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i-mn, we arrive at a total of 2 3 disturbance classes denoted by ECC, ESC, ECS, ESS, OCC, OSC, OCS, OSS
(14)
which can be analyzed separately. The letter E, 0,5, and C obviously stand for even, odd, sine and cosine like symmetry. Depending on the Prandtl number and the wavenumbers ax,ay of the steady knot solution, different classes of disturbances correspond to the highest real part crr of the growthrate cr. In the case P = 4, disturbances of the type ESC are the strongest grow ing ones. Their evolution to finite amplitude can be followed through time integration of the coefficients aimn(t) in the representation (10). Examples of such computations are shown in figures 4 through 6. Although there is already an increase in the heat
27
Fig. 4: Oscillatory knot convection with ax = 1.5, ay = 2.0 arising from the ESCtype instability in the case R = 36000, P = 4. Equidistant plots in time are shown in each column from top to bottom corresponding to a half period 0.020 of oscillation. Lines of constant vertical velocity in the planes z = 0 and z = —0.4 and isotherms in the same planes are shown (left to right). The x-coordinates run downward from 0 to 27r/ax.
28
Fig. 5: Isotherms (left) and lines of constant d cot 6. One can use this inequality together with the condition A < 1 to transform the general conditions (30), which leads to the local-validity conditions for the small-amplitude EE (31): max(W7L 3 , R/L,
1/L 2 , A) < 1.
(32)
The analysis of the derivation of the EE (31) shows that the third, destabilizing term has its origins in the inertia terms of the NS equations. The two stabiliz ing terms, the fourth and the fifth, are due to the hydrostatic and capillary (i.e.
50 surface-tension) parts of the pressure, respectively. Finally, the last, odd-derivative, dispersive term is generated by the viscous part of the pressure. Such a term also appeared in the EE which was obtained by Topper and Kawahara [93] who assumed the plane to be nearly vertical (as a result, the hydrostatic term was absent in that equation): They used the small angle of the plane with the vertical as their (single) perturbation parameter. Our derivation shows that this assumption is unnecessary. One can see that for an arbitrary inclination 0 (and any value of W), provided R is close to Rc = 5 cot 0/4 (so that 5 is sufficiently small), the equation (31) can be good, with its dispersive term being not small. At the same time the hydrostatic term can be large (but one needs a sufficiently large spanwise lengthscale Y for this, viz. Y ^> Z). If the dispersive term is omitted in the equation of Topper and Kawahara, it becomes the one obtained by Nepomnyashchy [77]. The one-dimensional version of the latter is just the Kuramoto-Sivashinsky equation [64, 87]. Thus, all of these equations—as well as the Zakharov-Kuznetsov equation [98] (see also [65])—can be obtained as certain limiting cases of the EE (31). Because the EE (31) is dissipative, the system evolves towards an attractor. Thus, it essentially forgets the initial conditions. On the attractor, there cannot be any systematic change in time (although fluctuations around the constant averages may have arbitrarily large amplitudes). Consequently (as was first argued in [6, 37]), the destabilizing inertia term and the stabilizing, capillary one should be of the same order of magnitude. Hence, an estimate of the (dimensionless) characteristic lengthscale at large times, L a , follows: La = (W/S)1/2. (33) Similarly, the balance between the nonlinear "convective" term and either the disper sive term or the capillary one (whichever is larger) determines the asymptotic magni tude of the characteristic amplitude: Aa = max(W r /L^, 1/1%). With these estimates, one transforms the condition (16) to the form max( W/L\, R/La, \/L2a) 5. Thus the global-validity conditions can be written as Eq. (17) again, but with the generalized definition of La given by Eq. (33). The transformation of Eq. (31) to a "canonical" form is accomplished by rescaling 3 3W/252. v = Nrj, z = Laz, y = Lay, and t = T a t,with N = W/(6L a) and Ta = The resulting small-amplitude evolution equation (with the tildes in the notations of variables omitted) is Vt + Viz + Vzz ~ Wyy + V477 + AV277Z = 0.
(34)
(with definitions K = cot 9/5 and A = 3/y/WS). This is the generalization of Eq. (18). Looking at the linear stability properties of Eq. (34), one substitutes the normal mode rj oc exp(s£ - iujt) expi(jy + kz) into the linearized version of Eq. (34). This yields the growth rate: s = -K (jf - (j)4 + [l - 2 (j)2] k2 - kA. From ds/dk2 = 0, we
51 find the streamwise wavenumber kmax corresponding to the maximum growth rate (at fixed j): k2max = 1/2 - j 2 . Hence, the maximum growth rate smax(K,]j) is Smax = 1/4 - j 2 ( l + *).
