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ADVANCES IN APPLIED MECHANICS VOLUME V

This Page Intentionally Left Blank

ADVANCES IN APPLIED MECHANICS Editors

H. L. DRYDEN

TH. VON

KARMAN

Managing Editor

G . KUERTI Case Institute oJ Technologlj, Cleveland, Ohio

Associate Editors

F. H. VAN

DEN

DUNGEN

L. HOWARTH

J. PkREs

VOLUME V

1958 ACADEMIC PRESS INC., PUBLISHERS NEW YORK, N.Y.

COPYRIGHT @ 1958 BY

ACADEMIC PRESSINC. 111 FIFTHAVENUE NEW YORK3, N.Y.

ALL RIGHTSRESERVED N O PART O F THIS BOOK MAY B E REPRODUCED I N A N Y FORM, B Y PHOTOSTAT, MICROFILM, OR A N Y OTHER MEANS, WITHOUT WRITTEN

PERMISSION FROM T H E PUBLISHERS

LIBRARY OF CONGRESSCATALOG CARD NUMBER: 48-8.503

PRINTED IN

THE

UNITEDSTATESOF AMERICA

CONTRIBUTORSTO VOLUMEV

H. N. ABRAMSON. Southwest Research Institute, S a n Antonio, Texas

H. DERESIEWICZ, Columbia University, New Y o r k , New York

J. FABRI,Office National d’Etudes et de Recherches Akronautiques, Paris EDWARD A. FRIEMAN, Princeton University, Princeton, New Jersey RUSSELLM. KULSRUD,Princeton University, Princeton, New Jersey L. M. MACK, Jet Propulsion Laboratory, California Institzcte of Technology, Pasadena, California H. J. PLASS,The University of Texas, Austin, Texas E. A. RIPPERGER,The University of Texas, Austin, Texas CHARLESSALTZER, Case Institute of Technology, Cleveland, Ohio, and General Electric Compan y , Electronics Laboratory, Syracuse, New York

R. SIESTRUNCK, Office National d’Etudes et de Recherches Aironautiques, Paris A. I. VAN DE VOOREN,National Aeronautical Research Institute, Amsterdam

P. P. WEGENER, Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California

V


PREFACE This fifth volume of “Advances in Applied Mechanics” contains reviews and surveys of the current state of research in selected fields of applied mechanics and two brief papers which the Editors hope will be of interest to readers of the series. Contributions to the “Advances” are, in general, by invitation, but suggestions of topics for review and offers of special contributions are very welcome and will receive careful consideration. The Editors are convinced that summary articles by research workers are important contributions to the advance of knowledge, that they are rewarding to the authors in giving a broad assessment of where we stand, and that they stimulate both author and reader to further research. When, as occasionally happens, the summary article grows into a book, the Editors applaud the result, while regretting the loss of the article for Advances. We are thus pleased that Professor W. D. Hayes, invited by us to review hypersonic flow theor$ became so engrossed in his subject that he completed a book which will appear as a separate monograph. THEEDITORS December, 1957

* W. D. Hayes and R. Probstein. “Hypersonic Flow Theory.” Academic Press, New York, 1958, in preparation.

vii


CONTENTS ....................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

CONTRIBUTORS TO VOLUME V

PREFACE.

v vii

Supersonic Air Ejectors A N D R . SIESTRUNCK. Office National d’gtudes et de Recherches BY.J . FABRI A ironautiques. Paris I . Introduction . . . . . . . . . . . . . . . I1. Aerodynamic Flow Patterns in Jet Ejectors I11. Experimental Verification . . . . . . . . . IV. Optimum Jet Ejector Design . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . .

1 2 13 30

. . . . . . . . . . . . . . . . . . . . . . . . . .

Unsteady Airfoil Theory BY A . I . VAN DE VOOREN.National Aeronautical Research Institute. Amsterdam I. I1. I11. IV . V. VI . VII . VIII . I X.

. . . . 36 Introduction . . . . . . . . . . . . . . . . . . . . . . . . The Fundamental Equations . . . . . . . . . . . . . . . . . . . . 31 The Oscillating Airfoil in Two-dimensional Subsonic Flow . . . . . . . 44 The Oscillating Airfoil in Three-dimensional Subsonic Flow . . . . . . 52 The Oscillating Airfoil in Supersonic Flow (Supersonic Edges) . . . . . 59 The Oscillating Airfoil in Supersonic Flow (Subsonic Edges) . . . . . . 65 Non-linear Approximations . . . . . . . . . . . . . . . . . . . . . 72 . . . . 77 Indicia1 Functions . . . . . . . . . . . . . . . . . . . . . . . . . 84 References . . . . . . . . . . . . . . . . . . . . . . . . .

The Theory of Distributions BY CHARLES SALTZER Case Institute of Technology. Cleveland. Ohio. and General Electric Company. Electronics Laboratory. Syracuse New York

.

I. I1. I11. IV . V. VI . VII . VIII . I X. X.

.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Theory of Distributions . . . . . . . . . . . . . . . . . . . . The Singularity Functions and the Finite Part of An Integral . . . . A Heuristic Approach . . . . . . . . . . . . . . . . . . . . . . . Fourier Series and the Poisson Transformation . . . . . . . . . . . Ordinary Differential Equations . . . . . . . . . . . . . . . . . . Applications to Fourier Transforms . . . . . . . . . . . . . . . . Fourier Transforms of Distributions . . . . . . . . . . . . . . . . Generalized Harmonic Analysis and Stochastic Processes . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . .

91 92 95 98 99 102 104

104 107 109 110

Stress Wave Propagation in Rods and Beams BY H . N . ABRAMSON. H . J . PLASS. A N D E . A. RIPPERGER. Southwest Research Institute. San Antonio. Texas. and The University of Texas. Austin. Texas I . Introduction . . . . I1. Longitudinal Waves . I11. Flexural Waves . . References . . . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

ix

. . . .

. . . .

. . . .

. . . .

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. . . . . . . . . . . . . . . . . . . .

111 113 151 188

CONTENTS

X

Problems in Hydromagnetics

BY EDWARD A. FRIEMAN A N D RUSSELL M. KULSRUD. Princeton University. Princeton. N e w Jersey I. I1. I11. IV . V.

Introduction . . . . . . . . . . . . . . . Fundamental Equations . . . . . . . . . . General Processes . . . . . . . . . . . . Stability of Hydromagnetic Equilibria . . Hydromagnetic Waves . . . . . . . . . . References . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . 195 . . . . . . . . . . . . . 196 . . . . . . . . . . . . . 198 . . . . . . . . . . . . . . 204 . . . . . . . . . . . . . 216 . . . . . . . . . . . . . 231

Mechanics of Granular Matter BY H . DERESIEWICZ. Columbia University. N e w Y o r k . N e w Y o r k 1.Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . I1. Geometry of a Granular Mass . . . . . . . . . . . . . . . . . . . . I11. Some Recent Results of Contact Theory . . . . . . . . . . . . . . . IV . Mechanical Response of Granular Assemblages . . . . . . . . . . . . V. Suggestions for Further Research . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

233 236 251 267 300 303

Condensation in Supersonic and Hypersonic Wind Tunnels BY P. P. WEGENERA N D L . M . MACK. Jet Propulsion Laboratory. California Institute of Technology. Pasadena. California I. I1. I11. IV . V. VI .

Equilibrium Condensation Limits . . . . . . . . . . . . . . . . . . 307 Condensation of Water Vapor in Supersonic Nozzles . . . . . . . . . 320 Condensation in Steam and Hypersonic Nozzles . . . . . . . . . . . . 343 Diabatic Flows and Thermodynamics of Condensation . . . . . . . . . 365 Kinetics of Condensation . . . . . . . . . . . . . . . . . . . . . . 403 Experimental Methods . . . . . . . . . . . . . . . . . . . . . . . 433 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 440

Author Index

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

449

Subject Index

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

456

Supersonic Air Ejectors BY J. FABRI

AND

R. SIESTRUNCK

Office National d ' l h d e s et de Recherches A kronautiques. Paris Page

I. Introduction. . . . . . . . . . . . . . . . . . . . . . . 11. Aerodynamic Flow Patterns in Jet Ejectors . . . . . . . . 1. Supersonic Flow Patterns . . . . . . . . . . . . . . . 2. Mixed Flow Patterns . . . . . . . . . . . . . . . . . 3. Mixed Flow Patterns with Primary Separation . . . . . 4. Theoretical Performance Curves . . . . . . . . . . . . 111. Experimental Verification . . . . . . . . . . . . . . . . 1. Experimental Set-up . . . . . . . . . . . . . . . . . 2. General Performance Curves . . . . . . . . . . . . . . 3. Operation without Induced Flow . . . . . . . . . . . . 4. Study of the Main Parameters . . . . . . . . . . . . . 6. Pressure Distribution along the Walls of the Mixing Tube IV. Optimum Jet Ejector Design . . . . . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . .

1 2 4 8 11 12 13 14 16 19 23 29 30

I. INTRODUCTION The industrial use of jet ejectors, though already very old, has been confined mainly to very particular cases of operation. The experimental results obtained with such ejectors were therefore insufficient to allow a safe extrapolation of the performance curves to more general cases. Large-sized modern jet ejectors, driven by high powered air-compressors and designed for large ranges of operating conditions, cannot be based on these earlier results, if one wishes to be sure of the final outcome. For this reason systematic experimental and theoretical researches have been resumed in many countries ([l], [2], [3]) on air/air jet ejectors with high $yesswe ratios, in which the primary air flow is supersonic. The tangential action and the turbulent mixing of air jets, made even more complicated by the proximity of the walls of the mixing tube, is one of the most difficult problems of gas dynamics. No correct solution has yet been given for this problem, and the performance of jet ejector operations cannot be predicted from such studies. The theories of jet ejectors, and in particular the one we proposed earlier [4] in an attempt to give theoretical interpretations of experimental results are all essentially aevodynamical. In such theories, the exhaust 1

2

J. FABRI A N D R. SIESTRUNCK

performance of jet ejectors is determined by the compatibility conditions of two air streams flowing in the same duct, called the mixing tube. The aerodynamic conditions of compatibility are written for a simplified geometric representation of the experimental set-up and do not take into account viscosity and diffusion effects. The success of the simple aerodynamic theories stems from the fact that viscosity and diffusion effects are essential in the building-up of the flow system, but, in an established movement, play a very minor part in maintaining the flow under nearly perfect-gas conditions. Of course, these aerodynamic conditions are incomplete. They do not show the effect of such important parameters as the minimum mixing-tube length necessary to induce secondary flows, or the best relative position of primary and secondary nozzles to give optimum ejector performance. However, the predicted rates of induced mass flows are very close to the experimental results obtained with actual ejector set-ups, if the mixing tube is long enough and the geometric configuration of the set-up is similar to the theoretical one. It is, however, necessary to take the friction losses into account, but this can be done by means of the classical results obtained for friction losses in smooth ducts. The resulting aerodynamic theory can be considered as giving very correct overall descriptions of the actual operation of jet ejectors. As the theory shows the existence of some similarity rules, the results of ejector performance calculations can be condensed into a small number of diagrams. By means of these diagrams the performance of any jet ejector with fixed air-supply conditions can be rapidly computed, and jet ejectors can be rapidly designed for any preassigned performance.

11. AERODYNAMICFLOW PATTERNS IN

JET

EJECTORS

Fundamentally, a jet ejector is designed to draw a given mass flow rate M" of induced secondary flow from a reservoir at a given stagnation pressure pirf into a vessel of higher pressure p , this latter pressure being generally the atmospheric pressure. Such a compression is obtained by means of a high pressure primary flow (mass flow rate M') which is expanded from the primary stagnation pressure pi' to the same pressure 9. A convenient representation of such an operation is obtained on the nondimensional p, &-diagram, where p = M"/M' is the ratio of the two mass flow rates and & = pi"/p, the secondary reservoir pressure divided by the exhaust pressure, is the inverse of the compression ratio. The characteristic performance curves ( p vs. &) are given for fixed primary conditions. Figure 1 shows a schematic representation of a jet ejector similar to those used in our experimental studies of ejector performance. Actually such an ejector differs from the industrial set-ups only in minor points.

SUPERSONIC AIR EJECTORS

3

A high pressure primary flow (stagnation pressure fi%‘, stagnation temperature T*’)is supplied to the primary nozzle. The supersonic primary flow leaves the primary nozzle and enters the mixing tube of larger cross section, on the axis of which this nozzle is placed. The induced secondary flow (stagnation pressure fi,”, stagnation temperature T,”) comes tangentially into contact with the primary flow, throughout the length of the mixing tube. At the exit of the mixing tube both flows are ejected into the surrounding atmosphere at pressure fi. Sometimes a diffuser is used a t the mixing tube exit, and the pressure condition fi is imposed only a t the exit of the diffuser. In this schematic description of a jet ejector, the induced secondary air is taken from the surrounding atmosphere through a duct equipped with a flowmeter and a control valve. As it passes through the control valve, the secondary flow suffers a pressure drop depending on the valve setting. The secondary stagnation pressure becomes subatmospheric, and as the induced flow expands from the secondary chamber to the mixing-tube entrance, its pressure decreases occasionally to very low values. In the industrial jet ejector facilities for aeronautical applications, the useful part of the secondary stream is between the control valve and the mixing tube (jet-engine combustion chamber for low-pressure combustion tests). It is not always necessary to have a secondary chamber slowing down the flow to stagnation conditions, and in the induced-flow supersonic wind tunnels the high speed secondary flow is drawn along directly by the primary stream. In some cases, jet pumps may also be used as injectors, and the useful flow is then the high pressure stream a t the mixing tube exit. The mixing tubes considered in this study are cylindrical. Their length L is measured from the exit section of the primary nozzle. For the sake of simplicity this primary exit section S is taken as unit section. The mixing tube section is then designated by AS, and A a S designates the exit section of the downstream diffuser. The throat section of the primary nozzle is ASIA, ; A* then represents the ratio of the mixing-tube section to the primary throat section. The dimensionless speed M * determines the local aerodynamical conditions. M * is the ratio of the local mean velocity of each stream to its respective critical sound velocity a * . A single prime (’) refers to the primary flow conditions and a double prime (“) to the secondary flow, if these two streams are still distinct; “no prime” means a homogeneous mixing of both streams. Subscripts 1, 2 and 3 designate the three successive sections shown in Fig. 1. Subscript 0 is reserved for a section of possible separation of the primary flow, when the primary stagnation pressure is too low for the primary flow to fill the whole primary diffuser. Subscript i refers, as already stated, to stagnation conditions. Specific heats of both gases are supposed to be independent of the temperature and keep the same values before and after

4

J . FABRI A N D R. SIESTRUNCK

mixing. The expression m2=(y+ l)/(y- 1) defines a fundamental combination of the ratio y of the specific heats a t constant pressure and constant volume. SECONDARY STAGNATION PRESSURE NDARY CHAMBER LINES TO MANOMETERS

MIXING

TUBE

STAGNATION COMPRESSED AIR

SECONDARY FLOWMETER

FIG. 1. Schematic representation of a jet ejector.

1. Supersonic Flow Patterns

For high values of the ratio pi’/$, the primary flow entering the mixing tube is supersonic and remains supersonic throughout the mixing tube (Fig. 2a)* and the jet ejector operates with a supersonic flow pattern. In this case, the downstream pressure conditions have no limiting effect on the mixing-tube flow and the induced secondary flow rate i s the m a x i m u m rate compatible with the coexistence, in the mixing tube, of the primary and the secondary streams. In this case the two streams remain distinct well after section 1. The subsonic secondary flow may be described sufficiently well by means of the

* The shadowgrams of Fig. 2 show a two-dimensional primary nozzle of average M’*l = 1.78 (opening angle of the diverging nozzle 10’). The two-dimensional mixing tube has a section ratio of 1 = 2.61; the four shadowgrams correspond to the same setting of the control valve.


5

classical quasi-one-dimensional assumption, but the primary expansion should be studied by the method of characteristics. However, when in section (1) the secondary pressure PI’’i s lower than the @%nary pressure # , I ,

FIG. 2 . a : b: c: d:

supersonic flow pattern, pj‘ = 69, p = 0.222, saturated supersonic flow pattern, pi‘ = 5 p , p = 0.266, mixed flow pattern, pi’ = 4 9 , p = 0.300, mixed flow pattern with separation, pi’ = 3 9 , p = 0.445.

6

J. FABRI AND R. SIESTRUNCK

the primary jet expands into the mixing tube, and a t the section of maximum expansion (subscript e) this flow is nearly uniform. Actually both flows may be described accurately by one-dimensional isentropic expansions between sections 1 and 2, although such an assumption does not imply that the mean pressures p,' and p," are equal. The wake issuing from the primary-nozzle trailing edge, very apparent on the shadowgrams of supersonic flow patterns (Fig. 2a and b) cannot be neglected in small-size set-ups, such as the one used in our experiments. This wake represents a certain fraction 1 - A' (Fig. 3) of the primary flow.

\ \\\\\ \

\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\,

\ \ \

, \

FIG. 3. Experimental arrangement for supersonic flow pattern (schematic).

It can be assumed quite safely that between sections 1 and e this wake is fed by neither flow, as the mixing process has hardly begun within such a short distance. Let v represent the maximum width of the expanded primary flow (section e) ; the mass conservation equations of both flows yield

The corresponding pressures are those of an isentropic expansion

When writing the momentum conservation of both flows between sections 1 and e, one notices that the pressure integrals on the wake interfaces


7

between primary and secondary flows vanish, since the local pressure is the same on both sides. Therefore one obtains by adding the two momentum equations

Finally, the condition of maximum induced secondary mass flow is expressed by the equation M Y , = 1. This completes the set of aerodynamic equations which can be solved for given values of the geometric and aerodynamic parameters of the ejector (A’,M i l ) and for given primary conditions (pj’, Ti’). For each value of pl“ one obtains the auxiliary set of values of v, p l , p,”, M i , , and the speed M i l of the secondary flow. Then, by means of p1” and M i l , the corresponding point of the (p, &) diagram is obtained:

-

Pl“

Saturated supersonic flow patterns. As pif‘increases and M Y l attains the critical value MYl = 1, the aerodynamic choking of the secondary duct in its minimum section (1’ - 1 ) limits the maximum induced flow. Thus the optimum operational condition becomes M t l = 1, and for the corresponding saturated supersonic flow pattern the p, &-performance curve is linear :

This straight line corresponding to the saturated flow pattern is tangential to the performance curve of the pure supersonic pattern, represented by (2.4). The slope of the lines (2.5) is inversely proportional to the primary stagnation pressure. I t is clear, however, that the secondary mass flow rate induced by a jet ejector operating with a saturated flow pattern is independent of pi’, for a given value of &. Such a result is not specific to parallelstream jet mixing; it occurs whenever a well-defined geometric throat limits the secondary flow. Similarity Laws of Supersonic Flow Patterns. Supersonic flow patterns of jet ejectors obey very simple similarity laws. The stagnation temperatures of both flows appear only in the ratiop of the two mass flow rates. Therefore,

8

J . FABRI A N D R. SIESTRUNCK

if the primary stagnation pressure is high enough to induce a supersonic flow pattern, the reduced performance curve [,u(Ti”/Ti’)1’2vs. G] is a thermal invariant of the set-up. The solution of either equation (2.4) or (2.5) depends only on the ratio pj”/p,’ of the secondary and primary stagnation pressures. The reduced performance curve [,u(fii‘/fii”)vs. &] is then the pressure invariant of the set-up. If these two similarity laws are correctly combined, the numerical calculation of jet ejectors with supersonic flow patterns is greatly simplified, a t least as long as the ratio y of the specific heats can be assumed to be constant. I t is interesting to note, that the operation of the jet ejector is described by the laws of perfect gases, when the primary conditions allow the establishment of a supersonic flow pattern. Viscosity and turbulence of the two air flows as well as the downstream geometric shape of the ejector do not interfere with the exhaust performance. Their only effect would be to advance or to retard the establishment of these supersonic flow patterns. 2. Mixed Flow Patterns

When the primary pressure decreases, the supersonic primary jet eventually breaks down, the downstream shock waves moving towards the primary nozzle exit (Fig. 2c). The subsonic secondary flow is then induced by a subsonic primary flow, and the induced mass flow rate is limited by the exhaust capacity of the mixing tube, whether or not it is followed by a diffuser, since the final exit pressure of the mixed and homogenized flow is the pressure of the surrounding atmosphere. In order to define the overall mixing of the two streams, the geometrical and mechanical characteristics of the set-up must be taken into account. The geometric characteristics have already been defined. The only interesting mechanical characteristic is the turbulent wall friction coefficient, f. The mean pressure drop between sections 1 and 2 is given by the classical one-dimensional representation 2

where p designates the density, 5 the ratio of the area of lateral surfaces to that of the transverse section AS, and M , represents a mean value of the speed in any intermediate section. The approximate overall expression

9

SUPERSONIC AIR E J E C T O R S

where 5 = 4 LID represents now the non-dimensional length of the mixing tube of diameter D ,gives a quite accurate estimate of that pressure drop. In (2.7), M e 2 is the mean value of M , in section 2 where uniform pressure and velocity are assumed. In the parallel-stream mixing considered here the momentum equation takes the form

m2 - M,,

= AP2

m 2 - M *2 2

If the stagnation temperatures are replaced by the corresponding critical sound velocities, a*‘, a*‘‘ and a,, the mass conservation equation Pl‘

+ (A’- 1 )

$1”

(2.9)

a

and the equation of total enthalpy conservation

take very similar forms, the last one replacing the condition (2.2) of isentropic flow. The ejector performance, directly derived from the secondary flow characteristics (pit', M i can now be calculated from (2.8), (2.9j, and (2.10), if the aerodynamic data (PI‘, M i of the primary flow, the thermodynamic data (a*’, a,”) of both flows and the mixing tube exit conditions (p2, M*2) are given. The values of the latter quantities are computed from the diffuser exit data (p3, M,3) by means of the one-dimensional flow equations. The mass conservation equation between sections 2 and 3, (2.11)

g=-

P2 P3

M*2(m2 - M2,3) M*3(m2- M2,2) ’

is independent of friction. The equation of motion takes into account the friction losses estimated as in equation (2.6) and is written in differential form : (2.12)

10

By elimination of obtaines


p

between (2.12) and the differential form of (2.11) one

In the particular case of a conical diffuser with apex angle u, Eq. (2.13) can be integrated [ 8 ] , and the numbers M*8 and M,, a t the entrance and the exit of the diffuser are related by (2.14)

where 5 = 2 tan a/f y. At the diffuser exit the subsonic flow pressure is necessarily the pressure of the surrounding atmosphere: (2.15)

P3

= P,

and M,, must be chosen so as to cover the whole mixed flow performance curve, where 0 and p are still defined by (2.4). Of course, a correct choice of f has to be made. The use of either von Kkrman's universal relation [9] (2.16)

f - 1/2 = 4 log ( Rf'") - 0.4

or one of its usual approximate representations yields sufficiently accurate estimates off, as the flow is always turbulent and the walls generally smooth. In (2.16), &! represents the transverse Reynolds number of the mixing-tube flow. Approximate Similarity Laws. Equations (2.8), (2.9), and (2.10) show that, using the same primary nozzle ( M , l ) for different mixing tubes (A) with similar geometry (c, u, and a) and in the same thermodynamic conditions (a,', a,"), one obtains nearly the same induced flow coefficient p, if the values of $,"(A - l ) / A and #,'/A can be kept the same. This approximate similarity law is valid only if the variation of the friction coefficient with the Reynolds number and the geometric imperfections of the actual set-up, i.e. the difference A - A', can be neglected. The influence of the stagnation temperatures cannot be represented in such a simple way. However, for small values of the induced flow coefficient p, the non-dimensional combination ,u(T,"/T,')''2 can be considered as an approximate thermal invariant of the ejector.

11

SUPERSONIC AIR E J E C T O R S

3. Mixed Flow Patterns with Primary Separation For primary pressures that are still lower, the exhausting capacity of the primary flow depends on the geometric shape of the primary nozzle. Actually the one-dimensional theory of jet ejectors yields a low-pressure limitation for the mixed flow pattern: this limit corresponds to the case where the local secondary pressure pl" a t the primary nozzle exit equals the pressure which would be obtained if, for the primary exit conditions (pl', M i a normal shock were to occur in the exit section 1, i.e.

pl"

-- Pl'

m2Mi21- 1 m2- M,, '2 ,

This would lead to a classical subsonic ejector. Such a flow pattern may appear in well-designed nozzles in which a quasi-normal shock in the divergent nozzle changes the supersonic flow into a nearly uniform subsonic flow. However, in the more usual conical nozzles such shock waves do not appear. Instead of emerging from the primary nozzle lips, the primary jet separates from the nozzle walls along a parallel situated inside of the divergent part of the nozzle (Fig. 2d). The corresponding section is designated by the subscript 0. It seems reasonable to assume that the subsonic secondary pressure pl" prevails in the whole part of the primary nozzle which is not filled with the primary flow, i.e. between sections 0 and 1. Then (2.9) and (2.10) still hold, but (2.8) takes the form a 0 Po'

(2.17)

+

a - (a' - 1: m2

"1

=

a P2

of course, the isentropic expansions -a* ao=7-( 1

(2.18)

A

M*o

Po'

m2 - M,2 2

4

m2 - 1

l/(Y- 1),

m2--*o

m2 - M , ,

Pi'

hold in the supersonic flow. According to a generally accepted rule [lo] the upstream pressure of the primary flow in the separation section represents always the same fraction

12

J. F A B R I A N D R. SIESTRUNCK

of the quasi-uniform downstream pressure. Such an assumption means that the oblique shock waves that separate the primary flow from the nozzle walls and deflect the jet toward the nozzle axis have a nearly constant pressure ratio. For the conical nozzles used in rocket motor tests* the experimental value of this compression ratio is close to 2.5. This value of the critical compression ratio agrees quite well with the theoretical predictions of flow separation due to shock wave-boundary layer interaction in the Mach number range considered [ l l ] . In the absence of a more accurate rule, the condition (2.19)

Pl"

= 2.5 p,l

represents the complementary equation. The number of unknown quantities being therefore maintained, the performance of the jet ejector in the low pressure range can now be calculated. This mixed flow pattern with primary separation disappears when (2.20)

plr'< 2.5 pl'. 4. Theoretical Performance Curves

It appears from Sections 1, 2, and 3 that for given primary conditions (pi', M i l ) and a given value of the secondary compression ratio (or a given value of the secondary pressure coefficient GI) each set of equations gives a value of the induced flow coefficient, and the actual value of p has to be chosen between these values. This choice depends of course on the downstream exhaust possibilities. We shall assume that the supersonic flow pattern appears whenever the mixed-flow assumption leads to a too optimistic value of p. That is to say, one chooses between supersonic or mixed flow patterns the one which, for a given secondary pressure, induces a smaller amount of secondary flow. Thus when the primary stagnation pressure is higher than a given critical value &', the supersonic flow pattern appears even for very small amounts of induced secondary flows, and the saturated pattern appears when p increases (Fig. 4a). When pi'< 3i'r the small values of p correspond to the mixed flow pattern; as p increases, the expansion of the primary flow decreases, and eventually the supersonic pattern may appear (Fig. 4b). In some cases, the flow pattern passes directly from mixed flow to saturated supersonic flow (Fig. 4c). * Generally these nozzles are designed for expansion ratios pi'/p,' of 15 to 30 (1.6 < M'*, < 1.9). Such a rule does not apply to nearly transonic flows (for M'*, = 1.37 the compression ratio 2.5 is obtained by means of a novmal shock).

13

SUPERSONIC A I R E J E C T O R S

When pif is still lower, this transition becomes impossible in the usual range of operation (& < 1) although the primary throat remains sonic.

0

ml

0

a

b

C

d

0 FIG. 4. Typical exhaust performance curves for diminishing primary pressures. S supersonic pattern, SS M Ms

saturated flow, mixed flow, mixed flow with separation.

It is usual to observe flow separation in the primary nozzle for such values of the primary pressure. The rule given above still applies though the mixed flow gives a higher mass flow rate with separation than without, since in this case the choice is to be made between two similar flow patterns each of which corresponds to a different primary pressure range (Fig. 4d). 111. EXPERIMENTAL VERIFICATION

In the schematic representation accessible to theoretical calculations, the exit section of the primary nozzle extends into the mixing tube. Actually such a geometric configuration is not the only one that can be described

14

J. FABRI A N D R. SIESTRUNCK

theoretically. The equation of momentum conservation, on which the theoretical description of the mixed flow pattern is based, has equally simple overall expressions in the case of a constant section mixing process and in the case of a constant pressure mixing process (i.e. with constant pressure along the wall of the mixing tube). Although the second process may show better theoretical performance, [2], it does not seem to have any physical significance in the supersonic flow patterns. I t is also very hard to verify whether such a process actually takes place, and it is difficult to build the various necessary geometric configurations. For these reasons all our experimental investigations were confined to constant-section mixing, the exhaust performances of which are interesting enough. 1. Experimental Set-up

The experimental results were obtained with a small-scale apparatus, schematically shown in Fig. 1. The whole ejector is built of metal parts, all of them rotationally symmetric and screwed together end to end. The easy exchange of parts allowed rapid transition from one geometric configuration to another. The primary nozzles generally consisted of two cones with common base (nozzles B , C, D,E , and F of Fig. 5). Although the flow in the exit section of such nozzles (section 1) is nearly conical, their easy and accurate manufacturing made them preferable to nozzles of the type E , (Fig. 5), which is a better design but difficult to manufacture. Two mixing tubes of different diameters (Dl = 11.7 mm and D, = 15 mm) were designed for this set-up. As all primary nozzles have throats of nearly the same cross section, the mixing tubelthroat ratio A* has only two different values. The actual enlargement ratios I' are easily computed from the data shown in Fig. 5. The stagnation pressures are measured by means of pressure taps placed in the primary and secondary chamber walls. These two chambers have large sections compared to the flow sections, and the flow conditions are therefore close to the actual stagnation conditions. The primary mass flow rate is deduced from the primary stagnation pressure after calibration of the primary nozzles. The pressure drop in the primary supply lin$& which is nearly independent of the mass flow rate as long as the p r i i a r y nozzle remains supersonic, is estimated to be four percent of the primary pressure in our experiments. Thus the actual stagnation pressure of the primary flow is 96 percent of the measured value. In all the numerical calculations made hereafter this correction was taken into account, in order to keep as basis of comparison the value of the stagnation pressure measured by the primary manometers.

15


As the secondary chamber is close to the mixing-tube entrance, there is no such problem for the secondary stagnation pressure, and the compression ratio of the jet ejector may be derived directly from the secondary-chamber pressure measurements. 8-C-D-E-F

El

int. ext.

10.95

11.26 FIG. 5. Description of the primary nozzles.

All our experiments were performed with air as the primary and the secondary fluid. The stagnation temperatures of both flows are the same and the mean friction coefficient f is nearly independent of the induced mass flow rate. The value f = 0.0053 derived from (2.16) gives a good correlation between the theoretical performance curves and the experimental measurements.

16

J. F A B R I A N D R. SIESTRUNCK

In order to show the correctness of these theoretical calculations, all the diagrams appearing hereafter were computed independent1y of the experiments by means of the equations of Part 11, and all the curves shown in the following figures are theoretical curves. The corresponding experimental points are marked on the same diagrams and generally agree quite well with the calculated predictions.

FIG.6. Comparison of theoretical and experimental performances for various primary pressures (Nozzle D, 1, = 5.45; 6 = 61.5; u = 1).

2 . General Performance Curves

Figures 6 and 7, though seemingly very intricate, represent the performance curves of two similar jet ejectors for various primary pressures. The first corresponds to the simple case of a mixing tube ejecting into the surrounding atmosphere, without a downstream diffuser (a = 1). I n Fig. 7, the mixing tube is provided with a conical diffuser, geometrically defined by

17


(a = 3.30°, CT = 2.92). In both cases the performance curves, corresponding to pi' = 5.5 p and consisting of a pure supersonic flow pattern followed by the saturated pattern, are identical, thus showing that the ejector operation is independent of the downstream conditions. For pi' = 4.5 p , with a downstream diffuser, the flow pattern is supersonic, then saturated; without diffuser it is mixed, then saturated. For pi' = 3.5 9, with a diffuser, the

FIG. 7. Comparison of theoretical and experimental performances for various primary pressures (Nozzle D. A,

=

5.45;

E

=

61.5;

IJ =

2.92, a

=

3.30").

mixed flow is followed by saturated flow; without diffuser, there is just a mixed flow pattern. Finally, for pi. = 2 . 5 p , the mixed flow pattern with primary separation can be observed in both cases. In general the theoretical calculations describe the experimental results well enough. The interpretation of the quite intricate practical performance curves becomes easy if one considers the theoretical discussion of the various flow patterns.

18

J . FABRI :\NU

a : siipersonic pattern, p,' = 7 p , b: supersonic pattern, pi' = 6.5 p , c : transition pattern, pi' = 6 p ,

I of the primary nozzle are not used in the performance calculations, and the base-pressure characteristics should be independent of the primary nozzle if the mixing tube is geometrically defined by the same value A, of the mixing-tube to throat section-area ratio. The results are represented in Fig. 9 where the same mixing tube is used with different primary nozzles having the same throat section. The theoretical base pressure characteristic is a single curve, and the experimental points follow this curve, a t least in the part where, for each primary nozzle, the conditions for the existence of mixed flow with separation are satisfied.

*

Geometric conditions similar to those of Fig. 2.

20


I n Fig. 9, the high-pressure limitation is represented for each nozzle by the straight line

-Pi” _

(3.2)

P

-2.5-(

pi’

m2 - M *’21

m2

9

)

Y / ( Y -1)

.

