)

(5)

ACOUSTIC INTENSITY DISTRIBUTION The derivative dx/d0i was computed by numerical Integraion (,iA-order Runge Kutta) ot the relation dr = (U acosO)dt, ii connection with Eq (2)

Of interest now isto fiad the sound intensit icceived by point (x,0) from aii acoustic source at (0,xi). '[lie sound latensity is commonly cdefiied as = t 2 /(pu) whle p' is the acouitic pressure fluciuation and p the deiisity of the medium A fundamental ceiurg) -onseration law i 111ol01pressible geometric acoustics is thdt the produt of intensit% aiicd rri-tube arei mciains (onstant. For the compressbile khiitzev case, the (oineration laa for I, deii',ed by 1l3.(

Computations of I(x) were carried out for the conditions corresponding to the ia) constructions in Fig 3, Tlhe intensity distributions are shown in IFig 6, where it is ohserved that they become narrower as All incieases At high M,l the intensity reach(- very low values not far fiori the It is Iistructive to connect these distributois to the ray constructions of Fig 3 The low intensities are generally associated with rays t hat cross the y = 0 plane after refle :tIg fmo1 the uppe or lower flmolegions, N ;thresulting large ray-tube areas

1916) and by Hayes (1968), is of tilie form A,- constant (1) 2 a here A,, is the ray-tube area, defined by the Iitelsect lo of adjacent ra%s i th wave fronts, and Q = d/( I - AIsInO) the sound frequency heard by an observer moving with the medim. with w,,th'e frequency of the acoustic source Note that the expression iii Eq (4) cannot le evaluated at the source Rather, an a.ssumption must be made as to the %alueof the acoustic energy flux la,,/f12 conservedh aitin each ray tube

12. 1.0 0 80. 1(x) 0.60.4--

a

"-0.2 0.0-

-10

0

-

-6

-4

-2

0

2

4

6

8

10

x

IA,, y

-8

Fig 6 Acoustic intensity distributions

-

Yx

INTEGRATED INTENSITY

Fig 5 Ray-tube geometry

To define a ray tub, we consider two adjacent rays ongimating from the source with wave-norial angles 0 and 01 + dO,, as shown in Fig. 5. We now make the assumption that the source is ominidirectional, that is, the energy flux Is umiformly distributed with 01 Consequently, each ray tube contains allenergy flux FdOl, where . is a constant. The

21-5-4

The intensity distributions of Fig. 6 indicate that, as the Mach number at the source location increases, communication between the source arid the receiver plane is confined to a &inallem region. To make quantitative comparisons anong the different distributions, we examine the integrated intensity along x E, I(x)dx (6) where L can be seen as a measure of the total acoustic energy flux received ou the y = 0 plane. From Iq (5), we

have

A E =

d

f

00 dx

]

dO

their energy flux twice If the receiver is chosen to be located

(7)

o

-

Note that the integral in x is converted to an integral in 0. The relation between 00 and 01 is known through Eq.(2) sin0

thus evaluation of the integral in Eq.(7) is straight-forward once the limits for 01 are determined. To find these limits, we must establish the range of 01 for which rays actually cross the y = 0 plane. Obviously, the rays that do not cross the receiver plane do not contribute to the energy flux received Also, attention must be paid to which rays cross the receiver plane twice. U+. . O /--

U

YYl

=

C13 and tile

last integral vanishes. It is interesting to note that the result given by Eq.(10) does not depend on the specific functions that describe U(y) and a(y) but only oil the free-stream, souice, and receiver conditions Calculations of E were performed for constant speed of

ao sin0 1 al - Ui sin01

0

at the lower edge of the shear layer, then 012

sound and for the following free-stream conditions: (a) U+_ = Ui, U_. = 0 (no reflections); (b) U+. = 1 5Uf .1__ = 0.5U1 (moderate reflections); (c) U+., = 3U 1 , U-__ = 2U(strong reflections). The integrated intensity E is plotted versus MI in Fig. 8 First, we note that for MI=0, E = oo. This is not surprising as we realize that the straight rays of the incompressible, uniform-density case, create a footprint of infinite extent on the x-axis. lowever, as M becomes finite, E also b.comes finite and decicases with increasing Mi, reaching an asymptotic value at high Mi. The magnitude of E is higher when reflections occur, as is expected from Eq (10)

-'

,,r,ty

/ _ 6-

a

U_

E 4 -

b

Fi-g 7 Ray reflections from shear-layei bounda es.

The sketch of Fig 7 helps us visualize the discussion that follows First, we note that no reflections can occui ii the un;f,)rm free stream outside the layer Therefore, all reflections will occur within the edges of the shear layer, defined by the velocities U+. and Uo. We now examine the 0 ra? paths as 01 increases from -7r Let 1, be the value of 01 corresponding to the first reflection on the upper boundary (U = (-r+o, a = a+,,sin0 = -1), which produces the rightmost ray that crosses the y = 0 plane As 0i increases beyond O1, the rays cross the receiver plane and exit to the lower free stream until the first reflection on the lower a = a-, sin0 = +1) occurs at boundary (U = -U_, the value 01 = 012 For 01 > 012, all rays that are initially reflected getplaiie directed twice,upwards. until the Consequently, reflection octhey crossdownwards the receiver curs above y = 0. The last ray crossing the receiver plaie (U = 0, a y = 0,Thenlat +1) corresponds to 0 = 013. For 01 > 013, no more rays cross the receiver plane. rays reflect from 0the under which Since theareconditions of the corresponding boundaries known,, the values 1 's are readily obtained from Eq.(2). We have,

Oil

(

+- Si

=

(a+m

012

= Szn

013

=

(

a

i + U+,

(9)

( ___\ Sin 1 (74a+j)

and Eq (8) becomes=

0

oil

2

(101

cosO0

+ 20 :13 dOi

Jo,2

_

0

I 1

Fig 8 Integrated intensity (a) no ray reflections; (b) moderate ieflections; (c) strong reflections

2 2 crsus All for

DISCUSSION The intensity curves of Fig. 6 warrant some discussion. noting that sound propagation gets altered Let us start by significantly even dt low values of the Mach mimbei This is further demonstrated by the integrated intensity plots of Fig 7 Hence d.ie velocity difference need not be sonic or supersonic for the Mach number effect on communication to become dominant

If we were located at the same streamwise position a~s the source (x = 0), we would feel a gradual loss of sound intensity as the Mach number at the source location increases, but we would not be completely cut off from the source even at high Mach numbers At MI = 1, for example, the intensity loss would be about 40%. The All = I case represents

- U)

ai + U_,_ + U,

O

(I0)

cos00

T lhe last integral is rultilred biy two bvcause, for 01 < 0v < , tue rays cross the receiver pslamie twice, hence deliver

21-5-5

communication between the edges and the center of a region bounded by relative sonic velocities. It is only when MI > 2 that communication at x = 0 becomes significantly restricted, but is still non-zero As MI increases, communication is inhibited both upstrearn and downstream of the source, with the upstream communication being attenuated faster with distance than the downstream communication. This may Provide some physical insight into the cause of stability at high Mach numbers. Since communication between fluid particles is vital for the onset and development of turbulent interac-

tions (perhaps best illustrated in shear-flow computations using vortex methods), lack or hindrance of coinmunication will lead to slower responses to a given perturbation, hence iiicreased stability

laige-scale structures in turbulent mixing layers. J Fhid Mlech., 64(4)., 775-781. CIIINZEI, N , MASUMA, G., KOMURO, G., MURAKAMI, T. & KUDOU, K 1986 spreading of two-stream supersonic

The present study has been restricted to constant speed of sound, primarily to highlight the cffects of Mach number uncoupled from efiets of density varftliTe equations variation 'approach presented here, however, are general enough that they can be applied to cases with speed-of-suond variation, provided a = a(y) only

mixing layers Phss. Fluids 23, 1345-1347. GRIMM, D W. &-HURST, C.. 1979 A geonetiic acoustics taiing shear flowvs J. Acous. Am. 66(6), ducts conin 1867-1875. piopagation of soundSoc. to the study

The central question is, of coimse, how applicable are the present results to a turbulent compressible shear layei. In ieality, the flow is unsteady and the disturb;,nce wavelengths may not be short enough for geometric acoustics to be valid. However, there is reason to believe that the underlying assumptions of this study, ic. steady flow and small %Navelength, may riot be as unrealistic as one might initially think The linear stability aiialysis of Mack (198,) has shown that, at high Mach numbers. the most unstable, higher-mode waves occur at fiequencies about three times those of the first-mode waves Experiments on shear layers by Papamoschou (1989) have indicated tl'at the large-scale tubulent structure is vey inactive at high Mach iumbers: there is imperceptible structure rotation and no evidence of pairing In recent schheren visualizations of supersonic shear layers at very high Reynolds numhbes by Hall ct.al (1991). large-scale structures are not discernible but smallscale turbulence is prevalent. Thus, there is increasing, albeit prelimnauv, evidence that small ,cale turbulence may be the dominant source of instability at high Mach nunib,,r. which kould be Inline with the smiall-wavelength assnp-

Experiments in non-reacting compressibleshear layers. AIAA91-0629 HAYES, W.D 1968 Energy invariant for geometric aeoustics in a moving mediumn. Phys Fluids 11(8), pp. 16541656.

tion of thus study Furthermore, since supersonic shear layers gow so slowlN. with fluctuation levels substatially loisei than ii tie incompressible case (Sanminy & Elliot 1990), they are much less unstable than their subsonic counterparts Note that ltayes (1968) has shown that ray theory is still valid in flow with small unsteadiness. 1,is hoped that time above findings and discussion will aid inthe construction of more refined compressible turbulence models that take into account the intricate ways in which sound can propagate ii a shear flow at high Mach numbers The analysis presented here represents a first step towards that goal, anl in the future needs to be expanded beyond the restrictions of geometric acoustics CONCLUSION Geometric acoustics theory has been applied to study the effel t of Mch numiier on the sunt,, fieldreared by a toure Msie a to-desmnaeidiso ar layco It s found that the YMacli niber seerely distorts thlie acotiC rays, ireating regions where little signal is heard Sound-intensity distributions on a streamwise plane become narrower with iicreasiig Mach number, indicating a dmnimshmng influence of the source on the surrounding medium'm The study suggests that the Mach number inhibits acoustic interactions in the shear layer, which is offered here as a fundamental explanation for the stability observed at high Mach numbers REFERENCES BLO'IIIINTZEV, D. 1946 The propagatio., of sound in an inhomogeneous amid moving medim I. J Acous Soc Am. 18(2), 322-330 BRO" N. G 1,& ROSHIKO, A 197.1 On denisity effects and

21-5-6

GROPENGTESSER, 11 1970 Study of tI a stability of boundary layers in compressible fluids NASA TT-F-12, 78. HAll, J.L., DIMOTAKIS, P E., & ROSEMANN, If.1991

KIM, S C. 1990 New mixing length model for supersonic shear layers AIAA J 28(11), 1999-2000 KORNHAUSER, ELT. 1953 Ray theory in moving fluids. J AIcois Soc. Am 25(5), 945-949. KRIEGSMANN G.A. & REISS,. E L 1983 Acoustic piopagation in wall shear flows and the foination of caustics J Acous Soc, Am. 74(6) pp. 1869-1879. LANDAU, L. 1944I Stability of tangential discontinmties in compresble fluid. Dokl. 4kad Nauk. SSSR 44(4), 1944, pp 139-111 bIN, C.C. 1953 On the stability of the laminar region between two parallel streams in a gas. NACA Report TA 2887. MACK, L M 1984 Boundary-layer linear stability theory. AGARD Report R-709. IMORKOVIN, M V 1987 Transition at hypersonic speeds. ASS-CR-1783i. ICASEInterim Report I PAPAMOSCHOU, D. 1989 Structure of the compressible turbulent shear layer AIA-89-0126. PAPAMOSCHOU, D. &, ROSI-IKO, A 1988 The turbulent compressible shear layer. an experimental study. J. Fluid Mech. 197, 453-477 RAGAB, S A aid WU, J.L 1988 Instabilities in turbulent free shear layers formed by two supersonic streams. AIAA88-00,8 LORI) RAYLEIGH 1945 'Theo, 0 Sm,, Vol 2, p. 1 3 2 Dover, New York, 19,15. SAMIMY M & ELLIOTT, G.S 1990 Effects of compressibiity on the characteristics of free shear layers AIAA J 28(3), 43q-445 SANI)IIAM, N &"REYNOLIDS, W 1989 The compressible ".ning avei' inear theory and direct simulation. AIAA89-0371 ,ll IX, M & SOLIGNAC, J.L 1966 Contribution a l'etude experiu ,ntale de Ia couche de melange turbulent isobare d'uiin ecoulenent supersonique. Symposium onl Separated Flow, AIGAID Conf Proc. 4(1),.241-270 ZIIi\aN(;, M of KUiiOTA, T & DIMOTAKIS l.E. 1988 On the iiistabmlity iniviscid, compiessible shear layers A11IA88- 953,;-'l)

EIGIlITH SYIPOSIUIM ON TURBULENT SHEAR FLOWS Technical Univcrsity of Munich September 9-1I, 1991

22-1 Analysis of turbulent mixing models. J.-Y. Chen*), W. Kollmann**)

*) Combustion Research Facility, Sandia National Laboratories, Livermore, CA 945510969, Research supported by the United States Department of Energy, Office of Basic Energy Sciences, Division of Chemical Sciences. *) MAME Department, University of California at Davis, CA 95616 where the density-weighted pdf A is defined by

Abstract. The mixing model based on pairwise interaction of notional particles in scalar space is analyzed The equations for the higher normalized moments show that the model based oln uniform probability for the outcome of the mixiiig interaction does not approach a Gaussian pdf in the limit of decaying or maintaned turbulence Further analysis of the mixing model generalized to conditional pdfs shows that mixing can be regarded as jump process, where selected notional particles undergo mixing and particles not selected remain unaTected. The jump rate pdf is shown to depend on the pdf itself reflecting the availability of mixing partners Flnally, it is shown that a modification of the pdf dcscribing the amount of mixing (luring pairwise iiteraction leads to a mixing model which is governed by a Fokker-Planck type

1p(m

-

'0,1)fl(v!, ,P)

and the scalar dissipation rates e,jin the conditional expectations are defined by

&P, a'.' (9" T

5

,j

(3)

with 1, = F, = r For homogeneous flows without chemical teaction the pdf equation describes turbulent mixing at a single point 0f

-2 = uOk) (-f 'F =p f

-_ -k

()

equation. 1.0 Introduction. The calculation of turbulent flows with chemical reactions based on pdf methods requires a closure model for mixing. Two approaches have been developed to deal with this aspect of pdf rnetnodb. First, simplified mechanisms for mixing in a turbulemit environineit were proposed a trbuentenvronentwer prposd and ad their teir statistical Aaistcal properties lead to closure models (Dopazo 1975,1979, Janicka et al 1979, Pope 1982,1985) and second, mappings were proposed to express the pdf in terms of a Gausian and the Jacobian of tme nonlinear napping (ChenChen amd Kraichian 1989) Following the first approach a class of closure models based on the notion of pairwise interaction will be analyzed It will be shown that either a jump process or the approximnation of a smooth process satisfying a FokkerPlanck type equation results depending on the shape of a

Any closume model for the mixing process described by (4) should share as many properties as possible with the exact term It should preserve normalisation and mean vah'es and decrease variances and covariances. The pdf should remain nonnegative and should not spread outside the domain of allowable states The presently available models for the nixing process in the pdf equation have several shortcomings have sever devshorm iigpes in te an delo of and Chen andolsmann, 1989 (e, Inent of 18 improved mixing models r qures careful heanalysis the physics of mixing and the mathematical properties of the corresponding terms in the equation for the pdf. The case of ingle scalar 'ariable will be considered for this purpose. 2.0 Moment analysis of the pair interaction model. The class of mixing models to be considered in detail is given by

pdf describing the degree of niixing

(O]J

The pdf transport equation for the set of Ithermo- cheiiical variables can be given for high Reynolds numbers in the form (see Kollmnann, 1987) "9f

at

afO t OX

Z

at .i,

1

ZE )=

d

ri

1'

O(9 The properties of this class of models can be assessed in terms of the normalized higher moments

+ =ii.-"Q)tpi,'" ci)fm))

(9)

T.)

-(P)

1

"'

02

O=P'k ((,kI'5

c,).m)

(1)

particular skewness p3 and flatness p4 will be considered. Inm The central part of the mixing model is the kernel T. It is time probability that the interaction of a material point T = V' with a point T = 'p" produces the values I = p

22-1-1

' + i" - V and is called the transition probability T',p,).T must satisfy the requirements

and %P P

)

T ( p', p , p)=

+ v1 T ( ', v, ,'

k n-k

a

=)) and T is pdf with respect to p

Ilk

fl:f0(2)

f) o

(k

q= P=O q=0

k= (2)

(10)

(22)

(i

Inspection of the right hand side shows that no moments of order higher than n appear. An easy calculation shows that

df0(12)

)

(

12)

t',he right hand side of (22) for n < 1 is indeed zero, hence

are normalisation and mean value preserved in time. The Conditions (10)-(12) lead to a class of transition pdfs constiucted as follows

wheze ,equation "1P i(v + 4")]

(14)

and t[r"',4"]

(15)

otherwise.

0

(T,) =

_-2)((T-')

N

(13)

', ", ) )= G(H ',

2

equation for the second moment -/(,))

(23)

4r

can be solved analytically for given G(() and time scale r independent of I The solution can be inserted into the for the third moment, which can then be solved This process can be continued up to the nth level and it becomes clear that the mixing model (8) corresponds to moment equations that are linear with time dependent coefficients

The segregation parameter for a binary mixture

follows from (2) and (22)

and G is a nonnegative symmetric function defined on [-1, 1) such that dCG(() = 2 (6 1

holds If G = 1 is chosen the mixing model of Dopazo (1979) and Jamcka et al. (1979) is recovered and for G(() = 26(C) Cul's (1963) droplet interaction model is obtained The function G(C) describes the extent of mixing and is analogous to the pdf A(a) introduced by Pope (1982) for the same purpose The moment equations follow from (8) by integration =r{JJ - o'"))

l

91 -jYA =

-L( 2 -

m2)aaB

ard shows that OAB decays with a rate determined by the time scale and the second moment of the transition pdf The earlier analysis of Pope (1982), Kosaly (1986) and Kosaly and Givi (1987) showed that the normalized higher moments of even order diverge and that the skewness (n = 3) ienains constant if the time scale r is constant Equation (22) repioduces these results. It follows from (22) and (23) that Ft

(17)

=

0

(24)

and

where

r

j

J,.j-=-d'

d"f(

')fol")G((("', "))

a =

(2 - 2 M,2 + 14)(44 + 3).

(25)

hold It follows fioin m2 !< 1 and in > 0 that J14grows exponentially for constant time scale r. The time scale r is, (18)

The triple integral can be expressed in terms of the moments of the pdf f and after lengthy manipulations the following result is obtained

however, not constant in homogeneous turbulence unless it is maintained by an external force field. If r = rot"' with an exponent larger than unity, then the noirmahzed moments (*ease to grow wit, time and the liiiting foim of the pdf depends essentially on the transition pdf The measurements of Conte-Bellot and Corrsin (1971) show that the time scale

= ",-

(19)

.M k=0(2)

where the summation is carried out with increment two (even terms only). .I, is defined by k ,k(_

)(1

= E (-l"Q (Ti

3.0 Analysi q) -

(11 7 +q}(

7

-p-5)

(20)

P O q=O

and G posesses finite moments of even order, because

jkd(('G(() T

formed with the kinetic eneigy and the integral length scale has an exponent In = 1.025 in their grid generated turbulence. The normalized moments would not grow over all bounds in this flow.


77, we have p)ropose(] new presumed p.cl.f. that resembles to the p.cl.f. of r, < T, but with a slight difference. T, represents the case when the line (C-C)/m' is adljacent to tli' curve W(C). For o < C < C, we assumied that there is no presence of thle probability for C since after thle ignition, C is constuned infinitely fast. Three 2 are needed to p~aramieters depending onl . Zt diefiletil p~~f.d1,bd0 fom o or he hap M~lid dl ,b). and C.k, for the shape (2) (see Fig.5). *dl do b

--

7er

-

C'111

= T-2 =

-do)

-

--

P(C)C 1 (---

-

C)

(14)

7rZr2e

2. CURL'S CLOSURE TeCr' oe sepesdb inpr euthen Cufor t delf Is tepomesse y u atinr flow considered here. the evolution of the probahility is given by

P,(2)

Ot-

(P(C)IV( C))

dO

b M_+9110]P(C

C' U,

C

+ C')P(C

-

C')dC'

-

cwP(C)

(15)

r

(See R.L.Curl (1963), S.Pope (1982), P.A.MN-cMurtry and P.Givi (1989)) where uwis the appropriate turFig.5. The 2 new shapes of the P.D.F. used whenl bulence frequency. This equation is valid only in a 7- >r homogeneous mnedicum, with mean velocity zero. In the Eq.(15), the r~' r.h.s. terni is the reactive termi Thle governing eqjuations for liomnogeliou-s. isotiojm which rep~resents a convection termn in the lprobainopesbeturbulent reactive flows are. bility space while the 2 "d r.h.s. termi is a molecuiinoprsibelar diffusion term which destro 'ys the fluctuations (IC in the probability spce The resolution of this T ' (9) equation is clone by the method of Monte-Carlo and we have used the method employed by C.Dopazo (IC1 + 1W(0 1979). CjrrIdtN particles are randomly selected d 2c, +"C1I'mIo) nt of the N particles representing the scalarn p.d.f. whre =f D.""Cl-'. Foi the modelling of the at time t and they are mixed by curl's p~rocedlure; where. ~ C,~ being a closure constant. Each of thle parice dissipation rate of C",.c,, we have assuinet a direct present onl the flow will evolve independantly by proportionality between the dissipation time scale the 1).(l~f. transp~ort equation That is the main for the turbulence kinetic energy and thle varia nce, difference between this model and the presumned of fluctulations of a reacting specie T)w:ch p.cl.f. which is based onl transport eqluations for yields, c, = 7''= FImean variables. -also

CDr7

1.2.) CALCULATION OF TV AND Cl ,

III RESULTS

the calculation of IV with f1 (C)P(C)dC is not so simple as it appears since irwe suppose C, and P(C) ;:z0 for 0 < C < C1 we will have' an indeterulination. In fact, for C = C, ,we have W(C.)=E=L. and as C,5 z:0, we have W(C);7I/r7, that resole O]V,,t, tie 1 indetermination in C=O. For the reaction rate, we have wnC

-c

d,

-

-C Tet

=

1 dc dobr~

C-C

We have uised the same initial conditions of Direct Numerical simlation and compared the mesults The initial condlitions of D.N.S and the presmimed p.d.f. model: 2 Z70.78, Z7 =.030, rj,, .0 Thle initial conditions of D.N.S and the curl's model: Z =0.78, Z7_=0.017, TTO= 2 .0 where 7rr 0 is the initial turbulence characteristic time.

22-3-3

The curve marked A represent the Direct Numerical Simulation and B the models. The conparison with the presumed p.d.f. model is shown at the left side and the curves at the right side correspond to curl's model (see Fig 6). Except the

015 "



evolution of the U with time for the presumed p.d.f. model, I(t) and W(C) seem to fit quite well

the evolution of Direct simulation. The agreement of coalescence/dispersion model with the Direct Simulation is very good. In this case, the better concordance is obtained for a value

0

of C=2.5.0

t

013B A

8.5

C t

0



o.8

8.5

AB

<w> 0

0

t5.5

0.8 C



the curves c and d, C means a reactive scalar which evolves without any turbulence. The set of curves (a,c,e) is for the evolution computed by the P.D.F. ignition while the set (b,d,f) is the results of the Curl's model.

A

B

0 .... .. .... ....

0 Ot

0

5.5

2) If we compare qualitatively (Fig.7) the shape of the P.D.F, computed by the direct numerical simuiation and the shape of the presumed P.D.F., we .an note that the presumed P.D.F. shape approximate fairly well the one's of the numerical simulation. The time evolution of the shape of the P.D.F. computed by the direct numerical simulation shows clearly that a hollow appears in a range

22-3-4

of values of C; this justifies the correc. prediction of < C >, < C' 2 > and < W >. This new )resumed P.D.F. depends more on the chemistry (with the introduction of C, in the shape)than the precedent P.D.F. This new P.D.F. allow to better represe the effect of the chemical reaction.

8

TIME- I.3 sec

.6

a)

01

01

)

,6

n

f

0

0.11

c

......M,

"

e)

01 b)

0

l

01 1

)

ME

1.6s

0

0.1

c

e

1

f)

0

2

0

C

1

0

C

1 6

Fig.7. P.D.F. of the reactive scalar for diffeent times computed by the D.N.S. compared with the P.D.F. constructed with rectangle and Dirac. .9

__56

,7

TIME.

3.ee,,c

22-3-5

S

9

i.0

e

,

.9

On the Fig.8, we have plotted the position of the particles which are used to solve the equation of transport of the joint P.D.F. PkC, o) by the method of Monte Carlo. We can see tile width of the P.D.F. by the spreading of the clots and its value by the density of the dots. As time goes on, the mean value of the inert scalar is conserved while the mean value of the reactive scalar decreases toward zero The joint P.D.F. computed by the direct numerical simulation are very similar to the one's computed by the Curl's model.

.

... ......... A.

.

1

Damk6ler number varies. In a turbulent reacting medium, the ratio r/r, where r, is a characteristic turbulent time of the reactive scalar field, has

TIME. 5.Isg.c

...

not always the same evolution with time. This fact explains why the constant is dependant of the Damk6ler number.

.

IV CONCLUSION "6

4

.3

.2

1

.2

3

.,

5

.4

.7

.6

.9

1.6

Fig.8. Joint P.D.F. of the inert scalar 0 and the reactive scalar C computed by the Curl's models for different times.

Acknowledgements: Computing time (on a CRAY 2) for the Direct Numerical Simulation was allocated the scientific commitee of the "Centre de CalculbyVectoriel pour la Recherche".

Cud

5-

The work described here give us interesting information on the agreement of models with regard to the D.N.S, The evolution of the mean scalar values ,C'2 , WVagree well in both the two models. In the two models, appear a characteristic exchange time. The investigation we have made,in order to study the influence of the Damnkh~ler number demonstrates that the characteristic exchange time is dependent of this number since we have to change the constant when this number changes. In the comparisons presented here, the conistants are not time dependent. With the D.N.S. we found that they are not really constant with time. Rather than using the characteristic turbulent velocity time, it would be better to use a characteristic time evaluated by the characteristic turbulent scalar field since the ratio between the two times vary with time for a reactive flow. For that, it is necessary to solve a scalar dissipation transport equation in the models.

t. POF Ignition

4

REFERENCES 3-

21

* 0

Da 0

23

4

Fig.9. Evolution versus the Damk6hler number of the constant of the 2 models.

3)The Fg.9 shows the evolution of te constant Cd used in the model of the P.D.F. ignition and the constant Cc used in the Curl's model for different, Damk6ler numler.The definition of the Damk6hler number is Da = 7-/ro where rn is the initial characteristic turbulence time . This characteristic time is the same in all the case in order to have the same evolution for the turbulence. During the comparaison between the models and the direct numerical simulation, the Nalue of the constant (Cd or Cc) is kept really constant. But, to obtain fairly good comparaisons between the models and the simulation, these constants must be modified when the

Curl,R.L.1963, AIChE J. 9,175 Dopazo, C. 1979, Phys. Fluids, 22, 20 Dopazo,C.,O'Brien,E.E. 1974, A probabilistic approach to the autoignition of reaction turbulent mixtures, Acta Astronaut. 1, 1239-1255 McMurtry, P. A., Givi, P. 1989, Combust. flame, 77,171 Oran, E.S., and Boris, J.P. 1987, Numerical Simulation of Reactive Flow, Elsevier, New York Moon, H.J., Picart,, A., Borghi, R. 1989, Joint Meeting of the British and French Section of the Combustion Institute, Rouen. April 17-21, 205 Orszag, S. A. 1972, in Statistical Models and Turbulence, Lecture Notes in Physics, 12, SpringerVerlag Pope, S.B. 1982, Combustion Science Technology,, 28,131

22-3-6

lItGliTll SYIPOSIUI ON T RI3ULE,N I SI EAR FLOWS Technical University of Munich

22 -4

September 9-11, 1991 A CONTINUOUS MIXING MODEL FOR PDF SIMULATIONS AND ITS APPLICATIONS TO COMBUSTING SHEAR FLOWS A.T. Hsu*) and J.-Y. Chen**)

*) Sverdrup Technology, Inc. NASA Lewis Research Center, Cleveland, OH 44135, USA **) Combustion Research Facility Sandia National Laboratories, Livermore, CA 94551, USA

ABSTRACT The problem of time discontinuity (or jump condition) in the coalescence/dispersion (C/D) mixing model is addressed in this work. A C/D mixing model continuous in time is introduced. With the continuous mixing model, the process of chemical reaction can be fully coupled with mixing. In the case of homogeneous turbulence decay, the new model predicts a pdf very close to a Gaussian distribution, with finite higher moments also close to that of a Gaussian distribution. Results from the continuous mixing model are compared with both experimental data and numerical results from conventional C/D models. 1. INTRODUCTION Accurate prediction of turbulent reacting flows requires the solution of an evolution equation for the probability density function (pdf) of the thermo-chemical variables using Monte Carlo simulation. Sin:e the pdf equation, like most equations describing turbulent motion, is not closed, closure models have to be devised. For the pai of scalars, the terms in the pdf equation that need modeling are molecular mixing and turbulent convection. The present work deals with the modeling of molecular mixing. Most of the mixing models are based on the coalescence/dispersion (C/D) model by Curl (1963). This model is known to have deficiencies, and efforts had been made to correct these deficiencies, for example, Janicka, et a. 1979 and Pope 1982. The most recent efforts have been devoted to the problem of coupling between mixing and chemical reaction. Chen and Kollmann (1991) proposed a reaction conditioned model that allows correct prediction of combustion in the flame-sheet regime. Norris and Pope (1991) proposed a new model based on ordered pairing that aimed at the same end.' All the existing models suffer in one respect, namely, they are discontinuous in time: once a pair of particles are chosen to participate in mixing, their properties will jump abruptly regardless of the step size of the timu integration. This phenomenon, clearly non-physical, could cause difficulty in coupling the processes of mixing and chemical reaction. In the present work, a new model that is continuous in time is proposed. With this new model, the procPesm.q of molecular mixing and chemical reaction can be fully coupled. In the case of homogeneous turbulence decay of a scalar, one expects a Gaussian distribution for the pdf, and finite values for the higher moments. Pope (1982) pointed out the modified Curl model could not produce the correct pdf for this problem, and the higher even moments from that model tend to infinity; he suggested an age biased sampling process to overcome these shortcomings. The present continuous

model, as we will show, predicts a pdf distribution very close to Gaussian for homogeneous turbulence decay, and gives finite higher moments wth values close to that of a Gaussian distribution. The continuous mixing model is applied to the study of both non-reacting and reacting flows, and the results are compared with earlier calculations by Hsu (1991) as well as with experimental data. 2. MOLECULAR MIXING MODELS The evolution equation of a single point probability density function of scalar random variables 0,.., &, - representing the species mass fraction and temperature - can be written as N

A0,P + PDAP + P =

-

(A < v' N

I10,(z)

= 'Pk > P)

N

, E 2 0,,(< fk10k(x) = ikk > P) (1) ,=1j=1 where the terms represent the rate of time change, mean convection, cbemical reaction, turbulent convection, and molecula- mixing, respectively; P is the density-weighted joint pdi: P = pp/p, (2) -

c is the scalar dissipation: C,= DZ,8,,

(3)

(where D is the diffusion coefficent), and < xly > denotes the mathematical expectation of a random function x conditioned upon y. The left hand side of the above equation can be evaluated exactly and requires no modeling; the right hand side terms contain the conditional expectation of the velocity fluctuation and the conditional expectation of the scalar dissipation, which are new unknowns and require modeling. In the present work we concentrate on the modeling of the second term, namely, the conditional expectation of the scalar dissipation, referred to as molecular mixing in the following. 2.1 The Modified Curl Model The simplest and most used mixing model is the modifled Curl model, which assumes binary interaction between sample fluid particles. As described by Pope (1985), in a Monte Carlo simulation, the continuous pdf is replaced by delta functions N

P*(0; t) =

E 6(0 - On(t)), n=1

22-4-1

(4)

where each delta function represents one sample fluid particle of an ensemble of N particles. The evolution of P*entails the movement of the particles in the 0-space, or the evolution of the individual values of O.'s.

The above equation states that the change of 0, due to mixing is proportional to tht difference between ',m and 0., and inversely proportional to t- turbulence time scale r.

With the modified Curl model, the change of 0, due to molecular mixing is acbieved by the following binary interaction process: divide the flow domain into small cells, each containing N sample particles. Given a small time interval St and a turbulent time scale r, select randomly N,, pairs of particles,

2.3 The Coupling of Mixii.g and Reaction The processes of mixing and chemical reaction are essentially decoupled when one uses the discontinuous C/D models. In contrast, with the above continuous model, coupling become natural since, for a given particle, mixing and chemical reaction can be described with a single equation:

o.5 7N,

(5)=-

+(,. ddt

Nm0.5N, =

(6) (7)

where A = 0.5, with a random variable uniformly distributed on the interval [0,1]. The remaining N - 2N,,, particles remain unchanged: 04(t+6)6 =

(t)

-

A(')(t)

to

.(t) .0)

(9)

zero exist because as 6t goes to The derivative 11 does not the right hand side of the equation becomes infinite since both A and the difference between 0, (t) and 0,(t) are finite. This means that there is a sudden jump in the value of the scalar quantities, which is typical of a Poisson process, but is non-physical in the present case since the flow properties of turbulence are continuous. 2.2 Continuous Mixing Model One can see from the previous section that the modified Curl model relies on the parameter N,. to control the extent of mixing. On the individual particle level, it assumes complete mixing once the particle is selected as one of the mixing pair, without considering the size of 6t. In order to achieve continuous mixing, we propose the following model: during a time interval 6t, we assume that all the particles within a cell participate in mixing. The extent of the mixing is controlled at the individual particle level. That is to say, the N particles within a given cell are randomly grouped into N/2 pairs; the properties of all the particles change according to eqs. 6 and 7. The extent of mixing now has to be controlled at the individual particle level through the parameter A, which is redefined aa 61 A = C'. 5-

(10)

7'

where C' = 2.0. With this new definition, eq. 9 can be written, in the limit 6t --+ 0, d- = C4 (0()r dt

0.0)).

(12)

Since the continuous mixing model allows full coupling of the reaction and miing processes, the C/D model with reaction zone conditioning by Chen and Kollmann (1991) can be easily implemented in the present model to simulate the fast reaction in the flame sheet regime. Here a modified finite difference vers'on of eq. 12 has to be used since w, is infinity in case ot tast reaction.

(8)

This model does not represent the true physical process since the properties of the sample particles change discontinuously regardless the size of the time interval 6t. This deficiency can be best illustrated by rearranging eq. 6 and dividing it by Jt (0.(t + 61) - 0(t)) 6t 6ti

W1) + w.,

where w,, is the chemical source term.

(C = 6.0) and let a pair, say, m and n, mix as follows O4(t + 61) = A2m(t) + (1 - A)O4(t) A.(t) + (1 - A)m(t) Om (t + 61)

-

(11)

3. RESULTS AND DISCUSSIONS The continuous mixing model described in the previous section has been validated using both non-reacting and reacting flow test cases. The results and their comparisons with earlier calculations (Hsu, 1991) using the modified Curl model as well as with experimental data are presented in this section. 3.1 Homogeneous TVurbulence Decay of a Scalar The case of decaying fluctuation of a passive scalar in homogeneous turbulence is used to test the continuous mixing model. The initial condition is N 1 [N/2 P'(1; t) = J [E 6(0 -1) + Y 8(o + , (13) I n=N/2+i n=i that is, in the Monte Carlo simulation, half of the particles are ascribed the value 1, and the other half -3. The pdf distribution of the normalized variable (0- < ' >)/a, where < ' > is the mean and a is the standard deviation, in the homogeneous turbulence decay problem converges to a single curve after certain time, and the correct distribution should be Gaussian, Fig. 1 and 2 are the pdf distributions from the modified Curl model and the present model, both compared to a normal distribution. One can see that the pdf from the modified Cur! model deviates considerable from Gaussian, while the result from the present model is fairly close to a Gaussian distribution. The evolution history of the rms and fourth and sixth moments of the scalar fluctuation are calculated using both the modified Curl model and the present ontinuous model. Fig. 3 shows the results from the modified Curl model. One can see that although the rms from that model behaves well, the fourth and sixth moments grow quickly out of bound, oscillating at a level several order of magnitudes higher than the value of Gaussian distribution. These results are similar to what Pope (1982) had observed. Fig. 4 shows the results for the same set of quantities from the present model. The rms behavts similar to that from the modified Curl model. The fourth and sixth moments, on the other hand, are quite different from those of the previous model; they

22-4-2

rise ~smoothly to the value predicted by Gaussian distribution. Although the values do not seem to converge, they remain finite, and are of the same order of magnitude as that of the Gaussian distribution,

ment between numerical predictions and experimental data is fairly good, and a comparison of the results from the coatinuous model and that from the modified Curl model shows that both performed well for this case.

The above shown results clearly showed the advantage of the present model over that of the modified Curl model. Pope (1982) had devised an age biased scheme that achieved the same end, which required an additional variable namely the age of the particles, and two extra adjustable parameters. In contrast, the present model needs no extra work or parameters.

3.4 Combination with Reaction Zone Conditioning Chen and Kollmann (1991) developed a mixing model based on reaction zone conditioning, aimed at the coupling of reaction and mixing. We have shown in Section 2.3 that with the present model, the processes of reaction and mixing can be fully coupled; therefore it is only natural to apply the

3.2 Heated Turbulent Jet

reaction zone conditioning suggested by Chen and Kolmann here.

Extensive experimental results for a heated turbulent plane jet have been reported by many authors (Bashir, et al. 1975, Browne et al., 1984, Uberoi and Singh, 1975, Jenkins, 1976, Antonia, et al., 1983). The turbulent jet has a slightly higher temperature than the ambient. Measurements of both the the mean temperature and the rms of the temperature fluctuations were given. We compared the 5olutions for the heated turbulent jet from the new model with experimental data as we'l as with previous solutions (Hsu, 1991) obtained using the ,iudified Curl model.

The H2 -F 2 diffusion flame problem is reformulated such that the chemical reaction is confined to a very narrow zone near stoichiometry. By applying reaction zone conditioning to the continuous mixing model, we were able to produce a scatter plot of the temperature vs. mixture fraction in which all the points reached the equilibrium temperature. This tesult is shown in Fig. 10. The mixture fraction here is defined as the molar concentration of fuel divided by the total molar concentration, and stoichiometry is located at f = 0.5.

In the present study, a combined CFD-Monte Carlo algorithm is used. The mean flow field is obtained by solving the Na-ier-Stokes equation and a two-equation turbulence model using a finite difference scheme. The temperature is treated as a conserved scalar and simulated by the pdf equation. Fig. 5 shows the comparisons of the the mean temperature distribution from the pdf Monte Carlo simulations and experimental data from various authors. The figure shows that both mixing models predicts the mean temperature distribution accurately. The results for standard variation, or rms, of the temperature distribution are given in Fig. 6. Although the two solutions do not show significant difference, the new model seems to agree s!ightly better with the experimental data. The skewness and the flatness, i.e., the third and fourth moments of the temperature fluctuation, are given in Figs. 7 and 8; the comparisons show that in the present case of a turbulent jet the statistical behavior of the new model is similar to that of the modified Curl model.

4. CONCLUSIONS A turbulence mixing model that is continuous in time has been introduced. The deficiency of non-physical jump condition in the mixing process is removed in the new model. It has been shown that the new model is superior to the existing modified Curl model (Janicka, et al., 1979) in that it can predict a Gaussian distribution and finite higher moments in the case of homogeneous turbulence decay; it has accomplished what the age biased samphng sciteme (Pope, 1982) is designed for, without the extra parameters required by that scheme. The numerical results from the present model compare well with experimental data ACKNOWLEDGEMENT The work of the first author is supported by the NASA Lewis Research Center under Contract NAS3-25266 with L. Povinelli as monitor. The second author is supported by the Unite 3tates Department of Energy, Office of Basic Energy Sciences, Division of Chemical Sciences. REFERENCES

3.3 Hydrogen-Fluorine Diffusion Flame The continuous model has been applied lo the H 2-

F2 diffusion flame studied earlier by Hsu (1991). The flow conditions are set according to an experiment performed by Hermanson and Dimotakis (1989). The flame consists two upper containsthe96% of Nstream of F2 , streams. 2 and 4% the flow The velocity is stream Ut = '2?me.s; lower contains 96% of N2 and 4% of H2 , with velocity U2 = 8.8 m/s. The estimated Damk~hler number is ranged from 25 to 130 (Hermanson and Dimotakis, 1989), and a fast chemistry model is deemed appropriate in the calculation. Again a modified version of eq. 12 is used to accommodate the fast chemistry. Fig. 9 shows the temperature rise due to combustion. In the figure, ST is the sheai layer thi, kness determined by 1%of the temperature rise, AT is the :tua temperature rise due to combustion, ( the two streams have the same temperatures initially,) and AT# is the adiabatic flame temperature assuming complete reaction. Details on the flow conditions can be found in Hermanson and Dimotakis (1989). The agree-

Antonia, R.A., Browne, L.W.B., Chambers, A.J., and Rajagopalan, S., 1983, Budget of the Temrerature Variance in a Turbulent Plane Jet. Int. J. Heat Mass Transfer, 26-1, 41-48. Ashir, J. and Ulberoi, M.S., 1975, lbxperiments on Turbulent Structure and Heat Transfer in a Two Dimensional Jet. Physics of FLuids, 18-4, 405410. Browne, L.W.B., Antonia, R.A., and Chembers, A.J., 1984, The Interaction Region of a Turbulent Plane Jet. J. Fluid Mech., 149, 355-373. Chen, J.-Y. and Kollmann, W., 1991, Mixing Models for Turbulent Flows with Exothermic Reactions. Turbulent Shear Flows, Springer-Vtrlag, 7, 277. Curl, R.L., 1963, Dispersed Phase Mixing, I, Theory and Effects of Simple Reactors. A.I.Ch.E.J. 9, 175.

22-4-3

Hermanson, J.C. and Dimotakis, P.E., 1989, Effects of Heat Release in a Turbulent, Reacting Shear Layer. J. Fluid Mech. 199, 333-375. Hsu, A.T., 1991, The Stddy of Hydrogen Diffusion Flames Using PDF Turbulence Models. AIAA Paper 91-1780. Janicka, j., Kolbe, W., and Kollmann, W., 1979, Closure of the Transport Equation for the Probability Density Function Scalar Field. J Non-Equilib. Thermodyn. 4, 47.

10

Jenkins, P.E. and Goldschmidt, V.W., 1973, Mean Temperature and Velocity in a Plane Turbulent Jet, ASME J. Fluids Eng., 95, 581-584.

0 8

Norris, A.T. and Pope, S.B., 1991, Turbulent Mixing Model Based on Ordered Pairing. Combust. and Flame, 83, 27-42. Pope, S.B., 1982, An Improved Turbulence Mixing Model. Combust. Sci. Tech., 28, 131, Pope, S.B., 1985, PDF Methods for Turbulent Reactive Flows. Prg. Energy Combust. Sci., 11, 119-192.

h

r

I

I

0 4 -A

I \t

0 2

Uberoi, M.S. and Singh, P.I., 1975, Turbulent Mixing in a Two-Dimensional jet, Physics of Fluids, 18-7, 764-769.

/

'

/

\V

00 0

4

8

12

16

20

Ttme 08

Figure 3. Evolution of moments from the modified Curl model. - standard deviation, - - - 0.01 x fourth central moment, - - 0.0001 x sixth central moment, o 0 01 x fourth for Gaussian distribution, 0 0.0001 x sixth moment for Gaussian distribution.

0 7 0 6

5

"moment

0-

4

03

1

.

0 0 020o

.. -2

3

-- 1 ('i'

0

-

'r

>"-

1

2

0'-'8 3

>)/u

-

06

Figure 1. Asymptotic pdf distribution for a scalar in homogeneous turbulence. - modified Curl model; --- Gaussian.

07---/

08

02-

0 7

1-

06 05

O00', 0

0.4

0 2 -

12

16

20

Figure 4. Evolution of moments from the present model. standard deviation, - - - 0.1 x fourth central moment, - 0.01 x sixth central moment, o 0.1 x fourth moment for Gaussian distribution, 0 0.01 x sixth moment for Gaussian

,

/""

0 1 -3

8

Time

0 3.

O.0

4

'

" -'"

-2

-1

, 0

V

1

2

13

distribution.

Figure 2. Asymptotic pdf distribution for a scalar in homogeneous turbulence. - present model; - - - Gaussian.

22-4-4

1

20[

0

18-

,""

k1 14

0

00

10+0

O 00

02-

0

0~

00~ 00

10

05

15

4

20

01~ 00

2.5

05

10

15

20

25

30

Y/Y 5T Figure 5. Mean temperature in heated plane jet. - contin-

Figure 8. Flatness of temperature variance in heated plane

uous mixing model, - - - modified Curl model, A Browne et

jet. -

Y/Y 5To continuous mixing model, - - - modified Curl model.

al. (1984), 0 Bashir and Uberoi (1975), o Uberoi and Singh (1975), o Jenkins and GoldsdImidt (1973). 0 40

10 0 8

:' 0308 00

Z0 30 --,025 -

0I6 0 20 0

"0 0 4

00
§ ,

00' .)-000 002

x

000

~-o

,

Omm 0z.emm M 9' O

-

-0-0

'-

.oo

.

p.0mm 0-4mm

-

U -00 a

.

/,,

\

LI -sOw

0rO6 Po2

Cmm) Spon..se tocotonfrom theCenltedme

'..

'

-20 - :so .o -so . -so o 'so'tsr'1o" iS

0~503430354

,oo o" 200 250jn

7ime Lapse (microseconds)

Fig. 7 RMS pressure fluctuations across the span of case 2 in the fully developed region at y* = (a) -0.6, (b) -0.15 and (c) 0.06.

Fig. 8 Spanwise space-time correlations for varied spanwise separations and for varied spanwise and lateral locations for case 2.

23-3-4

peak changes sign and the level drops significantly with relatively large time shift for dz/6, 2 0.25. At y* = -0.15 location, at small dzS6 w both positive and negative correlations are present, but when dz/6 w is increased the correlations is negative and the level of correlations first increases and then drops, Table 1 shows the correlation level, the time shift and the angle of structures from spanwise axis for y* = -0.15 and dz/Sw = 0.25. The angle is calculated using B = tan' (U,*T/dz), where U C is the theoretical convective velocity. More detailed results are reported by Reeder (1991). Tablo I cross corralation Data x=170mm y'=4.15 d=02M. Pk

such as pressure and temperature, the moisture may condense and form water particles in the test section (Wegener & Mack, 1958). In our case, the condensation occurs only in the high speed stream of case 2. These particles are so small that we cannot detect them with our LDV system in which case we use atomized oil particles on the order of 0.5 to 1 Am. Our estimate is that these condensed particles are in the order of 0.01 pm. These particles which are about an order of magnitude larger than molecules in the air provided scattering media for the flow visualizations. Therefore, in the visualization photographs, the bright regions mark the fluid from high speed side, and the dark regions the fluid mark the low speed side. To get a better contrast in the photographs we have

limited the grey scale to two. Figure 9

SendavPek

shows side view (x-y plane)

of

the flowfield for case 2 from about 40 to 350e 0 in (a) and from about 300 to 5400 0 in (b) where e is the incoming boundary layer momentum thfckness of high speed stream. One

z,(eM) ;(mm) -42 -36

Lewl,/I(sec)/B

•36 -30

-3

-24 0co21e.a/on-

-24

4.12 / .2 /7'

identifies very large structures with nearly

.24

-18

.008 /10

-21 -12 •9

-15 .12 -6 -6 -3

-010 1 .4 /IS07 / 4 /-L5 001 12 /.3r 005 /12 / Ir

-6

0

.00/ 4/15 -007 /.16 1,"

equal spacing in streamwise direction and many smaller structures on the edges of each large structure can be These photographs wf.re taken at twoseen. different two times. exposure time for all the photographs is The8 Figure 10 shows the spanwise view (y-z plane) of structures at an axial location of

-3

3

-013 /-40

0 3

6 9

412 /-12 / 4.13/8 /27-

-18

-12

.055/-12 /3

LeMl /I(sec)/

a

/33'

_ns.

/33'

6

12

.0.10/ 0 /

12

is

412/4

15

21

-010/-4

18

24

.0047/ 40-W9

21

27

4053/48

-7

24

30

006 /32

/-§4'

30

36

006

36

42

+004 /-40

30mm B1'

/

-90

/W5

52 /.74'

.0 045/0/0' .0045/ 0 /V

+0.04 /44

/71*

/69'

The picture of the structures arising from the results presented so far is a complex one. All the indications are that the most dominant structures are highly three-dimansional inclined both in the mean flow direction (x) and in the spanwise direction (z) and they are nearly spatially stationary. An element of structure may be a horseshoe type vortex with the head in the low speed side

15mm

and the legs, inclined both in the streamwise

and the spanwise directions extending toward the high speed side. The element of structure could also be a single vortex tube type spanning the mixing layer and inclined in both the streamwise and the spanwise directions. Most probably, a large structure is composed of a few of these elements with oblique angles of different signs connected at either edge of mixing layer. 2. Flow Visualizations As was discussed earlier, filtered Rayleigh scattering technique was used for visualizations. The background noise flow was removed by the molecular filter. As with the standard Rayleigh scattering technique, the signal is proportional to density an to the radius of molecules/particles to the sixth power. In a typical supersonic flow facility, the oil and water particles are removed by oil separators, dryers, and filters. Yet no matter how efficient these processes are, some water moisture will remain in the flow system. Depending upon the flow conditions

Fig. 9 FRS visualization of x-y plane for case 2.

-20mm I

o I

+20mm I"

Fig. 10 FRS visualization of y-z plane for case 2.

23-3-5

about 49000. The photograph shows various structures in the spanwise direction con-

However, case 2 shows totally different behavior. It seems that the higher compress-

firming very low spanwise correlation levels shown in Fig. 8. Figure 11 shows the plan view (x-z plane) cutting through the shear layer just slightly'above the splitter plate in the axial location extending from about 280 to 430e 0 . In these photographs, one

ibility case can support oblique structures; therefore, the structures are highly three dimensional and complex for case 2. All indications are that these structures are nearly spatially stationary in spanwise direction spanning the mixing layer and

easily observes oblique structures.

inclined

in

both

the

streamwise

and

the

spanwise direction. They have both positive and negative angles, and if they travel toqether they may connect at either edge of

:V

the mixing layers. Acknowledgements The research is supported by ONR (with Dr.

S. J.

Lekoudis)

and

NASA

Lewis

(with

Dr.

K.B.M.Q. Zaman). The authors would like to thank colleagues Diana D. Glawe and Stephen

A. Arnette and AARL staff for their help. REFERENCES

Bernal, L.P. and Roshko, A., 1986, "Streamwise vortex structure in plane mixing

15mm N1240.

Fig. 11 FRS visualization of x-z plane for case 2.

The technique utilized here is based on a pulsed YAG laser with 8 ns pulse duration and 10 pulse/second repetition rate. In taking

these photographs, the flow is frozen for about 8 no while a photograph is taken, but there is no time relation between two successive photographs or frames. However, we have taken hundreds of frames and observed them individually. The results of these observations are be summarized here. There are large scale structures seen in both streamwise and spanwise views. These structures grow both in the streamwise and spanwise directions. Sometimes they seem to have very well orga-

nized patterns in both directions. The plan view photographs show both spanwise and some photographs, oblique structures. In either there are no structures or the structures do not seem to have any pattern. A possible explanation for spanwise photographs cuts through different parts of the structures, e.g. the core or the braid section of structures. But the only explanation for the

streamwise and the plan views is that the structures are very poorly organized in time and are highly three-dimensional. This is what one would also conclude from coherence and correlation results discussed earlier, CONCLUSIONS The behavior

incoL.

of

case

layers," J. Fluid Mach,

170, 499-525. Bogdanoff, D.W., 1983, "Compressibility effects in turbulent shear layers," AIAM J., 21, 926-927. Bradshaw, P., 1977, "Compressible turbulent shear layers," Ann Rev Fluid Mech, 63, 449-464. Browand, F.K., and Troutt, T.R., 1980, "A note on spanwime structure in the two-dimensional mixing layer," J. Fluid Mech., 917, 771-781. Brown, G.L. and Roshko, A., 1974, "On density effects and large scale structure in turbulent mixing layers," J. Fluid Mach, 64, 775-816. Chinzei, N., Masuya, G., Komuro, T., Murakami, A., and Kodou, K., 1986, "Spreading of two stream supersonic turbulent mixing layer," Phys Fluids, 29, 1345-1347. Clemens, N.T. and Mungal M.G., 1990, "Two- and threedimensional effects in the supersonic mixing layer," AIAA-90-1978. Elliott, G.S. and Samimy, M., 1990, "Compressibility effects in free shear layers," Phys Fluids, 2, 1231-

1

ssible mixing layers

is

similar

to

in many ways.

Hussain, A.K.M.F. and Zaman, K.B.M.Q., 1985, "An experimental study of organized motions in the turbulent plane mixing layer," J. Fluid Mech, 159, 85-104. Ikawa, H. and Kubota, T., 1975, "Investigation of supersonic turbulent mixing layer with zero pressure gradient," AIAA J., 13, 566-572. Jones, G., Planchon, H.P., and Hammersly, R.J., 1973, "Turbulent correlation measurements in a two-stream mixing layer," AIAA J., 11, 1146-1150. Komine, H., Brosnan, S.J., Litton, A.B., and Stappa erts, "Real-time Doppler global velocimetry," AIAA91-0337. Messeromith, N.L., Dutton, J.C., and Krier, H., 1991, "Experimental investigation of large scale structures in compressible mixing layers," AIAA Paper 91-0244. Miles, R.B., Lempert, W.R., and Forkey, J., "Instantansous velocity fields and background suppression by filtered Rayleigh scattering," AIA-91-0357. Morkovin, M.V., 1961, "Effects of compressibility on turbulent flows," Mechanique do la Turbulence, Marseille, France. Papamoschou, D. and Roshko, A., 1988, "The compressible turbulent mixing layer: an experimental study," J. Fluid Mach, 197, 453-477. Papamoschou, D., 1991, "Structure of the compressible turbulent shear layer," AIAA J., 29, 680-681. Ragab, S.A. and Wu, J.L., 1989, "Linear instabilities in two-dimensional compressible mixing layers, Phys Fluids, 1, 957-966. Reeder, M.F., 1991, "A study of the effects of compressibility in mixing layers using two-probe pressure correlations," MS Thesis, Ohio State University. samimy, M. and Elliott, G.S., 1990, "Effects of compressibility on the characteristics of free shear layers," AIAA J., 28, 957-966. Sandham, M.D. and Reynolds, W.C., 1989, "A numerical investigation of the compressible mixing layer," Report No. TF-45, Stanford University. Shau, Y.R. and Dolling, D.S., 1990, "The detection of large scale structure in undisturbed and disturbed compressible turbulent free shear layers," AIAA Paper 90-0711. Wegener, P.P. and Mack, L.M., 1958, "Condensation in supersonic and hypersonic wind tunnels," in Advances in Applied Mechanics, Academic Press.

23-3-6

j

ill II IIIIlllI IIl lIl II lll 1l

23-4

EIGIIFIt SYMPOSIUM ON TURBULENT SHEAR FLOWS Technical University of Munich September 9-1I, 1991

THE EFFECTS OF VISCOSITY AND DIFFUSION ON A SUPERSONIC MIXING LAYER P. Vuillermoz* and E.S. Oran" Soci6t6 Europenne de Propulsion 27207 Vernon, France Laboratory for Computational Physics and Fluid Dynamics Naval Research Laboratory, Washington, DC 20375

ABSTRACT Time-dependent two-dimensional numerical simulations of high-speed, compressible, confined, temporally evolving sugas personic mixing layers between hydrogen and oxygen Eubetween differences the examine to streams are _bcd ler and Navier-Stokes solutions and the effects of convective Mach number. The computations show that the Euler solutions are reasonably accurate in the convective-mixing stage of the flow, when large-scale structures dominate. As the convective Mach number increases, the mixing and turbulent levels are reduced. INTRODUCTION The successful design of propulsive engines for hypersonic vehicles requires accurate prediction of the mixing and combustion efficiency in high-speed flows. In order to achieve this predictive capability, we need an improved understanding of the dominant processes. In this paper, we have begun

mine what is required to adequately resolve the flowfield in the different mixing regimes. In particular, we compare soluh toso h ue qain o hsfo osltoso complete set of Navier-Stokes equations that also contain descriptions of molecular diffusion and thermal conduction (referred to as the NS+ equatioi.s). PHYSICAL AND NUMERICAL MODEL General Formulation We solve the time-dependent, two-dimensional, compressible, Navier-Stokes equations for a multispecies gas, including the effects of molecular diffusion and thermal conduction. The balance equations for the densities, momentum, and energy are (1)

= -V.(pv),

i

n- = -V - (nv) - V. (nZvd)

v1

Opv at

to address several of these complex issues. In particular, we focus on the numerical solution of the mixing of a hydrogen and oxygen stream so that we are studying the features of layer between two gases of very different a high-speed shear denitesDi= densities

OE 0T

= -V. (pvv)

-

.

V. P

-V.(Ev)-V.(v.P)-

(3)

,

V.q),

Since the Brown and Rosko [1974] observations, many experimental studies have confirmed the existence of largescale coherent structures in turbulent mixing layers for both high and low Reynolds numbers. Papamoschou et al. [1986] and Clemens et al. [19901 showed the effects of compressibility on the growth of the shear layer, and recently, Dimotakis [1989] summarized experiments on the entrainment and mixing processes. There have been a number of theoretical stability analyses and concommitant numerical simula-

The mass fraction of species i is Y, defined by PY) , n: -

tions of both spatially and temporally evolving shear layers. Ragab and Wu [19901 and Jackson and Grosch (19891 have shown that for high Me, there are several unstable modes and three-dimensional modes become important. Sandharn and Reynolds 11989] and Lele [1989] have considered twodimensional and three-dimensional direct numerical simulations. Using vortex-dynamics methods, Soteriou et al. [1991] have studied the effects of density gradients. In the work presented below, we solve the equations describing a temporally evolving shear layer, where one stream is molecular oxygen and the other is molecular hydrogen. This approach allows us to focus on the structure of the flow and the details of the turbulent structures and deter-

and and-A

(2)

(4)

(5)

of the species i. The auxiliary equation for pressure P and heat conductive flux q are 2 P = P(p, T)1I-+o(V v)I - t,[(Vv) + (Vv)T] (6)

23-4-1

VT+pl hiYVd, + q = We assume an ideal-gas equation of state: P specific enthalpy h, can be written as h

(7) pJT. The

T

h,=oCtdT+h,

(8)

where at, and h are the specific heat at constant pressure and heat of formation of species i, respectively. We neglect the radiative fluxes and the Soret and Dufour effects (ther-

Diffusion Model The diffusion velocities Vd, are solutions of the following system of equations (see, for example, Oran and Boris j1987]), =

VX,

L(vd, - Vd,) +

D

(Y,

-

(9)

X,)

where, for each species t, X, is the mole fraction, Vd, the diffusion velocity, and D,, the diffusivity of the species i into the species j. These diffusion velocities must obey to the condition of mass conservation such that

solution of this problem provides initial conditions for the two-dimensional shear layer. The two-dime)'sional computational domain, shown in Figure 1, consists of a rectangle of length L and height H = L/2. The left and right boundaries are periodic, which means that (0, y) = W(L, y), and the bottom and top boundaries are slip wall conditions, pv, = 0 and &p/Oy = 0 where V = (p,pvo,{,),E) at y = 0 andy H. The computational cell size is always kept uniform in the x and y directions and are in the range Ax = Ay = (2.5-10) x 10' m, so that the timesteps are in the range (2-8) x 10- 3 ps. A typical computation described below requires about 10,000 timesteps. wall

w

(10)

YoVda= 0

Mc

We approximate the solution of the system of equations (9) and (10) by a gradient law, as proposed by Coffee and Heimerl [1981],

Mlxlng layer Mc

,

-

-TD"'VX,

i,

Dm =

/D

(

V, =

-

Y,!v,

and

Vd,

=

17d,

+

V, -

(12)

This approximation becomes rigorously correct in the binary case if we neglect the pressure effect on diffusion, Numerical Integration The different physical processes are solved separately and their effects are then coupled together by timestepsplitting methods (Oran and Boris [1987]). The convection is solved using a standard Flux-Corrected Transport (FCT) algorithm This is a nonlinear, monotone algorithm that is fourth-order accurate in phase. The integration is carned out by a two-step predictor-corrector procedure with, successively, a diffusive and antidiffusive step. The first step modifies the linear properties of a high-order algorithm by adding diffusion during convective transport to prevent dispersive ripples from arising. The added diffusion is removed in an antidiffusion step. The result is that the Lalculations maintain the high order of accuracy without requirig artificial viscosity to stabilize them. The physical diffusion terms are solved in conservative finite-volume form by secondorder centered algorithms. An overall global timestep is chosen by evaluating the individual stability criteria of each type of process in the computations and then selecting the minimum of these. THE MODEL PROBLEM In the one-dimensional problem, we compute the evolution of a mixing layer between two gases starting from a step profile for each variable in the y direction. We do this by solving the full set of equations (1) - (4), but assuming that there are no variations along the x direction. That is, cOV,]9x = 0, where V represents p, pv, {n,}, and E. The boundaries at the bottom and the top of the domain are open. The

wall

X

Then we correct these velocities in order to insure the mass conservation, equation (10),

wati

Figure 1: Computational Domain Computational Diagnostics A number of different global and instantaneous parameters are used to characterize the flows: 1) The convective Mach number M, and the convective velocity of the coherent structures U, can be approximated from known properties of the flow assuming a stagnation point in the center of each coherent structure. This analysis, suggested by Papamoshou and Roshko [1986], leads to ta

U

=

U2 + a 2 Ui)

(a

+ a2)

(U2 -UI) and M, =

+

(13)

where a, and a2 are the speeds of sound for the streams 1 and 2, respectively. 2) The energy transfer for two-dimensional turbulence can be described by the field of enstrophy, w'. Eirstrophy for two-dimensional turbulence has properties similar to vorticity for three-dimensional turbulence (Batchelor 11969]), tat is, there is a cascade process, independent of the viscosity, to higher wavenumbers where the enstrophy is dissipated. 1he global intensity of the turbulent flow may be characterized by M, = w2 >-', where the < > indicates an average taken over the computational domain. For an incompressible, homogeneous turbulent flow, M, is a linear function of time (Batchelor [1969]). 3) Mixing efficiency between different gases can be described in terms of mole fraction, X,, or mass fraction, ',. For the binary mixture, we define two global parameters, Vx = 4 < X1 X 2 > and My = 4 < Y1 Y2 >, that vary between 0 and 1. For the limit of fully mixed material, M' = 1 and My = 4W1 W2 /(W + W2 )2 . For the hydix gen and oxygen mixture we are considering, M ° = 0.22. The parameter Mvy =< VYH2 VYH2 > is related to the dissipation of scalar energy in the computational dow,,in

23-4-2

and describes the density of mixing iiterfaces devloping in the domain.

its dissipation by viscosity or diffusion, t. tcI

U

t

1

and td =-.

--

= fY

U "

THE ONE-DIMENSIONAL PROBLEM

td, where

=

2

(14)

Here e, is the dissipation of turbulent kinetic energy, k eL < U'2 >, and ey is the dissipation of turbulent scalar In the one-dimensional problem, one stream is molecular hydrogen and the other is molecular oxygen. Iritially the

energy, -1


E
=

D2 (16) 77Y where ily is the mixing scale, S is the tensor of deformation and r = rq/u. Note that the prime indicates fluctuation of the variable. If we define the Kolmogorov scale 77j as that scale where the molecular viscosity transforms the kinetic energy into heat (that is, the scale at which the Reynolds number is of order unity, Rek -pk u/P , 1), we obtain ey = D < VY'VY1

k = Re-

(17)

where Re = plU/lp is the Reynolds number of the large scale the same way, the equilibrium condition allows us to eaut y (18) T l = Pec-,

]where

P, is the Peclet number such that P, I U/D = = p/pD is the Schmidt number. When S= 1, is the Taylor scale. In our case, the large scales are bounded by the width

S, Re and S, H2 i ._.___...._____

0.

" 1 12 >,

y(cr,) 0.25 0. y(cm) Figure 2: One-dimensional Results

of the computational domain and their velocity scales with

0.25

the difference of velocities across the shear layer. Then, we can assume that,

Two-Dimensional Initialization

1H

The two-dimensional configuration is Kelvin-Helmholtz unstable. We use the o-c dimensional computation to provide initial condions, and then trigger the instability by superimposing a set of harmonic (SI = 27r/A) and subharmonic (Q/2) pressure disturbances. In the calculations presented below, we have chosen the wavelength of the instability, A, such that A/L = 2, 4, and 8 and a = 0.05.

and U-(U2 -U

)=M(a +a2)

(19)

This assumption allows us to asses the small scales. Several values computed for the hydrogen-oxygen mixing layer are reported in Table 1

Table 1. Estimations of Kohnogorov and mixing scales

THE TWO-DIMENSIONAL PROBLEM

l(inm)

M,

Re

7k(m)

qy(r)

Analysis of the Turbulent Scales The main feature of a turbulent flow is the wide range of its

5.0 5.0

0.6 1.2

1.85 x 10' 3.70 x 104

3.1 x 10' - 8 1.9 x 10

3.7 X 10- ' 2.6 x 10 - 5

energy spectrum. The largest scales may be characterized by a length I and a velocity U, and the smallest dissipative scales may be characterized by a length q and velocity u. If we assume that 'here is an equilibrium between the convection and the dissipation of turbulent energy, we can compare the large and small scales. This assumption means that the characteristic time tc of the transport of turbulent energy by the large eddies is equal to the times ,, and t d of

In order to isolate and examine the effects of molecular diffusion, we performed three NS+ calculations that were the same except for the values of the diffusion coefficients. These computations were performed for a system with L = Ay = 1.0 cm, 200 x 100 computational cells, and Ax

23-4-3

5.0 x 10 - m. In one case, the molecular diffusion is turned off completely. In another, the actual values of physical diffusion were used, as in the previous computations. In the final case, we multiplied the physical values by two. The results of the comparison are presented in Figure 3 that shows various diagnostics defined above as a function of a dimensionless time t/to, where t o = H/AU. 3 0

2.

.

...

mixing stage whereas 1, increases quickly and independently of the diffusion. When the diffusive-mixing stage begins, the turbulent structures broaden in proportion to the diffusion coefficient and the destruction of the turbulent scales is controlled by the amount of molecular diffusion. C;CO

3

0

00

...... 3. ~ 21

22

0. C5

D 0.

.

t/to 13. 0. t/to 13. Figure 4: Width and Length of Turbulent Structures (1. no diffusion, 2: 1 x diffusion, 3: 2 x diffusion)

t/to

Figure 5 shows instantaneous contours of mass fraction hydrogen during the three regimes described above. We note the apparent change from a very regular structure to the extremely mixed structure at the end of the computation. As time evolves, we see the shear layer expanding towards the lighter fluid on the bottom , an effect noted by Soteriou et al. [1991].

Xof

...... ...... ..... O. t/to 13.

I or Pr > I, can only be obtained if at least both a random age and a laminar flow with a time-dependent velocity gradient are ascribed to the fluid elements at the tube wall. Fortuin and Klijn (1982) were the first who developed a RSR model foi turbulent pipe flow to describe momentum transfer from fluid elements of random age and laminar flow with a time-dependent velocity gradient to the tube wall. This approach resulted in a relationship between ti.e mean age to of the fluid elements at the tube wall, the friction factor f, and the Reynolds number Re. The random distribution of the ages of the fluid elements at the tube wall has already been derived from velocity signals measured with a laser-Doppler anemometer (LDA) by Van Maanen and Fortuin (1982, 1983). A quantitative agreement between measured and calculated values of to has been obtained by Musschenga et a;. (1990). In the present study the RSR model of Fortuin and Klijn Is extcnded to describe heat and mass transfer in turbulent pipe flow. The resulting 'Extended Random Surface Renewal' (ERSR)

0t

Je Oa-'/o

-

2 THE ERSR MODEL Modelling of transfer phenomena in turbulent pipe flow should preferably be based on the observed behaviour of the fluid in the wall region. Hydrodynamic studies have shown that a fluid in turbulent motion may be considered as a mass of fluid elements of indefinite shape and size, which continually change their conformation and position. The fluid elements intermittently move from the bulk of the turbulent fluid towards the interface where they replace other fluid elements. The phenomenon of the inrush and ejection of a fluid element is called a 'burst'. The fluid elements are assumed to maintain their identity during residence time at the tube wall while momentum and heat or mass are exchanged simultaneously between the fluid elements and the interface. The time interval between two successive bursts, which is much greater than the short time required for the renewal process, is considered as the age of a fluid clement at the tube wall. For the mathematical formulation of the transfer mechanisms it is unimportant whether the renewal of the fluid elements at the interface is caused by the inrushing high-mominum fluid elements or, as suggested by Einstein and Li (1956), by the spontaneous breakdown of the viscous sublayer due to instabilities, In the ERSR model it isassumed that during turbulent pipe flow the exchange of mom-ntum, heat, and mass between the tube wall and the bulk of :!-- fluid is governed by the velocity profiles in the fluid elce%;,ts, and the age (distribution) of the fluid elements at the interface. 2.1 V1clocilty profiles in fluid elements In the fluid elements at the tube wall plug flow may be assumed in e.g. the description of heat transfer, if the penetration depth of heat is larger than the penetration depth of momentum (Pr < 1). However, if the penetration depth of heat is small compared with the penetration depth of momentum (Pr > 1), laminar flow has to be assumed with a velocity gradient at the interface, 2.2 Renewal of fluid elements If it is assumed that the chance of renewal of each fluid element at a fixed positior, at the tube wall is the same, irrespective of its age or position at the tube wall, it can be shown (Danckwerts, 1951; Fortuin and Klijn, 1982) that the chance p(o)d0 to find a fluid element with an age between 0 and e + dB can be obtained from:

(1/t )e~OeO d0

(2)

- to

2.3 Mean flut at random surface rewal If J(t) represents an instantaneous local momentum, heat, or mass flux in a fluid element at the interface of a turbulent flowing fluid, the value averaged during its residence time at the interface follows from:,

e :(e)

(1/0)f

-

(3)

J(t)dc

0 If the fluid elements at that position at the tube wall are renewed randomly in time, the local flux, averaged over all possible ages of the fluid e!ements (0 < 0 < m), follows from:

0[ {1/0)f J(t)d, 1 .-0/'o

- to derive the relationship between f, Re, and to; this relationship is verified experimentally using a LDA; - to calculate the mean axial velocity profile in the wall region; - to derive equations for a quantitative prediction of heat and mass transfer coefficients; - to elucidate the analogy between momentum, h-at, and mass transfer in turbulent pipe flow.

-

o)

The reciprocal value of the mean age to is defined as the characteristic burst frequency n.

model is described in the present paper, and will be used:,

O(O)d0

d(0/9

0

(1)

In Eq. (1) to represents a characteristic value of the age of the fluid elements at the tube wall; this value is dependent on the flow conditions. The mean age of the fluid elements at the interface follows from:.

-

-

0 e

t /to

(4) d(O/t0)

0 Rearrangement of Eq. (4) results in:, 0J

0

j

(IIt0)j

-

}

JJ(t)dt

'

" / t°

d(e '

)

(5)

Partial integration of Eq. (5) leads to:

J - (1/,t)[ e 0

-0

/t

°

(6)

J(O)dO

The result of this integral is not changed if 0 is replaced by t, so that:

-

(1/t)f J(t) e - t/t

(7)

dt

0 3. MOMENTUM TRANSFER It may be assumed that at a constant Reynolds number in a fully developed turbulent fluid flow through a straight, smooth tube, the time-averaged local shear stress Tw at an arbitrary fixed position at the tube wall is equal to that at each other position at the wall. In the ERSR model the momentui. transfer to the tube wall is approximated by that in a turbulent fluid flow along a flat wall. This approximation is only valid if the curvature of the tube wall is negligible, which approximately holds for turbulent pipe flow if Re > 104 (Fortuin and Klijn, 1982). Upon its arrival at the wall, an inrushing fluid element gradually looses momentum as a result of viscous interaction with the wall. In this fluid element, which originally travelled with a uniform The bulk velocity, a velocity profile starts to develop. development of this velocity profile, and the unsteady transfer rate of momentum to the wall, can be described by the following reduced equation of motion:

au/ag

-

2

p a U/ay

2

(8)

This equation, for a fluid with constant density and viscosity, can be solved using the following initial and boundary conditions: I.C. B.C. B.C.

t - 0; t > 0; i >0;

y yy

0; 0; co;

u - ub, u - 0, u- ub.

(9)

The solution to Eq. (8) under the conditions of Eq. (9) is:

u/ub - er2.,),

where

z -

3(10)

Using Eq. (10) the instantaneous shear stress rw(t) locally exerted on the wall by a fluid element is obtained from:,

24-1-2

au

'l s

7w(t)-

-UPub

y-0 y

-)i

/ 2

1

The time-averaged local wall-shear stress 7w for randomly renewed fluid elements at the tube wall can be derived from Eqs (7) and (11) by substituting 7w(t) for J(8): (12)

1/

vPub(t)o)' 2

w

Combining yt if Re > 10. it may be assumed that ub Eq. (12) with the definition of the Fanning friction factor: 2

.f- .T'w/(jp

(13)

)

results in the following equation:.

]

I

f-

-

Equation (14), first introduced by Fortum and Klijn (1982), gives the relationship between f, Re. and t o . Rewriting Eq. (14), and using t o = I/n o result in: -o.

(l/no)

(15)

t

. ( f.Re)-Z d

process. 5.1 Heat transfer into fluid elements with laminar flow (Pr>l) The heat flux from the tube wall into a laminar flowing fluid element encounters an approximately linear velocity profile if the penetration depth of heat is small with regard to that of momentum (Pr > 1). Upon its arrival at the wall, the fluid element gradually decelerates and the velocity of the laminar flow near the interface becomes smaller. The velocity profile in this fluid element can be obtained from Eq. (10). For small values of z the relative velocity may be approximated by:

(14)

2Fo /5 Re"

-

momentum and heat are exchanged by diffusion between the interface and the fluid element. After a certain residence time 0 the fluid element is replaced suddenly by a new fluid element from the turbulent core, and again momentum and heat are exchanged by diffusion between the new fluid element and the interface. The heat accumulated in a single fluid element is thus conveyed to the vrbulent core during this surface renewal

In the present paper Eq. (15) will be verified by results obtained from velocity signals measured with a LDA 4. THE VELOCITY PROFILE IN THE WALL REGION in a similar is obtained axial local The time-averagedwayasit sres. wit-her tie-verge asbee velocity dneforth way as it has been done for the time-averaged wall-shear stress, As a consequence the following equation holds:

(

u/ub

(21)

7[.z1-5

its arrival at the wall the velocity profile in the fluid element is assumed to be independent of the longitudinal coordinate x. As a consequence the velocity in the fluid element is only a function of the distance y to the wall and the time t. In this case the heat transfer can be described as a semi-infinite diffusion of heat from the interface into a laminar flowing fluid time =, whichof isx. only velocity element However, the c.rection change in(8ulay) not gradient anda does dependentwith After

low (Ta y)e cn the dient t te gradient (olay)y=o changes in the flow temperature the pe direction. Assuming a quasi-steady state, this heat transfer problem Is approximately governed by the folk ring differential equation:

i -

(1/to)J e

/t °

u aT/x

(16)

u(t)dt

a a2T/ay

-

2

(22)

0

into Eq. (16), and ino y(4vt)Substitution of Eq. (10) and z 0 taking into account that i is a function of y only, it can be obtained that. i(y)/ub

-

-

2f 0

(Z

2

2

2

Z 0 /Z ) dz

+

provided that heat conduction of the fluid in the flow direction is small compared with the forced convective transport in this direction, so that liquid metals are excluded. Further it is assumed that:,

aT/a1t

aT/ax - (aT/at)/(ax/at) -(1/ub)

(17)

(23)

Combination of Eqs (22) and (23) results in: After substitution of the solution to the definite integral in Eq. (17) and using Eq. (14), the following velocity profile for the wall region in turbulent pipe flow is obtained:

U(y)/u

b -I

-

e

1

This cquation can be rewriten in the common.y y+ notation, resulting in'

u-

I

St(

2

/

used u

+

versus

(f/2)

(19)

value of the friction factor is needed. A correlation giving sufficiently accurate values for f for Newtonian fluid flow through straight, smooth tubes is (Eck, 1973):

]-2

(2100

; Re

0;

I..

t -

B.C. B.C.

t > 0; t > 0;

108)

(20)

0;

T - Tb, T - Tw, T - Tb. P/a

(25)

= Pr into Eq. (24), it can under the conditions of

be shown that the solution to Eq. (24) Eq. (25) is:

(Tw

-

T)/(Tw

-

Tb)

-

"

l (4/)

e

"

(26)

d

0 where:,

4

1(27)

'r

•27t) . The instantaneous local heat flux qw(t) from the interface into a is: element laminar flowing fluid

aT"

5. HEAT TRANSFER It may be assumed that at a constant Reynolds number in a fully developed turbulent fluid flow through a straight, smooth tube, the time-averaged local heat flux i,"w at an arbitrary fixed position position at the tube wall :%equal to that at each other at the wall if the temperature difference between the tube wall and the fluid in the turbulent core has the same value. In the following it will be assumed that the tube wall has locally a constant temperature T w , and that the bulk of the fluid has locally a constant temperature Tb, where T w > Tb. When a fluid element with a uniform velocity ub and a temperature Tb comes from the turbulent core and arrives at the wall, both

(24)

y - 0; y - w;

y

After substitution of Eq. (21) and

From Eq. (19) it can be obtained that according to the ERSR model the velocity profile in the wall region is slightly dependent on the Reynolds number. Close to the wall, however, !he calculated protiles quantitatively agree with correlations for the semi-empirical 'universal velocity profile'. It may be noted that for the calculation of velocity profiles with Eq. (19) only the

f - 0.07725 1 log(Re/7)

2

The initial and boundary conditions are:.

Re) (18)

exo) pi.-

- a a2T/ay

(U/ub) aT/at

qw(t)

"

"h

XT.b(v)

( T

1/ 2

-

/3

1r

(28) (y-o

where: (/6)' C

/3 -

-

0.9026

(29)

r(4/s) The time-averaged local heat flux w, averaged over all possible ages of the fluid elements at the tube wall (0 < 0 < co), can be obtained from Eqs (7) and (28):

24-1-3

- / 2

qw - CX(Tw - Tb)('I)

/

(30)

Pr 1 3

o(Tw - Tb) and the dimensionless heat-transfer Defining w coefficient or Nusselt number, Nu = ad/),, Eq. (30) can be rearranged to: )

Nu - Cd(to -1/2 Pr

1/ 3

results in the following equation for

Combining Eqs (14) and (31) the Nusselt number: Nu - C(f/2)Re Pr

/

(31)

given by McAdams

Using the following correlation turbulent pipe flow: f - 0.046 Re-

"

(32)

(Pr > 1)

3

for

(1954)

(104

t;Re 4 2105)

(33)

we obtain from Eqs (29), (32), and (33):, 0

' ° Pr,t /3a ,Mu - 0.021 Re . 0 P

(34)

(Pr > 1)

Eq. (34) is approximately equal to the correlation presented by Colburn (1933): '

/3

following

well-known

(Pr > 1)

-

a a2T/8y 2

The solution to Eq

(36)

~ /2

where (pub - puw) is the axial momentum difference per unit of volume between the bulk of the fluid and the fluid at the pipe wall. As uw is zero it may be stated that. T

Combining Eqs (12) k, v//(Vto) Introducing

'

(47)

a dimensionless

momentum-tra,|sfer

(39)

kmA(pu)

cT ,, Nu -.- -

, and

Sh-

(40)

This result is in agreement with the fact that if Pr < I measured values of Nu fit better with an equation in which Nu 2. is proportional to Prii 6. MASS TRANSFER The ERSR model can in a similar way be applied to derive equations for calculating mass-transfer coefficients in turbulent pipe flow. After introducing the dimensionless mass-transfer coefficient or Sherwood number (Sh = kd/D) and the Schmidt number (Sc = j/D) it can be obta -d tha,:

Sb

-

C(f/2)Re Soi'

Sh - (f/2)Re Scl/2

kc -

(49)

DAc/d

XAT/d

Eq. (49) shows that each of these dimensionless numbers :s equal to the ratio between the total flux of momentum,

(Pr < 1)

called

(48)

PA(pu)/d

Combining Eqs (14) and (39) it may be stated that: Nu - (f/2)Re Pri/Z

number,

The three dimensionless numbers governing the transfer processes in turbulent pipe flow can also be defined by(38)

/ 2

and (46) gives:

Fa - kmd/vI- d/,/(vto)

The value of qw can be obtained using Eqs (7) and (38): ) -

(46)

w - kin(Pub)

Fa X(T, - Tb)(ato

(45)

Fw - km(Pub - PUw)

(37)

Con bination of Eqs (28) and (37) .esults in:. qw(t) - X(Tw - Tb)(Oat) '

7.2 The analogy according to the ERSR model A momentum-transfer coefficient for turbulent pipe flow can be defined by:

the Fanning number Fa = kindly, Eq. (47) results in:.

(36) under the conditions of Eq. (25) is:

(Tw - T)/(Tw - Tb) - er [ 21Ya

(44)

where 'd is the 'i-factor' for mass transfer. In the next section the vat. S of /h md id calculated with the ERSR model wili be given. Further, .. will be shown that the analogy between the transfer processes in turbulent pipe flow can be elucidated by introducing a momentum-transfer coefficient km.

(35)

5.2 Heat transfer into fluid elements with plug flow (Pr < 1) The heat flux from the tube wall into a fluid element encounters approximately a fluid flowing wih the bulk velocity if the penetration depth of heat is large compared with the penetration depth of momentum (Pr < 1). The heat transfer may then be approxima,'d by an unsteady diffusion of heat into a plug flow. This heat ransfer problem is approximately governed by Eq. (24) in which a = ub , resulting in:

aT/at

(43)

lb - Nu(RePri3) - f/2 where lh is the 'l-f actor' for heat transfer, and Jd - Sh/(Re SC1/3) - f/2

'

Nu - 0.023 Rea .0 Pr"

relationship between the friction factor and the Reynolds number. The heat and mass fluxes to or from the interface however, are described by using transfer coefficients. This difference in approach is traditional in engineering practice, and probab'y due to the fact that originally transport phenomena in turbulent flows were studied by engineers of diffcrent disciplines. Colburn (1933), and Chilton and Colburn (1934) introduced correlations based on the analogy between momentum, heat, and mass transfer in turbulent pipe flow. They proposed an empirical analogy for Re > 104 in long smooth pipes, which can be represented by:

heat, or mass

due to both molecular diffusion and forced convection and the flux of momentum, heat, or mass due to molecular diffusion only. From Eqs (14), (32), (41), and (48) it is obtained that according to the ERSR model the transfer equations for turbulent pipe flow for values of Pr > I or Sc > I can be written as:, (50)

Fa - (f/2)Re Nu - C(f/2)Re Pri/3 /

Sh - C(f/2)Re Sc"

(Pr > 1)

(51)

(Sc > 1)

(52)

key to make this set of equations operational for the calculation of transfer coefficients is a correlation between f and The

(Sc > 1)

(4.)

Re, see e.g. Eqs (20) or (33).

(Sc < 1)

(42)

It is interesting to note that according to the ERSR model factors Jh and id of Chilton and Colburn can be calculated From Eqs (43), (44), (51), and (52), it is obtained that for and mass transfer in turbulent pipe flow for Pr > I or Sc the following equations hold (where C is given in Eq. (29)):

where C is given in Eq. (29). 7. AN , )GY BETWEEN THE TRANSFER PROCESSES 7.1 Intr.Wuiuctlon In the foregoing sections it has been shown that the transfer of momentum, heat, and mass in turbulent pipe flow can be described in an analogou., way with the ERSR model, It is well known that the momentum flux to the interface, i.e. the shear stress exerted on the interface, is described by using a

-

C(f/2)

and

Jd " -hC(f/2)

the too. heat > I

(53)

8. SURFACE-RENEWAL MEASUREN., JTS USING A LDA According to the ERSR model, the exchang of momentum, heat, and mass between the tube wall and the co e in turbulent pipe flow is governed by the distribution and the mean value of the ages of the flu,- elements at the tube w,-1l The distribution and

24-1-4

the mean value of these ages can be derived from velocity sigrals measured with a laser-Doppler anemometer during turbulent pipe flow, followed by a discrimination procedure to detect the bursts. The time interval between two successive bursts is considered as the age of a fluid element at the tube wall. The mean value of the measured time intervals is indicated as "ex p. 8.1 Experimental set-up The set-up consists of a 51 mm inner diameter, smooth copper pipe with a horizontal, straight section of 8.5 m. There is sufficient entry and exit length for the turbulent fluid flow to be fully developed in the test section. The test section, situated 5.5 m beyond the entrance of the straight pipe, consists of a 1.4 m long glass tube with an inner diameter of 51 mm in which the velocity of light-scattering particles in the intersection volume of two coherent laser beams of a laser-Doppler anemometer is used to measure the instantaneous local axial fluid velocity Tap water is applied as measuring liquid. A laser-Doppler anemometer, operating in the reference beam mode, was used. The laser-Doppler anemometer consists of a Spectra Physics 15 mW He-Ne laser (wavelength- 632.8 nm) and quality optics, assembled by the Institute of Applied Physics TNO-TU Delft, The Netherlands, In the optical arrangement used, the angle between the two laser beams was 20.960, which resulted in a measuring volume with a length of 210 jpm and a diameter of 39 pm in air. 8.2 The burst detection criterion The burst-detection criterion of Blackwelder and Kaplan (1976), also called the 'VITA technique' was applied to the velocity signals measured with tne LDA. This criterion reacts on the crossing the fluctuating component of the axial velocity signal time-averaged local axial velocity a in a relatively short time. This corresponds to sharp accelerations or sharp decelerations. According to this criterion a burst occurs if:,

(u)-

9.1 Measured and calculated age distributions In Fig. 1 an example is given of the signals representing the instantaneous local axial velocities measured at Re = 15.0.103, showing two successive bursts. The time interval between the bursts is cnnsidered as the age 0 of a fluid element at the tube wall. In Fig. 2 an age distribution obtained from the LDA measurements at Re = 31.3.103 is represented as a histogram. In this figure the values of NIAO have been plotted against the residence time 0. The solid curve in F;g. 2 represents the age distribution calculated from f and Re with the ERSR model. This distribution is calculated with (Musschenga et al.,1990):. N1,16 - (N /to)e-0/t0

Fig. 2 shows a quantitative agreement between the histogram representing the measured age distribution, and the solid curve representing the distribution calculated from f and Re with the ERSR model. The results measured and calculated in the range 5.0 103 Re 43.4 103 show a similar good agreement. 200 180

Water: Re=31.3.10* Histogram:measured ASRe

160-Hitgrm: 140o 120N/IO O . Hz_ so-

40 20 00

a

(55)

02

04

06

0.8

2

a I en

(u )

10

1.2

14

16

18

20

/S

>

k,

where

u' - u - U

(54)

2

In Eq (54) - a denotes averaging over a (short) time of a seconds and ent means averaging over the entire signal; k is a dimensionless constant In the present study an additional condition of poste slope has been added, so that only accelerations will be counted as bursts. Application of this criterion requires the specification of two parameters, i.e , the a.,eragtng time a and the dimensionless threshold level k. A detailed description of the specification of these parameters has been described elsewhere (Musschenga et at., 1990). 9. RESULTS AND DISCUSSION Experimental results in the range 5.0-103 ( Re < 60.103 were obtained at temperatures ranging from 280 K to 25 K (the temperature was constant during each particular experiment). The distribution and the value of 0exp were determined from velocity signals measured at y+ = 36, applying the detection criterion represented as Eq. (54) and an additional condition of positive 3 slope, using a = (34.8.10- ).t 0 and k = k, = 1.13.