(35)
One can see that, for every fixed j , this smax is a (linearly) decreasing function of K. The results of the linear stability theory are useful for understanding certain results of numerical simulations (see [47, 48]) of EE (34). The same problem has been treated earlier by Krishna and Lin [63] with the LW approach (see also [84]). They derived a Benney-type equation in the series form (28). They had cited explicitly about one hundred terms of that equation, among which all the terms of Eq. (34) can be found. However, as the MP derivation establishes, only a small number of those terms are essential and should be retained in the equation. All the other terms can be neglected under conditions (32) or (17), i.e. when the equation can yield a good approximation. 3.1.3
Numerical studies of evolution equation
We [48] have numerically simulated the dissipative-dispersive equation (34) with pe riodic boundary condition. Below, we discuss some preliminary results. It seems plausible that the results will be insensitive to the exact form of the boundary condi tions and the size of the integration domain provided the dimensions of the domain are sufficiently large; namely it should include many "elementary structures" (we note that similar simulations give "surprisingly good results" [44] for certain problems of the boundary-layer wave transitions). Accordingly, we solved Eq. (34) on extended spatial intervals, 0 < y < 2Kp and 0 < z < 2irq, with p >> 1 and q ^> 1. We used spatial grids of up to 256 x 256 nodes and employed the Fourier pseudospectral method for spatial derivatives, with appropriate dealiasing. Time march ing was done (in the Fourier space) with a third-order Adams-Bashforth and/or Runge-Kutta methods. The results were verified by refining the space grids and time steps; by observing the volume conservation, / rjdydz = 0; etc. Some of our initial conditions were motivated by the experiments [72]: these initial conditions modeled the inlet conditions of their experiments [see Eq. (36) below]. Below, we discuss some results of our numerical simulations, such as (i) the effect of dispersion on the large-time behavior of the film surface near the attractor and (ii) the dependence of the transient states on the wavenumber of the initial "forcing", as compared to the experiments of Liu et al. [72]. 3.1.4
Unusual patterns on strange attractors for strongly dispersive falling films
In this section, we consider the most unexpected result of our numerical simulations. Namely, during our studies of the strongly dispersive cases of Eq. (34), A ^> 1, we
52 found highly nontrivial patterns (of the film surface) persisting for all large times. The system evolves toward these self-organized states starting from the small-amplitude (3-D) white-noise surface (whereas with the same, random initial conditions, look ing at the large-time states of the surface in the case of small dispersion, we see no ordered patterns). This contrasts the spatial patterns studied up to now in fluiddynamical experiments—as well as in solid state physics, nonlinear optics, chemistry, and biology—which invariably were almost periodic, at least locally (see e.g. [22]). Our results indicate that patterns of quite a different nature can exist on attractors of driven dissipative-dispersive systems. These patterns [48, 47] consist of two subpatterns of localized soliton-like deviations of the surface. One of these subpatterns is a V-shaped array of larger-amplitude bulges on the film surface. Those move in a "sea" of smaller-amplitude bumps which constitute the second subpattern. Each of the two subpatterns spans the entire domain of the flow and moves as a whole along the z coordinate axis. However, the velocity of the bulge formation is different from that of the bump subpattern, so one subpattern "percolates" through the other. Figures 1 through Fig. 3 illustrate these points. A snapshot of the film surface at t — 3200, for a vertical film (i.e., K = 0) and with A = 50 (and p = q — 16) appears in Fig. 1. (Note that in Fig. 1 the amplitudes of the structures—which in reality are all small-slope—are exaggerated, due to a different scale of the rr-axis.) The evolution of "energy" E oc / r}2dydz shown in Fig. 2 demonstrates the time t = 3200 the film waves have approached their asymptotic state in which there is no further systematic change (although chaotic undulations of the energy are evident). One can see the above-mentioned two-stream nature of the pattern in Figures 3(a) and 3(b). The V-shaped formation consisting of 13 large bulges (see also Fig. 1) moves down between the two snapshots shown [the earlier-time one in Fig. 3(a) and a later-time snapshot in Fig. 3(b)]. The small-amplitude background subpattern moves uniformly as well, but in the opposite direction (in the moving reference frame of the observer). One can see also that the bump subpattern slowly changes with time. (We have also made a computer animation in which one can clearly see these motions of the two subpatterns.) It had transpired that one component of this dynamical, spatiotemporal pattern, viz. the bulges, has already been observed by the authors of Ref. [92], who performed the simulations of the vertical-film equation with A = 25 for p = q = 64/(27r). (In fact, they simulated a dissipative-dispersive equation [93] in which the K term of Eq. (34) was missing, which happened because of a certain—unnecessary, as we discussed above—assumption on which their derivation was based.) However, as far as we know, the authors of [92] used only contour plots as their graphics tools, and overlooked the bump subpattern. Thus, the entire complex, dynamical character of the two-phase pattern remained undiscovered. Figs. 2 and 3 confirm the estimates of characteristic quantities obtained with the term-balancing considerations (see [48, 47] for more on this point). For the 1-D version of Eq. (34), a perturbation theory of weak interactions (e.g. [9, 8, 42, 27, 56]) of the "pulses" (see [55]) was developed. It is based on soliton
53
Figure 1: Snapshot of the pattern on an attractor of Eq. (34) with K = 0 (i.e. the scaled surface of a film flowing down a vertical plate, here—down the page, with illumination from the top left), A = 50, and periodic boundary conditions on 0 < y,z < 32n: a view in an oblique direction, for i — 3200.
F i g u r e 2: Evolution of the surface deviation "energy" Ji]2dydz from an, initial smallamplitude "white-noise" surface to an. attractor of Eq. (34). The snapshots of Figs. 1 and 3 were taken near the end of this run.
54
Figure 3: (a) Cross-stream view of the pattern shown in Fig. 1. i;;: Ji.-.m? as in (a) but after a time interval At = 0.25. Note that the vertical motion of t :>-■ ■'• ■■ h.-.ped formation of bulges from (a) to (b) is in the opposite direction to that of the n\n\u) lottem. solutions of the ODE describing the pulse in a certain reference frame, The soliton, or at least its small outer "tails"—which are responsible for the weak interactions— are found analytically. In contrast, no analytic solution is available for a solitary 3-D bulge. Therefore, the prospects for an interaction theory of the 3-D bulges appear to be uncertain at best. One observes that the bumps incessantly collide with the bulges. These interac tions are seen to be (almost) reversible, like interactions of KdV pulses (e.g. [970. This contrasts with the irreversible coalescences of 2-D pulses discovered in [38] and [591 for highly nonlinear dissipative equations. It has been implicitly assumed (see section 4.1) in the derivation of the globalvalidity conditions for the EE (34) that the solutions of that equation remain bounded. Our simulations justify that assumption. Also, the amplitude of the solutions turns out to be of the same order of magnitude as its estimate obtained by the painvise balance of terms in the evolution equation. These simulations also confirm that the large-time behavior of the solutions of (34) is essentially insensitive to the initial conditions: Every solution evolves toward an attraetor (whose nature is solely determined by the basic parameters of the XS problem). Similar to Eq. (34), we [29, 36] have derived an evolution equation for a film which lows down a vertical cylinder (see Eq. (46) in section 3.2.2). The only essential difference between the two EE is the opposite sign of the K term. However, this term disappears in the case of a vertical planar film, as well as in the limit of an ininitely large radius of the cylinder. According to our simulations, the corresponding #~~2~ term of the annular-film equation (in which the variable <j> is changed to y = 60, so that 0 < y < 27r6 = 2wp) is sufficiently small even for p as small as p = 55 so that the results essentially coincide with those of the K = 0 version of Eq. (34). So, physical
