Advances in Applied Mechanics, Volume 8

ADVANCES IN APPLIED MECHANICS

VOLUME 8

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ADVANCES IN APPLIED MECHANICS Ed itors

TH.

H. L. DRYDEN

VON

KLRMLN

Managing Editor

G. KUERTI Case Institute of Technology, Cleveland, Ohio

Associate Editors

F. H.

VAN DEN

DUNGEN

L. HOWARTH

VOLUME 8

1964

ACADEMIC PRESS

NEW YORK AND LONDON

COPYRIGHTQ1964, BY ACADEMICPRESSINC. ALL RIGHTSRESERVED N O PART OF T H I S BOOK

MAY B E REPRODUCED I N ANY

FORM, BY PHOTOSTAT, MICROFILM, OR ANY OTHER MEANS, WITHOUT WRITTEN PERMISSION

FROM T H E PUBLISHERS.

ACADEMIC PRESS INC. 111 Fifth Avenue, New York, New York 10003

United Kingdom Edition published b y ACADEMIC PRESS INC. (LONDON) LTD. Berkeley Square House, London W.l

LIBRARY OF

CONCRliSS CATALOG CARD

NUMBER:48-8503

P R I N T E D I N T H E U N I T E D STATES OF AMERICA

CONTRIBUTORS TO VOLUME 8 BERNARD D. COLEMAN, Mellon Institute, Pittsburgh, Pennsylvania HERSHELMARKOVITZ, Mellon Institute, Pittsburgh, Pennsylvania N. N. MOISEEV,Computing Centre of the U.S.S.R. Academy of Sciences, Moscow, U.S.S.R. E. L. RESLER, JR., Cornell University Ithaca, N e w York H. S. RIBNER,Institute of Aerophysics, University of Toronto, Canada V. V. RUMYANTSEV, Institute of Mechanics of the U.S.S.R. Academy of Sciences, Moscow, U.S.S.R. W. R. SEARS, Cornell Universit31, Ithaca, N e w Yurk

V

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Theodore yon Khrmhn: A Tribute The death of von KhrmAn in May 1963 has robbed this series of one of its editors. Even if he had not been so closely associated with the “Advances” it would nevertheless have been fitting that a tribute to one so distinguished should appear as a frontispiece to this volume. KBrmAn was one of the giants of our subject whose influence over more than 50 years has had incalculable effects on its growth. Thus he can be said to have been one of a select band present at the birth of real continuum mechanics in the early years of this century and who lived to act as guide, counselor, and source of inspiration to it throughout its graduate and postgraduate career. KirmAn went to Gottingen from Budapest in 1906 a t a time when, it is scarcely necessary to remark, Prandtl and his associates were developing boundary-layer theory and with it were injecting realism into fluid mechanics. During his 6-year stay there Kirmin developed the theory of the KarmLn street, contributed to the theory of the strength of slender columns, and collaborated with Max Born on vibrations in lattices and their relation to specific heat. From Gottingen Karmin moved in 1912 to Aachen where he remained, apart from service with the Austro-Hungarian Aviation Corps in World War I , until 1930. In 1930 Kirmin was appointed Director of the Guggenheim Aeronautical Laboratory at the California Institute of Technology which then became the Mecca of many a would-be scientist and engineer. If I discuss this period more than his time in Aachen it is because I know it better. No one could minimize the importance of the work he did in Aachen in building up the Aeronautical Institute or of the work he did there on the turbulent boundary layer (including the logarithmic law), on the momentum integral concepts, and on the rotating-disc solution of the Navier-Stokes equations all of which are still producing repercussions. For many of us, working under K i r m h ’ s direction in the 1930’s was an unforgettable experience - inspiring and in every way enjoyable. He was the most accessible of men and had built up a staff and school around him which was the envy of all. At that time Pasadena was home for Kirmin.. He lived with his mother and sister in S. Marengo, and together they did all they could to make the visitor welcome, providing appropriate passages of Mark Twain for the lady visitor, while the import of statistical theories of turbulence was examined under the stimulus of Tokay. KBrmAn having lived with the subject for so long had the insight to foster work in the many specialisms into which it was proliferating. Thus vii

...

Vlll

in this period there was work being done on structures, aerofoil theory (steady and unsteady), supersonic flow, turbulence, and meteorology, and in all these KBrmPn was entirely at home; wherever possible experiment and theory proceeded hand in hand. With the onset of World War I1 it was inevitable that KBrmrin’s services should be in great demand. This, together with his interest in post-war years in international scientific cooperation, meant that Pasadena saw less of him in person but continued to benefit from his guidance. He felt very keenly the death of his mother and later of his sister, and thereafter S. Marengo always held sad memories for him. G . I. Taylor has written of KBrmBn’s work as essentially that of a “broadminded and deepminded” engineer whose thought extended over the whole range of engineering science, of his work in founding the International Congresses, and his wider and later work on international cooperation in scientific matters as, for instance, in founding AGARD. I t would be supererogatory for me to attempt to add to what he has written. KrirmBn had a great gift of exposition whether it was of his own numerous original works or in presenting that of others. One could always look to him to go straight to the nub of any sound idea or expose a fallacy in the kindest possible way. His work will forever remain embedded in the literature but to us who knew him it is the loss of KBrmin - the friend, counselor, and source of inspiration - that we mourn above all else.

L. HOWARTH

Contents CONTRIBUTORS TO VOLUME VIII . . Theodore von Kbrmbn: A Tribute .

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

v

vii

Magneto- Aerodynnmic. Flow Past Bodies

.

BY W . R . SEARSA N D E . L. 'RESLER.JR., Cornell University. Ithaca New York Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I . Incompressible Flow . . . . . . . . . . . . . . . . . . . . . . . . I1. Compressible Flow . . . . . . . . . . . . . . . . . . . . . . . . . I11. Hall Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 4 6

23 60 64

Ineomprermible Second-Order Fliilds

.

BY HERSHELMARKOVITZ A N D BERNARD D COLEMAN. Mellon Institute. Pittsburgh Pennsylvania

. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . Relation to General Simple Fluids . . . . . . . . . . . . . . . . . . . Steady Simple Shearing Flow . . . . . . . . . . . . . . . . . . . . . Viscometric Flows . . . . . . . . . . . . . . . . . . . . . . . . . . Steady Extension of a Cylinder . . . . . . . . . . . . . . . . . . . . Relation to Classical Viscoelasticity . . . . . . . . . . . . . . . . .

I I1 . 111 IV V VI VII .

Nonsteady Shearing Flows . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. 69 71

77 79 87 89 96 100 100

The Generation of Sound by Turbulent JetR BY H . S. RIBNER.Institute of Aerophysics. Unioersity of Toronto. Canada Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . I1. RBsum6 of Major Features . . . . . . . . . . . . . . . . . . . . . A Simplified Physical Account . . . . . . . . . . . . . . . . . . . . . . I11 Physics of Sound Generation . . . . . . . . . . . . . . . . . . . . IV . Equivalent Aerodynamic Generators of Sound . . . . . . . . . . . V Sound Radiated from a Jet . . . . . . . . . . . . . . . . . . . . . B. Mathematical Development . . . . . . . . . . . . . . . . . . . . . . . VI Governing Equations

.

. . .

........................ ix

104 105 106 109 109

.

115 119 142 142

CONTENTS

X

V I I. VIII I X. X. XI .

Convection Effects for a Simplified Model of Turbulence . . . . Effects Due to the Mean Flow . . . . . . . . . . . Improved Model: Isotropic Turbulence Superposed on Mean Flow Sound Emission from a Complete Jet . . . . . . . . . . . . . Asymptotic Behavior at High Mach Number . . . . . . . . . . Acknowledgment . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. Refraction

. . . . . . . . .. .

. . . . .

.

151

In!) 163 169 115 118

178

Stabiilty of Motion of Solid Bodlea with Llqiild-Fllled Cuvitles by Lyupirnov’o Methods BY V . V . RUMYANTSEV. Institute of Mechanics of the U . S . S . H . Academy oj Sciences. Moscow. i J . S . S . R

.

Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I . Simplest Cases of Motion: the Cavity is Completely Filled . . . . . . . I1. Stability of Motion of a Solid-Liquid Body with Respect to a Part of the Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . I11. Stability of Steady Motion of a Solid Body with Liquid-Filled Cavity . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

183 184 186 203 214 230

Introduction to the Theory of ORelllatlonR of Liquid-containlnR Bodies BY N . N . MOISEEV.Computing Centre of the U . S . S . R . Academy of Sciences. Moscow. U.S.S.R. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I Survey of Special Problems . . . . . . . . . . . . . . . . . . . . I1 General Properties of the ]Equations . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . Selected Bibliography . . . . . . . . . . . . . . . . . . . . . . .

. .

AUTHORINDEX

SUBJECTINDEX

. .

.. . .

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

233 235 . 268 287 288 291

295

Magneto-Aerodynamic FJow Past Bodies BY W . R . SEARS

AND

E . L . RESLER,

JR

.

Cornell Universitv. Ithaca. New York Page Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1. Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . 5 2.Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1. Incompressible Flow . . . . . . . . . . . . . . . . . . . . . . . . . 6 The Magnetic Reynolds and Prandtl Numbers . . . . . . . . . . . . 8 1. Small-Perturbation Flow . . . . . . . . . . . . . . . . . . . . . . 8 Inviscid Perfect Conductor . . . . . . . . . . . . . . . . . . . . . 9 Effects of Viscosity and Resistivity . . . . . . . . . . . . . . . . . 11 2 . Aligned-Fields Flow . . . . . . . . . . . . . . . . . . . . . . . . 12 Ideal Conductor . . . . . . . . . . . . . . . . . . . . . . . . . . 12 . . . . . . . . . . . . . . . 16 Conductors of Large u and Small I’r, Small-Perturbation Flow at Arbitrary H m . . . . . . . . . . . . . . 19 3. Flow with Very Strong Magnetic Field . . . . . . . . . . . . . . . 21 4. S u m m a r y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 23 I1. Compressible Flow . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Aligned-Fields Flow of Inviscid Gases . . . . . . . . . . . . . . . . 23 Small-Perturbation Flow . . . . . . . . . . . . . . . . . . . . . . 23 Ideal Conductor . . . . . . . . . . . . . . . . . . . . . . . . . . 25 Relationship to Wave-Propagation Phenomena . . . . . . . . . . . . 30 2. Flow with Arbitrary Magnetic-Field Direction . . . . . . . . . . . . 34 Two-Dimensional Flows . . . . . . . . . . . . . . . . . . . . . . 34 Three-Dimensional Flows . . . . . . . . . . . . . . . . . . . . . . 36 3 . Magneto-Gasdynamic Shock Waves . . . . . . . . . . . . . . . . 38 4 . Compressible Aligned-Fields Flow with Arbitrary Scalar Conductivity . 41 Properties of Sinusoidal Solutions . . . . . . . . . . . . . . . . . 43 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 111. Hall Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 1 . Particle Motions . . . . . . . . . . . . . . . . . . . . . . . . . 50 2. Ohm’s Law for Ionized Gases . . . . . . . . . . . . . . . . . . . 52 3 . Aligned-Fields Flow of a Fully Ionized Gas or a Slightly Ionized Gas withoutIonSlip . . . . . . . . . . . . . . . . . . . . . . . . . 54 4 . Collisionless Plasma . . . . . . . . . . . . . . . . . . . . . . . . 56 Properties of Sinusoidal Solutions . . . . . . . . . . . . . . . . . 57 6. Generalized Wave-Speed Diagram . . . . . . . . . . . . . . . . . . 58 6.Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 NOTATION

a A

.4

Speed of sound. v y x p .4lfv6n.wave velocity Magnitude of the vector A

W. R. SEARS A N D E. L. RESLER, JR.

Components of the vector A Magnetic-induction vector, e.m.u. Magnitude of the vector B Components of the vector R Speed of light Wave-propagation speed Specific heat a t constant volume Mean thermal speed Electric-field vector, e.m.u. Charge of a positron, e.m.u. Frequency of oscillation, radianslsec. Probability that a particle exists for a time I without making a collision Magnetic-field vector, e.m.u. Magnitude of the vector H Components of H Perturbation vector of the magnetic field, H - H, Components of h Current-density vector, e.m.u. Components of j Magnitude of the vector J RmlZnM, a (See Eq. (3.26)) Typical length (See Eq. (3.24)) Wave length of siiiusoidal disturbance Mean free path Mach number U / a U / c (See Eq. (2.24))

- 1)W

(Atm*

Mass of particle Mass of an electron Mass of an ion Density of neutral particles Density of electroris Wave-normal vector Pressure Pressure perturbatlion Pressure in related non-magnetic flow pH4/8n Total pressure, p Magnetic Prandtl Number 4npoa Electron partial pressure Fluid-velocity vector Magnitude of the vector q Velocity vector in related non-magnetic flow

+

First and second roots of the equation for r Gas constant R,, R,, R,, R, Regions of the M c o , am-l plane Re Reynolds Number U L / v i e Reynolds Number of related non-magnetic flow Rm Magnetic Reynolds Number 4npaUL or 4npaUl t Time

MAGNETO-AERODYNAMIC FLOW PAST BODIES

1 T' U U'

U U -, U UP

Q

V V

U'

VP Y

ii W W,

x , Ys z YD a

Pa Y c' F

8 1 Id V

5 P P'

n 7

71

7e

7m

Temperature Perturbation of temperature, T - T, x component of q x component of v Free-stream velocity vector; i.e., value of q a t large distances from an obstacle Magnitude of the vector U Value of U in related non-magnetic flow x component of particle velocity V Average velocity component of particle along x Velocity vector of particle y component of q y component of v y component of V Perturbation velocity vector q - U Average velocity component of particle along y z component of q z component of Y Cartesian coordinates; sometimes x , y are cylindrical coordinates Distance t o damp t o I/e in y direction p H a / 4np U 2 (1 - Mm')l/2 Specific-heat ratio = 0 for plane flow, = 1 for axisymmetric flow z component of Complex constant in the expression $ a exp ( i l x - 8 y ) Heal constant in the expression # a exp ( i l x - @y) Magnetic permittivity of fluid, e.m.u. Kinematic viscosity of fluid curl H, therefore 4nJ Mass density of fluid Perturbation of density, p - pm Electrical conductivity of fluid, e.m.u. Mean time between collisions of electrons with ions Mean time between collisions of ions with neutrals Mean time between collisions of electrons with neutrals Mean collision time for electrons in collisions with ions and neutrals, (Tc-l

+ T-1)-1

Angle between n, and H m (See Fig. 3.1) Perturbation stream function (See Eqs. (2.19)) Parts of $ corresponding to Y , , Y ~ respectively . The vorticity, curl q Magnitude of the v e c t o r a Cyclotron frequency of ions. eB/mi Cyclotron frequency of electrons, eB/m, Plasma frequency, 4npn&g/mr eBy/mr Subscripts 8

1

3

Undisturbed condition Conditions a t outside edge of boundary layer in Chapter I .

4

W. R. SEARS AND E . L. RESLER, J R .

INTRODUCTION The class of magneto-fluid dynamic problems that involves streaming flow about solid obstacles has attracted attention by virtue of the interesting phenomena that are predicted by the theory, rather than its practical utility. Some of these phenomena ar,e analogous to the familiar features of conventional gasdynamics - standing waves, elliptic and hyperbolic fields, boundary layers, and wakes - and others present surprising contrasts. Such flows should be capable of realization in the laboratory, even if the great difficulties involved in any experiments with conducting gases are admitted. Furthermore, as will be seen in the examples that follow, some of the phenomena predicted for these flows are prominent and should be easily discernible and measurable. Thus, some of these configurations should be suitable for diagnosis of flows, determination of fluid properties, and experimental verification of theoretical predictions. This makes this category of flows important, for there is a dearth of such experimental results throughout the whole field of magneto-fluid dynamics. In this report we shall consider almost exclusively steady flows that consist of a uniform stream at great distances from the solid bodies, and upon which a magnetic field, also uniform at large distances, is imposed by external means at some angle to the stream. Whenever this angle is different from zero it is assumed that an electric field is also applied, so that the region of uniform flow and uniform magnetic field far from the body is current-free. Throughout the report, th'e approximations of continuum flow are made. This implies, of course, that the density is so large - or, more precisely, that the Knudsen Number is :SO small - that it is legitimate to assign to the fluid a t any point a pressure, density, temperature, and velocity. I t will also be assumed that thermodynamic equilibrium is maintained at every point, and the perfect-gas law will be used to describe the thermodynamic state. The assumption of thermodynamic equilibrium also implies that the mechanical stresses acting are adequately described under Stokes' Hypothesis ; i.e., that the mechanical equations of the problem are the Navier-Stokes equations. In the same spirit and with substantially the same validity and limitations, the electrical properties of the fluid are described, throughout the greater part of this paper, by assigning to it an electrical conductivity and magnetic and electrical permittivities (both usually equal to those of empty space). As is customary in magneto-fluid dynamics, that portion of the electric current that is due to the transport of charge by the fluid motion, the so-called convection current, is neglected. This does not mean that the fluid is everywhere neutral in charge, which it cannot be in general, but only that the charge density is always so small that the convection current is

MAGNETO-AERODYNAMIC FLOW PAST BODIES

5

much smaller than the conduction current. Also, relativistic effects are neglected throughout . Even with these simplifications, the equations describing the flow are found to be intractable in most of the interesting cases, and further approximations must be introduced to permit solutions to be constructed. In particular, the electro-magnetic properties of the fluid, such as conductivity and permittivity, are taken to be constants, rather than functions of the thermodynamic state a t every point. This implies that the temperature, in particular, does not vary widely - an assumption that may be of doubtful validity in some experimental situations. Rather than writing out the full equations appropriate to the fluid model just described - i.e., a viscous, compressible, conductive gas - we shall undertake here to expose and explain the principal phenomena of flow about bodies by beginning with the simplest model, an incompressible perfect conductor, and proceeding subsequently toward more complicated cases. To justify this approach, we shall argue that the main features are deducible from rather simple equations and that further elaboration of the fluid model results in modifications of these features rather than entirely new results. Thus we shall begin with a study of flows of incompressible fluids of infinite electrical conductivity and then proceed to investigate the main effects of electrical resistivity. These studies will permit us to investigate subsequently the effects of compressibility in perfectly conducting, inviscid gases with some knowledge of the modifications due to viscosity and resistivity. Some details of these modifications, at least those due to resistivity, will, however, be worked out as well. This results in a relatively clear understanding of the main features of flows in real, conducting gases in the absence of Hall Effect - i.e., a t high densities and moderate magnetic-field strengths. Finally, it will be pointed out that even perfect conductors may exhibit appreciable Hall Effect, and some effects of the Hall phenomenon - the drift of charged particles across the electric field due to the presence of the magnetic field - will be worked out. I . Boundary Conditions To solve problems in magneto-fluid dynamics consistent with the above discussion, solutions for the electromagnetic variables must be obtained throughout all space. Thus conditions on both the magnetic- and electricfield vectors at surfaces across which the material properties, including conductivity and permittivities, change, as at a solid-fluid interface, must be formulated consistent with the electromagnetic equations. The general statements of these interface conditions follow. In all cases the component of the magnetic-induction vector B normal to the surface and the component of the electric-field vector E along the

6

W. R. SEARS AND E. L. RESLER, JR.

surface must be continuous, since the former is divergence-free and the latter, in steady flow, curl-free. Similarly, the component of magnetic-field strength H along the interface must be continuous, because the curl of this vector, according to Ampbre’s Law, is proportional to the electric-current density j. To permit a discontinuity in the tangential component of H would mean that the current density becomes locally infinite, which is not realistic in actual fluids of finite electrical conductivity. However, in some cases the approximation of infinite conductivity will be introduced as a limiting case of fluids of large conductivity. When this is done it may be necessary to admit infinite current density in surface-current layers, i.e., to allow the tangential component of II to be discontinuous. When such an approximation is made, an effort will also be made to provide an explanation of the detailed nature of the current layer in a real substance. The boundary conditions on the velocity vector q at a fluid-solid interface are the usual ones; namely both normal and tangential components of q must be continuous, i.e., must vanish. If the approximation of vanishing viscosity is made, the condition on the tangential component must be relaxed; i.e., a vortex sheet must be permitted at the interface. The analogy between the current layer of infinite current density and the vortex sheet of infinite vorticity will be emphasized later in this report. 2. Units

Throughout this presentation electromagnetic units (e.m.u.) will be used. To facilitate changes to other units, however, the magnetic permittivity ,u will be retained in all the formulas. The conversion to M.K.S. units, for example, can be made by replacing the factor 7c in the formulas by 1/4. I.

INCOMPRESSIBLE FLOW

As has already been remarked, the mechanical equations of the problem are those of Navier and Stokes with the body-force and heat-addition terms written out explicitly for the magneto-hydrodynamical case. For incompressible flow the momentum and energy equations are uncoupled and the latter is not needed to determine the flow pattern. We have, for steady flow,

Continuity : div q = 0,

(1.1)

Momentum: (q -C7 )q +-g 1 rad p =v V2 q +-j P P

P

x H.

MAGNETO-AERODYNAMIC FLOW PAST BODIES

7

The electromagnetic equations, under the approximations mentioned, and assuming constant permittivity p are, in e.m.u.,

Absence of Magnetic Poles: (1.3)

div B = 0,

Amplre’s Law: 4nj = curl H,

(1.4) Faraday’s Law:

curl E = 0.

(1.5)

To assign a scalar conductivity u to the fluid means that the current flux j is related by Ohm’s Law to the local electric field experienced by the moving fluid particle: (1.6a)

j

= o(E

+ q x B).

I t is convenient to eliminate the electric-field vector E by means of Faraday’s Law; one has, for constant a , (1.6b)

u-l curl j

= curl

(q x B) = ,u curl (q x H)

or, in view of Eqs. (1.1, 1.3, and 1.4),

The equation of momentum, Eq. (1.2) can also be combined with AmpQe’s Law to eliminate the current density j ; the result is

The problem of steady flow of an incompressible ideal conductor is therefore reduced to the simultaneous solution of Eqs. (1.7) and (1.8) with the auxiliary conditions div B = div q = 0 and (1.9)

B = pH.

An important equation is obtained by operating upon Eq. (1.8) with the curl. For an incompressible fluid with constant v and p , the result is (1.10)

(q*V)rFz- ( S 2 . V ) q = v V 2 S 2 + ~ c u r l [ ( H V)H]. . 4ZP

8

W. R. SEARS A N D E . L. R E S L E R , J R .

The Magnetic Reynolds and Prandtl Numbers If the magnetic field is zero, we have in Eq. (1.10) the familiar equation [l] for the convection of vorticity and its diffusion due to viscosity. Thus Eq. (1.10) describes how vortex convection and diffusion are modified by the presence of magnetohydrodynamic effects. There is an obvious parallel between the vorticity equation, in the absence of magnetic fields, and Eq. (1.7) above. The latter states that the magneticfield vector H varies in much the same way as the vorticity as it is convected by the flow, except that its diffusion is controlled by the resistivity parameter 1/4npa instead of v . (Ry inspection of the equations, it is clear that these two quantities have the same dimensions.) The relative influence of these diffusive effects is given by their ratio, 4npav, which is a dimensionless material property and might be called the Magnetic Prandtl Numher, Pr,, in analogy with the conventional Prandtl Number, which measures the ratio of vorticity diffusion to heat diffusion. The analogy also suggests the definition of a new Keynolds Number ;2: (1.11)

Rm

= 4npaUL

where L and U are characteristic length and flow speed, respectively. This number then measures the ratio of magnetic-field diffusion-time 4npaL‘ to the characteristic flow-time LIU. I n view of the correspondence between the equations governing H in magneto-fluid dynamics and G? in conventional fluid mechanics, the various phenomena typically associated with ordinary aerodynamics a t different Reynolds Numbers Re can be expected to have their counterparts in various regimes of the Rm spectrum. Using the kinetic-theory approximation to v for gases, namely v = 0.499 d the magnetic Prandtl Number Pr, becomes 0.499 (4np)oEl; i.e., it is the magnetic Reynolds h’umber based on the mean free path and the mean thermal speed E. For ionized gases under typical terrestrial conditions, this material property is much smaller than 1, as substitution of typical values into this formula will verify. The same is true of Pr, for other fluid conductors, such as liquid metals, electrolytic solutions, and solutions of alkali metals in ammonia. Consequently, we may not expect that the Re and R m spectrums will coincide ordinarily, but rather that the large-Re approximation will frequently be appropriate for a wide range of values of Rm. We shall return to this point later in the report.

1. Small- Perturbation Flou: Suppose now that the only disturbances in the uniform, parallel currentfree stream are small perturbations, such as might be produced by flow past a slender body; i.e., that

MAGNETO-AERODYNAMIC FLOW PAST BODIES

(1.12)

q=U+v

and

H=H,+h

and

J h (<< (Hrnl.

9

where

(1.13)

Iv(

<(

IUI

Both equations (1.7)and (1.8)can then be linearized and assume the forms

(1.14)

1 V2h = (Ha V)V - (U * V)h 4npa

- __

and

(1.15) or if 51 denotes curl q, the vorticity, and times the current density,

g

denotes curl h, which is 4n

(1.16) and

(1.17) Eqs. (1.16)and (1.17)are clearly the linearized counterparts of Eqs. (1.7) and (l.lO), respectively.

Inviscid Perfect Conductor We shall postpone a general discussion of this interesting pair of equations in order to treat first the special case of an inviscid perfect conductor, i.e., y = a-1 = 0. The right-hand sides of Eqs. (1.16)and (1.17) are then zero, and by cross-differentiation the following fourth-order equation is found to be satisfied by both 51 and g:

(1.18) The solution of this equation is simple: both 51 and 6 consist of parts that are constant along lines having the directions sketched in Fig. 1.1, having the directions U A, and U - A,, where A, denotes the AlfvCnwave velocity Vpu/4npH,.

+

10

W. R. SEARS AND E. L. RESLER, J R .

The meaning of Eqs. (1.18) is now clear: vorticity and current are propagated at the AlfvCn speed in the direction of the magnetic field, relative to the moving fluid. The solutions for velocity and magnetic-field vectors are found by integration of and p. It is clear that they consist of rotational parts that are constant along the directions shown in Fig. 1.1 plus curl-free parts. By returning to the equation of momentum and Ohm’s Law, Eqs. (1.15) and (1.14) with Y = 0-1 0, one can relate the rotational parts of v and h and also their irrotational parts.

a

E

FIG. 1 . 1 . Diagram showing directions of stationary A l f v h waves for incompressible, inviscid, perfect conductor.

The complete flow then appears to involve an irrotational current-free disturbance field, which is, of course, elliptic in mathematical terminology and therefore far-reaching, superimposed upon a hyperbolic, rotational, disturbance pattern, i.e., a pattern made up of stationary waves. Since this system of waves is recognized to consist of Alfvbn’s famous radiation [3], their proper directions become clear from physical principles ; namely, they are the directions shown in Fig. 1.1,‘ and not their reverse. In other words, such radiation is produced at the body, by the flow disturbance there, and propagates outward from the body along the magnetic-field direction as it is convected downstream by the fluid. In an unlimited body of fluid such as we are considering, disturbances do not propagate in from infinity. This does not necessarily imply that the regions upstream of the body are undisturbed. It is interesting that the two irrotational fields are related by the equations of motion and that the same is true of the two rotational parts. The rotational and irrotational fields are coupled only through the boundary conditions a t the body. Since a number of problems in this category have been solved, [4], [ 5 ] , [6], [7:, [8], [9], and since our main objective here is to ascertain the general nature of the disturbance field, we shall leave this subject a t this point.

MAGNETO-AERODYNAMIC FLOW PAST BODIES

11

Effects of Viscosity and Resistizdy

By cross-differentiation of Eqs. (1.16 and 1.17), the fourth-order differential equation satisfied by both S and 5 in the case of incompressible, steady, viscous flow with finite conductivitv is obtained :

(1.19)

I t is difficult to discern the nature of the resulting flow pattern except in rather special cases. Hasimoto [lo] and Clauser [ l l ] have pointed out that the fourth-order operator in Eq. (1.19) can be factored into two secondorder operators if v = 1/4npa, i.e., if PY, = 1. Their result is highly instructive; it states that the cylindrical standing waves of the preceding section are diffused parabolically along their lengths. I t may be expected that the effects of v and u-l will be similar to this for other values of PI,,at least those near 1. This means that the introduction of these diffusive quantities will primarily cause diffusion of the standing AlfvCn waves. But, as has already been pointed out, realistic values of PY, are much smaller than 1. It is more useful to investigate the flow in the limit of vanishing viscosity. Unfortunately, Eq. (1.19) remains somewhat intractable even in this limit. The effects of small resisitivity have been studied by considering first the flow past an infinitely long, sinusoidal, wavy wall [a], [12], [13]. As expected, the effect of resistivity is to “smear” the AlfvCn waves, i.e., to cause them to diffuse and to attenuate in strength with distance from the wavy wall. This situation is analogous to the damping of sound radiation by viscosity. McCune [14] has calculated the analogous effect in flow past thin bodies in plane flow, using Fourier superposition of the wavy-wall results. His results disclose once again that the AlfvCnwave pattern is damped at moderate distances from a body. The rapidity of this damping increases as the electrical resistivity u--1 is increased. In [13] it is found that the damping distance, measured along the standing wave, is proportional to a for large a (large Rm). For small u a different approximation is valid: one assumes that the flow variables can be expanded in powers of Rm. The first approximation consists of neglecting all terms of order Rm2 and higher in comparison with quantities independent of Rm. I t is clear from Eqs. (1.6) that the first-order approximation to j is obtained by neglecting the perturbations of both E and B due to fluid motion, i.e., (1.20)

12

W. R. SEARS AND E. L. RESLER, J R .

where E, and B, are the electric and magnetic fields appropriate to the given boundary conditions with no flow. In the special case we are discussing here, where E and B are uniform at large distances, 8, and B, are constants, but the theory can be used to investigate many other geometries [15], [16], [171, WI, [191, [201. In this small-Rim approximation, therefore, the equation of momentum, Eq. (1.2), becomes independent of the induction equation, Eq. (1.7). The damped-AlfvCn-wave character of the flow is lost at this limit; the flow is everywhere rotational. The conductivity is so small that the magneticfield disturbances and accompanying vorticity are diffused to great distances within the typical flow-time. This type of approximation has been used by Rossow [21], [22] in boundary-layer studies, in addition to the authors mentioned above. Attention should also be called to a more general study by Stewartson [23] in which Rm is considered arbitrary in magnitude and it is shown how the various limiting cases such as R m - 0 , 00 (and also cases of very strong magnetic field, which are mentioned below) are obtained from a general formulation, within the small-perturbation approximation.

2. A ligned-Fields Flow A special case that has attracted some attention is the category of flows with U parallel (or anti-parallel) to B, ; these are called aligned-fields /lows and are in some respects singular in the family of flows with varying angle between the vectors. They are of some interest because they represent a particularly simple experimental arrangement, namely flow in a pipe or wind tunnel which is itself enclosed in a solenoid. For aligned-fields flow that is uniform a t large distances, E must vanish at large distances. Two cases of particular interest are plane and axisymmetric flows; for these, Eq. (1.5) requires that E be constant: hence it must be zero throughout the flow. The set of equations (1.7), (1.8) can therefore be simplified by replacing the former by its integral (i.e., by Eq. (1.6a)), viz., (1.21)

1

--

4npa

curl H = q x H.

Ideal Conductor Suppose now that u -+ co; Eq. (1.21) then states that q is parallel to H. The magnetic lines of force are said to be frozen into the fluid and therefore congruent with the streamlines. Moreover, in view of Eq. (1.21) and the divergence conditions it follows that q o( H, and evaluating the constant at large distances from the body we have (1.22)

qlU

= H/H,.

MAGNETO-AERODYNAMIC FLOW PAST BODIES

13

Although we have derived Eq. (1.22) only for plane and axisymmetric flow, this frozen-fields configuration describes a possible flow pattern for any aligned-fields flow of a perfect conductor, regardless of symmetry, as we shall see. Let us therefore carry out the analysis without any special assumption regarding symmetry. The momentum equation, Eq. (1.8) becomes, in view of Eq. (1.22), (12 3 )

where am2 denotes the ratio pHm2/4npU2,i.e., the ratio of the magnetic pressure to the dynamic pressure in the undisturbed stream. am is also the reciprocal of the Al/vLn Number of the stream; if a, > 1 the stream speed is less than the speed of Alfvkn waves in the undisturbed flow and may be called sub-AlfvLnnic; if a, < 1, it may be called super-Alfvkic. Moreover, if the “total pressure” p p H 2 / 8 nis replaced by the symbol I’, this momentum equation (1.23), takes on the form of the familiar momentum equation of ordinary incompressible, viscous flow. The additional relation needed for solution of Eq. (1.23) is the continuity equation, div q = 0. Hasimoto [24] has shown how solutions of ordinary viscous-flow problems can be transformed to solutions of Eq. (1.23). He introduces the fictitious velocity 4 and pressure $, defined by

+

(1 - aa2)q= ij

(1.24)

and (1 - a,2)P = j .

(1.25)

Eq. (1.23) then reads (1.26)

(q

*

V)q

1 + -grad$ P

= vP2q

and div q = 0. Thus any viscous, incompressible, steady flow involving an undisturbed uniform stream can be interpreted to yield a family of aligned-fields flows for various values of a,. If the undisturbed stream speed of the viscous flow is 0,the stream speed of the related aligned-fields flow depends on u,2, viz., (12 7 )

u = 0 / ( 1 - a,2).

This requires that the flow direction be reversed if a m 2 > 1; i.e., if the stream is sub-AlfvCnic. If the stream speed is equal to the AlfvCn speed the transformation fails.

14

W. R. SEARS AND E . L. RESLER, JR.

Some consequences of Hasimoto’s relation are sketched in Fig. 1.2. The flow sketched is of boundary-layer character; i.e., in the case sketched the Reynolds number Be of the related non-magnetic viscous flow is supposed to be large. Since Re = (1 - a,z)Re, this is not realistic for values of a , near 1 ; for such cases the related non-magnetic flow is low-Reynoldsnumber flow, and might require a different approximation, such as Oseen’s or Stokes’. I t is seen that whereas the effect of a relatively weak magnetic field (super-AlfvCnic flow) is to produce a flow pattern characteristic of a lower Reynolds number, the effect of a strong field (sub-AlfvCnic) is to reverse the direction of boundary-layer growth and to produce an upstream wake

H,

(C

1

FIG. 1.2. Related non-magnetic (a) and aligned-fields, viscous, incompressible flows according to Hasimoto’s theory. In (b) the stream is super-AlfvCnic (ama< 1 ) ; in (c) it is sub-AlfvCnic (a,* > 1).

or precursor. Such a wake, of course, attenuates parabolically upstream and does not disturb the stream a t great enough distances. The fluid is affected by Joule heating and viscosity as it flows in the upstream wake, and there must be a downstream wake of increased temperature and entropy, but in incompressible flow this does not affect the velocity field. Hasimoto’s flow pattern does, indeed, satisfy the boundary conditions of magnetohydrodynamics, which were discussed above, since and q vanish at the body surface as required, and according to Eq. (1.22), H vanishes there as well. Assuming only that the body has finite conductivity, we require that its interior be current-free; the unique solution for the interior magnetic field is therefore H = 0, and the continuity conditions on B at the interface are met.

MAGNETO-AERODYNAMIC FLOW PAST BODIES

15

It would be interesting to explore further the consequences of this category of flows, such as pressure distribution,* but we shall not do so here. This type of flow is of limited interest because it corresponds to an infinite value of PY,, rather than a very small value such as characterizes real conducting fluids, as mentioned previously. It is interesting, nevertheless, to take one further step before attacking more realistic fluid models; namely, to proceed to the limit of infinite Reynolds number, i.e., to put I’ = 0 in the formulas just derived. This yields the flow appropriate to very large Re and Rm for any value of Pr,. The result is very simple: the related non-magnetic flow now becomes inviscid and irrotational (in view of the uniformity a t large distances). The viscous boundary layer collapses into a vortex sheet a t the body surface and the wake disappears. The only effect of the magnetic field is therefore to produce a current layer coincident with this vortex sheet. Both the kinematic and the magnetic boundary conditions are satisfied by means of this discontinuity surface ; the magnetic field again vanishes within the body. There is no qualitative difference between sub- and super-Alfvhic flow. The current layer is not a trivial effect, however, for it alters the surface pressures a t the body. A calculation.of the body force on the layer [4] confirms that the total pressure P is conserved across the layer. We arrive a t the same conclusion from Eq. (1.25), for $ is continuous across the vortex sheet of non-magnetic flow in the inviscid limit. Thus the surface pressure is

(1.28) where the subscript 1 denotes conditions just outside the vortex-current layer. But can be calculated from Bernoulli’s equation, so that (1.29)

Thus, the pressure at the body surface and the resulting force and moment on a body are profoundly affected; their variations along the body contour

* The reader who undertakes to calculate the pressure should not be concerned by the fact that the pressures in the related non-magnetic flow, according to Eq. (1.25), are negative for sub-Alfvhic flow. The related flow is. of course, unrealistic to this extent, but it should be recalled that incompressible flow patterns (velocity and pressuregradient distributions) are unaffected by changes of pressure level, i.e., by adding or subtracting a constant pressure.

16

W. R . SEARS A N D E. L. RESLER, J R .

are opposite to those of conventional flow if a m 2 > 1 (sub-AlfvCnic flow). This result pertains not only to inviscid flow of an ideal conductor, but also, clearly, to Hasimoto's flow whenever (1 - am2)Reis large enough to permit boundary-layer approximations, and to flows with large Km and small PY,, as will be pointed out below.

Conductor of Large a and Small P r , I t has already been emphasized that most fluid conductors are characterized by very small values of the property PY, = 4npav. Unfortunately, Hasirnoto's elegant solution pertains instead to very large values of this quantity. An appropriate approximation can be made, however, for plane or axisymmetric flow at small PY, provided that the magnetic Reynolds number Rm is sufficiently large. This theory has been presented in Ref. [25] and will only be outlined here. I t is based on the assumption that current density and vorticity are large only within a boundary layer whose thickness vanishes as Rm + do. A study of the pertinent equations, Eqs. (1.8) and (1.21) and the divergence conditions, then discloses the orders of magnitude of the various unknown quantities, and the equations are simplified accordingly. The argument is analogous to Prandtl's concerning the conventional viscous boundary layer, but it is complicated by the fact that there are more unknowns, there are two Reynolds numbers, Rm and Re, and the orders of magnitude of the magnetic-field components H , and H, within the boundary layer are not known. Nevertheless, it is a powerful assumption that in the limit Rm = do the flow becomes identical with that discussed above, for that means that the boundary values of these components at the solidfluid interface must be o(1) as the limit is approached. The argument proceeds by assuming the dependent variables to be expressible in power series in some positive power of Rm and studying the consequences term-by-term in Eqs. (1.8), (lug),and (1.21). This process must be carried out, however, separately for two distinct layers, one (the inviscid boundary layer) in which viscous effects are negligible but current-density and vorticity are not, and a second (the viscous sublayer) in which viscous terms are essential. From consideration of the former, it can be concluded that its thickness is O(Rm-'/2)and that current-density and vorticity within it are O(Rm''2). One has boundary values for H, and u a t the outer edge of this layer but only for u (viz., u = O ( K e - 1 / 2 )m 0 ) at the inner edge. Four conditions are needed; the missing one must be deduced from the study of the viscous sublayer. This study leads immediately to the conclusion that the viscous-sublayer thickness is O(Re-'I2) while zc is 0(1), as in conventional viscous boundary layers. From this information the missing boundary condition for the

17

MAGNETO-AERODYNAMIC FLOW PAST BODIES

inviscid layer is deduced, for aH,/ay is a t most O(Rm'/') and therefore H , is a small quantity and may be neglected at the outer edge of the viscous layer, or the inner edge of the inviscid layer. The following description therefore applies to this type of flow : (a) The boundary layer consists of two distinct layers: an inviscid outer layer of thickness O ( R m - 1 / 2 )underlain , by a viscous sublayer of thickness O(Re-'/').

FIG. 1.3. Sketch of plane flow with verv strong magnetic field.

(b) In the inviscid boundary layer the following orders of magnitude prevail : 24 = 0(1),

Hz

= 0(1),

v = 0 (Rm- l/2),

H,

= O(Rm-1/2),

j = O(Rm1/2), Q = O(Rm1/2).

(c) A t the inner edge of this layer, v becomes o(Rm-1/2)and H , becomes o ( 1 ) ; hence the inner boundary values of these components for the inviscid

boundary layer are both zero. The tangential velocity component u remains

OP). (d) In the viscous sublayer the orders of magnitude, by contrast, are u = 0(1), 2'

= O(Ra-'/2),

+

H , = O ( R M - ' / ~ ) O(PY,'/~),

H,= O(Rm-1'2),

j

= O(Rm1/2),

Q = O(Re1/').

Furthermore, the momentum equation for this sublayer involves H , but not H,, because the product v H , is negligible, and H , can be taken to be constant. Thus the magnetic-field components need not be treated as unknowns in this layer.

18

W. R. SEARS A N D E . L. RESLER, JR.

(e) The total pressure P is nearly independent of the normal coordinate within the entire layer; i.e., its variation is O(Rm-l). Thus the static pressure within the viscous sublayer and a t the body surface is equal to P a t the outer edge of the inviscid layer, to this order. The procedure for detailed description of this type of flow is therefore to begin with the inviscid, irrotational flow, i.e., conditions outside the boundary layer, and to solve the inviscid-boundary-layer equations using these potential-flow values as boundary conditions. The results include the values of u and H, at the inner edge; these in turn are boundary values for the viscous sublayer. The equations of the sublayer are the same as those used by Rossow and others in boundary-layer studies of an entirely different kind [21], [22], [26] to describe boundary-layer flow with normal magnetic field a t small Rm. Here the equations are the same, not because Rm is small but because the magnetic-field strength is small. Finally, the abovementioned value of H, in the sublayer becomes the boundary value for the potential problem of the body’s interior. Unfortunately, the equations of the inviscid boundary layer are so complicated, involving as they do four scalar unknowns in a nonlinear fashion, that little can be said a t this time about their detailed behavior. Some studies of the inviscid boundary layer a t stagnation points have been carried out [27], [28]. These disclose that conventional leading-edge stagnation-point flow occurs a t super-AlfvCnic but is impossible a t subAlfvCnic stream speeds. On the contrary, trailing-edge stagnation-point flow, which is impossible at super-AlfvCnic speeds, appears to be possible a t sub-AlfvCnic speeds. Similarly, studies of wake flow at large Rm and small Pr, show that inviscid downstream wakes occur a t super-AlfvCnic, and inviscid upstream wakes at sub-AlfvCnic flow speeds. Thus, one is led to the conjectures that a t super-AlfvCnic speeds the distinct inviscid and viscous layers, the former P Y , , , - ~times / ~ as thick as the latter, both grow in the downstream direction and become, respectively, a thick and a thin wake, while at sub-AlfvCnic speeds the thicker, inviscid wake becomes a precursor. These conjectures are borne out by Greenspan and Carrier’s calculations [29:, [30] for flow past finite flat plates and also by Gourdine’s arguments and Lary’s calculations, referred to below. Nevertheless, the situation regarding sub-AlfvCnic flow is not really clear. Our calculations for sub-AlfvCnic trailing-edge stagnation-point flow have not yet led to acceptable solutions; it can be proved that if any solutions exist they must involve back-flow, and it is not at all clear how these could be fitted to the flow about a closed body. Some attacks on this problem have been made by considering the limiting case of very strong magnetic field; i.e., am 00. These will be mentioned below. --t

MAGNETO-AERODYNAMIC FLOW PAST BODIES

19

Small-Perturbation Flow at Arbitrary Rm For aligned-fields flows, Eq. (1.19) becomes (1.31)

(wU(l V -am2)

Gourdine 1311 has factored this operator in the form (1.32)

where (1.33)

1

11,2= 2L {(Re

___

+ Rm) f v ( R e + Rm)2- 4(1 - am2)ReRm}.

L being the characteristic length used in defining Re and Rm. The operators in Eq. (1.32) are of Oseen’s type; hence vorticity and current appear in wake-like regions. These wakes expand parabolically in the streamwise direction, with widths of order and respectively. The complete flow field consists of these rotational parts superimposed on irrotational, current-free fields.

v a

V./a,

The nature of the flow for small values of Pr, can be determined approximately from Eq. (1.33). Let

6 = Rml Re = Pr,

(1.34)

<< 1.

Then (1.35)

k 2

w

1 Re

2 r (1

+ 6 f (1 - 6 + 2am26)}

or (1.36)

1, w Re/L

A2 w (1 - aa2)Rm/L.

and

Thus, current and vorticity are confined to wakes whose lateral dimensions are, respectively, of orders

(F)

1/2

(1.37)

and

(

)”’,

4npc(1 X- am2)U

20

W. R. SEARS A N D E. L. RESLER, J R .

and it is clear that the latter, i.e., the thicker wake, expands in the upstream direction when a m 2 > 1. In other words, for arbitrary Rm and small P r , the flow appears to be of the character already deduced above, involving a narrow viscous wake and a much thicker inviscid wake or precursor. To be sure, studies of this nature do not cast much light on the question of how the wakes or precursors join the flow near the solid body, which for sub-AlfvCnic flow is the unsolved problem alluded to above. Lary [32] considered the limiting casePr, = 0, i.e., inviscid, smallperturbation, aligned-fields flow at arbitrary Rm. His results agree with what has been said above, and he was able to solve boundary-value problems of slender two-dimensional and axisymmetric bodies by distribution of singular solutions. Thus he did answer the question of the joining of the far-field to the flow near the solid body, but only within the accuracy of the linearized small-perturbation approximation. Lary’s results lend some support to the general picture we have deduced for aligned-fields flow but cannot be considered definitive. We shall not reproduce Lary’s detailed results here, but only draw upon them to assess the strength of the upstream wake in typical cases. For a body of revolution whose cross-sectional area is S ( x ) , in sub-AlfvCnic flow, Lary’s formula (62) of [32] gives the following expression for the streamwise velocity perturbation u’ at a distance - x ahead of the body nose ( x = 0): L

(1 3 8 ) 0

Here the quantity s denotes { ( x - x ’ ) ~+ Y ~ } in~ a/ cylindrical ~ X , Y coordinate system; hence s is to be taken equal to x‘ - x in Eq. (1.38) for points on the axis upstream of the body. L is the body length. It is clear that the largest effect, according to Eq. (1.38), at large distances x upstream is the magnetohydrodynamic term involving Rm, which is

source-like in comparison with the leading non-magnetic term, which is doublet-like, as is well known. For large values of - x Lary’s formula can be simplified by putting >> -- x’ and expanding s in a Taylor series and neglecting terms of order ( x ‘ / x ) ~compared to 1; the result is, for a closed body, -

x

L

(1.39) 0

MAGNETO-AERODYNAMIC F L O W PAST BODIES

21

where the integral is recognized as the body volume. It is clear that the r 3 term in Eq. (1.39) must be ignored when magnetohydrodynamic effects are present because comparable terms have been neglected in the magnetohydrodynamic term; we have retained it to show explicitly the leading non-magnetic upstream influence. Thus the upstream influence in the sub-AlfvCnic precursor, measured by the velocity deficiency, is an order of magnitude larger than the conventional effect. For example, if Rm = n and a, = 2, the wake velocity deficiency is x/2L times as great as the deficiency without magnetohydrodynamic effects, i.e., 5 times as great at a distance of 10 body lengths. The precursor should therefore be easily discernible in the laboratory. 3. Flow with Very Strong Magnetic Field

As we have already mentioned, attacks directed toward the question of sub-AlfvCnic flow properties have been made by considering the limiting M. These are, for example, case of very strong magnetic field; i.e., a, the investigations of Ludford [El, [33], [34], and Stewartson [35]. For the -+

FIG. 1.4. Sketch of aligned-fields flow with very strong magnetic field [38], [39].

general case where B, and U are not parallel, the conclusion is not unexpected in view of the results obtained earlier in this report: for either very large or very small Rm, the flow is undisturbed except by standing waves lying in the B, direction, and conditions are invariant along these waves (Fig. 1.3). This type of flow has been studied in more detail by Ludford and Singh [361, [371. For aligned-fields flow, in the same limit a, -+ bo, Stewartson [38], [39] finds an analogous type of flow (Fig. 1.4). This result leads Stewartson to the interesting conjecture that all steady, sub-AlfvCnic, aligned-fields flows are analogous to that sketched in Fig. 1.4; they involve wake-like disturbances both upstream and downstream of the body, whose widths

22

W.

R. S E A R S A N D E. L. RESLER, J R .

are not small, even for streamline bodies, but are of the same order as the body thickness. In view of our difficulties in finding acceptable solutions to the equations of the inviscid boundary layer, mentioned earlier, it seems possible that this conjecture is correct. But still other flow patterns have been suggested; e.g., by Tamada [a], and the correct solution seems to await more elaborate calculations or experimental observations.

4. Summary

In brief summary of what has been found thus far concerning the steady flow of incompressible conducting fluids past solid obstacles, we emphasize that for large Rm the picture is dominated by the phenomenon of AlfvCn waves. In real conducting liquids such waves are diffused by both viscosity and electrical resistivity; the latter is typically a relatively larger effect. With increasing resistivity (smaller Rm), this diffusion of the standing waves becomes so great that the waves actually become broad regions that expand parabolically away from the obstacle in the AlfvCn-wave directions. These are sometimes called “wakes,” even when they do not lie in the stream direction, because of their shapes. Finally, for small Rm, the diffusewave nature of the flow is lost, for magnetic diffusion is essentially instantaneous. The category of aligned-fields flows is singular, however, for here the AlfvCn-propagation direction coincides with the flow direction. For large Rm the flow is irrotational and current-free outside of thin boundary layers and the wakes to which they connect. Detailed study of these phenomena reveals that, for small Pi,, both boundary layers and wakes are essentially two-layered, with a thin viscous layer imbedded within a much thicker, nearly inviscid layer. The inviscid boundary layer appears t o grow in thickness from rear to front and to connect with an upstream wake or precursor, whenever the flow is sub-AlfvCnic. The detailed nature of this sub-AIfvCnic flow is not clear, however, and the conjecture has been advanced that an entirely different kind of flow pattern actually occurs a t sub-AlfvCnic speeds, viz., “slug” flow, in which there are dead-water wakes both upstream and downstream of a body. In the next chapter the effects of compressibility of the fluid are considered. It will be shown that only a few changes must be made in the descriptions given here in order to describe qualitatively the flow of a conducting gas past an obstacle.