Fig. 2. Age distribution determined in a turbulent pipe flow at Re 31.3.103. The histogram represents the age distribution measured with a LDA; the solid curve represents the age distribution calculated from f and Re with Eq. (55) of the 2 ERSR model (P = 1.04.10 6 m to 0.291 r 3 a = (34.8-10- ).t,; k, = 1.13; N o = 5739; AO = 0.01 s). 9.2 Comparison between t andis eu, plotted In Fig. 3 the ratio to/Oexp

versus Re. This figure

shows that for 5.0.103 Re 43.4.103 the values of to calculated from f and Re with the ERSR model agree quantitatively with those measured with the LDA in turbulent pipe 0 flow. The measured values exp at Re > 43.4.103 are larger than the calculated values to This is probably due to the measured values because at increasing Reynolds numbers the ratio between the length of the LDA measuring volume and the boundary-layer thickness increases. As a consequence the velocity signals obtained at these larger Reynolds numbers refer to instantaneous velocities averaged over a considerable part of the boundary layer. 2.0 a preseni study A Mussciengac tal. (1990)

0.5

1.5

0.4

A0

A

0

0.50.3 -

b

bur

000 0

----------------0 2 65 0

Fig. 3. The ratio tolGe,

" 65.2

21104

65 4 //s

t

65 8

66 0

Fig 1. Instantaneous local axial velocities plotted against time, measured at Re = 15.0-103 and y+ = 36, showing two successive 2 1 bursts (v = I 38.10-r m s- ).

... Re

.

4 10

4

. .6 104

measured in a turbulent pipe flow at

y = 36, is plotted against Re, wherea = (34.8.10k o = 1.13. The deviation of tole.t, from

3

).t, and one at

Re > 43.4.103 is discussed in Section 9.2. From the results represented In Section 9 it may be concluded

24-1-5

10.2 Heat and mass transfer coefficients In Fig. 5 values of Nu or Sh calculated with the ERSR model are compared with experimental values presented in literature. In th's figure values of Nu or Sh are plotted versus Pr or Sc for two different values of Re. The solid lines represent values of Nu or Sh calculated with Eqs (32) or (41) for Pr > 1 or Sc > 1, and calculated with Eqs (40) or (42) for Pr < I or Sc < 1. In these two values cases Eqs (20) and (29) were applied. The dashed lines represent of Nu calculated with Eq. (35). From Fig. 5 it can be seen that the values of Nu and Sh calculated from f and Re with the ERSR model closely agree with both the experimental data and the correlation of Colburn (1933).

that both the distribution and the mean vaiue of the ages of the fluid elements at the tube wall, measured with a LDA, closely agree with those calculated from f and Re with the ERSR model. The measured random age distributions confirm the investigations of Van Maanen and Fortuin (1982, 1983), and verify Danckwerts' (1951) assumptions concerning the random-age distribution of fluid elements at the interface in a turbulent fluid flow, 10. COMPARISON WITH LITERATURE DATA 10.1 Mean age of the fluid elements at the tube wall 1 Fig. 4 a compilation of experimental data concerning the mean ages measured in turbulent pipe flow has been plotted against Re. The crosses (+) refer to data presenttd by Meek (1972). 40 * prescnistudyandMusschega ctal. (1990) * Meek(1972) 30q +

S+

I

++ -

0+20 ----- -

-

10

Eq (s6)

*

,

. .... -comparable + PinczewskiandSidcman(1974) Loushlnctal. (1985)

0 10

to'

Re

0'

10

Fig 4. Comparison between our measured values of 0+ (solid + squares) and the 0 values (crosses) presented by Meek (1972) The solid line represents Eq (56) calculated with the ERSR model. The dashed lines represent the dimensionless mean ages proposed by Pinczewski and Sideman (1974), and Loughlin et al (1985). (Parameters in Eq. 54: a = (3 4.8.10-3).to; k o = 1.13). To compare our results with these data, 0+ = uREEexEN) i plotted against Re in Fig 4 According to the ERSR model the quantity u*I(exWy) can be represented by u*Xtdp). Combining Eqs (13), (15), and (20) results in:, - u*J(t 0/v) - 5.0882 log(Re) - 4.300

0

(56)

From Eq. (56) it is obtained that according to the ERSR model a linear relationship is expected between the %ariables plotted in Fig. 4. This linear relationship closely agrees with our experimental results in the range 5.0.103 4 Re ( 43.4.103. Further, the theoretical solid line 'agrees' with the 'clous2 of experimental data presented in literature. The dashed lines in + Fig. 4 refer to values of 0 proposed by Pinczewski and Sideman (1974), and Loughlin et al. (1985), who assumed that the value of u*A 0exp/') is independent of the Reynolds number. 10,

0 Re : 40 10 SR

O 10'0Eq

(35) A Rel- 10 10' EAA Eq. (32) or Eq (41) .

N

eS/h) Eq (40) or Eq (42) vISo (1st

',

_ \,A

Eq.(35)

'0

.q. (40) or Eq (42) IMizuushina, 10'

10'

derived from the ERSR model closely agree with correlations from literature, based on experimental data. - The introduction of a Fanning number for momentum transfer (Fa = kmdli) elucidates the analogy between momentum, heat, and mass transfer in turbulent pipe flow. The Fanning number is with the Nusselt number for heat transfer and the Sherwood number for mass transfer. - The Chilton and Colburn factors Jh for heat transfer, and Id for mass transfer follow from the ERSR model. -The relationship between the friction factor and the Reynolds number is the one and only correlation needed in the ERSR model for a quantitative prediction of the momentum, heat, and mass transfer coefficients. Acknowledgements - The authors wish to thank: the Netherlands Foundation for Chemical Research (SON) for the financial aid from the Netherlands Organisation for Scientific Research (NWO), and Messrs. E. Grolman, E P.S. Sch,.uten, and W. Visser for their contribution to the experimental part of this study. REFERENCES Blackwelder, R.F and Kaplan, R.E.,, 1976, J Fluid Mech 76, 89-112. Chilton, T.H and Colburn, A.P., 1934, Ind. Eng. Chem. 26, 1183-1187 Colburn, A.P., 1933, Trans. AIChE J. 29, 174-209. Corino, ER. and Brodkey, R.S., 1969, J. Fluid Mech. 37, 1-30. Danck-werts, P V., 1951, Ind. Engng Chem 43 (6), 1460-1467. Eck, B., 1973, Tech. Stromungslehre, Springer-Verlag, New York. Einstein, H.A and Li, H., 1956, 7rans. Am. Soc. Civil. Engrs 82, 293-320 Fortuin, J.M.H. and Khjn, P.-J., 1982, Chem. Engng Sct. 37, 611-623. Friend, W.L. and Metzner, A.B., 1958, AIChE J. 4 (4), 393-402. Hanratty, T.J., 1956, AIChE J 2 (3), 359-362. Harriott, P and Hamilton, R.M., 1965, Chem. Engng Sct. 20, 1073-1078. Higbie, R., 1935, Trans. Am. Chem. Engr. 31, 365-389. Kline, S.J., Reynolds, W.C, Schraub, F.A., and Runstadler, P.W., 1967, J. Fluid Mech. 30, 741-773 Loughlin, K.F., Abul-Hamayel, M.A., and thomas, L.C., 1985, AIChE J. 31 (10), 1614-1620. Maanen, H.R.E. van, and Fortuin, J.M.H., 1982, Proc. of the Int. Symp. on Appl. of Laser Tech. to Fluid Mechanics,

Lisbon, 1982.

or Eq. (41)

Eq.(32) o

11. CONCLUSIONS - The time intervals between successive bursts in a turbulent pipe flow are distributed randomly. - The mean age t of the fluid elements at the tube wall can be calculated from f and Re with the ERSR model. - The calculated time-averaged axial velocity profile in the wall region of a turbulent pipe flow closely agrees with correlations presented in literature. - Heat and mass transfer equations for turbulent pipe flow,

)

10'

Pr (SC) Fig. 5. Comparison between 6alues of Nu or Sh calculated with the ERSR model (solid lines), and experimental values of Nu and Sh (symbols) presented in literature by Friend and Metzner (1958), Harrtott and Hamilton (1965), and Mazushina et al. (1971). The dashed lines represent values of Nu calculated with Eq. (35) of Colburn (1933).

Maanen, H.R.E. van, and Fortuin, J.M.H., 1983, Chem. Engng S. 38, 399-423. McAdams, W.H., 1954, Heat Transmission, Edition. 3 rd McGraw-Hill, New York. Meek, R.L., 1972, AIChE J. 18, 854-855. T., Ogino, F., Oka, Y., and Fukuda, H., 1971, Int. J. Heat Mass Transfer 14, 1705-1716. Musscbenga, E.E., Hamersma, P.J. and Fortuin, J.M.H., 1990, Proc. of the 5th Int. Symp. on Appl. of Laser Tech. to Fluid Mechanics, Lisbon, 1990, paper 16.6. Plnczewskl, W.V. and Sideman, S., 1974, Chem. Engng Set. 29, 1969-1976. Sideman, S. and Pinczewski, W.V., 1975, Topics an Tranport Phenomena. Hemisphere Publishing Corporation, Washington. Thomas, L.C., 1980, Int. J. Heat Mass Transfer 23, 1099-1104.

24-1-6

LI4,I11 It 1 01

jI I UM J F "J")

IN

TURBULENT'SHE/AR FLOWS Technical University of Munich September 9-1 I, 1991

24-2

SCALING OF THE TURBULENT NATURAL-CONVECTION BOUNDARY LAYER ALONG A HOT VERTICAL PLATE R.A.W.M. Henkes Delft University of Technology, Department of Applied Physics, P.O. Box 5046, 2600 GA Delft, The Netherlands.

ABSTRACT By numerically solving the turbulent boundarylayer equations along a hot vertical plate for air the and water up to a Rayleigh number of 1025, are derived. functions wall and scalings proper Turbulence is modelled by different lowReynolds-number k - , models. In the outer layer the length, velocity and temperature are scaling with the height y, the velocity maximum vmax and 2 the temperature difference vmax/(g13y) respectively. In the inner layer proper scalings do not seem to exist. In particular the wall-heat transfer does not scale with the commonly used Ray/3 law. INTRODUCTION

indeed persist in the new calculations up to Ray = 105: wall functions in the outer layer do exist. On the contrary, the calculations up to scalings and wall proper thatlayer =1025forsuggest Ray do not exist. the inner functions MATHEMATICAL TREATMENT The turbulent boundary-layer equations under the Boussines aproximation read: q a Dt + av. = 0 Dy ax av av () + = g3(r-T) + VU 2 a V 2 TU

Vxax

aT

DT

a

V a2T

The proper scalings, giving the so-called wall functions, for turbulent forced-convection boundary layers are well-known and have the typical logarithmic profiles. On the contrary, the proper scalings and wall functions for the turbulent natural-convection boundary layer along a hot vertical plate are not well defined yet. Making some physical assumptions in their analytical approach, George & Capp (1979) gave a first proposal for the wall functions for the turbulent natural-convection boundary layer. Different scalings were formulated for the inner layer (i.e. from the wall up to the position of the velocity maximum) and for the outer layer (i.e. from the position of the velocity maximum up to the outer edge). Henkes & Hoogendoom (1990) studied the scalings by numerically solving the boundary-layer equations in the Boussinesq

ax Pr D2 ax Here x is the coordinate perpendicular to the plate, and y is the coordinate along the plate; (u,v) is the velocity vector in the direction (xy); T is the temperature and T., is the isothermal environment temperature; g is the gravitational acceleration; 3 is the coefficient of thermal expansion; v is the kinematic viscosity and Pr is the Prandtl number. The dimensionless solution only depends on x/y, Pr an(! the Rayleigh number Ray (= gP3ATy 3 Pr/v 2, with AT=T,-T*,). As a result of the Reynolds averaging, the Reynolds stress -u'v' and the turbulent heat flux -u'T' appear in the equations. These quantities are represented by the eddy-viscosity concept:

approximation, using low-Reynolds-number k-E

- u'v = V

models for the tuibulence. A shortcoming of this study was that the calculations were only performed up to Ray =1012 (Ray is the local Rayleigh number, in which y is the coordinate along the plate). At this maximum Rayleigh number transition effects were still visible in the numerical results. Therefore the present paper extends the calculations up to Ray= l025. Furthermore Prandtl number effects are studied here by considering

U"

both air and water. The outer-layer scalings as derived from the calculations up to Ray=l0 12

U -+

vya

V

-v-

-

aT

u/T= -

.

(2)

(T aX ax ' Here GT is the turbulent Prandtl number for the temperature (we take OT=0. 9 ). The turbulent viscosity v , is modelled by a k - e model (k is the turbulent kinetic energy (k=ui'ui'),and s is the turbulent energy dissipation rate): k

+ Pk -8+

24-2-1

= a a

+ak + Ty D

+ v

O

k

DX

+

(3)

De

+

F

_

+ (Cei fl

+

a

+

+

ax

a

ax+

vt Ia

ce2 f2 E

Pk

v]

with Pk = vt-h--

2

k

Here vo=(gPATv) 11 3 . The calc.ltions are started at a height corresponding to Ray= 10'. At Ystart the laminar similarity solution for the hot vertical plate is prescribed. Turbulence is intro-

E+

duced at the height Ytrans (corresponding to Ray= 1.5x109 for air and 1.5x10l0 for water) by switching on the turbulence model and by prescribing an amount of turbulent kinetic energy.

' v t = cp f9ax

The precise laminar-turbulent transition turns out The equations (3) contain both high-Reynoldsnumber terms (cgt, cel, CE2, 0 k, (e) and low- to largely depend on (i) the turbulence model, (ii) Reynolds-number functions (fV,f1, f2, D, E). The the numerical grid and (iii) how much and where standard k -e model (taking fjL=f1=f2=1 and energy is introduced. Figure 1 shows the dimenD=E=0) formally only holds if the local sionless wall-heat transfer (Nusselt number, Nuy = turbulence-based Reynolds number (Re, =kE/ve) decreasing -(y/AT)(aT/ax)w) a functiondecreasing of Ray the for introducedas energy: nergy: eneasinthe ecreas the is large. If Ret is small, for example close to a coic aproer of he low-Reynoldsow-Rynods-duced, solution remains onIf the introno laminar energy isbranch. the transition. energy the delays choice of the properwal, fixed wall, a fixe number functions is required to achieve the correct Fue, the t reno n h is rach. ortunately, once the turbulent branch is reached, damping of turbulence. Many different lowReynolds-number k-e models have been formu- the solution for increasing Ray does not depend on the transition process. Figure 2 shows vt for lated in the literature, different k-e models (denoted by the names of The equations are discretized with the finite- the authors who formulated the specific lowvolume method. Because of the parabolic charac- Reynolds-number k-e model) at Ray=1 5 . At ter of the equations, an y-marching numerical this Rayleigh number the transition is complete, method can be used to solve the system. because low-Reynolds-number effects in the outer The following boundary conditions are layer are now absent: differences in the profiles prescribed: (scaled with v max and y) are totally due to different choices for the high-Reynolds-number conY = Ystart : laminar v- and T-profiles stants.

Y = Ytrans : perturbation

in

ACCURACY

k and e,

. k2ec2 kav e.g. k=0.5 v, k x

400x400 points. Table 1 shows the results for air at Ray=1011 with the standard k-s model. The results indeed become grid-independent for k, e specified by model increasing number of grid points. For the results (k=0 £=o for standard k-e) to be presented in the sequel a lOOxlO0 grid was used up to Ray =1013 and a 200x200 grid for : v =k=e=0, T =T larger Rayleigh numbers.

x = 0

: u=v=O, T=Tw

x = x

(4)

10'

0.003 v,

2 k=05v

k =0.5 v-*

N,

The numerical accuracy of thefrom calculations verified by refining the grid 25x25 upwas to

klk'=0 - klk'=0.5 x-x klk*=0 75

Vm."Y

-

0.002 -

o-.Q kk'O 9j

°

o-kk -+klk*= -"I.x

Cebec, & Smith "-' standard k - - To & Humphrey

--

:

..

.- klk'=lO

/ N

Lam & Bremhorst

... Chien

Hasi d & Porch Hoffman

....Jones & Launder

- kk --'r"O

10"

}

:

0.001

Land

-://inner

layerOuelar

0.000 .10

Iu

01

5 10 t0'

Ra

FIGURE 1. Transition for air with different amounts of introduced energy (Chien model, 100xl00 grid).

FIGURE 2. Ray= 1015.

24-2-2

1010"4

|0"

l0 [0"?"

Turbulent viscosity

i010='xly

for

air

10t 10

at

Table 2 compares the turbulent results with different models for air at Ray= 1011 and for

at Ra y= 1013.

water

4gpATy

Vt.max v

0.3622

257.3

0.3570

lOOxl00

0.1750 0.1789 0.1806

0.3544

241.1 233.4

0.1814

0.3540

229.7

400x400.

0.1817

0.3537

226.9

Reynolds-number) k-s models were given by

Henkes & Hoogendoorn (1989) and by Henkes (1990).

Nuy Ra;y3

grid

VM X

The calculated maximum of the velocity

and the turbulent viscosity is compared with the experiments of Tsuji & Nagano (1989). The wallheat transfer is compared with a value fitted to Differences different existing experiments. between the models are larger for the wall-heat

25x25

50x5O

200x200

l11 and for water at Ray= 1013. (b) water

TABLE 2. Comparison of models for air at Ray (a) air

NuY a, Ra1

Vmax

Vt.max

4~g pATy

V

120 233 210

0.119 0.381 0.218

.... 0.243 0.208

544 459

0.320 0.320

241 240

0.088 0.088

0.166 0.166

436 435

0.138 0.161

0.329 0.323

261 231

0.137 0.161

0.184 0.184

509 461

Hoffman

0.083

0.422

26

0.082

0.218

60

Jones & Launder

0.114

0.335

175

0.085

0.175

373

mNt.y Ri3 3 /a~

Vmax 4g PATY

VI'max .

experiment standard k-_ To & Humphrey

0.119

0.344

0.181 0.164

0.355 0.346

Lam & Bremhorst (Dirichlet) Lam & Bremhorst (Neumant.)

0.132 0.132

Chien Hasstd & Poreh

model

transfer than for the velocity maximum. This is not surprising as the effect of the low-Reynoldsnumber functions is strongest in the inner layer, The low-Reynolds-number models of Lam & Bremhorst, Chien and Jones & Launder perform best. These models were also found to perform best for the forced-convection boundary layer in the study of Patel et al. (1981). The standard k - s model largely overpredicts for the wall-heat transfer. None of the models predict the maximum of the turbulent viscosity in agreement with the experiment; Vt,max for almost all k-s models is roughly twice too large.

.

SCALINGS To find the proper scalings, the calculations with the standard k-6 model and the low-Reynoldsnumber models of Chien and Jones & Launder are performed up to Ray = 1025. The Ray dependence is written as a Ray, with of a quantity y=(Ray/)[D/aRay]. The infinite-Rayleighnumber limit of y, in the case it exists, defines the proper scaling for a quantity. All the quantities in the outer layer give a constant y for increasing Rayleigh number. For example, figure 3 follows the scaled maximum turbulent kinetic energy, i.e. (2kmax/3) 1 1 2/Vmax, and the scaled stream function at infinity, i.e. 0.082

0.4

air, experimental.

air, experimental.

Tsuji &Nagano (1989) qj/(v,,.,y)0O 322 Ray-""~

.!! Tsuji & Nagano (1989) V '2 lv..w'O 509 R~ay00)5

0.04-,

0.2-

T

t

,"air -1A

\

ai

standard k

standard k-

/--

o-- chin

o-o Chien x-x Jones & Launder

,

-

10

--

S10o

(a)

101

Chien i

t

ti--(u

Ra

-

- sadr air:

/

9-x Jones & Launder

0

at

Ray = 1011 with standard k-s model

(low-

of the

Details

air

for

accuracy

1. Numerical

TABLE

o

_Jl

i10

e

10 9

O

i 10I t

10 01 a (b)

FIGURE 3. Scaling of differmnt quantities in outer layer; (a) maximum of turbulent kinetic energy, (b) stream function at outer edge.

24-2-3

wwater ater, i 03

10

I Ra

0iS

10

W

I

I

+1

A-& Ra) -l0

x-x Ra, =1012

NN'

\ \

\ 0

+\

"

0. 0

-o R

t

0

-

..

10

0-

10' 1020

014 s

Ra = 10

=1014

I!!

10"5

1013

o-o Ra +-+ Ra, - Ra

\

5.

0.5p

o-o Ra_

(a)

.

0.

-4

10

tb

10"

b) 10

to-, x/Y

t0°

FIGURE 4. Wall functions in outer layer (Chien model); (a) velocity, (b) temperature. XPi/(VmaxY) (the stream function is defined by a>ily=-u,xV=O at x=0). It is clear )aiI/x=v,

that indeed the proper scalings are used in this figure, because the lines become horizontal for increasing Rayleigh number: y and vmax are the proper length and velocity scale in the outer layer (the figure only gives the results up to Ray = 1015, but the lines remain horizontal up to at least Ray = 1025). Table 3 summarizes the outer-layer scalings for different quantities. Values of y are independent of the k- E model used. Differences in the values for the proportionality constant a are small and only due to differences in the highReynolds-number constants. There is no difference in the results for air and water (i.e. there is no Prandtl number effect). Because we calculate that y and v max are the proper length and velocity scale in the outer layer, the buoyant term in the boundary-layer equations (1) is of the same order of magnitude as the other terms if the characteristic temperature difference is taken as vmx/(gpy). The calculated outer-layer scalings lead to the following wall functions in the outer layer:

v

fo[x

tVyJ

Vmax

(5)

4xj

gAT

[T - Tly [ AT

2

The existence of these wall functions for increasing Rayleigh number is checked in figure 4 for air with the Chien model. These wall functions hold irrespective of the Prandtl number. In contrast to the outer-layer scalings, none of the inner-layer quantities give fully Rayindependent *y-values for increasing Rayleigh number. Also differences between the models in the inner layer, mainly differences in a, do not disappear. The dependence of a on the turbulence model is not surprising, because the lowReynolds-number functions become active in the inner layer. Figure 5 shows the wall-heat transfer (Nuy) for air. y increases from 3/8 at Ray= 1012 to 0.435 at Ray= 1025, showing that the difference with ,=1/3, as measured up to about Ray=5xl1O , becomes larger for increasing 0.3

turbulent

laminar quantity u.

-2.11

--

VMA

Y(2kmao/3)

Ra"

11-1 4

2 82 Ra-I1

4

-3.61

Ra

N Ray 3

A-A

X-

t 1 4

14

4.0,1 Ra- 1

0.2 -

-0.0702

1 'AA

Jones & Launder water. Chien

R

00497

Nuy=O' __1

2

0.

VMU0.

-

standard k-/

a-a Chien

air and water

water

air

-

A

air:

TABLE 3. Outer-layer scalings (Chien model).

air and water ,.-experimental, I

0.

0.183 0.0354

0.

Vl.m&X YVuMX

xv' 0 u.087

y

I

0.

0.00202

0. 10

0

I0

10

10

10

10"

R.,

10

FIGURE 5. Ray-dependence of wall-heat transfer.

24-2-4

TABLE 4. Ray-dependence in inner layer (Chien model). (a) air quantity

laminar

Nu, Vmax

0.387 Ra 1 4 1.76 Ray 0.555

0.0195 RaO.14 0.0948 Ray ° ° 311 0.734 Ray 0

Xvmax/Y

1.25 Ra' 1 4

0.0237 Ray

Ray= 1013Ray RaO.0404

cly

turbulent = 102

Ray=1025

R046a0.435

0.00819 Ra OlO 0.0129 Ray 0.499 Ray0.021S

o'0 4

0.00511

0.00562 Ray0°0 0.435

5

0.00680 Ray o'

Ra0.,t Ray

195

0.00503 Ray ° 48 2

(b)water

quantity

laminar

1turbulent

Ra,=102

Ra =10 °

Ray= I015

laina quatit

Nu

0.459 Ra1 1/ 4

0.0211 Ra0394

0.00310 Rao4.46

. 0.00266 Ra0 449

cly

2.08 Ray, 0.263

0.599 Ray 0.351 Ray 0 0239 0069.0-~580 ° I° 9

0.00547 Ray 0.251 Ray0 .0160

0.00251 Ray 0.247 Ray °'°017 R 0 0535

Vmax Xvm 0 axY

16

Xvm~x/Y

1.68

1 4

/

Ray

0.0696 Ra; '

Rayleigh number. Therefore the calculations suggest that the application of the commonly known wall-heat transfer law Nu yRa1 /3 is doubtful for y R Ray> 112,y. j012> where no experiments are known yet. Of course experiments in that regime are needed to give a final answer. The scaling with the lai-

inar velocity scale (gf3ATy) 1/2 for the velocity maximum in the experiments of Tsuji & Nagano (1989) is confirmed by the calculations in figure 6. According to table 4, which summarizes the Rayleigh-number dependence for different innerlayer quantities, only a very small Ray dependence in y persists. In this table the wall-shear stress coefficient Cfy is defined by 2v(Dv/x)/gATy. Because y-values in the calculations remain Ray-dependent even up to

0.00966 Ray

Ray = 1025 , perfect scalings and wall functions in the inner layer do not seem to exist. A best fit for the results up to about Ray= 1015 (figure 7) is given by the velocity profile V/Vmax vs (X/v max)(Dv/Dx) w and the temperature profile (T-T,0 )/AT vs (= xNuy/y). This velocity profile does not agree with George & Capp's (1979) analytical wall function for the velocity in the inner layer, giving v/(gPATv) 1/ 3 vs . The temperature profile does agree with George & Capp's wall function. George & Capp proposed the following wall functions for the outer layer, irrespective of the Prandtl number. v max -v (gI3AT8QT)

0.?5

r-T T

air:

V_1

A-A

Vo--

standard k-c

v

113

ri j

(6) -.

QT

Here 8 is the boundary-layer thickness and Q T is the velocity scale -v(aTlax)w/PrAT. The

Ar

--

13

_

AT

Chien

= f

I (gf3AT8QT)

--x Jones & Launder

0.50

0.00774 Ray

Chien

boundary-layer thickness is usually defined as 05..

-------- -elocities 0. 00t~ 10

air, experimental: Tsuji & Nagano (1989) v/V'jpAi)

lO 1

1010

l0"'

1012

o0 407 Ra; °0 103

R,

8=NA**/Vnax;

this indeed is proportional

according to table 3. The proportionality fortothey holds if o V max '3 /2 Nuy = C* Pr I1/ Ral/ y%

IgF-ATy

10

FIGURE 6. Ray-dependence of velocity maximum.

(7)

of (also independen, C* is aofconstant in which Pr). Comparison the calculations for Nuy a.,d

24-2-5

r

-

1.0-

1.0 A-A Ra, = 10 X-X Ray- 1012 o-0 Ra, 1013 0-0 RaR 10"

0.5--

0.0 10-2

X-

1

Ra) = 1012

-= T-T0,-

-

Ra R 10151 1020 R_ Ra1=0102/ ey 25 Ra 10

i0 -

- R

Ra.=101 = 1015 RRol)- 1020

0.5--

° l012

.L

1

"010-1 0.0avl2I

0

10-1

1,00

101

10z

10,

(a) V, a (b) FIGURE 7. Profiles in inner layer (Chien model, air); (a) velocity, (b) temperature. vmax in table 4 shows that this relation indeed holds for increasing Rayleigh number; the largeRayleigh-number calculations (Ray = 1025) with the Chien model give a Ray independent, and also practically Pr independent, value for C*, namely 0.0506 for air and 0.0529 for water. Therefore George & Capp's outer-layer length scale, 8, and velocity scale, (gAT6QT) 1/ , are similar to the calculated length scale, y, and velocity scale, v max:. this gives a full consistency between George & Capp's wall functions in the outer layer and the calculated wall functions. CONCLUSION Accurate solutions of the turbulent boundary-layer equations along the hot vertical plate in an isothermal environment were obtained for air and water up to Ray = 1025. Low-Reynolds-number modifications of the standard k-e model are needed to predict quantities in the inner layer (like the wallheat transfer) reasonably close to the experiments. The standard k-, model largely overpredicts the wall-heat transfer. The low-Reynolds-number models of Lam & Bremhorst, Chien and Jones & Launder give the best prediction for the wall-heat transfer. The k-e models strongly overpredict the maximum of the turbulent viscosity by about 100%. The proper scalings and wall functions can be derived from the k- s solutions by examining the Ray dependence of quantities in 4=oxRay. Model differences mainly influence ,:.3 cc values in the inner layer, whereas 7 is almost independent of the model used. y for quantities in the inner layer does not become fully Ray-independent. For example y for the wall-heat transfer (Nuy) for air increases from 3/8 at Ray=1012 to 0.435 at Ray = 1025, showing that the difference with ,y=1/3, as measured up to about Ray=5XlO1 1 , becomes larger for increasing Rayleigh number. Because , remains Ray-dependent, wall functions in the inner layer do not seem to exist. A best fit

*

for the results up to about Ray = 1015 is given by the velocity profile V/Vma, vs (X/v max)(av/X)w and the temperature profile T - T )/AT vs (= xNuy/y). This velocity profile does not agree with George & Capp's (1979) analytical wall function, but the temperature profile does agree. Th., y values in the outer layer become independent of Ray for Ray -4 o., implying that wall functions in the outer layer exist. The proper length scale is y and the proper velocity scale is v ax. Tht proper temperature scale is Vmax/(g3y). These outer-layer scalings are consistent with George & Capp's proposal. vmax itself approx:mately scales with the laminar velocity scale (gPATy)/ 2. REFERENCES George, W.K. & Capp, S.P. 1979 A theory for natural convection turbulent boundary layers next to heated vertical surfaces. Int. J. Heat Mass Transfer 22, 813-826. Henkes, R.A.W.M. & Hoogendoorn, C.J. 1989 Comparison of turbulence models for the natural convection boundary layer along a heated vertic,, -!ate. Int. J. Heat Mass Transfer 32, 157-169. Henkes, R.A.W.M. 1990 Natural-convection boundary layers. Ph.D. thesis. Delft University of Technology, The Netherlands. Henkes, R.A.W.M. & Hoogendoorn, C.J. 1990 Numerical determination of wall functions for the turbulent natural convection boundary layer. Int. J. Heat Mass Transfer 33, 1087-1097. Patel, V.C., Rodi, W. & Scheuerer, G. 1981 Evaluation of turbulence models for near-wall and low-Reynolds number flows. Proc. 3rd Symp. on Turbulent Shear Flows, Califoinia, pp. 1-8. Tsuji, T. & Nagano, Y. 1989 Velocity and tempefature measurements in a natural convection boundary layer along a vertical flat plate. Experimental Thermal Fluid Sci. 2, 208-215.

24-2-6

EIGllTH SYMPOSIUM ON TURBULENT SHEAR FLOWS Technical University of Munich September 9-1 I, 1991

24-3

THERMALLY DRIVEN TURBULENT BOUNDARY LAYER

T.Tsuji, Y. Nagano and M. Tagawa Department of Mechanical Engineering Nagoya Institute of Technology Gokiso-cho, Showa-ku, Nagoya 466, Japan

ABSTRACT

INTRODUCTION

The budgets of turbulent. energy, Reynolds shear stress, mean square scalar variance and turbulent heat fluxes have been evaluated in a thermally driven boundary layer in air along a vertical flat plate. In the near-wall region, the velocitygradient correlation plays an important role in turbulent energy production and thermal energy is directly converted into kinetic energy through this correlation, Consequently, Reynolds shear stress and streamwise turbulent heat flux near the wall, which are normally considered to have close relations with the energy production process, behave differently from those in the ordinary turbulent boundary layers. On the other hand, the energy transfer in the outer region of the boundary layer becomes a process similar to that observed in the usual boundary layer.

Turbulent natural convection boundary layer in air along a vertical plate is typical turbulence of thermally driven flow. The turbulence structure in this boundary layer has been generally considered not to differ significantly from that in forced convection, though the buoyancy effect is being added. However, when investigating the boundary layer, some marked characteristics dfferent from those in ordinary turbulent boundary layers have been found with respect to the viivcous sublayer structure and Reynolds shear-stress near Also, no the wall (Tsuji & Nagano 1988a, b). coherent structures such as intermittent bursts and low-speed streaks observed in the near wall region of ordinary wall shear flows were found to exist in this boundary layer (Tsuji, Nagano & Tagawa 1990). The present study aims to trace the origin of the structure peculiar to turbulence driven by buoyancy force.. We have investigated the budgets of turbulent energy, Reynolds shear stress, mean square scalar variance and turbulent heat fluxes on the basis of velocity and temperature measurements in the turbulent natural convection boundary layer, and have examined the essential factors associated with the formation of turbulent structure. The information obtained in the present study is very important to construct a turbulence model for turbulent buoyant flows.

NOMENCLATURE cp specific heat at constant pressure dissipation rate of mean flow energy E 3 Grx Grashof number, gft (T. - To)X /V 2 g acceleration of gravity k turbulent energy, (0 + V2 + _w)/2 p pressure fluctuation qw wall heat flux T mean fluid temperature t temperature fluctuation tr friction temperature, qw/pcpUT U, V mean velocities in x and y directions u, v, w velocity fluctuations in x, y and z directions ur friction velocity, /-Tw;7 x, y, z streamwise, normal and spanwise coordinates y+ dimensionless distance from wall, uTy/v a thermal diffusivity 6 volumetric expansion coefficient, I/T. e dissipation rate of k, e u + e v + e w 8u, v, ., E t dissipation rates of u2/2, v2/2, w/2 and P/2 SUV, a ut, Vt dissipation rates of T-V,UT and v-t v kinetic viscosity p density Tu time scale of velocity field, k/a 'rt time scale of thermal field, (t7/2)/a t SW wall shear stress f) time mean value Subscripts w wall condition o ambient condition Superscript + dimensionless quantities normalized by wall variables

EXPERIMENTAL APPARATUS AND INSTRUMENTATION The apparatus used for the present study is the same as that described by Tsuji and Nagano (1988a). The flat surface generating flow was a copper plate 4 m high and I m wide. Uniform surface temperature Tw= 60 'C was obtained by electrical heating. Ambient fluid temperature T. was about 16 *C . For the measurements of fluid velocity and temperature, two types of probe were used. One comprised a normal hot-wire and a coldwire of 3.1 g m diameter tungsten, and was mainly used to measure streamwise velocity and temperature. The lengths of the wires were 1.5 mm (n6v /ur) and 3.5 mm (c 14v /ur), respectively, and the cold-wire was located 2.5 mm (n 10v /ur) upstream of the hot-wire. The other consisted of a coldwire and V-shaped hot-wires (Tsuji & Nagano 1989) combined in an X-array, which were constructed by symmetrically bending 3.1 gm diameter and 1.5 mm long tungsten wires at the center. This probe was used to measure streamwise and normal (or spanwise) velocities and temperature. BASIC TURBULENT QUANTITIES IN VELOCITY AND THERM4AL FIELDS The profiles of mean velocities, mean tempera-

24-3-1

ture and intensities of velocity and temperature fluctuations in the turbulent boundary layer, which are obtained simultaneously with V-shaped hot-wire/cold-wire arrangement, are shown in Fig. 1. In evaluating uT and t-, the wall shear stress Tw and the wall heat flux qw were obtained from the ncar-wall profiles of velocity and temperature (Tsuji & Nagano 1988a). The results measured with a normai hot-wire and a cold-wire are also Excellent agreement presented 'or comparison. between measurements obtained in different manner for each profile verifies the high reliability of Mean velo-ity U4 becomes the present results. maximum at y* 40. The distinctive feature of this profile is that the relation U*-y+ does not hold even at y+:- l and the logarithmic velocity profile cannot be seen at all (Tsuji & Nagano 1988a). In the f* profile, however, the relation + T = Pry' holds for y+_ 5, and the logarithmic profile appears in the region y+> 30 as in the The maximum intensity of usual boundary layer. streamwise velocity fluctuation/ P occurs at yi 250 in the outer layer (beyond the maximum The intensities /" and /7 velocity location). are smaller than I in the entire flow zone and become maximum at almost the same location as The intensity of temperature the /P maximum. fluctuation I7 becomes maximum at y' 15 in the inner layer (from the wall to the maximum velocity

0

Gr =899x 7 Calculated fr

20

-- /ut -'-,

continuity equ-t-n

Normol hot-wire doto 44 (Gr 8 X1010)

2 " 3'

- i Uncertainty (95%Coverce)

5

D

2

10

5

IV.u ___

__

_V.

0 -

10 J 10

0

,i3 .j 1By

102

Gr

2'v/

10

-

using the boundary-layer approximation for the governing equations of the velocity mal fields in thermally driven flows Yaglom 1971), the transport equations flow and turbulent energy components, shear stress, temperature variance and heat fluxes may be written as follows:.

2 3 x

U

-

2 ay (a)-----c

1(:

U a11

Uvl -q

82 y

2Ox

d

+

(c)

gf8U(T - T.) + E

'Ce) Mf a 8--y TurbulIent_ energy p9ont§~.

~

ay

(b)

2 Ox.

,

i UG_ +

1

--

7tNr

10

V aU

V a 2 uz

*yg-

/ultuv SOUti :30

0

2

2

U 0U

---- Clculted from energy equation ° _ Norml hot-wire dOto(Grxz844x10 ) r3(a) I Uncertainty (95ycaeroge /

on

and ther(Monin & for mean Reynolds turbulent

Hean flow energy

9 9 x10i

Colculoted from momentum equton

-

.

*2

8

As demonstrated above, the natural convection boundary layer has unique characteristics in the turbulent structure, which are rarely seen in other turbulent boundary layers. Next, we examine the energy tr.,,fer processes which link the velocity and ' _rmal boundary layers.

y+

Fig. I Profiles of mean velocities, mean temperature and intensities of velocity and temperature fluctuations

3

uR distribution is also fairly different from that in the usual boundary layer in which R takes a negative value near the wall in the case of heated flow. BUDGETS OF TURBULENT ENERGY, REYNOLDS SHEAR STRESS, TEMPERATURE VARIANCE AND TURBULENT HEAT

5

-

location). The distributions of Reynolds shear stress ii-, streamwise turbulent heat flux R and normal turbulent heat flux vt obtained with the V-shaped In the hot-wire probe are shown in Fig. 2. figure, the solid and broken lines indicate the calculated values of i-v and iTt, respectively, which are estimated from mean momentum and energy equations with the measurements of the streamwise variations in mean velocity and mean temperature profiles. Both measured Ui- and VT agree very well Also, the streamwise with the calculated values. turbulent heat flux R measured simultaneously agrees very well with the measured result obtained by using the normal hot-wire probe. From these comparisons, the present measurements for Reynolds shear stress and turbulent heat fluxes are judged to be sufficiently reliable. The UT value near the wall is almost zero and increases in the positive direction as the maximum velocity location is approached. Consequently, in the inner layer of the natural convection boundary layer, Reynolds shear stress has no relation with the mean velocity gradient. The normal turbulent heat flux -t becomes smaller than the streamwise turbulent heat flux R in most of the boundary The vT distribution corresponds rather layer. well to the measured result for the usual boundary The Nagano & Tagawa 1986). layer (Hishida, streamwise turbulent heat flux R-, which takes about zero near the wall, increases rapidly in the location positive direction near the maximum /72 and reaches maximum at the location almost equivThus, the alent to the maximum velocity location.

(a)

0

uv

v 0z2

1+V

(e)

Fig. 2 Reynolds shear stress and turbulent heat fluxes

24-3-2

y

(g)

+Y + -

-

a

=U

+

1U

T-

x

p

(c)

(d)

_9-g+ EU (f)

(1)

n

2 i9y

(b)

0

(g)

=

0

(2)

u a v21v a__ v2 1I aP

UO 20x

V -O

uv

= Cv"20

+ 1 0y 5 +V 4OuOv +i--+ JV pp ay (c)

(a)

(d) V 0927 v+

2V

Ft

= 0 2 COy0yZ,

(3)

+ |1-.

OuOv Ox-x+yT y OuOv O-z -z of I'(ax) OX

Ct

=

+

OvS+Ct 7U 1 ay 2 a- VL 2 -2j'+W -- a)---(--

(e)

(g)

.olds ... shear sres aomitted, + Oy --

-

Ox

(a) S

-L ay

+ v2

y

(b) -

u

Ox 9

(c)

a 2T

--

aspectra y

y .

(d)

(e) - gA69 + (f)

Uv = 0

(5)

(g)

Temperature variance U OP 2 Ox

+ 7

Vt 2 3y

OT y

(a)

O vti 2

ay

(b) (c) a 0 2t + E 2 a yz (e)

0

(6)

(g)

Turbjjiejt, JeALf luxes

V 0a yU ++ -

u U ----

-

(a) t

(d) g,6 (f)

U-9 1 aX

-

V

7

Oay

(a)

-

+

C

ut = 0

O y1

(b)

velocity(or temperature)-pressure gradient correlation, molecular diffusion, buoyancy production and dissipation, respectively. The values of each term in these equations are estimated by using measured values and their balances are shown in Figs. 3- 5 and 8- 11. The budget of w2/2 is because of the difficulty in measuring the turbulent diffusion term 0 v--w2/a y. As commonly done, we assume isotropy of dissipated motion, we estimate dizsipation terms as e u= E Y =Ew- E/3 and Euv=Eut=evt=O. The e value is obtained by integrating the u fluctuation multiplied by square wave number. The velocity-pressure gradient correlation, temperature-pressure gradient correlation and dissipation rate s t were evaluated as the closing terms in the budgets from the measured values of all the other terms in relevant equations. To prevent confusion, less contributive terms are not shown in Figs. 3- 5 and 8-11. Figure 3 presents the balance of mean flow energy normalized by ur4 /v. The energy produced by buoyancy, which becomes maximum at yl- 10- 15, is transported to the near-wall region by molecular diffusion and to the outer layer by turbulent diffusion.. Also, part of the energy is convected downstream. In the outer layer, the mean flow energy is consumed for the production of turbulent energy through deformation of the mean motion by Reynolds shear stress, -ii(O U/O y), in a way similar to the case in the usual boundary layer. However, in the inner layer, the turbulent energy is countertransferred to the mean flow through Reynolds shear stress, which never occurs

(7)

appears in the u7/2 balance. In the outer layer, the sum of production te.ms due to buoyancy and deformation of the mean motion is balanced with the sum of turbulent diffusion, velocity-pressure

O-t-0.6 y aOy (c)

-04

)_

- 2

t 6 P-- 5 (vt ( t0-j-+ a(d)

-vOt) Oz 9 a

Y

(g)

ou OT 7t a + _V2 a a

+v Oy

in the ordinary turbulent boundary layer. The balances of the turbulent energy ui/ and 4 v-/2 normalized by UT /v are shown in Figs. 24 and 5, respectively, The buoyancy production g'6-

(e) -

a-

+ y -y, y

(b) (c) v--_-_t Y+ au- ) t y (y

Sy -

T + a iv-

+ uv

D-vt+

In the above set, the terms (a)- (g) represent advection, production, turbulent diffusion,

(d)

v a 2-2 2 a y2

Va y

zaz

vt

v= a)Ma xT

71

(c)

-

t)2 'y Oz T ~--oyut + + -----

ua t (v+a)( ,a x~Wx+ _a

U -:-+

_y-+

(g)

(e)

(a)

9__9 I II9I I

Grx=899x10' °

_uo UZ+ VaUi 2 ax 2 ay

" __a

ay

(e) + EVt=O

(8)

0-

(g) where, w here,

1-

Oy e U

V[(o

I + (-P-) + (1JU)] SO y aZ

i)

0.2

04 0.6

"'"-

/

g, U(T -T)

....

C w

x

~24-3-3

"Oy"

Oz

(9)

E

, il, I

10 V V [('- + T. 'Y' l + (_X)2] z t Ox + y O--z

---------- ay

m I

102

V0

y00 Fig. 3 Balance of mean flow energy U12/2

gradient correlation and dissipation terms. It is apparent that the energy is transferred to the other fluctuating velocity components through the velocity-pressure gradient, while being transported to the inner layer and outermost region by the turbulent diffusion. In the nearwall region, however, the marked energy production by the velocity-pressure gradient correlation u(,9p/Ox) is observed, and is balanced with the production due to deformation of the mean motion, molecular diefusion and dissipation. Consequently, turbulent energy is countertransferred

to mean flow through Reynolds shear stress.

possible energy transfer process in the outer layer. In this region, the energy transfer can be considered to be a similar process as seen in the usual boundary layer.

-006

Grx=8.99x1

o

E au -8x1 a -v ay I = 0 x

-004 -i Z
0 V>0

outward

1 interaction u

wallward 3 intera ction V

. ..

o-o. 0.14 0.25 0.36 0.43 0.50 0.57 0.75 0.86 1.0

0.5

()

luvi = H'u'v'

X

X X X

Figure 3

24-5-2

4 sw eep

UNZ500

-20 -40

1000o>N it750

-60,

1250>N-1000

-80--.--160

I.

10Z

20 mm

1, ________

,'

.

(b) Injection into the wake 0 Fig. 9

Clockwise

9

Wake

--

0

5

,(/U

.Xl2

(b) Profiles in the z-direction

Counter-clockwise

Revolution of the windmill and the flow ( x/d = 31.8 )

0

10

Z~~ MM

secondary

Fig. 10

25-3-5

Turbulence intensity profiles in thp central cross-section of the jet ( Uj/UO = 3 )

15 162z

50m

wake O:Uj...U2= Je

7:

10

z

-0m

4,

100 8002

600

400

0200

x/d

Fig. 11

Hz 2 10 k 0

1k

Fig. 13

Spectral distribution

behind

the

circular

cylinder ( Uj/U0 = 3

Variation of the center-line turbulence intensity of the jet and the wake

20

(b) xid =141 (a) x/d Fig. 12

=

31.8

of the turbulence Three-dmensional profles

s not found in the central However, the peak value ± 0.8, thereby, it can z/D z-drecton part of the vanish due to the vortex K~rm~n the that be sad interference wth jet.

COCUIN

nece

rm

lcua

oe

n a circular cylinder, even drilled in the rear of exflu id, quiescent into ion ject the case of an in flow spread, and a secondary h bits an asymetrical wake, the injection into the of case the In 06 exists. r: becomes rather remarkabe, the asymetrical spread than becomes even stronger flow and the secondary Although the fluid. quiescent into with injection velocity in relation to decay of the relative maximum velocity the injecton to 213 the wake differs according . to aoout x rat io,t decays in proportion the wake becomes low of intensity turbulence he the In the present flow field, by njecting the jet. upon the reladepends profile turbulence intensity intensity of the jet tionship between the turbulence from the power clarified was the and that of the wake. It vortex vanishes due to spectra that the Krm.n interference by the et.

intensty ( Uj/U

0 =

3)

REERENCE [I]

[2] 3

and Park, W. N., Cimbala, J. P. 479 1990, Vol. 213,

J. Fluid Mech., ..

and Okano, T., Rep. Inst. For example, Tuji, H., 196, Noy.1,ui 78u c Toky Kuni., .3, ., iAeronai

1615 P. Vol. 2, No. 9, 1990, P. Trans. AS E, N. V., and Hylton, L. D., N irmalan, 1990, P. 447 112, Vol. J. Turbomachinery, H., J. Fluid Mech., (5]Wygnanski, I., and Fiedler, 577 Vol. 38, No. 3, 1969, P. ASME, J. and Whitelaw, J. H., Trans. 0., [6] Dur~o, 3, 1973, P. 467 No. 95, Vol. Eng., Fluid No. J. Fluid Mech., Vol. 23, (7]Bradbury, L. J. S., , 1965, P. 31 Natl. [4]

[8]

I., and ani,

Kobashi, U.,

Japan 1st.

P. 465 Congr. Appl. Mech., 1951, Aeronautical Troucer, J., and B., H. Squire, [9] , 1944, P. 9 and R Council, Research Eng., Vol. Trans. Jpn. Soc. Mech. Isigaki, H., 10] Japanese in 1692, P. 1982, 48, No. 433, B, P. 63, Turbulent Jets, 1976, R ajaratnam,N., (ii] Elevier

25-3-6

ElGlrrii SYNMpOSIUNI ON TURBULENT SHEAR FLOW'-

25-4

Technical University of MunichSeptember 9-1I, 1991 TURBULENT MIXING OF MULTIPLE PLANE AND AXISYMMETRICAL JETS R. Karvinen, P. Saarenrinne and H. Ahlstedt Tampere University of Technology P.O.Box 527, SF-33101 Tampere, Finland

ABSTRACT The mixing of plane and axisymmetrical round jets with a fully developed turbulent velocity profile is studied theoretically and experimentally. It was observed that in the k-c model the e-equation must be modified if flow separation exists to the extent that mean velocity is r.tarded. The mixing of axisymmetrical jets with a hexagonal array can also be handled by

h

b

approximating them with a two-dimensional axisymmetrical jet.

U

The importance of correct boundary conditions, i.e. the profiles of velocity and turbulence energy at jet outlets, for mixing was found.

(a) Plane Jets.

udo x"'-

OMENCLATURE

b

channel width

colcc 2, Ce3

turbulence-model constants

dode. h k k1,k2

outside and equivalent tube diameter channel wall thickness turbulence kinetic energy experimental constants in self-preserving solutions rad;al coordinate equivalent tube radius = d/2

r r,

r umx

(b) Axisymmetr:al jets with a hexagonal array. Fig. 1. Jet configurations and arrangements.

Ue U.0 01Y

turbulence intensity k/Uoo streamwise velocity mean velocity components, tensor notation turbulent velocity components, tensor notation velocity defect Uoo - u maximum velocity defect bulk mean velocity cartesian coordinates

e u , et' Uk, or

dissipation rate of turbulence energy eddy and effective viscosities turbulence-model constants

T. u us~uj u',U" III

u(xly)

INTRODUCTION The mixing of parallel jets is utilized often in engineering applications. The origin of this work lays also in actual practice, namely in the mixing problem of a fibre-water suspension in the hydraulic headbox of a papermachine, where geometries of Fig. 1 are used to generate the desired flow and turbulence characteristics. The main facts affecting the paper quality are the decay of velocity differences and suc', turbulence that fibre flocs are broken. Turbulence and velocity profiles can be affected either by a wall friction or a flow separation from an abrupt expansion in a flow geometry. In addition, the spectral

distribution of turbulence should be such that the movement of fibres with the length of 1... 4 mm and the thickness equal to 20... 40 pm is affected by turbulence. There does not exist very much information concerning the mixing of parallel jets in the literature, althougth .ingle jets have been extensively analyzed, so that the informatiot, of published papers can be found even in the books (Rajaratnam 1976; Schetz 1980). When considering the situation in Fig. 1 the flow behaviour is affected by turbulence and velocity at the channel exits. There are some papers where the effect of turbulence in free-boundary flows, i.e. mixing of a single jet or the recovery of a wake has been studied (Patel & Chen 1987; Mohammadian, Saiy & Peerles 1976) but in the case where many jets are coupled together the information is thin. The effects of freestream turbulence on velocity boundary layer and heat transfer have also been studied considerably in cases where the fluid flow is past an obstacle or a wall (Simonich & Bradshaw 1978). In general, it has been noticed that free-stream turbulence does not appreciable affect the boundary layer near a fixed surface, but if we have a free boundary layer the initial turbulence level of ajet or a free-stream can have a significant effect on the flow behaviour. In this paper an analysis of the mixing of multiple plane or axisymmetrical jets is given as mentioned above. The mixing process of jets in Fig. 1 is affected by turbulence and velocity profiles of exhausted jets, and also by recirculation. Flow re-

25-4-1

circulation exists in the case of plane jets when wall thickness separating jets is not negligible and for axisymmetrical jets with a hexagonal array recirculation is always present. The research

o

1,r

procedure includes analytical, numerical and experimental ap-

UXr

proaches. The effect of various factors, i.e. velocity and turbulence profiles of exhausted jets and recirculation on the flovd behaviour is shown.

Uc

X

de

x)

U1

re

FORMULATION OF THE PROBLEM AND SOLUTION PROCEDURES

Fig. 2. Definition sketch of approximated circular parallel jets.

The flow configurations studied experimentally and theoretically are those in Fig. 1. In the case of mixing of plane jeb, the channel height is large compared with the width b. Taus, we have a two-dimensional problem which is governed by ellipdistance tic equations near the channel outlet, and at a certain from the channels the flow field is parabolic. In Fig. 1(b) a large number of axisymmetric jets are exhausted from parallel tubes with a hexagonal arr;y. In this case the flow field is everywhere three-dimensional, even if the tubes have hexagonal cross sections as shown by a dotted line.

thnt now r = 1in Eq. (1). The resulting self-preserved velocity profile takes the form

In the case of circular tubes there is always a flow separation and recirculation near the tube outlet even for very thin-walled tubes. In order to simplify the problem it is assumed that the flow field can be handled as a two-dimensional axisymmetrical jet, the cross sectional area of which is the same as a hexagonal tube drawn with a dotted line has in Fig. 1(b). Thus, an equivalent tube diameter d, is equal to 1.05 times the outside diameter d. of a tube (see Fig. 2). The mixing problem described above has been tried to solve theoretically by adopting an analytical and a numerical approach and the validity of the results has been verified experimentally.

b4 1 Y u(x,r) = Uo[1 - .-- cos(2rT)J

(3)

where k 2 is again an empirical constant. The result (3) has also been derived earlier using another kind of treatment in analysing the mixing of wakes (Schlichting 1979). The same kind of result as (3) can also be obtained if the eddy viscosity is proportional to X- ' (Eisner & Zielinski 1986). Ectuations of numerical modelling The numerical calculations of this study have been performed employing the eddy viscosity concept and the k-e model. However,, it was observed that the standard k-c model was unable to predict flow behaviour correctly when flow separation exists after channel outlets resulting in recirculating flow and in the retarding the mean velocity. For that reason the turbulence model was modified according to the suggestions of Hanjali and Launder (1980), so that the transport equations of turbulence kinetic energy and its dissipation rate are in cartesian coordinates

ok A~- k + V-)k

1

o L )+8(tO

o(Ok,

a, akOu

) P-- P(4) +14t(

Analytical approach Let us consider the mixing of parallel circular jets in Fig. 1(b). According to two-dimensional assumptions made earlier, the form of the velocity profile of a mixing jet is that shown in Fig. 2. At a certain distance from the jet inlet the velocity defect vi(z,r) = U,, - u(x,r) is small in comparison with the uniform mean velocity U,. It that case the momentum equation takes the same form as usually used in analysing wakes -Ueo

S

-

10 -,rv

ou, )

1 -.

(1)

Now, if it is assumed that the eddy viscosity is proportional to the maximum velocity difference vt = ku,(z) (k is an inspecified constant) a self-preserved turbulent mean velocity profile of mixing axisymmetrical jets with a hexagonal array can be derived using the method of ref. (Cebeci & Bradshaw 1977). The final result not known in the literature is (Karvinen ,991) -- )] u~x,,,) = U [I'+ Le-.1o(.832 2

-Jo(3 )k€1 x:

(2)

0 p(u-9 + V 0 9 Oo p~~~u+' v- ~ (( ) + --O9 OY Ox 0" Ox 9Y o" 09Y O e e Ou 2 - c1 2p-T - c.3pFX +ctP(y)

(5)

The empirical constant c,3 takes the value 4.44. In the standard k-c model the terms underlined in Eq. (4) and (5) are missing. These kinds of modified equations have also been used in the calculation of external boundary layers with adverse pressure gradients (Rodi & Scheuerer 1986). Boundary conditions for the governing equations are that gradients of u,v,pk and c must be zero both at the centerline of a jet and also at the separating iFne Letween two jets. No wall functions were needed. The set of g'werning equations was solved using the PHOENICS code (Spalding 1981), into which the modifications of the turbulence model were implemented. The PHOENICS code is based on the finite difference method 'in which an upwind difference scheme is adopted and the iteration is made using the SIMPLEST algorithm.

r,

where k, must be determined experimentally and J0 is the Bessel function. The mixing of plane jets far enough downtream from the channel outlet in Fig. 1(a) can also be solved as above except

When use is made of the finite-difference method, it also restricts the shape of control volumes. For this reason also in the numerical modelling of the mixing of circular jets an axisymmetrical assumption such as in the analytical treatment above was adopted.

25-4-2

Experimental research

1.50

Experiments were made using air as a flowing fluid. In order to get some generality of results the length of an experimental set-up in F:-. 1 was about 3 m in order to reach fully developed turbulent velocity profiles at jet outlets for both geometries. The width of a channel in the case of parallel plane jets was 14 mm the and walls with a different thickness h was used to study

u U.

x

x/b data I a 3 a 6

3 1.00 -

effect of recirculation, caused by flow separation, on the mixing process. Thr test installation of axisymmetrical jets consisted of the package of round tubes with the outside diameter d0 equal to 28.3 mm and wall thickness 1.7 mm.

waLL 0.50 0.00

0.20

0.40

0.60

0.80

10

2ylb Measurements of mean velocity and turbulent fluctuation constant temperavelocities were determined with a standard ture hotwire anemometer (Dantec). For measuring the streamwise mean velocity and its turbulence a straigth single wire probe was used. In order to get an idea about the isotropy of turbulence also other components of turbulence, in addition to the stream-wise one, were measured by using an X-wire probe. Velocities used in expariments were such that the bulk mean velocity Uoo after jet mixing varied from 15 m/s to 20 m/s.

(a) Velocity. 0.15 k U

1-

x/b dat a

6 -

0 6 i

k 7

005

0.20

000

040

WOO i

060

RESULTS AND DISCUSSION

0.80

1.00

2y/b

The mixing problem could be very easily solved if the analytical solutions shown above can be used. However, it was observed after comparing analytical solutions with the measurements that the constants needed in equations were dependent on the initial condition at jet outlets. Also, th-: distance were self-preserving solutions are valid is large and the detection of velocity differences is difficult. Thus, it can be concluded that analytical solutions are valuable in the respect that they -- nw the decay of the velocity difference to be proportional to x from the outlets for a certain geometry.

(b) Turbulence kinetic energy. Fig. 3. Mixing of fully developed turbulent plane jets with negligible wall thickness. - standard k-c model, - - modified model.

In Fig. 3 the decay of the profiles of mean velocity and turbulence kinetic energy is shown for plane jets when the thickness of a channel wall is very small, such that there is no flow recirculation. It can be seen that the effect of a turbulence model modification is very small and the standard k-e model predicts mean flow behaviour correctly. The required profiles of velocity, turbulence kinetic energy and its dissipation rate at channel outlets were also obtained numerically by calculating fully developed turbulent profiles in a channel. The Reynolds number was equal to 35800. These were in reasonable agreement with measured data in the literature (Hussain & Reynolds 1975). By measuring different components of turbulence it was observed that streamwise, lateral and transverse components have the same kind of shape and that when the distance from a channel outlet increased an isotropic condition was approached. Fig. 4 shows the measured profiles of different turbulence components. In order to evaluate the possibility of two-equation turbulence models to predict flow behaviour, it is enough to measure only the streamwise component of turbulence, of which the kinetic energy of turbulence can be obtained. The results showing the effect of flow separation on the mixing of plane jets are shown in Fig. 5. The geometry of a channel s the same as in Fig. 3, except that now the wall thickness between channels is 4 mm. The position of a wall is shown also in the figure. The ratio of a wall thickness h and a channel width b is equal to 0.286. Calculations have been made using three kinds of turbulence models: the standard k-e model, a modified model according to Eq. (4) and (5), and a third one, in which only the r-equation is modified. Velocity fields of Fig. 3 and 5 were computed on a 51 (streamwise) x 18 (cross-streamwise) grid.

015 u: -U"

1 u'

0.10

o W,

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2

0.05 wait 00 0

0

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, x/b,2 a a o 68

0.05 r

000

3

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1

i

2 a u' A v. ° 0 w

c,

02

0.4

wall

06

0.8 1.0 2y/b Fig. 4. Profiles of turbulent components for plane jets at different distance from jet outlet.

It can be seen that the model in which only the e-equation is modified agrees best with measurements. This kind of result was also obtained by Rodi and Scheuerer (1986) when calculating boundary layer flows with an adverse pressure gradient. Although the flow of our paper is quite different from that of Rodi, it seems that the same kind of modelling as in the case of wall flows is also valid for free shear flows. An interesting observation is made if velocity profiles of Fig. 3 and 5 are compared. It is seen that at the distance of z/b = 6 from jet outlets the nonuniformity of velocity is almost the same, although at the

25-4-3

1.5U"o

data

x/de

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0.75

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050

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030

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(a) Velocity decay.

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/de

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(b) Turbulence kinetic energy.

I -----040 0.60 2r/oe

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1 00

Fig. 7. Comparison of measured velocities and turbulence energy between different cross-sections in the mixing of round jets.

,jet

outlets in Fig. 5, maximum velocity is much higher than that in Fig. 3. Thus, the flow separation creates a turbulence which

T.

very effectively mixes the flow field and gradually eliminates the

u

nonuniformities of a velocity filed.

The last figure dealing with the mixing of plane jets shows effect of turbulence kinetic energy of jet outlets on the mix-

_the

ing. In Fig. 6 there are results for two different cases. Solid lines show the decay of velocity profile when at a channel outlet the profiles of velocity, kinetic energy and its dissipation rate are those of a fully developed turbulence channel flow'. Dot-

2 b

0.75

ted lines show similar results, but now it is assumed that the turbulence kinetic energy is uniform everywhere and equal to maximum value near the wall. The dissipation and velocity profiles are those of a fully developed channel flow. It can be seen that the turbulence kinetic energy has a great effect on mixing, and that the latter is also promoted by an increase in

Ithe

o.00.00

wall I 0.20

(b) Streamwise component of turbulence.

Fig. 5. Mixing of fully developed turbulent plane jets with flow separation. - k-e model, - . - the k- and c-equation modified, - - - only the c-equation modifi- :1. 1.25

/

* B-B

3,4

30.20

020

0.20

100

ta data

x/d e

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b

010 000

080

030

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_

-

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0.20

(a) Velocity profiles of mixing round jets.

1!k

T. Uoo

080

0.20

0.40

0.60 2y/b

0.80

10

Fig. 6. Effect of kinetic energy distribution at a jet outlet on mixing. energy of a fully developed channel flow, - - - uniform energy equal to the maximum value of a fully developed flow.

turbulence. In actual practice the turbulence kinetic energy and its dissipat;on rate cannot be changed freely; on the contrary, they are coupled together.

25-4-4

in numerical modelling the mixing three-dimensional jets has been approximated with a two-dimensional axisymmetrical jet, the cross-section of which is the same as that of an actual three-dimensional flow. In order to verify the validity of this assumption some measurements were made. Fig. 7 shows the behaviour of the profiles of mean velocity and turbulence kinetic energy at the distances of x/de equal to 0.8 and 3.4 from jet outlets. In this figure profiles are chosen so as to show the max. imum differences across the flow field. It can be seen that there are no essential differences in velocity and turbulence kinetic energy between the cross-sections A-A and B-B. Lines in Fig. 7 are not coraputed ones but they are fitted to the measurements in order to assist the comparison between measured data in different cross-sections. Thus, it can be concluded that results mixing of axisymmetrical jets with if theusing in error not mucharray are a hexagonal is made a two-dimensional approximation. This simplifies the numerical treatment and it is believed that the errors caused by turbulence models in three dimensions are of the same order as those caused by the approximation. Finally, a comparison between measurements and numerical calculation using a two-dimensional approximation is made in Fig. 8. It can be seen that if the standard k-c model is modified, such that there is a velocity gradient term in the c-equation, results are in reasonable agreement with measured data. The computation was made on a 51 (streamwise) x 18 (radial) grid data and the Reynolds number was 26300. In Fig. 8 measured the used are taken from cross-section A-A of Fig. 7. We also modification of the c- and k-equations, but the result was not as good as when only the c-equation was modified. This is a result similar to that noticed in the case of plane jets. Thus it seems that the mixing of decelerating plane and axisymmetrical jets for engineering purposes can be modelled by modifying the c-equation.

1. The standard k-c model is not capable of predicting flow behaviour if there exists flow separation resulting in recirculating flow after jet outlets. This kind of free shear layer can be modelled correctly if the c-equation of the standarm model is modified to take into account the deceleration of flow. 2. Axisymmetrical jets with a hexagonal array can be treated sufficiently adequately in engineering applications by approximating a three-dimensional flow with an axisymmetrical jet, the cross-section of which is the same as that in an actual flow field. 3. The values and profiles of the velocity and turbulence kinetic energy of jet outlets greatly affect the mixing of jets. In numerical modelling it is very import ,nt that these values should be correctly known and given.

REFERENCES CEBECI, T. & BRADSHAW, P. 1977 Momentum Transfer in Boundary Layers. Hemisphere Publishing Corporation. EISNER, J. W. & ZIELINSKI, J. 1986 Semi-preservation of in se-ake-flow a hEat ts n m J. Heat Mass Transfer 29, 293-300. HANJALIt, K. & LAUNDER, B. E. 1980 Sensitizing the dissipation equation to irrotational strains. J. Fluids Engng 102, 34-40. HUSSAIN, A. K. M. F. & REYNOLDS, W. C. 1975 Measurements in fully developed turbulent channel flow. J. Fluids Engng 97, 568-580.

R.1991 Some Self-PreservedSolutions of Turbulent Free ShearFlows. Tampere University of Technology, Engineering, Report.

1.50 -KARVINEN, te0Thermal

UU-

S., SAIY, M. & PEERLES, S. J. 1976 Fluid with unequal free-stream turbulence intensities. J. Fluids Engng 98, 229-235.

1.25 -0MOHAMMADIAN, 34 0mixing

PATEL, V. C. & CHEN, H. C. 1987 Turbulent Wake of a Flat

100

Plate. AIAA J. 25, 1078-1085.

modified mnodel 0.75

3,4 0.50 . 000

RAJARATNAM, N. 1976 Turbulent Jets. Elsevier Scientific Publishing Company.

x/\e,,,t. 0,8

020

0

RODI, W. & SCHEUERER, G. 1986 Scrutinizing the k-c turbulence modei under adverse pressure gradient conditicns. J. Fluids Engng 108, 174-179.

wall 0.40

0.60

080

1.00

2rid e Fig. 8. Comparison of measured and calculated velocities for the mixing of axisymmetrical jets. the standard k-c model, - - - the c-equation modified.

SCH ETZ, J. M. 1980 Injection and Mixing in Turbulent Flow. American Institute of Aeronautics and Astronautics. SCHLICHTING, H. 1979 Boundary Layer Theory. McGrawHill. SIMONICH, J. C. & BRADSHAW, P. 1978 Effect of free-stream turbulence on heat transfer through a turbulent boundary layer. J. Heat Transfer 100, 671-677.

CONCLUSIONS On the basis of theoretical and experimental studies of the mixing of parallel plane jets and axisymmetrical jets with a hexagonal array the following conclusions and observations can be made.

SPALDING, D. B. 1981 A general purpose computer program for multi-dimensional one- and two-phase flow. Maths. Comput. Simul. 13, 267-276.

25-4-5

EIG(;lTH SYMPOSIUM ON TURBULENT SHEAR FLOWS Technical University of Munich September 9-1I, 1991

25-5

EFFECT OF TABS ON THE EVOLUTION OF AN AXISYMMETRIC JET K.B.M.Q. Zaman NASA Lewis Research Center Cleveland, Ohio 44135, U.S.A. and M.Samimy and M.F.Reeder Department of Mechnical Engineering, Ohio State University Columbus, Ohio 43212, U.S.A.

ABSTRACT

I

'rn

The effect of vortex generators, inthe form of smail tabs at tl, nozzle exit, on the evolution of an axisymmetric je* was investlgated experimentally, over a jet Mach number range of 0.34 to 1.81. The effects of one, two and four tabs were studied in comparison with the corresponding case without a tab. Each tab introduced an "indentation' in the shear layer, apparently through the action of streamwise vortices which appeared to be of the "trailing vortex' type oriqinating from the tips of the tab rather than of the 'necklace vorttx' type originating from the base of the tab. The resultant effect of two tabs, placed at diametrically opposite locations, was to essentially bifurcate the jet. The influence of the tabs was essentially the same at subsonic and supersonic conditions indicating that compressibility has little to do with the effect.

The obvious technological significance of the ability to Increase mixing and reduce noise, even Insupersonic jets, provided a strong motivation to pursue the topic further. This led to the present investigation. A detailed flow visualization experiment was conducted together with quantitative measurements of the flow fie'd. Even though questions have remained unresolved, the results provided a clearer insight into the flow field changes caused by the tabs. This is what we would like to describe in this paper. Preliminary results of the experiment were reported by Samlmy, Zaman & Reader (1991). Only key results are discussed here due to space limitation, a more detailed paper is being prepared for Journal submissior EXPERIMENTAL FACILITY The experiments were carried out ina small supersonic jet facility at NASA Lewis Research Center. The facility Is schematically shown inFig. 1. Aconverging-diverging nozzle with throat diameter of 0.635 cm and design Mach number of 1.36 was used I he Mash number range covered was 0.34 to 1.81, representing subsonic and over- and underexpanded supersonic conditions. The flow visualization plutures were obtained by laser sheet illumination and Schieren photography. A 4 W argon-Ion laser was used for both techniques. The laser sheet Illumination was performed Inthe supersonic jets without any seeding. The cold supersonic jet core caused natural moisture condensation In the mixing layer from the entrained ambient air. Thus, with this technique, the mixing layer region was illuminated,

122

air sus e asp /

10 cm .d plenum -

Throat dameter Dt =0635cm -

Microphone ,._(4) Tabs End view nozzle exit plane

INTRODUCTION Tabs, or small protrusions in the flow at the exit plane of a nozzle, have long been known to reduce screech noise from supersonic jets (Tanna, 1977). Bradbury & Khadem (1975), to our knowledge, were the first to make flow field measurements for a subsonic jet under the influence of tabs The tabs were found to increase the jet spread rate significantly. Ahuja & Brown (1989) recently conducted a series of experiments on the effect of tabs on supersonic jets, and reached a similar conclusion in regards to the effect on the flow field. With support from NASA Lewis, the latter investigators continued to study the effect on rectangular jets as well as on the noise radiated from the jet (Ahuja et al., 1990). In terms of the plume reduction i.e., a faster spreading of the axisymmetric jet, the effect of the tabs was so dramatic that the technique has been at times referred to as the "supermixer' (E.J. Rice, private communication) However, the flow mechanisms, even the basic changes in the flow field caused by the tabs, essentially remained unknown.

122cm

Figure 1-Schemabc of flow facility

A gated double-intensified CCD camera was used to record successive images on a video tape. Standard hot-wire measurements were performed for the subsonic jets. At a jet Mach number of 0.34, hot-wire surveys Indicated a flat mean velocity profile with a 'nominally laminar" boundary layer at the nozzle exit. The boundary layer state for the supersonic regime remains unlnown, but is also likely to be "nominally laminar.' For the supersonic jets, a 0.8 mm (o) pitot tube was used to measure total pressure. A special 1 mm (od) static pressure probe, designed after Seiner & Norum (1979), was used to measure the static pressure only on the jet centerlne. Probe traverses and data acquisition were done remotely under computer control. RESULTS AND DISCUSSION The notations p, and p. are used to denote the stagnation pressure and the ambient pressure, respectively; p6 denotes sta Unation pressure Inthe plenum chamber. The design Mach number, 136 corresponded to a pressure ratio, pJp, - 0.3323. For any given plenum pressure p,the notation M, is used to denote the Mach number had the flow expanded to ambient pressure. In the supersonic regime, pressure ratios 0.3323 < pJpw, < 0.661 produced overexpanded jets and pjp,, < 0.3323 produced underexpanded jets. Thus, overexpanded condition existed for 0.79 < M, < 1.36, and the flow was underexpanded for M,> 1.36. In the range 0.7 < M1 < 0.79 (0.661 < pjp,. < 0.72), a normal shock would be expected to occur in the diverging section of the nozzle. The static pressure is denoted by P, P. representing that at the nozzle exit Flow Field Data Figure 2 shows the measured stagnation pres.suro for eight M, as indicated. In this and later figures, pairs of curves have been shifted by one major ordinate division, Inorder to present the data ina concise manner. Each pair In Fig. 2 contains data for the 2-tab case and the corresponding no-tab case. The data have been normalized by the respective plenum pressure, p6, In the supersonic regions of the flow, the measured stagnation pressure, p0, corresponds to the stagnation pressure behind the standing bow shock in front of the pilot probe. The oscillations

25-5-1

1M

- -,-

- - ---' ---

]

o.---

0.

147

--r'r

r-j :-+

8 8i

14

2 .

-1

-

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77

...

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---------

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05'- ----- .........

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a'.

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Figure 2 -Centedrine vanation of stagnation Pressurefor fhe eight M1 no tab, 2-tabSuccessive pairs of curves are staggered by one major ordinate division

.OS-

}--.'

d.5 8

0574

-_

A t

. . 2 1 WlOr Figure 3 -- Centerline variation of static pressure for the eight M, shown similarly as in

.

4

figure 2.

in the data near the jet exit are due to the standing shock structure Let us emphasize that due to probe interference there is measurement error and the data in the supersonic regions should be considered only qualitative. However, the data are accurate enough to capture the global features; for example, the number of shocks and their spacings are captured quite well as indicated by comparison with Schlieren photographs discussed later. It is evdent from Fig. 2 that the effect of the tabs Is similar over the entire M, range. Note that in the supersonic regime, the

___.___,_,_,_,_,_,___--

shock structure is affected drastically by the tabs This Isaccompanlod by elimination of screech noise for which data have been presented by Sanimy, Zaman & Reeder (1991). ,n order to calculate the local Mach number, one needs to measuso, besides p., either the static pressure before, P, or the static prossure after the 0hock, P2 In an over- or underexpanded supersoric jet, there is a complex shock/expansion system In the flow Itsel and itIsnot an easy task to measure either P, or P.. The double-cooe static pressure probe, de.-igned after Seiner & Non, m (!979), was fabricated and used only to measure P, on the jet centedine. Flqure 3 shows the P, distributions corresponding to the eight M, values of Fig. 2. For these data, essentially similar comments can be made as done for the p. data of Fig. 2 Including on the measurement accuracy. Note that even for the subsonic jet at the lowest M,, the static pressure iS negativo over a long distance along the centerline, a similar result was reported by Hussain & Clark (1977).

No-tab

" 2.5-

-4-tab

'

.

.

,

2-tab

0.

The P, and p, data were combined to calculate the local Mach number variation along tne jet axis. Data for Mi - 1.63 are shown in Fig. 4 comparing the 1-, 2- and 4-tab cases with the no-tab case. Again, it Is clear that the shock/expansion pattern Is drastically affected by the tabs. One also finds that while just one tab affects the flow field drastically, effects of two and four tabs are more pronounced.

6.

x/ot Figure 4.-Centedine varlaton of Mach number for Mi . 1.63 with and without tabs.

25-5-2 -

ti

l

II

I

I

I

I

I

I

ll

I

I

l

l

l

l

ll

l

II

I

I

ll

l

l

I

I

l

I

I

I

0.67

L21

Figure 5 -Centedrine variation of mean velooty forMt . 034 with and without tabs

Data for the centerline variation of the mean velocity, measured with a hot-wire at M,- 0.34, are presented in Fig. 5. Clearly, the potential core of the jet is reduced drastically and a similar inference is made, as with Fig. 4, In regards to the effect of the number of tabs. While the data of Figs. 4 and 5 show the mean velocity distribution only along the jet centerline, detailed data on radial profiles and far-field noise were also obtained but are not included here. These data showed that four tabs resulted In the most reduction of the far-fielk noise. However, although four tabs resulted in more Initial mixing, as evident from mass flux computed from the radial profiles, the effect of two tabs 1.3rsisted the farthest downstream. This also becomes evident from the flow visualization results.

Flow Visualization Figure 6 shows long exposure Schileren photographs of the flow field for five M as Indicated. The flow is from left to right and the pictures were obtained using a vertical knife-edge, visualizing the density gradients In the streamwlse direction. The flow fields approximately cover x/D, range of 0 to 13. The knife-edge position was changed for fine adjustments from case to case; thus, the visual jet width should not be considered as a measure of jet spread. In Fig. 6(a), flow fields without any tab are shown. The shock spacing iticreases with Increasing M, In the supersonic Note that the shock spacings, for example at M m 1.81, are found to be the same as observed in the p, and P,i meassurements (Figs. 2 and 3). About six shoci cells are observed consistently within a x/D, range of 0 to 12.

_regime.

Figure 6(b) shows corresponding photographs of the flow field with two tabs, when viewed parallel to the plane containing the two tabs. It Is clear that the jet bifurcates In the plane perpendicular to the tabs, at all M. In the supersonic regime, the shock spacings have also been significantly reduced (compare with Fig. 6(e)). When viewed perpendicular to the plane containing the two tabs, only one part of the bifurcated jet Is seen and thus the jet appears less, and these data are not shown. Note that the effect for the overexpanded case at M,- 1.14 (Fig. 6(b)) appears less than that at othr M; this point Is addressed In the following section.

1spread

The bifurcation of the jet observed in Fig 6(b) Is commen. 47

surate with the radial velocity profiles obtained Presently (Samimy, Zaman & Reader, 1991) as well as by Ahuja & Brown

(1989) and Bradbury & Khadem (1975). Ahuja at el. (1990) presented a set of Schiloren data for a&iunderexpandea case that appeared the same as those for the underexpanded cases of Fig. 6(b). The bifurcation effect resembles the effect achieved by dual mode helical acoustic

____________________________________excitation (Parekh, Reynolds &Mungal, 1987), or by simple lateral vibration of the nozzle issuing the jet (Andrade, 1941).

Laser sheet visualization of the mixing region was performed for the supersoni. jets. Figure 7(a) shows the jet cross section at four different axial locations for the natural jet at M, - 1.63. The bright and Initially narrow ring shows the mixing layer, which Is growing with the streamwlse distance eventually covering the entire

25-5-3

expenment using laser sheet illumination, Clemens & Mungal (1991) studied distortions in a plane, compressiule mixing layer, produce, by shocks originiting from the wind tunnel side wall. The distortione -eported had curious similarities with the present case, although it a )peared that the "bulging" occurred on the lower speed 3ide of their iiixing layer. Possible vorticity dynamics producing

2

the distortions nithe present case are discussed in the next section.

4

-

-

In Fig. 7. while the jet has regained the axisymmetrc shape for the 1- and 4-tab cases by 16D, it has remained quite elongated in the plane perpendicular to the tabs for the 2-tab case. In fact visualization at 30D, for the 2-tab case still shows a very elongated cross section; this is shown in Fig. 8 for M,- 1.81.

", .,

The initial evolution of the mixing layer under the action of two tabs is further shown in Fig 9(a) Itisclear that each tab produces a large distortion in the mixing layer which grows with downstream distance and results in a bifurcation of the jet by about 3D. Figure 9(b) further documents the effect of two tabs at covered, the effect three Mr the range x/D, - 1.5 for M,. With pronounced with increasing becomes moreWithin ppar,,ntly

3

C

Increasing M, the Jet core temperature becomes lower, thus, with the technique used for the visualization, the mixing layer may be expcted to appear sharper. However, there is also a diminishing effet ofthe tabs with decreasing M, as the overexpanded conditier Isapproached. A possible reason for this behavior is addressed Inthe following section.

W

DISCUSSION AND CONCLUDING REMARKS

An inference that can be made from this Investigation Is that compressibility may have little to do with the effect of the tabs The data show essentially the same effect all the way from incompres-

No lb

sible to highly underexpanded supersonic conditions. The tabs, however, do weaken the shock structu,,)drastically in the supersonic cases, which is accompanied by elimination of the screech noise. But the basic effect must originate from changes in the vorticity distribution caused by the tabs. crosction of the jet. The departure from axiymmetry In these thes to the camera anle r spondin pictures is mainly due the pictures an a angle. Corresponding frcthresflw filde for the flow field with 1,2 and 4 tabs are shown in Figs 7(b) to (d), respectively. Again, the camera needed to be adjusted from picture to picture and thus the visual cross sections do not represent the jet spread. The presence of a tab significantly distorts the mixing layer. The effect is to leave an "indentation' or a bulge into the high speed aide which grows and persists far downstream. In a recent

Perhaps, the most illuminating results are the laser sheet visualization pictures of Figs. 7 to 9. The 'indentation" In the may not be a "pa,iv' wake of the tab. Similar flow mixing layer visualization pictures (not shown here) in certain overexpanded cases, where the tabs were essentially ineffective, showed that the wake from the tab almost vanished by about two jet diameters. The Jet mixing layer in these cases appeared essentially oblivious to the presence of the tabs. In contrast, the flow field Is about 50 tab widths downstream, at 4D, In Fig. 7(c), yet the distortion appears to be growing and Influencing the entire jet cross section. The distortions introduced by the tabs are also not an artifact of the visualization technique. Radial profiles of Mach number confirmed

25-5-4

(a) Streamwise vortex pair from a tab

(b)Vortity distribution for 1-tab

(c)Vorticity dbution for 2-tab

(d)'Necklace vortex.

(a) 'Trailing vortex'.

the di-4trtions This leads us to believe that there must be a significant interaction between the flow field over a tab and the sheet of azimuthal vort-cty that emanates from the nozzle The following is a conjecture in regards to the vorticity dynamics

Figure 10 -- Likely vortioty dynamics

It is likely that a pair of counter-rotating streamwise vortices are shed from each tab as sketched in Fig 10(a). This should be a 'stationary" vortex and not like 'hairpin* vortices which are shed periodically (Acarlar & Smith, 1987) This is because the photographs in Figs. 7 to 9 are images with a long time exposure where any periodic structure would be averaged out The streamwise vortices interact and evolve with the azimuthal sheet of vorticity issuing from the nozzle. The resultant vorticity distribution on a cross-sectional plane may be expected to be as in Figs. 10(b) and (c) for one and two tabs, respectively.

of the tab is lifted up so that the approaching streamlines are at an angle of attack with respect to the tab producing a resultant force acting radially and away from the jet axis, In such a case, the pair of counter-rotating vortices as sketched in Fig 10(e) would be quite realiatic.

The strength of the streamwise vortices should eventually decrease to leave only the *iridentations" on the mixing layer as observed at x/D, - 3 in Fig 9(a). Farther downstream, the bifurcatIon of te let may take place in a similar manner as suggested by Hussain and Husain (1989) through what they called a 'cut-andconnect' process In fact, these investigators observed a very similar sequence of deformations leading to a bifurcatic. of a jet originating from an elliptic nozzle The question then arises is to the origin of the counterrotating vortex pair sketched in Fig. 10(a) Such 'stationary" vortex pairs, formed over protuberances in boundary layer flows, have been variously called a 'horseshoe' vortex or "necklace" vortex (Acarlar & Smith, 1987; Bandyopadhyay & Watson, 1988). However, a little scrutiny should indicate that the vortex sketched in Fig. 10(a) s not the same. A necklace vortex is sketched in Fig. 10(c;, following Bandyopadhyay & Watson (1988). The sense of rotation in this case is contrary to what Is sketched in Fig. 10(a). Lin, Selby & Howard (1991) recently investigated the flow over various vortex generating devices while studying their effect on boundar/ layer separation. ItIsinteresting that they reported pairs of streamwise vortices from several of these devices to be of the same sign as found here for the tabs. Itispose.)31e that the tab acts as a 'winglet' and produces a pair of *trailing ortices.' This is sketched in Fig. 10(e), which is similar to the trailing vortices originating from tha tips of a wing (e.g, Van Dyke, 1982, p. 51). It should be recognized that in order for the tip vortices to form, the wing should be at an angle of attack producing a resultant lift The tabs, however, are projected normally into the flow. It is plausible that the boundary layer upstream

One may further conjecture that the mechanism of streamwise vortex generation from the "ramps' of Bradbury & Khadem (1975) and from the "wishbones" or "doublets' of Lin, Selby & Howard (1991) are essentially the same as described for the tab. It seems that a triangular shaped tab with the base on the nozzle wall may act similarly (Rogers & Parekh, 1990) However, if this is placed like a "delta wing,' with the apex leaning upstream, it seems that vortices of sign opposite to what is sketched in Fig 10(e) will be produced Ifit is possible to produce such a vortex pair, the "Indentation' would be outward into the low speed side of the mixing layer. It is not completely clear if this would really be the case and what the resulting effect would be on the jet evolution The siggested strearnwise vorticity generation is therefore a pressure driven and inviscid phenomenon and not due to wrapping of the viscous boundary layer around the tab. In order for the tab to work, a favorable presture differential must exist across the tab. It was observed that in the overexpanded cases the effect of the tab was either less or absent. In the overexpanded case, there is an adverse pressure gradient within the diverging section of the nozzle prior to the exit. In severe cases of overexpansion this could even result in a boundary layer separation. It is, therefore, not surprising that the tab does not work In the overexpanded case becauso the pressure differential across the tab is either diminished or adverse. Questions have remained in regard to the optimum geometry of the tab. Further investigation is required to determine the optimum height of the tab relative to the boundary layer .hickness If the vorticity dynamics are correct, as conjectured above, then the width of the tab would also be a critical dimension as that determines the spacing of the two counter-rotating vortices; this needs to be studied further. It also remains to be demonstrated if the tabs will work in jets with a fully turbulent exit boundary layer and a highly disturbed core flow condition, as expected In a practical jet. These Issues need to be sorted out before the method could be understood clearly and applied successfully In practice.