55 experiments with films flowing down vertical cylinders can verify the theory which leads to the planar-film EE (34). However, the cylinder should be sufficiently long so that the waves propagating from the inlet could have enough time to approach the attractor. It is not difficult to find that with h0 ~ 1 mm, the radius ~ 1 cm, and with the restricting conditions (17), in order for the cylinder to be not too long, the liquid should be sufficiently viscous, in fact several hundred times as viscous as water. This can be easily achieved with e.g. a glycerin-water solution. [It is interesting that one can see a straight row of bulges in the photograph of a film flowing down a cylinder in Fig. 2 of Ref. [12]; however, the jRi-condition of validity (17) was not strictly satisfied there.] 3.1.5
Transient patterns: Qualitative agreement of simulations w i t h ex periments
As was mentioned above, the large-time behavior of the film is insensitive to the ini tial conditions. In contrast, the transient states do "remember" the initial condition: the film surface exhibits a variety of patterns as one varies the initial conditions. We studied the transient states primarily in connection with the recent experiments of Liu et al. [72], in which the flow (down an inclined plane) was perturbed at its inlet with sinusoidal pressure variations having a fixed frequency / . In some of their experiments, in addition to this single-frequency "forcing", they used a secondary forcing at the subharmonic frequency / / 2 , with an amplitude considerably smaller than that of the primary forcing. (The objective for the secondary forcing was to enhance possible broad-band subharmonic resonances). We modeled this inlet (pos sibly, two-frequency) forcing by the initial condition rj(y,z;t
=
0) = Af cos(nfz) + Ah[cos{2nfz
- fa) + cos(3nfz - fa)]
+Ah[cos(rifZ — fa) + cos(2nfZ — fa) + cos(3n/^ — fa)] cosy +Asub[cos(0.brifZ — fa) + cos(0.5n/£ — fa) cosy] H-small white noise,
(36)
(where y = y/p, z = z/q, and fa are independently generated random phases), with Ah +++)c_ ( c _ __++)] G LL{\ ( l - - -t)
(18) (18)
90 where <j> represents the degree of constitutional supercooling. The threshold values are obtained as the roots of Eq. (18). The numerical task of determining the zeros is accomplished by a bisection method that is based on Brent's method. This method is very efficient in detecting all zero crossings even for functions that are discontinu ous or whose curvature changes rapidly near a root. The root of interest is the solid ification velocity and its dependence on the bulk solute concentration with the rest of the physical parameters fixed. The following thermophysical properties, which are appropriate for a tin-bismuth alloy13, are considered: D = 1.8 x 10~ 5 cm 2 /s, k = 0.29, and m = —2.2. Two cases are considered. First the solidification ve locity is solved as a function of the bulk solute concentration with the rest of the parameters fixed. For S > 0, we have fixed two pairs of values for S and GL- The first pair is S — 0.01 and GL = 180 K/s, and the second pair is S = 0.0001 and GL = 180 K/s. The results, shown in Fig. 2, depict neutral curves that are qualita tively similar to those obtained by Hurle12 and Van Vaerenbergh et a/.13. However, a quantitative difference exists between their results and ours, namely, our neutral curve lies above theirs. This can be seen by comparing their asymptotic value for large c^ of 1.6^/m/s to our asymptotic value of about 10//m/s. The critical velocity is also found to decrease with decreasing values of the Soret coefficient. Fig. 3 de picts the neutral curve for negative Soret coefficient for S = —0.01 and GL = 180 K/cm. We note that, in this case, the nonlinear equation has two roots and only in a narrow range of values for c^. Our neutral curve differs significantly from the curve obtained by Hurle12 but shares some similarities with the curve corresponding to a linear thermotransport term. 13 Second, in order to clearly depict the modification of the threshold values for the onset of the constitutional supercooling by Soret diffusion, we have plotted the neutral curves in the (c^ — GL) parameter plane. Fig. 4 shows, on the same graph, a plot of the neutral curve for S = 0.01 and V = 200//m/s as obtained from the solution of Eq.(15) and a plot of the neutral curve without Soret diffusion which is described by 14 , GLDk C (19a) ~ = -(l-k)VmThe case for negative Soret coefficient is shown in Fig. 6 for S = —0.01 and V = 200/im/s. We note that, for positive Soret coefficient, higher temperature gradients are required for the onset of instability when Soret diffusion is included. Thus, the Soret effect has a stabilizing effect in this case. The opposite scenario holds, however, for negative Soret coefficients when realistic temperature gradients (GL > 20K/cm) are considered. The changes that occur in the concentration profile due to the inclusion of the Soret effect will also induce a change in the degree of supercooling (j>) Eq. (18),
91
woo
Figure 2: Neutral stability curve in the (c^ — V/D) plane for parameter values S = 10" 2 and S = 1G~4 (insert). which in turn affects the critical wavelength of the instability. This is given by \TM1
A = 2TTJ
(196)
where A is the wavelength of a sinusoidal deformation to the solid-liquid interface. We observe from Fig. 4 that positive Soret coefficient is associated with an increase in the degree of supercooling, and hence leads to a decrease in the wavelength at the onset of the instability. The opposite scenario holds for S < 0 and realistic tem perature gradients. Note that the regions of constitutional supercooling lie above the curves shown in the stability graphs. 3.
Soret-driven Convection Effects
The coupling between Soret-driven convection and solidification has been exam ined by Hadji and Schell15 and Zimmermann et a/.16. We refer the reader to other references in the latter paper. They investigated the influence of Soret induced con vection on the morphology of a solid-liquid interface. They utilized a Benard-like set up in which the top plate was maintained at a temperature that is low enough to allow for the formation of a thin layer of ice. The solid-liquid interface that appears in the process is assumed to be rigid and planar in the absence of convection, but is allowed to deform under the influence of the convective currents in the liquid.
92
Figure 3:
2
Neutral stability curve in the (c^ — V/D) plane for parameter values
s = -io- .
Figure 4: Neutral stability curve in the (Gi — CQQ) plane for parameter values S = 0.01 and V = 200^m/«.