23

MAGNETO-AERODYNAMIC FLOW PAST BODIES

11. COMPRESSIBLEFLOW 1. Aligned-Fields Flow of Inviscid Gases

To begin our discussion of gas flows past solid obstacles, we shall reverse the order of the preceding chapter and treat first the special case of alignedfields flows. We shall consider first inviscid flows, with some confidence that the principal effects of viscosity, viz., diffusion of waves, boundary layers, and wakes, are not qualitatively different from what occurs in nonmagnetic flows. The equations of the problem are Eqs. (1.2) to (1.6), inclusive, where now p is variable and the viscosity term in Eq. (1.2) is omitted, but Eq. (1.1) must be replaced by

div (pq) = 0

(2.1)

and therefore Eq. (1.7) by (2.2)

1 -V2H = (H V)q

4nP7

- (q * C7)H - H div q.

We are also in need of appropriate equations of state and energy. For the former we shall adopt the perfect-gas law as an approximation:

p

(2.3)

=9 p T .

The equation of energy conservation need only be a statement that heat is added to the flow in the form of Joule heating: i.e., for steady inviscid flow with no other forms of heat addition, (2.4)

-

cv(q P)T

+p(q - P)p-l=

j2./!a,

Small- Perturbatiori Flow We also introduce immediately the approximations of small-perturbation flows. Choosing the x direction parallel to the undisturbed vectors U and H,, we have, from Eq. (1.2), (2.5)

av ah p,U-----HH,-+grad ax 4n ax

where pm is the unperturbed value of the fluid density, once again v and h denote the perturbation vectors. The other pertinent equations become (2.6)

u

ax

+ p,

div v = 0,

p'

is

p - pa, and

24

W. R . SEARS AND E. L. RESLER, JR.

4nj = curl h,

(2.7)

1 av ah -__ V2h = H, -- U - - Hmdivv, ax

4nv

ax

and the energy equation. But according to Eq. (2.7) the current density is a first-order quantity, and therefore the Joule-heating rate is second-order and can be neglected. The energy equation becomes a statement that the flow is homentropic, and the temperature can be eliminated from the equation of state: (2.10)

P =py

or

P'1P.X

= yp'lp,.

The x component of Eq. (2.5) can be integrated immediately with respect to x : (2.11)

p,Uu'

+ P' = function of y

and z.

The arbitrary function of y and z is evaluated a t a large distance where the flow is undisturbed; it is clearly equal to zero, i.e., (2.12)

p

= Pa

i- p ' =

pco

- pmuu'.

Another important relation is obtained by operating upon Eq. (2.5) with the curl; the result is, upon integration with respect to x , (2.13)

p,UQ -

fi 8 = function of y and z 4n

where again 5 denotes curl H, or 4xj. Once again the function of y and z is seen to vanish by consideration of conditions a t large n. Thus, (2.14)

For all aligned-fields, small-perturbation, inviscid flows, therefore, the vorticity and current density are in constant proportion, regardless of compressibility and magnetic Reynolds number. The consequence of combining Eqs. (2.6), (2.10), and (2.12) is to give us a familiar relationship of conventional gasdynamics (Eqs. 10-16 and 10-18 of [41]): (2.15)

MAGNETO-AERODYNAMIC FLOW PAST BODIES

25

Eq. (2.8) therefore takes the form (2.16)

-

av ah aui 1 - U -ax - M,2H, -. 2&i V2h= H , ax ax

We shall postpone a discussion of Eq. (2.10) to a later stage in this chapter.

Ideal Conditctor At this point let us make the approximation of infinite conductivity in the equations just derived. Only Eq. (2.16) is affected; its left-hand side becomes zero and it can be integrated with respect to x :

H,v - M,’Hmu’

(2.17)

Uh

where again a function of y and 2 has been evaluated a t large distances. Eq. (2.17) is more meaningful, perhaps, in component form: (2.18)

where the parentheses enclose vectors and Pm2is the Prandtl-Glauert factor 1 - MWa. This is an interesting generalization of Eq. (1.22), to which it reduces a t small Mach number. In principle, Eqs. (2.14) and (2.17) are to be solved to determine the flow pattern. For simplicity let us illustrate this by assuming that the flow is either plane or axisymmetrical, so that a small-perturbation stream function $ can be used ( $ C , 11 of [41]), i.e., (2.19)

p,w

= y--Ia+lay,

VI =

- y-Ea+iax

where x , y are plane Cartesian or cylindrical coordinates, and E has the value 0 or 1 in plane or axisymmetric flow, respectively. This perturbation stream function assures that the linearized equation of continuity, Eq. (2.0), is satisfied. Eq. (2.14) then reads

(2.20)

and upon introduction of Eq. (2.18), this becomes

(2.21)

26

W. R . S E A R S AND E. L. RESLER, J R .

so that the equation for $ is (2.22)

This can be put into the Prandtl-Glauert form by introducing a new “Mach number” A,, i.e., (2.23)

4

-

c

CONVENTIONAL

a

f

I3

J5 P

I

ELUPTIC

1

*

0

%

I

0

FIG.2.1. Diagram illustrating the regimes of hyperbolic and elliptic flow problems for the aligned-fields case. Conventional aerodynamics occupies the extreme upper region.

where (2.24)

A w 2 =

8w2-1 aW2pm2 -1

-

M , 2a, -2

+ aW-2 - 1

MW2

*

The effective propagation speed of small disturbances is revealed by writing Aw2 as U2/cW2;Eq. (2.24) then requires that (2.26)

c,a

= am2

+ A,% - a , ~ A w 2 i U 2

where A , is the AlfvCn-wave speed of the undisturbedstream, (,uH,2/4zp,)1’2. The number A, plays the role of the Mach number of non-magnetic flow. The flow field is elliptical when A, < 1, parabolic a t A, = 1, and hyperbolic when A, > 1. It is interesting to locate these regions on a plot

MAGNETO-AERODYNAMIC FLOW PAST BODIES

27

of a,-l against M , , as in Fig. 2.1.* The most striking features are the existence of subsonic-hyperbolic and supersonic-elliptic regimes. As indicated, conventional (non-magnetic) flow occurs in this diagram at large values of a,-l and consists simply of elliptic-subsonic and hyperbolic-supersonic regimes. The parabolic loci are the straight lines M , = 1 and a,-l = 1. The transition at the quarter-circle M m 2 a,-2 = 1 does not have any counterpart in non-magnetic fluid mechanics, for the flow in the hyperbolic area adjacent to it is “hypersonic”: the ratio A, is very large, not because the flow speed is large but because the critical speed c , is small. The name hypercritical has therefore been suggested for this transition [a].

+

An important difference between the subsonic sub-AlfvCnic and the supersonic super-AlfvCnic hyperbolic regimes can be seen by considering the variation of the “Mach angle,” sin-l(I/A,), with flow speed U . In the supersonic super-Alfvknic regime this angle decreases with increasing U , as does the conventional Mach angle of supersonic nonmagnetic flow. In the subsonic sub-AlfvCnic hyperbolic regime, however, this angle increases with increasing U , from 0 at the critical semicircle (Fig. 2.1) to n/2 at either the sonic or the AlfvCnic transition. Now, standing waves of steady flow are the envelopes of propagating disturbances set up by the relative motion of body and fluid (as will be emphasized later in this report). For the inclination of such waves to increase with increasing speed means that they are envelopes of forward-running disturbances, and therefore that they are inclined upstream. This is the nature of the hyperbolic flow in the cross-hatched subsonic area of Fig. 2.1; it is analogous to conventional supersonic flow, but the waves that make up its patterns are inclined upstream. Details of the flow field, within the small-perturbation approximation, can easily be deduced from Eq. (2.23) by the methods of conventional gasdynamics. In the hyperbolic regimes the general solution for plane flow, for example, is (2.26)

$(x,Y) = f ( x

+ m m ~ +)

S(X

- mmy)

where f and g are arbitrary and rn, denotes l V d r n 2- 11. A typical, simple flow is sketched in Fig. 2.2 for a subsonic sub-AlfvCnic stream. It should be of interest that the streamline spacing is narrower than free-stream above the inclined plate, and wider below. In view of Eqs. (2.10) and (2.12), the relationships among speed, pressure, and density are conventional ; thus, Diagrams of this type were used in References [42] and [43];the latter is a study of the characteristics of the nonlinear equations of inviscid, perfectly-conducting. gas flow. This emphasizes the fact that the small-perturbation results derived here are locally correct in nonlinear problems.

28

W . R. SEARS A N D E. L. RESLER, JH.

since the flow is subsonic, the pressure is greater in the flow below the plate than above. For the case sketched, in fact, it is (2.27)

p

=pa

- pcoU2mmd/Pm2

in the area above the plate and (2.28)

p

=p

,

+ pmU%J/Pm2

in the area below the plate. But these are not the pressures on the plate, for the surface-current layers must be accounted for. This requires consideration of the detailed behavior of the magnetic field a t the body.

FIG.2.2. Sketc of plane, aligned-fields sub-AlfvCnic, subsonic flow over a flat pla .e at angle of incidence 6.

According to Eq. (2.18), the magnetic-field vector H is always parallel to the velocity vector q to a first-order approximation. In fact, this result is somewhat misleading because the parallelism of these vectors is not approximate but exact. To see this, we simply return to Ohm’s Law for infinite conductivity; for all cases where the electric field is zero, such as plane and axisymmetric flows, it is, again, (2.29)

qxH=O

so that q and II are everywhere parallel. From the divergence conditions of the present problem, i e . , div (pq) = div H = 0,it immediately follows that pq a H, or (2.30)

As indicated, Eqs. (2.17) and (2.18) are linearized statements of this result. A consequence is that, just as in the corresponding incompressible case, the magnetic field must vanish within any solid obstacle of finite dimensions. Once again there appears a current- and vortex-layer a t the fluid-solid interface, and the surface pressure differs from the fluid pressures calculated

MAGNETO-AERODYNAMIC FLOW PAST BODIES

29

above by the amount of the body force on this layer. I t is interesting to notice that this is not a first-order, but a zeroth-order amount: as in Eq. (1.28), the surface pressure is

P = p1= Pl +

HI2

P Z

(2.32)

to first order. Now It,, can be evaluated from Eq. (2.18)for small-perturbation flow; hence, (2.33)

FIG. 2.3. Sketch of plane, aligned-fields sub-Alfvhic, supersonic flow over an airfoil.

In the example of Fig. 2.2, therefore, the surface pressures are

The factor 1 - am2Pm2 is positive in the subsonic hyperbolic flow regime; thus the lift on the plate is positive for positive 6. In the elliptic regimes of Fig. 2.1, on the other hand, the solution is found by making a simple affine transformation of the well-known PrandtlGlauert type. Some consequences have been pointed out in Reference [46].

30

W. R. SEARS A N D E. L. RESLER, J R .

A particularly intriguing area is the elliptic, supersonic, sub-AlfvCnic regime. Since the flow pattern is calculated by an affine transformation of incompressible flow, its stream-tube characteristics are those usually associated with subsonic flow. In Fig. 2.3 for example, the stream-lines are more closely spaced above the obstacle than below it. But the flow is supersonic, and the relationships between speed, pressure, density, and stream-tube area are typical of supersonic flow. Thus the pressures in the fluid above the obstacle are higher than below. Nevertheless, because of the surface-current effect, the surface pressure on the upper surface of the body, given by Eq. (2.34),is lower than that on the lower surface, and the lift in the flow of Fig. 2.3 behaves conventionally.

But in sketching Fig. 2.3 we have implicitly assumed that the pattern can be carried over from incompressible flow without altering the circulation. In a kind of flow that involves both viscous and inviscid boundary layers, and in which some boundary layers appear to grow from rear to front on a body, it surely must be necessary to reexamine the principles - inherently viscosity-related - that fix the value of the circulation in actual flows.

Relationship to Wave-Propagation Phenomena Our principal objective here, however, is not to work out the details of aligned-fields flows but to arrive at general conclusions regarding the nature of all streaming flows past obstacles. To make further progress toward

FIG.2.4. Diagram relating wave-propagation speed c to the wave angle in steady flow.

this objective we now digress to present some well-known results concerning the propagation of infinitesimal plane waves in electrically conducting gases and to relate these to our steady flows. The propagation of such waves in the presence of a magnetic field has been studied by van de Hulst, Friedrichs, and others [MI, [47], [48] and their derivations need not be repeated here. Suffice it to say that van de Hulst's method consists of a small-perturbation attack upon the equations for inviscid ideal conductors, retaining, of course, certain unsteady effects that we have omitted in our Eqs. (1.2), (1.5),and (2.1). Upon assuming that the field consists of propagating sinusoidal plane waves, one obtains a formula relating the propagation speed c to the sound speed am, the AlfvCn

MAGNETO-AERODYNAMIC FLOW PAST BODIES

31

speed A , , and the angle e, between the wave normal and the magnetic-field vector. But the waves of steady flow patterns are just these magnetosonic waves viewed in a moving system. Since they are stationary in this system, their propagation speed c must be - U cos e,, i.e., the component of U in the normal direction must be the propagation speed c (Fig. 2.4). Thus the propagation speed c can be identified with c, of Eq. (2.25), which clearly has the same meaning. Equation (2.25) can then be written as

+ A m 2- am2Am2 cos2 (p C2

(2.36)

c2 = a m 2

~

or (2.37)

c4

- (a,2

+ Am2)c2+ am2Am2cos2q~ = 0.

This is the formula referred to above, giving the propagation speed of plane waves for plane and axisymmetric flow. In the references cited, however, the restriction to plane and axisymmetric flow is not made and an additional wave speed is found; the formula (2.3'7)is replaced by (2.38)

+

+

(c2 - Am2)[c4- ( a m 2 Am2)c2 a , 2 A , 2 ~ ~ ~= 2 0. ~]

Further study of the modes of the respective waves discloses that the flow produced by the waves whose speed is A , is not plane but involves velocity components normal to the plane of H , and the wave normal. I t is for this reason that this root was eliminated from our formulas for plane flow and (if azimuthal velocities are assumed not to occur, for reasons of symmetry) from axisymmetric flow as well. The roots of Eq. (2.38)are usually plotted in a polar diagram of c versus e,, as in Fig. 2.5. There is also a companion diagram to Fig. 2.5, due to Friedrichs and Kranzer [47], giving the shape of a self-similar disturbance that expands from a point according to the speed-inclination relationship of Fig. 2.5. It can be constructed from Fig. 2.5 by a graphical process [49] that is equivalent to Huygens' construction. The result is shown in Fig. 2.6. In geometrical terminology, Fig. 2.5 gives the pedal curves of Fig. 2.6. This figure has the property that is is shape-preserving if every element propagates according to Fig. 2.5; it is therefore the shape of the disturbance that propagates from every point along the path of a moving point-disturbance and is the magnetogasdynamic analog of the spherical disturbance of radius a, in ordinary gasdynamics. Figure 2.6 is a figure of revolution, symmetrical about the B, direction. The disturbance front caused by the fast waves of Fig. 2.5 is an oblate spheroid. The slow waves produce two cusped figures of revolution that

32

W. R. SEARS AND E . L. RESLER, J R .

sLowKd

WAVES

0-

-

FIG.2.5. Friedrich’s Wave-Speed Diagram. drawn for the case

FIG.2.6.

a, = V Z A , .

(The dimen-

MAGNETO-AERODYNAMIC FLOW PAST BODIES

33

lie of course, within the spheroid; it can be shown that the volume inside of these figures is undisturbed [50]. The intermediate waves produce only two point-disturbances propagating at the Alfvkn-wave speed along the B, direction. The innermost parts of the slow-wave curves of Fig. 2.5, being approximately circular, also produce point-disturbances ; these are the points propagating with speed (urnd2 Am-2)-1/2 at the inner ends of the cusped figures. A three-dimensional sketch of the disturbance diagram is given in Fig. 2.7. Now the standing-wave results derived above can be understood in relation to this diagram (Fig. 2.6) and the general nature of more general, aligned-fields or non-aligned steady flows can be predicted as well.

+

FIG. 2.7. Three-dimensional sketch of the disturbance pattern of Fig. 2.6 (not to scale).

First, all of the elliptic and hyperbolic regions of Fig. 2.1, which pertains only to plane and axisymmetric flow, can be understood. Stationary waves are produced whenever the sequence of disturbances produced by a moving body forms an envelope. This can be ascertained by drawing the body's velocity vector in Fig. 2.6 and asking whether a tangent can be drawn from the head of this vector to any point of the disturbance figure. If the flow is plane, the envelope is a plane; if the flow is axisymmetric, the envelope is a cone. Thus, body-velocity vectors that terminate in OA (Fig. 2.6) produce no tangents and the flow is elliptic. Those that terminate in A B produce tangelits to the cusped, slow-wave figures; these represent the forward-facing waves of subsonic sub-Alfvknic flow. When the flowspeed vector terminates in BC, no tangent can be drawn; this is the elliptic regime of subsonic super-Alfvknic flow. When the speed U is greater than OC, tangents can be drawn to the fast-wave diagram, giving correctly the hyperbolic, supersonic, super-Alfvknic regime.

34

W . R. SEARS

AND E. L. RESLER, JR.

The preceding discussion has been based on the assumption that a , > A , or M , < a,-l. Study of Eq. (2.38) will disclose that when a, < A , the fast- and slow-wave curves in Fig. 2.5 remain unaltered except that the dimensions a, and A , in the figure must be interchanged. The intermediatewave circles, of course, must still have diameter A , . Therefore, when a, < A , , Fig. 2.6 is altered only by interchanging the dimensions a, and A , and moving the intermediate-wave point to point C. Then the regime
O x

BC

2. Flow with A rhitrary Magnetic-Field Direction

Having related the phenomena of aligned-fields flow to the disturbance diagram, let us use this diagram to work out the properties of steady flow with arbitrary angle between the vectors U and If,, still assuming the fluid to be inviscid and perfectly conducting. Now we must carefully distinguish between two-dimensional (plane) and three-dimensional flows ; this is in contrast to the aligned-fields case where plane and axisymmetric flows, at least, could be treated together, with differences only in details. Two-Dimensional Flows Suppose that the disturbance is caused by a cylindrical body whose axis is perpendicular to the direction of the undisturbed magnetic field H,.

FIG.2.8. Formation of stationary waves in planc steady flow at high speed.

Let the stream vector U have any arbitrary direction. By viewing the flow in a coordinate system moving along the cylindrical axis with a speed equal to the axial component of ti, one can always convert the flow to one

MAGNETO-AERODYNAMIC FLOW PAST BODIES

35

in which both H, and the stream vector lie in a plane perpendicular to the axis. We shall therefore confine our discussion to this case. Now, the disturbances set up by the relative motion are cylindrical; i.e., they are the envelopes of the three-dimensional disturbances of Fig. 2.6 and consist of cylindrical wave fronts having the cross section of Fig. 2.6. By symmetry the flow pattern can be expected to be plane. The nature of the resulting flow patterns can again be ascertained by drawing the body-velocity vector in Fig. 2.6 and looking for tangents, which represent plane envelopes. Figure 2.8 illustrates one such case: the bodyvelocity vector - U is so great that its arrowhead lies outside the fast-wave front ; hence, four plane, stationary, wave families are produced. The flow pattern about an obstacle, when the stream speed is U and its magnitude is greater than that of the fast-wave front, is therefore doubly hyperbolic, made up of four families of waves whose directions are given by the construction of Fig. 2.8. The perturbation stream function $, for example, is given by an expression of the form (2.39)

where the mi are constants. The four functions f j are determined uniquely by the boundary conditions as follows: (a) At each surface of a slender body in a uniform stream two of the f’ vanish, because waves extend outward only. (b) For the remaining two functions for each surface we have a relationship provided by the kinematic condition of given normal velocity component. The tangential velocity component is not prescribed because the flow is inviscid, i.e., a vortex sheet is permitted a t the interface. (c) The required additional conditions to determine the solutions are formed from the requirement of continuity of both magnetic-field components at the body surfaces. (d) The interior solution in this case is a potential solution only if the body is a nonconductor; otherwise it is a solution for given current density determined by the body’s electrical conductivity and the value of E = - U x B,. In either case it relates the upper- and lower-surface values of the field components. Problems in this category have been solved by McCune and Resler 11451 and by C. K. Chu [51]. The requirement that both components of H be continuous a t the interface may require explanation, since it was not imposed in alignedfields flow of the same perfectly conducting fluid. We cannot permit a current layer in crossed-fields flow, however, for the body force on such

36

W. R. SEARS A N D E. L. RESLER, J R .

a layer acts in a direction perpendicular to the magnetic field. In alignedfields flows this is in equilibrium with the pressure jump across the layer, which we discussed and evaluated a t an earlier point in this report. Hut in crossed-fields flows the body force has a tangential component, which cannot be balanced. It is interesting to notice that, according to this discussion, there is one less boundary condition at each surface in aligned-fields flow than in more general geometries. But the aligned-fields case is just the singular one in which one family of waves is lost (cf. Figs. 2.7 and 2.8), so that the flow pattern remains determinate.

FIG.2.9. Formation of stationary waves in plane steady flow at low speed.

In Fig. 2.9 is sketched another two-dimensional crossed-fields case, namely one in which the vector - U lies within the fast-wave front. In this type of flow there are only two families of waves, but there is also an elliptic disturbance field provided by the fast waves. This is in analogy with conventional subsonic flow, where an elliptic disturbance field occurs whenever an object travels a t a steady subsonic speed. The functional form of the solution, instead of Eq. (2.39), is therefore 4

(2.40)

+(x,

Y) == C / i ( x

+ WY) +

+e

r-1

where zh,C satisfies an elliptic differential equation. The situation regarding boundary conditions and uniqueness of the solution is substantially the same as described above. Once again we permit a vortex sheet but not a current layer. Problems of this type have been solved in Reference [45j.

Three-Dimensional Flows Without going into any more detail concerning two-dimensional flows, let us proceed to the more complicated, general case. I t has already been remarked that the Friedrichs disturbance diagram, Fig. 2.6, is a figure

MAGNETO-AERODYNAMIC FLOW PAST BODIES

37

of revolution. In general, a point disturbance - a small body - moving at an angle to the magnetic-field direction H, create3 a sequence of these disturbances and a complicated arrangement of magnetosonic “Mach cones.” For example, if the body-velocity vector penetrates the fast-wave front, a “Mach cone” of more-or-less conventional appearance will be formed by the envelopes of these fronts, lying behind the disturbance. (To say that the vector penetrates the front means, of course, that the body’s speed is so great that the oblate spheroids do lie behind the body and form an envelope.) But the body also produces a trail of the cusped, slow-wave fronts, which lie behind it (in general) in an orientation determined by the magneticfield vector. Two narrow “Mach cones” are formed by the envelopes of these slow-wave fronts, as has been pointed out recently by Cumberbatch 1,521. These cones have sharp edges, as Cumberbatch has also pointed out; i.e., their cross sections are always cusped. The reader may be able to visualize these by studying Fig. 2.7. The consequence of this is that the flow field of a three-dimensional body in crossed-fields flow is rather complex, and will not be easy to describe analytically. The wave pattern set up by the fast-wave cones does not seem to be greatly different from the supersonic-flow pattern of conventional aerodynamics. The narrow, cusped, cones discovered by Cumberbatch, however, must produce nearly-cylindrical disturbance patterns extending out from the body roughly in the plane of B, and U. If the cones are very slender - note that Fig. 2.7 is not drawn to scale; consult instead Fig. 2.6 we might approximate to them by lines; such lines then coincide with the influence lines of the intermediate waves, i.e., the loci of the points C of Fig. 2.6. The resulting disturbance patterns of three-dimensional compressible flow then resemble the cylindrical standing-AlfvCn-wave patterns of incompressible flow (cf. page 10). At high speeds these lie within a zone of influence bounded by a slightly distorted Mach cone: at lower speeds they lie in an elliptic field. On the basis of our previous discussions, we can anticipate the effects of electrical resistivity on the flow patterns deduced here. For large but finite Rm, the effects will be to diffuse all of the wave patterns described and, in the special case of aligned fields, to introduce boundary layers in replacement of current sheets. For moderate Rm,we expect these tendencies to increase to the point where the various magnetosonic waves and wavecones become broad, diffuse, and wake-like rather than wave-like. These effects must be analogous to the effects of resistivity in incompressible flow, complicated mainly by the multiplicity of wave families and complexity of their three-dimensional geometries. For small Rm,however, we may expect some interesting new phenomena, at least in supersonic flow, for when Rm vanishes we are left with stationary sound waves (Mach waves), diffused only by viscosity. We may ask: if the

38

W. R. SEARS AND E. L. RESLER, JR.

magnetosonic wave patterns of infinite-Rm flow are increasingly diffused by increasing resistivity, how do they finally pass over into the familiar patterns of conventional supersonic flow ? To verify some of the statements made here and to approach an answer to this question, we shall take up, below, a brief study of the equations for compressible flow at arbitrary Rm. 3. Magneto-gasdynamic Shock Waves

Refore doing so, however, we should interpolate some remarks about magneto-gasdynamic shock waves. These will occur when disturbances are not everywhere small. Magneto-gasdynamic shock waves have been studied by several authors [53], [54], [MI, [56], [57], [583. They are in direct analogy with ordinary gasdynamic shocks, namely, thin fronts across which the flow and state variables change very rapidly, so that they are often approximated by surfaces of discontinuity. In this approximation the fluid is considered to be inviscid and perfectly conducting both upstream and downstream of the discontinuity, and the shock equations are derived from conservation principles that apply across the discontinuity. In this sense the shock equations pertain to an idealized gaseous conductor, but of course the same relations apply to the regions upstream and downstream of real shock fronts in viscous gases with electrical resistivity. Magneto-gasdynamic shocks differ from conventional shocks in essentially two respects: (a) the surface of discontinuity is generally a current sheet, so that a body force, as well as a pressure jump, acts upon the fluid a t that surface, and (b) there are therefore additional terms in the conservation relations, namely the body-force terms in the momentum equations and an additional term in the energy equation. There are, of course, two more parameters characterizing the shock than in ordinary gasdynamics, namely the magnetic-field strength and its angle to the stream. The multiplicity of defining parameters has led the various investigators to use different \;ariables and to present their findings in different sorts of graphs, so that it is difficult to relate their results. I t should not be surprising, perhaps, that the above-mentioned set of conservation equations has a greater number of solutions than the corresponding Rankine-Hugoniot equations of gasdynamics. There are, in fact, six different kinds of magnetogasdynamic shocks, differing from one another by the Mach-Number and AlfvCn-Number regimes in which they fall upstream and downstream. These six types can be defined in relation to the graph of infinitesimal-wave speed versus orientation, Fig. 2.5, which is redrawn in Fig. 8.10 with certain notation added. This diagram consists of four distinct regions, R,, R,, R,, and R,, as shown. The six different types of shock wave can be designated by six transitions R, Ri,j > i,

-.

MAGNETO-AERODYNAMIC FLOW PAST BODIES

39

where the normal velocity component lies in K i before the shock and K, behind. (Note that, since the processes being considered involve finite increments, all of the quantities defining the diagram, such as a, A , and B are different on the two sides of the shock, in general.) The requirement > i results in increasing entropy across all of these shocks; they are therefore thermodynamically acceptable,. and all of them are compression waves.

FIG.2.10. Same as Fig. 2.5 but with notation added to define the six possible kinds of magneto-gasdynamic shock waves. The left half (a) is drawn for A , > a,, the right half for a,,., > A , .

Three of these types, namely R, --* H i + r , clearly include the finite counterparts of the fast, slow, and intermediate infinitesimal waves of Fig. 2.5 and Eq. (2.38), to which they reduce when the flow deflection through them goes t o zero. This would imply that in flow past bodies of appreciable thickness, which produce compression processes of finite strength, the descriptions given above, in which stationary-wave patterns were emphasized, should be modified t o account for the presence of stationary shock waves of these families. There are also “strong shocks”, like the conventional gasdynamic strong shocks, which go over into normal shocks in the limit of vanishing flow deflection. The other three families of shock waves, R, -. R,, R, -. K,, and R, -. K,, do not have infinitesimal counterparts (for given upstream conditions).

40

W. R. SEARS AND E . L. RESLER, J R .

T o make the situation more difficult, there are grave doubts about the “stability” of some of the six families, i.e., whether they will actually be produced in real-gas flows [59], [60], [61]. There seems to be general agreement among these investigators, that four of the families, namely all those that cross the dashed line in Fig. 2.10, are “unstable”. It is difficult to predict how and whether the stable strong shocks will occur in flows past bodies. Cabannes [58] has shown that attached strong shocks, namely those of the families R, -+K, and R, .-+ R4,can appear in the calculated flow patterns for a very simple type of flow, namely steady sub-Alfvknic flow past an infinite wedge. One more investigation of steady, plane, inviscid flows about obstacles, involving shock waves, has been undertaken by Geffen [@L]. The particular case studied is sub-AlfvCnic, supersonic, plane flow. As has already been pointed out, this flow is paradoxical in that its streamline patterns are elliptical but its speed-area relationships are supersonic. As the streamlines widen to pass around the body, the local speed should increase. How, then, does such a flow negotiate the leading edge of an obstacle ? This question was posed in References [63j and [64], and the flow of Fig. 2.11, involving a strong shock of the family R, R,, was proposed as a conjecture.

-.

FIG. 2.11. Conjectured flow pattern for a case of sub-Alfvhic supersonic flow. References [63], !64].

Geffen’s method of attack on this problem is based on numerical solution of the flow equations. For simplicity the upstream conditions are assumed to be only slightly supersonic, so that the approximations of transonic flow are permissible; in particular, the flow is homentropic. Her conclusions are in rough agreement with the conjecture of Fig. 2.11, but differ from it in one essential feature : the leading shock wave cannot be a single, strong shock as sketched, but must be replaced by a sequence of two or more shocks. This conclusion is based upon numerical calculations for the flow problem sketched, but it is also confirmed by analytical results obtained for flow about the same body at sonic stream speed (Fig. 2.12). For this case the conclusion is definite: the shock wave is of the weak family, i.e., the flow ahead of it is hyperbolic (subsonic), as well as the flow behind it. There is no strong-shock solution. We believe, therefore, that at supersonic stream speed the shock at the leading edge will be a weak shock and that it will be preceded, upstream, by one or more additional shocks. At sonic speed the latter have moved infinitely far upstream, giving the flow of Fig. 2.12.

MAGNETO-AERODYNAMIC FLOW PAST BODIES

41

I t is interesting to note, however, that in the analogous axisymmetric sonic flow Geffen finds only a strong-shock solution. I t would appear that two- and three-dimensional flows may be grossly uniike, when shock waves are involved.

FIG.2.12. Sketch of calculated flow pattern for a case of sub-AlfvCnicsonic flow. After N. Geffen [62].

In the absence of further studies of this subject, our conclusions regarding the appearance of magneto-gasdynamic shock waves in flows around bodies of arbitrary shape must remain very vague indeed. 4. Compressible Aligned-Fields Flow with Arbitrary Scalar Conductivity

We shall now undertake a brief study of compressible flow fields with arbitrary scalar conductivity. The equations derived will be used to indicate quantitatively the effects of finite resistivity on the flow patterns, and also to suggest what conditions must be met in the laboratory if flows of real gases are actually to exhibit the features already predicted by our studies of flows with infinite conductivity. This discussion will be based upon the approximation of small-perturbation flow. Previously, before making any assumptions about conductivity, we derived a relation between the perturbed magnetic- and velocity-field vectors in aligned-fields, small-perturbation flow, Eq. (2.8). Taking the curl of this equation so that we may use the general relation between the vorticity and the current, Eq. (2.14), we have

42

W. R . SEARS AND E. L. RESLER, JR.

To satisfy the compressible continuity equation identically we can, as before, define a stream function as in Eqs. (2.19), restricting ourselves here to plane flows. In terms of the stream function we then have

(2.4'2)

Let us rewrite this equation in terms of the previously defined symbol Vry, (Eq. (2.24)); the equation is then

If the gas is not electrically conducting, u

=

0, Eq. (2.43) beconies simply

The term inside the brackets is the Prandtl-Glauert operator, long familiar to aerodynamicists. The only appropriate solutions to (2.44) are the Prandtl-Glauert flows; i.e., the expression in brackets must be put equal to zero, since, when u vanishes so do the currents and therefore the vorticity Q. The expression in brackets in Eq. (2.44) is just - Pmz1;2. For very large u, for which the left side of equation (2.43)can be neglected; we have (2.45)

Thus if we consider flows where all streamlines originate in an undisturbed region we can put the terms in brackets in Eq. (2.45) equal to zero and we have "Prandtl-Glauert phenomena" based on the number Am,as in Eq. (2.23). Suppose now that 0 is large, but not so large that the left-hand side of Eq. (2.43) can be neglected everywhere. In regions where the y-derivatives are large compared with the x-derivatives (i.e., near boundaries), we have (2.46)

I his is the linearized form of an inviscid-boundary-layer equation for compressible flow, for suppose we approximate the y derivatives by the I .

43

MAGNETO-AERODYNAMIC FLOW PAST BODIES

differences across a layer of dimension 6, assumed small, and approximate the x derivatives by dividing by a typical x dimension, say L ; then equation (2.46) gives, as an estimate of 6,

6

- = {am2Rm(Mm2+

(2.47)

L

- l)}-’‘*

where Rm = 4npaUL and a,-l is the free-stream AlfvCn Number. If M , = 0, 6 is the thickness of the inviscid boundary layer discussed in Chapter I. Thus the compressibility of the medium appears to reduce the thickness of this layer for given Rm.

Pro#erties

01

Sinusoidal Solutions

It is very difficult to work out a general flow problem that can be interpreted for all conductivities. Instead, we shall discuss here the simple solutions of Eq. (2.43) that represent flows due to sinusoidal boundary conditions. From the properties of these simpler solutions we shall infer the general properties of flow fields involving obstacles of more general shape. We shall find that the simple, periodic solutions consist of damped waves, and that the damping is dependent upon the wave-length. This is actually the reason for the intractability of more general flow problems, mentioned above. Assume, then, that $ varies as exp (iAx - s y ) , where A = 2n/l and 2 is a wave-length. This presupposes that we shall be interested in flows about thin bodies lying substantially along the axis and whose shapes can be made up by Fourier superposition of sinusoidal elements. By replacing the x-derivatives by iA and the y-derivatives by - 6, the differential equation (2.43) becomes an algebraic equation for the ratio ??/A; namely, if we let r =

s/A,

(2.48)

r4

+ r2 IM m 2- 2 - _iRm ___ M_m 2+ } 2 n “4,= 1

pm2

{+

where now Rm is based upon the wave length 1. quadratic in r 2 ,

{

(2.49) r2 - 1 = 1 Mm2 i R m 2 - 1) 2 2nAm ~

1

kvq

R ;-(1 2 .

-am2)

Since this equation is

1----- iRm

)’+

iR;nm2

2n”#,2

This equation in its present form is unwieldy, and more insight can be gained by making various approximations first. The general case is treated in the literature j13]. Using the definition of Am,Eq. (2.24), this formula can be rewritten as follows:

44

W . R. SEARS A N D E. L. RESLER, J K

(2.50)

In this result, notice that wherever Km appears it is divided by M m 2 . Suppose we now define a parameter K as (2.51)

We can then treat equation (2.50) for both large and small K . Making an expansion for small K , i.e., Rm << 2 x M m 2 ,in equation (2.50) we find (2.52)

r2 = 1

+ iKM,2,

1 - M,2

+ iKM,2(Mm4 - l ) a m 2 .

The first root corresponds to the positive sign in equation (2.50) and the second to the negative sign. (We will consistently adhere to this convention in what follows, referring to the roots as the first and the second, respectively.) Substituting for K we have, after solving for r ,

for Rm << 2nMw2. The first root gives us a first-order effect of the electromagnetic field on the flow pattern, for the case Rm << 2nMw2. The corresponding part of the perturbation stream function is (2.64)

41 a exp i A

(E) x =F

~

y exp

Ay.

(The subscript 1 indicates that this is only the contribution to I/J from the first root.) This part of the flow therefore consists of wavy disturbances set up a t the boundary and damped rapidly with distance from the boundary. The signs in Eq. (2.54) must be chosen to result in positive damping; thus the negative signs are appropriate for positive y , and the lines of constant phase are inclined downstream from vertical a t an angle proportional to Km. This root represents the effect of currents generated a t the boundary, which diffuse into the stream and are carried downstream by the fluid. Note that the damping with y depends on the wave-length; longerwave disturbances extend further out into the stream. When Rm = 0,

MAGNETO-AERODYNAMIC FLOW PAST BODIES

45

this root would indicate a harmonic disturbance pattern; this is the solution corresponding to putting Vz# = 0 in Eri. (2.44). As mentioned, it actually disappears in the limit Rm =- 0, for a compressible flow. The second root is the one that, in the limit Rm = 0, reduces to the ordinary aerodynamic solution, namely either Prandtl-Glauert or Ackeret flow. l h e part of (I, appropriate to this root is

Consider first the supersonic case, i.e., M , greater than one. Note that for Rm == 0 we have Ackeret flow: the streamline pattern is propagated -~ unchanged along lines of slope d y / d x = f 1/l/Mm2 - 1, the Mach lines. Since the slope is independent of wave-length, this result is true of the whole disturbance pattern after Fourier superposition. For small Rm the slope of the lines along which the disturbances stand is not changed, but they are damped. As y increases the upper sign must be chosen in order for the disturbance to disappear at large y , while if y decreases the lower sign must be chosen. Thus the waves are inclined downstream. The amplitude of the disturbance carried by these waves is damped to l/e a t a distance y equal to

Suppose we call this distance y o ; then since yo is not wave-length dependent the damping of the whole pattern is uniform. Thus one can easily draw streamlines for a small-Rm supersonic flow in terms of a conventional supersonic flow pattern by calculating yu. must also Note, however, that this is only part of the solution, as be computed. But the solution damps in a wave-length and in many circumstances y D is much greater than 1. The damping length yo is smaller the larger the magnetic field ( a , varies directly with B,) and instead of being governed by IJ alone, also depends on B,. This is due to the term j >( B in the equation of motion. The damping is least for waves in flows where M , is closest to one. This is easily understood, as the magnetic field tends to suppress velocities across it and the steeper the standing-wave angle the smaller the component of velocity across the field lines. The behavior exhibited by q52 answers, a t least in part, the question posed earlier in this Chapter: how do the damped magnetosonic waves go over into undamped Mach waves in the limit Rm + O ? We see that, as this limit is approached, Mach waves are formed gradually by reduction of the um2Rm-dampingof Eq. (2.55). Presumably there is a regime where

46

w.

R. SEARS AND E. I-. RESLER, J K .

magnetosonic and Mach waves are undistinguishable, because of diffusion due to resistivity. As Rm decreases toward zero, Mach waves gradually appear in this diffused field of flow. Consider now the case where M , is less than one. Now equation (2.55) can be written

(2.57) Of course, as Rm -0 we have the Prandtl-Glauert solution, which can be related to flow fields with M , = 0 by the usual transformations. The effect of increasing Rm in this case is not to damp the waves but to incline the lines of constant phase. Since the lower sign must be chosen for large y , it is seen that these lines are inclined upstream. In this case the angle of inclination, measured from the cross-flow direction y, is proportional to the product aW2Km,which is the so-called “interaction parameter”. Qualitatively, this result tells us that the Prandtl-Glauert disturbance pattern of subsonic flow past a body, which extends symmetrically fore and aft in the case of a symmetrical shape, will be deflected upstream by magnetogasdynamic effects, for small Rm. The reader is cautioned that we cannot discuss the case M , = 0 in the above context. That is, the phenomena of incompressible flows have been excluded in this approximation. I t is permissible to consider small Mm2, but only if Rm is still smaller; for M , = 0, Rm has also disappeared. Making now an expansion for large K , i.e., Rm >> 2 n M m 2 , in equation (2.50), we obtain

(2.58)

1

.u,4

iK M m 2 ( 1-A:?

Mm2-”U,2

Mm2

1

or making the substitution for K ,

(2.59) I n discussing this expression we must not let Rm approach zero for a finite M,, but we can now discuss incompressible flows, M , = 0, and in this limit we can make Rm as small as we like. I t is important to realize that incompressible flow a t arbitrary Rm is essentially different from flow

MAGNETO-AERODYNAMIC FLOW PAST BODIES

47

at moderate M, and small Rm. This is brought out by the fact that Eqs. (2.59) are completely different from their counterparts in the preceding discussion, Eqs. (2.53). The first root for the present case can be written (2.60)

yl =

in(1

f

+

A , 2

-M,2)l,p]

RmMm2

with n(l

Ql oc e

x p i l { x F 2 E ( l -

(2.61)

+ Am2 - Mm2)A?a2 RmMm2 n(1

+ Am2 -M m 2 ) d , ") I y . RmMm2

To orient ourselves with respect to the properties of the solution let us consider first incompressible flow with R m large enough so that the = Mm2a,-2/(M,2+ terms proportional to lfRm can be neglected. Since Am2 a,-2 - l ) , the ratio M,2/A?,2 is equal to 1 - am2for M , = 0. For this case i,hl becomes

for am2< 1 . Thus, for am2< 1 (pmU2/2> ,uH,2/8n, i.e., dynamic pressure greater than magnetic pressure) we must choose the upper sign to insure that the disturbance vanishes at large y. The slope of the lines of constant phase in the region above the body is therefore (2.63)

The larger Rm, the smaller the slope for given am, and for given Rm, the stronger the magnetic field the less the slope. Thus, again, the effect of increasing conductivity is to inhibit diffusion, while the effect of increasing magnetic-field strength is to channel the disturbance and confine it. However, if the flow is sub-Alfvknic; i.e., am2> 1 (pmLi2/2< ,~~CcH,~/87c), this part of Q becomes

for a m 2 > 1 .

48

W. R. SEARS AND E. L. RESLER, J R .

Now for positive damping with increasing y the lower signs must be chosen; thus the lines of constant phase lie along the slope (2.65)

above the body. Thus these waves slope forward a t a slope decreasing with increasing Rm, and again the larger the magnetic field the stronger the channeling of the disturbance in the magnetic-field direction. The phenomena discussed here are the phenomena covered by Lary and Gourdine and mentioned in Chapter I. For large Rm this root produces the inviscid boundary layer already discussed in Chapter I.

If we now consider a compressible gas but again have an Rm large enough so that the terms proportional to 1/Rm can be neglected, we have essentially the same phenomena except that the ratio M m / J m = [ a m 2 ( M m 2 am-2-l]'/2 determines whether the lines of constant phase are inclined forward or backward. The disturbances tend forward for Mm2 am-2< 1, which means inside the semi-circle of radius unity in the M,, amd1plane, Fig. 2.1. This tells us that such phenomena as inviscid boundary layers and wakes, which reverse their directions of growth a t the A l f v h speed in incompressible flow, actually reverse their directions a t this critical speed, instead, in compressible flow.

+

+

If the 1/Rm term is included, the discussion must be modified in an obvious way. Now consider the second root, which contributes for Am> 1 the following to the perturbation stream function : (2.66)

$2 a

exp i A ( x

_VAma - 1 y ) exp f YAW2 -1

7d

Aw4 -___

Rm 1 - a,-2

~

Iy.

Thus, as before, if A m 2 > 1 and a m 2 < 1, we must choose the upper signs for increasing y and the lines of constant phase are inclined backward, while if a m 2 > 1 they are inclined forward. These are the waves discussed previously, i.e., our general predictions concerning the effects of resistivity in compressible aligned-field flows are confirmed. Suppose in an experiment we wish to observe these waves inclined at an angle p with the stream; i.e., sin a,= l/Awand tan p = - 1 / y A m 2 - 1. The amplitude damps to l / e when y = y,), or (2.67)

-1 n tan a, R m sin4

1 - ___._

AnYt, = 1. l(1 - a,*-2)

49

MAGNETO-AERODYNAMIC FLOW PAST BODIES

If we ask what magnetic Reynolds number Rm is required for yD = 1, we have Rm

(2.68)

=

- 2n2 cos 'p sin5'p(l - u,-2)

This formula can be used to estimate the Rm necessary to see forwardfacing waves in the laboratory. Although the other solution, is also superimposed on the field, it is damped out much closer to the body for a given Rm. In the case that A,< 1 the form of $2 becomes (2.69)

$,

(

a exp zl x

----,11

__ n "uw4 f 1/1- Am2 y)exp f Rm 1 - u r n ~

-___

-d m 2 2 y

for A,< 1. To damp with positive y, we need the lower signs, and then (2.70)

Therefore the slopes are backward-facing if am2 is greater than one and forward-facing for urn2less than one, in contrast with the case discussed before, and the larger Rm the steeper the slope in any case. The distance these waves extend out from the body, measured in wave-lengths, depends upon d wand not upon Rm or u,. In the incompressible limit this root represents the ordinary irrotational flow outside the inviscid boundary layer mentioned previously. Sulltmar y

In summary, the results of this brief study of compressible flow with aligned fields at arbitrary Rm might be outlined as follows: (1) Our general remarks (page 37) concerning the diffusive effects upon magnetosonic wave patterns are confirmed. This includes the appearance of inviscid boundary layers in aligned-fields flows. (2) We find that a definitive parameter in the flow of compressible fluids with finite conductivity is K = R m / M m 2 . It follows that small-Rm flows of incompressible fluids and of gases at low M, may be completely different in character. (3) For example, the large-K approximation exhibits the typically large-Rm phenomena of the inviscid boundary layer and wake, for both incompressible and compressible flow, and also the phenomenon of resistivitydamping of magnetosonic waves. All these are the effects discussed in Chapter I and anticipated for compressible flow by analogy. (4) The small-K approximation brings out some new phenomena, however. These include (a) effects of current diffusion from the boundary, rapidly

50

W. R. SEARS AND E. L. RESLER, JR

damped even a t small Rm,(b) the appearance of Mach (sonic) waves, diffused by resistivity, in supersonic flow, and (c) a tendency for Prandtl-Glauert flow patterns to be skewed upstream as Rm increases from zero in subsonic flow. For flows with arbitrary angle between U and B,, we may anticipate that these effects will have their counterparts, except, of course, that the boundary-layer phenomenon will be replaced by another family of diffuse standing waves. To understand in more detail the effects of varying Rm on these patterns in compressible conductors will require further investigation. 111. HALLEFFECTS 1. Particle Motions

Since charged particles in a magnetic field do not move in straight lines, it is an approximation to assume that the current density j is in the direction of the electric field for all conditions. To make clear the phenomenon involved, consider the motion of a single charged particle due to the presence of electric and magnetic fields. We consider a Cartesian coordinate system, choose the z-axis along the direction of the magnetic-induction vector B, and then rotate the coordinate system about this axis to make E , = 0. Let the velocity of the charged particle be V. Thus

(3.2)

dV m -itt

= e(E

+ V x B)

Taking the scalar components of this vector equation one finds that the motion of the particle in the z or magnetic-field direction is independent of the presence of B. One also finds

t:)

u p = k,cos - t

(3.3) f’p = - k ,

sin t:t)

t:)

+&sin - t

,

+ k, cos t:t) + C

where k,, k,, and C are constants. The quantity eB/m appearing in equation (3.3) is called the cyclotron frequency and is denoted by w . A particle with velocity V circling in a magnetic field has a centrifugal force balanced by the Lorentz force:

MAGNETO-AERODYNAMIC FLOW PAST BODIES

(3.4)

m Va -=

R

eVB

V

51

eB

--- - = cyclotron frequency = w R m

or

where R is the radius of the particle’s path. If we assume that at t = 0 the particle started from rest,

eE, up = -sin mw

ot,

(3.5)

eE, mw

vp = ---cosot

- _eE, _. mw

Note that eE,/mw = E,/B = IE x B ( / B 2 ,so that the particle drifts in the E x B direction while circling about the magnetic lines a t its cyclotron frequency. Suppose we now consider a collection of charged particles for which the mean time between collisions is t. Then F(t), the probability that a particle exist for time t without making a collision is given by

F(t) = e-:I7.

(3.6)

Define now the average velocity of the particle, as follows:

a = total distance traveled by all particles in the x direction before collision (number of particles)

t

(3.7)

or if n, is the number density of charged particles, m

m

up(t)neF(t)dt = t up(t)e- dt 0

0

and similarly for 5. Using the expressions given in Eq. (3.5) for u&) and vp(t), we find

Since the current is, by definition, the transported charge, and if we wish t o preserve Ohm’s Law (j = aE), in this particular case

52

W. R. SEARS AND E. L. RESLER, J R .

j x = axxEx= npzi, (3.10)

j y = ayxEx= np6,

In Eqs. (3.11) u is the scalar conductivity used previously. Thus, to retain the form of Ohm’s Law we must give conductivity the properties of a tensor. 2. Ohm’s Law for Ionized Gases

Usually a plasma consists of ions, electrons, and neutral particles. The ions can be assumed to have the same mass as the neutrals. There are, in a sense, three fluids: the electron gas, the ion gas, and the neutral gas, whose motions and interactions must be taken into consideration. The simplest treatment for a gas such as envisioned here, consistent with our previous assumptions, is given by Cowling [ 6 5 ] . Essentially, one must now distinguish between three different collision times,

(3.12)

t = mean

time between collisions of electrons with ions

ti = mean

time between collisions of ions with neutrals

t,= mean

time between collisions of electrons with neutrals.

It is then assumed that the electrons experience a drag force, proportional to their relative velocities, from both the ions and the neutrals. There are similar drag forces acting on the ions and neutrals. If it is assumed that the electrons are light enough to be very quickly accelerated to a terminal velocity, so that the driving and drag forces on them are in equilibrium, and further that the velocity of the ions relative to the velocity of the center of gravity of the whole gas is small, so that the ion acceleration term is almost the same as the acceleration of the center of gravity, one finds an Ohm’s Law for a partially ionized gas as follows:

(3.13)

MAGNETO-AERODYNAMIC FLOW PAST BODIES

53

Here* K

= l/W,t,

Kc

= IlWeTe,

Ki

we = eB/m,,

m, = mass of electron,

w, = eB/mi,

m; = mass of ion,

+ n),

n = neutral density, n, = electron density,

+

9, = electron pressure.

f = n/(n,

p

= K,/(K;

= 1f2WiTi

KJ,

--

Note that in all cases K , / K ~ = Vme/m,;this is a very small number and will be neglected in what follows. It is not often that the equation above, Eq. (3.13), must be used in its entirety. The following forms are often used: Slightly Ionized Gas with Ion Slip: Here the electron pressure gradient Vp8 may be neglected; also n,/n (< 1.