25-5-5

Hussain, A.K M.F., and Clark, A.R., 1977, Upstream Influence on the Near Field of a Plane Turbulent Jet. Phys. Fluids, vol. 20, no. 9, pp. 1416-1426.

ACKNOWLEDGEMENT MFR wishes to ttaenk the Ohio Aerospace Institute for a graduate student fellowship. Helpful discLa.ssions with Mr T D Norum, Dr. P.R. Bandyopadhyay and Dr. J.M. Seiner of NASA Langley, Prof. C.R. Smith of Lehigh University and Dr. James E. Bridges of NASA Lewis are acknowledged.

Hussain, F., and Husa;n, H.S., 1989, Elliptic Jets. Part 1. Characterlstics of Unexcited and Excited Jets. J. Fluid Mech., vol. 208, pp. 257-320.

REFERENCES Acarlar, M.S., and Smith, C.R., 1987, A Study of Hairpin Vortices in a Laminar Boundary Layer. Part J. 1.Fluid Hairpin Vortices Generated vol 175, Mach, Protuberance. a Hemisphere by pp. 1-41. Ahuja, K.K., and Brown, W.H., 1989, Shear Flow Control by Mechanical Tabs. AIAA Paper 89-0994. Ahuja, K.K., Manes, J P., Massey, K.C. and Calloway, A B, 1990, An Evaluation of Various Concepts of Reducing Supersonic Jet Noise AIAA Paper 90-3982.

Lin, J.C., Selby, G.V., and Howard, F.G., 1991, Exploratory Study of Vortex-Generating Devices for Turbulent Flow Separation Control. AIM Paper 91-0042. Parekh, D.E., Reynolds, W.C., and Mongal, M.G., 1987, Bifurcation

of Round Air Jets by Dual-Mode Acoustic Excitation. AIAA Paper 87-0164.

Rogers, C.B., and Parekh, D.E., 1991, Noise Reduction and Mixing Enhancement by Using Streamwise Vortices In a Compressible Air Jet. Bull., 43rd Annual Meeting, American Physical Soc, (DFD)

Andrade, E.N. Da C., 1941, The Sensitive Flame. Proc Phys. Soc, vol. 53, pp 329-355

Samimy, M., Zaman, K.B.M.Q., and Reeder, M.F., 1991, Superonic Jet Mixing Enhancement by Vortex Generators. AIAA Paper 91-2263.

Bandyopadhyay, P.R, and Watson, R.D., 1988, Structure of Rough-Wall Turbulent Boundary Layers. Phys. Fluids, vol. 31, pp. 1877-1883.

Selner, J.M., and Norum, T.D., 1979, Experimer.'s of Shock Associated Noise on Supursonic Jets. AIAA Paper 79-1526.

Bradbury, L.J.S., and Khadem, A.H., 1975, The Distortion of a Jet by Tabs. J. Fluid Mach., vol. 70, no. 4, pp 801-813.

Tanna, H.K., 1977, An Experimental Study of Jet Noise, Part I1 Shock Associated Noise. J. Sound Vibr, vol. 50, no 3, pp. 429-444.

Clemens, NT, and Mungal, M.G, 1991, Effects of Side-Wall Disturbances on the Supersonic Mixing Layer To appear in J Propulsion and Power

Van Dyke, M., 1982, An Album of Fluid Motion The Parabolic Press, 1982.

25-5-6

EIG1fTI! SYNMPOSIUM11 ON TURBULENT SHEAR FLOWS

26-1

Technical University of Munich September 9-1 I, 1991 TURBULENCE MODEL TESTING WITH THE AID OF DIRECT NUMERICAL SIMULATION RESULTS N. Gilbert *) and L. Kleiser DLR, Instilute for Theoretical Fluid Mechanics D-3400 G6ttingen, Germany

ABSTRACT A database obtained by direct numerical simulation of turbulent channel flow is used for testing and improving turbulence models and modelling assumptions With respect to low-Re k-t models, wall damping functions are checked. The gradient models for the turbulent diffusion of kinetic energy and dissipation are analysed Modelling of the dissipation-rate terms in the Reynolds-stress equations is discussed. Attention is given to the possibility of calculating certain turbulence model constants directly from the simulation data INTRODUCT:ON To predict turbulent flows for complex geometries turbulence models of different categories (one-, two- and Reynolds-stress-equation models) are used The subject of turbulence modelling has been reviewed quite often, e g. by Rodi (1980), Patel et al (1985), Speziale (1991) and, with particular emphasis on aerodynamic flows, by Rubesin (1989) and Bushnell (1991) Improvements of current models and recent developments in turbulence modelling are discussed e.g by Hanjalic (1990), Launder (1990), Rodi (1988) and Rodi & Mansour (1990). The status of an international project on testing of turbulence models is reported by collaborative Bradshaw et al. (1991). Until recently turbulence model development has been based mainly on indirect methods, i.e. testing the closure models by comparing the computed results for different applications with experimental or theoretical data for the specific problem. This is because until now there has been no way to measure pressure and velocity fluctuations simultaneously with sufficient spatial accuracy. With the advent of sufficiently powerful supercomputers and highly accurate numerical (mostly spectral) methods, direct numerical simulations (DNS) of inhomogeneous turbulent flows in simple geometries and at moderate Reynolds numbers have now become possible (cf Reynolds 1990). These data bases can be used to test closure models by direct comparisons of the closure formulae with the terms being modelled. The data bases will also provide guidelines for model developers Considerable work on this subject has been done at the

NASA Ames / Stanford Center for Turbulence Research where some of the first direct numerical simulation databases of turbulent flows were generated. These databases are being evaluated continually by turbulence researchers during workshops such as reported by Hunt (1988) and Moin et al. (1990). Mansour et al. (1988) published one of the first papers on direct testing of turbulence closure models for wi-bounded flows In the present work a drtabase of fully developed turbulent channel flow (generated by the first author In 1988)

has been used for testing various turbulence models and/or model assumptions. Only a few results are presented here; a detailed report is given by Gilbert (1991). Similar results based on an earlier simulation are published in Gilbert & Klelser (1989) and have partly been reported in a paper by Rodi (1988) In the present paper, some DNS data are presented and compared with existing numerical and experimental data. The near-wall behaviour of the c, .f,-coefficient in the eddy-viscosity expression used in k-r models is investigated. A critical evaluation concerning the modelling of diffusion of turbulent kinetic energy and turbulent dissipation by gradient assumptions is given. Finally modelling of the dissipation-rate terms in the Reynolds-stress equations (Launder et al. 1975) is discussed DIRECT NUMERICAL SIMULATION OF TURBULENT CHANNEL FLOW As in our earlier simulation (Gilbert 1988, Gilbert & Kleiser 1990). the turbulent channel flow data has been obtained by starting a simulation with slightly disturbed laminar flow and continuing all the way through the transition process up to developed turbulence. The mass flux is kept fixedpressure at all times by imposing an appropriate gradient. The numerical resolinstantaneous ution in the late transitional and turbulent stages is N, x N2 x N3 = 1602 x 128 gridpoints in the streamwise (x,), spanwise (x2) and normal (X3) directions, Again we employ a spanwise symmetry condition, which arises naturally for the transition problem and cuts the computing time and memory by a factor of two. This does not appear to adversely effect the obtained turbulence statistics. The turbulent mean wall Reynolds number (based on channel half width h, wall shear velocity v, and kinematic viscosity v) is Re, = 210. The centre-line Reynolds number is RecL m 3830. The fully turbulent flow is calculated over a time interval of AT' _ATv,2/v-,1000 wall units. The box lengths in the streamwise and spanwise directions are L, 2360 and L2 z 1180 and are twice as large as in Gilbert (1988). The sampling time Interval of our running time average, which includes horizontal averaging, Is At,',,,, 1. Averages over horizontal planes, both channel halves and time are denoted by ..

All correlations up to third order moments appearing in the exact k, t and Reynolds-stress equations were calculated. The overall agreement of the computed turbulence statistics with existing numerical and experimental data at comparable Reynolds numbers is good. Some discrepancies occur In the near-wall region which have also been observed In the turbulent channel flow simulation of KIm et al. (1987) at the somewhat lower Reynolds number Re, = 180. Our present results are quite close to the Kim el al. data (Oilbert 1991).

1 Now with BASF AG. Technical Development, ZET/EA. D-6700 Ludwigshifen

26-1-1

In Fig 1 the mean velocity profile is given, which shows a well-developed viscous sublayer and buffer region, Unlike in our earlier simulation, the logarithmic part of the profile is now clearly visible The differences between our data and those of Kim et al. are small Both simulations give a log-law with an addit w; constant slightly above that given by Dean (1978) In Fig. 2 the computed root-mean-square values of the three velocity fluctuations normalised by the local mean velocity are shown. Whereas the two sets of DNS data nearly collapse, the experimental data show considerable scatter. for u,'/ fit our The results of Alfredsson et al. (1988) simulation data very well, whereas the recent results of Niederschulte et al (1990) clearly differ BUDGETS OF TURBULENT KINETIC ENERGY AND DISSIPATION Fig 3a shows the balance of the kinetic energy k, using for (see Mansour et al 1985) except usual notation the >jax (Here and dissipation rate = 4 the viscous (ead wal unit) given in re intheviscous digussatn

in the later figures data are given in wall units.) Away

fro m the w all there is an a pproxim ate balance betw een the production PAand the dissipation r. Closer to the wall turbulent and viscous diffusion transport energy from the region of main production towards the wall and towards the core region of the channel In the r-balance, shown in Fig 3b, the turbulent production P, and dissipation r, are the dominant terms in the core region of the channel. Near the wall the rate of production by the mean flow 5 At this position F', P: + Pf increases and peaks at y' reaches its minimum value Gradient production P, and turbulent diffusion T, are relatively small compared with the other terms Nevertheless, as will be seen later, these terms are important in turbulence modelling since they balance the net effect of all other source and sink terms

(;e. the sum of P, I P,' i- P,

--

derived from a (high Reynolds number) second-moment closure of Gibson & Launder (1978). This result confirms the suggestion of Launder & Tselepidakis (1990) that the rapid reduction of the Reynolds stress -- as the wall is approached is due to the damping of rather than to direct viscous eifects.

GRADIENT DIFFUSION HYPOTHESES in the k-equation is the gradient usual A the turbulent diffusion term model forassumption .(

a (1)

=

"t (?.

k

k) 0Xk

To test this assumption, we calculate both the exact term (I h.s.) and the model term (r.h s,) Since diffusion is very small away from the wall a more critical way of testing the model is to compare the integrated terms, which are shown in Fig. 5a. Over the whole channel the two quantitities t;

T2v< k

k

-u>

with the expression for sufficiently high Reynolds numbers, where the concept of local Isotropy holds 2 u *r > -2v< a--(4) " 3 OXk aXk

26-1-2

The blended form of the t,, model (cf Hanjalic & Launder 1976) reads (5)

£

c,,.,-+

(1

-2 )

E0.8)

Hanjalic & Launder inferred that the blending function f, is a function of the turbulence Reynolds number Re, = k 2/(vc) and determined it by computer optimization to be (6)

f,

0+

ReT 1

-and

The coefficient c, in the eddy viscosity expression is calculated from the DNS data as 0.11 instead of the usually applied value of 0.09. A possible explanation Is the dependence of c. on the ratio P,/e which is below 1 (about In the log-law region where c, is found to be constant. None of the tested damping functions t, used in low-Re k-& models shows acceptable agreement with the simulation data over the whole near-wall region. Some functions rise too fast whereas others approach unity too slowly. The turbulent diffusion of turbulent kinetic energy dissipation is usually modelled via a gradient-type model. The agreement between such a model and the

In a more recent proposal by Fu et al. (1987) 4, itselfs deend mone niropsaly ensor l.using8thefanisty depends on the anisotropy tensor, using the anisotropy

DNS data in case of the turbulent kinetic energy is good over the whole the channel depth. The model works well also in the case of turbulent dissipation except in the immediate vicinity of the wall.

(7)

In Reynolds-stress models the dissipation rate terms s,, have to be approximated. Modelling of these terms is based on blending the non- isotropic behaviour near the wall (Rotta's model F,, = )/k) with the expression for local isotropy at high Reynolds numbers, e,, = 26j./3. Most models do not work well since the blending functions restrict the influence of the wall term to the viscous sublayer. From the simulation data it follows that a blending function decaying more gradually with the wall distance leads to acceptable agreement with the DNS data for all components. The blending function of Fu et al (1987), 4, = 1 - /? where A is the anisotropy factor, beAiur de show sc g

f, = 1 -

rA.

The anisotropy factor is defined by A = 1 - 9(A 2 - A3)/8 where A2 = a, - a, and A3 = a,. ak a, are the second and third invariants of the Reynolds-stress anisotropy tensor a,, = Ik - 26,,/3 The value of f, in Eq. (6) changes from unity to zero as the Reynolds number Rer varies from zero to infinity This behaviour is also attained with Eq. (7) because A is zero in 2D turbulence (near walls) and A = I in isotropic turbulence Fig 6 shows the two , functions Eqs. (), (7) computed from the simulation data The difference between the two functions is quite big The function Eq (6) quickly decays within the viscous sublayer (y' < 10) to its minimal value (which is non-zero because Re, is finite), whereas Eq. (7) decays more gradually, and roughly linearly, with the wall distance and reaches its minimal value at y' z 60 - 80. The third curve shows an attempt by Launder & Tselepidakis (1990) to use an exponential function depending on ReT instead of Eq (6). This curve also decays very rapidly The reason is that Rer rises quickly (within y4 < 20) to a value of Rer - 180, which is maintained over most of the channel (see Fig. 7) A more gradual behaviour of f, - which is needed as we shall conclude -- would be achieved by using Re v = Y.JW/v as the independent variable. A comparison of modei predictions with the DNS data, shown in Fig 8, reveals the striking observation (made already by Mansour et al. 1988) that using the wall model Eq. (3) of Rotta throughout leads to better agreement than the blended model Eqs. (5), (6) The blending function of Hanjalic & Launder rpstricts the influence of the "wall"-terms to the viscous sublayer whereas the blending function Eq (7) of Fu et al (1987) extends it up to the log-law region. We therefore propose to use the Hanjalic & Launder model but with their f, function replaced by that of Fu et al (Fu et al. used their f, in conjunction with a new wall model which captures the proper near-wall behaviour of tj but gives pooi results away from the wall). The results of the "new" model are also included in Fig 8. They are clearly superior to those of the original model for E,, , E,2 and E33and still acceptable for c,3.

ACKNOWLEDGEMENT We would like to thank R Kessler and N Sandham for reviewing a draft version of this paper.

REFERENCES ALFREDSSON, P.H, JOHANSSON, A.V., HARITONIDIS, J H & ECKELMANN, H. 1988 The fluctuating wall shear stress and the velocity field in the viscous sublayer Phys Fluids 31, 1026-1033. BRADSHAW, P., LAUNDER, E. & LUMLEY, J.L. 1991 Collaborative testing ofB turbulence models. AIAA Paper No. 91-0215 BUSHNELL, D.M 1991 Turbulence modeling in aerodynamic shear flows - status and problems AIAA Paper No. 91-0214. DEAN, R.B. 1978 Reyncids number dependence of skin friction and other bulk flow variables in two-dimensional rectangular duct flow. J. Fluids Eng. 100, 215223 ECKELMANN, H. 1970 Experimentelle Untersuchungen in einer turbulenten Kanalstr6mung mit starken viskosen Wandschichten. Mitt. MPI f. Stromungsforschung und AVA G6ttingen Nr. 48. FU, S, LAUNDER, B.E & TSELEPIDAKIS, D.P. 1987 Accomodating the effects of high strain rates in

CONCLUSIONS

modelling the pressure-strain correlation. Mech. Eng. Dept. Report TFD/87/5.

Turbulence data from direct numerical simulations has been used to compute the terms in the budgets of the turbulent kinetic energy, the Reynolds stresses and the dissipation rate. Testing of turbulence closure models has been done by direct comparison of the exact terms with the model expressions. Special attention has been given to the possibility of calculating turbulence model constants directly from the simulation data (Gilbert 1991).

GIBSON, M.M. & LAUNDER, B.E. 1978 Ground effects on pressure fluctuations in the atmospheric boundary layer. J. Fluid Mech. 86, 491-511 GILBERT, N. 1988 Numerlsche Simulation der Transition von der laminaren in die turbulente Kanalstromung. Dissertation, University of Karlsruhe, Report DFVLR-FB 88-55.

26-1-3 i

imw

mr

mml

mlml

UMIST

mew

GILBERT, N. 1991 DLR Report in preparation. Turbulence model GILBERT, N. & KLEISER, L. 1989 data derived fiom direct numerical simulations. In: Advances in Turbulence 2 (eds. H.-H. Fernholz & H.E. Fiedler), Springer, Berlin, pp. 329-333. GILBERT, N. & KLEISER, L. 1990 Near-wall phenomena Turbulence in transition to turbulence. In: Near-Wall New York, Hemisphere, (eds. S.J. Kline & N.H. Afgan), pp. 7-27. HANJALIC, K. 1990

Practical predictions by two-equaTurbution- and other fast methods In: Near-Wall Hemisphere, lence (eds. S.J. Kline & N.H. Afgan), New York, pp. 762-781.

Contribution HANJALIC, K. & LAUNDER, B.E 1976 towards a Reynolds-stress closure for low-Reynolds-number turbulence J. Fluid Mech. 74, 593-610 HUNT,J.C.R. 1988 Studying turbulence using direct numerical smulation 1987 Center for Turbulence Research NASA Ames / Stanford Summer Programme J Fluid Mech. 190, 375-392 KIM, J.,MOIN, P & MOSER, R D. 1987 Turbulence sta-

ReynoldsMANSOUR, N 14, KIM, J. & MOIN, P. 1988 stress and dissipation-rate budgets in a turbulent channel flow. J. Fluid Mech. 194, 15-44. MOIN, P, REYNOLDS, W.C. & KIM, J. 1990 (eds.) Studying Turbulence Using Numerical Simulation DataSummer Program. bases - III, Proceedings of the 1990NASA Ames / StanCenter for Turbulence Research, ford University. NIEDERSCHULTE, M.A., ADRIAN, R.J. & HANRATTY, T.J. flow in aIn channel of turbulent Measurements 1990 Reynolds 9, Fluids at Experiments numbers. low 222-230. PATEL, V.C., RODI, W. & SCHEUERER, G. 1985

Turbu-

lence models for near-wall and low Reynolds number flows: a review. AIAA J. 23, 1308-1319 REYNOLDS, W C. 1990 The potential and Ilmitations of Direci and Large Eddy Simulations. In: Whither Turbulence? Turbulence at the Crossroads (ed. J.L. Lumley), Springer Lect 313-343.

Notes in Physics 357, pp.

tistics in fully-developed channel flow at low Reynolds number J Fluid Mech 177, 133-166

Turbulence Models and Their Application in Hydraulics. International Association for Hydraulic Research, Delft.

Behavior of KREPLIN, H.-P & ECKELMANN, H. 1979 the three fluctuating velocity components in the wall region of a turbulent channel flow Phys. Fluids 22, 1233-1239

Recent developments In turbulence RODI, W. 1988 modelling. Proc. 3rd Int Symp. on Refined Flow Modelling and Turbulence Measurements, Tokyo, July 26-28, 1988.

Phenomenological modelling LAUNDER, B E. 1990 and future? In,Whither Turbulence? Turpresent bulence at the Crossroads (ed. J L. Lumley), Springer Lect Notes in Physics 357, pp 439-485. W. 1975 Progress LAUNDER, B E, REECE, G.J & RODI, in the development of a Reynolds-stress turbulence closure J. Fluid Mech. 68, 537-566

RODI, W. & MANSOUR, N.N. 1990 Low Reynolds number k-t modeling with the aid of direct simulation data. Proceedings of the 1990 Summer Program. Center for Turbulence Research, NASA Ames / Stanford University, pp. 85-106.

ContribLAUNDER, B E & TSELEPIDAKIS, D P 1990 ution to the second-moment modelling of sublayer turbulent transport. In Near-Wall Turbulence (eds II Kline & N H Afgan), Hemisphere, New York, pp. b 8-833

RUBESIN, M W. 1989 Turbulence modeling for aerodynamic flows AIAA Paper No. 89-0606

RODI, W. 1980

Statistische Theorie nichthomogener ROTTA, J. 1951 Turbulenz. Zeitschrift fur Physik 129, 547-572.

SPEZIALE, C G. 1991 Analytical methods for the development of Reynolds-stress closures in turbulence. Ann Rev Fluid Mech. 23, 107-157.

26-1-4

0.2 D, (a)

Pk

0

1050

Y+

100

20F_

0.02

V

De

P+ P' C C

(b)

0 0

ig 1.e

10C,++100 1 10 100-0.02 Y Y+

25

rfl nw l nt Fi.10envlct ecty prflDNS wlu t ... DNS Kim el al. (1987), A~ Experiment Eckelmann (1970). -v2.44. In y*+ 5.17

50

Fig. 3. Terms in he budget or (a) the tu bulent kinetic energy k and (b) the dissipation rate c

0,2

06. 0T

0.1P.l

[0 061

0

005

10 Figctutio 2. RMS svelcit I thfl ner-w tt egiokn nomaiedb te oalm avoIt ) . ofD Srsls(Xrsn Kim per et menal t. 987ata wih ~e cke man Co prio

Fg.

02 a.(98,x N e es(19 h0),leo eta.(9 nK(epln 97),&Ecklma 0.L Alred sonS:

26-V,

.

b aiu

a pn

ucin

E dy-~sc sit reati n i lo -Re k-t modls (a N aafr ,f n At x: exp rim nta daa (urv f~ by Patl e at 195) NS ata wit : m Re .2(~I mo el f : Rmynods dlo a c B= e.11 h rt dlo G'2bson & andr

0.2-(a

-02-

-0 .6 -~

-0.

(t 3( u, / 2 + p)) 0.5

0

0 1.0

Y

100lo

50 _____

_____________

-0005 b)0

I, d

-0010 0

00

y1100

~-005--*-

0* 0

1001VI Fig. 7.

so

0100

Fig. numbefrs ulandsrbto Vareain fnturuncfrte Rydispto Re, acrss theannell

Y100

1

fth

ispto

-e26-dt,-1-6'

ae em

oars

(91 mdl E.()

EIGUiTih

SYMPOSIUM ON

26''

TURBULENT SHEAR FLOWS Technical University of Munich September 9-1 I, 1991

K - s Turbulence Model for Low Reynolds Number Wall-Bounded Shear Flow G.B. Deng, J.Piquet CFD Group, LHN-URA 1217 CNRS ENSM 1, Rue de ta Noe, 44072 NANTES Cedex FRANCE

with Abstract : Direct simulation data for a channel flow and a flat plate boundary layer are used to develop a new k-c model which predicts the dissipation, the turbulence kinetic energy level and budget correctly, in contract with classical k-e model and presently available Reynolds stress models.

P(K) = -,uij tJl : production term TK) = fu i j],] : turbulent transport term D(K) = V Kjj : viscous transport term n(K) = - i i jui C= WV

INTRODUCTION

: pressure-velocity correlation term : "dissipation"

The overall success of turbulence models is determined in a large measure by the treatment of the vicinity of solid walls. The underlying assumption of the so-called wall function approach, namely the universality of the law of the wail and the related concept of turbulence in equilibrium, severely restricts the range of validity of turbulence models and excludes a possible use in cases with separation or high skewing of velocity profiles. For such complex flow situations, the turbulence model must be introduces, solved including the viscous sublayer. This in turn apart from computational difficulties due to large grid aspect ratios, a modelling problem usually associated with the so-called low Reynolds number corrections : weighting functions are introduced in standard models and are empirically determined, at best, on a correct asymptotic basis, and in such a way that mean velocity and friction velocity predictions agree with experiments, at least for simple flows.

The equation for Eis as follows

The recent availability of direct simulation channel flow [1][2] and flat plate boundary layer data [3] makes it possible to assess the most representative models [4][5116][7] which share two main common deficiencies. (i) Although the mean velocity and the Reynolds shear stress are correctly predicted, there is a systematic underprediction of the turbulence kinetic energy. (ii) Close to the wall, the prediction of dissipation fails completely.

ku jlj : turbulent transport of dissipation [ Tkt) D(,) n v e.,: viscous diffusion of dissipation

Direct numerical data are used in the following to suggest new two-zone K-c model, on the basis of the mixing length hypothesis. The most important modifications are the following: (i) Instead of introducing a universal weighting function, a two-zone mixing length model is used with an algebraic expression, independent on K and E,close to the wall. (ii) A new closure for turbulent transport of K is introdoced. (iii) Turbulence dissipation, in the inner zone, consists in two parts. One part, independent on kinetic energy destruction, is modelled by an algebraic term. The other, E, results from K and a Taylor scale. In the outer zone, Eis given by a new transport equation model.

DIe = 1 - Pt) with: I P(e)

+

2'

V-

P;,)

+

1()

., dk.j

+

P(," Ti)

+

D(,)

+

n(,) - YF

(3)

S&, : mixed production term

.'

p2 ) -2v 7 i iak im' n: generation by the mean velocity gradient (E) 3 Pt) '= " 2vWi tJi. : gradient generation of dissipation 4 i s n io of dissipation P(c) E. 2V UikUimUkm : tre turbulent generation u k

2v [

kp()k]j : pressure transport of dissipation

2

: destruction of dissipation YE s 2VXF u The closure assumption for Reynolds stresses is the eddy viscosity assumption (4): i -6--E2 K4 uI . 3 Far from wall boundaries, this closure appears correct 'vih CA = .09 ; f = 1. However, close to the wall, the eddy viscosity level tends to be strongly overpredicted ; the aim of f. is to correct this effect. Possible choices for ft have been discussed in [5] and the most recent contributions [6][7)[8][9] try to follow the asymptotic behavior :-7' - y3 when y, the normal distance to the wall, vanishes. However, it appears hopeless to use direct simulation data to improve further the fA shape. Fig.I plots e+2 U-'v+ fT = A OW (with C. = .09) with respect to y+

1.THE MEAN FLOW EQUATIONS The Reynolds-averaged Navier-Stokes equations for an incompressible flow are : .i=0;

DO.-

["'i'l -j+ =-pi P +v

The turbulence kinetic energy is also solved: DK P(K)- T(K) + D(K) + rl(K) - e

iJjj

for Spalart's data at R9 = 300, 670, 1410 and Kim et Al data at RH = 3250,7890 and demonstrates the lack of universality of f.

(1)

2. THE MIXING LENGTH MODEL

(2)

The mixing length model considers a fluid particle A with velocity VA which moves transversally to point B (AB = 1) where

26-2-1

:

_

_

__._ EGENDE

Ci-

I. FIH-78o

- I--R 3250 [

-

-

1g

R_-._

0 .5.-

implies: (i) that the mixing length hypothesis is valid close to the 3 and involves the Van wall. (ii) also I does not scale with K /2/e Driest factor fv.In order to explain this result, we may consider:

log10 (I +)

LEGE -...300 .-.... 7890

"

0.5 ,

Fi..Lcko

nveslt

0

---- 32,50

C+orsvea vlesorR, 30 of10 40 0 50 6077

-1lo

h

Y+

T+

-,

1

i

0.O0...

I

I I I

20

10

0

'

40

30

i

I

l i

60

50

70

Fig. L Lack of universality of Cp for several values of Ri

channel Reynolds number and R0 , the flat plate Reynolds number. the velocity is UB. Then: (5) 6-v - (CB-OA)V where v is the characteristic velocity fluctuation in the direction /2

normal to the wall: [v'T]1 .A Taylor series gives: 2 +B--a + u (6) 1 + 1 + +I The first series of terms accounts for the influence of the mean velocity, the second series accounts for the influence of the fluctuation. Substituting (6) into (5)and truncating gives:

0[

V I j where v = [vT"ifl Direct simulation data (fig.2) indicate that: -

(7)

-d

= .63 fv

log 10 (Y+)

.2e-2

-, -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 Fig.3. Inner length scale (10). c1, case C = 0. +, present model. Other curves are direct simulation data with RT or Ro indicated.

El I

;22 C33 19 ; C 11 which are compared with I. Close to the wall, the best agreement with direct simulation data is provided by I, without algebraic correction (fig .4), confirming that [R is not the correct characteristic velocity. This is because several types of eddies exist, each of them having their own characteristic velocity and length scales in given directions. with Cu = .065 ; CC

We thus introduce the characteristic velocity and length vectors:

3 vie, ; I= 1__ 3 1, e U= 2;, I LEGENDE -0-o (9 )

1

2r

-

' " 're 0.1 (which approximately corresponds to the plume half width), most of the production is maintained by the mean velocity and buoyancy gradients and the turbulent buoyancy pro duction is only a small fraction of these two. The closing term in the heat flux balances is labeh%d as II, and represents the sum of the pressure correlation and the molecular destruction terms i.e. =la cp _ - 9ui Ot (2) tL = pI The molecular term in (2) is thought to get weaker with increasing Reynolds and Peclet numbers, eventually approaching a value of zero in the limit of local (small scale) isotropy. This term was not measured and, therefore, its magnitude relative to others can not be established.* However, in turbulence modeling, it is customary to combine this term with the pressure correlation term (Lumley 1978) and, therefore, from that point of view not knowing each term separately does not reduce the usefulness of these budgets. Notice that the shape of this term is very similar to the shape of the heat flux w-and its magnitude remains large throughout the flow field. The equation for the radial heat flux is u

+'WCO'

U- + War az

=

-

i1 a

-

=

-

[W ,i

+

13i0-+

-

(u-i" Ujk + u-I 1

a~r

1 pt'

(3)

Ui,k)

j-7t + 1(uipj + ujpi uP, (4)

-

ujku(4)

where the viscous diffusion term has been neglected since it will be small as compared to the turbulent diffusion. For reasons of convenience, turbulence modelers do not model the pressure correlation term in the form as it appears in the above equation but rewrite it in a different form by separating it into a deviatoric and a non-deviatoric part. Two ways of doing this have been suggested in the literature and we will look at both of these before deciding which one to use in the present study. The traditional way (see e.g. Launder et. al. 1975) of writing this term is

1 - (Uip,j + ujPi) "-p(U,,j - uji) -(ibik (5)

ik),k

ra Trttt az _ aT -OU -au -

Uk u),k

az a"

-

-

- Tw T

makes up a substantial part of the budget and its shape is similar to the radial heat flux. We also note that this budget can not be divided into any subregions, where some phenomenon are more dominant than others, because the relative magnitude of each of the terms in equation (3) remains the same across the flow field. Reynolds Stress Budgets The transport equation for the Reynolds stress, within Bussinesq approximation, is

where the first term on the right hand side is the

Wtzdeviatoric part. The second term ibthe so called

-

-

- (v+ I)t,"u,,

The balance of this equation is shown in figure 2. Again, we note that the advection term is quite small as compared to the other dominant terms in the equation. Unlike the wt heat flux balance, the shear production is extremely small here. This is because the gradients of mean radial velocity are much smaller than the gradients in the mean vertical (streamwise) velocity. There is no turbulent buoyancy production in this equation and all the production is due to the mean buoyancy gradients. We note that the term reprcscnting sum of the pressure correlation and the molecular destruction * Temperature and velocity derivatives were not mensured to check if the local isotropy existed. The spectral measurements did show a -5/3 range but it is not clear that whether this range was long enough to insure local isotropy.

pressure diffusion term. Lumley (1975) has instead suggested the following separation

-(up, +

Uj p)

=-[

(Uipj + ujPi-()

(2- )k6iJ

-(2)(

36

' )k6(6)

where the term in the square brackets is the deviatoric part and the last term on the right hand se is essurdiorus e tr ion Rg dl which separation is employed a correction or model . The model (178 has to be used for the correlation used here isdue to Lumley (1978) and isgiven by uk/5. This study indicates that the use puk = -q of this model with (5) produces so much pressure diffusion that it negates the velocity diffusion (i.e. due to UTU-juk). On this basis it was concluded to use the separation given by (6) in the present study. Further details about this issue are given in the appendix.

27-1-2

Theretbre, using (6) the equation for the Reynolds stress can be re-written as

6ij] ,

+

Uk([U + 1

-

-

(+-7"uUj,

and represents the sum of thc pressure correlation

i'jt - /'it

=iik Ui,k) -

term and the anisotropic part of the dissipation. This term is a loss for the u2 budget and we note that beyond r/z = 0.08 this term and shear production approximately balance each other.2 u is given by The equation for the radial component

3p 2(.....+ (\'k)J + ujp,i) P 2 2+ 2 (7) 6i./ LUj,ktuk + jE6 ij (1978), Note that, following Lumley

where c = cii. the anisotropic part of the dissipation part has been combined with the pressure correlation term. The term in the curly parenthesis has a zero trace and will be denoted by 'Dij in the rest of the paper. It is this term whose models have been proposed. Note that the above equation is exact since no approximations have been used so far. Now we introduce the model for the pressure diffusion term, as given above, and with this approximation the above equation becomes

152(q--

79 -

+"U7ii

(V.- Uik

over the rest of the flow field the shear production is much larger than the buoyancy production. It is also interesting to note that the buoyan,'y production and dissipation rate approximately balance each other. The closing term in this balance is 4,

U " + w2U2 "9z Or

_

1 r Or 2 [rWl + 15 r rq 2 b -2u O- 2 V

Ui,k)2

- k)

is a loss near the plume center and is a gain after about r/z = 0.04. This is because the radial gradi-

ent of the radial mean velocity is positive near the plume center. The mechanical production term is not large. The diffusion term is a loss over most of the flow field and becomes a gain toward the outer edge of the flow field. The sum of the pressure correlation term and the anisotropic part of the dissipation rate is obtained a.- the closing term 2 in the budget and represents a gain for u . We further note that beyond r/z = 0.08 it approximately balances the dissipation rate. Finally we look at the budget for the shear stress TU' as shown in figure 5. Its equation is given by ."

W au-w

SOz 2

OW

-

uw Or

-

-

-

2OO

--

10

0

rOr

OZ

-

) w ) (r u WWU rr Or +-2[Ir'!(ruq2) + -( --

(10)

The production due to velocity gradients

of w 2 .

$j.

-W + W -W Or Oz

q

advection of u2 has the same form as the advection

k

Note that due to the model for the pressure diffusion term this is no longer an exact equation and - has been used to emphasize this fact. It is this equation which will be balanced out with the experimental data and the term D,, will be obtained as the closing term. It should be reminded that in addition to the measurement errors, any uncertainty in the approximation of the pressure diffusion will also be lumped into The equation for the streamwise Reynolds stress 2 w is given by U

)

and its balance is shown in figure 4. Obviously the

+ IDj - 2 61 (8) + # iu17t + 61 )i7t

2

. Oz + ) "z az

-uw.T-

-OW2 w - + 2g)3ii

_a- O .- m O9 r

OzwT -u Or (11) +.ift- s,

Both advection and the turbulent buoyancy production are of very small magnitude and over most

(9)

The balance of this equation is shown in figure 3. Advection is the smallest of all the terms. Diffusion term is a gain near the center of the plume and a loss in the rest of the flow. Also, its magnitude near the center is comparable to the other dominant terms in the balance. We note that the buoyancy production is comparable to the production due to mean velocity gradients near the plume center but

of the flow field these approximately balance each other., Neglecting these two terms would not cause any significant change in the shear stress balance. We note that the diffusion term is not negligible in this budget. The term $r, is essentially balanced by the difference between the shear production and diffusion processes. The shape of 4Pr is obviously similar to that of the shear stress and its peak approximately corresponds to the peak in the shear production.

27-1-3

IMPLICATIONS ON MODELING Local Equilibrium Assumntion The assumption of local equilibrium implies that the evolution of the turbulence is slow enough to ignore the effects of advection and diffusion (or that these two phenomenon balance each other). For the turbulence kinetic energy it means that the production and dissipation rates balance each other whcreas for the heat fluxes and individual Reynolds str(sses it implies that the sum of the production and pressure correlation terms balances the respective molecular destruction terms. This idea forms the basis of the algebraic stress models (see e.g. Hossain and Rodi 1982) and we will use the present balances to assess the validity ot this assumption. From the budgets of vertical heat flux (figure 1) it is obvious that the advection and diffusion terms are small and ignoring these will have no significant effect on the heat flux. In the radial heat flux budget the advection and diffusion terms are about 9% of the production term. This may not be a small amount to ignore but one can argue that the neglect of these two terms will introduce an error which is within the other approximations used in the algebraic stress models. Therefore, it seems that for the heat fluxes the algebraic stress model can provide a reasonably good approximation if not an excellent one. However, success of these models will depend on the accuracy of the closure expressions for the pressure temperature-gradient correlation, The situation in regard to the Reynolds stresses is not as encouraging. From their balances it is clear that the sum of the advection and diffusion terms in the radial and shear stress components is not small as compared to the other terms and neglecting these will introduce a much bigger error. This is specially true near the plume center. So we conclude that the assumption of local equilibriumn is not a good approximation for the Reynolds stresses. This will become more evident when we iook at the ratio of mechanical to thermal time scale next. Ratio of Mechanical to ThermalTmL e .cales The time scales q2/c and P/ct represent the eddy turn over times for the mechanical and thermal fields respectively and their ratio is widely used in turbulence models. A universal value of this would un-necessitate a transport equation for the thermal dissipation rate although it is now realized that this ratio can widely vary from one flow to -nother and that a better approach would be to use a, ansport equation (see e.g. Newman et al 1981). Full line in figure 6 shows that for. the present experiment this ratio is around 3.3 over most of the flow field. This value starts to sharply increase as the outer edge of the plume is approached. The measurement er-

rors are quite large at this location and it is not clear whether this rise is due to the measurement errors alone or also because the mechanical dissipation rate approaches zero faster than the thermal dissipation rate. This value is substantially different than the commonly used value of 2.0 which was recommended by Beguier et. al. (1978) after analyzing several experiments on wall bounded (turbulent flat plate boundary layer, fully developed pipe flow) and boundary free shear flows (heated wake and mixing layer). The dissipation rates were not available for some of these experiments and were obtained by invoking the local equilibrium assumption. The authors further noted that the turbulence was found to be in local equilibrium for near wall flows and that although this was not true for the free shear flows, the production and dissipation rates were stil. the dominant processes for these flows. The implication was that this was the reason the different experiments and the different methods used in calculating the time scale ratio gave more or less the same result. In the present experiment the production and dissipation processes are certainly the dominant ones but advection and transport are large enough to account for the differences with the study of Beguier et al. Therefore, it is concluded that the reason the current experiment is not in accord with the above mentioned study is that its kinetic energy and temperature variance balances do not satisfy the local equilibrium assumption. This can be further illustrated by calculating the time scale ratio by using the local equilibrium assumption. With this assumption the expressions for the dissipation rates for the buoyant plume reduce to c = - -u U,,j - fliuT and ct = -u'itTi The resulting time scale ratio is shown in figure 6 as a broken line and interestingly enough its value is about 2.0 over most of the flow fie.d. It might be of interest to compare the present results with the homogeneous shear flow experiment of Tavoularis and Corrsin (1%,). The time scale ratio for their experiment is about 3.0 which is very close to what is found here for the buoyant plume. Again, if local equilibrium assumption is used, the time scale ratio for their experiment reduces to 2.0. It should be noted that the buoyancy effects are absent in their experiment and that the temperature behaves as a passive scalar. The similarities between the time scale ratios for the two experiments could be due to the fact that the relative magnitudes of the production and dissipation rates are similar in these experiments. It should be emphasized that the purpose of this comparison is not to advocate a new universal value of the time scale ratio.

27-1-4

MON MIL TO TCMIIERATUREGRADIENTS POI 15001-k )v'V

0.

II 0 ,

~~ OC ~N

~

R DE TSS

~

~ ~ ~ C ;NDUTO

s s MOrPRESSURE CORRELATIO NS A NS RkTE ACISOTROI'ICPART OFDISSIPATIONJ

iU00iCTO DIFFI 10'

PROM1 MTol DUE 10 VELOCITY

-

-----

-0.0--

SUM1V, P6SURE.O6RELATIO'. All) MbLEC '.AR TERIS

oba

.

-0

DSIAINRT

Figure 1. Budget for TO equation.

0080/

i

Figure 4. Budget for W- equation.

DUE TO VELOCITNCRAIDIESTS

II~2CLIS 00011'.~PRODUCTI0?" DI T TtII~~IKl.P

r

=

~

500

11101 CsUII.Ts dO 0 TOSEl.O1

-502 L

ADSECTIO

-)f

-

-

-

--

-5 001

04

016

000S

20 00-

---------- ----

8IOIANCN PRODUCTIOC

-s__

- -AVCTO

5

01

00

00

IV15 0 SI

U 'PP

SLE

'OLEC..1A3 TERSI

O S

UM0OF PRESSUREC0RRLATIO : AND

ASISOTROPIC PART Of DISSIPATION LATE

AN

0016

0

0IU

.C C"'LtSTIOI. AND

0 24

000

008

'1./

0 16

0 24

r1z

Figure 2. Budget for ulequation.

Figure 5. Budget for iiw equation.

PRODI'CTIOI. DU, TO SELOCITYCRADIENThb

5

0

PRODUCTION

- (for a given value 0o of 0) Is strongly dependent on go for small times corresponding to a bimodal pdf of 9, and becomes Independent of 0o at long times when the 0 distribution tends to a gaussian. In addition to the scalar field dissymetry, a relevant feature may be the Intermittent character of some regions of the flow, so that another question arises: what Is happening when Intermittency is strong, I.e. y 10), a linear development can be accepted, with a spreading rate a - 0.25, independent of the density. The speed of the motion in its initial phase did not permit direct measurements in regions close to the origin, that is, direct evaluation of a,.

1 (33)

Fig. 3 allows the acceptance of this explanation, though for values of p0 so different as 1063 Kg/m 3 and 1458 Kg/m 3 the "ddcalage" is so small that can be ignored for practical purposes. 4.CONCLUSIONS This paper reports a theoretical and experimental studyon the motion ofthethermals. The consideration

27-4-4

of a zone of flow establishment as a transition to the zone of established flow is the main novelty of the theoretical model. It wascomparedwith experimental data, with the particularity that very heavy fluids have been used. Principal conclusions:

We also would like to thank NATO Scientific Affairs Division, Instituto Nacional de Investigaoo Cienttlfica and Instituto de Hidriulica e Rerursos Ilfdricos / Universidade do Porto.

(i) In the zone of established flow, and independently of the density, a constant spreading rate a - 0.25 was found, as well as KM 0 0, which means that the added mass isirrelevant in this zone. However, the virtual origin seems to depend on density, the greater origin; the density, the farther from the real

6.REFERENCES ESCUDIER, M.P. & MAXWORTHY, T. 1973 On the motion of turbulent thermals. 1. Fluid Mech., 61, 541-552. KORCY.18Ocadipslfmucpl wastewater: impacts on the coastal environment. National Oceanic and Atmospheric Administration, vol. 1, 129-175.

(ii) The length of the zone of flow establishment is relatively small, though increasing with density, and a maximum value of Sbo was deduced from the experiments. In this zone it seems plausible K ,.v 0.5, which is the representative value for a sphere. The spreading rate a. was found to increase with density. As the spreading rate is related to the entrainment coefficient a' by Eq. (10), it seems possible to deduce a' by analising the geoiiietry of the cloud. We hope to do that in

MORTON, B.R., TAYLOR, G.l. & TURNER, 1. S. 1956 Turbulent gravitational convection from maintained and instantaneous sources. Proc. Roy. S A 234, 1. NEVES, MJ.V. 1985 Estudo de jactos turbulentos. PhD thesis, Universidade do Porto, Portugal. SCORER, R.S. 1957 Experiments on convection of OERS 97Eprmnso ovcino isolated masses of buoyant fluid. J.Fluid Mech., 2, 583.

the future, as an effort for better understanding the relationship between entrainment and densimetric effects.

TURNER, J.S. 1986 Turbulent entrainment: the development of the entrainment assumption, and its application to geophysical flows. J. Fluid Mech., 173, 431-471.

5. ACKOWLEDGEMENTS The experimental work was done at the Arizona State University (USA) and the facilities provided by Professor Fernando are gratefully acknowledged.

WANG, C.P. 1971 Motion of an isolated buoyant thermal. The Physics of Fluids, vol. 14, 1643-1647.

i t rltg. I. t3,,vqdopr,,ent or

a th,,, r,,nI

30

1

nlun Symbol

2 o

4

3

0

5 a

6 a

7 x

9 4

10 5

kg/n 33

1,'458

n20

1.

es0

10

20

1

0

30

50

40

P. = 10 63 kg/m 3

60

70

90

90

100

z

Ve=0.26 Fig. 2. iNodius vs. distance

r 2

63 (%=10O kg/rn3

10

45

-

0

21

Theorefical solution

Symbol

6

10

4

0

a1

7

40 I,

m

160 I/1 b./a.1/

Fig. 3. Distance vs. tItye

274-

640

2560

EIGhrm SVNIPOSIUIJ

ON

28-1

TURBULENT SIEAR FLOWS

Technical Unversity of lMunich September 9-11, 1991

A MODEL FOR BURSTING OF NEAR WALL VORTICAL STRUCTURES IN BOUNDARY LAYERS Paolo Oriandi* and Javier Jim6nez+ * Dipartimento

di Meccanica e Aeronautica, Universiti di Roma

+ Departamento de Termofluidodinimica, Universidad Politkcnica de Madrid

SUMMARY In the last decade, much effort has been devoted to the understanding of near wall turbulence, both through experimental and numerical observations. Different scenar-

nal variation is represented only through a variable forcing term in the equations. This permits the use of grids which are unaffordable in a three-dimensional simulations, and the observation of small details which would be unavailable oth-

ios have been proposed, but a broad consensus on a basic model for the near wall events has still not been achieved, We present here a simple numerical model for some of the events occurring during the bursting phenomenon. In particular we study the behaviour of longitudinal vortices as they are brought near a wall by an x-dependent external shear, which is the near wall equivalent of a plane strain. We also study the effect of the interaction on the local wall stress, and we calculate the distributions of turLalent stresses to analyse the zones of maximum production in the transversal y-z plane.

erwise. On the other hand, the longitudinal variations of the flow are only represented in an approximate manner. As a consequence we expect to find correspondence with experiments in the development of the small spatial scales, and for short time intervals, but not necessarily for large scales or for long times. Ersoy and Walker (1986) were probably the first ones to investigate numerically two dimensional vortex motion near a wall in connection with the bursting phenomenon, although the basic model was proposed much earlier (Blackwelder and Eckelmann, 1979). They considered the viscous boundary layer induced by a pair of counter-rotating point vortices impinging on the wall at an arbitrary angle, and concluded that the wall layer always separates and generates a violent eruption as a consequence of the pressure field induced by the pair. They concluded moreover that, in asymmetric cases, the effect of vortex nearest to the wall dominates the motion. Their study was done in the context of non-interacting boundary layer theory and, as a consequence, the wall layer became singular in a finite time. The present study tries to do a more realistic simulation by using the full two dimensional Navier Stokes equations and non singular vortices, and by using parameters derived from direct numerical simulations of the boundary layer. The aim is to study the formation of compact longitudinal cores from initially fiat vorticity distributions, to follow the subsequent behaviour of those cores, and to quantify the average (streamwise) wall stress produced by them.

INTRODUCTION The direct simulation of the channel flow by Kim et al (1987), using visualisation techniques similar to those in the experiments of Kim et al. (1971), showed the existence of streamwise vortical structures near the wall, very similar to those observed in experiments, and with similar spacings and lengths. Their computational box was 2300 x 1500 wall units in the streamwise and spanwise directions, and contained many structures, making it difficult to isolate the details of individual events. This is even harder in experimental studies. To achieve a better understanding of individual events near the wall, Jimdnez and Moin (1991) introduced the use of numerical simulations in "minimal" channels. In that case only a few structures are present in the computational domain and it is easier to follow them in time and to study how they interact among themselves and with the wall. Because of the intrinsic three-dimensional character of the flow, it is difficult to obtain simple models for the near wall region, but both the full channel and the minimal channel simulations show that the structures are very elongated in the streamwise direction, and that the key event occurs when they come near the wall. Their interaction with the wall induces a layer of secondary wall vorticity of opposite sign, which thcn diffuses and interaLb with the primary structure, During this interaction a very rapid growth of the wall shear stress is observed. A similar mechanism was observed by Robinson (1991) analysing Spalart's (1988) turbulent boundary layer direct simulation. He came to the conclusion that, in the inner zone, single unpaired near-wall quasi-streamwise vortices generate persistent low-speed streaks, and that counter-rotating vortex pairs are rare. The fact that the whole structure is elongated suggests that a quasi two-dimensional model may be built, in the y-z transverse plane, in which the longitudi-

28-I-

PHYSICAL MODEL We consider an infinite streamwise vortex subject to a variable shear that increases linearly with the streamwise coordinate, x. From continuity there is a downwash towards the wall which is independent of x. This mean flow is intended as a first approximation to a sweep event, in which a streamwise vortex is pushed towards the wall, and presumably initiates a new bursting cycle. The assumption is that the spatial and temporal scales of the pressure gradient that produces the sweep are large with respect to those of the evolution of the vortex. Under those conditions the mean flow can be assumed to be steady, and a good model is Hiemenz flow (Schlichting, 1987, p. 87), which has the advantage of being itself a steady solution of the Navier-Stokes equations. It can be expressed in terms of a dimensionless function 0(y) as UH(z,y) = az4'(y), VH'y) = -aft). The function 0(y) behaves quadratically in y -tear the wall, so

between consecutive sublayer "streaks" in the spanwise dithat the horizontal velocity drops linearly to zero in that rection but excludes the logarithmic region in the vertical region while the norms velocity vanishes quadratically, as one. All the simulations are performed in a 256 x 256 grid. in real profiles. Assuming the same x dependance for the perturbations as for the mean flow, we write the streamwise INITIAL CONDITIONS velocity perturbation as u(X,y,z) = ax q(y,z) Robinson (1991), analysing Spalart's (1988) turbulent The equations for the transverse and longitudinal moboundary layer data base from direct simulations, computed tion are weakly coupled through the vortex stretching term. The transverse velocity (v, w), is incompressible, and can be distributions of diameters, strengths and wall distances for the quasi-streamwise vortices. He found that the radii have characterised by the streamwise vorticity component w = + a distribution with a maximum around r = 5- 20, the wall aw/Oy - av/Oz, and by a stream function 0(y, z). From distance, around y+ - 10 - 50, and the Reynolds number, now on we will use dimensionless quantities normalised with + Re = r/v = 60 - 250. The value r = 10 chosen in our r, the circulation of the streamwise vortex, and with ro, the radius of the vortex, and a, related to the variable shear. is computations is intended to represent an average of these The dimensionless equation for the longitudinal vorticity is vortices. preliminary computations using those values ]Some w 2 showed that the vorticity was rapidly dissipated. Since the T.. + j(, ) = Ru 1(0,(Y) + q) w + 0(y) TY + Re V w, goal of the study was to investigate the vortex dynamics, we (1) . with Re = n/v , Ru = ar /l' and J(w, 4) collects the usual convective terms. The first term on the right hand side repvaria ser, et frtx ste o g the contiv ter. resents the effect of vortex stretching by the variable shear, while the second one represents advection of vorticity by the

decided to increase the Reynolds number to allow for longer and all the simulations presented otpoal here are done uslifeimes, igU=re+0.Tedsrpnyi u ing r+= Re = 500. The discrepancy is most probably due to the three dimensional effects that are neglected in our model. In a strictly two dimensional approximation, like ousthvrtcyrtaintrs U yad x /z

resulting normal velocity,

ours, the vorticity rotation tc-.'ws wa8U/Oy and w5 ~9U/8z

With the assumptions for the perturbation velocities, tpresthe perturbation for gether with a corresponding one sure, we can also write a transport equation for the perturbation streamwise velocity, q, ,q)+

qRu[(q2+2 + J(q, 0)+

cancel identically, and there is no way to regenerate the streamwise vurtices from the transverse vorticity in the mean shear. As a consequence any streamwise vortex is evntually dissipated. On the contrary, it was shown in (Jim6nez and Moin, 1991) that those reorientation terms play a significant role in the evolution of the streamwise vorticity in the real

+

boundary layer. +0(y) t L+

Re V2 q.

(2)

This quantity behaves approximately as a passive scalar evolving under the effect of viscous diffusion and of advection by the (q, v, w) field. Note that, due to the assumption of two dimensionality, these two equations are exact, except 8 for a "quasi linear" approximation in (2) tii, ap/ y - 0. We integrate them to evaluate how the vortex evolves under its own induction, coupled to the stretching of the variable shear, and how the streamwise structires affect the longitudinal velocity distribution, including the mean wall stress averaged over the spanwise direction. Note that the limit Ru - 0, which is reached by letting a -- 0 and x - oo, is that of constant shear and no strain, but satisfies formally the equations. We will present later some simulations in this limit to separate the effects of self induction from those of stretching. In our calculations, the equations are solved in a rectangular domain with periodic conditions in the spanwise direction. Free-slip conditions are imposed at the upper wall and a very efficient discretisation of the wall vorticity, described wall. used at the lower,arenon-slip in (Orlandi, main features1989), of the isnumerical method described in The that paper. It is a scheme with second order time and space accuracy, and with global conservation of encrgy, enstrophy, and vorticity in the inviscid limit. The dimensionless units used in the numerical calculations can be related to the boundary layer "'all"units by fixing the vortex radius and the kinematic viscosity. The latter is defined as v+ = 1, while the + former, for reasons given below, will be taken as r = 10. In the following all results will be expressed in those wall units. All the calculations are performed within a domain of size + equal z = 100 in the spanwise direction and y+ = 60 normal to the wall. This corresponds to the average full spacing

The initial streamw;se velocity profile, Uq, has two effects. First, through ,he scale factor a, it generates the external strain and the normal velocity field that brings the streamwise vortices closer to the wall. Second it is itself convected by the transver.e velocity field and generates the shear layers and the wall stress that, although essentially passive in our approximation, are such important features of real flows. Both effects can be studied approximately by using an initially linear profile near the wall, although the consistency of the equations forces us to use a more realistic model such as the Hiemen flow. Still, all that is required is * oo, and the detailed profile is quite that aUy/Oy -- 0 as y -irrelevant. In particular, the vertical scale of the Hiemenz profile is arbitrarily, and was adjusted in our simulation so that 0' = 0.9 at y = 4.5 (y+= 45), giving an essentially linear profile over most of the region of interest, but levelling to 0k = 1 at ' ,e upper edge of the computational domain. The strain parameter, a, was estimated from direct simulation data. In those, the wall shear has a spotty distrituheay+ = 0 - 3, with a streawise bution, in the range length scale of the order of x+ = 50 - 100 (Jim nez and fo stain ard oeond -vf Momn, 1991). The corresponding values for the strain are,

a+ = (82U+/Os,+Ox+)/(d4'/dyl+),. 0.5

-

2.0. From these

data the dimensionless strain in Eqs.(4) and (5) can be approximately evaluated as Ru e 0.1 - 0.5. Because we are interested in investigating t e effect of the strain, we considered two cases: the first one with constant shear, Ru =0, and thc econd one with a strong strain, Ru = 0.243. The choice of the initial vorticity distribution and location for the longitudinal vortices influences in h,large measure the evolution of the flow. Rather than introducing circular vortices ab initio we decided to study whether they might form naturally by self induction, and we used vortices whose initial shapes were very elongated in the spanwise direction,

28-1-2

40-

rr

, A

Y + 30 -

•+

(

)i

Y+ 20-z

r-0

0.

0

-0

20

40

+1

b

Y,

z+

Fig. 2 - Contour plots of vorticity at t + = 4.8 for a symmetric initial condition. (wm, = .52, Aw increment = .10, Re = 500, Ru = 0).

(

Z+

jected to the shear of the main vortex, are dissipated quickly. When an external strain is applied, the vortices are pushed

Fig. 1 - Time sequence of w. maps in the transverse section of a minimal channel, moving with the approximate convection velocity of the vorticity structures at the wall. Flow is minimal channel from Ref.[3]. Time from left to right and top to bottom.

against the wall and become more compact under the effect of the stretching, but tile wall interacticn mechanism does not change appreciably. When the two counterrotating vortex sheets have the same initial circulation magnitude, but their shapes and distances from the wall are different, the evolution is asymmetric for a while. An example is given in Fig. 3a-c, in which one of the vortices is relatively compact and close to the wall, while the other one is more elongated and located

described by the expression -)=%-(

exp (WM/

y

sin (7r X - X___, x E (0,a) (3) ,

a

40

-

------------.

/'(a)

30-

where WM was adjusted so that I' = 1, and with a' = 0•75, a + = 10 - 40, and y+ = 7.5. A typical initial condition is shown in Fig. 3.a. These vortex sheets are found often in the near wall region of real boundary layers (see Fig. 1), and it is not immediately obvious whether they result from the deformation of initially circular vortices, or whether they are precursors to them. Our goal in choosing them as initial conditions was to see whether they would evolve spontaneously into circular cores, and how fast. In the case of the simulations presented here, a vortex sheet with r + = Re = 500 and a + = 20 rolls up into a compact core in t + ; 1, which is shorter than other relevant time scales in the flow.

y+ 20 ,=

a

tO o 40

(b) 3 3o

Y+20

RESULTS AND DISCUSSION

10

We were also interested in investigating whether ourquasi-two-dimensional model would predict the type of events depicted in Fig. 1, whose most characteristic feature is the presence of intense compact vortices of one sign and weaker, more diffuse, vortices of the opposite sign which seem strained by them into thin sheets. Approximately symseveral calculations with different initial configurations to

_ 0 40 (c) 30 y 20

study their evolution. With counterrotating vortices of similar initial circulations and shapes, either with or without external strain, both vortices roll up in approximately half an eddy turnover time. The final configuration is that of Fig. 2, with a vortex Configurations similar to pair receding away from the wall. those of Fig. 1 cannot be achieved. The layers of vorticity of opposite sign which are formed at the wall play an important role in the roll-up process. They are initially very intense, but they later diffuse away from the wall and, sub-

28-1-3

J,

10

0

..-.. -4o

-20

-0

20

__

_

40

Fig. 3 - Contour plots of vorticity with asymmetric initial conditions. (Aw increment = .1, Re = 500, Ru = 0). a) t + = 0; b) t + = 2.4, + .90; c) t = 4.8, w . = .55. Wio

40

4 20-

-40

-20

0

1+

20

40

40

Fig 4 - Contour plots of streamwise velocity corresponding to Fig. 3.b. (Aq increment =.025)

301 2 Y+ 20-' 101

I

-

0.70, (c)

! 0.6-, i

30j

0

20-1

.I -y+

o.5

1.

2.

3.

4.

!$

i.

0

-

;-

-40 Fig. 5 - time history for flow in Figures 3-4.

Fig. 6

at a greater distance. The small core on the right, having a higher vorticity, assumes a circular shape in a very short time and prevents for a while the roll-up of the more diffuse sheet. The results at t+ = 2.4 (Fig. 3.b) is quite similar to some of the structures in Fig. 1. The main difference is the behaviour of the secondary vortex sheet at the wall, which looks weaker in the three dimensional case. This Aieet tends to protrude in between the two members of the vortex pair, shielding the diffuse vortex and allowing it to roll up even in the presence of the strong core. In fact, in the two dimensional case, in which that secondary vorticity is strong, the two cores have essentially rolled up at t+ = 4.8 (Fig. 3.c) and the final result is quite symmetric, similar to that in figure 2. Therefore, a first conclusion of our simulation is that the single structures observed Robinson (1991) have to be caused by very non-symmetric events, e.g. legs of hairpin vortices lying at different distance from the wall, and that the vortex closest to the wall is always dominant. Itis genorally ;r,-pted that these longitudinal structures are responsible for the formation of the low velocity streaks in the wall region. Fig. 4 shows that this process is caused by the convection of the streamwise velocity field, U = Uy + u, by the longitudinal vorticity. In the region of updraft, the initial shear is convected away from tne wall, generating a low velocity streak and a detached layer of intense spanwise shear. These are regions of negative u' and positive e (II quadrant) which generate negative u'V and production of turbulent energy.

-20 -

-

0

20

40

z+ Normal stresses distributions for Fig. 3.b. a) q' , b) v'", c) w'". (A increment = .00125)

I eoi three dimensional flows these shear layers probably become unstable and lead to the formation of new longitudinal vortices and of new turbulent cycles, but these effects are absent in our simplified two dimensional model. To the sides of these low velocity regions the downwash pushes the background shear towards the wall and generates a high yelocity streak. It is not a priori clear whether the two effects would cancel at the wall and whether the global effect would be a increase or a decrease in the wall stress averaged over the span. Figure 5 shows that the former is the case and that the averaged wall stress increases by 50% in t+ = 5, which is a time scale comparable to that estimated for the growth of a burst in (Jimdnez and Moin, 1991). Since the eddy turnover + time can be estimated as t, = (27rr+)2/r+ ;s 8, the increase in wall stress is very rapid, and occur in time which are short compared to an eddy rotation, although longer that those associated t-) the formation of the compact cores. Fig. 6 shows the spatial distribution, in the y-z plane, of th, velocity perturbations at the particular moment in time which corresponds to Fig. 3.b. These perturbations would contribute to the turbulent intensities u', v' w' in the three dimensional boundary layer, but can only be related directly to them after a randomising and averaging step that was not attempted here. Still, the location of the different contributions is interesting. The fact that the u' perturbation that defines the low velocity streak is wider in the spanwise direction, and stronger, away from the wall than close to it, is well known from experiments. The association, in the cen-

28-1-4

~~0.9 --

__

40 T...

..

...

.

_

__

.

30

0

+

y+ 20-

time history for flow in Figures 7-8.

tre of the streak, of v' > 0 and u' < 0, defines an "ejection" event, and suggests once again that ejections are not transient events in the boundary layer but long lasting states that pass by the stationary probes used in experimental observations (Moin, 1987, Jim~nez and Moin, 1991) From this first simulation, without external strain, we see that our simplified model reproduces some of the features of the bursting event: it is very rapid, it is governed by the presence of longitudinal vortices located near the wall, and during the bursting event there is a large growth of the wall stress. To analyse how the external strain modifies the flow structure, a simulation was run with Ru = 0.243 and with the same initial conditions as those in the previous calculation. In Fig. 7 we show the vorticity distribution at a time comparable to that of Fig. 3.b. The main difference is that now the whole structure is closer to the wall, being convected by the normal velocity created by the strain. This is true both for the original vortex cores and for the secondary vorticity layers at the wall Also, the general intensity of the vorticities has increased, due to the stretching term. As a consequence, all the interactions are stronger and, in particular, the effect of the wall vortex sheet becomes more important than before. The resulting shielding effect prevents the straining of the left vortex by the right one, and the pair becomes more symmetric, even for the severely unsymmetrical initial conditions used here. The end result is that a symmetric pair is always formed, which flies away from the wall even in the presence of downwash created by the strain. The resulting longitudinal velocity distribution is shown in Fig. 8, and also shows the formation of a low velocity streak surrounded by high velocity regions. Fig. 9 shows the growth of the mean wall shear, which is initially faster than in the unstrained case, but levels off

in a shorter time. This levelling corresponds to the drifting of the symmetric pair away from the wall under its own induction. These characteristics are common to most of the simulations that were carried out including a strain. Asymmetric evolutions could only be achieved in those cases under the most extreme initial conditions. Because the whole boundary layer is contracted vertically towards the wall by the strain, the general level of the longitudinal vorticity gradients is higher, but the structure of the streak is the same as in the unstrained case. Robinson (1991) found both single vortices and counterrotating pairs in the wall region, but he notes that the probability of the former is larger than that of the latter. The same was observed in the minimal channel by (Jim~nez and Moin, 1991). The present simulations suggest that events with a single vortex are associated to weak streamwise strains while events with vortex pairs correspond to intense strains. At longer times our quasi-two-dimensional simulations always tend to produce symmetric configurations, but it was pointed in the introduction that the model can not be trusted in that range. More interesting is the possible relation of our strained model to the boundary layer in a positive pressure gradient. The results suggest that symmetric pairs should be more common in that case, but we could not find any independent experimental evidence for or against this prediction. CONCLUSIONS We have seen that the present simplified quasi-two-dimensional model for the near wall region of the turbulent boundary layer reproduces several of the aspects of the burst-

ing phenomenon, as well as of the structure of the low velocity streaks. !n particular it has been shown that flat sheets of streamwse vorticity near the wall roll into compact cores, th-.t this process, which is a prerequisite for the appear-

______and

anLe of strong longitudinal vortices, is fast and relatively 30

pendent of the presence of a longitudinal strain. Morevoe,, the roll-up mechanism is essentially two dimensional

-inc,

3

-(;n

y" 20

" e y-z plane) and unrelated to the formation of hairpins by longitudinal tilting of the spanwise vorticity. While the latter process has been shown repeatedly to be important in transition of wall flows into turLulence, this new mechanism provides an alternative way for the regeneration of turbu-

___--

10 0-____

lent structures once fully developed turbulent flow has been

0

0

0

Fig. 8 - Contour plots of streamwise velocity corresponding to Fig. 7. (Aq increment = .025)

40

established. This new mecha.nism also makes less surprising the relative rarity of symmetrice! vortex pairs that has been reported in several recent investigations of the wall region. On the other hand, our simulation. show that even very

asymmetric vortex pairs would eventually roll-up into a more or less symmetric configuration, which then lifts away from the wall under its own iuductiop, and that this tendency is greater in the presence of a positive strain. This suggests that symmetric hairpins should be more common in the outer region of the boundary layer than near the wall, and that the same should be true in layers subject to a favourable pressure gradient. While there is some experimental support

REFERENCES

for the fi'st trend, we know of no experimental or numerical evidence relating to the second. The effect of the longitudinal vortices on the average streamwise shear also reproduces well the general structure of the longitudinal streaks found in the wall region. While it has been clear for some time that low speed streaks are associated with the advection of slow fluid away from the wall by the vertical velocities induced by the longitudinal vortices, we believe that this is the first time that it has been shown quantitatively that the net effect of this transport is to increase the averaged wall stress. In other words, that the transport of high speed fluid towards the wall is more important than that of low speed fluid away from it. The generation of a high average wall shear is, of course, one of the most characteristic properties of the turbulent boundary layer, and oae of its most technologically interesting ones. On the other hand, most of the three dimensional effects are absent from our model, and the main consequence is that our turbulence will not self sustain. The Reynolds number that we had to use for our vortices to survi , tor a reasonable time were at least twice larger than those observed in direct numerical simulations. We believe that, even if the present

Jimdnez, J. and Moin, P. 1991 The minimal flow unit in near wall turbulence. J. Fluid Mech. 225, 213-240.

model does give interesting indications of the behaviour of bursting structure for short times, their long time behaviour, and their genesis, depends on the correct understanding of the three dimensional effects.

Schlichting, H., 1968 Boundary layer theory, 6th Edition, McGraw-Hill, pg. 87.

ACKNOWLEDGEMENTS

Blackwelder, R.F. and Eckelmann, H. 1979 Streamwise vortices associated with the bursting phenomenon. J. Fluid Mech. 94, 577-594. Ersoy, S. and Walker, J.D.A., 1986 Flow induced at a wall by a vortex pair, AIAA Journal, Vol. 24, No. 10, 1597.

Kim, J., Moin, P. and Moser, R. 1987 Turbulence statistics in fully developed channel flow at low Reynolds number. J. Fluid Mech. 177, 133-166. Kim, H.T., Kline, S.J. and Reynolds, W.C. 1971 The production of turbulence near a smooth wall in a turbulent boundary layers. J. Fluid Mech. 50, 133. Moin, P. 1987 Analysis of turbulent data generated by numerical simulations, AIAA Paper 87-0194. Orlandi, P. 1989 Numerical simulation of vortices motion in presence of solid boundaries. Proceedings of 8th GAMM Conference on Numerical Methods in Fluid Mechanics edited by P. Weeseling (Vieweg, Braunschweig), 436-441. Robinson, S.K. 1991 Coherent motions in the turbulent boundary layer. Annual Review of Fluid Mechanics, 601-639.

Spalart, P.R. 1988 Direct numerical simulation of a turbulent boundary layer up to R, = 1410. J. I luid Mech. 187, 61-98.

This work was started while the authors were visiting the Centre for Turbulence Research, whose support is gratefully acknowledged. We are especially indebted to the fruitful discussions with 0. Sendstad, who also produced figure 1. Subsequent research of P. Orlandi was supported with funds by the Ministero Pubblica Istruzione. That of J. Jim~nez, by a grant from United Technologies Corporation.

28-1-6

EIGII'll SYNIPOSIUJ[I ON TURBULEN'r SlIEAR FLOWS

28,2

28-2

Technical University of Munich September 9-11, 1991

COHERENT STRUCTURES IN THE TURBULENT WAKE OF A POROUS BODY

M. Matsumura*, Z. Huang', J.G. Kawall*" and J.F. Keffer" Applied Mechanical Engineering, Kitami Institute of Technology, Kitami, Japan Mechanical Engineering, University of Toronto, Toronto, Ontario, Canada M5S 1A4

ABSTRACT

early stages of its evolution and in its eventual self-

The coherent structures within the near region of the plane turbulent wake generated by a porous body are investigated by means of hot-wire anemometry and digital data analysis. The results serve to establish that the coherent structures in question are generally similar to those within the near region of the plane turbulent wake generated by a solid body, such as a vertical flat plate, being essentially two-dimensional vortices, with dominant spanwise vorticity, that occur in a quasi-periodic fashion. There are, however, fundamental differences between the coherent structures within the two types of wakes. To be specific, the relative contributions of these structures to the total Reynolds stresses are definitely not the same. Moreover, the porous-body wake structures are more diffuse than are the solid-body wake structures and, unlike the latter, do not appear to be inter-connected via 'ribs'.

preserving state, upon the type of body generating tbwake. Moreover, it is now generally accepted that coherent structures play a central role in the dynamics of a turbulent flow. To be specific, these organized motions contribute, along with the random turbulent field in which they are embedded, to the statistical properties of the flow (e.g., Kiya and Matsumura, 1988; Hussain, 1986). This suggests that different types of wake generator give rise to coherent structures that differ significantly in their detailed characteristics, such as, shape, size, orientation, circulation strength, etc., although their general effects on flowbehaviour are similar. If such differences exist, then they should manifest themselves in terms of the relative contributions of the organized motion and the random motion to the statistical properties of wakes produced by different types of bodies. As well, the detailed spectral composition of the turbulent field is likely to depend upon the type of wake generator involved. The present work concerns an experimental investiga-

NOMENCLATURE f - frequency, Hz. Su

- length of the wake generator in the z direction, m. - autospectrum of the fluctuating streamwise velocity component, m 2 s - 1 .

S

-

L

autospectrum of the fluctuating lateral velocity component, m 2 s -1 .

S,,v - co-spectrum of the fluctuating streamwise and lateral velocity components, m2 s - 1. T - time period, s. I - timep, s - time, s. U,V - stredmwise and lateral velocities in the x and y directions respectively, m u.v x,y,z

-

-

-1

.

fluctuatng sreamwise and latera! velocity compo-

to the coherent structures and those due to the random turbulent field.

nents n - 1 .

EXPERIMENTAL DETAILS

streamwise, lateral and spanwise coordinate directions, mm.

The experiments for this investigation were performed in the variable-speed low-turbulence recirculating wind tunnel situated in the UTME (University of Toronto, Mechanical Engineering) Turbulence Research Laboratory. This tunnel has a test section 4 m in length and 0.91 m by 1.2 m in cross-section and a speed range of about 1 m/s to about 15 m/s. The wake was generated by means of a mesh-strip with a solidity of about 60%, mounted horizontally at the beginning of the test section. The width (W) of the wake generator was 18 mm and its length (L) was about 0.91 m, so that the aspect ratio (L/W) was about 50. A

angle, deg. spanwise vorticity, s-1 .

-phase

o

-

tion of the near region of the plane turbulent wake generated by a porous body, viz., a vertical mesh-strip. The main objective of the work is to try to establish whether or not the "near-wake" coherent structures in a porous-body wake differ significantly from those in a similar solid-body wake. To this end, certain structural features of the present flow are compared to those of the (solid-body) wake generated by a vertical flat plate, reported by Kiya and Matsumura (1988). The signal-analysis technique of the latter, which is based on the triple-decomposition method (Reynolds and Hussain,1972) is employed here to separate the contributions to the statistical properties of the flow due

INTRODUCTION Recent studies (e.g., Zucherman, 1988; Louchez, Kawall and Keffer, 1987; Wygnanski, Champagne and Marasli, 1986) have indicated that the detailed behaviour of a two-dimensional turbulent wake flow, based on its measured statistical properties, is strongly dependent, both in the

free-stream velocity (U,) of 9.8 m/s was used, so that the Reynolds numbe based on W was about 11,000. A Cartesian coordinate system was employed, with x in the streamwise direction, y iii the lateral or vertical direction and z in the spanwise direction ( parallel to the axis of the wake-generator). The origin of the coordinate system was taken to be in the centre of the wake-generator. Two streamwise (U) velocity signals and two lateral (V) velocity signals were measured simultaneously at various positions in the flow by means of two DANTEC 450 X-wire probes, in conjunction with four DANTEC 55M constant-temperature-anemometer systems. These signals were low-pass filtered at 5 kHz and digitized by means of a 12-bit A/D converter attached to a PDP-1 1 computer. The digitization rate was chosen to be 12,500 points per second (to prevent aliasing), and the digital signals were processed on a DEC MICRO-VAX workstation. The length of each signal processed was approximately 28 seconds. The foui velocity signals were measured in the vertical centre-plane of the flow (z = 0) at a streamwise location corresponding to x/W = 24. One X-wire probe was kept fixed near the centre of the a.-e (y = 0) while the other probe was moved laterally acro.. the wake in increments of approximately 0.1 W between y/W = -2.4 and y/W = +2.1. (Both of these y/W locations were in the free-stream.) Streamwise velocity (u) autospectra, measured by means of a normal hot-wire probe. established that for x/W minant locations between about 10 and about 40, a quasi-periodic motion occurred in the flow, sign..ying that large-scale coherent (i.e., vortical) structures were present in the near region of the wake ( - 10 < x/W < - 40 ). Moreover, these spectra showed that the quasi-periodic motion was most pronounced at x/W = 24. Accordingly, the simultaneous measurements of the two U signals and two V signals were taken at the later x/W location. A typical lateral velocity (v) autospectrum ob-tined at x/W = 24 and y/W = 0.1 is presented in Fig. 1, where a distinct peak is clearly evident at a frequency (fto) of about 100 Hz. This frequency is analogous to the Karman vortex shedding frequency in the case of a solid-body wake.

001-

0 O

,

r>

r0 r

,--,,0

f (Hz) Fig. 1. Lateral velocity autospectrxzn at x/W=24 and y/W0.1. Each V signal measured by the fixed X-wire probe was (digitally) band-pass filtered, with the centre-frequency of the filter set at of f,. The filtered V signal (v,) was used to condition each pair of U and V signals measured by the movable probe so that these signals could be decomposed into overall time-averaged components, U and V, coherent components, and i, associated with the organized (vortical) motion and random or incoherent components, u' and v', associated with te .andom turbulent motion. A typical v. signal is presented in Fig. 2. As can be seen, this signal has a distinct periodicity ( with a period T, = lfo ), but it is also significantly randomly-modulated. We note that the conditioning signal (vo) obtained by Kiya and Matsumura (1988), in the case of a solid-body wake, was not randomly-modulated to any significant extent. The implication here is that the large-scale vortical structures within the present porous-body wake exhibit much greater dispersion ( with respect to size, shape, orientation, circulation strength, etc.) than do those within the solie-body wake examined by Kiya and Matsumura.

0.4-

0.2-

0.0 -0.2

-0.4

0.10

0.15

t (s) Fig. 2. Band-pass-filtered lateral velocity signal.

28-2-2

0.20

With the U and V signals decomposed, appropriate averages, based on the overall fluctuating components u = ii+u', v = i4v' and uv, the coherent components, i, and 0, and the incoherent components, u', v' and uv', were determined. These statistics enable the contributions to the statistical properties of the flow due to the organized motion and those due to the random turbulent motion to be obtained. Isocontours or phase-averaged patterns of various quantities were also determined and plotted in terms of y/W versus the phase angle 4, which is given by 4 = 360 tITo, where t is time and T = 1/f,. The phase-averaging was accomplished as follows (Kiya and Matsumura, 1988): consecutive instants tI and t2 = tj ± T, were determined such that the conditioning signal, vo,satisfied the conditions v. = 0 and dvo/dt > 0 at t , the r2 - tl intervals were aligned and then averaging was performed. The number of realizations used in the phaseaveraging process was approximately 2,800, a value sufficiently large to enable stible phase-averaged patterns to be obtained, In addition, spectral analysis of the u and v signals was performed, yielded u & v auto-spectra, uv co-spectra, pe rfre which wiyeeu& anduv atspectra, co-spectraunci uv quadrature spectra and uv coherence functions, RESULTS AND DISCUSSION An isocontour plot of the phase-averaged spanwise vorticity (o, - a=/ax - Qi /Oy), obtained with the aid of Taylor's hypothesis, is presented in Fig. 3. It is evident from this figure that there are large-scale coherent structures present within the porous-body wake at x/W = 24. These structures are vortices possessing a dominant spanwise vorticity (co,); also, they occur in a quasi-periodic fashion, with a period T0 = lIf,, and alternately on either side of the horizontal centre-plane of the flow,, with their vorticity concentrated near this plane at approximately y/W = ± 0.5. Isocontour plots of the phase-averaged streamwise and lateral velocities (i and i) are presented in Fig. 4 and 5 respectively. As expected, these figures show that (a) the

centres of the

u

while the centres of the isocontours are located in the vicinity of y/W = 0.0, and (b) the spacing between the centres is essentially constant, corresponding to To/2. At this streamwise location (x/W = 24), the present porous-body wake is thus qualitatively similar to a plane solid-body wake in the early stages of its evolution (e.g., Kya and Matsumura, 1988). It should be emphasized, however, that there is no evidence of large-scale coherent structures in the present flow prior to x/W = 10. In contrist, in the case of a solid-body wake, "Karman" vortices (with dominant spanwise vorticity) o'cur almost immediately downsur:sm of the wake-generator, i e., by x/W = 2 (Zucherman, 1988). This difference between a solid-body wake and the present porous-body wake stems from the fact that the coherent structures in the litter flow are the product of the formatio'i and evolution of two shear layers at the upper and lower edges of the wake-generator (i.e., the mesh-strip) which do not roll-up until x/W = 10. Another difference between the two "wake types" is that, in the near region of the porousbody vake (x/W < - 10), before the large-scale vortices are completely formed, the central core of the flow is similar to a homogeneous turbulent flow, containing tuibulent fluid with a relatively wide range of scales of motion; whereas, tie central core of the near region of a solid-body wake contains essentially unstable laminar (vortical) fluid.

2.0

1.0

0.0

>,

-1.0

2.0

isocontours are located on either side of

270

9.

ISO

90

0

1.0

-1.0

.10

.270

0

-l.UfL1 2.

9

90

Phase(deg.) Fig. 4. Phase-averaged streamwise velocity iso-contours.

cenre-plane of the flow in the vicinity of y/W = ± 1.0,

OW3

-2.0

3

"0

In

-2.0

270 270

IO

90

0

90

.1so

1W

90

0

90

.1S0

Phase(deg.)

270

Phase(deg.) Fig. 3. Phase-averaged vorticity iso-contours.

Fig. 5. Phase-averaged lateral velocity iso-contours.

28-2-3

270

Moreover, the fluid constituting 1Ia t.-'herent structures in the porous-body wake is a mixtu -.of laninar fluid and fine-scale turbulent fluid, since, as the two aforementioned laminar shear layers develop, they entrain both turbulent fluid from the core of the wake and potential fluid from the free-stream. On the other hand, the fluid constituting the coherent structures in a solid-body wake is, by comparison, essentially laminar fluid only. The detailed characteristics of the (near-region) coherent structures within a porousbody wake can therefore be expected to be considerably different from those of the (near-region) coherent structures within a solid-body wake. The results presented below serve to establish this difference. Lateral (y) distributions of the streamnwise and the lateral mean-square velocities and the Reynolds shear stresses due to the coherent motions and the random turbulent motions (Fig. 6(a) and (b), Fig. 7(a) and (b), Fig. 8(a) and (b)) indicate that the relative contributions to the overall statistical properties of the flow from the two types of motions are substantially different with respect to the present porous-body wake and the solid-Lody wake exam-

0.014

(a) 0.012

G

of the maximum value of 2 to the value of v'2 at the corresponding y/W location is about 3.3, whereas, for the solid-body wake, this ratio is about 1.4. And, for the present wake, the ratio of the maximum value of iii to the value of u'v' atthe corresponding y/W location is about 2.0, whereas, for the solid-body wake, this ratio is about 0.2. Further evidence of the difference between the near-

"00.0 08 P 0.006

respectively pertaining to y/W = 0.6, where the magnitude is close to its of the overall Reynolds shear stress (v)

. , 0

0.004 0

0.002

0 0

0

.0 0.0 -2.0

2

1

0

-1.0

1.0

0.0

2.0

y/W I

I

0.006--

I (b)

"

°°'00°4 =.0.004

0*****.

I

* o

.

-2.0

0*

-1.0

0.0

2.0

1.0

Y/w Fig. 7. Lateral distributions of lateral

mean-square velocities: (a) due to coherent mtion, (b)

due

riotion.

to randof

maximum value. (The u' and v'signals on which the spectra in Figs. 10-12 are based were obtained via digital bandpass filtering of the corresponding u and v signals, with the band-pass filter centred at approximately fo.) As can be seen from Fig. 9, the Fourier components that contribute to

0.006

region coherent structwes within the present porous-body wake and those within a solid-body wake is provided by Figs. 9,10,11 and 12, which show the uv co-spectrum, the u'

autospectrum, the V autospectrum and tIe u'v' co-spectrum

o

0.010

ined by Kiya and Matsumura (1988). To be specific, for the present porous-body wake, the ratio of the maximum of value of i 2 (coherent) to the value of u2 (random) at the corresponding y/W location is about 0.4, whereas, for the solid-body wake examined by Kiya and Matsumura (1988), this ratio is about 0.2. Also, for the present wake, the ratio

)

0

(a)

0.004 0

o

0.002 -

"o i

0.0

_--_-

--000-

--

----------0

V -0.002

000*

-0.004 0.004 .

.....

-0 0 6

1

-1.0

0.0

1.0

(b)

0.004 0

0.002 (b) .

Do

o

000.

0.0 .---

--- -

---

o

00

-0.002

0.004 o

0.0 ' 0.0

y/W

0.006

0.008 000000000

2.0

1.0

0.0

2.0

y/W

0.006 0.004

-1.00

0

O 0 1*

-2.0

L,

2.0

(a)

****

0.002

-0.0 04 0

1. 1____0___-0.006______

-2.0

-20 -10

0.0

10

-1.0

2.0

0.0 -2.0

Fig. 6. Lateral distributions of streamwise nean-square velocities: (a) due to coherent motion, (b) due to random motion.

28-2-4

-1.0

0.

1. .0

20 2.0

Fig. 8. Lateral distributions of Reynolds shear stresses: (a) due to coherent motion, (b) due to random motion.

0003.

0 0

cosetuNa

/=2

aayW06

Fig. 9. Strea and .Fwise lateral velocity co-spectrum at x//W24 and y//W.0.6.

the coherent motion are confined to a relatively narrow

6

band of frequencies centred at to = 100 Hz. On the other hand, it is evident from Figs. 10-12 that the Fourier corn-

.o

to

Fi.1.

adm

11. atg

ponents contributing to the random motion are spread over a wide band of frequencies extending from 0 Hz to the cutoff frequency of 5000 Hz used in the present work and that there is no frequency centred activity below fo. In contrast, in the case of the solid body wake examined by Kiya and Matsumura (1988), the u'v' co-spectrum displays a distinct

f(Hz

60

ow

(aeaHvlctyz)o~etu utsctu

=2laerl n a~no d elciy

0001

frequency-centred activity in the vicinity of fo/2. As well, these authors established that the Fourier components associated with the latter frequency make the major contribution to u'v" (the random Reynolds shear stress) and that there is little contribution from Fourier components associated with frequencies greater than fo. From this, they inferred that the coherent structures within the near region

o

of their solid-body wake are physically inter-connected via

0

'ribs' which are themselves organized structures arranged in the spanwise direction, with dominant vorticity perpendicular to the dominant spanwise vorticity of the main structures. The present results serve to show that there are no dominant Fourier components contributing to uv' in the case of the porous-body wake. Thus, it can be concluded that the near-region coherent structures of this flow are not inter-connected via 'ribs'.

-,

-

--

f,(Hz) Fig. 12. Random streawuise and lateral velocity co-spectrUrn at x/W=24 and y/W=0.6.

ACKNOWLEDGEMENT This research was supported by the Natural Sciences and Engineering Research Council of Canada through Grant A-2746. wl

REFERENCES

-a

HUSSAIN, A.K.M.F. 1986 Coherent structure and turbulence.J. Fluid Mech. 173,303-365. KIYA, M. and MATSUMURA, M. 1988 Incoherent turstructure in the near wake of a normal plate. J.

1.bulent

Fluid Mech.

\

,-

-r.r.rT

--.-

f(Hz)

r-Trrr----r-,

10, 343 - 356.

LOUCHEZ, P.R. KAWALL, J.G. and KEFFER, J.F. 1987 Detailed spread characteristics of plane turbulent wakes. Proc. 5th Symp. on Turbulent Shear Flows, Lectures Notes in Physics. Springer Verlag, 98 - 109. REYNOLDS, HUSSAIN, A.K.M.F. 1972 shear The mechanismW.C. of anandorganized wave in turbulent

Fig. 10. Randomn screamwise velocity .autospectrum at x/W=24 and y/=O.6.

flow. Part 3. Theoretical models and comparisons with

experiments. J. FluidMech. 54, 263 - 288.