93
Figure 5: Neutral stability curve in the (G, - Coo) plane for parameter values S = -0.01 and V = 200f/m/$. 3.1
Formulation
Consider a thin layer of a dilute binary mixture contained between two rigid and isothermal plates that are separated by the distance d. Fig. 6 shows a schematic diagram. The temperature gradient across the plates is selected so as to allow for the solidification of an upper portion of the fluid mixture. The solid-liqid interface that separates the lower liquid phase from the solid layer is assumed to be planar in the conduction state but deformable under the influence of the convective currents. The mathematical model is based on the conservation equations for heat, solute, mass and momentum, which in the Boussinesq approximation, take the form16, ^
01
^
+
v
• VTL = /c L V 2 T L ,
+ v • VC = DV • [VC - SC(l - C)Vr L ], V • v = 0,
?£ + v • v = — V p + i/V2v + [aT(TL - TR) + as{C - CR)]gk,
(20a)
(206) (20c) (20rf)
where TL, C and v are the temperature, concentration and velocity in the liquid phase, respectively; aT and as are the thermal and solutal expansivity coefficients, respectively; g is the gravitational constant, k is a unit vector in the upward vertical
94
Figure 6: Sketch of a solid-liquid interface that forms in a binary mixtures between two plates. direction and TR and CR are reference values. Only heat conduction is assumed to take place in the solid, and this is accounted for by dTs di
KSV2TS.
(21)
Eqs, (20-21) are supplemented by the following boundary conditions. The kinematic condition and the no slip conditions at the interface yield v « n = v x n = 0,
(22a)
where n is the unit normal vector to the interface. The interface temperature is obtained from an idealized phase diagram and is described by TL=TM
+ mC.
(226)
The conservation of mass which takes into account the thermotransport due to the Soret effect takes the form C^l
= -D[VC
+ SC(l - C ) V I i ] ■ n/|n|
(22c)
where |n| represents the magnitude of the vector n. The balance between the production of latent heat £ and the interfacial heat flux yields £jt
=
lKsVTs^KLVTL].m\m\
(22d)
95 The continuity of temperature at the interface implies TL = TS}
(22c)
and the assumption of a lower plate that is impermeable, rigid and isothermal leads to T, = T0,
v = 0,
and
— + 5(7(1 - C)-^-
= 0,
(22/)
At the upper plate, located at z = d + h) the temperature is fixed TS = TU
(22g)
where Xi is assumed to be lower than the freezing temperature of the mixture. The governing equations are then made non-dimensional by using h^ for length scale where hi is the height of the fluid layer bounded above by the planar solidliquid interface and below by the lower plate; time is scaled by the thermal dif fusion time. Several parameters appear in the nondimensionalization, namely, the Rayleigh number R, the separation ratio S, the Lewis number r, the dimensionless solid layer height A and the dimensionless liquidus slope 6. The results of the linear and nonlinear studies are discussed in the next sections. 3.2 Linear Stability Results Zimmermann et a/.16 carried a detailed linear stability analysis of the conduc tion state. They consider a non-linear relationship for the thermotransport term in the concentration equation which led to an expression for the concentration profile that is non linear in the vertical variable. As expected, their results show that the bifurcation to a state of convection depends on the separation ratio, namely that a critical value for the separation ratio 5* exists below which the bifurcation is to oscillatory convection and above which it is to steady convection. The initial concentration C 0 appears in the results for the effects of diffusion and conduction. It is found that the critical Rayleigh number for the onset of convection /? c , the critical wavenumber kc and the critical frequency UJC depend on Co- For the case of steady convection, S < 5*, both Rc and kc decrease with Co. However, for the case of oscillatory convection, RC) kc and u>c all increase with Co- The dependence of the critical parameters on the initial concentration can be explained by the fact that an increase in initial concentration leads to an increase in the effective Soret coefficient. The effect of the solidified layer is represented by the parameter A. For S > S«, both Rc and kc are found to decrease with A in agreement with the results of Hadji and Schell 15 . An explanation of this result is stated as follows: As the parameter A increases, the solid phase begins to act as the upper boundary
96 for the fluid layer. Since the thermal conductivity of the solidified layer is lower than that of the plate, a temperature fluctuation carried in the upward direction is less easily released through the effective upper boundary. Zimmermann et a/16 have also noticed that the critical Rayleigh number and associated wavenumber are only slightly affected by increases in latent heat. The impact of an increased latent heat is, however, noticeable on the value of the angular frequency at critical. Hadji17 considers the case of plates that are both nearly impermeable and poor conductors of heat compared to the fluid. The reformulated problem is characterized by two coefficients: a heat transfer Biot number 7 and a mass transfer /?. For S > 0, which corresponds to steady convection, it is found that the variation of the critical wavenumber kc with the separation ratio S depends on 7 and /9. Namely, it is observed that kc decreases with S for 7 > P and increases with S for 7 < /?. The magnitude of kc becomes vanishingly small as both 7 and /3 approach zero. This case corresponds to a system that is completely insulated and impermeable. The critical Rayleigh number is found to decrease monotonically with decreasing values of 7 and /3. 3.3 Nonlinear results Hadji and Schell15 have carried a weakly nonlinear investigation of the interac tion between steady Soret driven convection and solidification. In order to make their mathematical analysis tractable, they assume that thermal diffusion due to the Soret effect is independent of the concentration. The basic state is, in this case, linear in the vertical variable. Their linear stability analysis shows that, for a relatively large positive separation ratio, the conduction state is unstable to long wavelength disturbances and that the bifurcation to convection is stationary. In this limit, an expansion in the small wavenumber is possible. Hadji and Schell15 consider a double expansion in the small amplitude and the small wavenumber in order to analyze the stability of the interfacial patterns. The analysis leads to the prediction of a bifurcation diagram that is characterized by two bistable regions. First a stable structure consisting of down-hexagons appears through a subcritical bifurcation. This convective structure coexists with the static state in a very small range of Rayleigh numbers. Second, squares become stable at higher Rayleigh numbers and coexist with the down-hexagons in a small parameter range. It is also shown that these convectives structures, down-hexagons and squares, are imprinted as patterns on the solid-liquid interface through the melting action of the convective currents. The nonlinear stability results demonstrate that the appearance of the down-hexagons is due to the coupling between the convective flow in the melt with the deformations in the solid-liquid interface. Hadji17 relaxes the small amplitude assumption and derives the following evolu-