(3.14) where a again denotes the electrical conductivity of the gas; for this case, where there are three kinds of particles, it is given by n,e2t,,,/m,where 5, is the mean collision time for electrons in collisions with neutrals and ions, viz., (3.14a)

T.=(;+

);

-1

The last term in Eq. (3.14) represents the phenomenon called ion slip, which arises from the fact that the ions and neutrals are not perfectly coupled. Thus, if the mean time between collisions of these species is appreciable, there is an imperfect transfer of electromagnetic effects from the charged particles t o the neutrals that make up the bulk of the gas. Slightly Ionized Gas without Ion Slip: This is the case in which the number of charged particles is so small compared to the number of neutrals that collisions between electrons and ions are very rare (t oo),whereas collisions with neutrals are frequent. The ion-slip term is then negligible compared to the j x B term in Eq. (3.14). The result is

-.

(3.15)

OeT,

j = a ( E + q x B ) - - j xBB ,

* We retain the factor 1/2 in the definition of K; to conform with [65]. Some authors do not retain it, and care must be used in comparing results.

54

W. R. SEARS AND E. L. RESLER, JR.

where now u is equal to nee2tc/mc.A study [66] of numerical values discloses, however, that Eq. (3.16) is appropriate only for very slightly ionized gases ( e g , less than O.Olo/, ionized, for argon). In many technically interesting cases Eq. (3.14) is therefore needed. Fully IonizedGas: Here there are no neutrals (n = 0) and therefore

tc=ti=00.

(3.16)

where u denotes n,e%/m,. This is the same result as Eqs. (3.10) and (3.11), provided that the electric field due to electron pressure gradient is included in the definition of E. Note the similarity between Ohm’s Laws for fully ionized and partially ionized gases : they differ by the presence of the electron-pressure gradient term and the definition of u. Although the term “tensor conductivity” is used quite freely, these equations (3.14)-(3.16) are vector equations and should be used as such. If this is not done the “tensor conductivity” changes with the boundary conditions. The tendency for the electric current to flow across the direction of the electric field, under the influence of the magnetic field, is called the Hall Effect. The electrical conductivity of an ionized gas is a measure of one of the transport properties. Other transport properties are similarly affected by the magnetic field. The flux of the transported property is reduced if it is carried by charged particles in the presence of a magnetic field. As in charge transport, the transport in a direction across a magnetic field is reduced by a factor (1 ~ , ~ int a~fully ) -ionized ~ gas, and an induced ~ perpendicular to both the flux proportional to o,t(l4-w , T ~ ) -appears, gradient of the property driving the flux and the magnetic field. For example, if R is the usual heat conductivity of electrons in a fully ionized gas with no magnetic field applied, the heat flux in the presence of a magnetic field is given by [67]

+

(3.17) 3. Aligned-Fields Flow of a Fully Ionized Gas or a Slightly Ionized Gas Without

Ion Slip In Chapters I and I1 we treated only flows of fluids having scalar electrical conductivity u. We shall now develop the corresponding theory for ionized gases, in particular for fully ionized gases and for slightly ionized gases without ion slip. It has already been noted that their Ohm’s Laws, Eqs. (3.16) and (3.16) are nearly the same. Under conditions such that the tensor

MAGNETO-AERODYNAMIC FLOW' PAST BODIES

55

character of the conductivity disappears, i.e., when the Hall Effect terms in Eqs. (3.15) and (3.16) are negligible, the results will reduce to those previously obtained. Conditions under which the Hall Effect dominates the flow will also be defined. Once again we shall consider small-perturbation flow with uniform aligned fields I! and B, a t large distances from a slender body. As before (equation (2.14)) the proportionality between the vorticity and the current density in small-perturbation flow still holds true as well as the linearized Bernoulli's equation. These were both derived from the momentum equation. Taking the curl of the appropriate Ohm's Law and using the isentropic energy relations, which are still appropriate, one obtains (3.18)

av aui aB a B , - - uB,M,2-- all - = curl j ax

ax

ax

aj + W,T ax

where we have written T as in Eq. (3.16), but t, as in Eq. (3.15) is also appropriate; the results of this section apply to both cases. Taking the curl of this equation and using the relation between j and 51 to eliminate j, we have

(3.18)

At this point we shall assume that the flow is two-dimensional, i.e., that a/& = 0. This does not mean, of course, that the perturbation velocity component in the z direction, w',is equal to zero, since the Hall term in the Ohm's Law destroys the symmetry with respect to the x,y-plane. For example, where previously the vorticity had only one component, this is not now the case, and the equation for 51 must retain its vector form. From the x- and s-components of this vector equation one finds (3.20)

V 2 - 4np0U(1 - U,

where ZL" is the perturbation velocity component in the z-direction. Let 5 be the z component of the vorticity: using the z component of equation (3.19) to eliminate w' we obtain

56

W. R. SEARS A N D E . L. RESLER, JR.

Thus, we have a scalar equation for the velocity components u‘ and v ’ . Defining as before a stream function (CI we have finally the equation governing linearized aligned-fields flow for fully or slightly ionized gases with arbitrary a and w e t [cf. 6Sj :

where L is a typical length on which the magnetic Reynolds number is based. If w e t = 0 we have in Eq. (3.22) the case discussed in the last section, namely aligned-fields flow with arbitrary scalar conductivity (cf. Eq. (2.43)). In the previous section we discussed, inter alia, the case of R m = 00. Let us now reconsider this limit. The magnetic Reynolds Number is 4n,uaUI., where a is given by ne2t/m. In the present context t is either the mean time between collisions of electrons with ions (for the fully ionized gas) or the mean time between collisions of electrons with neutrals (in the slightly ionized gas without ion slip). The term on the right-hand side of Eq. (3.22) disappears as R m --* 00 if wet remains finite; thus the results presented for perfect conductors in the previous chapters correspond to making Hm large by making the electron density n large. 4. Collisionless. Plasma

One often hears the term “collisionless plasma” used to denote highly conducting plasmas. Rut a collisionless plasma is one in which a is large because the time between collisions t is large; hence it is not the same as the ideal conductors considered in Chapters I and 11. Suppose we now consider this case, t -+ 00. From Eq. (3.22) we have, noting that Km is proportional to t, a = ne2t/m., and p = nm,,

(3.23) where again w , denotes the ion cyclotron frequency eB,/m,.

MAGNETO-AERODYNAMIC FLOW PAST BODIES

57

Eq. (323) is written here in three different forms to emphasize the importance of the different parameters. To make the Prandtl-Glauert operator based on Amimportant, as in Eqs. (2.23) and (2.45),requires the right-hand side to vanish. If we stay away from the singularities am2= 1 or M m 2 + amp2= 1, the right-hand side can be made small for small enough U2/wi2. The quantity U / w i is the distance the flow moves in the time it takes an ion to circle a magnetic-field line. If this distance is small compared to L , a typical length in the flow field, the infinite-conductivity results cited before are appropriate. This is the physical meaning of the magnetic lines “frozen” to the fluid: the ions are held close to the field lines. The combination U2am2/w,2 = 1,2 is also mentioned frequently in the literature. Writing out this expression in terms of basis fluid properties, we have (3.24)

The quantity li so defined has been called the “characteristic ion Larmor radius”; it is seen to be actually a material property. The quantity up is the plasma frequency. The quantity 1, depends on fundamental constants as well as mi and n ; substituting for these constants, (3.25)

Zi2=

Z -5.15 x

n

IOl4

(cm2)

Z in this equation is the atomic weight of the ion. Thus this distance is small for high-density plasmas.

Properties of Sinusoidal Solutions To obtain some insight into the phenomena governed by Eq. (3.23) we can again resort to the technique of separation of variables. Thus, again assuming t,h oc exp { i l x - 6 y } , r = all, and with (3.26)

we have for (3.27)

7

kl2r4

- [kA2(%- M a 2 ) + l ] r 2+ k12Pm2+ 1 -Am2 =0

or, since this is quadratic in

y2,

W. R. SEARS A N D E. L. RESLER, J R

For high-density plasmas, which are the plasmas usually discussed in this paper, the quantity k is small. Of course, Eq. (3.28) could be used for a complete discussion of the properties of the equation, as before, but space does not permit it here. Instead we shall consider only small k, for which one finds

or, using the definitions of k and l i ,

(3.30) I .

l h e first root represents a phenomenon that is damped in a distance comparable with I,. Near its surface a body interferes with the natural orbiting of the ions about the magnetic lines, and this effect extends out into the flow a distance proportional to I,, the constant of proportionality depending on the detailed conditions. The second root can be interpreted in terms of waves. For 1, = 0 we have the waves given by the Friedrichs diagram, u = m, W,T = 0. For d , > 1 and small I, we see that the effect of I, is to tilt the waves back in the flow direction. Thus forward-facing waves are not inclined quite so far forward, and in the region A, > 1, am< 1, where the waves are inclined downstream but always forward of the ordinary Mach waves, the tendency is for them to move back toward the Mach-wave position as I , increases. We see that the effects of the Hall phenomenon in a collisionless highdensity plasma are twofold: first, a highly damped disturbance in a layer of thickness comparable to I, near the surface of the body, and second, a change of the inclination of standing magnetosonic waves. The former is the effect pointed out by Sonnerup [69 . 5. Generalized Wave-Speed Diagram

We return now to the flow of ionized gases with arbitrary conductivity and Hall Effect. The equations are very complicated, as we have seen. I t would be possible to predict phenomena qualitatively if one knew the effects of resistivity and Hall Effect on the Friedrichs Diagram, i.e., the diagram of wave-propagation speed versus wave orientation, and therefore on the shape of disturbances set up by a moving object. From a generalized acoustic analysis one can again draw phase-velocity plots analogous to Fig. 2.5 and directly make observations about propagating

MAGNETO-AERODYNAMIC FLOW PAST BODIES

59

and standing infinitesimal waves in ionized gases. One could, in principle, proceed directly from the steady flows discussed above to the phase-velocity plots, but this process becomes cumbersome when there is damping, as the steady-state considerations give damping in the direction of the y coordinate while the acoustic diagram is concerned with damping from the origin of the source of disturbance; the relations between the two are not convenient. We have seen that when there is damping present the wave angles are also affected; thus in an acoustic treatment we might espect a frequencydependent phase velocity. Consider again the Ohm's Law for a fully ionized gas, or a slightly ionized gas without ion slip, Eqs. (3.15) and (3.16). Again take the curl of this equation and neglect second-order terms, but now write curl E = -- aB/at, since we are treating a time-dependent problem. The result is

In deriving this equation we ha\-e assumed that initially the plasma was at rest and current-free. Kow take the curl of equation (3.31), and find

(3.32) Again we can eliminate div v by using the mass-conservation equation and the isentropic relation between p and p to eliminate the density: (3.33) laking the curl of the linearized equation of momentum, including the time-dependent term, we have (3.34) Substituting these results into Eq. (3.32), after differentiation with respect to time, one has

(3.36)

60

W. R. SEARS AND E . L. RESLER, J R .

To eliminate 9, take the divergence of the linearized equation of momentum and find (3.36) Finally, eliminating fi between (3.35) and (3.36),we have as an equation for j,

To make use of this equation, assume that the disturbance is a plane wave moving along the y axis, the magnetic-field vector H, making an

X

FIG. 3.1. Geometry of wave-propagation problem.

angle with respect to this axis and therefore with the plane-wave normal n,,, (Fig. 3.1). Suppose that j is proportional to exp {ill - z f y / c } ; since the x - and z-derivatives are zero, we find

where the expressions in parentheses are vectors and (3.38)

AY2= pHY2/4nprn,

A x 2 = pHX2/4np,,

miy

= eHyp/m,.

If the scalar components of this equation are written out, the y component is uncoupled : (3.39)

MAGNETO-AERODYNAMIC FLOW PAST BODIES

61

The x- and 8-components give two homogeneous relations in i xand is,whose solution requires

For any given situation, i.e., given am2, A2, pa, and oiy, the wavespeed diagram can be used to construct a disturbance-shape diagram analogous to Fig. 2.6, but, in general, such a diagram now pertains only to a given frequency f. In other words, it tells us the shape of the propagating constant-phase loci due to a steady-state, oscillatory, point disturbance of frequency f at the origin. The motion of a solid body is equivalent, through Fourier superposition, to combined oscillatory disturbances of many frequencies, and each of these has a different diagram. Obviously the result of superposition requires a good deal of calculation, but one expects qualitative results to be similar to the effects at some dominant frequency, probably of the order of magnitude UIL. Thus it can be expected that the “Mach cones” of three-dimensional large-% flow (Chapter 11) will be broadened as well as rendered somewhat indistinct by diffusion. The two narrow, cusped cones of Cumberbatch will be connected by a thin, flat sheet of diffusion lying in the plane of lJ and B,. In the first two chapters of this presentation the treatment was for frequencies / small compared with the ion cyclotron frequency wiy, so that the right-hand side of equation (3.40) can be put equal to zero; thus we obtain (unless

(3.41)

and

+

am2 Ay2

if ) + + 4npa Ax2

c2

+ am2Ay2+ ifam2 -= 0.

~

In terms of the angle p, these can be rewritten: (3.43)

and

(3.44)

where A 2 = Ax2

+ Ay2.

iy = 0)

4npua

62

W. R . SEARS AND E. L. RESLER, J R .

Of course, for uam2 -+ bo we get only the van de Hulst roots mentioned before. For other limiting cases, e.g. A = 0,one gets ordinary sound waves 00 or incompressible and current diffusion (skin-depth phenomena) ; for a, flow, A l f v h waves and again current diffusion. Notice that again it is the quantity ua,2 - proportional to Rm/Ma2 of Chapter I1 - that appears, and care must be taken in going to various limits. In any but the limiting cases named, c as given by Eqs. (3.43) and (3.44) is complex. The phase velocity to be plotted on a Friedrichs-type diagram is then the reciprocal of the real part of c-l. If the phase velocity is plotted as a function of the angle pl for different values of fluam2,one finds that, as the quantity f/uam2goes to infinity, the outer curve, of om1 shape, always

-

FIG.3.2. Wave-speed diagrams (fast, slow, and intermediate phase speeds vs. wave inclination) for plasma without Hall Effect for steady oscillation at frequency /. The numbers on the curves are values of //4npa&. When this parameter is infinite the slow- and intermediate-wave diagrams are circles of infinite radius. urn = 1/2A .,,,

approaches the sonic circle, while the inner curves, having the shape of the cu sign, approach a circle of infinite radius. The circle of infinite radius occurs because we have not retained relativistic terms in our equations ; if these terms were retained, the figures M would approach a circle with radius of the speed of light. In Fig. 3.2 are plotted the phase-velocity curves for a gas without Hall Effect but with several values of the resistivity parameter / / 4 y ~ u a , ~ . These will give the reader some idea of how the Friedrichs Diagram is affected by resistivity. (See also [70].) As a second example, let us consider a so-called cold coIlisionless plasma. This is the case represented by uoo= 0 and u = OQ in Eqs. (3.39) and (3.40). In this case the equations become (3.45)

c 2 - A Y 2 -- 0

MAGNETO-AERODYNAMIC FLOW PAST BODIES

63

and

or in terms of the angle p, (3.47)

c2

- A2 cosz f$ = 0

and (3.48)

c4 - c2A2(1

+ C O S ~p) + A 4

C O S ~p

f 2 A 4 C O S ~p = 0. -w'lY

In this case, even though u = do, the fast and slow phase velocities are frequency-dependent. Comparison with other references can be made by noting that (3.49)

FIG. 3.3. Wave-speed diagrams (fast, slow, and intermediate phase speeds vs. wave inclination) for cold collisionless plasma, for steady oscillation at frequency /. The numbers on the curves are values of ( f / ~ i ~ )The * . intermediate-wave diagram is the same for all values of this parameter.

in which case equation (3.48)becomes, using c = f L ,

(3.50)

64

W. R. SEARS AND E. L. RESLER, JR.

This equation can be compared with the results given in Reference 1711. In Fig. 3.3 are plotted phase-velocity curves for the cold collisionless plasma for several values of the frequency parameter ( f / w i J 2 . 6. Summary

In this chapter we have shown that the Hall Effect may be important even in gases with very large electrical conductivity. If the high value of u occurs because the plasma is rarefied, i.e., the collision time t is large, the result is that the Hall phenomena remain appreciable. They are manifested, for example, in alterations of the flow pattern near the solid-fluid interface and deflection of magneto-sonic waves towards ordinary Mach angles, as well as the appearance of cross-flow. On the other hand, if large conductivity is achieved by making the electron density large in a gas of high density, the Hall term becomes negligible, and the model used in the earlier chapters is appropriate. The case of an ionized gas with arbitrary conductivity and Hall Effect is relatively intractable. Some qualitative information can be obtained by considering the propagation of sinusoidal plane waves in such a fluid, leading to a generalization of the wai.e-propagation and disturbance-shape figures. The relationship of these to the disturbances set up by the steady motion of bodies is not so close as before, however, since the pertinent effects of resistivity and Hall phenomenon are frequency-dependent.

References 1. LAMB,H., “Hydrodynamics“. Cambridge, 1932. 32R. Eq. ( 8 ) . 2. ELSASSER, W. hI.. Hydromagnetic dynamo theory, Rev. Mod. P k y s . 28, 135-163 (1956). 3. A L F V ~ N H., , On the existence of electromagnetic-hydrodynamic waves, A rkiv /or Mafenzatik Asfronomi och F y s i k , 2BH, No. 2 (1943) ; see also Nalure 160, 405-406 (1942). 4. SEARS,W. R . , and RESLER,E. L., J R . , Theory of thin airfoils in fluids of high electrical conductivity, J . Fluid Mecli. 6 , 257-273 (1959). 5. LUDFORD, G. S. S., Note on airfoil theory in hydromagnetics, J . Aero. Sci. 2H. 511-512 (1961). 8. LUDFORD, G. S. S., Further note on airfoil theory in hydromagnetics. J . .4ero. Sci. 28, 741-742 (1961). 7. SUNG.K. S., “Linearized Magneto-gas Dynamics for Arbitrary A l f v h Number, Mach Number, and Magnetic Reynolds Number.” Ph. D. Thesis, Harvard University, Cambridge, Mass. (1903). 8. CUMBERBATCH, E., SARASON, L., and WEITZMER, H., Magnehohydrodynamic flow past a thin airfoil, A . Z . A . A . Journal 1, 079-690 (1963). 9. STEWARTSOS, K., Magneto-fluid dynamics of thin bodies in oblique fields I, Zetfs. 1. angew. M a t h . und Phys. 12, 261-271 (1961).

MAGNETO-AERODYNAMIC FLOW PAST BODIES

65

11). HASIMOTO. H., Magnetohydrodynamic wakes in a viscous conducting fluid, Rev. Mod. Phys. 82, 860-866 (1960) : also “Magneto-Fluid Dynamics” (F. N. Frenkiel

and W. R . Sears, eds.), Natl. Acad. of Sci.-Natl. Kes. Council, Publ. No. 829, 1960. 11. CLAUSER,F. H . , Concept of field modes and the behavior of the magnetohydrodynamic field, The Physics of Fluids, 6 , 231-253 (1963). 12. RESLER,E. L., J R . and MCCUNE. J. E., Electromagnetic interaction with aerodynamic flows, i n “The Magnetodynamics of Conducting Fluids” (D. Rershader, ed.), pp. 120-135, Stanford, 1959. 13. RESLER,E. L., J R . and MCCUNB,J . E., Some exact solutions in linearized magnetoaerodynamics for arbitrary magnetic Reynolds numbers, Rev. Mod. P h y s . 82, 848-854 (1960); also “Magneto-Fluid Dynamics” (F. N. Frenkiel and W. R . Sears, eds.), Natl. Acad. of Sci.-Natl. Res. Council, Publ. No. 829, 1960. 14. MCCWNE,J . E., On the motion of thin airfoils in fluids of finite electrical concluctivity, J . Fluid Mech. 7 , 449-468 (1960). 1.5. LUDFORD, G. S. S., Inviscid flow past a body at low magnetic Reynolds number, Re?,. Mod. P h y s . 82, 1000-1003 (1960); also “Magneto-Fluid Dynamics” (F. N . Frenkiel and W. R. Sears, eds.), Natl. Acad. of Sci.-Katl. Res. Council, Publ. No. 829, 1960. 16. LUDFORD, G. S. S. and M U R R A Y ,J . D.. On the flow of a conducting fluid past a magnetized sphere, J . Fluid Mech. 7 , 516-528 (1960). 17. CHESTER, W., The effect of a magnetic field on Stokes flow in a conducting fluid, J . Fluid Mech. 3, 304-308 (1957). 18. CHESTER,W., On Oseen’s approximation, J . Fluid Mech. 13, 557-569 (1962). 19. LUDFORD, G. S. S., The effect of an aligned magnetic field on Oseen flow of a conducting fluid, Arch. Rational Mech. 4, 405-41 1 (1960). 20. T A M A D AKO., , Flow of a slightly conducting fluid past a circular cylinder with strong, aligned magnetic field, Phys. of Fluids, 6 , 817-823 (1962). 21. Rossow, V. J . , On the flow of electrically condocting fluids over a flat plate in the presence of a transverse magnetic field, N A C A T N No. 3971 (1957). 22. Rossow. V. J., On magneto-aerodynamic boundary layers, Zeits. /. angew. Math. und P h y s . Bb, 519-527 (1958). K., Magneto-fluid dynamics of thin bodies in oblique fields 11. 23. STEWARTSDN, Zeals. f . angew. Math. und P h y s . 18, 242;255 (1962). 24. HASIMOTO, H., Viscous flow of a perfectly conducting fluid with a frozen magnetic field, P h y s . of Fluids 2, 337-338 (1959). 25. SEARS,W. R., On a boundary-layer phenomenon in magneto-fluid dynamics, Astronautica Acta 7 , 223-236 (1961). 26. LYKOUDIS, P. S., On a class of compressible laminar boundary layers with pressure gradient for an electrically conducting fluid in the presence of a magnetic field, Proc. I X t h Internatl. .4stronautical C o n t r . , Amsterdam (l958), 168-180, Springer, Vienna (1959). 27. SEARS,M:. R. and MORI, Y,,Studies of the inviscid boundary layer of magnetohydrodynamics, i n “Progress in Applied Mechanics (The William Prager Anniversary Volume).” (D. C. Drucker, ed.), Macmillan, New York, 1963. 28. LEWELLEN,W. S., “An Inviscid Boundary Layer of Magnetohydrodynamics”, M. Aero. E. Thesis, Cornell University, Ithaca, New York (1959): also A F O S R TN-59-927. H. P. and CARRIER, G. F., The. magnetohydrodynamic flow past a 29. GREENSPAN, flat plate, J . Fluid Mech. 6 , 77-96 (1959). 30. GREENSPAN, H. P., Flat plate drag in magnetohydrodynamic flow, Phys. of Fluids 8, 581-588 (1960). 31. GOURDINE, M. C., “On Magnetohydrodynamic Flow over Solids”, Ph. D. Thesis, v Calif. lnst. of Tech., Pasadena, Calif. (1960). See also On the role of ? J ~ S C O S l t and

66

32.

33. 34.

38. 36.

37.

38.

39. 40.

41.

42.

43. 44. 45.

46.

47.

48. 49,

W . R . SEARS AND E. L. RESLER, JR. conductivity tn magnelohydrodynamtcs, Tech. Rept. S o . 32-3, Jet Propulsion Lab., Calif. Inst. Tech., Pasadena. Calif., 1960. LARY,E. C., “A Theory of Thin Airfoils and Slender Bodies in Fluids of .Arbitrary Electrical Conductivity”, Ph. D. Thesis, Cornell University. Ithaca, N e w York (1960), Available from University Microfilms, Ann Arbor, Mich. See also A theory of thin airfoils and slender bodies in fluids of finite electrical conductivity with aligned fields, J . Fluid Mech. 12, 209-226 (1962). LUDFORD. G. S. S., The effect of a very strong magnetic cross-field on steady motion through a slightly conducting fluid, J . Fluid Mech. 10, 141-155 (1961). LCDFORD,G. S. S., The effect of a very strong magnetic cross-field on steady motion through a slightly conducting fluid: three-dimensional case, Arch. for Hatianal M ~ c h .and Anal. 8, 242-253 (1961). STEWARTSON. K.. Motion of a sphere through a conducting fluid in the presence of a strong magnetic field, Proc. Camb. Phil. Soc. 52, 301-316 (1956). LUDFORD, G. S. S. and SINCH. M. P., The motion of a non-conducting sphere through a conducting fluid in a magnetic cross-field, Proc. Camb. Phil. SOC. 69, 6 1 5 6 2 4 (1963). LUDFORD, G. S. S. and SISGH.M. P., On the motion of a sphere through a conducting fluid in the presence of a magnetic field; Proc. Camb. Phil. Soc. 69, 625-635 (1963). STEWARTSON, K . , Motion of bodies through conducting fluids, Rev. Mod. P h y s . 80, 855-859 (1960) ; also ‘Magneto-Fluid Dynamics” (F. N. Frenkiel and W. K. Sears, eds.), S a t l . .\cad. of Sci.-Xatl. Res. Council, Publ. No. 829, 1960. STEWARTSON, K., On the motion of a non-conducting body through a perfectly conducting fluid. J . Fluid bfech. 8 , 82-96 (1960). TAMADA. KO, “On the Flow of Inviscid Conducting Fluid past a Circular Cylinder with .4pplied Magnetic Field”, Grad. School of Aero. Eng.. Cornell University, lthaca, S e w York, XFOSR No. 1087, 1961. SEARS,W. R., Small perturbation theory, i n “General Theory of High-speed .Aerodynamics”, pp, 61-122 (W. R. Sears, ed.), Princeton 1954. Also “SmallPerturbation Theory”, Princeton Xero. Paperbacks. Princeton, 1960. SEARS,W. K., Magnetohydrodynamic effects in aerodynamic flows, A . R . S . 11. 29, 397-406 (1959). T A X I U T IT., , An example of isentropic steady flow in magnetohydrodynamics, Progr. l’heor. I’kys. 19, 749--750 (1958). SEUBASS,A. R . , On “transcritical” and “hypercritical” flows in magnetogasdynamics, Quarf. A p p l . Math. 19, 231-237 (1961). MCCUNE, J . E., and RESLER, E. L., JR., Compressibility effects in magnetoaerodynamic flows past thin bodies, J . Aero. Scz. 27, 493-503 (1960). See also correction in Sears. W. R., Sub-Alfvhic flow in magnetoaerodynamics, J . ‘4ero. S c i . 28, 249-250 (1961). \‘.as DE HULST,H. C., Interstellar polarization and magnetohydrodynamic waves, i n I’voc. Symposium hfotioii of Gaseous Masses, Paris, Central Air Documents Office S o . 1103347, 1949. FRIEDHICHS, K. 0. and K R A N Z E RH., , “Notes on Magneto-Hydrodynamics V l I I . Son-linear Wave Motion”, Inst. of Math. Sci., S e w York University. NYO-6486, 1958. HERLOFSEN, S . , Magneto-hydrodynamic waves in a compressible fluid conductor. Nature 165, 1020-1021 (1950). SEARS,W. l t , , Some remarks about flow past bodies, Rev. M o d . P h y s . 82, 701-705 (1960); also “Magneto-Fluid Dynamics” (F. S . Frenkiel and W. I<. Sears, eds.). S a t l . .\cad. of Sci.-Satl. Kes. Council, Publ. S o . 829, 1980.

MAGNETO-AERODYNAMIC FLOW PAST BODIES

67

50. WEITZNER.H., “On the Green’s Function for Two-Dimensional Magnetoliytlrodynamic Waves. I I ” , Inst. of Math. Sci.. New York University, NYO-948!). I!J60.

51. CHU, C. K . and L Y N N ,Y. M . , Steady magnetohydrodynamic flow past a nonconducting wedge A . I . A . A . Journal 1, 1062-1067 (1963). .id. CUMBERBATCH. E., Magnetohydrodynamic Mach cones, .I .4ero. . Sci. 89, 1476 1471) (1 962). .53. HELFER.H . L., Magnetohytlrodynamic sliock waves, Asfrophysical J . 117, l77-1!1!) ( 1 953). 54. LUST, H . , Magneto-hydrodynaniische StoUwellen in einem I’lasma rinentllicher Leitfahigkeit, Zeits. /. Nafurforsch. 8 A . 377-284 ( 1953). 5 5 . BAZER,J. and ERICSON,W. B.. Hydromagnetic shocks. .4strophysacal ,/. 1811. 758-785 (1959). 56. ERICSON,W. B. and BAZER,J., On certain properties of hydromagnetic shocks, Phys. of Fluids 3, 631-640 (1960). J . A., One-dimensional magnetogasdynamics in oblique fields, ,/. F l u i d 57. SHERCLIFF. Mech. 9, 481-505 (1960). 5 8 . CABANNES, H., Attached stationary shock waves in ionizetl gases, R e v . M o d . I’hys. 32, 973-976 (1960) ; also “Magneto-Fluid Dynamics” (F. N. Frenkiel ancl W. R. Sears, eds.), Natl. Acad. of Sci.-Natl. Res. Council, Publ. No. 82!), 1960. 59. GERMAIN, P., Shock waves and shock-wave structure in magneto-fluid dynamics. Rev. Mod. P h y s . 32, 951-958 (1960) ; also “blagneto-Fluid Dynamics” (F. N. Frenkiel and W. R. Sears, eds.), Natl. Acad. of Sci.-Natl. l<es. Council, Publ. No. 829, 1960. G. I . . and POLOVIN, I<. V., The stability of shock 60. AKHIEZER, A. I . , LIURARSKII, waves in magnetohydrodynamics, J . E x p t l . 7‘heoref I’hys. 36, 731-737 ( I 9 R H ) ; Translated in Soviet P h y s . J E T P . H. 507-511 (1959). 61. TANIUTI, T., A note on the evolutionary condition on hydromagnetic shocks, Progv. Theor. P h y s . 28, 756-757 (1962). 62. GEFFEN,N . , “On Aligned-fields Magnetogasdynamic Flows With Shocks”, I’h. I). Thesis, Cornell University, Ithaca, New York, l!)63; also available from University Microfilms, Ann Arbor, Michigan. 63. SEARS,W. R.. High-field effects in magnetohydrotlynamics, i n “High Magnetic Fields” (H. Kolm, B. Lax, R Bitter and I<. Mills, eds.). 652-658, MIT Press and J . Wiley, New York, 1962. 64. SEARS,W. R., Some paradoxes of sub-Xlfvenic flow of a compressible conducting fluid, i n Proc. Symposium Electyomagnetics and Ffuid Dynumics of Gaseous Plusma 363-372, Polytechnic Press, Brooklyn, 1962. T . G., “Magnetohydrodynamics”, Interscience, New York, 1957. 65. COWLING, A , , Electrical conductivity of highly 66. LIN, S. C., RESLER,E. L. and KANTROWITZ, ionized argon produced by shock waves. ,journal of A p p f i r d Physics 28, 95-10!) (1955). 67. CHAPMAN, S. and COWLING,T. G., “The Mathematical Theory of Non-Uniform Gases”, Cambridge University Press, p. 337, 1952. 68. RESLER,E. L., J R . , Aerodynamic Aspects of Magnetohydrodynamics i n Developments in Mechanics”, Vol. 1, pp. 503-509, Plenum Press, New York, 1961. Proceedings of 7th Midwestern Mechanics Conference, East Lansing, Mich ., Sept. 1961. (Lay, J. E. and Malvern. L. E., eds.). 69. SONNERUP, B., Some effects of tensor conductivity in magnetoliydrodynamics, J . Aevo. Sci. 28, 612-621, 643-644 (1961). 70. KESLER. E. L., JR., Some remarks on hydromagnetic waves for finite conductivity, Rev. M o d . P h y s . 82, 866-867 (1960); also “Magneto-Fluid Dynamics” (F. N. Frenkiel and W. R. Sears, eds.). Natl. Acad. of Sci.-Natl. Res. Council, Pnhl. No. 829, 1960.

68

W. K. SEARS AND E. L. KESLER, JR.

71. FISHMAS, F. J., KANTROWITZ, .I. H. antl PETSCHEK, H . E., Magnetohydrodynamic

shock wave in a collision-free plasma, Heo. M o d . Phvs. 32, 059-966 (1960); also "Magneto-Fluid Dynamics" (F. S . Frenkiel and U'. It. Sears, eds.), Satl. .\cad. of Sci.-Satl. lies. Council, l'uhl. So. 829, 1960. Note: The references listed above are not interitled to constitute a complete bibliography of the subject, and many important papers have undoubtedly been omitted. For more extensive bibliographies the reader is referred t o !42J antl t o the following:

72. XAPOLITANO, L. G. and COXTcRSl, C;., "Magneto-Fluid Dynamics: Current Papers and .Abstracts", AC.4 RI) Bibliography I, Enlarged Edition. I'ergarnon, Oxford, London, S e w York. antl Paris. 1962. 73. Par, S H I H - I ."Magnetogasdynamics , and Plasma I)!~namics". Springer, Vienna. 1962.

Incompressible Second-Order Fluids BY HERSHEL MARKOVITZ

AND

BERNARD D. COLEMAN

Afellon Institute. Piltsburgh 13, Pennsvlvaniu

I. Introduction . . . . . . . . . . . . . 11. Relation to General Simple Fluids . . . 111. Steady Simple Shearing Flow . . . . . 11'. Viscometric Flows . . . . . . . . . . 1. Steady Couette Flow . . . . . . . . 2. Axial Flow between Concentric Pipes 3. Approximate Results . . . . . . . . Steady torsional flow . . . . . Steady cone and plate flow . . 4. Appendix . . . . . . . . . . . . . V. Steady Extension of a Cylinder . . . . VI. Relation t o Classical Viscoelasticity . . V I I . Nonsteady Shearing Flows . . . . . . 1. Sinusoidal Oscillations . . . . . . . References . . . . . . . . . . . . . .

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I, INTRODUCTION An iiicompressible simple material is a substance whose mass density never changes and for which the stress is determined, to within an arbitrary hydrostatic pressure, by the history of the gradient of the deformation. An incomfiressible simple fluid is an incompressible simple material with the property that all local configurations are equivalent in response, with all observable differences in response being due to definite differences in history. * In recent years much has been learned about the mechanical behavior of general incompressible simple fluids. It has been found that there are several boundary-value problems which can be solved for incompressible simple fluids in general, using no special constitutive assumptions beyond the definition of fluidity. Yet there are many important hydrodynamical problems which can be discussed in great detail for incompressible Newtonian fluids, sometimes even to within explicit calculation of the velocity field, ____-

* This is, in essence, an expresson in words of Noll's [ I ] mathematical concept of fluidity. 69

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HERSHEL MARKOVITZ AND BERNARD D. COLEMAN

but about which little can he said for general simple fluids.’ Indeed, when one considers that whereas a Newtonian fluid is defined by a linear isotropic function of a tensor, a general simple fluid is defined by an isotropic functional (which need not be linear) of a tensor-valued function, it is rather surprising that any problems can he solved for general simple fluids. If one adds to the definition of a-simple fluid a smoothness postulate which makes mathematical the physical notion of “gradually fading memory,” i.e. the notion that deformations which occurred in the distant past should have less effect on the present value of the stress than deformations which occurred in the recent past, then one can prove theorems justifying the intuitive idea that for slow motions the local behavior of a simple fluid should be approximated by the behavior of a Newtonian fluid [2]. I n other words, on assuming fading memory one can prove that the theory of Kewtonian fluids gives to the theory of perfect fluids a correction for viscoelastic effects which is complete to within terms of order greater than m e in the time scale. The theorems which yield this result also enable u s to find a constitutive equation which gives corrections for viscoelastic effects that are complete to within terms of order greater than two in the time scale :* in Cartesian components that constitutive equation reads as follows:t

Here SIj is the stress-tensor ; b,, is the Kronecker delta; p is an indeterminate hydrostatic pressure; ?lo, P, and y are constants. The tensors A(,,,),, are called Rivlin-Ericksen tensors ‘31 ; they are related to the velocity gradient by the following recursion formula tensor zl,.,

+

(1.2a)

-4 (1)1, = U f , ,

(l.2b)

A(NJr7 = A ( N - 1 J r k U k . j

t’j.1

+ A(N

-1)lkvk.i

+

A(N-l)l7;

the superimposed dot denotes the material time derivative. A fluid obeying the constitutibe equation (1.1) is called an (incompressible) seemid-order flriid. More hydrodynamical problems can be solved for second-order fluids than for simple fluids in general. In several situations where the theory of Newtonian fluids gives rise to linear partial differential equations for the \docity field so also does the theory of second-order fluids.** + Since this article is concerned with incompressible materials only, i t is understood that whenever we iise the word f l u i d we mean incompressible fluid. This assertion is made more precise in Section 11. + We m e the summation convention for indices repeated in a given component or 111 products of components. ** Cf. (7.11)-(7.14) of L3,.

INCOMPRESSIBLE SECOND-ORDER FLUIDS

71

Here we discuss the theory of second-order fluids, placing emphasis on its relation to the general theory of simple fluids and the special theories of linear and second-order viscoelasticity. In Section I1 we make more precise the concept of an incompressible simple fluid with fading memory and outline in more detail the mathematical ideas leading to the constitutive equation (1.1). If one ignores the density, then the (incompressible) Newtonian fluid has one material constant: the viscosity l;lo. In addition to yo, the corresponding second-order fluid has two new material constants, p and y . In Section I11 we give these constants physical significance through a detailed consideration of simple shearing flow. Experimenters know how to measure yo; in Sections IV and V, we derive several formulae which describe methods for measuring p and y . In Section \‘I we discuss the relations between finite second-order viscoelasticity and the theory of second-order fluids. There we show that the material constant y is determined by a knowledge of the shear relaxation modulus G = G ( t )familiar from linear infinitesimal viscoelasticity (cf. (6.11b), or (6.22) relating y to the storage modulus). This result is of interest because y governs certain measurable normal stress effects, such as the difference in radial thrusts on the inner and outer cylinders in a slow Couette flow. By relating y to G, which is positive for thermodynamic reasons, we are able to predict the sign of the normal stress differences determined by y . In Section VII we discuss a flow problem for which the velocity field can be calculated for a second-order fluid but not for a general simple fluid. The constitutive equation of a second-order fluid arises often [4] in the course of perturbation procedures for solving boundary value problems for general Rivlin-Ericksen fluids [3]. That application of the second-order fluid is not treated here. Concepts of functional analysis are used in Sections I1 and VI only, and there only qualitatively, The reader who is interested only in the hydrodynamical behavior of second-order fluids, and not in the relation of (1.1) to the various extant theories of viscoelastic behavior, can skip Sections I1 and VI.

11. RELATION TO GENERAL SIMPLEFLUIDS Consider a material point P in a deformable body and let x i , j = 1, 2, 3, be the j’th Cartesian coordinate of P at time t. Let ~ ( ~ ) t~ ( xs )~ give , the i’th Cartesian coordinate of P at time t - s, 0 s < 00; ~ ( ~ ) i) (is called the relative deformation function for P . The tensor FM)ij(t- s ) , defined by

<

72

H E R S H E L MARKOVITZ A N D B E R N A R D D. COLEMAN

is the spatial gradient of the deformation a t time I - s rclative t o thc configuration a t time t ; for short Z;(r),,(t - s) is called the relatiw &/ormation gradient. A necessary and sufficient condition that the mass density a t P be the same at times t and t - s is that (2.2)

det

(F(t)ij(t- S ) ) =

1.

For an iiicompressible simple material (2.2) holds identically in t and s ; and the stress .S,,(t) is related to F(t)rithrough an equation of the form m

(2.3) where p ( t ) is the indeterminate hydrostatic pressure and Gjj is a symmetrictensor-valued functional of the function F(t)al(t- .). In (2.3) Ck&) is some tensor-valued measure of strain from the configuration a t time t to a special time-independent reference configuration. The functional Gii is said to characterize the material a t P. If Gii is independent of the strain parameter Cii(t), then we say that the material described by (2.3) is a simple fluid. Hence, for a simple fluid

where G,, is a functional which depends on only the point P under consideration. In words, (2.4) states that if the deformation gradient, relative to the present configuration as reference, is known for all past configurations, then the extra-stress, S,j psi,, is determined; the operator G,,, which so determines the stress involves no “preferred” configurations and is the same operator whatever the present configuration may be.

+

The principle of material objectivity is the assertion that a constitutive relation, such as (2.4), should be independent of the frame of the observer. Using this principle one can show that (2.4) can be written in the form t2.6) where G(t)kl( ) is a symmetric-tensor-valued function defined by

and aij is an isotropic functional; i.e., for every orthogonal tensor Qii, g,j obeys the identity

INCOMPRESSIBLE SECOND-ORDER FLUIDS

73

for all functions Gkl( ) in its domain.* We call the function G(t)kI( ), defined in (2.6), the history of P up to time t. When no confusion can arise we drop the subscript t on G,l)kl. The domain 9 of B,i is usually taken to be an (infinite dimensional) vector space of symmetric-tensor-valued functions ; it follows from (2.2) and (2.6), however, that the physically meaningful part of 9 can be only a set of functions GkI( ) obeying the condition (6.8)

det [ (GkI(s))

+ &I]

= 1,

0

<s <

do.

To render mathematical the concept of fading memory, Coleman and No11 [2, 51 introduce into 9 a norm.+ To define their norm we need the concept of an iiifluence function h = h(s). This is a positive, continuous function of s which goes to zero rapidly as s ca. For the validity of the results to be quoted here it is not necessary to specify the form of h(s). It suffices to assume that, for large s, h(s) approaches zero monotonically and faster than s - ~ . Let us suppose that some such influence function has been selected; we then define the norm l/Gkl( ) / I of Gkl( ) b).

-

We assume that .9 is the set of all histories Gkl( ) with finite norm llGkl( ) / I ; then we can say that 9,together with the norm (2.9),is a Hilbert space, H. We regard the norm of a history in H as a measure of the “total intensity” of all the past strains (relative to present configuration) at a material point P. I t will be observed that jlGkl( )I1 is defined in such a way that it places greater emphasis on the values Gkl(s)for small s (recent past) than on values for very large s (distant past).

If a neighborhood about P has never been distorted, i.e. has been in its present configuration (to within a rigid rotation) a t all times in the past, then the history Gkl( ) of P reduces to (2.10)

GhI(S)

= 0,

0

<s <

~ i ) .

No11 [ I ] showed t h a t the constitutive equation (2.5) [with (3.7)] is equivalent to his definition of a simple fluid. t The implications of gradually fading memory for compressible fluids. with emphasis on physical applications, are outlined in (61.

74

HERSHEL MARKOVITZ AND BERNARD D. COLEMAN

The function (2.10) is called the rest history. We note the norm of the rest history is zero and the norm of an arbitrary history G k l ( ) may be interpreted as the distance of Ghl( ) from the rest history. A functional $j,,is said to be a memory junctional (of order n ) if $,i has n) a t the point in X corFrCchet functional derivatives (of all orders responding to the rest history (2.10) and if these functional derivatives obey certain technical requirements of continuity. For our present purposes it suffices to put n = 3, and we say that a simple fluid has gradually f a d i q memory if $j,i in (2.5) is a memory functional of order three. Not all simple fluids have gradually fading memory. In fact there is an important class of simple fluids, called Rivlin-Ericksen fluids, which do not. To discuss these fluids we need to consider the Rivlin-Ericksen tensors which were defined in (1.2) but which can be shown to be also given by*

<

(2.11) In 1955 Rivlin and Ericksen 131 considered the theory of isotropic materials for which the stress depends on only the spatial gradients of the velocity, acceleration, second acceleration, . . . , (M - 1)th acceleration. They showed that for an incompressible material of this type, the extra-stress Si,f pS,, must be given by an isotropic function of the tensors N = 1, 2,. . . , M ; i.e. (4.12a)

sij

+ psi,

=

( l j k d (2)kl.

. . ., A ( M ) k l )

where, for every orthogonal tensor Qii,hi, obeys the identity QimQjnhij(A (1)kh A ( 2 ) k L .

(2.12b)

*

, A(M)kI) = hii(QrmQlnA

(ljmni Q k m Q d (2)mn.s

,QkmQlnA (Mjmn).

Many important constitutive equations previously discussed are special cases of (2.12). A perfect fluid corresponds to the case Sii psi, G 0. For a Newtonian fluid

+

(2.13)

Sbj

+ PSij = qoA(l)ij

where qo is a constant. For a Reiner-Rivlin fluid [S,!)I, the extra-stress depends on only A(lji, but is not necessarily linear in for that fluid (2.12) and the representation theorem for isotropic tensor functions of one symmetric-tensor variable yield (2.14)

sij f p&j = "'l(l),j

+

aA(l)tkA(l)ki

* Equation (2.11) Seems to be due to Sol1 [I]; Hivlin and Ericksen start with (1.2) as their definition. Other kinematical formulae for the tensors A(,)ij, a summary of their history, and a proof of the equivalence of (2.11) and (1.2) are given in the encyclopedia article of Truesdell and Toupin [7].

INCOMPRESSIBLE SECOND-ORDER FLUIDS

75

where N and 3 are scalar functions of the invariants A(l)ijAfllj, and A(,,ijA(I)jJ(1)Ri. Although the general Rivlin-Ericksen fluid (2.12) accounts for “sheardependent viscosity” and “normal-stress effects,” it shares with its special case, the Newtonian fluid of classical fluid dynamics, the serious shortcoming of not accounting for the phenomenon which gave rise to Boltzmann’s theory of linear viscoelasticity : that is, gradual stress relaxation. When A ( M l i= i 0 for M = 1,2,. . , N , the extra-stress, Sii pSii, on a RivlinEricksen fluid cannot change in time, but it does in actual stress-relaxation experiments on viscoelastic materials such as high polymeric fluids.* Stated differently, when a Rivlin-Ericksen fluid is brought to rest its memory does not fade gradually, but rather precipitously.

.

+

A theory based on the general equations (2.5) and (2.7) of a simple fluid is compatible with stress relaxation. In fact, it appears to us that (2.5) and (2.7) express, for materials which can be considered incompressible, a concept of fluidity sufficiently general to cover almost all purely mechanical effects in real fluids.+ The Rivlin-Ericksen fluid corresponds to that special case of (2.5) and (2.7) in which $jijis such that its value is determined by a finite number of derivatives of G&) a t s = 0. If we restrict our attention to only those histories Gki( ) which are polynomials in s, then it is true that (2.5) and (2.7) imply (2.12) for all functionals $ i j . We cannot, however, make such a restriction, because there are many important physical situations in which Gkl( ) is not a polynomial (for example, stress-relaxation experiments). Yet this observation does not give the whole story of the logical relation of Rivlin-Ericksen fluids to general simple fluids. For, in the limit of very slow motions, general simple fluids with fading memory are approximated, albeit in a rather drastic sense, by certain particular Rivlin-Ericksen fluids, even when Gki( ) is not a polynomial. We now make this remark more precise. Let Gij( ) be any history; from it we can construct for each a, 0 < a a new history G$’( ) as follows (2.15)

G$’(s) = Gjj(as),

0 \< s <

< 1,

oc,;

* It is worth noting that gradual stress relaxation seems to be particularly striking in those same substances for which shear-dependent viscosity and normal-stress effects are easily observed, and the time intervals for which stress relaxation is most pronounced are usually the same order of magnitude as the reciprocal shear rates at which the shear dependence of the viscosity is most severe. The only exceptions which occur to us involve surface tension or those very special substances called liquid crystals. Of course, it is only by restricting ourselves to incompressible materials that we are able to make such a strong statement.

76

HERSHEL MARKOVITZ .4ND BERNARD D. COLEMAN

we call G!P1( ) the relardation of GI,( ) by the factor a. In rough language, C:;)( ) corresponds to a kinematical history essentially the same as that which gives rise to G,,( ), with the exception that the new history is carried out at a slower rate. I t follows from (2.11) and (2.15) that the KivlinEricksen tensors corresponding to Gl;’( ) are related to the A(,,,, corresponding to G,,( ) by the formula (2.16) Coleman and No11 121 have shown that when s,, is a memory functional, then Siq’ corresponding to G:;)( ) obeys the approximation formulae (2.17) (2.18)

+ ps,,

(a) +

= 7?0’4(h,

p ~ ; d ~ 1)rk

+ YAI$r, + O(a3).

( 2 1

(IJkl

Equation (2.17) tells us that, for a simple fluid with fading memory, the equation (2.13) of a Newtonian fluid furnishes a complete first-order correction, for viscoelastic effects, to the theory of perfect fluids in the limit of slow motions. Equation (2.18) exhibits all second-order corrections for viscoelastic effects.* In (2.17) and (2.18), q0, ,8, and y are material constaizts, i.e. numbers depending on only the functional sj,, defining the simple fluid under consideration. The constitutive equation obtained by striking out the error term O(a3) in (2.18) is just our equation ( l . l ) ,i.e. the constitutive equation of a secondorder fluid.’ I t is, of course, a special case of (2.12) and hence does not describe (exactly) a fluid with gradually fading memory, yet it approximates such fluids in sufficiently slow flows. It is now clear that the theorem leading to (2.18) justifies our assertion that ( 1 . 1 ) “gives corrections for viscoelastic effects that are complete to within terms of order greater than two in the time scale.” Kote that the theory of Reiner-Rivlin fluids (2.14) would not yield the term in (2.18) with coefficient y ; hence that theory cannot give a complete second-order correction to the Xewtonian fluid.

*

In L%: and [lo: a procedure is outlined for obtaining a higher-order correction. To our knowledge the first mention of a constitutive equation of the form ( 1 . 1 ) is in N‘.E. Langlois’ unpublished study (ca. 1957) of the slow motion of a “slightly viscwlastic liquid” between rotating concentric spheres. We believe that Eq. (2.18) gives Eq. ( 1 . 1 ) a clear physical interpretation. Note added in proof: Langlois’ study has been published recently: Langlois, W. E., Steady flow of a viscoelastic fluid between rotating spheres, Quart. A p p l . M a f h . ‘11, 62-71

(1963).