28-2-5

WYGNANSKI, I., CHAMPAGNE, F. and MARASLI, B. On the large-scale structures in two1986 dimensional, small-deficit wakes. J. Fluid Mech. 168,31-71. ZUCHERMAN, L. 1988 An investigation of coherent structures within plane wakes using a patternrecognition technique. Ph.D. Thesis, University of Toronto.

28-2-6

EIGIITIh SVIOSL

ON

2 -

TURBULENT SHJEAR FLOWS Technical University of Munich September 9-11, 1991 On the Quasi-Coherent Turbulence Structures in the Two-Dimensional Channel Flow Koichi NISHINO "1 and Nobuhide KASAGI " ) " Department of Mechanical Engineering and Materials Science,

Yokohama National University, Tokiwadai 156, Hodogaya-ku, Yokohama 240, Japan Department of Mechanical Engineering, The University of Tokyo, Hongo 7-3-1, Bunkyo-ku, Tokyo 113, Japan ABSTRACT The quasi-coherent turbulence structures have been studied using an experimental data base of a fully-developed two-dimensional turbulent channel flow. The data base was obtained by Nishino & Kasagi (1989) using the threedimensional particle tracking velocimeter (3-D PTV). The stochastic estimation proposed by Adrian (1975) is used to approximate the flow patterns that most contribute to the Reynolds stress generation. The two flow patterns being estimated reveal the streamwise vortical structures, which are respectively associated with the ejection and the sweep. They are inclined to the wall at 10' with spanwise and streamwise extent of about 50v/u, and 300v/u,, respectively. Analysis of vorticity fields reveals that typical hairpin-like vortex lines are seldom created near the wall. Based on these results, a conceptual model of the vortical structures is proposed. NOMENCLATURE H channel width L, estimation coefficients ,I length scales in the x- and z- directions P(4,, 0) joint probability density function of 4 and 0 Re, Reynolds number w uH/v Re, Reynolds number w uHi/2v R, R-, R._ . Two-point correlation functions u', v, w' velocity fluctuations in the x-, y- and zdirections u center-line velocity u, friction velocity x, y, z streamwise, wall-normal and spanwise directions v kinematic viscosity mean vorticity in the z-direction w', aw,' (o', vorticity fluctuations in the x-, y- and zdirections 4, 0 angles of velocity vector ( ),. :root-mean-square value ( ) :conditionally averaged value < > ensemble-averaged value INTRODUCTION The quasi-coherent structures in turbulent wall shear flows have been studied by many researchers over the past three decades, and various types of the structures have been reported (see, e.g., Kline and Robinson, 1990). Using the direct numerical simulation (DNS) data base, Robinson et al. (1990) extensively studied interrelation among various instantaneous turbulence structures perceived in a simulated turbulent boundary layer. Since those structures show very large variation in their size and strength between realizations, ensemble averaging is necessary to capture their common and universal characteristics that are essential to the better understanding of their role on the turbulence mechanism.

The stochastic estimation devised by Adrian (1975, 1978) is advantageous for obtaining averaged features of the turbulence structures. Unlike the conditional sampling techniques such as the four-quadrant and VITA analyses, this technique is based on unconditional statistics, i.e., correlation functions of fluctuating quantities of interest. This property makes the technique very attractive in providing wellconverged results from relatively small number of data. In addition, direct linkage between the estimated flow patterns and the correlation functions allows objective interpretation of the role of the turbulence structures. Despite these advantages, however, it has rarely been applied to the experimental data of turbulent shear flows; a few exceptions are studies by Adrian et al. (1987a), Guezennec and Choi (1988) and Guezennec (1989). This is because it requires full information on the two-point correlation functions which are hard to measure with the conventional velocimeters. Hence, the stochastic estimation has mostly been applied to the analysis of the DNS data base, e.g., Adrian et al. (1987b), Moin et al. (1987) and Adrian & Moin (1988). These studies have revealed in detail important characteristics of the turbulence structures, most of which, however, have not been confirmed experimentally. One of the objectives of the present study is to provide an experimental evidence that can be compared with those DNS results. The authors have developed a three-dimensional particle tracking velocimeter (3-D PTV) that can measure an instantaneous distribution of all the three velocity components in a measuring volume. Details on this technique can be found in Nishino et al. (1989), Sata et al. (1989) and Kasagi & Nishino (1990). Using this velocimeter, Nishino & Kasagi (1989) measured a fully-devloped two-dimensional turbulent channel flow; the flow conditions are compiled in Table 1. The measuring volume was 40x40x40 mm3 (200x200x200 v3/u,3), which covers the region from the wall to the channel center. An average of 140 instantaneous vectors were simultaneously measured at random positions in the measuring volume, and a total of 32,000 3-D vector distributions were obtained. The data base was qualified in Nishino & Kasagi (1989) through detailed comparison with the previous experimental and DNS data. In the present study, this data base was further analyzed to investigate the quasi-coherent turbulence structures.

28-3-1

Table 1. Basic Flow Conditions

H = 80 mm, v

z

0.831x10

u, = 4.27 mm/s, Re,

6

m2/s, u = 78.0 min/s,

7510, Re,

205.

LINEAR STOCHASTIC ESTIMATION The linear stochastic estimation is briefly described below; readers should refer to Adrian (1975, 1978) for full information. This technique approximates a conditionally averaged velocity field, O',(x+r), as a linear combination of given conditions, u'(x), as follows: (1)

i = 1, 2, 3,

i',(x+r) = L,(x,r)u'(x),

where repeated subscripts are summed. As the velocity conditions, the present analysis uses three fluctuating velocity components at a prescribed position in the flow field, i.e., (u', v', w') at y'=14.5. L,,(x,r) are determined so that the mean square error between the estimated velocity field and the instantaneous realization is minimized, < {',(x+r) - Lt(x,r)u,(x)} 2 > = minimum.

(2)

Nishino & Kasagi (1989) are used to solve Eq. (3) for LI.(x,r). The velocity conditions (u'(x), v'(x), w'(x)) are tage of linearity of the linear determined by taking r stochastic estimation. Ab described by Moin et al. (1987), the estimated velocity field does not alter its pattern with the proportional changes in the intensity of the velocity conditions. For example, the conditions of (u'(x), v'(x), w'(x)) and (ku'(x), kv'(x), kw'(x)) produce exactly the same flow patterns with difference only in their strength by a factor of k. The proportional change preserves the direction of the velocity vector, thus a joint probability density function (PDF) of threedimensional velocity angles is investigated as below. The two velocity angles, 0 and 0, are defined as shown in Fig. 2. With this definition, each quadrant of the 4-0 plane, shown in Fig. 3, corresponds to that of the u'-v' plane used in the four-quadrant analysis. In both cases, the first to fourth quadrants respectively represent the outward interaction, the

L,(x,r) as follows: This equation is formally solved for (3)

L'(x,r) = -',

where are the one-point, second-order correlations and are the two-point, secondorder spatial correlation functions. It is obvious that O',(x+r) represents the most probable velocity field in a sense of the least square error under the given conditions. This feature of the estimation is exploited in order to approximate flow patterns that most contribute to the Reynolds stress generation. Full components of the two-point, second- order spatial correlation functions are calculated from the data base in the region of -150sxs 150, 0y's150 and -150 10 so the influence of the term is automatically confined to the viscous sublayer, as desired. With c¢ = 0.05 a much improved distribution of u2 and u2 results. While the term is redistributive, it is unlikely that the phenomenon arises from oij since, in the exact process, both 41 and 3 vanish at the wall. It seems more likely that the term is really providing a small correction to the model of e., . Apart from the effect shown the term seems to be passive and it has thus been retained in all subsequent computations. Figuie 3 and 4 show the computed rms velocity profiles and turbulent shear stress for this case; vety close accoid with the data is ach~eved. Tile corresponding profiles at tle higher Reynolds number of 13750 appear in Fig 5 and 6: agreement is again acceptably close though, by comparison with the previous case it is clear that the computations exhibit rather less sensitivity to Reynolds number than do the simulations. This is a little surprising since the present model seems to repioduce correctly the shift in the additive constant in the logarithmic law foi the mean velocity [4).

-.nd u. Nevertheless, consideration of the profiles of c,j - and in particular the distribution of e12 - suggests that considerably more effort is needed to arrive at a satisfactory modelling of the component processes. Acknowledgements We gratefully acknowledge support of this research by the UK SERC under grant GR/D/8285.2. Our thanks also go to Mrs J. Buckley for her help in preparing the camera-ready manuscript. Authors are listed alphabetically.

References 1. Craft, T J. and Launder, B.E., Paper 17-1, Proc. 7th Symp. Turb. Shear Flows, Stanford University, 1989.

In fact the best choice of cl and c ' is hard to determine because both act in time same sense. Thus a reduction in one can be neaily offset by an increase in the other, at least in a simple shear flow. To illustrate this point, Fig 7 shcws the computed rms fluctuating velocity profiles that result from taking c = 0.4 and cu' = 0, i.e. with the wall reflection term eliminated entirely. The agreement with the DNS piofiles is still very close though not quite as good M in Fig 3. Nevet theless, the prospect that one may be able to devise a widely apphicable 2nd moment ch'suie that makes no lefalcn' to wall dist ante is ei tainly intee~tiig fiomi a ptu tlmal poit of view.

2. Cresswell, R.W., Haioutuman, V., Ince, N.Z., Laundel, B.E. and Szczepuia, R.T., Papei 12-4, Pioc. 7th Synip. Tuib Sheu Flows, Stanford University, 1989.

Finally Fig 8 tonsideis time noinialized dmsbipatmoii pfiod eso k/ Iuo: (no s,mmation). Oue notices that the DNS dat, Fig 8a, clearly does asymptote to the theoretical values given by 2q (7) at x + = 0. The comparison serves to make two further points. Firstly, while the Reynolds stress field itself is very accurately predicted by the closure, the same cannot be said of the dissipation components (or, for that matter, the other piocesses) in the stress budget. Secondly, the armtion of the e12 component is so complicatel and so at variance with current concepts of the dissipative pioce.ses that, if the DNS data are reliable, the modelling of the dissipation tensor will require a great deal of attention in future.

5. Kim, J., Moin, P. and Moser, R., J. Fluid Mechanics, 177, 133, 1987.

.

Launder, B.E. and Tselepidakis, D.P., "Contibution to the second-moment modelling of sublayer turbulent translort%, Invited paper, Zoran Zari Memorial, International Seminar on Near-Wall Turbulence, Dubrovnik, Yugoslavia, 1988.

4. Launder, D.E. and Tselepidakis, D.P., AIAA Paper 91-0219, 1991.

6. Kim, J., Personal commutication, 1991. 7. Lee,. M J. and Reynolds, W.C , Proc. 5th Symp Turb. Shea Flows, p. 17.7, Cornell Univeisity, 1985. 8 Lumley, J L., Adv. in Applied Mech, 18, 123. 1978. 9. Slih, T-H. and Lumley, J.L., "Modeling of Pesmuc Corielation Terms in Reiolds Stiebs and Scala Flux Equatom", Report FDA-85-3, Sibley School of Mech. and Aerospace Eng., Cornell Universty, 1985. 10. Chou, P.Y., Qnart. Applied Mathematics, 3, 38, 1945.

4

Summarizing Remarks

A second moment closure for the near-wall region has been . veloped that in1ims vCiN closely the distitbutmon of the Reynolds simi(,l(,ll' t .loll, of stress conllpononits, obtalined bN\the (3(C 1 /C2 ) + 3)/2 then E -*0 for k - 0 as t- 0,,

OO l

where n1 is the initial low wavenumber dependence of the energy spectrum. For the present application Ct/C2 = 2, therefore, {3(C /C 1 2 ) + 3)/2 4.5. Figures ", 8, and 9 show the differences in the spectral development for ni less than 4.5. The figures show that the energy spectrum quickly approaches its

00001

Figure 8.

29-2-4

.

01t

......

01

t

10

........ 10

1000

k/m, Turbulent energy spectrum as a function of time with ni = 4.0

02D0

o0011

n =al

010-

256

ooo, 0001

o

00

000 1

00001 00t

1

o1

020

100015

to

kAMAXY/U~t

Turbulent energy spectrum as a function of time with ni = 2.0

Figure 9.

Figure 11.

Not only does the initial low wavenumber dependence affect the final shape of the turbulent energy spectrum but also the final self-smular forms of the mean velocity and spectrally integrated turbulence quantifies. This is shown in Figs. 10 and 11. Here are plotted the mean velocity and streanwise tur-

CONCLUSIONS The present spectral transport model sufficiently predicts the self-similar behavior of the mixing layer within the variations

bulent intensity with a given set of constants except the initial low wavenumber dependence is varied. It is shown that this dependence can greatly affect the final self-similar shapes of the profiles. This is a possible explanation of why there is such a large variation in the self-similar profiles as given by the. different experimental investigations (see Figs. 2 through 5); that is, the initial low wavenumber dependence will affect the final selfsimilar form. I n

09-

found in the experiments. Also predicted is the power law dependence of the energy spectrum in the transition to turbulence. Some universality of constants is believed since the present ones fit the data of mixing layers as presented here and anisotropic strain presented in Clark (1991), although questions do arise on thu appropriate value of CD. Many predictions of the nature of the turbulent energy spectrum can be made with this model, but the most interesting would be the initial low wavenumber dependence of the final self-similar forms of the spectra and mean flow quantities.

..-

-

Experimental verification is hampered by the fact that in general I-D spectra is commonly measured not the 3-D spectra which the model predicts. The low wavenumber portion of the 1-D

.? -

08 0.81

spectra does not correspond to that of the 3-D spectra due to

0.7-i..;

Further experimental evidence of the nature of the

/aliasing.

0.-

spectrum at low wavenumbers would be very useful in further validation of the model.

06OA

Streamwise turbulent intensity versus crossstream direction for various initial low wavenumber dependence of the energy spectrum

/

03.

02-

REFERENCES

0.t 0_

Besnard, D. C., Harlow, F. H., Rauenzahn, R. M., and Zemach, C., 1990, "Spectral Transport Mode) for Turbulence," Los Alamos National Laboratory report LA-1 1821-MS.

0,

0.00

o 0.36

010

0.15

y/Uot Figure 10.

Nondimensional velocity versus cross-stream direction for various initial low wavenumber dependence of the energy spectrum

0.20

Clark, T., 1991, Ph.D. Dissertation, University of New Mexico, in progress. Huang, L. S. and Ho, C. M., 1990, "Small Scale Transition in a Plane Mixing Layer," J.FluidMech. Vol. 210, pp. 475-500. Jimenez, J., Martinez-Val, R., and Rebollo, M., 1979, "On the Origin and Evolution of Three Dimensional Effects in the Mixing Layer," Internal Report DA-ERO-78-G-079, Univ. Politec., Madrid.

29-2 -5

Liepmann, H. W. and Laufer, J., 1947, "Investigation of Free Turbulent Mixing," NACA Tech. Note No. 1257. Patel, R. P., 1973, "An Experimental Study of a Plane Mixing Layer," AIAA J., vol. 11, pp. 67-71. Riley, J. J., Metcalfe, R. W., and Orszag, S. A., 1986, "Direct Numerical Simulations of Chemically Reacting Turbulent Mixing Layers," Phys. Fluids, Vol. 29, pp. 406-422. Taulbee, D. B., 1990, "Engineecing Turbulence Models," in Advances in Turbulence, W. K. George and R. Arndt, Eds. Wygnanski, I. and Fiedler, H. E., 1970, "The Two-dimensional Mixing Region," J. Fluid. Mech. Vol. 41, pp. 327-361. Zohar, Y., 1990, "Fine Scale Mixing in a Free Shear Layer," Ph.D. Dissertation, University of Southern California.

29-2-6

El1Tilll SYMPOSIUM ON TURBULEN i' SiEAR FLOWS Techn'ical University or Munich September 9-1I, 1991

29-3

TURBULENCE MODEL FOR THE DISSIFATION COMPONENTS OF REYNOLDS STRESSES

M Tagawa, Y. Nagano and T. Tsuji Department of Mechanical Engineering, Nagoya Institute of Technology Gokiso-cho, Showa-ku, Nagoya 466, Japan

ABSTRACT

()

This paper develops a new model for the nonisotropic dissipation rate tensor c,, to close the Reynolds-stress transport equations. The previous models of e, are all algebraic, based on a hypothesis of either the isotropic relationship of the dissipation tensor or the one-to-one relationship of the dissipation amsotropy tensor with that for the Reynolds stresses. However, the recent direct simulation data of the turbulent boundary layer and channel flows have revealed the strongly nonisotropic nature of the dissipation process, which differs considerably from that of the Reynolds-stiess components. The present model provides nonisotropic dissipation-rate components through the solutions of approximated transport equations for e,,, and hence, the conventional prescription of an ad hoc value of -,, is not needed The proposed model is tested by direct comparisons with the reported full simulation data to validate the performance of the model predictions

(),

NOMENCLATURE CI, C2, Cl',C , = model constants for P,' CiC,p2 = model constants for N), Cs, C, = model constants for the diffusion term, in U," and c,, equations C.1 , C,2, C, 3 = model constants for the producticn and dissipation terms in e,, equation 'D,), PD, V Dj = diffusion terms in V equation [Eq.(8)] "D,,,P'D,,, "'D, = diffusion terms in e,, Pn.f ion [FPq.(10)] =dissipation term in equatmo Iq.(10)] E, f ,, f,,ff.= turbulence model functions k = turbulent kinetic energy, ,/2 T, p = mean and fluctuating pressures F.', lep,, 2ep",ap,', 4ep) = production terms in =,z and C,, equations [Eqs (8) and (10)] term inl k-equation, Pk R, = production puroductine termolds n-eatio, == P,12 P/ R,= turbulemice Reynolds number, = k0Ive time scales [Eqs. (16) and (11)] T, T, U,, u, = mean and fluctuating velocity components in z, direction it= friction vplorty, = .c/r/ X. = coordinates in tensor notation = coordinates X, y = dimensionless distance from wall, = ny/v y+ 6,, = Kronecker delta 6, e. = dissipation rates of k and u-n.m/2 i, sw,, = kinematic viscosity and turbulpnt diffusivity p = density % r = time and wall shear stress (P,4, = redistributive terms in U-u,- and c,-equations [Eqs. (8) and (10)]

= time mean value = differentiation by x, (= 0/Ox,) = tensor notation without summation convention

INTRODUCTION The current transport models for the Reynolds-stress equations have the following thr, e challenging problems: modeling of the redistribution te,rn (the pressure-strain correlation) 0, = (p/p)u,,; modeling of triple velocity correlations U~u,uk ; and modeling of the dissipation-rate tensor of Reynolds stress components e, = v U,,kU,,k. All of these exert an important influence on the universality of the Reynoldsstress equation model. There have been many proposals for modeling ou. At present, the model proposed by Launder, Reece & Rodi (1975) (hereinafter referred to as LRR) is most often used for turbulence predictions, although several more sophisticated models have been proposed (e.g., ILaunder & T. 3lepidakis 1990). As for the triple products 7u-,u-k, Nagano & Tagawa (1990a; 1991) have developed a -ew structural model based not on the conventional assumption of gradienttype diffusion but on the statistical and structural characteristics of turbulence (Nagano & Tagawa 1988). On the other hand!, the turbulence model for the dissipative correlation e,, which is an important sink term in the Reynolds-stress transport equation, is still rather primitive. The previous models of c, are all algebraic, based on a h pothesis of eithei the isotropic relationship of the dissipation tensor, so that (1) e, = (6 ./3) e or the one-to-one relationslip of the dissipation aiiisotropy tensor with that for the Reynolds stresses at lower Reynolds numbers, i.e., C,, = [(V-u, /2)/k] c (2) It is assumed that there is a gradual changeover from this limit, Eq.(2), to local isotropy, Eq.(1), as the turbulence Reynolds number R, = k'/ve becomes large. Thus, Hanjali & Launder (1976) modeled c, as = (1 -

.(3),

e+

,

with (4) fs = 1/(1 + R-/10) Here, the scalar trace of the dissipation-rate tensor, e ( c,, = vu,7ku), is the dissipation rate of the turbulent kinetic energy k (= u,/2). However, the recent direct simulation data of the turbulent boundary layer and channel flows have revealed the strongly non-isotropic nature of the dissipation process, which differs considerably from the anisotropy of the Reynolds-stress

29-3-1

components (Mansour, Kim & Mom 1988). In view of all these, this paper develops a new model for the non-isotropic dissipation-rate tensor e,, to close the Reynolds-stress transport equations. The present model provides non-isotropic dissipation-rate components through the solutions of approximated transport equations for c,,. The proposed model is tested by direct comparisons with both reported full simulation data and measurements of boundary I-i) r flows,

Here, the terms " P,, -. 4'P,; denote the production rates. 'E,, represents the dissipation (or destruction), 'F,, is the redistributive term, and "D,,, P'D. and I'D,, respectively represent the diffusion terms due to turbulent, pressure and viscous effects. Again, all the terms except for Dc,/Dr, "P,, and I'D,, should be modeled in Eqs. (9) and (10). In what follows, we take the xi, X2 and X3 axes in the streamwise (x), wall-normal (y) and spanwise (z) directions, respectively

GOVERNING EQUATIONS

CLOSING THE

The equations governing the flow in an incompressible fluid

1

'37EQUATION

Turbulent Time Scale

are

U,, = 0 D,/Dr = -Th,/p +

(Ul-)"

-

(5) (6)

and a comwhere the Einstein summation convention is used ma followed by an index indicates differentiation with respect to the indexed spatial coordinate, In the Reynolds-stress model, the unknown Reynolds stress u-3 in Eq (6) is determined from the solutions of the following transport equation: Du-'5/Dr= P, - 2c,, + 'F,, +'tD, + 5'D, + "D,

where

"

P,

-,i-

, U,, k +

(7)

We reflect the non-isotropic nature of both turbulent transport and dissipation processes in the present model Thus, we define the "directional"time scale T, as follows: '1' 2/(, (11) where the indices in parentheses do not obey the summation convention rule. Obviously, Eq.(11) gives a time scale equivalent to the relative lifetime of the energy-containing eddies in the x, direction. Reaistribution (Pressure-Strain Interactions) In principle, we use the LRR model for the pressure-strain term ',, in Eqs. (7) and (8). Thus, we write

U, k

4') = I (D" + 24'. +

V u,,kuZ,k

6,,

'FI,= (p/p)(u,., + us.,) ,=

-(P0,7),

(8)

t.,s = -(p/p~),k,

wh

+(u1)3)

P,, = -(PP)(u ),k

W ,

(12)

with

+ u

5

)],

F,, = -C[1u--2I

= -[(C

(2/3)6,,k]/T

(13)

+ 8)/Il] [P, - (2/3)6,sP k ]

-[(3002 - 2)/55] k (-,, + U,,)

In the above equations, P,, represents the production term, 2 e,, denotes the dissipation rate, 4I,, is the redistribution term (i e , the pressure-strain term), and 'D,,, PD, and ID,, represent the terms of the turbulent, pressure and viscous diffusions, respectively. We need to model all the terms except for e,, aid 'D,,. The existing Reynolds-stress DU,./Dr, P,,, equation models need the modeling of e,, as given by Eqs. (1)-(3) In the present study, however, we determine c,, from the solutions of the following e,, transport equation, without modeling e,, algebraically: Dc,,/D7

-

-[(8C

2)/11] [Q., - (2/3)6,,Pk]

=[C{-

(14)

(2/3)b.,k}/T

+C2(P

-

Q,,) + y'k(U,,, +U,,)If,

(15) (15)

where Q,, = -(UU7 k,, + VuUk,,), aid Pk = P.,/2 is the production rate of turbulent kinetic energy 1. The nondirectional (scalar) time scale T in the above equacan be defined from Eq (11) as:

lp_ + 2, i>.) + 3 PJ + 4tP,, - ction

+"D l,, + P' D.+ If,,

2-

T = u /-2= k

(9)

(16)

As will be described later, some substantial .iodifications have to be made for the constants in the wall term 'FD, of

where

the LRR model so as to satisfy the physical requirements of P,= P,, 41P, p)

(U7Uk, U,k + 7,,,'Uk,, Uk)

-V (U

+

_'~ V (Ui!' ,), ,_

_V (ij~kiA

1

= -v (us,,,.,,,,

the wall limiting behavior of turbulence. Turbulent Diffusion

)mU'--, k,.

+ +ItUk,

,

U-,,.k)

was found to be unfavorable to apply a gradient-type diffusion model to the triple correlation u-.7u (Nagano &

+ u ,,u,,,,u5 ,.,)

'E,= 2v2 ua,,n,km

(10)

4",, = v (pm/P)(u,,,m + u,,,,)

= -V( -u"-M),h P'D.= -v[(p,,/p)(u,,d6sk D ; = V 2 (i'i, ),s

+ u,,.6,k)],k

Tagawa 1988, 1991). However, we adopt the LRR's closures for the H i equation in the interest of brevity. Hence, we use a gradient-type approximation that is implicitly taken to account for both pressure contributions and velocity-driven transport The main difference is that we reflect the nonisotropic nature'v,,of turbulent lent diffusivity and write transport process on the turbu-

29-3-2

t

D,, + D,, = [Csf,

(17)

Vk,,(uOj).,] .

As to other lower-order terms representing the production rate of e6,, we apply the followii~g ap')roximation First, p does not require modeling, since we may rewrite lIp,, as

with

w= v(,)

(,)

The turbulent diffusion is strongly damped by fluid viscosity near a wall as the scales become progressively smaller, and thus the damping function f, is introduced in Eq.(17) (Prud'homme & Elghobashi 1983). CLOSING THE Ezj EQUATION

Since, by continuity u,,, = 0 and, at the wall (x2 = y = 0), ui,1 and u3,3 both vanish, it follows that u2,2 = 0 at the wall. quantities Consequently, fromwall a Taylor-series expansion, as follows'turbunear thle are represented lence =

ce yj+ J1y2 + 'Yi 3 +

u3

--

a3Y +

2

2y2

S

=

=

a

+ ',1,i

4

+

02'i')Y'

(

+

+

,T3')y' + .

,, = C,(',,12)/T

2

+ 12---?y +....sour, = 2,'72y - ,

e 33 /1

=

a3+ '4ta3--3 y + (,t3-3 + 61-T,)y2 +.

e1,1v

=

2aT-y + (3-

e,

=

c,,/

2+

4a2

non-isotropic nature between U-7u7 and e,, near the wall (Man-

Kim & Mon 1988) Thus, iii place of Eq.(23), the rational modeling of 'D,, may be written as

)y' +

',,

= le,, +

'i,,, + u'9,.

(24)

= (a1 + a3) + 4(TIFj+ -3-3)Y +

1(),, = -C,

From the above equations, we obtain the following relationship of the dissipation tensor e,, in the liut as a wall is approached (Launder & Reynolds 1983): e

(23)

However, we have to take into account the difference in the

3

e 2/u

1 u2

0), this term

nomena of the turbulence energy components, so that-

y + (4Ir + 6U--TS)y +

2

,i 0)2ye

occurs, accompanied by the corresponding redistribution phe+

k = [(T&+ oT.)/2]y + cn/'v

Note that toa function in Eq.(20) for is the function innear-wall limiting fulfill thef conditions the model troduced behavior of turbulence mentioned previously. Redistribution of ei;

not appear in the transport equation of e. Thus, '(D, represents the redistributive process among the anisotropic dissipation c mponents e,, . The production and dissipation rate of the turbulent kinetic energy are very nearly in balance in near-equilibrium shear flows. Accordingly, it will be legitimate to conjecture that the redistributi-e process of C,,

+TI 2-,T-i)y' +...

y3 + (j. + 27--)y

3 P1k2Y + l2

(22)

=C,.3V Vkm(U,,tU.rk + U

,p

Since the trace of ",F is zero (i e

+ ' -does

+ 2 #V' + 2 (+ +

u2= c 3 y + +2 U iZ2

3

+13y "P

(21)

However, with c,k -. /u'/T and e , U-k/T (see Eq.(2)), this term can be considered to be involve. in the modeled production term on the right side of Eq.(20) The term 2'p, , on the other hand, is negligible compared with the other terms, since mean velocity can be significant only in the direction parallel to the wall (Kebede, Launder & Younis 1985). The gradient production term 3 , can be modeled by using the

,

"-y_ + 27--2 y' + .

U-

C,kg,)

+

0

ui

,P j = -(Ciki:;,

gradient-type diffusion representation:

Anisotropy of Dissipation Process Near a Wall

u l

1

(18)

1 C2

2 4 J 12

6 33

12

_

1

2 F,7-2i' /"',I,,

C

(25) (26)

+C.,{C(P,, - Q.,) +-y'k ('7,,, + U,,,)}/27]f, Diffusion of

(27)

E,

Coiresponding to Eq.(17), the diffusion of -,, may be apby the following equation: D +r- 'D,= (C

Ord.r-of-magnitaidp analyik indiratpq that) in the high

"P,, - 'E,, = [C,,(P,,/2) - C 2 f,,,]/T

(1/3)6,, c]T

eq)"(19 = [C,piCl f,{ ,, - (1/3)6,, c}/T

Production and Destruction of Eproximated bulence Reynolds number regime, the turbulent production rate "eP,, and the dissipation rate 'E,, dominate the balance equation. These two terms collectively represent the net effect of the production of e,, due to vortex stretching of turbulent filaments and its destruction by viscous action. Thus, we model collectively these terms as

-

2'),, = C,:("F,j/2)/T

(19)

The above equation demonstrates the strongly anisotropic nature of the dissipation process in the near-wall region, and thus the models for ,, given by Eqs (1)--(3) are all inappropriate. Recently, to overcome this drawback, some attempts have been made by Launder & Tselepidakis (1990), Lai & So (1990) and Hallbick, Groth & Johansson (1990) lowever, all of these models are still algebraic.

iCif,[e,,

f

,,

(28)

From these considerations, we finally obtain the governing equation for the transport of dissipation components e,, as follows: De,,/Dr = [C.(P,1/2) - C.2 fe,,] 1T 77 4.3 / ( ,Jm77tk +7*

(20)

29-3-3

+9(D,j +"[(C-;d 'u,

+"5b, ) C,,']k

(29)

.

MODEL CONSTANTS AND FUNCTIONS In accordance with tie reassessmunt made by Morris (1984), the model constants C, and C2 in 4,, are set to Cl = 1.7 and C 2 = 0 5. On the other hand, the wall limiting behavior

k Ilishida 1987. Nagano & Kim 1988), the coordinate for regiois of very large gradients should be expanded near the wall. Thus, the following nonuniform grid (Bradshaw, Cebeci & Whitelaw 1981) across the layer is employedy, = AYl(K

- 1)/(K - 1)

(35)

yieldstb,,+

I)',

=-(1/p)(u,p, + up,,) - 0-0)

(30) (30)

The above relation requires that 4), -- 0 for y 0, since Eq.(17) gives that the turbulence diffusion and pressure-driven diffusion both vaish at the wall In order to meet the requirement that ),, - 0 for y - 0, the following ielation should be satisfied because of the vainshment of P, Q') and k in the limit as the wall is approached (P,

P +''ttJ

[1I - (CIVfJ 14, 0 (51) = 1. rhus, we finally ob.. (C'I/Cl)f have we (31), Eq From Froni Eq (317, ihae tCilCe modeled . 1.fuctionwe isi ob-n =

ta b C c= = 1.7, sincte the modeled function fo is chosen to become unity at the wall amid zero far away from the wall The above matter is extremely important. The LRR model has not taken into account this requirement, and hence the physically inappropriate behavior is brought about in the resuits of near-wall region (Mainsour, Kim & Moin 1988). We adopt the simplest form for the model function f, , which meets the aboxe-mentioned condition, as +

= exp(-0.0065y )

(32)

In this case, the values of C, = 0 055 and Y 0 are determinied as being optimal The constant in the diffusion term, Eq (17), is assigned to a sidtndard value of Cs = 0 21, and the following simplest form is adopted as tile model function f,' = - exp(-y+/5) (33) (29) is reduced Oii the other hand, the contraction of 1,(1 to the standard c equation (e.g , bee Launder & Tsekpidakis 1990) when the local isotropy is absumed fur tile s ,all scale

issipati.e mutions Therefore, most of model constants i the c, equation (29) are put to agree with those ii the conventional s equation' C,, = 1 44, C,2 = 1 9 and C, = 0 17 For the sake of smnplicity, however, we put C,3 = 0. 'rie model constants for '4,, are bet as C0,1 = 3 3 aid C,, = 1 44 In addition, in view of the noar-wall limiting behavioi of turbulence (Nagano & Tagawa 1990b), we set a function f, in Eq.(20) as 1

f, = 1 - exp[-(Th../18)' ']

(34)

where Ayl, the length of the first stop, and K, the ratio of two successive steps, are chosen as 10' and 1.03, respectively. To obtain grid-independent solutions, 201 cross-stream grid points were used. The first grid point was normally located well into the viscous sublayer. The maximum streamwise stepsize was restricted to a sublayer thickness, i.e , Ax < v/u, The initial profiles of the Reynolds-stress and dissipationrate components, U and c,,. were provided using the algebraic stress model (ASM) together with the low-Reynoldsnumber k - e model (Nagaio & Tagawa 1990b) The boundary conditions are

U, UU,= 0, el = u(tV/::/y)', e3 = V(Or/u/8y) and e, == ei = e,, = 0 2 = 0aty = 0(wall); Ui= Uanrd 1=U

wl)

0a

c2=c2

,,=c

n

at the free stream. Note that the above boundary conditions for elIand c33 satisfy exactly the wall limiting behavior up to the first order of y The predictions of the velocity profiles of a flat-plate boundary layer are shown in Fig. 1. File results are compared with the DNS data (Spalart 1988, Re = 1,t10) and experiment (Gibson, Veiriopoulos & Nagano 1982, R = 2750). Although slight dependence oii the Reynolds number is seen in the predicted wall-region velocity profiles, particularly at a low Reynolds number case, agreement with tile standard log-law profile, 7+ = 2.44 In y+ + 5.0, is generally good. To check the performance of the present model in more detail, we have compared the predictions of the Reynolds shear stress, -7U2-/u2, and turbulent kinetic energy, k/u2,with the DNS data (Spalart 1988) and the experiments (Verriopoulos 1983, Re = 2750, (lebanoff 1955, Re 7500) ii Figs. 2 and 3. As shown in the figures, a response of these quantities to the change of the Reynolds number is predicted correctl). To perform the critical evaluation of the present model, we have compared the predicted profiles of the normal stress components with the DNS database of Spalart (1988) at the same momentum thickness Reynolds number of R = 1.110 Fiu res 4 and 5 show the overall and near-wall profiles of /! l/u,, 4_/ u, and V/"/u,, respectively As seen in Fig. 4, general behavior of tile DNS data is well predicted by tile present model over the entire region of the boundary layer As the wall is approached '(y+ < 40), the strong anisotropic behavior of the normal stress components is reproduced by

30I Prewuit triodei

1110

MODEL TESTING AND CONCLUDING REMARKS

20

Calculations with the proposed model are performed for flat-plate turbulent boundary layers since tile dissi pation -rate components of the Reynol., stresses are quantitatively available from the recent direct numerical simulation (DNS) database of Spalart (1988). To investigate the Reynolds number dependency of the present model predictions, we have varied the momentum thickness Reynolds number, Re, as Re = 1410,2750 and 7500. The numerical technique used is a finite-volume method (Patankar 1980). As described in the previous studies (Nagano

V" Y./

tZ

29-3-4

DNS Experment 0 Qbson eta] (R=2750)

0

. 1

10

Fig. 1

101

y+

10"

Mean velocity profiles

101

the present model (Fig. 5). However, the difference in the profile between the DNS data and the present predictions becomes large in the near-wall region (y+ < 20). This shortcoming is also seen, for example, in the results of the recent second-order closure model of Launder & Tselepidakis (1990). Lai & So (1990) have pointed out that to reproduce t!.e cotrect near-wall anisotropic behavior of the normal stresses, the velocity pressure-gradient term in the ?-'),-equation should be modeled without splitting it into P, and PD, as done in the LRR model. Using the DNS database of a channel flow

(Kim, Moin & Moser 1987), Mansour, Kim & Moin (1988) have also shown that, from the standpoint of modellers, the conventional split of the velocity pressure-gradient term iVnot a good way near the wall and that a judicious choice may be necessary. At the present stage, the model of -i-equation is based on the LRR model because of its simplicity and general acceptance. It would be, however, preferable to revise the model of the velocity pressure-gradient term to reduce the predicted level of u2 profile near the wall (Lai & So 1990).

Present model

(U 0 1-7 .K ebo

,5-0

5t

7u

(R3 1410)

0 Experieneite Klebanoff (Re

0

0.2

0."t

7500)

.

/

._

0.6

0.8

DNS (Spalart)

(R# 1410) (R=1410)

Present model

1.0

to

1,.2

...

020

40

Y+

60

80

V/6 Fig. 5 Turbulence intensities near the wall

Reynolds shear stress, profiles

ig 2

Present model

/ %\ Vrriopouios (Re = 2750)

0 Prt

3

(1 7500 lMebanoff

(

020

+

1410)

33

model

.... ... .........

00.10

.110

-

DNS (Spalart)

(Re

1410)

2750

'

\

/to 'Pl \ Spalart

00,______0

),)\NS

(R= 1410)

0

10

102

+

&

t10

-0.05

Vig. 3 Turbulent kinetic energy profiles

' -, '0\_ 0

__,

10

20

30

40

50

Fig. the 6 Dissipation-rate tensor components near wall 0.4

-Z

DNS

Present model

DNS (Spalart)

(R,= 1410)

(R,

Spalart (Re = 1410) Mansour et al (R# =287)

14 10)

2

Present model

-

Re

0.2

0.2 N

-,,7500 ... I

01

0

00 0.2

,/ I 01 0.4

0.6 y/6 0.8

.. .

1.0

1.2

Fig. 4 Turbulence intensities

j

1410

--

Jz*.~

0

20 Fig. 7

29-3-5

-

40

y+

Dissipation rate of k near the wall

60

The predictions of the dissipation components e (= ue,/ u'.) are shown in Fig. 6. These are the solutions obtained for the first time from the transport equation of e,. It can be seen that the present solutions reproduce the strongly nonisotropic nature of the dissipation process near the wall. Overall agreement of e'jwith the DNS result (Spalart 1988) is very good. Significant improvements may be made in the predic4 tions of 4+ aid s through a revision of the model for (D, to reduce the level of u7 and u3as discussed above. General behavior of e+ 12 near the wall accords with the DNS result, though the reduction of the peak level in the vicinity of the wall may be required. The most important and interesting feature of the present results is the realization of the peak values of e+ and c' at the wall. As for this point, an attempt to improve the existing models has just started (e g., Mansour, Kim & Mon 1989; Launder & Tselepidals 1990, Shili & Mansour 1090, Lai & So 1990) to create a maximum of c at the wall as evidenced in the DNS database (Kim, Moin & Moser 1987, Spalart 1988) are shown The present model results of e+ (= e+1+c+ +c3+3) in Fig. 7compared ith theexistimg DNSdataof Kim, Moir & Moser (1987) (Re = 287) and of Spalart (1988) (Re = 1410). It is obvious that the profiles of c depend greatly upon the flow Reynolds number as pointed out by Mansour, Kim & Moin (1988) The present predictions of , for Re = 1410 well tr., -e the behavior of Spalart's result. Thus. the present approach to the modeling of the dissipation-rate tensor of Reynolds stress components has, we believe, good prospects in developing a full second-order closure.

KLEBANOFF, P. S. 1955 Characteristics of turbulence in a boundary layer with zero pressure gradient. NACA Report 1247 LAI, 'Y G & So, R M. C. 1990 On near-wall turbulent flow moielling J. Fluid Mech. 221, 641-673. LAUNDER, B. E., REECE, G. . & RODI, W. 1975 Progress i the development of a Reynolds-stress turbulence closure. J. Fluid Mech. 68, 537-566. LAUNDER, B. E. & REYNOLDS, W C 1983 Asymptotic nearwall stress dissipation rates in a turbulent flow. Phys. Fluids LAUNDER, B. E. & TSELEPIDAKIS, D. P. 1990 Contribution to the second-moment modeling of sublayer turbulent transport. In Near- Wall Turbulence (ed. S. J. Klne & N. 1I. Afgan), pp. 818-833 Hemisphere MANSOUR, N N., KIM, J. & MOIN, P. 1988 Reynolds-stress and dissipation-rate budgets in a turbulent channel flow. J. Fluid Mech. 194, 15-44. MANSOUR, N. N., KIM, J & MOIN, P 1989 Near-wall k - c turbulence modeling. AIAA J. 27, 1068-1073 MORRIS, P. J.1984 Modeling the pressure redistribution terms. Phys. Fluids 27, 1620-1623. NAGANO, Y & HISIHIDA, M 1987 Improved form of the k - e model for wall turbulent shear flows Trans ASME J. Fluids Engng 109, 156-160. NAGANO, Y. & Kmi, C 1988 A two-equation model for heat trani-ort in wall turbulent shear flows. Trans. ASME" J Heat r,?ansfer 'IQ, 583-589.

ACKNOWLEDGMENT This research was partially supported by a Grant-in-Aid for Scientific Research on Priority Areas from the Ministry of Education, Science and Culture of Japan (No 01613002)

1988 Statistical characteristics of wall turbulence with a passive scalar J Fluid Mech 196, 157-185

NAGANO, Y. & 'IAGAWA, M

NAGANO, Y. & TAGAWA, M. 1990a A structural turbulence model for t:iple products of velocity and scalar. J Fluid Mech. 215, 639-657. NAGANO, Y. & TAGAWA, M. 1990b An improved k - e model for boundary layer flows. Trans. ASME: J. Fluids Engng 112, 33-39.

REFERENCES BRA-

Iaw, P., CEBIcI, T. &IWInIrLAW, J. 1 1981 Engineernq Calculation Methods for Turbulent Flow, Academic

GIBSON, M. M., VERRIOPOULOS, C. A & NAGANO, Y 1982 Measurements in the heated turbulent boundary layer on a mildly curved convex surface. In Turbulent Shear Flows 3 (ed. L J S Bradbury et al), pp. 80-89 Springer. M , Guont, J & JOiiANSSON, A. V. 1990 An algebraic model for nonisotropic turbulent dissipation rate in Reynolds stress closures. Phys. Fluids A2, 1859-1866 E. 1976 Contribution towards IIANJALI, K & LAUNDER, B. a Reynolds-stress closure for low-Reynolds-number turbulence J. Fluid Mech 74, 593-610. IIALLB. CK,

, & YOUNIS, B A 1985 Largeainpitude periodic pipe flow A tiomd llimi,.ent Josurc study. In Proc 5th Symp on Turbulent Shear Flows, pp 16.23-16.29 Cornell University

KEREDE, W , LAUNDER, B

KIM, J.,MOIN, P. & MOSER, R 1987 Turbulence statistics in fully ieveloped channel flow at low Reynolds number J Fluid iM!ch. 177, 133-166.

NAGANO, Y. & TAGAWA, M 1991 Turbulence model for triple

velocity and scalar correlations. In Turbulent Shear Flows 7 (ed. F. Durst et al.), pp. 47-62 Springer. PA'rANKAR, S V. 1980 Numerical Heat Transfer ad Fluid Flow,

McGraw-Hill PRUD'HOMME, M. & ELGttOASII, S.1983 Prediction of wallbounded turbulent flows with an improved version of a Reynolds-stress model In Proc. 4th Symp. on Turbulent Shear Flows, pp. 1.7-1.12. Karlsruhe. Slim, T. fl. & MANSOUR, N. N. 1990 Modeling of near-wall turbulence. In Engineering Turbulence Modelling and Experiments (ed W. Rodi & E. N. Gani6), pp. 13-22. Elsevier. P. R 1988 Direct simulation of a turbulent boundary layer up to Re = 1410 J Fluid Mech. 187, 61-98.

SPALART,

VEmtRIOPOULOS, C. A. 1983 Effects of convex surface curvature on heat transfer in turbulent flow. Ph.D. Thesis, Imperial College.

29-3-6

TURIIULENT SIIEAR FLOWS Technical University of mticlich September 9-II, 1991

29-4

STRUCTURE AND MODELLING IN STRONGLY SHEARED TURBULENT COMPRESSIBLE FLOW Y. LEBRET, D. VANDROMME,

H. HAMINH

t

LMFN,, URA CNRS 230, INSA de Rouen, BP 8, 76131 Mt S'Aignan, France. t IMFT,. URA CNRS 005, Av du Prof. C. Soula, 31400 Toulouse, France.

ABSTRACT

Recent works ha\ e shown the existen e of large scale orstenrues o inrk gase t wae fmotions compressible f ws Coherent structures were found in comipressible ,hcai layers (PAPAMOSCHOU and ROSHKO [1986]). These structures are unsteady, but play a significant iole in the mixing layer spreading late. Introducing semi-detemninistic" turbulence modelling, a standard k - emodel was used to calculate unsteady flows The main result is that unsteady features can be resolved with this type of turbulence model A rather good description of the unsteadiness does not g e a good estimate of the cotnpressible mixing layer spreading rate. This emphasizes the need for a better statistical ticatmuent with a specific modciting of compresible effects

Indeed, coherent structures are recognized as the major agent of large-scale mixing in turbule;t shear flows. In incompressible flows, the vorticity field is a suitable idicator to visualize these structures, while compressible flows require different criteria. Interest for the mixing mechanisms is growing with the to-day needs of designing new propulsion devices for high speed transport aircrafts, since the efficiency -of combustion systems is depending mostly on mixing phe-

In this perspective, the present contribution is devoted to. Characterzatn of the coherent s uctures i compressible turbulent si-ea flows. * Analysis of the filtering features due to the numerical integration

List of symbols

* Evolution of the mixing layer growth with respect to a characteristic Mach number.

p density

U,V velocity comp.,nliti,

TURBULENCE MODELLING

a Speed of sounid

To iepresent rroperly the large-scale mixing, HUSSAIN and more iccently IIAMINH and VANDROMME [1983], [1980], have lt oposed In tile frainework of turbulence mod-

k turbuleit kinetw enmgy turbulent kim'tim e,iiQ di,,smpation rate

elliiig, a different fomi of variable decomposition, similar to

tie splittiig used i Laige Eddy Simulations Let f(.F, t) the instant amneous value of any, dependent vaiable

Pk production of tmuhct ktic energy rd tni m e Al Mach ii mibci a,

k and= Tibulent Piandtli/Schmundt numbers for Tn

C1, C2 , C,, Tm bul(enIc(, model com-aants p Lamninar

The first term ii the RHS is the steady state component. second term is the coherent o organized contribution

~The

vi,;cosit

which depends on space and time in a dcetermimistic manner, and the last pait is the iandon fluctuation.

pT turbulent viscosity Ilk = /1 -

,= p +

INTRODUCTION Thai.k to efficient numnical tools, it is now possible to siunulate ursteadv turbuient compressible flows which present a great interest for basic understanding and practical applications as well These iunerical predictions can be obtained either by direct numerical simulations, or by using adequate turbulence modelling. However, the numeuios recent sir-ulations do not highlight basic physical aspects of the phenoinmi. , since the analysis of nunierical data is strongly dependent of the interpretation. The compressible mixing layer is probably one of the most interesting cases, with well identified unsteady coherent structures resulting from Kelvin-Helmholtz instabilities.

Thus the difference with the clacsical L E.S approach comes fmoin the melative inpottance of the three components If the coherent pal tis small, then the decomposition leads to classical Reynolds aveiaging If the yandom pat is small, we have the basis for the conventional L.E.S. with subgrid scale modelling. The original contribution in the present work is to consider that random fluctuations are not constrained in scale and frequency. Theiefore it is not possible to account completely for them with a space filter (such as the mesh iesolution) like subgrid scale modelling does. To represent ail random motions (whatever their chliamtt:istic scales are), we choosed to use a statistical model i.e. k - c model. Following the original terminology of HAMINH et al.[1989], this approach is called the Semi-Deterministic Modelling. The variable decomposition can be writter, as:

29-4-1

f(i,0

< f(Z 0 > +f'(it)

(2)

RESULTS in which < f(iF, t) > represents thle ensemble avcrage of f. Under some circunistances, this quantity could be replaced by the phas~e average Since this splitting preserves a similar fotim to the clasbial tuibilence modelling, formally identical mod~els call be dciixcd to lepiesent the ianidominmotion effects. The key dilffeICeicc is that thle IeSUltling bsoltion Wouldl be stiongly dependent onl the flow haractcristics and the numllet ical mlet hod as wvell To illustiate the difference between this new approach anid thc classical tuibuleiice modelling, it suffices to coimpare the soltitionis wvhich would be given by atpmalbofic ,,plice ituching code (steady state boundary hayei approximation) and a time marching NaN icr-Stokes solver, when calculating at flow as simphle as a p~lanie mixing layer. . Both solutions will be true, but they will show tlifferents views of the reality. Voi tex shedding oi no voi tex shedding, the tlirutli is iii bo0th approachies. only the tepiesentation is dhiffiereint.

Unisteady featuics The first flow wich is examined is mnade of two air streams The two fiestream velocities aie U1 = 432 ni/s, U2 =137 oils, total temiperattuc Tr and total pressure P'j, are identical in thle tAxo sticamls (TT = 300*K, PT = 3.67 106 Pa). The Macli numbers of the two streams ate, tespectively M,

1 5 andI Al2

= 0J.4

VOlT'ICIFY FIELD)

..

35

17

6

0

ii1

___

aIS, t - 15 2

Cq

-d

Thus, in this seiini-detexiinistic appicoach context, a twoU1g> equat ion hbulcii e mdlfoi high Reynolds nuniber flows would be writ tell a,

oa

(3)

a

a t-

a

(41) Pk

>

u',>

500.

An experimental and numerical investigation of the turbulence field in the vicinity of a rotating cylinder in a quiescent fluid has been carried out. Radial distributions of mean velocity and non-vanishing Reynolds stress components have been measured using hot-wire technique, and corresponding components have been calculated with a low-Reynolds number second-moment closure model. From the measurements it can be concluded that the turbulence field is established in the present set up at a Reynolds number which is roughly twenty times larger than earlier measurements have shown. Comparisons between calculations and measurements yield some deviations. However, a qualitatively good agreement is obtained. In the paper, different methods for the determination of the friction velocity are also described and discussed

An objective of the present study is to provide further measurements of the individual turbulent stress components at as low Reynolds numbers as possible. Such data are essential for the verification of low-Reynolds number extensions of second-moment closure models, and, in particular, to demonstrate their ability to mimic curvatureinduced changes in the turbulence structure. Another objective of this investigation is therefore to compare numerical predictions based on a low-Reynolds-number second-moment model with the measured radial distribution of the individual stress components. EXPERIMENTAL SET UP

INTRODUCTION

A PVC tube, 1000mm long and 500mm ir diameter, was used as a cylinder. Fixed end plates were fitted in order to minimize the influence of the end effects. An electrical motor, which could be continuously adjusted in the range 200 through 3000 rpm, was used for the accomplishing of the rotation of the cylinder. The lowest number of revolutions corresponds approximately to a wall velocity of 0.6 m/s. A dial test indicator and a vernier s( " were used in the near wall region and further out, respectively, for the determination of the distance between hot-wire and the cylinder surface.

Turbulent flows in curved channels and over convex or concave surfaces are of obvious engineering interest Many flows in industrial equipment exhibit strong centrifugal-force fields, and it is well known that the centrifugal forces exerted on a turbulent flow field have important implications not only on the mean flow pattern but also on the turbulence structure, e.g Bradshaw (1973). To further ivestgate curvature effects on turbulent To frthe inestiatethe shear flows, a centrifugal-force field is created in a very simple and fundamental geometry The one-dimensional boundary layer flow around an infinitely long and constantly rotating cylinder in an otherwise quiescent fluid is studied, In this particular flow case the mean velocity vector is in the azimuthal direction, and the mean velocity and the turbulent stresses vary only in the radial direction, i.e. with the distance from the curved wall.

Velocity measuements were performed using a constant temperatura anemometer system, Dantech 5600, with standard single- and cross-wire probes, 55P01 and 55P61, respectively. A computerized version of Siddal & Davies' (1972) calibration law was employed for conversion of the anemometer voltage Into velocities, and a samoling frequency of 3 kHz was used throughout the measurements. Conventional methods used for the evaluation of the mean velocities and were Reynolds stress

Whil Thodosen Reier(194) masued hr' frictional drag on a rotating cylinder, the radial distributions of the mean velocity and the turbulent stresses have been reported by Kasagi & Hirata (1975) and Nakamura ot al. (1983, 1986), respectively. The latter data were obtained

components. All these acquisition methods have been tested, and compared to measurements of others, in a twodimensional turbulent flat plate boundary layer at a

for Reynolds number Re = 41000 and above, where Re = UR/v, R is the radius, v is the kinematic viscosity of the fluid, and U. = o)R is the wall velocity. Kasagi & Hirata

Reynolds number of 4.2-106, see Lbfdahl (1986) and

Preseril address VERITEC A/S P 0 Box 300. N-1322 Hovik, Norway

30-1n

i

m

m w

mu w w

w

m

where the dimensionless drag coefficient cd is defined as

An important question in this kind of measurements is whether a disturbance caused by the wake of the hot-wire probe will be transported by the flow along the cylinder circumference and interfere with the measurements. Therefore, a dummy probe was inserted at the same axial position as the measuring probe, but located at different angular positions No influence on the measurements were recorded when the dummy probe was positioned more than 45' upstream the "active" measuring probe. It might thus be concluded that the wake of the measuring probe does not affect the measurements

u.:

TW

c S

PG,2R2

=2-U

(2)

2

and u' The linear and logarithmic curves u' = y' 1/0.41 In(y') + 5.0 have been shown as a reference. As can be seen, the expected deviation from the log-law due to streamline curvature is present at all Reynolds numbers studied. The experimental resolution in the linear region is good for Reynolds numbers 20000 and 33000. For the highest Reynolds number Re=1 00000 the boundary layer becomes thinner and the number of measuring points in the neawall region is therefore limited by the size of the hot-wire probe. Nevertheless, the friction velocity has been estimated from the slope of the innermost mean velocity esiadfrmteloeothinrotmanvocy data points and compared with u. computed iteratively from Eqs (1) and (2), see Table 1.

MODELLING APPROACH Kasagi & Hirata (1975) and Williamson & Koukousakis (19671 used an algebraic eddy-viscosity model to represent the tuioulence field inthe vicinity of the rotating cylinder thetuioulncefied i th viiniy o th roatig cliner mean velocity distribution deviated resulting the However, significantly from the measured velocities When the turbulence structure is directly modified by a body-force, like the centrifugal force, a model for the individual Reynolds stress components is needed. In the present approach the recently proposed low-Reynoldsnumber second-moment transport model of Launder & Shima (1989) is adopted This model is essentially a lowReynolds-number extension of the Reynolds-stress closure due to Gibson & Launder (1978) The essence of the Launder & Shima model is that some of the model constants are made functions of four dimensionless parameters, namely a turbulent Reynolds number, the ratio of production to dissipation of turbulent kinetic energy, and two Reynolds stress invariants Kristoffersen et at (1990) demonstrated that the Launder & Shima model, with rotational stress generating terms included, accurately captured the most striking Coriolis-force effects induced by system rotation In the present study the model has been adapted to cylindrical rolar coordinates, while the empirical"values of the model coefficients are retained For the onedimensional flow under consideration in the present study, the transport equations for mean momentum, Reynolds stresses and dissipation rate reduce to a strongly coupled set of 6 ordinary differential equations, which is solved numerically with a significantly higher degree of accuracy than generally achieved for solutions of systems of PDE's

Re

0

**** Mt A

00 21000

V 75000 - 0o0X0 - 100000 20000

08

-

3300

1o000

06 X .

04

04

-

-.

02 -

""'" 0

oo

4

-

.

10

0 0 ------------------------ r--------T--------00 02 0.4 06 0.8 1.0 y/R 1 i

EXPERIMENTAL RESULTS

_0

._ ,\\

0

08

Measured mean (azimuthal) velocity profiles are shown in Fig. 1 for different Reynolds numbers Compared to the exact analytic laminar flow solution U = UJ(1t(y/R)),, the profiles clearly exhibit turbulent character. A significant

.

--A..--

....

,...

o

-

8

02

o

00

000

~y/R

)

*

8

0 0.4

Fig 2 shows the mean velocity profiles in inner variables for threc different Reynolds numbers. Here, the inner variables are defined as u'=Ud/u. and y'=yu./v, where U =U,-U is the velocity defect and u.u,-I-is the wall friction velocity To obtain this scaling velocity a correlation formula of Theodorsen & Regier (1944) has been used:

. --

i,

D

innermost part of the boundary layer, i.e. for y/R < 0.02, where y a r-R denotes the radial distance from the wall.

I

0

D

Reynolds number dependence can be observed only in the

.. -Figure .0.6+4.071Og,o(Rev ,,

"I, \ "

1

(1)

Mean velocity profiles for some different

Reynolds numbers.

30-1-2

0.02

0100

The measured turbulent intensities u' (azimuthal), V (radial) and W'(axial) are shown in Fig. 3. Considering the0.1 4tw

intensity in the main flow direction, u', clearly dominates in

the innermost part of the boundary layer. Further out,

0.10

y/R=0.06, the profilles for the axial and radial components mreinto each other at anintensity level below u'. The higher level of u'can, however, be explained by the "extra" stress generating terms which appear in the Reynolds stress budget due to the streamline curvature. 30~~~~

AA ii

A1

08

-8AI 3!006 -

-

O~

,

-------

~~O( O 0

002

.--

-

-

---

c

6c6

0

.8

0

00

0.0

P-700000

0

ii

A

C 0

-----A

A AA

A 000

0.0

-90.00

0.0

0 30

500

/

1

01

-2

A0

00

30

08

006

I004-

6

f)

o

0.02

0

20

: -----r------~-------------

"A00

0

0

00

00

00

.0

12

----

to

Figure 2 Mean velocity profiles in inner variables for three diffperent Reynolds numbers

0i

0 10

-

A

----

C

A

1

000

0.8

1:1.

v/U, 'U

0

0

06

6 0

Table 1 Estimated friction velocities u/U0 [Method \ Re l~erative solution of Eq. (1)

C,057

0.053

Modified constants in Eq. (1) 0.040 Extrapolated froii measured niean velocity

/-

133600100000 11

120000

0.047

004

1 002

0.0380031 0.046

0.066

0.6

1

0

i--

ooo

Fixtranolated from measured tcrbulent shear stress

---

Numerical prediction

0 019

0 034 II

0.027

10.045

0.042

_________________________________

6 0

3 0 060

-

0.6i'0.64 '066

0.68

0.10

y/R

Figure 3 Radial distribution of the turbulence intensities for Reynolds numbers 20000 (upper), 33000 (middle) and 100000fl

30-1-3

(lower).

Reynolds number (also at a constant y'=50). According to Fig. 8 the turbulence intensity In the main flow direction reaches an almost constant value above Re = 25000, thereby indicating a fully develop6"i turbulent regime in agreement with Fig. 7. The two oihe" normal components increase more gradually with Increaslng Reynolds number. It can be noted, however, that these components tend to zero at a Reynolds number of about 10000, which is much higher than the transition Reynolds number Re = 500 reported by Kasagi & Hirata (1975). Autocorrelation coefficients for the streamwise fluctuating component are shown inFig. 9. Here the time delay h is

InFig 4, the radial variation of the kinematic turbulent shear stress U-q is shown, from which it can be observed that the experimental data are consistent with the azimuthal momentum balance (i.e. rT = constant) away 2 can thus be from the wall A fairly constant level of -uvr used as an alternative way to estimate the wall shear stress T , and the friction velocity u.. The intermittent nature o* the flow at the lower Reynolds number complicates the situation and the wall friction can not be accurately obtained from the measured data The two different techniqueq for obtaining u. are compared with the correlation (1) due to Theodorsen & Regier in Table 1.

Nakamura et al

copnnarshwinFg9.Hethtmedlyhs to one revolution

(1983) replaced the dimensionless

normalised by the period T corresponding ofteclidrhhefilelrevlusorh of the The out r p ar aut c l in indicate the existence of large-eddy structures visualized

(3.09,4d79 1)e fit their by (3.09,4 79) to fi in Eq (1) (-0 6,4a07) 1983in constants cnstat this on based velocities Friction data experimental alternative set of constants are also included in the table. data increases It is readily observed that the scatter in the a general However, number Reynolds with increasing tendency of decreasing u. for increasing Reynolds number can be noted, fully in agreement of what can be expected The dimensionless ratio between the shear stress and

recently by Nakamura et al. (1986).

00030

(twice) the kinetic energy is shown in Fig 5 The observed level of the structural paarneter uv/q2 away from the cylinder surface is consistent wii the experimental findings of Nakamura et al (1986) rhe present measurements of the individual stress components in Figs. 3 5 cover, however, the innermost part of the turbulent boundary layer more completely than any of the previously published data,

o.oo2,)

.

.

-

-

00020o

,. i

'ooo

0-

0

0

In Fig 6, the intermittency factor y is plotted Here, y has been calculated from the flatness factor F(u) for the streamwise velocity component as y=3/F(u), according to Klebanoff (1955) It can be seen from these data that the intermittency factor decreases below one for a certain distance from the cylinder surface for both the studied R6ynolds numbers Especially for Re-22000 this means

,ooot

00

o

0

00 o o

0

N > 0 0001

o

0

that a highly intermittent flow is prevailing, and this can

-' o 2

o0o

0.06

064

oo8

0.

y/R

considerably influence the measured turbulence intensities. If the laminar part of the signal is excluded in the measurements, this will increase the turbulent intensity This effect is clearly pronounced in the outer part of the boundary layer, where the largest deviations between measurements and calculations occur according to Fig. 3 Of course the influence of the intermittency is strongest at the lower Reynolds number and weaker at higher The results for Reynolds number 100000 are in agreement with the data of Nakamura et al (1983), while the results for Reynolds number 22000 are significantly different Only the innermost part of the boundary layer (y'), ) R(o-= oo, 11, 5, 2, 1, 0 56, 0 25, a) Cyclonic case; b) Anticyclonic case. ') Numerical simulations are performed at R( = oo, 11, 5, 2, 1, 0.56, 0.25 for both cyclonic and anticyclonic cases. In Fig2 2, we show the time development of < w' >, variance of

y tanT

the spanwise velocity component (z-component), which in-

and a gaussian profile for the wake

dicates the degree of three-diinensionality. The brackets denotes the average on the whole computatienal domain. Fig-

i

TI(y)

=

tire 2 a) is for cyclonic cases and 2 b) for anticyclonic cases. Figure 2 a) shows that a cyclonic rotation is more and more stabilizing as the initial Rossby number decreases from in-

U,,, exp(-Ln2l,)

m

finity, up to 0.5. Afterwards the energy exhibits oscillations which are the signature of inertial waves The presence of the waves leads to a reduction of the dissipation rate. Figure 2 b) shows extremely different behaviour in the anticyclonic 2 case. The initial growth rate of < w' > is higher than in the ) - 5,the destanon-rotating case for R(I = 11,5,2. At R bilization reaches its maximum. The Rossby number based upon the instantaneous mean-velocity profile decreases with R(-) - 2U 120,16, time Thus, the rotation could have a destabilizing effect at and a stabilizing one when the r te mthe beginning of the evolutionrislwrhnaciial-Je inttaeuRosvum for thme mixing layer. For thme wake, the iiaximuin vorticity instantaneous Rossbv number is lower than a critical J' le. 2 this could explain the decay of < w' > for R(o'= 11,5,2 0 7V,/r,,,. Thus, associated with the gaussian profile is Time initial fluctuations consist in the superposition of tworandom perturbations. the first one is three-dimensional 2 f of kinetic energy 3D U (or U,2), the second one is two2 dinensional (z independent) of energy 020 U . In both cases, (3D = (2D = 10-'. The Rossby number is defined with the initial relative vorticity'

' 07 V R(o

0 -,,, r 1201rm

The numerical inethod used is of pseudo-spectral type For the mixing layer, a mapping is used in the y direction ill order to reject the boundaries at infinity. The results shown here are obtained with 48 x 48 x 24 nodes. The calculation involves 2 fundamental Kelvin-Helmoltz billows. The initial Reynolds number U,5,/v is 150. For the wake, 48 x 48 x 48 modes are used The Karman street is composed of two pairs /v = 180. of opposite sign vortices, Re = Ur

at later time. When the Rossby number goes from 2 to 1, a sharp transition takes place, since, from the beginning of the evolution, the rotation is stabilizing as far as the growth 2 of < uw' > is concerned. At a lower Rossby number, the evolution is very similar to the cyclonic case, with evidences of inertial wave propagation. Plane wake

STATISTICS

Cyclonic and anticyclonic eddies are now simultaneously present in the computational domain. Figure 3 is the ana2 logue of Figure 2 in the case of the wake. < w' > designates a spatial average of the spanwie velocity variance on the points at which the vorticity i3 cyclonic for Figure 3 a) and anticyclonic for Figure 3 b). The Rossby number considered

Mixing layer We fdimensionalization We first consider the mixing layer. The results of these simulatins are described in details in Lesieur et al., 1991.

here are R(" = 00,7,2.,2.5 0.7. On the anticyclonic side, similar effects to those described for the anticyclonic mixing layers art observed. However, maximum anticyclonic threeoccums at R (0 = 2.5. On the cyclonic side, the rotation, in an initial phase, is stabilizing for all considered values of the Rossby number. Afterwards, for

30-2-2

' = 7, 2., 2.5, the st.ing three-dnusionalization leads R to the generation of cyclonic vorticity oil the side of the wake, which was originally anticyclonic. Furthermore, the cyclonic eddies can be--ome more tridimensional because of their interaction with the highly three-dimensional antiLyclonic ones. As opposed to the mixiag layer's cdse, this im2 pkes a growth of < w' > in cyclonic regions stronger than without rotr tion. 1/2 < w

>

1/2 < u,

cyclonic

.

. -.

a)

anticyclonic

>

b) 2.

b-i.].

'

2.5

2.5 2

W

Figure 5 is the analogue of figure 4, but tile total kinetic energy is now integrated over k, and k, wavenumbers and we consider its time evolution for wavenumbers k = 0, 2, 6 and 8. The wavenumber k. = 2 is the fundamental mode coiresponding tu the 2 pairs of alternate-sign primary vortices of the Karman street. The energy coriesponding to this mode strongly grows in the non-rotating case. Its growth is slightly reduced in the presence of rotation. In the rotating case, the most striking feature is the rapid growth of the energy at k, = 0. The destabilization of the anticyclonic eddies will give rise to structures of small spanwise extent but very elongated in the streamwise direction. This is confirmed by linear stability analysis performed by Yanase et al. (1990).

4°'

2

//a)

R() = oo

E(k.)

b)

2.6

E(k.)

/.2

2

"

to-,0

1

Soo0'

25

,

" x..0.7

"07 00

4

Soo

250

00

time

to-,

4 6

'.

. -

t0o,.

•|

l0/

time

Figure 3: same as figure 2 for the plane wake (R( ' ) r0, 7, 2,2 5,0.7)• < tu/2 > designates a spatial average of the spanwise velocity variance on the points at which the vorticity is cyclonic for Figure 3 a) and

.

./

" 250

/ 00

00

t timel

In figure 4, we present the time evolution of tile total 2 kinetic energy (1/2 < u'2 + v12 + WO' >) for wavenunber k, = 0,4,8,12 and 16 integrated over all wavenumbers k. and k.. Figure 4 a) corresponds to the non-rotating case and figure 4 b) to the most destabilized case (RO(' = 2.5). The growth of the two-dimensional energy is sligitly reduced by the rotation. On the other hand, in the rotating case, the energy growth is more and more pronounced as k, gets larger. This indicates a strong three-dimensionalization of the wake (on the anticyclonic side) with generation of small

/

i0'_

00

anticyclonic for Figure 3 b).

/

.

500

250 timle

Figure 5: time evolution of the total kinetic energy at wavenumber k, = 0,2,6 and 8 integrated over all wavenumbers k, and k.. Figure 4 a) non-rotating case; figure 4 b) rotating case (R/') = 2.5).

FLOW STRUCTURES

Mixing layer

scales in the spanwise direction

We have investigated the three-dimensional structure of

the eddies at t = 25 4. We use low pressure iso-surfaces to a)

R', = o

.k,)

b) E(k.)

25

R,'

..-.

identify the Kelvin-Helmoltz vortices:

0

a) R(') = oo: one observes two quasi-two-dimensional bil-

4

lows, slightly distorted in the spanwise direction Thin Iongitudinal vortices are also present shown by longitudinal vor-

12

ticity isosurfaces (not plotted here). b) R (' ) = 11 (antcyclonic): figure 6a) shows the pressure field of the anticyclonic case. The eddies are greatly distorted by a strong three-dimensional instabil:,y due to the solid-body

•8,". • /' 12

"...... 00

• ... ' !structure

.'

220

20

00

20

20

time

tine

Figure 4: time evolution of the total kinetic energy (1/2 < v'+v+w 2 >) at wavemunber k. = 0,4, 8, 12 and 16 integrated over all wavenumbers k, and ky. Figure 4 a) non-rotating case; figure 4 b) rotating case (R ' ) = 2.5).

. .... . ..

....

..

.

.

. ... d1

c) R( ' ) = 5 (anticyclonic): there is no large cylindrical eddy anymore, but the layer is highly three-dimensional, as shown on figure 6b). The pressure contours show now very anisotropic structures of small spanwise extent and very elongated in the streamwise direction. One can check that they correspond to concentration of vortex lines of near zero absolute vorticity which are stretched by the ambiant shear. At low Rossby number, in both cyclonic and anticyclonic cases, a very strong two-dimensionalization is observed.

...

. ..

. .

..

..

Figure 6: b) mixing layer; constant pressure surfaces ') (anticyclonic case) for R, = 2.5 at t = 25.4

Figure 6: a) mixing layer; constant pressure surfaces ) for (anticyclonic case) for R,( = 11 at t = 25 4

Figure 7. a) plane wake: vorticity contours for the cyclonic spanwise vorticity (light gray) and for the

Figure 7: b) plane wake: constant pressure surfaces for R( ' ) = 2.5 at t = 30

anticyclonic spanwise vorticity (white) for R'. ) = 2.5 at t = 60

Plane wake

DISCUSSION

Figuire 7a) shows vor'icity contours for the cyclonic spanwise voz.licity (light gray) and for the anticyclonic spanwise vorticity (white), for the critical Rossby number (R( ' ) = 2.5, t = 60). Both cyclonic eddies exhibit spanwise oscillations due to tl.e strong interaction with the three-dimensional anticyclonic eddies. Figure 7b) is low-pressure isosurfaces at t = 30. The anticyclonic edc" -s are in the foreground and ................... b... a, A for thcmixing As .d. laycr,

The numerical simulations qualitatively confirm the predictions of our theoretical model: the cyclonic eddies are stabilized, while the anticyclonic eddies are three-dimensionally

very elongated streamwise structures are visible on the anticyclonic side. At low Rossby number, both sides are twodimensioralized.

number- R. ; 1) will be surroundered by less-destabilized and also by stabilized lines: the proportion of lines with maximum destabilization will change with varying Rossby number. This explains why there is only an approximate correspondence between the local Rossby number R. of the

destabilized upon a critical Rossby number. Since we considered here inflexional unidirectional mean velocity profiles, the vorticity associated to these profiles varies with y (with extrema at the inflexion point(s)). Consequently, strongly three-dimensionai absolute vortex lines ( at a local Rossby

30-2-4

thcory and the initial Rossby number R (') based on the mean profile: in the anticyclonic case, Ro = 1 corresponds approxinately to R ( ) 5 for the mixing layer, and to R ( 2.5 for th, wake. and the crossover value for the restabilization ") I The striking feature is that the cx(R 0 =:0.5)to R plosive anticyclonic three-dinensionalization leads to very

Bidokhti A.A. and D.J. Tritton, 1990 The str" 'ure of , ub. a turbulent free shear layer in a rotat'ng milted to the 1. Fluid Mech..

anisotropic absolute .,orticity structures, the vorticity originally oriented in the span rotates by an angle of 450 and is ultimately ditected in the streamwise direction. The results oblained here present soesim; 1,1ri ties with what isobserved ineoundary h yer oresent s cure witehen ha obserquium is in boundary layer over a curved pla e. When the plate is concave, the angular rotation induced by the plate curvature and the vorticity associated with the boundary-layer velocity piofile are of opposite directioi' there is formation of Gbitler streniwise vortices (which can be explained in teins of centiifugal instability) Furthermoic, for convex plate, the boundary layer is st-ibilized Recent calculations (Bartello. private coinunication) of rotating homogeneous isotropic turbulence show that the rotation acts uo)n ed(lies whose vorticity is parallel to the rotation axis and leads to the foimiation of quasi- two-dinienbiinal cyclonic edldies. from the original three-dimensional chaotic flow emerge very

Chabert d'Hihres G. , P.A. Davies and H. Didelle, 1988: Laboratory studies of pseudo-periodic fcrcing due t0 vortex shedding from an isolated sold obstacle in an homogeneous rotating fluid. 20th 1at Liege Colloon Ocean Hydrodynamics, 2-6 May, Elsevier Publisher. Lcsieur, M., 1990, Turbulence in Fluids, ,luwer Publishers, Dordrecht, 1987, revised edition 1990) Lesiur, M., S. Yanase and 0. Mtais, 1991' Stabilizug and destabilizing effects of a solid-body rotation upon quais-two-diniensional shear layers, Phys. Fluids A, 3, pp 403-407. Rothe, P.H. and J.P. Johnston, 1979: Fiee shear layer behavior in rotating systems. J. Fluids Eng., 101, pp. 171-120. Witt, H.T. and Joubert, P.N., 1985: Effect of rotation on a turbulent wake. Proc. 5th Symposium on Tur-

well-organized cyclonic coherent structures.

References

bulent Shear Flows, Cornell, pp 21.25-21.30.

CONCLUSION We have shown ,,using Kelvin's theorem in a Yotating frame, that the straining of absolute vorticity is reduced for a cyclotic rotation leading to an inlbition of shear-flows three-dimensionaiization. I the anticyclonic case, rotation is three-diniensionn;ly destabilizing for a local Rossby number larger than 0.5, and stabilizing at smaller Rossby numbeis At a.Rossby nuinober of the order of one (anticyclonic case) , the spanwise component of the absolute voiticity is cancelled by the solid-body vorticity, ai,.l the coresponding absolute vortex lies aie highly three-dimensional. Mixing layer and plane wake three-dimensional numerical siniulations suport the theory, and aree qualitatively with experimental results For the rotation rate coriesponding to an explosive anticyclonic three-dimensionalization. lowpressure and high-vorticity stiuctures as well as the statistics show that the absolute "ortex lines of near-zero vort:city are rapidly strained by the basic flow This leads to the forination of strong hairpin vortices of small spanwise extent and elongated in the streamwisc direction. The authors are grateful to J.J. Riley for many enhghteiing discussions.

30-2-5

Yanse, S., J.J. Riley, M. Lesieur and 0. Mdtais, 1990: The effects of rotation on transition in a two-dimen sional shear layer, Bulletin of the anterican physical society, 35, No 10, p. 2277

lIGlI II SN MIPOSITI ON

30-3

TIRBULENI SIHlEAR FLOWS I echnical UnIversily of Mkiiuch Septeniber 9-11, Ig9 EFFECTS OF CURVATURE ON THE SPECTRA OF SHEARED TURBULENCE A.G.L. Holloway* and S. TavoularisDepartment of Mechanical Engineering, University of New Brunswick Fredericton, New Brunswick, CANADA, E3B 5A3 Department of Mechanical Engineering, University of Ottawa Ottawa, Ontario, CANADA, KIN 6N5

ABSTRACT One-dimensional spectra from seven uniformly-sheared nearly-homogeneous flows

with various degrees of mean flow curvature are presented and analyzed. The data suggest that the flow curvature affects the large-scale motions most strongly, altering the partition of the kinetic energy among its three components and the coherence of the streamwise and transverse fluctuations. For relatively mild curvature, the coherence of large-scale motions diminished in cases where the mean velocity increased away from the center of curvature and increased in cases where the mean velocity decreased away from the center of curvature. For "strong" curvature, the sign of the coherence, was reversed and momentum was transported up the gradient of mean velocity and away from the center of curvature, NOMENCLATURE C (k 1) dU/dn

the coherence function gradient of the mean velocity

F (k,) Fvv(k,) F_ (k,) F, (k,)

streamwise components of the onedimensional spectral densities of the velocity flucLations

ki

streamwise component of the wavenumber vector Luu the integral length scale of the streamwise fluctuations n transverse coordinate R, radius of curvature of the tunnel centerline s streamwise coordinate S =(U,/R,)/(dU/dn) - curvature parameter u streamwise velocity fluctuation 6 Fourier transform of u Uc streamwise component of the mean velocity on the tunnel centerline v transverse velocity fluctuation w spanwise velocity fluctuation z spanwise coordinate u v

1.

impact upon sizes. For measurements (1978) show

turbulent eddies of different example, the boundary layer of Ramaprian and Shivaprasad that both convex and concave

surface curvature affect the integral length scales more than the Taylor microscales. The interaction between shear and curvature in boundary layers is complicated by the presence of wall effects, the entrainment of irrotational flow and inhomogeneity. For the present investigation an effort was made to design a flow to evince the effects of flow curvature on the different scales of a sheared turbulence in relative isolation from wall and entrainment effects. An ideal flow for studying this interaction would be one where turbulence, curvature and shear are homogeneous. Such a flow can be conceived by extending the notion of rectilinear, homogeneous, shear flow to a flow with mean streamlines which are not parallel but form concentric rings. The shear should be assumed uniform in the radial direction but a fluid particle would on the average move along a curved path. It is easy to see that inhomogeneity of the turbulence would inevitably develop under such conditions, as a result of the transverse nonuniformity of curvature and strain. Nevertheless, it seems worthwhile to explore the plausibility of an approximately homogeneous curved shear flow by introducing additional constraints. Consider a flow field bounded on average by two coaxial circular cylinders with a difference in radius that iE large in comparison with the scales of the t rbulence but small compared to the mean radius. The mean streamline curvature can be considered as nearly constant within the volume and the turbulence structure in the core of volume can be assumed as independent of wall effects, at least over a sufficiently small time interval. If the turbulence within this volume were homogeneous initially, it would likely remain homogeneous for some time. An approximation to that flow was materialized by extending previous studies on nearly homogeneous shear flows (Tavoularis and Karnik, 1989). Uniformly sheared turbulent flows were allowed to reach a ciliasi-self-preserving structure in a straight rectangular duct and then were

the rate of viscous dissipation Kolmogorov velocity kinematic viscosity Kolmogorov microscale INTRODUCTION

passed tangentially into a curved duct, also

It has been well documented that curvature has a strong effect on the structure of sheared turbulence (for a recent review of the literature see Holloway and Tavoularis, 1991). It has also been documented that curvature has a different

of rectangular cross-section. The near homogeneity of the turbulence and the near uniformity of the shear were preserved, while the structure of the turbulence was modified by the flow curvature.

30-3-1

Details of the experimental apparatus

and measurements of the development of the turbulent stresses and length scales for the present flow have been reported by Holloway and Tavoularis (1991). A primary conclusion of that work was that the effects of curvature appeared to scale with the parameter S=(Uc/Rc)/(dU/dn)*, where U c was the centerline velocity, n was in the direction normal to the mean streamlines and Rc was the radius of curvature at the Flows with centerline of the duct. different values of S were generated by adjusting both the mean shear and R . Flows with different values of shear, eac h having a self-preserving structure, were produced and various using a shear generator combinations of screens and grids. Shears of opposite direction were produced by inverting tha shear generator. In addition, two curved ducts were built, corresponding The to R. = 2 m and 5 m, respectively. combination of curved sections and shears produced 16 distinct flows with S ranging from -.27 to .65. Measurements of the turbulent stresses indicate that the growth of the turbulent kinetic energy was enhanced for negative values of S. and suppressed for positive values of S. Whether the kinetic energy of turbulence grew or decayed, it seemed to do so exponentially and to approach a selfpreserving structure, with the components of the Reynolds stress tensor approaching constant ratios. Seven quasi-self-preserving flows of the type described above, where S=-.27,-.1, -.07,0,.07,.l and .65 have been chosen as representative cases. Some of their main characteristics are shown in Table 1. These flows have been ranked by their value of S, under the assumption that this is the determining factor, but the wide range of shears, RL and R, for these flows should not be overlooked. For example, the flow with S=-.1 was being sheared at nearly twice the rate of the flow with S=-.07 and hAO nearly three times the Reynolds number. The energy and length scales of the first four flows grew, while those of the last two decayed. The flow with S=.65 was of particular interest, because it exhibited a turbulent momentum flux, in the same direction as the mean shear, opposite to the implications of gradient transport. However, conclusions derived from this flow must be treated with some caution, because its structure near the end of the tunnel might not have reached an asymmptotic state. 2.

assumed to be small compared to Rc . The z coordinate is normal to both s and n and has a direction following the right-hand rule. The local, instantaneous, velocity vector can be decomposed into means, U,V,W parallel, u,v,w, fluctuations and respectively, to the s,n,z, directions. In flows with uniform mean shear in the n direction alone, the mean velocity can be represented as (U,V,W) =(Uc+(dU/dn)n,0 0) (2.1) The one-dimensional spectral densities, F (k,), F,(kl), F (k,) and Fu(ki) of the flctuating kinetrc energy components and the shear stress are defined as usual (PApoulis, 1984), for example (asterisks indicate complex conjugates) F, (kl) = E{(

(2.2)

}

where k, is the component of the wave vector in the streamwise direction, E( ) denotes the mean value and i(kl) is the Fourier streamwise velocity transform of the fluctuation. F (k,), F,(k,) and Fw(k.) are real and even unctions of kl, while F (kl) is complex with an even real component and an odd imaginary component.' Because the imaginary part of Fu(ki) is odd, only the real part of Fu(kl) contributes to the shear stress. The streamwise integral length scale, L , is defined from the limiting values of at zero spectra te corresponding wavenumber, as, =

3.

9

F F(k klim 1 -0

1

)

(2.3)

AND DATA ACQUISITION INSTRUMENTATI1'

The velocity was measured using a custom-made, cross-wire array (TSI 1248bJT15) and conventional hot wire anemometry procedures. The sensing elements were made of tungsten and had a diameter of 5 Am and a length of 1.2 mm. They were separated by 0.5 mm and had nominal inclinations of ± 45" with respect to the axis of the probe body.

MATHEMATICAL DESCRIPTION OF THE FLOW

The present flow can best be d'scribed in a curvilinear orthogonal system of coordinates, similar to those commonly used in studies of curved boundary layers. These will be denoted by s, n and z, as shown in Figure 1. The s coordinate is the length of a circular arc, with radius R , coincident with the flow volume centercline. The n coordinate is normal to s, has its origin on the volume centerline and is positive when directed away from the centre of curvature. The radial extent of the flow volume is This parameter arises naturally from the Reynold stress equations and has been identified by several authors,

In the present flows the Kolmogorov microscale ranged from .14 to .3 mm, which was at most a quarter of the length of the hot wire. The analysis of Wyngaard (1968) suggests that measurements of fluid motions with sizes less than the wire length would require corrections for the size of the hot wire array. The anemometer signals were low-pass filtered to 10 kHz, converted to velocities using a modified form of King's law, digitized at two different rates, 2 kHz and 25kHz, and divided into statistically independent records, each of 1024 points. The Fourier transforms of the fluctuating velocity components were estimated using the discrete time series and Taylor's frozen

30-3-2

flow approximation.

only modestly more energetic than predicted

The corresponding spectra computed from the two sets of time series collapsed in the overlapping ranges. However, only the high frequency part of the spectra computed from the 25 kHz sample and the low frequency part of the spectra computed from the 2 kHz sample were retained. The Nyquist criterion was not strictly observed for the 2 kHz sample but the portion of the spectrum which was significantly distorted was removed, The energy estimates for the longest wavelengths were subject to higher uncertainty, due to the finite record length and deviations from flow stationarity. Calculation of L from equation (2.3), was done approximatefy by extrapolation of the spectra at the lowest measured wavelengths to zero.

by the Karman formula. One should note, however, that these spectra were forced to agree at the lowest wavenumbers and to cover the same area by the way in which they were scaled.

4.

In each flow F (k,) and F (k ) were different for kjLtu < 5 and generally less energetic than in isotropic turbulence of equal energy. These spectra approached one another at kv - 10 u and then they diverged again in tho viscous range. F,(k,) for the flow where S=-.27 appeared to have the greatest proportion of the energy and the flow with S=.65 the least, while F,(k,) showed no similar trend. It would appear that shear and curvature make the distribution its three components at oftheenergy lower among wavenumbers more anisotropic.