97 tion equation that is valid for larger supercritical region,
^
= -TX(S,
A, T) W
- (/ir/*o)V 2 „/ + VHr, • Vuf -0f + y r[VHf ■ VH\VH\2f
+V2„/|V„/|2] - ^0^\yHf
■ V„(VV) + (V2„/)2] -
15V2H(»?/),
(23a)
where Xlb>A>T>-lm+
2315
4 5 ( 1 + .1)
and
'-why***"-
J
'
(236)
0, both the liquid depth and the temperature difference are larger than those required for convection onset. The opposite scenario holds in re gions where 77 < 0. Thus, a periodic array of unstable regions separated by stable regions forms. See Fig. 7. The convection currents that initiate in the unstable regions penetrate the neighboring stable regions resulting in the formation of wide
38
Figure 7: Plots of the cell pattern for parameter values S = 0.035, r = 0.001, A = 0.1 and 0 = 1.4 for ( = 0 (top) and ( = 500 (bottom), regions of cold fluid separated by narrow regions of warm luid. Jin and Hadji 18 re-examine the solution of Eq. (24) using a fully implicit finite difference scheme combined with Newton linearization with coupling. They have. obtained results which were unobtainable when a fully explicit scheme was used. Mainly, it is found that, for large values of the parameter (", the solid-liquid interface exhibits a cellular morphology consisting of thin fingers that extend deep into the liquid. See Fig. 8. This morphology is similar to the thin interfacial fingers which have been observed experimentally by Jackson 19 during the directional solidification of tetrabromomethane, CBr^, Jin and Hadji18 have shown that this morphology bifurcates subcritically from the planar state. 3.4 Experimental
validation
Zimmermann ti el.16 have undertaken an experimental verification of their analyt ical results. They have selected to use alcohol-water mixtures in their experiments for several reasons. Besides the fact that they exhibit strong Soret effects and are suitable for experimental visualizations due to their transparency, they also offer the added advantages that their physical properties remain nearly constant within the range of temperature used and that all alcohol is rejected upon freezing. The latter requirements are needed in order to eliminate any non-Boussinesq effects and for the experimental conditions to be in agreement with the assumptions made in the derivation of the analytical results. Their linear stability results agree with their experimental findings when the dependence of the separation ratio on concentration is taken into account. Indeed, Marcher and Miiller20 have shown that the variation
99
Figure 8: Plot of the steady state solution of Eq. (24) for 5 = 0.035, r = 0.001, A — 0.1, /? = 1.4 and £ = 500. Dotted line represents the initial condition 0.1 cos a;. of concentration plays a dominant role in the stability analysis and must be included in the calculations. Zimmermann et al.16 have also noticed that in the presence of solidification (A ^ 0) and for sufficiently low mean temperature in the mixture, the state of travelling waves that is observed when ice is absent, becomes unstable and gives way to a state of stationary convection. The authors attribute this result to the presence of the solid-liquid interface and other possible non-Boussinesq effects. The visualizations of the solid-liquid interface using interferometry and photogra phy reveal that the morphology consists of hexagonal-like patterns. For a thin solid layer, they report observing an interface that consists of ice-regions surrounded by ice free regions. This scenario is consistent with a convection flow in the form of hexagons with down flow at their centers. These experimental observations, even though valid for negative separation ratio, seem to corraborate the results of Hadji and Schell 15 , namely, that the coupling between Soret driven convection and the deformations of the solid-liquid interface yields a flow pattern consisting of downhexagons. 4.
Concluding R e m a r k s
In this paper, an attempt was made at summarizing existing results dealing with the role of thermotransport in solidification processes. The main focus was on the
100 influence of the Soret effect on the instabilities usually encountered in solidification systems, namely, interfacial instabilities during the unidirectional solidification of a binary alloy at constant speed and fluid instabilities induced by Soret driven con vection in the presence of freezing. The summary of results has shown that many aspects arising from the Soret effect are nt>t yet fully resolved. These results also show that in order to bring to light the significance of Soret diffusion in solidification processes, the nonlinear form of the thermotransport term must be considered. We are currently attempting the complete solution of the stability problem taking into account the nonlinear form of the thermotransport term and the coupled morpho logical and convective modes of instability. 5.
Acknowledgements
I am grateful to Dr R.F. Sekerka for pointing out to me references 12 and 13. I also thank Dr L. Nastac, Mr A. Catalina and Mr J. Leon for many stimulating discussions.
6.
References
1. 2.
M.C. Soret, Arch. Sci. Phys. Nat. Geneve 2 (1879) 48. S.R. DeGroot and P. Mazur, Non-equilibrium Thermodynamics (Dover, New York, 1984), Chap. XI, Sec. 7. D. R. Calwell and S. A. Eide, Deep Sea Res. 32 (1985) 965. D. Gutkowicz-Kruskin and J. Ross, J. Chem. Phys 72 (1980) 3577. A. Rice, Geophysical Surveys 7 (1985) 303. J. F. Smalley, R. A. Mac Farquhar and S. W. Feldberg, J. Eleciroanal. Chem. 256 (1988) 21. S. A. Akbar, M. Kaburagi and H. Sato, J. Phys. Chem. Solids 4 8 (1987) 579. C. R. Carrigan and R. T. Cygan, J. Geophys. Res. 91 (1986) 1451. G. R. Longhurst, J. Nuclear Materials 131 (1985) 61. D. T. J. Hurle and E. Jakeman, J. Fluid Mech. 4 7 (1971) 667; D. T. J. Hurle and E. Jakeman, Phys. Fluids 12 (1969) 2704. P. Bigazzi, S. Ciliberto and V. Croquette, J. Phys. (Paris) 51 (1990) 611; E. Knobloch, Phys. Rev. A 40 (1989) 1549. D.T.J. Hurle, J. Crystal Growth 61 (1983) 463. S. Van Vaerenbergh, S.R. Coriell, G.B. McFadden, B.T. Murray and J.C.Legros, J. Crystal Growth 147 (1995) 207. R.F. Sekerka, in Crystal growth: an introduction, ed. P. Hartman (NorthHolland Publ. Co., 1973) p. 403. L. Hadji and M. Schell, Phys. Fluids A 2 (1990) 1597.
3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.
101 16. 17. 18. 19.