.8

.7 .6 .5

.4

.3

I

2

3

4

5

FIG.9. Base pressure vs. primary stagnation pressure for mixed flows with separation ( A , = 8.93; = 48; u = 1).

The low-pressure limitation of these flows is less well defined. Of course, the theoretical calculations lose their significance as soon as the separation line reaches the transonic part of the primary nozzle, where the breakdown of the flow is never simple. The transition from mixed flow (with or without separation) to subsonic flow seems therefore difficult to describe theoretically, and as the latter does not have much industrial application, we shall not study it here. Supersonic Flow Patterns. For mixed flows the base pressure decreases as the primary pressure increases, and the difference between the primary exit pressure and the base pressure increases as well. Eventually the primary exit pressure becomes high enough to allow an enlargement of the primary

21


jet up to the mixing tube walls (the flow configuration is very similar to Fig. 8c). The flow configuration may then become independent of the downstream part of the set-up, i.e. one may obtain the supersonic flow pattern. Thus, with the simplifications (3.1), the supersonic flow pattern may appear when

1 -- tmonly.This critical value 5, actually depends on the primary stagnation pressure. For example, Figs. 2c and 2d show that for higher primary pressures the supersonic part of the primary jet extends further

FIG. 12. Influence of downstream diffuser on performance curves (Nozzle D , 1, = 5.45; 6 = 61.5; pi’ = 4.5 p ) .

downstream and the actual subsonic mixing length becomes shorter, increasing therefore the critical value tm. Similarly, for values of $ / / $ that are high enough to allow the supersonic flow pattern, the value of Em decreases and becomes independent of pi’/$. As a general rule, the theory represents always correctly the effect of mixing tube length for > 50. The downstream exhamt diffuser (parameter a) has an effect on the jet ejector performance only in the mixed flow patterns. However, the critical primary pressure for which the supersonic flow pattern appears depends on u. In Fig. 12 the performance curves of the same ejector with three


25

different exhaust diffusers are compared for the same primary conditions (see Fig. 10). It can be seen that a well-designed diffuser facilitates the appearance of the supersonic flow pattern arid improves therefore the theoretical and experimental performance of the ejector.

FIG. 13. a: influence of primary Mach numbers on performance curves (nozzles B , = 61.5; u = 1 ; pi’ = 4.5p). C, D , A, = 5.45; nozzle E , 1, = 5.43;

The mixing tube cross section, characterized by the ratio A, to the primary throat section, is a fundamental parameter of the ejector operation. The influence of A, is well known by the theory. Large values of A, give high induced mass flow rates but necessitate high primary pressures in order to maintain low secondary pressure conditions. The Primary Mach number has also a great effect on the performance of a given set-up. Large values of M ; , give low secondary pressures if high primary pressures are available. For moderate primary pressures, primary nozzles with weak expansions are more appropriate. Some examples of

26


performance curves as functions of the primary Mach number are shown in Figs. 13a and b. The effect of the geometrical shape of the primary nozzle is shown in Fig. 14 where the performances of nozzles E and E , are compared. These

Fig. 13 b : influence of primary Mach numbers on performance curves (nozzles B , C, D , 1, = 5.45; nozzle E , 1, = 5.43; 6 = 61.5; u = 1 ; p$' = 3.5 p ) .

two nozzles differ only in the shape of their divergent part and thus in their ability to give primary flow separation. The supersonic flow pattern remains therefore unchanged, and the high-pressure mixed-flow performance curves are also identical in both cases. For lower primary pressures, the conical nozzle gives mixed-flow performance with separated primary flow, while the aerodynamically better designed nozzle E , still gives pure mixed-flow performance. Actually the performance is better with separation than without. I t may happen for such low-pressure performance curves that the experimental verifications are not as good as usual: this is mainly due to the


27

fact that the flow patterns are less stable for partial or total separation of the primary flow. The empirical rule (2.19) defining the separation criterion does not hold universally either, as was assumed in the numerical calculations for the sake of simplification.

FIG. 14. Comparison of mixed flow performances with a n d without separation. (Nozzle E , I , = 8.93; nozzle E,, 2, = 9; 6 = 48; (I = 1; pi’ = 3 . 5 9 ) .

The elementary aerodynamic theory of jet ejectors cannot describe the effect of all parameters on the performance of the ejector. The effect of the location of the 9rimary-nozzle exit section along the mixing tube axis can only be determined experimentally. In the small-sized set-up described in section 111.1, the relative positions of the primary and secondary nozzles could be slightly changed. In particular it was easy to place the primary nozzle farther back, i.e. into the secondary chamber (Fig. 1). Although in this case the geometrical configuration of the set-up differs from the theoretical configuration assumed in the calculations, the measured perform-

28

J . FABRI AND R. SIESTRUNCK

ance curves show very little deviation from the theoretical predictions. Of course, this is true for large values of I only, that is, for primary nozzles having external dimensions which are small compared to the mixing tube

2

%i

I

0.527

'd

0

"

10

20

30

40

sox

FIG.15. Mixing-wall pressure distributions in supersonic flow pattern for various induced flow coefficients. (Nozzle D, I , = 5.45; 5 = 61.5; u = 1 ; p,' = 5.88p).

section. If the space between the outside of the primary nozzle and the inlet of the mixing tube is small, any displacement of the primary nozzle may change the geometrical configuration of the secondary duct, and in particular the geometry of the secondary throat section. Thus the performance curves can be profoundly changed, an effect which is most noticeable for the saturated supersonic flow patterns.


29

For some larger industrial set-ups designed for combustion research, it was possible to show that the energy level of both fluids (i.e. the stagnation temperatures Ti' and Ti") have the effect predicted by the similarity rules.

FIG.16. Mixing-wall pressure distributions in mixed flows for various induced flow coefficients. (Nozzle D, 1, = 5.45; = 61.5; 0 = 1 ; p,' = 3.54p).

5. Pressure Distribution along the Walls of the Mixing Tube

Basically, the one-dimensional theory of supersonic flow patterns in ejectors assumes that the induced secondary fluid accelerates as it flows along the primary jet. Thus the formation of a sonic section limits the amount of secondary flow which can be induced. Such an assumption can be justified by means of wall pressure measurements along the mixing tube.

30

J . FABRI AND R . SIESTRUNCK

Some longitudinal pressure distributions +"/pirr are shown in Fig. 15 as functions of the axial distance X from the primary nozzle exit*. These pressure distributions correspond to different induced mass flow rates a t the same primary stagnation pressure, which was chosen high enough to insure a supersonic flow pattern throughout the whole operating range of the ejector. For zero induced flow, the nearly periodic pressure fluctuations along the wall are due to the structure of the primary jet. A final pressure rise appears upon the subsonic breakdown of the jet. The pressure distribution remains periodic as long as the induced mass flow rate is small enough to retain boundary layer properties. For more larger induced flows the secondary pressure fluctuations disappear, and the secondary pressure attains the sonic value 9" = 0.527 pi' as predicted. For lower primary pressures the continuous increase of the wall pressure is characteristic of a mixed flow pattern (Fig. 16).

IV. OPTIMUMJET

EJECTOR

DESIGN

By means of appropriate diagrams [5], in which the similarity rules of Sections 11.1 and 11.2 were used to condense the performance curves into a small set of useful characteristic curves, the design of jet ejectors for given tasks becomes easy. As a great number of geometric configurations remains possible, it is still necessary to choose among them the one which gives the best results. The designer has generally at his disposal a given primary air supply with a fixed stagnation pressure and a maximum mass flow rate. He has then to choose the most satisfactory combination of primary nozzle and mixing h b e . The primary throat section is determined by the available air supply. The remaining free parameters are the primary nozzle speed M ; , and the ratio 1, of the mixing tube section and the primary throat area. However, this last parameter is very often determined by the available space, and can be considered as fixed. The above theoretical and experimental study of supersonic jet ejectors shows that, among the various flow patterns which can be observed during ejector operation, the supersonic flow pattern gives the best exhaust perfomance. But this very general condition only determines the minimum pressure necessary for a given set-up to operate under economical conditions; beyond that, it is desirable to determine the best ejector configuration for a given job.

* The reference length is one quarter of the mixing tube diameter. This same unit was used in the caIcuIation of the non-dimensionalmixing tube length E.


31

This optimum corresponds to the lowest secondary pressure for a fixed primary pressure and a given secondary mass flow rate; or to the highest secondary mass flow rate for a given secondary pressure, and of course, a given primary pressure.

FIG.17. Optimum performances of jet ejectors with supersonic flow patterns.

If one considers, for a given value of the parameter A*, the various performance curves corresponding to various primary nozzles, the set of such curves possesses an envelope which constitutes the optimum operation curve. The analytical definition of that envelope is derived from the analysis of Section 11.1 together with the contact condition

- _ac;, _ - - _ap_ _ _ a6 (44

aiwg, aa

aa

and can be written in the simple form (44

p,'

=

pl".

ap

aM;,

--o

32


I t will be recalled that this is also the limiting pressure condition of saturated supersonic flows, but the corresponding operation point is not on

FIG. 18. Pressure in the secondary exit section for supersonic flow patterns. (Nozzle D, a, = 5.45; E = 61.5; = 2.92; JI = 3.30’; pi’ = 5 . 3 3 ~ ) .

the envelope, since it is a singular point of the (p, &) parametric representation. The numerical solution of (4.2) shows that there exists on each performance curve a point of operation where plf = p i f and M i < 1. In the operating range between that point and the saturated flow limit, the ratio Pif/pl’ passes through a maximum value, larger than 1. A single p, &-diagram gives the set of optimum performances for different values of A, and M i l (Fig. 17). The theoretical and experimental variation


33

of the induced mass flow coefficient p and the pressure ratio 6 = pl”/p1’ are represented as functions of the secondary compression ratio 6 in Fig. 18 for a jet ejector having a secondary section of moderate width, which permits pressure measurements at the wall of the secondary duct, giving a valid indication of the mean value of pl”. The external wall of the primary nozzle is here parallel to the mixing-tube wall in order to avoid the formation of a secondary throat upstream of the primary exit section. There is good agreement between the theoretical predictions and the measured values of 6 in the vicinity of the optimum operation point (6 = 1). At this point the experimental primary and secondary pressures are nearly equal as predicted by the theory. This agreement no longer holds for the transition to saturated flows. However, the disagreement between theory and experiment can be attributed to a transverse velocity distribution in the secondary duct which becomes progressively less uniform. It is to be expected that for such flows the sonic line no longer coincides with the section 1 of the secondary duct; the wall pressure measurements then fail to give correct indications of the actual mean pressure in the fluid. On the other end of the 6 vs. &-curve where ,u is very small, it is interesting to note that the experimental values of 6, corresponding to an overexpansion of the primary jet, lie on the continuation of the theoretical 6, &-curve. Unfortunately, no physical explanation can be given for that. These comparisons of theoretical and experimental studies of supersonic jet ejectors show that the optimum design of a jet ejector for a given program presents no particular difficulties. Supersonic jet ejectors can now be designed with a relatively small amount of calculation with good accuracy in performance prediction. At the same time, the seemingly intricate performance curves of ejectors can be interpreted and simplified, if the various flow patterns which govern their operation be considered.

References 1. MELLANBY,A. L., Fluid jets and their practical applications, Trans. of the Institution of Chem. Eng. 6, 6G-84 (1928).

E. P., and LUSTWERK, F., An investigation of ejector 2. KEENAN,J . H., NEUMANN, design by analysis and experiment, Journ. A p p l . Mech. 17, 299-309 (1950). 3. JOHANNSEN, N. H., Ejector theory and experiments, Trans. of the Danish Academy of Techn. Sciences 1, (1951). Copenhague. 4. FABRI,J., LE G R I V ~ E., S , and SEESTRUNCK, R., Etude akrodynamique des trompes supersoniques, in Jahrbuch 1953 der Wissenschaftlichen Gesellschaft fur Luftfahrt, F. Vieweg u. Sohn, Braunschweig, pp. 101-110 (1954). 5. LE GRIVES,E., FABRI,J., and PAULON, J., Diagrammes pour le calcul des Bjecteurs supersoniques. O.N.E.R.A. N . T . No. 35, (1956). Paris.

34

J . FABRI AND R. SIESTRUNCK

6. FABRI,J.. and SIESTRUNCK, R., Etude des divers regimes d’Bcoulement dans l’elargissement brusque d’une veine supersonique, Revue GLnLrale des Sciences Appliqubs 2, 229-237 (1955), Bruxelles. 7. ROY,M., TuyBres, trompes, fusees et projectiles, Publ. Sc. et Techn. du Ministire de Z’Air No. 203 (1947). Paris. 8. FABRI, J., MBthode rapide de determination des caractBristiques d‘un Bcoulement gazeux B grande vitesse, O . N . E . R . A . , N . T . KO. 17, (1949). Paris. 9. VON KARMAN,TH., Mechanische Ahnlichkeit und Turbulenz, Nach. Ges. Wiss. Gottingen, Math.-Phys. Klasse 68 (1930). 10. SUMMERFIELD, M., FORSTER, C. R., and SWAN, W. C., Flow separation in overexpanded supersonic exhaust nozzles, Jet propulsion 2.7, 319-321 (1954). L., and LEES,L., A mixing theory of the interaction between dissipative 11. CROCCO, flows and nearly isentropic streams, J o u m . Aeron. Sc. 19, 647-676 (1952) 12. WICK,R. S., The effect of boundary layer on sonic flow through an abrupt crosssectional area change, Journ. Aeron. Sc. 20, 675-682 (1953). 13. KORST,H. M., and WICK,R. S., Comments on ref. 12. Journ. Aeron. Sc. 21, 568-569 (1954) and 22, 135-137 (1955).

Unsteady Airfoil Theory BY A . I . VAN DE VOOREN National Aeronautical Research Institute. Amsterdam

Page I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 I1. The Fundamental Equations . . . . . . . . . . . . . . . . . . . . 37 1. Conditions for Linearization . . . . . . . . . . . . . . . . . . . 37 2 . The Linearized Equations . . . . . . . . . . . . . . . . . . . . 39 3. Harmonic Oscillations . . . . . . . . . . . . . . . . . . . . . . 41 4. Edge Conditions and Radiation Condition . . . . . . . . . . . . . 42 42 5. Reciprocity Relations . . . . . . . . . . . . . . . . . . . . . . I11. The Oscillating Airfoil in Two-Dimensional Subsonic Flow . . . . . . . 44 1. Method of the Velocity Potential . . . . . . . . . . . . . . . . . 44 2 . Method of the Acceleration Potential . . . . . . . . . . . . . . . 47 3. Method of the Integral Equation . . . . . . . . . . . . . . . . . 48 4 . Semi-Empirical Methods . . . . . . . . . . . . . . . . . . . . . 50 I V . The Oscillating Airfoil in Three-Dimensional Subsonic Flow . . . . . . 52 1. Analytic Solutions . . . . . . . . . . . . . . . . . . . . . . . . . 52 2 . The Integral Equation . . . . . . . . . . . . . . . . . . . . . . 54 3. Approximations for Wings of High Aspect Ratio . . . . . . . . . . 55 4 . Wings of Very Low Aspect Ratio . . . . . . . . . . . . . . . . . 56 5. Wings of Low Aspect Ratio . . . . . . . . . . . . . . . . . . . 57 6. Swept Wings . . . . . . . . . . . . . . . . . . . . . . . . . . 58 V . The Oscillating Airfoil in Supersonic Flow (Supersonic Edges) . . . . . 59 1. Method of the Moving Sources . . . . . . . . . . . . . . . . . . 59 2 . Riemann’s Method . . . . . . . . . . . . . . . . . . . . . . . . 63 3. Operational Method . . . . . . . . . . . . . . . . . . . . . . . 63 4 . Calculation of Pressure ..................... 64 VI . The Oscillating Airfoil in Supersonic Flow (Subsonic Edges) . . . . . . 65 65 1. The Rectangular Wing Tip . . . . . . . . . . . . . . . . . . . . 68 2. Oblique Tips . . . . . . . . . . . . . . . . . . . . . . . . . . 3. The Delta Wing with Subsonic Leading Edges . . . . . . . . . . . 69 4 . Numerical Approach for Arbitrary Planform . . . . . . . . . . . . 70 71 5. The Integral Equation . . . . . . . . . . . . . . . . . . . . . . VII . Non-Linear Approximations . . . . . . . . . . . . . . . . . . . . . 72 1. Supersonic Flow . . . . . . . . . . . . . . . . . . . . . . . . . 72 2. Hypersonic Flow (Piston Theory) . . . . . . . . . . . . . . . . . 76 VIII . Indicial Functions . . . . . . . . . . . . . . . . . . . . . . . . . 77 1. Incompressible Flow . . . . . . . . . . . . . . . . . . . . . . . 78 2 . Compressible Flow . . . . . . . . . . . . . . . . . . . . . . . 79 3. Calculation of Flutter Derivatives from Indicial Functions . . . . . 81 4. Calculation of Flutter Derivatives for Large Subsonic Mach Numbers 82 from Indicial Functions . . . . . . . . . . . . . . . . . . . . . 5. Indicial Functions for Finite Wing . . . . . . . . . . . . . . . . 84 I X . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

35

36

A.

I . VAN DE VOOREN NOTaTION

Coordinates fixed in the airfoil ( x chordwise, y spanwise, z downward) Coordinates fixed in the undisturbed fluid Time Lorentz coordinates, see 11, 2 Velocity potential Acceleration potential Reduced values of and see 11, 3 Pressure Pressure difference between upper and lower side of airfoil Air density Velocity of main stream (body) Downwash Frequency Reduced frequency = v l / U hM/I1 - M21, see (2.19) Mach number Speed of sound Semi chord Ratio of specific heats.

#.

A coordinate as subscript indicates partial differentiation with respect to t h a t coordinate. A subscript 0 denotes free stream values or, in case of harmonically varying quan. tities, the amplitude.

I. INTRODUCTION Unsteady airfoil theory as treated in the following review refers to that part of aerodynamics which considers the calculation of the pressure distribution over an airfoil moving with constant speed while performing an unsteady motion in the direction perpendicular to its plane. Since the publications in the middle 1930’s of Theodorsen [ l ] and Kiissner [2] on the oscillating airfoil in incompressible, two-dimensional flow, the subject has been extended widely by taking into account both compressibility and finite-span effects. Being in itself an interesting branch of applied mathematics, the technical importance of this field of aerodynamics lies in the fact that it permits the calculation of flutter speeds. These are defined as the speeds where the forces on the oscillating and deforming airfoil are such that the overall damping just vanishes and, hence, the oscillation tends to become unstable. In order to determine the pressure distribution over an airfoil performing an arbitrary motion normal to its filane, use is made of the indicial functions. These functions describe the lift or moment, or more generally the pressure distribution, as functions of the time, after a unit-step disturbance of either the angle of attack or the rate of pitch a t constant angle of attack has been applied. Although the forces for the oscillating airfoil can also be obtained by means of the indicial functions, the direct approach is often more convenient in this special case.

UNSTEADY AIRFOIL THEORY

37

The results obtained in the analysis of arbitrary normal motions are of importance for the calculation of gust loads. For this problem the variation of the pressure distribution with time, caused by the entrance of the airfoil into a gust front, should also be known. Nearly all available theoretical results are based upon certain simplifying assumptions, viz. (i) the neglect of viscosity (although the Kutta condition at the trailing edge reflects, in a certain sense, the influence of viscosity), (ii) the neglect of heat conduction, (in) the absence of shock waves of finite strength. A further simplification made in the greater part of all investigations in this field is that of linearization. The conditions under which linearization is justified are considered in Section 11. The assumptions mentioned above are the cause of discrepancies between theoretical and experimental results, but little progress has as yet been made in the elimination of these restrictions (with the exception of the second-order theory given in Section VI). However, if these assumptions are accepted, the theory has now entered into a fairly complete state, and it is the object of the present paper to give a review of what has been obtained. Lack of space prevents the inclusion of systems of airfoils which are of importance for compressor flutter as well as for wind-tunnel wall corrections, nor was it possible to consider slender bodies. The treatment is essentially restricted to single airfoils. Completeness of the already large list of references has not been intended; the references should be considered as illustrative rather than exhaustive. 11. THE FUNDAMENTAL EQUATIONS 1. Conditions for Linearization

The conditions for linearization of the equations governing the nonsteady motion of two-dimensional planar bodies have been established by Lin, Reissner and Tsien [3]. Later, Miles has extended these conditions to the cases of three-dimensional planar bodies [4] and to slender bodies [5]. Recently, Landahl, Mollo-Christensen and Ashley performed a similar investigation, including viscous flows in their considerations [6]. Let it be assumed that the flow can be described by a velocity potential y . This excludes viscosity, heat conduction, and shock waves of finite strength. The equation of continuity and Euler’s equations imply that p? satisfies the non-linear equation

38

A. I. VAN DE VOOREN

where the Cartesian coordinates x,y,z move with the flight velocity U of the body and c denotes the speed of sound. The speed of sound c as well as the pressure p are determined by Bernoulli's equation, which follows from Euler's equations and the isentropic equation of state as

here p denotes the density, y the specific-heat ratio, and the subscript 0 indicates free stream values. In the case of three-dimensional planar bodies (thin wings of finite span), the positive x-axis taken opposite to the direction of flight, the y-axis positive to starboard and the z-axis positive in downward direction, the potential y should satisfy the following boundary conditions : (i) at the wing surface, z = h(x,y,t), the condition of tangential flow (2.3)

pl*=k+pL+V,h.Y,

(ii) a t infinity the conditions v x =

u,

yy=o,

qx=O.

In order that the theory of small disturbances should be applicable, it is required that the departure of the velocity components and the pressure from the corresponding free stream quantities remains small. As is shown in [3] and [a], this is only possible under the following conditions : (2.4)

6 d2/3it is the sum of the “unsteady terms” p t t / c 2 2 M p X r / c ; while for R 1.

x2@

=0

42

A. I . VAN DE VOOREN

Solutions of these equations are sought which satisfy the boundary condition of prescribed GZ at the airfoil as well as the conditions to be mentioned under II,4. Both for M < l and M > l the relation between yo and @ is (2.22) Introducing a reduced acceleration potential Y which is connected with (2.16) into

4 in the same way as @ with p, see (2.18), we can transform X

(2.23)

@ ( X , Y , Z )= -exp U

(

-- i

. 1-M2

k

Fat.

--oo

for M 2 1. 4. Edge Conditions and Radiation Condition The physically important solutions of the boundary value problems formulated in II,2 and II,3 are, in general, not regular. In case of the velocity potential, the wake forms a discontinuity plane, the strength of which is determined by the Kutta condition. This condition prevents the pressure at the trailing edge from becoming infinitely large. For a subsonic trailing edge this.means that the pressure becomes equal to the free stream pressure & i.e. the acceleration potential vanishes (strong Kutta condition). For a supersonic trailing edge the acceleration potential may have a finite value (weak Kutta condition) in which case a shock will occur a t the trailing edge. In the wake the acceleration potential is zero since the pressure will be continuous there and should be antisymmetric in z. It follows from (2.14) that in the wake ytl vanishes which means that all vortices (discontinuities in y ) are carried off with the flow under retention of their strength. At a subsonic leading edge the pressure a t the airfoil surface becomes infinite as where s is the distance from the leading edge. At a supersonic leading edge the pressure remains finite, but may be discontinuous in the presence of a shock. Sommerfeld's radiation condition requires the solution to correspond to waves which are expanding from the wing toward infinity. This condition is meaningless in incompressible flow.

l/vL

5. Reciprocity Relations

Reciprocity theorems give relations between the aerodynamic properties of wings of the same planform in forward and reverse flows. They have been developed for steady flows by a number of authors including A. H. Flax,

43


who extended them to unsteady flow by considering a harmonically oscillating wing [lo]. Heaslet and Spreiter [ll] treated the case of indicia1 functions, while Timman [12] gave a unified treatment for arbitrary unsteady flows. The reciprocity relations are based upon Green’s theorem

/I/

(2.24)

(udv - vdu) d t = /-/-(24?&-

vg)ag.

The function u is identified with a solution @ for the reduced velocity potential in forward subsonic flow satisfying (2.20). The function v is identified with a solution for the reduced acceleration potential in reverse flow, which is easily seen to satisfy also (2.20). It is required that for the reversed flow the Kutta condition is satisfied a t what was originally the leading edge. The volume of integration is the half space 2 > 0. All integrals in (2.24) appear to vanish except that over the surface Z = 0. Hence

/‘/-(@E-F@z)dxaY =o.

(2.25)

plane 2 = 0

It follows from (2.14), (2.15) and (2.18) that

The minus sign of the second term is due to the reversal of flow direction. Upon substituting (2.26) into (2.25) and performing a padial integration in X , it follows that

where the integration may be confined to the airfoil surface, since !P = = 0 outside the airfoil in Z = 0. In physical coordinates the relation may be written as (2.28) wing

no

wing

where and wo are the amplitudes of the pressure difference and the downwash for forward flow and and Go the same quantities for the reverse flow, which should have the same frequency and speed.

170

A.

44

I. VAN DE VOOREN

The reciprocity theorem (2.28) may be extended to arbitrary unsteady flows by remarking that 17,(x,y,z;v) may be considered as the Fourier transform of a function n ( x , y , z , t ) , viz.

Ilu(x,y,z;v) =

.i

e - i y ' 1 7 ( x ,Y&) dt.

--m

170,

Similarly w,,and G,, are the transforms of of the convolution theorem, (2.2s) becomes

17, w and E.

Hence, by means

where 17 and w now denote instantaneous values of pressure difference and downwash for arbitrary unsteady motions. By a slightly modified derivation the same relations can be shown to hold also for supersonic flow. The importance of the reciprocity theorems is that they allow the calculation of overall forces or moments, if a well-known flow is taken as reverse flow. For applications see [lo], [ l l ] and [12].

111. THE OSCILLATING AIRFOILI N TWO-DIMENSIONAL SUBSONIC FLOW This problem can be solved by any of the methods described in 111.1, 111.2 and 111.3. However, if x is large (eg. larger than 2) an alternative method given in VIII, 4 yields results more quickly. 1 . Method of the Velocity Potential

The complete solution for the reduced velocity potential Q can be separated into Q, Q2, where both @, and Q2 satisfy the two-dimensional form of the wave equation (2.20). Q1 is the regular solution corresponding to a flow without circulation and satisfying the boundary condition of prescribed normal velocity at the airfoil, i.e. Qlz = Qz = f ( X )if - 1 < X < 1. G2 has a vanishing normal derivative at the airfoil. This potential denotes a flow with circulation and, hence, is discontinuous in the wake. For the oscillating airfoil the wake is a vortex sheet of harmonically varying strength.

+


45

The boundary value problem for @, can be solved by means of a Green's function G(X,Z;X1,Z,) of the second kind. Physically, this function denotes the potential a t the point X , 2 due to a source at X,, Z, (or reversely) in the presence of the boundary, where the normal derivative of the unknown potential is prescribed. I t will be clear from this definition that the normal derivative of G at the boundary vanishes. A source a t X,, Z, is defined as a solution of the pertaining equation, which becomes infinite as log R ( R = distance from X,,Z,). In a three-dimensional flow, G becomes infinite a t X,, Y,,Z, as R-l. The boundary for the present problem being the slit - 1 < X < 1 of the X-axis, it is convenient to introduce elliptic coordinates by

X

= cosh 6 cos q,

Z = sinh 5 sin q.

The solution for @, then can be written as*

(3.1)

@,(Ed

=

i

@l~(O,vl) G(E,q;0,qJ sirlq,dq,.

0

In incompressible flow the function G is a solution of the Laplace equation and can be obtained by conformal transformation (Theodorsen [l]) as

I n compressible flow where G should be a solution of the wave equation, no solution in closed form is known. However, it follows from work of Haskind [13], Reissner [14], Timman [15], Timman and van de Vooren [16] and Billington [17], who found the regular solution of the boundary value problem by separation of variables, that

(3-3)

where se,(q) and NeL2)(5)are solutions of the Mathieu and the modified Mathieu equation respectively (see Mc Lachlan [18]). The potential GiZhas been obtained by Theodorsen [l] by considering a vortex distribution in the wake. The corresponding field is easily obtained if the airfoil is mapped onto the unit circle. Here and further on, G has the slightly different significance of being the potential due t o the combined action of a source a t 0, ql and a sink of equal strength at 0, 2 n - q1 in presence of the double line segment - 1 < X < 1.

46

A. I. VAN DE VOOREN

An alternate procedure for determining G2 which is also possible in compressible flow is due to Haskind [13]. The reader is referred to [la] for a detailed description of this method. The Kutta condition is used to determine an as yet unknown factor which still could be added to G2. The resulting pressure distribution for the oscillating airfoil with flap in incompressible flow has been calculated by Theodorsen [l] and also by Kussner and Schwarz [19] in a different way. In order to illustrate the result it may be mentioned that the force K due to a harmonic translatory motion A e x p i v t is equal to

(3.4)

K

= (npl2V2- 2npUlivC) A ,

where K and -4 are positive in the same direction. The first term denotes the inertia of the air. This concept of virtual mass keeps its simple meaning only in incompressible flow. But for very high frequency (vl/c is of order 1) the fluid may not be considered as incompressible even if U = 0, since the time required for a disturbance to travel along the chord corresponds to an appreciable phase difference. Then the airfoil behaves as an acoustic radiator. It follows from (2.19) that for U = 0 the parameter vZ/c = kM determines whether the potential or the wave equation should be used. Therefore, physically, the first term in (3.4) is not valid for very high frequencies (see also Sec. 8.3). The second term contains Theodorsen’s C-function, given by

(3.5)

C(k) =

Hi2)(k) iHr’(k) Hi2’(k)’

+

where Ha) and Hi2) denote Hankel functions of the second kind. This function, the modulus of which lies between 0.5 and 1 for all real and positive values of k, gives the reduction of the circulatory part of the force compared with its quasi-steady value (C = 1). The function C appears also in the results for forces and moments due to harmonic motions with arbitrary downwash distribution along the chord.

It may be added that the theory is not only valid for an airfoil performing harmonic oscillations since t = - 00, but also for an airfoil performing oscillations of exponentially increasing amplitude. In this case v is complex but Im v < 0. However, for damped oscillations (Im v > 0) the theory does not hold since then the disturbance due to the initial phase of the motion is always large in comparison with that due to the oscillatory motion itself as has been shown by van de Vooren [20] and others.

47


2. Method of the Acceleration Potential The complete solution for the reduced acceleration potential Y can be separated into Y1 Y,, where both Yl and Y, satisfy the two-dimensional wave equation. Yl is the regular solution corresponding to a flow with zero pressure a t leading and trailing edges and satisfying the boundary condition of prescribed normal acceleration. In general, Yl will correspond to a flow with circulation. Ylis obtained in exactly the same way as @j1, see (3.1). The way in which the singular solution Y2can be determined has first been given by Timman [16]. !Pzhas a vanishing normal derivative a t the airfoil. For incompressible flow, YZmust be identical with the acceleration potential of the steady flow past the flat plate a t incidence, since for this flow the normal velocity along the plate is constant and, hence, the normal acceleration is zero. In the case of compressible flow, however, Yzshould be a solution of the wave equation instead of Laplace’s equation. This is achieved by adding to the result of YZfor incompressible flow a series of correction terms. It has later been recognized by Kussner [21] and Timman 1221 that this singular solution is nothing else but

+

where G again denotes the Green’s function of the second kind. In order to explain this, it is first remarked that G is a solution of the wave equation and that its normal derivative a t the boundary vanishes. However, G has a logarithmic singularity a t q = ql, 6 = 0 which vanishes for q = q1 = n, as is seen from (3.2), valid for incompressible flow. Since the singularities of the wave equation and the Laplace equation are of the same type, this holds also for compressible flow. The proper singularity a t the leading edge is produced by differentiation with respect to ql, and since Gql is also a solution of the wave equation and has zero normal derivative a t the boundary, this function satisfies all requirements. The determination of the arbitrary factor a,, by which the singular solution may still be multiplied follows from the condition that the normal velocity a t the airfoil, corresponding to the acceleration potential Yl a,, Y,, should agree with the prescribed value. By the method of the acceleration potential numerical results for forces and moments acting on the oscillating airfoil in compressible flow have been obtained by Timman, van de Vooren and Greidanus [23] and [24]. A variation of this method is due to Hofsommer [25], who writes

+

48

A. I. VAN DE VOOREN

where @, and Y2have already been defined while Y3is a solution similar to Y2but with the singularity at the trailing edge. Although Q), is the flow without circulation, it will have pressure singularities at both edges. The coefficient a3 should be determined in such a way that the singularity a t the trailing edge disappears. Next, a2 follows from the condition that the normal alone yields the true normal velocity due to a2 Y2 a3 U3vanishes since velocity. Hofsommer then obtains the pressure distribution over the airfoil in incompressible flow as

+

n

(1

+ cosq‘) ~ ( 7 ‘dq’)

0

(3.6)

*

+ (ik - A?) sin q aq flog Jog

+

1 - cos (17 17‘) sin __ 1 - cos (7 - q’)

ql

w(r’) q,

0

where T = 2 C - 1, see ( 3 4 , while 17 and w are positive in the same direction. (3.6) has been generalized to compressible flow in [22] and [26]. 3. Method of the Integral Equation

In this method use is made of Green’s function of the first kind with the X-axis from - 00 to w as boundary. This function has similar mathematical properties as Green’s function of the second kind, but it vanishes at the boundary (instead of its normal derivative). Then, for compressible flow

+

a-

@(X,Z) = - - J@(X,,O+) [&Hi2’ {. V(X - X,)2 + ( Z 2

-

Z1)2)]

-m

z,

=0

ax,

if 2 > 0.