77

INCOMPRESSIBLE SECOND-ORDER FLUIDS

111. STEADYSIMPLE SHEAKINC FLOW

Steady simple shearing flow is a flow for which there exists a Cartesian coordinate system xl,xz,x3 in which the components of the velocity field zli have the form (3.1)

v1

= 0,

v2 = v ( x , ) ,

v3 = 0.

This flow is isochoric. Now, for steady simple shearing flow the history G ( l ) k l ( ) defined in (2.6) is a quadratic function of s and is given by [lo, 111 (3.2)

- A(1)kIS

G(t)kZ(S) =

+

4A[2)kfs2,

where the matrices of the Cartesian components of A(,)kland to x1,xz,x3, have the forms

A(.L)k[,

relative

Here K = d v ( x l ) / d x l and is called the rate of shear. Using Eqs. (3.1) and (3.3), and the isotropy of the functional 4jij in Eq. (2.5), the following result can be proved 1111: For a steady simple shearing flow in any simple fluid, the components of the stress S,,, relative to the coordinate system x1,x2,x3 which gives z', the form (3.1), must have the form

(3.4)

(Sij) ==

0

s33

where (3.5a) (3.5b) (3.5c) Let us put (3.6)

q(K)

==

T(K)/K;

7,ul, and a, are all even functions of (3.7)

q(K) =

?(-

K),

K:

Ui(K)

=a;(-

K).

78

HERSHEL MARKOVITZ A N D BERNARD D. COLEMAN

Moreover, the functions q ,oz,q (and hence t) depend only on the material under consideration, i.e. are determined by the functional B,,of Eq. (2.5). For this reason (I,, u2, and q are called material junctions. Of course, q is just the “shear-dependent viscosity” of the rheological literature, while u1 and gZ govern the “normal stress effects.” We turn now to the behavior of a second-order fluid in steady simple shearing flow. On substituting (3.3) into (1.1) we find that for a secondorder fluid (3.8a) (3.8b)

Thus for a second-order fluid q ( ~ reduces ) to the constant q0, while ax(.) and u,(K) reduce to quadratics: (3.9a)

V(K)

=

vo,

(3.9b)

Ul(K)

=

(p -k 2 Y ) K 2 ,

(3.9c)

Dz(K)

= 8.’.

i.e.

T(K)

=V

~ K ,

We note that for a second-order fluid ;‘(y) = y/qo,where is the inverse of the material function t. If follows from Eq. (3.8a) that the shearing stress in a steady simple shearing flow of a second-order fluid is given by the same formula as in a Newtonian fluid. In other words, in such a motion the second-order terms, /L4(,,,,A(I,kiand yA,,,,,, do not affect the viscosity function q but instead bring in new phenomena: the normal-stress differences, S,, - S, and -5-22

- .s,

It follows from Eqs. (2.18) and (3.!1) that whenever a general simple fluid with fading memory is approximated in a slow flow by a second-order fluid, the material constants qo, P , y, of the second-order fluid must be related as follows to the material functions q , u,, of the simple fluid [lo; :

(I,

(3.10a)

. --= d4 qo = Iim r+O

K

~(0)

INCOMPRESSIBLE SECOND-ORDER FLUIDS

79

Equations (3.10) tell us that a knowledge of 17, ul, and u, for a simple fluid with fading memory suffices to determine the constants yo, p, and y and hence suffices to completely determine the second-order fluid which approximates the simple fluid in slow flows. Because of Eq. (3.10a), ljro is called the zero-shear-rate viscosity. Since there is a large literature on the experimental determination of V ( K ) for small K, and hence of q(O),*having now identified q,, with q(0) we can leave consideration of this material constant and turn to a detailed discussion of p and y . IV. VISCOMETRICFLOWS

A motion is called a (global) viscometric flow j15] if, for each material point P and each time t, G,,,( ) has the form (3.2) and there exists an orthonormal basis e i ( P , t ) , i = 1 , 2 , 3, relative to which the components have the forms (3.3) with K independent of t and depending A(l)kt, only on P; here e , ( P , t ) may, but need not, be the physical basis of an orthogonal coordinate system. Examples of viscometric flows are steady simple shearing flow, steady Couette flow, steady Poiseuille flow, and steady helical flow in an annulus. Viscometric flows are of interest to us here because they constitute the class of flows for which the stress Sii in a simple fluid is determined by the same three material functions t, (I,, and 0, which determine Sii in a steady simple shearing flow. In fact, for any simple fluid in a viscometric flow the matrix of the components of the stress tensor relative to the basis ei(P,t) has the form (3.4) with elements obeying (3.5).+ Because of this the theory of viscometric flows is a manageable theory in which one can actually solve hydrodynamical problems for simple fluids without specializing the functional $jij [II, 171. We now consider some results in this theory which permit the determination of the material functions u, and u2 and hence of the material constants and y. 1. Steady Couette Flow

This is an isochoric flow in which an incompressible fluid is sheared between two coaxial circular cylinders of radii K , and R,, R,< R,, which are rotating with constant angular velocities ill and Cl2. respectively. We use a cylindrical coordinate system r,O,z, with the z axis coincident with the See, for example, [12. 13, 141.

t The proof of this assertion is essentially that given in fi 3 of reference [ I l l .

80

HERSHEL MARKOVITZ AND BERNARD D. COLEMAN

axis of the cylinders. The physical components v i of the velocity field relative to r,O,z are assumed to have the form (4.1)

v' = 0,

YO

= YW(Y),

v' = 0 ,

where O ( Y ) is the angular velocity of a material point. For a simple fluid (4.1) is compatible with the equations of motion only if do

M is the torque per unit height required to maintain the motion. Assuming that the fluid adheres to the cylindrical walls (which are taken to be very long), we have the boundary conditions

w(R,)= ill,

(4.3)

w(R2)= Q2.

Steady Couette flow is a viscometric flow with K = r d o / d r and for which e,(P,t) is the basis used in computing physical components relative to r,B,z. Hence, Eqs. (3.4) and (3.5) hold for general simple fluids in steady Couette flow, and in the case of the second-order fluid we have dw S,O = TOY-&

(4.4a)

I

(4.4b)

(4.44

where S,, S,,, So,, and SIB are physical components: all other physical components of the stress tensor are zero. It follows from (3.9a) that for a second-order fluid (4.5)

Equation (4.2) may be integrated, subject to (4.3), to obtain w as a function of Y : I

(4.6)

w(Y) = ~[R2R22- ftlK12 - (it2- Rl)R,2R2%-2].

RZ2- R 1

We note that in deriving (4.6) we use only the function t which is the same for Newtonian fluids and second-order fluids. Hence, we are not surprised

INCOMPRESSIBLE SECOND-ORDER FLUIDS

81

that (4.6) agrees with results known to follow from the Navier-Stokes equations* and yields the familiar formula (4.7)

Q2

- 62, =

-i'&)' M

47%

R,

h'ormal stress measurements can be made in Couette flow by determining the difference of thrusts normal to the cylindrical walls in an instrument of the type described by Padden and De Witt [19j. For a general incompressible simple fluid it has been shown [ I 1 1 that, when there are no body forces in the I direction,**

(4.8)

where p is the mass density. It follows from (3.9) and (4.5) that for a secondorder fluid (4.9a) and (4.9h)

On substituting (4.6) and (4.9) into (4.8) we find that the integrations in (4.8) can be performed explicitly. The result is

where 9 is the contribution of the inertia of the fluid to the stress difference. For given values of R,, Sl,, R,, R2 and p , 9 is the same for h'ewtonian fluids and second-order fluids and is independent of ?j:'

(4.10b)

*

- 4R12R22In (R2/Rl)(R2- C2,)(S2,KZ2- R,RI2)/(K2*- R12)}< 0.

See, for example, [18]. There is a misprint in Eq. (4.27) of reference [lo]. t This formula is derived for Newtonian fluids in reference [18], [Eq. (4.20)]. There is, however, a misprint there.

**

82

HERSHEL MARKOVITZ A N D BERNARD D. COLEMAN

For a general simple fluid with fading memory, Eqs. (4.10) give an expression for S,(R,) - S,,(R,) which is valid for small angular velocity difference & - R, and which involves an error of order O((R, Hence if we measure S,,(R,) - S,(R,) as a function of R, - R, for a simple fluid, in an apparatus of fixed geometry, we can evaluate y :

We remark that when the gap between the cylinders is small, or, more precisely, if we keep R, and L2, fixed and let K , K,, R,, in such a way that R,(R, - LZ,)/(R, -- K , ) remains fixed (which means, approximately, that the rate of shear is kept fixed), then Eq. (4.10) is approximated by -+

-+

Sw(R2) (4.12)

where the first term on the right arises from the inertia and vanishes i f either of the cylinders is stationary.' The second term is written in its present form to emphasize that, for a second-order fluid, S,,(R,) - S,,(R,) goes to zero linearly in ( R , -- K , ) / K , if (R,-- H , ) / K , is made to approach zero a t fixed R,(R, - iZl)/(R2 - Rl).

2 . A xial Flow hetween Concentric Pipes Here we consider a steady flow of an incompressible fluid between two fixed coaxial circular cylinders of radii R, and K,, R, < R,. We use a cylindrical coordinate system r,z,O with the z-axis coincident with the axis of the cylinders. The physical components of the velocity field are here assumed to have the form (4.13)

z" = 0,

1''

= V(Y),

LIB = 0 ,

and the boundarv conditions are taken to be (4.14)

/I(

H,) = U( R,) = 0.

Like steady Couette flow, this viscometric flow is a special case of steady helical flow in an annulus [16] which has been shown, in the theory of simple fluids, to be compatible with the equations of motion 1171. This term was not included in the otherwise more general Eq. (4.28) of reference [lo].

INCOMPRESSIBLE SECOND-ORDER FLUIDS

83

For a simple fluid in axial flow between concentric pipes, the difference in the radial thrusts per unit area on the inner and outer cvlinders is [lo]

(4.15)

when there are no body forces in the radial direction. Here* (4.16)

((Y)

= by-'

1 - -UY.

2

The constant I) is chosen so that

(4.17a) and a is the driving force per unit volume in the axial direction: (4.18)

a = [n(R,2 - R12)]-1/;

/ is the applied force per unit length in the z direction exerted on the annulus of fluid. For a second-order fluid, it follows from (3.Ya), (4.16), and (4.17a) that

(4.17b) on doing this integration and using (4.18) one finds that (4.19)

b=

i 4n In (R2m '

Substitution of (3.9a), (3.9b), and (4.19) into (4.15) enables explicit evaluation of the integral in (4.15), and we find

For a general simple fluid with fading memory, Eq. (4.20) gives an expression for S,,(R,) - S,,(Rl) which is valid for small f and which involves an error There are misprints in Ey. (4.14) of reference [lo] and Eq. (3.7) of reference [17].

a4

HERSHEL MARKOVITZ AND BERNARD D. COLEMAN

of order O(f3). Hence, if we measure S,,(R2) - S,(R,) as a function of / for a simple fluid, in an apparatus of fixed geometry, we can evaluate /? 2y:

+

/? + 2 y = - 8n2q02(R22- K 1*)

We remark that if ( R , .- R,) << R, then (4.20) is approximated by (4.22) 3 . Approximate Results

The viscometric flows which we have considered so far satisfy the dynamical equations exactly. We now consider two examples of viscometric flows, steady torsional flow and steady cone and plate flow, which satisfy the dynamical equations under physically reasonable body forces only after certain terms are neglected. I t is believed that these two flows approximate, under evident limiting conditions, flows which an exact analysis would yield for the same boundary conditions. The reader is cautioned, however, that a rigorous theory to this effect does not yet exist. Steady Torsional Flow. This is a flow whose velocity field in a cylindrical coordinate system z,O,r has the physical components (4.23)

u' = 0,

= rw(z),

u' = 0.

I t is assumed that the flow takes place in a right cylindrical region with axis z, height 1 and radius R . We imagine two rigid discs, perpendicular to the z axis and rotating about it. The disc a t z = 0 has angular velocity R, and that at z = 1, R,. The boundary conditions expressing adherence to the d'iscs are (4.24)

w ( 0 ) = 4,

w(1) = Q,.

We assume that the body forces are zero. In an incompressible simple fluid, a flow with velocity field of the form (4.23) can satisfy the boundary conditions (4.24) and the equations of motion only if inertia is neglected, i.e. only if the centrifugal term is neglected in the dynamical equation involving balance of force and inertia in the I direction. Under this neglect of inertia, (4.24) and the three equations of motion are satisfied if

INCOMPRESSIBLE SECOND-ORDER FLUIDS

(4.25)

w(z) =

a, - Qo 1

85

+ Q,.

Here we are interested in the normal thrust S,, on the upper disc. This is determined by the equation [IS, 20, 211 (4.26a)

where (4.26b)

For a second-order fluid Eqs. (3.9) and (4.26) yield (4.27)

and (4.28)

Equation (4.27) has been obtained on a less rigorous basis by Markovitz and Brown [22] and illustrative data are given in their Figure 7. The correspondence between notations is (4.29)

B =~ I I I kr,

y = (kr - kIr)/2.

Equation (4.27) suggests the following method of determining the sum (3p 4y) from measurements on a general simple fluid with fading memory in a torsional flow apparatus with fixed 1 :

+

(4.30)

Steady Cone and Plate Flow. This is a flow whose velocity field has the physical components (4.31)

v e = 0,

v1

= r(sin 8 ) o ( 8 ) ,

vr = 0

in a spherical coordinate system 8,$,r. I t is assumed that the fluid lies between a rigid cone, rotating about its axis with angular velocity Q,, and a rigid disc, rotating about the same axis with angular velocity Q,. We take the axis of the cone to be the polar axis of the spherical coordinate system. We assume that the disc lies in the plane 8 = 742 and that the

86

HERSHEL MARKOVITZ AND BERNARD D. COLEMAN

equation of the cone is 0 = x / 2 - I) where I) is the angle between the disc and the cone. The boundary conditions are (4.32)

.,(x/2) = no,

o,(n/2- I))= ill.

For a simple fluid a flow field of the form (4.31) cannot satisfy both the boundary conditions (4.33) and the equations of motion unless certain approsimations are made. One such approximation, believed to be satisfactory in many cases, is to neglect the inertia and to assume that I) is small, i.e. t o put PY[w(e)] = 0 , and also cos 8 = 0 , sin 0 = 1 forn/2 - 4 < 0 < x/X, in the dynamical equations. We are here interested in the radial dependence of the normal thrust on the cone. Under the assumptions made above, it can be shown that [SO, 21 j (4.33a) where (4.33b)

K

=

(0,- flo)/I).

Thus, by (3.9), for a second-order fluid we have (4.34)

1he normal pressure should be a linear function of In Y (at least for r small enough so that the experimental complications due to uncertain “edge effects” are ignorable). For a general simple fluid with fading memory, one should be able to obtain (p + y ) from measurements of a.Seeia In Y as a function of n, - Ro in a cone and plate apparatus of fixed angle $: (4.35)

4. Appendix

Markovitz and Brown have recently determined values of p and y for a 5.4% solution of a polyisobutylene in cetane a t 30°C. For this material qo = 18.5 poises and p = 0.773 g/cm3. From normal-stress measurements in steady torsional and Couette flows, they obtain, via Eqs. (4.30) and (4.11), ,f?= 1.0 g/cm, y = - 0.2 g/cm. Within experimental error, these numbers agree with the value of p + y which they calculate from normal-stress measurements in steady cone and plate flow using Eq. (4.35).

87

INCOMPRESSIBLE SECOND-ORDER FLUIDS

V. STEADYEXTENSION OF

A

CYLINDER

A steady extension is a flow for which the velocity field has the form (54

v1 = alx,,

v 2 = a2x2,

213

= a3x3

in some fixed Cartesian coordinate system x,,x2,x3. Here the a' are constants obeying the relation a'

(5.2)

+ a2 + a3 = 0,

which makes the flow isochoric. It is known that for steady extension of a simple fluid the components of the stress relative to the Cartesian system x1,x2,x3 obey the equation [23] Sij

(5.3)

+ pS+ = [ a i r + (ai)2A)S,,

(no summation)

_-

the indeterminate hydrostatic pressure; r = r ( Z Z , Z 1 1 ) and where p-isA = A(ZI,ZII) are functions of only the two quantities n

n

(5.4) 1-1

1-1

The functions r and A depend OR only the material under consideration; they are material functions. For incompressible simple fluids in general, a knowledge of the material functions 7, al,and a, does not suffice to determine the material functions r and A . For a second-order fluid, however, r and A must be determined by yo, /I,and y. Indeed, it follows from (5.1) and (2.6) that G,,,,(s) = (e-2a's

(5.5)

- 1IS',

(no summation)

and, hence by (2.11), that (5.6)

A ( I ) ,= ~ 2a'&,,

A(2),1= 4 ( ~ * ) ~ 6 , , . (no summation)

On inserting (5.6) into (1.1) we find (5.7)

S,

+ pS,,

= S,,[a'27j0

+ ( u ' ) ~ ~+( Pr ) ] ,

(no summation)

and on comparing this with (5.3)we see that for a second-order fluid and A are given by (5.8a)

r = 27j0 = const.

(5.8b)

A = 4(p

+ y ) = const.

r

88

HERSHEL MARKOVITZ A N D BERNARD D. COLEMAN

It follows from Eqs. (2.18) and (5.8) that whenever a general simple fluid with fading memory is approximated by a second-order fluid, the material constants qo, ,!?,and y must be related as follows to the material functions of the simple fluid: (5.9a)

70

P

(5.9b)

= tT(O,O),

+ y = iA(0,O).

Steady extension is compatible with the equations of motion for general simple fluids under arbitrary conservative body forces r23'. A special example of steady extension is realized by considering a cylindrical fluid mass which is being continually elongated, or shortened, at a rate proportional to its length. We let the axis of the cylinder coincide with the %,-axis and assume that (5.1) holds with (5.10)

a' = a,

a2 = a3 = - & a ;

(5.1) is then automatically fulfilled. If R = R(t) and L spectively, the radius and length of the cylinder then

d d a=-lnL=:-2--1n dt dt

(5.11)

=

L ( t ) are, re-

R.

We let the two bounding cross-sections of the cylinder be the planes x1 = 0 and x, = L(t) and put Y = V x z 2 -!- xa2. Assuming the applied body forces to be zero, the normal stresses Sll(O,r) and S,,(L,v) on bounding crosssections are 1231 (5.12a)

S,,(O,r)

(5.12b)

SlI(L.,y) = UT

=a T

+ a2A + pa2r2/8 + a2A + p(aZL2 + a2r2/4)/2.

These equations hold for simple fluids in general. The last terms (involving p) in Eqs. (5.12) arise from inertia and should be negligible in most applications. We now assume that they are negligible, and hence have Sl1(O,y) and .S,,(L,Y)not only equal, but independent of Y . For a second-order fluid Eqs. (5.12) and (5.8) then yield

For a general simple fluid (assuming fading memory and neglecting inertia) Eq. (5.13) gives an expression for S,, which involves an error of order O(a3). Hence for such a fluid l;lo and ,!? y may be determined as follows from measurements of S,, versus a :

+

89

INCOMPRESSIBLE SECOND-ORDER FLUIDS

1 . dS,, qo = - lim __ 2 a-0 da

(5.14a)

1

(5.14b)

P+y=-lim-. 8

a-+O

dZS,, da2

VI. RELATIONTO CLASSICAL VISCOELASTICITY We here consider the behavior of general simple fluids with fading memory. For such fluids it follows from the definition sketched in Section I1 that if &i be a memory functional and if GkI( ) be a history with small norm, m

then to within an error of order three in the norm of Gkl( ), gjii(GRI(s))is s=0

given by the sum of a h e a r and a quadratic term in G k l ( ). When this observation is combined with the known isotropy of gj,,, it can be shown that (2.5) yields the following approximation formula r.51: S,j

+ PGij =

r

m(s)Gij(s)ds

0

(6.1)

+

I

+

[a(S~,SZ)Gik(si)Gkj(sZ)b(s,,s,)Gkk(si)Gii(s,)1dsidsz

00

+ o(llGkI(

)It3).

Here m, a, and b are material functions determined by the functional b,,; the function u is symmetric in the sense that a(s,,s,) = a(s,,s,). The equation obtained by striking out the term O(llG( )\la) in (6.1) is called the equation of finite second-order wiscoelasticity (for incompressible fluids). That equation gives a good approximation for the stress whenever 1IGkr( )I\ is small. can be When IIGkl( )]I i s so small that terms of order o(llGkj( neglected, the equation

)/I2)

(6.2)

which we call the equation of fanate linear viscoelasticity (for inconipressible fluids), approximates the stress.* The sense in which the theories of finite linear and second-order viscoelasticity should be independent of the choice of strain measure is discussed at length in 5 7 of reference [ 5 ] . I n this connection, we remark that the approximation formulae (6.1) and (6.2) remain valid if the strain measure C k j be replaced by any symmetric tensor related to CkJ by a smooth one-to-one transformation.

90

HERSHEL MARKOVITZ A N D BERNARD D. COLEMAN

For finite second-order viscoelasticity to hold, i.e. for IIGkI( )I1 to be small, it is sufficient that the deformation relative to the present configuration be small in the recent past. In particular, if Gg)( ) be a retardation of an arbitrary history Gv( ), then, for small a, //Gk)()I1 will be small [ 2 ] . This is true for any history Gkl( ), even a history describing a steady flow for which the deformation in the distant past, relative to the present configuration, was very large. Of course, when a is small, Eq. (2.18) must apply. This is consistent with our present remarks because retardation is but a particular way of constructing histories with small norm, and when this way is used Eq. (6.1) itself reduces to (2.18). If a is very small, one can start with (6.2) which will be approximated by the equation of an incompressible h’ewtonian fluid. Another way of constructing histories with small norm is to consider cases in which E

= SUP

[G,~(S)C,,(S)]’’*

m >s>O

is small: in other words, to consider motions for which the “strain” relative to some fixed configuration is always small. Such steady flows as those considered in Sections 111, IV, and 1-do not have this property, but oscillatory motions of “small amplitude” do. So do histories occurring in stressrelaxation experiments involving small imposed deformations. Let us consider now the linear theory (6.2). When IIG(t,u( )I1 is made very small by making e of (6.3) small, (6.2) is approximated by Boltzmann’s theory [24] of infinitesimal linear viscoelasticity ; i.e., for a general simple fluid with fading memory [ 5 ] m

(6.4)

Si,

+ pS;i = 2G(O)E;i(t)+ 2

1 ~

E,,(t - s)ds

+0 ( c 2 )

0

where Eil(t - s) is the infinitesimal strain tensor at time t - s relative to a fixed reference configuration, and the material function G is related to the function m appearing in (6.1) and (0.2) as follows:

s

In a shearing stress relaxation experiment, i.e. when E,, has the special time dependence

INCOMPRESSIBLE SECOND-ORDER FLUIDS

91

Eq. (6.4) yields, for 1 > 0, S12= 2E&G(t).

(6.7)

The material function G is called the shear relaxation modzllus.* We now show that, for a given simple fluid with fading memory, the material constants qlo and y , occurring in its second-order fluid approximation (2.18), are determined by G. To do this we evaluate (6.1) for the steady simple shearing flow (3.1). On putting Eqs. (3.2) and (3.3) into (6.1) we find that, to within an error of order O ( K ~ )the , nonzero components of stress are given by a

cc

(6.8d)

mm

s,, = - p .

Comparing this result with Eqs. ( 3 4 , which by Eq. (2.18) must hold for the same fluid also to within an error of order O ( K ~ we ) , see thatt

(6.9a)

i

qo = - sm(s)ds, 0

y

X

(6.9b)

y = - s2m(s)ds, 2

* Cf. H. Leaderman [25]. Note that the symbol c: used in that committee report represents, in our present notation, 2E,,. The Eqs. (6.9) can be obtained directly by considering Eqs. (5.14) and (7.4) of reference [2] and Eqs. (6.1) and (6.4) of reference [5]. without recourse to a special kinematical situation. Such a derivation was carried out by Coleman and Markovitz [26] : it shows that our Eqs. (6.9a) and (6.9b)arise from the fact that the terms in (2.18) which are linear in Rivlin-Ericksen tensors involve, a t bottom, only the first Fr6chet derivative of 8;j a t the rest history (2.10). The derivation we give here has, however, the advantage that it does not require familiarity with the theory of functional derivatives, if one grants Eq. (2.18).

9s

HERSHEL MARKOVlTZ A N D BERNARD I). COLEMAN

(6.!k)

Equation (6%)tells us that r], is related to m and hence to the shear relaxation modulus G, a known result. Equation ( 6 . 9 ~ )tells us that one of the “normal stress coefficients” is not determined by functions occurring in the Roltzmann’s theory of infinitesimal linear viscoelasticity, a not surprising result. Equation (6.9b)is quite surprising, however; for in relating y to m it tells us that normal stress differences occurring a t low shear rates are sometimes determined by G. In particular, it follows from (6%) and (4.10a) that to within an error of order O((f1, - QJs) the results of the normal stress experiment in Couette flow should be predictable when G is known.* I t follows from an argument similar to that which led to (5.22) of reference [ 5 ] , that

1

m2(s)k-Z(s)ds

must be finite. Since we here assume that as s - - bo, h(s) ---c 0 faster than it follows that m(s) also goes to zero faster than s - ~ ,and hence, by (6.5) that

s-3,

(6.10)

lim sZG(s)

=:

0.

Therefore, through an integration by parts we can write Eqs. (6.9a) and (6.9b) in the forms

(6.1la)

(6.11b)

j

y = - sG(s)ds.

On the basis of a similar argument, Coleman and No11 LJ71 have shown t h a t in an oscillatory simple shearing motion of small amplitude, to within an error of order o(e*). the normal stress difference S,, - S,, is determined by G and so is the difference in radial thrusts on the inner and outer cylinders in a periodic Couette flow. The material function p of references [ 5 ] and [27] is equal to our m while the function # of those references equals l j 2 our G. There is a misprint in Eq. ( 8 ) of reference i27]; the lower limit on the integration should be s, not 0.

INCOMPRESSIBLE SECOND-ORDER FLUIDS

93

We expect, on the basis of thermodynamic intuition, that G(t) in (6.7) should be positive for all t. It follows from this assertion and Eqs. (6.11) that rlo> 0

(6.12)

and

y

< 0.

That y be negative has an interesting consequence. From it we deduce the result that in Couette flow the contribution of the term in Eq. (4.10a) involving y (the “viscoelastic” contribution) to S,,, that is S,,(R,) - S,,(RI),must be opposite in sign to the contribution of the term 9 (the “inertial term”). This means that, when the theory of second-order fluids is applicable, the viscoelastic effects should tend to make the pressure on the inner wall be greater than that on the outer wall. This is, in fact, the sign of the effect observed by Padden and De U’itt [19], albeit their experiments were conducted in a range of shearing rates which is probably too high for application of the theory of second-order fluids. An important type of experiment in linear viscoelasticity is that involving sinusoidal stress and strain. Let us consider a motion such that the infinitesimal shear strain is a sinusoidal function of time with circular frequency w , i.e.

El&)

(6.13)

= El2 sin wit.

Here Ei2 is a real constant. From (6.4) we see that for such a strain, according to infinitesimal linear viscoelasticity, the shear stress is given by

Through integration by parts and use of (6.10), Eq. (6.14) can be recast into the form m

(6.15)

m

S,,/2E:.L = 0

The material function

(6.16)

0

G’ defined by the equation*

i

G’(w) = w G(s) sin wsds 0

*

Cf. B. Gross [28].

94

HERSHEL MARKOVITZ AND BERNARD D . COLEMAN

is known as the shear storage modulus [25]; it determines the part of the stress S,, that is in phase .with the strain E12. The material function G“, defined by the equation [28]

(6.17)

i

G”(w) = w G ( s )cos Wsds n

is called the shear loss modulus; it determines the part of the stress S,, that is in phase with the rate of strain dE,,/dt. The dynamic shear viscosity, ~ ‘ ( w ) , defined as (6.18)

i

y ’ ( ~= ) G”(w)/w = G(s) cos wsds n

is also frequently used as a material property. It follows from (6.18) that

lim f ( w )

(6.19)

=

u+o

i

G(s)ds

0

and thus, on comparison with (6.11a), we see that (6.20)

lim $ ( w ) = ?lo; w+o

i.e. that the dynamic shear viscosity a t zero frequency is equal to the steady shear viscosity measured at low shearing. It follows from (6.16) that (6.21a)

lim G’(w) = 0 w-+o

(6.21b)

and (6.21c)

Thus, from (6.11b) (6.P2)

lim

s+o

dG’(w) -0 dw ~

INCOMPRESSIBLE SECOND-ORDER FLUIDS

95

It thus becomes possible to calculate, from the results of sinusoidal experiments a t low frequencies, the normal stress difference across the annulus in Couette shearing at low speeds of rotation, using (4.10a) and (6.22). I t is frequently assumed that G(t) can be written as the Laplace transform of some function* N ( s ) , i.e.

1

G(t)= e-”N(s)ds.

(6.23a)

0

It is common practice to write this equation in the form

i

G(t) = e- ‘I7F(z)dt,

(6.23b)

0

or

i

G ( t )=

(6.23~)

e-’17H(z)d(lnt).

J

-m

Here F(s-l) = s2N(s)= s H ( s - l ) . The term relaxation spectrum [25]? is used for both F ( t ) and H ( t ) . It follows from (6.23) and (6.16)-(6.18) that

1+ 1 m

(6.24)

G’(w) =

s2

a

X

w2

ds =

n

0

H (t) w 2t2

F (z)w 2t2 2dt= l+wt --a

and

(6.25) m j ( w ) = G”(w) =

[----+ N(s)ws w2

F(t)wt l f w t

s2

0

0

,dt= -a

From (6.20) and (6.25) we then find that

(6.26)

i

0

t

i

s-lN(s)ds = t F ( t ) d t =

See, for example, [29]. Cf. J. D. Ferry [29].

0

i - x

tH(t)d(lnt)= qo,

96

HERSHEL MARKOVITZ AND BERNARD D. COLEMAN

while the relations

result from (6.22) and (6.24). Rheologists usually assume that G is completely monotone, i.e. that N , F , and H have only non-negative values. Our equations (6.26) and (6.27) are independent of this assumption.’

V I I . NONSTEADY SHEARISC FLOWS The viscometric flows discussed in Sections I11 and I V and the steady extension treated in Section V are all special cases of a class of motions called substantially stagnant motions [15, 301 or motions with constant stretch history [311. All those boundary value problems involving substantially stagnant motions for which exact solutions are known (under conservative body forces) for second-order fluids, or even for Newtonian fluids, can also be solved for simple fluids in general. Furthermore, all of the hydrodynamical problems for which exact solutions are known for genelal simple fluids yield subst ant ially stagnant mot ions. Yet there are some hydrodynamical problems which can be solved exactly for Newtonian fluids but which, by their very nature, involve motions which are not substantially stagnant. Although solutions to such problems for general simple fluids seem beyond our present grasp, it is of interest to see whether solutions can be found for second-order fluids. Consider, for example, an isochoric rectilinear flow which in Cartesian coordinates x1,xz,x3 has a velocity field with components (7.1)

v l = 0,

212 = v ( x , t ) ,

v3

=:

0,

(XI=

x).

When v(x,t) is constant in t such a flow reduces to the simple shearing flow discussed in Section 111. Here we assume that v is not constant in t , and we call a motion obeying (7.1) a nonsteady simple shearing flow. Such a flow is not substantially stagnant. In the case of a Kewtonian fluid, substitution of (7.1) into the NavierStokes equations yields the observation [32: that v(x,t) is determined by a We note that the integral appearing on the right of Eq. (6.11b)has been related by Fujita (as quoted on page 56 of [29:) to the “steady-state shear compliance” J ( c ) m

of linear viscoelasticity through the formula qo2J(c) = I s G ( s ) d s . Hence, we now have I)

the interesting formula y = -

tlOp](c).

INCOMPRESSIBLE SECOND-ORDER FLUIDS

97

linear second-order diffusion equation which is solvable. Nonsteady simple shearing flows in Newtonian fluids were much studied in classical hydrodynamics 1331. For second-order fluids, Eqs. (7.1) and (1.1) yield the following expressions for the nonzero Cartesian components of the stress: (7.h)

%,= - p

+ (P+

dy)

(:y

,

--

(7.db)

(7.84

(7.8d)

s,

= -p.

In the absence of body forces the equations of motion read (7.3) Substitution of (7.2) into (7.3) yields (7.4a)

a2v

81’

at1

t l ax o ~ + y - -w=ap t- -

(7.4b)

aP _

(7.4c)

--

ax

ap az

at

+

ap a~

ax

= 0.

I t follows from these equations that fi has the form (7.5) and hence (7.4a) reduces to (7.6) We shall not develop here the mathematical theory of this linear third-order partial differential equation, but we will discuss one special solution, that corresponding to “steady-state’’ sinusoidal oscillations.

98

HERSHEL MARKOi‘ITL ANI) HEHNAHD L). COLEMAN

1 . Sinusoicial Oscillations We consider a second-order fluid, free from body forces, undergoing a nonsteady shearing flow between two parallel plates. We assume that the fluid adheres to the plates and that a t all times t , - 00 < t < m, one plate lies in the plane x = 0 and is at rest, while the other, located at x = I , undergoes sinusoidal oscillation in the x 2 direction with velocity cos rut. We further assume that there is no pressure gradient in the direction of flow. It follows then that d ( t ) zz 0 in (7.5) and that (7.6) becomes (7.7) We now seek a solution of (7.7) with the form ~ ( x , t= ) Kt1 {I*’(x)eiylfj

(7.8)

where Re stands for “real part of,” and C‘(x) is a complex-valued function of x obeying the boundary conditions

l.(O) = 0,

(7.!))

l’(1) = C‘,

Such a solution esists; it is given by (7.10a) where

Denoting the real and imaginary parts of

/I

by a and h, respectively, we have

(7. I la)

(7.1 Ib) Equation (7.8) is equivalent to the assertion that u ( x , f ) may be written in the form: (7.12)

v ( x , t ) = b’,(x) cos xt

- F 2 ( x )sin wt.

Here Fl and L‘, are real valued functions giving, respectively, the real and imaginary parts of C ’ ( x ) . It follows from (7.10) that (7.13a)

I’, = I’,q[sinh

QX

cos h x sinh al cos bl

+ sin b x cosh ax sin bl cosh a1

INCOMPRESSIBLE SECOND-ORDER FLUIDS

(7.13b)

99

V , = Voq[sin bx cosh ax sinh a1 cos 61 - sinh ax cos bx sin 61 cosh all

where (7.13~)

= sinh2 a1 cos2 hl

q-1

+ sin2bl cosh2al.

From (7.2), (7.5)’ and (7.12), it is seen that p - l and the normal-stress differences [(S,, - S,,), etc.] oscillate with twice the circular frequency m of the velocity gradient & / a x . It is also true that for a general simple fluid in certain periodic simple shearing flows, including sinusoidal oscillations, the normal-stress differences oscillate with twice the frequency of the velocity field [ 101. A special characteristic of a second-order fluid is that a sinusoidal velocity gradient gives rise to sinusoidal normal-stress differences which are in phase with If the spacing between the planes is so small that at<< 1, expansion of the hyperbolic functions in (7.10a) yields the result (7.14)

V ( x ) = V,x/l

+ O((al)2),

and therefore, by (7.l2), v ( x , t ) = ( V0%/l)cos wt

(7.15)

+ O((al)2).

Hence, when a1 is small, a v / a x is independent of x and

p

(7.16)

m

( p + 2y)(1/,/1)2cos2wt+ [ ( t ) .

We may use Eqs. (7.2) to calculate the values of the stress components in this limiting case. They are also independent of x and are given by: (7.17a) (7.17b)

s,,m - p + (P + 27)( V,/l)2 cos2 mt S,, = S,, m ( Vo/l)(qocos wt - yw sin wt) =

(7.17~)

S,, w - p

(7.17d)

Sa3% - p .

+

+ tan-l[ym/q,])

(Vo/l)(qo2 y2ze~2)1’2cos (wt

+ /?( VO/J!)~

C O S ~~t

It will be noticed that the shearing stress S,, = S,, oscillates with the same circular frequency w as the velocity but is out of phase with the velocity by an angle equal to tan-’ (yw/qo). This result for S,, is the same as that which would be obtained by solving the present boundary-value problem using the constitutive Eq. (6.4) of infinitesimal linear viscoelasticity, taking the limits al-. 0 and zl-. 0, and then using Eqs. (6.20) and (6.22). However, on working within the framework of infinitesimal linear viscoelasticity, one does not obtain information about normal-stress differences,

100

HERSHEL MARKOVITZ A N D BERNARD D. COLEMAN

while our present calculation yields the interesting result that, for a secondorder fluid and small separation I, the normal-stress differences oscillate with a circular frequency twice w and are in phase with the square of the velocity of the moving plate. We also note that the particular normal-stress difference (7.18)

is determined by that material constant y which enters into Eqs. (4.10) and (6.11). ACKNOWLEDGMENTS We wish to thank Professor Walter No11 for helpful discussions and for showing u s some of his own unpublished calculations similar to those presented here in Sections 4 and 7. We are also grateful to Professor Tsuan W-uTina for showing us his unpublished manuscript “Certain Non-Steady Flows of Second-Order Fluids.”

References 1. SOLL, W.,

h mathematical theory of the mechanical behavior of continuous media, Arch. IZafional Mech. Anal. 2, 197-226 (1958). 2. COLEMAN, R. D., and TOLL,m’.,An approximation theorem for functionals, with applications in continuiim mechanics, Arch. Rational Mech. A n a l . 6, 3.55-370 (1960). 3. RIVLIN,R . S., and ERICKSEN.J . L., Stress-deformation relations for isotropic materials, J . Rational Mech. A p a l . 4, 681-702 (195-5). 4 . R I V L I NI<. . S., Second and higher-order theories for the flow of a viscoelastic fluid in a non-circular pipe, Proc. Intl. Symposium on Second-Order Etfects in Elasficity, Plasticity and Fluid Dynatnics. Haifa. 1962 (in press). 5, COLEMAN, B. D., and NOLL, W.. Foundations of linear viscoelasticity. R e v s . Mod. P h y s . 88, 239-249 (1961). 6. COLEMAN, B. D., and SOLL, W., Simple fluids with fading memory, Proc. f n t l . Symposium on Second-Order Effects i n Elasticify, Plasticit,y, and Fluid I)ynamics, H a i f a . 7962 (in press). 7. TRUESDELL, C., and TOUPIN,R . A . , The classical field theories i n “Handbuch der Physik” (S. Fliigge, ed.), v. IIIIl. pp. 326-793. Springer, Berlin, 1960. 8 . KEINER,>IA ., mathematical theory of dilatancy, A m . J . Math. 67, 350-362 (1945). 9. RIVLIN,l<. S., The hydrodynamics of non-Kewtonian fluids. I., Proc. R o y . SOC. (London) A198, 280-281 (1948). 10. COLEMAN, B. D., and YOLL,W., Recent results in the continuum theory of viscoelastic fluids, A n n . New York Acad. Sci. 89, 672-714 (1961). 1 1 . COLEMAN, B. D.. and NOLL,W., On certain steady flows of general fluids, .4rch. Rational Mech. Anal. 8 , 289-303 (1959). 12. TSUDA,A , , Zur Kenntnis der Strukturviskositat einiger Sole polymeren Kohlenhydrate, h’olloid-Z. 46, 325-331 (1928). 13. EISENBERG, H., and FREI,E. H.. Precision rotation viscometer with restoring torque for the centipoise range, J . Polymer S c i . 14, 417-426 (1954).

INCOMPRESSIBLE SECOND-ORDER FLUIDS

101

14. MARKOVITZ, H., ELYASH,L. J., PADDEN, F. J., J R . , DEWITT,T. W.. .4 cone-andplate viscometer, J . Colloid Sci. 10, 165-173 (1955). 16. COLEMAN,B. D., Kinematical concepts with applications in the mechanics and thermodynamics of incompressible viscoelastic fluids, Arch. Rational Mech. Anal. 9, 273-300 (1962). 16. RIVLIN,R . S., Solution of some problems in the exact theory of visco-elasticity, J . Rational Mech. Anal. 6 , 179-188 (1956). 17. COLEMAN, B. D., and NOLL, W.. Helical flow of general fluids, ,I. AppZ. Phys. 30, 1508-1512 (1959). 18. PAI.Shih-I, “Viscous Flow Theory. I. Laminar Flow.” Van Nostrand, Princeton, 1956, p. 54. 19. PADDEN,F. J., and DeWITT, T. W., Some rheological properties of concentrated polyisobutylene solutions, J . APPI. Phys. 26, 1086-1091 (1954). 30. NOLL, W., and TRUESDELL, C., The non-linear field theories of mechanics i n “Handbuch der Physik” (S. Fliigge, ed.) v. VIII/2. Springer, Berlin. To be published. 21. MARKOVITZ, H., Normal stress effect in polyisobutylene solutions. 11. Classification and application of rheological theories, Trans. SOC.Rheol. 1, 37-52 (1957). H., and BROWN,D. R., Normal stress measurements on a polyiso22. MARKOVITZ. butylene-cetane solution in parallel plate and cone-plate instruments, Proc. f n t l . Symposium on Second-Order Effects i n Elasticity, PEasticity and Fluid Dynamics, Haifa, 1962 (in press). 23. COLEMAN, B. D., and NOLL, W., Steady extension of incompressible simple fluids. Phys. Fluids 6 , 840-843 (1962). 24. BOLTZMANN, L., Zur Theorie der elastischen Nachairkung, Sztaber. Kaiserlich. Akad. Wiss. ( W i e n ) , Math.-Naturwiss. KI. 70, Sect. 11, 275-306 (1874). 65. LEADERMAN, H., Proposed nomenclature for linear viscoelastic behavior, Trans. SOL. Rheol. 1, 213-222 (1957). 26. COLEMAN, B. D., and MARKOVITZ,H.. Normal stress effects in second-order fluids, J . A p p l . Phys. 86, 1-9 (1964). 27. COLEMAN, B. D., and NOLL, W., Normal stresses in second-order viscoelasticitv. Trans. SOC. Rheol. 6 , 41-46 (1961). 28. GROSS,B., “Mathematical Structure of the Theories of Viscoelasticity,” Hermann and Cie, Paris, 1953. 29. FERRY, J . D., ”Viscoelastic Properties of Polymers.” Wiley. New York, 1961. B. D., Substantially stagnant motions, Trans. SOL. Rheol. 6, 293-300 30. COLEMAN, (1962). 31. XOLL, W . , Motions with constant stretch history, Arch. Rational Mech. .4nal. 11, 97-105 (1962). 3’7. STOKES,G. G., “Mathematical and Physical Papers.” Cambridge University Press, Cambridge, 1901, vol. 111, p. 19, Eq. (8). 33. LAMB,H., “Hydrodynamics,” Cambridge University Press, Cambridge, 1895. $ 5 298-300.

This Page Intentionally Left Blank

The Generation of Sound by Turbulent Jets BY H . S . RIBNER lnstitutr of Aerophysics. I :nii:ersity of I‘oronto

1. Introduction . . . . . . . TI . Resume of Major Features

I’age 105 i06

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

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

A . S I M W F I E I I PHYSICAL xcoux‘r 111. Physics of Sound Generation . . . . . . . . . . . . . 1 . Simple Source or Monopole . . . . . . . . . . . . . 2 . Force Source or Dipole . . . . . . . . . . . . . . 3 . Stress Source or Quadrupole . . . . . . . . . . . 1. Equivalent Patterns of Simple Sources, Dipoles and I c’. Equivalent Aerodynamic Generators of Sound . . . . I . Reynolds Stresses (Quadrupole Sources) . . . . . . 2 . Vortex Stretching (Dipole Sources) . . . . . . . . 3 . Fluid Dilatations (Simple Sources) . . . . . . . . . 1’. Sound Radiated from a Jet . . . . . . . . . . . . . . I . Structure of a Round Turbulent Jet . . . . . . . 2 . Sound Radiated from a Volume Element . . . . . 3 . The Ujs, yo. y-’. f z and f - 2 Laws . . . . . . . . . 4 . Self-noise and Shear-noise . . . . . . . . . . . . . . .iEffects of Convection and Refraction . . . . . . . 6 . Shifts of the Spectrum Peak . . . . . . . . . . . 7. Power and Efficiency vs . Mach Number . . . . . . X . Effects of Density and Temperature . . . . . . . . !I . Reduction of Turbojet Noise . . . . . . . . . . .

.

. . . .

. . . . . . 109 . . . . . . 109 . . . . . . 111 . . . . . . . 112

Quadriipoles

. .

113

. . . . . . . . 115 . . . . . . . . 115 . . . . . . . . 116 . . . . . . . . 117 . . . . . . . II!) . . . . . . . . II!) . . . . . . . . 122 . . . . . . . . 123 . . . . . . . 125 . . . . . . . . 126 . . . . . . . . 131 . . . . . . . . 133 . . . . . . . . 136

. . . . . . . .

138

R . MATHEMATICAL DEVELOPMEN1 \‘I . Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . 142 I . Wave Equations for a n Inviscid Fluid . . . . . . . . . . . . . . . 142 2 . Solutions for Radiated Sound . . . . . . . . . . . . . . . . . . . 143 3 . Spectrum of Radiated Sound . . . . . . . . . . . . . . . . . . . 146 4 . Developments of Lighthill and Proudman . . . . . . . . . . . . . 147 \.I1 . Convection Effects for a Simplified Model of Turbulence . . . . . . . 151 I . Moving vs . Stationary Axes . . . . . . . . . . . . . . . . . . . 151 2 . Directivity of Sound from Unit Volume . . . . . . . . . . . . . . 153 3 . Spectrum of Sound from Unit Volume . . . . . . . . . . . . . . 155 1. Moving Jets . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 V11I . Refraction Effects Due to the Mean Flow . . . . . . . . . . . . . . 159 I . Convected Wave E(Iuation . . . . . . . . . . . . . . . . . . . . 159 2 . Form of Solution . . . . . . . . . . . . . . . . . . . . . . . . 161 3 . Qualitative Effects on Directivity . . . . . . . . . . . . . . . . . 162 IS. Improved Model: Isotropic Turbulence Superposed on Mean Flow . . . . 163 1. Remarks on Lilley’s Approach . . . . . . . . . . . . . . . . . . 163

n. s.

104

RIBNEH

2. Self-Noise and Shear-Noise . . . . . . . . . . . . . . . . . . . . 164 3. Directivity and Spectrum of Sound from Unit Volume . . . . . . . 168 X. Sound Emission from a Complete Jet . . . . . . . . . . . . . . . . 169 1. Integration Over a ‘Slice’ of J e t . . . . . . . . . . . . . . . . . 169 2. Idealized Source Distribution: the Y o and Y-’ L a w s . . . . . . . . 170 3. Idealized Spectrum Shape: the f a and f-* Laws . . . . . . . . . . 172 XI. Asymptotic Behavior at High Mach Number . . . . . . . . . . . . . 175 1. Inferences from Convected Reynolds Stresses or Dilatations . . . . . 175 2. 0. M. Phillips’ Fundamental Investigation . . . . . . . . . . . . . 175 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

SOTAT 10s The more important symbols are collected here for convenience. Other symbols of limited use are explained in the text. Sozzle cross-section area Convection factor (C = [ ( I - Mc cos 818 + a*kfCaj14 Speed of sound Nozzle diameter (D = 2 H ) Frequency .4 scale of turbulence

u/c Eddy convection speed/co (often taken as U,/Zc,) Flight speed/c, m

z.

Total acoustic power (I’ JJ~(y,ru)daydw= m

d

=

n

P(y)dsy =

P(yl)dy, =

z-ye)zx sin W O ,

m

P(w)dw = 0

etc.)

Instantaneous pressure ( p = p o I pW) + # I ) ) Ambient pressure ‘Pseudosound’ pressure ( c near field) ‘Acoustic’ pressure ( far-field) Two-point space-time correlations ; also, It‘ = nozzle radius H (Part A) radial and axial coordinates in jet; in Part R.y --c V1, and t + y, in mixing region and + y o in developed jet Time Retarded time (e.g., t = I - 11 - yl/c,,) Lighthill quadrupole strength (puiu, viscous terms ( p - co*p)btl) Local time average velocity ( l l y in Part .A, jly, in Part R) Value of L‘ a t jet nozzle Resultant fluid velocity or turbulent increment according to context (components 14;) Component of u in I-direction Effective volume of region generating flow noise Position vectors (components x i , y i , zi with 1-direction taken along jetaxis in flow direction) Eddy length/decay distance (a = f . / U L x time scalc) (Part A) in mixing region, = (I - I ? ) / y ; in developed jet, = t / v Angle between observer vector x and jet axis Vector separation in two-point correlation (components :,) Instantaneoiis local density ( p 5 + pfo) + pP1) Density associated with pseudosound pressure (p(0)R= p(Wji2)

-

-

+

+

105

T H E GENERATION OF SOUND BY TURBULENT J E T S

Density associated with acoustic pressure ( p ( l )w p('J/>) Difference in emission times for two source points of separation value for simultaneous reCeption at x given by (6.15) Arbitrary increment t o the T of (6.15) Radian frequency Typical radian frequency in the turbulence, defined by correlation of form ~ X P- wt171 Peak of 'slice' sound spectrum (wp = 2yFw,/C)

p(')

5;

t t'

w wt WP

Ouevbars, eic. In Part A, effective time-space average over jet: in Part B, local time average Double partial time derivative Associated with single-frequency source pattern Used with different meanings: defined where used ~

..

-

SubscriPts

.4mbient value in quiescent fluid outside jet Tensor index with values I to 3 In Uj. c,, pi, value at jet nozzle; otherwise tensor index with values 1 to 3

0 i

i Other

[ I (

)AV

Denotes retarded-time integrand, where appropriate Average over unit sphere, with or without weight factor according to context

I. INTRODUCTION The purpose of this article is the explanation of major features of jet noise. The framework has evolved from the fundamental work of Lighthill [I, 21 and has been contributed to by a substantial number of workers (see list of References). A further extension has been made in the course of preparing the present survey; it takes the form of an advance in the handling of spectra and directivity based on a new treatment of 'self-noise' and 'shear-noise'. After some preliminaries, a single coherent analysis is developed in detail, drawing in the main from [l-131. Alternative approaches are touched on but not treated exhaustively. A number of excellent review articles and a bibliography have been written on this and the more inclusive subject of aerodynamic sound, [3, 14-24, 901. Of these [lF] and [18] are comprehensive surveys of experimental results. The present article should be regarded as complementary in nature. I t provides a state-of-the-art account of the theory as it has now been extended. Ideas from these earlier reviews have been drawn on where appropriate, and the writer would like to record here his debt to the authors concerned. Of particular help have been the notions of Powell [2l] and \t-estervelt [25] for Section 111 and of Lighthill [3] for Section V.9. As in Lilley's work [4] no mention is made of the noise emission from choked jets or supersonic jets that are under- or overexpanded. This noise is dominated by certain resonance phenomena [26, 271 excited by the interaction of turbulence with shock waves [28-30]; the generation process

106

H.