G. Zimmermann, U. Miiller and S.H. Davis, J. Fluid Mech. 238 (1992) 657. L. Hadji, Phys. Rev. E 4 7 (1993) 1078. X. Jin and L. Hadji, Phys. Rev. E 50 (1994) 2361. K.A. Jackson, in Solidification, eds. T.J. Hughel and G.F. Boiling (American Society of Metals, Metals Park, OH, 1971), p. 133. 20. C. Karcher and U. Miiller, Int. J. Heat and Mass Transfer 3 7 (1994) 2517.
103
THE INSTABILITY OF FINITE AMPLITUDE WAVES IN STRONG VISCID AND INVISCID SHEAR
W. R. C. PHILLIPS Department of Mechanical and Aeronautical Engineering Clarkson University, Potsdam NY 13699-5725, USA
ABSTRACT Finite-amplitude initially spanwise-independent two-dimensional rotational waves and their nonlinear interaction with unidirectional 0(1) shear flows are reviewed. Of particular in terest is the instability of the flow to longitudinal vortex form. The wave-mean interaction is described by a generalized Lagrangian-mean formulation in combination with a separate theory to account for the back effect of the developing mean flow on the wave field. Both the inviscid and viscid eigenvalue problems relevant to the instability are outlined, along with detailed results for uniform shear in the case of the former and periodic Poiseuille flow in the case of the latter.
1. Introduction Shear flows that comprise both mean and fluctuating parts are commonplace in Na ture and engineering, and often interact to form longitudinal vortices. Examples include Langmuir circulations12, which occur beneath wind driven surface waves, stream wise rolls in the atmosphere over surface waves 23 or mountainous terrain13,21, and quasi-streamwise vortices in laboratory boundary layers (both laminar10 and turbulent3), free shear layers2 and Stokes layers8. Important in each case is an understanding of the nonlinear processes that couple the mean and fluctuating motions, and moreover the secondary and possible tertiary phenomena attributable to the nonlinear rectification of those oscillatory motions: e.g. modifications to the mean flow as a result of the waves; and the back effect, if any, of those mean flow modifications on the wave field. Quantities that follow individual fluid particles are crucial to such studies and such quantities are poorly described by the Eulerian equations of mean motion. Indeed, Eulerianmean vorticity as defined by Reynolds averaging has no simple conservative properties, even when viscosity is ignored and thus acts to conceal the role played by nonlinear recti fication (in its guise as Stokes drift) in vortex line deformation. A more rational way to separate 'wave' from 'mean flow' and to define wavemean interactions was developed by Andrews & Mclntyre 1 in the form of the generalized Lagrangian-mean (GLM) equations. These equations describe the back effect of oscillatory
104
disturbances upon the mean state and are exact provided the mapping between the true Lagrangian and the reference GLM remains invertible. Of course GLM still describes mean motions and is therefore conceptually equivalent to Reynolds averaging, but it describes Lagrangian aspects of the motion from an Eulerian framework and is consequently able to capture structural aspects of the flow. The object of the present work is to review the application of GLM to describe mean structures in unidirectional 0(1) shear layers (be they viscid or inviscid) owing to the pres ence of neutral finite-amplitude rotational waves that are initially spanwise independent. Wave-mean interactions of this ilk were first investigated by Craik4 and Leibovich 11 , who each sought to model Langmuir circulations. They considered 0(e) neutral irrotational waves interacting with an 0(e 2 ) unidirectional Eulerian mean shear flow and found the interaction unstable to longitudinal vortex form via an instability now known as CL2, or Craik-Leibovich type-2. CL2 continues to operate in 0(e) mean shear flows with only minor modifications to the theory, but that is not the case for strong 0(1) shear flows, where the back effect of the mean flow modification upon the wave field must be explicitly calculated (Craik6). In essence, waves do not drive CL2 (Mclntyre & Norton14) but act through the pseudomomentum as a catalyst. This means that the magnitude of the mean flow modification is bound not by the strength of the waves but by the magnitude of the pre-existing vorticity in the initial state. With sufficiently strong pre-existing vorticity, therefore, the mean flow modification acts to distort the waves. Of course the detailed kinematics of the instability mechanism are less clear with 0(1) shear than with 0(e 2 ), though the seminal idea of the CL2 instability remains within the theory. In order to construct an inviscid theory for 0(1) shear flows in the presence of 0(e) rotational neutral waves, Craik6 employed the GLM equations and found that the resulting eigenvalue problem for longitudinal vortices is far more complex than its counterpart for weaker shear; requiring inter alia, a further differential equation to account for wave distor tion. That notwithstanding, he was able to obtain definite results analytically to demonstrate the existence of longitudinal vortex instability when the spanwise spacing of the vortices is small, and this same technique was extended to a different, wider class of flows, by Phillips & Shen18, who found that CL2 continues to operate. Numerical results by Phillips & Wu 20 and Phillips et aP1 concur, and further indicate that wave distortion acts (i), to diminish catalytic action for all but the shortest waves; and (ii), to suppress the instability markedly if the waves are sufficiently long. We begin with a brief review of GLM (§2) and then specialize the GLM-equations to the problem of 0(e) neutral waves interacting with an 0(1) unidirectional shear flow (§3). The waves are initially two-dimensional. An appropriate numerical scheme is sketched in §4 and detailed results for the case of uniform shear in §5. The relevant viscous eigenvalue problem is outlined in §6. 2. The generalized Lagrangian-mean equations Andrews & Mclntyre's 1 generalized Lagrangian-mean equations are an exact and very general Lagrangian-mean description of the back effect of oscillatory disturbances
105 upon the mean state. The formulation is based upon an exact Lagrangian-mean operator ( ) , corresponding to any given Eulerian-mean operator ( ), through an exact disturbanceassociated particle displacement field £(x, t) and is valid provided the mapping X H X + { is invertible. In consequence dependent variables are given an Eulerian description with position x and time t as independent variables. The Lagrangian-mean velocity u L is then the velocity field describing trajectories about which the fluctuating particle motions have zero mean, when any averaging process is applied. For homentropic flows of constant density p in a non-rotating reference frame the GLM momentum equation is DL(uf - p{) + u^(uLk - pk) + 7T, = -Xu
(1)
where repeated indices imply summation, commas denote partial differentiation and (xi, x2, x3) = (x,y,z). The operator DL is defined as DL = d/dt + vJ-d/dxj, and the vector wave property pi, the pseudomomentum per unit mass, is Pi = -tj,iu£3,
(2)
with DL(j = uly Finally Xi are dissipative terms while