(3.7)

Since @ is antisymmetric in 2 and continuous ahead of the airfoil, the range of integration may be taken from - 1 to m. Upon differentiating the whole equation with respect to 2 and letting 2 approach 0, (3.7) becomes m

a2

(3.8) @z(X,O)= lim

z+o 2

@(X,,O+)z2

H!) { x v ( X - X,)2

+ Z2}dX,.

-1

If follows from the condition of zero pressure in the wake that

Q)(X,,O)= @(1,0)exp

49


Since QZ is known for - 1 < X < 1, (3.8) is an integral equation for @ ( X l ,O+) in the interval -1 < X,< 1. Approximate solutions of this equation as well as results for forces and moments have been presented by W. P. Jones, [27] and [28]. For incompressible flow it can be shown that (3.7) becomes m

-m

if

z

> 0,

which by suitable reduction can be brought into the form m

(3.9)

pz(x,O) = 2 n

J’*

Y ( 4 = px(x1,0-) - p.(x1,0+).

dx,,

-1

Equation (3.9) expresses the downwash due to a vortex distribution at the airfoil and in the wake. From this equation exact solutions for the incompressible case have been obtained by Kussner [2], Schwarz [29] and other authors. Equation (3.7) holds also in terms of the acceleration potential Y instead of @, Then

(3.10)

Y ( X , Z )=

Y(Xl,O+)

a

Hi2’ { X V ( X - X J 2

+ z2}

-1

if Z

> 0.

By means of (2.23) the reduced velocity potential @ can be obtained. Differentiation with respect to Z yields then the famous Possio equation [30], viz.

(3.11)

a

*a -zH b 2 ’ { ~ V ( X - X l ) 2 + T 2 } d l d X l . This is an integral equation determining the pressure distribution. Various approximation methods have been based upon this equation. Collocation methods have been applied by Possio [30] and Frazer [31]. Schade [32]

50

A. I. VAN DE VOOREN

reduced the problem to the solution of a set of linear equations which are obtained by an expansion in Legendre functions. The method of Dietze [33] is probably the one most used. It is an iterative method using the known results for incompressible flow. Turner and Rabinowitz [34] have calculated further values using Dietze’s method. Fettis [35] has approximated the kernel in Possio’s equation and then succeeded in giving a solution in closed form. Convergence of the iterative methods becomes slower with increasing x. The analytic methods of II1,l and 2 are then also handicapped by the slow convergence of series such as (3.3). In this case an asymptotic solution of the wave equation (2.20) (asymptotic in X ) is possible as will be described in VIII,4. 4. Semi-Empirical Methods There exists a certain discrepancy when measured pressure distributions are compared with theoretical results. This is due to the idealizations which have been introduced in the theory, viz. the neglect of viscosity (save for the Kutta condition) and the linearization. In order to obtain pressure distributions which are closer to the experimental values a number of semiempirical methods have been proposed which make use of measured data for steady flow. Schwarz [36] and W. P. Jones [37] have introduced the concept of a “skeleton line” z(x), which is defined by aid of the measured pressure in steady flow by distribution n(5)

(3.12)

The pressure distribution as well as the skeleton line depend on the angle of incidence. For the oscillating airfoil, it is assumed that the skeleton line varies harmonically between the two skeleton lines corresponding to steady flow past the airfoil in the positions of maximum and minimum angle of incidence. After calculation of the normal velocity w of the flow for the oscillating skeleton line the pressure distribution over the oscillating airfoil follows by solving (3.12) for IT([). In this way, W. P. Jones [37] was also able to obtain results for the oscillating airfoil a t high mean angles of incidence. By expanding the pressure distribution along the airfoil in a Fourier series and taking into account only the first terms contributing t o lift and moment, the same procedure can also be applied approximately if only lift and moment are known.


51

In general, formula (3.12) will lead to skeleton lines whose angle of incidence is smaller than the angle for which the pressure distribution has actually been measured. Another method for reducing the pressure distribution is to reduce the circulation, which requires the introduction of a trailing edge singularity. In this case the skeleton line is kept identical with the mean camber line of the airfoil. Rott [38] has proposed a method based on this idea. If, in steady flow, the Kutta condition is replaced by the condition that the total lift should agree with the measured value, the flow pattern is also completely determined. Leading-edge singularity S , and trailing-edge singularity S , then follow as functions of the angle of incidence a, Elimination of a leads to a relationship

St

= f(S1).

For unsteady flow, where both St and S, are functions of time t it is assumed that st(t t)= f{Sr(t)),

+

thus introducing a time lag t,due to the fact that it may take some time before the viscous effect of the change in the leading edge singularity is felt at the trailing edge. The time lag t is taken in the form t = Cl/U,

where c = 2 would mean that the influence of a variation in S, is carried along the chord with the speed U . Other assumptions may be made concerning c and a general form is

St

= M(K)

s,,

where M is a complex function of the reduced frequency k. A third theory has been presented by Woods [39], who besides viscosity effects takes also into account the potential-theoretical thickness effect. The latter effect multiplies the lift curve slope by 6. I t then follows that K should be replaced by an effective reduced frequency k6, while the various derivatives are also to be multiplied by powers of 6. Furthermore, Woods assumes that the wake vorticity moves downstream with the local velocity of the mean steady flow, which gives an important correction to the pressure for large values of k. The viscosity effects are introduced by assuming that the rear stagnation point performs a small chordwise motion in phase with the local angle of incidence a t the trailing edge and with such an amplitude that the lift-curve slope 2 7 4 1 - E ) is reproduced. The factor 1 - E is called the Joukowski efficiency of the profile. The measured value of the moment curve slope, ac,lau, is used for a correction to the velocity distribution.

A.

52

I. VAN DE VOOREN

Although these semi-empirical theories may in some cases give a n impressive improvement of the results, the agreement is still far from complete and the problem of the introduction of viscosity effects in unsteady flow is still open to further research. This is especially important for controlsurface oscillations.

AIRFOIL IN THREE-DIMENSIONAL SUBSONIC FLOW IV. THE OSCILLATING 1. Analytic Solutions

The method of separation of variables can be used as a first step in obtaining an analytic solution for the three-dimensional problem. Recently, Miles [40] has recalled the theorem that separation of (2.20) is only possible in eleven coordinate systems, whose coordinate surfaces are confocal nondegenerate or degenerate quadrics. Lifting surfaces of practical interest for which the boundary value problem can be described in one of these coordinate systems are, in subsonic flow, the two-dimensional wing and wings of circular or elliptic planforms. The circular lifting surface in incompressible flow has been considered by Schade [41] and recently by van Spiegel [42]. The orthogonal coordinate system for the circular wing is given by _____

(4.1) X = v 1 +q21/1 -p2cos 6,

8,) = 00

n=l

with Em=

1 if

m>O

and

E,,,=&

if

m=0.

Consider now

This is a solution of Laplace's equation having zero normal velocity a t the airfoil and a singularity at the point q = 0, p = 0, 6 = 6,,which lies either a t the leading or a t the trailing edge. The singularity is of the type ~(COS a)/r3, where r is the distance from a point on the airfoil to the singular point and il. the angle between the vector Y and the radius of the circle. The singular solution Y2is then given by

1

3 n/2

Y2(q,p,6)=

a(@,) Gpl(7,p,8;0,0,6,) d6,,

n/2

where the integration is along the leading edge only, since no singularities are admitted at the trailing edge. As in the two-dimensional case, the where Y is the distance from the leading singularity in Y2is of the type edge. The function ~ ( 6should ~ ) be determined from the condition that the normal velocity at all points of the leading edge due to the acceleration potential Yl Y2should agree with the prescribed function. In general, this will lead to an integral equation for a(6,) which can be solved by expanding a(&,) in a Fourier series. I t may be noted here that Kiissner 1261 has proposed a different singular solution which, however, fails to have the proper singularity a t the leading edge. Schade [all has treated the problem of the circular lifting surface in a different way, but only for the six downwash distributions up to the second degree in X and Y . The problem of the oscillating circular wing in compressible flow can be solved in an analogous way by means of spheroidal wave functions. Also the oscillating elliptic wing can be treated when Lam4 functions are used.

v$,

+

A.

54

I . VAN DE VOOREN

The main difficulty will be that for this purpose the knowledge of these functions has to be extended.

2. The Integral Equation All approximate treatments use the integral equation in two variables as a starting point. Using Green's function of the first kind for a plane, which is taken as boundary of the half space 2 > 0, one has in analogy to (3.10)

2-

Y ( X , Y , Z )= - 2 n / / - Y ( X l , Y I , O + ) (4.3)

wing

{&);t}

z,

=0

dX,dY,

if 2 > 0 , where Y =

V ( X - X,)2

+ ( Y - Y,)2 + (2- 2,)2.

A result which is the extension of Possio's equation to the three-dimensional case can be derived in the same way as (3.11) has been obtained from (3.10). This yields

(4.4)

-cc

which, by (2.22), can also be written in the form

11

(4.5) w ( X , Y ) = -

PU wing

D(X,,Y,) K ( X , - X , Y , - Y ;K,M) d X , dY,.

The kernel K in this equation has been expressed by Watkins, Runyan and Woolston [43] in terms of known functions and is being evaluated numerically at Harvard University. This will lead eventually to a completely numerical method. (4.3) holds also for the velocity potential @ or for QX, provided the region of integration is also extended over the wake. Such equations, too, have been used as starting points of numerical methods, e.g. by W. P. Jones [44] who uses the equation for @. The unknown @-distribution corresponds


55

physically to a doublet distribution. As in two-dimensional flow, Jones uses a set of fundamental doublet distributions corresponding to fundamental downwash distributions. This correlation, simple as it is in the two-dimensional case, is much more intricate in three-dimensional flow. Although the downwash distribution can in principle be calculated to any desired accuracy by analytical methods, Jones recommends the use of Falkner’s method with corresponding tables. Applications of Jones’ procedure have been made by Miss Lehrian [45]. Work along similar lines, but restricted to the circulatory component of the flow, has been published by Dengler and Goland [46] as well as by Dengler [47]. 3. Approximations for Wings of High Aspect Ratio

Wings of aspect ratio larger than 3 or 4 are characterized by the fact that the inequality ly - yl(>> Ix - xll holds for most points ( x , y ) and (x1,y1) of the wing. The transformation (4.6)

6= X / l ,

=E Y~I?,

where E is a small parameter denoting ‘the ratio between root chord and span, introduces coordinates 6 and 7 of the same order of magnitude for most points of the wing. The expression for the downwash w may then be expanded in a series of increasing powers of E. There is always a term independent of E , and if the integral equation is solved by taking into account only this term, the results of “strip theory”* are obtained. In the case of straight wings in incompressible flow there is no term linear in E and the &2 log & &2 next terms are of order c210ge, E % , -- and - , see Eckhaus [48]. k k The terms of order loge and e2 are neglected in a theory for wings of high aspect ratio. The terms with a factor k in the denominator are the correction terms for oscillating wings of high aspect ratio. Hence it follows that these corrections increase in importance with decreasing frequency. If k becomes smaller than E , the estimate of the orders is no longer correct. Reissner [49] has presented a theory which is based more or less on the idea mentioned above. The method starts from (4.3)with QX instead of Y , which yields Biot and Savart’s equation after some transformations. Terms of order E~ log E and ~2 are again neglected, while in the terms of order & 2 log & &2 and - a certain approximation is introduced (viz. the replacement k k

* I n strip theory the flow in any chordwise strip is assunied to be two-dimensional, as i t would be if that strip were part of a two-dimensional wing.

56

A. I . VAN D E VOOREN

of the incomplete by the complete Cicala function). This approximation makes the method workable, the result being that the formulae for the aerodynamic forces in two-dimensional flow may be used provided a correction factor is added to Theodorsen’s C(k)-function. The correction factor, which depends upon the spanwise coordinate, is determined by an integral equation that resembles, but is less simple than, the integral equation for lifting-line theory in steady flow. Extensive numerical tables and results are given in [50]. Kussner [9] has independently come to the same results as Reissner by using (4.4) and replacing the upper boundary X by 0 in the integration over 6. Later, Reissner has extended his theory to compressible flow [51]. 4. Wings of V e r y Low Aspect Ratio

Wings of very low aspect ratio (less than 1/2) are characterized by the inequality 1% - xl/ >> Iy - y,l, valid for most points of wing and wake. The transformation (4.6) for high aspect ratio wings can be used again, but now E is large, and the expansion of the downwash must be made in increasing powers of 1/&. The first approximation to the integral equation is in the case of incompressible flow [48] blxl

(4.7)

which is the usual two-dimensional slender wing approximation. This equation can be inverted in the usual way for all spanwise sections lying ahead of the section of maximum span. The corresponding pressure distribution, which is of order 1 / ~ has , been calculated by Garrick [52]. One so obtains the complete solution for wings with a straight trailing edge, which is a t the same time the section of maximum span. Merbt and Landahl [53] have presented a theory which under certain conditions is also valid for compressible flow. They simplify (2.6) to

which is allowed under the conditions

The first condition annihilates the term (1 - M2)px, and the second condition makes the term with pxt in (2.6) small compared with the term ptt/c2. However, for incompressible or nearly steady flow, the term with p r t


57

need not be small compared with tptt/c2, since this latter term can then itself be neglected. Therefore the results obtained from (4.8) are also valid if

the latter condition denoting that the difference between the Laplace and wave equations becomes unimportant, see eq. (2.20). The solution of (4.8) is obtained by means of elliptic coordnates and Mathieu functions. For incompressible flow, the results are equivalent to those of Garrick [52], but this method also permits one to find the pressure in sections aft of the maximum-span section provided these sections are simply-connected (unswept wing). The pressure there is given by the regular part of the solution of (4.8) written in terms of the acceleration potential. 5. Wings of Low Aspect Ratio

Solutions for wings of low aspect ratio (we mean here the range between 112 and 3) are the most difficult to obtain. Besides the general methods of Sec. IV,2 there exists a method due to Lawrence and Gerber [54] for calculating the force and rolling moment in a spanwise strip of a low-aspect ratio wing with straight trailing edge in incompressible flow. This method, which yields very reliable results a t least for rigid wings, starts from the same equation as Reissner’s high-aspect ratio theory, viz. from an equation expressing the downwash as a double integral over wing and wake, containing yx. This two-variable integral equation is reduced to a single-variable integral equation by multiplication with the weight function v b 2 ( x ) - y 2 and integration over the span b ( x ) . A further approximation is made regarding the distance, viz.

This makes it possible to perform the integration over y in closed form. The resulting integral equation in x is solved by collocation. One of the reasons for the success of this theory is that it yields the correct results in the limiting cases of both zero and infinite aspect ratio. The main drawback is that no pressure distribution is obtained, but only spanwise pressure integrals. It is interesting to consider the various approximations used for 7 in different theories. Reissner’s theory appears to be equivalent to replacing 7 by ly - yll on the wing. In the wake a similar approximation is of course impossible, and the wake integral is obtained by means of the complete Cicala function. In Garrick’s very-low-aspect ratio theory Y is replaced by Ix - xl/. The approximation used by Lawrence and Gerber, both for wing and wake, is given above. Laidlaw [55] has proposed another approximation, viz.

58


V ( X

- X1I2

+ (Y- Y J 2 w 1 , ( m 1%

-

4+

IY - YlL

to be used for the wing integral only. The functions 1, and ,I2 are determined by the requirement that for any value of the aspect ratio the squared error {V(X

- X1)2

+ (Y-

Y1I2 - 1,(4

1%

- x11 -

1 2 w

IY - Y11I2

after integration over the 4 variables x,y,x,,y, should be a minimum. The function A, and A2 have been evaluated by Laidlaw for rectangular wings. The relation

1,(W) = In(A4-l) exists. If W = 0, one has 1, = 1 and A2 = 0 (Garrick) and if W = co, then I , = 0 and A, = 1 (2-dim.). The wake integral is evaluated exactly by Laidlaw using the incomplete Cicala function. The resulting equation for the vortex distribution is solved by a collocation method that uses Fourier series in both chordwise and spanwise directions.

A theory for swept wings of high aspect ratio can be obtained by the procedure described in Sec. IV,3, viz. by using the transformation (4.6) and expanding the expression for the downwash in powers of E . While there is always a term independent of E , there exists in the case of swept wings a term proportional to E sin A ( A = angle of sweep). This term is due to the fact that a linearly varying circulation a t the wing, which produces streamwise vortex lines in the wake whose strength is independent of their spanwise position, gives no contribution to the downwash in the case of straight wings, but does contribute in the case of swept wings. Van de Vooren and Eckhaus [56] have given a theory that takes this term into account. The strength of the streamwise vortices in the wake is put equal to the spanwise rate of change of the two-dimensional circulation, which is known. This introduces an error of order e2, but such terms are neglected. The same integral equation is obtained as in the two-dimensional case, but the downwash is corrected by the influence of the streamwise vortices in the wake. The result is again a strip theory, containing now also terms proportional to the local rates of change of the translational and rotational amplitudes and of the chord. In this way the strip theory for swept wings is brought to the same accuracy as that for straight wings. The evaluation is possible in closed form and [56] contains correction terms for lift and moment to be added for swept wings.


59

By expanding the downwash as given by the three-dimensional Possio equation (4.4)in a series of increasing powers in E , Eckhaus [57] has succeeded in extending the above mentioned theory to the compressible subsonic case. For this case no numerical results are yet available. Swept wings can also be dealt with by the general theories valid for any planform, which were mentioned in Sec. IV, 2. In particular, the theories of Dengler and Goland [46] and of Dengler [47] have been presented with swept wings in mind.

V. THE OSCILLATING AIRFOILIN SUPERSONIC FLOW(SUPERSONIC EDGES)

If all edges are supersonic, there are no disturbances in regions outside the airfoil which affect the flow at the airfoil. Examples of such purely supersonic planforms are the two-dimensional wing and the delta wing with leading edges ahead of the Mach lines from the apex. In these cases the solution for the velocity potential is directly given by an integration over a part of the wing. The most general method (valid for arbitrary unsteady motions) of arriving at this solution is that of the moving sources (Sec. V,l). Two more mathematical methods, valid for harmonic oscillations and solving eq. (2.21), will shortly be considered in Secs. V,2 and 3. 1. Method of the Moving Sources

This method, originally due to Possio, has been brought into a more easily accessible form by Garrick and Rubinow [58] and [59]. Recently Heaslet and Lomax [60] used this method in a modified form by introducing the concept of the acoustic planform. The method is based upon the physical fact that a disturbance occurring a t a certain point is felt at another point only after a time rlc, r being the distance of the two points. Consider a source of variable strength Q(t) moving in the direction of the negative XI-axis with a speed U and situated a t the instant t a t the point A ( r , q ' , [ ' ) . All primed coordinates are fixed in the undisturbed medium. The potential a t P(x',y',z') a t the time t is then given by

where r1 and r2 denote the distances C,P and C2P (Fig. la), C, and C, being the centers of the spheres through P which are tangent to the Mach cone

60


through A . Clearly AC, = M Y , and A C , = MY,, and it follows by simple geometry that

FIG.la. Plane through P and the x’-axis.

Let now A not be a single source but a point of the leading edge of a wing (= source distribution). The influence of points of the wing aft of A , but ahead of B ( B P = Mach line through P ) is given by a formula similar to (5.1) with Y denoting the distance from P to points between C, and C,. To B corresponds the point D ( D P 1B P ) . The action of points aft of B is not yet observed at P a t the time t. If the leading edge is perpendicular to the x‘-axis, all points from which P receives a signal a t the time t were situated within an ellipse with P as a focal point (Fig. l b ) when these signals were emitted. This area is called the acoustic planform. The total potential a t P becomes equal to

61


where F is the acoustic planform, Y,, is the distance from P to any point of F , and q the strength of the source distribution per unit area. Since it can be shown that the normal velocity at a point of the distribution is exactly equal to half of the local source strength, the result can also be written as

’i

I

ACOU s T Ic PLAN FORM

FORWARD MACH COME O F P W I T H X‘ ,y’- PLANE

Fig. l b . Projection on the

%‘,

y’-plane. The acoustic planform for supersonic flow.

It may be added that for subsonic flow the acoustic planform is limited by one branch of a hyperbola, corresponding to the fact that only one “propagation”-sphere would pass through P. In the result of Garrick and Rubinow the integration is not over the acoustic planform but over the part S of the wing lying within the forward Mach cone of P (Fig. lb). The correspondence between the area element d t F dr]F of the acoustic planform and the element d5 dr] of s is given by dr]F = dr],

(5.4) d5F =

M(x’ - 5’) F V ( x / - 5 ’ ) 2 - ( M 2 - 1) { ( y ’ - r ] ’ ) 2 ( M 2 - 1)V ( x / - 5 ‘ ) 2 - ( M 2 - 1){ ( y ’ - r ] ’ ) 2

+

+

(2’

(2’

- 5’)21d5*

- 5’)2}

The preceding relation is found in the following way: if the point A (coordinate l‘) corresponds to C l , 2 (coordinate 5‘ MY^,^), a point with

+

62

A. I. VAN DE VOOREN

+

+

coordinate 6' df will correspond to points with coordinates f' dE + M ( r t 2 d ~ ~ where , ~ ) , d r , , can be expressed in d5 by differentiating (5.2) with respect to 5'. Since d f F = IdE 1, (5.4) follows. Upon substituting (5.2) and (5.4) into (5.3), introducing coordinates without primes (which are fixed in the wing), putting

+

+

and noticing that the transformation gives a double mapping of the acoustic planform onto the region S, the potential at P turns out to be

The values of vEused in (5.3) and (5.6)are exactly the same. Note also that

are fundamental solutions of the equation for q in x,y,z-coordinates, i.e. Eq. (2.6). For harmonic oscillations (5.6) becomes, by (2.17),

(5.7)

if

z>O.

Transforming to Lorentz-coordinates and introducing reduced potentials produces the final result @z(X',Y',O)cos x R d X ' d Y ' , R

(5.8) @ ( X , Y , Z )= -

~

S

where (5.9)

R

= V ( X - X')2 -

(Y - Y ' ) 2- Z 2

if Z > O ,

63


In two-dimensional flow where QZ is independent of Y' the integration over Y' can be performed with the result n

(5.10)

@ ( X , Z )= -

I

@z(X',O)J o { x v ( X - X ' ) 2 -?z}dX'

if

Z

> 0;

the integration has to be carried out from the leading edge to the point X ' = x - 2. 2. Riemann's Method

This method, described in many textbooks on mathematical physics (eg. [61]), has first been applied to the two-dimensional oscillating wing in supersonic flow by Temple and Jahn [62]. It gives the solution of (2.21),viz. (5.11)

L(@)r @ x x- @zz

+

x2@ = 0

with prescribed values of QZ by making use of Green's theorem. A Riemann function Z ( X , Z ;X,,Z,) is introduced which may be considered as the hyperbolic counterpart of Green's function for elliptic boundary value problems. It has the properties of being a solution of (5.11) and assuming the constant value 1 if the points X,Z and X,,Z, are on the same characteristic. It appears that

z = J , { x V ( X - X,)2 - (2 - Z,)2}. I t can then be shown that (5.10) is the solution of the boundary value problem. Although the concept of the Riemann function cannot be extended to three-dimensional flow, W. P. Jones [63] has shown that a solution can still be obtained by application of Green's formula. This method leads to Eq. (5.8). 3. Operational Methods Equation (2.21) with the pertaining boundary condition can also be solved by means of Fourier transforms, This method which has been introduced by von KBrmBn, was followed by von Rorbely [64], Gunn [65] and Stewartson [66]. Miles [67] was the first to apply double transforms, once to the streamwise and once to the spanwise coordinate. The Fourier transformations reduce the partial differential equation to an ordinary differential equation in Z of which the solution satisfying the boundary condition can be written down immediately. The transformation back to the original X and Y coordinates appears to be possible, and the

64


same results as derived in V,1 are again obtained. The method is treated extensively by Temple in his contribution on Unsteady Flow in Modern Developments in Fluid Dynamics [SS]. 4. Calculation of Pressure

According to (2.14) and (2.18) the pressure can be obtained from the formula

p -Po

C

=

1

(ix@ - M @ x ) e x p i x T .

For the two-dimensional case (5.10) should be substituted. Taking X a t the leading edge, the result is after some reductions

=0

X

$(X,O+) - p - - Po[ix@z(X’,O)- M@ZX~(X’,O)] O1c { /0- (5.12)

I

Jo [x(X - X ’ ) ] dX‘ - M @z(O,O) J o ( x X ) exp i x T . On the other hand, formula (5.10) also holds for Y instead of @, since Y satisfies the same differential equation as @. However, it should be kept in mind that the normal acceleration Yz is infinite a t the leading edge, since there the normal velocity jumps from zero to a finite value. It follows from (2.14), (2.15) and (2.18) that (5.13)

C

Y z = - - ( i x @z - Max).

The discontinuity in

1

QZ

as function of X a t X

=0

permits one to write

where S ( X ) denotes the Dirac &function. Substitution of (5.14) in (5.10), written for Y, leads again to (5.12), where the integration over X’ should be taken from O+ to X (i.e. not including the &function in Qzx/, since this is already taken into account by the term with Qz(0,O)). It follows from (5.12) that the pressure takes a finite value a t the leading edge as well as at the trailing edge, which is in agreement with II,4. The same considerations also hold for three-dimensional flow with a supersonic leading edge.


65

VI. THEOSCILLATING AIRFOILIN SUPERSONIC FLOW (SUBSONIC EDGES) In the case of a subsonic edge the potential in points of a part of the wing is influenced by a region outside the wing, but in its plane. In such a region the upwash is not known a priori, but is determined by the condition that the potential (pressure) should vanish there. This leads to an integral equation for the upwash which, unlike to the steady case, has not yet been solved in closed form. Other methods not requiring the explicite determination of the upwash outside the wing have led to the solution in the case of a rectangular wing tip (V1,l). By aid of a suitable transformation the case of a raked-out or a raked-in wing tip as well as that of a delta wing with one supersonic and one subsonic leading edge can then be obtained (VI,2). A method using pseudo-orthogonal coordinates succeeds in solving the case of a delta with subsonic edges (VI,3). Besides these analytical methods numerical methods have been developed which are valid for any planform. These are the methods presented in VI, 4 and 5. 1. The Rectangular Wing Tip

This problem has been treated by Miles [69] using the Wiener-Hopf technique, by the same author [70] also by using a Laplace transformation, by Stewartson [66] with the aid of a different Laplace transformation method, by Goodman [71] using Gardner’s method, and by Rott [72] by means of a generalization of a method used by Lamb for the half-plane diffraction problem. In the following, in accordance with Goodman, Gardner’s method will be followed, and the result will be given in a simple form valid for an arbitrary downwash distribution a t the wing [73]. The idea of Gardner’s method, which has been described a.0. by Heaslet and Lomax [74], is to separate the four-variable boundary value problem (2.12) into two boundary value problems of three variables each, viz.

nxx

n,

- QTT -

nc ,

= 0,

- QYY - Qzz = 0.

It will be clear that any function Q(X,Y,.Z,T,() satisfying both (6.1) and (6.2) will also be a solution of (2.12) provided (, is kept constant. Accordingly, the function to be identified with the solution v ( X , Y , Z ,T ) of (2.12) will be taken as (6.3)

v ( X , Y , Z ,T ) = (Qdt =o.

In order to solve (6.1) and to comply a t the same time with the boundary condition of prescribed vz, a new variable (6.4)

a(X,Y,T,t)= (Qz)z=o

66


is introduced. (6-5)

The equation for OXX

G

becomes

- aTT - GCE = 0.

By (6.3) and (6.4),the boundary condition which a should satisfy is that (6.6)

(VZ)Z=O

= (Qz&=o,z=IJ = (a&=o

has a prescribed value at the wing ( Y > 0, X > 0 ) , while a and uE vanish ahead of the wing, that is for X < 0. For Y > 0 the determination of a is identical with the solution of an Evvard problem. Its solution is obtained immediately from (5.8) by substituting x = 0 and replacing Y and 2 by T and 6 respectively. Hence

dX, dT, . v ( X - X,)2 - ( T - T J 2 - E2

t>o.

v(X- X,)2 - t2 and for X , The limits of integration for T , are T 0 and X - 5. The integration over T , for harmonic oscillations gives

This determines the boundary condition a = (Qz)z=o for all Y > 0 for equation (6.2). Thus we have again an Evvard problem with the flow in the direction of the negative &axis (Mach number X appears here as parameter (Fig. 2 ) . The wing occupies the area Y > 0, E < X , and here Qz = a is given by (6.8). For Y < 0 the condition Q = 0 is assumed since, by (6.3), this leads to g~ = 0. Hence the solution Q of this problem yields a function v satisfying all conditions. In analogy to (6.7), the solution is

vq;

The region of integration S is the triangle lying in the plane Z = 0 within the forward Mach cone of the point P ( Y , ( ) , Fig. 2. If Y E < X , this triangle contains a part of the upwash region a: the, side of the wing, viz. ADE. According to a well-known property of the steady flow problem, the

+

67


contribution of d A D E in the integral of (6.9) is completely cancelled by that of AABC and hence, S is equal to the region PABC. Substituting (6.8) in (6.9), differentiating with respect to 5, putting E = 0, and performing the integration over 5, leads to the following result for v, as has been shown by van de Vooren in [73]. If Y > X , i.e. without tip effect, the usual formula reappears

where R = v(X- X , ) 2 - ( Y - Y , ) 2 , and the integration should be performed over the triangle cut by the forward Mach cone in the plane 2 = 0. I f 0 < Y < X , the integration in (6.10) should be performed over the region PABC (Fig. 3), while the triangle AOB gives a contribution to v(X,Y,O+,T) equal to

1 n12

x7c/ ~ v z ( X l , Y , , T )

(6.11)

J l ( x Rsin6,) d6,dX1dY,,

where

/

/

/

FIG.2. The plane 2 = 0 for the second problem (6.2).

FIG.3. Regions of integration for (6.10) and (6.11).

68


This gives a confirmation of Stewart and Li's result [75] that the contribution of A O A B may be neglected when considering only zero and first order in frequency. The same solution but with a trivial change in the area of integration holds also for a supersonic leading edge that is not perpendicular to the flow direction. I t follows from (6.10) and (6.11) that pl -0 if P approaches the wing tip (Y .+ 0). Moreover, p l y becomes infinite as Y-'I2. The complete solution for a wing with supersonic leading and trailing edges and with streamwise tips can be built up from this solution, provided the Mach lines from the foremost points of the tips do not cut the tips at the other end. A solution for the rectangular wing with B V M 2 - 1 < 1 (Mach lines from leading-edge wing tips now intersect the opposite tips) has been given by Miles [76] using a Laplace transformation to eliminate the X-coordinate in (2.21). This leads to (6.12)

FYY

+ Fzz = (s2 + x 2 )F ,

where

dX. 0

Equation (6.12) with its boundary conditions is then solved by introducing elliptic coordinates instead of Y and 2. The function F is obtained in terms of Mathieu functions. The difficulty is to determine @ from F. This has been performed in [76] by an expansion in powers of A I v M 2 - 1, which suffices for the low-aspect-ratio wing (except for M near to 1, when the expansion is in powers of k Az). 2. Oblique

Tips

With the aid of a transformation originally given by Lagerstrom and applied to unsteady flow by Miles [77], it is possible to transform an oblique tip into a streamwise tip. Let the equation of the tip be Y = - m X, 0 if it is a where 0 m < 1 if the tip is a leading edge and - 1 < m trailing edge. New coordinates are defined by

3.33 To/2 only one wave group appears. On the basis of the lowest dispersion curve alone, Davies concluded that no flexural displacements will occur at the cross section x until t’ 1.56 To/2, i.e., t’ 3 1.56 x/co. However, on the basis of the second dispersion curve, flexural displacements may be expected to occur at considerably earlier times, i.e. t‘ = 0.9 To/2; and since there are an infinite number of higher branches, and the maximum group velocity is the dilatational velocity, it is clear that the first arrival will occur at t’ = 0.887 T,,/2, followed by an infinite number of wave arrivals. At this place it may be worth-while to comment on the physical nature of these vibrations. An important question concerns the existence of phase velocities greater than the dilatational velocity, although no disturbance can be propagated with a velocity greater than that. A disturbance propagated into an elastic medium, in the general case, originates as one composed of both dilatational and distortional waves.

>

STRESS WAVE PROPAGATION I N RODS A N D BEAMS

161

These waves reflect from the boundaries in a very complex manner [92], thus making it difficult to follow the mode interactions that take place. In any event, the situation is closely analogous to that obtained in electrical wave guides and may be discussed in similar terms. The essential point is, of course, that energy is propagated at the group velocity rather than the phase velocity, and group velocities greater than the dilatational velocity will not occur. The phase velocities spoken of earlier represent the velocities of propagation in the axial direction of loci of constant phase in the mode pattern.* One of the principal assumptions of all approximate theories of beam bending is that of plane sections remaining plane. The exact theory may provide the means to judge the validity of that assumption. With the relationships between phase velocity and wave length established in Fig. 17, it is a fairly straightforward procedure to study the cross-sectional distortion and, for that matter, the distribution of normal and shear stresses, as well. For a given combination of values of a/A and c/co, the ratios A / C and B/C may be solved for from the three homogeneous algebraic equations comprising the boundary conditions. The three functions U , V , W may then be evaluated from (3.9) in terms of a single amplitudinal constant, e.g. C, and the radius Y. The displacements u", uo,u, can then be obtained from (3.8), and the corresponding stresses from (2.10). I t may be noted from (3.8) that 26, is essentially dependent upon the function W; further, from (3.10), W is a complex quantity dependent upon the constants A and B. Therefore, the results are presented in Fig. 20 in the form of curves with the modulus of W as abscissa and the ratio Y/a as ordinate. The curves shown [89] have been normalized by means of the largest value of W . The three curves at the top of Fig. 20 show the distortion variation, for three values of a/A, corresponding to the first branch of the phase velocity relation (curve 1 of Fig. 17). For the smallest value of a / A the distortion is nearly linear, confirming again the validity of elementary beam theory for long waves; however, for 0.4 < a/A < 0.8, a nodal plane appears as the curve departs from linearity.? The lower three curves of Fig. 20 show the distortion variation for corresponding values of a/A for the second branch (curve (2) of Fig. 17). Three features are of interest here: (a) the maximum displacement occurs at about Y = 0.8 a for a/A = 0.2, but occurs at the surface Y = a for a / A = 0.8; (b) the slopes of the curves are quite different from those based on the first branch; (c) if a nodal plane occurs at all, it will be for a/A > 0.8.