S. RIBNER

is quite different from that for the normal shock-free jet. A simple brief account is given, e.g., in [8!. The following section gives a short summary of the major features of jet noise with a qualitative explanation. The casual reader may stop here. The remainder is divided into two parts. Part A is a greatly amplified account of the material given in the earlier resume. The emphasis is on the physical explanation and the mathematics is kept to a minimum. Moreover, analogies are used from time to time in place of genuine derivations. Part €3 is the serious mathematical development documenting the conclusions of Part A, with extensions. There is some overlapping in the expository material of the two parts, but an effort has been made to keep this within bounds. Thus the material is presented on three successive levels of elaboration. I t is hoped in this way to serve the interests of a wider range of readers. Perhaps, also, the mathematical development will be clearer if there is a qualitative account to fall back on for perspective. 11. RESUMEOF MAJOR FEATIJHES A turbulent jet flow involves inertial forces associated with the fluctuations of momentum flux. Equivalently, the forces arise from the movement of vorticity. These forces are effective in generating sound (Figs. L(b) and 2). Since the resultant force field must be zero, the fluctuating forces may be thought of as occurring as opposed pairs as in stresses. A fluid element subject to such a force pair suffers a fluctuating quadrupole deformation: this is a second mechanism of sound generation (Pigs. l(c) and 3). Associated with the inertial forces will be pressure gradients in the flow. A region of high pressure will be slightly compressed and conversel).. The transient compressions and expansions (dilatations) constitute a third mechanism of sound generation (Figs. l(a) and 4). Mathematically, all three are equivalent in the sound they produce when summed over the region disturbed by the flow. The sound power generated from a volume element d l ’ of turbulence of effective scale L , mean square velocity and dominant frequency f in low speed flow comes out to be

(2.1) where po is the ambient density and co is the ambient speed of sound. In an idealized model of jet flow we assume 3 4 U 2 , the square of the local U / L with the result mean velocity, and f

(‘2.2 )

dP

dVpoU8/Lcos.

107

THE GENERATION OF SOUND BY TURBULENT JETS

For the jet as a whole we may take dl.' total power as

P

(2.3)

-

4

U 3 ,L

r/.

D,U

4

Ui to obtain the

poUj8D2/~,5

where U j is the nozzle velocity, D the diameter. This is Lighthill's famous US law, which agrees well with experiment over a power range of a million to one. It holds for U, up to about twice the ambient speed of sound. With certain assumptions of flow similarity, taking one form in the jet mixing region (A) and another form in the developed jet (R), (8.2) leads for subsonic jets to the approximation (2.4)

dp

dy

*

Iyo )yP7

in mixing region in developed jet

for the distribution of sound power emission with distance y measured along the jet axis (Fig. 10). Furthermore, if the 'slice' of jet dy a t y emits as a simplifying assumption -. only a single frequency f(y) it follows under the assumed similarity that (2.5)

-4 dp

f2

df

/-2

low end of spectrum high end of spectrum

gives the slope of the frequency spectrum on either side of the peak (Fig. 10). A more detailed analysis indicates that the sell-izoise due to the turbulence is accompanied by shear-noise due to cross-coupling with the mean flow. The shear-noise spectrum is peaked an octave below the selfnoise spectrum and has a relative factor c0s4 0, where 0 is the angle of emission relative to the flow direction (y-axis). The low frequency shearnoise spectrum dominates at small values of 8 (e.g. 30") and the high frequency self-noise spectrum dominates near 90' where C O S ~8 0 (Fig. 17). Convection of the eddies by the mean flow (when subsonic) crowds the sound waves in the downstream direction (Fig. 11). This causes an effective Doppler shift of frequency in the ratio C-l and an associated amplification C'-* (Figs. 16 and 17) where

-

--+

(2.6)

C = [(l - M , cos 8)2 4-a2M,2]l/Z,

M, is (convection speed)/c,,, and

a is inversely as the lifetime of an eddy. A progressive leftward increase in the vertical shift C-4 is hypothesized, arising from a variation of M , along the jet. This would distort the spectrum so that the peak lies further toward the left, despite a Doppler shift C-l of the individual spectrum points toward the right (Fig. 17). An "anomalous" leftward shift of this kind is observed experimentally as well as the difference shown between spectra at 30" and 90". In all this the abscissa scale is f / U , 4

108

H . S. H I R N E K

to allow for the increase of frequencies in the turbulence with velocity. Thus the leftward shift of the peak implies the peak frequency increases more slowly than U j . Empirically fWak varies about like I/i1'2. The overall effect of convection, obtained e.g. by integration of the spectrum, is given by the directional amplification C-5. This modifies the basic 1 C O S ~0 directivity to beam the sound into a broad fan pointing downstream. Opposing this, the mean jet flow refracts the rays outward to give a pronounced valley of low intensity at the heart of the fan (Fig. 13). The resultant directional pattern peaks strongly a t an oblique angle ( e g , 40" for turbojets, 20" for cold air jets) to the fIow direction. For smaller angles the sound intensity falls sharply below the factor ( 1 c0s4 0)C--5. (See Fig. 14 for comparison with experiment.) The larger the extent of the jet (region of sensible velocity) compared with a typical wave length of sound the more refractive effect it should have. This is only qualitative because the finite jet has not proved amenable to analysis. Experimentally we do find an increase in the refraction valley - an outward rotation of the peak intensity - with increasing frequency. An outward refraction of the sound rays is also to be expected when the speed of sound in the jet is above ambient. Experimentally a large outward rotation of the peak intensity is found for a three-fold increase in the speed of sound. If the local jet density p differs from ambient the theory gives a multiplicative factor j 2 / p O 2 for the noise emission (2.1) from unit volume. Experimentally, the emission is closely proportional to pi2/po2 where pi is the density at the nozzle; this ratio differs more from unity than p2/po2 because pi is not diluted by mixing. We refer here to a jet of one gas issuing into another more or less isothermally. When the jet consists of heated air issuing into air the change in jet density or temperature appears to have no measurable effect on total noise power. The writer suggests that turbulent heat transport generates additional sound from entropy fluctuation, offsetting the reduction associated with reduced density. The low-speed derivation of a U; noise power law was based on an inaccurate model of jet flow. A reevaluation should employ rms turbulent velocity 4 U?'' approximately (from subsonic experiments) to yield close to lJj7. This must be multiplied by the convection factor C-5, averaged over direction. This average (Fig. 18) exhibits a slow rise to a moderate peak for convection speed U j / 2 near sonic. For a2 w 0.3 the product with Ui7 is not far from U; over most of the subsonic range. On proceeding to supersonic convection speed the mean convection factor decays like 1J,--5. ('I his reflects a change in the generation process wherein eddy Mach waves are dominant.) If the Uj-5 is multiplied by a hypothetical Ui8basic emission the result is a Ui3 law. Experimentally, the data for subsonic convection

+

+

109

THE GENERATION OF SOUND BY TURBULENT JETS

speeds (model jets, turbojets) very accurately follow a U; law, and there is a transition at supersonic speeds (afterburning jets, rockets) to something approximating a U j 3 law for the limited region of the data (Fig. 19). Division of the two-slope sound power law U? - U j 3 by the kinetic power of the jet Ui3 gives a two-slope efficiency U: and constant. Thus the steep rise for subsonic jets levels off to a constant limiting efficiency (of 0.3 to 0.80;,) for rockets. The experimental rocket data do not extend beyond UiM Sc, (the effective eddy speeds are half this) and the use of the convected quadrupole (or dilatation) deductions for higher speeds is purely speculative, as has been indicated. The approach of 0. M. Phillips - an asymptotic theory for high values of Ui/co- predicts that the efficiency must widely used diminish like U , - 3 / 2 . Substantial reduction in jet noise can be accomplished, for a given thrust, by the reduction in velocity associated with a larger jet diameter according to the U; law. This is exploited in the increasingly popular bypass or turbofan engines. In the pioneer Rolls Royce Conway the noise reduction is 7 dh in one case cited. Corrugated and multi-tube nozzles are the most widely used means for quieting existing turbojets. Their development has been motivated by conflicting interpretations of the theory, and the explanation of their behavior remains a matter of controversy and speculation. It is generally agreed that reduction in shear plays an important role, presumably reducing overall turbulence levels : these jets entrain external air into a restricted region, giving it some forward velocity. Another notion is that the sound from one small jet (or corrugation) is reflected and refracted (‘shielded’) by the temperature and velocity field of nearby jets so that the aggregate sound has a less peaky directional distribution. A group headed by R. Lee of General Electric has had some success with a semi-empirical computerized method for predicting muffler behavior (and flow development of interfering jets) in some detail. Their results suggest that the shielding principle cited above may play only a minor role. 4

A. SIMPLIFIED PHYSICAL ACCOUNT 111. PHYSICS OF SOUND GENERATION 1. Simple Source or Monopole

A balloon caused to pulsate by an oscillatory air supply is often cited as a model for a simple source of sound. The sound waves arise from the radial acceleration. [Ye illustrate this by means of a sphere with signs to indicate radial acceleration (Fig. l(a)). An outward acceleration of fluid

+

110

H. S. RIRNEH

+

with the balloon rempved will have the same effect. Thus the signs have a dual interpretation as radial acceleration (of the balloon) and as a time rate of injection of sources of matter. The sound pressure radiated from the balloon is proportional to the product of surface area times radial acceleration and density; this is a measure of the total strength of +Is. The value at a given distance is proportional to p u s / , where p is the fluid density, u the rms velocity, S the surface area and f the frequency. The corresponding radiated sound power is

P,

(3.1)

4

(puS)2/2/pr

(a) Pulsating sphere as model of simple xxlrce of sound

Oscillating rigid sphere

Oscillating force

Dipole

Sources ond sinks wl sphere

(b) Alternate models of dipole source of sound

c

Rigid spheres Force pair (stress)

Dipole pair

Deforming sphere

(c) Alternate models of oblique quodrupole source of source

FIG. 1 . Elementary sources of sound.

where c is the speed of sound. Since p u s is also the rate Q a t which mass is supplied, an alternative form is (3.2)

P,

-

Q2pyPc

(aQ/at)2/pc.

Here % / a t is to be regarded as the strength of a single acoustic source whose magnitude is the total of the +'s. This mav be recast in terms of mass flow rate m per unit volume as

(3.3)

P,

-

rn21d6/2/pc

-

(am/a1)21a6/pc

with the volume taken as - L 3 , the diameter being L .

111

THE GENERATION OF SOUND BY TURBULENT JETS

Now suppose we compare the sound power radiated from dynamically similar systems of such simple sources. We may for example specify that each system be characterized by a typical velocity U , dimension D, and frequency f U / D (common in fluid flows), and take u [J, S D2. Then bv (3.1) [el]

-

- -

(3.4)

This U4 law is characteristic of the sound power from systems of simple sources with the type of similarity defined above. But this is not the only similarity possible. If the strength integral m in (3.3) is taken proportional to, say, pU2 and is also a time derivative (giving a factor f in sound pressure is replaced by the later and f 2 in power) then a Ua law results. In fact (3.4) equation (3.10) applicable to quadrupoles having the first-mentioned similarity. This is the case corresponding to the dilatation theory of flow noise (see Sec. IV.3). 2. Force Source or Dipole

The model for a dipole source of sound is a rigid sphere that oscillates back and forth along a line ( e g , a pendulum bob). The outward or normal acceleration of the forward and rearward hemispheres is now opposite. A t a given instant we may put +Is on the face with positive normal acceleration and -'s on the face with negative normal acceleration. The spherical sound waves that are produced are likewise 180" out of phase in front and rear. The intensity pattern is a figure of eight. and are equal. This would seem to imply a The sum total of mutual cancellation : the outflow from the sources tends to circulate around the sphere to be ingested by the sinks. However, occurrences at one face will be transmitted at the speed of sound to the other, i.e., at a time L/c later, where L is the diameter. The transmitted pressure will fall short of cancelling the local pressure disturbance by a fraction given by the phase shift f L / c ; the transmitted velocity will behave similarly. Hence the acoustic power acquires a factor (fL/c)* compared with that for the sources or sinks separately, --Is

+Is

4

P,

(3.5)

-

-

(puS)2L2f4/pc-

where u is taken as the velocity of the center of the sphere !21j. Another approach [31, 251 is to consider the force F required to oscillate the rigid sphere, considering its density to be that of air. For L sufficiently small (2nL << wave length) the reaction of the surrounding air can be taken into account by increasing the mass of the sphere by a factor 1.5; this is well known. On equating the force to the mass acceleration, (3.6)

F

= 1.5p

(volume) (auli3t)

pSLfu.

112

H. S. RIBNER

Thus (3.5) may be rewritten (3.7)

Here the force F replaces the dipole strength 4 p u S L f in (3.5). We may compare the sound power from systems of dipole generators characterized by a typical velocity U , dimension D,and frequency f U/D. With the same further similarity assumed before, u U, L D , and (3.5) gives 4

4

P2

(34

4

4

pUSD2/c34 pU3D2M3.

This result may be applied to Aeolian tones: these are the sounds generated by a cylinder in an airstream when there is an oscillating lift associated with an eddying wake. The cylinder (unlike our assumed sphere) may be perfectly stationary, the force being a reaction. Curle's analysis [32] shows that (3.7) and (3.8) are still applicable to the radiated sound when supplemented by a term corresponding to quadrupole generators. The quadrupole power has, however, a relative strength's Mach number squared; it tends to be much smaller a t the subsonic speeds of the reported measurements. These measurements lie very close to the U S law of (3.8) (e.g., [96-991). 3. Stress Source or Quadrupole

The force source or dipole is a vector (Fig. l(b)). If a pair of opposed forces are displaced as in Fig. 1 (c) the result is a general type of quadrupole. An equivalent effect (at large distances) is produced by the parallel oscillations of two spheres with opposite phase. Still another model is a sphere deforming without change of volume ; this approximates best the behavior of a fluid element. All these cases are shown in the figure. The quadrupole is characterized by two axes: that of the force vector and that of the separation vector. If the axes are the same (e.g., coaxial forces), the quadrupole is longitudinal, if they are perpendicular, the quadrupole is lateral. A more general quadrupole has oblique axes, but it can be made up as a superposition of the other two kinds. A sphere being constricted all around a t the equator and bulging a t the poles is a model of a longitudinal quadrupole. A sphere being deformed by a shearing motion is a model of a lateral quadrupole. The opposed forces or dipoles will not cancel in their effects. The disturbance transmitted from one to the other (from one side to the other of the deforming sphere of diameter L ) will again suffer a phase shift fL/c for both pressure and velocity, yielding a factor ( / L / C for ) ~ power. Then from (3.7) 4

4

P,

F L ~ / L2(a2F/at2)2/pCti. ~ C ~

I13

THE GENERATION OF SOUND BY TURBULENT JETS

The opposed force-pair F (Fig. l(c)) separated by a diameter L A L creates a mean stress in the sphere cn FIL2. Thus (3.9)

In Lighthill’s quadruple approach to the generation of flow noise [ 1-31 the stress in (3.9) is taken as the fluctuating Reynolds stress or momentum flux of the unsteady flow; this point will be amplified later. For the present we note that for flow systems characterized by a typical velocity U , dimension D, and frequency f UID the Reynolds stress * pU2. Correspondingly the basic generators are taken in effect as deforming spheres or quadrupoles with u U , L D. With this type of flow similarity (3.9) gives

-

(3.10)

~1

P3

4

-

pUSD2/c5 pU9D2M5.

This resuit contains the famous U8D2 law that has been widely confirmed by experiment for the noise of subsonic jets. The efficiency of sound generation may be defined as the ratio of the acoustic power to the mechanical power or flux of kinetic energy pU3D2 in the flow. These conversion efficiencies for systems of sources, dipoles, or quadrupoles are proportional respectively to M ,M3, and M 5 (see (3.4), (3.8), (3.10))for systems with the assumed similarity. It is evident that for M < 1 the sequence of decreasing efficiency is source : dipole : quadrupole for equal efficiencies at M = 1. The decreasing efficiency results from the ‘Stokes’ effect : the partial cancellation of the output of the two sources of the dipole, and, carrying the process a step further, of the two dipoles of the quadrupole [ 1, 31. In more physical terms, the fluid pushed out in front of the oscillating sphere ‘dipole‘ is sucked in again at the rear, thus greatlv reducing the tendency to create compression waves or sound. The effect is further compounded in the deforming sphere ‘quadrupole’, with a further reduction in efficiency.

-

4. Equivalent Patterns of Simple Sources, Dipoles, and Qiiadrupoles

It is always possible to replace a distribution of dipoles by an equivalent distribution of simple sources. In this case the total strength (volume integral) of the sources is zero. Similarly a quadrupole distribution may always be replaced by an equivalent dipoIe distribution, and this in turn by an equivalent source distribution. In this case the total strength of either dipoles or sources is zero.* The phase cancellation effects discussed Source strength = - divergence (dipole strength). Dipole strength = - divergence (quadrupole strength). If everything vanishes outside a certain region, the total strength of sources and dipoles so defined must vanish by the divergence theorem. An illuminating discusion is given in [23] (p. 59).

114

H. S. RIBNEH

earlier for the individual sources and sinks making up a dipole or quadrupole will likewise apply to some degree in a spatial source distribution whose total strength is zero. Lighthill has therefore made the point [l-31 that the emission from such a source distribution has the character of quadrupole radiation. However, this characterization is oversimplified and some further discussion is in order. We can imagine a continuous random distribution of sources and sinks in two dimensions (for convenience) as a random system of hills and hollows: the hills represent sources and the hollows, sinks, and the height or depth the local strength. As the pattern fluctuates in time, the hills and hollows change in an erratic way; however, they have an average size, or radius. ,. 1his ‘source radius’ is a measuret of the average effective size of coherently radiating source or sink patches. Only regions of sources whose extent is small compared with a typical wave length of sound can be regarded as emitting simultaneously: for larger regions sources more distant from an observer must be counted earlier because of the travel time. If the ‘source radius’ is very small compared with a sound wavelengthlh, then such regions each contain a number of sourcesink patches that may be treated as emitting simultaneously. If the and --Is essentially cancel in these regions, the emission figured in this way will be essentially zero. Then if one wants to retain the simultaneous emission approximation, a better approach is to replace the source-sink pattern by the equivalent dipole pattern. If this, too, virtually vanishes the replacement is by the equivalent quadrupole pattern. + I s

l h i s last is the state of affairs implied in Lighthill’s analysis of flow noise in terms of quadrupoles [l-31; it follows in part from the vanishing of the volume integral of the equivalent sources. A necessary second assumption is ‘source radius’ very much less than a sound wavelength/Pz. In a jet flow a t the higher subsonic speeds this criterion is approximated but not fully met. If the ‘source radius’ becomes comparable with or greater than the sound wavelength then the time delays will affect what an observer hears: the output of a group of + - source patches will no longer cancel for him. In this last case there is little effective cancellation of the sound from the sources and sinks despite a zero net strength. In fact, the essential ‘quadrupole’ character attributed by Lighthill to such a source distribution is no longer in evidence: the distribution behaves like a simple source/sink distribution of nonzero net strength. (Quantitatively, this situation occurs t The more standard ‘correlation radius’ is defined in terms of a volume integral of the two-point correlation of source strength; this will vanish when the integrated simple source strength is zero. In this case the correlation function has balancing positive and negative regions. .4 non-zero ‘source radius’ may be defined as the (directional) average radius at which the correlation function first passes through zero.

THE GENERATION OF SOUND BY TURBULENT JETS

115

when cotL/covG (which is fi source radiuslsound wave length) is of order unity or more in the factor C of (3.18); M, in M,cos 8 is set equal to zero to suppress the convection effect.)

IV. EQUIVALENT AERODYNAMIC GENERATORS OF SOUND 1. Reynolds Stmsseg ((juadrupole Sources)

In a turbulent flow the velocity u and hence the momentum density pu varies from point to point. One would imagine that these differences in momentum would tend to distort the fluid elements. In first approximation the volume of an element would be unchanged. Thus we seem to have arrived at the defotming sphere or quadrupole noise generator of Fig. I(c). To be more precise, it is the rate of momentum flow piio (per unit volume) that is effective: its gradients give rise to inertial forces that supply the required acceleration. The inertial force on a fluid element is given by the excess of the rate of momentum flowing in over that flowing out. (Mathematicallv, this excess (per unit volume) is the negative divergence of the

FIG. 2. Random configuration of inertial forces associated with a hypothetical turbulent flow. Arrows denote resultant inertial force in each box.

rate of momentum flow.) These forces can be regarded as virtual or teal depending on the point of view: they have as much reality as centrifugal forces in other areas of mechanics. The unsteady inertial forces add up to zero when summed over the entire region of a free flow or a jet: there can be no net outflow of fluctuating momentum flux without unsteady applied forces. (The constant thrust of a jet is balanced by a constant component of momentum flux.) The random configuration of inertial forces might look something like Fig. 2. The resultant force in each small cube is indicated by an arrow: the force per unit volume is actually distributed continuously. A fully equivalent distribution may be made up of pairs of opposed forces which characterize a stress field (cf. Fig. 1). Such an arrangement

116

H . S. RIBNER

seems to ensure the vanishing of the resultant force. The magnitude of the local stress is the rate of momentum flow pnu from which the inertial forces are derived. This is closely related to the Reynolds stress in the theory of the turbulent boundary layer. K e shall retain the name 'Reynolds stress' for puu but observe that here it is an instantaneous stress and moreover it now includes the contribution of the mean flow. The Reynolds stress may also be written pupj (i = 1, 2, or 3: i = 1, 2, or 3) to show that it has nine components: e.g., the component pula2 represents the force in the u1 direction acting on unit area normal to the ug direction. This is a shear stress and corresponds to a lateral quadruple. A term like puI2 is a longitudinal stress and corresponds to a longtudinal quadrupole. These are illustrated in Fig. 3 [l-31.

is epuivolent to

-+ FIG.

+-

is equivalent to

+-

+

3. Longitudinal and lateral quadrupole sound sources associated with the put' and pu,ul Reynolds stresses, respectively (after Lighthi11 r3:).

The Reynolds stresses p i u j are in a sense artificial compared with the inertial forces (divergence of p u p i ) . Suppose we imagine the sphere of Fig. l ( b ) to consist of air forced to oscillate rigidly by combined body and surface forces. The interior of the sphere is quite stress free - and neither does it generate any sound. Yet the interior appears on the quadrupole theory to be generating sound like a stress distribution pu12,which is not zero. This 'virtual sound' when added to the sound attributed to the external quadrupoles gives the correct total radiation from the sphere. The inertial forces, on the other hand, arise solely outside the sphere where the sound is actually generated. (The primary sound field generated by the postulated driving forces must be added to the aerodynamic sound field of this paragraph to obtain the total radiated sound.) 2. Vortex Stretching (Dipole Sources)

I t is well known '311 that a vortex loop may be replaced by an equivalent layer of dipoles. The loop constitutes the perimeter of the area covered by the dipoles. If the vortex stretches the integrated dipole strength changes with time (area of loop changes). It takes force to do this. This brings us back to the force-source or dipole as a generator of sound. The new feature is the notion, due to Powell in [33], that vortex stretching may be an important source of sound (cf. also i34J).

THE GENERATION OF SOUND BY TURBULENT JETS

117

He cites as an example the Aeolian tones produced by a rod or wire in a wind. The sound may be regarded as due to the oscillatory lift associated with the periodic shedding of vortices. Alternatively, one may consider loops formed by ‘image’ vortices bound in the cylinder and connected by trailing vortices to convected cross-stream vortices in the wake. Each loop grows and new ones are being formed in succession, with alternating sign. Powell explains the sound generation in terms of the stretching vortex loops. In the case of a free flow like that of a jet there can be no net fluctuating force (the constant thrust is balanced by the mean momentum flus). Thus the local forces associated with vortex stretching may be regarded as matching up in opposed pairs : these constitute stress’ quadrupoles of the kind shown in Fig. l(c). Powell illustrates these and other features mathematically. He ultimately proves an equivalence of his vortex formulation with a form of Lighthill’s quadrupole integral. The approach has not been developed further for the treatment of jet noise. 3. Fluid Dilatations (Simple Sources)

The fluid accelerations in a turbulent flow will be balanced primarily by pressure gradients. (Stated another way, the inertial forces described in Section 117.1 will give rise to pressure gradients.) Thus the local static pressure will varv from point to point. Since the fluid is compressible the

0 dilatations in an eddying flow. The local pressure field (essentially ‘pseudosound’) has t h e opposite sign.

FIG. 4. Positive @ and negative

local density must follow the pressure fluctuations. The fluctuations in density are associated with inverse fluctuations in the volume of a fluid element. The volume pulsations or dilatations (Fig. 4) simulate the pulsating balloon described earlier and are effective in generating sound [9-111. We are concerned here with a turbulent flow of limited extent imbedded in fluid at rest. For acoustic purposes we replace the turbulent flow by

118

H. S. RIBNEK

a n equivalent acoustic medium at rest containing the appropriate dilatations. For such a medium (4.1)

-

5, dilatation rate =

-

ap/at

where ji is the mean value of the local density p . On a certain understanding (see below), either side of this equation has the same local sound-generating effect as a fluctuating source of matter injected a t a rate m rn - +/at. But it is the time rate of m that constitutes the effective acoustic source strength (cf. (3.3)): this is - a2pjat2. We must exclude that part* of the density fluctuation associated with propagating the sound waves; this does not contribute to the effective W E and is ineffectual in generating sound. If the remainder is called p(O) the result is [35, 36, !4-11, 92, 931 (4.2)

effective source strength,

am

at

-

-

azp(o) at2

I azp(o) c2

at2

where p(") is the pressure perturbation associated with p(O). This equation relates the acoustic source strength to the pseudosound pressure [37] that produces the effective dilatations. The pseudosound

FIG. 5. Relative strengths of pseudosound

p(0)

and sound field

p(I)

vs. distance froni

flow.

field dominates within and near the turbulence a t subsonic speeds (Fig. 5 ) , constituting what is known as the acoustic near field. Further out it is overriden by the acoustic radiation field $ ( I ) , which decays more slowly with distance. The p(") field has virtually the characteristics of the pressure field in an incompressible flow, being dominated by inertial rather than compressional effects; in particular, it exhibits no wave propagation : hence the name 'pseudosound'. Relatively small at subsonic flow speeds

119

THE GENERATION OF SOUND B Y TURBULENT J E T S

There is no conflict between the dilatation picture and the Keynolds stress or quadrupole picture of flow noise generation. The pseudosound pressure field that produces the dilatations is actually the acoustic near field r38, 3, 111 of the quadrupoles. The dilatations can be regarded as an alternative intermediate step in the sequence quadrupoles far field (radiation field). In the quadrupole theory of Lighthill 11-31 this step is bypassed. Hy way of summary, the effective volume of a fluid element in an unsteady flow fluctuates inversely as the local pressure: part of this dilatation generates sound and another part propagates the sound. The local pressure consists of a quasi-incompressible ‘pseudosound’ $CO) plus the sound field $(I) (relatively weak in subsonic flows). The effective part of the dilatations appears in a virtual acoustic source term @$o)/iV.

-.

-

V. SOUNDRADIATEDFROM

A JET

1. Stucture of a Round Turbulent Jet

Lilley [4] has organized the available experimental data on the round turbulent jet (especially that of Laurence [39]) into a form suited to the estimation of jet noise. More recently the data have been extended by Davies, Rarratt and Fisher [40]. \Ve shall summarize briefly the relevant material, following Lilley’s plan in the main.

“f

Self -similar profiles

I

I I I

I I I Mixing I

I I I 1

I

Nozzle

Y region

Transition region

I

Mean

1

Fully deve loped jet

FIG. 6. Idealized model of a round turbulent jet. In a real jet similarity of turbulence profiles is not attained in H until a t least 50 diameters from nozzle.

Figure 6 shows the development of a round initially laminar jet issuing at subsonic speed from a nozzle of diameter D. The annular mixing region becomes turbulent almost immediately downstream of the nozzle unless great care is taken (e.g., [41, 421) to suppress upstream turbulence and other disturbances. The turbulent mixing region spreads linearly outward and inward, progressively encroaching on the inner laminar flow in the form of a conical boundary about 4.5 diameters long: hence the name ‘potential cone’ or ‘potential core’.

120

H. S . RIBNER

The radial profiles of some major mean and turbulent quantities are approximately self-similar in the mixing region (A), changing only in scale. For example (Fig. 7(a)), the profile of turbulent intensity, normalized to unity at the peak, fits a universal function of = ( I - R ) / y a t all axial stations ( R is the diameter).* The profile of the mean velocity (7 (not shown) is likewise a single function of 5 [43],

(b)

(0)

FIG. 7. Quasi-self-similar profiles of turbulence intensity in a real jet. S o t e difference of ordinates of (a) and (b). (Curves (a) after Davies, Barratt, and Fisher [40!: curves (b) replotted from Laurence r391.)

-!

UIU,= 1/A/2n exp (- A(s - To)2/2)ds where A M 300 and C0 M 0.03 according to Lilley 141. The integral longitudinal scale of turbulence L, and the typical radian fluctuation frequency’ oq do not vary much across the mising region and are given roughly by [39, 401.

L,

(5.2)

(5.3) WID -= 0.71,0.46,0.37,0.33

ui

0
M 0.13 .y

for

< 611

’

- - 1.5, 2.0,3.0,4.5 resp. D

(The relative constancy in the radial direction is supported by unpublished measurements of W. T. Chu at the University of Toronto, kindly supplied

+

I n Part B t h e notation is changed so t h a t y + y1 and r 4 y, R in mixing region and + y, in developed jet. t Proportional to the reciprocal time scale of the space-time correlation when transformed to a reference frame moving with the convection speed. Such data appear in j40’.

THE GENERATION OF SOUND BY TURBULENT JETS

121

to the writer. Laurence [39], on the other hand, reported a pronounced variation for Ll.) The transverse scale L, is about one third of L,. The mean velocity profile undergoes a gradual transition from the form (5.1) characterizing the mixing region to the form (5.4)

U/Uj= 6.39/[(1 + 52/0.0157)2y/D];

[=r/y

characterizing the fully developed jet. The transition region extends from about 4.6 to 8 diameters. Here 5 has been redefined as r l y , r is the local

!:;$&Dovies 0

et all

M=0.3 (Laurence)

0.05 I

YdD

I

radius and the origin of y is taken 1.0 diameter upstream of the jet exit. Figure 7(b) shows that the turbulent intensity profiles are not fully selfsimilar in terms of r/y as early as 8 diameters; in fact, Townsend [43]

I

-0.1

I

0 Radial distance

0.I

0.2

Y2-R Yl

FIG.9. Profiles of mean-flow speed (dotted) and convection speed (solid) across mixing region (reproduced from Davies, Barratt, and Fisher [40]).

122

H. S. RIBNER

remarks that similarity is not attained up to at least 50 diameters from the exit. The data on turbulent scales in the developed jet are in conflict. Townsend gives L, m 0.065 y, whereas Laurence finds L, diminishing for y > 8 diameters Even approximate determinations are very difficult a t these large distances : the instrumental low-frequency cutoff introduces an error that becomes increasingly hard t o correct for. As for the typical fluctuation frequency (JJ, we have only a single measurement a t y / D = 8, r / D = 0.3 from the unpublished work of W. T. Chu, showing a reduction of 25% compared with the value a t yID = 2.3, r / D = 1. The data on turbulent intensity is completed by Fig. 8 which gives the intensity at the peaks of the profiles of Fig. 7, and connects up the three regions of the jet. The space-time correlation measurements of [401 show that the effective convection speed of the turbulence pattern varies considerably less across the mixing region than does the speed of the local flow (Fig. 9). The two speeds match a t the 0.5 Ui locus. 2 . Sound Radiated from

cl

Volume Eiemeiit

Three alternative mechanisms of aerodynamic sound generation have been discussed in Section IV: they all lead to (3.10) for the comparative sound power emitted by dynamically similar flows. (This has been shown explicitly for the dilatation and Reynolds stress approaches.) For the sound power from unit volume it will be convenient to return to (3.3),which refers to the power PI from a single simple source corresponding to the dilatation mechanism (Fig. l(a)). The total sound power d P from the N effective sources in a volume dV of turbulence is N times P , or (5.5)

dP

4

dV(am/at)PL3/poco

where N equals the kolume ratio dV/L3 and we are being a little more precise now, designating p and c as the ambient values po and c,,. In this equation L3 may be approsimated as (average the effective volume of the source scale)3 of the turbulence - in crude terms the volume of an eddy. A more precise measure is the ‘correlation volume’ defined in Part B. In the dilatation theory the effective source strength (in terms of a fictitious rate of mass injection m ) is given by 4

(4.2)

amiat =

-

co-Wp(o)iat2

where pcO)is essentially the local pressure fluctuation in the turbulence. Suppose the ms turbulent velocity in dV is 2 and a typical frequency is f : then it can be shown that p ( O ) p o d and azp(o)/at2 f 2 p O d so that (5.6)

dP

-

CT

d V po(U2)zf4L3/~06.

4

123

THE GENERATION OF SOUND BY TURBULENT JETS

(When the jet is hot or consists of a foreign gas, po in p(O)+ poGmust be replaced by the local (time average) value p ; this leads to p2/po in place of po in (5.6). The speed of sound, here c,, must also be reexamined. Cf. Sec. V.8.) This same equation is readily derived from (3.9) based on the Reynolds stress or quadrupole model of flow noise. It can also be obtained from the vortex model, a qualitative argument being that leading from (3.7) to (3.9). 3. The U;, yo, y-’, fa and

Laws

In the mixing region (A) of a jet (cf. Sec. V . l ) the profiles of mean and turbulent quantities are self-similar or invariant with axial distance y when expressed nondimensionally : along certain rays the turbulent velocity and scales and the mean velocity maintain a fixed proportionality. A second region (B) of self-similar mean velocity profiles starts about 8 diameters downstream of the nozzIe defining the fully developed jet (Fig. 6). Full self-similarity of the profiles of turbulent velocity and scale is not attained until some 50 or more diameters downstream [43](Fig. 7). In our analysis we shall deal instead with an idealized model of jet flow in which full selfsimilarity of the mean and turbulence profiles throughout region B is postulated, starting at 8 diameters. The differences are to be borne in mind in applying the results to a real jet. According to the assumed behavior the ratio G/U2is the same for ‘similar’ elements d V . We assume a further similarity in the scaling of frequency,* f cc U I L ; this is roughly approximated by the limited experimental data covering region (A) (Sec. V . l ) . With these relations (5.6) becomes

dP

(5.71

4

dVpOU8/Lco5

for low speed flows. This equation may be applied to obtain the sound power emission from successive slices of a jet taken normal to the axis; in this case U and L are taken as typical for the slice. The effective region of the slice occupied by the peak turbulence is d V . The assumed similarity for the mixing region (A) gives dl; + yDdy (annulus), L + y, lJ M U j / 2 = nozzle velocity/2. From (5.7) then (5.8)

-

yo Law:

dP dY

poUjaD - constant,

CC--

co5

mixing region (A)

The assumed similarity for the developed jet (B) is dl,‘ L y, U,+ U j D / y . Insertion into (5.7) gives

~1

y2dy (disc),

We refer here to the typical frequency o ~ / / 2 of n turbulent fluctuations as seen by an observer convected along with the turbulence. Measurements of this kind are quite recent [40].

124

(5.9)

H. S. RIBNER

y-’Law:

(;)-’

dp po;;? - - dY

-

developed

’

jet (B)

These results [8,4,6] are exhibited in a curve* of (sound power emission)/(unit length) vs. distance along the jet (Fig. 10). Slices of jet within, say, four diameters of the nozzle are predicted to emit the same sound power ( y oLaw); and beyond eight diameters the emission falls off very abruptly (y-7 Law).

5 Y

t

Y

FIG.10. Determination of jet-noise spectrum d P / d / from noise emission d P / d y from successive slices of jet. Each slice is assumed to emit a characteristic frequency f ( y ) .

The area under the curve is the total sound power emitted by the jet. By (5.8) and (5.9) this must be of the form (5.10)

Ui8Law:

P -poU?Da/cos

( U , = nozzle velocity)

in agreement with (3.10) obtained by less detailed considerations. The righthand side of (5.10) with De replaced by the nozzle cross-section area -4 (to allow for noncircular nozzles) has been termed the Lighthill parameter by a number of writers. For simplicity we shall imagine that a given slice of jet emits just a single characteristic frequency : the actual rather peaked spectrum of the slice (cf. Part B) is considered squeezed into a single line. Then the spectrum emitted by the jet as a whole can be approximated as a continuous array of these lines side by side by writing [6, 211 (5.11) This is not a curve of quadrupole strength distribution along the jet as might be inferred from remarks in [3]. The (quadrupole strength)’ differs by a factor c (frequency)-‘ which varies strongly with y.

THE GENERATION OF SOUND BY TURBULENT JETS

125

-

since by hypothesis f = f ( y ) . The similarity assumption f U / L becomes U j / y in the mixing re'gion (A) and f * U j D / y 2in the developed jet (B) by virtue of the relations preceding (5.8) and (5.9). Therefore

f

f-2

(5.12)

f2Law:

(5.13)

d P *PoUi0D f-2,

Law:

~

~

Y

dP df

Cob

--

poUjbD6 Cob

f2*

mixing region (A); developed jet (R)

after some simplification [el. These results and their derivation are illustrated in Fig. 10. Equations (5.12) and (5.13) may be combined into (5.14)

in terms of a reference frequency f* rn U J D ; here H(f/f*) is the bell-shaped spectrum shown in Fig. 10. This is not yet the final spectrum of the overall jet noise, although it exhibits the proper general shape. The results of this section (perhaps excepting the yo and f-' laws) constitute approximations limited to Ui below about twice the speed of sound outside the jet. This follows from an implicit neglect of eddy convection effects (see Sec. V. 5), together with the use of low-speed jet mixing data. 4. Self-Noise and Shear-Noise

The foregoing rather oversimplified development is refined in Part B. The more detailed treatment exhibits two distinct components of noise with separate spectra. One is due to the turbulence alone and may be called self-noise [4]. The other arises from a cross-coupling of the turbulence with the mean-flow shear and may be called shear-noise. The peak amplitudes of the two noises appear to be roughly equal ; hence, we prefer to avoid the term 'shear-amplified noise' of [4]. Briefly, the two terms arise in writing the flow velocity in direction x as u, U,, the sum of turbulence plus a mean flow. Then, on squaring to obtain momentum flux as used in one form of the quadrupole theory, one obtains uX2 2U,u, as effective noise generators. The first term yields the self-noise and the second term the shear-noise. It turns out that the magnitude of the integral of the latter term over the flow depends primarily on the gradient of the mean flow ; hence the name shear noise. Here x refers to an arbitrary direction of emission which makes some angle 6 with the mean flow U. The initial shear-noise term 2U,u, thus

+

+

126

H. S. RIBNER

contains a factor cos 0. It turns out that this leads to a factor cos48 for the shear-noise emission. The basic directional factor for the overall noise may therefore be written as 4

(5.16)

1

+ COS~O

since the relative coefficient of cos40 appears to be near unity. If the shear noise in a slice of jet emits a frequency f , then the self-noise must emit a frequency 2f because of the squaring of u,. This implies that the single spectrum ~h H ( f / f * ) of the last Section must be replaced by two spectra peaked an octave apart (5.16)

where the factor of 2 equalizes the integral over frequency of the two H-functions. This result does not include the very important effects of convection and refraction which are introduced in the next Section. 5. Effects o/ Convection and Refraction

We shall now consider how the foregoing results are modified, first by convection of the eddies, and next by refraction due to the “wind” - the mean jet flow. Hotwire studies employing correlation techniques [40] show that, roughly speaking, an eddy in a jet is convected about three times its length before it has decayed, i.e., before it has lost its identity due to fluctuation. Thus we have the picture of the moving eddy emitting sound pulses during a short lifetime. This is illustrated qualitatively in Fig. 11. For subsonic convection speeds the sound waves are crowded in the downstream direction and more widely spaced in the upstream direction. This powerfully enhances the intensity in directions making an acute angle with the flow. The polar plot above the sketch shows the variation in intensity (at a fixed radial distance) with emission angle 0 , for a continuous succession of such wave trains. For supersonic convection speeds (Mc> 1) the effects are more extreme in that the sound waves can coalesce t o form a concentrated envelope or annular segment of a Mach cone. The sound intensity shows a strong peak normal to this cone; the peak direction is defined by M , cos 8 = 1 from aerodynamic considerations. The wave pattern grows even after the eddy or sound source has died, enveloping both upstream and downstream points. Thus there is no zone of silence as in steady supersonic flow. (The difference lies in our use of a

THE GENERATION OF SOUND BY TURBULENT JETS

127

stationary frame of reference that does not follow the moving source.) Further, the finite lifetime of the source accounts for the truncation of the Mach cone In a jet the successive passage of eddies will provide a succession of wave trains leading to a distribution of such truncated Mach cones, one within the other, expanding outward with time. A given point will receive a continuous fluctuating sound pressure instead of a short burst of pulses. Further, the randomness in time - but not the randomness in space - will tend to smear out or otherwise impair the sharpness of the envelope at M , cos 6 = 1. This will moderate an otherwise infinite intensity peak which would occur for moving eddies emitting a single frequency (or even zero frequency (d.c.)) [ll].

‘--

Subsonic

Peak sound pressure

Supersonic

FIG. 1 1 . An eddy moving from left to right emits a series of sound pulses during its short lifetime. Upper curves show rms pressure variation provided by crowding of sound waves, given by factor C 5 I 2 = [ ( l - M c cos O)* aaM,a]-5/4 where speed is Mcco and eddy length/decay distance is a.

+

In the language of aerodynamics the sound wares associated with the intensity peak a t M , cos 0 = 1 are Mach waves - in this case fluctuating Mach waves. Their generation is . . .somewhat analogous to their formation by thin bodies moving supersonically” [ 5 ] . This ballistic analogy is apparent in Fig. 11. Lighthill [I] obtained an approximation to the directivity of Fig. 11 in the form of a convection factor “

(5.17)

(1 - M , cos 6 )-5

that multiplies the sound intensity obtained in the absence of convection. (The exponent was corrected from - 6 to - 5 by Williams [lZ].) The

128

H . S. RIBNEH

analysis in effect postulated very large decay times for the eddies (or frequencies approaching zero). An allowance for finite decay time via a characteristic radian frequency w1 and scale L was made in the work of Ribner [9-111 and Williams [13], giving finally*

c-6 = [ (1 - M , cos e ) 2 + o

+

~ ~ L ~ / x= c ~[ ( I~ ] -M, ~ cos / ~€q2 O L ~ M , ~ ] - ~ / ~

(5.18)

in place of (5.17) (equating the two forms serves to define a, with M , = Uc/co). The derivation according to the method of [ l l ] is given in Part B herein. This is supplemented by a more detailed physical argument than that given for Fig. 11. The early work of [9] and [lo] dealt with “simple” sources and yielded C-1 in effect as the convection factor for intensity (cf. [ l l ] ) . The phase-shift

FIG. IS. Refraction and reflection of sound rays by a semi-infinite plane flow. Plane waves are assumed, the rays being normal to the wave planes.

cancellation effects discussed in Sec. IV.3 for the constituent sources and sinks of stationary quadruples show up again in the convection process; these give an additional factor C-2 in sound pressure or C-* in (pressure)2*intensity, so that the overall factor is C-s. The double time derivative in the equivalent dilatation source behaves similarly, likewise providing the additional C-4 factor. It is well known [44]from theory and experiment that sound rays are refracted by velocity gradients - that is, by wind. For the case of plane waves in a semi-infinite plane flow the angle of refraction for sound leaving the jet (Fig. 12) and the refracted intensity can be accurately predicted [45, 461. There exists a minimum refraction angle given by (5.19)

+

sec 8 = (Uj/cj) (co/cj)

-.

which corresponds to grazing incidence (0, 0 ) . This value defines a ‘wedge of silence’, since no sound rays can leave the jet at smaller angles. (See also Sec. VIII.) This embodies also a correction of the basic sound intensity (that for no convection, = 0) for neglect of retarded time differences in the source region (or, equivalently, neglect of w~*L*/nc,* compared with unity). Cf. remarks in Sec. 111.4.

M, cos 0

THE GENERATION OF SOUND BY TURBULENT JETS

129

In practice, the region of sensible velocity in the jet is of only limited spatial extent. This is difficult to allow for analytically. Oualitatively it must have the effect of weakening the refraction. The wedge of silence is presumably reduced to a valley - rather deep as judged from experiment in the polar directivity plot [15, 9-11]. Figure 13 shows such an anticipated effect of velocity refraction on the directivity. An outward refraction of the sound rays is also to be expected from theory when the speed of sound in the jet is above ambient. Equation (5.19) shows that a sound-speed increment co - ci is comparable with jet velocity Ui in producing refraction. Here again analysis is difficult for a jet configuration. Experimentally, Lassiter and Hubbard [47] found a progressive

---- Convection only Convection plus refraction FIG. 13. Outward refraction of sound rays by jet flow, giving a valley of low intensity in the polar plot.

outward rotation of the peak sound intensity (peak of the polar of Fig. 13) with increasing speed of sound: the sequence was 15", 30", 42" for air (ci = 1030 fps), turbojet (cj M 1800 fps), helium (ci = 2910 fps). We can now summarize the overall directivity of the jet noise approximately as follows. The directivity is governed by the combined factor (5.15) and (5.18), namely, (5.20)

(1

+ C O S ~ ~ )=C(1- ~+ cos48)[(1- M , cos 0)2 + a2Mc2]-5/2

from near 180" to near 40". Below roughly 40" (turbojets), 20' (model jets) refraction provides a deep dimple (reduction) compared with this factor. In the above the two terms of the factor (1 cos40) respectively describe the basic directivity of the self-noise and shear-noise, and the factor C-5 describes the directive amplification provided by convection of the eddies. In Fig. 14 the directivity factor (5.20), with GC chosen as 0.55 for best fit, is compared with measurements for turbojets and with the factor (1 - M , cos 13-5. (The experimental a from space-time correlations of turbulence in the mixing region 1401 is nearer 0.33 for unheated air jets; there are no measurements on turbojets known to the writer.) It is seen that (5.20) with the chosen a is quite good above the refraction cutoff. The

+

130

H . S. RIBNER

-.- M,= 0.76 (J33-A-I0 engine) -&=0.78 (J34-WE-34 engine)

\.

- .\.

---- h4pO.W(JSe-P-Sengine)

30-

4.0.82 ( I - M,

,,.'.

m

COS

e)*

-".- t&=0.62[(i-~mSe)*+o.3ragl-s/* (Itcos*e)

t ~

L"

FIG. 14. Relative intensity of turbojet noise versus angle 8 from flow direction (referred to zero db at 90'). Experimental data from Pietrasanta [91j is compared with the theoretical factors. (Allowance is not made for the cutoff below 40a, presumably due to refraction.)

8 0.I 0.3 0.5 07 09 10 Mc

+

FIG. 15. Separate display of thedirectivity factors C-6 = [ ( l - M , cos O ) * a * M c 2 ] - 6 ~ 2 and (1 COS'O) in decibels vs. 0 and convection Mach number M c ( w Uc/2c,). For combined directivity add dashed curve with reversed sign to solid curves for 8 < go", but with no sign reversal for 0 > 90'. ( N o allowance for refraction cutoff.)

+

The directivity contributions of the two factors of (5.20) are displayed separately in Fig. 15 for a = 0.55 and convection Mach numbers M , up to unity. (For M,> 1 (with a = 0.310) see Fig. 3 of [ l l ] . )

THE GENERATION OF SOUND BY TURBULENT JETS

131

These remarks have referred to the directivity of the overall sound. The directivity of specific frequency bands varies because of shifts of the spectrum peak with direction, described in the next Section. The refractive aspect which accounts for the valley in Fig. 13, however, is not discussed there. Theoretically, the outward refraction - whether due to jet velocity or to an elevated speed of sound in the jet or both - should increase with frequency; that is, as the wave length diminishes in comparison with the jet dimensions. The increased refraction would be expected to deepen the valley. This effect in conjunction with the spectrum shifts treated in the next section (which are applicable outside the valley) would tend to rotate the lobe of peak intensity progressively further outward with increasing frequency. Lee et al. [48] empirically correlate these and other effects in the equation (5.21)

omax =

+

17 loglo [(fo/uj)(rj/TJ2.11

where T,/Tois the ratio of jet temperature (at nozzle) to ambient in "R, and emaxis the angle of the peak of the noise polar in degrees. The points show considerable scatter from the line (5.21), but it does exhibit the general influence of the principal variables. The equation applies to both round and corrugated-nozzle jets, D being the diameter of a round nozzle of equivalent cross-sectional area. Jet velocity makes itself felt in two competing ways: via the convective factor C-s and via refraction. The former, which involves velocity to the fifth power, is presumably more sensitive to speed. In the writer's view, with increasing jet velocity the convection strongly enhances the downstream amplification to encroach even more on the refractive valley of Fig. 13. This rotates the peak intensity inward (smaller emax)in qualitative conformity with the equation. 6 . Shifts of the Spectrum Peak

The dual spectrum introduced in Section V.4 may be generalized by the results of the last section to allow for convection as

by insertion of the factors C and C4. The convective amplification C-5 contained in the sound power from the slice of jet with frequency f appears here as a vertical shift C-4 plus a Doppler shift C-l. This combination is derived in Part B and illustrated in Fig. 16. (The bell-shaped 'slice' spectrum of Fig. 16 has been squeezed into a single frequency f (b-function) to simplify the present sections. A progressive reduction of the spectrum with increasing

132

H. S. RIBNER

frequency pushes the peak slightly to the left of the true Doppler shift (1 - M , cos O ) - l giving the rather smaller effective shift C-1= [ (1 - M , cos 0)2

+ a2Mcz]-1/2

Radion frequency

(cf. [ill)).

w

FIG. IS. Sound spectrum radiated by unit volume of a simplified model of turbulence. Convective peak amplification and 'Doppler' shift are given by factors C-4 and C-l, respectively, to give power amplification C6.