RESULTS

F, (k)

Typical measurements of the spectral density F (kl), normalized with the Kolmogorov scales, have been plotted in Figure 2. The corresponding flows had a range of turbulent Reynolds number, R = u'A/v, from 115 to 370. The theoretical spectral form of Pao (see Hinze, 1975) has been plotted for comparison. For > 0.005, F.(k,) wavenumbers where k1 appeared to be unaffected by the flow curvature or shear as the data from each flow essentially collapsed into a universal shape, close to Pao's estimate. At the lower wavenumbers, F (k,) varied widely from flow to flow, due to the different velocity and length scales. These differences may not be entirely attributed to flow curvature, but also too differences in the history of each flow. For example, the flows with S = ±.l initially had similar scales but, after sustained flow curvature of opposite sense, the kinetic energies of these two flows deviated exponentially and differed by a factor of two at the streamwise position where these spectra were measured. The length scales were affected in a similar way. To assist in the evaluation of the effects of mean curvature and shear on the spectra, the present results will be compared to the spectra of isotropic turbulence. Among the several approximate forms which have been proposed for the three- dimensional spectrum of isotropic turbulence (see for example Hinze, 1975), one leading to a simple expression for F,,(k) and having a good agreement with data from grid- generated turbulence is Karman's interpolation formula 2F(k 1 ) = .63[l+(l.33kL) I lu7L,

22/6

/

(4.1)

The spectral densities F (kl), F (k,) and F,(k1 ) have been plotted in Figures 3, 4 and 5. They have been normalized with the streamwise comnonpnt of the kinetic energy, u2 , and the streamwise integral scale, L . The Ka.man interpolation formula has also been plotted for comparison. The fine-scale structures of each flow were different, so the spectra differed significantly for k.L > 10. For k1L < 10, the curves for diflerent flows were close to each other and

and

F. (kl)

showed

little

evidence of inertial subranges (certainly much less than F (k,) did), but this could be partly due to he lower energy levels of these components. The one-dimensional cross spectral density, F.(k,), has been normalized with F.(k I) and Fv,(kl) to form the coherence function, CIk1\=F.(k1\/&.(k1\F \

/k ))

(4.2)

In isotropic turbulence, C,(k,) is zero at all wavenumbers while in shear flow, -I. < C v(k,) < 1. Cu(kl) should have a sign opposite to that of the shear, because momentum is on average transferred down the gradient of mean velocity. For the present experiments, flows with positive and negative values of S were generated by reversing the direction of the shear. This would have the result of reversing the sign of C,(kl), a result which does not reflect a true change in flow structure but is an artifact of the experimental method. To accurately portray the relationship between C.(k,) and the shear we have plotted Cu(k,) sgn (S) where sgn (S) is -1 if S 0 and +1 if S < 0. The real part of the coherence is shown in Figure 6 as a function of the normalized wavenumber, k1LY. It was relatively weak for motions with k1 L > 10, which is consistant with the observel local isotropy of F (ki), F (kl) and Fw(k,). The coherence o? the motions with low wavenumbers was a strong function of S, ranging from .9 to -.4. The flow with the strongest, negative S (S=-.27) had the greatest coherence. For the straight flow, where S=0, the coherence exceeded 0.7 at its maximum, which is considerably greater than the values of the correlation coefficient u/u'v = 0.45 in this flow. Cur(k,) became less positive with increasing S and the strongest affects appeared to be at a wavelength equal to ten times the integral length scale. In the flow with the strongest positive S (S=.65) the coherence reversed its sign so that it was in the same direction as the shear. It would appear then that stabilizing curvature, for which S is positive, does not necessarily reduce the coherence of the streamwise and transverse motions but rather makes them more likely to transport momentum away from the center of curvature.

30-3-3

The imaginary part of the coherence, shown in Figure 7, was relatively weak and This part does not highly scattered. directly influence the turbulent shear stress but does make a contribution to the it in particular covariance, spatial contributes to its asymmetry. This part was. largest in the flows with positive S in the' wavenumber range where the real part of the coherence changed sign. 5.

DISCUSSION

The experimental data seem to indicate that, although the shear and the curvature act uniformly on all turbulent motions, it is the larger motions which are the most strongly affected. In the higher wavenumber 10) the distribution of energy range (kiL differed little from that in isotropic turbulence and the coherence was small. At the low wave numbers, the spanwise and transverse components of the energy spectrum were, in most cases, an order of magnitude lower than the streamwise component and the coherence of the streamwise and transverse motions varied widely from values as high as .9 to as low as -.4. In an attempt to understand the observed effects of curvature we will first separate the spectra into a low wavenumber range (kLu < 1) and a high wavenumber range (k1L. > 1). Hinze (1975) suggested that the mean vorticity, dU/dn, of the flow could only be expected to have a significant effect on the distribution of energy in wave number space if it is of comparable magnitude to the root mean squared fluctuations of the vorticity, j (e/v). Weak mean shear would only serve as a supply of energy and one might expect a nearly isotropic spectrum, similar to the Karman interpolation formula and including a -5/3 slope in the inertial range, if the Reynolds number were large enough. This prediction would presumably be most relevant for relatively small-scale motions. However, at the lower wavenumbers, the spectral density of the dissipation is relatively small and the turbulence distortion could be weak compared to that due to the mean shear. In this wavenumber range the distortion would be strongly influenced by the mean shear and the turbulence could perhaps be considered as passive. Similarly, the importance of mean flow curvature would depend on the instantaneous curvature of the turbulent motions which presumably would increase with The present increasing wavenumber. experiments were designed such that %u .05 R,. However, rough estimates of the spectra for the present flows are presented in Figures 3 to 7 for wavelengths up to .5 Rc . Attempts have been made to explain the observed development of turbulence in terms In rapid flow distortion. of mean distortion theory, the development of the turbulence is assumed to be completely determined by the mean distortion but, in other approaches, some model of the effects of turbulence on its own evolut3.n is useu. If we neglect the direct effects of the mean flow curvature on the distortion of the turbulence and concentrate on the low wavenumber motions, then the mean distortion can be considered as the sum of a mean

strain rate, Dsn , and a mean rotation rate, n., which take the foims Dsn

(1d-S) 1 Id 2 (d- -

nz = -1 (dU 1(l+ S) 2 duI

(5.1)

(5.2)

Equations (5.1) and (5.2) show that flow curvature explicitly affects the rates of mean flow rotation and strain. An irrotational flow would have S = -1 while an unstrained rigid body rotation would have S = 1. Equal rates of strain and rotation, as in a rectilinear shear flow, result in S = 0. Rapid distortion analyses have been performed by other investigators for these three flows and some of the results may be relevant to the present flows. Townsend (1976) has calculated the redistribution of isotropic energy amongst the three components of kinetic energy due to an irrotational strain whch would appear to correspond, in the present context, to a flow with S = -1. The overall trend is a decrease in the streamxise component of the kinetic energy and an increase in the transverse and spanwise components, The transverse one. especially the predicted shear correlation coefficient, and the streamwise between uv/u'v', transverse velocity fluctuations reaches a maximum of .45 after a small amount of strain. None of the present experimental flows were irrotational on the mean, but the flow with the least rotation had S = -.27. Figure 4 shows that the strength of F (k,) increased relative to F (k.), while F W(k,) showed little change from the rectilinear The coherence for shear flow (Figure 5). this flow (Figure 6), was as high as 0.9 at its maximum and the self-preserving nature of this flow seems to suggest that this coherence would not decline with further strain. When S = 0 the rates of flow rotation and strain are equal and the vorticity is turned away from the principal axis of strain in such a way that the material lines of the flow become parallel with the streamwise direction. Analyses of this flow by several authors (Hinze, 1975) predicted a rather complex development for the an increasing turbulence, but, also concentration of the kinetic energy in the streamwise component and a corresponding reduction of the transverse and spanwise components. The overall shear stress correlation coefficient, uV/u'v', was found to reach a maximum of .6. Each of the present flows at the start of curvature was a rectilinear shear flow (S = 0) with a self preserving structure. For these flows, Fu(kl) was nearly an order of magnitude greater than F (k,) and Fww(k) and the coherence had a maximum of .7 without any indication of decline with further strain. When S = 1 there is no straining and energy cannot be extracted from the mean flow by the turbulence. A partial analysis of this flow is reported by Townsend (1980)

30-3-4

who suggests that the kinetic energy of the flow is constant but that components of the velocity fluctuations oscillate about their initial values at a rate which depends on the orientation of the waves and the mean rotation rate. The present flow which is the closest to rigid body rotation on the mean is that with S = .65. The overall kinetic energy of this flow was decaying because the viscous the turbulence exceeded dissipation production, however, Figures 4 and 5 show that F.(k ) and F6w(k) were declining faster than F (k,) and the coherence shown in Figure 5 had the opposite sign compared to that in rectilinear shear flow. Measurements of this flow were not made for sufficiently large total strains, so that it is uncertain whether this structure was self-preserving or it was a result of oscillations of the type suggested by Townsend (1980).

Townsend, A.A. (1976) "The structure of turbulent shear flow", 2nd Ed., Cambridge University Press, Cambridge. Wyngaard, J.C. (1968) "Measurement of small scale turbulence structure with hotwires", J. Sci. Instr., 1, 1105-1108.

S

sgns

R

Symbol

--

The distortion of the other four flows could be considered intermediate between the extreme cases discussed above, each having different proportion of flow rotation and

strain. The structure of the low wavenumber motions showed a trend of increasingly negative coherence of the streamwise and transverse motions with increased rates of flow rotation but the effects were not proportional to S; the structure was more strongly affected for S > 0 than for S < 0. 6.

'd', u

k.

27 - 1 - 07 0 + 07

20

2

4500

370

63

a

52

2

2900

340

280

49 51

0

+ 1 +

53

2100 470 880

160

115 170 270

45 32 30 0

30 64 27

2 2

____

Table 1:

770

5 0 5

68

x 0 5

Characteristics of the flows, (Uc 10 m/s).

CONCLUSIONS

Neither shear nor curvature appear to have a direct influence on the fine structure ot turbulence, while they both tended to increase the aaiisotropy of the large-scale structure. Negative values of S tended to enhance the relative strength of the large-scale transverse motions. Flow curvature modifies the coherence of the streamwise and transverse fluctuations so that momentum is more likely to be transported away from the center of curvature. For flows with negative S this enhances the coherence while for small positive S it diminishes the coherence. For flows with large positive S the curvature effects overcome the tendency for momentum to be transported down the gradient of mean velocity and there is a net transport away from the center of curvature.

\

R,

R, s

REFERENCES Hinze, J.0. (1975) "Turbulence", 2nd Ed., McGraw-Hill, New York. Holloway, A.G.L. and Tavoularis, S. (1991) "The effects of curvature on sheared turbulence", J. Fluid Mech. To be published. Papoulis (1984) "Probability, random varioabl, and stvuhdtLiv processes", 2nd Ed., McGraw Hill, New York. Tavoularis, S. and Karnik, U. (1989) "Further experiments on the evolution of turbulent stresses and scales in uniformly sheared turbulence", J. Fluid Mech. 204, 457-478. Townsend, A.A. (1980) "The response of sheared turbulence to additional distortion", J. Fluid Mech., 81, 171-191.

30-3-5

S >

0

Figure 1:

S < The flow configuration.

0

1)100

10

i2 00 ±

101

10"3

3

100 100 1 10"

10=2 10"5

3

10

+

Q

100

"

3

t

10"2

m ,ll

10"1

1

I

100 k, nl 101

10"2

Figure 2: Fj(k) normalized with the Kolmogorov scales: symbols in Table . 100

-

"

Figure scales. 3:

10-1

102 103 k Luu

-_

1

0

0t O-k v1o4

101

F (k1 ) nomalized with the energy

100

--

10

o.a10

10 101

10 3

1

10

T 4

30

at•qI

..

1.

0-2

Figure 4: scales.

10-1

10-0

Kluu103

10

F(k) normalized w th the energy

fC{Cuv(kcin

0

Figure 5: scales.

I

Z

101

100

101

102K1

03

F,4R1 ) normalized with the energy

c((

20

0 20+

2

k,.

O303--

Y

-0.9C.M....

10-2

10

1

100

101

102k

102

U1003

Figure 6: The real part of the coherence function c~r(k 1 ) -coherence

Figure 7:

30-3-6

10o1

100

101

102 k U 0

The imaginary component of the function C~v(k 1 ).

EIGI lii

3-3

SNIPOSI II'N ON

I1, URIULENT SHEAR FLOWS lvechnical Univcrsity of Mun ich September 9-11, 1991

-

Experimental Investigation of the Three-Dimensional Boundary Layer on a Rotating Disk Howard S. Littell and John K. Eaton Dept. of Mechanical Engineering, Stanford University, Stanford CA 94305

ABSTRACT The ,icocit\, field above a laige spirnnnig dsk has beeni shludied osiiig fliessure probes and fiotisie aiieiionicei s The) lomi eld (onsists of a thu ce-dimiisional boundary layei due to a ci o.silo%%caused by ceiitiifugal forces. Thel(mnean flow in bothi thec laninar andl turbulent tegines (onupaics well with pi evions st ild es ol~f i ite' stioothIiot atinig disks, and all six II cy nold, st i sses and~ thle teii tiile pio Iots has e becen naso i ci usinhg estaabIis Ished rossNs ite a I cii on iet , tek liiques. Selci ted rcvult aile comipared to pres Ious t Ill cc-dil ieniisioiial boun da)i I s i :::casieieiits aiid ses ci al imp11):taut dilrcieiics ate inoted

an extenisive data set The appropriate refe: ence fi aim foi analysis of the data is not apparent because tile flow direction Vles ini all three (lilections. \,e chose to investigate the flow on a rotating disk to shnplIify the analysis of the resulting data. There is no extei nally ieti ic so at imposed pies., it(e gradient and the flow is axisymm ihe thie (lesci to needecd are coordinates space ':wo miost only ex peiinient. Ini add ition, variations in the radjial direction at(e vi y slow i.vid call be neglected excepst heal t iansi tioii. 'I'llic d isk flow does develop a substantial skewing across the houndlm:y laiyer. At its peak the radial velocity component real iie, a level (of 11% of the local dlisk speedl. Ani added advantage of using a dlisk to investigate 3DTI3L's is thatt It(leIeytiolds unilbei cain he lianged easily 1,y Viiyi g the disk speed. Tlhci e ai key difliences, though , between a dlisk lsoiiulaiy layer ex per-

INTRODUCTION DTM)'Fhl) 'lh Ic-dilliensioinal tilt bilfeni hoiiidi s layvl: o in ivwd esct~l ineiet, s nodale e~oingthefocs ef ci s ict og i ze their pi acti al i 119)01taii It an d the key' di ffamiliar couiuii, *tle ci cinccs bet weein thein andl thiir n110: two- (himilnslolial boll Tidal N avers 'fhe tei in t ht:c-ddI ilie; ISloil al ye1r geiierally mnealis a wail- at td(bed sliva i fay in :da la 1)1)1 swhichi thle floss directioni sal is with distance from thle wcall T11l1s. iega d less, o~f the (hoice of coordlin ate system. hothIi51lo( ity om~poinents5 parallelil to the wall valyV 5it hi ist aiice fioiii the alaidbt wall-psi ahlel '.orti(ity onpoiieits al igi::icanit Ile tudiiriicie in tilice-dimleuisional bonidliy layel s hias 1we10stuidiedin a variety of geomiletries ilInluinlg flow af)plioaclinlg a swepit foiward-facing step), (Johinstoni, 1970), flow Elsenaar anlilovisiina, in ducts stimulaiting ilifiiiite swept wiings (1 aishass aiid Pont ik-os, 1985). 1974I, van deni Bet g t ofl. 1975, 131 loss%appiollching obstacles (DIechow aiid Felseli, 1977, Aiidcison1 aiid Eatonl, I 989) and axial flow along a iotating cyliiidci (Di 1ver a11(1Johniston, 1990). Ini additiin, thicie have been several ieci ii attempts to expflore tliice-diniieiision-I effeints oil tulrble1cme using dIirect nu1merical simulation (Spal ai t, I 9N9, Colemani et al. 1I990). Ili cases whei e an inmiti ally twoidary layer is tumiid, thuis becoin g tI ireediiimensional boun1 d1111elisional, it is geniiially found that the vectoi foinicd by 11)1 the shevar sti ess coim)potients piarallIl to the wall lags behin Vt'LLUi. 1115, 4 is fitiniiity founid d01 t tim,; WU 1,1% Lidi04 tul bulent sheai stress levels are reduced relative to the tiirbulent kinetic energy. It is not clear if this latter effect is caused bsy sudd~eni turninig of the flow or if it is a general feature of all boilli dat y layers coiitaiing mnean flow thIlCe dimi ensioniali ty. It is dIifficult to extract general conclusions froni the suth of the pies'ions experiments because they are often uonplik ated by extraneous effeu ts such as streatow ise pressure gradient Ailnd spaliwise iniotnogeiieity In miany 31DI1 e x per ients all thiree slmc coon inotes ai e signifhicanit aiid aiiy aiialysis of thle IHeynolds stress ti.1-.rnot equlatioiis, for inistanice, requires

initent aiid pies'ious 3DT3L expei imient . Fiist , filie skewing

is always piesent rather thaii developing as a prt-sire-iiiduced tt ciutr~ieiema viIual ec peit 1irbatioiin. i iisoaIsiai e-d ipsdt nwt the ii iiI o Fin ally, flow lotation adds tile coimplexi ty of the Col 1111isel fects to the basic eqleatioli set. lhuiecivsotepestiasrlipg.iiaetoiiply detailedl meanl velocity data and ti blilce ii csui lenlivts wisi. to assess if thiis 3DIT'Ill,, ill) to) tliid(-or(er quantities. WVe v iiiI iesumnably has leachied aii equiilibriumi state, is fiiidameintally different iii its tillhoilence piopert ius I hall weli uiiolstomud21Tl.

EXPERIMENTAL APPARATUS Thie expertIilienital l Altll sConsists of a I iilictel 111, lictel aliiniinim disk iiiotcd oii a vei tical spindle The, 1 i un1it k dhisk was p1ciision miachiined and lapped to allow onily ±20 /tinl vel tial dcviat ioiis whlen the dhisk iit atcs. 'Ile disk spill is t Speeds up1to 1100 IP NI which col ic5 11)11s In a1ip h-yiiolds n iiiei cf 1.8 m~illIion.- It is dtiveii by a 2 11i) D) tli iougli a belt dti e to avoid vibraitions. Th Iiin11oio is intli equlip ped withi solid state speied conitrol ailobwyinig simlooth val i&a tin of the disk 5see-I uiidvr luauial or comipiuter (oiitriil Tife Iotat inig assemibly was cai efllly halani d by vii tili' of its wei 61q, il 1 lhifactli(" cinid tlicii is no inic5ahishlc vihi ation a an opelitiiig speed~. aproii appi oxiinat,!y 0 3:1111~ sull il I oiiiih A fixed aldi tI e disi, to cli mi iate aiiy in teractimi hetwee ii tie( bou)1111 iy layers onl the top) alld hottoini surfaces. TIhe iottoll sil iifv i hnisiii a closed cavity to avoid (rea,,iig aiiy idistili haiti iii thle inuti d urn]dwi test cell. A set of 14 spiral-sI a~m! vaties aie 111i upphil siirfmc of till, ap oii to reii ove thet alignl lo sni -1liiin oh thle flow leav inig the disk. We oo)ser veil lo sigii tilt, siviI i V.Lli 5 ill iii the 2.4 fin by 2.7 iii by 3.4 iii isol ationi telI widthe

thut

plaice.

30-4-1

All experiments are conducted under the remote control of a data acquisition system consisting of an IBM PC-AT with a Metrabyte DAS-f I multifunction card and an SSH-4 simultaneous sample and hold. The computer controls the disk speed and all the measurement systems. Probes are postitioned by a stepper motor traverse mounted on a 1" X 3" beam located 1 meter above the disk surface. The traverse can rotate the probe around its stem axis with a resolution of 0.90 and translate it normal to the disk with a resolution of 1.6 pm. Mean flow data are acquired using a three-hole probe in a nou-nulling mode as described by Anderson and Eaton (1987). The dynamic and differential pressures are measured using a Setra model 239 (± 1.0 "120 range) or a model 264 (010 "H20 range) for high speed dynamic pressure. The transducers are thermally insulated to minimize drift and are calibiated against a micromanometer before each run. Three different hotwire probes are used. Each probe is mounted in a gooseneck stein allowing yaw of the probe without moving the measurement volume. The single-wire piobe

of each wire are acquired in each of the four probe orientations. The measured Reynolds stress and triple product tensors are then rotated into appropriate coordinates. The uncertainty involved in the Reynolds stress data are similar to those reported by Anderson and Eaton (1989) near the disk surface, but farther out in the boundary layer the local turbulence intensities became quite large. Therefore the measurements in the outer part of the boundary layer are subject to question. The analysis of Tutu and Chevray (1975) suggests that the crosswire is subject to very large errors for turbulence intensity greater than 35%. However, their analysis predicts a strong disagreement between single-wire and crosswire measurements for large turbulence intensity. Figure 1 shows a comr arison of 10 1 08

uses a Dantec 55P05 boundary layer tip strung with Dantec

06

gold plated 5 Imiplatinum coated tungsten wire with an active P, length to diameter ratio of l/d = 250. The cross wire probe was . custom built. It uses 2.5 pin platinum coated tungsten wire 00 which has been copper plated and subsequently etched for an S active length to diameter ratio of l/d = 200 The wire spacing 02 oC in 450 is 0.35 mi. The crosswire can be rotated about its axis iiceinents allowing determination of all six Reynolds stresses. 0 I I The third probe is a custom dual wire probe which has two 14 12 10 a 6 4 2 0 wiles at ±45* to the stem axis in the same plane, parallel to the disk surface. This probe is used to provide redundant ¥/52 measuiemnents of the flow direction, and was strung with 5 /in ligure I: Comparison of measured turbulence intensity beplatinum coated tungsten wire, copper plated and etched for tween single boundary layer probe and crosswire. an active length to diameter ratio of l/d = 200. All hotwes are operated in constant temperature mode the longitudinal turbulence intensity as measured by the two by a TSI IFA-100 unit modified to eliminate the built-in linhe agrenal turbulence inten eaiizers. The signals are DC shifted and amplified to utilize pioes. The agreement is excellent out to a turbulence intenthe fullwith ±5aVolt range on the Metrabyte A/D.low-pass Noise filter is fil- aie sity of of the at least 60%, sign and than even beyoid the differences tered Frequency Devices niodel 901F1 opposite predictedthisbypoint the analysis Laser Tteswith cl te ue Deies mctolele s01Flow-isssaltor Doppler measurements now in progress should shed light on The test cell temperature isnot controlled so itisnecessary to this pioblem but for now,. the Reynolds stress measurements calibrate the hocwmres for both velocity and temperature varishould be tieated with caution inthe outer part of the boundation. This is accomplished by a portable jet apparatus which shouldyete is placed on the disk and allows precise control of the flow tempeiature and velocity. A temperature dependent calibration is obtained by varying both the temperature and velocity and RESULTS AND DISCUSSION fitting King's law in a modification of the temperature correcThe mean velocity and turbulence measurements are pretion suggested by Cinbala and Park (1990). In this imple' sented in a reference frame rotating with the disk so the vementation, the original form of King's law, Nu = a + b.Re locity profiles appear similar to those in a typical wind tunnel is used, with fluid properties evaluated using the film temperaexperiment. This also gives direct mathematical formulation tuie T7= (T + T,). The calibration is then of the following of the Coriolis effects, and allows the centrifugal forces to be foi 111, included into the pressure term. The frame of reference is [B 2 21 /n / defined with its origin at the radius of interest and is rightSV(P(m,) k7. (T.- T.) E01 handed with +Y directed upwards from the disk surface, +Z pointing at the axis of rotation, and +X directed in the cirThns calibration is implemented using an empirically obtained wie temperature specific to the type of wire, then B and E.' aie found with a least squares fit. The typical RMS deviation of the fit from 20 calibration points varying from 1 to 40 m/s was 0.2% at a single temperature, and < 1.5% for a temperature which varied 10°C. The mean flow velocity and direction are measured using tliee different techniques, the three-hole probe corrected for shear and wall proximity, the custom dual wire probe, and finally by yawing the single-wire probe in an automated version of the Cham and Head (1969) technique. For mean flow measurements, 4000 samples are acquired at 100 Hz. The crosswire probe is aligned with the previously measured mean flow direction at each measurement point then 10,000 samples

cumferential direction. One unwanted consequence of the transformation is that small uncertainties in measured flow angle close to the disk lead to very high uncertainties in the flow angle viewed in the rotating reference frame. This is due to the fact that once the local disk speed is subtracted from the measured velocity, the measurement uncertainty is retained as a percentage of the (large) measured value. For this reason the experimentally measured mean vales Lend,to show much more scatter close to the disk than the scattc. mIshe turbulence statistics, which are not subject to this ,ftct. Both laminar and turbulent cases were examined. The laminar cases are not shown here because they were used piimaril for instrumentation qualification. Agreement with an

30-4-2

Disk Flow Turbulent Cases Table 1:, Sym'bol Radius / Re U. [m/si 699[nmn] = 0.235 in lItRe = 400,000

I]

(

R = 0.356 in Re = 650,000 Re = 9410,000 R = 0.421 in Re = 650.000 Re = 1,000,000 Il e = 1,300,000

62 [mm]

H

Res2

U [m/s

V/U. [Im]

26.13

11.60

0.978

1.342

1657

1.206

12.8

28.22 41.01

19.22 20.05

1.457 1.456

1.290 1.269

2662 3839

1.255 1.750

12.3 8.9

23 72 36.72 4814

21.04 20.61

1.712 1.641

1.298 1.282

2645 3903

1.051 1.556

14.6 9.9

] 20.39I

1.270

1.594

7.7

1.997.

4969

analytic solution was excellent for all three probes especially

is in the -Z direction. These measurements resemble previous

at the inner radii where the disk appeared to the flow as if it weie indeed infinite. The turbulent cases to be presented here aie listed inTable 1,along with integral properties of the flow in the X diiection The integral properties were calculted by diect numerical integration of the measured velocity profiles and the skin friction was calculated using the two-dimensional = 0.41. It is apparent from this infoilaw of the wall with tc mation that iclatively high Reynolds numbers are produced

3DTBL experiments plotted in freestream coordinates. It is always difficult to gain a physical understanding of the p. crn of the turbulent fluctuations from single point statistics, and even more so to compare results between experiments. In the process of Reynolds averaging the Navier Stokes equations, quantities (Reynolds stresses) result which are the con07

0.6

with a small experiment, due to the high speeds involved.

0 Re -650,000 r.0 356 aRe - 1,000.000 r-O356 ORe - 650.000 r.0 421 0 Re. 1,000.000 r.0421

o

05 25

Re .400,000 -"235

o-

-

04

[o

20•0

20

03 --

tRe- 1,300,000 r.O421 a

02

*15-

Re - 4O.000 r-0235 oRe 650,000 r-0356

tq 0

O Re - 1,000,000 r.O356

10

/

1

01

10

103

. ..

.

CA '-

...

0

II Me . 650,000 r=0 421 0 fie..1,000,000r-.0421 ARe . 1.300.000 r-O421 102

. .

0

10'

2

4

8

Figure 3: Primary Reynolds stress shear velocity. 07 -=-r--TrT--1---

Figue 2 Me nci cucumferential velocity in wall units

yrT-

10

12

14

-

'v , normalized by the

,

06

Figuie 2 shows the mean flow nofihes in the +X diiection plotted using wall scaling. One immediately notices the lack of a wake component, which is likewise absent in the data of Chain andllHead (1969). This is not the result of a favoable pressuie giadiemnt, since the X direction will not support

Re -400,000 r-0 235 C Re - 650,000 r.0.356

05 04Re1,000,000r.0356

oRe -650,000 r-0 421 ARe, 1,300,000 r.0 421

04 7

a icn pressuie gradient. but instead may be taken as in in-

"Y03 7

dication of structural differences between this 31)TBL and the typical zeio pressure gradient 2DTBL. -The primary Reynolds stress u7 is shown in Figure 3, noi malized with the circumferential shear velocity. It appears much the same as a zero pressure gradient 2DTBL, and totally unlike the 3DTBL flows in Table 2 which all exhibit a peak away fioni the wall The effect seen in other 3DTBL's can not

3: - 02-

be ascribed only to aii adverse pressure gradient, but is also

0

the result of some type of structural change in the turbulence which will be discussed more fully below. The secondary Reynolds stress 7w7 is shown in Figure 4, normalized with the circumferential shear velocty. This quantity is a measure of the effect oii the turbulence of the spanwise stlain, so it is identically zero in a 2DTBL. It is interesting to note that ; has a nearly constant value in the locations the crosswire could acquire data, but must change sign closer to the wall in order to approach the wall shear stress value as expected. The sign change is necessary because the spanwise mean velocity profile has a maximum, and therefore the outer layer sees the opposite shear than the inner layer. The sign of the stress is positive in the region plotted because the crossflow

6

O Re - 1,000,000 r-0 421

0 01

0

.

-01 .

, 2

4

P

8

I , 10

12

14

Y/52 Figure 4:Secondary Reynolds stress ;'--, normalized by the siear velocity. sequence of average momentum transport by the fluctuations. In attempts to form 'closure' models for these terms, several quantitative relationships between the various turbulent quantities have been found to have a predictable value or behavior in a certain type of flow. These 'structural parameteis' allow similarities and differences in fluctuating fields and thus the physics of flowflelds to be compared. Selected structural indicators from the present flow w2:l be compared to previous experiments listed in Table 2. It is important to note

30-4-3

that three major categories of 3DTI3Ls are represented. infinite swept wing, obstacle, and shear driven. These combined with Kiebanoff's classic 2DTBL should represent the current state of knowledge of the effect of three-dimensionality on an initially 2DTBL from wind tunnel experiments. Table 2: Reference Experiments Experiment

Flow Type/Designation

P

Bradshaw/Pontikos (1985)

Infinite Wing, X=1092

R

Driver/Johnston (1990) Elsenaar/Boelsma (1974) Klebanoff (1955)

Shear Driven, St 9 Infinite Wing, X=1095 2DTBL Reb2= 8000

_Symb. _A

El K

Anderson/Eaton (1989)

foi the outer radius. The consideration of this effect combined with that in Figure 5 leads one to the conclusion that irrotational (inviscid) motion plays an even bigger role in the outer region of the disk boundary layer than in a wind tunnel layer. 0.20

Obstacle, Case I St 4

°O o

0 15

M 0 10

-

-

The ratio of streamwise normal stress to vertical normal stiess appears as Figure 5, along with comparisons to the expenniets listed in Table 2. This ratio is a measure of the boundary layer's preference for streamwise fluctuations com-

exper

ts,

a

g wh t

E

. 'n, 1 Cb A d 0 oA

0 1A

E, P 0 0

0

00

o

,

, 2

0

4

6

8

I 10

I , ,

12

14

16

y/82 Figure 6: Ratio of shear stress magnitude to twice the turbu-

C),

-

o

0A

0

molecular viscosity when inserted into the mean momentum equation. The ratio of the relative effectiveness of the turbu-

0

0

-

K P

0 a0~ A

lence in transporting streamnwise and tie spanwise momentum

0

0 'A 0 .'

Z,

R

is defined as:

A, n

E

r: K U I

E E if K

E

2

4

6

KE

8

N, =,

10

12

14

normal stress to vertical noi mnal 5: Figure ts (of Ratio of streamwise stics. (Symbols as in Tables I amid 2) 2DTI3L experiment of Klebanoff, show a nearly cotistant or mootonic decrease in this ratio moving away from the wall. E,6iy case from the present experiment shows a clear peak in the outer region of the boundary layer. The height of this peak is seen to decrease and perhaps move outward with increasing Rlc iuolds number at a given radius. This implies the structure of the boundary layer is substantially different from the other experiments, but has not reached an asymptotic state. Another commonly investigated structural parameter is the.o ratio of the vector magnitude of the shear stress to twice the tuibulent kinetic energy:

±w -

yb)

angle. This may also be thought of as the ratio of local spanwise to local streamwise eddy viscosity. The values obtained foi N, are shown in Figure 7. These data exhibit more scatter than most of the other plots because they contain derivatives expeiimental data. As concluded by Anderson Eaton (1989), low values in N, (as low as 0.2) are observed and in sharply tu mid experiments such as shear driven or obstacle flows, and a value closer to one in slowly turned flows like the infinite wing. This implies that the disk flow more closely resembles a slowly turned flow or equivalently one which has more time to relax to a new state after the imposition of crossflow. The values near unity indicate that an isotropic eddy viscosity should perform well, and this was indeed the case for a simple cross12 o 10

0

A. ;%,

A/ 12

-

The angle -y, is that of the local mean velocity, 7 , is the angle of the velocity gradient ,ector, and y, is the shear stress vector

Y/5,

A_

(-,tanb,-V

IK

0

0

One of the most common approaches to closure is the assumption that the Reynolds shear stress responds linearly to imposed strain rate. The constant of proportionality is known as the eddy viscosity, because it is analogous to the

6

0L

P

1,

lent kinetic energy. (Symbols as in Tables I and 2)

8the

2

P

0

R

005 -

paied to fluctuations normal to the wall. A high value implies

0seenin each of te

o

R

R

- 6$- % nRke ER r5 P A A . 0 0 A AA

that stieamwise fluctuations are dominant over wall-normal events, pointing to a decrease in eddy coherence. The behavio1 seen in each of the 3DTBL experiments, along with the

R

-2

08

q2

A

D OO0

~OAA

O

0

06

This parameter is approximately constant at 0.15 in a 2DTBL evem when the layer is strongly distorted by pressure gradient. The wind tunnel experiments plotted in Figure 6 show the

04 A

typically observed trend that three-dimensionality depresses A near the wall, and this depression diffuses outward. As noted above, A, is not strongly affected by pressure gradient alone, and the general agreement among 3DTBL experiments flow structure in imllies that three-dimensionality affects the

this qualitatively predictable fashion. The disk flow is vastly diffeent, showing A, to decrease almost linearly with distance from the wall for the inner radii, and at least monotonically

02 0

2

4

6

8

10

12

14

2 Figure 7: Ratio of eddy eiscosity in coordinates aligned with the local mean velocity. (Symbols as in Table 1)

30-4-4

flow calculation for the disk flow reported by Chain and ]lead (1969). One might suspect that the outer region of the boundary layer is dominated by Coriolis effects. The Reynolds stress transport equations for a rotating coordinate system show that Cotiolis effects act to redistribute the Reynolds stress among the various components of the tensor. This could lead to the icreasig values of n' 2/v' 2 observed in the outer layer. If the outer region were dominated by Coriolis effects, then w' and ' fluctuations must be negatively correlated. lowever, the correlation coefficient for 177 was calculated and is strongly pusitive in the outer layer, indicating that Corioli Ifects are not a major factor. This then supports tire conclusion that the outer region is dominated by irrotational (inviscid) motions, a conclusion also supported by Chain and Ilead's (1969) measuietnent of reduced entrainment. It renains to address the cause of the reduced level of shear stics in the disk boundary layer. Bradshaw and Pontikos h pwthesized that turbulent eddies formed iua 2D boundary layer ate tilted out of their preferred orientation by the inpositiomu of three-dimensional skewing Implicit in their discussion was the assum)tion that normal boundary layer turbulence would eventually de'elop if the 31) strain field remained constarit foia long enough distance. Anderson and Eaton (1989) theuized that the turbulence is stabilized in the region 6f peak turbulence production by tire presence of crossllow Eaton (1991) took this further, stating that production is reduced because a fraction of the los speed streaks inthe boundary layer at eliminated b) the crossflow and because longitudinal vorti(es ate attt ruated by the crossflow. Implicit inthis argument is the assumption that boundar) layer turbulence developed i a three dimensional strain field is inherently less efficient at extiacting energy from the flow field The present results suppot the latter hypothesis. The value of A, is suppressed below norral 2D levels even though the entire boundary layer (levelopincnt occurs in a nearly constant three dimensional strain field. CONCLUSIONS Tile disk boundary layer, while subject to Coriolis effect,, al)pears to be a good configuration for studying the fundaineital liysics of 3DTBL's. Many of the effects observed m wind tunnel 3DTBL's are also observed here. The A, structuial parameter is less than in comparable 21) boundary layers and a small lag in tile shear stress direct;o behind the mean velocity gradient angle was noted, behaving not unlike a slowly-turned three-dimensional boundary layer. Some of the results nmust be treated with caution since the hmotwire measurements may be subject to considerable uncertainty. Laser Doppler measurements now underway should resolve such questions. Additional length scale measurements should cast further light on the structural features of the boundary layer.

REFERENCES Anderson, S.D. and J.K. Eaton, (1989) "An Experimental Investigation of Pressure Driven Three- Dimensional Boundaiy Layers," JFM, Vol. 202, pp. 263-294. Bradshaw, P. and N.S. Pontikos (1985) "Measurements in the Turbulent Boundary Layer on as 'Infinite' Swept Wing," JFM, Vol. 159, pp. 105-130. Coleman, G.N., J.H. Ferziger, and P R Spalart (1990) "A Numerical Study of the Turbulent Ekman Layer," JFM Vol. 213, pp. 313-348. Chai,. T-S. and M. R. Head, (1969) "Turbulent BoundaryLayer Flow on a Rotating Disk," JFM, vol 37, part 1, pp. 1291,17. Ciibala, J M., W.J. Park (1990) "A Direct IHot-Wire Calibiation Technique to Account for Ambient Temperature Drift in Incompressible Flow," Experiments in Fluids, Vol. 8, pp. 299-300. Dechow, R. and K.O. Felsch (1977) "Measurements of the Mean Velocity and of the Reynolds Stress Teisor in a ThreeDimensional Turbulent Boundary Layer Induced by a Cylinder Standing on a Flat Wall," Proceedings of Symposium on Turbulent Shear Flows, April 18-20, 1977, University Park, Pennsylvania, Vol 1 Diiver, D.M. and J.P. Johnston (1990) "Experimental Study of a Three-Diniensional Shear-Driven Turbulent Boundary Layer with Streamwise Adverse Pressure Gradient," NASA TM 102211, and Thermosciences Division Report MD-57 Eaton, J.K. (1991), "Turbulence Structure and Heat Transfer in Three-Dimensional Boundary Layers," 9th Symposium oil Energy Engineering Sciences, Argonne National Laboratories Elsenaar, A. and S.HI. Boelsma (1974), "Measurements of the Reynolds Stress Tensor in a Three-Dimensional Turbulent Boundary Layer Under Infinite Swept Wing Conditions," NLR TR 74095 U. Johnston, J.P. (1970) "Measurements in a Three-Dimensional Turbulent Boundary Layer Induced by a Swept, Forward-Facing Step," JFM Vol. 42 pp 823-844. Johnston, J.P. (1973) "The Suppression of Shear Layer u'llbulence in Rotating Systems," J. Fluids Eng., June, pp. 229-236 Klebano(f, P S. (1955) "Characteristics of Turbulence in a Boundary Layer With Zero Pressure Gradient," NACA Report 12,17. Spalart, P.R. (1989), "Theoretical and Numerical Study of a T,ee-Diiensronal pou.dary Turbulent Layer," FM, Vol. 205, pp.). 319-340. Tutu, N.K. and R. Chevray (1975) "Cross-Wire Aneiometry in High Intensity 'urbulence," JFM Vol. 71, pp. 785-800.

ACKNOWLEDGEMENTS This work is supported by the Department of Energy Ofihe of Basic Energy Sciences Grant DE-FGO3-86ER13608. '[le numerous discussions with Prof. James P. Johnston and Prof Peter Bradshaw are especially appreciated.

30-4-5

EIGHTTI1 SYMPIOSIUM ON

TURBULENT SIHEAR FLOWS Technical University of Munich

30-5

September 9-1I, 1991

Effects of Solid Body Rotation on the Transport of Turbulence Shao L., Michard M. and Bertoglio J.P. Laboratoire de MI6canique des Fluides et d'Acoustique,URA CNRS 263, Ecole Centrale de Lyon, 36 av. Guy de Collongue, 69130 Ecully, France

ABSTRACT.

Concerning triple velocity correlations : when

The mechanism of turbulence transport is studied using a Large Eddy Simulation of a shearless turbulence mixing layer, subjected to solid body rotation. The behaviour of both transport by triple velocity correlations and transport by pressure-velocity correlations is found to be completely different from what is found without rotation. A modified onepoint model, that accounts for the effects uf rotation, is proposed for the triple correlation transport terms. 1. INTRODUCTION In a previous study, the mechanisms responsible for transport of turbulent kinetic energy and transport of the components of the Reynolds stress tensor were investigated in a simple case of inhomogeneous turbulence : the case of a shearless turbulence mixing layer, in which two nearly homogeneous fields, having different energy and length scales, were put side by side and were interacting (see the sketches in figures 1 and 2). The study was made using a Large Eddy Simulation (Shao, Bertoglio and Michard,, 1990). Results were found to be in qualitative agreement with the experimental data of Veeravalli and Warhaft (1989). Comparisons with results of the classical onepoint closure models, like k-s and Reynolds stress models, were presented in another paper (Shao, Le Penven and Bertoglio, 1990). The main conclusion of the study was that triple velocity correlations were found to be responsible for the energy transport and that pressure-velocity correlations were found to tend to counterbalance the triple correlation transport, their effects remaining small. T L

Fig. 1

Exp.

2

~studied

1

.2that LMxnL2 Fig. 2 box

a; 1

the Launder, Reece and Rodi (1975) second order model was used, the value Cs=0.11 usually retained in the expression for the triple correlation =-Cs(u

-

C a + 2 '-Z--" Z -

4 + Viscous term

in which <wq 2 > = <w3 > + <wv2 > + <wu 2>. The important terms for the transport

0 Li L1

The equation for twice the turbulent kinetic energy is: D Dt- =

cube

.

1Z

transport. The viscous terms essentially account for molecu!ar dissipation. Viscous effects also contribute to turbulence transport, but their contribution is not discussed here since it vanishes for high Reynolds number flows. The last terms are Coriolis terms, whose trace is zero, since Coriolis forces are not producing energy.

(4)

of

turbulence are therefore <w3>, <wu 2 >, <wv 2> and .

Fig. 3<w>

The results of the L.E.S. clearly show that

mixing layer

rotation does reduce the triple correlation term as shown in figure 4b where the component responsible for the transport of <w2> in the Z direction, <w3>, is plotted as a function of Z. In the present computations, turbulence first evolves without being subjected to rotation between t=0 and t=0.04 s. Then rotation is suddenly switched on at t=0.04 s. The value of 12 is 50 rd/s, corresponding to an initial value of the Rossby number approximatively equal to 0.7 on the high turhulence side of the mixing layer. In figure 4b the normalized quantity, <w 3 >/<w 2 > 3 /2 , is found to first increase up to a value of about 0.4, reached at the end of the nonrotating phase (t=0.04 s). For larger values of time, this quantity decreases drastically : at

At the beginning of the computations, on the left side of the box, a turbulent field with strong intensity and large length scales is introduced, On the right side, a turbulent field with lower intensity and smaller length scales is prescribed. The ratio of turbulent kinetic energies between the two sides is nearly equal to two, and the ratio of length scales is appi ,.imatively two. This case corresponds to test case 64-02 in the steady frame study proposed in Shao, Bertoglio and Michard (1990). Results are averaged in planes of homogeneity, that is to say in (X,Y) planes. They are also averaged over different realizations in order to improve statistical sampling.

,

The equations for the double velocity correlations at one point, or Reynolds stress tensor, can be easily written in a rotating frame. The important terms clearly appear in the equations for the diagonal components. Taking :,ito account the symmetries of the problem and homogeneity in the X and Y directions, we obtain :

ii-, t=0 08 s, it is nearly equal to the statistical noise (in this case, results are averaged on 3 realizations only). In the same case without rotation, larger values of <w3>/<w 2>3/2 were obtained (case 64-02 in Shao, Bertoglio and Michard, 1990) as shown in figure 4a where this quantity continue to increase between t=0.04 s and t=0.08 s.

D

Dt

2

-<wu

2

D

Sp

D<w I 2> =

-

>

2

>

0 0wv

2

In

Du

+-+vicoustenn P +cases +Coriolis term

+viscouste y

(3)

<w3 > 2o 2 Dw z-"-+ zp->+viscusterm + Coriolis term

The first two terms on the right hand sides are respectively, transport terms by triple velocity pressure-velocity by and correlations correlations. The third terms are for pressure redistribution, whose effect is the interchange of energy between components of the Reynolds stress tensor, and which do not affect energy

figure

5, the evolution with ti, 9 of the

turbulent kinetic energy profiles is plotted. The with (figure 5b), and without (figure 5a) rotation are compared. It immediately appears that higher levels of turbulent kinetic energy are obtained at the end of the simulation on both sides of the box when rotation is present. This is due to the inhibition of the energy transfer between eddies of various sizes, an effect which appears to be consistent with what is found in homogeneous turbulence. A more interesting remark can be made concerning energy distribution along Z : despite the fact that triple correlations are reduced by rotation, the diffusion of turbulent kinetic energy towards the low turbulence side of the box is still comparable to what was found without rotation. The only possible explanation is therefore that,

30-5-3

in the rotating case, a large part of the turbulence transport must be due to the pressure term. When pressure velocity contributions are evaluated, this conjecture is corroborated by the fact that they are much larger than in the case without rotation. In the steady frame case, to tend pressure-velocity correlations counterbalance transport by triple correlations, while they are found to contribute positively to the diffusion mechanism in the rotating case (see figures 6a and 6b). In the non-rotating case, turbulence transport is due to triple velocity correlations, the effect of pressure being to slightly reduce transport. In the rotating case, transport is mainly due to pressure velocity correlations. In a recent study, Riley (1990) has found very similar results in the case of turbulence subjected to stable stratification : when a slab of stratified turbulence is immersed in a nonturbulent field, diffusion of turbulence appears to be comparable to the diffusion obtained in the non-stratified case, even though triple correlations are strongly reduced. In Riley's work, this effect is attributed to the presence of waves in stratified turbulence. An interesting analogy is possible, since it is known that in the case or rotating flows, waves also exist : inertial waves. If results not averaged over different realizations are plotted as functions of time, irregular profiles are obtained when rotation is present, while, in the non rotating case, smoother behaviours are found. This is, in particular, clear in figures 7a and 7b where profiles of <w2 > are compared. This behaviour can not be considered as evidence for the existence important to ofbe waves, noticed. but it is sufficiently 4. ONE-POINT MODELS The aim of the present section is to include in one-point models the reduction of the triple correlation induced by rotation. A model is devised by taking into account the linear term due to Coriolis forces in the equation for the triple velocity correlation. In a second step, a simplified model is proposed. In the one-point model approach the equation for correlations the fourthHanjalic order 1972),, isandwritten the (see and triple Launder

The quantity -L- is a damping term aodamping erm t Ck introduced in order to model the effect of pressure. It can also be regarded as taking into account the effects of the fourth-order cumulants. When the asymptotic values for deduced from equation (5) are used in Reynolds stress computations, results show that triple correlation levels are reduced by the presence of rotation - as expected. Consequently, the inhomogeneous transport of turbulence appears to be reduced when compared to the non-rotating case (see figure 8). A simpler model is also proposed, in which triple correlations are expressed using the classical formulation corresponding to the Hanjalic and Launder model (equation (1)), with C, replaced by Cs'=Cs/(1+Ro'l). R, is here the Rossby number defined as Ro=C/kn. Comparisons between results obtained with the simple model and with the complete form are very satisfactory, as appears in figure 8. However, it has to be pointed out that more work is needed to model the pressure-velocity correlation in this case, since one of the conclusions of the Large Eddy Simulation study is that pressure appears to play a very significant role when rotation is present. In order to determine whether the pressure effect was due to the rapid part of pressure or to non-linear effects, a Rapid Distortion study was made, by simply discarding the nrn-linear terms in the numerical code. The conclusion is that the effect is present in R.D.T. and is consequently a linear effect, which is consistent with the fact that transport is due to inertial waves. At the present time, it seems difficult to model this effect at the level of one-point closure. The problem involves quantities like group velocities that are difficult to take into account in classical models. The case of the axis of rotation parallel to the inhomogeneity direction OZ could also be investigated. In this situation, effects more similar to the mechanisms observed in the experimental study of Hopfinger et al. (1982) are likely to be found. ES

R -AUPOIX

B., COUSTEIX J. and LIANDRAT J., 1983; Effects of

Rotation on Isotropic Turbulence, Fourth Int. Syrup.

moments are expressed by introducing QuasiNormal assumptions. The approach is here

Turb. Shear Flows, Karlsruhe, Sept. 83. .

extended to the case of rotating turbulence, in

which we do not neglect the effects introduced by the presence of rotation. Thus : . C, it oD

u

-

+ -

+ +

R1, a

2

(5)

Ckinfl2 + 2Ejifi)

In equation (5), Cijk is the permutation tensor.

30-5-4

BARDINA J., FERZIGER J.H. and REYNOLDS W.C., 1983; improved Turbulence Models based on L. E. S. of

Homogeneous, Incompressible Turbulent Flows, Report

T.F. 19, Stanford Univ., May 83.

BERTOGLIO J.P., 1982; Homogeneous Turbulent Field within a iRotating Frame, A.I.A.A. Joum., Vol. 20, no 9.. CAMBON C., TEISSEDRE C. and JEANDEL D., 1985; Etude d'effets couples de D~formation et de Rotation sur une Turbulence Homog~ne, Journ. M6c. Th~or. Appl., 4. CAMBON C., JACQUIN L.and LUBRANO J.L., 1991; Towards a New Reynolds Stress Model for Rotating Turbulent Flows , submitted to Physics of Fluids. CHOLLET J.P. and LESIEUR M., 1981; Parameterization of

Small Scales of Three-Dimensional Isotropic Turbulence

J., 1978; Computational Modeting of Turbutent Flow, Adv. AppI. Mech., 18, pp. 123-176. -MALIKUS, 1990 ; Europ. Turb. Conf., Stockholm, July 90. -RILEY, 1990, Seminar on Stratified Turbulence, Ecote Centrate de Lyon, Sept. 90. -SHAO L, BERTOGLIO J.P. and MICHARD M., 1990: Large Eddy Simulation of the Interaction between Two Distinct Turbulent Velocity Scales, Europ. Turb. Cont., Stockholm, July 90, to appear in Springer-Verlag[ -SHAO L., LE PENVEN L.and BERTOGUiO J.P., 1990, Study of the Modelling of the Transport Terms in One-point Closures using L. E. S, of Turbulence, Int. Symp. on Eng. Turb. Modet. and Measur., Dubrovnik, Sept. 90. -SHAO L. et BERTOGLIO J.P., Effet dle ItIntermittence sur les Termes de Transport Turbulent 10e) Cong. Mdca. Fr. Sept. 1991 -SPEZIALE C.G., GATSKI T.B. and MHUIRIS N.M G., 1990; A

SET

171I

///

C

-LUMLEY

Critical Comparison of Turbulence Models

. ..............

.

utilizing Spectral Closures, Journ. of Atm. Sci., Vol. 38.. *HANJALIC and LAUNDER B.E., 1972 A Reynolds stress model of turbulence and its application to thin shear flows , J.F.M. 52 pp 609-638 *HOPFINGER E.J., BROWAND F.K. and GAGNE Y, 1982;1jl Turbulence and Waves in a Rotating Tank. J. F.M.125. -JACQUIN L., LEUGHTER 0., CAMBON C. & MATHIEUUJ., J.F.M. 220, pp 1-52, 1990 -LAUNDER, REECE and RODIt 1975 Progress in the lvlpenn of Reynolds stress closure J.F.M. 68.

o

~in

0

s

(m

fig 53

for

L'.E-E1,

Homogeneous Shear Flows in a Rotating Frame, Phys. Fluids, A2, 9, Sept. 90, pp. 1678-1684. -VEERAVALLI

and WARHAFT, 1989 The shearless turbulence-___

mixing layer

J.F.M. 207, 191.229

t008.-

___

c

ER'p

C4-32 CWEGA-',

___

___

3 REAL

.

ICI 0

fig

FVOL T I ON vi'

~

i-

s

4a~~im

32'

tO

o

00$fig

/ LVol1) UTIONW3

fig 4h

Profiles of the normalized triple Fig. 4: correlation <w 3 >/<w 2 >3 / 2 as functions of the Inhomogeneity direction Z for different times. Results averaged over 2 realizations. Fig 4a: Fig 4b:

without rotation. with rotation 92 = 50 rd/s (rota~ion switched on at time f=0.04 s).

30-5-5

5b

Fig. 5 : Evolution with time of the turbulcnt kinetic energy profiles. Results averaged over 3 realizations. Fig. 5a : without rotation. Fig. 5b: with rotation 12= 50 rd/s (rotation switched on at time t=0.04 s)-

02 PP

t2Z 2 CU/2.'0

fig~rsut

Vo

Fig

0

6: Comparison of the d~~~~~~(lffin rnpr

Inspn

2. I Frot

sc

0 ,.

rtto

?4

wthd

moel

>tlio0

obaie coresonin

-LJ

50rd/ ttm

coprsnewe wit

the>opae comlet

the V..eime

32

ffig

o

and

(pefrae

Z

7b

-

-

V

30g-:5vltinwt-tm6f w>prfls

EIGIHTrH SYMPOSIUM ON TURBULENTSHEAR FLOWS

31-1

Technical University of Munich September 9-1I, 1991 INTERMITTENCY AND CONCENTRATION PROBABILITY DISTRIBUTION IN TURBULENT SHEAR FLOWS

V. A. Sabelni kov Central

Aerohydrodynamic

Insticute

Zhukuvtkv-3. roskow region,

ABSTRACI

,F

A self-similar solution of the equation density probability concentration for the function in free turbulent investigated. using

140160 USSR

shear

flows

is

equation

independence of

scale motions in fully

of

new type

sin(ular

self-similar

the

probability density function

large-

turbulent

developed for

equation

flows. The

fine and

the

+or

de-

4y' 11i12G,

(I

=XI ex(-.2), a2 = 1.0

and C' 3 = 1.0. In the third term, dIV is a Wiener process, with the properties (01V)= 0 and (d~lV) = d. The stochastic model for the velocity following a fluid partit.le is

The model roblei used in this paper is the piloted methane flame of Masri and Bilger (1986). It consists of a central jet of methane, radius R = 3.6mm, surrounded by an annular pilot flame, width 5.4mm, of stoichioinetric composition. This flame is situated in a uniform coflow of air. The flow velocities chosen are 27.0 m/s bulk flow for the methane jet, 24.0 m/s for the pilot and 15.0 m/s for the coflow. These correspond to the K flame in Masri and Bilger (1986). This set of flow conditions is chosen as it shows the least local extinction, and so an equilibrium assumptioi for the thermochemistry can be justified. With such an assumption, all thermochemical quantities can be represented by functions of D t)ensity, p, is given as a piecewise function of ,,.

(I' =

+ Ddl + (c~w,)'1'dW,

- =

(5)

p ax.

P

where-

D), =

I_

+ ,iu (21 43 C'o)I(kk )u,* + -3Co(A kSiu;

, = U"

p

(

u

-

(,mi,) )

-

(6)

1(-j-1)+

(10)

0 'L-i

-i) )+-,

(I1)

where p, = 0.65, p, = 0.11 and p, = 1.20, referring to the jet, stoichiometric and coflow initial densities respectively. , is the stoichiornetric mixture fraction and is given the value of 0.055, oresponding to that of methane.

,; A,, is the normalized Reynolds stress tensor,

A,, = 3(u,u,)/(uiu1);

(7)

and A,, and &i are the p and pw weighted conterparts of A,,. T['he constant Co is given as 3.5. and we set Gf = 0 as in Pope(! 991'a). The dIV, term z an ;sotrop~c Wiener proccss, independent of that in Eq.(2). A stochastic model for a passive scala is also needed. We adopt the simple relaxation model of Dopazo (1975) which gives the evolution of a scalar as -tical d "= I dT w(( - 0, where Q,' = 2.0.

(8)

SOLUTION PROCEDURE We assume that the flow is statistically axisymimetric and stationary, allowing us to solve Eq.(9) by a parabolic Monte Carlo 5cheme, which marcimes muthe axial (x) direction. At each axial station, f is represented by a large number ef fluid particles, with each particle having a radial position, (r), a velocity vector, dissipation and scalar value. The number of fluid pa. ticles, N is taken to be 80,000 to yield small statiserror, while still providing reasonable CPU times. The width of the solution domain expands to encompass the evolving radial profiles, and the length is restricted to 100 jet radii

31-2-2

fromn the jet exit. Profiles

of mean quantities are obtained

by the method of cross-validated least squares cubic splines

2

------------------

(Pope and Gadh 1988). Forty eight equidistant basis funcIions are usedl to represent these splines. A variable step size, based onl the profile spreading rate is used, and the scheme

14-

1

10

takes approximately 150 steps to reach x/li = 100. Comiputa3090S, With eachI r11n taking6 tionS were performed on an IBNM approximately 20 minutes CPU time.

a

2

-2

INITIAL AND BOUNDARY CONDITIONS

-

At thle jet exit plane, mean profiles of velocity and mixture fraction are specified. in the same mianiier as those for the K flame in Nlasri and Pope (1990).

i

'1the rmns velocity profiles

-2

3

however, differ slightly from tlioqe of Nlasri and Pope (1990), being exponential fits to (data from linize (1975) for turbulent pipe flow and soudarv layers, The coxarianI(

(MI) iZ gis IIi with Ithe Constant C",, = 0.1. as

bx (iii) =(,(i)()P.

4

5

6

7

r/11 Figure 1. Initial profiles of

Uand]

suiggestedl bN Tenneckes and Lunde\v( 1972). Mixture fraction is gi\ en as 1.0 in t he iiet liaiie jet

erow. as InIN\LaSri

0.0 inthieco

=

0.0,33 in the pilot and

aid POIe( 1990), that the

mean turbiileint freqiienc\ Z is initiaiv u-nforni across, the

4U

~j1

/

3 5

Ceil-

J

ter of the flowv and we adopt the value of L' = 8.0 for the jet

--

t

aiid psilot flow, coniisteiiI with the value used bN Nlasri andL

Pope( 1990)

Hlowever we expsect thiat at a large radial dis-/j

tance from the axis of the jet. the value of w will be '/ero,

2

/

correspondiiig to iioii-turbulciit flow ('akiilatiiig ,'fioiii the dissipatioii aiid kiiietic eiiergy profiles of the boundary layer gix en In lTMnz( 197-5)), results in a p~rofile that dlecays as the distance fromn the surface increases. This piofile is used for the

'

.

profile in the coflow

The distribution of _/L is lognoiinal, with the variance of

2

1

2

culations. Boundary conditions are iiimplied by thle the coflow

7

3

li(Lw/w ) being uinity (consisteiit with o,, 1.0). Figures I and 2 how the iiiitial profiles used in the cal-ni

/ Figure 2 liiOtia profiles of u', v' and I,'

pirofiles. For r/J? > 7, the turbulent flow quantities are ap1)r(xiinatel) zero. with other qiianiituc approxiiiately constaiit

where p,(r) is the i-th sample profile at rdlial position r. TIhe 99 pe'rcent conifidence interval for these ;uiofiles canl be approxiinatcd by

e

STANDARD RESULT In t his section, we present a standard soluitioii that is coii-

3[

1

A)Jf~(

-

Figure 3 shows the crossflow profiles at x/H = 100 of: t lie

psared to other solutions in which the rmodel constants or initial (oniditioiis have been varied. We also estimate the accuracy

meaii axial velocity, (1: the miean relaxation rate c; andi the( rins of the fluctuating axial velocity, denoted -u, ?I'=

of tile solutioin,

Figure 4.

TIaking the model constants as given by Pope ( 1990a) and iiial conditions as described above, we perforimed M = 10 ulls. ideintical except for the randoin number seqiuenice. Fromn these riiis we evaluate a set of mean profiles, (p(r)), given by (p(r))

IAf

T p, (r),

shows the mean scalar profile

the scalar comiponeiit, denoted as

'

aiui the riiis Pof

at the saii-ic streainwise

location.

rhe size of the error bars iii both Figs, 3 anil .1shows that an accurate solution is being obtained, even for the highier order mometits. The exp~eriinental data of Masri ai(J IHilger

(12)

(1986) is also included as a comparison

31-2-3

A discussion of the

between experimntal awl] calculate(] firoflies is

(OmfpearisciI

defei red to thle end of this papet

8

i i

7

10

5.0, curresponding to (7x

1.0 and 2.2 respectively. Ini Fig. 6

sse show the tiean & and

profiles at x/R = 100. as well as' the

case dlata. rhe trend to a reductioni inipiofile widIth with increased values of the constant agrees with the trend olbscrvcd in Chen aiid Pope (1990). The profiles of it', (' and

,standard

r

show negligible change to variations of these constants.

1

20

5T 4

it~'x 2

0

0

'

-]

value~ of C, is also varied while holding Co~ constant, and the

j

results display negligible variaioi, froiii the standard result,, indicating that the imiportant constant in this group is ,. Pope (1991a) obtained the value of GCi = 0.0.1 by considering the ease of a turbulent boundary layer and adjusting Ciso the model yielded the von Kirriiin constant. Dowever a different 'ralue of C.i = 0.09 is required if tie model

2

0

is to he consistent with the k-c model. Figire 7 showvs the

%

-,

00 0 0

profiles at x/R = 100 for 0 .A = 0.090 aiI(] 0.0641. WVe observe that the magnitude of the & and profilc s increases

0/"andl '

2

1

with larger values of Q' The p~rofile oft:; shows the .'anie -;oif response, while the profiles of ?t'and ~'remain relati . Jy

-6of

r/R

7iia

Figure 3. Standard proiles of ('. and Open syinbols are C~ perii ut ,saldat a of MIasrl anid 3ilgel (1986). ________

!L 'ged ' 26

-

_____________24-

22

20

.'~-6

2j

04

112 a2

2

0

1

4

1

Fiur

--

2

5 Prfie

-

of

-

S-

an

C rsutn

)

fro art

10

1

of C,

r/Hi

Figure 4. Standard profiles of and ' Open symbols are experimental dlata of Masri and lBilger (1986)

VARI~ O-N OF MODEL CONSTANTS

The final tonstant varied i a'. Pope and Chien (1990) showed that experimental data inidicate that a' had a weal, Reynolds numbi er (dependlence. aiid usedl the value or' = 1.0. corresponding to a Reynolds number based( oii the Tlavloi scale of 110 . Figure S shows profiles of 0' and foi length 2 equal to 0.3 and 1.5. It is seen that an increase in a' re-

cq

The first constant varied is the mixing model parameter

C4. Figure 5 shows the and C'profiles at x/R = 100 for C = 1 and 4. It is seen that an increase in C( results in aii increase in the C profile, while the corresponding $' profile decreases. '[lie effect on, the U, it' atid iSvprofiles is neglegible. Next we vary Co amid C,. Pope and Chen (1990) showed that these two constants are approxirmately linearly dependent for a fixed value of (7' and integral time scale ratio of velocil 'y and Inuw.We use two -,ahies for these constants, Co = 2.0 and

sults in a taller, narrower profile for both (' and . r[le same response occurs for Ca,while u' and ' show little variatioin. The remaining constants, C.~2 and Cl,, were not varied. U.2 is obtained from grid turbulence results anmd can he (onsidered fixed and the value of Cj1 has been shiowni by Pope (1991a) to have a negligible effect on the solution.

31-2-4

\

J..

2

iJ

\\a

" -

'N'' 0

4.

',R.>

7'

-2

f

-t

...

-s+

I igme 6 ,11d C,

Profiles of I" and

.....

,...~

7

q

5 8

.

11

.....

-"

1

_+

12

0

resulting from vaiiation of ('U

1

,

7

IL. 10

0

......

-2-

7

1,

8

9

12

11

12

resulting fron; varlati of e2

7

F

,2

4

Figure S. Piofiles of 0r and

\

-

2

._ __

..

.

.

.

4

,

1

1-

12

-

r

of I" and Prof~les

Figure 7 I sam

aeBthIh

m-I1tMfo

r/ of resulting from variation it

n

CONDITIONS \-RITION.OF INITIAL Three

cl main

sliylvelocity profiles, s tipan iop ro fi le s ,

-

Figure 9

,

,

I.

of 11ofiles

I and

from "i,UhMlOi svliiig

r

,Of

and axial pilot veocty.

;

polliles Ithe to corresponds is that, It found s hat speciication latter (1990). Tt,losri Pope and other. the by used otfles, t l ce beteen negligible dill'elc x/l = 100, tiiele is t -c~;b i l n s i e t In lt l c h e in it ia l p ro fi h,o f tng s u g g es t i

Ihc den(ondition are varied: of initial asses I and the disthe turbulence quantities,

icl. -,L~vs 100 that e ( Iwo (w).T of profiles at x/Hl = 0 ad vary the inilnal profile we Figure 9 shx s tile IPinaly wlid density and a se prole, he he standard stoiciometric of t equal to half t result from the variation ' considered are g the 1;; mainltain the to lcmpih fi;, ' scpro in the pilot jet velocity hange corresponding c tol twif,; the tatnd,.rd qua J, e . !,, /lH at , -1,, .olll~~ .tJ.. the results D. i in the intial profile, chanige ra-Iohth same~ nitude of the value of Fsig ure 10 lower a se with p to incre ion be t wee n t he ro fi l . of ctileprofiles is seen tu = 100 sho w li t t vaeria t f o s fi le or , sunthe o r p f e h t o t s axial mst oation c again t h t resnd e xt e nd of shows the axial value , , a ridt o a le s se r e xt e n t

~~ ~ ~ ~~~~~~~

u',

and'

.

w ' are varied rm s pro files of u ', v' an d itia l vin e -dry he in T det U' which is an experimentally in tilefollowing way: p one nt com he r e d . T he t wo ot q ua nt ity i s no t c ngha m i d ne

i-

a l to it' r fo a n d ellu o f- u ' fo r o n e ru n , a re g ive n th e va l e

s

th at the va lue sven two varian ts. It an (be Ithe d ard case an d history. une st, al aino idvil quickly relax to an slt e we e th w, a re not va ried

T he in it ia l p ro fi le s o f

n

and

asu red e x p e rim en ta lly .

N

31-2-5

s

=,

-- --

1.0

.

,x20

,=s80 =

c

16.0

I

o 0

-I

,

1

25

30

40

55

63

70

8

9

1

1

4

r/R

x/lR Figure 10. Evolution of centerhne,

for -ariation in initial ,

Figure It.Profiles of "and

for C, = 0.065. Open symbols

are experimental data of Masri and Bilger (1986). The results of this section show that the sensitivity of the solution to changes in initial conitioins is small. The lack of

flame. This results in a large enthalpy excess at the pilot,

sensitivit

which cannot be modeled by a single scalar equilibrium ther-

to initial profiles of u',v',w'and , is reassuring

as often these quantities are difficult to specif

accurately.

mochemical model. In addition the experimental data do ex-

The variatio, of 0j profiles with densilt 3 variation is just a

tibit some local extinction for this flame, which is not treated

reflection of the effect of reaction on the flow, and as density is relatisel\ ea.sy to determine is of no cause for concern

by our chemical model I lie uncertainty in the value of C, suggests that the model for dissipation ploduction, S, ma" be at fault. Figure

CO)MPAR:SON WITH EXPERIMENT-L DATA

I shows experimental data plotted against profiles of 0 and generated by the model with Cli = 0.065 [he agreement with experinient is seen to be superior to that shown in 'gs.

While the agreement between the experimental data and computed profiles ii Figs 3 and I is diappomting, there are

3 and 4. Pope (1991a) indicated that the expression for S., was tentative, and the present results support this argument

st,eral ,nit igat ig factors to be (onsidered The ax'l location of "/1R= 100 was chosen to einphasive

tie differen( e between profiles obtained with lfferent iiodel constants and Initial Conditions. I'hie area of interest in this flow is at x/lH = 10. where chemical/turbulence mnteraction.

CONCLUSION

are ,ausing lon al extinction ( Masri and Pope 1990 ) At this location the agreement between our calculated profiles and the experimental data is good We should also emphasize

pdf model caii produce accurate solutions for the piloted nonprenixed turbulent methane flame problem. 3y varying the

that the ability of the model to duplicate experimental results

to the choice of a few constants

is no worse thamn that of other schemes.

restricted to changes ii the mean axial velocity and mean scalar profiles: rms quantities being relativ!y unaffected. The

The agreement of

ra!h-J,t(d profiles to experimental data of Nlasii and Pope (i*J9t,) .t x, R = 100 shows similar magnitudes of error as our esuilts. Th., results of Chen, Kollmann and Dibble (1989)

We hase demonstrated that the velocity -(issipation-scalar

model constants we have shown that the scheme is sensitive However this sensitivity is

effect on the model due to choice of initial conditions has been shown to be small, justifying the use of approximate

.h,,w rc-atively good agreement at x/r = 100, but exhibit a poor match closer to the jet. despite the benefit of finite-rate

initial profiles for certain quantities

thermochemistry employed in their scheme,

dissipation production needs to be adjusted, to improve the

There are also several differences between the model conditions and the experimental conditions that could cause such

performance of the model. Future work will focus oii improving the dissipation production term, and the inclusion of finite

a lack of agreement in results. In the experiment the pilot

rate theimochemistry.

A comparison of experini itai datd to solution profiles, suggests that the model for

is composed of acetylene. hydrogen arid air, made up so the atom balance is the s,,me as for a stoichiometric methane-air

31-2-6

ACKNOWLEDGEMENTS Tis

ivoi k was supported in part by the National S( mice

Fouiidation Grant CIIT-SSI 1655. during the ze's(a-.cl were

Computations conducted

1 ,erfoinied onl the Cornell National

Superconmputer Facility, which is supported in part by the National Science Fouindation1, New York State. Tfhe 113NI Corporat ion and members of the Corporate Rleseai chl Iit it

it

e.

REFERENCES CHlEN. .J )

IKOLLNIAN N. W auid 113BBLL. It XW.1989

..

1Pcm nical JRcport SA A'D89-8403 Combustion Research Facilit.

Sandia National Laboratories. Livermore, CA 94155

DOPAZO, C 19)75 Plips Fluids 18(2):389-394.

IIINZE, J. 0. 1975 Turbulcnicc NIcGraw hfill. Second edition 'dASH!, A. Ht.and BILGER. It NV 1986 'Jwenly-FI'vst Synmposiuu

(lntcmiational) on Combustion. 1511-1520

MASH!. A. IH DIBBLE. I{ XW.anid IILGEH, IH XX 19,,a C'ombustion and Flamc 71-24-5-266. 1988 MIASI,

A. Ht., DJIBBILE, It XX and IIILGER. IH WXl9881

C'ombustion and Flamn , 73.261-285 NIASH I. A IH..D)IBBLE, R. \X and lhILGEH C'ombustion and Flamf

.

I? XW1988(

74 267-25.

MIASH!. A It and~ POPE. S. B3 090 Comibustion and FMaom

.81 13-29 POPL. S. 13 1983 hoy Encigy Combust .SczM.1119-10

POPE. S. B3 1991a 1'hys Ilud,. in PReS, POPE.> S B31991 t, f r y hidSjp~ on~ Comiiiua

an(In liunlion ol)

on ii.ii press

POPE. S B1 and CHEN, Y 1, 1990 Pliys Iaidq A. 2(8). 11371119! P015 . ' Ht Md GAXI!

H. 1988 ('oiimun Statist

iiiiui/a

17(2) 319. 3763 'I ENNEKES

If and FUNIhEX) , J 1,I

19)72 A First ( oiisf in

lurbubiiner, XIII Press.

31-2-7

EIGlI'll SYMPOSIUMN ON

3 -

TURBULENT S1EAR FLOWS Technical University of Munich September 9-11, 1991

An Experimental Study of Laminar, Transitional, and Turbulent Jet Flames of Air-Diluted Methane R. S. Barlow*, S. H. StAimer* and R. W. Bilger** *Combustion Research Facility Sandia National Laboratories Livermore, California 94551 "Department of Mechanical Engineering The University of Sydney NSW 2006, Australia

ABSTRACT In previous studies the mass fractions of OH and H2 measured in turbulent methane jet flames near extinction were significantly higher than those measured in laminar flames and predicted by steady strained laminar flame calculations. The present work investigates frames over a range of Reynolds numbers (laminar, transitional, and turbulent) to determine when the measured chemical states begin to deviate from the laminar flame conditions. Simultaneous point measurements of temperature, major species concentration, and the hydroxyl radical concentration are obtained in nonpremixed jet flames of air-diluted methane. Results suggest a gradual increase in OH mass fractions through transition from laminar to turbulent conditions. Conditional pdfs show no further increase in OH mass fractions as the turbulent flame is pushed closer to extinction, so that the elevated OH mass fractions cannot be attributed to extinguishing and reigniting samples. Possible mechanisms for the observed effect of turbulence on chemical states in these flames are discussed, :NTRODUCTION In recent years, it has become increasingly recognized that finite rate chemical kinetics can have significant effects on the structure of turbulent nonpremixed flames. Modelers of turbulent flames, making use of expanding computational capabilities, have incorporated chemical submodels with increasing detail and complexity in attempts to accurately account for the effects of finite rate chemistry. Pope (1990) has reviewed recent progress in the computation of turbulent reacting flows. Some modelers, such as Haworth et al. (1989) and Rogg et al. (1989) have made use of libraries of strained laminar flame calculations based on detailed reaction mechanisms. An alternative approach taken by Peters & Kee (1987), Chen & Dibble (1991a), Chen (1991), Bilger et al. (1991), and others has been to develop reduced mechanisms, which include only three to five composite reactions that preserve the important chemical kinetic features of detailed reaction mechanisms. These reduced mechanisms are then incorporated into Monte Carlo or pdf simulations of turbulent flaimes, such as those by Chen (1991) and Chen et al. (1991). There has been some debate as to the applicability of the flamelet model to turbulent nonpremixed combustion, Peters (1988) argues that the flamelet approach is valid for many turbulent flames of practical impo'tance. Bilger (1989), on the other hand, argues that the width of reaction zones in turbulent flames is often broader than the smallest scales of turbulent mixing, such that reaction occurs in distributed zones that may be affected by internal turbulence structure. Previous experiments at Sandia (StAtner et al., 1990a,b and Barlow et al.,1990a, 1991) have shown that the measured concentrations of OH and H2 in turbulent flames of air-diluted methane, nitrogen-diluted methane, and undiluted methane are significantly higher than predicted by steady strained laminar flame calculations. These previous experiments were conducted using the piloted burner developed at Sydney University by Stirner & Bilger (1985), and all of the cases

considered were highly strained turbulent flames that were approaching the blowoff condition and exhibited local extinction. Subsequently, OH concer ,ation profiles were measured by Barlow & Collignon (1991) in laminar opposedflow (Tsuji-type) diffusion flames and in laminar jet flames using the same OH fluorescence excitation/detection strategy as in the turbulent flame experiments. Results showed good agreement between measurements and laminar flame calculations of the maximum OH concentration. The present experiment was conducted to determine where, within the regime between laminar low-strain-rate flames and turbulent flames near extinction, the measured mass fractions of OH begin to exceed the range of mass fractions predicted by steady strained laminar flame calculations. Four jet flames of air-diluted methane are investigated. The Reynolds number based on fuel nozzle exit conditions is varied from Re=1,770 (laminar flame, low strain rate) to Re=20,700 (highly strained turbulent flame exhibiting local extinction). Simultaneous point measurements of major species concentrations, temperature, and the hydroxyl radical concentration are obtained using a combination of spontaneous Raman scattering, Rayleigh scattering, and laser-induced fluorescence (LIF). Results suggest that OH mass fractions increase gradually through transition. High OH mass fractions are attained at a Reynolds number well below that which causes local extinction, indicating that the 'super-flamelet' OH is not associated with the local extinction/reignition process and is due to changes in the structure of the reaction zone caused by turbulence. Possible mechanisms for the observed influence of turbulence on the distribution of chemical states in these methane flames are discussed below. EXPERIMENTAL METHODS Facility and Laser Diagnostics The flow facility, piloted burner, and diagnostic techniques used here have been described in detail by Barlow et al. (1989, 1990b) and StArner et al. (1990a,b). The axisymmetric burner is mounted just above the contraction section of a vertical wind tunnel that supplies coflowing zir at low turbulence intensity. The wind tunnel assembly, including the burner, is mounted on a stepper-motor-controlled threedimensional traversing mechanism, which allows the flame to be positioned relative to the fixed optical diagnostic system. The piloted burner consists of a central fuel tube of diameter D=7.2 mm surrounded by a premixed pilot that extends to an 18 mm diameter. The pilot flame serves to anchor the main jet flame in the transitional and turbulent cases, preventing liftoff from the nozzle. Measurements are obtained above the pilot flame at a distance of 20 nozzle diameters from the burner. Spontaneous Raman scattering is used to measure the concentrations of N2, 02, CH 4 , CO, C0 2 , H2, and H2 0.

31-3-1

Temperature is determined from the Rayleigh scattering signal and a species-weighted Rayleigh scattenng cross section based on the Raman measurements. Both the Raman and Rayleigh measurements are performed using the beam from the Combustion Research Facility's flashlamp-pumped dye laser (532 nm, 2.5 pS, 1 J/pulse). A spherical return mirror doubles the laser energy ir. the probe volume. The laser probe volume is approximately 0.7 mm in diameter by 1.0 mm in length along the laser beam. OH concentration measurements are obtained using linear laser-induced fluorescence. The doubled output of a Nd:YAG-pumped dye laser is tuned to the S21(8) (0,0) transition of OH, and this ultraviolet laser beam is combined onto the axis of the Raman laser beam using a dichroic mirror. The LIF probe volume coincides with the Raman/Rayleigh probe volume and has dimensions of 0.5x0.7xl.0 mm. (The laser beam has an elliptical cross section.) The smallest dimension is in the radial direction relative to the axis of the jet flames. The time delay of 3.5 ts between the LIF and Raman laser pulses is small compared to the fluid-dynamic and chemical time scales in the flames of interest. Thus, the measurements may be treated as simultaneous. OH measurements are referenced to the conditions over a calibration burner, where the OH concentration has been measured by absorption. The Raman/Rayleigh data for temperature and major species concentrations can be used, as described by Barlow et al. (1991), to correct the linear OH fluorescence signals for shot-to-shot variations in the collisional quenching rate and the ground state population fraction. The effect of these quenching and population corrections on the measured OH mass fractions in methane-air flames at atmospheric pressure has been analyzed by Barlow & Colligr.on (1991), based on collisional quenching cross sections provided by Garland and Crosley (1988). In laminar flames at low and intermediate strain rates the corrections for quenching and Boltzmann fraction tend to offset one another, such that the net change in OH mass fraction is only a few percent. As the strain rate approaches the extinction limit, the calculated quenching rate increases, and the net effect of the corrections on the OH mass fraction approaches 20 percent. However, recent measurements by Jeffries et al. (1988) and modeling of collisional quenching cross sections by Paul et al. (1991) suggest that the Garland & Crosley model overpredicts the temperature dependence of the cross sections at flame temperatures. Thus, the net corrections for quenching and Boltzmann fraction are expected to be small even in highly strained methane flames. Because there is uncertainty due to shot noise in the experimental measurements of temperature and major species concentrations, the application of corrections to the OH fluorescence data in these methane flames increases the standard deviation in the OH mass fractions, even when the average mass fraction is unchanged. For these reasons, the OH mass fractions presented in this paper are uncorrected values Inclusion of the quenching and population corrections would increase the OH mass fractions in the turbulent flai,.es by a few percent, pushing them further from the laminar flame results. Therefore, application of the corrections would not alter the conclusions of this paper. Flow Conditions and Flame Descriptions flow conditions Four flaidic are investigated, with summarized in Table 1. In all cases the central jet fluid is a mixture of 25 percent CH4 and 75 percent air, by volume. Airdiluted methane has two useful characteristics from an experimental perspective. First, it has a higher extinction limit than undiluted methane or nitrogen-diluted methane. Consequently, laminar flames are easier to stabilize, and the piloted flames can be pushed to relatively high Reynolds numbers before local extinction and blowoff occur. This means that changes in the measured chemical states due to the influence of turbulence on reaction zone structure can be

separated from changes resulting from the extinction-reignition process. The second advantage of air-diluted methane is that flames of this mixture are essentially nonsooting. Therefore, energy loss due to particle incandescence is minimized, and fluorescence interference from soot precursors, as described by Masri et al. (1987) is reduced. Table 1 Flow Conditions for the Air-Diluted Methane Flames Re=U D/v Ucf (m/s) Up (m/s) Flame UI (m/s) 1,770 I 0.8 0.0 A B 12.3 0.75 1.9 4,420 I C 24.5 1.5 3.7 8,830 I D 57.3 3.0 8.4 20,700J Uj, Up, and Ucf are unbtrnt jet, pilot, and coflow velocities. The four flames listed in Table 1 are designated A, B, C and D in order of increasing Reynolds number. Flame A is laminar (Re=l,770) with unsteadiness due to buoyancyinduced instability apparent only near the flame tip. Flame B (Re=4,420) is transitional. It is visibly unsteady, and the reaction zone moves radially (wavers or flaps). However, this flame is not fully turbulent. The reaction zone in flame B is not strongly distorted or curved, and :here is no indication of local extinction in the data. Flame C is turbulent, with a modest Reynolds number of 8,830. Local extinction, which is indicated in the measurements by low temperatures and near zero OH mass fractions within the reactive range of mixture fraction, occur; for a szdll fraction of the samples in flame C. Local extinction occurs when local rates of turbulent mixing or scalar dissipation exceed the rates of the chemical reactions that produce radicals and thermal energy. Flame D (Re=20,700) displays a significant probability of local extinction, is audibly unstable, and has an overall appearance similar to the turbulent flames near extinction reported by St/rner et al. (1990a,b) and Barlow et al. (1991). As shown in Table 1, the velocities of the coflowing air and the pilot gases (when used) are scaled with the velocity of the central jet. This is done to maintain flow similarity and to keep the contribution of the pilot flame to the total energy release up to the measurement location approximately constant. (Note that he pilot bums a mixture of C2H2, H2 , C02, N2 and air that produces the same calculated equilibrium composition and adiabatic flame temperature as a stoichiometrc mixture of methane and air.) In the turbulent flames C and D the pilot is estimated to contribute 12 to 15 percent of the integrated energy release up to the measurement location. Therefore, while the pilot has the important effect of stabilizing the transitional and turbulent flames, we do not expect the pilot to contribute to the elevated OH mass fractions discussed in the following section. Flame A was operated without a pilot because the premixed gas flow rate could not be scaled back sufficiently without causing flashback. OH fluorescence profiles were measured in two piloted laminar flames to determine whether the OH concentration at the measurement location was affected by the pilot. One of these laminar flames had the same pilot flow rate as flame B, and the other had a small hydrogen diffusion flame as a pilot. No significant differences in the peak OH fluorescence signals were observed for these three laminar flames, and the complete Raman/Rayleigh/LIF measurements are reported only for the laminar flame with no pilot. RESULTS AND DISCUSSION Measurements were obtained at a fixed axial location 14.4 cm or 20 nozzle diameters above the burner (x/D=20) at the radial location in each flame corresponding to the maximum average OH fluorescence signal. Results for measured temperatures and species mass fractions are plotted vs. the mixture fraction fin Figs. 1, 2, 3, and 4 for flames A, B, C, and D, respectively. Each scatter plot for flame A includes 800 points, while 2000 points are plotted for flames B, C, and D.

31-3-2

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Fig. 1.

(Re=l1,770). Curves show results of steady larmar flame calculations for values of the strain parameter of. t a 50 ,1 . , 200 s- ( . ),400s - ( and 800 s "1 ( .............. ). 2500

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Fig. 2. Scatter plots of temperature and species mass fractions vs. the mixture fraction f in the transitional flame B (Re=4,420). Curves show results of steady laminar flame calculations for values of the strain parameter as in Fig. 1.

31-3-3

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9.

EIGIITII SYMIPOSIU[I!