* Some additional comments of this general nature may be found in [91], [93], and [94].

t

A similar result, in the case of longitudinal waves, was noted by Davies [ I l l .

162

H. N. ABRAMSON, H. J. PLASS, AND E. A. RIPPERGER

These few curves give some idea of cross-sectional distortion for the two lowest dispersion branches as a function of ./A and also for the change with c/co as a/A is held fixed.

(0)

BASED ON CURVE

I

0

OF FIG. 17.

t

I.

-=0.2 A

(

iJ

a A

'=0.4 A

BASED ON CURVE

0

0.8

:

OF FIG. 17.

FIG. 20. Cross-sectional distortion curves associated with the first two branches of Fig. 17 (From Abramson [SS]).

Timoshenko Theory. The Pochhammer-Chree theory just described is too complicated mathematically to be useful as an engineering theory for predicting stresses and displacements in rods and beams subjected to bending impacts. On the other hand, the elementary theory is too limited to describe adequately the transient behavior resulting from sharp bending impacts. A comparison of the dispersion curves in Fig. 17 makes this evident because the (single) curve of the elementary theory agrees with the exact theory

STRESS WAVE PROPAGATION I N RODS AND BEAMS

163

curve for the lowest mode only for very long waves. No higher branches are predicted by the elementary theory. The gap between the over-simplified and the too highly complex theories is very well bridged by an intermediate theory due to Timoshenko 182, 831. His theory adds terms which correct for shear strains and for rotatory inertia of beam elements. Whereas the elementary theory contains the assumption that plane sections remain plane and perpendicular to the deformed middle curve of the beam, the Timoshenko theory is based on a less restrictive assumption : the sections remain plane but can be inclined a t different angles to the deformed middle curve. Also, whereas the elementary theory assumes the conventional equilibrium relation between bending moments and shear forces, the Timoshenko theory includes the inertia term arising from rotation of beam elements in the plane of vibration of the beam.

I

d

x

-!I

d

FIG.21. Element of a beam in bending.

A discussion of the essential details of the Timoshenko theory now follows. The forces, moments, and directions of motion of a beam element are shown in Fig. 21. From the figure it can be shown that the equations describing the behavior of the element are as follows:

164

H. N. ABRAMSON, H . J. PLASS, AND E. A. RIPPERGER

aK am -_- _ at

= EIK

M

(bending)

at

(bending)

(3..16)

Q

= As7

(shear)

i

1

of continuity)

(equations of material behavior),

where

M Q K

= moment = shear force = axial rate of

change of section ahgle

y = shear strain = awlax - # o = angular velocity of section ZI

E p p

I A

A,

=

-

= -

a#/ax

a#jat

= transverse velocity = awlat = Modulus of Elasticity = Shear Modulus = density = section moment of inertia =

section area

=

z ) dA s s y ( true variation of y over the area. area parameter defined by

= yA,,

where y(z) is the

When all but one variable, say M , is eliminated from the six equations above, a single fourth-order partial differential equation results. This equation is

where (3.18)

This is the form usually designated as the Timoshenko equation. The quantities Elp and p A , / p A have the dimension of a velocity squared. The quantity \ / E T has appeared before, as the velocity of propagation of longitudinal waves according to the elementary theory. The other quantity, ,uA,/pA, is a modified form of the square of the shear velocity, which is ,u/p. The modification results from the assumption of plane cross sections.

STRESS WAVE PROPAGATION IN RODS AND BEAMS

165

The steady state solution to the Timoshenko equation is found by substituting into it the following form of the solution: (3.19)

M = Moexpi[ar(x- ct)].

The resulting equation is

or (3.21)

The soiution of this equation for c/co vs. a/A, where a is the radius of the bar, and M = 2n/A, yields two separate branches. They are plotted on Fig. 17 and labeled “Timoshenko Equation”. One of the branches agrees quite well with the solution given by Hudson [85] using the PochhammerChree theory discussed previously. Better agreement can be had if the constant A , in the Timoshenko equation is adjusted to produce a velocity CQ = V p m i n agreement with the asymptotic value of the lowest mode of the exact theory. Such adjustment is not contrary to the basic physical assumptions in the Timoshenko theory, as the distribution of shear strains over the cross section is not an ingredient of Eqs. (3.14) - (3.16), and this distribution is precisely what affects the value of A,. The second branch, however, agrees only in form with the corresponding branch of the exact theory. The asymptotic value of the phase velocity in the Timoshenko theory is the bar velocity; however, the asymptotic value in the exact theory is the velocity of Rayleigh surface waves along the cylinder (approximately the same as the shear velocity, Other attempts to develop similar theories have been made by Mindlin [95] (actually for plates), Volterra [27], with his method of internal constraints, and Plass [29], using a method similar to that of Mindlin. In all of these theoretical approaches the exact equations are used as a starting point, assumptions about displacements are introduced, and the equations subject to these restraints are found. In the work of Volterra and Plass it is found that the velocities appearing in the final equation are the dilatational and the shear velocities instead of those mentioned above. The reason for this difference is that in the displacement assumptions used by Volterra and Plass, no account was taken of the Poisson’s ratio effect, that is, of the motion of particles perpendicular to the plane of bending. An interpretation ____ of the velocities co = V E / p and cQ = VpAJpA, appearing in the Timoshenko

v&).

166


equation, was given by Fliigge [96]. He showed that co is the velocity with which discontinuities of moment or section angular velocity are propagated, and that cQ is the velocity of propagation of discontinuities of shear force, or transverse velocity. Fliigge also pointed out that the disturbance occurring a t a particular station bears no resemblance, timewise, to the applied disturbance because of the complicated mixing of waves of both bending and shear types. The Timoshenko equation has received a great deal of attention in recent years. Solutions for different kinds of impacts have been found by Uflyand [97], Dengler and Goland [98], Miklowitz [99], Boley and Chao [loo], Zajac [loll, and Plass and Steyer [19]. The solutions of Uflyand and Dengler and Goland are for transverse velocity inputs in the form of impulse functions, and each contain certain errors, later corrected. Miklowitz [99] has solved the impact problem of a step function moment by Laplace transform techniques. Boley and Chao [loo] have presented a rather complete set of solutions for step-function type impacts of transverse velocity, moment, shear, and angular velocity. They again use the Laplace transform methods. An important part of their paper is the table showing the interrelation between solutions corresponding to various boundary conditions. Zajac [loll studied in detail the problem of response to a step moment impact. Asymptotic solutions for large time or great distance from the impact end are developed. Also a series solution is developed for the behavior of the beam just behind a discontinuous moment wave front. The work of Plass and Steyer [19] is aimed directly at obtaining solutions for a problem easily duplicated experimentally. The problem is that of a moment impact whose time variation is in the form of a single half sine wave. The problem is solved, for a limited range, by evaluation of integrals arising from a Laplace transform solution. A large number of solutions, for half sine impacts of moment, shear, angular velocity, and transverse velocity, are obtained by means of the method of characteristics. Some of these solutions are compared with experimental results of Ripperger [102], to be described in more detail in the next section. Some solutions for a bar of finite length were obtained by Leonard and Budiansky [103]. In their work they compare modal solutions (i.e., superposition of vibration modes) with corresponding solutions obtained by the method of characteristics. Other investigations relating to this problem are presented in [104-1141. The solution of the Timoshenko equation by the method of characteristics is very similar to the solution of the Mindlin-Herrmann equations discussed in Section 11. The characteristics are solutions of the differential equations (3.22)

167


These equations have as solutions the families of straight lines, in the x - t plane, (3.23)

where C, and C, are constants. The equations (3.14) - (3.16) can be rewritten as differential equations along the characteristics as follows : along

d x - codt = 0

along

dx

dQ - pAcQdw = p A c Q o d t ,

along

d x - cQdt = 0

+ pAcQdw = p A c i o d t ,

along

dx

dM - PIC,&

dM

= COQdt,

+ p I c o d o = - coQdt,

+ codt = 0

(3.24) 2

dQ

+ CQdt = 0.

EXPERIMENT

Iz W

I 0 I

TIMOSHENKO THEORY

22

+0.4+0.2 -

=z

--

5

-0.4

oz

&-

0

-v.'c

A 2

v 4

-

6

10

8

I

1

12

14

16

CO +

(0)

D

x = 2 DIAMETERS, D.0.516 INCH. PULSE DURATION = 6 MICROSECONDS (HALF SINE WAVE FORM).

TIMOSHENKO THEORY

Iz W

EXPERIMENT

0 I

t0.2 I

5

5

-0.2 -0.4

CO'

a

D

El%-

(b) x = 2 DIAMETERS, D = 0.516 INCH. PULSE DURATION= 24 MICROSECONDS (HALF SINE WAVE FORM).

E

FIG.22. Moment vs. time-graphs for two different impact durations (From Plass and Steyer [19] and Ripperger [102]).

168

H. N . ABRAMSON, H. J . PLASS, AND E. A. RIPPERGER

It is seen from the above equations that waves involving M and w are associated with the propagation velocity c,, while waves involving Q and ZJ are associated with cQ. The two g;ioups are not isolated from one another, however, as can be seen when the right-hand sides of the above equations are examined. The changes in the M - w waves depend upon Q, and the changes in the Q - v waves depend upon w. Physically, this is to be expected, as moment and shear are closely connected to each other through the equations of motion. The solutions of the Timoshenko equation for a semi-infinite beam subject to Cnd-moment impacts in half sine wave form, given in the report of Plass and Steyer [19], have a small initial wave train, the front of which travels with the velocity co. The main part of the moment wave occurs, for a particular station, at a later time than that required for a wave traveling at velocity c, to reach that station (see Fig. 22). I t is not possible to assign a unique velocity to this greater part of the traveling wave, as its shape continually canges. This main part of the wave can be thought of as a group of smaller waves, some faster, some slower than the speed of the peak. As the wave travels along the rod, the membership of the group changes from a population corresponding to one velocity to that of another velocity. Experimental Results. Experimental investigations bf the propagational characteristics of short bending wave pulses have not been nearly so extensive as the investigations of the propagation of longitudinal waves. Most of the early investigations were concerned with the impact of a mass on the beam and the measurement of the force acting between the beam and the mass, or the deflection of the beam [115 - 1171. The results of such investigations as these do not elucidate the problem of pulse propagation and therefore will not be discussed here. Shear and Focke [32], in the investigation previously mentioned, carried out a few measurements of phase velocity vs. wave length which fall on the lowest branch of the family of dispersion curves shown in Fig. 23. These points are so closely grouped that they do not verify the accuracy of the theoretical dispersion curves, but they do suggest that there is some confirmation of theory by experiment, a t least for the lowest branch of the curve. The first account of an experimental study of bending wave pulse propagation appears to have been given by Dohrenwend, Drucker, and Moore, in 1944 [118]. The stated purpose of this paper was “to demonstrate that usable experimental techniques exist for the determination of transverse impact transients.” A series of oscillograph records show the manner in which a bending wave pulse changes its form as it propagates along a beam away from the impact point. The pulse was about 300 microseconds long and hence would be considered in the range of long pulses. Nevertheless, there is a very appreciable change in form of this pulse as it travels along the beam. The anti-symmetrical nature of these pulses was shown by

169


measurements made on opposite sides of the beam. Propagated moments were computed from measured strains and were compared with computed values based on the elementary beam theory and an assumed initial distribution of lateral velocity in the beam. The agreement is very good but the significance of the agreement is questionable since the initial velocity distribution was selected arbitrarily to give the best fit to the experimental data. 1.c 0.1 -

0.E

FLEXURAL WAVES.

-

X

-DIAMETER OF ROD = 4.615MM. " " = 5.895MM. THEORETICAL CURVE (FOR u = 0.25). ,I

0 -

0.i -

0.E C 0.e CO

0.4

0.; 0.2 0.I

C

I

I

I

I

I

1

I

I

I

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 -

1.0

h

FIG. 23. First branch of phase velocity curve for flexural elastic waves in a solid circular cylinder, with experimental results (From Kolsky [2]).

Even more questionable was the assumption of an initial velocity over a finite length of the beam. This assumption violates the basic concepts of wave propagation in that the effects of the impact are required to be transmitted instantaneously to remote parts of the beam. The results of this investigation are of a qualitative nature, but significant in that they show that these bending strains can be satisfactorily measured. Quantitative data concerning the propagation of bending wave pulses were presented by Dengler, Goland and Wickersham [84] in 1952. Bending wave pulses were generated in their experiments by the transverse impact

170

H. N. ABRAMSON, H. J. PLASS, A N D E. A. R I P P E R G E R

of steel balls on steel beams. The durations of the impacts ranged from 11.5 to 37 microseconds. Peak forces developed by the impact ranged from 8 to 64 pounds. The maximum strains developed were of the order of 0.5 microinches per inch. In order to measure these small strains the authors found it necessary to use piezoelectric barium titanate strain gages similar to those previously discussed. Measured values of bending strain were compared to computed values, the computation based on the Timoshenko equation. The agreement is in general very good as may be seen in Fig. 24. .I

f r ao EZ

1

0

-

‘i.-.I

..2

. EXPERIMENT

..3

-COMPUTATION

- .4 -3 0

TIME

- MICROSECONDS

FIG.24. Measured and calculated strain vs. time-record a t 4 diameters away from station impacted transversely. Solid line is measured curve, broken line is Calculated curve from Timoshenko equation (From Dengler, Goland, and Wickersham [84]).

The Timoshenko theory indicates that a sharp pulse would have its front propagated at the bar velocity and the experimental results presented appear to indicate that this is true. However, the more exact theory (PochhammerChree) predicts a first arrival traveling at the dilatational velocity, followed by waves traveling at all velocities between the dilatational and about 30 percent of the bar velocity. The first arrivals in the experimental work seem to have been propagated at a velocity equal to or slightly greater than the bar velocity, but the amplitudes of the arrivals are too small to allow accurate velocity measurements. It must be pointed out here that the gages were not arranged to eliminate the symmetrical strain components, generated along with anti-symmetrical components by the impact. Hence, the form of the pulses observed may have been altered by the presence of symmetrical components. The authors have shown, however, by means of a set measured outer fiber strains on the top and bottom surfaces of the beam, that the strains are essentially anti-symmetrical. These curves are almost mirror images of each other except at the later times (see Fig. 25). At the later times a high frequency component appears which the Timoshenko theory does not predict and which is not completely anti-symmetrical. This component either represents transmission in one or more of the higher branches which are not provided by the Timoshenko equation, or it arises


171

from the symmetrical strains previously mentioned. In any event, these results indicate that the Timoshenko equation predicts transmission characteristics adequately enough for engineering purposes.

1

1

l

1

1

1

1 "1

1

1

1

1

Measurements of short bending wave pulses were reported by Ripperger [102].' These pulses were generated by the eccentric impact of small steel balls on the end of the beam. The strains were measured by means of wire resistance gages (i"gage length) connected to cancel out the symmetrical strain components. Pulse durations ranged from 3.5 to 30 microseconds, and measurements were made at 4, 6, 10, 16, 22, and 28 diameters from the impact end on a 3" diameter bar. These measurements, like those of previous investigators, showed that the change of pulse form during the propagation along the beam is essentially independent of the pulse duration. This continual change, which is shown in Fig. 26, makes it impossible to determine a definite velocity of propagation. The time of arrival of the initial disturbance was too indefinite to be measured; however, if the first positive peak in the pulse is used as a reference point, propagation appears to be a t about 10,000 ft/sec. Comparisons of Ripperger's data with computed results based on the Timoshenko theory were made by Plass and Steyer [19]. The agreement was good for all but the shortest pulses. The elementary theory did not predict the propagational characteristics satisfactorily for any of the pulses (see Fig. 22). In this investigation [lo21 a series of measurements was also made of pulses propagating in beams that were pinned at the impact end. I t was expected that this end condition would alter the pulse forms that appear as a consequence of the effect of the support on the shear in the beam.

172

H. N . ABRAMSON, 13. J . PLASS, A N D E . A. RII’PEKGER

The recorded pulses for the beam with the pinned end appear, however, to be practically identical with those for the free end beam. This may be an (a) Applied Mornen t

Transmitted Moments

(b)x= 2d

(C)X =

6d

( d ) x= 10d

(C)X

=

16d

(f)x = 22d

( g )= ~ 28d

t

FIG. 26. Oscillograph records of bending strains in a simply supported cylindrical steel bar (From Ripperger [102]).


173

indication that the end of the beam was not restrained sufficiently to prevent lateral movement, or the shear loading introduced by the support is negligible under the conditions of the experiment.

-

time

10-6 secs

FIG.27. Oscillograph records of pulse transmission of a bending wave in a cylindrical steel bar (From Ripperger and Abramson [119]).

These experiments were later extended by Ripperger and Abramson [119] to pulses having a duration of about 2 microseconds applied to a 4 inch diameter rod. These pulses were generated by the eccentric impact of a 1 le inch steel ball with an impact velocity of 460 ft/sec. Parts of the disturbance were detected traveling at (a) the dilatational velocity, (b) the bar velocity and (c) the shear velocity. These results are shown in Fig. 27. The dilatational disturbance disappears by the time the pulse has traveled 8 diameters. The conclusions drawn from these measurements, when compared with the Pochhammer-Chree theory [89], were that the Timoshenko

'0-20 -

--TIME AFTER PULSE ARRIVAL

M

---

I

x.0

x.+

I 1.

I

It

I

I X.

l

3

l

I x.4f

I

1

I X I

l 6

t

.

I

b (zfz

f*

I

22' CLAMPED DRILL ROD $.3O'SIMPCY SUPPORTED MILD COLD ROLLED STEEL I

x=ri

x = DISTANCE ALONG B E A M ,

I x=g

.

"

I

,

x.12

INCHES

FIG. 28. Progress of major portion of bending pulse in a circular cyIindrical rod (From Cunningham and Goldsmith [IZO]).

i


175

theory predicts arrival times quite accurately with the exception of the first arrival traveling at the dilatational velocity. This is not serious since the amplitude of this portion of the disturbance is very small. A detailed investigation of the manner in which the form of a bending wave pulse changes as it propagates along the bar was reported by Cunningham and Goldsmith 11201. In this study an effective gage spacing of inch was obtained by making a series of measurements in which the

A

point of impact was shifted inch farther from the gage point for each impact. The results of these measurements are shown in Fig. 28. Note how the pulse, which is initially positive, gradually changes, until a t a point 2 inches from the impact end it is almost completely negative. These results confirm the observations of other investigators qualitatively but add more details. From measurements of the time of arrival of the maximum strain at successive gage stations Goldsmith and Cunningham [121] concluded that the average velocity of propagation of the predominant group is 13,500 ft/sec. Measurements of the earliest arrivals in the pulse indicated that this part of the pulse had been traveling a t the bar velocity. Since the technique used did not eliminate the symmetrical strain component, these early arrivals could well have been influenced by them. Vigness [122] studied the bending waves which were generated in a beam by suddenly putting one end of the beam in motion. The results which he obtained largely confirm what other investigators found.

Concluding Remark. Recent experimental and theoretical investigations [84,119] haveservedtoestablish the adequacy of the Timoshenkobending mechanism for describing the response of beams to sharp impacts. Other investigations, e.g. [98 - 1011, have provided solutions of the Timoshenko equation for a wide variety of impact loads and boundary conditions. In view of these two advances it would appear, as in the case of longitudinal vibrations, that the theory has attained sufficient strength so that attention may be directed in the near future to studies of more complicated beams, such as those containing cross-sectional or material discontinuities 1491. 3. Flexural Plastic Waves

Introduction. Interest in the problem of plastic deformation due to impulsive loading is of comparatively recent origin. Investigations of the longitudinal plastic deformation in a rod under end loading were undertaken in the early 1940’s by Taylor, KBrman, and others as discussed earlier. The first attempt to deal with transverse plastic deformations seems to be that of Duwez, Clark and Bohnenblust [123], reported in 1950. In the ensuing six years this area of investigation has been very active; more than a score of papers have appeared in the unclassified literature.

176

H . N. ABRAMSON, H. J. PLASS, AND E. A. R I P P E R G E R

Efforts in the transverse deformation problem have been directed, with but one exception, towards analyses based on simple beam theory with the moment-curvature relationship being considered generally as elastic-plastic

k ( a ) MOMENT - CURVATURE RELATION:

ELASTIC- PLASTIC.

k (b) MOMENT-CURVATURE RELATION:

RIGID -PLASTIC.

lV'E~~E~~~~l M

N

PERF ECT LY PLAST IC

k ( c ) PERFECTLY PLASTIC AND LINEAR STRAIN HARDENING CHARACTERISTICS. FIG.29. Moment vs. curvature-curves.

(Fig. 29a) or rigid-plastic (Fig. 29b), and the plasticity being described generally as either perfectly plastic, as in Fig. 29a and b, or as linear strain hardening (Fig. 29c). The single exception is a paper that considers shear deformation and rotatory inertia effects and also the effect of strain-rate; it will be discussed in detail later.


177

I t is indeed unfortunate that the majority of the papers on this subject are simply solutions of special problems or, at best, of special classes of problems; this of course is evidence for the difficulty of the general problem. Because of the difficulty involved in obtaining solutions on the basis of elastic-plastic analyses, it is not yet possible to compare directly the relative usefulness and accuracy of the elastic-plastic and rigid-plastic theories. Further, the importance of the higher order bending effects and strain-rate is not known, and experimental evidence is meager. Practically all of the work to date has emphasized the permanent deformation resulting from the impulsive loading rather than the propagation of the deformation along the beam. The concept of plastic hinge formation has been employed, and while some discussion has been made of the case of a moving hinge, the question of wave propagation alone has not yet been studied to any great extent. Nevertheless, the analyses available at present should be discussed briefly here, not so much for their direct relation to the wave propagation problem, but rather to complete the general picture of the impulsive loading problem, in order to provide an introduction to the transverse plastic-wave propagation analysis based on a more exact theory which follows later. Elastic-Plastic Analyses. As stated previously, the first investigation concerned with the plastic deformation of beams under transverse dynamic loading seems to be that of Duwez, Clark and Bohnenblust [123]. The problem considered is that of an infinite beam subjected to a dynamic transverse constant-velocity loading. The early solution of Boussinesq [124] for elastic beams is extended to the plastic problem by replacing the elastic bending moment-curvature relation by a more general expression. An elastic-plastic analysis is developed which considers both perfect plasticity and linear strain hardening. The analysis is valid only for the time during which the impulse is acting. We repeat here the essentials of the theory, which is due to Bohnenblust, as it is basic to analyses presented in other papers. The equations of motion, neglecting rotatory inertia, are (3.25)

Q=

aM ax

_ - I

(3.26)

The curvature relation is (3.27)

M

= Elk.

178

H. N . ABRAMSON, H. J. PLASS, AND E. .4. KIPPERGER

Using Eq. (3.27), Boussinesq found for the elastic beam that ujt is proportional to x2/t; this is true also in the case of the plastic beam. Therefore, a new function r j is defined by (3.28)

A solution of the form u =tfh)

(3.29)

is assumed, so that

(3.31) Also, (3.32)

Now, a quantity S is introduced by (3.33)

Eq. (3.26) thus becomes (3.34)

S'

+ l/17(2aak - f ' )

= 0,

and by differentiating and using (3.31) and (3.33), (3.35)

S"

dk + E I -S dM

= 0,

is obtained, which is the fundamental differential equation of the theory. From the previous relations, the following expressions may be obtained : m

(3.36)

(3.37)


179

(3.38)

(3.39)

This relation makes it possible to show that the constant impact velocity is m

(3.40)

The corresponding impact force P and energy W are given by (3.41)

(3.42)

EI P=-S(0) a3

w = 2EI a3 --

1

-,

V t

VJ(0)

v.

Since all of these quantities depend upon S ( q ) it is clear that a solution of (3.35) is a solution of the problem. Further, it is shown in the paper [123] that these relations satisfy all boundary conditions of the problem so that the assumed form of the solution u = t f(q) is valid. The essential point is the solution of Eq. (3.35), which is complicated because of the nonlinear function d k / d M . A procedure is given in [I231 for a numerical solution of this equation. The authors' conclusion that strain is not propagated along the beam at a constant velocity is consistent with our earlier observation that no definite velocity can be associated with the propagation of an elastic bending wave pulse. I t should be noted, however, that the analysis presented neglects the effects of shear and rotatory inertia. Neglecting these factors in the elastic case led to a physically unrealistic result, and the final equation did not predict propagational characteristics satisfactorily under any circumstances. Under these conditions it should not be expected that the simplified theory for filastic propagation would lead to a satisfactory representation. In an experimental investigation designed to test the theoretical results, cold rolled steel and copper beams 10 ft. long with rectangular cross sections were used. These beams were pinned at the ends and struck at the midpoint with a heavy hammer moving at a measured velocity. The deflection curve at the end of the impact was photographed for comparison with calculated deflection curves.

180

H. N. ABRAMSON, H. J . PLASS, AND E. A. R I P P E R G E R

The relationship between bending moment and curvature required for the solution of (3.35)was obtained by calculation based on experimentally determined stress-strain curves. The calculated curve was then approximated by two straight lines for convenience in solving the equation. The calculated and approximate curves for the two materials are shown in Fig. 30. A comparison of measured and calculated deflections is shown in Fig. 31. For steel the calculated curve based on the elastic theory gives the best

M

?-k COLD-ROLLED

M

STEEL

k ANNEALED COPPER

FIG.30. Actual and approximate moment-curvature curves.

i

EXPERIMENT.

----THEORY. --THEORY (ELASTICI.

I

COLD-ROLLED STEEL

1

ANNEALED COPPER

FIG.31. Theoretical and experimental deflection curves for a long bar subjected t o a transverse impact sufficiently large to produce plastic strains (From Duwez, Clark and Bohnenblust [123]).


181

fit but for copper the plastic theory gives the best fit. The authors interpret this to mean that the influence of plasticity is much greater for copper than for steel, and point out that in the copper beams plastic strains are found some distance from the point of impact, whereas in the steel the plastic deformation was localized near the impact. The calculations show that the distance from the point of impact to the point where u = 0 varies as a relation which is independent of the shape of the moment-curvature curve. Experimental results indicate that the variation of xo with V t i s linear but the slope of the line representing this variation is much greater than had been calculated. By taking a different approximation for the moment-curvature relationship a better agreement could have been obtained as the authors point out, but this would have made the fit for the deflections much worse. It seems doubtful that much significance can be attached to any agreement, or lack of agreement between the computed deflection curves and the experimental results in view of (a) the neglect of shear and rotatory inertia in setting up the basic equations, (b) the strong possibility that the measured deflection curves were affected by reflections, not accounted for in the theory, from the pinned end, and (c) the approximate nature of the momentcurvature relationship used in the computations. A somewhat different type of elastic-plastic analysis has been developed by Bleich and Salvadori [125]. The analysis is applied to free-free beams moving under dynamic forces and possessing perfect plasticity (Fig. 29a). The motion is expressed in terms of the normal modes of the beam, when the beam is subjected to a given symmetric initial velocity distribution. The final deformation is determined on the basis of the formation of a single plastic hinge at the impact point. By use of the proper continuity relations between displacements and velocities the solutions are established, and the final deformation, in terms of the plastic angle a t the impact point, may be determined for a particular problem. Conroy [126], by a method similar to that of [123], has treated the problem of a semi-infinite beam, subject to a constant bending moment and a transverse force of magnitude inversely proportional to vtapplied a t the free end. Under this type of loading the free end of the beam moves a t a constant velocity, which accounts for the similarity of solutions. A solution for the problem of an elastic-plastic beam (Fig. 29a) subjected to a velocity impulse at the middle cross section, when the acceleration time is small but finite, has been given in [127]. The permanent angle of rotation at the struck cross section is computed on the basis of a single plastic hinge. The permanent deflection predicted by this theory exceeds that observed in recent experiments [128] by an appreciable amount. In [129] a simply supported uniform beam of ductile material, subjected t o impulsive loading such that the initial velocity is a half-sine wave, is

vc

182

H . N . ABRAMSON, H. J . PLASS, AND E. A. RIPPERGER

considered. The elastic-plastic analysis is based on that of Bleich and Salvadori [125] and retains the assumption of a single plastic hinge. I t is pointed out that the principal difficulty of the elastic-plastic problem is that of satisfying the moment-plasticity condition throughout the beam. The authors of [129] suggest that one way of obtaining an adequate approximate solution to elastic-plastic problems might be to insert, at some stage of the deformation, a central plastic zone whose length is chosen arbitrarily. Thomson [130] has employed a normal mode superposition method to several elastic-plastic beam problems, the solutions being obtained on an analog computer. The plastic action is again simulated by plastic hinges.

1'

I

I

FIG. 32. Schematic description of rigid plastic bending.

Rigid-Plastic Analyses. Following a suggestion by Prager, an essentially different type of analysis was developed by Lee and Symonds [131]. If the final strains in a beam are large compared with the elastic strains, it may be sufficient to consider the behavior characterized by a rigid-plastic beam (Fig. 29b). The basic assumptions of such an analysis are: (a) negligible elastic strains, (b) segments of the beam undergoing rigid-body motion are joined by plastic hinges where the entire relative motion is assumed to take place. This analysis, in contrast to the elastic-plastic analysis of Bohnenblust [123], includes also the motion after impact. The motion is described in three phases. In phase I, M < M,, so that there is only rigd-body translation. In phase 11, M = M , at the central impact point so that the two halves of the beam rotate with respect to each other as though the beam were hinged in the middle. In phase 111, the bending moment at cross sections a certain distance from the mid-point


183

attains the value M , with sense opposite to M , at the midpoint. Thus there are three plastic hinges to be considered; one remains at the struck point while the other two move outward. A pictorial representation of the three phases of the motion is shown in Fig. 32. The analysis consists essentially of writing the dynamical equations for each stage of the motion, as described above, writing the kinematic conditions at the hinge, and finally writing the equations of motion of beam segments on either side of the hinge. Continuity conditions at a traveling hinge are also discussed. The theory is applied to a free-free beam of finite length under the action of a triangular impulse. It may be noted that the solution is to be obtained by successive approximations, although the analysis presented does not go beyond the first such approximation. A major objective of the paper is an attempt to formulate a criterion for the use of the rigid-plastic analysis for finite beams. The criterion given is (3.43)

where t is the fundamental period of elastic vibration of the free-free beam, 8, is the final central angle of deformation, T is the duration of the impact, and m is the beam mass per unit length. For a given beam and a given value of maximum force P,, this relation sets a lower limit for T , above which the present analysis can be expected to give satisfactory results. Pian [132] employed the same basic analysis to determine the plastic strains in a simply supported beam subjected to a concentrated impulsive load at the center of the beam. For a rectangular impulse, the solution is obtained in closed form. The problem treated by Bohnenblust [123] has also been studied by a rigid-plastic analysis [133]. This leads to a discussion of the two outward traveling hinges in the case of an infinitely long beam. Both perfectly plastic and linear strain-hardening materials are studied, although no analytical results are obtained for the latter. Symonds [134] also considered the rigid-plastic analysis [131] when a force function of square wave type is applied. An empirical formula is offered by which the plastic deformation can be estimated from knowledge of the impulse shape and the peak value. This same analysis was further extended [ 1351 to provide numerical results for concentrated force pulses of rectangular, half-sine, and triangular shape. The calculations show that the central angle of permanent deformation for all three cases can be obtained from the empirical relation

8, = C J 2 P?, (3.44) where J is the impulse and C depends on the dimensions and proportions of the beam and a numerical factor (slightly different for each pulse shape).

184

H. N. ABRAMSON, H. J. PLASS, AND E . A. RIPPERGER

This result is valid as long as M does not reach M , at any section other than at the center or the two traveling hinges. A true impact problem was studied by Symonds and Leth [136] who treated a finite beam whose mid-section acquires a given velocity in zero time. The basic method of Lee and Symonds [131] was again employed. The case of a distributed load studied by a rigid-plastic analysis was presented by Seiler and Symonds [137], the pulse being rectangular in shape. The same problem was treated by Salvadori and Di Maggio [138], for a smoother but more complicated load distribution function ; the normalmode method [125] was employed. Several cases of simply supported and built-in beams, loaded by uniform pressure or by a concentrated central force that is a specified function of time, were solved by Symonds [139]. A problem involving an impact velocity variable in time, with consideration of the motion following unloading, was presented by Hopkins [140]. An interesting study of the case of linear strain hardening by the use of the rigid-plastic theory was made by Conroy [141]. Considering finite beams, the author noted that if the rate of change of curvature caused by the loading is of the same sign (or zero) along the entire beam, the differential equation has the same form as in the elastic case. With this idea in mind, three problems are discussed : (a) initial-motion problem using superposition, (b) initial-stress problem for a simply supported beam using superposition, and (c) a free-boundary problem for a uniform beam. In the first two problems the beam was initially plastic and remained entirely plastic for all time so that no rigid-plastic boundaries were present. The third problem typifies a case in which the beam is initially rigid; the application of a load produces a moving rigid-plastic boundary. The determination of the free boundary in the third problem was made by the inverse method. Cotter and Symonds [142] treated an earlier problem [125] by considering various zones of plastic distortion along the beam. I t was found that these precede the ultimate rigid-body motion a t constant velocity. A rigidplastic analysis, considering only a single plastic hinge, was also made for this problem by the original authors [125]. The method of Bleich and Salvadori was also applied to a simply supported rigid-plastic beam by Symonds [143] ; the specific problem was one treated previously [lag]. The solution of Symonds and Leth [136] was extended in [144] to include finite acceleration time. Mentel [145] applied the analysis of Lee and Symonds [131] to simply supported and built-in beams with an attached mass and subjected to a uniform pressure pulse of rectangular shape. Rigid-plastic analyses presented in connection with elastic-plastic analyses are contained in several papers [125 - 1271.