1

I

02

1

0.5

I 2 5 Relative frequency/flow speed

10

FIG. 17. Variation of theoretical idealized jet noise spectrum with emission angle 0. Relative heights of "bass" and "treble" without allowance for convection are in approximate ratio 2 c0s4 0 t o 1. Convective amplification coupled with Doppler shift yields final (upper) 30" curve. (This figure is oversimplified compared with (5.22) or (10.22). The separate convection factors (C-* in the equations) for the bass and treble spectra are lumped together.)

133

THE GENERATION O F SOUND BY TURBULENT J E T S

In (5.22) the shear-noise and self-noise spectral contributions have been labelled 'bass' and 'treble', respectively, because the second is an octave above the first (Fig. 17). The proportions vary with direction 0 from the jet axis being dominated by the bass spectrum at small angles and the treble spectrum in a broad range about 90". This is accomplished by the C O S ~factor. ~ For small angles (e.g. 30") points of the resultant curve are amplified (vertical shift) and Doppler shifted. These shifts approach zero a t 90" because of the nature of the convection factor C. I t is suggested that an increase in amplification toward the left may move the amplified peak in that direction as indicated in Fig. 17. This could occur if somewhat downstream of the potential core the dominant noise emitters were located sufficiently close to the axis to be convected substantially faster than those in the mixing region. Recent calculations by the writer (unpublished), based in part on the turbulence measurements of Laurence [39], are inconclusive on this point. On the other hand, the comparisons made by Gerrard using convected quadrupoles [49] support the notion. The two effects - one calculated, the other speculative - exhibited in the figure result in good qualitative agreement with experiment, e.g., [49-511. They predict the observed 'paradox' that the sound spectrum a t a small angle with the flow (30") peaks perhaps an octave lower than the sound radiated a t right angles (90"). This is just the reverse of the expectation from a single spectrum subject to Doppler shift as a whole at small emission angles. Comparisons at low speed and high speed for a single angle (30") reinforce the point. The figure predicts the middle curve at very low speeds and the upper curve a t high speeds: the peak of the high speed curve lies to the left of the peak of the low speed curve, despite Doppler shifts of the individual points. This is again in agreement with experiment, when the abscissa scale is * frequency/velocity or * Strouhal number fD/Ui [49, 501. The present explanation of these 'paradoxes', particularly the second, gets away from the notion that the peak of the spectrum must scale directly with a typical frequency in the turbulence as the speed changes. The experimental results cited above show the spectrum peak frequency varying somewhat like U:/*, with no allowance for Doppler shift. If typical frequencies in the turbulence varied this slowly (instead of like Ui as assumed in the earlier analysis herein), the predicted noise power would fall far short of the measured Uia-dependence.* 7. Power and Efficiency

Number The low speed analysis (Sec. V.3) predicted a variation of sound power with jet nozzle velocity Ui proportional to U T . We shall now reexamine this, taking account of the modifications due to convection and other effects. * This effect alone would yield U,'.

11s.Mach

-

A measured turbulence intensity u14 c Uj312 roughly instead of Uj2 must also be considered, but its effect is approximately offset by the convective increase provided by C6.

134

H. S. RIBNER

The simple factor (1 - M , cos f3)-6 predicts infinite sound intensity at the condition for Mach wave emission ( M , cos 0 = 1). Moreover this factor, when modifying an otherwise spherically uniform radiation pattern, augments the total power in the ratio (5.23)

(1

+ Mc2)/(1- Mc2)4

for

M,

< 1,

00

for M,> 1,

which results from integration over a unit sphere. With the effective convection Mach number M , taken as Uj/2c,, (mixing region of the jet) (5.23) predicts infinite augmentation at jet speeds Uj of twice the speed of sound

-10

Convection speed Speed of sound outside jet

n

FIG. 18. Directional average of convection factor, ( C d S ) ~ v= (1/2) sC-S sin Ode, 0

in db relative to value at M c = 1. This factor multiplies low-speed power law (e.g. Uj*). (Weighting factor ( 1 ~ 0 ~ 4 has 8 ) not been included in average.)

+

or more. Suppose we apply this to our basic U t speed dependence: the result is a predicted catastrophic rise above Ui8 at the higher subsonic convection speeds that is not observed in practice. The improved convection factor C-6 (Section VII.2) is better behaved. Its average over a unit sphere (normalized to unity a t M , = 1) is (5.24)

135

T H E G E N E R A T I O N OF S O U N D BY T U R B U L E N T J E T S

which provides a slow rise (with appropriate choice of a) to a moderate peak a t M , near unity, followed by a decay like Mc-6 or Ui-5 (Fig. IS), where M , is taken as Uj/2c,. If we ignore (or cancel out in some wajr) the small rise, the product of Uj8 with the mean convection factor (5.24) varies like U? below M, = 1 with a transition to U j 3above M, = 1 [13j. Such a transition appears to be supported by the experimental data for afterburning jet engines and rockets (Fig. 19)

Rockets t engines

terburners)

)

I

I

I

I

500

loo0

2000

5000

uj (ft/sec)

I(

100

- -

FIG. 19. Noise power vs. jet speed, showing transition from U,a to U , S , approximately, when convection speed L’j/Z exceeds the speed of sound (Reproduced from Powell [31]). X/D Lossiter-Hubbord 3

I

0.02 0.04

0.1

0.2 0.4

r/D 0.5

1.0~10~

Reynolds NO.(pUjD//d

FIG. 20. Dependence of turbulence intensity (in yo of U,) on jet nozzle velocity Uj (after Lassiter and Hubbard [87], with data from Laurence [39] replotted on a Reynolds number basis).

The deduction of for subsonic nozzle speeds (before multiplying by the convection factor) is actually faulty: a reconsideration of the turbulence data indicates instead more nearly Ui7. (This tends to cancel out the subsonic rise in the convection factor, Fig. 18.) The Ui7 results because the rms turbulent velocity increases about like U?I4 rather than linearly

136

H. S. RIBNER

with U , as first assumed (Fig. 20). I t is not clear whether the relative intensity levels off again at higher speeds to restore a basic U? law as seems to be implied by the noise data. A marked deviation from an eighth-power speed dependence is to be expected eventually from considerations of efficiency. Division of noise power (- Ui8) by mechanical power (- U,”) of the jet gives an efficiency Ui5 for subsonic values of Ui/2. The corresponding division of noise power (- via)by jet power (m U,”) for the supersonic range of Uj/2 gives a constant efficiency. More precisely (according to experiment for air jets, turbojets, and rockets)

-

?I = K1Mj5

(5.25) = K2

< uj < 2c0 2~< 0 Uj < 8 ~ 0 0

where K , M x~ po/pj [5l], K , = 0.3 to 0.8yu [52]. The point is that the efficiency cannot continue t o increase like Mi5 without exceeding 100%: it must level off somewhere (or even decrease), corresponding to a marked reduction of the speed exponerit. The experimental rocket data do not extend beyond Mi(= Ui/co)M 8, which corresponds to an effective convection Mach number M, M 4. The use of the convected quadrupole (or dilatation) inference of constant efficiency a t higher speeds is purely speculative because the Uis law on which it depends is just an assumption there. The approach of Phillips [5] - an asymptotic theory for high values of Ui/co - predicts that the efficiency must ultimately diminish as Miw9l2for constant assumed turbulent intensity. 8. Effects of Density and Temperature

When the jet consists of one gas exhausting into another the jet density may differ substantially from the ambient value po. We can generalize from remarks following (5.6) that the sound power may be expected to vary as p2/po2, where i; is an effective average density in the jet. If the density of the jet gas is pi at the nozzle, then the ratio p2/po2 will be nearer unity than pi2/po2 because of the mixing. Experimental measurements [47] show the sound power depends more strongly on density than the theoretical j2/po2, in fact, the more extreme pj2/po2fits the data very well over a range from 16.7 (Freon 12/air) to 0.027 (helium/air). The reason for the discrepancy is not clear. I t is tacitly assumed in the theory that the mixing dynamics are unaltered by change in density. This is false for a hot jet [53] and unproved for an unheated jet of a foreign gas.* (Added in proof.) More recent information discussed in 1941 shows a dependency of jet spreading on density ratio (and on Mach number).

THE GENERATION OF SOUND BY TURBULENT JETS

137

The pj2/po2 variation of sound power breaks down when the density change pi arises from heating. Indeed, measurements on air jets heated to as much as 1000"F [47, 541 and turbojets at much higher temperatures [55] indicate no sensible effect of temperature : the proportionality between sound power and the Lighthill parameter p o A U t / c 2 (5.10) "holds for a wide range of jets from very small jets, both hot and cold, up through several sizes of jet engines" [54].

A 1000" F air jet has about 1.3 times the rate of spread of a 90' F jet [53], and this appears to be associated with a 1/1.3 reduction in typical sound frequencies [54], presumably due to the reduced shear. This would not account for the failure to scale with p2/po2, as the effect would be in the wrong direction (a further decrease in emission). Furthermore, the frequency effect is probably largely offset by increases in turbulence scale and effective volume of the noise-emitting region. The writer suggests that turbulent heat transport generates additional sound from entropy fluctuation offsetting the reduction associated with reduced density. The appropriate entropy term can be found e.g. in [lo, l l j (Appendix A). The entropy spottiness arises from the turbulent nature of the transport of hot fluid from the interior to the cool exterior. Furthermore, the entropy fluctuation in time may be expected to be rather similar to that of the turbulent velocity. The foregoing has left out of account differences between the mean square speed of sound inside (2) and outside (co2) the jet. Lighthill [2] argues that a ratio c o 2 / s large compared with unity, e.g. 5.3 for a Freon jet in air, may substantially augment the sound emission (especially at the higher frequencies), aside from the direct density effect. Experimentally, only the density effect - mentioned above - appears to be in evidence [47]. We may reexamine this in terms of the source strength on the dilatation theory. In terms of the local mean square speed of sound 3 the source strength is - ($)-1a2fi(o)/at2 (4.2). But this source is radiating into a medium characterized by .? nearby with a transition to co2 outside the jet. Moreover, a t low speeds the jet dimensions << typical radiated sound wave length. A crude analogy is an oscillatory point source Q, centered in a sphere of imbedded in ambient gas stationary gas of density ii, speed of sound (S)ll2, of density po and speed of sound co. We ask, what is the source strength Qo that will generate the same far-field sound when p = Po, 3 = co2. A simple analysis shows that, for a sphere with circumference <( wave length, Qo = (po/p)Q,. If the inner and outer gases are the same, but at different temperature, then p o / p = $/co2. Thus the effective source strength in the equivalent unheated jet is

138

H . S. RIBNER

in which the speed of sound co outside the jet appears. This means that jet temperature should have no direct effect on the sound power radiation for wave lengths >> the jet dimensions. (The associated density effect from f~@l remains, however, in theory.) I t will be noted that (5.26) still holds approximately when the gas sphere of the analogy is helium or Freon 12 imbedded in air, with no heating. That is, for these gases po/p M c2/co2within f 20%. Thus it is not surprising that no effect of speed of sound on noise power was found experimentally [47]. (Sound speed differences have, however, a powerful refractive effect ; this was discussed in Section V.5.)

Fs

9. Reduction i n Twbojet Noise

The basic IJ8D2law for the noise power from similar jets may be compared with the U 2 D 2 relation governing the thrust. Thus an increase in jet diameter a t constant thrust will provide a net reduction in noise: the increase in D will be offset four-fold by the corresponding reduction in U . (We overlook here the modest effect of jet density.) This powerful effect of reducing the jet velocity by going to a larger diameter engine has been exploited in the bypass or turbofan engines. In this approach the enlarged compressor constitutes in part a fan to provide an annular flow of air that bypasses the combustion chamber. In the pioneer Rolls Royce Conway the bypass ratio is 40% and in one case cited [56] the noise is reduced 7 db. Coupled with the reduction in jet noise, the bypass principle has other virtues that make it attractive. The thermodynamic efficiency is good, and the increased diameter increases the propulsive efficiency. There is a net gain in economy. The favorable features have led to a trend to engines of much larger bypass ratio, but with the penalty of fan noise emerging as an increasing offender. The first generation of jet engines did not employ bypass and even the bypass engines require further quieting as the power goes up. Thus since the early 1950’s engineers have been faced with the urgent requirement for quieting the engines. The mathematical theory has not, unfortunately, provided a clearcut guide. The techniques have been motivated by conflicting interpretations of the theory, but have nevertheless met with a fair degree of success. Furthermore, the explanation of successful muffler behavior is still a matter of controversy and speculation. In view of the very limited understanding of the subject (as it appears to the writer) no very detailed discussion will be attempted here. The inference has been drawn that reduction in the mean-flow shear should be beneficial in two ways. First, it should reduce the expected direct noise enhancement due to mean shear [2]. It is now thought, however, that the amplifying effect of shear is approximately offset by an associated reduction in scale [3, 111. Second, reduced shear should reduce the

T H E G E N E R A T I O N OF S O U N D BY T U R B U L E N T JETS

139

intensity* of turbulence with a consequent noise reduction, presumably according to the 8th power law. This view is still supported by current thought and some data [36, 571. The earliest device to show any promise was the toothed nozzle of Westley and Lilley [58], motivated by the notion of shear reduction. More complete tests [50] disclosed some serious flaws, for example, reduction of engine thrust and increased fuel consumption. The device evolved then into a family of corrugated nozzles, (see e.g. [59, 60]), which have been reasonably successful. For example, one of the best nozzles - one with six deep corrugations - achieved an average noise reduction of 7 db with a peak reduction of 12 d b [59]. The corrugated nozzle reduces shear by aspirating an external flow into the convolutions where it is given some forward velocity: the mechanism is somewhat like that of a jet ejector pump. The corrugated nozzle suppressor is used in different variations on a number of Rolls Royce turbojet engines, for example the Conway powering versions of the Douglas DC-8 and Boeing 707. An alternate arrangement is the multiple-nozzle muffler as developed by Boeing [61] and used on the Pratt & Whitney powered version of the 707. In this scheme the jet exhaust is emitted through a cluster of small nozzles in place of the single large nozzIe. Here again the jet entrainment effect aspirates air into the space between the jets; there the flow acquires some forward velocity, reducing the shear. Here the effect is especially easy to visualize. The space between jets available for inflow diminishes as they spread to .zero where they meet. Because of this space restriction entrainment of the inflow by the jets imparts a forward velocity prior to entering the jet. \I.'ith both the corrugated and multi-nozzles the direct sound reduction due to reduction in shear is thought to be complemented by the so-called shielding effect [60]. This is a modification of the directivity of the noise due to acoustic interference between multiple jets or the convolutions of a corrugated jet. I t is argued (cf. [3]) that the sound from one jet is reflected and refracted by the velocity field of a neighboring jet, so much so that the peak intensity of the second jet is scarcely augmented by the sound from the first jet. Thus although the total sound intensity i s unchanged, the sound is redistributed so as to have a broader, lower peak - a desirable effect. The major part of the scattering process must be due to the mean jet flow as discussed in Secs. V.5 and VIII.3; however, the higher fre-

* A more precise statement is t h a t the volume integral of turbulent intensity (to a suitable power and suitably weighted, cf. (10.4)) should be reduced by reduction in mean shear. Dr. G . T. Csanady has pointed out [95] that the experimental data (his own, and the measurements of Corcos [36j) indicate no reduction in peak turbulence levels, but rather in the effective volume of such levels. .4 theoretical argument for the failure to reduce peak levels is given in Csanady's paper.

140

H. S. RIBNER

quencies can be scattered by the larger eddies in a jet, the criterion being wave length comparable with or less than eddy size [62]. In the direction of peak sound a corrugated nozzle produces a bellshaped curve of sound reduction vs. frequency to be subtracted from the spectrum obtained with a standard nozzle. The peak reduction shifts toward higher frequencies as the width of the convolutions is decreased (their number increased) [59]. Multiple nozzles behave similarly [61]. It is argued [63] that the convolutions tend to eliminate turbulent eddies of dimensions comparable with the convolution width; thus sound of the associated band of frequencies is minimized. An alternative speculation of the present writer is that the shielding effect may be optimized with a certain ratio of corrugation width to sound wave length. The spectral curve of sound reduction provided by these nozzles goes somewhat negative at the upper end; i.e., the high frequencies are increased, rather than attenuated. This is to be expected from the greatly increased perimeter, compared with that of a round jet, of the initial mixing region responsible for the higher frequencies (cf. the derivation of the f - 2 spectrum law). Unhappily, the higher frequencies are substantially more irritating to people, it is now realized [64]. Thus although the corrugated and multinozzles substantially reduce the noise intensity, there is little reduction in degree of annoyance. That is to say, there is little cHange in the PN db (perceived noise level, db), a rating scale that weights the frequencies according to annoyance [64-681. Despite the pessimistic tone of the early remarks of this Section, R. Lee et al. [48] have had an impressive degree of success with an ambitious semi-empirical formalism for predicting nozzle-suppressor behavior. It is unfortunate that there has not been time in preparing this review to give their voluminous report the study it deserves. Several alternative formalisms are proposed and the results compared with relevant parts of a comprehensive set of scale model tests which included, besides the conical-nozzle jet, a rectangular jet, a pair of interfering rectangular jets, and the jet from an %lobe corrugated nozzle. One simple notion therein is, in effect, that the sound power radiated from a slice of jet is related solely to the local mean velocity according to (5.27)

with the frequency of the slice emission given by an empirical relation (5.28)

THE GENERATION OF SOUND BY TURBULENT JETS

141

with constants a and @ obtained from source tracing (cf. e.g. [67]). (Also included is an alternative to (5.27) that replaces Us by U s x (turbulent shear this is somewhat closer to the accepted theory.) Experimental patterns of U for e.g. the interfering rectangular jets and the 8-lobe nozzle, are used to evaluate (5.27), and this with (5.28) yields the predicted spectrum for each case. The agreement with the measured acoustical spectrum data varies from fair to very good, with the spectral distortions being well predicted. Thus although (5.27) would appear to be a gross oversimplification, its success in conjunction with the easier-to-rationalize (5.28) is significant. I t may well be that the interaction between the turbulent quantities and the mean flow provides a measure of validity to (5.27) as an integral. Another important development in the report of Lee et al. [48] is a digital computer procedure for computing the development - the mean velocity U (and shear stress) as function of position - of one or more parallel jets of arbitrary initial cross section. Good agreement with experiment is obtained in the example of two interfering rectangular jets. This computation provides the data for (5.27); it appears, however, to have been a parallel development and was applied only in the alternative form cited under (5.28).* The method is a development of a principle of mixing due to Reichardt. Here again the assumptions are oversimplified with respect to accepted theory, but substantial success may be claimed. The results for overall power are extended to directivity by this hypothesis based on the experimental data: “the directivity characteristics of jet noise are functions only of frequency and are essentially independent of nozzle (or suppressor) configuration.” Satisfactory agreement of overall directivity predicted on this basis with experimental values was found for a 6-tooth nozzles, an %lobe segmented nozzle, the same with a shroud, and a 21-tube nozzle. These results suggest that the shielding principle put forward above may play only a minor role. The apparent dependence of directivity primarily on frequency despite changes in nozzle configuration can perhaps be explained. Nozzle configuration will probably not alter convection speed in the dominant noise generator, the mixing region, so the directivity due to convection will be unaltered. The refractive effect may depend primarily on frequency via the ratio wave-length/mean diameter. Finally, the proportions of bass and treble emitted from unit volume on the ‘two spectrum’ theory vary in a definite way with direction (Secs. V.4 and V.6).

* The relation for frequency was also somewhat modified. The agreement between predicted and measured spectra was good for a conical nozzle and fairly good for an 8-lobe nozzle.

142

H. S . RIBNER

R. MATHEMATICAL DEVELOPMENT VI. GOVERNINGEQUATIONS 1. Wave Equations

In a turbulent jet flow the viscous stresses are negligible compared with the inertial stresses; molecular heat conduction is likewise negligible. Thus the main acoustic features are exhibited when the fluid is treated as inviscid and isentropic.* For such a fluid the conservation equations for mass and momentum read

where c2ap/ax, has been written in place of the pressure gradient aplax, by virtue of the isentropy. Elimination of pui by cross-differentiation and subtraction yields

approximately, on replacing the square of the local speed of sound c by its time average 3, (In these equations p is the fluid density, p the pressure, zc, the velocity component in direction x I , where z = 1, 2, or 3. A repeated index i or i implies summation; thus a2p/ax,2= a2p/ax,ax,= V 2 p in (6.2)). Equation (6.2) is an inviscid-flow approximation to Lighthill’s equation governing sound generated aerodynamically. Consider the momentum transport term or inertial stress pu,ul on the right hand side: in a medium without flow u,, ul are the velocities associated with acoustic waves and the nonlinear right-hand side is negligible in the range of linear acoustics. The expression (6.2) then reduces to the wave equation for a quiescent acoustic medium free of sound sources. On the other hand, a medium with flow is equivalent to a medium at rest containing a spatial distribution of virtual sound sources: the strength per unit volume is given by a2(pu,u,)/ax,ax,. Thus the equation states in effect that an aerodynamic flow provides a forcing term for generating a sound field

* The case of the hot jet requires consideration of entropy fluctuation (cf. Sec. V.8) : see .Appendix A of [ l l ] for appropriate equations.

THE GENERATION OF SOUND BY TURBULENT JETS

143

The acoustic waves in the flow may be neglected in the forcing term if they are weak. That is, the aerodynamic flow may be approximated as incompressible at low speeds. Stated in another way, the back reaction of the sound field on the flow generating the sound is negligible in the quasiincompressible range of flow speeds. This should cover jet turbulence in the schockfree range. It is equally valid, if the fluid is unbounded, to treat the sound field as generated by dipoles of strength a(puiui)/a‘ci or by quadrupoles of strength puiui (cf. Sec. 111.3). This follows from the fact that the source strength has the form of a double divergence and puiui approaches zero outside the region of flow. The implications of the equivalence of such patterns of monopoles, dipoles, and quadrupoles are discussed in Sec. 111.4. We may split the pressure perturbation into the ‘pseudosound’ pressure p ( O ) and the ‘acoustic’ pressure # ( I ) :

p

(6.3)

- #lo = p ( 0 )

+ $(‘I.

The pseudosound pressure is defined (up to a constant of integration) by

which accounts for the quasi-incompressible features of the flow (Fig. 5 ) . For comparison the Lighthill equation is rewritten in terms of pressure as 1 (6.5)

azp

7 at2

v2p =

a2(puiuj) axiaxi ’

-

with use of the isentropic relation S p w c V p . from ( 6 4 , there results

Lighthill Equation ’ Upon subtraction of (6.4)

by virtue of (6.3). This equation relates the acoustic pressure $‘) to the pseudosound pressure pc0); it has the form of the wave equation for an acoustic medium at rest containing a distribution of virtual acoustic sources. The interpretation of the virtual source term - a2p(n)/&%2 in terms of fluid dilatations was fully discussed in Part A. 2. Solutions for Radiated Sound

Since the dilatation equation (6.6) has the simpler source term - the complexities are buried in the pseudosound pC0) - we shall deal with it first ; later we shall exhibit the corresponding solution of the Lighthill equation. Let a volume element in the source field be d3y at point y. Then the solution

144

H. S. RIBNEK

for the acoustic pressure

p‘l)

at point x and time t reads

(6.7)

where the brackets [ ] indicate evaluation a t the retarded time i-= t - Ix - yl/co and we have specialized to a uniform speed of sound (c2 = coz). Strictly speaking, the integral is over all space, but in practice it may be limited to an effective volume V containing the disturbed region the aerodynamic flow. A t distances x large compared with the dimensions of V the approximation Ix - yi m x may be made in the denominator (but not in the retarded time t). If in addition 2nx >> (typical wave length), then p ( * )so approximated refers to the acoustic ‘far field’: Far (6.8) Field

I:I’[

#(l)(x,t)= - -4nco2x

d3y

___

(Dilatation).

Component sound waves too long to meet the criterion must be excluded. The corresponding solution of the Lighthill equation is discussed in Section 1’1.4; a convenient form due to Proudman [7] is

(6.9)

Far Field

’

p(l’(x,t)= ___ 4nco2x

1[

]

____

d3y

(LighthillProudman)

where u, is the component velocity, including the mean flow it any, in the direction of x. The equivalence is proved in [ I l l . The two equations are of parallel form. Further developments will refer at first to (6.8): they can be converted to refer to (6.9)by replacement of the pseudosound pressure p ( O ) by the x-momentum flux density puX2. The acoustic intensity at x is given by pc’,2/poco. The power radiated in the direction of x (per unit solid angle) is therefore (6.10)

p(e)= x 2 p ( x , e ) / p o c o

where x makes the polar angle 8 with the jet flow direction. (This and what follows applies to a round jet: for the more general case where there is an azimuth dependence P(8) becomes P(O,$)). The square of (6.8) may be written in a special way for insertion into (6.10) to yield (6.11)

11-

P(8) = (16n2poco5)-l

p(0)(y‘,i‘)p(o)(y‘’,i‘’)d3y’d3y’’,

m

the average being over time. The integrand is the correlation of P ( O ) a t the two points y’ and y” a t the different retarded times i’ and 2“. This space-

THE GENERATION OF SOUND BY TURBULENT JETS

145

time correlation will be a function solely of the space separation y' - y" and of the time separation 1' - t". Thus it will be convenient to make the definitions

(6.12)

g = y' - y",

y = (1/2)(y'

+ y"),

t=

k - P,

and call the integrand R(y,g,t). It will be noticed that y lies halfway between the two correlated points. The integration limits in y and 5 are again infinite; however, in practice the limit on y may be reduced to the effective flow volume I/ while retaining for convenience in calculation the infinite limits on g. Then the power in direction 8 is

(6.13) u

r

n

Since we consider only statistically steady jets (stationary random processes), (6.13)can be written in the alternative form, (e.g. [ll])

(6.14) where Ro(y,g,t) = p(o)(y',?)p(o)(y'',~')is the space-time correlation of pseudosound pressure. The difference (6.12)of the sound travel times from y' and y" to the observer a t x is given by t in cot = (x - y"l -

Ix - y'I

according to the definition of f under (6.7). When, as has been assumed, x >> y' or y", then x, (x - y") and (x - y') are essentially parallel vectors, and a close approximation is [9-111 (6.15)

Cot

=5*

x/x

that is, the projection of the point separation = y' - y" on the vector x. This relation is to be inserted into (6.13)or (6.14); in the latter case the ?Plat4 operation must be carried out first. The integral over 5 in (6.13)may be interpreted as the value of (p(o))2 at point y multiplied by a certain volume, the 'correlation volume'. This volume is an effective eddy size, or region within which values of p ( O ) are well correlated. Because of the time delay (6.15)in the integral, the correlation volume may depend on the direction 6 of the emission vector x. Thus the integral states in effect that

146

H. S. RIBNER

is the contribution of a turbulent volume element dqy= d l i to t h e sound power radiated in direction 8. This relation averaged over a unit sphere has been exploited in a number of developments in Part A. 3. Spectrum of Radiated Sound The contribution to P(0) from radian frequencies in a unit band width a t w may be designated P(8,w). Thus

(6.17)

where P(8) is the acoustic power per unit solid angle in direction 8 and P(8,w) is the spectral density of this power. An expression for P ( 0 , o )can be obtained from the theory of stochastic processes as m

that is, as x2/poco times the Fourier cosine transform of the sound pressure autocorrelation p(')(~,t)p(~)(x,t z') at point x. (The factor x2/pocuarises in the conversion (6.10) from mean square pressure to power.) The autocorrelation is the correlation of the value at a given time with the \ralue a certain time increment t' later. The far-field sound pressure autocorrelation appearing in (6.18) mav be written as a generalization of (6.14) in the form (cf. (6.10))

+

where t satisfies (6.15) and t' is constant for the integration. l'he Fourier cosine transform (6.18) of (6.19) may be carried under the y and integrals to be performed on Ro first, if convenient. This would give

(6.20)

i

Fourier cosine transform of KO = (2/n) Ro cos o t ' d t ' . 0

The cos w t ' may be replaced by exp (- z w t ' ) in (6.18) if the limit zero is replaced by - 03 and (2/n)by (l/n). Correspondingly (6.20) may be replaced by

THE GENERATION OF SOUND BY TURBULENT JETS

(6.21) Fourier (exponential) transform of

147

i

KO = (l/n) R,exp (- ion’)dt’. -a3

This is known as the cross-spectral density of the two components of R, (cf. (6.11)-(6.14)) and is complex whereas (6.20) is real. The imaginary part is odd in { and integrates out in the transform of (6.19). More information can be extracted from four-dimensional Fourier transforms wherein exp (- iwt’)dt‘ is replaced by exp - i(k * { wt’)dskdt‘. The chief architect of this approach was Kraichnan [68] and his methods have been developed by Mawardi [69], Lilley [4], and Williams [13]. The latter has developed the formalism in detail to account for the effects of eddy convection a t Mach number M. The manipulation and the formulas obtained are elegant but complex. It is the writer’s view that equivalent results for power and spectral density are obtained more simply by the one-dimensional autocorrelation procedure of (6.18) and (6.19) [9-111. However, a particular success of the four-dimensional transform approach was the discovery that a given wave number k in the turbulence radiates sound of the identical wave number: wave direction and wave length both coincide [68, 69, 4, 131. This can be explained in physical terms as follows. The Fourier transform or integral involving exp - i(k 5 w t ’ ) implies that the turbulence pressure pattern #(’) is built up from plane waves of pressure of form exp - i ( k x a t ) . The argument k x wt implies that a wave travels with speed w / k in the minus k direction only; moreover, waves with all orientations of k and all wave numbers k are included. Although the turbulence pattern is assumed limited to an effective volume V these Fourier component waves extend to infinity: they mutually cancel outside V . I t follows that such waves of the turbulence field individually coexist throughout space with the sound waves they generate. Thus there will be cancellations between source waves and sound waves that do not match up. Only those sound waves will survive that match up exactly with corresponding travelling source waves.

+

+

- + - +

4, Developments of Lighthill and Proudmati

?’he Proudman form of the Lighthill equation for radiated sound, already introduced as (6.9), will be central to many of the developments herein. This equation and underlying developments of Lighthill on which it rests will be derived in the present section. Lighthill’s treatment of the ‘amplifying effect of shear’ will also be sketched, although it will not figure in the approach followed in the present article. This will be a convenient place to generalize the conservation equations (6.1) to allow for viscosity, as Lighthill did. The exact equations for mass and momentum may be written

148

€I S. . RIBNER

(6.22) (6.23)

where co2ap/axi has been added to both sides of the momentum equation such that (6.24)

T81. .- puiuj

+

tij

+ ( p - co2p)dij

in which t i jis the viscous stress tensor and hi, = 0 or 1 according as i # j or i = j . The elimination of pui between (6.22) and (6.23) yields (6.25)

The right-hand side is an effective source strength and the solution for p(x,t) is of the form

(6.26)

where [ ] again denotes evaluation at the retarded time (cf. after (6.7)). Two applications of the divergence theorem yield (see e.g. [23],pp. 62-64) (6.27) m

if the surface integral of Tii at infinity is taken to vanish. The formal differentiation in (6.27), taking account of the retarded time, yields integrand terms showing r3, r 2 A - l , and Y - ~ A - dependence ~ on the source-observer distance r = ] x - yI and typical wave length A (e.g. [70, 111). (A time differentiation of Tii a factor co/A.) Only the last term survives when r is large compared with A/2n: VI

If further r is large compared with the dimensions of the flow this reduces to the far-field relation (6.29)

THE GENERATION OF SOUND BY TURBULENT JETS

149

In the far-field the left-hand side p - po is indistinguishable from $(')/co2 as used in earlier sections. From (6.10) the sound power radiated in the direction x (per unit solid angle) is (0.10)

P ( x / x ) -- x ~ $ ~ ~ ( x ) / p o c o

The square of (6.29) may be written in a special way for insertion into (6.10) to yield

(6.30) The integrand here, which is a summation of correlations as i, j , fi, 1, take on different values, is developed further in [I]. The term Tii can be identified as a generalization of the quadrupole strength discussed earlier. In a jet flow, if we esclude cases with density greatlv different from ambient, Tii will be dominated by the momentum flux or Reynolds stress pu,uj. Equation (6.29) (with the left side replaced by p(*)/co2)thus reduces to (6.31) W

The summations xiuc/x and x i u i / x are each merely the component of u in the direction of x, which may be written u,. The equation may therefore be written in the very neat form (6.9) due to Proudman r7]. Proudman employed this formula in the analog of (6.30) to calculate the noise generated by decaying isotropic turbulence. \V:ith use of techniques from the statistical theory of turbulence (e.g., [71]) he was able to obtain finally (6.32)

P

* 38p0f(U,2)5/?co-2

as the total acoustic power generated by unit volume in terms of the mean rate of dissipation E per unit mass. This was, of course, radiated uniformly in all directions. Lilley [a] has applied this in the calculation of the socalled 'self-noise' generated by the turbulence in a jet - the part not

150

H. S. RIBNER

contributed to by mean shear. However, the noise power is dependent on (frequency)4 (cf. the four differentiations in (6.30)), and the frequencies governing (6.32) are those of decaying isotropic turbulence [7]. The frequencies in a jet are relatively higher - because of the effect of shear in all of the regions explored (compare the space-time correlations in [40] and [72]). Thus (6.32) must seriously underestimate the ‘self-noise’ emitted by a jet. Lighthill [2] has reexpressed (6.31) to put the fluid shear into evidence. A transformation using the exact equations of motion gives

+

where pii = tii psi, (cf. (6.24)). He argues that the last term is a space derivative and so represents an octupole field, which is a relatively poor generator of sound, particularly at the lower Mach numbers. Moreover, the viscous stresses in the free turbulence are relatively small, whence the dominant term in (6.33) reduces to (6.34)

according to the argument. Equation (6.34) states that the time fluctuations of momentum flux (or Reynolds stress) are dominated by the product of pressure and rate-of-strain. (The latter defines the distortion of a fluid element .) In [2] - and this is further developed in [4] - a strong presumption is made that the mean shear d,, overrides the fluctuating shear within the mixing region of a jet. Thus the dominant quadrupole is T,, w pu1u2,and (6.31) reduces to

p ( x , t )w

(6.35)

xlxz

__I

4nc02x3

5 [z] E,,

d3y.

The interesting feature of this is the directivity factor, which for the mean square sound pressure for a round jet takes the form [2]

+ x ~ ~ ) /=z +sin20~os20= (1/4) sin220

(6.36)

which resembles a four-leaf clover. It is implied by Lighthill [2, 31 and stated explicitly by Lilley [4] that the directivity of the overall sound generated by a jet should be of the form (6.37)

[1

+ (constant) sin2201 x convection factor

T H E GENERATION O F SOUND BY TURBULENT JETS

151

compounded of (6.36) and an omnidirectional part (e.g., from (6.32)). In Fig. 12 of [3] the constant is in effect 4, whereas a value unity* is taken in [a]. The agreement of (6.37) with experimental directivity curves varies from only fair with the value unity to poor with the value 4 (after allowance for refraction effects at small 0).

VII. CONVECTIONEFFECTS FOR

A

SIMPLIFIED MODEL OF TURBULENCE

1. Moving us. Stationary Axes Consider first a pattern of turbulence unconvected by any mean flow. The fhctuating eddies will have a spatial decay and a temporal decay. A two-point space-time correlation pattern for u1 will exhibit this decay and

Y‘

semi--froz& pattern (a)

Ur

semi-frozen

pattern (b)

__

FIG.21. Correlations of turbulent velocity, ulul’,at two points with streamwise and with time delay 7 . Sound intensity is maximum (C-s peaks) when separation path of integration lies along 45’ ridge line [ l l ] .

may be expected to look like R,, Fig. 2l(a). If, now, the turbulence is convected with velocity Uc the correlation pattern will appear like Fig. 21(b) to the stationary observer. The transformation is

Figure 21(b) resembles the curves obtained by measurements in a jet flow with a pair of stationary hot wires [40]. Figure 21(a) is in effect derived

* Actually, Lilley’s analysis [ 4 ] appears to suggest 4 rather than unity. The constant must be 4 in order that the sound power associated with sine 20 be about four times that associated with the omnidirectional part, the ratio stated by Lilley.

162

H. S. RIBNER

therefrom by the inverse of (7.1): the reference frame therefore moves with the mean convection speed of the turbulence. The true temporal fluctuations of the turbulence are responsible for the generated sound [ l ] : these are exhibited by the decay along the U,t axis in Fig. 21(a). The longer the contours, the slower are the typical fluctuation frequencies. (A ‘frozen’ pattern would show an infinitely long hill or ridge of high correlation along the U,t-axis.) The decay along the U,t-axis in Fig. 21(b) is much faster, implying much higher frequencies. These are the apparent frequencies recorded by the stationary hot wire : the convection of a space pattern past the wire yields a signal that fluctuates rapidly in time. In this case the true frequencies are almost completely obscured. The operation a4/at4 in Fig. 21(b) would bring the excessively high apparent frequencies strongly into evidence in equation (6.14). For this reason Lighthill introduced a transformation [l] to the moving reference frame of Fig. 21(a); with this transformation (6.14) becomes

where a correction in the convection factor (1 - M , cos O)-s from power - 6 to - 5 is due to Williams [ 121. Here the spurious convective frequencies no longer appear because a4/at4 is carried out, in effect, in the moving frame. Lighthill further argued that for eddy sizes << (typical wave lengths of sound) - a condition met to some extent in turbulent jets - the emission phase differences within an eddy could be neglected (t set equal to zero) in the moving frame. This would complete the elimination of any apparent convection effects from the integral proper, the sole effect appearing in the factor (1 - M , cos O)-s. This convection factor is physically incorrect near the singularity 1 - M,cos 0 = 0. The difficulty arises from setting t equal to zero in (7.2). Actually, the moving-axis transformation exaggerates the time delays within an eddy according to (7.3) for large observer distances .v (unpublished work of the writer, and [13]). The zero t approximation clearly fails near the critical 8 and the actual value (7.3) must be used in evaluating (7.2). This has the effect of eliminating the singularity in the convectioii factor, as will be seen later. The moving-axis relation for sound pressure (7.2) has the virtue of exhibiting the main effect of low-speed convection in the simple factor moreover, it permits neglect of the time delay t to a (1 - M , cos consistent approximation. However, we have seen that both simplifications are invalid near the singularity (1 - M , cos 0) = 0 and there is then no escape from the use of the correct time delay.

THE GENERATION OF SOUND BY TURBULENT JETS

153

The alternative approach of using the stationary-axis relation (6.14) is quite valid despite the presence of the spurious convective frequencies. Actual calculation (to be given later) shows these frequencies do not appear in the radiated sound spectrum. It is, however, necessary to use the correct nonzero value of the retarded time (6.15). Since this is ultimately necessary in the moving-axis approach to obtain the complete picture, the pros and cons for the two approaches become rather blurred. We proceed now to the determination of a more accurate convection factor as the dominant element in the directivity of the radiated sound. 2. Directivity of Sound from Unit Volume

The acoustic power radiated in direction 8 (per unit solid angle) from unit volume of turbulence at y in a round jet is designated P(8,y). From (6.14) this is (7.4)

5

P(e,Y)= ( 1 6 7 ~ 2 p ~ c ~ 5 ) -at4 1a4 R0 (Y*6*t.)d3E a3

where on the dilatation theory (7.5)

+

Ro = Rp = pC0)(y e/2, t

+ t ) p ‘ ’ ’ ( ~- 6/28 -___-

t)

and on the quadrupole theory --

(74

Ro = Rx = pux2(y

+ E/2,t + ~ ) p ~ x 2-( yc12, t )

in the version due to Proudman. (These alternate forms of the correlation Ro are different functions that may even lead to different values of P(8,y); however, the respective integrals of P(8,y)over a ‘slice’of jet must agree [I 11. The form R, may show a dependence on the direction x not permissible in R$.) Neither of these correlations is known experimentally for a turbulent jet (but see [4]). In order to obtain useful results we shall follow [lo] and [ll] in postulating a convected Gaussian form. This is an improvement over the moving square-box correlation with zero time delay first used by Lighthill [2]. Later we shall see how a still more plausible form can be inferred in the case of (7.6). For the present we assume that (7.6) has the form

wherein p has been replaced by its local mean p. This equation exhibits the general character of a convected fluctuating pattern shown in Fig. 21(b). A typical radian fluctuation frequency is of. An effective ‘eddy volume’ or low-speed correlation volume is the value L3 of the volume integral of (Rs/p2T2’)d36 when t is set equal to zero. L may be interpreted as a

154

H . S. RIBNER

directional-average scale of turbulence. (The scale anisotropy of jet turbulence may be simulated by the use of longitudinal and transverse scales L,, L,, L, [13, 111 but the refinement seems unwarranted in view of the basically oversimplified form represented by (7.7).) The integration of (7.4) with Ro given by (7.7) and with the time delay therein specified from (6.15) as (Fig. dl(b)) cot = t1cos 8

(7.8)

+ & sin 8

yields the sound power radiated in direction 8 (per unit solid angle) from unit volume at y as [ l l ] (7.9)

where C is a modified convection factor (7.10)

c = [(I - M,cos

+

0,2~2/nC02~1/2.

An alternative form is (7.11)

C EZ [ ( l

- M,cos 0 ) 2+ a2Mc2]1/2

where a is defined by comparison of (7.10) and (7.11). (A constant a implies frequency cc velocity/scale, L.) The nonsingular convection factor C-6 replaces the idealized factor (1 - M , cos introduced by Lighthill [ l , 121. The directivity or &dependence of P(O,y), is dominated by this factor (Figs. 14, 15). The simple expression (1 - M , cos 0)4 exaggerates the variation in directivity by up to ten decibels (factor of ten) at turbojet flow speeds (Fig. 14). These matters were discussed in detail in Part A. The same result (7.9) can be obtained from the moving-axis formulation [131. In that case the correlation R, is referred to the moving frame as Ro,mby omission of U , t in (7.7), and equation (7.6) must be integrated with the time delay (7.12)

cot = (1 - M , cos O)-1(E1 cos 0

+ & sin 0 )

obtained from (7.3). Upon evaluation, the integral is proportional to (7.13)

(1 - M,cos 8)6C-5.

By virtue of the (1 - M , cos 8)-5 factor outside, the final convection factor is jtist the C-5 of (7.9). Thus the moving-frame integral with time delay possesses a zero that exactly cancels the (1 - M , cos /3)-s singularity. (This zero follows from (7.3) which implies that the path of integration in the E, - U c t plane lies along the Uct-axis for 1 - M,cos 8 = 0. Since the integrand is a t-derivative that vanishes at infinity, the integral vanishes.)

THE GENEHATION OF SOUND BY TURBULENT JETS

155

3. Spectrum of Sound from Unit Volume The contribution to the acoustic power P(B,y) from radian frequencies in a unit band width a t UJ has been designated P(B,y,w). The procedures in Sec. VI.3 for computing this spectral density, given the correlation function R,, are valid and straightforward. However, to simplify the integrations involved it has proved expedient to employ the modified approach that follows. The Gaussian correlation R, assumed in (7.7) corresponds to a convected turbulence pattern random in space and time, the time correlation being exp - w f 2 t 2 . We define a corresponding correlation that represents a single-frequency oscillation when viewed in a frame moving with the eddy convection speed U, : - 3 (7.14) R , ( y , r , t ) = j2(%

+ + t32]} cos coot.

)2 {exp - x L - ~ [ ( &- U C V FZ2

It follows that R, can be written as a weighted integral of

R,,

[ ( w t v3-1exp - ~ U , ~ / ~ U J / ~ I R A U J ~

(7.15)

where the weight factor in square brackets is the Fourier cosine transform of esp - o ~ , ~ t 2 . Suppose now the spectral density P(B,y,cn) associated with the singlefrequency correlation (7.14) has been determined. Then the spectral density associated with the random pattern described by (7.15) is

(7.16)

i

p(e,y,w) = [(o,1/7c)-1 exp - U

J,~/~UJ~~~P~B,~,UJ~~UJ~

0

-+

The spectral density P(f?,y,OJ) is related to the autocorrelation B ( x , t ) j ( x , t t’) by a transform of the form (6.18). The required autocorrelation, for generated by unit volume of turbulence at y, is

according to (6.19), with c o t = x/x. The integrations involved in (7.17), (6.18) and (7.16) have been carried out in 1113 for the special case of a line distribution of ‘turbulence’ ([, = & = 0). The procedures can be generalized to nonzero E2 and t3by use of the transformations given in Appendix E of [I13 up to the analog

156

H. S. RIBNER

of equation (E 14). The results may be expressed in terms of P(8,y) (equation (7.9)) as

with the abbreviations (7.20)

0 z 1 - McCOS8,

(7.10)

The &function in (7.18) confirms an expected result: the source pattern of single frequency oo moving with speed U, = Mcco generates sound a t the Doppler-shifted frequency wo/@ 14, 10, 11, 131. The spectral density (7.1!1) of the sound power in direction 8 from unit volume may be recast in more convenient form as (7.21)

which may be abbreviated to (7.22)

P(e,Y,w) = P ( e , Y ) w P - w o i o p ) .

This exhibits the spectrum as a multiplier of P(8,y) having a unit integral and peaking at w =: wp where (7.23)

wp = 2

V5o,,/c.

A plot of the spectrum function P(B,y,w) versus o is shown in Fig. 16. Comparison of the curve C = C << 1 with the curve C = 1 illustrates the effect of convection contained in C. The peak is increased by the factor C-4 and the breadth by C-l; thus the power in direction 8, corresponding to amthe area under the spectrum, is augmented in the ratio C-5. This plification has already been noted in equation (7.9) for P(8,y), which is just the integral of P(B,y,w)over w. The shift of the spectrum peak is C-l, which is less than the Doppler shift 1 0 1 1= 1(1 - M, cos 8)l-l according to (7.10). The reason lies in a progressively decreasing efficiency of radiation as the ratio wave length/‘eddy size’, L, decreases. This modifies the Doppler-shifted spectrum shape so as to move its peak somewhat to the left [lo, 111.

157

THE GENERATION OF SOUND BY TURBULENT JETS

4. Moving Jets

We have been treating jets in an ambient medium at rest. The flight of a jet aircraft provides a relative motion parallel to the jet flow. For a given jet velocity this modifies the mixing, lengthening the potential core [73-751. The changes in the jet dynamics are incompletely known, and it will be expedient to ignore them. Thus we shall discuss only how the outer mean motion (due to flight) modifies the sound emission from a specified unit volume, following [ll]. The governing equation (6.6) refers to a frame at rest in the medium. We transform to a frame attached to the aircraft relative to which the medium has a uniform velocity U, toward the right. The governing equation becomes

where again a uniform speed of sound ($)*/' = co has been assumed. When the stream speed Uo is limited to subsonic values this has the solution (e.g., [37], Eqs. (3.5.0) and (3.5.2)) (7.25)

where

,.

t^ = t + [M&l (7.26)

9

- Yl)

-~1/C0PO2~

= {(xl - yJ2 + PO2[(x2- y2)2 + (x3 - y3)2]}1/2

with

Po2 E 1 - U0'/c 0 2 -= 1 - M 0 2. Evaluation of the mean square pressure involves the two-point spacethat figures in u. We shall time correlation of pseudosound pressure assume the convected Gaussian form

R, = p e x p {-

n ~ - 2 [ t,

(v,+ ~ , ) i ] 2- ~

-

-w

~ - 2 6 ~n ~2 - 2 ~ , 2

, ~ )

(7.27)

which is a modification of (7.7). The velocity of the eddies relative to the fluid outside the jet is U,; the fluid has an additional velocity U , relative to the reference frame. The effective convection velocity is therefore U, U,. The formalisni in terms of a general R, is set up in [ll] and the calculations are carried out for the specific form (7.27). The result for the mean square sound pressure radiated in direction 0 from unit volume is

+

-

(7.28)

p y x , e )=

+

3014p72L3 1 M , cos 0, 4n2c04x2 (1 - Mo2sin20)C6

158

H. S. RIBNER

where 8, is an angle related to 8 by (7.29)

cos 8, =

cos 0 - M,(1 - MOasin2 r3)1/e (1 - Mo2 sin20)1/2- M , cos 0

and C is the familiar convection factor (7.10) or (7.11). In a numerical example we compare the case M , = 0 (.- jet noise in jet noise in a Mach number 0.8 fluid at rest) with the case M , = 0.8 (4

Stationary 'jet'

FIG. 22. Comparison of directional sound patterns of simulated stationary and moving supersonic jets. M , = Mach number of external uniform stream ( - motion of jet nozzle). M + M,, = Mach number of convection of eddies through effective jet volume V (based on external sound speed c"). Local normal to Mach cone

origin

FIG. 23. Construction to explain sweepback of noise peak of moving jet.

stream) for a constant relative eddy speed M , = 2.0 (.- excess of jet speed over stream speed). The results are shown in Fig. 22 as curves of rms sound pressure versus 8 for a fixed radial distance x .

THE GENERATION O F SOUND BY TURBULENT JETS

159

The sweepback of the noise peak results from convection of the wave pattern. The peak at 8, is normal to the Mach cone and satisfies M , cos 8, = 1. The angle 0 of the M , = 0.8 peak is related to the angle 8, of the M , = 0 peak by the construction shown in Fig. 23. 1Jpon deletion of the cross-hatched Mach cone and its normal Fig. 23 applies to directions other than the noise peak. The figure then shows the relation between the angle 8 measured from the actual origin (in a frame attached to the moving aircraft) and the angle 0, measured from a certain effective origin, at the timr of signal reception. A t the earlier time of signal emission the jet nozzle was located back at the effective origin. By the geometry of Fig. 23 it can be shown that (7.30)

+

x2(1 - MO2sin2 0) = xC2(1 M,cos 1 9 , ) ~ .

If the left-hand factor is replaced by the right hand factor in (7.28) the latter takes a form given (for the directional part) by Williams 1131. Williams’ formula thus applies to the geometry a t the time of signal emission and our formula (7.28) to the geometry a t the time of signal reception. The former gives a ‘snapshot’ of a transient sound pattern whereas the latter remains statistically steady in time (for fixed .Y) relative to an origin in the moving jet. A feature associated with the moving origin is the sweepback of the sound polar exhibited in Fig. 22. A corresponding polar reckoned relative to the stationary ‘effective origin’ of \Villiams (for fixed x,) is unswept. VIII. REFRACTION EFFECTSDUE TO

THE

MEAN FLOW

The material of this Section was developed in [!)-11]. itative notions were given in [15] and [76].