ON

" UIlBI.INT ,SI lEAR FI,O\VS T [Iniversity iechmcil of M11ch

1-4

September 9-11. 1991

VORTEX SHEDDING IN A DRAG REDUCING SURFACTANT SOLUTION Joachim Dohmann Lehistuhl Energieprozebtechnik Fachbereich Chemietechnik. Universitat Dortmund Postfach 500500, W-4600 Dortmund, Germany

o 30

ABSTRACT

1

In turbulent shear flows some cationic surfactant

C 1 4TSai+ NaBr,

solutions show a number of peculiar effects In this experimental investigation the wakes past circular cylinders were studied by using Laser-Dopplervelocimetry based conditional sampling techniques and flow-visualization Interesting details of the Kdrmn- vortex- street and the occurence of vortex shedding were obtained

25

A Roshko (1953)

water / /

/

INTRODUCTION

0 d . 2 mm mm A 0 .d'd,2 4mm

Some aqueous solutions of cationic surfactant solutions

-

show a peculiar behaviour in turbulent shear flows

. 5

Most famous is the effect of drag reduction in turbulent pipe flows (Toms effect) which could be observed e g in rough pipes [I] Other investigations 121 indicate a dramatic change in the mechanism of turbulence in the presence of these surfactant additives Manly the production the developement and the decay of coherent structures a influenced by the addiives The observed effects cannot be understood if only simple rheological propeit.es are made rcsponpible for them The solutions show a steep increase (factor 4) in vicosity if a critical shear strain is exceeded (3) Therefore an experimental investigation of a simple system of vortices was performed the study of the Karmln vortex street under the influence of a drag reducing additive (24 wM aqueous solution of tetradecyltrimethylammoium salicylate with sodum bromide C 4TASaINaBr)

R, Re =iu..d

02

2

5

2

5

Fig 1 The Strouhal number as a function of the Reynolds number obtained from Fast-Fourier-analysis The cttical Re-number is shifte-' An increased Snumber indicates a reduced size of tht ,-ortices Another simple geomedty parameter is the lateral spacing h/l as calculated in the theory of v Kcrma'n At last parameters are interesting describing the intensity of the coherent structures which is related to the input into the cascade of the turbulence decay EXPERIMENTAL SETUP

In the Newtonian case the Karman vortex street is a well known system of two rows of altenated counterrotating vorices it. the wake of a circular cylinder It can be dscrrbed by a set of non- dimensional numbers The rate of vortex shedding is represented by the Strouhal number S (S = f d/U, w'ere f is the shedding frequency. a the cylinder diameter and U the mean flow velocity), Tne 5trouhal number is related to the distance between two vortices Vortex shedding occurs if a critical Reynolds nu,. ber of 30 is exceeded Ifthe Reynolds number is above the value of 300 the Strouhal number becomes a constant (S = 0 21) [4) A collection of these data is shown lit Fig I as an envelope

In the present investigation different cylinders with diameters of 1mm, 2mm and 4mm were installed in a test channel with a mean-flow velc' ,ity between 3 m/s and 125 m/s, which led to Reynolds numbers in the range between 300 and 4800 The upstream turbulence level was of about 1%in the Newtonian case and about 4% if the surfactant solution was used The flow fplid itn the tesisection was examined by using two Laser-Doppler-velocimeters (LDV) which are operated simultaneously with two trackers and additional with two burst--pectrum analyzers Different methods were used to describe the Karman vortex street at different flow speeds and different distance- downstream the cylnde The vortex street was visu-!ized (d= 5mm) by dye injection through a hole at the front stagnation point

I..1-1

EXPERIMENTAL RESULTS To determine the shedding frequency the Fast-Fourieranalysis of the velocity signal was used At 'ell choosen positions the frequency spectra show very clear peaks In the Newtonian case the obtained Strouhal numbers show good agreement with other published data (see fig 1)A surprising effect shows the surfac'ant solution The critical Reynolds number of about 50 is shifted up to a value of about 3500 This cannot be explained only with an increased viscosity measured by Ohlendorf et al 131 Strouhal numbers of S= 29 (at Re=4800) could be detected, not having reached a saturation value A simple influence of an increased viscorty would only affect the critical Re-number but not the Strouhal number S

' ,'Fig 2 Visualization of the cylinder wake Left The Krm~n vortex street in the water flow at high Reynolds number (Re = 5000). xiSght the wake in the surfactant solution at similiar conditions In both cases the same dye injection rate was choosen to demonstrate not on!y the changed turbulent diffusion but the molecular diffusion as well The molecular structure of the aqueous surfactant solution can be made responsible for the observed effect in turbulent shear flows If a critical micelle concentration is ex,:'eded slender rodhke micelles are formed from the monomeres In turbulent shear flows the micelles orientate iro flow direction and the fluid becomes anisotropic This behaviour was determined from small angle neutron scattering [6] From own thermodynamical experiments it can ne concluded that the micelle -monomere equilibrium depends from the shear stran Under deformation conditions the equilibrium is dropped to lower values of the critical micelle concentration according to an increased length scale of 'he anisometric micelles or an increased volume fraction of this phase The shear inducLd state of the material as discussed in lierature (eg [31) could be confirmed by measurements of the refractive index and optical turbtity indicating a shear induced phase transition Under shear strain the material becomes more dnisotropic than expected only from orientation effects of the micelles

Furthermore two conditional sampling l.chniques weie used to obtain the momentary spatial ,elocity fcld ,U(x)> The velocity signal of a reference LDV was examined for characteristics of the vortices In the first case this was done by using the window- averagegradient algorithm described by Antonia et al [5i The second method used was a hardware application consisting of a signal conditioner and a differentiator circuit It was installed to shorten the detection scheme If a vortex was detected the measurement of the second LDV was triggered By variation of the position of this second LDV the momentary" conditional velocity field was obtained This leads to the knowledge of most of the interesting details of the flow The geometrical parameters (size and spacing) of the voitex street were determined from the spatial distribution of the vorticity which could be calculated from the velocity field In the same way the local contributions to the kinetic energy density or the circulation of the vortices could be calculated

The peculiarities ooserved in turbulent shear flows of the aqueous surfactant solutions are related to the occurence of this shear induced phase transitions

The obtained data indicate some important and surprising changes in thf mechanism of turbulence production in the pr.ance of surfactant additives The flow field becomes anisotropic with reduced turbulent motions in lateral directions The kinet'c energy density of the vortices is reduced even if there are only small changes in the strength of voticity Fig 2 gives an illustrative impression of the cylinder wake

REFERENCES [1 Thiel, H. Widerstandsverminderung be tu~bulenten Stromungen in kunstlich rauhen Rohren 1990 Dissertation Universitat Dortmund [2] Riediger. S. Untersuchung der Turbulenzstruktur in einer ebenen Mischungsschicht und hinter einer Kanalerweiterung be. '4-'r Stromung zweier nicht newtonscher FlussigkeiteF-m 1989 Dissertation Universitat Dortmund (31 Ohlendorf, D et at. Rheoiogica Acta. vol 25. 1986. pp 468.486 [4] Roshko. A. NACA techr,,cal note 2913. 1Q53 (5] Antonia. R A et al. Proc 6th symposium on turbulent shear flows. 1987, Toulouse. France [6] Bewersdoff. H W et al. Physica B vol 156 & 157 '89, pp 508-61! ACKNOWLEDGEMENT The assistance of M F Hibberd during the flowvisualization experiments is greatfully acknowledged

1-4-2

EI(;il Ill SVI'1POSIUlN1 ON TURBI

ENT SI lEiR FLOWS

J

rechicel University of Munich September 9-11, 191 PRESSURE DRAG ON OBSTACLES IN A TURBULENT BOUNDARY LAYER

Stefan Einers Son der forschungs beretch 210 Institut fur Meteorologie und Kliiaforschuing Kernforschungszeiitruin Karlsruhe / riniversitit Karlsruhe Postfach 36,10, W-7500 Karlsruhe 1 Gerniarix

A BS L'RC('

P~ressure drag oil obstacles iii a turbulent boundary layer of a %iscous, and stratified flow conies due to deformation of the tho\% oer obstacles and generation of turbulence inear obstacles, due to the generation of internal waves, and clue to blocking of heavier air miasses onl the upstreami side of the obstacle Tlhis leads to aIdecomposition of pressure drag iinto forii drag, Nkave drag, arid hydrostatic drag, respectively The decoripositror is a prerequisite for paranieterimig the drag lII this paper atscale analysis of tireequation of motion a ill ,t rodrece nondiriiensioiral riirrbers a lich govern tilie drag Out put fromt a numerical miodel a ill show the dependence of le draLg onl these nunilbers T[le results for formn drag will be tuparel to thou , oser rough plate, and riuipliatmons for tile /at ion of pm u nre ((rag IIiI larier- scale mlod el" wil paralleri h' ,dhre-e'd

\l

'~

F XNALX''~lSdrutg'),

d~e~ue(rag 1,,.r unit lenigthI of an obstacle (2L ) is de-

~ ~~~

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'tnit rcl %oluri cont airning the obstau'lt (21)-flow I1/i

1I

2 466(vtU 2 /L)' 1/H/L 2 13 2,66(vtU /L) 1L 2 p* 0 IU 11L

UHIL ziu L1HIL v* it/=(1 JIlL IV* p

it

-

v

=

-v

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It"

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.

-

where t le asterisks denote nondimensional values which should be of order uiiity if the scaling is correct, yields for the pressure drag coefficient Cl$' IV of a siiiglc obstacle c H2 L

rid

12 -

02~t'i'~ 3

I f

21

f

e

n-ff]'rd)()

(3

,

Tlhe termn with e 12( u-) is present Only if we have gzravity waves (for stable s-tratification _7 , 2 zthle usual Froude num iiber) awhichi t rainsport iioiieiitiimi vcrtically ('gravrity wave lie errii with tile tuirblent Rleynoldls num nber Re, = 41L describes 'viscous forrii drag' due to deforrmation of the l1,001s flow, anid lie tern wrth thle Rossby numruber Ro. ffUdescribes a redhistribantloi of miomniitumu front the I'- to the ~~i uopo tiort (' iertinal wave drag') F-or neutral stability and ri~) o rotatiorn (3) reduces, to

X )

8 [ 2 Ref/3

h (x) is the urugrapl., L is half of the a idth of tire obstacleInldgtienfuceobokngodr rmassy air1 p(x. z) is the des iatr'rr uf thle pressure frotr a basic statc alnicudmrigmtere riere of b"(ltokigh colistnderi mse b %ilut obstade a rueruibrF TF'(atog r 5 i o ndpn 5 ('nt froint (Y, it indicates a difiereit. physical process), tire For stationiarity, thle horizontal equaliori of -mrotron (inteiritlnitce of a rough surface iby a surface Rossby numrrber Rb, igled ring horizontal advectiori, 4rated ,ker thc 21) sline 71.7(nrurmerical results sioa'ed that for Ho, beic w a cert aini lii rzo nt ii tur bulierrt fluiixes, mid pulre sirrface frictiron ter iris) threshold value, tire drag be ai larger with clecrec.,smg Ho, roads Ibis eflect cannot be separated frontr viscous formi drag by / ~~1 .r ffsiale analysts, but only by mnmerical experiments ), anid tire dx ii~ 'J 2 in rfhueiice of tire (detity of obstacles by A _s1 (it is till' numiibler of obstacles iii L,), we have Iher u, v, arid tVare ve'locity dleviatiorns fronti tire basic state d ue to tire presence of tile obistacle, resolved b) tile numerical H1 (4ai) ('isf I( 0 " He,, Ro., Io,, Fri,, A) imodel lised later, z4' arid zu are siibgrrdscale fluctuations rn thn numerical mrodel 'Fire Integrals over uw, arid in'to' are

Jf

J~

Scaling the varrables arthr a basre stante flow velocity scale IU, a turbulent velocity scale 2 166(vU'l L)'I' (I,, is tuirbiulent

viscosity set to 10 nit

irntrodhuced iii analogy to mrolecular v'iscosity althrough it is a.ir internal variable of tire boundary layer), a buoyant velocity scale ti'/21 (a - N 2 ). arid tire height and length scales of the obstacle 11 arid L (EMEIS, I

I

~-

(4b), R2e,, Ro,, A) Keeping U arid v, fixed, ! Ict, arid A describe tire dcforIlationi of thle viscous flow ('viscous formi dlrag'), Ro, describes tire additional produlctioni of turbulence at a rough surface of tire obstacles ('run ro-trirbulent forim drag'), arid L, CiV

ad ditionrally

M-

rr t rol ' flow 'i'jarLtioii ('miacro-t iirbulent forii

Pressure

Wae" drag

Micro-turbuient form drag srf. roughees

Mar.-turbulent form drag flea 6eParation

drag') The distinction between iniicro-turbulent and macroturbulent form drag is artificial and caused only by the grid resolution of the numerical model uied to simulate the drag or by the design of a field experimeint FigLre I shows schemiatically the different parts of pressure drag

Figure 1

over a rough plate corresponds to a turbulent flow with flow separation over obstacles The dependence of viscous form drag on Ret72 1 3 for small Ll (no flow separation) shows that both viscous and inertial forces play a role For larger further numerical have shown that viscous form drag more dependsexperiments on Re- 1 / 2 showing that we approach flow Iseparation (turbulent flow over a rough plate)

distinction betwcen friction over rough surfaces and formThe drag ove: obstacles is again artificial and is due to the 'limited resolution of a numerical grid or an experimental netNumerical simulations were carried out with a two-dimensional~iwork which capnot resolve the single roughness elements of mesoscale, prognostic, non-hydrostatic, priiutive-equation gridthe rough surface For grid sizes which on the other hand canpoint model (DORWARTIt, 1986), with a turbulence closure ,not resolve mesosiale obstacles, form drag is part of surface using a two-equation model for turbulent viscosity It is iterIfriction and can be parameterized via the roughness lengthated until the solution becomes stationary oncept The numerical results confirm (3a) and (4b) We obtain This leads to the introduction of an effective roughness by empirical evaluation of the numerical data for form drag length for parameterizing form drag of mesoscale orographic features in larger-scale models As hydrostatic drag and wave Ro, 6 082 .1 C )1 (2 5-( 3 + 0 15. - )(I - 3 5(A - A)) drag are no boundary layer phienoriena they cannot be ir (5) chided in an effective roughness length-concept but must be NUMERICAL SIMULATION'

ijth Ru, - 5 10' jor io, < Ro, ihis additive term is not lt aA, = 02ro tsdtvtrvertical ~-resent) and ,\, = 0 02

Iparamcterized by effective drag coefficients and respective momentum source distributions

2 5 ( 11 )217 in (5) instead of (M)2 in (3a) includes the effects of flow separation The difference is the macro-turbulent form drag The term containing Ro, represents the microturbulent form drag. The factor 3 in front of Ree"2/3 shows that the scaling of the viscous form drag in (3) was not conpletely correct pleel crrctwiderstands The effect of obstacle density has been assumed to a firsto-der approximation to be independent of the three is different to likel departs of form drag although flow separation isKaserstr pend considerably oi A

REFERENCES

Dorwarth, G., 1986:

Numenrische Berechnung des Druck-

typischer Gelindeformen Ber Inst Mew erst andt rsr h e or6, Ber pp. Meaero lo 0, unid 52 KLtnaforschunmg, ipA vailable frommmnaf lumstituit fiiarlsr r Meteorologie 12, W-7500 Karlsruhe 1, Germany) Emeis, S., 1990:,

Pressure Drag of Obstacles in the At-

mospheric Boundary Layer 461-476

COMPARISON TO FLOW OVER1 ROUGH PLATES Pi,ca be comard *, Viscous form drav nr,orrti,nnl flow ressteice over rough plates proportional to Re (Reynolds number formed with molecular viscosity) For laminar flows dominated by viscous forces the resistence scales with Re -i, for laminar flows dominated by inertial forces with Re - 1/2 , and for turbulknt flows with Re - ' / ' (PRANDTL et al ,1984) The analogy we found is that a laminar flow over a rough plate corresponds to a turbulent boundary layer flow without flow separation over niesoscale obstacles and a turbulent flow

1-5-2

J Appl

Meteorol,

29,

Prandtl, L. et al, 1984: Fuhrer durch (lie Stromungslehre Vieweg, Braunschweig, 622 pp

ACKNOWLEDGEMENT

The Sonderforschungsbereich210 is sponsored by the Deutsche Forschungsememnschaft

1-6

TURBULENT SHEAR FLOWS Technical University of Munich September 9-1I, 1991 SIMULATION OF TURBULENT VELOCITY PROFILES APPEARING ON ROTATING AND STATIONARY PARTS OF SUBSONIC, COMPRESSORS AND TURBINES J. K. Kaldellis The Institute for the Development and Management of Natural Resources, University of Piraeus. Pontou 58, Hellinico Athens, Greece 16777

ABSTRACT The successful analysis of tileviscous flow field through any type of turbomachine is based on the use of a closed analytical formulation in the primary direction. An extended and improved form of the classical turbulent shear layei theory is developed to take into account the c implex flow pattern through rotating and stationary parts of real turbomachines. 'lhe resulting model is successfully applied for a large variety of subsonic compressors anc turbines, N(IMNCLATURE F Rer WiS 6shear r]

THEOREITICAL ANALYSIS Among the numerous analytical velocity laws the most commonly used is the one based on the law of the wall and the law of the wake, proposed by D. Coles in his early work (Coles D.,1956a & 1956b). The proposed by D. Coles standard equation for the description of the velocity profiles combines the two previously mentionned similarity Jaws and is based on a detailed analysis of an enormous volume of experimental data for a large variety of f %o-dimensional turbulent shear flows In the present work, the appropriate modifications to the classical turbulent shear layer theory in order to take into account the complexity of the flowpattern through the configurations existing in real machines, are briefly described. For this purpose the longitudinal velocity profile (i.e. in the primary direction) can be expressed (see also Kuhn and Nielsen, 1973) as:

friction parameter (--uT/W, e Reynolds number ( Wrei,.S/v) analytical velocity distribution longitudinal velocity layer thickness non-dimensional (by 8) normal distance Cole's wake paramet er

W1/W..-F{2.439"1n(1+ijRe8'jF1 +5.1 - (3.391tRe6 IF+5.1).exp(-0.371Re.IF

INTRODTION The successful analysis of the viscous flow field through any type of turbomachine using the circumferentially averaged Navier-Stokes equations (see for example Ka)dellis et al., 1989, Kaldellis et al., 1990, Ktenidis and Kaldellis, 1991, Kaldellis, 1991), is based on two dilferent approaches for the flow directions, tle peripheral primary and respectively. In the peripheral direction, the velocity component is computed using the corresponding meridional vorticity distribution (Kaldellis and Ktenidis, 1990), resulting from the solution of the complete form of the meridional vorticity transport equation, written in a partial differentia' form. On the other hand, in the primary direction, i.e. the direction of the external (inviscid) flow field, al integral formulation is adopted. Foi this purpose a non-Inear system of integral equations, including the meridional momentum and the total kinetic energy equation, is used in combination with a two-parameter veloity'profile laanly in order to describe the char -teristics of the existing turbulent endwall shear layers. More precisely, tilesolut ion of the non-linear system of integral equations plovides the necessary shear layer parameters to be u,ed by the analytical low in order to reproduce the real flow velocity distributions. Special attention is also payed to estimate the correct external-primary flow directionitilied as the basis for the representation of the velocity profiles. For this rcason, the necessary communication between the external and the shear flow field is established via a fast viscous inviscid interaction scheme ,(haldellis et al. 1990).

or in a more generalized form as:,

A detailed parametrical study is carried out by varying one of the shear layer parameters at a time in order to demonstrate its influence on the shape of the corresponding velocity profile. Only some very characteristic results are given here (see figures [1],[2]) since this study is beyond tilescope of the present work. However it is interesting to mention that the choice of the appropriate primitive parameters, to be used for the representation of the corresponding velocity profile, may be sometimes crucial These parameters (e.g. the shear layer thickness and tilefriction velocity at the wall) must lulfil the additional physical constraint to take 'normal" values both for strongly decelerated and highly accelerated shear flows. Accordingly. the classical two-dimensional theory must be extended to take into accouvt the extreme pressure gradients (figure [31), the strong flow deceleration or acceleration through the bindings, and the relative EO

100000.

LONGITUOIN;L VELOCITTPROFILES -

-I*

*.

31kIT11"kII T

I

,.

--

(2)

W/W 1 r=f(i,Re,F,ff)

while the correspoding friction law (see also fig. [lj and Nash J., 1965) can be written as: =0.205/F-O.5"in(I+Ro9"[ F )-1 0455 F (3) +0.205.(3.39.Re8 .IFI+b.1)exp(-0.37.Re*lFI)

PECj *F- 30

.flTdfi

(I)

+2.439-ft-[1-cos(nT-)])

E v'

f.lFLl,

iii~~~

t,it Pitt

,,1D

NI

VELOC[ITY(Wl/sl e

l

*.M

onl

lo

O'nl

'.0

1.8i$

iI,,, XNSIONP.'EkLOCT P"A'VILES (W.I.11 figure 1:Presentation of thereltiOI fzfjIle§ according the eqation (1) for -1O0.1-i f 01OO,

Figure2: Pirulerici1 poreseolihoo.,of toe Veto'Ityprofiles variousiofe lre

i~~,P s a functionof F.

1-6-1

f:igure 3: Schemaic representationl of /sogtlu-

dana/ ve/ocily grofi/es rith pressure gradient.

motion of the successive blade rows. For the velocity component Ws, expressed in the relative to the blades of the machine system of coordinates,' the .xpression y MllorG.(966)maybe propsed proose byMelorG.(966 my b ued:to usd:choice (4) W(01)

realistically represented, for the hub region. rsls ncmaio ncmaio The quality of the theoretical rsls the experimental data, is greatly improved by the of teapropriate primitive shear layer parameters. Only under this conidition is almost always =W~.t(~) -fWY~f,,-W~o1)]possible to obtain the correspondin*eoiypoie uising the results from the solu Ion of the above CAI('UTION RESULIS - aCul11SIONS mentioned non-linear system of integral equations. If Thie results from the analysis of a large variety of this is not taken into account, severe problems may appear. are systematically configurations industrial investigated, in order to check the limits and the Smaiig ecnsyta codn oteaa i ral tetacasesgt the implysisw wievret ofn ofmarig applicability of the simple similarity theory based ral tetrbulentshea fmlowsa classicf tha wdiate on) the fundamental physical laws. Fo this purpose the uffientshaccralo thea wthe sia beaextenedtoie longitudinal velocity distributions, concerning beetne odsrb ihsfiin cuayth typical selected stations, from several test cases velocity profiles appearing through various industrial including a highly loaded compressor cascade wit n without tip clearance, two hig turning axial turbine cofgrtns if heppsdmdfcaosad improvements are taken into account. Of course, for cascades, a highly accelerated turbine stator, an is needed, investigation cases additional specialtip-clearance someinclude impeller along with a modern axial inutilradial ae en wall ha effects, to industrial ~~-. fan, are carefully studied. merging and the existence of strong shock waves. 'Themajority of the results obtained show that the REFER~IES main physical laws, concernin* the turbilent shear flow, are verified for the large variety of the ~Ir.Foluiecs ,Ila T~Lo h aei h ubln ona ae' ehe26teTublet.ufdjLa Vol'D191o-2 I0h investigated geometries.. The slightly decelerated, ar 'Tue La f teFl n ubln SerPo', 2. luid,tech., highly accelerated and jet-like velocity profiles, Vtitcis trunSer r,101 era Fir FBrcku, GrCenshii appearinf at the specific inlet-exit 2lanes of a moderate y loaded (fig. [4]) and a high turning .7.idellis I., hu8rlian A FlciettHf., tapailio, 11, 1. (fig.[5) axial turbine cascades, covering a wide SendrFb Caclto thdfrOetgeAilrnsnclo Axa ti Ai Pper Foe nter Scoarysor culigSok range of Reynolds (S0O.011eaS23O,0OO.) and Mach Pipe Inerctp, 2-6 r. of Turboinry Vlo.12 aso AnldigSh Co-Gt-21 Me 10 71) numbers, are successfully reproduced (0.u5 V. IIID,n pp y chan5e4and Fr.rosails lod 89UlaldesoJ.,IteF bthe proposed theoretical model. Unusual values of n ECoa ie 'nr an RaIal Iedioni I,g pdoxil SecondryLoss Pr fozmshapefactor H112at the exit of the stator & ScnayLse rdcini ihSedA~ n da oitsosW of an Industrial axial turbine (fig. [61) are closely paper 9o-01-229.prsrsIE related to the existence of typical jet- Iike turbulent let Like Profiles S llellis I 'Iteaidisf IN1 'Nigh Turn Lisit and shear layers with very small or even negative values Scoodarl YlorField Of/Pral turbines', I4lE Paper YO-OT-izi. int see also thickness, lacement is ridng ofcor~e te of te corespndin d 8l990" 6 laldellis I., utrato, P., Itesidis P., lyf, 'Effects of the Tip Kaldellis andKteis Clearance Flot Field onthe Secondary Losses', ASIR Paper 9bG-S8 4. Di'e velocity profiles, typical for a strongly 7. llellis I., III/, 'Paratetrical Investigation of the Interaction Settceen decelerated shear layer throu',;h a highly loaded Turbulent Shear Layers anid Shock Faves, Including Separation', Initernaitional compressor cascade are also sat-.sfactori ly described Conference onthe Analysis of Therial aninergy Systeis, ATIENS' 91, Athens, for cases excluding clearance (see for example fig Greece., I Iteoidis P., laldellis J., 1111,'1-D11isyssetric Viscous Internal Floes [7]). However, this is not the case when remarkable

-eho.dfry

Ont Rotating andStationary Surface soTurbotachines', ASIR Paper Yi-G1-lY6 9.yeru, A.114 Paer IN 17.1PedcinofTruletSpaae Budr /9. Ilter 0.1., 1166,'TheEffects of Pressure Gradients onTurbulent Floe near a Skooth Fall' Jr of Fluid lechanics, Vol.24,pp. W5-274, /1, lash J.F.. IYS, 'Turbulent Bloundary Layer Behaviour andthe Auxiliary Equations', A~lRgrph 97,part I, pp. 205-279.

clearance exists, L~eeKaldellis et al. (1991). The effect of the rotation of the blades does not invalidate the proposed model, since the velocitTr profiles at the inlet and the outlet of a radia i impeller (fig.[8]) , along with 'he corresponding velocity profiles at the inlet and the exit of tlhe rotor of a modern axial fan (fig.[9]) are again En.AXIALTW~INEi

MESLa.TS ATSTATIONS 549

TAMAXIALTUINE~ i MERLTS ATTHE INL.ET - EXIT

04I2..0N) S - REAL FLOW I STATION .. 0. S TLTSO , Ii..E STTI-

PALFLW 042-1.040)SM.TSON..~ EXMM SM0820.M-.0t.F(19LLTI

ETXTRNAL FLOW

do."

.#DDo 0.10

-EXTERNAL

. O.K

IOnO

NONCIICN51ONK VELOCITY (14SIW#4l

Figure 4. Lnnitudital velocity profiles at tie inlet aid tie exit of tie WL axial Wuhite. AXIA.COMPRESOR CA$=Ci MEW FfT IT4 - NA FLOW (Ptd'UiS.. F..0.IIS) tv PEWLTS. 012.161 (Iwo 01.4ncoj WLFO W%6K.H9.11)EXIT, Wf. FLOWIS"1 iI0. 4.'0 ITS

[

.

EXIT,

~~

lgt.la

Me& WOO0. , No.0.71) FLO"

~."010

O.W

O.K6

1.60

Figure 5: Loagitial relocity profiles atthe aulet -eit otf V111 igtraing Axial turbine. COWMESSO i C!SLTS FORTIC MW. 1A. FLOW 512.1.26) UP~£. PSST d.N. H9-.02) RIELFLO 0121,3 & o. MILS0" RON031 ENLFO XTERNAL FLOW E i 7 f1.9S~S5,le0O )

NEW WIS

-INITt

~

51104O AXIALTU4IBINE i FCSXTSAT TIC EXIT .

14I ON I NRL~ VELOCITY WN/WelI

~.

let

REA FLON 0112-1.0 INLET. XII .m O O(Il2.4.S i

-

~

~

REAL Ff LOW 0412.070) #tw UK .. . LOW0412-1,1112) lw the TIP 00*700. i ,1 ,. ed- SM.O. me 0.3v EXToE.A FLOM

340 . . 01.60 s 1.6 NOtWI1CNSIONAI. VELOCITY Wor44a.l

Figure 6: Eit Ioatitudiaal velocity profiles for

the hub andthe ip of the Ala thor axial turbine, AXIA. Fimds MEWLSFORTIE MOTOR K6 ILET, REAFLOW 0112.0.97) £0. to E!KTSOI0.dIIIC.. Ke.0.66I IILETi EXTERNAL FLOW XT LFLO 0412.1.56) X.MSLLSM0M.V-.1 0? le01.12..6111 . EXIT,. IANFLOW 5215

d4,

CU

.KO.

1,0K18

IKN1 EOIYIH#1*4I

Peguse 7:Exit Inc itudisal velocity profiles, FLOT axial' coupressor cascade, ith aid rithout clearance.

-

N~OICeNIM

K

: 0.6 .6 VELOCITY(Wo/Wtfl

Figure I.: le-eit longitudioal velocity profiles. lelative sy.ate, of coordinates, 112radial ispeller.

1-6-2

a

0.49al"

HINSIONKS

. LN

10

VELOCITYWN/NMO'

Figure 1. tutor inlet-exit longitudinal velocity profiles, Relative syotes of coordintes. tiaxial fan.

rI taI I III

Z-jYi I I -VIUIVI OiN

TURBULENT SHEAR FLOWS-7

Technical Universi(y of Munich September 9-1I, 1991 THE CALCULATION OF A COMPRESSIBLE BOUNDARY LAYER PAST A POINTED BODY AND A DELTA WING E.Krause*), T.V.Poplavskaja and V.N.Vetlutsky**) *)Aerodynamisches Institut, RWTH Aachen Wullnerstras!-e 5-7, 5100 A'chen, Germany **)Institute of Theoretical and Applied Mechanics, USSR Academy of Sciences Institutskaja 4, 630090 Novosibirsk, USSR

ABSTRACT The three-dimensional compressible, laminar and turbulent boundary layers past a pointed body and a delta wing by supersonic gas flow at the angle of attack are studied. An implicit absolutely stable difference scheme for a solution of the boundary-layer equations is suggested. Laminar' and turbulent boundary layers are calculated for the ogive-cylinder combinations and for the delta wings. The skin-friction coefficients and the Stanton numbers are compared with experimental data. PROBL.;Dd FORMULATION The pointed body surface is given in the cylindrical coordinates r=r(, ). The a-coordinate coincides with axis of the body, the ;-coordinate is a meridional angle in a cross section. For the boundarylayer equations the non-orthogonal coordinate system ( is introduced, which is orientated with respect to the body surface. Here the i-coordinate is a normal to the surface (Vetlutsky & Krause 1990). The three-dimensional compressible boundary-layer equations are considered in the region n( t O)1/

and C,

4

amplitude variation (Fig.S, e,

by a factor of more than two, i.e they are not universal The nonuniversality is due to the external intermittency since all data are generalized by introducing a !,ingle parameter, namely, the intermittency factor -iymmtric the notaton Fig 1-3, wake, Re,=380-1080, 2-three-dimensional wake, Rex=340-520, 3 -mixing layer, Re,=1040-1660; 5 -boundary Ca-nncl,

3000,

Rc\

corresponds

4 -plane wake, Rex=160-200..

layer, to

the

Rex\70-50; the curve dependence

6 -return on

Fig 1

C = const

the curveon Fig 3 - to the dependence

0

<en>to,

/

t , 2

-1/2

, The intermittency field has a scale similarity in a wide scale range which is indicated by extended exponential subrange inntknc the intermittency spectrum Er u

and the cons-

change

over a sphere of the diameter r which underlies the deduction of

tU 1

9,

2> (r/L)~

dissipation e

is detected according to the hypothesis in a rather wide range of the fluctuation

in the energy dissipation

correlation function <e (x, )e (x, + r)>= Cc

energy

the relation p = 2-q(6) did not found a convincing confirmation either. The hypothesis of a statistical independence of small and large-scale motions in the turbuient fluid has been verified. At all points investigated no dependence of conditionally averaged energy dissipation moments -"> on the conditioned velocity

0 r .xfO

In half-power region, the constant (2A+3B) depends on the shear gradient ?Ls=0(s /u# . Here A equalsi/x and B indicates turbulent diffusion coefficient. Figure 4 compares the present results with others

Fig. 5 Hysteresis change of defect law coefficient along stream-wise direction.

The quantitative agreement for the case of Ds>0. can not been obtained although the general trend is the same.

4 Re 8000

C

3

300

(x/I)

0aProduction ~)gEAtAA

-0--

0

ElTelbany&Reynolds

100

0.125

o"---

,

,

Advection

i

---

Sa

-

0.125

50 5

* 0.75

6 0325

:

0375

o 10 Re =8x10

,1 -1 1 0.1

Fig. 4

1 05

I

1o-- 1 2A,

Coefficient 2A+3B in Townsend type description of half power law.

0

10 y+20

-3

%

I

-

_'' 7 30

40

Fig. 6 Distribution of turbulent intensity at various sections.

1-13-2

1-

'

0,875

0025

10

01

+ 0625

0 0i

Fig. 7 Production and advection of turbulent kinetic energy.

EIGHTH SYMPOSIUM ON

TURBULENT SHEAR FLOWS Technic .1 University of Munich September 9-11, 199, The Coherent Structure in a Corner Turbulent Bounuary Layer by I. Nakamura*, M. Miyata**, T. Kushida* and T. Yamaguchi*** * Department of Mechanical Engineering, Nagoya University Nagoya, 464-01 Japan ** Department of Mechanical System Engineering, Yamanashi University, Kofu, 400 Japan *** Matsushita Electric Industrial Co. Ltd.

1. Introduction As a paradigmatic flow having the secondary flow of Prandtl's second kind, much research has been conducted on the turbulent shear flow along a corner formed by tao perpendicular plates. Although a theoretical explanation has not yet been developed on the coherent structure in a flat plate turbulent boundary layer, many experimental and some CFD results have convincedly documented such a structure. Bursting is detected even in a rough wall turbulent bouneary layer. It is therefore interesting to investigate how and to what extent the coherent structure will be modified by the secondary current. in what follows we will describe experimental results of the coherent structure in a corner turbulent boundary layer, shown schematically in Fig.1. 2. Measuring Method Fig.,1 also contains the coordinate system used, where y' denotes the corner bisector and z' is perpendicular to both x and y' axes. Uo is a constaat free stream velocity 'which was so adjusted to give - a unit Reynolds

number of' U /v=3 .5 Y 06m 1 where v is the kinematic viscosity of air. The working section of Ln Eiffel type wind tunnel is 0.5 x 0.3 m2 and 2 m long.

U. y Corner Bseto V

C" XL Sew F1oV

Z Flg.1. Flow field and coordinate system. -16 -14 -12 -18

-

80 60

I

40 2 2-

'

Figs.2a the

0

0

8

0

8

ejection period 4

6

8

1

12

14

(me)

0 88 68

velocity

distribu-

£

,,

40'"

measured 20

0

velocity

2

100

of

time

0 8.2 8.4 0.6 8.8 1.8

8

u/uQ a).

bursting, as well as the conventional mean

a

0

and 2b show the variations of

conditional mean

tions during one bursting cycle, at

---

-

08

Blackwelder and Kaplan.

1.as)

180

Measurements were performed at a section 1.9 m downstream from the leading edge, using a conventional hot-wire anemometer set. 3. The Effect of a Corner on Burstirp To detect the bursting event ti short time variance is defined accor ,.Lng to

-6 -4 -2

-8

elapsed from tne

onset

profiles denoted by dotted line,

both obtained in the two-dimensional region. Well known decelerated and accelerated profiles can be seen clearly just before or after T =0. The results obtained nt 7=25_m o.. mila r profilc as Figs. a and 2b. A typical ejection process was seen to occur at nearly T=-4%-2 ms and then a sweep process follows. The ensemble averaged velocity profiles differed from those in Figs.2a and 2b. At T=-2 ms in the section of 25mm, a strong shear layer develops near the region y+=30 and the velocity above it shows a larger value than the conventionally averaged profile, which leads us to infer that a strong clockwise vortex is generated there. The

1-14-I

0

0

0

_ 0

8

8

0

8

8.2 0.4

8.6 8.8

1.8

U/Uo b).

sweep period

Fig.2a,b. Variations of conditional mean velocity distributions during one bursting cycle at z=100 mm. spanwise vortex parallel to the wall is lifted by the effects of both ejection and secondary current and may cause the high shear layer. Figs.3a and 3b show the bursting process at the position nearer to the corner. Ejection and sweep are obvious but the velocity profile shows a

more complex pattern. There are two changes, the one occurring in a small scale found at the region near y+=50 and the other in a larger scale at near y+=20 --30. Specifically, in the sweep stage of T=O 4ms, there is a definite dent in the ensemble averaged profiles, which implies inflection point instability, In order to examine more details of t'he relation between bursting and the secondary current, space time correlations conditioned by bursting were measured. An averaging time T, i.e. the 2 bursting period, was estimated by Tu T /v =140. Typical results of the conditioned space time correlation at z=25 mm are shown in

Figs.4axA

= XB)

and 4b(xe =

clear evidence of bursting in a turbulent boundary layer along a corner and that the bursting process is evidently modified by the interaction with the secondary current near the corner. The effects of the secondary current appear also in the space time correlation curves and the correlation contours conditioned by bursting show a significantly smaller eddy scale than that of a conventional one. znn,

z.Zsmn

I

where

XA+6),

the burst detecting probe A is fixed at y+=15 in both figures and the values of y+ indicate the heights of probe B. In Fig.4a, the pattern of the bursting process changes significantly with the height of probe B, but it is not found in Fig.4b. Also in Fig.4a the peak value of -he conditioned correlation shows a rapid decrease near y+=58, indicating a localized character of bursting. From the results mentioned above, iso-correlatio contours conditioned by bursting are produced in Figs.5a, b and c. Comparing with the corresponding results in the two-dimensional region, it is suggested that the eddy conditioned bursting has a smaller scale than that of the conventionally correlated eddy. A similar feature can be found in Figs.5b and 5c and particularly Fig.5 c(z=15mm) shows a significantly small spatial extent of the coherent eddy, compared it with 2-D region.

Y.,117

Y'.117

y'.se _____

_

9.58

--

__i______

9.2 .29.29

9= 00

Y.

. -1 J

a).

u./64 xA

Fig. 4a,b.

-2 0 = X8

4

b).

xB

2, +

Distributions of space time correlation conditioned by bursting.

0.5 I

Z=1OOmm

4. Concluding Remarks From the experimental results described above, we can say that there is

=

-IC -14

-12 -I1

-8 -6 -4

-2

0

(,s

.,

-20

-10

0

a).

1.0

tU16 20

I

z=100 mm

05

20

mm

N,,Z=25

210 41

1

i

1

1

'1 1

70

,

N

151411.15'

U/UoN

a).

ejection period -2.0

11

2

4

6

8

10

12

14

-1.0

0 b).

(.s

1.0

z=25 mm

I

811

60

20

Z 15 mm

40

'5

0

0

0

08

b).

0

0 80.2

0.4

0.6 0.8 1.0 U/Uo

0''

-20

sweep period

Fig.3a,b. Variations of conditional mean velocity distributions during one bursting cycle at z=15 mm.

1-14-2

-I.0 c).

Fig. 5a,b,c.

0 z=15 am

10

TJ/d

2.0

Eddy shapes expressed as iso-correlation curves conditioned by bursting.

EIGHITH- SYNIPOSIUNI' ON

5

TURBULENT SHEAR FLOWS[ Technical University of Munich September 9-Il, 1991

THE EFFECTS OF REYNOLDS NUMBER ON TURBULENT WALL LAYERS. Ronald L. Panton*) *)Mechanical Engineering Department University of Texas Austin TX, 78712, USA

ABSTRACT This paper displays the theoretical form of composite expansions for mean flow properties and shows how they depend on Reynolds number. It also gives semi-empirical equations that have been fitted to channel flow results . Reynolds number trends of the experiments are well represented by composite expansions. INTRODUCTION There has been a interest in the effects of Reynolds number on turbulent wall layers. Much of this comes from the need to interpret flow visualization experiments and direct numerical simulations, Spalart(1988) and Kim et. al. (1987). It is accepted that turbulent wall layers have a two layer structure; a classical singular perturbation situation. Mathematically the twolayer structure has an analogy with the theory of inviscid flow and boundary layers. One practical difference is that boundary layers and inviscid flow have a thickness ratio of 1/100 (or smaller) while turbulent wall layers the thickness ratio can be as high as 1/4. Consider a layer of thickness h, centerline velocity U, viscosity v and friction velocity u.. The outer layer variable is y/h , the inner y= y/u.v and the ratio is the Reynolds number Re,=u. h/v . An inner function for any variable is fi(y + ) - f(y+,Re.= oo) Similarly the outer function is f(y+- h Re-, Re-= cc) hu(Y+)composite Matching between fi and fo is aptly called the common part;, fcp =fo(0)=fi(co). Neither fi nor fo are uniformly valid for ail y. What is not discussed in textbooks on turbulence is the idea of a comoosite expansion. Such an expansion is not only uniformly valid, but it contains the first dependance of the profile upon the Reynolds number. An additive composite expansion is fn(-)

-n

=

-O-+-Yf(y+) + f]composite = fO(Re. )-fcp The mean velocity law of the wall plus law of the wake is a composite expansion, Panton(1 990a). Now, we apply this scheme to the Reynolds stress. For pipe or channel flow theory shows that the Reynolds stress in the outer layer is

-Y u. 2 '

1 h

From this one can see that the common part is The inner layer function g(y+) is Gcp = G(0) =1. not known. The composite function is uv> composite = g(Y+) + G( V+f) -G e

Re-

Re-)

= g(y+) + ReIn order to explicitly display the trends with Reynolds number a curve fit to experimental data was performed. The equation assumed was + .+.Cz g(Y) + C[exp( )- eXp( yexp( 2 A fit of this equation to the Re* 860 data of Tiederman & Harder(1990). yielded coefficients; Cl = 0.517, C2 - 136, and C3 =2.51. Figures 1 and 2 show the composite expansions and the experimental data. The constants for all of these curves are the same. The fluctuation u' was treated similarly. Since at the centerline u' is symmetric, a hyperbola was used for the outer behavior. U'o

-

C1 +

{ C22 + C3 2[1 - (y)2]}1/2

The common part of this is the constant u'cp = C 1 + { C22 + C3 2} 1/ 2 = Ccp The inner function was taken as u'i(y+)=Ccp[1 -exp(-C4y+)]+C5[exp(-C6y+)-exp(C7y+)] The composite expansion is - U'o + u'i- ucp This equation was fitted to the data (Re*=860) with values: C1-0.527; C2= 0.369; C3- 1.57; C 4 =0.0697; Cs= 90.8; C 6 = 0.0905; C 7 = 0.0948. The results are compared to data in Fig. 3 and 4. A m.del equation for the vertical velocity fluctuation v'was also a hyperbola for the outer behavior, The common part of this is another constant. According to theory the inner function 2 should rise away from the wall - y . A n equation with this property is v'i(y+) - Ccp [1 - exp(-C4y+) + C5 exp(- C6 Y+) [ exp(-C7y+) -1 + C 7 y+] The composite expansion is v(y+)composite = V'o + vi - Vcp This equation was fitted to the data with values: C1 =0.447; C 2 - 0.226; C3 - 0.691; C4 -0.0509; C5=

1-15-I

F~g.2 Reynolds Stress at Various As'

Fig. 1 Reynolds Stress at Various Re' Data ot Tioderman & Harder 08

*0l6

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Fig. 4 Re' Effect on u' Fiuctions

Fig. 3 Re* Effects on u' Fiuctions

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y+

Distance

1110.2

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-

-

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.2

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05

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0

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A

Re'.1855

o0tO1

_i to'vo'1,10

tooO

C

10

Distance y+

Distance y4

Fig. 5 Effect of Rcoon V' Fluctuations

Fig. 6 Effect of Re* on v' Fluctuations

A

as Os.8 ataof Wei&Wimarth

a

00

S08--4 Re'*750

0

So.0

i

0L

.t

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06-

>~0

0*

04-

~0

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A

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0

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-

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0202-

0

0_1I

--

-

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to'

~Distance

y+

o

o

-

I.

10

0

Distance y+

Here the Re*-380 22.0; C~a 0.539; 07 -0.0850. data was used. Figures 5 and 6 show the results. From the figures it can be seen that the composite expansions exhibit considerable Reynolds number effects, even for y+values less than 100. The ability of first order expansions to track these trends to such low Re* indicates that the physics of these layers is much the same as those at higher Reynolds numbers. This paper was written while the author held a visiting position at the Acoustics Center of Ecole Centrale de Lyon. A fellowship from the REGION of Rhone-Alp is greatly appreciated.

1-15-2

REFERENCES Spalart, P. R.(1988) J. Fluid Mechanics.187, 61. Kim, J., Momn, P., and Moser, M.(1987) J. Fluid Mechanics.177, 133-165. Tiederman, W. and Harder(1990), private communication. Wei, T., and Willmarth, W.W., (1990) J. Fluid Mechanics.187, 61. P3nton, R. L.,(1990a) Proceedings of 11th Annual Conf of Canadian AppI. Math. Society. Computational Mathematics Publications, Southhampton, UK, 1990. Panton, R. L.(1990b) J. of Fluids Engr.112,425.

EIGh111 SYMIPOSIUMI ON

1-16

TURBULENT S1IEAR iLOWS inich Technical University of September 9-11, 1991

A Comparison of Temporal and Spatial Temperature Derivatives in a Strained Turbulent Flow Hamid R. Rahai Department of Mechanical Engineering California State University Long Beach, Ca. 90840.

and John C. LaRue Department of Mechanical Engineering University of California, Irvine Irvine, Ca. 927i(. Introduction: Both Taylor's hypothesis and the assumption of local isotropy are routinely applied to flows with a range of mean velocity gradients and velocity intensities. It is well known that the use of Taylor's hpothesis leads to increasing un-ertainty with increases in the mean velocity gradient and velocity intensities (cf. Taylor (1938) and Townsend (1976)). The result. presented herein quantify this uncertainty and also indicate that the uncertainty depends on the integral length scale of the flow. The recent studies of Sreenivasan et. al. (1977) and Antonia et. al. (1986 a k b) show flows with moderate to small mean velocity gradients are not locally isotropic. The present study complements those of the previous studies and quantifies the effect of mean veloclength scale on departures from local ity gradient and integral isotopybridges isotropy. Based on the rapid distortion theory of Batchelor and Proudman (1954) and subject to the same constrains, Cauchy's equation can be used to relate the scalar gradient before and after the application of a mean strain as Oq' Ox, Oq Z, aj= 37

2 J)-

(3)

2

_

b6

(5)

Results: The present results indicate that in a heated gridturbulent flow approaching two dimensional objects, along the mean stagnation streamline, for if 0.75 and increase to about 1.40 for . = 0.15. This increase in dissipation is higher for the experiment which has larger i5 ratio. For large -g ratio, the dissipation obtained based on Cauchy's equation and from the temporal derivative and Taylor's hypothesis agree with the corrtsponding results obtained from direct measurements when

is respectively larger than 0.75 and

1.85. Near the stagnation pot, resuitsbased on Cauchy's equation and from the temporal deivative and Taylor's hypothesis differ from the corresponding result obtained from direct measurements by 60% and 560% respectively [Fig.31.

Similarly, it can be shown that

Al

00 09

ex, 6j3(5 where 60 is the time resolved difference in the temperature at the two sensor positions which are separated by a distance 60.

decreases in the spanwise, and remains constant in the transverse directions [Fig. 2]. For example, the variance of the streamwise

U

-

_(0

They are

The time resolved spatial gradient, 0-, is approximated

(1)

bining Eqns. 1 and 2, squaring and averaging, leads to the following equation which relates the variance of the spatial gradient of the scalar in the strained flow to that in the unstrained flow-

('j

which have a frequency response of 8 kliz.

supplied with a constant r.m.s. current of 1801iA which makes their velocity sensitivity negligible (cf. LaRue et al. (1975)). The probes are directly calibrated using a calibration jet over the range of 19 to 241C as

where q corresponds to the spatially resolved scalar, s refers to measurement in the strainea field, and x, and a, are the conponents of the position vector, respectively, before and after the application of a strain The strain experienced by the fluid is equal to If tl strain in the downstream or x, direction is e . fgenerated assumed to be equal to the mean strain, then it is easy to show ( that a, = m ani c

Two different ranges of mean velocity gradients are produced by placing one of two cylinders downstream of the heatd wire screen. The integral scale is varied by using two grids of different mesh size. Table ! provides flow parameters for the experiment. Two special, two sensor probes are used to measure the time resolved temperature. Probe 1 consists of two platinum wires with a diameter of 0.625 Mn which are separated in the downstream direction. The length of the first wire is 1.0 mm and the second is 0.75 mm. The spacing between the wires is 1.0 mm. Probe 2 is similar to Probe 1, except that the wires are separated by 1 mm in either the transverse of sppnwise directions. The lengths of both wires are 1.0 mm. Both wires are operated using two a.c. temperature

*) u(

( )

For small ! ratio, the dissipation obtained from all methods agree with each other when -. >2.0. However, for 1.

.

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1.5

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1-16-2

1.5

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EIGIlTII SYMPOSIUM ON 'I URBULENT SIEAR FLOWS Technical Unrversity of Mumch Septembe' 9-II, 1991

l-17

The preliminary quantitative measurements of the scalar field in the rectangular jet mixing flows

Gong Xin Shen,

Jian in

Fluid Mechanic Institute Beijing University of Acronaulcs & Astronautics

Abstracts The nrelmimary quantitative measurements of the scalar field in the mixi.,g flows of the rectangular jet using Laser Induced Fluorescnce (LIF) and Digital Image Capture & Proceassng tcchncques arc presented. Introduction Recent dcade, it r,, well known that the Laser Induct d Fluorescence iechneque could be and must be a very important and potential approach for attack on deep understanding the complex and turbulent flows . The contribution which has been made by 11 t many researchers [ Q 2J[3l4 werc involved with itself development of the techcques and its application in various flow field Also, the non-axisymmctrical let flows were studied broadly for its background on many industrial application even the flows in practce could be more complicated Certainly, it would be interesting if the experimental observation on the scalar field could be made directly in the rectangular let mixing flows Experimental Arrangement The schematics of the experimental arrangement is shown in Fig. I The water channel works like as a water tank (water is static here). The jet flows was supplied by a water storage tank in which water was premixed with fluorescence species. The critical diameter of the rectangular nozzle exit is 5 sm and its width / height ratios are 1,3,5, repectively A light sheet of laser, which crossed the axies of et, illuminated a vertical or horizontal section view field of the jet flows The inner view field of flows was captured by a CCD camera (12 X 512 pixel, 8bit darkness level, 25 frames/ see, the spatial reso2 lution 122 x 0 92MM / pixel here), then digittzed, stored by a im. age card (DT-2861, 16 frames Luffer) and processed by a PC-386 microcomputer Measurement & Calibration According to Bear' law, the flourescent intensity (Hr) mainly depended on the incitent intensity of laser beam or sheet (lo) and the concentration of the fluorescent species in the solution (C) Several fluorescent species (Rhodamine 6G, Rhodamie B and Sodium fluorescein) were tested in different conditions (the power of laser (W), as shown in Fig 2, .he maximum concentration of the species in the solution (Co), the apaturc of the CCD camera) It was found that among them. only sodium fluorescin was quite suitable to have fine linear calibration curve at low maximum concentration ,'o) and certain power of Lasser. Also the H1- C / Co curves were cilibrated . -,pectivcly in vertical (V) and horizontal (H) section "lcw fields as shown in Fig 3 Fc- a 2-D view field, a fluorescent intensity field H1(x,y) depends on not only a concentration field c(x,y), but also the local incident intensity field I(x,y) jy -,(x,y) Q[I - e-y

when e

Ay x + Ax C(x~.s)dxdy ]l(xy)

tain degree, as shown in Fig 4, if the concentration of used species was very low and the intensity resolution of the optical system was enough high The maximum error of concentration value due to decay of incident intensity without alterative correction in our case was estimated roughly about ± 4% (y and z direction) and ± 7.5% (x-direction) for the maximum concentration Results & Discussion The preliminary investigation on concentration and scale field for the rectajigular jet mixing flows took place in the same condition for the calibration. Renolds numbers is 1000, 2000, 3000, 4000, and 5000 The spatial and temporal resolution could be down to the Taylar scale (for Re = 5000, X / dz 50-100 ). The typical images of instantaneous concentration field in a rectangular jet mixing flow were shown in Photo I and 2 The dankness level or pseullo colour correspanded to the dimensionless concentration C/Co (C / Co =I at the exit of the nozzle). Some image processing techneques were used for better displaying the structures and emphasizing main feature of the flow patterns. It is found that one kind of unsteady and blended large-scale structures (from symmetric pattern to spiral pattern or inverse) exists in the ractangular jet mixing flows as shown in Fig.5(c), except the cxisting symmetric and spiral-larie scale structure which have been found in round jet flows. The instantaneous profile of the concentration along the axis y and z, as expected, were strongty intermitent , and the instantaneous quantitative distribution of concentration along the X axis is very intcrmitent too Obviously, all these kinds of the intcrmitency are connected directly with the scalar interfaces or scalar structures (Certainly, including thelarge-scale structure). Also the results of present investigation show that the entrainment mechanism in the rectangular letflows is dominated by large-scale motions. These large scales are engulfing unmixed ambient fluid and transporting it, in significant quantities, deep into regions in the flow interior where it can be subsequently niixed by smaller scalar motion (Kolmogorov and Batchelor scales). Fig 5 show the typical instantaneous average concentration distributtons along the axis x with W / H = 1,3,5 at vertical and horizoutal section respectively By comparison with all the results, it could be found that the average mixing rate and the concentration pattern in rectangular jet flows did not simply monotonously depend on the width / height ratio, also the average mixing rate along the x axis in the vertical section is considerable different with that in the horizontal section. Obviously, the 3-D effect on the concentra. tion field in the rectangular jet flows isconsiderable strong. For the further work,the observasion would be down to the smaller scalar field. Reference [1] Koochesfahani, M M. (19861JFM vol.17, 83-112 (2]

cAxAy an, axAy v'/ r

uv/~ +20

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+.5,

EIGHT11 SYMNPOSIUNI ON TURBULENT StI lEAR FLOWVS Technical University of Munich September 9-11 1991

EFFECTS OF MULTIPLE EXTRA STRAIN RATES ON THE STRUCTURE OF WALL TURBULENCE H. Yoshida*), Y. Horiuchi*), R. Echigo*) and K. Suenaga**) *)Department of Mechanical Engineering, Tokyo Institute of Technology Meguro-ku, Tokyo 152, Japan **)Mitsubishi Heavy Industries, Ltd., Chiyoda-ku, Tokyo 100, Japan

ABSTRACT Effects of multiple extra strain rates on the structure of wall turbulence have been investigated for the flow with lateral divergence and streamline curvature. The turbulence structure changed by tie extra strain rates were discussed in relation with the bursting phenomena. The most striking feature of the present turbulent shear flow is the nonlinear interaction between lateral divergence and concave curvature. INTRODUCTION The response of shear-layer turbulence to extra strai. rates is very important as summarized by Bradshaw (1990). This paper is the second report of our work dealing with turbulent shear flows with lateral divergence. While the previous paper (Yoshida et al 1989) described the experiment for the pure effect of lateral

S = 1: Turbulence has an ordinary structure because there exists no extra strain rate. S = 1.2: The effects of streamline curvature are dominant, and divergence is negligible. Consequently, turbulence structure becomes highly antisymmetric around the centerline. S = 2: According to the duct geometry, curvature vanishes, and alternatively divergence becomes large. The effective extra strain rate for curvature, however, is comparable with that of divergence. Hence, these two extra strain rates and the basic strain rate strongly interact among them. S = 3: The effects ot' curvature becomes negligible, and the pure effects of divergence appear. urulence -r - promoter

divergence, the present one focuses on the interaction between

p

1

multiple extra strain rates, i e., lateral divergence and streamline curvature. This is because the authors were stimulated by the study (Smits etal. 1979), which discussed the interaction between these two extra strain rates in a turbulent boundary layer

r,

i

196 ID

716

'

I

1 -

c

over an axisynunetnc cylinder-flare body.

In the present riaper, an axisymmetnic annularhas flowanwith lateral divergence is investigated This flow system advantage that since both stabilizing and destabilizing curvatures are included, two kinds of tnteraction between divergence and curvature can be simultaneously observed. The effects of such extra strain rates on the turbulence structure are examined not only from time-averaged quantities but also from conditionally averaged velocity signals obtained by the Quadrant method (Lu &

O

r

ifi H w

pre - pr-be -

Blower

'

Z E

-

Willmarth 1973).

, j

_

(Dimenslons inmm) 1030

APPARATUS AND EXTRA STRAIN RATES

The air channel built for the present investigation is shown in Fig. 1.The flow passes first through an annular entrance section (length le3000 mm, outer diameter do 196 mm, inner diameter d, 116mm, hydraulic diameter dh = do - d,= 80mm), and is fully developed before entering the distorting section. Figure2 shows the flow geometry of the distorting section. The centerline between the inner and outer walls is expressed by a sine curve and a straight line. The sine curve (from 0 to n)was used so that the radius of the curvature changes continuously from the entrance to the distorting section. The reason why a circular curve wss avoided in the present work is that such an impulse was too strong to fairly interact with divergence. The channel height h vanes inversely proportional to the

Fig.l. Schematic of experimental facility 4

e40(Lterol divergence)

3 2

_ a ec (Curvature) ectf to

0

St

section). Hence, the cross-sectional area remains constant, and the stretching ratio S = r/rI= hI / h increases up to 5 in the

_

10

_i

-.2

0

400

0

(ri and hi are the channel dimensions at the start of the distorting streamwise direction. Turbulence measurements by a hot-wire anemometer were made at the stations of S = 1, ;.2, 2, 3. Figure 3 shows the extra strain rates e calculated from the channel geometry; subscripts d and c denote divergence and curvature, respectively. To estimate substantial contribution by these extra strain rates, the effective extra strain rate eeffia calculated by the lag equation (Smits et al 1979), and the results are plotted also in Fig.3. As demonstrated in the previous paper (Yoshida et al. 1989), lateral divergence destabilizes turbulence. On the other hand, streamline curvature exerts destabilizing effects along concave walls, but it exerts stabilizing effects along convex walls. As a result, influences by these extra strain rates on turbulence are summarized as follows:

Fig.2. Flow geometry of the distorting section

I

distance from the symmetric axis r, namely, h = h i rl/ r = c /r

[

S

8

ise dostooce x neoo __

2.0

__

30

40

Stretchingrotio S Fig.3. Streamwise variations of extra strain rates RESULTS The following data were obtained for Reynolds number Re = Umdh / v = 50000, where Um is the bulk-mean velocity. The coordinate origin is taken on the centerline at the start of the distorting section; x is distance measured along the curved centerline, and y is measured along straight lines normal to the centerline and toward the center of curvature. Hence, y /h = - 0.5 is on the inner concave wall, while y /h = 0.5 is on the outer convex wall. U and V are the components of mean velocity in the x and y directions, and u and v the corresponding fluctuating components.

i-1I9-I

rL

Time-Aveaed- ,undu Figures 4-7 show the streamwise variations of nieai vdocity, turbulent intensities, Reynolds stress, turbulent k, wtic energy, respectively. Contributions to R,ytolds itress from Different Events Figure 8 shows the contri!tudonq o"each quadrant to Reynolds stress, Here, it should be noted that unlike the expression by Brodkey et al. (1974) we display the resuss using absolute values for each qradrant. Conditionally Averaged Ylocity Signals Figures 9 and 10 show the conditionally averaged uv signal obtained by the Quadrant method. Figures 9a and 10a are the results aty /h = 0.35 (outer convex side), while figures 9b and 10b are those at y /h = - 0.35 (inner concave side). Each signal is normalized using the maximum value. DISCUSSION S I Although in the foregoing chapter we mention that turbulence is ordinary at this station, somewhat antisymmetric nature is observed (especially in Fig.8a). This antisymmetry is considered to be ascribed to the anomaly associated with a turbulent flow in annuli. We now use these data as the base state to assess the effects by extra strain rates. S =1.2 At the inner concave side turbulence is markedly amplified, while it is attenuated at the outer convex side. These tendencies agree well with those reported on a curved channel (Kobayashi et al. 1989). Corresponding to this turbulence behavior, the con-tributions of 2 and 4 quadrants drastically changes as shown in Fig.8b. The conditionally averaged signal for ejection at the concave side (Fig.9b) becomes broad, while that for sweep at the convex side (Fig. lOa) becomes narrow. S=._2 Compared with the profile at S = 1.2, the streamwise turbulent intensity Urms further increases at the inner concave side, and recovers at the outer convex side. At the inner concave slightly decreases, side, however, the normal component Vrras and much more remarkable decrease is seen in the Reynolds stress profile. This fact is very surprising because both divergence and curvature are expected to amplify turbulence; the their measured results demonstrate that a simple summation ofresults separate effects is not valid even qualitatively. (Since the reported on the similar flow system (Smits et al. 1979) do not show such tendencies, we first doubted our experiment. The repeated measurements for two stations which are apart in the circumferential direction, however, showed good reproducibility.) Whereas the mechanism of this nonlinearity is not clear at the present stage, relatively small contribution of the 4 quadrant shown in Fig.8c corresponds to decrease in Reynolds stress; the 004 . .

000

REFERENCES BRADSHAW, P. 1990 Effects of extra rates of strain -Review. InNear-Wall Turbulence (eds. SJ.Kline &N.H. Afgan, Hemisphere Publ. Corp.) 106. BRODKEY, R.S., WALLACE, J.M. & ECKELMANN. H. 1974 Some properties of truncated turbulence signals inbounded shear flows. J.Fluid Mech 63, 209. KOBAYASHI, M., MAEKAWA, H., TAKANO, T., UCHIYAMA, N., KUBOTA, M.&KOBAYASHI, Y.1989 Two-dimensional turbulent J,32, 324. flow ina curved channel. JSME Int LU. S.S. &WILLMARTH, W.W. 1973 Measurements of the structure of the Reynolds stress ina turbulent boundary layer. J.Fluld Mech. 60,481. SMITS, AJ., EATON, J.A. & BRADSHAW, P. 1979 The response of a turbulent boundary layer to lateral divergence. J.FluidMech. 94, 243. YOSHIDA, H., FURUYA, T. &ECHIGO. R. 1989 The effect of lateral divergence on the structure of aturbulent channel flow and its heat transier. Turbulent Shear Flows6 (Springer-Verlag) 269.

1o0

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....

.

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.

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50%

-

0%.

-0.5

a S =1

5

0

100%

00%

1

3 R

-os 05

5

,

0.5 -05 dS=3 0 y/h 05 0 y/h cS=2 Fig.8 a-d. Contributions of each quadrant to Reynolds Stress

S-2 0 -

0

10

as

-

... 5 0 -5 t U,21V a convex side 0a-o0

10

A

~

b Vrrs 0 05

5 Yh 5

Fig.5 a and b. Profiles of turbulent intensities

-05

I

05

-05 05 Fig.7. Profiles of turbulent kinetic energy

~

0

1-19-2

[

b concave side Iul Fig.9. Ensemble-averaged

uv using Quadrant (ejection)