185

The foregoing represents a rather formidable array of special solutions ; they have all been summarized in Table 2 for the convenience of the reader. TABLE2. type of analysis Ref.

SUMMARY O F SOLUTIONS

plasticity

beam

loading elasticplastic

rigid- perfect strain plastic hardening

123 infinite 125 finite 126 semi-infinite

x

X

x

127 finite

X

X

X

129 130 131 132 133 134 135 136 137 138

finite finite finite finite infinite finite finite finite finite finite

X

X

X

139 140 141 142 143 144 145

finite finite infinite finite finite finite finite plus added mass

X

X

X

X

X

X

X

X X

X

X

X

X

X

X

x

X

X

X

X

X

X

X

X

X

X X

constant velocity velocity impact constant moment plus force velocity impulse with finite acceleration time half-sine wave velocity concentrated force rectangular pulse square pulse constant velocity square pulse distributed load variously shaped pulses constant velocity distributed load, rectangular pulse variously shaped pulses various time-dependent velocity velocity distribution pressure pulse half-sine wave velocity finite acceleration time

Comfiarisons Between Elastic-Plastic and Rigid-Plastic Analyses. It is to be emphasized at the outset that both the elastic-plastic and the rigidplastic theories have their usefulness. For those cases in which the final strains are not large compared with the elastic strains, the elastic-plastic theory must be employed. Several attempts to establish criteria for the applicability of the rigid-plastic theory have already been mentioned; here we will mention only those few cases in which a direct comparison of the two theories was attempted. A direct comparison was made in [125] but no definitive statement could be made regarding the comparison. Again, no conclusive statement was formulated in the comparison of Conroy [126].

186

H. N. ABRAMSON, H. J . PLASS, AND E. A. R I P P E R G E R

Alverson [127] compared his elastic-plastic solution with the rigidplastic solutions of [136] and [144], but again the comparison was not conclusive. A direct comparison was also attempted by Seiler, Cotter and Symonds [129], but it was found again that the solutions have regions of validity which do not overlap so that comparison is difficult. Strain-Rate Theory. The theories of plastic bending deformation under impulsive loads are all inadequate, when the effects of suddenly applied transverse or bending loads are to be investigated, because of their dependence on elementary beam theory. I t was pointed out previously that the elementary theory predicts an infinite propagational velocity while physical observations reveal finite propagational velocities. A theory which considers these significant influences of rotatory inertia and finite shear rigidity on plastic wave propagation has been given by Plass [25]. In addition to the above mentioned effects, the theory also includes the dependence of the material properties on the rate of strain. Although this last inclusion might at first appear to complicate the theory, it really has the opposite effect. Calculations of stresses and velocities are made easier because of the fact that straight-line characteristics independent of the state of strain are predicted, with consequent simplification in the numerical work. The assumptions about the strain-rate effect are generalizations of the linearized form used by Malvern 1241 for longitudinal waves. It is shown by Plass that the equations connecting moment and shear with the strains and velocities are as follows (see Fig. 21) :

EJ

aK aM at = __ + k ( M - M ) , at material behavior)

(3.45) /@s

ay

-=

at

aQ %

+ k(Q

-

Q);

}

(3.46)

at

(equation of motion)

ax

(equation of continuity),

(3.47)

at

ax

187


where

E,I ,uoA, k _M,Q

= static flexural rigidity = static shear rigidity = strain-rate constant = static

values of moment and shear for a state of strain corresponding to the one existing dynamically; the yield condition of von Mises is assumed,

and the other symbols are as previously defined. Equations (3.45)apply to loading; for unloading the terms containiiig k are set equal to zero. The above equations can be solved by means of the method of characteristics in very much the same way as the Timoshenko equation was solved. The characteristics are integrals of the following differential equations: d x = 0, dx =

(3.48)

[twice) codt,

dx = f CQdt.

I t is easy to see that the characteristics are all families of straight lines in the x - t plane. Corresponding to each family of characteristics is an ordinary differential equation as follows: EoIdK - dM

= k(M - M ) d t ,

along

dx=O

p o A & - dQ

= k(Q - Q ) d t ,

along

dx=O

along

d x = codt

coQ - k ( M - @))it,

along

d x = - codt

dQ - pAcQdv = [ p A c t w - k(Q - Q)]dt,

along

d x = cQdt

along

d x = - cQdt

dM - plcodw = [coQ - k ( M - M ) ] d t , (3.49)

dM

dQ

+ plcodw = [-

+ pAcQdv = [ p A c i w - k(Q -a]&.

Again, the above equations are used for loading; for unloading, the terms which are multiplied by k are set equal to zero. Details of the boundary conditions and a method of solution for the problem of a step moment and step angular-velocity impact on the end (assumed supported by a pin to a rigid base) of a semi-infinite beam are found in the paper by Plass [25]. It is of interest to point out here that the zones of plastic flow, i.e., permanent deformation, occur only near the impact end and in a thin zone near the bending wave front. The remainder of the beam behaves elastically.

188

H . N. ABRAMSON, H . J. PLASS, AND E. A. RIPPERGER

In the case of the applied step moment, essentially no plastic strains occur, as unloading occurs almost immediately after the passage of the wave front. However, in the case of the step angular velocity, a zone of plastic strain forms near the impact end and lengthens with time, but a t a slower rate than the velocity of propagation c,, of the bending wave front. The theory just described would need to be applied only when very sudden loads are involved. When the time of rise of the load or moment, from zero to maximum value, is of the order of the time required for a bending wave (velocity c,,) to travel from the impact end to the far end and back again, an elementary theory similar to that of Duwez, Clark, and Bohnenblust [123] should be adequate to describe the behavior of the beam. Concluding Remark. The inadequacy of experimental data makes it very difficult to judge the present state of development of plastic bending wave theory. Even the theoretical analyses of the final permanent deformation in a beam resulting from an impulsive load, discussed a t some length here, have not been subject to critical appraisal in the light of experimental results. Progress in the development of an adequate theory of wave propagation has been made [25], but this has come about almost exclusively from knowledge of plastic longitudinal and elastic flexural wave propagational characteristics. Further progress must rest on the availability of reliable and adequate experimental data.

ACKNOWLEDGMENT

Portions of the work reported in this paper were supported by the Bureau of Ordnance, Department of the Navy, under a contract with the University of Texas, Defense Research Laboratory. The authors’ institutions generously provided funds and services to assist in the preparation of this paper.

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Problems in Hydromagnetics BY EDWARD A . FRIEMAN

AND

RUSSELL M . KULSRUD

Princeton University. Princeton. New Jersey Page I . Introduction . . . . . . . . . . . . . . . . . . . . I1. Fundamental Equations . . . . . . . . . . . . . . . 1. The Equations and Their Validity . . . . . . . . 2 . The Boundary Conditions . . . . . . . . . . . . I11. General Processes . . . . . . . . . . . . . . . . . . 1 . Motion of Magnetic Lines of Force . . . . . . . . 2 . Validity of Approximation of Infinite Conductivity . 3 . Conservation of Energy . . . . . . . . . . . . . 4 . Three Limits . . . . . . . . . . . . . . . . . . 5 . Hydromagnetic Equilibria . . . . . . . . . . . . IV. Stability of Hydromagnetic Equilibria . . . . . . . . 1. Normal Mode and Energy Methods . . . . . . . . 2 . The 6W Formalism . . . . . . . . . . . . . . . 3 . Applications of the SW Formalism . . . . . . . . 4 . The Pinch Effect . . . . . . . . . . . . . . . . 5 . Minimization Technique for Expansion Problems . 6. An Axisymmetric Problem; Coordinate System . . . 7. An Axisymmetric Problem; Stability . . . . . . . 8. Physical Interpretation . . . . . . . . . . . . . . V. Hydromagnetic Waves . . . . . . . . . . . . . . . . 1. Introductory Remarks . . . . . . . . . . . . . . 2 . Three Modes . . . . . . . . . . . . . . . . . . 3 . Equation of Propagation . . . . . . . . . . . . . 4 . Energy and Wave Generation . . . . . . . . . . 5 . Group Velocity . . . . . . . . . . . . . . . . . 6. Reflection and Refraction ; Boundary Conditions . . 7. Angles of Reflection and Refraction . . . . . . . . 8. Transmission and Reflection Coefficients . . . . . . 9 . Damping . . . . . . . . . . . . . . . . . . . . 10. More General Waves . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . .

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. . . . . . . . . . . . . . . . . . . . .

. . . 195 . . . 196 . . . 196 . . . 197 . . . 198 . . . 198 . . . 200 . . . 201 . . . 202 . . . 202 . . . 204 . . . 204 . . . 205 . . . 207 . . . 207 . . . 209 . . . 210 . . . 212 . . . 214 . . . 216 . . . 216

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216

. 217 . 220 . 221 . 222

. 223 . 225 . 227 . 229 . 231

I . INTRODUCTION The subject of hydromagnetics is concerned with the interaction between magnetic fields and the motions of a highly conducting fluid . This field of physics is governed by classical equations of motion. and its phenomena 195

196

EDWARD

A.

FRIEMAN AND RUSSELL M. KULSRUD

could have been investigated by the 19th century physicists following Maxwell since all the necessary concepts were known at that time. However, it is only in the past 20 years that hydromagnetics has been actively considered. A possible explanation for the late development of this field is that one impetus for its study has come from modern astrophysics where the importance of magnetic fields in astrophysical phenomena has only recently been appreciated. Hydromagnetics has also become important in applications to such varied phenomena as controlled thermonuclear reactors, the theory of the aurora, terrestrial magnetism, and flow of liquid metals. There are five excellent review articles in this field; Elsasser [l, 21, Lundquist [3], Chandrasekhar [a], and a book by Spitzer [5]. In consideration of these review articles, no attempt at completeness has been made in this paper. We have chosen rather to treat certain branches of the subject with some detail indicating the approach to the solution of problems. Section IV on the Stability of Hydromagnetic Equilibria and Section V on Hydromagnetic Waves contains much previously unpublished material and material which has not previously been reviewed.

11. FUNDAMENTAL EQUATIONS 1. The Equations and Their V a l i d i t y We consider a highly conducting fluid in which magnetic fields may be present. The equations governing such a fluid can be taken to be

C7

dv dt

x

B

p -= -

4n

=-j, C

x B op + jC

dp =--pV.V, dt E + - v- -x, B - j C

CT

197

PROBLEMS IN HYDROMAGNETICS

We use Gaussian units; E is the electric field, B the magnetic field, j the current density, p the mass density, v the fluid velocity, the pressure, c the velocity light, (T the electrical conductivity, and y the ratio of specific heats. We use the convention that d/dt is the total time derivative following the motion (“material derivative”). The first three equations are the usual Maxwell equations with displacement currents neglected. This approximation is valid if the fluid velocity is small compared to c. The term in the momentum balance equation (2.4) arising from the action of a field E on the charge density is also relativistically small and is neglected. Therefore, the Maxwell equation E = 4 n ~where E is the charge density, serves only to determine E . We have further neglected the viscosity terms in (2.4) and have assumed that the pressure # is always a scalar. Equation (2.5) is the usual equation of continuity. Equation (2.6) is Ohm’s Law for a moving fluid. The term v x Blc arises fiom the fact that the current is produced by the electric field seen by the moving matter. Equation (2.7) states that the change of state is isentropic and assumes a perfect (compressible) gas in which heat conduction is negligible. For an incompressible fluid no such equation is needed to complete the system. For a highly ionized gas these equations apply in many situations. See, for instance, the book by Spitzer [ 5 ] .

+

v*

2. The Boundary Conditions

The system is described by fifteen scalar equations in fifteen unknowns. In a vacuum region adjacent to a hydromagnetic fluid (2.1) - (2.3) apply with j = 0. At the interface the two sets of equations are connected by the following set of boundary conditions. Let n be a unit normal on the interface pointing into the vacuum and let {X}= X,, - X,,,. Then

(2.9) (2.10) (2.11) (2.12)

I-“,“

{ n . { B } =o,

n x B --K,

{#

+} ;

= 0,

n . { v }=o.

The above boundary conditions are obtained by integrating the respective equations (2.1) - (2.5) across the interface. Relations (2.6) and (2.7) yield

198

EDWARD A. FRIEMAN AND RUSSELL M. KULSRUD

no boundary conditions. K is the surface current density. We have assumed no surface distribution of mass. Therefore, if B has a component normal to the surface, K must be chosen to be zero since otherwise infinite accelerations would arise. Thus, in this case, by (2.9)and (2.10)B must be continuous and no refraction of the lines of force is possible. If n * B is zero, on the other hand, a jump in the tangential component of B is allowed. I n this case K need not be zero, and (2.9) merely serves to determine it. Equation (2.12) is not significant at a fluid-vacuum interface since v is undefined in the vacuum. It should be noted that the above set of boundary conditions apply equally well to any interface in the fluid. 111. GENERALPROCESSES 1. Motion of Magnetic Lines of Force

Almost by definition, the subject of hydromagnetics, is concerned with these equations in the limit of large or infinite electrical conductivity. This is because the interaction between the fields and the fluid mass motion only becomes strong in this limit. I n the opposite limit, of small conductivity, the motion of the system is treated by ordinary hydrodynamics, while the behavior of the fields is treated by electromagnetic theory. If the conductivity can be taken to be infinite, the matter can be regarded as strictly tied to the lines of force of the magnetic field. Thus the motion of the fluid completely determines the behavior of the field. To see this we combine (2.1) and (2.6) to obtain

Using (2.2) we obtain

-1-a-B_-_ 1 c

at

0

c

1

4n

B x B x B- -V C

x

(V

x B),

and by (2.3) we finally get (3.3)

aB =

at

vx

(V

x B) +-

C2

4na

V2B.

Assume that (T is so large that the second term on the right hand side of (3.3) may be neglected. Let us consider two fluid particles separated by a distance A f on the same line of force. Thus (3.4)

df x B = 0 .

199


We wish to show that a t any later time the line of force which passes through one particle will pass through the other. That is, that (3.5)

d -AZ at

xB

= 0. \

+d

Note from Figure 1 that

\

From (3.3)

(3.7)

& at

U

(v+Avldt,

aB at

- B = -+v.VB

FIG. 1. Diagram illustrating the change of distance A1 on a line of force.

=B.VV-BV-W. Thus

(&Al)

x B + A Z x dB dt

= ( A Z * V V )x B + A l x [ ( B * V ) U - B V * W ] . From (3.4) we see that (3.8) vanishes, which proves that adjacent particles stick to the same line of force and, therefore, that any two particles originally on the same line of force will remain on that line. This result enables us to identify lines of force at different instants, giving the lines of force a type of physical reality which they do not in geqeral possess, since there is usually no meaningful way to label them from instant to instant. The elementary definition of lines of force demands that the number of lines crossing a unit area perpendicular to them, i.e. the flux, is equal to the magnitude of the field. A sensible labeling of the lines should preserve this definition in the course of time. Such a labeling is called flux-preserving. The labeling given by the particles on the lines is of this type. This subject has been elaborated by Newcomb [6], and he gives the conditions that a general electromagnetic field must satisfy to have a flux-preserving labeling. To show that our labeling is flux preserving, consider a small area A S which follows the motion of the fluid. If A@ is the flux through this area we must show that (3.9)

d -A@ at

d

= -B

at

*

AS

= 0.

200


To calculate d A S / d t consider the volume dt conservation 1 d 1 dP O=--pddt=-----dt+-Az. P dt P dt

(3.10)

a

-AT=

(3.11)

= A S . AZ.

Then from mass

d

at

V.YAz=dZ.(VY).dS+dZ.-~S, d

at

at

where we have used (3.6) for dAZ/dt. Thus from (3.3) and (3.11) (3.12)

a

-B * A S = [B * VY - BV

at

*

V]

+

* A S B * d SV

*

Y

- B * ( VV)* d S = 0,

where we have used the fact that B and AZ are parallel. 2. Validity of Approximation of Infinite Conductivity

We now ask when the conductivity is large enough for the above considerations to apply. Or alternatively, when can the third term in (3.3) be neglected ? Define T to be a characteristic time and L to be a characteristic length for the system. We can then write (3.3) in order of magnitude as (3.13)

where (3.14)

The criterion for large conductivity is that T ) =sinP"Ay.

Hence, for the reflected wave (5.49)

and for the refracted wave (5.50)

By use of (5.13),Z, simplifies to

(5.51)

z+=

- u2 cos2 8 - c:

sin 0 cos 8

Near a critical angle for total reflection c'+' + u2, 0" + 0, and 2; 00, while Z, remains finite. Hence A; -+ 0 and A: A+. For normal incidence ,8 --*go", 8 - 4 2 , both 2 ' s become infinite and (5.49) and (5.50) become indeterminate expressions. Their values are obtained by passing to the limit. Let 8 = 4 2 - p where p is small. Then -+

---+

(5.52)

while (5.40) gives (5.53)


227

so that

(5.54)

(5.56)

The expressions for reflection and refraction when a slow mode is involved in either medium are obtained by replacing Z, by Z- in that medium. Z- is obtained by replacing c- by c+ in 2,. p is replaced by (90"- p). 9. DamPing

So far we have neglected any terms retarding the wave motion. If viscous terms are present in (2.4) they will slow down the motion while still allowing the lines of force to follow the matter, and will damp the wave eventually either in time or space. If we consider the motion to be sinusoidal in space, it must damp out in time. The presence of a non-infinite conductivity will likewise produce damping. The lines need no longer follow the matter exactly and can move relative to the matter getting out of phase with the motion and retarding it. We will assume in this subsection that both viscosity and resistivity are finite but so small that the wave is not appreciably damped over many wave lengths or during many oscillations. We compute the damping effects to first order neglecting products of viscosity and resistivity terms. It will be found that the velocity has an imaginary part but that its real part is unchanged. If 6 is the skin depth, I the wave length of the wave, and the viscosity is zero, it will be found that the wave damps appreciably in A2/b2 wave lengths or oscillations. The equation of motion (2.4) with viscosity included is (5.57)

where p is the viscosity. In Ohm's Law (2.6) the term ]/a must now be included. With these modifications (5.6), (5.7), and (5.8) become (5.58)

i

i =~iwVx ~ cos2 ~ e,

228

(5.59) (5.60)


+ Z2k2sin 6 cos Ov,, W2v, = Z2k2sin 8 cos Ov, + ( G2 + Z 2 sin2 6) k2vz, W2v, = Z2k2cos2Ov,

where (5.61)

(5.62)

(5.63)

Thus (5.6), (5.7), and (5.8) take on the same form in these new variables. (5.15) becomes (5.64)

-c 2 = 7

W2

k

1 =-{Z2+ 2

&'&((a2+

% 2 ) 2 - 4 Z 2 & 2 ~)' ~I 2 I~. 2 8

The quantity p is also modified but we do not include this change since it merely rotates the plane of polarization. Let (5.65)

w -

k

=

5 + ir.

Expanding (5.64) we find is just the velocity for zero damping. That is, the velocity is unchanged to first order. The expression for q is

(5.66)

where c+ and c- are the velocitiesof the fast and slow modes for zero damping. If a and u are roughly equal and p = 0 (5.67)

2

c&q+ M c,k2d2,

where 6 = ~ / ( 4 n o w )is~the ' ~ skin depth. If one regards w as real, the amplitude of the wave is proportional to (5.68)


229

and the wave damps in a distance (5.69)

If k is real, w is complex and the damping time is (5.70)

If c is infinite and p finite, the damping length is (5.71)

the damping time is (5.72)

2u2p 1 Ti=-------. "P

10. More General Waves

There are many other different types of waves that may exist in situations in which the main field is no longer uniform but varies in space, or in which the boundary conditions are different. Because of the increased complexity, it is difficult to treat such waves in as much detail as in the plane-wave case. In investigating these more complicated waves, the dW-formalism of Section IV leads to the equations for them in a simple and direct manner. One case of such waves is the situation in which uniform field, pressure, and density are present in an infinitely long cylindrical tube with rigid infinitely conducting boundaries. The magnetic field is parallel to the axis of the tube. The boundary conditions are: v * n and b * n must vanish on the boundary. If the propagation direction k is parallel to the axis of the tube, there are three simple modes. For the mode in which the mass velocity is along k, an undisturbed sound wave propagates with velocity a. For the mode in which the velocity is radial, no wave can propagate if its wave length is much greater than the radius of the tube. Thus, a cut-off frequency exists for this mode in analogy with microwaves in a wave guide. A third mode has its velocity in the 8 direction and propagates with velocity u. This is a torsional wave. Since each cylindrical shell oscillates independently of its neighbors any radial distribution of velocity is possible. No cut-off exists for this mode. For propagation in other than axial directions the waves become more complicated.

230

EDWAHD A. FRIEMAN AND RUSSELL M. KULSRUD

To illustrate the application of the GW-formalism in determining the equations of motion let us consider the radial pulsations of a cylinder of radius R and length L in which the field is parallel to the axis and the pressure and field are functions of the radius. Introducing cylindrical coordinates we find t o= 6, = 0 and from (4.9)

Qr = Qe = 0.

(5.73)

Since everything depends only on the radial coordinate, 6W can be written R

sw=?C,srdr{(g

+YP)[($)"$]+(f

+.p);,}.

1

at2

0

(5.74)

We have dropped the subscript r on 6,. Since t vanishes a t term may be integrated by parts to give

Y =

R, the last

R

t 2 (W- YW') &t'2+ 2 7

(5.75) 0

where primes denote differentiation with respect to r and (5.76)

W =-+ y p . B2

4n

The motion is determined by the condition that (5.77)

must be stationary, and the motion is proportional to exp [iwt]. Carrying out the variation we find (5.78)

W'

If i;, is constant the equation simplifies and (5.79)

1


w is determined

231

by the condition 5 ( R ) = 0, to be

(5.80)

where yi is the i-th zero of

J1.

References 1. 2. 3. 4. 5. 6. 7.

8. 9. 10. 11. 12. 13. 14. 15. 16. 17.

ELSASSER,W. M., Hydromagnetism I, a review, A m . J. Phys. 23, 590 (1955). ELSASSER,W. M., Hydromagnetism 11, a review, A m . J. Phys. 24, 85 (1956). LUNDQUIST, S., Studies in magneto-hydrodynamics, Arkiv For Fysik 6 , 297 (1952). CHANDRASEKHAR, S., Problems of stability in hydrodynamics and hydrodynamic astrophysics, Mon. Not. Roy. SOC. 113, No. 6, 667 (1953). SPITZER,L., JR., “Physics of Fully Ionized Gases”, Interscience Publishers, New York, 1956. NEWCOMB, W. A., Motion of magnetic lines of force, Princeton University Observatory Technical Report No. 1, 1955. LUNDQUIST, S., On the stability of magneto-hydrostatic fields, Phys. Rev. 88, 307 (1951). BERNSTEIN,I. B., FRIEMAN, E. A., KRUSKAL,M. D., KULSRUD,R. M., t o be published (1957). LORDRAYLEIGH, “The Theory of Sound”. KRUSKAL,M. D., and SCHWARZSCHILD, M., Some instabilities of a completely ionized plasma, Proc. Roy. SOC.A, 223, 348 (1954). KRUSKAL, M. D., (Private Communication). A L F V ~ NH., , “Cosmical Electrodynamics”, Clarendon Press, Oxford, 1950. VANDE HULST,H. C., Chapter VI, in Problems of cosmical aerodynamics, Central Air Documents Office, Dayton, Ohio, 1951. KULSRUD, R. M., Effect of magnetic fields on the generation of noise by isotropic turbulence, Astrophys. J. 121, 461 (1955). FERRARO, V. C. A., On the reflection and refraction of Alfven waves, Astrophys. J. 119, 393 (1954). ROBERTS,P. H., On the reflection and refraction of hydromagnetic waves, Astrophys. J. 121, 720 (1955). BERNSTEIN,1. B., (Private Communication).


Mechanics of Granular Matter *

BY H . DERESIEWICZ

.

Columbia University. New York N . Y . Page 1.Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 I1. Geometry of a Granular Mass . . . . . . . . . . . . . . . . . . . . 236 A . External Shape . . . . . . . . . . . . . . . . . . . . . . . . . 236 237 B . Internal Configuration . . . . . . . . . . . . . . . . . . . . . . 1. Arrangements of Equal Spheres . . . . . . . . . . . . . . . . . 237 2. Densest Packing of Equal Spheres . . . . . . . . . . . . . . . 239 3. Effect of Interstitial Spheres on Density . . . . . . . . . . . . 245 4 . Packing of Non-Spherical Bodies . . . . . . . . . . . . . . . . 249 111. Some Recent Results of Contact Theory . . . . . . . . . . . . . . . 2.51 A . Normal Forces . . . . . . . . . . . . . . . . . . . . . . . . . . 251 253 B . Tangential Forces . . . . . . . . . . . . . . . . . . . . . . . . 1.Increasing. . . . . . . . . . . . . . . . . . . . . . . . . . . 253 256 2. Decreasing . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Oscillating . . . . . . . . . . . . . . . . . . . . . . . . . . 258 C. Oblique 'Forces . . . . . . . . . . . . . . . . . . . . . . . . . . 263 D . Twisting Couples . . . . . . . . . . . . . . . . . . . . . . . . . 265 I V . Mechanical Response of Granular Assemblages . . . . . . . . . . . . . 267 A. Continuous Model . . . . . . . . . . . . . . . . . . . . . . . . 267 R . Discrete Model . . . . . . . . . . . . . . . . . . . . . . . . . . 270 1. Rigid Granules . . . . . . . . . . . . . . . . . . . . . . . . 270 2a . Elastic Granules. Normal Contact Forces . . . . . . . . . . . . 271 2b. Elastic Granules. Oblique Contact Forces . . . . . . . . . . . . 285 V . Suggestions for Further Research . . . . . . . . . . . . . . . . . . . 300 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303

I. INTRODUCTION Perhaps the most striking characteristic of a granular material is its dual nature which prohibits rigid classification as either solid or fluid . For. although the individual particles of the medium are solid. the tendency of the entire mass to flow readily is a property shared with fluids . That is. in contrast to the essentially fixed geometric form of a solid. both granular

* This study was supported by the Office of Naval Research under contract Nonr-266(09) with Columbia University . 233

234

H . DERESIEWICZ

and fluid media assume a form dependent on imposed external boundaries: each is capable of indefinite shear. On the other hand, when a liquid is subjected to no active (as opposed to reactive) forces other than the force of gravity, its traction-free surface is a horizontal plane. Under similar circumstances, however, a granular medium may take on a variety of shapes, the particular one assumed depending on the geometry of the supporting surface. Before proceeding further it will be well to set down a few basic definitions. By “granular medium” we will understand an aggregate of discrete, solid granules in contact. A given volume of material will thus consist, in part, of contiguous granules, the remaining portion being taken up by voids. The granules may be free to displace with respect to their neighbors and the voids may be filled with gas, as in a sample of dry sand, or the voids may be filled with liquid, as in a sample of wet sand. Or it may happen that some or all of the voids are filled with a binding material which holds the granules in more or less fixed positions, as, for example, in sandstone or concrete. Following the geologists, we call such material “porous granular.”* Finally, we shall denote a material “porous” if it consists of a continuous solid matrix which surrounds either isolated or interconnected voids filled with gas, liquid, or a solid binder. Although the study of the mechanical behavior of porous materials is of great interest and a good deal of progress has recently been made in this field, we will not concern ourselves with it here. What are the problems peculiar to granular media and what are some of the applications and uses of such materials ? In attempting to answer these questions we uncover an enormous variety of technological fields in which granular matter plays a major role. The oldest and perhaps best known of these is soil mechanics, with its problems of foundations, grades, retaining walls, etc. Such problems have been treated on a more or less scientific level for a t least two and a half centuries, in the course of which they have attracted the attention of a large number of architects, engineers and physicists; among the most prominent of these we may cite Coulomb, Navier, Poncelet, Rankine, Boussinesq, St.-Venant, Flamant, and Winkler. A detailed account of the early work in this field will be found in a review article by Kotter [l]; for later work, reference may be made to Terzaghi [ZIT. Similarly, problems of computing static

* It is to be noted that no stipulation was made concerning the size of the individual granules but, for our purposes, we may assume that we are dealing with essentially convex particles whose mean diameter exceeds lW4 cm. Thus, we exclude colloidal particles from our definition, although some questions touching upon granular media are pertinent even to problems of molecular configuration.

t Added at press-tima: For an exhaustive treatise on theoretical soil mechanics, see the book by Caquot and Kerisel [84].

MECHANICS OF GRANULAR MATTER

235

pressure exerted by a granular mass arise in the design and construction of silos (Janssen [3]). The dynamical response of granular materials has been much investigated in connection with the propagation of seismic waves and with petroleum prospecting (a bibliography on these subjects may be found in a paper by Brandt [4]). The problem of the vibration of a granular assemblage arises in the analysisof the performanceof awidely used type of microphone (Sell [5]) of which an aggregate of carbon granules is the essential constituent. The internal configuration of a granular mass is of interest wherever problems of ordering and packing appear. Thus, it is found that the most favorable results are obtained with densest packing of an aggregate in such widely separated fields as the manufacture of ceramics, refractories, and grinding wheels (Westman and Hugill [6]), of cement and mortar products (Anderegg [7]), in powder metallurgy (Wise [8]), and, in connection with the strength of asphalt, in highway engineering (Horsfield [9]). The study of the packing of granules is intimately connected with problems in crystallography (Schonflies [lo]) and cytology (Marvin [ll]),and belongs to the branch of pure mathematics called extremal problems of geometry (Fejes T6th [12]). Problems of hydrodynamics and aerodynamics arise, among others, in the analysis of the movement of ground water, petroleum, and natural gas through sand and rock (Muskat [13]), in studies of drainage, seepage, and filtration (Polubarinova-Kochina and Falkovich [14]), in attempts to predict distribution of hydrothermal mineral deposits (Fraser [15]), and in questions pertaining to combustion in a solid fuel bed (Bennett and Brown [16]). Finally, granular matter has been employed in an attempt to explain formation of islands (Terada and Miyabe [17]). The purpose of the present article is to discuss some recent progress made in the investigation of the static and dynamic response of granular media to external forces, that is, of small deformations and vibrations of such materials and of the propagation of elastic waves through them. To facilitate their task, the authors of most modern theories assume such granular matter depicted by means of models in which the individual granules are represented by spheres or other convex shapes in contact in, for the most part, regular arrays. Accordingly, Section 11. of this review is devoted to a summary of the geometry of packing. The contiguous granules in many of the models are considered elastic, so that the theory of contact of elastic bodies must be invoked to furnish a description of some of the phenomena. However, the classical theory of contact due to normal forces does not, in general, suffice for the purpose. The extension of the theory to account for contact due to oblique forces and due to twisting couples is quite recent; it is discussed in Section 111. The account of studies of static and dynamic behavior of granular media constitutes the content of Section IV. I t was found desirable and convenient

236

H . DERESIEWICZ

in this portion of the article to group the various investigations not by subject matter, but rather by the types of models used in approaching the subject matter. A brief outline of some of the unsolved problems in the fields treated in Sections 11.-IV., given in Section V., and a list of references conclude the paper.

11. GEOMETRYOF

A

GRANULAR MASS

A . External Shafie Mention has already been made of the diversity of forms which a granular aggregate may assume when it is subjected solely to gravitational forces. An extensive experimental investigation of this subject of “equilibrium figures” was carried out by Auerbach [18] who came to the following conclusions: a) The traction-free surface of a dry granular medium bounded by a horizontal base and a vertical wall, both of indefinite extent, is a plane whose slope depends on the size, shape, density and surface roughness of the constituent granules.* If the pressure in the interior of the medium were transmitted by all the granules through the height of the mass, we should expect the free surface to be concave rather than plane, that is, its trace in a plane normal to the fixed bounding planes would be a curve whose slope increases with elevation above the base. Accordingly, we must conclude that the pressure is transmitted only by a limited portion of the granules.? b) The equilibrium figure over a circular base may be approximated by the lower sheet of a hyperboloid of two sheets; the figure over a base with a circular hole is a crater generated by rotating a portion of the upper half of one branch of a horizontally oriented hyperbola about the vertical axis through the center of the hole. c) The figures over bases having the shape of regular polygons are pyramids with rounded edges and vertices. The rounding of the vertex is greatest over a circular base, smallest over a triangular base, and varies directly as the size of the granules. Further, the steeper an edge, the greater the amount by which it is rounded.$

* Somewhat related experiments were performed by Terada and Miyabe [19] with a different end in mind. They started with a rectangular box of sand with a free horizontal surface and, by slowly receding one wall, were able to simulate step-faulting. This phenomenon was found to be strongly sensitive to the coefficients of friction of the granular mass. With the same setup as above, an investigation was also made of the effect, on the medium, of the wall pushing in on the sand and, further, of reciprocating motion of the wall [20]. t We shall have occasion to return to this question in Section IV. B, 1. t I t may be noted that such equilibrium figures, obtained with infinitesimal granules, represent the stress surface for the case of complete yielding of a twisted bar of the same cross-section as the base of the heap (Nadai [21]).