p
z
where u^ is the actual fluid velocity and $ L is the force potential per unit mass. We shall restrict attention to fluids in which all mean quantities except possibly the mean pressure V are independent of the stream wise direction; then, with $ L = 0, IT reduces tCPL. 3. In viscid 0(1) shear and 0(e) waves Consider the interaction between an inviscid 0(1) primary shear flow and two di mensional straightcrested periodic waves that propagate in (or opposite to) the direction of the flow. Then in a reference frame that moves in the x-direction with the phase speed of the waves and with space coordinates (x, y, z), the primary shear flow is [u(z), 0,0], where u is the Eulerian-mean velocity profile in [z\ ,z2]. The waves are initially independent of the span wise coordinate and are of constant amplitude with slope characterized by the small pa rameter e; moreover they induce, in most circumstances, an 0(e 2 ) pseudomomentum field [pi,0,0]. Envisage now small spanwise-periodic perturbations with streamwise averaged Eu lerian velocity components of the form (u, v, w) = £Re{e^eWl/[u(;z), -eiv(z),
ew(z)]}
(3)
which, provided the amplitude field of the waves is steady, satisfy continuity correct to 0(e 2 ) as Iv + u>,3 = 0. Here a is the growth rate of the spanwise perturbation and 6 is a
106 second small parameter that measures the strength of this motion relative to the primary shear flow; 6 is assumed sufficiently small that linearization with respect to it yields a good approximation to the equations governing the spanwise periodic disturbance. Observe that velocity perturbations in the y and z directions are weaker, by a factor of e, than the x velocity perturbation; this is necessary in order to ensure the GLM equations yield nontrivial solutions for boundary conditions of the form6 w = 0 at z — z\, z2. Accordingly a = eov In such cases the GLM equations reduce to cj\U — —wu, and
10,33+* [—i
l\w =
(4) pj.
(5)
Here prime denotes d/dz and pi has been expanded as Pl
= e2P? + c 2 «te{e' t e i , y p 1 (^)} + 0(e\ e36, e2S2).
(6)
So P® is the 0(e2) component of pseudomomentum5
A° = 4 { I A T + " 2 6 2 } ,
(7)
l u u in which <j>(z) and a denote the eigenfunction and wavenumber of the primary wave field which together satisfy the Rayleigh equation, u(" - a2) - u" = 0.
(8)
Note that P® is unbounded in the vicinity of critical layers (u = 0) unless 4> oc um (m > 1). Such behavior is an indication that the mapping upon which GLM is based has broken down and thus that, except perhaps at a rigid boundary, critical layers should be avoided. The 0(e2S) spanwise-periodic perturbation of pseudomomentum, R e { e 1 ^ } , arises because the emerging secondary Eulerian velocity field distorts the primary wave field. But because the GLM formulation provides no direct means of evaluating p1, a separate examination of the wave field is necessary. In carrying out that examination, Craik6 notes that the 0(6) spanwise periodic z-velocity, u, causes the significant part of distortion of the wave field. Then n = A(z)u(z)
+ B(z)u'(z)
+ Rt{C(z)j>(z) + V(z)4'(z)},
(9)
where A, By C and V are functions which are independent of a while (z) relates to the 0(e6) spanwise periodic wave field modification and satisfies the Rayleigh-Craik equation, u[^
- (a2 + l2)]$ - u"$ = - u [ ^ - (a2 + l2)] + u»*.
(10)
Thus given the primary Eulerian-mean shear flow u(z), the primary wave-field eigenfunction <j>(z) and appropriate boundary conditions, the eigenvalue problem for <jx is completely specified by the coupled system (4), (5) and (10) together with (7) and (9).
107
Craik6 has further shown that when P > a2 and a = 0(1) non-trivial solutions to (4), (5) and (10) with homogeneous (i.e. = w = 0 on each wall) boundary conditions exist provided Im{/ (1 + \k)1/2dz} = —
(fe=lor2),
(11)
for some non-negative integer N. Here + H) ± [(z) can be expanded in linearly independent complete sets of basis functions U{(z) and <j>i(z) as M
A
Hz) = J2 biui(z)>
M
kz) = Y, bM+ii(z),
( 14 )
i'=l
i=\
where 6, and &M+* are the expansion coefficients of u(z) and <j>(z) respectively, and M is a positive integer. The basis functions were chosen to satisfy the boundary conditions of u and , viz u(^i), u(z2) = 0; <j>(zi), ^>(z2) = 0. Substitution of (14) into (4), (5) and (10) then leads to the residual functions Ru(z) and R(z) which must satisfy the inner products (Ru(z),uj(z))=0]{R^(z)^j(z))=0. A 2xM order linear eigenvalue problem for A2, where A = l/a\, results as C = \2M,
(15)
where C and M, are 2Mx2M order matrices with components that are MxM order submatrices; specifically, the components are functions of a and /, but not a\. Of interest is the largest real value of cr\ for each pair (a, /) and the eigenfunctions u and ' '"' (~. .i constant). Note that two very different classes of Rayieigh waves are admissible: those for which ao diverges and those for which ao converges with jo.:| as :n: ! — ?-,:. Phillips & Shen18 denote the former 'type-1' and the later, 'type-2* waves. SJ The limit I'2 > n'2, n = (){ 1). In order to proceed analytically. Craik1' employed (12) and (13) to consider the case ■i = 0, and showed that type-1 waves destabilize the flow for n.;- t ( — 1.0) and accordingly type-2 waves for ail n;; > 0. Phillips & Shen have applied the same technique to oilier combinations of -. and .1 There findings are plotted in figure 1. in which the shaded portion denotes instability. Note that points of symmetry at - / J = n 1 are clearly evident, as is the obverse {i.e. ^ / .i = 0) of Craik's result. Note also that ~ / .j = —1 is stable for all ^ : | greater than zero. Interestingly. - j3 — — I is the only situation that admits, at least in the limit an —^ 0, the amplitude ao —r 0. giving rise to what is tantamount to plane Couette flow subject to infinitesimal waves, an inviscid configuration long known to be stable (see Drazln & Reid'). in short, instability to longitudinal vortex form occurs whenever the local wave amplitude \ao\ everywhere (Ucn-ast s in the direction of increasing \u\. but occurs only for
109
Figure 2: Curves of G\ against a in the limit I2 —► oo, both with and without wave distortion (Phillips & Wu20). sufficiently long waves when \a<j>\ everywhere increases in the direction of increasing \u\. Indeed, waves are stable to longitudinal vortex form wherever in the direction of increasing mean flow, the relative increase in wave amplitude exceeds the relative increase in mean flow18 and this is so for all wavelengths when 7//? = — 1. But although bounds deduced from (12) and (13) are useful to determine whether the flow is stable or unstable to longitudinal vortex form, these equations alone do not yield the growth rate or resolve whether wave distortion acts to enhance or inhibit the instability. Phillips & Wu 20 determined to answer both questions and, because (3 — 0 depicts instability over the widest range of az, chose it as the test case. They then considered the problem both with and without wave distortion. When wave distortion is ignored, only (4) and (5) with A = 0 need be solved; moreover from (11) the largest upperbound for G\ is then seen to occur when 01
A2 e - c *(l + 2a + 2a 2 + -c*3)2 o
(a ^ 0,
I2 -> 00).