Earlier qual-

1. Convected W a v e Equatioii

The source term in the Lighthill wave equation is (8.1)

a 2 ( p ~ ~a>fiaYl t~~)/

in the inviscid approximation, where the space coordinate has been changed from x to y. This same term also governs the pseudosound pressure pi0) that figures in the dilatation theory (cf. ((5.4)). In the low-speed approximation the density p of (8.1) is treated as a constant po, an incompressibleflow assumption. Actually, however, the density derivatives can be important at the higher flow speeds, and even at low speeds they account for refraction effects. Thus (8.1) may be expanded as [lo, 11, 23, 411

160

H. S. RIBNER

which shows how the flow velocity weights the density derivatives. If this is inserted - into the wave equation (6.2) together with the isentropic relation d p M c2dp there results (8.3)

where

DiDt E ajat -+ uaiay, and the subscript U = 0 implies evaluation in a frame moving with the local mean velocity U ; in this frame density derivatives are negligible. (U is to be set equal to zero after applying the differentiation : ui, ui are still referred to a frame stationary with respect to the jet nozzle.) Equation (8.3) is the wave equation referred to a medium with local velocity U : a convected wave equation. It is well known that if U = U ( y 2 ) this equation will predict refraction of sound waves. (More generally, U may be taken as U ( y 2 , y , ) . ) The quasi-incompressible assumption p = p, in (8.1) effectively suppresses this refraction by elimination of the convective terms, i.e. the density- (or pressure-) derivative terms, from the wave equation. The right-hand side of (8.3) does not lend itself to reduction to quadrupole form by application of Gauss’ theorem. Therefore we resort to a modified dilatation formalism to simplify the source term. The definitions

inserted in (8.3) yield the convected-wave form

of the dilatation equation. The source term on the right-hand side is the time derivative of p:) following the mean motion. Here p:) is a modified pseudosound pressure defined by (8.5). It can be argued [ll] that p:) approximates the local pressure field within the flow up to much higher Mach numbers than does the original pC0) of (6.4). Moreover, the replacement of the right-hand side of (8.5) by incompressibleflow values is here a much better approximation than for pcO). The remaining pressure )#! in (8.8) may be called the acoustic pressure. Whereas the pseudosound p:) is dominant in the near field, the less rapidly decaying is dominant in the far-field. Equation (8.6) is thus an equation for the far-field pressure.

pi)

THE GENERATION OF SOUND BY TURBULENT JETS

161

2. Form of Solution

The corresponding equation for an acoustic source of unit strength a t y emitting impulsively a t time to is

(C“)-lDg/Dt2 - P g = 6 ( x - y)6(t - to), The solution, the Green’s function g(x,tly,t,), gives the sound pressure a t point x and time t due to this point source. The source strength on the right-hand side of (8.6) can be expressed as an integral of the right-hand side of (8.7). Because of the linearity the solution of (8.6) for the acoustic pressure fi2) can be expressed as a similar integral of the Green’s function t+n

On physical grounds we expect the Green’s function to express a spatial distortion of the spherical sound pulse characteristic of the ordinary wave equation. Thus we write it in the form of a time pulse

in terms of a known retarded time t’(x,y,to)obtained in the solution of (8.7). The integral (8.8) reduces to (8.10)

1-

&)(x,t) = - (4n)-’ [ ( ~ ~ ) - * D ~ p / D t ~ ] ~ , t f G ( ~ l y ) d ~ y V

where .? may be a(y)as in a hot jet or jet of foreign fluid. Now what can we say about the Green’s function G short of actually solving for i t ? Outside the jet where U has fallen to essentially zero (8.7) reduces to the homogeneous wave equation. By Kirchoff’s formula the sound pressure radiated outside some control surface S‘ enclosing the jet may be determined by (8.11)

![

1

i 1 ap ar p 1 ar ap dS‘ +(x,t) = - - -- +--+--4n r an an r2 cor an at S‘

that is, as the emission into air at rest from a certain virtual distribution of sources and dipoles on S’. This equation may be applied to the emission from a point source a t y within S’ to obtain G ( x / y )for an external point x. (Kirchoff’s formula is applicable as well to the entire aerodynamic sound field generated by the jet [77] ; this is a point of some practical significance.)

162

H. S. RIBNER

In the far field, defined by x >> dimensions of S' and 2nx >> wave length, the distance I = IX -- yl in (8.11) approaches x . The pressure p radiated by the surface distribution approaches asymptotically an x - l dependence and depends also on the direction of x. Thus we may write Far

(8.12)

Field

where 0 is the polar angle between x and the jet axis and # is the azimuth angle. For a circular. jet x may be restricted to the zl,x2 plane with no loss in generality because of the axisymmetry. TheJ! , I dependence then disappears, 3. Qualitative Effects on Directivity The factor K(O,i,h,y) of the Green's function describes the directional distribution of sound pressure in the far-field radiated from a point acoustic source at point y in a specified jet flow: the refractive and diffractive effects of the flow are embodied in this directivity function. The analytical determination of K(e,#,y) is a very formidable task (see [78] and [79] for similar problems) and only very idealized cases have been treated [80-83]. M.1.2

o0

1

M

-i -I \Source

FIG.24. Idealized example of refraction of sound by a flow field: transmitted directivity pattern for oscillating line source near a velocity discontinuity. P. Gottlieb from ms of [80].

(Supplied by

Thus consider a two-dimensional oscillating acoustic line source lying cross-stream in an infinite plane jet (jet width >> wave length). Such a source can be built up from plane waves disposed radially like the spokes of a wheel [84,851. Previous papers [76, 45, 461 show that these waves are refracted outward and forward (upstream), leaving a wedge-like zone of silence opening downstream. Gottlieb [80] has obtained an analytical solution for the closely related problem of an acoustic line source in a semi-infinite plane flow: the source lies cross-stream and parallel to the interface (cf. also [81-831). Figure 24 shows the directional distribution of the refracted sound pressure at large

THE G E N E R A T I O N OF SOUND BY TURBULENT JETS

163

distances from the source. The curve segments labelled A = M, 1 = 2nh, 1 = nh lie in the wedge of silence; they represent diffractive “leakage” of sound into this zone for small source-to-interface distances h. It is expected (in the absence of further curves on Gottlieb’s figure) that the penetration must approach zero as 1 + O . In a real jet the effectively short axial length of the jet measured in wave lengths must greatly weaken the refraction: certainly the null at 8 - 0 will be modified to just a minimum [15, 9-11]. The qualitative directivity provided by the refraction factor K(O,$,y) will presumably resemble a figure of revolution developed from Fig. 24 with a deep dimple instead of a null as 8 0. For off-center source positions the pattern will be distorted, but the average for all source positions will be symmetric in 8 for a round jet. Once the refraction factor K(O,$,y) has been determined analytically or numerically (this goes beyond the present state of the art), it is possible to modify the jet noise computations to allow for the refraction. Formulas in terms of K for computing the mean square value of the far-field pressure, its autocorrelation, and spectrum are given in [ I l l . -3

IX. IMPROVED MODEL: ISOTROPIC TURBULENCE SUPERPOSED ON MEAN FLOW I . Remarks

011

Lilley’s Approach

The most comprehensive and illuminating development of the Lighthill theory (except for the convection effect) has been made by Lilley [4]. A particular feature is the derivation of the sound radiation formula in terms of measured and estimated properties of a turbulent jet (cf. also [SS]). The resultant expressions can be integrated across and along the jet to give the radiated intensity without empirical constants. “The calculation is divided into the contribution from ‘self-noise’ in the turbulence and the interaction between the turbulence and the mean shear. For the first part of the calculation Proudman’s results are used while for the second an approximation is developed for the +/8t covariance in incompressible shear flow turbulence. These estimates for the sound intensity from a low speed jet are then modified to include the effect of eddy convection velocity so that they can be applied to the case of a high-speed jet” [4]. Of special interest is the extensive analytical work in the two appendices. The first one develops the formalism for the ap/at covariance and the second derives the spectrum of the noise from the mixing region. A four-dimensional Fourier-transform technique is used. The reason for adopting here an alternative approach lies in part in the discovery of a coherent treatment that deals with the ‘self-noise’ and ‘shear-

164

H. S. RIBNER

noise' together. I t results in a theoretical directivity that agrees much better with experiment (with allowance for refraction), eliminating the marked dip at right angles to the jet that characterizes the shear-noise derived from +/at times shear. Furthermore, the a+/at approach incurs an undetermined error in omitting a space derivative that is asserted to radiate little sound (cf. Sec. VI.4). Another feature of the new approach is the resolution of a paradox concerning the shift of the spectrum with direction of emission. There is further motivation coming from specific objections to portions of Lilley's work. His spectrum from unit volume (for which the theoretical basis may be questioned) is less peaked that the one obtained herein; a superposition of these will not be convex enough to fit the observed shape of the spectrum from a complete jet. Moreover, the peak frequency in the unit-volume spectrum is in effect assumed rather than related by derivation to the dominant frequency in the turbulence. Still other points are brought out in Section VI.4. The perspective provided by Lilley's work and certain of the ideas have been of considerable help in the development of the alternative approach reported herein. In particular, the notion that the formulas can be expressed in terms of, for the most part, existing types of hot-wire data for a turbulent jet has been of much value. The notion of 'self-noise' and 'shear-noise' has been carried over but reevaluated on a different basis. 2. Self-Noise Plus Shear-Noise

We return to the formula for the directional power from unit volume at y, (7.4) where R, is taken as the Proudman form (9.1)

R,(O,y,g,t) = F2ziX2zixt2

+

+

in which zi, refers to the point Q = y 512 a t time t t and 4,' refers to the point Q' = y - ?$2 at time t. This has been modified from (7.6) by over u, with the approximating p by its local mean p and by placing meaning A

(9.2)

zi, = u,

+ ux

where U , is the local time-mean velocity component in the direction of x and u, is the turbulence component. The expansion of (9.1) gives (9.3)

+

__

+

R,(B,y,e,t)= p2[ux2ux12 4UxUx'uxuxl other terms].

165

THE GENERATION OF SOUND BY TURBULENT J E T S

Three of the ‘other terms’ are time-independent and differentiate out in (7.4). The others are zero or virtually zero, depending on the assumed statistics. Hence in what follows the ‘other terms’ in (9.3) may be neglected. To make progress we shall replace the real nonuniform turbulence by turbulence postulated as homogeneous and isotropic within a correlation volume. The statistical values may, however, vary from one correlation volume to another; that is, they may vary with the position y, the centerpoint of the correlation volume over which Q and Q’ are allowed to range. The isotropy ensures that u x u i is independent of the direction of X, when expressed relative to-axes one of which lies along x. If this axis be chosen _ _ as the 1-axis, then uxu; = ulul’. We require further that the joint probability density of u, and u,’ be Gaussian. Then it can be shown that

(9.4)

-2

uZ2ui= 2 u12

+ 2R12,

where

R,

~ G

_

_

_

u X u l G ~ulul’.

Also U , = U cos 0, where 8 is the angle between the observer vector x and the local mean velocity U. Thus with the assumed flow model and the ‘other terms’ neglected

(9.5)

jF2Rx(8,%,t)=

q2+ 2R12(S,t) + 4R,(P,t)UU’(Q cos28 self-noise shear-noise

where the y-dependence is, to simplify the notation, no longer explicitly written in. The significance of the descriptive terms (adapted from Lilley [ 4 ] ) will emerge later. The kinematics of homogeneous isotropic turbulence dictate, in a broad way, the general form of R,. Equation (9.5) with such a form for R , inserted then replaces the earlier Gaussian form assumed for R,. This is definitely is a pure an improvement, since a Gaussian correlation of hX2and assumption with no physical basis. Equation (7.4) for P(8,y) refers to a stationary reference frame. I t will be more convenient, however, to employ a frame moving with the local convection speed U , = M,co, in which R, takes its simplest form. The Lighthill transformation (e.g. as in (7.2)) allows the a4/at4 operation in (7.4) to be carried out in the moving frame and applies a multiplicative factor (1 - M , cos 8)-5 and an exaggerated time delay. We shall reduce this factor to unity by allowing M , to approach zero so that (7.4) is formally unaltered, being changed only in interpretation. (The effects of convection a t finite M , will be approximated later by an artifice.) With this low-speed stipulation it can be argued that the ratio (eddy size)/(wave length of sound) is small compared with unity. This implies that the time delay is negligible throughout the volume - approximately the correlation volume - that makes the major contribution to the integral.

166

H. S. RIBNER

Accordingly we shall examine (7.4) in the moving frame with zero time delay. Moreover, in this frame it would seem reasonable to assume that R,(g,t) is factorable into a space function and a time function; more specifically we shall assume that exp - wltl

RlK4 =

(9.6)

which approximates experimental data ; the parameter wf is a typical radian frequency of the turbulent fluctuation. With these assumptions the directional power P(8,y) from unit volume of turbulence is proportional (with factor to

m

W

shear-noise

self-noise

upon carrying out the differentiation in (7.4) and afterward setting t = 0. The theory of isotropic turbulence [71] gives the general form of Rl(Q as (9.8)

R I ( ~= ) ?[f 4- (5z2

h2)f’/261

where f = /([) is a spherically symmetric function. We shall assume the kinematically possible form (9.9)

t 2= El2

/ = exp (- nE2/L2);

+ 622+ t32

adjusted to a turbulence scale L ; equation (9.8) becomes (9.10)

R,(Q = ~ ~ - (nE2/L2) ~ [ 1 (nE12/L2)] exp (- nE2/L2).

+

I t is now possible to assess the two terms (which are proportional to noise power) in (9.7). If the mean flow were uniform VV’(5)would be a constant. Then the ‘shear-noise’ term would reduce to a constant times the volume integral of Rl(g),uhich vanishes. It follows that the contribution of a uniform flow to noise generation is zero. The contribution is nonzero only when the mean flow is nonuniform - i.e., possesses shear. Hence th.e use of the expression ‘shear-noise’ term. In comparison the ‘self-noise’ term in (9.7) does not vanish, but with the assumption (9.10) comes out to be (9.11)

self-noise term

-2

= 2- 3/2014u12L3.

I t remains to approximate the comparative magnitude of the ‘shear noise’ term with UU’(g) appropriate, for example, to the mixing region of

THE GENERATION OF SOUND B Y TURBULENT JETS

167

a jet. To this end we rotate the axes of (9.10) so that the 1-axis lies along U with which x makes the angle 8:

(9.12)

A crude approximation to UU'(8) for points near the middle of the mixing region may be made in the form (9.13)

U U ' ( 8 )= U 2exp (- unE22/L2)

if we ignore the wedge-like shape of the mixing region and treat it as a parallel flow with the correct width at y. This should be a reasonable approximation over the limited effective volume (the volume of an 'eddy') covered by the integrals in (9.7). Upon carrving out the integration there results (9.14)

+ 0 ) - 3 / 2 w , 4 U 2 ~ L 3 13

shear-noise term = 8-%(l

COS~

I t follows that the two kinds of noise from a volume element in the mixing region are in the ratio (9.15)

1/% V 2 cos48. + 7

shear-noise - a self-noise 4(1

u)3l2

Laurence's data [39] for shear in relation to the lateral scale suggest an effective value for u of the order of 0.45. If this is applied together with u12/U2= (0.28)2there results (9.16)

shear-noise = 1.16 cos4e self-noise

on the basis that the zone of peak turbulence in the mixing region dominates in sound generation. The numerical factor is not sensitive to the value of a. The writer regards this result, aside from the variation, as indicating no more than an order-of-magnitude agreement between the shear-noise and the self-noise. The uncertainty arises from the joint uncertainties in the forms taken for R,(Q (e.g., the deviation from the assumed isotropy) and U U ' ( 8 ) : the fact that the volume integral of R,(Q by itself is zero makes the nonzero integral of the product UU'R, quite sensitive to the assumed forms. The experimental noise directivity curves (Fig. 14) - the details were discussed in Part -4 - appear to be compatible with a co& content of the order of (9.16).

168

H . S. RIBNER

3. Directivity and Spectrum of Sound from Unit Volume

The improved model of turbulence has led to a better estimate of the sound power P(0,y) radiated in direction 8 (per unit solid angle) from unit volume of turbulence of y. The results of the last section suggest that equation (7.9) should be modified to read

where B ( y )U2/uT is estimated as about unity for a point in the center of the mixing region. Because of the difficulty and uncertainty in estimating the it is thought that the assumption value of B(y) 3 a1/2/4(1

+

(0.18)

is a sufficient approximation for the present state of the art. The convection factor CS, where (7.10)

c = [ (1 - M,cos 0 ) 2 + w,2L 2/nco2]1'Z

has been carried over without change from (7.9); it provides a generalization of the low-speed results of the last section (wherein C is replaced by unity) to allow for convection at nonnegligible Mach number M,. This C was calculated on the simplified model of turbulence of Section VII, and a new calculation with the improved model of the present section would be expected to show some change. I t is thought, however, that the change would be minor, as (7.10) seems already to exhibit the major features expected on physical grounds and has other evidence (Part A) to recommend it. The factor (1 cosW) accounts for the basic directivity in (9.18) and the factor C-s for the downstream beaming due to convection. The effects of refraction, discussed qualitatively in Secs. V.5 and VIII.3, are not included. We may approximate the spectrum of the shear-noise term of (9.18) by the bell-shaped curve (7.21) peaking at

+

(7.23)

wp

=2

vw,/c.

The corresponding spectrum of the self-noise term will peak an octave higher at 2wp. This frequency doubling arises from the presence of the square of the turbulent velocity component in the self-noise (cf. (9.6) and (9.7)). The spectral density may be written out as

in the abbreviated notation of (7.22).

1ti9

THE GENERATION OF SOUND BY TURBULENT JETS

Like C,this spectrum was derived for the simplified model of turbulence, not the present model with correlation (9.6). The overall spectrum of the jet, however, is not sensitive to the shape of the spectrum from unit volume, so long as it is sufficiently peaked. It suffices if the peak frequency is properly located and the area is normalized to unity. In fact, it will appear later that the bell-curve (7.21) and the &function spectrum of Part A yield the same f 2 and f-’ asymptotic laws for the complete jet spectrum.

X. SOUNDEMISSION FROM

A

C.OMPLETE JET

1. Integration Over a ‘Slice’ of Jet The acoustic power emitted in direction 8 (per unit solid angle) from a volume element d3y a t y has been approximated in (9.18) as (10.1)

FIG. 23. System of coordinates for mixing region. For the developed jet the y,-axis is shifted to the jet axis.

We shall choose coordinates as in Fig. 25 [4j with y1 measured downstream along the mixing region centerline and yz radially outward. Then an annular volume element appropriate to a round jet may be written (10.2)

d3y = n(D

+ 2yz) dy2dy, = dS dy,.

-

It will now be convenient to place a over the variables in (10.1) to designate values at arbitrary radial positions y2. Values without will refer to that value of yz where P(8,y) peaks for given y l . Then (10.1) mav be rewritten in terms of ratios

-

where the proportionality factor is (2s”%2)-1. The frequency oris a typical radian frequency in the turbulence; it is defined by approximately matching

170

H. S. RIBNER

a function exp - W t l t l to the experimental autocorrelation of u1 in the convected reference frame. This frequency will vary with the position y of the volume element of turbulence dSdy,. Approximate experimental values at several axial stations are given in Sec. V.1. For a given jet operated a t a given nozzle velocity U j and temperature etc., f and g are functions of position alone: f = f(y,), g = g(y2,yl). The function g(y2,yl) is a measure of the radial distribution of the effective sound sources. Lilley first introduced a similar function and showed that it is a narrow function of yz peaking sharply a t the center of the annular mixing region 141 ; further downstream the peak moves toward the center of the jet. For fixed y1 the integral of g(y2,yl)dS measures the effective annular (region A) or circular (region B) area S(yl) occupied by the sound emitters in a ‘slice’ of jet d y , a t yl, considered as replaced by emitters of the (radially) uniform strength f(yl). Thus we may write

where P(O,y,)dy, is the power emitted from the slice dy, in direction 8 from the jet axis. The variables on the right-hand side are functions of y1 alone, being taken a t that value of y 2 for which g(y2,yl)peaks in the radial direction. 0

2. Idealized Source Distribution: the

Y oand Y-7 Laws

For the idealized model of a jet discussed in Part A the function g(y2,y,) is of the form g(y2/y1) (with suitably chosen origin for yl) in regions A and B. (Strictly speaking, d / c cannot scale with y1 in B, but this will be ignored in the present development.) With dS taken as n D d y , (an approximation) in A and 2ny2dy2 in B the integral of g(y2/y1)dSover a ‘slice’ of jet gives (10.5)

y,D in A y12 in B

(mixing region), (developed jet).

With a function differing somewhat from g(y2,y1) Lilley obtained in effect roughly 0.15 as the proportionality factor for region A. Thus he showed that the chief noise-emitting eddies are confined to the central annular mixing region of the jet [4]. The idealized jet has an assumed similarity given by * U 2 ,uf cn UIL, L * y1 with perhaps different proportionality factors in regions A and B ; and, additionally, U U , in A and U i D / y , in R, and C is taken as constant within each region. With these relations and those for S(yl) equation (10.4) yields

2

4

171

THE GENERATION OF SOUND BY TURBULENT JETS

('Oe6)

p(8,y,) ~1

+ cos48

p2ui8D { Y o in A poco6Cs Y-7 in B

(mixing region), (developed jet),

where Y has been written for the distance along the jet axis in nozzle diameters, y J D . When P ( 6 , y l ) is integrated over a unit sphere the factor (1 cos48) yields a weighted average convection factor (C-')Av. Thus the total noise power emitted by a slice of jet at y , satisfies the proportionality (10.6) with (C-S)Av instead of Cs. This may be written in the form

+

(10.7)

where the directional average convection factor (C-s)Av is taken to be a constant. (In a real subsonic jet (C-s)Av is constant in A , probably rises someY 12, and thereafter decays to near unity.) Equawhat in the range 5 tion (10.7) shows that the noise source distribution for the idealized jet is constant along the mixing region (like Y o ) with a progression to a very rapid fall-off (like Y-') in the developed jet (Fig. 10) (cf. [8, 4, 61 and Part A herein). The major contribution to the total noise power P comes from region A where (C-5)AY is constant with Y . Thus P is approximated well by integration of (10.7) with (C-s)Av assumed constant at the value for region A,

< <

where G is the proportionality factor. G (which depends e. g. on jet spreading [94]) is constant for a given jet specified by particular parameters : e.g., nozzle velocity Ui, temperature, density (aside from Pa/po2), initial turbulence and swirl, combustion effects, etc. Thus G may vary as these parameters vary. This is more apparent if the assumed are restated in more precise form; e.g., - similarity relations write ula/U2= F, in place of u12 U2,where F, is independent of y , within A and (with a different value) within B, but may depend on the above cited parameters. Experimental evidence C39, 871 (Fig. 20) suggests that F , m (Uj/Uref)-l'z at the higher subsonic speeds. (It is not clear whether this dependence is dominated by Reynolds number or Mach number; hence the reference velocity Urefis not identified.) Evidence [40] is sketchy as to ofLIU. I t would seem that F , may dominate G, giving G m (Uj/Umf)-l. This would Uj7 in (10.8) in place of Ui8. This provide an effective speed dependence would tend to offset the rise above Uj8 provided by the factor (C-')Av at subsonic values of M , as discussed in Sec. V.7. 4

4

172

H. S. RIBNER

3. Idealized Spectrum Shape: the

f2

and f - 2 Laws

The spectral density of the sound power emitted in direction 6 (per unit solid angle) from a slice of jet at y1 will be obtained as the first step. the time scale V. &,-l of the turbulence correlation Experimentally [a] function is nearly constant through a slice in the mixing region. Thus when the integration of (9.19)over a slice of jet is carried out, as was done for (10.4), the spectral shape for the slice approximates the spectral shape for the element. The slice spectrum is therefore a generalization of (10.4) (10.9)

+(we,Yltw/~)

P(e,Y,,w) = W , Y , , ~ ) coS4e

in terms of the basic spectrum

where wpLC/U has been written in place of w,LIU with a change in the proportionality factor to (215%2)-? The spectrum P(8,w) from the entire jet is the integral over the slices along y1; the corresponding @-integral or basic spectrum may be designated @(O,w). With the similarity assumptionsp U 2and (in effect) w p U / L C that have been made for bath regions A and B of the jet this is V.

V.

(10.11)

The variable of integration may be changed from y1 to wP-l and the slice spectrum F(w/wp)may be inserted from (7.21) and (7.21)

The contribution of the mixing region A to the spectrum covers the integration range 0 to a in y, and correspondingly 0 to a, say, in w/wp.The U j , C = constant similarity assumptions for region A are L V. yl, U together with S(yl) ylD from (10.5). These relations give VI

V.

(10.13)

@(e,w)A

fs2UjsD w-2 [(E,”.xp(POCOSC6

2$)d(E).

~

0

The integration yields (10.14)

@(O,w), VI

P0Co6C6

+ 2a2 + 1) exp (-

w 2 [ 1 - (2a4

2a2)]/8.

173

THE GENERATION OF SOUND BY TURBULENT JETS

The bracketed expression is virtually constant (< 1.5% variation) in the Thus we may write 00 (a2 2). range 2wp(a) w

< <

(10.15)

The contribution of the developed jet, region B, to the spectrum covers the integration region b to 00 ia y1 and correspondingly p to 00 in w/wp. The similarity assumptions are L yl, U VI U j D / y l , C = constant (the incompatibility of a variable U with a constant C is ignored for the present purpose). Application to (10.9) gives +-

(10.16) P

The integration yields (10.17)

The bracketed expression is nearly constant (< 8yo variation) in the range 0 w 0.2 wb(b) ( p 0.2). Thus we may write

< <

<

(10.18)

Evidently the contributions of regions A and B can overlap, if a t all, only in the middle zone 0.2wp(h) w 2wp(a). In the low-frequency range @(O,w) reduces to @JB and in the high frequency range to @". The results (10.15) and (10.18) can be expressed more symmetrically by introduction of a reference radian frequency w* = 2n/* UJD:

< <

~1

(10.19)

( w c / w * ) 2=

qe,w)

(/c/f*)2,

(wC/w*)-2 = (/C/f*)-2,

0

< < 0.2 wp(b)} < < .

w 2wp(u)

0

00.

This w* is really a special value of C o p = 2v%0,; hence w* = 2V%wf*. These /2 and f 2 laws were first derived by Powell [6] by simplifying the mathematics as though each slice of jet generated a single characteristic frequency and a corresponding &function spectrum (cf. Part A). Here we have obtained the same laws on the basis of the bell-shaped slice spectrum arrived at analytically. A semi-empiricp1 spectrum with the correct asymptotic behavior is (10.20)

H(Y)

= Y2/(1

+

@)2;

Y = w c / o * = fC//*

174

H . S. RIBNEH

which may be used in place of { ] in (10.19). In terms of (10.60) the spectrum of the complete jet may be written (cf. (10.8)) (10.21)

as a tentative approximation to cover the entire range of w . The reference frequency /* = (V2/n)w,* appearing in v identifies the peak of the spectrum H ( v ) and is associated with a certain slice of jet y1 =: .y,*. The jet spectrum function (10.21) embodies two spectra peaked an octave apart as shown in Fig. 17. The relative proportions depend on direction of emission in the proportion 2 c0s4 0 to one. The convective amplification C-6 contained in the directional sound power from a slice of jet appears as a vertical shift C-4 plus a Doppler shift C-l in (10.21). This follows from the corresponding behavior of the elementary spectrum from a single slice of jet exhibited in Fig. 16. Equation (10.21) has been derived under the assumption that C is constant along the jet. This assumption is met only in the mixing region A and fails further downstream. It would seem that we may tentatively allow for this by allowing C to vary with y,, and hence with frequency via 71. This would entail rewriting (10.31) in the form

The implications of this equation as to the shift of the spectrum peak with 0 and jet speed have been discussed in Part A for subsonic jets and illustrated in Fig. 17. In conclusion we note that the total radiated sound power is related to

p(e,u))by n m

(10.23)

By virtue of this (10.22) may be written in the alternate form

(Direct substitution of (10.24) into (10.23) yields an identity and serves, with use of (10.20), to define the weighted average convection factor ( C - 5 ) A , . . )

THE GENERATION OF SOUND BY TURBULENT JETS

175

XI. ASYMPTOTIC BEHAVIOR AT HIGHMACH NUMBER 1. Inferences from Convected Quadrupoles or Dilatations

When the Mach number of the flow is small, Lighthill's quadrupole strength Tii (6.24) may be approximated as pouiuj. Equivalently, the pseudosound pressure +(") figuring in the dilatation approach is of the order of p a s . Either approximation leads to the Ujslaw, in the absence of convection, for the noise power emitted by a simplified model of jet turbulence. The effect of convection a t subsonic speeds is approximately offset by the error in the assumed model of turbulence to recover approximately the Ut law. Upon transition to supersonic eddy speeds the convection factor behaves like Ui-5. Thus if we can postulate a basic U: law of emission for these speeds, the effect of convection will yield a net acoustic power emission rn Ui3, or equivalently a constant acoustic efficiency. The weak point is that the basic Uis law must be postulated for the supersonic convection speeds: it is not a derived result. Variations in the density p have been neglected in the low-speed analysis; they may be expected to become important a t high speeds. In fact it can be shown [5, 10, 11, 411 that the density derivatives in a2puiuj/ayiayiaccount for convection of the sound waves. Furthermore, time derivatives of p have a sourcelike effect [lo]. A t high speed where p may not be approximated as po Lighthill's relation (6.26) becomes an integral equation for p of a rather intractable kind [68, 69, 51. 2. 0. M . Phillips' Fundamental Investigation

We have seen that with the Lighthill or dilatation integrals it is necessary for solution to suppress variations of the density in puiui. For this reason 0. M. Phillips [5] decided to make a new start and reformulate the basic wave equation in terms of pressure (and entropy), eliminating the density. Under the neglect of viscosity and heat conduction the entropy and viscous stress terms drop out, and he obtains (11.1)

The convective derivative is simplified to (11.2)

DlDt = alat + n,a/ax,

where 4, is the local mean velocity and the speed of sound is ultimately required to vary but slowly with distance (gradients are neglected) to yield (11.3)

ax, ax,

176

H. S. RIBNER

approximately, where 3 = ?(x3) is the time average and only what is considered to be the dominant term has been retained on the right-hand side: the assumption is that the mean shear aG,lax3 is much greater than the fluctuating shear. Equation (11.3) is a convected wave equation that allows for both generation and refraction of the sound. Our earlier equation (8.3) - the governing equation for treating refraction [9-111 - is in fact just a linearized version of this equation. Phillips applies this equation to an idealized shear layer separating two semi-infinite plane flows. The lower flow has a velocity - U (- x,-direction) and the upper flow a velocity U (+ x,-direction). The local mean velocity til(x3) varies monotonically between the - U , U asymptotes in a semiS-shaped profile. (A mirror image of this flow is shown in the inset of Fig. 26 ; it is intended to represent qualitatively a jet of nozzle velocity 2U toward the right so far as the upper boundary sound emission is concerned.) This mixing zone thickness is proportional to a length scale L. The paper is devoted to obtaining asymptotic solutions for large Mach number describing conditions just outside the shear layer. (The pressure field there will be dominated by Mach waves, not quasi-incompressible pseudosound, because of the compressibility associated with high speed.) The first step is to make a three-dimensional Fourier transform (time and the x,,xz space dimensions) of (11.3). This in effect describes the generation and propagation of waves characterized by a wave number k = (k,,k,) in the plane of the shear layer and a frequency n. A Green’s function technique is used to find the asymptotic solution for hypersonic speeds. The solution shows that waves of given wave number k and fluctuation frequency ta can be associated with a particular layer within the shear zone. A major finding is that eddy fluctuation contributes relatively little to the sound generation for supersonically moving layers; the contribution approaches zero with increasing Mach number. The mechanism resembles the generation of Mach waves by supersonic flow over a slightly irregular wall; the wall itself may be rigid and free of vibration. Hence, the phrase ‘eddy Mach waves’ for the radiated sound suggests itself [5]. Integration over all the wave numbers k leads finally to the mean square sound pressure in the form

+

(11.4)

where 1 is a length scale of the velocity fluctuations u3 normal to the flow direction. The relative mean square sound pressure / ( a ) in Mach waves radiated in a direction at an angle a with the (negative) flow direction is plotted in

THE GENERATION OF SOUND BY TURBULENT JETS

177

Fig. 26 for several Mach numbers M . For large M the bulk of the radiation is nearly normal to the flow, the peak a t a,,,lying just a little below the cutoff at cos a = (2M)-l. This property suggested to Phillips the possibility "of a simple estimate of the acoustic energy flux N per unit area from the shear zone using expressions derived by Ribner [88]". He wrote (11.5)

approximating a,,,as 90". The corresponding acoustic efficiency is __ 2 1 2 L2 (11.6) 7 = acoustic energy flux/poU3 M - 3/23- -. u 2 I' 4

10-

8-

f(a)

64-

a

FIG. 26. Relative mean square sound pressure / ( a ) in Mach waves radiated in direction a from supersonic turbulent shear layer. (Replotted from Phillips [ 5 ] . )

If the turbulence relative intensity and scale are essentially constant with M (as assumed in deriving the low-speed U8 law), then this equation implies an efficiency decreasing as M-3/2. This is an even greater change from the subsonic M 5 law than the constant supersonic efficiency estimated on the convected quadrupole or dilatation models. The experimental data for afterburning jets and rockets (Fig. 19) suggests a constant supersonic efficiency ( U3 law) ; however, the range is too short for extrapolation to the very high Mach numbers for which (11.6) is most applicable. Other comparisons cited by Phillips deal with the directivity. Lassiter and Heitkotter [89] measured the directional distribution of the sound from a rocket whose exit Mach number (based on external sound speed) was 3.16. They found a directional peak a t about 50", compared with the 57" predicted by the theory for 2M = 3.16. Moreover, as a increased above 60" the experimental intensity dropped some 60 db, approximating the cutoff shown in the theoretical curves of Fig. 26.

178

H. S. RIBNER ACKNOWLEDGMENT

This review was supported in part by the Air Force Office of Scientific Research under Grants AF-AFOSR-62-267 and AF-AFOSR-223-63. Unpublished experimental work of W. T. Chu (briefly touched upon) was supported also by the Defence Research Board of Canada. The author wishes to record his appreciation to both organizations and to Dr. G. N. Patterson, Director of the Institute of Aerophysics, for their support over many years of the UTIA program of research on aerodynamic noise which provided the background and motivation for the present study.

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Circular Cylinder in a Subsonic Stream, and of the Associated Sound Field, UTIA Rep. 76 (AFOSR 2147) (1961).

Stability of Motion of Solid Bodies with Liquid-Filled Cavities by Lyapunov’s Methods

BY V . V . RUMYANTSEV Institute of Mechanics of the U.S.S. R . Academy of Sciences. Moscow Puge

Introduction . . . . . . . . . . . . . . . . . . . . 184 I. Simplest Cases of Motion; the Cavity is Completely Filled . . . . . . . 186 1. Irrotational Motion . . . . . . . . . . . . . . . . . . . . . . . . 186 2. Two Examples of Irrotational Motion . . . . . . . . . . . . . . . . 191 3. Uniform Vortex Motion of a Liquid in an Ellipsoidal Cavity . . . . . 196 11. Stability of Motion of a Solid-Liquid Body with Respect to a Part of the 203 Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . 203 2. Stability with Respect to a Part of the Variables . . . . . . . . . . 206 3. Two Examples (Viscous Liquid, Partially Filled Cavity) . . . . . . . 209 111. Stability of Steady Motion of a Solid Body with Liquid-Filled Cavity . . 214 1. Some Relationships . . . . . . . . . . . . . . . . . . . . . . . . 214 2. A Stability Theorem. . . . . . . . . . . . . . . . . . . . . . . . 217 3. The Problem of Minimum . . . . . . . . . . . . . . . . . . . . . 224 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 NOTATIOK Space-fixed system of coordinates Body-fixed system of coordinates Cosines of angles formed by axis C with axes x,y,z Velocity vector of point with coordinates x.y,z Velocity vector of point 0 Angular velocity vector of solid body Density of liquid Viscosity coefficient Kinematic viscosity coefficient Unit vector along the external normal to the surface of liquid u consisting of wet walls q of the cavity and free surface u Region bounded by surface (I Moments of inertia of solid (s = 1) and liquid (s = 2) about axes x,y.z Coordinates of the centre of mass of solid (s = 1 ) and of liquid (s = 2) Mass of solid (s = 1). and of liquid (s = 2) Mass of system Acceleration of gravity Kinetic energy of solid (s = l), and of liquid (s = 2) Kinetic energy of system

183

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INTRODUCTION The problem of the motion of a solid body with cavities completely filled with a liquid began to attract the attention of scientists over a century ago. Stokes [ l , 21 appears to have been the first to draw attention to this interesting problem of mechanics; it was also taken up by Helmholtz [3], G. Lubeck [4], and Lamb [5] who investigated a number of special cases: F. Neumann [6] became interested in it when studying the motion of solid bodies in a liquid. N . E. Zhukovskii 171 for the first time made a thorough general study of the motion of a solid body whose cavities are completely filled with a homogeneous incompressible liquid. He proved that the irrotational motion of the liquid in the cavity is determined by the rotation of the body and does not depend on its forward motion, while the motion of the body proceeds as if the liquid had been replaced by an equivalent solid body. Simultaneously with these studies interest arose in the question of stability of that motion. Kelvin who made experiments with a thinwalled spheroidal gyroscope containing a liquid [8] found that the gyroscope is stable if the cavity is sufficiently oblate and if the gyroscope revolves rapidly. If, however, its shape is slightly elongated, it becomes very unstable irrespective of its angular speed. The mathematical explanation of this phenomenon was the object of studies by A. G. Greenhill 191, F. Slutskii [lo], S. Hough [ l l ] , Poincare [12] and A . B. Basset [13], who analyzed the homogeneous vortex motion of a liquid in an ellipsoidal cavity. Hough [I13 studied small oscillations about the state of uniform rotation of solid and liquid as a whole, the axis of rotation being the principal axis of inertia. He obtained necessary conditions of stability and applied them to a shell of negligibly small mass. PoincarC [ 121 likewise investigated this problem, considering also the elasticity of the shell and the non-uniformity of the liquid. Considerable interest in this question arose again in our time, especially in the past 20 years, when the new problem of a solid with cavities tzot completely filled with liquid came up and was extensivly explored. Numerous papers of many researchers were devoted to this topic: G. E. Pavlenko [14], S. L. Sobolev [15], N. G. Chetaev [16], L. N. Sretenskii [17], D. E. Okhotsimskii [18], S. G. Krein [19], N. N. Moiseev [20], G. S. Narimanov [21], S. V. Malashenko [22], A. Y. Ishlinskii, and M. E. Temchenko [23], K. Stewartson [24] and many others (see reviews [25, 261). Two theoretical approaches to the problem of stability are known : 1) the study of the linearized equations of the perturbed motion of the system, with application of the theory of small oscillations and of the spectral theory of operators; 2 ) the study of the full non-linear equations of the perturbed motion with application of Lyapunov’s methods [27-291.

STABILITY OF MOTION OF SOLID BODIES

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The linear theory for this problem has been substantially developed and achieved considerable success. I t provides a quite adequate description of the motion of the system and its conclusions have been confirmed experimentally to a certain extent. Lyapunov’s criticism of certain investigations by PoincarC [30] concerning the stability of the equilibrium figure of a rotating liquid may, however, be fully applied to the linear theory also in the present case. Lyapunov 1291 emphasized that the conclusions about stability, reached by studying linear equations that have been obtained by deleting some terms considered as small in the differential equations of the problem, “cannot be regarded as rigorously established”. In linearizing the equations of motion, one problem is in point of fact replaced by another with which the former may in some cases have nothing in common. There are no criteria at present for the correlation of these two problems in the theory of small oscillations of solid bodies containing a liquid; the establishment of such criteria would be of considerable importance for substantiating the linear theory. The only exception is provided by the conditions obtained according to the linear theory [20] from the Lagrangian theorem: since the latter is valid by virtue of the full equations [47], these conditions may be regarded in first approximation as sufficient conditions of stability. But the complexity and clumsiness of the characteristic equation, for which the signs of the real parts of the roots must be investigated, are also typical for the linear theory of the stability problem in our case [24]. In this respect Lyapunov’s non-linear methods compare favourably with the linear theory. These methods are mathematically rigorous and permit in a number of cases to obtain effective sufficient conditions of stability. When we pose the problem of stability of motion of a solid body containing a liquid (a system with an infinite number of degrees of freedom) the question arises: In what sense and in relation to which variables does stability interest us in this problem? In his famous book [27], Lyapunov, defining stability for a system with a finite number of degrees of freedom, emphasized the necessity of specifying those quantities which are supposed to remain small, since the term stability in itself contains nothing absolute [29]. The present review of non-linear methods in the stability of stationary motions of a body with enclosed liquid has three chapters. The first deals with the simplest case, in which the motion of the liquid is fully defined by a finite number of variables. Such a case is evidently possible only when the cavity is completely filled with an inviscid liquid. Cases of irrotational motion and of uniform rotation of the liquid are investigated. Here the problem would naturally be posed as a stability problem according to Lyapunov for a system with a finite number of degrees of freedom [27, 311. The second chapter is devoted to the study of the stability of motion of a body with a cavity partly or completely filled with an inviscid or viscous liquid, no assumptions being made regarding the nature of the motion of

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the liquid. The state of the system is now described by an infinite number of variables, and the problem of stability becomes considerably more complex. However, in this case, too, it is possible to pose the problem of stability with respect to a finite number of variables by introducing certain quantities which integrally define the motion of the liquid. The third chapter outlines another approach to the problem of stability. It develops Lyapunov’s ideas [28, 291 proceeding from the theory of stability of the equilibrium figure of a rotating liquid. Proof is supplied that our problem reduces to a minimum problem for a certain expression, and a solution of the latter is given. The Lagrangian theorem is also established for a body with a liquid. All three chapters supply examples illustrating the application of the theory. It will be noted that the solid body with a cavity completely or partly filled with a liquid is treated as a single mechanical system. The equations of motion of such a system can be derived [32, 331 from the HamiltonOstrogradskii principle. 7 hese equations are mathematical expressions of the general theorems of mechanics about momentum and moment of momentum of a system and of Euler’s or Navier-Stokes’ equations of the motion of a liquid. Under certain conditions, the equations of the motion of the system have integrals of energy and areas as well as some other first integrals whose existence is important in applying the methods under consideration. In most cases we confine ourselves, for the sake of brevity, to motions of the solid body with a single fixed point. I. SIMPLEST CASESOF MOTION;THE CAVITY IS COMPLETELYFILLED We consider cases in which the motion of the liquid in the cavity can be fully described by a finite number of variables. Evidently, this is possible only when the cavity is completely filled (no free surface). It is well known [ri] that if in this case the liquid is in irrotational motion or uniformly rotating (quasi-solid), the motion of the system can be described by ordinary differential equations. Such cases are analyzed below. 1. Irrotational Motion

We suppose an inviscid, homogeneous, and incompressible fluid (liquid), which fills completely the cavity of the body, is subject to a body force derivable from a potential, and performs an irrotational motion. Let us designate the velocity potential of the liquid expressed in the mobile system of body-fixed coordinate axes Oxyz by +(x,y,z,t); then the vector of the velocity of the liquid v = grad I$. By div v = 0, the function d ( x , y , z , t )

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187

is harmonic in x,y,z; on the wall of the cavity, 0 , its normal derivative equals the normal component of the wall velocity.* Hence the determination of the velocity potential is reduced to solving Neumann’s problem; and the irrotational motion of liquid in an n-fold connected cavity is completely determinate, if one gives the normal component of the velocity for every point of the boundary of the cavity and the value of the circulation for each of the n - 1 independent irreducible closed curves (circuits) which can be drawn in the cavity [5]. Following Zhukovskii [7], let us resolve the motion of the solid body into a translational motion proceeding with the velocity of some body-fixed point 0 and a rotary one about that point. The progressive motion has a single-valued potential wlx w,y w3z; it is directly passed on to the liquid and does not affect its internal motion. The latter is changed only by the rotary motion about point 0 and will be the same if we consider this point as space-fixed and rotate the body about it in the same way as it rotates about the point 0 moving in space. The velocity potential d(x,y,z,t) is expressed as

+

+

d=x+v

(1.1)

where the harmonic iunctions

x

and q satisfy the conditions

on the walls of cavity u. Function x produces along the principal circuits the preassigned principal circulations k , = const. If the cavity is simply connected, then x = 0 ; in the case of a multiplvconnected cavity, (13)

where 6 i ( x , y , z ) are harmonic functions whose normal derivatives on the walls of the cavity equal zero. The function 6j(x,y,z) decreases by unity when the argument point passes through the jth partition in the direction of the jth principal contour, and changes continuously on passing through the rest of the partitions. If the liquid is at rest initially, all k j equal zero. We set 3

(1.4)

q(x,y,z,t) =

2 wi$i(x,Y,z) i= 1

Note that v is measured in the space-fixed (inertial) system. Formally, therefore, the conditions for 4 are the same as in the space-fixed system.

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where the single-valued harmonic functions i,hi on the walls of cavity u satisfy the conditions

I t follows from the formulae that I$ depends on the time t only by means of the mi. If, after some period of time the body comes to rest, then 4 becomes equal to x, i.e. the liquid inside the cavity makes the same motion it made initially [7], when the body was a t rest. We now turn to the kinetic energy and the moment of momentum of the liquid which makes an irrotational motion with potential I$. According to Kelvin's generalization of Green's theorem [5] for a multiply-connected region

where ui is the area of one of the jth partitions which can be put in the cavity without destroying its connectivity. Taking into account the boundary conditions (1.2), we have

By substituting expressions (1.3) and (1.4) in this formula, we obtain (1.7) 2T2 = A*wI2

+ B*w,, + C * O , ~+ SDO,W, + ~ E W , W+, BFUJ,CU~ n-1

r.j= I

where the following notations have been introduced for brevity :

1

A* = p $ ~ ~ (-y zzm)du, U

B* = p

STABILITY OF MOTION OF SOLID BODIES

189

Ry (1.7) the kinetic energy of the liquid, T,, is the sum of the kinetic energy T,(')(q,w2,u3)of the motion with potential p and of the constant kinetic energy T2(*)(kl,. . .,kn- ,) associated with potential x. The moment of momentum of the liquid relative to point 0, given by pJ,r x grad +It, is likewise made up of the moments of momentum associated with p and The projections of the moment of momentum belonging to are P,Q,R; they are represented by linear functions of the principal circulations ki and do not change in time. The projections of the resultant moment of momentum of the solid-body motion and the motion of the liquid with potential 9 are equal to aT/aw, (i = 1, 2 , 3 ) . Equation T,(')(x,y,z)= 1 represents some ellipsoid in the body-fixed system. Zhukovskii established that the latter always includes the inertial ellipsoid of the liquid. Let us now consider some solid body with an ellipsoid of inertia for some fixed point

x.

x

+

T , ( X , Y , ~ ) TZ'')(x,y,z) = 1.

(1.9)

The same inertial ellipsoid can be obtained by joining to the given solid body, instead of the liquid in the cavity, another solid body with an ellipsoid of inertia

T,("(x,Y,z)= 1 about the same fixed point 0. Its mass equals that of the liquid to be replaced, and its centre of gravity coincides with that of the liquid. Stokes called such a body the equivalznt solid body; it does not depend on the location of the fixed point 0 and has smaller moments of inertia in relation to any axis than the corresponding liquid mass, i.e. A* < A , , B* < B,, C* < C2 [7]. If axes x,y,z are directed along the principal axes of the ellipsoid (1.9), its equation is

AX2

(1.10)

+ By2 + c z 2 = 1

where A,B,C denote the sums of the moments of inertia of the solid body and of the equivalent solid body, i.e. (1.11)

A =A,

+ A*,

B = B,

+ B*,

C = C,

+ C*.

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We obtain the equations of motion of the system from the theorem about the moment of momentum: dw

A2 dt

+ (C - B ) 0 2 0 ,+ w2R - w ~ Q= L,

~ d w +, ( B - A)w1w2f w @ dt

wzP= N .

Here the moments of the external forces in relation to the mobile axes x,y,z are L , M , N . These equations are identical with those of the motion of a solid body to which a rotating gyroscope is joined. The direction cosines of the axis of rotation of the gyroscope are in the ratio P : Q : R , while the product of the initial angular velocity (with the solid body at rest) and the moment of inertia about the axis of rotation equals (P2 Q2 R2)'/'. Consequently, the mechanical effect of a liquid in the cavity withozd initial velocity is identical with that of some equivalent body. If the liquid in the multiply-connected cavity has some initial velocities then its additional action is similar to the action of a rotating gyroscope joined to the body. This important theorem was given by N. E. Zhukovskii [7]. Hence, for the irrotational motion of liquid in the cavity, the motion of the solid body together with the liquid is fully defined by a finite number of variables, with respect to which the stability problem should be posed according to Lyapunov [27]. The solution of this problem for a number of cases was given by N. G . Chetaev [16]. It should be noted that the moments of external forces, L , M , N , applied to the system, may be functions not only of the time and of the angular velocities mi, but also of some other variables. In such cases the differential equations for these variables must be joined to the equations (1.12). For example, in many practical cases we shall have [35]:

+ +

(1.13)

where U = U(y,,y2,y3)is the force function of the external forces acting on the system; U is assumed analytical in its arguments. The direction cosines yi of the fixed axis C in relation to the mobile axes x,y,z satisfy (1.14)

STABILITY OF MOTION OF SOLID BODIES

191

For example in a homogeneous gravity field

if, axis ( is directed vertically upwards. When a projectile moves in a very flat trajectory, the moment of the overturning couple depends on the angle between the velocity of the centre of mass of the projectile and the axis of symmetry z [l6]. In this case (1.16)

W 3 )=

- ay3

if axis ( is directed along the vector of velocity of the mass centre, and a = const. For the case of a central force field

if axis O,( is directed from the centre of attraction 0, along the straight line OIO. Here f is the constant of gravitation, p ( x , y , z ) is the density at point x,y,z of the solid or liquid, 11 is the volume of the solid and the liquid, and R is the distance 0,O. If R is much larger than the dimensions of the body, then one can take [36] the approximate value of the force function instead of (1.17)

(1.18) 2. Two Examples of Irrotational Motion

Consider two problems dealing with the stability of motion of a solidliquid body with the liquid in irrotational motion.