237

B. Internal Configuration 1. Arrangements of Equal Spheres. Two basic questions arise in connection with the arrangement of granules in an aggregate: (a) What is the actual packing of granules in a well-agitated mass ? (b) How can densest packing be attained? The first of these questions can, clearly, be answered solely on the basis of experimental evidence; the second is susceptible to analytical formulation.

(a) Simple Cubic

(c)TetMgonal Sphenoidal

(b) Cubical -Tetrahedral

(dl Pyramidal

(el Tetrahedral

FIG.1 . Modes of regular packing of equal spheres.

Let us first consider the type of model in which the granules are represented by equal spheres in contact with each other in such a way that they form a regular pattern in space.* The simplest of all such arrays is formed by the sim9le cubic packing (Fig. l a ) in which the centers of spheres of radius R form a space lattice generated by a cube of side 2 R . Thus, every sphere is in contact with four spheres with centers in one plane and with one sphere in each adjacent parallel layers, i.e., a total of six spheres. The number of spheres in contact with any given sphereis called the “coordination

* The first to represent symmetric structures (crystals) by regular packings of like spheres was Barlow [22]; for a bibliography of the early work on this subject, see the article by Schonflies [lo]. A d d e d at press-time: It appers that Barlow had been anticipated by Wollaston [85], who employed not only spheres but oblate and prolate spheroids in his models, and, to some extent, by Hooke [86].

238

H. DERESIEWICZ

number” of the packing. The density, D ,of the packing is defined as the ratio of the volume of space occupied by solid matter to the total volume. Since each primary cube of volume V = (ZR)3 contains one octant of each of eight spheres, the density of simple cubic packing is D = n / 6 m 0.5236. In cubical-tetrahedral, or single stagger, packing (Fig. l b ) each sphere is in contact with six others in the same layer and, as in the simple cubic packing, the highest points of spheres in one layer are in contact with the lowest points of spheres in the layer above. Hence, each sphere is in contact with eight others. The volume of a unit prism containing segments equal in total volume to one full sphere is V = ( 2 R ) 2 ( R the density of the packing is D = n / 3 v F m 0.6046. Tetragonal-sphenoidal, or double stagger, packing (Fig. lc) is similar to single stagger packing in that each sphere is in contact with six spheres of the same layer, but now each sphere in a given layer rests in the depression between two adjacent spheres in an adjoining layer. In this case, then, the coordination number is ten; the volume of a unit prism is V = (2R)(RV3)2 and the density of the packing is D = 2x19 M 0.6981. In pyramidal packing (Fig. Id) each sphere is in contact with four neighbors in the same layer (as in the simple cubic arrangement) and each of the spheres in a given layer nestle in the hollows formed by four contiguous spheres in the next layer. Accordingly, each sphere is in contact with four others in each of three adjacent layers, or a total of twelve spheres. The unit volume is V = (2R)2(Rv5jand the density of the packing is D = ~ / 3 V $ m0.7405. Finally, in tetrahedral packing (Fig. le) each sphere is in contact with six others in one layer (as in the cubical-tetrahedral array) and is lodged in the hollows formed by three adjacent spheres in each of the layers immediately above and below. Each sphere is, therefore, in contact with twelve others. The unit volume is V = (2R)(RVg(ZRv2/3), so that the density is D = z/3vFm 0.7405. We may note that in each of the packings described above, the density is independent of the size of the spheres, provided that the packing is extended over a very large region so that the effect of boundaries is negligible.* The results are summarized in Table 1, in which the column marked “porosity” indicates the percentage of voids in a unit vo1ume.t

vg;

* The problem of packing of spheres in a finite region is considerably more difficult: packing in a circular cylinder has been discussed by Supnik [23] ; inside a hollow sphere, by Hadwiger [24]. t A detailed study of these packings may be found in a paper by Graton and Fraser [25].

239


2. Densest Packing of Equal Spheres. It isseen that, among the arrangements listed in Table 1, the greatest density is attained in the pyramidal and tetrahedral structures. That these represent the closest of all regular packings* of like spheres may be shown by the following construction. TABLE1

Coordination number

Spacing of layers

Simple cubic

6

2R

Cubical-tetrahedral

8

2R

Type

Of

packing

Volume of unit prism

4v3Rs 6R3

Density

Porosity

(%I

(431j3) 0.6046 (2749) 0.6981

39,54

Tetragonal-sphenoidal

10

Pyramidal

12

RvB

4v2R3

(ni3Vz)

25.95

Tetrahedral

12

2R1/2/3

4v2R3

(n’3v2)

25.95

0.7405 0.7405

30.19

Arrange the spheres, their centers a distance 2R apart, in rows in a horizontal plane and order these rows in such a way that the lines connecting the centers in each row are parallel and as closely spaced as possible. In this position the spheres in a given row will fit in the notches between two contiguous spheres in an adjacent row, so that the generating lattice of centers in the plane is a rhombus whose short diagonal divides it into two equilateral triangles. This layer .represents the densest packing in the plane, each sphere being in contact with six others.? Now take a similar layer and place it on top of the first so as to cause the planes of the centers in each of the two layers to be the smallest possible distance apart; that is, the spheres in the second layer will just fit into the

* Proof that the largest number of spheres in contact with a given sphere is twelve, regardless of the regularity of the packing, has been given only recently by Boerdijk [26], Schiitte and van der Waerden [27], and Leech [28]. In addition, Boerdijk demonstrates configurations which, locally, are denser than the close-packed systems. A discussion of loosest packing of like spheres (in which the mobility of the spheres is completely restricted) is given by Hilbert and Cohn-Vossen [29]. The coordination number of such a packing is four, but the density is a matter of conjecture ( D 0.123). Added at press-time: A further study of arrays of spheres having large porosity is due to Heesch and Laves [87]. t Hence, the density of closest packing of the horizontal great circles (packing in 0.9069. two dimensions) is D = 4 2

485 O K no condensation effects were present. Finally, many observations of light scattering have been made to detect the presence of air condensation [53, 57, 58, 59, 63, 64, 651. Stever and Rathbun used this technique extensively to determine the onset of condensation as a function of varying supply conditions. The method depends on a small ray of collimated light, and is usually less accurate than pressure but it offers the practical advantage of rapidity measurements in finding Mk,

360

PETER P. WEGENER AND LESLIE M. MACK

and simplicity of operation. Quantitative investigations on the nature of the condensation process were made by Durbin [65] and McLellan and Williams, and we shall discuss these results later.

1

cn

..

il6-

; ; - ISIS 'c)

P 14: o 14P 0

a? 0 o

{ 1312-

-G No condensation

I

I

1

1

I

I

I

FIG. 32. Measurement of shock angle on a 11' included angle wedge as a function of supply temperature in a M = 6.7 nozzle from Stever and Rathbun [64]. 9, = 68.6 atm.

The preceding observations have shown that air condensation is likely to occur when the Mach number of equilibrium condensation is exceeded. The air condensation takes place gradually, just as observed in nitrogen, and is a process different from that of moist air and steam condensation. We shall next attempt to estimate the thermodynamic state at the onset of condensation, again choosing a state where the stagnation enthalpy has changed sufficiently to show deviations of the three Mach numbers determined from (1.14), (3.3), and (3.5). We have tried to apply similar criteria of divergence of these Mach numbers as indicative of condensation for six different series of experiments. One series [64] based on detection of condensation by light scattering will also be shown. After M Rhas been determined in this fashion, supercooling, pressure, and temperature a t M kcan be found from isentropic flow tables. Figure 33 shows that in general M k> M,, and the difference increases with increasing Mach number. A substantial Mach number increase without condensation beyond M , occurs only a t the highest Mach numbers. The experiments shown in this and the next two figures were made in a large range of environmental conditions. The nozzle heights varied from 2.5 to 36 cm, and the distance from the throat to the point a t which M , occurred was very dependent on the nozzle design and size. I n two cases plane diverging walls [54, 601, which resulted in a continuous expansion, were used. Some observations were made a t a fixed location in the flow and the supply temperature was varied, while in other cases the supply conditions remained constant and the observation point was moved along the centerline or wall. Supply pressures ranged from 3 to

36 1

CONDENSATION IN WIND TUNNELS

70 atmospheres, and supply temperatures from 285 to 760 O K . The highest pressure of about 70 atm. was used by Stever and Rathbun, and at their lowest temperatures the application of the perfect gas law and y = constant is not strictly permissible (Table 1). However, the uncertainty of determining the onset of condensation accurately by light scattering in a small nozzle

0 0

A

v 0 0

A A

-0

5

Arthur Grey Hansen and Nothwang Kubota Mckllan and Williams Stever and Rothlwn

10

MC

FIG.33. Mach number at the onset of air condensation as a function of Mach number of equilibrium condensation from experiments in a wide range of environments.

prompted us to evaluate their data like the others. The moisture content of the air supplies was low compared with that in a conventional supersonic tunnel, but it varied greatly. In some cases the air had been stored in highpressure vessels at, say, 200 atmospheres which resulted in very low dewpoints; in other cases, continuously operated power-plants with driers in the circuit were employed. Even the carbon dioxide content differed, since in compressing air the normal fraction is reduced by the increased solubility of carbon dioxide in water at high pressures [58]. The highest purity with respect to traces of vapors was probably obtained by Arthur [54],

362


who produced “synthetic” air by mixing 0, and N, from bottles. Various types of electric heating equipment may perhaps have produced foreign nuclei by oxidation of the heating elements; powdered desiccants may well have been present in some plants: and in some experiments oil vapor from the flow machinery was present. The uniformity of the supply-temperature distribution ahead of the nozzle throats may have been poor in some experiments. Finally, considering that it is difficult to apply uniform judgment to the raw data in the estimate of the onset of condensation, we are not surprised to see the scatter exhibited in Fig. 34, where the pressure and

‘F

/

A

10

0

0 0

n

0 I .o

0.31: 0

0.51

0

0

0.2

0.11

I 35

I 40

/

/ I 45 T(”K)

V Arthur

IGrey 0 HonsenondNothwang 0 Kuboto A McLellan and Williams A Steverand Rothbun 0 Wegener et ol I

I

50

55

a

FIG.34. Thermodynamic state at the onset of air condensation from experiments in a wide range of environments.

temperature at M R are indicated. The corresponding isentropes are not shown (they are much steeper than the coexistence line). When we determine the supercooling for the same group of experiments, we arrive a t Fig. 35. The fact that M , was determined graphically from Fig. 5 contributes to the scatter. Also, with the inaccuracv of the experimental Mb.the cluantitv

363

CONDENSATION I N W I N D TUNNELS

ATk as determined from (2.2) and (1.15) is only known within a few degrees. For 5 < M k< 8, the range that includes the largest number of experiments, no trend is discernible, and for all observations 0 < AT, < 24 "C. The largest supercooling is below that observed for moist air (Fig. 19) and steam (Fig. 20) and is of the order of that found for pure nitrogen. 0

V Arthur

0

-

0 A A

0

Grey Hansen and Nothwang Kubota McLellan and Williams Stever and Rathbun Wegener et at

A

0 O.

0

0

0

0.

0 0 1

1

I

I

1

-

1

5 -

I

I

I

I

I

10

Mk FIG. 35. Supercooling as a function of the Mach number at the onset of air condensation from experiments in a wide range of environments.

It is difficult, if not impossible, to correlate the experimental evidence on air condensation in other than qualitative terms. The supercooling is likely to be a function of several parameters, just as in our previous studies. We expect the temperature and pressure at condensation, the nozzle temperature gradient, and the availability of surfaces of condensation to be dominant factors. I t is just these three conditions that varied over such a wide range in the experiments shown in Figs. 33, 34, and 35. I n general the larger tunnels showed less supercooling. The temperature-gradient effect appears to be the only variable that can be studied separately,

364

PETER P. WEGENER AND LESLIE XI. MACK

thanks to the experiments of Kubota [66]. In Fig. 36 we show the supercooling observed by him in two nozzles, one short and one long, with the same exit height of about 13 cm, but with throat radii of about 0.3 and 8 cm, respectively. Fortunately, both nozzles were tested in the same

0 4

5

7

6

8

9

Mc FIG. 36. Effect of temperature gradient (or nozzle length) on supercooling of air for a 13

x 13 cm nozzle a t fixed initial conditions from Kubota [66].

powerplant under presumably identical conditions of traces of vapor impurities. We see that the short nozzle with the small throat radius exhibits a much higher supercooling. This is comparable in effect, though not in magnitude, to our experience with moist air subjected to large temperature gradients. In general, however, as a result of the experiments with nitrogen and traces of vapors, we expect the impurity level to be the dominant factor governing any possible supersaturation in the case of air. In fact, if the impurity level is high enough to produce condensation right a t the equilibrium Mach number, we might use the supersonic tunnel as an instrument to study the condensation line of air. In qualitative terms, the experimental evidence on air condensation suggests that for the larger hypersonic wind tunnels it will be necessary to preheat the supply air to a temperature such that the highest Mach number expected in the flow field about some model in the test section is less than M,.


365

Iv. DIABATICFLOWS A N D THERMODYNAMICS OF CONDENSATION In this section we investigate to what extent one-dimensional fluid mechanics and thermodynamics can describe the condensation phenomena without recourse to kinetic considerations. When condensation and vaporization occur in a flow, we have an example of what is called diabatic flow. Many authors [e.g., 67, 68, 69, 701 have analyzed one-dimensional diabatic flows; Oswatitsch [34] has derived a set of equations specifically applicable to the condensation of steam and of water vapor in air. We derive here a different form of these equations which can be applied either to the condensation of a vapor trace, or a large amount of pure vapor. However, even at the outset we make the simplifying assumptions that the liquid (or solid) condensed mass is uniformly distributed throughout the gaseous components and has the same speed and temperature as the stream. The equations which result from the application of the conservation laws of mass, momentum, and energy, and the equation of state are not sufficient. The actual computation of a condensing flow requires an additional equation for the condensed mass, or else an experimental measurement of one of the flow variables. Even when the equations are supplemented in this manner to permit a particular flow to be calculated numerically, they do not allow many general conclusions to be drawn. For the purpose of a general discussion we consider what may be thought of as two limiting situations. The first of these is shown in a $ - v diagram, Fig. 37 (for simplicity we have drawn the diagram for a pure vapor). The flow continues along a dry isentrope into the coexistence region, until a t E the supersaturated state collapses instanteneously to state B. This picture leads to the analysis of the condensation shock, which can be carried out in some detail, but does not lead to a complete solution of the condensation problem as the location of the shock and amount of heat released must be specified separately. The other limiting situation is the complete absence of supersaturation. Condensation starts a t the coexistence line (point C, in Fig. 37), and the flow continues along the saturated isentrope, C, - F,. The analysis of this saturated isentropic, or equilibrium, expansion is particularly simple and valuable, as fluid mechanics and thermodynamics are sufficient to describe the flow completely. Most actual condensation processes lie somewhere in between these two idealized situations. For instance, the supersaturated state may start to break down a t D. The whole collapse region then extends to the point where the saturated isentrope C, - F , is reached. Here the vapor is again saturated, and if thermodynamic equilibrium is maintained, the subsequent flow is another saturated isentropic expansion a t a higher entropy level than C, - Fl. When the vapor behind the condensation shock, point B, is saturated, that flow may also continue along an isentrope C, - F, a t a still higher entropy level than C, - F,.

366


When a shock wave occurs in a condensed flow, the temperature rise behind it vaporizes some or all of the condensate. As a result the conditions behind a shock differ from their value in dry flow. The effect of this vaporization on the pressure measured by a pitot tube is of particular interest. Also, the angle of an oblique shock will change, which affords an experimental tool for the investigation of condensed flow. These matters are briefly discussed in the last part of this section.

I

v = I/P FIG. 37. Schematic

p -u

diagram of condensation processes.

1. Derivation of One-dimensional Condensation Equations We derive here a set of one-dimensional, steady-flow equations which can be applied to either the case of an inert carrier gas with a condensable component, or to a pure vapor (or a mixture of vapors) which itself condenses. We assume that the gaseous components are calorically and thermally perfect and allow the molecular weight and specific heat of the mixture t o change as condensation takes place. There are five equations in all: the equations of continuity, momentum, energy, and state, and the condensationrate equation. The latter, however, can only be formulated here; its complete derivation must await Section V. Three .terms enter the continuity equation: the mass of inert carrier gas, ma, which passes a given location in the nozzle in unit time; the mass


367

per unit time of the condensible vapor, mu; and the mass per unit time of the condensed phase, mL. The condensed phase can be either liquid or solid, but for convenience we shall usually refer to it as if it were liquid. The sum of the three mass rates must be constant, which gives (4.1)

ma

+ m, + mL = m = constant

Since the mass of inert carrier gas remains constant, and the condensate grows at the expense of the vapor, we have dma = 0, and dm, = - amL. For water vapor in air, all three terms are present, but for steam and air condensation there is no inert carrier, and ma = 0. The density of the liquid phase is much larger than that of the vapor phase, and we can, to a very good approximation, neglect the volume occupied by the liquid. Further, we can define a density pL’, where the condensed mass is referred to the same volume as the carrier gas and vapor. Equation (4.1) implies for the density of the mixture at any cross section (4.2)

+ pv + PL’.

p = pa

Using the density p we can write the continuity equation in terms of the velocity, w , and area, A, pwA = m,

(4.3) or, in differential form, dp

- + - + -dw= o .

(4.4)

P

W

dA A

The equation of state for a mixture of two perfect gases is (4.5) When we introduce the molecular weight of the mixture, p, we can write (4.5) as

For ,u we have (4.7)

1ma p ma+m,

-

-1+ ,ua

mu

1 ma+mu pv

where we have replaced the ratios of the densities by ratios of the masses.

368


Let us consider a fixed mass, m, of mixture. The sum of the vapor and liquid mass is constant and equal to the vapor mass in the nozzle supply condition, muo. Then, from (4.1) we have

or, (4.9)

In (4.9) we have introduced the initial specific humidity, w,,, from (2.4), and the mass fraction of condensed phase, g = mLlm. With reference to Fig. 37 we recall from thermodynamics that g at, say, point B equals the ratio BC’IAC’. By (4.2) we can write p a pv in terms of p and g to obtain

+

(4.10)

or, with (4.9), (4.11)

P=P

(7+ u) RT.

1 -0 0

Pa

The differential form of this equation of state is (4.12)

With the assumption that the condensate moves with the stream velocity, the momentum equation is simply (4.13)

- pwdw = ap.

The energy equation can be obtained from the first law of thermodynamics, which, applied to a unit mass of the mixture, is (4.14)

0 = du

+ pav.

The left hand side is zero since there are no external heat sources. In terms of the specific enthalpy, and with (4.13), we have (4.15)

0 = d (h

+) ;

.


369

When we apply (4.15) to the mass m of the mixture and write out the enthalpy in terms of the enthalpies of the separate components, we have (4.16)

1

+ maha+ m,h, + mLhL = 0.

I n general, the liquid and vapor are a t different temperatures. When the two phases are in thermodynamic equilibrium the enthalpy difference of the phases is simply equal to the heat of vaporization [e.g., 71. However, when the temperatures are different there is an additional internal energy term for the liquid phase. Therefore, we have for the enthalpy of the liquid, (4.17)

h L = h,

-L

-

cL(T,- TL),

where cL is the specific heat of the liquid. Since the last term is generally only a few percent of the other terms, we shall consider the liquid and vapor to be a t the same temperature in order to simplify the analysis. Substituting (4.17) in (4.16), we then obtain (4.18)

or (4.19)

where

the specific heat of the vapor mixture in the supply. Actually, the liquid phase is present in the form of droplets, and a mixture of vapor and droplets is a dispersive medium for the propagation of sound. Therefore, the speed of sound is a function of frequency. Oswatitsch [71] has made a detailed analysis of sound propagation in fog, and Buhler, Jackson, and Nagamatsu [72] have considered the same problem for a pure vapor fog. In both cases the droplets act as inert particles in the limit of a very high frequency sound wave, and the sound speed is given by the perfect gas formula in terms of the vapor density. It is this sound speed which we shall introduce here. We have, therefore, (4.21)

370

PETER P . WEGENER AND LESLIE M. MACK

and we define the Mach number in terms of this speed of sound. Then (4.19) becomes y _ H _

(4.22)

CP,

M 2 -w

P

d(gL)

dT

dw +

T

-

~

CPoT

= 0.

The specific heat of the vapor mixture is, from (2.15), (4.23)

(ma

+ m&p

+ mucpu,

= macpa

or, in terms of the initial specific humidity and the mass fraction of condensate, (4.24)

With RIP

= cp - cu,

(4.23) becomes

CP dw dT -(y-1)M2-+----=00. w T cpO

(4.25)

d(Lg) cpoT

The four equations, (4.4), (4.12), (4.13),and (4.25) involve six unknowns, T , w ,g, and A . Considering the area as given and g related to the other variables in a manner yet to be determined, we can solve for the pressure, density, velocity, and temperature differentials in terms of dA and dg. The results for d p and dp are

9, p,

9-

YM2 [yM' - ( 1 - g) - CP ( 1 - g) (Y - 1 ) M 2 CP0 - gL'

(4.26)

x dP

I-?+[ 1

1

L (CP,

Pv 1 - g

-

1 - ( 1 - g) CP,

- .@'

(4.27)

x

{-

Cp(1

[yM2 -

-

1

g) (y - 1 ) M 2 d A ]Af[(cptF;:))T

where L' = dL/dT. The velocity differential can be found from the continuity or momentum equations, and the temperature differential from the energy equation or equation of state.


371

The stagnation temperature can be defined in the usual manner and is given by (4.28)

~p,To= cp,T

+ 21 w2. -

When we introduce (4.28) into the energy equation (4.19) we can integrate to obtain (4.29)

where To, is the stagnation temperature before condensation starts. The stagnation pressure can also be defined as usual from the isentropic relation (1.10). From an experimental viewpoint, the static pressure distribution of the flow is easily measured when it is assumed that the pressure at the nozzle wall is equal to the free-stream pressure. Then if the area distribution is known in some manner, as described in the previous section, all of the other flow quantities can be obtained from the preceding equations once the pressure has been measured. In fact, when we solve for dg, we find

dg =-

1

Therefore, with a measured static pressure distribution, known area and heat of vaporization, the entire flow can be computed from (4.30), (4.27), (4.4), and (4.12) by a step-by-step numerical calculation. This has been carried out for the experimental results given by Wegener [29] for moist air, and is shown in Fig. 16. The heat of vaporization was assumed constant in the small temperature interval under discussion, and equal to 688 cal/g, which is a value suitable for the vapor-ice transition. Further, since the amount of water vapor in this experiment was small, the molecular weight and specific heat were assumed constant and equal to their values for air alone. The missing equation for g can be obtained in two ways. The simpler one is to assume that thermodynamic equilibrium exists a t all times, with the result that the Clausius-Clapeyron equation (1.7) can be used. This leads to the analysis of the saturated isentropic expansion which is presented in

372


Section IV, 4 for a pure vapor. However, in many instances some supersaturation is present, and the assumption of thermodynamic equilibrium is not correct. We must then turn to the kinetics of droplet formation and obtain what is called the condensation-rate equation, which relates g to the other flow variables. First we formulate the condensation-rate equation where condensation occurs as a result of self-nucleation. The number of droplets of critical size which are formed per second in volume Ad5 at position 6 in the nozzle is

where J is the nucleation rate, that is the number of droplets of critical size produced per unit time and volume. Once a droplet is formed which is able to grow, its further growth is determined by a droplet-growth law, G ( t , x ) . The growth law expresses the mass increase of a critical-size droplet as it passes downstream with the flow. We consider each droplet to grow separately, and not to coalesce with other droplets. The quantity G(E,x) is the mass a t position x of the droplet which was formed at 6. Hence, the mass fraction of condensed phase a t position x must be

-m

This is the condensation-rate equation 1341; it sums up the contributions to g from all critical-size droplets formed upstream of x. When self-nucleation is unimportant and condensation takes place on foreign nuclei, the number of droplets does not change through the condensation region, when it is again assumed that the droplets do not coalesce. The condensation-rate equation simplifies to

where N is the number of foreign nuclei per cm3, and G ( x ) is the mass of vapor condensed on a nucleus a t position x. In order to have such a simple equation, N must refer to some average-size nucleus. Thus, with G and J expressed in terms of the other flow variables, and the area known, we are able to calculate the whole flow when the condensation occurs as a result of self-nucleation. This is true for steam and water vapor in air as will be seen in Section V. For the condensation of air, where the foreign nuclei are important, N must be determined in addition. When these nuclei are themselves the result of condensation of vapor impurities this means, for a complete calculation from first principles, that the computation of another condensation process must be carried out previously.


373

We can also write a continuity equation for the number of droplets per cm3, N , present in the flow. The difference between the number of droplets flowing into a volume A d x in unit time and the number flowing out of it, must be equal to the number created in the volume per unit time, provided that the droplets grow only a t the expense of the existing vapor and do not combine into larger droplets as a result of collisions. Thus, neglecting the volume occupied by the droplets, we have (4.34)

~ ( N w A=) J A d x ,

where J is the nucleation rate. When N refers to foreign nuclei, J and (4.34) gives N = constant, as it should.

=

0,

2. Analysis of the Condensation Shock

We have seen in Section I1 that the supersaturation that occurs during the expansion of steam or moist air collapses in a short distance in the flow direction (e.g., Fig. 11). The disturbance marking this collapse is customarily called a condensation shock. Because of the short distance involved, the area change in the nozzle is usually small. Also, the mass of condensate is often small enough so that the mass of the gaseous part of the flow can be considered unchanged. With these simplifications, the only effect of condensation on the flow is the release of the heat of vaporization, and we can simplify the analysis to the consideration of a discontinuity front with heat release. This type of analysis, which is called condensation-shock theory, was first presented by Hermann [25], Heybey [32], and Oswatitsch [35], and later by many authors, including Lukasiewicz and Royle [27], Charyk [73], and Samaras [74]. The governing equations, which are the equation of state, and the three conservation equations of mass, momentum, and energy, can be written in integrated form as (4.35) (4.36) (4.37)

PlWl=

Pl + P1W:

P2W2

= P2

+

2 P2W2

(4.38)

where q = Lg, the heat added, or subtracted, per unit mass. Subscripts 1 and 2 refer to conditions in front of and behind the shock respectively.

374


There are several possible solutions to these equations, corresponding to the various cases of supersonic or subsonic initial and final flow, and heat release and heat extraction. Two graphical representations are very useful in organizing the various solutions: the Rayleigh lines and the Hugoniot lines. A Rayleigh line connects states that satisfy the continuity and momentum equations, and it is specified when the mass flow and the initial thermodynamic state (pl,pl) are given. A Hugoniot line is obtained by eliminating the velocity from (4.36)-(4.38) ; thus it represents the possible thermodynamic states which can be reached from the given initial state.

v/v, FIG.38. Rayleigh lines and Hugoniot curves in

p - v diagram.

The intersection point of the Rayleigh and Hugoniot lines in a particular case satisfies all of the flow equations and initial conditions and gives the actual final state.

375


From (4.36) and (4.37) we have for the Rayleigh line

1 P - P = - p2l 2q =- y k M : ,

(4.39)

v - v,

V1

where v = l/p, the specific volume. The subscript 2 has been dropped since the final state is as yet unspecified. We see that the Rayleigh line is a straight line in a P - v diagram, and for all M,, except zero and infinity, has a negative slope (Fig. 38). Therefore, a pressure increase is always accompanied by a density increase, and a pressure decrease by a density decrease. In an enthalpy-entropy diagram, the Rayleigh line always has the general appearance shown in Fig. 39, drawn for a supersonic initial flow, M,. The point of maximum entropy correspnds to M = 1, the lower branch to supersonic Mach numbers, and the upper branch to subsonic Mach numbers. When M , > 1, heating leads to a lower supersonic Mach number, and cooling to a higher Mach number. Correspondingly, when M,< 1, heating leads to a higher subsonic Mach number, i.e., heating always drives M towards unity.

-1.6

-1.2

-0.8

-0.4

0s-sO.4

I

0.8

1.2

1.6

2.0

R

FIG.39. Rayleigh lines and Hugoniot curves in h - s diagram.

We see immediately from Fig. 39 that there is a maximum amount of heat that can be released, and this amount of heat results in the maximum possible entropy increase and sonic flow downstream from the shock. Since all of our experience has been that condensation shocks are observed only a t supersonic speeds, with supersonic or sonic flow downstream, it is the lower branch of the Rayleigh line which represents the condensation-shock

376


solutions. A normal adiabatic shock causes transition from the lower to the upper branch (point S in Fig. 39). Subsequent heating or cooling would then move the final state along the upper branch. An example of this type of flow is provided by a normal shock in a stream in which condensation has already occurred. The temperature rise through the shock vaporizes some of the condensate and cools the flow, with the result that the Mach number behind the shock is lower than with an adiabatic shock. When we eliminate the velocity from (4.36), (4.37),and (4.38),we obtain for the Hugoniot curve

using the equation of state to eliminate the temperature, we have

y f l (pv Y--l

(4.41)

(PV,

- P1V) - 2

q = 0.

In the - v diagram the Hugoniot curves are a family of hyperbolas with q as the parameter and common asymptotes given by #J

(4.42)

p=

2Ylp l , +l

v =Y - l v , .

Y+l

A family of Hugoniot curves, with y = 1.40, is plotted in Fig. 38 for q 2 0, together with some typical Rayleigh lines. The ratio of the slope of the Rayleigh line to the slope of the q = 0 Hugoniot curve a t p,, vl is easily found to be (4.43)

Therefore, the Ml

=1

Rayleigh line is tangent to the Hugoniot curve a t

PlJV l .

The meaning of the various intersection points in Fig. 38 can perhaps be seen more clearly by examining Fig. 39, where the Hugoniot curves are drawn in the h -s diagram for the initial state, M,. The intersection point with the lower branch of the Rayleigh line, 3, is the condensation shock solution, and M2 > 1 ; the intersection point with the upper branch, 3', is the solution for a normal shock followed by heat release, and M 2 < 1. The Hugoniot curve which is just tangent to the Rayleigh line a t C - J corresponds to the maximum heat release, and M 2= 1. This state can be attained either by a condensation shock, or by a normal shock with heat release. We can now locate these intersection points in Fig. 38. We start at PI, v1 and follow the Rayleigh line upward. By reference to the correspond-

377

CONDENSATION I N W I N D T U N N E L S

ing path in Fig. 39, we see that points 1, 2, and 3 represent the condensation shock solutions. C - J is the maximum entropy point, where M , = 1. The points l', 2', and 3' represent the solutions for a normal adiabatic shock followed by heating. Point S represents the solution for the adiabatic shock. Similar results are true when the initial flow is subsonic, except that in this case transitions to supersonic speeds must be excluded because they would require an entropy decrease. Therefore, the only permissible exothermic subsonic solutions are those which correspond to the condensation shock in supersonic flow, i.e., subsonic initial flows, which, upon heating,

0

2

I

3

MI

FIG.40. M ,

-

M , diagram for constant area flow with heat addition or subtraction [67].

become subsonic flows a t a higher Mach number. In Fig. 38 the permissible solutions are represented by points A and D ; the excluded solutions by A' and B. If the Hugoniot curves for q < 0 are drawn, still more solutions show up. One such curve is drawn in Fig. 39, and the upper branch intersection point, V, represents a normal shock with vaporization. The analytical results which correspond to the intersection points can be obtained from (4.35)-(4.38), and have been given by many authors.

378

P E T E R P. W E G E N E R A N D LESLIE M. MACK

With all of the flow variables expressed in terms of the initial Mach number, the results [27] are:

FIG.

(4.46)

(4.47)

_ -- --, P1P1 M;

P 2 $2

379


where (4.48)

(4.49)

7

I

2

3

4

5

6

7

MI FIG. 42. Pressure ratio across condensation shock as a function of initial Mach number.