(16)
Unfortunately (11) proved quite a challenge to solve for the wave-distortion case and Phillips & Wu chose instead to solve (4), (5) and (10) in the appropriate limit. It transpires that the distortion and non-distortion results are markedly different, as we see in figure 2. Indeed the two cases are asymptotically equivalent only as az —> +00 and it is evident that wave distortion plays an increasingly greater role as az decreases from +00.
110
F i g u r e 3: Contours of constant growth rate cr\ in a-l space with stability boundaries along wnicn G\ is not constant. The boundary connecting lR to aR separates instability to longitudinal vortex form (I) with same of alternating sign (II); the dashed curve separates II from a region which is stable to longitudinal vortex form (III) (Phillips &Wu 2 0 ).
5.2 The long-wave long-wave limit Phillips & Wu further show that a\ increases monotonically to an upperbound with increasing /, except in the long-wave long-wave limit, figure 3. Here they find that of the 2N roots for cri, only 23 are real and 2K are imaginary (J, K integer); the remainder are complex conjugate pairs. Specifically, J = 0 for a < aR and K = 0 for a > a/, where aR « 0.76 and cti « —0.61. Moreover 3 = 1 (K = 1) at a = aR-\- (a = a/—) and increases with increasing (decreasing) a until eventually J — N (K = N), at which point all the eigenvalues are real (imaginary). Finally, while the real parts of the complex conjugate pairs are positive when N > J > 0, they change from positive to negative in «/ < oc < aR and remain negative for N > K > 0. So on writing <jx — OLCV{ q= iacj!, the right-hand-side of (3) should be interpreted as 6Re{eac> V ^ a c ^ [ u ( ; z ) , -eiv(s), ew{z)]}. The curve connecting lR to aR is shown, with impinging contours of constant (T\, in I-a space in figure 3. Note that the lR-aR-cxxrvG is not a line of constant cra but rather a demarcation between eigenvalues for GX for which J = 0 and J = 1. The /^-a^-curve also depicts marked suppression in catalytic action beyond that for J > 1; so although the flow remains unstable to longitudinal vortex form on the long wave side of the curve, the vortices are relatively weak rolls. On the J > 1 side of the curve, the instability is dominated by eigenmodes which are real; the real parts of the complex conjugate pairs, on
111 the other hand, are tertiary at best. But only complex conjugate pairs occur on the J = 0 side of the curve where the instability not only grows, so long as OLCV{ > 0, but is subject to a standing oscillation owing to the ±acvT contribution: that is, the longitudinal vortices stand in space and alternate in sign; there is also the possibility that a +acvr or —acvr alone each give rise to spanwise propagating rolls with equal and opposite phase speed. Of course on fixing / on the lR-aR-curve and further decreasing a, we ultimately reach a point at which OLCV{ < 0 for all eigenmodes; here the flow is neutrally stable to longitudinal vortex form. The neutral curve is depicted by dashes and indicates that az ~ —0.5 as P —> oo and that az = a\z when / = 0. 5.3 Non-uniform shear Phillips & Shen18 extend the analytical techniques introduced by Craik6 in the limit I > a, a = 0(1) to deduce that type-1 and/or type-2 waves are unstable to longitudinal vortex form in the presence of a wide variety of mean velocity profiles, from boundary layers to mixing layers. Indeed the implication is that CL2 is ubiquitous to all wave-mean interactions that satisfy the criteria stated in §5.1. Further, in order to deduce more detail, several cases of particular physical interest were treated numerically: to wit, an exponentially decaying velocity profile beneath surface gravity waves 20 and power-law and logarithmic-law velocity profiles as would occur in atmospheric boundary layers over random terrain21. The gross features in each case are much as those in the case of uniform shear, although of course details vary. One point of particular interest is that the fastest growth rate occurs (in both cases) when a = 0(1) for all/. 2
6. The viscous eigenvalue problem Although the CL2 instability is inviscid, it can at times be modified by viscosity and the appropriate eigenvalue problem that accounts for viscosity is given by Phillips 17 . Here it is best to work in terms of the associated velocity field Qi = u + Ji —pi, which must be determined as part of the problem; here d is the generalized Stokes drift. Indeed the problem comprises two parts: (i), a two-dimensional primary instability composed of equilibrated waves interacting with a mean flow often vastly different from its unperturbed laminar forbear. And (ii), the three dimensional secondary instability which manifest as longitudinal vortices. Here we shall outline only the secondary linear instability. 6.1 Secondary flow Here the perturbation equations take the form [D2-l2-a?]u
=
faw,
[D2 _ /2 _ < ] [ D 2 _ /2]t a = - ^ [ / f u - Qifc]
(17)
(18)
112 where av = <XJ\, D = d/dz and to avoid annihilation of the 0(e) growth rate predicted by inviscid theory, i/a/C = 0(e), where C is a typical phase speed; appropriate boundary conditions at rigid boundaries are: u = w = Dw = 0
on
(z =
zi,z2).
Accordingly px is described through the 0(e6) spanwise periodic wave field modi fication,