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0

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-1

-0

Fig.6. Profiles of Reynolds stress

~~~

1. 2 0:V2

0 S-20

es

>1'

S-1.2

a *0

f

.7o

.o

xA

Fig.4. Profiles of mean velocity 0 S.. 0 .. : -s-. 1,2 \•s-

4

0 .-

Re- 50D00I

05

0

.

-05

-10

.

.

0

. .

L. . .. o0.

-S-20. S0

.

0(;",

. 450x

50% -

2 quodron

" °os. -0002 -0

0.5

,(,)30 (w),

8

3.0r

/00

W

q

1

V(,)2

a-1 o~0.

o s-700o

05

1.20

bS bS

V(,,0 o(,,), 0(-)4 a(-

..

.

-A

o02

conditionally averaged signals also show marked reduction both in ejection (Fig.9b) and in sweep (Fig. 10b). S--3 At this station, all the turbulent intensities and Reynolds stress increases owing to lateral divergence, as reported by the previous paper (Yoshida et al. 1989). The conditionally averaged signals, however, are not so different from those without extra strain rates (S=I).

-10---S-3 . .e50 .1.__ .__ .__.

-5

b concave side

5

0 t uV2/v

Fig.1O. Ensemble-averaged uv using Quadrant (sweep)

EIGIIrll SYMPOSIUM ON TURBULENT SIIEAR FLOWS

1 2

Technical University or Munich September 9-1 I,1991

EFFECT OF LONGITUDINAL VORTICITY ON THE TURBULENCE STRUCTURE IN THREE-DIMENSIONAL BOUNDARY LAYERS: MODELLING AND PREDICTION G. Zhang and

P. Zhao

Department of Engineering Mechanics Tsinghua University, Beijing, P. R. China

ABSTRACT

uuk [

k

A series of cxperiment (Ref. 1-4) revealed that the tur-r bulenice behaviour in a pressure-driven 3D boundary layer may differ considerably from what one would imagine with the available knowledge about 2D turbulent boundary layers. Departure of the turbulence behaviour in 3D boundary layers form that m 2D boundary layers is attributable to the effects of longitudinal vorticity on the turbulence structure in 3D boundary layers (Ref,5). A survey of experimental data uncovered that, for a boundary layer which is initially 2D and is driven into 3-dimensionality by a lateral pressure gradient, the effects of growth of longitudinal mean vorticity can be summed up as follows. (I) to increase the dissipation rate of turbulence energy; (2) to suppress the turbulent fluctuations normal to the wall, hence to reduce the shear-stress magnitude and the turbulent diffusion of various quantities; (3) to increase the streamwise fluctuations keeping the level of crosswise fluctuation almost unchanged, hence to cause the momentum transport normal to the wall to be anisotropic. The basic second-moment closure (with IP model for the rapid part of the pressure-strain correlation) is not available for modelling the above mentioned effects in 3D boundary layers. It may predict neither the lag of the shear-stress direction behind the velocity-gradient direction nor the reduction of the shear-stress magnitude.

aU,

U

[uUkx

+ r[ A 2(u lu kf

U/U

k

+

U/Uk

1

x,JJ k)

____0

+ 3a,,IaW,(u,

+u-U111)

where f)k is the vorticity tensor, A2 is the second invariant of the stress-anisotropy tensor, and r is a freely assignable parameter. Computational results show that the velocity fluctuation normal to the wall aud the magnitude of the shear stress are sensitive to the value of r. In the present case, we assume r to be a linear function of the tangent of streamline-skewing angle, i.e. r = Cl +

C' 2 tan(q,

- (p,)

where q is the angle of direction of the velocity in boundary layer, (p, is that at the outer edge of the boundary layer, and C,,C, 2 are constants. 2. A new source term which represents the increase in dissipation rate of turbulence energy du,- to vortex stretching is added into the dissipation equation. This term is supposed to be simply proportional to (5 • V)V, the rate of increase in mean vorticity oue to vortex stretching. Thus, the following modified form of dissipation equation is suggested: DeCa-L

Refinements of the basic closure are introduced mainly in two aspects:

Dt

'aXI L k + C, [(fi -

k at I+1C' I

ax,] ) k

2

Ep-C

"k

kk

L 2

k

1.A rationally-developed model originally proposed by where C& is a positive constant to be determined. ObviousFu et al (Ref.6) is adopted for the rapid part of the ly, the last term on the right of the equation leads to a greatpressure-strain correlation. This model contains the er dissipation rate when the boundary layer is becoming quadratic and cubic terms of the Reynolds stress and the latmore 3D. ter explicitly represents the effect of 3D vorticity field on the Reynolds-stress transport. The resultant expression is Bradshaw and Pontikos'experiment (Ref.3) with a simwrittensexermnt( as)wthasie written as ulated infinite swept wing was taken as the test case for the refined second-moment closure. The optimized values of (P'J2 - ,[ jP r ] + 0 "3 c,,(kk -_0.6[pil-3 newly-introduced constants are -0.2 UkUIkUIU,

k

[.aUt+ ax

1U'1

C,l =0.06,

C, 2 =2.5,

Cd =0.18

axk J

Other constants involved remain as those in the basic

1-20-I

second-moment closure. Predictions of the profiles of mean-velocity components, Reynolds-stress components, turbulence energy and its dissipation rate using the refined second-moment closure and the basic second-moment closure are compared with the measurements as shown in Fig.l (at the 4th measurement station of Ref.3 for example), where U, W and u, w are components of the mean velocity and the velocity fluctuation normal and parallel to the leading edge of the swept wing respectively, Q, is the magnitude of mean velocity at the outer edge of the boundary layer, y and v are the coordinate and the velocity fluctuation normal to the wing surface. Other symbols in Fig.1 are the same as those in Ref.3. It is obvious that the prediction of the basic second-moment closure greatly overestimates the magnitude of every component of the Reynolds-stress tensor, whereas the corresponding prediction of the refined second-moment closure is quite agreeable with the measuredment data. The refined second-moment closure may also qualitatively pre. dict the lag of the shear-tress direction behind the velocity- gradient direction.

REFERENCES 1. Johnston, J. P., 1976, Experimental studies in threedimensional turbulent boundary layers. Rep. MD-34. Thermosciences Div., Stanford University. 2. Van den Berg, B., 1982, Some notes on threedimensional turbulent boundary layer data and turbulence modelling. Proc. IUTAM Symp. on ThreeDimensional Turbulent Boundary Layers, 1-18. 3. Bradshaw, P. & Pontikos, N.S., 1985, Measurements in the turbulent boundary layer on an 'infinite' swept wing. J. fluid Mech. 159, 105-130. 4. Anderson, S.D. & Eaton, J.K., 1989, Reynolds stress development in pressure-driven three-dimensional turbu. lent boundary layers. J. Fluid Mech.202, 263-294. 5. Bradshaw, P., 1987, Turbulent Secondary Flows. Ann. Rev. Fluid Mech. 19, 53-74. 6. Fu, S., Launder, B.E. & Tselepidalds, D.P., 1987, Accommodating the effects of high strain rates in modelling the pressure-strain correlation Rep. TFD / 87 /5, Dept. Mech. Eng., University of Manchester.

COo

60

60

"

6'

-

° " ,,,

,,

0.5

6

,

,I

(.5

0C

"'"

Cs.

1,,

4'.C '0

0

40

5.5

Fig.l

5.0

.4

,t

'.

0

..

5.0

C..5

Comparison between predictions of bas:, and refined second-moment

clo~ures and measurements ( 4th mea-osrement station of Ref.3 ) -

refined closure 0

basic closure

measurements

1-20-2

1.0

EIGHTH SYM'POSIUM~ ON

11-1

TURBULENT SIEAR FLOWS Technical University of Munich September 9-11, 1991 LARGE EDDY SIMULATION OF RAYLEIGH-BENARP CONVECTION A Critical Look at the Wall Function Boundary Condition

D. Angirasa and F.T.M. Nieuwstadt Laboratory for Aero- and Hydrodynamics, TU Delft Rotterdamseweg 145, 2628 AL Delft The Netherlands

ABSTRACT

LES MODEL

A numerical study of Rayleigh-Benard convection with large eddy simulation is preened.Th gadinttrnsfr yptheisis presented. The gradient transfer hypothesis is beentperformedeusinga 4 grid foria 145 cm x e cmen pe20cmed cutinal grid or S5m of te cm x 20 cm computational grid box Some of the results are presented in this summary. A comparison with the experimental data of Adrian et. al. (1986) shows good agreement. Some of the features of the higher moments are discussed. The wall function for tomperature boundary condition

The LES model used in this study was described by Nieuwstadt and Brost (1986), which was meant for the study of the decay of the convective turbulence in the atmosphere. The model is modified for the laboratory situation that is presently under study. In this model the subgrid terms are parameterized by gradient transfer hypothesis. For example, the deviatoric subgrid Reynolds-,;tress T is given as

r = K

T

J

j5eyamined

u a-

ax

+

u -J

ax

(11

where

the tilde denotes the resolvable scale field. The exchange coefficient K is expressed as

INTRODUCTION The type of turbtient convection that arises by heating a fluid layer from below is called Rayleigh-Benard convection. The warm fluid from the bottom will then raise and is replaced by cooler fluid drawn from the top. This pattern of hot updrafts and cool downdrafts is referred to as coherent structures in atmospheric convection. Much knowledge about the structure of turbulence has been gained by large eddy simulation of such coherent structures in the atmosphere Large eddy simulation (LES) is an important computational tool in the study of turbulen fluid flow. It has been particularly successful in understanding the structure of the turbulence in the atmospheric convective boundary layer because the atmospheric turbulence comprises of large scale eddies in the form of thermals. Its application to laboratory models and turbulent flow in devices is lagging far behind because wall-bounded turbulent flows pose their own computational problems distinct from the atmospheric convection. In LES, turbulent eddies larger than the grid-size are explicitly resolved by solving the time-dependent three-dimensional Navier-Stokes equations However, eddies of a size smaller than the numerical grid cell have to be modelled These are called the subgrid scales of the motion and we must filter them to solve for the larger eddies. This spatial filtering leaves additional terms in the equations, which are parameterized. Nieuwstadt (1990) presents a review of the LES method and the possibilities of its applications

In the present study, we are concerned with the LES of turbulent buoyant convection in the laboratory situations. The major difficulty in this problem is that we need a wall-function to determine the temperature drop across the first grid cell adjacent to the wall in terms of the heat flux at the wal. Most of the mean temperature variation normal to the horizontal heated wall takes place within the first grid

cell, hence the need for a proper wall-function.

!1I-1I-1

K = c E

12

1

where E Is the subgrid kinetic energy, were we solve ithsbionkietin, ana additional equation, and mixing length, which is related to the

I = (AX Ay Az)

(2) for which for 1 is hih the grid-size

0)

The constant c is Inte-preted is proportional to the ratio of the filter width to the grid-size. In the present study its value is 0 12. The calculation domain is a rectangular box. The horizontal dimensions are 145 cm x 150 cm, and the vertical 20 cm. The number of grid points placed in each direction is 40. For all fluctuating variables the horizontal boundary condition, are prescribed to be periodic. At the top vertical velocity and gradients of horizontal velocity components, and temperature are zero. These satisfy adiabatic boundary conditions at the top.

WALL FUNCTION FOR TEMPERATURE BOUNDARY CONDITION AT THE BOTTOM For prescribing temperature boundary condition on the bottom side, we need a wall-function that gives a functional relation between the wall heat flux and the mean temperature drop across the heated layer in the vertical direction. The horizontally averaged mean temperature drops sharply with height near the wall, and it is not possible to resolve it numerically. In fact, most of this temperilure drop takes place within the first numerical .i id cef, We have faced several difficulti , finding proper wall functions.The thermal structure of the convective layer is quite complex and multiple layer theories have been proposed, with each layer exhibiting its own thermal ciaracteristics. Adrian et al. (1986) discussed the models.

00

A matching analysis of a two layer model has been performed to obtain a wall-function. In this analysis the inner layer is assumed to be dominated

by

conduction and the

outer

layer by

75

10-

500

.,....10 .0

075

"05

00.5

-/3

2m_= _ C T1- '10

+ C z

(4)

20 o

/

CC

5

In the above formulation, the subscripts o and conduction and convection scales to refer respectively as given in Adrian et. al. (1986). T

0

C

z

1.0

and C

-

0

(10.0).

25

75

5.o

1o

S

Root mean square horizontally averaged temperature at various time-levels

Fig. I

Extensive is the mean surface temperature. comparison is made with the available experimental data to obtain the values of C and C . The values suggested are

50

015 .

convection. The following functional forms of the temperature profile in the overlaying region have been obtained after matching of the two layers TT

25

00

0

5

0 75

0

C

0 75

075

(6)

0

5

.5

appears in the calculation of Nu as follows. Nu =

[[C- C (Ra Nu Pr) 0

3

4/3 Ra /

3

1 Pr /

1/12

We have calculated Nu from Eq (6) and found that the calculated values agree well with the measured values. Table 1 presents a comparison. Since Eq (6) is an implicit expression for Nu, Iterative procedures have to be employed for calculating Nu In any case, the immediate utility of Eq (6) is in checking the validity of the model constants C and C N . Nu

is

not sensitive

to the

L2 9 t 00

00

00 Fig. 2

value of C as can

0I

.5 1 0 51.0

Time evolution of the vertical velocity skewness

In Fig

I we present

tiie vertlca

protiiLe

of

be seen from table 1, but its dependence on the value of C is rather pronounced. We are carrying N out further calculations to fix the value of C

A iuvels. at various time RgdS temperature comparison with the experimental data of Adrian et. al. shows good agreement Near the lower boundary the RNS temperature starts at the RMS

that matches the experimental data.

value of the plate,, which is zero in this case, and reaches a maximum, before beginning to decrease. The decrease can be fitted into a power law variation. This behavior is similar to the one observed by D~ardorff and Willis (1967). Above 0.5z. the power law variation is no longer valid since the profile becomes linear.

In the LES, most of the mean temperature drop above the surface takes place within the first grid-cell. Hence we need the above empirical law to prescribe the drop in the temperature across the grid-cell, as also the temperature on the surface in relation to the heat flux. The Pe'i'it ins 4 and 5 together provide this :f', tion 0f Table 1 Pr = 6 0 Ra

Nu

Nu

1 18 x 108 9.49 x 10

C

features.

Present

Expt.

1.5 x 107

C

19.43 38.64 77.42

21.15 21.28 18.7 37.03 78 73

As a final point of interest in this summary, we present the evolution of skewness of vertical There Is considerable velocity in Fig. 2 disLrepancy between the ,casuroments in the laboratory and the conputations of this higi.er moment. There is also wide disagreement among different measurements and various computations. Moeng and Rotunno (1990) recently discussed these

0.75 1.0 1.0 1.0 1.0

10.0 10.0 11.0 11.0 10.5

REFERENCES Adrian, R.J., Ferreira, R.T.D S. & Boberg, T. 1986 Turbulent thermal convection in wide horizontal fluid layers. Experiments in Fluids 4, 121-141.

_

NUMERICAL RESULTS Computations have been performed for the following set of conditions: Qn = 0.0153 cm/sec, w* = 0.45 cm/sec, z. = 20 cm, e= 0.034 'C. 004skewness

Deardorff, J.W. A Willis, G.E. 1967 Investigation of turbulent thermal convection between horizontal plates. J. Fluid Mech. 28(4), 675-704. Moeng.

C.-H. & Rotunno, R. 1990 Vert ical-veloc Ity in the boundary layer. J. Atmospheric

Here Qo is the kinematic heat flux at the surface,

Sol.

w. is the convective velocity, z. is the distance

Nieuwstadt, F.T.M. 1990, Direct and large eddy Proc. IX free convection. of simulation Conference Transfer Heat International Jerusalem, Hemisphere, Washington,D.C.

between the two horizontal surfaces and e. Is the temperature scale given by Qo/w 9 . From the convective velocity w., and the length z,, we obtain a time scale to given by z./w.

1!-1-2

Nieuwstadt, F.T.M. & Brost, R.A. 1986 The decay of convective turbulence. J. Atmospheric Sci. 43, 532-546.

EIGhVI'l SYMIPOSIUM! ON

11-2

IURIULENTSEAR FLOWS Technical University of Munich September 9-11, 1991 LARGE EDDY SIMULATION OF A TUhBULENT FLOW WITH SEPARATION M. Arnal and R. Friedrich Lehrstuhl fir Fluidmechanik Technische Universitiit Miinchen Federal Republic of Germany

ABSTRACT

exactly with the use of Chorin's (1968) projection method and a fast elliptic solver. A fast Fourier transformation in the spanwise direction reduces the 3-D problem to a set of decoupled, 2-D Helmholtz problems which are solved in parallel employing the cyclic reduction technique Further details of the numerical procedure as well as a complete discussioi of the boundary conditions is given in Friedrich and Arnal (1990).

Results of one of , seies of numerical simulations of the turbulc:.t flow over ;. ,eaiward-facig step aie presented and discussed. The large eddy simulation technique is used to simulate the statistici lly-stationary, fully-tuibulent, sep iating and ieattaching dow. Statistical quantities are coin paled with the results of experimental investigations of the same flow at similar Reynolds numbers. In the piesent contribution the instantaneous flow field piedictions are compared with the mean flow field to illistrate the laige fluctuations occurring in the separating sheal layei.

RESULTS Because of space limitations, only a few examples taken from the extensive data set of the rearward-facing step flow is presented. The computational domain consists of an Inlet channel upstream of the ste) and an expansion chamber downstream. The Reynolds number, based on the stepheight h and the maxinmum inlet velocity Uo, is Reo = 1.55x 10. In Figures 1 and 2 a comparison is given of our simulations with experimental studies at similar Reynolds numbers from Tropea (1982) and Durst and Schmitt (1985). Two streamwise locations downstream of the step are shown for the mean btreamwise velocity and Reynolds stress profiles. The predicted mean reattachment length of XR = 8.6 compares well with the exl)erimental values of XR = 8.6 (Tropea) and .CR = 8.5 (Durst & Schmitt), where the distance is meantLred from tile step location. In general, the mean velocity profiles show good agreement at all streamwise positions in the flow. The Reynolds-stress profiles show regi,s of larger disagreement however, the uncertainty i',the mne,surements is also remarkable. We note that tl,e location of the predicted maximum stress is further from the reattachment wall than experimentally observed and that the decrease in the stress levels downstream of reattachment occurs more slowly than in the measurements. In Figure 3 the mean streamwise and cross-stream yelocities are compared with instantaneous profiles of the two velocity components. In the figure the mean reattachment length is at .r/h = 12.6. In this region we note the large fluctuations in the instantaneous profiles, particularly in the cross-stream velocity component. The region of large departures from the mean corresponds roughly with the location of the shear layer. In Figure 4 the instantaneous, fluctuating velocity vector field downstream of the step is compared with tile fluctuating pressure field. A comparison of the figures shows that the instantaneous low-pressure points are the locations of large vortices in the shear layer. As the vortices move downstream of the step they become more inten. (stronger negative values of pressure) and interact with one another and with the reattachment wall. The turbulent production of these large, energy-beaing structures in tile shear layer far exceeds the production which occurs in the wall regions. This leads to a maximum in the turbulence energy iii the region of the shear layer which decays slowly downstream of the reattachment region.

NOMENCLATURE

h L

Reo TO U0 U. V,, III X,

chalacteli.tic length, step height coiputational donmain length, L = 24/i global Reynolds No , Rt 0 = Uoh/ii characteiistic tine, To = L/Uo upstrean channel centerline velocity velocities in the x, y,z directions ddhmeiiionless reattachment length btreaiwise, tangential, and vertical coordinate directions.

INTRODUCTION The fully-turbulent flow over a backward-facing step has been the subject of many experimental investigations becauseof its simple geoiietry and a well-defined separatioii point. It has also served as a benchmark test-case for the turbulence modellhng and siulation coniniun ties, In the present study we apply the large-eddy simnulationi (LES) technique to predict the developing turbulent flow. In this technique the large, energy-rich structures are directly simulated while the smallest, dissipati ve scales are modelled Siice the tinie-dependent, three-dimensisonal flow field is simulated, instantaneous distributiois of tile resolvable flow quantities can be obtained. In addition, statistical distribut, us of the mean velocity field and higherorder correlations may be computed. MATHEMATICAL MODEL Tile governing equations are derived by applying a spatial, lowpass-filter to tie Navier-Stokes equations for an icomliessible, constant-viscosity fluid. Among the various filter methods in use, we employ Schumann's (1975) 'volune balance procedure." In this case filtering is accomplished by integration of the governing equations over a (ontrol volume of amiequidistant, Cartesian, staggered grid. Thi leads to a set of urdinaiy diffeenLiai equations in time describing the resolvable (grid-scale, GS) flow quantities. In the present study a subgrid-scale (SGS) model of the Schumann-type (1975) is adopted to close the equations. This two-part eddy-viscosity model relates the SGS-stres3 to the fluctuating and statistically-averaged portions of the filtered deformation tensor. Details of the complete model are given in Friedrich (1988).

CONCLUSIONS The present study has focussed on the simulation of the fully-turbulent flow over a backward-facing step. Comparisons of the mean flow results with experimental studies show good agreement. Improvements are still possible with new SGS models and higher numerical resolution. Of particular value is the availability of the instantaneous veloc-

NUMERICAL PROCEDURE The momentum equations are integrated in time using the second-order leapfrog scheme with an initial Euler-step to start the simulations. The pressure field is computed

-I

II-

ity and pressure fields for inproved understanding of the physics of separating and1 reattaching flows,

FRIEDRICH, R. & ARNAL, M. 1990 Analysing turbulent backward-facing step flow with the lowpass-filtered

REFERNCESNavier-Stokes

equations. J. Wind Eng. and Ind. Aero.,

PIOMELLI, U., FERZIGER, J., MOIN, P. & KIM, J. 1989 CHORIN, A.J. 1968 Numerical solution of the Navieromptatons22,New approximate boundary conditions for large-eddy euatons Stoks Mahemticsof simulations of wall-bounded flow. Phys. Fluids A, 1, Stks4 qaios5Mteatc7f6osptt2n.2 1061-1068. TROPEA, C. 1982 Die turbulente Str6mung in Flachkaniilen und offenien Gerinnen. Dissertation, University of Karlsruhe, Karlsruhie, Germany. SCHUMANN, U. 1975 Subgrid scale model for finite difference simulations of turbulent flows in plane channels and annuli. J. Comp. Phys., 18, 376-404.

DURST, F. & SCHMITT, F. 1985 Experimental study of high Reynolds number backwaid facing stcp flow. Proc. 5ths Symp. on Turbulent Shear Flows, Aug. 7-9, Cornell University, Ithaca, NY. FRIEDRICH, R. 1988 Simulation of turbulent flows. Comiputational Methodls in Flowv Analysis (H. Niki and M. Kawahara, eds.), 2, 833.843. UTrope&A

13-

.e

Se4

.

.2

.1

.

5

.,

sea~~

6, a

.2

.3

A

6

.9

Trapa

ml*Schzrnitt

*Schmitt Simulation

-.

U / UO

1

/

Simulation

Trope&..pe

Schmitt

*e*Schmitt

iua~n'

Simulation

11U

III t1I

Fig 1 Mean streamrwise velocity profile, ut/U0 . a) xnh b) x/h =10. Position measuredl from step location

c"

2

0

4

10

=4.

(a)

5

8

10

14

12

16

IS

.

.........

.

..

-------------

CV

22

24

7C

-,....................
20

x/h

Fig. 3 a) Streamwise velocity, u. b) Cross-stream velocity, w. Comparison of mean and instantaneous profiles.

S..............

prssr

2

Fig. 2 Reynolds-stress profile. -0",01UO - a) x/h .r/h =10. Position measured from *.location.

=4.

7

~

10 - 9-

U

in anx-zplnt=)1A

...................

.

.

------

-2

b)

EIGHTH SYM~vPOSIUM ON TURBULENT SH1EAR FLOWS

11-3

Technical University of' Munich September 9-11, 1991 ON THE MODELLING OF TURBULENT SHEAR FLOWJS UNDER STRONG BUOYANCY EFFECTS

D.V.Bayandin"I, G.2.Fainburg*

I, I

.~rg

Perm Polytechnical Institute 29 A,Komsomoisky Ave.,614007,Perm,USSR Mining Inst. Urals Branch USSR Acad. Scl. 78A Karl Marx Street,614004,Perm,USSR IInstitute of Continuous Media Mechanics Urals Branch USSR Academy of Sciences. I.Acad-Korolyov Street,6l4061,Ferm,USSR ABSTRACT

field

This paper presents some results of wider programme of worl,

of turbu-

to obtain an understanding

lent shear rlows

under strong buoyancy effects

to develop mathematical

The purpose of this paper is to comare signi-

tion.

of

ficance

different

approximations

cf turbulent

heat flux and various descriptions of turbulent ternperature field.L.Inendnt

by only eddy Prandtl number

ekonapir.Tedsrpino

wihms

turbulent heat flux uses gradient approximati~on. The second model is four-parameter one of 'eddy

and

techniques for their predic-

is characterized

The turbulent energy E, tempera-

diffusivity' type.

and their dissipation rates b2and -.

ture variance e

are determined by differential equations.This allows to obtain the expressions for turbulent viscosity v t ,and turbulent thermal diffusivity t needn

description of turbulent temperature field and estiTURBULENCE

M'ODELS

mation of eddy Prandti number became possible.

Turoulent shear flows under strong buoyancy ef-Th fects are common

in both environmental sciences z,.d

irdustrial engineering but a general understanding of their chysics and adecuate modelling are still

thr moe isfr-amtrdieena-algebraic one with non-gradient approximations of ae isisptc uru1enhalfIxI-tn F -'(-

far from complete.

d

The :rrrncrpai turbulence models in use were dexeloped

for high Reynolds number

fl o ws

In

h ehscs two-parameter

.

model

.- r

exhibited a suirprising degree nf prediclive ability. F ,Inn urtcu i At buoyancy d riv en flIows how ev er the sit,,,'ion is quite different. due

the influence

ton

nhang ing

characreristics. flows

The main difficul ties are

of buoyancy

and

the flow behavior

sality

fact recuires univer-

for practical compuitations

purpose

of computers.

of physical and

and

approximation

heat

flIux,

Three different models were used. Tn'sfirst model is modified -Reynolds-number

I~model.

ensri

Turbulent temperature

cipation

-

at,

vertical vector,

tin

late

M-Vle

n

nte5aIIg-gaittoa

Tio

f < ut',YT

exp 4-5-

ceea

unit normal vec-

-

dx

X 6-boundary layer thickness,, s

-

normal coordinate.

Source terms in the heat dissipation equation are: -c

2

C 1,

.,

1

i' 1-

7,Tl.c e- al C,

~

n

,

i,

ci~)B2 Ti21

kI~

-empirical constants,,

AllIthe models are derived by the reduction method

from a full second-order mode,

the transport equations ments of velocity

two-pagimeter low-

/3)

-

for of solid wall, (4 -volumetric expansion coefficien.Teifuceowalsdsrbdbyuctn

two problems of dlo-

of turbulent

13 =b IR/E

tion, y-unit

with the present

sure technique: description of turbulent temperature field

of 41ad1

lX n

r 5ke, ,,

Reynocs stresses, L

cL

and zvv

For the attainment of our

we analyzed in detail

-6,

of-n. -

and high predictive ability

generation

D /D

transfer

commutational models, but also simpl icini bility

0ZU(Z

the heat

ciosed rectangular cavi'/ there are the narrow boundarl layers nownr the vvrtical walls anc uinder (over) the horizontal plate5, large stagnant core in.the middle of cavity, regions w.th stable, neutral and unstabe stratification. This

n r 2nk'.,

Here F.-

in particuliar, in a en-

c9D'eI+gI? a

ttE

strongly

differen, tvoes of interac:tions between furbuient

fields5 exist sinuitanerusly.

Q, -~IS

forces

Furthermore,

with various

I..e

n I

nnn 7n

In

-Z

forced convective

their gradients:

which includes

for second single-point mo-

and temperature fluctuations

and

the Reynolds-stress tensor, turbu-

lent heat flux, temperature variance

for fluids and

~liii

r

~~e -2I

__

2-Pr-7O

05

-7Yfl!T

0.5 Fig.

1.

Fig.

surrounding sol id and relative dissipation functions 111 the finite thermal conductivity of

In this model

the influence of the lat-

and

the surrounding walls

taken into account.

fluctuations are

ter on piessure

data. For the horizontal

flcows irnplane hcriiorta; anc

etn

Eaaios

ethc-is.

of stationary and ncn-staticvar/

were made ('23.7he inmerical

and

layers

enclosur es %.ere inv.estrgatvd by r.Jme,i-:

flows

enabled us

evperimvrta

of numrerical

7he rcsults

that a 1tine nydols tie

features of

c-alI ,ta-

latec b

t rar sf er

and

for

the rate

es 'or

if its grcwtt with hor,-con-

the increase of the Grashof number (Fig.: 'aver

tal

heated from beiow) ,

racterist_:s ,9!,,e as

o

U

are

wnEreav

Frandti nunzer, .-: , fy4.

he

slent temperature field descriotion Froduces the opee ent approximation

-) "re

wh il:e in csse

wes)kening and eden

thermal under

horizontal

possible

Onlv 'his

temperature gradient.

adiabatic

Reg ions

with heat transfer

are olealy

layer

advective flow

ideal ly

between

prediction

of turbulent

shear

f Ic.s

under

of

differentia;-lgebraic

few-parameter

models.

low-Reynolds /Feclet-number

to

REFERENCES

In the to

FAINSURG, G.Z.

1976 Towards natural

convection tur-

bulence equations of incompressible fluid /1 Hvdrodynamics (PGPI Scientific Report N 15:),

model

Ferm.

100-112 tin Russian). BAYANDIN, D.V,

turbulent

FAINBURG, G.2. & WERTGE!M,

Mathematical modelling

Russian).

This makes

-

in fluid layer

1. .

1938

of turbulent naturA] on-

section. Preprint fN83)881,

layers

horizontal solid boundaries (31.

between

growing direction

leads

in horizontal

to predict advective flIows

in near-

The most reasonable a,,-roach to problem of adequrate

of turbulent

'negative'

conductivity existing

maxima

1,2 - horizontal

tine gra-in-

3trs'.fwr-ation -

its dissipation rate.

alIlows to describe the effect

,prcai

be the use

in secono. model!

modPi wth non-gradient approsimatin)n heat flux and

with

thernocorductive and insulated boundaries).

are yielded by third

rsl

4,6 -

friom aside;

t:) tc'al damping of rirlbulenre.

Considerably better

third model,

region are presented

strong buoyancy effects at the corient time seems to

Pmoersture gradient

of stable

the

the

r* 15 the zanre

furojilent heat flujx

strong increase of mean

if

on Fig.2

number smace distrioutrons, caicu-

rn'e,-.dent tujr-

Furthermore

results. of

cnl!y

criteria,

heated from below, Pr=0.7.7; 3 -vertical slot heated

tnc other cha-

in good accord iitlf,

1'5 mean valve in rea

the

-wall

e'ample,

For

svace coordinates.

is non-

on flow type,

significantly

value of similarity

coundary conditions,

th

How-

for the

number

its value

that

data

but deuends

turbulent Frandtl

convectionf icws.

inra

the models used.

turbulent Frandtl

eacerimental

constant,

Indicate

e,,er t -,efir:stmodEl ov Er c;ed icts tie val nea t

existing

sru.atrcn,

redirt correc'lv

the natura

~

second and third models show in good accordance with

and

to test ciffcevnt c; suref.

was obt.Aired

simulation for all

Evaluationis of

convecti.'-

minima near 'hie

Rayleigh number values,

moderate

~two-drmensronai naztural

have an ex-

wich was observee in exneriments at the

CA:CLAT:N REPLT turbulent

layer heated from below, the

pressed maxima,and L profile Maxima of E.

exnerimen~al data for homogeneous turbulence.

Tne different

lie close to experimental

orofile in the third model simulation

-ompujtational 3nr

of

in the flow

temperature gradients

problems yielded

Soime closure coafficients airc deter-

mined from tinebest comparison

Inrall

the layer,, despite a stable stratification.

core regions are small and

is the basis of rlo-

The tenso)r apoxioation method s'lre procedire.

2.

The turbulence intensity is sufficient everywhere in

verdlovsk.

62 p. (in

Translation submitted for Heat Transfer

Soviet Research tin the press).

SAYANDIN,, D.V.

1988 Modelling of turbulent advective

toward

the temperature

flows in fluid horizontal

seen

near the walls.

ture Thermophysics, USSR Acad.Sci. 25, 1128-1134.

11-3-2

layers ?/ High Tompera-

11-4

TURBULENT SHEAR FLOWVS Technical Univeisity of Munich September 9-11, 1991

LARGE EDDY SIMULATION OF CUMULUS CLOUDS

J.W.M. Cuijpers *) and P.G. Duynkerke **) *) Royal Netherlands Meteorological Institute, P.O Box 201,

3730 AE DE BILT, The Netherlands **)University of Utrecht, Institute of Meteorology and Oceanography, P.O. Box 80 000, 3508 TA UTRECHT, The Netherlands

MODEL DESCRIPTION

ABSTRACT

The model we use is based on the large eddy model of A large eddy model used for studying the dry convective Nieuwstadt and Brost (1986), who simulated the dry boundary layer, has been extended with an equation for the convective boundary layer. In our model the liquid water specific humidity and acondensation scheme, to simulate the has been made A simulation layer moist convective boundary earPueto Rco n~potential temperature 01 is used as a conservative variable in onthe bsevatons basea tio based o n th e observ ns gaathred th ered near Puerto R ico on 15 t e t e m d n mc l e u t o .B c u e o h o l u o e the thermodynamical equation. Because of the low cloud cover December 1972. Starting from a clear air situation the model Demer 1972.totartingrom a wtth situation smdclearair e l it does not seem necessary to use adetailed computation of the evolves to a situation with small cumulus clouds. Verticalthermodynamical agreement profiles of vanances and fluxes show satisfactorily equation. Instead we fixed the radiational cooling to a with the expenmental data c,-nstant, which was chosen such that the radiational cooling balnces the vertical flux of potential temperature. This means INTRODUCTION that the surface heat flux will remain constant and a stationary situation will be reached atmospheric In buoyant updrafts of the convective 2000 boundary layer (ABL) the temperature decreases with height Provided that there is enough water vapor this leads to Z(m) condensation and the formation of clouds The latent heat that is released during this f.rmation enhances the buoyancy of the 1500 updrafts and thus the production of turbulence. So, turbulence generation and cloud formation are strongly coupled. 1000 Furthermore, due to the enhanced buoyancy clouds are able to penetrate into the stable layer capping the ABL. This causes mixing over greater depth than would be possible in a Soo situation without clouds. It is the purpose of this study to investigate the influence of non-precipitating cumulus clouds on the turbulence structure. 0

0005

0010

0015 q (kglkg)

Fig 2 The honzontally averaged profiles of the m,, 1humidity

0020

0025

total water

specific humidity at t=0h(full line) and t=3h(dash-dotted) with the observed values (circles). The dotted linc is the horizontally averaged profile of the saturation specific at The model has been extended with an equation for the

"

specific humidity (with the total water q, = qv + ql as conservative variable, where qv 's the specific humidity for --

10

"-

-o*

1s

20

25

30

vapor and qj is the liquid water specific humidity). The condensation scheme we use is the one described by Sommeria and Deardorff (1977), in which it is assumed that a grid volume contains no liquid water until the total water specific humidity exceeds the saturation value q. This means that a grid volume is either entirely spvirated or entirely unsaturated.

'

3s

time virs) Fig. I The temporal evolution of the maximum value of the horizontally averaged cloud water max (full tine) and the total amount of cloud water averaged over the whole domain 0.006. CONCLUSION The visualization data presented in this paper confinm the previous results obtained from hot film signals on the response of ejections and bursts to forced oscillations._When the forcing frequency changes from quasi-steadiness to Te4 the amplitude of the ejection frequency droos by a factor four to five from the quasi-steady value and the amplitude of the BSE's by nearly as much. The respon~se of the BME's on the contrary varies relatively little with the forcing frequency. The difference in tht; response of the BME's and BSE's suggests that they result from

REFERENCES 1. Bogard, D. 1982 PhD., Purdue U%. U.S.A. 2. Bogard, D. & Tiederman, W.G. 1986 J.FlaldMech. 179, 1 3. Cousteix. J, & Houdeville. R, 1985 5th Symp. Tu*rbulent Shear Flow, Cornell Univ., USA 4. Kobashi. Y, & Hayakawa. M, 1981 In Unsteady Turbulent Shear Flow, Ed. R. Michel, Springer 5. Tardu. S, Binder. G & Blackwelder.R, 1987 6th Symp.Turbulent Shear Flow, Toulouse 6. Tardu. S, & Binder. G 1989 7th Symp. Turbulent ShearFlow, Stanford U.S.A 7. Tardu. S, & Binder. G, 1990 Submitted to thel.Flwd Mech. 8. Tardu.S, 1988 Doctoral Thesis

different mechanisms.

_______________

ACKNOWLEDGMENTS 1o

Financial support from the European Research Office of

-

Group under contract DAJA 45-87-c-0015 is gratefully

acknowledged.0. 0.000 f-

0.002

0.004

0.008

0,008

Figure 4. Relaive arnipiiude of ejection lift-up modulation vsforcing frequency.U &Ci=0.2O;* &a--0.13; - j.I8(qs)

0.015.

I

a

0.010. 0

0,000

0,002

0.004

0.006

0,008

0

Figure 1. rtme-mem ejection frequency vsforcng frequency.

0 aacrO2O:9

aa7-O.I3;A

VrTAaa-O2O

301f

0.000

0,002

0.004

0.008

0.008

Figure S. Phase shift of ejection lift-up modulation vs forcing frequency. for legend seeFigure .6 1.2

1.8

O's-

1.4

0.4 0,000

0,002

000,4 0.0 * 0,004

&

& BMltMasts

1,0

a

A

0.6

A

+

BSE Measurements

f0.2 0.006

0.008

0.0 0.000

Figure 2. Relative amplitude of ejcirxifrequency moodulation vs forcing fr~uency For legend seeFigure 1.

.

0.002

0,004

0,0

0.0'08

Figure 6. Relaiivc amplitude of burst frequency mlodulation vs forcing frequency. A BEla 0.l3; * BMEa-O.20; * BSE sc-O.13; 0 B5a.-O.2O, x BMETardu&Bindcr * BETsrdu&Binder

200. ISO.U 100.

100

01

.10o030 -00r+f

000002

004

0.006

0.080000

Figure 3. Phane shift of ejection freqluencynmodulation vs forcing frequency.For legend seeFigure 1.

0.002

0.204

0.008

0.008

Figure 7. Phase shiftnf but freqluency modulation vs forcing frequency.For legend seeFigure 6.

11-6-2

EIGHT! SYNIPOSIUM ON TURBULENT SHEAR FLOWS Technical University of Munich September 9-1I, 1991

1

SIMULATION OF THE CONVECTIVE BOUNDARY LAYER IN COMPARISON TO AIRCRAFT-MEASUREMENTS DURING LOTREX-EXPERIMENT

.1.Graf, A.M. Jochum, U. Schumann DLR, Institut fur Physik der Atmosphiirc W-8031 Obcrpfaffenhofen, Germany ABSTRACT The convective atmospheric boundary layer during a specific case of Lhe field experiment LOTREX is investigated by Large Eddy Simulation (LES). The numerical simulations are nitialized using measured profiles of wind and temperature. The lower thermic boundary condillon is obtained by extrapolating measured heatflux profiles down to the surface. The calculated vertical profiles of turbulent quantities compare generally well with the corresponding measurements. INTRODUCTION During the last years several LFS-models have been developed and applied to various problems, but the simulated turbulent quantities are not very often compared with measurements In Deardorff (1974) the simulation results are compared with data from the Warngara-experiment. Moeng and Wyngaard (1989) found good agreement of the calculated velocity variance with corresponding field- and laboratory measurements in the upper boundary layer, whereas in the lower boundary layer differences have been obtained. The model used in this paper is validated for pure convective situations (Schmidt, da88The foru e octve suton (Shmidt 1988). The influence of the results on the parameterization scheme is discussed in Schmidt we simulate and Schumann (1989). In this paper BL or da. day. Daa Data fr for mdel model the CBL for the a speifi specific been have initialization and intercomparison collected during LOTREX. The series of field experiments LOTREX was conducted in North Germany to study land surface processes (Roth et al., 1988). The effect of Inhomogeneous heatflux, sheared mean-flow and vertical resolution on the turbulence statistic is investigated in Graf and Schumann (1991).

METHOD AND PARAMETERS

For the present study we use the LES-model as described in Schmidt and Schumann (1989). The model integrates the equations for mass and momentum balances and the first law of thermodynamics in three space dimensions and in time. Here, we are using 100 x 100 x 40 grid cells. The subgrid-scale fluxes are parameterized using a second order closure model. The computational domain extends horizontally over an area of 10 x 10 km 2 and vertically to a height of 2500 m ( 3 z,) The horizontal gridspace is 100 m in all simulations. Vertically, the gridspace is either 100 m or 50 m. The results after a total integration-time up to 5 t., when the turbulence is quasi stationary are compared with available measurements (Jochum et al., 1990). The experimental area is 15 km by 15 km, almost flat with varying landuse: essentially agricultural with field sizes on the kilometer scale. For the model initialization and comparison data from the quasistationary period around noon on July 13, 1988 are chosen. This day was characterized by light wnds and no clouds. The convective boundary layer has a height z of 800 m. In addition to 5 surface stations 5 research aircraft were taking measurements. Three powered gliders and a jet all equipped for turbulence measurements were simultaneously flying along the same horizontal flight legs with fixed vertical spacing. Vertical profiles of mean meteorological parameters and of selected concentrations were obtained fr'om a twin-engined aircraft. The data from the differica rto yil v are cmie aircraft en are combined to yield vertical proaircraft ent fieofmaprmtrsflxsndpcrl characteristics. The measurement uncertainty is estimated to be around 30% for fluxes, and 20% for variances. Scales included in the turbulent fluxes and variances range from 17m to 3.5km.

RESULTS Generally, good agreement Is found with respect to vertical profiles of the turbulent healflux, of temperature and vertical velocity fluctuations.

!1!1-7-1

-'

05

The sim-,.ated profiles of the turbulent quintities correspond with the measured profiles up to the error ot the experimental data. In Figure 1 the simulated and measured profiles of the vertical velocity variance are shown. Model and field data agree closely in the lower and upper thirds of the CBL. Around 4z, the model seems to underpredict the data In order to assess the sensitivity of the model two different surface heatfluxes (80Wm', 10OWm -1) have been used. Better agreement has been obtained using the larger value, which is closer to observed values.

2 78

;

2 11 I 1 89

Simulation

o

Zi 10-

05-

X×I° x°

1 0

X

253

!$

-

i -- . . . " ' "" zI 2 56

w

1.0

Soo

1000

~

1500 o

of The the the spectra wavelength as waoffunction Fig 2.: Maximum of zlz,. velocity fluctuations vertical dots denote the calculations (constant heatflux of o1.67 80Wmn ), the crosses the measurements.

Messung

.

CONCLUSIONS

144 0225g

10 078

\LES

0:6

\

033 011

A case study of a comparison between field and model data for an undisturbed CBL over

moderately

x

2

\ .the 0 05

10

inhomogeneous

terrain was

per-

formed. Generally, the model reproduces well observed turbulent fluxes and variances and

X

the spectral structure.

15 20 .25 30 35 .40 .45 50 .55 .60 .65 70 w 12

w* 2

REFERENCES

Fig 1. Vertical profiles of the nondimensionalized

vertical velocity variance in comparison to measurements for two different surface heatfluxes.

From the power spectra of the vertical velocity fluctuations one can see, that the smaller eddies are predominant in the lower boundary layer and that there is a second minimum at the inversion height (Figure 2) Obviously surface and inversion have a similar effect on the vertical motions. This reproduces well the same features seen in the field data. With respect to inhomogeneities, the measurements reveal a horizontal variation of the heatflux by 20 %. Corresponding to a unidirectional trend suggested by some of the data, we model the inhomogeneity by assuming a sinusoidally varying heatfiux, which varies in only one direction at a wavelength of 10 km Our simulations show a weak decrea, e of vertical variance over inhomogenous surfaces. The sensitivity studies show very little effect of the mean wind with a maximum of 5 m/s. This was to be expected, because the resultant friction velocity is still very small in comparison to the convective velocity.

ARDORFF,,.T.W., 1074: DF,

'Three-dirmncsional

numerical study of turbulence in an cntraining mixc(] layer. 3ound.Lay.Metcotl.7, 199-226.

GRAF, J., SCIIUMANN U., 1991:

Simulation der konvektiven Grenzschicht im Vergleich mit ilugzcugmessungen beim ,O'FRlO,X-xperiment. in press: Met. Rundschau. JOCIIUM, A., F NT1'S'I'RASSIFR, N., FIMP1, I, If., MORI,, P., ROSI,ER, F., WILI,IKF, 11., 1990: Fvaporation and energy fluxes in the Iayer during ,OTRIX. atmospheric boundary Proc. Symposium on FIFF, Fcbr. 7-9, 1990 Anaheim, Calif., Amcr. Mctcorol. Soc., Boston, 177-180. MOENG, C.]H., WYNGAARD, J.C., 1989: Fvaluation of turbulent transport and dissipation closures in second-order modeling. J.Atmos.Sci. 46, 2311-2330. ROT111, R., et al,, 1988: ,0''RFX 10 F /I 11I1F, 88. Fxpcrimentplan (in German). Report, Metcor. Inst. Univ. hlannover.. SCITMIDT!', I1., 1988: ',,obstruktur-Simulation konvektivcr Grenzschichtcn. Forschungsbericht I)FVLR-FB 88-30, obtainable from )IR. SCIIMII)'1', H1., SCUMANN, U., 1989: Cohercnt structure of the convective boundary layer derived from large-eddy simulations. .1. Fluid Mch., 200, 511-562.

11-7-2

TURBUILENT SHtEAR |rLOWS

]1]-8

Technical University of Munich September 9-11, 1991

LARGE EDDY SIMULATION OF TURBULENT FLOWS AROUND FLAT PLATE ROWS I.HARADA Energy Research Laboratory, Hitachi, Ltd. 1168 Moriyamacho, Hitachilbaraki 316, JAPAN

ABSTRACT A LES(Large Eddy Simulation) program based on a finite difference method is developed to compute the turbulent flow around or ina complex geometry. Aturbulent channel flow is computed to verity our LES program Amean velocity profile inthe streamwise direction is excellent agreement with the logarithmic velocity profile. Flows around flat plate rows are computed to understand the mechanism of aerodynamic sound generation. Results of both mean velocity and turbulent intensity agree with theory and experiment.

because other method such as FFT used by Horiuti(1987) and Moin & Kim(1982) is not able to applicable to the complex geometry due to its numerical method. Adams-Bashforth and central difference schemes are used for time and space derivatives using the staggered grid arrangement. An algorithm of the present solution procedure is based on the SMAC (Simplified Marker and Cell) method. Pressure is obtained from the Poisson equation solved by ICCG(Incomplete Conjugate Cholesky).

iNTRODUCTION Sound is generated by movement of vorticity predicted by Powell(1964) Inthis situation, the two-equation turbulence model is not adequate to simulate the flow due to its time averaged model. LES or DS(Direct Simulation) is applicable as higher turbulence models, because these models do not include time averaging in models. DS takes much more CPU time than LES due to a requirement of a fine resolution at high Reynolds number flows, so that LES is chosen as an adequate turbulent model. On the other hands, the let noise generated by high speed compressible flow has been investigated to clarify the mechanism of aerodynamic sound generation, but sound generation by low speed incompressible flows in such as air conditioners is different from that incompressible flows. Flat plate rows are chosen to simplify the geometry of wings LES is,therefore, made on aturbulent flow over flat plate rows as a first step to understand the mechanism of sound generation that is closely related to organized structure of turbulence

CHANNEL FLOW Aturbulent channel flow is computed to verify our LES program FLOWLENS (FLOW bv Large Eddy Numerical Simulation). The comr,, tatio;al c.ondition is the same as that inboundary layer splitting model by Harada (1988). A comutational domain IsLx(=3.2L), Ly(.0.SL) and Lz(-L) in x,y,z directions, where the channel width is taken as the characteristic length L.62mm The horizontal plane is divided into uniform mesh sizes of 64 gnd points ineach x ,ydirection The mesh sizes increase from the minimum mesh size near the lower or upper wall to keep a ratio of mesh sizes tobe constant without using hyperbolic function by Horiuti(1987) and Moin & Kim(1982). A set of values is chosen such as charactenstic velocity U=6.67nVs, friction velocity ur -0.309rn/s, Re=27800, C-0 10, and atime mesh t=0.008. Non-slip and cyclic cunditions are imposed on both plates z.0, L and planes x=0, Lx, y=0, Ly as boundary conditions. An initiai velocity is given by superimposing the uniform streamwise velocity U to fluctuations produced by random number smaller than 0 05U. The long term time integration is made until t=496 to reach astatistical equilibrium state. The iile integration of 62000 cycles required 762

BASIC EQUATION

minutes CPU time with 102MB on HITAC S820/60 super computer

Let us take characteristic velocity U iength L,time LU, nondimensional SGS(subgrid scale) eluations given by Deardorff (1970) are

(15 GFLOPS at the maximum speed).

au,

21 'Wi

X7 =020

ap_ a a-" + -,(ului + QX1 at SiJ=1 ,auL, u, ,3 au,

2 ax-

ax

a

1 a~u ex dXJ

.. ,,Z

2

'()2

2v:fl

"

,, (CA)'/s, , ,

(4) (.

where, uIs the velocity component of the coordir.ate x 1,p the pressure, I the time, A (= rA,A2&I)the averaged SGS grid interval, C(-0.1) the universal constant, Re(-UIJ Y)the Reynolds number defined by characteristic length(channel width or plate pitch)L , and , the dynamic viscosity. The turbulent viscosity Isgiven by the Smagorinsky model(1967). Van Driest(1956) wall damping function (A-25) Isused as the boundary condition of the wall. NUMERICAL METHOD A finite difference method ischosen for solving basic equawions,

/

wO

too

i0o0

Fiv.1 Astreamwise mean velocity. Figure 1shows the streamwise mean velocity averaged inx-y plane. The profile agrees well with both viscos sublayer a id logarithmic layer. The Karman constant K =0.40 and a constant 1-' 5 are good agreement with an experiment B-5.0-5.5. The same result x -0.40

11-8-1

and B=5.5 isobtained by Moin & Kim(1 982) using the residual stress model that is limited to application than the damping model. FLOWS OVER FLAT PLATE ROWS Flows over flat plate rows are computed using a similar condition to that of a channel flow. Figure 2 bhows a computational domain offlat plate rows The flat plate is placed at the center of computational

symmetric

Figures 3(a)and (b)show streaks visualized by marer particles generated at upstream in the vertical planes y/Ly=0.20 and 0.47. This corresponds to hydrogen bubbles visualization in experiment Fluctuations are observed behind the rear edge in the wake. The structure is similar to Karman vortex street growing in between flat plates. Figure 4 shows streaks in honzontal plane zLz=0.51 near the upper surface of the plate. The streaks show the typical structure of

cyclic condition

yondition

h

width

Fig 2 A computational dcmain of fat plate rows domain Lx x Ly x Lz. Tihe plate is defined by the length Cx, the.," thickness Cz and the infinite width in y direction hhe periodic boundary condition is imposed on x=0,Lx in the streamwise direction and on y=0,Ly in the spanwise direction The symmetry condition is given on upper and lower planes z=0,Lz. Table 1 shows parameters used for present computations The thickness ratio Cz/Cx is varied Table 1 Parameters used for present computations. Lx/L Ly/1 J --/ -L Cz_ C . Xstructure ( (-)I (On) (-) H (am) () direction CASI-i 3 i 2.36 63 1.58 2.36 1.0 ;. 6 ,,64 4 8 CAI 2

3 20 2.40

i2

1i61

(AS)4

3.1012.32

61

i 5G 12.32, 2.0

(

2.40

0

3 _4.8lZ2.36/ 63 1s 582.301.0

0

640W4x6

3 2

6 x48xG6

'

"

' Fig 4 Streaks in horizontal plane z+=28 from the upper surface peak and valley in turbulent boundary layer Figures 5(a) and (b) show streamwise velocity profiles averaged in horizontal plane of the wake The mean velocity of present results agrees well with theory. The Is kept until the rear edge where shear force reaches maximum in horizontal plane. This fact explains that the turbulent intensity becomes large near the rear edge in experiment[8]. The maximum turbulent intensity averaged in the wake region is about 4.0 that agrees roughly with 5 2 ofexperiment

1 6{ 98x,,64 4

1---~ .0

1.0

j

from zero o3 2/6 The resuls are essentially similar to each other so that the CASE-1 ischosen as the typical case The computational domain is divided by 64x64x68 grid points in x,y,zdirections. CPU lime required 789 minutes for a cycle number 46000 at 1-276 to reach

h 0.5

statistically equilibrium state on HITAC S820/60 super computer with 107MB using a time mesh A t=0.006 From a view of aerodynamic sound generation, velocity fluctuations

-

,!

0.5

--... ,,r 0

0.

in the wake play an important role,i.e Fukano et at 18] obtained by the experiment that the turbulent intensity in the wake well corresponds to the sound intensity.

(

25

uu Mean Velocity

0

2. 2.0

.0 4.0

T (b) Turbulent intensity

Fig.5 Streamwise mean velocity profiles of the wake As is shown by present computations, results of LES provide us a clue to the mechanism of sound generation produced by turbulence.

REFERENCES

-...

(a) .,

1 I,

y/LyO. 20 .Co -c ---..------

j (b)

,

..

,~~ I-

J

Deardorff,J W 1970 J.Fluid Mech.41,427-453 Fukano,T.et.al 1985 JSME(B) in Japanese,51,2505-2514. Haradat. 1988 Proc.3rd Int.Sym.on Refined Ftaw Modelling Van. 19 Measurements,69.73. 56 3.100 and Turbulence

HoriutiK.1987 J.Comput.Phys.71,371.39o Moin,P and Kim,J. 1982 J.Flud Mech.118,341-377 1967 Mon W.,ather Rev.91,99-164 Powell,A 1964 J.Acoust.Soc.Am33,177-195

~SmagorinskyS.

..[ . .L .,

:'j

"

1

"1Van

y/Ly=O 47

Fig.3 Streaks visualized by marker particles inthe vertical planes,

!1-8-2

Driest,E.R 1956 J Aero Soc.23,1000-1007

II

uuu

al'

i,t'

1 -9

"l-VNj

I~

Technical University of Munikh September 9-1I, 1991

NUMERICAL SIMULATION OF DENSITY AND VISCOSITY EFFECTS WITHOUT GRAVITY INFREE TURBULENT JETS G. Harran and P. Chassaing Institut de mqcanique des fluide *do Toulouse Laboratoire de Banlbve INPT-iNSEEIHT Avenue du Professeur Camille Soula, 31c)0 Toulouse (France)

Second order models were implemented by ChassaingH6rard-Chibat/3/, considering the Lumley's development

ABSTRACT A numerical experimentation is used to find out some density and viscosity effects In round turbulent heterogeneous isothermal and non reactive gas jets. The values of the jet/outer-flow density ratio vary from 0.4 to 4.88. NOMENCLATURE Ui Instantaneous velocity (I . 1,2,3 u'i Instantaneous fluctuating velocity (i = 1,2,3 ) p' mass fraction Instantaneous density C Instantaneous CI Mass fraction fluctuation x Longitudinal coordinate

technique. The treatment is restricted to the situation Ui., =0 which results from the high turbulent Reynolds number assump !on. The transport equations are solved numerically using the Patankar and Spalding's procedure. The additional terms, resulting from the choice of the Reynolds decomposition, have been included by new treatments of the "source" terms. COMPARISON WITHEXPERIMENTALDATA The code was first tested by comparison with the experimental results of Chassaing/4/ for a 80% of CO2/air jet discharging Into a quiescent atmosphere. Gravity acts in the same direction as the mean momentum and the exit conditions correspond to a fully developed pipe flow.

R Radius of the jet Subscripts: ax Values along the axis 0 Values at the exit ext Values outer of the jet

Reo - 55100 Re - 0.018 m

INTRODUCTION

Uo

To our knowledge, the effects of global density differences remain poorly characterized. References collected about different nature of gases, give conclusions cross-checking experimental and bibliographical results. Due to the difficulties of controlling the outlet conditions when dealing with experimental studies, a numerical approach is chosen, which allows us to study the influence of one parameter only.

uax 6

of CO2) Fro -35 Z0,,80% (concentration Vo - 0.85 10- 5 m2 /s Po/Pext-l, 39

4

CALCULATION PRESENTATION Because of the non linearity of the convection terms, new correlations are introduced when averaging the instantaneous equations of a variable density turbulent motion. Thereby, the

quantity pUjU3 becomes: M1,= p U,UJ=puiU,+p,u,

+p'u jUj+pu

0

100 80 40 x/Ro 60 20 0 Calculalion,O Experience) Fig 1: Velocity along the axis(-

u,U. + pu-

Except Lumley/l/, most of the authors use the mass-weighted averages as proposed by Favre in 1958. In 1985, Chassaing/2/ suggested an alternative to the formulation of turbulent motion equation for a variable density fluid, using the conventional averages. This leads to a similar formulation as in the incompressible case. The new proposition regroups Mij as: +Cij A+ Mi=A~ A= pUTican be considered as characterizing the mean contribution, considering that only mean values of the function appear. Bj=pu'iu'j Is a contribution which implies the fluctuating

su Ro 8

4

motion through a correlation related with a flux term by

classical advective non linearity. i = pu'iUJ +pu'jUi is a contribution that appears as the main component that singularizes the variable density fluid motion. It is henceforth considered as an "external flux" to recall that the equations have no longer a conservative form. Due to the thermodynamical state equation, the pu'i and F' correlation are exactly linked with the transportable fluxes, when considering the isothermal turbulent mixing.

00

2 Fig 2

Half width velocity radius.

According to the Imprecisions Of the parabolic

the calculation calculation, the and the calcuation and the Actinthe tagreement between

measurement Is fairly correct. Usir., this model and this numerical procedure, we shall now Investigate some effects of the physical properties of the jet on the turbulent mixing.

!!-9-1

:

decrease as Rp increases. This tendency cannot be observed with conditions the literature values because of the flows discrepancies(Fig.5). 0,15[

j..2

U0 0,12

B

I

B? 11 0,07

J

+

,.

+

0,10 I+ I

0,02 0

4

20

100

80

0 x/Ro 60

0,05_

Fig 3

longitudinal turbulent intensity a!ona !tie axis GLOBAL DENSITY VARIATIONS EFFECTS ON THE MiXiNG

,

with

RNq

and

....

"_____..

.

....... P /P ext 10 Density effects on the constant B

1

0,14 2 U0 0,10

0,0

x1, et Xc : vriual origins

Ro

R_

BB0,06 R

C3x

.

1 Fig 5

A strict analysis shows that, in a variable density flow. self-similarity is never reached. But, tar from lhe exit, when transversal density gradients are small ano if grdvity effects are negligible, the self-similarity condition is dpp cched. Introducing an equivalent diameter, Thring and Newtuy5/ deduced the following relations for the velocity and the concentration respectively: U = A U7.x

',

' values Present study I 0iLiterature

.-

..

.

Pitts

Chen and Rod/6/ give relations for the halfwidth of the velocity and the concentration profiles respectively: 8U = A' (x-x ) 8C = B' (x-x$) An other important quantity is the entrainment rate R:cou aind Spalding/7' snov that. with K = 0.28 for gas jets in ai Q = K x Q0 .vJn-R,

Fig 6 shows that lothigh density jets, the turbulent intensity maximum is blunted and retarded. Moreover, we have investigated the viscosity effects

From the experimental data of CO'.assaing/4/, we impleinent numerical experiences where aii parameters are fixed, except the density at the exit. We choose to negletL Lrayt. in computations to ensure the zimlr;1ty laws validity.

which are often neglected. We observed that increasing the molecular viscosity, tends to increase the similarity constants values and we noted also that the turbulent intensities grow up wi.,the viscosity.

-

15

p

-

compared with experience, some spec;ic density and viscosity effects were numerically found out without boya:scy For the density, we have shown that simiitude constants in good agreement with averaged literature values but decrease for hiqh density jets, And more generalt/, we obseived ihat the mixing is niuch better when the density ratio Rp is less than ore.

'C Iare

5 -

_,_,__

40 x/Ro 60 80 100 Fonthe mean velocity along the axis Fig 4 : Denit effects onhe180, The table 1 regroups the constants calculated with our results20

Table 1: Density effects on the similitude coants

R P _0,40 1 1,54 2.44

100 20 40 x/Ro 60 80 Density effects on longitudinal turbulent intensity

Considering a mudo, which results were previously

19! 3 6

I --

.

WUSO

05

3

10

0

0 Fig 6

(kg/m3)

UaX

0

0,02

A_' IB A 0.090 0.0976 0.125 0.086 0.096 0.0944 0.123 0.085 0.0912 0.116 0.083 00.0 4,88 0.0780 0.2 L 0 .9 Experiences = 0.075 = 0.11 I= O.0,,

0 152 0.140 0.131 0. .0 12

0 325

0314 0.297 0.292 0.292 0.250 0.282

First, we can see that results are in good agreement with the similarity laws since, from 40 Re, the evolutions are linear for the concerned quantilies(Fig.4). But, as experimentally observed by Pitts/8/ for the constant B(Fig,5), the values

REFERENCES 1/ SHIH T.H , LUMLEY J.L, JANICKA J. 1987 Second-order modelling of a variable density mixing layer J. Fluid Mech. 93-116. 2/ CHASSAING P. 1985 Une alternative A laformulation des Tquations du mouvement turbulent d'un fluide A masse volumique variable.J. do Mic, Th. et Ap. 4, 375-389. 3/CHASSAING P., HERARD J.M 1987 Second order modelling of turbulent shear flows 17-8. 41 CHASSAING P. 1979 M6lange turbulent do gaz inertes dans un jet de tube libre. Thbse INPT Docteur bs-Sciences . 5/ THRING M.W.,NEWBY M.P. 1953 Combustion length of enclosed turbulent jet flames Fourth Symp. of combustion, Piltsburg,The Standing committee of combustion 789-796 6/ CHEN C.J,RODI W. 1980 Vertical turbulent buoyant jets-a review of experimental data. Pergamon Press, New York 7/ RICOU F.P, SPALDING D.B 1961 Measurements of entrainment by axisymmetrical jets. J. Fluid Mech. 11, 21-32. 8/ PITTS W.M. 1986 Effect of global density and Reynolds number variations on mixing In turbulent axisymmetric jets. Rapport NBSIR, 86-3340, Dpt of commerce Washington.

11-9-2

EIGitT

SYmPOSIuM ON

TURBULENT SIIEAR FIOWS Technical University of Munich September 9-I I,1991

AN ASYMPTOTIC ANALYSIS OF TWO-DIMENSIONAL CURVED TURBULENT FLOWS AT HIGH REYNOLDS NUMBERS BY USING REYNOLDS-STRESS EQUATIONS B. Jeken Institut ffir Thermo- und Fluiddynamik Ruhr-Universitit Bochum W-4630 Bochum 1, Germany

method of matched asymptotic expansions and leads to a

SYMBOLS

four-layer structure(Fig. 1):

Re

A. Inviscid outer flow region B. Viscous superayer C. Inviscid defect-layer D. Viscous sublayer. Each layer has its own scales with respect to thickness, velocities and stresses and hence its own system

outer flow velocity chord length mol. viscosity friction velocity

U, L v u, U,0 L =

Reynolds number

of equations. The main analytic results of the analysis are as follows, see also Table 1.

INTRODUCTION Turbulent boundary-layers over curved surfaces have been calculated by many authors applying the so and various called boundary-layer 'approximation' turbulence models. Since the governing equations are free of curvature, and because of substantial disagreement with measurements, curvature corrections of different kinds have been implemented. However, no information is available,

51

,,d5 as

when and how curvature has to be taken into account for the calculation of turbulent boundary-layers. The purpose of this paper is therefore, to show by a systematic asymptotic analysis how curvature effects have

, s's ,\)?

"

-B

to be taken into consideration. ASYMPTOTIC ANALYSIS OF TURBULENT FLOWS This asymptotic analysis is an extension of the work by Mellor(1972). Consideration of surface curvature and the necessary turbulence model in the analysis are the main further developments of the above mentioned basic work. The closed system of timne-averaged Navier-Stokes equations and modelled Reynolds-stress equations after Gibson et at. (1981) are used to describe turbulent flows over curved surfaces. For high Reynolds numbers this system leads. to a singular perturbation problem. The perturbatioli parameter c= vcf0/2

Fig. 1:Four-layer structure of curved turbulent flow INVISCID OUTER FLOW REGION (scales: L , U10 , PU102 ) Assuming that the free-stream turbulence level is zero, the first order of the outer flow region describes the potential flow subject to the kinematic flow condition at the A.

wall. Since there is no contribution of the second order, the third order again describes a potential flow but with a nonzero normal velocity V3(S,0) at the wall, corresponding to the displacement effect of the defect-layer up to second order.

tends to zero wher, the Reynolds number Re goes to infinity, where cf 0 is the skin-friction coefficient at a fixed location So on the wall (e.g. start of the fully turbulent regime). The singular perturbation problem is solved by the

B.

VISCOUS SUPERLAYER , (scales : 1"= v/ve*, ve1 , pve )

Non-turbulent inviscid fluid of the outer flow region can become turbulent only by viscosity. While the outer

!!I-11

flow region and the defect-layer are not influenced by viscosity at high Reynolds numbers, both layers cannot be matched without the occurrence cf singularities The introduction of a viscous layer is therefore necessary to overcome this mismatch. The characteristic length 1"is proportional to the Kolmogorov length scale and the velocity vei* is the entrainment-velocity, which appears to be constant across this layer. The viscous superlayer is not influenced by curvature and can be solved universally for given entrainment-velocity and Reynolds number. The singularity defines a finite thickness 6(S) of the defect-layer. INVISCID DEFECT-LAYER 2 2ching (scales: b = L.-O() ) ,U 0 1 1 pu0 ) The first order solution coincides with the first order

C

of the outer flow solution for N -, 0. The values u,(S,n) and p,(S,n) are constant across the layer, whereas vi(S,n) varies linearly.all lInearly, In the second order the S-momentum equation is

linear and the normal velocity v2 (S,n) occurs only in the continuity equation. Curvature enters at this stage via the turbulence model. An essential requirement to the model is the capability to cope ,vith curvature effects. The dominant parts of the stress-equations are the production terms, which are, ir. case of a Reynolds-stress model, exact.

Now the second order of this asymptotic expansion is comparable to classical boundary-layer theory Without taking curvature into account this theory is incomplete as far as turbulent flows are concerned. In the laminar flow case curvature is a higher-order efftt and not included in classica. boundary-layer theory. Tv c ntrast to the different influenc of curvature, the displaccr.it effect is a higher -order eVect in both flow regimes. A Iharcteristic feature of the governing equations in the defect--layer is the linearity of the momentum equation The normal velocity v2 can be calculated separately after solving the momeitum- and stress--equations. The boundary conditions as n -4 0 are deduced from the macondition with the sublayer, which leads to the logarithmic law of the wall and the local equilibrium of production Pq and dissipation f in the overlapping region. At the outer edge (n = 6) the defect velocity u,and turbulent quantities go to zerk. or to their free-stream

values respectively, leading to a local discontinuity. Therefore,

following

Rotta(1983), the

,.,.,, LAYsR

production of the shear-stress and consequently the velocity

A

.. ..

field

,,,

,..,

... c

models) must therefore lead to unsatisfactory results.

D .

VISCOUS SUBLAYER D. (scales. i = v/u, 0 , u,0 , pu' 02 ) The universal sublayer solution for the streamwise velocity fi 2(Sfi) is linear for fi-,0 and logarithmic for . - oo, as has been shown by many authors

....

,.

J'!,

,|.,.

...

, ...

'"

.

.. 11

,.,,WI

.

.

,.."-*.;l'l, ,5,

models unable to describe the interactions between all

.,,,,,,, ,..

...

'-..',

..

All turbulence

-

sEcoo

free of any normal-stresses the Reynolds-stress equations account as far as curvature is concerned

is

The results show good agreement with measurements and with other Reynolds-stress model calculations

and together with the ss-component both determine the

clearly state that normal-stresses must be taken into

domain

scheme is based on Keller's box method.

Curvature alters the nn-component of the stress-tensor,

Although the momenturr equations themselves are

solution

d int oa ta ngular gri b latin do toi the fore numerica ( ofct he d t yer e ins l ol es bs o Kle bofethod. sch

..

,,,,,,,,

. lt.

..

f

........ ...114

ITable 1:

Asymptotic-Solution Scheme for Curved Turbulent Flows

The sublayer

as well is the superlayer, both very thin at high Reynolds numbers, are not influenced by curvature.

REFERENCES MELLOR,G.L. 1972 The large Reynolds number asymptotic theory of turbulent boundary-layers. Int.J. Engng

SOLUTION OF TtlE DEFECT-LAYER EQUATIONS The main interest is focussed on the solution of the governing set of equations in the defect-layer to second order, while the first order is completely solved without any turbulence modelling. As a result of the foregoing asymptotic analysis curvature effects only appear in the combination K(S).U(S,0), where U,(S,0) is constant with respect to this layer. Hence curvature has to be included if K(S) = O(1) and can be dropped if K(S) = O(c).

Sci.10,851-873. GIBSON,M.M.,JONES,W.P.,YOUNIS,B.A. 1981 Calcula-tion of turbulent boundary-layers on curved surfaces. Phys.Fluids 24,386-395. ROTTA,J.C. 1983 Einige Gesichtspunkte zu rationeller Berechnung turbulenter Grenzschichten. Z.Flugwiss. Weltraumforsch. 7,417-429.

11-10-2

EIGHTI SYmPOSIUM ON

II-11

TURBULENT SHEAR FLOWS

Technical University of Munich September 9-11, 199i

DIRECT NUMERICAL SIMULATION OF THE LOW PRANDTL NUMBER SCALAR FIELD IN A TWO-DIMENSIONAL TURBULENT CHANNEL FLOW N. Kasagi

Y. Ohtsubo

Y. Tomita

Department of Mechanical Engineering The University of Tokyo

ABSTRACT A direct numerical simulation (DNS), of the fully developed thermal field jn a two-dimensional turbulent channel flow was carried out The simulation was made at a molecular Prandtl number of Pr = 0 025 to examine the low Prandtl number effect on the thermal turbulence

NUMERICAL PROCEDURE

statistics. The cmpnutation was executed on 1,589,248 grid points using a spectral method. The statistics obtained include rms temperature fluc-

4580. Referring to the numerical procedure used by Kim et al (1987), _fourth-order partial differential equation for v, a second-order partial oifferential equation for the wall-normal component of vorticity w. an , the

tuations, turbulent heat fluxes and so on. They are compared with those obtained at Pr = 0 71 Instantaneous thermal fields are also examined to investigate the structure of the scalar field.

The flow geometry and the coordinate system axeshown in Fig. 1. The Reynolds nimbei based on the wall friction velocity u, and the channel half width 6 is set to be 150. The resultant bulk Reynolds number is

continuity equation are used to solve the flow field. A spectral rm thod, i.e., Fourier series in the x- and z-directions and a Chebyshev polynomial expansion in the y-direction, is used. For the time integration, the

NOMENCLATURE Pr Pr,

molecular Prandtl number turbulent Prandtl number

second-order Adams-Banhforth and Crank-Nicolson schemes are adopted for the nonlinear and viscous terms, respectively Once the velocity field is calculated at each step, the scalar field is obtained by integrating the energy equation. This is done also by the spectral method. Mereury is

Re,

Reynolds number, 2u,.6/.,

choser, as a test fluid, and the Prandtl number is 0.025. Iso-flux bound-

Re, u,,

Reynolds number, u,.6/v bulk mean velocity

ary conditions are imposed so that the bulk mean temperature lir, arly increases in the streamwise direction The local temperature iq defined

Ur u, v, w X,y, z

friction velocity velocity components in the x-, y- and z- directions streamwise, wall-normal and spanwise coordinates

as the difference from the wall temperature so that periodic boundary conditions are used in the streamwise and spanwise directions

o,

channel half width temperature , friction temperature

V

;

6 0

d

"

kinematic viscosity

;fluctuating component )+ normalized by the wall variables, u,, Y and 0, ),,. , ; root-mean-square value )'

z

INTRODUCTION

Fig I Flow geometry and coordinate system.

In recent numerical simulations of turbulent heat transfer in a channel flow of air (Kim & Moin, 1989; Ksagi et al., 1991), it was found that the temperatur- fluctuation was hifsly correlated with the stream-

20

wise velocity fluctuation. Furthermore, it wa. shown that the turbulent

Present

heat fluxes and the Reynolds stresses were also highly correlated, and

15

controlled by the same transport mechanisms throughout the flow field. As for low Prandtl number fluids, however, the balance in the transport

- --

Kim et o.(987)

to 1+

equations seems to be different from that of moderate Prandtl number fluids. Although this is an important question in turbulent heat transfer, the current knowledge is not satisfactory mainly owing to difficulties in performing exp-riments (especially, of liquid metals). The aim cf this investigation is to offer new information on turbu. lent heat transfer in very low Prandtl nimber fluids. Foe, s, we simulate

0. too

to,

t0

Fig. 2 Mean velocity profile.

a fully-developed turbulent thermal field in a two-dimensio.:al channel. Mean flow parai,.eter- ar- well as various statistics of the velocity and temperature fields ere,.culated and compared with previous experimental

RESULTS AND DISCUSSION

and numerical result- Ir particular, extensive comparison is made with the DNS data for Pr = 0.71 (Kasagi et al., 1991) in order to examine the

of y+(= u,y/i/)

lrcndtl number dependence of thermal turbulence statistics,

Y

The dimensionless mean velocity profile is shown as a function in Fig. 2, where good agreement is obtained with the numerical result at Re, = 180 (Kim et al., 1987). Figure 3 shows the

!1-1 1-1

distribution of dimensionless mean tkrperatuie The previous DNS data

(1)The effect of molecular conduction on the mean temperature profile

of Pr = C71 (Kasagi et al,1991) and the K vier's formula (1981) are also included n the figure. In the case of Pr = 0 71, the logarithmic region is visible, but for Pr = 0 0 5 it does nu exist owing to thickening of the ,onductive sublayer D1e calculated lNu3selt number agrees well with the value reccrnmended by Kays & Cr,iwfoid (1980) ."igure 4 shows the root-

is appreciable even in the channel central region and the thermal logarithmc region does not exist. (2) As the Prandtl number becomes very low, the magnitude of temperattre fluctuation isdecreased with its peak value moving away from the wall, and the anisotropy of turbulent heat fluxes isalso decreased

mean-square temperature fliu-tuation 0,+,. It is found that the lr-ation

(3) The turbulent Prandtl number for Pr = 0.025 reaches its maximum of about 3.27 at y+ = 37. It is much larger than that for moderate P,andtl number fluids.

of the peak value moves away from the wall as the Prandtl number is decreased Figure 5 shows the distribution of streamwise and wad-noimal turbulent heat fluxes. In the case of Pr = 0 Z1,the wal-normal heat flux is by an order of magnitude smaller than the streamwise one However,

(4)Much difference is observed between the thermal streaky structures

the difference between them becomes relatively small when the Prandtl number isvery low. The turbulent Prasdtl number Pr is shown in Fig.6

visualized at Pr = 0.71 and Pr = 0.025. Thse in the low Pr fluid are thick in size and not so much elongated in the streamwise direction. The authors acknowledge the financial support through the Grant-

For Pr = 0 71, Pr1 varies moderately across the channel and reaches its maximum of about 1.1 near the wall. For Pr = 0 025, however, it

in-Aid for Cooperative Research (No 02302043) by the Minitry of Education, Science and Culture.

increases rapidly from the wall and gives a maximum value of 3.27 at V4 = 37. Figure 7 shows contour surfaces of 0' = 0.15 at Pr = 0.025 in

REFERENCES KASAGI, N., TOMITA, Y & KURODA, A. 1991

an instantaneous computational subvolume, while Fig. 8 shows those of

"Direct Numerical

Simulation of the Passive Scalar Field in a Two-dimensional Tur-

'+

0 = 4.0 at Pr = 0.71. Although the so-called streaky structures clearly exist in the temperature field for Pr = 0.71, similar structures appear

bulent Channel Flow", ASME/JSME Thermal Engineering Proc. 3, 175-182.

much different in the case of Pr = 0 025. They are thichker in diameter and not so much elongated in the streamwise direction It is also noted

KADER, B A. 1981 "Temperature and Concentration Profiles in Fully Turbulent Boundary Layers", Int J Heat Mass Transfer 24, 15411544. KAYS, W. M. & CRAWFORD, M. E. 1980 Convective leat and Mass

that the angle between the high temperature mass and the wall islarger than that for Pr = 0.71. CONCLUSIONS A direct numerical simulation of a fully developed two-dimensional channel flow at Re,,, = 4580 and Pr = 0.025 has been carried out. Various turbulence statistics are calculated and the following conclusions are derived

Transfer, 2nd Ed., McGraw-Hill, New York. KIM, J , MOIN, P. & MOSER, R. 1987 "Turbulence Statistics in Fully Developed Channel Flow at Low Reynolds Number", J. Fluid Mech. 177, 133-166. KIM, J & MOIN, P 1989 "Transport of Passive Scalars in a Turbuhtnt Channel Flow", in Turbulent Shear Flows VI, Andr et al eds., Springer-Verlag, Berlin, 85-96 4 -P=.2

30

30 Pr=O 025 =Pry+ ...... Koder: Pr=0702

" cs2 0

-"

/

N,

0 20c

s:=

71

---

70

.

1.0

3Pr=O.7

2 0.0~

07 Fig. 3 Mean temperature profile.

0 0

0.2 Pr=0.025

0

50

y+

700

150

Fig. 6 Turbulent Prandtl number.

'

0

. . ' ' 0 50 + 00 150 Fig. 4 RMS temperature fluctuation.

/

0.2

-- j

Fig. 7 High temperature regions in the lower half volume

6

of the channel (Pr = 0.025).

2 O.'' ' " 6

. 50

yf

t0

150

Fig. 8 High temperature regions in the lower half volume

Fig 5 Turbulent heat flux profile.

of the channel (Pr = 0.71).

!!-11-2

EIGhI'Vh SYMPOSIUM ON

11-12

TURBULENT S1tEAR FLOWS

Technical University of Munich September 9-1I,

1991 TURBULENCE

IN

K. Kikuyama*),

ROTATING

CURVED

CHANNEL

K. Nishibori**), and T. Maeda*)

*)Mechanical Engineering Department, Nagoya University Chikusa-ku, Nagoya, Japan 464-01 **)Mechanical Engineering Department, Daido Institute of Technology Daido-cho, Minami-ku, Nagoya, Japan 457

ABSTRACT Effects of channel rotation and curvature on the turbulent motion are examined experimetally. Flows in the channel are stabilized or destabilized corresponding to the exerting direction of the Coriolis or centrifugal forces.

for the channel rotation Ric =

(2U3(Ur)/By/(r3U/ay) 2

(2)

and for the channel curvature, Rir =

-2Q(aU/ay - 2Q)/(3U/3y) 2

(3)

NOMENCLATURE where the positive or negative Richardson number corresponds to the suppression or promotion of the turbulent motion, respectively. Then, the mixing length in the rotating curved channel was assumed to be a linear summation of these two Richardson numbers as(Johnston and Eide, 1976)

1 mixing length lo value of I on a flat stationary plate N rotation rate(Eq. (1)) r local curvature radius Rir: Richardson number for channel rotation Ric:, Richardson number for channel curvature Ue : velocity outside of boundary layer Um : longitudinal mean velocity u,v,w:, turbulent velocity components Q : angular velocity of channel

I/lo = I -

6c Ric

- $r Rir

(4)

and the validity of the above relation is examined in this paper.

INTRODUCTION

RESULTS OF EXPERIMENTS AND DISCUSSIONS

In curved rotating channels such as impeller passages of turbomachinery, turbulence exhibits a quite different nature between the pressure and suction sides due to the Coriolis and centrifugal forces exerting on the flow. This paper concerns the stabilizing or destabilizing effect of these forces on the boundary layer inside a rotating curved channel and the relation between the mixing length and Richardson numbers is discussed,

ROTATION RATE AND RICHARDSON NUMBER

Time-mean and turbulent velocity components Figure 2 shows the profiles of time-mean velocity and turbulent components in the convex side of the section of F " for N = 0 and 0.15. The turbulence intensities are largely suppressed when N = 0 and 0.15, but increases when the channel rotates at N = -0.15. The stabilizing andldestabilizing effects of the Coriolis force are moie clearly seen in the distribution of Reynolds shear stress component of (-5 ) as shown in Fig. 3. In the concave side the tangential stress (-v) for N = 0 and 0.15 exhibits large values by the destabilizing effect of the centrifugal force but is suppressed by the Coriolis force when N = -0.15. On the other hana in the convex side the turbulent motion is largely suppressed by the centrifugal force in the stationary state but destabilized when the channel rotates in the counter-clockwise direction, N = -0.15. Mixing lngth The values of the mixing length 1 determined from the Reynolds stress and time-mean velocity are plotted in Fig. 4, which shows that thmiiglnhdensonhewlcuaurad the mixing length depends on the wall curvature and rotating direction of the channel. The relations Richardson between the mixing length and the numbers are examined using the relation of Eq. (4).

The magnitude of the channel rotation wa expressed by the following dimensionless p6rameter

1/lo + BrRir) and (1/lo)r (= 1/1o + 6cRic) against Ric and Rir, respectively. The values of Br and Bc are determined experimentally to be 2.4 and 3.0.

EXPERIMENTAL APPARATUS AND METHOD Figure 1 shows the configuration of the channel, which has an aspect ratio of 4 and is mounted on a rotating table. Time-mean and turbulent velocity components were measuied at the mid-height of the section by traversing hot wire probes. The outputs of the hot wire anemometer were amplified in the rotating system and transmitted to the stationary system through slip rings. In order to make the boundary layer turbulent, tripping wires of 1.5mm diameter were located at the inlet of the curved channel. t expnlerimfe wrved coanced. agradient Experiments were conducted at the Reynolds numbcr of Re(= UmD/v ) =104 and rotation of N(= OD/Um) = 0, and ±0.15.

N N -

number

Dm D/Um

where N is defined to be positive or negative according to the clockwise or counter-clockwise rotation of the table, respectively. The effects of the centrifugal and Coriolis forces on the turbulence are expressed in terms of Richardson numbers given by the following equations(Bradshaw, 1969):

Figures 5(a) aiud(b) :ahuw the changes of (iilo)c (=

As seen relation

from these figures, the validity of the of Eq.(4) is better in the concave side

rather than in the convex side.

CONCLUSIONS (1) The effects of the Coriolis force on the flow become dominant when the channel rotates counterclockwise so as to destabilize the boundary layer ll-lZ-

in the convex side and stabilhze in the concave one. (2) The modified mixing length model is found to be feasible on the concave surface when the flow is

subject

both

to

the

Coriohs

and

x10-1 1.5

o0

Concave Surface

centrifugal

forces. REFERENCES

1.0

.

r Bradshaw, P., J., Fluid Mech., Vol. 36 Pt-1,(1969), 177.

J. ASM,

Jphnston, J.P., and Eide, S.A., Trans. ASME, Fluids Engeg., Vol. 98-1,(1976), 374.

I F' oConvex Cove

0.5

0.150 0.075

0 0.0 0.07 a

r

-0.075 m -0.150

0.5 1.0 1.5 Distance from Wall yi/6, y2/6

S-I(x/D-1.9y 2

-

S- (x/D-3.6) Fig.4 Changes in mixing length

Y ROTATING DIRECTIONI

Q

2.0 -

cR tc

(k/Jt0)cFig.1

N

o

0.0 0.0

TRIPPING WIRES x/D-1.4 AIR

to

J

A

Configuration

of rotating channel

O

u

"Convex "

Surface

Pa

a 0.150 a 0.075 0 0.0 & -0.075 EN-0.150

1.5 -

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ifo

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0

I

g-

02

.0

~

Concave Surface

150

1.0

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h r

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0.0

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rm

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A A &nf

m

0.0

-u.0

0

-2.0 0.0 2.0 Richardson Number for Centrifugal Force Ric;

a~

~

Co 0 (0

C

4.0

(a) Changes of (l/lo)c

0a

.0.0or o3

bl e c x0

§_1P0000

men a0s n

150

05

10

C- 1

Distnce romWall yji

1.5

2.0 CU

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A

00.0 1 aej.r-0.150

~

0~ .0

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0

A

.0

C- 0.150

.5

-0150

AA

N

~rhj-N

0.5

- 9os30,o

Cce Surfaceo N is0.150 A 0.075

A

A

0

A

w-0.150 0.0

Srae-4.0

I -2.0

I

0.0

2.0

4.0

Richardson Number for Coriolis Force Rir

-2.

0.0

& -0.075 2m

-

I

Q)

a0.7 0.0

1.5 -A

~1.0

X10-1

4.0 0

U

a

(1/ter-1- rRir

6

Distributions of time-mean velocity and turbulent components in the convex sideA

Fig.2

Convex Surface N

0

0.5

1.0

1.5

(b) Changes of (1/lo)r

1

Distance from Wall y/6, Y2 6 Fig.3 Distributions of Reynolds stress

component

Fig.5 Relation between the mixing Richardson number

11-12-2

length

and

EIGTH1 SYMPOSIUNM ON

11-13

TURBULENT SHEAR FLOWS Technical University of Munich September 9-1lI, 1991 LARGE SCALE VORTICES OBSERVED IN RIVER FLOW R. Kinoshita*), T. Ptaui":

5

) and T. Ueno**)

5-30-11 Asagaya-Kita, Suginami-Ku, Tokyo 166, Japzn *9 Ujigawa Hydroulics Laboratory Disaster Prevention Pescarch Institute, Kyoto University Yoko-Ohji, Fushimi-'u, Kyoto 612, Japan

ABSTRACT Aerial photographs of a flood flow were analyzed using picture processing to obtain two-dimensional distrihution of velocity vectors on the water surface. Using the obtainpd data, the structure of a train of large scale vortices appearing near the edge of a flood plain and its relationship to sediment deposition which results in a spanwise shift of a river channel are examined. INTRODI'CTION Kinoshita (I°'F4) surveyed the sediment deposit'cinon the- !iod Dlain f th-' ! hi'0ri Riv-r -.st aft-r a major flood in 11l. Some of the results ar? shovn in Fig.l. It is noted that th-re is hsavy sediment dcposition elo,,g the bank of the L-ep river course (main channel) downstream of the convex bank. A train of large vortices was found in the ae-ial photographs (Fig.2) of the flood flow ,-t that pnint. In Fig.2 broken lines mark the bank )f the deep river course, Trains of large vortices are generated by the ,rtraction ketween the main channel flow and the flow on the flood plain. When the river is meandering, the point where the vortices are gen' eratod is fixed en- the vnrticos are sometimes enhanced to reach the scale of the river width. These vortices cause a large amount of sediment deoosition along fhe benk of the main cnannel, which results in a spanwise shift of the river channel. In t is paner, the aerial ohotograohs of flood flow /er- Analyzod using picture processing to ohtain .o-diriensional distrihution of velocity vectors on the water surface. The structure of a train of large vortices appearing near the edge of a flood plain is elucdated and its relationship to sedimnt depositon is discussed, PICT!'RE PROCESSING Fig.1 shows the land configuration of the Ishikari River npar Uraus, -here moandering of the main chann.=l is noticeahle. A rectangle outlines the analyzed area. Tie density distributions of a pair of ohotograohs of the flood flow taken over a short san of time (about 4 sec) w.re digitizec using a POqSmirro-densitometer. Two-dimensional crossu

ir:Lkr

Crej~ f the

den

it

u-t

bu

lines and the boundary of the main channel by solid lines. Streamline are noticeably wavy around the vortices in phase with the vort'ces arrangement. Fig 4 shows the streamlines observed from the frame moving in the mean velocity in the main channel. The regions where the absolute value of divergence exceeds 0.01 (1/s) are also shown in this figure. It is noticeable that the streamline patterns coincide fairy well with the outline patterns of vortices, which suggests the vortices are convwct-d downstream ith the mean velocity of the main channel. It is also noticeable in Fig.4 that div-r-,nce has a positive large value inside the vortices, and around the outline of vortices where strp-c linns convorse, divergence is negativc. This means strong bols exist inside the vortex and sinks around the outline. Fig.5 shows the map of vorticity calculnt-d using the obtained velorliy. The vorticity is concentrated locally at sc,.e points, suggesting the existence of small srpli vortices. As the small scale vortices are convected downstream, those which are adjacent each other gather in groups and the peak value of vorticit decays, resulting in a larger scale vortex. LARGE SCALE VORTICES AND RIVER CHANNEL FORV:ATICM The conceptual patterns in Fip.4 and Fig.5 can be given as Fig.6, which sugges' mechanism of growtn of the deposition on thslope of the main channel. The hishily concent, - d flow carryinp a suspended load (seliments) near the river bed is lifted up by boils inside the vortices. Near the water surfe-e, they are transported townrc's thi outline of the vortices and then sink down. As flow velocity npar the bank slope is low due to large scale vortices, sediments are deposited there. Thus, deposition on the bank slope of the main channel would increase. This prccess occurs on the bank just dowr- 'eam of the convex bank. The bed of the opposite bank is scoured as flow velocity is highir. Thus, tne river channil shifts spanwise. Considering that, on the flood plain, the flow velocity is low and the flow is clear, Fig. 6 also shows why large vortices become visible in the river flow. REFERENCES

-o

over a small area of the pair of photographs to calculate a two-dimensional vistribution of volocity vectors (Utamri,Blackv,ider P, Ueno 1991, 'inoshita, Utami 9 1leno 190). The obtained vectots are shown in Fig.3 together with the streamlines that were calculated using the data. CP'P-rCTFRTSTICS (F TP' 'ORTFXRTrICTURE Tn Fig.3 and the followin; fi,,,ures, the cutlin,. of lare scale vorices are marked by shadowed

KI,;C'

1

TA

.

1i

034 Po

.

itrE,

!!.34

11-1,

(in Japanese). KINOSHITA, R. 1988 In 'Control of flood flows in fluvial rivers and improvement of safety of river channels during flood (ed. T. Kishi)', 55-68 (in Japanese). KINOSHITA, R, UTAMI, T 9, UENO T. 1990 J. Jaoan Soc. of Phntogrimmetry ane Ppmote Sensing, Vol.29, No.6, e-117. UTAI, T, BLACVIELER, .. F 8. UE;,C-, T. 'S91 Experiments in Fluids, 10, 224-229, 1991.

PtS

Lft

t-

LO ,e.. t

r , .1

**#

a

Ifl

~

I

to C

.,a**

I

a44

40~~~~~~~,

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l4

o)0*

pgpppLh~~ k.L ,.,ep##P~~pit

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PDtt

C CtPP

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40

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>

S0C a,~

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C)

C.- C >0o

-

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11- 13-2

OO

ncIGrIF SYVMP0SIUM~ ON TURBULENT SHEIAR FLOWS

- 4

Technical University or Munich1

September 9-11, 1991

DOPPLER RADAR INVESTIGATIONS OP INTERMITTENT MESOSCALE TURBULENCE IN THE TROPICAL ATMOSPHERE

I.N.Klepikov, '.S.MoiSeev,

I.V.P krovskaya

and L.A.Sharkov Space Research Institute Profsouznya str., 84/32 117810, Moscow USSR

Investiaiations ot convective atmo-I sphere turbulence in 'he scale range above that of the small-scale turbulence are presented.I Experimental measurements of well developed convection in the tropical atosphere ( in Western par t of Pacific Ocean) by' Doppler radar set based on the ship were made. The computer-oriented experimental instrumentation gives the ou tpu t informations in the forms of lields of Doppler velocities on cloud turbulent drops trapped by atmospheric flows. Moment,, up to order six, of the longitudinal structure function of the stre,,mwise Doppler veiocity fluctuations3 for two-dimensional have been measured Doppler velocity maps made in operating tine 05cer 1. sin. Investigation of the variaoilities of temporal s~patial And the structure function due to mesoscale

cided evolution from its Appearance to 1,.e cIou c: collapse il1,5-2 hoirs ) are presen ted .

An. important Special detail cf tnetubln str Ct,.re Tu nctions cons.s' in nonreQular svat~al an d temrporalI variabil ities. TnL slIouce of thei structure fnrctiur (moment, uIP to the order Oix) doe,, nct agree with Kolnogorov s model I k-l!. log-normal model, b-model anid/ nul*,-fr,.ctal models. Distircui5sed Cor'elte0 scales (Spatial r" ,a1u, ga In st a ba c a0 our.d ont ilc .t I ,o variability, t he ir/ cervr Aco r, and t re-1 mevo ti10., do)tsh *oar 'lc-e scolec re Sc a~cadel mO 1toc, l cae (Kclmoqorov s Qa;Scodvl nay be observed. During of theI )roi 'h c' mc-oscale clIoud and the ir colla3pse the structure ucto cocome ,p,.cal of the like 'whi~e nos? random process. Analysts of the aitt tude Ceoendtences of the structure function ;*1ows that mesoscale turbulence has a lit-layer Lharacter. The reasons of such strong intermittexicy of atmosphere turbulence 'Aill1 be discuss.ed. It will be shown that multiplicative noise And its intluence on large-scale kinematics are of great

importance. It results in faster variabilities of high-order veloci ty structure

functions

owing

to

tuations of helicity and viscosity,

fluc-

aI

r2 o

'

4 103 10,-

4 2

R,m

'00

Fg

tutr

(n=6),

multiplied ucin by

ucino

truetsaeR

-drD

5i.1.Srctr iodr the 5,o with

I

sce R in 4mi.Vria .V rca lines Indicate 90%, confidence limits.

6

S

3

Ahcei' 7

Cnr

2

/ C -

--

-

o--

.

G*

i-.Vra

on

fscl

ex

nnt

n

s raino cl xoetr ni..V a a function of the oder n, I - Kolmo, orov model, 2 - log normal, 3 - P-model when D=2.83, 4 - multifractal when D=2.83, 5-7 - experimental results for R=5 km and In 4, 31 and 71 min.

EIGIlTil SYMIPOSIUNM ON TURBULENT Sh1EAR FLOWS Technical University of Munich September 9-1 I. 1991

11-15 THE COMC!NED WAKE OF AN ARRAY OF OSCILLATING RIBBONS D. E. Parekh*) and P. Pulvin**)

*) McDonnell Douglas Research Laboratories

Saint Louis, Missouri 63166 USA *') Ecole Polytechnique F6d6rale CH-1015 Lausanne, Switzerland

ABSTRACT This work considers the combined wake of an array of oscillating ribbons oriented parallel to a uniform flow. The characteristics of this wake are documented with flow visualization and hot-wire anemometry for various excitation frequencies and amplitudes, INTRODUCTION The wake behind multiple bluff bodies exhibits characteristics and vortex-shedding patterns that can be similar to or quite different from characteristics of the wake from a single body. An understanding of the wake behind multiple bluff bodies is important for analyzing flows about heat exchangers, skyscrapers, marine structures, and ribbon parachutes. Williamson (1985) showed that the symmetry of wake patterns and the synchronization of the vortex shedding behind a pair of cylinders depends critically on the gap size between the bodies. In their studies of flows behind arrays of flat plates oriented normal to the flow, Hayashi, Sakurai, & Ohya (1986) found that the gap flows are biased to one side or the other in a stable manner. Higuchi & Takahashi (1989) demonstrated that arranging flat plates so that the plates form a porous curved surface normal to the flow has a stabilizing and drag-reducing effect on the wake flow relative to an arrangement with no curvature. In this work we consider the combined wake behind an array of oscillating ribbons oriented parallel to the flow. This work is particularly motivated by the need to understand and control multiple wake interactions and by the need for a well-defined vortical disturbance for boundary layer transition studies. In his analysis of the receptivity of a laminar boundary lyer to free-stream turbulence, Kerschen (1990) models the free-stream turbulence as a superposition of periodic vortical disturbances of varying wave number and orientation. Parekh, Pulvin, and Wlezien (1991) found that an array of oscillating ribbons can produce a controllable vortical disturbance useful for receptivity experiments. This work focuses on the far-field characteristics and vortex patterns of the combined wake of an array of oscillating ribbons, APPROACH The facility used in this work is a low-disturbance, closed-return, blower-driven wind tunnel. The test section of this tunnel measures 0.91-m wide by 0.66-m high by 5.5-m long, and its inlet contraction has a 9:1 area ratio. The typical streamwise velocity fuctuations from D.C. to 10 kHz are about 0.1% of the freestream velocity, and the transition Reynolds number on a polished flat plate in this tunnel is over 2.6 million,

A schematic of the array of ten nbbons is shown in Fig. 1. The 0.051-mm-thick stainless steel ribbons span the entire width of the tunnel. The vertical spacing, s, between adjacent ribbons is 4.76 mm, and the total height, H, of the array is 43 mm. The local area blockage associated ,Nith this array is 1.07%; however, the total blockage is only 0.077% of the tunnel cross-sectional area. The relative streamwise position of the ribbons can be adjusted to produce periodic gusts of various inclinations. When all the ribbons are aligned vertically, as they were in this work, a 0-degree inclination of the vortical disturbance is produced. A pair of electro-magnetic shakers drove the ribbons in phase. The fundamental resonance frequency of the ribbons, which was set by adjusting ribbon tension, ranged from 90 to 110 Hz. By using excitation frequencies, f, below this fundamental resonance, nearly uniform motion of the ribbons was obtained over their entire span of the tunnel. The tension in the ribbons was sufficient to avoid aerodynamically induced oscillations at flow velocities up to 10 m/s. Velocity was measured by an x-wire probe mounted on an automated traversing system utilizing microstepper motors and a precision linear position indicator. Traverse control and all data acquisition were performed on a VAXIab data-acquisition and processing system. The hot-wire signal was directly digitized and linearized using a fourth-order polynomial and temperature compensation. Accelerometers placed on top of each drive-block assembly provided a detailed measurement of the motion of the ribbon array. Visualization of the wake behind the ribbon array was achieved by strobe illumination of a compact stream of smoke introduced upstream of the iolet contraction. RESULTS With no excitation, the flow behind the ribbons appears undisturbed. With excitation, the individual wakes of the ribbons rapid.y merge to produce a combined wake. The general shape of the combined wake resembles a sine wave as seen in Fig. 2. Visualizations at different excitation amplitudes and fixed Reynolds number, Re, indicate that the wake possesses a similar shape for ribbon acceleration amplitudes, A, up to 78 m/s2. Reducing the flow velocity below about 4 m/s, while keeping the frequency and absolute amplitude fixed at f=70 Hz and A=49 m2 /s, respectively, results in vortex shedding patterns similar to a Karman vortex street (Fig. 3). To characterize the combined wake, a set of velocity profiles spaced 20 mm apart in the streamwise direction was measured. The mean wake profile has a symmetric

II-15-I1

velocity deficit which decays with x, resulting in the

Dnve-b k

acceleration of the flow surrounding the wake (Fig. 4). The amplitude of the fundamental component of both the streamwise and normal velocity fluctuations, u' and v, remained nearly constant with x up to the end of the measurement domain, which was 1 m downstream from

the ribbons.

bly

_

_910,_,_

OuK S-s

The. variation in y, however, was not

mm

_

nl ep =1yer nc elr,bi,,

mm)6Wm (0051 by12.7

identical. The u' component has peaks at the edge of the wake and a 180-degree phase shift across the wake. The v' component is constant in amplitude within the core of the wake and is essentially uniform in phase across the wake at any given streamwise position. Combining the mean flow and the u' and v'piofiles, one can compute the

[]amsc T

streamlines of the velocity field as shown in Fig. 5. The characteristic wavelength, )., of the wake remains

j= =i =

b3om k

Ek=r-",,cUs

essentially constant in x in the domain considered. The phase speed (f./Uo) of the wake over a range of velocities, U0, (6 to 10 m/s) and excitation frequencies, f, (50 to 70 Hz) and amplitudes, A, (9.8 to 49.0 m/s2) is 0.94.

91,7

Fig. 1. Front view of gust-generating device and test section.

CONCLUDING REMARKS This work considers the combined wake of an array of oscillating ribbons oriented parallel to the flow. Over a range of Reynolds numbers and excitation amplitudes and frequencies, the wake flow exhibits a stable sinusoidal pattern with an invariant phase speed of 0.94. The visualizations are consistent with the velocity measurements which also indicate a sinusoidally oscillating flow with a characteristic wavelength dependent on excitation frequency. Other wake patterns can be obtained by appropriate adjustment of the flow velocity and relative excitation amplitude.

Fig. 2. Smoke visualization of excited wake for Uo=8 m/s, f=70 Hz, and A-49 m/s2.

ACKNOWLEDGEMENTS This work was conducted under a McDonnell Douglas IRAD program in collaboration with the Ecole Polytechnique F~d~rale. The authors thank Edward Kerschen of the University of Arizona and Rich Wlezien of the Illinois Institute of Technology for many useful discussions. The authors also thank Joseph Kroutil and William Faith for their assistance in designing the ribbon array device.

i.. Fig. 3. Smoke visualization of excited wake for Uo=3 m/s, f=70 Hz, and A=49 m/s2. The magnification is 1.5 times larger than that of Fig. 2.

REFERENCES

75

HAYASHI, M., SAKURAI, A. & OHYA, Y. 1986 Wake interference of a row of normal flat plates arranged side by side in a uniform flow. J. Fluid Mech. 164,

*

1-25. HIGUCHI, H. & TAKAHASHI, F. 1989

-75-

Flow past two-

dimensional ribbon parachute models. J. Aircraft 26, 641-649. KERSCHEN, E. J. 1990 Boundary layer receptivity theory. Appl. Mech. Rev. 43, S152-S157.

-7o

x-X'(,mM) -8-&"9 -. -7-7,1,,1.2 0

-7,

"4121 ->72

Fig. 4 Mean velocity profile of multiple ribbon wake for 2 Uo=7 m/s, f=70 Hz, A=49 m/s , and X0 =400 mm. 75

PAREKH, D. E., PULVIN, P., and WLEZIEN, R. W. 1991 Boundary layer receptivity to convected gusts and sound. To be presented at 1st ASME-JSME Fluids Engineering Conference, Portland, OR.