When we choose the lower sign in (4.44) and (4.45), we find that M , = MI for Q = 0. This is the condensation shock solution, since in the absence of condensation we must have continuous flow. When we choose the upper sign. we find that for Q = 0, the equations reduce to those for a normal adiabatic shock. It is evident that Qmax is the maximum amount of heat that can be added to the flow, since imaginary results are obtained for Q > Qmax. This corresponds to the fact that no intersection points exist in

380

PETER P. WEGENER A N D LESLIE M. MACK

the diagrams when Q > Qmax. The release of Qmaxresults in the state given by the maximum entropy point in the h - s diagram, where the Rayleigh and Hugoniot lines are tangent and M , = 1. An instructive way to represent the solutions of these last equations is afforded by (4.47) when lines of constant Q are drawn in an M,- M, diagram. Such a diagram is given in Fig. 40 [67]. The shaded area is the region of physically impossible solutions (entropy decrease). All condensa-

I

2

3

4

5

6

7

MI FIG. 43. Temperature ratio across condensation shock as a function of initial Mach number.

tion-shock solutions lie between the line Q = 0 (no-shock line) and the line M, = 1 ; the solutions for the normal shock with vaporization are between the normal-shock line and the M,-axis; and the solutions for the normal shock followed by heat release are on the other side of the shock line up to the line M , = 1. In Fig. 41 Qmax is plotted as a function of M I , from (4.49), for y = 1.40; we see that there is an asymptote as M , m,

-


381

and only a finite amount of heat can be added to a steady, supersonic flow regardless of how large the Mach number becomes. In Figs. 42, 43, 44, and 45, we show pressure, temperature, Mach number, and stagnation pressure ratios as functions of M , for a few values of Q/Qmaxfor y = 1.40. In Fig. 44 we have for Q/Qmax= 1, M , = 1 for all values of M , . The condensation-shock analysis leaves two questions unanswered : the location of the condensation and the amount of heat released. In Section V we shall see that although kinetic theory offers qualitative information

FIG.44. Mach number behind condensation shock as a function of initial Mach number.

about the location of the condensation, it does not allow a quantitative prediction of condensation of water vapor in air. In the absence of any definite information, the correlation presented in (2.35) may be used as a guide. To obtain Q, the most reasonable assumption is that the vapor behind the shock is in equilibrium saturation. Lukasiewicz and Royle [27] found that this was a satisfactory assumption for some of their experiments, and that it resulted in a value of Q very close to Qmx. However, when Keenan [7]

382


analyzed Yellott’s experimental results with steam on this basis, he found that the theory gave too large a pressure rise. Only by the assumption of a definite, small droplet size and, thus, a certain amount of supersaturation retained at the end of the collapse zone, could he fit the experimental results. Also, it is found that, when M , is near one and Qmaxis small, only a portion of the water vapor in air can be condensed in the condensation shock. The remainder, which can easily be over half of the original content, must condense downstream from the shock, where the flow will be a saturated expansion of the type considered in meteorology [75]. We have indicated in Fig. 46 the results of applying the condensationshock theory to the experiment of Wegener for which the detailed calculation

FIG.45. Stagnation pressure ratio across condensation shock as a function of initial Mach number.

of Fig. 16 was carried out. We have assumed a dry isentropic expansion up to the shock, then applied the shock equations, and finally assumed another isentropic expansion from the new pressure and temperature level. Two assumed positions are shown for the shock. The first is at x = 1.58 cm, where M I = 1.28. At this location the release of Qmax also satisfies the condition of saturation behind the shock. However, the total amount of heat released is only q = 2.69 cal/g compared to the value of q = 3.06 cal/g

383


given by the step-by-step calculation (Fig. 16). We see that the pressures behind the shock fall slightly below the experimental points. The second shock was placed at the experimental peak-pressure point, where, from the isentrope, M , = 1.36. Enough heat was released to saturate the flow, which meant Q/Qmax = 0.724. For this condition, q = 2.96 cal/g, which is close to the result of the detailed calculation, and we see that the pressures a short distance behind the shock are very close to the experimental points. In summary we can say that the condensation shock theory can be used to fit experimental data, but that it is not too helpful in predicting details of the flow. However, it can be quite useful in estimating the order of magnitude of condensation effects. 0.6

-

0 .5

-

Condensation shock at MI = I. 28 q = qmox= 2.68 caI/g

P Pol

0.4

Condensation shock at MI = 1.36 q = 0.724 qmm= 2.96 . d / g

-

0.3 -

@0=53'/0 Calculated q = 3 . 0 6 calh I I

0

0.2

I

I

2

I

3

3. Condensation as a Weak Detonation The preceding analysis of the condensation shock can be placed in a more general framework when we realize that the same type of analysis is used in the theory of flow processes in which chemical reactions occur across sharply defined fronts [76}. The flow in front of the shock corresponds to the unburnt gas, the shock itself to the detonation (considered stationary),

384


or slow combustion front, and the downstream flow of vapor and droplets to the burnt gas. As heat is released by the chemical reactionstheseflows must be among the solutions given by the various intersection points in Figs. 38 and 39. The solutions with subsonic initial velocity are the slow combustion processes, sometimes called deflagrations. When the final velocity is subsonic [point A in Fig. 38) they are called weak deflagrations; when it is supersonic (point A ’ ) , strong deflagrations. The strong deflagration can be ruled out on physical grounds, but none of these processes is of interest to us because they occur in subsonic flow. The reaction fronts with supersonic initial velocity are called strong detonations when the final velocity is subsonic (points l‘, 2‘, 3‘) and weak detonations (points 1, 2, 3) when it is supersonic. The detonation front that releases the maximum amount of heat, Qmax, and has a sonic final velocity (point C - J)isof special importance. It is called the Chapman-Jouguet detonation and can be shown [77] to be the only detonation that is expected to occur in practice. All observed chemical detonations appear to be Chapman- Jouguet detonations. From Fig. 38 we clearly see that the condensation shock appears to be an example of a weak detonation, since both processes axe transitions from a supersonic velocity to a (lower) supersonic velocity. It was formerly thought that weak detonations were impossible, being excluded on the basis of the shock-ignition model for the reaction. In this model, a normal shock is supposed to precede the detonation and raise the temperature sufficiently so that the reaction can take place. That is, the sequence of states within the reaction zone is assumed to be as follows: first, a shock which changes the state from PI, v1 to the state given by point S on the q = 0 Hugoniot curve; then, as the reaction proceeds, the state moves back along the Rayleigh line to the Hugoniot curve for the final amount of heat released, qf. This processes is a strong detonation, and with this model the weak detonation can never occur because it would be necessary to traverse states for which no Hugoniot curves exist (q > 4,). However, Friedrichs’ [78] analysis of the detonation process did include a weak detonation solution, which was linked to a requirement for a “very fast” reaction rate. Later, Burgers [79] speculated that the condensation shock might be an example of a weak detonation, and more recently, Reed [80] and Heybey and Reed [Sl] established that it is indeed a weak detonation. Oswatitsch [82], and Hall [83], also had similar thoughts. The analysis of Heybey and Reed will be followed here. In Fig. 38 we now consider the various Hugoniot curves to represent intermediate stages within the condensation zone of finite width. The q for each curve is the total amount of heat released up to that stage of the process. The reaction proceeds along the Rayleigh line from fil,vl to the final state, say 3, and is therefore a weak detonation. The strong detonation point, 3’. cannot be reached from 3 since this would require passage through states in

385


which more heat than qf is released. Instead, as previously described, 3' can only be reached by a normal shock transition to S, followed by successive steps along the Rayleigh line to 3', where the reaction must stop. As qf is made continuously smaller the weak detonation solution approaches the initial state; the strong detonation approaches S. The Chapman-Jouguet detonation can be approached from either side and is a stable state, i.e., the reaction cannot proceed past it.

I

fi

1.30

0.901

I

I

I

I

"I

FIG. 47. Experimental demonstration of water vapor condensation as weak detonation.

The fact that the condensation of water vapor in air is a weak detonation has experimental backing from the previously discussed experiment of Wegener (Fig. 16). When we convert the actual flow process as measured in a diverging nozzle into a fictitious one in a constant-area channel by a series of isentropic compressions, we obtain the sequence of points shown in Fig. 47 [29]. We find that the points start from p,,v, and then move up the Rayleigh line towards the Hugoniot curve for qmax. This establishes the condensation process experimentally as a weak detonation.

386


Instead of considering a family of Hugoniot curves, each for a fixed q as in Fig. 38, we can introduce the idea of an equilibrium Hugoniot curve. That is, we add the integrated Clausius-Clapeyron equation, (1.9), to our system of equations, (4.35), (4.36),(4.37), and (4.38), and require that it be satisfied behind the condensation region. Then we have only a single Hugoniot for the given initial state, and each point on it represents a state of thermodynamic equilibrium. Such a Hugoniot is shown schematically in Fig. 48 for a pure vapor, although the situation is similar for water vapor in air. Outside of the coexistence region, the Hugoniot curve is the usual one with q = 0. At the coexistence line there is a discontinuity in slope since within the coexistence region the amount of condensate g must be appropriate to an equilibrium state. The amount of heat that must be released for the transition from pl,v, to each point on the curve is q = Lg.

Coexistence line P

Isentrope, no condensation

V’

I/P

FIG. 48. Equilibrium Hugoniot curve in p - v diagram and condensation process in pure vapor.

In Fig. 48 we have also drawn a family of Hugoniot curves to represent the intermediate non-equilibrium states. We assume that each curve can be characterized by a parameter E , where 0 < E < 1, and represents a fixed departure from equilibrium. The flow follows the dry vapor isentrope (shown dotted) to Z,, where the supersaturated state collapses. It then

CONDENSATION I N WIND TUNNELS

387

proceeds along the Rayleigh line through the non-equilibrium states 1, 2, and 3, until the equilibrium Hugoniot is reached a t the weak detonation point 2,. The strong detonation point 2,’ cannot be reached, nor can Z, be reached from Z2’ in a continuous manner, because the necessary intermediate states do not exist. Heybey and Reed have also pointed out that an adiabatic shock at Z,, and a weak detonation a t Z, followed by a shock a t Z,, both lead to the same final state 2,’.

4. Saturated Equilibrium Expansion

In a saturated equilibrium expansion condensation occurs as a gradual process that starts a t the condensation line and maintains thermodynamic equilibrium throughout. We can expect this type of condensation process to occur when supersaturation is practically absent, as in some experiments on nitrogen and air condensation (e.g., Figs. 27 and 30). We shall see later that some of the experimental results obtained with nitrogen and air do actually agree with the theory of the saturated equilibrium expansion within a few percent. Since the flow during such an expansion is always in thermodynamic equilibrium, the analysis is greatly simplified. We do not have to obtain a condensation-rate equation from kinetic theory, but can use instead the Clausius-Clapeyron equation to complete the system of equations. Thetheory along these lines has been developed by Reed [59, 631, and in particular detail by Buhler [61]. The equations for the flow of a pure vapor in equilibrium with its condensed phase axe the conservation equations of mass, momentum, and energy, the equation of state, and the Clausius-Clapeyron equation. From our previous results, they are

(4-4)

(4.50)

(4.51)

(4.52)

(4.53)

dp

dw -+-+-=o ,

P

dp

w

-

_

P

dw (y - 1 ) W -

w

dA *

y M 2 dw 1--g w

_

_

dT +T

-

p

L c~T

-dg

-

= 0,

388


The momentum equation is obtained from (4.13) by the introduction of the Mach number defined in terms of the speed of sound at infinite frequency, (4.21). The energy equation is a simplified form of (4.25) for constant heat of vaporization and with c - c,, = constant since we have a pure vapor. Po -

FIG. 49. Ratio of zero-frequency speed of sound t o infinite frequency speed of sound ih pure vapor as a function of temperature.

The equation of state is likewise (4.12) with p = p,,. The last equation is an alternative form of the Clausius-Clapeyron equation (1.7). These equations can be solved in terms of the area, which we can regard as known. The results are

389


(4.55)

dA A '

dP ---

-

P

T/ Tc FIG.50. Mass fraction of condensate in pure vapor as a function of temperature from saturated equilibrium expansion theory.

(4.66)

390

P E T E R P. W E G E N E R AND L E S L I E M . MACK

I .o-

0.0 0.6 -

0.4

0.2

-

P PC

0.1 0.00 0.06

0.04

0.02

-

-

-

0.010.6

0.7

0.8

0.9

I .o

T/ Tc FIG.51. Pressure ratio in pure vapor as a function of temperature from saturated equilibrium expansion theory.

The entropy of the mixture must be constant since thermodynamic equilibrium exists throughout the expansion. Thus, (4.54) and (4.55) give dpldp at constant entropy,

391


This, however, is the square of the zero-frequency speed of sound, i.e., the speed of sound when the frequency is sufficiently low so that thermodynamic

0.6

0.7

0.0

0.9

1.0

T/Tc FIG.52. Density ratio in pure vapor as a function of temperature from saturated equilibrium expansion theory.

equilibrium can exist a t all times. Both Reed [84] and Buhler [61] have obtained (4.58) ; the former by means of purely thermodynamic arguments.

392


The square of the ratio of this speed of sound, sound speed, Vy(R/,u)T,is given by (4.59)

i2

the infinite frequency

1-g

-

___ R

i, to

~

0 0.68

0.76

0.84

0.92

I .oo

T/T, FIG. 53. Velocity ratio in pure vapor as a function of temperature from saturated equilibrium expansion theory.

We have plotted (4.59) for y = 1.40 in Fig. 49 as a function of TITc,where Tc is the temperature of equilibrium condensation (Fig. 8). The ratio has a discontinuity at the coexistence line, where TIT, = 1. Equations (4.56) and (4.57) yield a simple differential equation for g as a function of T whose integral is (4.60)


393

as was first found by Buhler [ S l ] . This means the saturated expansion theory can be given completely in terms of the integrated relations: (4.61)

FIG.54.

(4.62)

(4.64) Figures 50, 51, 52, and 53 give g, p/p,, pc/p, and W as functions of TITc for y = 1.40 and several values of the parameter cpTc/L. This parameter,

394


which occurred previously in (1.13), is plotted for air as a function of To for several Po in Fig. 54. We see that for large values of M,, say 5 or 6, the velocity is nearly constant during the expansion, with the result that the area ratio is closely approximated by the inverse of the density ratio. This 0.01c 0.OOE

0

Arthur, Nitrogen with 0.26 O/O C02

p, = 8.33 atm

O.OOE

T =292 O K 0.004

0.00:

Saturated Expansion Theory

0.002

P Po

0.00I 0.0008 0.0006

Isentrope for no condensation from measured impact pressure 0.0004

0.0003

I

I

1

I

I

x(in1 FIG.55. Comparison of saturated equilibrium expansion theory with Arthur’s experiments [54] with nitrogen.

means that Fig. 52 can be used to find TIT, for a given position in a nozzle; we are then able to obtain rapidly all the other flow variables. We can now compare the saturated expansion theory with a few experiments. In Fig. 55 are shown the experimental points of Arthur [54] for nitrogen with 0.26% by volume of CO,. In this experiment, nitrogen condensation began very close to M,. The measured pitot pressures were assumed to be those that would exist for no condensation, as discussed in Section IV, 5. This defines the effective area ratio as well as the constant stagna-

395


tion enthalpyisentrope. Theintersection of thisisentrope with the vapor pressure curve of solid nitrogen establishes 9, and T,. The value of the parameter cpTclL is 0.2576. With this parameter known, all of the flow quantities follow from the preceding equations or plots. The pressure ratio is shown in Fig. 55 in comparison with the experimental measurements. The agreement is within about 7% and demonstrates that the highly idealized saturated isentropic expansion theory bears a close relation to the actual expansion in this particular experiment. 0.030

0.026

Condensation shock plus

0.022

-

P -

pd 0.018

-

0.014

-

0.010

-

I

I

o Kubota,Air po = 14.6atm To = 394 OK I

I

FIG. 56. Comparison of saturated equilibrium expansion theory with Kubota's experiments [66] with air.

In Fig. 56 we compare the saturated expansion theory with experimental results obtained by Kubota [66] in air. We have used the Buhler method of plotting (e.g., Fig. 31). This example is of interest because some supersaturation was present. We see that the experimental points follow the zero condensation isentrope for some distance past M,, and then move toward the saturated expansion curve and finally above it. However, the collapse of the supersaturated state is not an isentropic process, and there is no reason why the experimental points should ever follow the saturated isentropic curve. We have attempted to include an entropy increase in the theory by

396


st. supposing a process in which the expansion follows the zero condensation isentrope to a point where a condensation shock occurs. The shock returns

4.0

4.5

5.0

5.5

6.0

65

M(pitot) FIG. 57. Calculation of flow parameters from saturated equilibrium expansion theory for conditions of Hansen and Nothwang’s experiment [53] with air.

the flow to saturation conditions, and the subsequent flow i s again a saturated isentropic expansion. We see from Fig. 56 that such a curve lies above the simple isentropic curve, and in the direction to which the experimental points appear to tend. Finally, we have in Fig. 57 a comparison of the theory with the experiments of Hansen and Nothwang [53], previously shown in Fig. 30. These experiments were camed out with air in a 25 x 36cm tunnel, which is larger than the ones used by Arthur and Kubota. We see that almost no


397

supersaturation was present ; the experimental points lie above the theoretical curve with a somewhat different slope. Actually we cannot expect experimental results obtained with air to follow exactly the theory as calculated. What we have taken as the vapor-pressure curve of air is really the condensation line, i.e., the locus of states where condensation first occurs (see Section I). As condensation continues, the proportions of the components in the air will change, and the actual vapor-pressure curve will depart from the condensation line. Also in Fig. 57, we have shown all of the other flow quantities calculated by the saturated expansion theory for the conditions of the Hansen-Nothwang experiment. We show for comparison the same quantities as calculated from the measured pitot-stagnation pressure ratio, assuming no condensation. An interesting point is that the density is almost identical in the two cases. The velocity changes only a few percent during the expansion, as mentioned previously, and is little different from that in the expansion without condensation. 5. Shock Waves with Vaporization

When a normal or oblique shock wave occurs in a flow in which condensation has taken place, the temperature rise behind the shock will cause some, or all, of the condensate to vaporize. For this discussion we assume that vaporization will take place instantaneously, and, therefore, the vapor and condensate are in thermodynamic equilibrium behind the shock wave. The vaporization will change the flow variables from their usual values in condensation free flow. In addition, the angle of an oblique shock wave for a fixed flow deflection will be changed. We have shown in Fig. 39 the intersection point, V , representing a normal shock wave with vaporization. The Mach number behind the shock is smaller, the pressure is higher, and the temperature lower, than for the corresponding adiabatic shock. This solution is obtained from the equations (4.35), (4.36), (4.37), and (4.38), when q < 0, in accordance with the physical fact that heat must be supplied to vaporize the condensate. The equations are suitable for moist air and steam because the condensed mass is usually small. However, for nitrogen and air condensation the condensed mass fraction may be as large as 10-20%, and we must account for the change in the vapor mass across the shock. Such a generalized analysis has been performed by Buhler [61] and Ross [GI. We first discuss the simple situation where the mass change can be neglected. The effect of condensation on the measured pitot pressure, a problem first considered by Taylor [86], is of considerable interest. If we assume that complete vaporization occurs, Eqs. (4.44), (4.45), (4.46), (4.47), (4.48), and (4.49) provide, when we choose the plus signs, a solution of this problem for M , and amount of condensate being fixed. However, at a fixed

398


position in a wind tunnel downstream from a condensation region, the Mach number and other flow variables differ from their values in condensation free flow. To find the combined effect of the condensation and subsequent vaporization at the pitot tube on the measured pitot pressure, we must first consider the effect of a condensation shock on the downstream flow. This is done in Section‘VI, and we defer the discussion of the pitot pressure until then. A second point of interest is the effect of vaporization on the angle of inclination of an oblique shock for a fixed flow deflection. For a definite amount of heat absorbed, Head [lo] calculated the apparent Mach number obtained from the measured shock angle and the usual oblique shock theory, as a function of true Mach number for several different flow deflection angles. He found that the shock with vaporization is inclined at a smaller angle to the flow than otherwise, and that the apparent Mach number for sizable flow deflections can be much larger than the true Mach number. This same result can also be obtained from the work of Samaras [74] and the more general analysis of Ross [85], where a sample shock polar diagram is given. We turn now to normal and oblique shock waves in partially condensed nitrogen and air and shall follow closely the analysis of Buhler. The equations are just those used for the condensation shock (4.36), (4.37), and (4.38) except that the equation of state must be in the form (4.10),and q = - Lg. We consider only those cases where complete vaporization takes place. At low Mach numbers it is possible that the temperature rise is not sufficient to satisfy this condition. It is convenient in this analysis to introduce a new Mach number which is proportional to the ratio of dynamic to static pressure, (4.65)

According to the saturated expansion theory, this Mach number happens to be very close to the Rayleigh Mach number found from (3.5) for the measured pitot-static pressure ratio [61]. From the continuity, momentum, and state equations, (4.36), (4.37), and (4.10),we can obtain (4.66)

where subscripts 3 and 4 refer to conditions in front of and behind the shock respectively. The state and continuity equations give (4.67)


399

Then, from (4.66) and (4.67) we find (4.68)

T/ T, FIG.58. Shock theory parameter as a function of temperature.

When we solve (4.67) for ua/u3,use (4.68), and substitute in the energy equation (4.38),we obtain

where (4.70)

400


For complete vaporization behind the shock, g, = 0 and M , = M,. Therefore, when the conditions in front of the shock are known, (4.69) can be solved

FIG. 59. Effect of vaporization on m e pressure ratio across a normalshockinair [61].

for M,, and the pressure ratio across the shock follows from (4.68). In Fig. 58, we have plotted E in the usual manner as a function of TIT, for several values of c,T,/L. In Fig. 59 [61] we show the ratio of the pressure ratio across a shock with vaporization to the pressure ratio across an adiabatic shock as a function of M3. The dotted lines represent the limits above which the assumption of complete evaporatiQn is invalid. We note that a t high Mach numbers, evaporation has only a slight effect on the pressure ratio. Information on the assumption of vaporization immediately behind a normal shock is supplied by an experiment of Grey [62] with air. He


401

established a normal shock in front of a cylindrical duct of large diameter in which the flow was choked. A pitot survey in the region behind the shock showed that the pitot pressure was constant. If vaporization did not occur until some distance behind the shock, a change in pitot pressure should have been observed. Light scattering experiments of Hansen and Nothwang [53] also indicated that vaporization occurs right a t the shock. We are now in a position to find the pitot pressure in a saturated expansion, and compare it with the pitot pressure at the same location in a nozzle with no condensation. We may write this ratio as (4.71)

We specify the effective area ratio and find the corresponding flow conditions from the saturated expansion theory. Then the first factor on the right is obtained from isentropic flow tables once M , has been determined from (4.69). The pressure ratio across the shock is found from (4.68). The third factor is known from the saturated expansion theory. The fourth is determined by the nozzle supply conditions and the vapor pressure curve. Finally, the last factor is obtained from isentropic flow tables for the given area ratio. We have carried out this calculation for the conditions of the Hansen and Nothwang experiment presented in Fig. 57. The result is that a t an area ratio of A / A * = 70, the pitot pressure for the saturated expansion is 1.1% higher than for the expansion without condensation. At smaller area ratios, this number is much smaller. Calculations for other conditions indicate that even a t area ratios which correspond to M = 10, the saturated expansion produces a pitot pressure only about 3% higher than in the absence of condensation. Therefore, experiments performed with heated and unheated supply conditions as well as equilibrium saturated expansion theory agree in the result that for the same effective area ratio the measured pitot pressure is almost unaffected by condensation. This validates our previous procedure of determining experimental isentropic and constant stagnation enthalpy expansions from pitot-pressure measurements in condensed flow. The theory developed for the normal shock wave can easily be extended to oblique shocks by the usual device of considering the flow normal and tangential to the oblique shock. The normal flow is governed by the normal shock relations ; the tangential flow requirement is that the tangential velocity component be equal on both sides of the shock. An interesting question concerns the strength of the oblique shock necessary for complete evaporation. The results of a calculation by Grey for a fixed supply temperature and two supply pressures are shown in Fig. 60. We see that for the larger pseudo Mach numbers % a large flow deflection is needed to produce an oblique shock of sufficient strength to cause complete vaporization.

402

PETER P. WEGENER AND LESLIE M . MACK

In the limit of very small flow deflection, 6, the oblique shock reduces to a Mach wave. From the equality of the tangential velocity components, and the momentum equation for the normal components, we obtain for an infinitesimal oblique wave [61],

9= Y M tan ~ eas,

(4.72)

P

4.0

4.5

5.0

-

5.5

6.0

M

FIG. 60. Supply conditions required for complete vaporization behind oplique shock waves [62].

where 8 is the angle of the wave. This equation permits an experimental determination of the effective speed of propagation of infinitesimai waves. Since is very nearly the Rayleigh Mach number, a measurement of dpld6 for dp -+0 gives tan 0, and, therefore, the effective Mach number. If the


403

proper Mach number is the one defined in terms of the low frequency speed of sound i, (4.72) can be written (4.73)

This relation can be used to predict the pressure coefficient on slender bodies in condensed flow.

V. KINETICSOF CONDENSATION

We have seen in Section I V that purely thermodynamic considerations are not adequate for a complete description of condensation except in those flow processes where thermodynamic equilibrium exists at all times. This equilibrium does not prevail in the supersaturated state and subsequent collapse region (C,DF2 in Fig. 37), which makes prediction of the onset and course of the condensation process by thermodynamic methods impossible. The description of this part of the condensation problem, as well as the prediction of whether or not supersaturation is present, is of necessity a problem outside the scope of thermodynamics. In this section we shall first show that condensation phenomena in high speed flows, like other condensation processes, are linked to the availability of surfaces of condensation. These surfaces, or condensation nuclei, come principally from three different sources: a) inert impurities in the nozzle supply such as dust, etc.; b) small drops (or crystals) formed in the flow by condensation of vapor impurities of higher vapor pressure than that of the main constituents; and c) small drops (or crystals) formed in the condensing vapor itself by statistical cluster formation due to fluctuations. We must first investigate which of these three possibilities is applicable in each typical process, and determine the available number and rate of production of the nuclei. The prediction of the availability of nuclei will provide a determination of the point in the expansion where condensation should occur (point D in Fig. 37). Once condensation nuclei are present in sufficient numbers, it is necessary to understand their further growth in order to predict the thermodynamic state in the collapse region. We shall find that the prediction of nucleation rates and droplet growth still defies exact calculation except in the simplest cases. The predictions are difficult not only because the detailed condensation mechanism is still the subject of some speculation, but also since many of the thermodynamic properties required in the calculations are in doubt or unknown. However, the kinetics of condensation during expansions in nozzles is qualitatively well described,

404


and approximate answers for the functions J ( l and ) G ( E , x ) in the condensation rate equation (4.32) can be given. Because of its limited application in flow problems, it is unnecessary to discuss the details of the nucleation theory, which has been summarized by Volmer [87], Frenkel [6], and recently Becker [88]. 1. Condensation Nuclei and Classification of Condensation Processes

Surface phenomena play a decisive role in processes of phase transition. Vapor in a nozzle cannot condense upon reaching saturation with respect to a plane liquid phase if either this phase is absent or no “cold” container walls are available to serve as surfaces of condensation. Owing to the presence of “hot” boundary layers surrounding the flow, the nozzle wall is ruled out as a surface of condensation. Thomson [89], von Helmholtz [go], and Gibbs [91] found that the pressure of a vapor in equilibrium with a small spherical drop of the liquid a t a given temperature increases with decreasing droplet radius Y. Therefore, a vapor may be supersaturated with respect to a drop of large or even infinite radius, and not saturated with respect to small droplets. For the mechanical equilibrium of a spherical droplet with the surrounding vapor [e.g., 61, we have p,-p,=2-.

(T

Y

The pressure in the drop, p,, is larger than the surrounding vapor pressure, $, by the amount 2017, where (I is the surface tension of the liquid, with dimensions forcellength, or work/length2. The surface tension is generally a linearly decreasing function of temperature, but, unfortunately, it is not known experimentally at low temperatures for the highly supercooled liquids of interest to us. To extrapolate the available experimental results to lower temperatures, we may resort to the semi-empirical expression of Eotvos and Ramsay-Shields [e.g., 6, 141,

where vL is the specific volume of the liquid and Tcrit.the critical temperature of the vapor. The surface tension vanishes when T = Tcrit,;for many liquids the ‘constant is 2.12. Although many factors such as impurities, solvents, etc. affect surface tension strongly [92], it is little changed by the nature of the surrounding vapor. A t T = 20 “C, the surface tension of water in water vapor is 70.60 erg/cm2,while in air it is 72.75 erg/cm2. For 0, and N, a t 70 OK, the values are 18.43 and 10.72 erg/cm2, respectively, according to the newest work by Keilly and Furukawa [93]. Extrapolations to lower


405

temperatures based on (5.2) or obtained by other methods are available for H,O [e.g., 27, 281, and for 0, and N, [eg., 61, 641. Application of the stability criterion that the thermodynamic potential of the system of the spherical drop of liquid and the surrounding vapor must be zero leads, with (5.1), to the familiar Thomson-Gibbs equation [eg., 61 for the vapor pressure of the droplet, (5.3)

where p L is the liquid density [8,9, 931. From (5.3) we obtain for the radius of a droplet in thermodynamic equilibrium with the surrounding vapor (5.4)

y*

2up 1 ~ L R lTn v

=____,

where y is the saturation ratio, (2.1). In the limit of pv = fim,or y = 1, = 0 0 , (the droplet is infinitely large). At constant temperature, decreasing droplet size leads to increasing equilibrium vapor pressures, as shown for water vapor in Fig. 61. The solid curve was computed from (5.4) (1 Angstrtim = 10-8 cm). We have applied (5.4) down to very small droplet radii without concern about its region of validity. The range of very small droplets, with radii of the order of the molecular radius, is of concern to us in extreme cases of supersaturation, as shown in Fig. 62. In this diagram the moist-air condensation experiments [39, 401, previously shown in Fig. 17, are plotted as saturation ratio a t condensation against the temperature a t condensation. Also shown (solid lines) are values of stable droplet radii computed from (5.4), which indicate that a t condensation the droplets were of the order of the size of the molecule. The molecular radius of H,O is about 2.3 8,as determined from viscosity measurements [eg., 941. However, we cannot expect (5.4) to hold for Y < lO-'cm, say since the Thomson-Gibbs formula was derived on a continuum basis and with a constant value of surface tension. Actually, for very small droplets, surface tension is a decreasing function of the drop radius. Much effort has been expended to account for surface tension correctly on thermodynamic grounds or to determine the equilibrium droplet conditions statistically in order to find expressions corresponding to (5.4) for small clusters of molecules. Tolman [95] extended earlier ideas of Gibbs [91, p. 2321, and arrived by thermodynamic methods a t a first approximation for the surface tension of a drop of radius r p , , which is smaller than the surface tension of the plane surface urn: Y*

(5.5)

406

PETER P. WEGENEK AND LESLIE M . MACK

where 6 is a constant for a given liquid. This constant is of order 1 A, which is approximately equal to the intermolecular distance in the liquid. Kirkwood and Ruff [96] showed qualitatively by statistical methods that the trend of (5.5)is correct, as was earlier indicated for the free surface energy of crystals by Kossel [97]. For greater accuracy higher approximations must be found when only a few molecules are in one drop; Gilmore [98] discusses such results. Also shown in Figs. 61 and 62 are dashed lines for the equilibrium I00

I

I I I

---Ir Pa, 0

0

- P” --

p,

with surfoce tension correction

------------I 50

I I00

r(A)

FIG. 61. Vapor pressure of a water droplet in water vapor at 30 “C as a function of drop radius.

vapor pressure of a water drop and its radius, respectively, with surface tension based on (5.5); and large deviations from the result computed with (5.4) for small radii are seen. Stever and Rathbun [64] arrived at qualitatively similar answers by considering spherical molecules packed into spherical drops. All of these treatments are approximations which extend continuum notions to small droplets. Reed and Herzfeld [99, 1001 made calculations of equilibrium droplet conditions by building up nitrogen drops from individual molecules and determined the molecular cluster arrangement of lowest energy under the assumption of a Lennard- Jones potential. It is interesting to note that for nitrogen in the range of 2 to 8 molecules per “drop”, the results fair reasonably well into those obtained from macroscopic

407


considerations without a surface tension correction. Finally, Kuhrt [loll recently demonstrated that very small clusters, even when surface tension corrections are applied or treatments like those of Reed are used, cannot be considered as drops at rest in an infinitely extended vapor space. Small clusters are similar to very large molecules, thus they are subject to Brownian motion. When the translational and rotational motion of the extremely small drop is included, the Thomson-Gibbs formula (5.3) changes to

0

210

2 20

230

240

2 50

T(OK) FIG.62. Droplet size in the range of water vapor condensation with large supersaturation from Wegener [39]. Droplet radii from the Thomson-Gibbs formula (solid line) and with a Tolman surface tension correction (dashed line).

where n denotes the number of molecules in the drop. A substantial correction appears if n is small. It has long been known from Wilson’s classical experiments in cloud chambers [30] that fog formation in vapors occurs at varying degrees of supersaturation, depending on a variety of conditions in the cloud chamber. Also von Helmholtz, as early as 1886 [go], observed that a jet of saturated steam issuing into the atmosphere remained clear for some distance before

408


clouding. Whenever condensation is actually observed, it takes place on condensation nuclei whose size approximately corresponds to that given by (5.4)for a saturated equilibrium condition. For condensation phenomena in high speed flows we must, therefore, investigate the nature of the condensation nuclei that initiate immediate or delayed condensation. The slightly delayed condensation in cloud chambers takes place on particles such as dust, ions, etc. The same process is found in the atmosphere, where cloud formation is possible only in the presence of condensation nuclei; in fact, clouds may be produced artificially by introducing just such nuclei (e.g., silver iodide or solid carbon dioxide) into air layers mixed with slightly supersaturated water vapor. The number of foreign particles ordinarily present in the atmosphere is variously found to range from, say, lo2 to lo6 particles per cubic centimeter [31], depending on many extraneous factors. Wind-tunnel air is first drawn from the atmosphere, and may in addition carry small particles produced by the dessicants of the drier systems. I n hypersonic wind tunnels, oxidation of heating elements may contribute small pieces of foreign matter. Oswatitsch [33] first demonstrated that these customary condensation muclei must be discounted as being responsible for the collapse of the supersaturated state in nozzles operated with moist air because of their insufficient number. A conservative estimate would be that N = 105/cm3particles are present in a nozzle supplied with laboratory air. In a typical supersonic nozzle an element of the flow may travel 10 cm between saturation and condensation in about 3 x 10-4sec. We assume that during this flight time enough water vapor condenses on every particle cm. At the end of this to result in the unlikely large drop size of Y = flight path, there will be lo5 spherical droplets of Y = cm per cubic centimeter with a total amount of water equal to about 4 x 10-'g/cm3. The heat of vaporization released in the flow by this phase transition is not sufficient to change the stagnation enthalpy of the flow measurably, and no flow parameters could be affected, in contrast to the actual observations. Therefore many more nuclei are needed for condensation than are provided by impurities in the supply. This is even more true for nitrogen or air expansions owing to the lower heat of vaporization. Indirect evidence for the fact that foreign particles do not play the customary role of condensation nuclei in moist air expansions rnay be seen by inspection of water vapor condensation data obtained in a great number of experimental facilities. Correlation of the results in Fig. 19 was possible without reference to the foreign nuclei present. A discussion of the size, distribution, and number of particles measured by the light scattering technique in a 'hypersonic air nozzle by McLellan and Williams [58] provides evidence that foreign matter nuclei do not enter the air condensation process. The situation is somewhat different for steam [31]. In expansions with small temperature gradients of industrial steam containing many foreign nuclei foreign nucleation may occasionally


409

affect the onset of condensation. This is also apparent in cloud chamber experiments, whose time scale is much “slower” than that in nozzles [30]. The situation is different when we consider the number of nuclei produced by the condensation of vapor impurities which are present in hypersonic wind tunnel nozzles. Figure 6 shows that possibly H,O, CO,, and A will condense before the Mach number of equilibrium condensation for air is reached in a nozzle. The mere fact that liquid droplets or crystals of some substance are present when another vapor enters the coexistence region is not an obvious condition for condensation. A particular combination of the geometrical structure and size of the condensation nucleus and the molecular force field of its surface layer with respect to the condensing substance is needed to initiate the adherence of the condensate [6]. These conditions are difficult to predict in detail, and we must resort to our previous empirical experience concerning the effectiveness of carbon dioxide and water vapor droplets as condensation nuclei for nitrogen as shown in Figs. 25 and 26. We shall apply this experience directly to air since nitrogen is its major component. In particular, we would like to explain those observations where air condensation began at or near M,. At an early stage several investigators [59, 53, 61, 581 suspected that previous condensation of impurities initiates air condensation. Simple considerations indeed show that an extremely high number of nuclei can be produced. As an example, we can estimate the number of water droplets present a t the point of air condensation. By the definition of the mixing ratio, (2.3), we may write for the vapor density pv (5.7)

If we assume all water vapor to have condensed into N spherical drops per cm3, we obtain 4 pv = % n r 3 p ~ N .

The air density a t M , is (5.9)

and, with (5.7), (5.8), and (5.9), we find for the estimated number of water vapor droplets at M,, (5.10)

410


For a given nozzle, x is known, and M , can be found for given supply conditions from Fig. 5. Figure 63 shows the results obtained from (5.10), with

x ~ 2 . 1x

2.6 x

10-3

10-4

,2.4 x 1 0 - 5

FIG.63. Estimate of number of water droplets present in a nozzle as a function of radius and mixing ratio a t M , (air) for Po = 7 atmos, To = 288 "K.

10 8

6

4

P" 43

2

1

FIG.64. Critical droplet radius as function of saturation ratio for air at 64 OK from the Thomson-Gibbs formula.

Po = 7 atm,

To = 15 "C, and a typical range of low mixing ratios. Application of (4.44) shows that condensation of such a small amount of water


411

vapor cannot affect the pressure of the flow measurably in the water-vapor condensation range, 1.2 < M , < 2. The further growth of water droplets up to M,-4.2 cannot possibly be calculated accurately. For instance, Hansen and Nothwang [53] estimate the droplets at M , to have a radius of about 30 Angstrom, and Buhler [Sl] and Arthur [54] give such calculations in more detail. However, light scattering observations of McLellan and Williams [58] indicate droplet sizes of the order of 200-400 Angstroms a t M,. In their experiment carbon dioxide, whose normal concentration x = 0.46 g CO,/kg air, was present and would provide a number of droplets equal to the number shown in Fig. 63 for water vapor when x = 2.6 x In any case, Fig. 63 shows that from lo7 to lo1, centers of condensation are present a t M,, a number that vastly exceeds the number of nuclei due to foreign matter. Figure 64 gives the critical droplet radius of air at M,, obtained from (5.4),as a function of saturation ratio [63]. Since we know empirically that water vapor droplets are effective nuclei a t least for nitrogen, we can now match McLellan and Williams’ observed radii of 2 to 4 x cm with those of Fig. 64, and see that such drops already correspond to practically plane condensation surfaces. Therefore, we are not surprised to find very little supercooling of air in the larger nozzles with ordinary supply conditions. We are now able to classify the previously described experimental results for all types of condensation from the nucleation point of view:

A. Process Absence, or unimportance, of all foreign impurities and traces of vapors of lower vapor pressure. Collapse of supersaturated state due to self-nucleation in the condensing vapor. Degree of supersaturation controlled by initial conditions and time scale (nozzle geometry).

Examfdes: Water vapor in air (nozzle Mach number M Pure steam. Pure nitrogen.

< M , for air).

References: Moist air: [lo, 25, 26, 27, 28, 29, 33, 34, 37, 39, 40, 431. c Steam: [7, 31, 47, 48, 49, 50, 511. Nitrogen: [52, 53, 54, 551. B. Process: Large numbers of condensation nuclei present in droplet or crystal form because of previous condensation of vapor impurities such as H,O and CO,. Little or no supersaturation observed.

412

PETER P. WEGENER AND LESLIE &I MACK .

Examples: Air in the larger hypersonic nozzles with ordinary drying methods of supply air. References: [e.g., 53, 57, 581. C . Process Varying numbers of condensation nuclei present in flow, depending on vapor purity, self-nucleation, and time scale (nozzle geometry). Supersaturation controlled by processes A and B may range between both limits. Examples. Air in the smaller hypersonic nozzles. Nitrogen with varying degrees of impurities. References: [54, 55, 59, 60, 62, 63, 64, 661. 2. Spontaneous Nucleation

In the absence of foreign condensation nuclei of all kinds centers of condensation must be created in the supersaturated vapor itself. A new phase may appear first in microscopic dimensions without any change of the thermodynamic parameters of the macroscopic system. The phase change is then due to a local parameter change only. On the molecular scale, the number of molecules in any small volume of a gas will continually fluctuate as molecules enter or leave the volume. In fact, the very basis for a state of thermodynamic equilibrium is the transport of molecules and energy from one place in the system to another. This continual exchange is needed for the adjustment to equilibrium after each change of the macroscopic parameters. From this molecular interplay the steady state of a large volume of gas at uniform density results, with the entropy having a maximum value. This maximum value of entropy, however, may never be assigned to a very small subvolume in which we observe entropy fluctuations as a result of the changing number of molecules. Molecular collision processes in the presence of entropy fluctuations may then lead to accidental cluster formation in which two or more molecules may form an embryonic drop for some period of time. We call such a cluster a nucleus, or droplet of critical size, if it has reached a stable size in its given macroscopic environment. This stable size is given by the Thomson-Gibbs formula (5.4), or its modifications. When the Boltzmann relationship between entropy and the probability of a certain state is solved for the probability, we find it to be proportional


413

to the Boltzmann factor, exp (- A S / k ) , where A S is the entropy decrease and k is the Boltzmann constant. Volmer [87] reasoned that this relationship, well-known in kinetic theory, is also applicable to the problem of formation of critical-size droplets in a supersaturated vapor. We can write the entropy decrease, A S , as

W A S = -,

(5.11)

T

where W is the work required to produce a droplet of critical size reversibly and isothermally. Our interest in the formation of a new phase in the vapor concerns the rate of production of stable nuclei per unit volume, J , which we may now write (5.12)

J

=K

exp

(- $)

=Kexp

(- % : ).

The Boltzmann constant is given by k = R/NA, where N A equals 6.02 x loz3 molecules per mole. The exponential term describes the probability of the formation of one stable drop, and K is a constant of proportionality which we expect to depend on the thermodynamic state of the vapor. Next we derive, on thermodynamic grounds, the work needed to produce a spherical liquid drop from the vapor phase, as first shown by Gibbs [91]. In the process of producing a droplet, work must be done against the surface tension to increase the drop surface area from zero to 4nr2. This work is

W , = 4nr20,

(5.13)

where (T is taken to be independent of the radius. Furthermore, during droplet formation the volume is increased from zero to 4nr3/3. From (5.1) we remember that the pressure inside the drop, p,, is higher than that of the surrounding vapor p,, by the amount 2alr. The work expended in the growth is, therefore, equal to the product of the final volume and the pressure differential between the inside and outside of the drop, (5.14)

8 w,= (p, - p") 45 ?zr3 =,nrz(T.

The net work required for the formation of the droplet is then (5.15)

W

=

4 W1 - W - -nr%. ,-3

The work needed to produce one stable droplet of critical size is, by (5.4), (5.16)

W=

16na3,u 3R2T2pLzln2y '

414


We observe that (5.15) may also be written as 1 W=-Ao. 3

(5.17)

Volmer [87] has shown that this result is also true for the formation of a stable solid cluster of molecules, or crystal of quite general shape. The surface tension must then be replaced by the free surface energy of the crystal in contact with a vapor. Some assumptions must also be made on the shape of the cluster [38]. Becker and Doring [lo21 point out that the droplet, or crystal, growth in the preceding analysis must be regarded as expressing the mean growth of the surface. None of the above considerations are applicable to single molecules, or even very small clusters, because of the thermodynamic nature of the reasoning. If there are only a few molecules in one cluster, the surface growth proceeds in a large discrete step when a single molecule is added or removed. Only a difference equation, such as that applied by Reed [99], can describe this initial cluster formation, while all other analyses treat a cluster as a uniformly extended mass of some arbitrary shape. Mathematical investigations of phase stability [e.g., 61 cannot be discussed here beyond a simplified physical model. If a drop of exactly the size prescribed by Thomson’s formula, (5.4), accidentally absorbs just one more molecule, the surrounding vapor will be supersaturated with respect to the drop, which will then grow. Conversely, if the drop loses one molecule by accidental evaporation, the surrounding vapor will be superheated with respect to the drop, which will then evaporate completely. Droplets of critical size, or larger, can grow, which leads to a collapse of the supersaturated state. I t is this metastable condition of the supersaturated state that leads to the collapse of the equilibrium with respect to the small nuclei formed according to (5.12) and explains the sudden condensation observed in those high speed flows where no particles of foreign matter or previously condensed vapor impurities are initially present. The rate of formation of clusters of critical size from (5.12) and (5.17) is (5.18)

where A* is the .surface of a cluster in unstable equilibrium with a given environment. For the special case of spherical liquid droplets of critical size, whose work of formation is expressed by (5.16), we obtain as nucleation rate (5.19)

J=Kexp

[

16n N A

-__ 3-

0

1

Rd(FY($)21nY(p[ipmll


415

or

J

(5.20)

=K

[

($(:)&]

exp - 17.49 -

According to (5.20) there is a completely defined nucleation rate for a given supersaturation. The very strong dependence of J on the supersaturation q.~may be illustrated, as done by Oswatitsch [33], for a moist-air expansion in a supersonic nozzle with an atmospheric supply. In terms of the supercooling rather than the supersaturation, we compute (for some assumed value of K ) that for A T = 30 "C less than one nucleus is formed per cm3 for one centimeter of travel in the flow direction. For A T = 40 "C, this number is lo6, and for AT = 50 "C, it is loll. In other words, the moist air would have to travel about 10 meters at slightly supersonic speeds to contain lo3 nuclei/cm3 if A T = 30 "C, while for a supercooling of AT = 40 "C, the same number is formed after millimeter travel. For AT = 50 "C, 108 nuclei would be produced for every millimeter travel. In Section I1 we have seen that this is indeed the range of supercooling observed. Obviously, the number of 106 foreign condensation particles per cm3 is insignificant with respect to the nuclei formed statistically in the vapor itself. Any uncertainty in the knowledge of K in (5.20) simply shifts the location of rapidly increasing nucleation rate to a slightly different flow temperature. This shift will be small with respect to the extension of the collapse zone of the supersaturated state in the flow direction. The same uncertainty appears in the prediction of delayed condensation in cloud chambers [%I. Becker [88]proposes the following simple physical picture for a rough estimate of the determination of K . Let us consider nucleation as a game in which each collision of two molecules can lead to the formation of one nucleus. The Boltzmann factor may then be regarded as an expression for the chance to win, i.e., the probability that a given collision actually results in a cluster formation. K must then simply be the number of gas-kinetic collisions per unit volume and time (this result has also been stated by Volmer [87]). To gain insight into the magnitude of the various terms, we estimate on this basis the minimum saturation ratio q ~ ,below which no condensation in pure water vapor could take place in the absence of foreign nuclei. In Figs. 20 and 21 we saw that steam condensed in Binnie and Woods' nozzle 50 "C and a temperature slightly above room temperature. At a t AT, normal pressure and temperature, the collision frequency of the molecules is p 1O1O sec-1 and NL = 2.69 x 101g/cm3is the number of molecules per cm3. From Becker's argument we find K = pNL loas sec-l ~ r n - ~ .Inserting the proper numerical values for water vapor into (5.20), we obtain N

N

(5.21)

-

416


-

We may now assume J(lng,) 2 , say, as the beginning of nucleation, since we saw before that a “reasonably” arbitrary choice of J would result only - ~ in (5.21), in a small error in this estimate. To obtain J = 1 ~ m sec-l we must have ln2g,= 1.7, or loglog, -0.6. Indeed this value of the 300 OK supersaturation, g, 4,is in the range of the observed values a t T in Fig. 21. Historically, the first successful attempt to solve (5.18) for the case of liquid droplets formed in the vapor was made by Volmer and Weber [103]. In this treatment the simplifying assumption was made that the macroscopic state of the vapor is maintained in the system by adding a number of molecules equal to the number absorbed by every drop that has reached t h e critical size given by (5.4). Furthermore, it was assumed that every molecule striking a small cluster is absorbed when its center of gravity has penetrated a spherical shell about the drop in a stand-off distance of the order of the molecular diameter. Finally, all evaporation was prohibited. For the first assumption to be valid, the droplet growth must be slow with respect to the time scale of molecular collisions. Ordinarily, this assumption of a quasisteady state is valid in flow processes, since the time scale for molecular collisions is of the order 10-lo seconds. The thermodynamic treatment requires first the knowledge of the Boltzmann distribution of molecular aggregates of all sizes. Throughout the entire system of droplets and vapor, there is then steady transport of molecules from the vapor phase into clusters, which are removed after they reach the critical size and the number of molecules in the clusters is replaced by vapor. Volmer’s formula was improved by Farkas [lo41 on the basis of kinetic, rather than thermodynamic, calculations, and his results are equal to Volmer’s for the first instant of production of nuclei as long as the boundary conditions have not changed. This work was again improved by Volmer [87]. Volmer’s final expression, in which he has left out a number of terms that approximately cancel each other, is

-

(5.22)

-

K

ZDv

(-)

=2

r*

3W nkT

[cm-3 sec-11.

2 is the number of molecules per unit volume, kept constant in the system,

v, is the volume of one liquid molecule, and D is the number of molecules impinging on unit area per unit time. In Volmer’s treatment the condensation coefficient, a, which gives the ratio of molecules absorbed in the liquid drop on impact to those not absorbed has been set equal to one, although physically it is possible that 0 a 1. W has previously been obtained in (5.16)Volmer shows further that

<
> Y. Farther downstream, 1 Y,and the droplet growth must be computed from macroscopic considerations.

-

-

-

-

424


The following remarks are first restricted to pure vapors, and the nature of the initial droplet of radius r0 remains unspecified. It may have been produced by self-nucleation, previous condensation of a trace of vapor, or it could be a nucleus of foreign matter. Although we shall first follow the growth of a single droplet, we must bear in mind that the expressions to be derived will be valid only in the average taken over an actually large number of droplets. Finally, the laws of droplet growth are strictly valid only if the given thermodynamic environment is unchanged during the period of growth. The application of these laws to the rapidly changing states in an expanding flow results in a quasi-steady analysis. For free molecular flow we can determine the mass added to a droplet from the rate of impingement given by (5.24) [e.g., 33, 1111. With (5.38) and the equation of state, we find for the mass impinging on unit area in unit time (5.40)

Dmn= f p v c ,

where m, is again the mass of one molecule. We can write (5.40) in terms of the speed of sound as (5.41)

The mass increase of a spherical drop with volume v (5.42)

=

4nr3/3 is given by

dG -

dv dv dr dr - P L - = p~ - - = p~ 4nr2 - .

dt

dr dt

dt

We can express the same quantity by (5.43)

dG - = ctDmn4nr2, dt

where ct, the condensation coefficient expresses again the fraction of the impinging molecules that actually stick to the drop surface. From (5.41), (5.42), and (5.43), we obtain for the increase of the drop radius per second,

(5.44)

With the quasi-steady assumption, dt Mach number, we have finally (5.45)

= dx/w,

and the definition of the

425


suitable for calculations in a nozzle (the quantities on the right hand side are now functions of x ) . In obtaining (5.45),we have used the condensation coefficient and thus avoided discussion of the detailed molecular processes a t the drop surface. Oswatitsch [33] and, particularly, Buhler [61] have given a refined treatment of droplet growth. Buhler’s quasi-steady solution, which includes an expression for u, is (5.46)

5

.

-

I

II

I

5

6

,

I

I

7

8

9

M FIG.67. Droplet growth rates in air for r < 1, as a function of Mach number from (5.46) as given by Buhler [61].

where T,, the surface temperature of the drop, may be chosen as the equilibrium saturation temperature of the vapor with respect to the drop. Droplet growth in air, computed from (5.46) by Buhler, is shown in Fig. 67. Buhler also derived an approximate solution for the non-steady case, which results in appreciable differences in the initial phases of growth. Once the droplets are of the order of the mean free path, further growth is governed by macroscopic processes. The drop absorbs the heat of vaporization when condensation takes place on its surface, and its growth will be limited unless this heat can be carried away by conduction, since radiation

426


is negligible for the small temperature differences involved. balance may then be written as

The energy

(5.47)

where k is the heat conductivity of the vapor, and 7 is a radial coordinate. At the drop surface 7 = r , and if we assume a linear temperature gradient a t the drop surface, we may set Ts - T r

(5.48)

which is in accordance with the accurate solution [ l l l ] . With (5.42) and (5.48), we can then write (5.47) as (5.49)

or also (5.50)

The important difference between the molecular (5.44) and the macroscopic (5.49) growth laws is that in the latter case the drop radius grows with the square root of the time in a constant environment. For a complete calculation of the condensation process, we must decide which growth law to use in (4.32) by comparing the order of magnitude of r with 1 a t every position in the nozzle. However, the x-coordinate a t which the changeover is made may be somewhat in error without seriously affecting the final result because of the small size of the droplets a t this point. When we apply the droplet-growth laws, on the other hand, to condensation of air, which takes place on nuclei produced by previous condensation of vapor impurities, it will be much more important to know the number of nuclei available a t M , than precise knowledge of the changeover point [61]. When we estimate the droplet growth for moist air, rather than a pure vapor, we use pv and y for water vapor in (5.45). I n the macroscopic growth law (5.50),the heat conductivity of air is used for k. More discussion of both laws and other variants for moist air condensation are given by Oswatitsch [33] and Wakeshima [115]. There is considerable uncertainty in choosing a suitable value for the condensation coefficient a in (5.45). I n the formation of clouds, it is found that a 0.03 [e.g., 1171, but there is reason to suspect that the value is 0.3 or higher in situations where extreme supersaturation is observed [log, 114, 1151. On the other hand, a may be adjusted to fit the data for a given

-


427

experiment. In particular, this must be done when the molecular growth law must be applied without knowing whether we are dealing with liquid drops or crystals. There are no known direct physical measurements on droplet growth in the molecular regime. Drop growth in the macroscopic region has been indirectly measured in cloud chambers [30]. The velocity of the drops after the expansion is measured; and, assuming that they fall with their terminal velocity, Stokes' law (5.33)is used to compute the radius.

FIG. 68. Droplet growth of water droplets in air in a rapid expansion from (5.45) and (5.50).

Hazen [118] found good agreement of (5.49) with experiments on water and alcohol droplet growth in permanent gases, and Barrett and Germain [119], in particular, studied water drops in air. They found that from 0.2 to 0.6 seconds after the expansion in their cloud chamber, r 2 varied linearly with time. They predicted, from an expression similar to (5.49), a growth value of 4.4 x 10" cm2/sec, and found experimentally that 7.2 x 10" < dr2/dt < 7.8 x 10" cm2/sec. During the initial 0.2 seconds r < 2, and the drops grew more slowly; however, this may have been because they had not yet reached their terminal velocity. Frossling [117] also found good agreement cm. As an of (5.49) with the growth in clouds of large drops of r 2 example for an extremely rapid expansion we show in Fig. 68 the results

428


of a calculation with both laws for water drops in a small moist air nozzle (Fig. 17). The expansion started from atmospheric supply conditions, and the mixing ratio was 12g/kg. The starting point of the growth estimate was arbitrarily chosen at J = 1 in (5.31), and ro = r* 7 A. We took a = 1, and used L for ice. The location of observed condensation is indicated,

-

and up to that point all flow parameters could be computed from an isentropic expansion. The droplet radius from (5.45) does not reach ilbefore condensation sets in. We shall see later that because of this and other difficulties no quantitative prediction of the onset of water vapor condensation in moist air expansions is yet possible.

4. Comparison with Experiment

Self-nucleation: In the aeronautical literature attempts have been made to predict the onset of condensation in high speed flows by determining a so-called “critical nucleation rate”. From the definition of the saturation ratio and (1.9), we may write (5.51)

where p, is the partial pressure of the condensing vapor, or p, = p in pure vapor expansions. Using (5.31)and (5.51),we can plot curves of J = constant in a p,T-diagram and compare thern with experimental results of observed condensation, such as those in Figs. 23 and 34. Criteria like Jcritical = lo3 were then sought after to define the thermodynamic state of the onset of condensation quite generally. Such criteria have no physical meaning whatsoever. The point in a nozzle where a vapor has condensed to an extent that measurable heating effects occur, can only be determined by the set of equations, (4.4), (4.25), (4.26), (4.27), and (4.32); and, therefore, only a complete discussion of the history of the flow is physically meaningful. We have further seen in our discussions of various condensation processes that use of (5.31) or its variants in the condensation rate equation (4.32) is justified only in expansions of pure vapors where no condensation nuclei from any other source are present a t saturation. However, we may use (5.31) and (5.51) to help us determine the validity of application of selfnucleation rate calculations. To demonstrate this, we show in Fig. 69 nucleation rates for 0, in N, [a] in comparison with air condensation experiments from Fig. 34. The pressure has been converted to oxygen partial pressure by Po, = 0.2 p. The nucleation theory is obviously not applicable, since condensation was observed for 0 < J < 1 ; we again conclude that self-nucleation plays no role in the air condensation process.

429

CONDENSATION I N W I N D T U N N E L S

Stever and Rathbun [64] computed nucleation rates, including a surface tension correction. When we apply these results or those of Charyk and Lees [110] without such a correction to condensation experiments in pure nitrogen (e.g., Fig. 23), we find that the experimental location of condensation coincides with high nucleation rates. Therefore, we are justified in believing that condensation in pure nitrogen expansions occurs as a result of selfnucleation, and the application of (5.31) in (4.32) is warranted. Unfortunately, such calculations have not been performed. However, we have seen, for these two contrasting examples, that the application of nucleationrate expressions to observed condensation states assists us to classify the physical nature of the processes, even though they do not produce quantitative predictions of the location of condensation.

30

40

50

60

70

T(OK) FIG.69. Comparison of nucleation rates of 0, in K, from Volmer’s equation with experimental results on air condensation.

Steam condensation: The only successful complete kinetic and thermodynamic prediction of a condensation process due to self-nucleation in supersonic flow was carried out for steam by Oswatitsch [34]. He employed a set of equations similar to those in Section IV, Becker and Doring’s nucleation rate expression, and suitable droplet-growth laws. In Fig. 70, we show his results as computed for Yellott’s experiments [48]. The arrows indicate the extent of the condensation zone, xk to xe, found from pressure distribution measurements. Oswatitsch calculated the rate of change of the number of nuclei, i.e. essentially the nucleation rate, as a function of distance from (4.34) and the nucleation rate equation. In the first part of the condensation zone, the production rate increases rapidly and then falls to zero because, with condensation, the supersaturation‘ and, therefore, the J-rate drops. It is this first steep increase in the nucleation rate which is decisive, since these

430


are nuclei that grow to appreciable size and contain the major part of the condensed liquid. Additional large numbers of nuclei formed further downstream contribute little to the condensed mass fraction, owing to their small size [lo]. Also shown as a function of distance is the actual number of stable drops present. This number likewise increases enormously in the initial phase of the process and remains constant after the production rate has dropped to zero. Finally, the total mass fraction condensed, g, computed from the amount of liquid in all droplets, increases rapidly throughout the condensation zone. After x, has been reached, the existing drops are unchanged in number and they grow in thermodynamic

I 0.4

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01

0.6

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rk

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xkm) FIG.70. Kinetic calculations of Oswatitsch applied to Yellott’s experiments on condensation in a steam nozzle. (-) Production of critical droplets per unit length. (- - -) Number of stable droplets present per unit volume. (- .-) Mass fraction of liquid phase.

equilibrium as the expansion proceeds. This stepwise solution of the kinetic and thermodynamic equations also yields the complete pressure distribution. The result calculated by Oswatitsch [34] for Binnie and Woods’ [50] experiment is shown in Fig. 71. The agreement between theory and experiment is excellent. We have previously linked these experiments with those by Head in moist air in Fig. 21 by a curve for log J = 26 (calculated from Becker and Doring’s result with the Tolman surface tension correction by Head). The success of the kinetic theory in the steam case is due to three factors. First, we see in Fig. 21 that Tk is relatively high, and we expect only water droplets (rather than ice crystals) to be formed. This permits us to use

431


immediately (5.31) or a similar expression. Furthermore, we saw previously that for the steam experiments the critical size drops, according to (5.4), have diameters of the order of 10 hi and contain about lo2 molecules. Therefore, surface-tension corrections, with all their inherent uncertainties, are unimportant. Finally, again because of the relatively large initial droplet size and the high density and short mean free path of the steam, the major part of the drop growth follows the well-established macroscopic dropletgrowth law, (5.50). The heat conductivity of the steam like all other properties needed is well known in this temperature range. 1.01

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x (cm) FIG.71. Predicted pressure distribution in a steam nozzle with condensation. Theory: Oswatitsch [34]. Experiment: Binnie and Woods [ 5 0 ] .

Condensation of water vapor in air: Comparisons of experiments on condensation in moist air with the kinetic and thermodynamic theory of self-nucleation have so far been quantitatively unsuccessful. Oswatitsch [34] has computed pressure distributions, similar to those in Fig. 71, for his experiments in a small nozzle. He was able to correlate the point of deviation from the isentropic expansion for three different relative humidities when he adjusted the surface tension in the nucleation rate expression to fit the experimental results. The further trend of the collapse, however, could not be calculated. He had to choose values of the surface tension, or free surface energy, in the range expected for ice crystals. From Fig. 18 we see that for higher degrees of supersaturations condensation occurs a t Tk< 210 OK,

432


a temperature below which Sander and Damkohler [38] believe they observed the first formation of ice crystals in their cloud chamber experiments. Wegener [29] applied (2.31) to a condensation zone and found that agreement with the experiment could only be obtained if the latent heat for ice was chosen, which gives another hint that ice crystals are probably first formed in the nozzle. Under these circumstances, the application of nucleation rate and crystal growth expressions is indeed extremely difficult.

Theory 0.50

0.40 I

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0

5

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10 15

,

0.34 I

,

20 25 30 Microsec.

FIG. 73. Pressure rise through the condensation zone in moist air. Theory: Wakeshima [115]. Experiment: Head [lo].

Wakeshima [ 1151 applied kinetic and thermodynamic calculations to one of Head’s experiments (Curve 3, Fig. 11) and computed the pressure increase with respect to the isentropic pressure distribution. He took the surface tension as independent of the droplet radius and u equal to one; it appears from Fig. 72 that the errors due to these simplifications roughly cancel each other. Wakeshima also applied unsteady solutions for the nucleation rate to this experiment, and he found that about 2 x seconds elapse before steady-state nucleation occurs. This time is short with respect to the duration of the condensation process, and we see again that the quasi-steady nucleation equation is sufficient. Oswatitsch [33] obtained a semi-empirical equation, including nucleationrate and droplet-growth considerations, which determines, for a given initial state, when condensation will first be noticeable. His final equation, in our notation, is (5.52)

(- )-.i.

1039pn5exp

WNA = const.

(g) 4

(CGS units).

The individual terms of (5.52) may be plotted as a function of temperature, and in this manner Tkcan be found. From his own experiments, Oswatitsch chose the constant equal to 2.2. The interesting feature of (5.52) is the strong dependence of the condensation temperature (or supercooling) on the temperature gradient of the flow. We previously noted this dependence in empirical terms, and now find it verified by kinetic considerations.


433

Air and nitrogen Condensation: Buhler [61] has given a simplified theory of the collapse of the supersaturated state that is applicable for air and nitrogen expansions, such as the one shown in Fig. 31. This theory requires a knowledge of the number and average size of the nuclei present a t the point in the flow where the observed expansion deviates from the condensation-free isentrope. Once this starting condition has been determined by optical measurements [58] or by a self-nucleation calculation for the vapor impurities from (5.30) or by an estimate of the nuclei present from the measured mixing ratio and (2.3) and (5.10), the further growth of these nuclei must be calculated. Buhler was able to derive an equation for this growth, essentially based on the droplet-growth law (5.46); hence he could compute g ( x ) and the pressure. The calculated pressure was found to be in general agreement with the experimental results [54, 551.

VI. .EXPERIMENTAL METHODS

1. Effect of Condensation on Measurements i n Wind Tunnels

The effect of water vapor condensation on the flow in a wind-tunnel test section can be estimated from the condensation shock analysis of Section IV, 2. For this one-dimensional analysis to be applicable, the condensation region must be nearly normal to the flow direction. In addition, we assume that almost all of the water vapor is condensed out a t the condensation shock in order to have an adiabatic flow between the shock and the test section. Such a treatment was first given by Heybey [32]. Because of the entropy change in the condensation shock, the throat area downstream from the shock that is required for sonic flow is changed. We can determine this new area from the continuity equation

(A*p*a*)l = ( A * ~ * u * ) ~ the equation of state, and the speed of sound relation (4.21), and obtain

Since po2< pol, and Q > 0, A,* is larger than the actual nozzle throat area A,*, and the Mach number at a fixed position in the nozzle behind the condensation shock is, consequently, reduced. The sonic-throat area ratio, (6.2), is a convenient parameter to represent the strength of the condensation shock, and we have plotted it against M , for several Q/Qmaxin Fig. 73.

434


We can obtain an expression for M3, the Mach number in the test section when there is condensation upstream in the nozzle, n terms of A2*/A1* from the isentropic area relation (3.4). The result is

1.0

I .4

I .8

2.2

2.6

MI FIG.73. Sonic throat area ratio across condensation shock as a function of Mach number.

where M is the Mach number when there is no condensation. From (6.3) we have plotted, in Fig. 74, the ratio M 3 / M as a function of M for several values of A2*/A,*. We observe from the figure that in a given nozzle with a fixed condensation shock, the Mach number change due to condensation becomes less with increasing distance downstream from the shock. We can obtain the static pressure with condensation, p3, from (1.14) and (6.2). The result is


435

where p is the pressure with no condensation. We have plotted the left hand side of (6.4) in Fig. 75 as a function of M for several values of A,*/A,*. The same effect as noted for M3 is evident. With increasing distance along the nozzle, the effect of the condensation shock on the pressure becomes smaller.

M

FIG. 74. Effect of water vapor condensation on test-section Mach number.

We now have a method to determine the flow in the test section for a given humidity of the supply air. From the supply relative humidity and nozzle temperature gradient, the supercooling can be estimated from Fig. 19, or the shock position obtained directly from (2.37) in terms of the mixing ratio. This establishes the Mach number M,, which we may take equal to M k , at which the condensation shock occurs. Some of the available experimental information on the shock location has been used by Wyker [120] to construct a chart for the prediction of M,, knowing Po, To,w,, and dTldx. For q, we can take either Lw, (approximately L x ) , or qmX, whichever is smaller. However, if L x is appreciably larger then qmax,or if T2is sufficiently high to require the presence of a large amount of vapor for saturation, the analysis cannot be valid since a significant amount of vapor will condense downstream from the shock. With q known, and Qmax found from Fig. 41,

436


we obtain p,,/fi,, from Fig. 45, and A2*/A,* from Fig. 73. Then (6.3), or Fig. 74, gives the Mach number in the test section, and (6.4), or Fig. 75, the static pressure. Condensation effects are completely avoided in a wind-tunnel test section when M , is greater than the test section Mach number. The previously mentioned chart of Wyker and a nomograph constructed by Smolderen [lSl] can be used to find the allowable humidity of the supply air to meet this requirement. However, for M > 2 , practically impossible degrees of dryness are necessary, and a more important quantity is the maximum humidity which results in a negligible effect on the measurements in the test section.

FIG.75. Effect of water vapor condensation on test-section static pressure.

This humidity is expected to be quite low, with the result that the condensation shock occurs at a high enough M , for qmx to be much larger than Lx. Also the temperature T , is low (for unheated supply air), and the assumption that all of the water vapor condenses a t the shock is well satisfied. With Q

Advances in Applied Mechanics, Volume 5

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Advances in Applied Mechanics