WILLIAMSON, C. H. K. 1985 Evolution of a single wake

behind a pair of bluff bodies. J. Fluid Mech. 159, 1-18.

_

0

___

16

200

30

lm0

s0

".=19 XX'(mm) Fig. 5 Streamlines correspondirg to the mean velocity plus the fundamental component of the perturbation velocity at a fixed phase of the excitation signal for Uo=7 m/s, f=70 Hz, A=49 m/s2, and XO=400 mm. The v' component is amplified by ten for clarity.

11-15-2

EIGIITU1 SYMPOSIUM ON

I1

TURBULENT SHEAR FLOWS Technical University of Munich

September 9-11, 1991

-1

LARGE EDDY SIMULATION OF SPATIALLY DEVELOPING PLANE MIXING LAYERS C Pianesel and C Benocci von Karman Institute for Fluid Dynamics Chauss&e de Waterloo, 72 B - 1640 Rhode Saint Gen~se - Belgiums

ABSTRACT Results are presented of a Large Eddy Simulationi (LES) of thle 2) 2) spatially dcselopitig planie inig lat~er Thieiinmcrical approach is based upoll a finite differences numerical solution of the Navier-Stokes equatitis for the large scales iii primitive variables foriiulatioii Simulatioiis -ire perforied for buth natural and fsrced iiixinig layors and tile results comspared with experinients results aitd Direct Numerical Simulations (DNS) INTRODUTCfIION

along x, and a centered scheme (4th order accurate oii an uniform mesh) for the ailvection along X2; all other terms are discretised using 2nd order centered formulae. Eq (2) is advanced in time with the Adamns-Bashforth predictorcorrector schiemse and a Direct Poisson Solver is used to solve the pressure equation siill finld the pressure distribution for which the , verify Eq. (1) BOUNDARY CONDITIONS On the boundary x, = 0 the mean streamwise velocity 'U1 is as-

Tile interaction betweeni two co-flowing fluid streams having differeiit velocities gives birth to the Plane Alizing Layer, %Inch liss been definled iii a recent weork (Lowery & Reynolds 1986) as "one of thle buildingblock flows upoit wich the field of fluid mechanics is based' B~esides their importanlce for a niumber of practecal applicatioiis plae, inixiiig layers are of paraimouiit interest to the rei~arclier for thle insigi~t they offer into lie physics of turblileiic and(]i,t particular, into the development of large Scale vortical structures and their interaction through the varier pairing process

suned to follow a taith profile

One attractive characteristic of the plane mixing layer is its geometrical simplicity, wehich miakes it amenable to numserical inivestigatioits using DNS or LES technique!. Loweer) & Reynolds amidSandim & Rleynolds 1989 have performed h)NS of mixing laye.s subjected to a periodic forciitg at thle inlet to promote the grow th of periodic large StULKtares (fsrccd muixintg layer), %%hile Comte & al 1989 have sltidied with LES the c,. where the vertical structures are allowed to grow naturally frott a raiidon nitt al perturbation (natural iinig layer) fIn the present stutdy both natural and forced cases are considered in order to attaiti a miore comuplete utiderstandimig of thle physical aspects of mixing amiuassess the perfornmances of LES for this class of flows

frnthwaequio

LARGE EDDY' MNODEL

~~ ~ ~ ~

th lag sclFaibes 7 The transport eutions fo th for ag cl aibe wAritten, withI obvious meaning of thle syitbols, as

cte alb

e9IJ, -

= 0

.

(2)

+V

--

ax,

dr

a, U

aP at

(1)

ar,

0XX, c

X

ishicl ius consistent with a finiite differences discretisatioii over a staggered mteshi (Sciuinanii 1979, 1%lasoit 1989) 'rile conitributioni =,uI of the smiall scales is miodeled using thle w'ell kniowii Smagoritisky siibgriul viscosity approach 77u

-21,,S, U,

C~A'(2,,S,

U,

+ tah22 2 [A

5

(5 J

where A is the ratio between the difference and the sum of the two freestream velocities Up~ aiid Vmn Onl the bouindary xi = L both velocity comfponeints are obtained

ai+ c-u at

= 0

0

(6)

where c is the advection velocity of the wvave comsputed from the values of the velocity at the previous timestep Oit the two traiisversal boutidaries U1 is determsined using a stressfree condition, while the tranisversal velocity 7 2 is founid applying Eq (1). RESULTS contditions for thte calculation, chosen to allow comparison with Lowery k Reyniolds aiid the experimnital results by Dater &XWygnanski 1982, are r = 0 5, where r is the lower stream to upper stream velocity 'ile

ratio, aiid Re = 100, where Re is the Reynolds nutmber based upon the velocity 6 difference between the tvwostreants and the inlet vorticity thiickniess s T[le compuutationial donmaiin was 130 uiiits loug atid 35 high, iicretised with a 340x40 grid The siibgrid stress coefficient C, was takeni equal to 0 06 Forced miixinig layer ile forced miixing layer is obtatined superuimposing to the meani inlet velocity a perturbation of the type it. = a

(i(Wkt),

4-COS(Wkt))

(7)

(3)

where a is a coefficient ane? wkuthe fuiidamnttal frequeiicy obtained fronti

(4)

lincar instaility thiewiy anld its filit twv aiffiviiila Formationi of periodic vortical structures aitd vortex pairinig were observed, as shown in Fig 1, structure wavelength atid position of the two pairings agree well with thle boundaries of the different regions ideittiby Oster & Wygiianski in their experimental study of a forced layer. rrns values of the streamwise velocity fluctuation (Fig. 2) show tile

where S, is tile strains rate of the large scales , A thle typuical mesh size anid C, the coefficienit relating isiesh size to filter size (see Mlason). N01EICA TECNIQ;ESfled NUMEICAL'rEIINIUESThe

Eqis (1) to (3) are discretised over a staggered grid with uniforn lack of self-similarity typical of the forced mnixintg layer, computed values mieshi 5pscing in the lonigitudinal directioni xi aiud non uniform spacing in are sensibly higher than correspondintg onies from experimenits and direct simtuilation the tranisversal directioni x2 using finite differences.: a 3th order upwiiid biased discretisation (Kawainuri & al l9uts) is used for thn advectioiiNnrlmxn ae 1'reseiit address . Facolt d'Itgegneria, lDipartimento hngegiieria MI ecaitica pcr l~nergetica, Uiiiversiti~ Federico 11, 1-80125 Napoli, Italy

The ntatural inixiiig layer is obtained suiperimiposinig a rat'doni perturltation with divergence equal to 0 to the initial flowfield

Formation of large scale vortical structures was observed at randons locations in space and time (Figs 3 and 4); vortex wavelengths agree well with the ones found by Oster & Wygnanqki The rms values

X X

1

*i

of the streaiiwisc velocity fluctuations are in good agrt-ment with the CONCLUSIONS The present LES Isss shown that it is psossible to reproduce numericahly the mails physical features of both forced and natural plane niixing layer :the process of formation and evolution of vortical structures is in good agreement with the experiiients performed by Oster & Wygnanski A significatihe disagreement however subsists for the forced case, uhiere the numerical solution does isot reproduce the typical two-peaks profile of the longitudinial rnss fluctuation Causes of this discrepancy aire psresently iiider iivestigatioii.X

Xl Fig 3. Isovorticity contours t -162 5. Natural layer.

~~A.~~~

REFERENCES

Coiiste, P , Lesicur, NI ,Larochie, II., Normnand, X., 1989 Nuimerical Simsilatioii of Turbulent Plane Shear Layer. Turbulent Shear Flows

~~j

63,Spriiiger-Verlag, 441-4541.

-

New Higher Order Kawamura, T , 1)skaisi, If , Kuwaliara, K ,1985 Upwiind Scheiie for Incoispressible Navier-Stokes equations Ninth Iiternational Coiiference on Numerical Methods in Fluid Dynamics, Spri nger- Verlag, 291-295

'",'

Fig 4 Isovorticity contours t=245 Natural layer.

Lowery, P S & Reyisolds, WVC ,.1986. Numerical Siiiulatioii of a Spatially IDe~elopziig, Forced, Plane Mixiiig Layer TF-26 Tlierinoscience Divisions, Dept. Mlcl Engineeriiig Staiiford University. Masoin, P)J 1989 Large Eddy Sinsulation ofTurbuletit Flow. Turbulent Shear Flow, Von Karmsaii Institute Lecture Series 1989-03 Oster, D & Wyguisuski, 1, 1982 The Forced Mixiig Layer between Parallel Streaiss J Fluid IMcl 123, 91-130.

0

Sanudliu, N.D & Reyinolds, WVC., 1989. Somsefiulet-Plane Effects on the Niiimerically Simisii~ted, Spatially- Developinig Mixiing L~a~er Trurbulenit Shear Flows 6, Spriiiger- Vcrlag, 4,11-454 Schumuuann, U;. 1975 Subgrid Scale Model for Finite Differeiices Sinnula. tuoiss of Iiurbuleuit Flows iii Plane Chuaiinels and Auinul J C'.isp Physics I8, .376-40.1

.s

67 6Ea 1801

*W1 3 iz2q 2

.

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x

is X2

,-,

.sx

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5

4.

X A.

.

- 4P

0.0

-2

6

a

5

A

I

F ig I Isovortucity coiitours t=269 5. Forced layer Fig 5 Resolved longitudiiial velocity fluctuations Natural layer.

067 6

- £0 73 4

x

e6 0 98s 6

a

.,~42,

1, +

I

A1 x1I:? x

5

is

I-

.5

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Fig. 2. Resolved longitudinal "ehocity fluctuations Forced layer.

11-16-2

EIGHTH1 SYMPOSIUM ON

11-17

TURBULENT SHEAR FLOWS Technical University of Munich September 9-11, 1991

SIGNATURES OF QUANTUM-LIKE MECHANICS AND DETERMINISTIC CHAOS

IN ATMOSPHERIC FLOWS A. Mary Selvam, J.S. Indian

Pethkar and S.M. Sholapurkar

Institute of Tropical Meteorology, Pune 411 008, India

ABSTRACT for system model A cell dynamical aLAor!setiC flows is summarized. The model pred. Lo quantum-like mechanical lavs for atmosihere flows manifested as the obsercorrelaspatiotemporsl Long-range ved tions, now identified as signatureu of self-organized criticality or deterntnistic chaos. Thle model predicLtons are in agreement with power spectrum analysis of annual mean COADS surface pressure time series for 100 grid points 0 round the Indian region. The power spectra follow the universal and unique inverse power law form of the statistical normal distribution. INTRODUCTION Long-range spatioteeporal correla tons fractal as the selfaimilar manifested pattern conthe spatial geometry to comitant with inverse power law form for the power speccram of temporal fluctuations .s ubiquitous to real world dynaas is identified and systems3 mica2 signatcres of self-organized criticality or 1988) Wiesenfeld, and Tang (Ba, non-Jocal Such chaos. deteLiciniTc connections are manifested in atmospheric flows as the selfsimilar fractal geometry to the global cloud cover pattern conconitant with inverse power law form for spectrum er.ergy eddy atmospheric the Schertzet and Lovejoy by documented Mathe1986). Schertzer, \LqGoy and matical models ot atmosoheri'. flows are on tne nonlinear by convention based ajier-Stokes touations which do not have precision Vinite solutions. analytical computer realizations of such nonlineat atmospheric flows are not moo&Is for totally realistic oecause of deterministic on dependence sensitive i.e., chaos, initial conditions. Such sensiti-;e depennamely, conditions, initial on dence long-range spaLi':emporal correlations are also exnibited by redl world dynamical s'ystems and is now identified as selfmentioned as criticality o nized ;f d e rminstic phyic hC earlir. in zr selr-orqanized criticality chaos ?Al wori, and model dynamical systenw, is identified. In Lhis paper a yot n2t dynamical system non-det.ermristic cell u-odel for atnospheric flows (Mary Selvan 19goal is summarized. MODEL PREDICTIONS The mignatues of determinictic (I) flows, namely in aLcospher)c chaos

long-range spatiotemporal correlations are inherent to the quantum-like mechanics governing atmospheric flows. The apparent wave-particle duality is attributed to the bimodal, namely, formation and dissipation respectively of the phenomenological form for energy display in the bidirectional energy flow intrinsic to eddy circulain clouds of formation e.g., tions, updrafts and dissipation of clouds in downdrafts giving rise to the discrete cellular geometry to cloud structure. (2) flow structure atmospheric The global consists of a nested continuum of vortex roll circulations with ordered energy feed larger and smaller the between back scales. A complete quantitative description of the atmospheric flow structure, can attractor strange the namely, given in terms of the therefore be component periodicities and their phases. Such a concept of an eddy continuum for strange the of structure fine the attractor has recently been put forth by Cvitanovic (Cvitanovic, 1988). The model predictions are in agreement with the results of continuous periodogram analysis ot the annual mean surface pressure daa for 28 years (1961-1988) available for about 100 grid points over the Indian OceanComprehensive the in region Atmosphere Data Set (COADS, Release 1, Climate Research Program, ERL, Boillder, Colorado). The percentage contribution to total variance and the cumulative percentage contribution to total variance for normalized standard deviate t values for 20 representative grid points are shown bance the model predicts in Figure i. logarithmic spiral circulation pattern for atmospheric flows, the t values are obtainP6 as t = (Iogx/ log 50)-1 where X is the the period at wnich the period and A50

to contribution percentage cumulative total variance is equal to 50 and t=0. The normal and cumulative normal probability density distributions are also plotted in Figure 1. It is seen that the percentage contribution to total variance follows The distribution. normal the closely o.dness of f-t' was tasted using the stanoacd statistical cni-square test. The arrows at the top and bottom in the figures indicate respectively the values of t up to which the fit is good at 95% level of significance for the normal and cumulative normal probability density distributions. Incidentially, the power spectrum of the chaotic Lorenz attractor (Lorenz 1963) also exhibits the iniversal inverse power normal the 6tatistical form of law

!!-17-1

dist ibution as shown at Figure 2. The cell dynamical system model concepts described in this paper may therefore be applied to quantify deterministic chaos or self-organized criticality in model dynamical systems also (Mary Selvam 1990b).

REFERENCES BAK, P., TANG, C. and WIESFNFELD, K. 1988 Self-organized criticality. Phys. Rev. A38, 364-374. BLACKADAR, A. 1990 Computer butterflies : an adventure in chaos. Weatherwise, Aug., 210-21.. CVITANOVIC, P. 1988 Invariant measurement of strange sets in terms of cycles. Phys. Rev. Lett. 61; 2729-2732. LOVEJOY, b. and SCHERTZoac, D. 1986 Scale invariance, symmetries, fractals and stochastic simulations of atmospheric phenomena. Bull. Amer. Meteorol. Soc. 67, 21-32. MARY SELVAM, A. 1990a Deterministic chaos, fractals and quantum-like mechanics in atmospheric flows. Can. J. Phys. 68, 831-841. MARY SELVAM, A. 1990b A cell dynamical system model for turbulent shear flows in the planetary atmospheric boundary layer. Proc. Ninth Symp. Turbulence and Diffusion, April 30-May 3, 1990, Roskilde, Denmark, Amer. Meteorol. Soc., 262-265t

CONCLUSION The signature of self-organized criticality or deterministic chaos, namely, the inverse power law form for the power spectrum of nonlinear fluctuations in real and model atmospheric flows is the same as the universal and unique statistical normal distribution with the normalized standard deviation of the fluctuations expressed as a function of the corresponding wavelengths (eddy). Such a universal quantification for the nonlinear variability of real and model atmospheric flowo is a natural consequence of quantum-like mechanics governing the dynamical evolution and implies predictability of the to.al pattern of fluctuations. ACKNOWLEDGEMENTS

Figure 1 : Power spectra of annual mean surface pressure for 28-years (1961-1988) for 4 grid points whose locations in degrees latitude and longitude are indicated at the top.

The author expresses her deep gratitude to Dr. A.S.R. Murty for his Keen interest and encouragement during the course of this study. Thanks are due to Shri M.I.R. Tinmaker for typing this manuscript.

6NT6' 10 00

.

--

6'NTO*

'

6*S T4"

-0

-...

CUMULATIVE

NIORMAL DISTRIBUTION \ 100-

~

Ai

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0.

OISTRIBUTION

'

U

00

':

... "

---

+2

+

0

0

0

NORMALISED STANDARD DEVIATE ......... CUMULATIVE % CONTRIIUTIN

t

• (log

TO TOTAL VAAAMCE

'/.

0

A / log Xo) CONTRISUTION TO TOTAL VAA, ...

100-

Figure 2

9

The power spectrum of the Lorenz follows the universal power law form of the statistical normal distribution, data series consists of 1000 successive means of 25 y comomenta values starting from the 1001 value. The program written by Blackadar (1990) was used for computing the Lorenz attvactor. The crosses refer to the cumulative normal distriattractor

Lo

Ainverse

z "The

Z4

50

*

Ze 25 .. NORMALISED STANDARD DEVIATION

Iordinate

bution.

|ui(;i I l'I SNN1NPOSIUNI ON "I.{IRILINTV SI liAR F1OIVS

F~lRILlLF'F

S

Technical t~ niversity of Munich

It,11-18

September 9- II , 1991

DISPERSION IN A TURBULENT BOUNDARY LAYER OVER A LOW HILL H. Stapountzis University of Thessalcniki Department of Merehanical 7nglneering Thessaloniki 540 06, Gr-.,-ec.a

D0spersion in a turbulent boundary layer was measured in a wind tunnel downstream of a long thin heated wire placed normal to the flow. The boundary layer was formed on the floor of the wind tunnel and was tripped at the entrance of the working section. Part of the tunnel floor was occupied by a low symmetric two-dimensional hill of the Jackson and Hunt (1975) type, y = H/(1 + (x/I)'), where H is the height of the hill, see Fig. 1. Thus the boundary layer experiences changes in curvature, from concave to convex, separating in the leeward side of the hill before reaching the convex part. The importance of curvature in a similar type of flow was discussed by Baskaran et al. (1987), who also noticed the formation of an internal, 'sub-boundary layer*. Our case is different in that the boundary layer height to the heiaht of the hill near the upstream foot of the h;ll is about 2.0 while in their case was 0.4. However the boundary layer characteristics are qualitatively the same i.e. the skin friction reaches a minimum near the foot of the hill, rises to a maximum before the top and drops to a minimum close to zero near separation. The boundary layer displacement thickness, denoted here by 6 (and by 6 its value at the source), changes in a reverse manner. The positions of the line source with respect to the hill are denoted by the letters A, B, C, D in Fig 1. The source was placed at various distances, Y., from the wall, resulting in dimensionless wall units y+, between 5 and 90. At position D, where the4 flow has separated ysiH - 0 017. In all cases U0H/v 1.2x10 , where U o is the free stream speed we:! upstreairi of

again a rapid decay after the "end', E, the hill. For source position D, within the recirculating separated region, concontrations are roughly constant up to the end of the hill. Beyond that point significant drop Inthe concentrations occur, as ais, ohserved by Castro and Snyder (1982) for three dimensional ni:Is The distancu fhOm the surface where the maximum occurs (not shown here) generally decreases as the top of the hill is approached and then Increases. The ground !evel concentrations, Cw, shown in Figure 3 drop with increasing x. for all but the mostly elevated, y./b. = 0.9, source position where a maximum is observed at the top of the hill. The intensity of concentration fluctuations, presented in Figure 4, Initially exhibits a dependence on y, and source size. Our data are In qualitative agreement with the predicted values from the Random Flight Model 'jr a fiat boundary layer (Sawford 1984) the main difference being in the location of the maxima. The streamwise position where the intensity is maximum depends on the ratio of the source size to the turbulence length scale at the source elevation, y.. n the region of the top of the hill the intensity tend to increase and drop again in the leeward side. pY The centroid of the mean concentration profiles, vCdy/TCdy) which isa measure of the mean heated particle -isplacemint from the wall (Chatwin, 1978), Is initially of the order of y. and then increases an 0 at di/dt < 0-

(2)

3. The theoietical explanation of the self-similaity of thlemocline in case of the mixed layer growth dh/dt > 0 was given by Turner (1978). ie noted that the heat tiansfer equation 0' 0 OT at

1 Two stiiking empircal facts iepresent a starting point of the discusion (i) vertical temperature profile in a thiermochne, a iegion of supercritical stability adjoining a timbulized layer, piove to be self-smnila in a seiies of laboratory experimnents and, though with much less accuracy, in natural watei reservoits (Kitaigoxodskii & Miropolsky 1970), (ii) comparison of the experimental data oii kinematic heat flux Q and vertical temperature gradient dT/dz shows tl,,At the effective heat conductivity K = -Q/(dT/dz) is much higher than the molecular one, and, rather unexpectedly, m' higher the higher is -dT/dz, it contrast with the well-knmown inverse dependence of K on -dT/dz wich takes place li a weakly stable shear flows (Gill & Tiefethen, unpublished; Turner 1978)

-

0:

a~lOT(3) aK 0:

wheie K is the effcctivc heat conductivity, has the following self-similar solutioi of a piopagating-wave type in the ieg;oii z > h(t) at dlh/dt = coast > 0. (4) where z' = z - h(t) , T(z, t) = T(z') wiiich is easilx reduced to the form given by Eq (1). Turner took K = -[11(dh/dt)/AT]dT/dz , where 1 is a length scale, and examined the thermochne development in a two-layer fluid, when the boundary conditions are 7' = '., at z = h,

TF= T1.,. dT/dz = 0 at z = h + Ah (5)

Tie thud one allows to detei mine the thermocline thickness Ah. At T/ = coast, A'!' = coust and 1 = const the solution of the problem belongs to a class given by Eq.(4) and can be written as the second line of Eq.(l), where 0 = I - (1 - C)2 , Ah = 21 If T, varies slowly with time such that dT/dt > 2(dh/dt)AT/Ah, the above expression is no longer exact but an approximate solution In support of his formula for the effective heat conductivity, Turner referred to Gill and Trefethen, who discoveied tuat K in a thermocline increases with the increase of -di'/dz (in

contrast with the usual inverse dependence of K on -dT/dz in weakly stable shear flows). 4. The direct dependence of K on -d7'/dz in thirmocline can be caused by the following mechanism. The disturbances at the lower boundary of the mixed layer generate internal waves which tiansfer kinetic energy downwards. The occurrence of breaking waves manifests itself in generation of tur-

bulence spots, i.e. the waves expend a part of their energy for maintenance of intermittent turbulence. It is precisely the same mechanism as was implied in Turner (1978). Nevertheless in his formula for K the buoyancy parameter # = gaT (where g is acceleration due to gravity and aT is thermal expansion coefficient) was somehow overlooked. Thus, the list of parameters governing the intermittent turbulence in a thermocline must include f, dT/dz and 'he length scale 1,in which capacity Zilitinkevich et al. (1988) assumed the amplitude of disturbances at the mixed-layer/thermocline interface. Then the following formula for K is obtained due to the reasons of dimensionality: K = C'(-/3dT/dz)"/2

,.

)3 (1 ,Ah

16

=-C,2 (3AT)-(dh/d)

2

(6)

(7)

1980 Laboratory studies of the entrainment zone of a convectively mixed layer. J. Fluid Mech. 100, 41-64. KITAIGORODSKII, S.A. & MIROPOLSKY, Yu.Z. 1970 On tie theory of the open ocean active iayet:-v. Akad. Nauk SSSR, Fizika Atmosfery i Okeana. 6,178-188.

where the half thickness of the thierinochine h serves as the vertical scale I for disturbances at tie external boundary of the mixed layer. The first formula of Eq.(7) coincides with the upper line of the empirical Milkki and Tamsalu Eq.(2). The second one is conflimed by the data of laboratory experiments on deepening of the mixed layer due to tle current shear (Kreiman 1982) and penetrative convection (Deardorff et al. 1980). At (3ATh)-'(dh/dt) > 10' these data are reliably approximated by the expression Ah/h = 7600(3A'h) - (dh/dt)2 , which gives C = 10-'. 5. To include I into the set of unknown quantities, we make another step by supplementing Eqs '(3),(6) with the balanmu equation of tuibulent kinetic energyOe/ot = -aF/Oz - /3Q - c,

(8)

where e is the turbulent kinetic energy per unit of fluid mass; /3Q = -3KdT/dz is the energy loss for overcoming of the buoyancy forces; c and F are viscous dissipation rate and vertical flux of this energy. We use the hypothesis, according to which 0, dT/dz and I are the governing parameters of the turbulence regime under consideration. Then, due to dimensionality reasons, the following expressions for e, 3Q , e and F hold true: e = CEI2 N2 , =

C,12 N 3,

REFERENCES DEARDORFF, J.W. & WILLIS, G.E. 1982 Dependence of mixed-layer entrainment on shear stress and velocity jump. J. Fluid Mech. 115, 123-149. DEARDORFF, J.W., WILLIS, G.E. & STOCKTON, B.H.

where C is a dimensionless constant. Using Eq.(6) and solving Eqs (3),(5), we obtain I-

tion experiments, which allo' -d to determine the dimensionless constants. Using these constants, numerical simulation of the temperature profile evolution in Deardorff & Willis (1982) shear flow experiments in annulus showed the qualitative agreement between theoretical and laboratory results.

KREIMAN, K.D. 1982 Laboratory modelling of turbulent entrainment in thermocline. Trudy LGMI. N77, 147-149. LINDEN, P.F. 1975 The deepening of the mixed layer in a stratified fluid. J. Fluid Mech. 71, 385-405. MLKKI, P. & TAMSALU, R. 1985 Physical features of the Baltic sea. Finnish Marine Research. N252, 110 p. MIROPOLSKY, Yu.Z., FILYUSHKIN, B.N. & CItERNYSHKOV, P.P. 1970 On the parametric description of temperature profiles in the active ocean layer. Okeanologiya. 10, 1101-1107. OZMIDOV, R.V. 1983 Small-scale turbulence and fine structure of hydrophysical fields in the ocean Okeanologiya. 23, 533-537. RESIlETOVA, O.V. & CHALIKOV, D.V. 1977 On the universal structure of the active layer in the ocean. Okeanologiya. 17, 774-777. TURNER, J.S. 1978 The temperature profile below the surface mixed layer. Ocean Modelling. Nl1, 6-8. WYATT, L R. 1978 Mixed layer development in an annular tank. Ocean Modelling. N17, 6-8. . .li, NE7, eaC M S.S., LT KREIMAN, K.D. & FELZENZILITINKEVICH,

#IQ = CQI 2N3,

F = -CFI2N

2

PN2,

(9)

where N (-,dT/dz)'/2 is the buoyancy frequency; CE, CQ, C, and CF are dimensionless constants; I is the turbulence length scale. The same expressions for e and Q were obtained earlier by Ozmidov (1983) from consideration of the energy budget in a turbulent spot. The third expression of Eq (9) presents the vertical flux of energy in a traditional form as a pioduct of a turbulent exchange coefficient (cx lN) and the vertical gradient of turbulent energy. The system of Eqs.(3),(8),(9) is closed. With respect to the vertical coordinate it is of the second order relative to both T and I. 6. The suggested theoretical model is applied to calculation of the vertical structure of thermocline. The analytical solution of the propagating-wave type is obtained and compared with the results of Deardorff et al. (1980) penetrative convec-

11-20-2

BAUM, A.I. 1988 Turbulence, heat exchange and selfsimilarity of the temperature profile in a thermoclne. Dokl. Akad. Nauk SSSR. 300, 1226-1230.

EIGIhTH SYMPOSIUM ON TURBULENT SIEAR FLOWS Technical University of Munich September 9-1l, 1991

Two-Scale Double-Layer model in Wall Bounded Turbulent Flow Eran Arad and Micha Wolfshtein Technton, Israel Insttute of Technology

Abstract is presented. Two scale model nd tiee equation turbulence A pfr eted. s alie. Ateeequationeused, turbulage equations are used, foi the large and for the small eddies.

only. In the inner region we follow the practice of Chen and Patel("), and use a single equation model 0). The two scales are calculated in the inner layer from algebraic relations and matched with the external solution at the interface. The

The viscous wvall sub-layer is treated by a reduced oneequation model, which produces correct asymptotic behavquantities ior near the wall. The predictions of turbulent

algebraic scales are so devised as to ensure that they exhibit tne correct asymptotic solution on the wall.

near walls, are better than those obtained by previous twoequation models.

2. Presentation of Model 2.1 Outer domain Two-Scale Model

1. Introduction 1.1 Scale Formulation

The variables of the triple equation model are the turbulent kinetic energy k, the turbulent time scale r and the dissipation rate e The energy equation is written in the following form:

Most two equations models use the dissipation rate as a measuie of scale. Modeling of this equation is often considered unsatisfactory (1) Thus J3ernard (-) reported poor modeling of the pressure diffusion term, while Mansour et al. 3) and Patel et al. ") reported incompatible LowReynold number damping functions, and wrong asymptotic behax ioi of turbulent quantities neai walls 0). In particulai these drawbacks cause poor prediction of the nearwall peaks of the turbulent kinetic energy and dissipation

+ Pk Pt ) -k D" (!/ - pDk + p +"o P"t = x Pk is the production term and depends on the mean strain. The turbulent viscosity pt is calculated by the large-scalemodel k -- -, while the decay term is related to the dissipation rate model: jt

rate 0)

A possible cause may be that single-point two-equation niodel requires the assumption of spectral similarity The problem may be oveicome by a solution of the energy equation in the wave number space. Hanjalic, Launder and Schitel ("). suggested a sunplification by the use of a small nu ibei of domains in the wave number space, and a typical scale foi each domain Following (5) we use in the present xoi kAtwo mayn ~gacuh the rapdforchange qpectral is in flow domains, the large and chaacteristic small eddies s

xNith the distance from solid walls. Wall-functions or solution up to the wall using Low Reynolds number coirections weie setexe! criticized by several authors (2,3 4,6) On one hand it was shown that the existing Low Reynolds-number iiodels (ho not provide correct asymptotic behavior near solid Iklls (). and on the other hand, ir. many flows the validity of the Wall Functions assumptions is questionable (5). An alternative appioach, the use of double layer model (6), enables the diffusion of turbulent energy down to the wall, while the known length scale profile near the wall may be easily retained. 1.2 Present contribution In this work we use two-scale model, a time scale equation for the big, eneigy carrying eddies and a dissipation equation for 'Jhe small dissipative eddies. However, as the energy contet of the small eddies is negligible, only one turbulent energy equation, for the big scales, is solved. The turbulent diffusion and production of energy are related to the energy and scale of the big eddies, (the time scale r in this paper (7)), while the decay term in the energy equation is obtained from the high wave number scale (the dissipation rate ). The triple equation model utilizes partial differential equations in the external part of the boundary layer

(2)

Dk = 6

= Cvpkr

The time scale equation is written after Zeierman and Wolfshtein ('): 9 . k (3) j +aC-'irjk-pCd, - a ,+ ,t pD

Dt

axrO ]

aX,

The dissipation rate equation is written in high Reynolds number form. D De

0[

Dt

ax,'

,)O] 00

ax,] 09- +

,,

, -

(4)

using (p, the energy transfer rate related to the large scales, defined as follows: (5) ep = Cg,k C The energy dissipation rate now responds only slowly to the applied mean strain, giving a better representation of a rapidly changing turbulence field. The decay term,, on the other hand, depends only on the high wave number region. 2.2 Inner Domain In the thin wall layer, a single equation for the turbulent energy is solved (eq(1)). Algebraic scales are used for calculation of the turbulent viscosity pt and the decay term:, kI (6) c= k = pL~k At = p (Cfy) L, Like in the external two-scale model, different length scales are specified for pt and c. It is also consistent with asymptotic analysis of. scales near the wall. It can be shown that the scales behave in the following manner for y --+ 0: 2 - BY, + 0(Y) = Ay + = (Y) (7) 3 y r 4 L, = A,y - Bey + (y )

''I-I-

It may be noted that at very small distances from tht wall

9. Israeli M. and Livne A., Aerodynamic Laboratory Re-

LI, varies as Y2 while L,varies as y3 . In the outer part of the innei layer the scales are continuously approaching the values of the outer layer. The scheme is not sensitive to the location of matching interface, as long as it is kept beyond the %iscous dominated zone. 3. Numerical Method

port No. 160-073. Technion, Israel, 1977 (in Hebrew) 10. Aral ,. and Wolfshtein M., Proc. Turbulent Shear Flow 7, 1989, pp. 5.4.1-5.4.6. 11. Arad E. and Wolfshtein M., Int. Symp. or. Comp. Fluid Dyn., Sydney, Aust., 1987. 12. Arad E. and Wolfshtein M., 16th. Cong. of Int. Council of Acro. Sci., Jerusalem, Israel, 1988. 13. Klebanoff P. S., NACA TN-3133, 1954.

The piesent papei is limited to parabolic boundary layers flow The solver used belongs to the OCI family and was developed by Israeli and Livne (9) The scheme is of fourth older accuracy, and can produce rather accurate results with coalse mesh, The method retains the tri-diagonal forni of the matrices, and is easy to solve. Implementation of the scheme to boundary layers was described in previous publications (iO 1,12)k

Flat Plate LEGEND Rt,

4. Results and Discussion The tet cases drsplayed below are a flat plate boundary layer t'), and a 2D diffuser AF0141 (1)(with strong ad-

verse pliessure gradient). The flat plat boundary layer is

0

a case which allows reasonable agreement with experimentaldata In the diffuser flows, large discrepancies have existed between computation and measurements (10),Consideing first the flat plate case, the solutions obtained by the two-scale double-laer method are compared witl solu-

I

tioris obtaine(i bN two-equation models. The later are performed with either wall functions (WF) or Low Reynolds

Fig 1:. Turbulence kinetic energy profiles for a flat plate flow (13)

-r

number solution t to the wall (SW). The measurements of Klebanoff ("i) are also plotted. It is shown in figuie 1, that the quality of the pIe(iction of the energy peak near the wail is inproted The improvement is mnore significant in the preence of adveise pressure gradient: Plots of the kinetic turbulence energy for two x stations (Figures 2-3)g show good agreement with experiment of the present model, while tIre k-I Low-Reynolds model fails to predict the energy peak. Good agreement was also obtained for other turbulent vaiiables, like shear stress or c,but the results are not displayed here because of limitation of space.

0.0

0A

OA

a..

O.A

1.0

_Y

-

--

AFOSR 0141 .. 6

LEGEND model experiment

d

5. Conclusions The two-scale model is highly flexible intreating wall flows Yet it does not require much more computational resources than two-equation model

Computational results presented

show that this model predicts well the turbulence quantities near the wall. where other models performance is poor. The model is now evaluated for other types of wall flows.

6 o0 00 005 -o 005 00000

. 0 0o 0 6 5 0.6 .o16 0l

00 001

0.0o30

y X 1.04 Fig 2' Turbulence kinetic energy profiles for a 2D diffuser flow, X=l.04 (AFOSR0141 (i)

6. References 1. Kine S. J-. Cantwell B. J and Lilley G M (Ed.), The

1980-81 AFOSR-HTTNI Stanford Conf Turbulent Flows, Stanford Univ,

F

AFOSR 0141

on Complex

1981

_

9 2 Bernard P S , AIAA J., Vol. 24, No 4, 1986, pp.61 622. 622 3 Mansour N. N , Kim J. and "oin P., J. Fluid Mech., Vol 194. 1988. pp. 15 -4 4 .

F

--

_______nLEGEND

-resen

m a / .......... h.-.I...9091 .e ........ *0 epeimnt

.

4. Pateh V. C., Rodi WV.and Schecuerer G., AIAA 3 , Vol. 23, 1985, pp 1308.1319. 5. Hanjalic E, Launder D. B. and Schiestel R., Proc. Turbulent Shear Flow 2, 1979, pp. 10.31-10.36. 6 Cien H C. and Patel V. C., AIAA J , Vol 26, No. 6, 6 4 1 64 8

1988, pp.

-

.

7 Zeierman S. and Wolfshtein I' , AIAA J., Vol. 24, No. 10, 1986, pp 1606-1610. 8. Wolfsliteii M., int. J. of heat and Mass Trans., Vol. 12, 1969, pp. 301-318

-001 0.00 001 0.02 0.03 o.04 o.05 0.06 007 X -2.38 m y Fig. 3 Turbulence kinetic energy profiles for a 2D diffuser

L

flow; x=2.38 (AFOSR0141

!!1-1-2

())

EiGHTh1 SYMIPOSIUN1 ON1 TURBULENT SHEAR FLOWS

1111-2

Technical University of Mt.nich September 9-1 ,v 1991

Analysis of turbulence models for hypersonic boundary layers B. Aupoix

J. Cousteix

ONERA/CERT Department of Aerothermodynamics 2, Avenue E. Belin BP 4025 France Toulouse Cedex 31055

ABSTRACT The predictions of various turbulence models are cornpared for different compressible boundary layer experiments Algebraic models seem to be superior in the wall region while transport equations are required in the outer part of the boundary layer A two-layer model is thus proposed to improve the prediction INTRODUCTION knowledge aircrafts, a good too hypersonic For pesey s dvelpent i ipraftat son c!a airbreathing of he of the boundai-, !ayer development is important to precisely determine the boundary layer thickness at the engine inlet as well as the friction drag or the wall temperature Such turbulent hypersonic boundary layers have some specific features the wall region can be thick and fill an important part of the boundary layer, compressibility effects on the turbulent motion are to be expected and should be accounted for in the turbulence model while significant heat transfer occurs in the boundary layer and must be accurately represented Turbulence models commonly used to compute compressible boundary layers have first been reviewed. The key conclusion is that these models are often crude extensions of incompressible models and that compressible turbulence effects, when accounted for, are generally modelled by us ng approximations which are not Galilean invarients COMPARISON WITH EXPERIMENTS idens and too ieacapabilities the In order to evaluate oof mod mo~lsand ealute In oderto he cpablites algebraic, various drawbacks, tify their main advantages and on a set of relmodels have been tested one- and two-equation exerimnts abounarylaye aproe h t hav a evan usig evant experiments, using a boundary layer approrch to have a fast, cheap and accurate numerical prediction Tne drawback of a boundary layer approach is that only a few test cases are available which really satisfy boundary layer hypothesis In addition, data on turbulent stresses and heat fluxe, are rare The tested turbulence models are. * A simple mixing length model, using Michel's mixing length formulation, Van Driest damping function, and a constant turbulent Prandtl number is used as a reference, e The Cebeci-Smith model, with Pletcher's correction for !ow Reynolds number effects, " The Baldwin-Lon,,A model, " The CLIC model, which uses Cebeci-Smith model in the inner region and Michel's mixing length in the outer region, * The one-equation model of Rubesin, in the compressible version used by Viegas and Horstman, * The Jones- Launder k - c model, * The compressible version of this model, as developed by Galm~s and Dussauge, with a somewhat different set of coefficients tuned at CERT/DERAT, @The Wilcox q - w model

The prediction of three cases, which exhibit characteristic features encountered in the hypersonic boundary layers discussed above, are mainly detailed, i.e. 9 An experiment performed by Owen and Horstman, of an hypersonic boundary layer , at a Mach number about 7.2, without pressure gradient, on a cold wall, to investigate the influence of high Mach numbers and compressiblity effects on turbulence, e A supersonic boundary layer experiment performed by Carvin, at a Mach number about 2, without pressure gradient, on a hot wall, to investigate the influence of a hot wall and hence a thick wall region, o A supersonic boundary layer experiment performed by Lewis, Gran and Kubota, with adverse and then favourable pressure gradients and Mach numbers ranging between 2.3 and 4, on an adiabatic wall, to investigate the influence of severe pressure gradients in compressible flows The main conclusions of this comparison, which are now used as guidelines in the development of hypersonic turbulence models, are the following ' e As in transonic or incompressible flows, algebraic models are well suited to compute the wall skin friction or the wall heat flux, as the turbulence is in equilibrium in the wall region but give poor predictions of the velocity or temperature profiles for boundary layers submitted to severe pressure gradients (Fig. 1). On the contrary, transport equations models of the but fair predictions wall values give pooror predictions layers submitted for boundary temperatureofprofiles velocity 2). (Fig. to severe pressure gradients o a aFge d as Invte pers t In the hypersonic case, a large discrepancy is observed between the experimental and computed turbulent stresses and heat fluxes A part may be due to experimental uncertainties The best prediction is achieved with the k - c models The scatter between the various predictions evidence the importance of compressibility corrections. * A model which accounts for the increase of the turbulent Prandtl number in the wall region is needed to better predict the wall region over a hot wall. Viegas model, which predicts a decrease of the turbulent Piandtl number in the wall region, behaves poorly. The turbulent Prandtl number variations affect s;gnificantly the shape of the enthalpy/velocity diagram for adiabatic walls but have little influence on cold walls @In the outer region of the boundary layer, the enthalpy/velocity diagram is poorly reproduced as the decrease of the turbulent Prandtl number in the outer region and the sharp intermittency profile in hypersonic flows are not well represented by the models (Fig. 3). * Many models are calibrated to reproduce the slope of the logarithmic law but usually fail to reproduce the intercept value. It is however possible to extend Granville's analysis to compressible flows to improve the ntercept value prediction for algebraic models

*10

*10 -I 0.12 -

Van Driest .... Cebeci.Smith .... Clic Baldwin-l.omax

0l-i

E

0 •.... 10

Experiment .... Launder Galmes WilcoX

0.10-

0.081

0.08-

'006-

-0.06

0.04-

0.04-

0.02 -/

0.02/

0.000.00 0 100

'

0.12-

Experiment

200

300

400

•

500

600

700

0

U (M/S) Fig 1 - Lewis et al experiment - Prediction of the velocity profile at the beginning of the favourable pressure gradient region - Algebraic models

100

200

300

400

U (M/S)

500

600

70

Fig 2,-, Lewis et al experiment - Prediction of the velocity profile at the beginning of the favourable pressure gradient region - Transport equations models

1Fxperiment Van Driest ---- Cebeci-Siuiili 0ic

1.25

Mach

60 0.75

2 3 5

/.

8

-.

Om-0.25 _050_0

(120

0.60 11.401 U/UC

10'Ct (Model) 2.018 1.582 0.9960 0.5706

Ratio 1051 1.062 1.092 1.132

Table 1 - Two-layer model Evolution of the skin friction coefficient versus Mach number for an adiabatic flat plate flow (Re= 10000.)

............ ...

(100

103C (Van Driest) 1.92 149 0.912 0.504

i.00

1 HO

1

T./T..

1O3C!

CfICI,.

2St/C

0.2

(Model) 1.458

1.464

1.204

4

1.311

1.316

1.208

0.6 08

1.186 1.082

1.191 1087

1.211 1.213

Lewis et al experiment - Prediction of the enFig 3 thalpy/velocity diagram - Algebraic models

Table 2 - Two-layer model -Evolution of the skin friction coefficient versus Mach number for a Mach 5 flat plate flow (Re = 10000 )

* The Baldwin-Lomax model, which is often presented as a derivation of Cebeci-Smith model to allow an easier implementation in Navier-Stokes codes, does not include some features of the Cebeci-Smith model such as a variable turbulent Prandtl number or a pressure gradient term in the wall damping function and behaves quite poorly at high Mach numbers. The numerical properties of the model, especially for transport equation models, are important to allow an easy implementation.

gether with a revised form of the Galm s and Dussauge k - c model in the outer region A constant turbulent Prandtl number is presently used but this must be improved. Results of first tests, on the skin friction coefficient and Stanton number variations for a flat plate flow at Re = 10000, are presented on tables 1 and 2

TWO--LAYER MODEL From the above conclusions, a two-layer model is proposed. Ir, the wall region, a simple one-equation model, which is a revised version of the Norris-Reynolds model, is used to.

111-2-2

EIGHTH SYMPOSIUM ON

111-3

TURBULENT SHEAR FLOWS Technical University of Munich Septembec 9-11, 1991

Characteristics of the Velocity Field in the Central Region of a Spirally Fluted Tube by D. S. Babikian and J. C. LaRue Department of Mechanical and Aerospace Engineering University of California, Irvine Irvine, California 92717

Hia. R. Rahai Mechanical Engineering Department California State University, Long Beach Long Beach, California 90840

section is about 142. Thus, the flow should be fully developed at

Abstract

the measuring station.

An experimental study of the effect of swirl on the turbulent flow in a tube is presented herein. The mean rotation is produced by means of small spiral flutes on the tube wall. The flutes are found to lead to a mean fluid rotation rate which, except very near the fluted surface, is much less than that which would correspond to the mean velocity and flute angle The flutes also have an unexpected effe.t on the mean axial velocity. For example, near the flute valley, the radial gradient of the mean axial velocity is significantly reduced relative to that for a smoothe tube at the

to measure two time-resolved velocity components. In addition,

same Reynolds ,uniber and relative distance from the tube wall while that near the crest is unchanged. As expected, the effect of swirl on the turbulent -eIocity fluctuations is to reduce the radial component. The azimuthal turbulent velocity component is also reduced while at some radial locations, the axial turbulent velocity component is increased.

Introduction The goal of the present study is to investigate the effect of swirl on the turbulent velocity field in a tube flow where the swirl or mean azimuthal velocity does not vary with downstream distaitce. This type of flow is of both fundamental and applied interest. From a fundameital point of view, it is expected (cf. Bradshaw (1973)) that the swirl should lead to a reduction in the root mean square of the turbulent radial velocity component. Tubes with swirling flows have been produced by means of tangential injection, twisted tape inserts, wall slots, rotating tubes, and spiral flutes, lowever, only tube flows with swirl produced by means of small spiral flutes are found to have pressure drops that are flows nearly the same as found in corresponding smoothe tube (cf Yampolsky (1979)) For this reason, a spirally fluted tube, simular to that used by Yainpolsky is used in the present study.

Apparatus and Procedure A centrifugal air compressor is used to supply filtered laboratory air at a steady flow rate. The air, which is heated to about 30'C above ambient by the air compressor, is sent through a set of flow straighteners located in a 5.1cm diameter pipe that is just upstream of the 4.0m long test section. Thc downstream 2.4m of the test section is centrally placed in a 7.0cm diameter tube. Water at a constant flow rate is passed through the annulus formed by the two tubes so as to maintain a constant test section wall temperature of about 38.4°C. The sensing volume of the probe is placed 0.5mm downstream of the exit plane of the test section and is mounted on a vertical traverse which has a radial position resolution of 0 254mm. A sketch of the spiral multi-fluted aluminum tube is shown in Fig. 1. The mean radius, R, of the tube, which is measured from time centerline to a position midway between the crest and the valley, is 1.4cm. The flute height, h, is 0.168cm and the flute spacing, S, is 0.235cm. There are 30 flutes equally spaced around the tube circumference. The angle of the flutes relative to the tube axis, Of, is 300. The length to diameter ratio of the test

The average or bulk velocity and corresponding Reynolds num2 ber are, respectively, 29.8m/s and .64x104. Comparisons are made to the smoothe tube data of Laufer (1953), Sandborn (1955), and Schildnect et al. (1979) which corespond to bulk Reynolds 4 numbers of 4.49,4.48, and 1.72x10 . consists of three sensors. the measurements used for The probe Two hot wire sensors, 1.25mm in length and 3.8pm in diam., in an "X" configuration which are sepaiated by 0.5mm, are used

mounted normal to tile plape of the "X" array, 0.6mm upstream of the center of the "X" array. The frequency responses of the hotand cold-wires are 30 and 8kHz respectively. All are calibrated both before and after the collection of data. The voltages from the two constant temperature anemometers and the A.C. Wheatstone bridge are passed through calibrated buck and gain amplifiers which remove the mean voltage and amplify the fluctuating portion of the signal so for recording on an FM instrumentat:on tape recorder which has a frequency response of l0kHz. The recorded signals are played back, low passed filtered at lOkHz, and digitized at a sample rate of 10240 samples/sec by means of a Tustin 14 bit AD. converter which is equipped with simultaneous sample and hol- circuits. The data triplets corresponding to 30sec of data are analyzed an LSI 11/23 computer. Two types of data sets are obtained and analyzed. The first set starts at a flute crest, called hereafter crest data, and extends past the center of the tube The second set starts at the flute valley, called hereafter valley data, and extends past the center of the tube

Results and Discussion The radial profiles of the mean axial velocity normalized by the centerline mean axial velocity plotted as a function of distance from the centerline are shown on Fig. 2. For (1 - rm/R) > 0.15, the mean axial velocity profiles for the crest and valley data are nearly the same. llow,ver, at the same position relative to the centerline, the mean axial velocities for both the crest and valley data are generally lower than the values for the corresponding smooth tube flow. For (1 - riR) < 0.15, the radial gradient of the mean axial velocity for the flute data is less than that for the crest data. The radial distribution of the measn azimuthal velocity normalized by the centerline velocity for both the crest and valley data are presented in Fig. 3. As expected, the azimuthal velocity is nearly zero at the centerline and rises to a maximum at a circumference corresponding to the crest. The apparent differences in the mean azimuthal velocity for the valley and crest data in time range 0.2 < (1 - r/R) < 0.6 are small and probably correspond to the uncertainty in the data. The gradient in the mean azimuthal velocity in the radial direction varies as a function of position. Thus, the fluid is not in solid body rotation. The intensities of the axial, radial, and azimuthal velocities

normalized by the local mean velocity along with the corresponding smooth tube data are shown in Fig. 4. For both the crest and valley data, the distributions of the root mean square (rms) axial velocity normalized by the centerline velocity are a few percent higher than the corresponding smooth tube data of Schildknecht et al. (1979) and for the mid-radius positions are about 12% higher than the corresponding smooth tube data of Laufer (1953). This slight increase in the axial rms velocity may be due to the swirl which is anticipated tc lead to a transfer of turbulent kinetic energy from the fluctuating radial velocity to the fluctuating axial velocity component (cf. Bradshaw (1973)). Consistent with the hypothesized stabilizing or reducing effect of swirl on the radial veloci.y fluctuations, the normalized rms radial velocity to,(1 - r/R) > 0.1 for both the crest and valley data is found to be about 14% less than the corresponding smooth tube data. The normalized rms azimuthal velocity for (1 - r/R) >_0.1 is about 20% less tha:i the corresponding smooth tube data and about 60% of the rms - .al velocity. The reason for the reduction in the rms azimuthal velocity is not clear.

on o

,

0 "' 0 2 ,o 4 0

o

0,2

o .,,

0

0,

0

0'

0

0,0 0,

0l

R

a,

D

Figure 2: Radial variation of mean axial velocity normalized by the centerline velocity plotted alongaradius asafunction of radial position measured relative to the centerline: crest to centerline, A; valley to centerline, 0; smooth tube data of Laufer (1953), 0; smooth tube data of Schildknecht et al. (1979), V; and, smooth tube data of Sandborn (1955), 0.

,,

Conclusions The mean axial velocity profiles in the spiral fluted tube differ from those in a smooth tube. This difference is especially obvious in the valley data where it can be seen that the mean axial velocity can be redured by as much as 40% compared to the smooth tube values at the same relative distance from the tube surface. Away from the fluted surface, the mean velocity profiles for both the crest and valley data are nearly the same. The radial gradient of the mean axial velocity forthe valley data isabout 40% of tile corresponding smooth tube value while the radial gradient of the mean axial velocity for the crest data is only slightly less than the corresponding smooth tube value. The presence of the flutes does induce a mean swirl to the flow but the flow is not in solid body rotation. Near the crest, the rotation rate is only about 35% of that which would correspond to solid body rotation. The rotation of the flow is seen to lead to a transfer of turbulent kinetic energy from the radial to the axial component.

0

60 0 _ 00 08 a 0

00

,

o 0

% 0

F

00

0 0

2

, 0,/

Figure

,

0

0, 00

i

00

0

Radial variation of mean azimuthal velocity normal-

ized by the mean axial velocity on thp centerliie. Symbols have the same meaning as in Fig. 2

References Bradshaw,P., 1973, AGARDograph NO. 169, N.A.T.O Laufer, J , 1953, "The Structure of Turbulence in Fully Developed Pipe Flow," N.A.C.A., Technical Note 2954.

000

0 Sandborn, V.A., 1955, "Experimental Evaluation of Momentum Terms in Turbulent pipe Flow," N.A.C.A Technical Note 3266.

OR -ii

Schjldknecht, M., Miller, J.A., and Meier, G E.A., 1979, "The Influence of Suction on the Structure of Turbulence in Fully Developed pipe Flow," J. Fluid Mech., Vol. 90, Part 1, pp. 67-107.

0

- 0

'20

% ON

'00 0 60

0

V_

. 0

Yampolsky, J.S., 1979, "Spirally Fluted Tubing for Augmented Ileat Transfer," General Atomic Report GA-A15442.

,.V

0

V0V

o,t

ii ,

j

1, ,, , 0

..... .... Figure 1: Sketch of spiral fluted tube.

0 00 I

,-

0

6

I

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,

0

00

00

II]

0

Figure 4: Radial 'ariation of rms axial velocity (top), rms radial velocity (middle), and rms azimuthal velocity (bottom) normalized by the local mean axial velocity. Symbols have the same meaning as in Fig. 2.

111-3-2

EiGh1TH SY.,'POSIUN! ON TURBULENT SiIEAR FLOWS Technical University of Munich September91 I, 1991

111-4 A new model for premixed turbulent combustion based on direct simulation results 2 By Boudier P.l, Dillies B. ,

Veynante D., Lacas F. and Poinsot T. Laboratoire EM2C, Ecole Centrale de Paris Chatenay Malabry 92295, France IFP, Rueil Malmaison, FRANCE 2 PSA, Velizy, FRANCE

1. Introduction Turbulent combustion models based on the flamelet concept assume that the turbulent reaction zone may be viewed as a collection of laminar flame elements. Under this approach, a central parameter for combustion modelling is the stretch of a flame surface A which is defined by (Candel and Poinsot 1990):

The Coherent Flame description combines three elements: 1) a model of the turbulent flow. 2) a local model providing WL which is a function of temperature, local stretch rate, local equivalence ratio and may be obtained from numerical computations through a laminar flamelet

library (Darabiha et al 1987). 3) a balance equation for the flame area density which, following Small flam stretch modrkt crete arctdiel flame suyberritc: itioa the initial work of Marble and Broadwell (1977), may be written while high stretch leads to flame quenching. Different expressions -. OE OEu, a (,1 DE may be found in the literature for the mean flame stretch (av(5) - + - = O x,(' - ) + IE - Q , eraged along the flame front) hi.Considering that small scales (1 1 dA K Small to moderate flame stretch creates active flame surface

where k is the mean stretch rate and Q, corresponds to flame

control flame stretch, Cant and Bray (1987) propose "= V/7, (2) where c is the dissipation of turbulent kinetic energy and v is the kinematic viscogty. Candel et al. (1990) assume that large scales are more relevant to estimate flame stretch and use

surface annihilation by mutual interaction of flame fronts.

(3) K = t/k. energy. where k is the turbulent kinetic These expressions do not take into account any of the flame characteristics and neglect partial quenching of the flame front. A more precised analysis is proposed by Meneveau and Poinsot (1991) who combine a spectral approach with results of direct simulations. The resulting model for the stretch 7 is implemented here in the Coherent Flame Model (CFM) and used to compute a turbulent premixed flame stabilized by a stream of hot gases.

Meneveau and Poinsot (1991) propose an estimate of the flame mean stretch by combining different approaches: (1) Use of direct simulations of flame / vortex interactions to predict the effect of a given isolated structure on a laminar flame front (Poinsot et al 1990). (2) Use detailed experimental data about intermittent turbulence to detcmine the distribution of stretch along the flame front (Meneveau and Sreenivasan 1987). (3) Define a net stretch of the flame surface by substracting the rate of destruction by quenching from the rate of increase due to hydrodynamic straining. This ITNFS model accounts for viscous, curvatore and transients effects but neglects non-linearities and flame generated turbulence when deducing the net stretch rate. However, it represents a substantial improvement on classical estimates of the flame stretch. For the present work, it is combined with the CFM to provideseh (see Menevea et Poinsot 1991) an expression for the flame

2. The CFM for premixed turbulent combustion In the Coherent Flame Model (Candel et al 1990), the reaction zone is viewed as a collection of flamelets embedded in the turbul-nt flow. The mean turbulent raction rate is axpressed as the E by the consumption product of the flame area per unit volume rate per unit of flame surface WL. *

3. A model for flame stretch based on direct simulation

-L)

se =

(6)

w=f(.-)((6

where u' is the RMS turbulent velocity, L is the integro! length scale in the fresh gases, 3L is the laminar flame speed and IFis .he laminar flame thickness. A typical set of ITNFS results is given , in Fig. 1. IR is scaled by the small scales stretch vc//. Regions where 7?' is iqgative indicate global flame quenching. ._ m 111-4-1 I

m

w

m~wm =

=ml

mj

4 Numerical validation

2300

The model is tested in the case of a two dimensional turbulent premixed flame stabilized in a square duct (100 mm x 100 mm) by a parallel flow of hot gases.

0a -

- .0

0.0. A0. *

The experimental setup (Fig. 2) was built by r. Moreau at ONERA (France). The combustion of a premixed air-methane flow - 60 m/s fast and 600 K hot - is ignited and stabilized by a

- - -'-

0

-

flow of stoechiometric burnt products - 120 m/s and 2000 K hot The turbulence level of the premixed gases is about 10 percent • and the equivalence ratio is 0.8 . The experimental data which were obtained are two transverse profiles for axial velocity (Laser Doppler Velocimetry) and two transverse profiles for temperature

00._

(Cohereit Anti Stoke Raman Spectroscopy).

Fig 4 . Comparison of axial velocity profiles.

MM cx I ~~~~Xffi43 MM CALCULATIONS IESRMNS MM _x50 MEASUREMENTSJ

00

0.1

Y-AXIS M

The agreement between numerical results and experimental data is fairly good both for temperature profiles (Fig. 3) and axial velocity profiles (Fig. 4). This approach combining the ITNFS

References

analysis of Meneveau and Poinsot and the Coherent Flame Model teems to be promising but needs, of course, further validations,

CANrrl. S.M. FT POINSOT T. 1990 Flame stretch and the balance equation for the flame area. C'ombus. Sci. and Tech., 70, 1-15 CANDE.I

03-

S.,

VEYNANTE

D., LAtAS F., MAISTRF'

E.,

DANA-

N. AND POINSOT T.: Coherent flame model: applications and recent eztensions, Recent Adv. in Combustion Modelling, B. HilA

0.1

Larrouturou ed., Series on advances in mathematics for applied sciences, World Scientific, Singapore, 1990.

01

oo

K.: Twenty Second Symposium (Interno. tional) on Combustion, Strained laminar flamelet calculations of premixed turbulent combustion in a closed vessel, 1988 p. 791, The Combustion Institute. D,\ARAllA N., CANDFI, S. AND Mlauvr F 1986 The effect of strain rate on a premixed lamnnar flame Combus. and Flamr, CaN't R. AND

-01 02 -03

0

1

°

0 1.

1,

o

,

',0

BRAY

64,203.

Fig I Typical set for the mean ITNFS stretch.

MAnim, F.E. AND BROADW.LL .I,: Prect SQUID, The coherent flame model forturbulent chemical reactions, 1977 Report TRW-9-PU. - ,....

INFV A

.......................

C.

AND

PoINsoT T. 1991 Stretching and quench-

ing of flamelets in premixed turbulent combustion. Combus. and Flame, ,to appear.

.1 WK

S1.... - ?7 .7 .

Fig 2

.

Al C. AND SRHENIVASAN K.R. 1987 The multifractal MFNFVr, spectrum of the dissipation field in turbulent flows. Nucl. Phys. B Proc. Suppl., 2, 49.

- ....

N1OEAII P. AND IOUTIFt A.:, Sixteenth Symiposium (International) on Combustion, Laser velocimeter measurements in a

Experimental setup.

turbulent flame, 1976 p. 1747, The Combustion Institute.

OAngeles,

MOR.AU P : Fifteenth Aerospace Sciences Meeting, Turbulent flame development in a high velocity premixed flow, 1977 Los AIAA paper 77-49.

0

MoRAU P.:, Eighteenth Symposium (International) on Com. bustion, Experimental determination of probability density func-

21ise0

"'-0 0

0 -e

tions within a turbulent high velocity premixed flame, 1980 The 0

7k----.... - .Mech., 34NTS

Mx=4 A MEASUIM •.X-'22 MM CALCULATIONS 0 X-322 MM MEASUREMENTSJ

0.0.oo0.1

Y-AXIS

H

Fig 3 . Comparison of tempetature profiles.

Combustion Institute, p. 993. POINSOT T., VFYNANTE D. AND CAND, S : J. of Fluid Quenching processes and premixed turbulent combustion diagrams, 1990 228, p. 561. POINSOT T., VEYNANTP' D. AND CANIW, S.:. Twenty Third Symposium (International)on Combustion, Diagrams of premixed turbulent combustion based on direct simulation, 1990 'rheConbustion Institute.

111-4-2

EIG!IhTt! SYMPOSIUM ON TURBULENT SHEAR FLOWS Technical University (,f Munich September 9-1I, 1991 INVESTIGATION OF NO, FORMATION IN TURBULENT NONPROMIXED FLAMES: HYDROGEN-AIR JET FLAMES J.-Y. Chen*), W. Kollmann**)

*) Combustion Research Facility, Sandia National Laboratories, Livermore, CA 945510969, Research supported by the United States Department of Energy, Office of Basic Energy Sciences, Division of Chemical Sciences. **) MAME Department, University of California at Davis, CA 95616 As a first step toward a bettet understanding of NO, foimation in turbulent flames, a partial equilibrium chemical model for hydrogen-air combustion is incorporated into a hybrid pdf method, in which the flow field is calculated based on a second moment model. Production of NO, is assumed to be dominated by the Zeldovich mechanism A 2 + 0 ,: NO + N

N + 02 N + OH

NO + 0

NO + H.

(RI) (R2) (R3)

With the assumption of a steady state for the N atom and NO/NO, q,, < 1, th NO formation rate can be approximated by SNO ; 2k/.ri "- 2 - 0

the lrge Damkohler regimes as has been observed in the experiment by Chen and Driscoll (1990). Howevel, the model predicts a flat trend when the Damkolfer becomes small. Figure 3 presents the predicted NO, emission ndex profiles obtained with and without the radiation heat loss for a fixed jet diameter. Du_ to the complex nature of radiation transfer, the current radiation model represents the maximum possible heat loss that would occur in the flame With the radiation heat loss, the NO, emission index reduces significantly for low velocity jet flames. Moreover, the NO, emission index trend is totally opposite to that for adiabatic flames in the low jet velocity range. This reverse trend has been observed in turbulent hydrocarbon jet flames (Turns and Lovett, 1989) where substantial radiation heat loss may occur due to the presence of soot. As the jet velocity increases (tile flame residence time decreases), the effect of radiation heat loss become less important but it still has

ueof te Noremiss idx ifluenceton heat Other possible routes for NO formation, such as reactions inof the NO, emission index. magnitude the on Finfluence evaluate To considered. not are 0, N intermediate volving 2 with the nonfFig . 4 shows the emission index obtained e thep tposb eaesf radiation heat proimates the Nmaxio equilibrium model and the equilibrium model for round jet heato, asraditin del e.,wth rx the mssumptionofopa i flames (upper part of the figure) and for plane jet flames heat loss is included (i e., ith tme assumption of optically (nonequilibrium model only, lower part of the figure). In thin limit), contrast to the nonequilibrium jet flames, EINo,/Tr for performed for turbu-

have been Numerical calculations lenujetflmei alcuelinsave ee perforad fu rr lent jet flamies with a fuel mixtue of 78%H 2 and 22%Ar. In the past, extensive measarements of these flames have been performed (e.g., Magre and Dibble, 1988) providing us data for compaiisons with predictions for flow and scalar fields. Figure 1 presents a comparison of predicted and measured centeiline profiles for mean velocity and density showing reasonable agreement. For adiabatic flames, Figuic 2 shows the predicted NOA emission index profiles versus 1 (Uf - UJo)/Df, which is proportional to the recipiocal of the Damkohlcr number if the chemical time scale is constant Two different normalization parameters are used. In Fig 2a, the NO1 emission index has been normalized by a group of variables which scales with the global flame residence time ifthe flame volume scales with the cubic of the flame length. However, the normalized NO1 emission index profiles do not collapse to a single curve indicating that in the present unconfined coflowing turb,,lent jets, the flame volume does not remain similar in shape as the jet diameter is changed. In Fig. 2b, the global flame residence time suggested by Becker and Liang (1978) is used for normalization, and all the results collapse to a single curve. More importantly, the current model predicts the 1/2 slope trend over

cnrs

otennqiirtmjtfaeE~,7

o

equilibrium remains almost constant (within the statistical uncertainty) for all jet velocities. As suggested in the scaling parameter section, in the limit of equilibrium chemistry, EINo. /7R approaches a constant value only if the flame geometry is not a function of flow parameters. This implies that the one-half power dependence is due to the dependence of thermal NO on the flame Damnk6hler number. References Becker, H. A. and Liang, D., "Visible Length of Vextical Free Turbulent Diffusion Flames," Combustion and Flame, 32, 115-137 (1978). Chen, R.-H. and Driscoll, J. F, "Nitric Oxide Levels of Jet Diffusion Flames: Effects of Coaxial Air and Other Mixing Parameters," Twenty- Third Symp. ium (Internatmoal) on Combustion, Orleans, (1990). Magre, P. and Dibble, R. W., "Finite Chemical Kinetics Effects in a Subsonic Tuttrbulent Hydrogen Flame," Combustion and Flame, 73, 195-206 (1988). Tuirn, S. R. and Lovett, J. A.,"Measurements of Oxides of Nitrogen Enmssions fiom Turbulent Propane Jet Diffusion Flames,"Combust. Sci. and Tech., 66, 233-249 (1989).

111-5-1

0.50.

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300.0 200.0 Uf (mis)

S

X/Df Fig. 1 Centerline velocity and density profiles for a 78%H2 +22%/Ar turbulent jet flamne with Uf=225 in/s, Uco= 9.2 fills and Df=5.2 77m. Symbol: experimental data; linespredictions

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Fig. 3. Prediced NO, emission index profiles vs fuiel jet velocity and 1/Damk6hler number for flames with 5.2 min

-2

jet diarnetei. Q adiabatic flames; 6 with maximum radiation heat loss. Sgyi ficant reduction in the NO, formation is observed when the radiation heat loss is included, and, foi low jet velocity flames, the NO, tiend becomes opposite to

0

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that of adiabtic flames.

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k6hler number with various fuel jet dliamneters: U 10.4 mm; 5.2 Trm; 7 2.6 mm: co 1 3 mm? (Is'. normalized by flame length, jet diamt.tcr and jet velocity. (2a) normalized by the residence time for jet flames (Decker aid Liang, 1978). Resuits show an approximate 1/2 slope .lependence for large Damk6hler numbers consistent vith t , xperimnental observation by Chien and Driscoll (1990).

o

.

10

1

(us)

Fig. 4. Comparison of the NO1 emission index scaling trends between axisymmctric jet flames (78%H2+22%Ar) and planar jet flames (47%H 2 +53%Ar) for various jet diameters. U 10.4 mm, 0 5.2 mm, V 2.6 mm, o 1.3 mm: round jets with a partial equilibrium chemistry model; (B5.2 mm: round jet with the equilibrium chemistry; U, 2.6 min; Q 5.2 mim planar jets using the same nonequilibrium chemistry but with a different fuel mixture. Different power dependence of EIlvo, R on (U1 - Uc,)/Df is observed consistei-t with the results from analyses based on self-similar laws foi jets.

111-5-2

EIGHTh SYNlPOSUNI ON

1111-'1-

'URBULENT S"EAR FLOWS

Technical University cf Munich September 9-I 1, 199'1 AN EXAMINATION OF PERIODIC TURBULENT PIPE FLOW USING A LOW-REYNOLDS-NUMBER k~e TURBULENCE MODEL

M. A. Cotton and J. 0. Ismael Department of Engineering University of Manchester Manchester M13 9PL U.K.

ABSTRACT Turbulent pipe flows in which the mass flow rate varies in time in the form of a sinusoid imposed upon a steady level are computed using the turbulence model of Launder and Sharma (1974). Comparison is made with the experimental data of Tu and Ramaprian (1983) and Burnel et al. (1990). INTRODUCTION The present study investigates the performance of the low-Reynolds-number k-c turbulence model of Launder ana Sharma (1974) in the calculation of periodic, spatially fully-developed isothermal pipe flows. The mass flow rate varies as a pure sinusoid of frequency w imposed on a steady flow (at amplitudes such that the flow remains uni-directional). The cases we examine from the two sets of data were obtained at approximately the same mean Reynolds number,, Re = 4.7 x 104. To a limited extent (because of different bulk velocity amplitude modulations, y) the data allow a study of the effect of a single parameter, the imposed frequency, characterized here in terms of the Stokes number, D = D (w/v)1/2 (Table I). The experimental data (and present calculations) raise interesting questions in relation to the occurrence of 'frozen' turbulence patterns at higher oscillation frequencies: it is seen from Table I that Case 2 of Tu and Ramaprian and Case 2 of Burnel et al. are at similar Stokes numbers, however very different Reynolds stress distributions are reported in the two papers. MODEL FORMULATION The phase-averaged momentum equation for periodic fully-developed pipe flow may be written as follows:

meanings. Reynolds stress is supplied by the turbulence model, the k- and c- equations being written in appropriate form with all advective te:ms omitted. An implicit finite volume/finite difference discretization scheme following Leschzine- (1982) has been employed. The radial gridnode consists of 100at nodes and the wall-adjacent is located y+ < 0.25. RESULTS AND DISCUSSION In the following presentation of results, phase positions are defined in terms of a cosine variation of bulk velocity (which necessitated a shifting of the time origin in treating the data of Burnel et al.). W is the long time mean bulk velocity and v,' is the phase-averaged periodic component of Reynolds stress (i.e. the long time mean is subtracted from the simple phase average). y is distance from the wall and R is pipe radius (=D/2). Experimental measurements are represented as points in the figures below and present computations are shown as lines. Comparison with the data of Tu Pnd Ramaprian Reynolds stress profiles at two frequencies examined by Tu and Ramaprian (1983) are shown in Figs. I and 2. In the lower frequency case (Fig. 1), the present calculations are in reasonable accord with the experimental measurements of strongly phase-varying Reypolds stress distributions. In contrast, Fig. 2 shows computed profiles which are frozen over the cycle (except close to the wall), whereas the experimental points exhibit considerable variation with phase. It should be noted, however,, that the experimental points on Figs. 1 and 2 do not represent direct measurements of Reynolds stress, but are obtained instead from integrations of equation (1) using measurements of wall shear and velocity. Tu and Ramaprian report direct LDA measurements of the streamwise normal Reynolds stress which reveal a frozen structure, similar to that apparent in the present computed results for v,,

I dp I a aW = --+ [r(7- )] (1) at p dz rr r where r and z are radial and axial cylindrical coordinates, W is axial velocity, is phase-averaged Reynolds stress and other symbols have their usual TAbiAL Conditions of the e.ements of Tu and Ramaprian (1983) and Burr.el et al.(1990) Authors fluid Re D v f Y) 2 (m) (m /s) (Hz) %) f20*el aW

-

Case 1

Tu and Ramaprian

water Case 2 Case 1

Burnel et al.

0.0508

1.0x10 6

, air

Case2-

4.9x10' 4.4x10 4

0.092

l.5xl0-5

0.5

64

90 - -

-

3.6 1.65

"

15 26

242 76 -

180

30

-

253

ACKNOWLEDGEMENTS We are grateful to Professors J.D. Jackson, B.E. Launder and B.R. Ramaprian for helpful discussions in the course of this work.

Comp12arison with the data of Burnel. Raelison and ThomAWfas The experimental data of Burnel et al. (1990) were obtained for different dimensional conditions than those of Tu and Ramaprian. It is seen from Table 1, however, that the Reynolds and Stokes numbers of the low and high frequency tests of Burnel et al. are close to the corresponding Re and a values of Tu and Ramaprian's experiments.

REFERENCES BURNEL, S., RAELISON, J.C. & THOMAS, J.M. 1990 Radial distribution of the Reynolds stress in a turbulent pulsating flow in a pipe. Engineering Turbulence Modelling and Experiments, eds. Rodi, W. and Ganic, E.N., Elsevier, 419-427. LAUNDER, B.E. & SHARMA, B.I. 1974 Application of the energy-dissipation model of turbulence to the calculation of flow near a spinning disc. Lett. Heat Mass Transfer. 1, 131-138. LESCHZINER, M.A. 1982 An introduction and guide to the computer code PASSABLE. Department of Mechanical Engineering, UMIST., TU, S.W. & RAMAPRIAN, B.R. 1983 Fully developed periodic turbulent pipe flow. Part 1. Main experimental results and comparison with predictions. J. Fluid Mech., 137, 31-58.

Burnel et al. measured Reynolds stress diiectly using cross hot wire probes. Experimental data and present compulational results for phase-averaged periodic Reynolds stress profiles in the two test cases are shown in Figs. 3 and 4. At D = 76 (Fig. 3), data and calculations show variation with phase over the entire pipe radius. Results are therefore qualitatively in agreement with Tu and Ramaprian's Case I (.o = 90). In the higher frequency case (sj = 253, Fig. 4) the measurements of Burnel et al. and present calculations reveal Reynolds stress distributions that are essentially frozen in the core region. This result is in marked contrast 'o Tu and Ramaprian's Case 2 (o = 242, Fig. 2). CONCLUSIONS Computational results for periodic pipe flow are compared against the experimental Reyaolds stress data of Tu and Ramaprian (1983) and Burnel et al. (1990). Agreement is acceptably close, except in the higher frequency case of Tu and Ramaprian. In the light of the Burnel et al. data and considering the indirect

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thought likely that the source of these discrepancies lies principally with the data.

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111-6-2

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Phase-averaged periodic RevnoldSL9.0_

EIGHITH S~'N'lPOSIlUrl ON

7

TURBULENTSEAR FLOWS1 Technical University of Munich

=1

September 9-1 1, 1991 COMPUTER SIMULATION OF SEPARATED FLOW IN A BRANCHED DUCT H Fu*, M J Tmdal*, A P Watkins + and M. Yianneskis* *Cel

for Heat Transfer and Fluid Flow Measurement School of Physical Sciences and Engineering King's College London Strand, London WC2R 2LS, UK + Mechanical Engineering Department

UMIST, P 0 Box 88, Manchester M60 IQD, UK

This paper presents a combined experimental and numerical investigation of the flows in internal combustion engine manifolds. The flow pattern thiough a single-branch manifold (T-junction) was calculated using a three-dimensional finitedifference numerical code incorporating the k-c model of turbulence. The flowfield was measured in a steady water flow rig using laser-Doppler anemometry. Detailed measurements of the mnen velocity components and correspoading tuibulence levels were obtained in order to characterise the flow and provide benchmark data appropriate for the development and assessment of the accompanying calculations. Calculations were also performed foi two- and three-branch manifolds. The flow configuration consisted of a square duct, 50 8 mm x 50.8 mm in cross-section,, with four branches, 30 mm x 30 mm in cross-section. For some of the work presented here three of the branches were blocked, so the geometry corresponded to a T-junction with a main duct/side branch area ratio of 2.87. The flow Reynolds number, based on the main duct bulk flow velocity and hydraulic diameter, was 60,000. The simulation was performed with a computational mesh size of 77 x 18 x 10 in the streamwise and two cross-stream directions respe tively in the main duct. In the branch the corresponding grid size was 42 x 12 x 7. The streamline pattern in the symmetry plane of the duct is shown in Figure 1. The results show that the flow is very uniform up to 30 mm upstream of the junction and an elongated recircuiation region of 110 mm length and 18 mm width is formed in the side branch, Figure 2 shows the predicted mean velocity vectors in a cross-stteam plane in the side duct. A single vortical motion occupies the duct crosssection. Figure 3 shows a comparison of prcdicted (solid lines) and measured profiles of kinetic energy of turbulence, k, in the plane of symmetry of the flow. The highest values are found in the vicinity of the recirculation region in the branch, as expected. The comparison of the measured and preuicted profiles of k shows excellent agreement between the data and predictions in the main duct

and good agreement downstream in the side branch;, however in the first profile in the side duct there is some disrepancy with the data in the reverse flow region where the k values are overpredicted. In the branch, the cross-stream flow comprises two vortical motions, as indicated by the profiles shown in Figure 4. The predicted profiles of velocity are shown as solid lines and are compared with the LDA measurements: notwithstanding the small magnitudes involved,, the agreement is very good acrors most of the duct cross-section. Figures 5-8 show a characteristic sample of results from the three-branch calculations. The velocity vectors in the plane of symmetry are shown in Figure 5 and a three-dimensional diagram of the flow configuration in Fig. 6. The pressure at the exit of each branch was set to the same value in the calculations, but the predictions showed that about 18% more mass flowed through the middle branch; such findings show the suitability of three-dimensional calculation methods for the prediction of manifold flows. The predicted velocity vectors in a cross-section of the side and of the main duct are shown in Figs. 7 and 8 respectively. In the upstream plane (Fig. 7) the flow pattern is similar to that in Fig. 4 in that tNo senses of circulation are evident, whereas in the main duct there is only one vortical motion. In general the simulations yielded results which showed excellent agreement with the pressure drop data (to within 2 Pa or 0.5 % of the total pressure drop along the duct). The streamwise mean velocity values in the main duct are predicted to within 2 - 3 % of the experimental values. In the side branch the streamwise velocities are in very good qualitative and quantitative agreement with the experimenta, data,, except for the reverse velocities in the recirculation region near the junction which are overpredicted. The comparison of the experimental and numerical results indicates clearly the suitability of the combined approach used for the study of branched flows.

!11-7-1

VELOCITY ON X-Z PLANE

-

-0

_________________

II

4 SIDE DUCT

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zwz

Figure 1 Single-branch configuration: streamline contours in the plane of symmetry .velocity

Figure 2 Single-branch configuration: predicted vectors in side duct.

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________________________________________

VELOCITYONX-ZPLANE 27

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SIDE DUCT

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MEASURED

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igre3 Single-bianch configuration- comparison of Ineasuredl and piedicted piotiles of kinetic envigy ol turbulence iii thle plane of symmetry,

Figure 4 Single-branch configuration: comparison of predicted and measured mean velocities in the side duct. * MEASURED -PREDICTED

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Three-branch viw.ct

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thricee-

EIG IITlI SYMI'OSIUM ON

111-8

TURBUILENT SIEAR FLOWS

Technical Uniersrty of Munich September 9-11, 1991

EXPERIMENTAL INVESTIGATION OF THE FLOW STRUCTURE NEAR THE NOZZLE Motoaki

KIMURA,

Kiyoaki ONO and Atsushr

SAIMA

Department of Mechanical Engineering College of Science and Technology Nihon University, Tokio, Japan

ABSTRACT

2. EXPERIMENTAL APPARATUS

The experiments are carried out to investigate the flow structure of jets in the near-nozzle region, rime mean static pressure and pressure fluctuatoon are measured bv the newly developed static pressure probe with a hot-wire anemometer. A schheren photograph technique is applied to the flow visualization. Axial and radial velocities are measured bi LDV. The following results are obtained. The mean static pressure and the pressure fluctuation distributions and the velocity distribution are gotten by the measurement. The mean static pressure distributions in the cross sections have a remarkable transition toward the stream-wise. The static pressure lower than the at mospheric pressure around the jet is one of the important causes of the jet entrainment.

Air, carbon dioxide and helium are separately jetted into still an from a vertical round nozzle. These gases are supplied from gas cylinders or a compressor tank. Their flow rate are controlled by pressure reducing valves and measured b the calibrated orifice. The nozzle diameter (d) is 10 [mm]. The exit flow Reynolds number is between 200 and 20000. For measuring the velocity, the laser doppler velocimeter (LDV) is used as back-scattering mode. Its light source is a He-Ne laser with a power output of 10 [mWl. The condensed oil is seeded with mean diame ers of 0.5 to I [U m] and a density of 840 [kg/In ]. A new probe to measure the static pressure was developed by the authors. The static pressure probe is designed to measure the mean static pressure and the pressure fluctuation at the same time. The static pressure measuring system is composed of a Prandtl-type static pressure probe (Fig.l), an air supply system with a constant pressure and a constant temperature hot-wire anemometer. The static pressure probe consists of a static pressure tube and a hot-wire spanning the cross section of the static pressure tube inside as shown in Fig.2. The probe is connected to the constant pressure air supply system. Th~s system supplies the constant pressure (Pi). The constant pressure (Pi) is slightly above the atmospheric pressure (Po) in order to create a bins flow at the static pressure holes through the hot-wire section. The pressure difference (Pi-Po=1l [Pal) can be kept constant with this device. The static pressure probe can be used for any gas, because of the bias flow. The air velocity at the hot-wire is settled by a pressure difference (Pi-Ps) between a static pressure (Ps) and the inside pressure (Pi). Air flow velocity from the state pressure holes L ver slow. So the effect of it I. neghgtil small 'or the measured flow. When pressure fluctuations are incident on the static pressure holes, induced velocity fluctuation is uniquely determined arid a voltage is produced. The relation between the pressuro difference and the probe output voltage is linearized from -10 to +10 [Pal under the atmospheric pressure.

1.

INTRODUCTION

Many experimental and analytical studies of turbulent free jets and the~a diffusion process have been presented (Kataoka, 1986)[l]. There is the cc m plex structure as the entrainment and the coherent eddy both spatnall and temporally in the near-noAzle region (Kimura, Ono and Saima, 1990)[21. 1he coherent structure has been approached by the flo" visualization (Roshko, 1975)[3] and the analysis o measured velocity component (Yule, 1981)[4]. Savas and Gotlahali (1986)[5] inrestigated the reciprocity of the coherent structure arid jet flames. Higaki and Saima (1983)[6] pointed out the importance of the investigation on the ilow structure neir the nozzle for the stabilization mechanism of lifted diffusion gas flames. It can be considered that the momentum and the mass diffusions relates to the mean static pressure distrrhution and the pressure fluctuation. It is important to investigate the jet behavior on the nearnozzle region. This experLmen tal stuly alims at approacuing a jet diffusion process In the near-nozzle region, from the potential core to the transitional region whuch sits up stream the fully developed region. 'e Investigate the relatior-_hp of the velocity field and the distribution of the time mean static pressure and the pressure fluct ation. This paper reports t~e first stage of the in, stugaLion.

Po.

Atmospheric pressure

static presre tube

Ps: Static pressure

Fig.1 Static pre,,:nure probe.

Ht

2

Fig.2 Static pressure tube with hot-wire.

nnu-8-

3. RESULTS AND DISCUSSION

4. CONCLUSION

Schheren photograph of carbon dioxide jet int.o still air is shown in Fig.3. The flow field is quite complex. The jet flow is laminar near the nozzle at small Reynolds number. The break point where the free boundary layer undergoes a remarkable transition from the laminar to the turbulent flow, approaches to the nozzle as Revriolds number increases. The Stream-wise variation of the time mean radi l velocity distribution is shown in Fig.4 in the case of the carbon dioxide jet at Re=5000. X is th axial coordinat". r is the radial coordinate. V is the mean radial velocity. Um is the mean axial velocity on the jet axis. The mean radial velocity LS about zero (V/Um=O) under the jet center region in the cross section from X/d=0.06 to 3. This range overlaps the potential core. Thus, the condition, which mean radial velocity profile is flat and almost zero, is one of the potential core characteristics. There i3 mean radial-flow out side the potential core, and its flow is toward the surrounding. The further outer region is figured by hatching, and there is mean radial-flow from the surroundings toward the jet center. The further outer flow region overlaps the jet entrainment. The spread of this region shows that the entrainment ranges largely. The Stream-wise variation of time mean static pressure and pressure fluctuation distributions are shown in Fig.5 (a) and (b) in the case of the cArbon dioxide jet. p' is the rms pressure fluctuation. Three remarkable characteristics are observed. (1) The mean static pressure distribution at the limit of jet csnter is the positive static pressure (Ps>Po) in the cross section X/d=3 and 4. Out of this limit region, the mean static pressure distribution is the negative pressure (PsI

:

4

r.in s -value of the component 1 * SE

From these a turbulence ntcnsity Tup can be derived:

1 T

2

2

W

> 1

>2 +

-C >

1, the stochastic particle motion is chiefly determined by, the non steady-state character of the negative corona, for NEHD values < 1, however, de puticle velocity fluctuations are governed by the turbulence of the neutral fluid. This behaviour is clearly demonstrated by Fig. 4 which shows the measured turbuleice inteisities Tup of the particles.

Fig. 2 shows the results of the measurements wiihih can

RefereceC:

be related to the residence time and the saturation charge iespectively. When compared with calculated values it can be shown that the migration velocity of the particles is approximately three times higher tha.l theoretically expected, provided they receive at least 90 % of their saturation charge. -

IIl Leonard G.L., Mitchner M, Self S.A.: An experimental study of the electrohydrodynalic flow in electrostatic precipitators; J. Fluid Mech. 127 (1983) 123-140 121Yamoinoto T.:Some aspucts of efficiency theory for r pp.d.n e or. .... tro,,a,,c precipitators; Send. rInternafino1,2onfercnco

The results of the measurements (Fig. 3) distinctly verify

Electrostatic Precipitation, Kyoto (1984) 523-521 131Shaugnessy E.J., Davidson J.H., lay J H.:The Fluid

that the ionic wind influences the parti(le motion At a voltage of 35 kV, the velocity profiles change and, its one can obscrve, this change is more dominant for 0,71 m/s thtan for 1,41 n/s. In contrast to the 0 kV state, the particle velocity increases at

Dynamics of Electrosritic Precipitators; Aerosol Science and Technology 4 (1985) 471-476 141Yabe A., Mori Y . Nijikate K.. El ID study of the corona wind between wire and plate electrodes, AIAA J. 16 (1918)

the central channel regions and decreases substantially at the wall zones. This phenomenon can be seen as an indication for the existence of secondary flows as already observed ii the

340-345 151Yamamoto, T., Velkoff, Il.R..Electrohydrodynamnics in ain Electrostatic Precipitator; J.Fluid Mech. 108 (1981)1- 18

case of positive corona fields 14,51. The ionic wind does induce eddy regions as a result of continuity, which causes the observed particle velocity decrease in the vicinity of the channel walls and the particles can hardly follow such a back-flow of the fluid phase due to the influence of the electrical force.

111-15-2

E11G111i SYN1!POSIUNI ON FURBULENT S11EAR FLOWS

Technical University of Munich September 9-11, 199 1 hOME FURTHER EXPLORATION OF TURBULENCE MODELS FOR BUOYANCY DRIVEN FLOWS

S. Vasid

and

K. Hanjali6

Maginski fakultet, University of SaraJevo Omladinsko setaliste bb, 71000 SaraJevo, Yugoslavia

the interface of turbulent and non- turbulent fluid The penetration of the mixed layer incurs a reversal of sign of the heat flux in the region close to the interface (compensating cooling of the stable fluid and heating in the second incase). caseother firstand in thethese the recorded features All pose a se eeres and other mented serious a pose experiments, other and mentioned challenge to turbulence modelling.

ABSTRACT Based on truncation of general second order closure methods, a simpler form of model is searched which would enable sufficiently faithful reproduction of specific features of buoyancy dominated turbulent flows, but still retain an eddy-viscosity form which general be incorporated easily into the could 3-dimensional Navier-Stokes computer code for complex

Second group of considered cases concerns again simplified geometries with side heating and cooling so that the gravitational vector is perpendicular to the mean temperature gradient. The representative of Cheesewright and cases are the experiments coworkers, and of some other authors. A specific feature of these cases is the concentration of steep temperature and velocity gradients as well as high turbulence intensity in the thin boundary layers quiescent and almost along the vertical walls isothermal core region occupying the majority of the channel cross-section.

geometries. Validation of the model was carried out buoyancy characteristic several by considering dominated flow situations INTRODUCTION are flows turbulent dominated Buoyancy characterized by several features which can not be modelled successfully by popular k-c or other eddyviscosity type models. A strong coupling and a mutual feed-back between the velocity and temperature field, as well as inadequacy of gradient hypothesis by which turbulent flux is usually represented, are some of most widely used However, such specific features. for not suitable codes Navier-Stokes incorporation of second-order closure scheme which would overcome most of the deficiencies of simple models. Hence, for time being, a compromise is eddy basic retain the which would searched diffusivity formulation, but modified as to account for mot important buoyancy effcts.

The enclosed diagrams illustrate the quality of predictions obtained by a simplified version of full stress/flux Lodel as well as by a modified version of k-c model which produces almost the same accuracy.

The present paper focuses on the analysis of "clean" in which test cases relatively simple buoyancy effects can be isolated and various modeling ideas explored in details. For that purpose two groups of test cases have been examined, one In which the density stratification vector is collinear with the gravitational vector (bottom heating) and the other with two vectors being perpendicular (side heating and cooling).

measured by Deardorff et al. The presented results were computed by a simplified form of the second order model which entails the solution of t-ansport equations for vertical turbulent heat flr.x and temperature variance in addition to the energy and dissipation equations with the standard LaunderSharra [4] low-Re-number modifications. The vertical profiles of mean temperatures and total heat flux at different time instants show excellent agreement in full details, including the sign reversal of the heat flux just below the interface. The development of the (two different the mixed layer thickness of

RESULTS AND DISCUSSION Figures 1 to 2 present measured and computed values

We consider first the two test cases which , of almost one-dimensional, simulate by buoyancy. mixeddynamIcs layers, driven prrel unsteady, the The first is the well known experime',.t of Deardorff et al. (1969) [3] of the development of a turbulent mixed layer in the lower region of initially stably stratified quiescent fluid by heating from below, and its upward penetrative convection into the stable is the case The second test It. layer above laboratory model of Markofski and Harleman (1971) [5] of the development of a surface mixed layer due t) the sutrfecA cooling (si ilation of diurnal variation in the of a ternocllne) and advancement ian donwr advanngen ofha and its itsedownward both of feature common A fluid. underlying stratified cases is the creation and growth of highly turbulent mixed layer in which the transport of momentum and the turbulent diffusion, while heat occurs by uniform, basically remain profiles temperature demonstrating clearly the failure of the simple local gradienL -A

~~

~

~

important feature of these caseb is the entrainment of adjacent overlaying oi underlaying non- turbulent consequent layer and a into the mixed fluid development of a sharp density jump (termocline) at

the comparison of the of two major parameters

definitions, one corresponding to the position of sign reversal, and other to the minimum turbulent heat flux), both show very good agreement, except for the initial period which Is uncertain because of predicted time initial conditions. The unknown development of the fluctuating temperature variance At different heights show a (not shown here) qualitatively similar behaviour to those found by the of scatter large a though experiments, experimental data brings in a dose of uncertainty, ntb so soa differenee that some difference in rates of change can not be explained satisfactory at present.

~

Figures 3 presents a comparison of measured and computed mean temperature profiles for one of the experiments of Harkofski and Harleman [5] for the period of one diurnal cycle with successive heating f ufc .. "---,; layor by -zdiction= -_nA - ,,fa: oll~

~

convection (here also the simulation was performed oZ the reservoir inflow and outflow, with changeable water level and variable radiative heat exchange). The computations show a good agreement including the

111-16-1

compensating heating of the entrained fluid below the

heat fluxes

mixed layer, during the period of surface cooling.

Figs.

Figures 4 and 5 show the comparison of predicted profiles of vertical mean velocity and mean temperature at the mid height of a 5:1 rectangular channel heated and cooled from sides, compared with measurements of Cheesewri~ht and King [2]. The presented results have been obtained by a kind of low-Re-number algebraic flux model which could be reduced for computational convenience to a form of k-c model in which a tensorial "Prandtl-Schmidt" number accounts for additional terms originating from truncated full stress model. This procedure generates a more general invariant- instead of simple gradient hypothesis for turbulent heat flux. In addition the length scale correction of Yap [6] was also applied.

qualitatively similar behaviour as measured data but predictions, athe tuned to smulate fully symmetric situation, could not follow a vsible asyemietry of the experiments. The paper demonstrates that a form of k-c model, suitable for incorporation into a general NavierStokes code for a complex 3-d flow, but modified on the basis of a truncation of the full stress/flux second order model, can generate a reasonably faithful reproduction of several experiments of distinct features with buoyancy flux being aligned or perpendicular to the imposed external heat flux.

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visible improvements of the predictions. The wall shear stress shown on Fig. 6 agrees reasonably well with the experimental data for both walls (which should have been equal If full symmetry had been attained). Profiles of turbulent shear stress and

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III I Ills study Wo' tnicitse~ (in Slow Term motiel of tilt, where 11 ai Ill are defined by l1oi~ klibkfi anod IlI* hie prorcsure- sI~trnl Corteinltione, It Is well1-knon that Sloew kbk.i.hj bl ni ore cal led as the sec.'d and the third Term uil neI,'es fitrr-n rntuyI below RTI I phertnomenoeninvariant of anisotropy tenber b 1.1. respr tlvely. andt Hi II ;ets oendns on Itile tir d Invar lntid. Illi(if tlitr~ op y leisl n~rNr toogey. or exi1st ing mode li. (II Rot l 3. lRED IfTABI ITY OF EXI STING MODELS (I tiinuley aend Newmno ii iShimn (ivlSenrlnr aind Spenla. At first we Investigated the predictability or exisiting mitt onin-chLe 1111dtine. ollimriceil results were compared with~ modelI m. Four modelrs were use~d for thin purpose. Aeccording ex pet-ivmit Ik ill t it. We' forund that each model Call Out to Eli, (6) tiwy can tbe suomar iz7ed as fall ows. repi othit til, peieeante crrn'c IIy in ithler tile cease vith modnel Iil Rattio poit ive Ill orc th(, ote wi thn egat ,4~i Ill. Than we trn ied If -3.0 to 4etn.irtt Ict an ie mno- Ilinear Slow Term modul by akng it~ .0 totnt;t ntis inc (rnet i uis nof Inavariaints. It is tshon thant our MadloI (ill [aIn ey and Newman o)Int m uait ein' tine Hi I tInltttIiOMn'ti more reasonablIy. Far of .2. Fig exp (-7. 77/Ret II.-, stnuty i n ricnintm ati, tint! effect or low-Heynoldfl-,euuber. * 172/Rct .80.1I ln( I 62.4* (11/2.2.3*111/3111 ci 0.0 1. INiTRODU GiT I ON whnere F-1-9/2*ll.9*lll and Rtq/(9fiesyl It-e',toi ab stress trannspnort modl I be low RSTM )IN on! Model (il111 Sb lma or time-avc'rnined turbulnace modeels. RSTM Is superior to a 2#7.297*Fri*Fsllods vis:ost ty monelis ( typicnally k- c model )Iin the mniif .0 Vtromti bly bVN'c113.'t! it Lenknv accounts of titl! Reynolds-stresswhere FrI = 1-cxpi- (0. I50Ret) I I ratnreuat pirltmn'ttn. A ioat nr f IST~s have been proposed anti Model (ivi Spez iale and Sarkar i rivent i Intneed. ranin n its uutndeu'-livcc- Hodi tind Siis . of -3.4 RSTietS loVeV bVet coteSXtruCnla ntn ine basit Of Maty woUH~J1pai -3. 00 (a -2. 0) tianats. Wethink thnat ties It shoald be dotneto cheeck again Mntdel (I) Is linear in bij. Modacel (ii) and (Ill) arc, whnether nude term is modelnani reasonably or not. Presenet quasi-linear. Model 0i0 Is non-linear. Model (I)Is Lte study is reganrdedn ens tile' first step (if such Investigations, most Popular In conventional RST% and Nuceeds whe the We foucmea oie Hieturn-Lo-lsotrrpy ( below RTI I problem. It turbulence is not strongly aninotropic. Mandel(ili) was ts vol I -known Lnt tis procesti depends on whether tife third developed lin the utear-wall calculation. irnvaerita Ill ise post~itve ncrnegative. Ontepurpose ort Calculations have been performed by four step Rungetia studny in tLt chieck soer models far Slov Term ni Kutta method which In second order in the time marching. Art jnrnssarr-tmin cuiirelationt. Four nfuiting models were thte initial es are not measured. we set thom so that the selected o tnini le rnntmericuel resuilts wore compared witih chtanes of (I are watce~hd with the experimental data. 4Ae.rrltannl dtatu It i4s rhown that they can not reproduce Time increments were selected as nearly sameci magtnitude an HITI piennrt correctly inn either tile case with positivet 1/500*( final time In experiments 1. In Oder to chock tile I II aIrItie onto wfi tit negat ive lI1. The other purpose Isi to appropriatness of this time Increments, calculations were carrnre~tncI% new tem otin-linnear Slow Term model by making perrfoirmed with i/IDDolvI final time 1. Hloweve~r meaningful nnnrl-intr toe fntct,innns nof intvarianets. It Is shoewnt that our discrepancies could not be recognized. mandt'i is ablie to predict RTI process more reasonably for Four experimental data were adopted in this study. mantle 111>0 fold IM< ( IIlI.e Pnven. Goence anld Comte-Ifllot, (111>0 and 1110). W3Tucker undt Reynolds Z.GFiNERtAt. EXII:XHSSION1 OF UF,*TJRN-rO- ISOTROPY (111(0). Exact Ileynnaldr stress transort equations intclee ale Fig. I to 4 show the temporal decay of normal Recynoldst unnknonu presantre-litraln cmortrlation. Because this Ie restresses. Numerical results are compared with the cnrrespamasituie for Line enerloy transfer between each Reynolds spending experimental data. Because of the limited space stress n:ermtntitett, it. Ii very Important to modlel this terms theser figures Include the results obtained by our model cnrrectly in HT~re. Conventionally this correlation is I Modtel (v) I which will be described In the next section. mindeinni by tinl! feel lowing expression, From these figures following characteristics are evident. II'inuntrc-Strairt Correlation)- eli. Ill# 1.1.24 C.w (I) Par 111)0 Moee (1) and 00v reproduce the correct UtTl miere 0 Ui. 1. 0 .2. etnel0 i.i~w reprueent. Lte contribution process. but for 111