Example 1. Permanent helicoidal motion maintained by inertia 1371. The cavity is simply connected. Let the kinetic energy of the system equal

+ M~~ +

2T = M(w12

+ A o I 2+ B

+

w ~ CO,~. ~

Then the equations of motion of the system have integrals

2T = const,

Aw,w,

+ Bw2co, + Cw3w3= const,

7u:

+ w: + w:

= const.

We now analyze the stability in relation to wi,oj of the particular solution of the equations of motion w1 = w,

w2 = w3 = 0,

w1= 0,

0 2

= u 3 = 0.

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Assuming for the perturbed motion w,=w+x,

ru,=w+5

and retaining the previous designations for the rest of the variables, write down the first integrals of the equations of the perturbed motion

+ 2wx + wz2 + w3,) + A ( t 2 + 206) + BwZ2+ CO,~= const, V2 z A ( o x + w6 + x t ) + Bw2w, + Cw,o, = const, V , E x 2 + 2wx + + w32 = const. V, z M(x2

w22

Now construct Lyapunov’s function as [37]

(1.19)

I’ = v1 - 2AV2

+

XV,

+ v,2

+ x + 4 ~ ’ ) x ’ - 2AAx5 + A t 2 + (A4+ x)(W2’+ w - ~ I ( B w , <+u CW,O,) ~ + Buz2+ C W , ~

= (M

~ ~ )

where

1 = o/w,

M

+x =Ao2/w2.

If

A>B,

(1.20)

A>C,

then function 1.’ will be positive definite in all the six variables wj,wi and its time derivative V’ will vanish as a result of the equations of the perturbed motion. Proceeding from Lyapunov’s theorem about stability [27], it is possible to state that the inequalities (1.20) are sufficient conditions of stability (with respect to w ~ , o J ~of) the permanent helicoidal motion of a solid body with a liquid-filled cavity. Example 2. Uniform rotation about a fixed axis. First consider the solid-liquid body moving about a fixed point under the action of forces possessing a force function U ( y 3 ) . Suppose the cavity is multi-connected and the initial motion of the liquid is such that P=Q =0, R # 0. For example, let the cavity h a r e the shape of a torus obtained by rotating about z some circle with centre not on the z axis. Then [7] R = M2k/2n, where k is the circulation. I t can be easily seen that the equations of motion (1.12) and (1.14) in the case under consideration have first integrals of energy Awl2

+ Bw,, + Cwa2- 2U(y,) = const

Aqy,

+ Bo2y, + (Cw, + R)y3= const.

and of areas

1!13

STABILITY OF MOTION OF SOLID BODIES

+

+

Further y12 y22 y32 = 1, and we have constancy of the projection of the instantaneous angular velocity on the axis of symmetry, w3 = const,

if the ellipsoid of inertia (1.10) is an ellipsoid of revolution, i.e. if

(1.21)

A

= B.

The equations of motion have the particular solution

(1.22)

w1= wz= 0,

y1 = yz = 0,

w, = w ,

y3 = 1

which describes the uniform revolution of a solid body about the lixed axis of symmetry z, coinciding with axis 5. Let us assume that in the perturbed motion

y3=1+q1

w,=w+E,

retaining the previous notations for the remaining variables. The equations of the perturbed motion [27] have first integrals

Vl

Am?

~

V, =A q y , V , zz y12

zl)l (

+ BwZ2+ C(E2+ 2wE) - 2 (

q-

+ Bw,y2 + C ( r q + E + Eq) + Rq

__

q2

+ . ..

= const,

= const,

+ + q 2 + 2q = 0 y22

as well as the integral

V, if condition (1.21) is fulfilled. (au/aY3)1

6 = const

Here the dots designate terms o ( q 3 ) ,and

= (au/aY3)y,=l.

First consider the case A the form

=

B and construct Lyapunov's function in

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V. V. RUMYANTSEV

where 1 is an arbitrary constant. The condition of positive definiteness of this function reduces to

This is possible, provided the polynomial to the left has two different real roots: (1.24)

M'ith this condition satisfied, the constant 1 can be so selected that c' is positive definite with respect to all variables q , y i (i = 1,2),5',17; in addition, I/' = 0. Consequently, according to Lyapunov's theorem [27], inequality (1.24) is a sufficient condition of stability. Let us now demonstrate that (1 2 4 ) is also necessary for stability. Consider the function r38j (1.25)

W = W l Y 2 - %Yl

and its time derivative, taken by means of the equations of the perturbed motion,

-

L(%) (y12 + y22) + . . . .

A aY3 1 According to Sylvester's criterion, W' will be positive definite in the variables q , y , (z' = 1,2), if (1 26)

holds. Here it is assumed that the variable 17 preserves all the time the order of smallness of the variables q , ~ otherwise )~: the motion would evidently be unstable with respect to y3. When condition (1.26) is fulfilled, the unperturbed motion (1.22) is unstable since the function W meets in this case the conditions of Chetaev's instability theorem 1311. Hence condition (1.24) is necessary and sufficient for the stability of the unperturbed motion (1.22). If there is no initial motion of the liquid in the multi-connected cavitv, then R = 0, and we obtain from (1.24) the condition of stability (1 2 7 )

195

STABILITY OF MOTION OF SOLID BODIES

For the case of a homogeneous field of gravity, if x o (1 2 8 )

= y o = 0,

U(Y3) = - MgzOy,,

and condition (1.24) appears as (1.29)

(CU

+ R ) 2- 4 A M g z 0 > 0

[3H],

and with K = 0 as Maievskii’s condition [lti, 311 C2w2- 4AMgzO > 0.

(1.30)

Now let us consider the case A tion as

3B

and construct Lyapunov’s func-

+ (Cu2 + (g), + Rw) (y: + y; + q2)+ where the constant

This function will be positive definite if the following condition is satisfied: (C - A ) w 2

(1.32)

+ RU +

- >O. (

3

1

As 1;’ = 0 again by virtue of the equations of the perturbed motion, ( I 32) will, according to Lyapunov’s theorem, be a sufficient condition of stability of the unperturbed motion (1.22) relative to o,,y,. In the case of a homogeneous field of gravity (1.28), condition (1.32) appears as (1.33)

(C- A)w’

+ RW - MgzO>

0

[39],

and with K = 0. as (1.34)

(C - A ) d > MgzO

(381.

I t is worth noting that for cavities in the shape of bodies of revolution, the axial moment of inertia the equivalent solid body [7] C* = 0. The equatorial moment of inertia A is determined by the shape and dimensions

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V. V. RUMYANTSEV

of the cavity and is proportional to the density p of the liquid. For instance, for a cavity in the shape of a round cylinder with radius a and height 2h [16]

where [, are the roots of the equation dJ,(C)/dS = 0, zo and z, are the z co-ordinates of the cylinder faces, and M , = 2npa2h. For a cavity in the shape of an ellipsoid of revolution with centre on the z axis at zo

M (a2- c2)2 A*=> 5 a2+c2

+ M2z02.

3. U n i f o r m Vortex Motion of a Liquid in an Ellipsoidal Cavity

This is another simple case, in which the motion of the solid-liquid body is fully determined by a finite number of variables. Let the cavity have the shape of an ellipsoid (1.35)

The motion of the liquid in the cavity can be described by the formulae [9-111

(1.36)

where y(x,y,z,t) is a harmonic function of the spatial co-ordinates and the Q,(t) are functions of time only. It can be easily seen that the harmonic function [40]

(1.37)

+ [(c2- a2)x.z + 2a2z0x - 2c2x,-,z] c2 +- a2 w2

Q2

a3 + [ ( a 2- b2)xy + 2b2xoy - 2a2yox] a2 + b2 0 3

satisfies the boundary condition for an ideal liquid on the walls of cavity

6.

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STABILITY OF MOTION OF SOLID BODIES

To determine the functions R,(t) we use Helmholtz’ equations for the vortex motion of a liquid [5, 341. Since in the present case the projections of curl v on the mobile axes are 2Ri, Helmholtz’ equations appear as [40] a2(c2- b2)

(a2 (1.38)

dn, = 2b2(* dt dt

,+c2

a 2 + b2

+ a2

c2

)

w3R2, - 2R3Q1 - ___

b2

+ b2)(a2+ c2) ’

b2(a2- c2)

(b2

+ c2)(a2+ b2) ’

c2(b2 - a21 (a2 c2)(b2 c2) .

+

+ c2

+

It should be noted that the motion of a liquid in an ellipsoidal cavity can be described by formulae differing in appearance from (1.36). For example, Zhukovskii [7] set up this motion as a superposition of a potential motion with potential and an elliptical rotation; the components of the latter are U‘ = a2[q’(z - z0)

- ~ ’ (-y y o ) ] ,

V’

= b2[r’(X - x0) - P’(z

- zO)],

w’ = C2[P’(Y- Yo) - q’(x - x0)l

where p‘, q‘, and r’ are functions of time only. PoincarC [12] assumed that the velocities of the liquid particles are linear functions of their co-ordinates. He termed such a motion of a liquid a simple motion. Proceeding from Helmholtz’ theorem, he pointed out that if the motion of the liquid a t the initial instant was simple it would remain so all the time, provided the cavity has the shape of an ellipsoid. In this case the projections on the mobile axes of the relative velocities are

a

24 = - Ql(Z C

- 20) -

a

r,(y

-

b

Yo),

z, = C r,(x

b - xo) - a P,(Z

- zo),

where P I , q,, and r, are functions of time only. It can be easily proved [40] that in either case the projections of the absolute velocities of the liquid can be represented by (1.36). For the projections on the mobile axes of the moment of momentum of the liquid, G, we obtain (1.39)

G,

= A*wl

+ A’R,,

G,

=

B*u, + B‘R,,

G3 = C*O,

+ C’R3

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V. V. RUMYANTSEV

where

M (b2 -. c2)’ A*=-? 5 h2 + c2 (1.40)

M

(c2c2

B* = L 5

+ M2(y,2 + zO2),

a2), + a2 + M,(zZ +

(a8- c , ) ~ c*=M2 + M&,2 5 a2 + b2

A’

4 M,b2c2 5 b2 c2 ’

+

-I__

R’ = 4 M 2 a 2 c 2

XZ),

5

+ Y$),

c2

+ a2 ’

4 M,a2b2 C’ = - ___ A a2 f b2

In this case A’,B’,C’ denote the differences between the moments of inertia of the liquid and of the equivalent solid body, M , = $npabc. We obtain the equations of motion about the fixed point of the solidliquid body from the theorem of moment of momentum of a system as

where A,B,C are determined, as in formulae (1.11). It can be easily seen that in the particular case of a spherical cavity, when a = b = c , the motion of the liquid does not affect that of the solid body; the system moves then as a single solid body whose principal moments of inertia at point 0 equal (1.42)

A

=A,

+ M2(yo2+ zo2), c c, + =

B

M,(XO2

=

B1

+ M2(zo2 + xo2),

+ yo2).

[The equations of motion (1.38) and (1.41) can also be obtained from PoincarC’s equations [12] in group variables [33].] Thus in the present case of a solid body with an ellipsoidal cavity filled with an ideal liquid, the motion of the system is fully characterized by a finite number of variables. Therefore the problem of stability can, in this case too, be posed and solved as one of stability according to Lyapunov. Exanifilc 3. Let us consider a solid body moving about a fixed point 0, = 0, and with a cavity of the kind (1.35), where x, = yo = 0. Suppose the system is under the action of forces with a force function of the form C(y,) and assume that the moments L , M , N are determined according to formulae (1.13).

199

STABILITY OF MOTION OF SOLID BODIES

In this case the complete system of equations of motion will consist of the nine equations (1.14), (1.38), and (1.41). These equations have first integrals of energy and areas

+ A’R12+ B’R22+ C’R32- 2U(y3)= const,

+

Awl2 + B w , ~ Cw32 AwIy,

+ Bw2y2 + Cw3y3+ A’R,yl + R’R2y2+ C‘R3y3= const ;

in addition, y12

+ + y22

732 =

1;

the integral b2c2RI2

+ a2c2R22 + ~ 2 b 2 5 2 ,=~ const

expresses Helmholtz’ theorem about the constancy of vorticity. When the body and the cavity have an axis of symmetry, let it be the z axis, hence a = b and A = B ; then the equations of motions have still another first integral w3 = const.

The equations of motion of the system have the particular solution w1 = wz = 0,

(03

= (0,

(1.43)

R,

= R2 = 0, y3 =

R,

= R,

y1 = yz = 0,

1,

describing the uniform rotation of the solid body about the z axis (coinciding with the fixed C axis), and a relative elliptical rotation [7] around the same axis with components

Let us assume that this is the unperturbed motion and analyze its stability with respect to variables w,,R,,y,(i = 1,2,3). It is worth noting that in the particular solution (1.43), magnitudes w and R may have arbitrary values. Of practical interest are values R in the interval 0 < E R w , where E is infinitesimally small, for, as shown by experiment, during the uniform rotation of the body about a fixed axis passing through the centre of the cavity, the liquid, unless it has been put into vortical motion earlier, first makes an irrotational motion (it is at rest), and then is increasingly involved in the motion of the body by friction until it moves along with it as a single solid body. In what follows we shall confine ourselves precisely to this case E Q oi.

< <

< <

V.

200

V. RUMYANTSEV

We obtain the equations of the perturbed motion from equations (l.L4), (1.38) and (1.41) assuming that in the perturbed motion r=w+5,

Q3=R+lj7,

y3=1+5

and retaining the previous designations for the rest of the variables. I t is evident that in the general case the equations of the perturbed motion will possess the following first integrals: I/,

E Awl2

+ B w , ~+ C ( t 2+ 2wt) + A‘R12+ B’RZ2

To construct Lyapunov’s functions, we apply N. G. Chetaev’s method. Let us analyze the function (1.45)

1/

I/,

- 2wV2

= Ao,2

+ AV4 + pV3 $- ~ X L ’ , ,

- 2Aow,y, + ( A ’ + lb2C2)R12 - 2A’woR,y,

+ py,2

+ Bw,’ - ~ B w w , ~+, (B‘ + A u ~ c ~-) R2B’wQ2y2 ~~ +/ ~ y 2 ~ + CE2 - 2Cwtc + (C’ + i l ~ ~ b ~ ) l j2C’wvC 7~ -

+ (x+,+J)P+

...

where the following abbreviations have been introduced : (1.46)

A=C’-

o-R Ra2b2 ’

11

= (CW

+C’Qw +

for the time being, coefficient x remains arbitrary. Function (1.45) represents the sum of three quadratic forms, each depending on three variables; the omitted terms are of third and higher order with respect to y1,’y2,C. The conditions of positive definiteness of these forms can be obtained by using Sylvester’s inequalities; with w 2 R and x > (a2U/ay,2),- (aU/ay,), and with regard to (1.46), the latter are reduced to

201

STABILITY O F MOTION OF SOLID BODIES

(1.47)

+ C'WO+

(C - B)w2

By Lyapunov's theorem, the inequalities (1.47)are sufficient conditions of stability of the unperturbed motion (1.43)with respect to variables mi,Qi,yi. Let us look at these conditions when O = E (i.e. the liquid is almost in potential motion), and when O = o (liquid and body rotate as one solid body). In the first case, if E is considered as infinitely small, they evidently will be satisfied, provided

where b ( ~is) small. Conditions (1.48)practically coincide with the sufficient conditions of stability (1.32)for R = 0 of the rotation of a solid body, the cavity filled with liquid in irrotational motion. In the second case conditions (1.47)appear as

These conditions coincide with the sufficient conditions of stability of rotation of a single solid body with moments of inertia A A', B B', C C' equal to the sums of the moments of inertia of the body and the liquid. Inequalities (1.47)with U = 0 are sufficient conditions for stability of the unperturbed inertial motion. With O = 0 ) and U = 0, they appear as (1.49). We now investigate the stability of motion (1.43)in the case a = b, A = B. The equations of the perturbed motion of the system have now in addition to the first integrals (1.44)also the integral V , G E = const. I t can be easily seen that conditions (1.47)of stability of the unperturbed motion are now reduced to the singIe inequality

+

+

a2 - c2

4a2c2 ( @ 2 + ,2)2

+

C ' d > 0.

202

V. V. RUMYANTSEV

Other sufficient conditions of stability can likewise be specified. For this purpose consider the function

v = v, -!-2 1 v 2 -

Cwl

+ C’QA - (E)) v,+ pV4 1

C(C

+

(1.50) = A(wI2

+ -C2 p+ A

02*)

-A)

+ 2 A l ( ~ , y +,

(

- CwR -

~ 2 ~ 2 )

( ) (Y12+ ~

aY3 1

Y22)

2c1tg -

+ (A‘ + pa2c2)(Q,Z+

Q22)

3- 2A’I(Q,y,

+

Q2y2)

- C’QA(y12

+ (C’ + pa4)172 + zc’nq2‘- C ’ m p + . . . where 1 is some constant and ,u = - C’(A + R)/Qa4.

+ Y22)

For simplicity, (1.50) may be regarded as the sum of six quadratic forms of two variables each*. The sum of the first three quadratic forms in (1.50) is similar to Lyapunov’s function (1.23); it will be positive definite when condition (1.27) is satisfied. The constant coefficient 1 may have any value in the interval

I , < 1 < 1, where &,A2 are the roots of AA2 + CwA - (aU/ay3)1= 0. (1 51)

The fourth and fifth quadratic forms in the right part of equality (1.50) will be positive definite when the following condition holds for negative 1: (15 2 )

Finally, the last of the quadratic forms in (1.50) is easily seen to be sign-constant positive if 1< 0. I t is evident that in the case of very small Q = E condition (1.52) will be satisfied; condition (1.27) is then a sufficient condition of stability of the unperturbed motion (1.43) when the motion of the liquid is very close to a potential one. When i2 is finite, (1.50) is clearly satisfied for cavity (1.35) if a > c. If, however, a - b < c, then sufficient conditions of stability of the unperturbed motion (1.43) are the inequalities (1.27) and

2A

n.

STABILITY OF MOTION OF SOLID BODIES

203

Some particular cases of this problem have been analyzed in [41]. For inertial motion about the centre of the cavity (1.35), when a = 6 and the shell is so thin that its moments of inertia may be neglected in comparison with the liquid, the sufficiency of the conditions (15 3 )

a>c

or

c>3a

Ell]

has been shown by construction of Lyapunov’s function for the stability problem of motion (1.43) with respect to o,,sl,.

11. STABILITY OF MOTION OF A SOLID-LIQUID BODYWITH RESPECTTO PARTOF THE VARIABLES

A

This chapter provides the formulation and solution of the stability problem for the motion of a solid body with cavities partly or wholly filled with liquid with respect to variables determining the motion of the solid body and with respect to some quantities which define integrally the motion of the liquid L42-441. When so posed, the problem may be regarded as one of conditional stability of motion of a system; stability now refers to a part of the variables, but not to all variables which characterize the motion of the system which has an infinite number of degrees of freedom. To solve this problem, one can again use Lyapunov’s methods for systems with a finite number of degrees of freedom. 1. Equations of Motion

We consider a solid body with a cavity partly or wholly filled with a viscous or ideal liquid and shall confine ourselves to the case of the body moving about a fixed point, which will allow us to reduce the calculations without affecting the generality of method of approach. By applying the theorem about the moment of momentum of a system we obtain the following equations of motion for the solid body referred to mobile axes x,y,z that coincide with its principal axes of inertia at the fixed point 0 :

(2.1)

204

V. V. RUMYANTSEV

where the G, (i = 1,2,3) denote the projections on the mobile axes of the moments of momentum of the liquid. As these quantities are not known beforehand, equations (2.1) must be taken together with those of the motion of the liquid

dv,

___

dt

1afi + v A v ~ , + w3vl - w1v3 = Y - P aY

.

+ wlvz - w2v1 = 2

dv 3

at

avl

av,

1

afi

P

a2

- --

+ vdv,,

av3

-+-+--=0, ax ay

a2

where P(x,y,z,t) is the pressure, d the Laplacian operator, and X,Y,Z the projections of the body force per unit mass acting on the liquid. The projections of the absolute velocity of the liquid on the mobile axes may be represented as (2.3)

~ 1 =~ 2 2

+ U,

+

V%= W ~ X 012

V,

~3

=W

+

~ Y W ~ X W,

where u,v,w designate the projections of the velocity of the liquid relative to the solid body. The solutions of equations (2.2) should meet the following boundary conditions on the boundary of region T , occupied by the liquid. The velocity of the liquid on the wall a1 of the cavity with which the viscous liquid is in contact at a given moment of time should be equal to that of the corresponding point of the wall, i.e. u=v=w=O

(1.4)

In the case of an ideal liquid (v components can be required : (2.5)

on

= 0 ) only

al.

the equality of the normal velocity on

ul+vm+wn=O

al.

The kinematic and dynamic conditions must hold on the free surface a, of the liquid. The first implies that the velocity of the displacement of any point on surface ac and the velocity of the particles of the liquid adjoining the surface a t this point should have the same projections on the normal to the surface, i.e. (2.6)

af

af

af

af

ax

ay

a2

at

-u+-v+-w+-=O

where f(x,y,z,t) = 0 is the equation of uc.

on

a;,

STABILITY OF MOTION OF SOLID BODIES

205

The dynamic condition implies the continuity of the stress vector [34] pn on a free surface, i.e. (2.7)

pn = - p0n

on

a,,

where Po is the constant air pressure on the free surface. In the case of an ideal liquid, we have instead of (2.7) only the constancy of the hydrodynamic pressure (2.8)

p

=Po

on

uc

Hence, the problem of motion about a fixed point of a solid body with a liquid in its cavity is formulated by equations (2.1) and (2.2) with boundary conditions (2.4)-(2.8). Let us now give some integral relationships and first integrals for the equations of motion (2.1) and (2.2). If the forces acting on the solid body and the liquid possess a force function U , the following equation of energy dissipation is valid [ 5 ] :

(2.9) It follows that motion of a viscous liquid (p # 0) without energy dissipation is possible only when everywhere in the liquid

(2.10) which implies that there is no lengthening or shortening of linear elements anywhere [5]. This is possible only when the liquid and the solid body in whose cavity it is located, move as one solid body. In the case of an ideal liquid ( p = 0 ) we obtain from (2.9) the integral of energy (2.11)

T - U = const.

If the moment of the external forces about some space-fixed axis 5' equals zero, equations (2.1) have the integral of areas

which expresses the constancy of the projection on the 5' axis of the moment of momentum of the system. If, however, the moment of the external

V. V.

206

RUMYANTSEV

forces about the point 0 equals zero, then we have the integral of constancy of the moment of momentum of the system (2.13)

(Aim1

+ GI)' + (Bim,+ Gz) + (C103+ G3) = const.

If the liquid is ideal, if the ellipsoid of inertia of the body for point 0 and the cavity have the shape of bodies of revolution about the same axis, and if the moment about that axis of the external forces acting on the system vanishes, then the projection of the instantaneous angular velocity of the body on that axis remains constant throughout the motion [32]. Let for example A , = B, and let the above conditions be satisfied for the axis z, then the equations of motion have the first integral m3 = const.

(2.14)

2. Stability with Respect to a Part of the Variables In the general case of a body, free or bound by some constraints, with a cavity filled with a liquid, the equations of motion represent a system of ordinary and partial differential equations with corresponding initial and boundary conditions. The position of the solid body and the liquid relative to a space-fixed frame O,E$ can be determined by the co-ordinates of the body qi (j' = 1,. . . , a ; % 6) and the absolute t,q,C or relative x,y,z coordinates of the particles of the liquid, whose number is infinite. The velocities of the points of the system express themselves by the time derivatives of the co-ordinates of the system. Supposing the equations of motion have some particular solution

<

qj' = /j'(t)

(i = 1,2,3),

( j = 1,.

p

. .,%),

= F.,(x,y,z,t),

which satisfies certain initial conditions (2.16) .

qio = / i (to)1

qjo' = /j'(t0)

I

vi(x,y,z,to)= pi(x,y,z)

( j = 1,. . .,%),

(i = 123)*

as well as boundary conditions of the form (2.4)-(2.8). Solution (2.15) defines some motion of the solid-liquid body; we consider this motion as the unperturbed motion and raise the question about its stability. Suppose the initial conditions for the perturbed motion are (2.17)

* The

ranges of z and j indicated here will be omitted from now on.

STABILITY O F MOTION OF SOLID BODIES

207

where the .ci,ei' are real constants and the E ; ( x , Y , z )are real functions satisfying the continuity equation. We shall term the E'S perturbations. By giving the perturbations, the perturbed motion of the system is fully determined since the forces acting on the system are assumed to be invariable [31]. We shall consider the absolute value of the perturbations as sufficiently small. The equations of the perturbed motion are obtained from the original equations by substituting for qi,qj',vi,p the quantities

If the absolute values of the deviations (or variations) ~ ~ ~ ~ , ~ x ~ ' ~ , ~ u remain arbitrarily small for any t 3 to and for any sufficiently small perturbations I ~ ~ l , l & ~ ' l , l & , ( x ,then y , z ) ~ ,the unperturbed motion (2.15) is stable with respect to qi,qi',vi,p. However, the problem so posed involves tremendous difficulties, and at present the methods of the theory of small oscillations are the only ones known for its solution. These difficulties can be by-passed by changing the approach to the problem. The point is that in the problems which occur in the applications the interest is centred chiefly in the question of stability of motion of the solid body, while the stability of motion of the liquid in the cavity is of interest only inasmuch as it affects the stability of motion of the body. Of course, these two aspects of the same stability question are interconnected: the motion of the body depends on the motion of the liquid, and vice versa. In view of the foregoing, it is natural to pose this problem in the sense of Lyapunov, i.e. with respect to the variables determining the motion of the solid body and to some quantities p , = ~ T Q ) S ( ~ , y , ~ , v 1 , v 2which , v 3 ) d tdefine integrally the motion of the liquid. Here the functions @,(x,y,z,v1,v2,v3) are real, continuous, bounded functions of their variables. For example, it can be seen from (2.1) that with given external forces the motion of the solid body depends on the moment of momentum of the liquid and its time rate of change which, in turn, depends on the motion of the body. In this problem, the projections of the moment of momentum of the liquid are therefore a suitable choice for p,. Note that quantities #, such as momentum or moment of momentum of the liquid etc. do not determine fully the motion of the liquid (which depends on an infinite number of variables). As a result, the stability of motion of the liquid with respect to quantities p , is a conditional stability, i.e. the stability of motion of a continuous medium with respect to certain fuitctionals of the velocity distribution.

208

V. V. RUMYANTSEV

By this approach, our stability problem, originally of a system with an infinite number of degrees of freedom, is reduced to the investigation of stability with respect t o a finite number of parameters qi,qi’,ps, which ensures stability with respect to qi,qi’. Such an approach permits especially the application of Lyapunov’s second method. To construct Lyapunov’s functions, we shall use the relationships obtained for the perturbed motion from (2.!)), (2.11)-(2.14). Consider real continuous functions Q, of qj,qi’,ps and of C. For the unperturbed motion, the functions Q, will change after substitution of (2.15) into some known functions of time F,(t); for the perturbed motion (2.18), the Q, (for which we retain the previous notations), will be functions of t and of the variables xi,xj‘,ui (deviations), depending in turn on the (initial) perturbations E ~ , E ~ , E , ( x , Y The , z ) . differences

y,

(r = 1, . . . ,K )

= Q, - F ,

also depend on t and xi,xi’,ui. Let L, be arbitrarily given positive numbers. If for any L,, however small, positive numbers Ei,Ei‘,E,* can be so selected that for all perturbations E ~ , E ~ , E ~ ( x , satisfying ~,z) Ei, J E ~ ’ ~ Ej’, / e , ( x , y , z I) Ei* the inequalities

<

<

IQ,

<

- F r J< L

hold for any t 2 to, then the unperturbed motion is stable with respect to the quantities Q,, otherwise it is unstable [ Z ? ] . We now prove a theorem [44] which may be useful in some cases as a modification of Lyapunov’s theorem about stability. Let us assume that there exists a real, continuous, bounded, one-valued function p(t,xi,xi’,ui)which vanishes when the variables xi = xi’ = u, = 0, and whose time derivative satisfies cp’ 0 owing to the equations of the perturbed motion. Let the inequality

<

(2.19)

Y ( t , y , ,* .

< p(t,xj,xj‘>ui)

, ~ k )

hold for all values of the variables t,xi,x,’,2ti under consideration and the corresponding values of the variables y,, where !P(t,yl,. . . , y k ) is a real, continuous, bounded, one-valued function of its variables, which disappears when all y , = 0 ( I = 1,. , .,k).

Theorem 2.1. If the differential equations of the perturbed motion are such that it is possible to obtain a function p(t,x,,xi’,ui)whose time derivative y’ 0, and if there exists a positive definite function Y ( t , y 1 , .. .,yk) such that (2.19) holds, then the unperturbed motion is stable relative to Q, ( I = 1,. . . , k ) .

<

Proof [31] : By the definition of a positive definite function it is possible to obtain a positive definite function W ( y , , . . . ,yk) independent of t such that

809

STABILITY OF MOTION OF SOLID BODIES

-

F(t,y1,* * * 9Yk)

w ( Y 1 , . .Yk)

is valid. Suppose A > 0 is an arbitrarily small number and 1 > 0 is the exact lower boundary of W on the sphere Z f = 1 y s 2= A . The function Q ) ( ~ ~ , X ~ , X ~ ’ ,being U,) clearly independent of time, admits an infinitely small upper limit;* consequently, for given 1 > 0 a number 1 > 0 can be found such that for all values of the variables xi,xj’,u, satisfying

< 1,

(2.20)

lxj’j

Jxjl

< 1,

<1

the values of functions Y(t,,y,) and ( P ( ~ ~ , x , , x ~ ’ ,will u , ) satisfy

F(io$yi,*

*

< Q)(fo>xj,xj’,%) < 1.

#Yk)

If the initial values of variables xi,xj‘,ui are selected according to (2.20), we have, in conformity with the conditions of the theorem, the following inequalities : W ( y 1 ,*

*

,Yk)


*

< y(t,xj,xj’.ui)<

*Yk)

We may thus conclude that in their changes variables ys satisfy condition Z:=,ys2< A since 1 is the exact lower boundary of function W on sphere ( A ) . This proves the theorem. 3. Two Examples (Viscous Liquid, Parlially Filled Cavity) Example 4. Problem of stability of rotation about a fixed point of a solid body with a cavity completely filled with a viscous liquid. Suppose such a solid body moves about a fixed point in the force field U ( y 3 ) . Let us assume for simplicity that the principal axes of inertia of the body for the fixed point 0 are also the principal axes of inertia of the cavitv. The equations of motion have the particular solution w1 = w2 = 0,

wg = w ,

Gl= G2 = 0,

G, = C2w,

y1=y2=0,

y3=

1,

(2.21) u = II = ZE, = 0,

which describes the uniform rotation about axis z coinciding with axis 5 of the solid body and the liquid in its cavity. Let us investigate the stability of this motion relative to the projections of the instantaneous angular velocity of the body OJ,, the projections of the moment of momentum of the liquid G,, and the direction cosines y,, assuming that in the perturbed motion < a determines * Since 1~1(1,,0,0,0) = 0, the set ( x j , x j ’ , u i ) defined by q~(l,,xj,xj’,ui) for a n y positive a a non-empty neighborhood of 0.

210

V. V. RUMYANTSEV

and retaining the previous designations for the rest of the variables. First we obtain an expression for the kinetic energy of the liquid in terms of the G, [45]. During the motion of the system, the G, will be functions of time. Now introduce instead of G,new unknown functions of time x,, defining them by (2.22) If the velocities of the liquid ui are known, this in itself determines functions x,. Let us also introduce u(i)(x,y,z,t),defined by (2.23)

~ ( 1= ) vI+ x3y

- x#,

~ ( 2= ) ~2

+

-

X ~ Z X ~ X , ~ 1 3= ) 23

+

X ~ X XI)’.

It can be easily seen that owing to (2.22)

By making use of equalities (2.22) and (2.23), the kinetic energy of the liquid can be represented as

This, incidentally, entails at once the validity of the inequality r28 (225)

2T2K >, G12

+ Gz2 +

G32,

K

= max

(A2,B2,C2).

Proceeding to the analysis of the stability of the unperturbed motion (2.21), we note that in this case (2.26)

x1 = x, = 0,

x3 = 0

,

V(l)

= V l 2 ) = V ( 3 ) = 0.

By (2.9) we have for the perturbed motion of the system (227) where

d

-
STABILITY OF MOTION OF SOLID BODIES

211

I t can also be easily seen that the equations of the perturbed motion have first integrals

T

where the constant A > (a2U/ay32)l- (aU/ay3)1. According to Sylvester's criterion the inequality (2.29) is necessary and sufficient for V to be positive definite if A 3 B. The time derivative of V , taken by means of the equations of the perturbed motion wiIl not be positive as a result of (2.27). Consequently, under condition (2.29) function V satisfies all conditions of Lyapunov's theorem about stability, which proves the stability relative to w,,G,,y, of the unperturbed motion (2.21) of a solid body with a cavity filled with a viscous liquid. In the case of a heavy solid body, when U ( y 3 )= - MgzOy,, condition (2.29) appears as (2.30)

(C - A ) W , - MgzO> 0.

The problem of stability of a symmetrical gyroscope with an axisymmetrical cavity completely filled with an ideal liquid making a vortex

212

V. V. RUMYANTSEV

motion, has been studied in a linear analysis by S. L. Sobolev [15], who proved in particular that the operator eiBt characterizing the perturbed motion of the system is bounded if

Evidently the condition of boundedness of this operator coincides with the condition of stability (2.30) of the unperturbed motion (2.21) relative to w;G;,yi.

I t will be noted that in the case of inertial motion of a solid-liquid body about the centre of mass (zo = 0) condition (2.30) appears as

C>A>B. Hence the permanent rotation of a solid body with a cavity filled completely with a viscous or inviscid liquid about the small axis of the central ellipsoid of inertia of the system is stable. This result may be regarded as supplementary to Zhukovskii’s theorem [7] about the motion of a solid body with a cavity completely filled with a viscous liquid.* In a similar approach, paper [46] provides the solution of the stability problem for a free solid body with a cavity completely filled with a viscous liquid, if the body is attracted by a fixed centre according to Newton’s law.

Example 5. Stability of equilibrium of a pendulum containing liquid. Let us consider a heavy solid body with a fixed point 0 and with a cavity 9artly filled with a homogeneous heavy liquid. We suppose the surface of the cavity closed and convex, and the vector of the surface normal a continuous function of position. The equations of motion (2.1) and (2.2) have a particular solution (2.31)

w1 = w2 = w3 = 0,

G, = G,

= G, = 0,

y1 = yz = 0,

y3 = 1

if the centre of gravity of the system is on the z axis. Solution (2.31) means equilibrium of the body and liquid, whose free surface is the plane z = zl. Let us analyze the stability of the equilibrium (2.31) in relation to wi,G,,y, setting in the perturbed motion y3 = 1 7 and retaining the previous notations for the rest of variables. The equations of the perturbed motion have the integral relationship and the first integral

+

* According to Zhukovskii’s theorem, the limiting state of any motion of such a system (starting from arbitrary initial conditions) is a rigid-body rotation of the whole system about one of its principal axes of inertia (say a ) with constant angular velocity. The final direction of a coincides with the direction of the initial angular momentum h, and w = h/moment of inertia about a.

213

STABILITY OF MOTION OF SOLID BODIES

I/, G

(2.32)

+ T , + M,gzloq - U , - M,gz,O) + + q 2 + 2q = 0.

2(T,

v, = y12

= const,

y22

In virtue of inequality (2.25),

where K denotes the largest of the principal moments of inertia of the liquid a t anyone instant [28]. It can also be easily seen [4Y] that the potential energy of the liquid in the perturbed motion

K

where - Uz* denotes the potential energy of the liquid when it has the plane xyl yy, zy3 = z1 as free surface. It is easily calculated that

+

+

where A,,B, are the principal central moments of inertia of the segment Q of plane z = z1 limited by the walls of the cavity. Let us now consider the function

Hi = 2 T 1 +

1

(GI2

+ G: + G3') + 2Mgzooy - pg(A*yl ,+ B*yZ2)+ . . *

(2.33) It clearly follows from the above that

v, 2 H,,

(2.34)

and if we construct the function

!?'

=H,

(2.35)

1

- MgzooI/, = 2 T 1 + - (G,2 K

- (MZOO

+ G,2 + G,2)

+ A*p)gy12 - (MZO0+ B*p)gyz2 - Mgzo0y2+

it becomes evident that

y
q) = v1- Mgz0OV,.

'*

*3

214

V. V. RUMYANTSEV

If the inequality (2.36)

MzoO+A*p
(A*>B*)

is satisfied, function Y is positive definite and satisfies all the conditions of theorem 1.1. Consequently, (2.36) is a sufficient condition of stability of equilibrium (2.31) relative to cui,Gi,yi [49]. 111. STABILITY OF STEADY MOTION OF A SOLIDBODYWITH LIQUID-FILLED CAVITY This chapter develops Lyapunov’s ideas p28, 291 proceeding from the theory of stability of equilibrium figures of a rotating liquid. Proof is supplied [48]that the question of stability of uniform rotation of a body containing a liquid-filled cavity can be reduced to the problem of existence of a minimum for a certain expression W , and a solution for the latter problem is given [50]. 1. Some Relationships

Consider a rigid body with a simply-connected cavity of arbitrary shape, filled with an incompressible homogeneous ideal liquid. We shall treat simultaneously the cases of the cavity being completely and partially filled with a liquid: in the second case the liquid inside the cavity has a free surface on which the hydrodynamic pressure remains constant throughout the motion. Stationary workless constraints are imposed on the body; the given forces applied to the body and the body force acting on the particles of the liquid possess force functions U , and U , clearly independent of time. We suppose as before that the motion of the body is continuous; the liquid moves as a compact deformable mass so that the co-ordinates of every particle of the liquid are continuous functions of their initial values and of the time. Under these conditions the equations of motion of the system have the integral of energy [32]

where 1; is the potential energy of the forces acting on the system and h an integration constant. Let us determine the position of the solid body relative to the inertial frame O,&[ by Lagrangian co-ordinates q,.. . .,qn(n 6). The potential energy of the system I.’ will, generally speaking, be a function of the coordinates q,,. . .,q, (r n) and of the shape of the liquid inside the cavity. We also assume that the constraints permit rotation of the whole system as a single solid body around some fixed straight line, and that the forces acting on the system have no moment about that line. In this case the

<

<

STABILITY O F MOTION OF SOLID BODIES

215

potential energy of the system, V, will evidently be independent of the angle of turn qn of the body about the straight line. Under these conditions there exists an integral of areas for the plane orthogonal to the straight line. Taking that line as axis 0,C of the fixed frame, we can put down the integral of areas as

Gc = const

(3.2)

where G, denotes the projection of the moment of momentum of the system on axis 5'. which Along with the fixed frame we introduce the frame O,t,q,[ rotates about axis C at some angular velocity 0). Designating by v(v1,vz,v3) the vector of the absolute velocity of some point of the body or the liquid with radius-vector r(t,q,C), and by u ( u , v p ) the vector of velocity of the same point in its motion relative to the rotating frame, we have

v=n+oxr. The kinetic energy and the projection on axis ( of the moment of momentum of the system may be represented as (3.3)

T = 2 + n)Gc'

+ -21 0 2 S ,

Gc = Gcl

+ oS.

Here

Y

Y

denote the kinetic energy and the projection on the axis of the kinetic moment of the system in its motion relative to frame 0161q15;

s = 2 m 1 ( t u 2+ q u 2 ) I'

is the moment of inertia of the system about asis 5. (In these formulae the summation refers to all mass points of the system.) The angular velocity LO of the frame O l ~ l q l [ may be taken arbitrarily; let us choose it in such a way that the projection on axis 5 of the moment of relative momentum of the system, Gcl, vanishes at anv time, which by (3.2) and (3.3) is equivalent to (3.4)

WS L'k = const

With such a determination of

(3.5)

(IJ

1291.

the integral of energy (3.1) can be written as 1

k2

X+--+V=h.

2s

216

V. V. RUMYANTSEV

With the above assumptions about the constraints and the acting forces, the system under consideration can indeed carry out a “wheel rotation,” that is a uniform rotation as a single solid body, about the fixed axis 5. In this case the system will be in equilibrium relative to the frame Ol(,q,~, rotating about axis [ with the angular velocity of uniform rotation ooof the system. Formally, this follows from (3.6)

C {(mv5vlf- FiV)6tv + (mvqvlf - F z , ) ~ ++ (mVC,” - Fav)6Cv}= 0 V

which expresses the D’Alembert-Lagrange principle in the usual notation. In the case of “wheel rotation” about axis [ a t angular velocity coo, If

5v

=-

0025v,

q v f f = - w02qv,

C‘ = 0,

as a result of which (3.6) appears as 1

- wo26

2

2mU(tv2+ qv2)+ 2 (FlVSEv+ F ~ ~ +6 F&,) q ~ V

=0

V

or as (3.7)

6U=O

if the notation

is introduced and 6U is considered as the change of U in a virtual displacement of the system, compatible with the constraints and preserving the volume of the liquid. But U in (3.8) may be regarded as a force function of given and fictitious (centrifugal) forces. Thus (3.7) represents the equilibrium condition for our system relative to the frame 0,~,ljil[. Let us now introduce the function (3.9)

1 ko2 w=--+v 2 s

where KO is the value of the constant k in the law of areas for the present case of uniform “wheel rotation”. The change in this function on a virtual displacement of the system is

where So is the value of S for steady motion.

STABILITY OF MOTION OF SOLID BODIES

217

By comparing 6W with 1 2

6U = - w026S - 6V and using woS, = k,, we see that equation (3.7) is equivalent to

6W = 0.

(3.10)

Thus, in this case of steady motion of the system, expression (3.9) is stationary. According to definition (3.9), W depends on those co-ordinates of the body ql,. .,qn- on which S and V actually depend, as well as on the shape of the liquid and on the magnitude of the constant k,. The quantity k, can be regarded as a variable parameter; we may then apply to our system the results of the general theory of “equilibrium” of material systems with potential energy depending on a parameter [31!. Condition (3.10) leads, as can be easily seen, to equations

.

(3.11)

for co-ordinates qi of a solid body in steady motion, as well as to equations for the pressure in the liquid, which lead to

for the free surface, if the liquid does not fill the cavity completely. Here U,(t,q,C) denotes the force function of the body force per unit of mass acting on the liquid, and the potential energy of the liquid is given by = - pJrU&t. For a fixed value of the parameter k,, equations (3.11) and (3.12) determine the co-ordinates of the solid body and the shape of the free surface of the liquid in this steady motion.

v,

2. A Stability Theorem

Consider steady motion of the system corresponding to a given value of the constant of areas k,, and assume (without restricting generality) that the solution of equations (3.11) with the given value of KO is q, 1=: 0. The liquid has the shape of relative equilibrium f,; it is bounded by the free surface a,, determined by (3.12), and by the walls of the cavity with which the liquid is in contact. We shall analyze the stability of this steady motion using the integral of energy (3.5).

218

V. V. RUMYANTSEV

+

Our mechanical system possesses n 60 degrees of freedom, and we must first agree on what we mean by stability of its motion. When the liquid fills completely the cavity of the body, stability of motion as understood by Lyapunov [27] refers to those of the non-cyclic co-ordinates ql,. . .,qn - of the body, on which the potential energy V and the moment of inertia S clearly depend, and to the generalized velocities q,‘,. ,,q’n-l and the kinetic energy Z, of the liquid. When the cavity is not completely filled and the liquid has a free surface, the matter is more complicated. As found by Lyapunov [28], in the case of a liquid the integral of energy is in general insufficient to disclose, after the state of (absolute or relative) rest has been disturbed, those traits which are accepted as a sign of stable equilibrium in the mechanics of systems with a finite number of degrees of freedom. Lyapunov has established that the difficulty is wholly eliminated only, if the stable shape of equilibrium is defined as that one, from which the shape of the liquid after a sufficiently small perturbation differs arbitrarily little, a t least until thin thread- or leaf-like projections are formed on the surface of the liquid. Such projections may be large by linear dimensions but small in volume, hence they are incapable of carrying much of the energy. Let us accept this definition and, in line with Lyapunov’s idea, introduce several relevant concepts pertaining to our problem. Compare the shape of relative equilibrium of the liquid fo with its shape f at some moment of the perturbed motion, ignoring the motion of the liquid particles themselves, but taking into account the magnitude of the kinetic energy of the liquid. Shape f is limited by the free surface uc of the liquid and the walls of the cavity with which the liquid is in contact at the given moment. For a perturbed motion, which is sufficiently close to the unperturbed one, in a frame rigidly connected with the body, the shapes fo and f differ only by the free surfaces a, and 0,. The volumes belonging to f and lo are of course equal. Consider some point P on ac and the point Po, the closest to it on a,. As the position of P on a, changes and, consequently, that of Po on u,, the distance PP, changes too, and for some position of P it will become the largest possible distance a t the given instant. This maximum was termed by Lyapunov separation. We designate it by I:

.

I = max (PP,) . Introduce also V as a measure of the deviation of f from f,:V is the volume of that part of f which is outside ,/ or, which amounts to the same, the volume of that part of fo which is outside f. With a separation of given value I, the deviation V will have some maximum which may be written in the form 1#(Z): max V = Z#(Z).

1= const

STABILITY OF MOTION OF SOLID BODIES

219

Here $(l) is a positive function, no value of which exceeds some definite bound. If I does not exceed some given number A , the function $(I) will have a minimum different from zero: min +(I)

> 0.

I g A

But the minimum of the deviation V for the given value of separation always equals zero [29j. For continuous motion of body and liquid, 1 and I7 will evidently be continuous functions of time. We are now ready for the following definition of the stability of motion of our system when the cavity is not completely filled: Consider the perturbed motion of the system following an initial perturbation. If the initial separation and the initial relative velocities of the particles of the liquid as well as the initial desiations and relative velocities of the body can be selected so small that 1qiI,Iq,'1,1221, and I remain smaller than some arbitrarily small preassigned bounds, either throughout the motion or a t least until the deviation has itself become smaller than some arbitrarily small preassigned de\+ation, then the motion of the system under consideration is stable; otherwise it is unstable. According to this definition, the following is a necessary and sufficient condition for the stability of the unperturbed steady motion: To any small given numbers L , and L, a positive number 1 can be so selected that for initial values of generalized co-ordinates q,(')) and velocities q i ' ( 0 ) of the body, of separation of deviation VO)and of relative velocities of the liquid U ( ~ ) , V ( ~ ) , Wwhich (~), satisfy conditions (3.13)

at t

= to,

lq,(O)l, l q i f ( 0 ) l lI(O)l, , Iu(O)I,1v(O)1,jw(O)I all for any t

(3.14)

> to, or

< 1,V(O)> do)

a t least until

v >El,

the inequalities (3.15)

/q*1$ < L,:

1 4 ~ ' I 1 1 q< L?

are satisfied. Here E denotes a positive number smaller than the minimum of the function $(1) under condition lIj L,, while €1 may be regarded as a possible deviation of the liquid. I t is worth noting that the condition related t o (3.14) should be omitted, if the cavity is completely filled. \Ve shall also need the concept minimuni o i expression W . If u' represents function IY(q,,. . . , q n - I , k o ) , we mean by minimum of this function (at fixed k,) an isolated minimum with respect to those variables q l , . . . ,qn- ,, on which it actually depends. If the liquid does not fill the cavity completely, then we accept with Lyapunov 1281 the following definition of the minimum of W : If for the steady motion under consideration (when qi = 0, I = 0,

<

220

V. V. RUMYANTSEV

V = 0)the number W , is a minimum of the expression W , then there exists a sufficiently small positive number E such that for every system of values qi, of separation I, and of deviation V , satisfying /qil

< E,

d E,

V > €1

[where E is a positive number smaller than the minimum of $ ( I ) under condition 111 El all values of the difference U’ - W , remain positive and vanish only with vanishing qi, I, and V . Note that with any given value of I the difference W - W , can be made as small as desired by choosing such a position of the body and such a shape of the liquid for which /q,J and V are sufficiently small. But the limiting case, when with 1 # 0 all q,,F, and consequently W - W , vanish, is evidently impossible if only such shapes are considered which can be assumed by the liquid. To eliminate this inconvenience, condition V > PI has been introduced.

<

Theorem 3.1. If in the steady motion of a solid body with liquid, filled cavity, the expression

formed for all slightly perturbed states near the state of steady motion, has an isolated minimum W,, then the unperturbed motion is stable. Proof [29]. Let us impart to the points of the system some sufficiently small initial deviations and velocities. If left to itself, our system will move on further in conformity with the integral of energy (3.5)which we rewrite as

where again (0) denotes the initial perturbed value and k is the magnitude of the constant of areas in the perturbed motion. Suppose A is an arbitrarily small positive number which does not exceed the number L , and which we shall assume a t any rate smaller than number E above. Let us designate by IY, the least possible value which W can assume if the separation I or one of the co-ordinates q, are absolutely equal to A , while the remaining quantities and the deviation satisfy the conditions lQll

d A,

111


v 2 El.

Since according to the conditions of the theorem IY has a minimum W,, we must have

w,> w,.

STABILITY OF MOTION OF SOLID BODIES

221

But in choosing /I) and 1qil sufficiently small, and B > EE, expression W can be made to differ from Woas little as desired. Let us choose A so small that

IW -

(3.17)

L,

is satisfied. The initial values qi(O) and

E(O)

can be chosen so small that

W(O)<W,.

(3.18)

In selecting such an initial perturbed state of the system, let us assume with regard to the initial values of q, and the initial shape of the liquid that they also satisfy Iqi'O)l


1E(O)j


V(O)> d(0).

Whatever the initial position and the shape of the liquid, the initial velocities of the points of the system can be so chosen that the constant magnitudes

are as small as desired. Let us choose these constants so small as to satisfy the inequality (3.19)

for all values that S can assume when the conditions (3.20)

IQil < A ,

Ill B A

are fulfilled. Now choose R, which appears in the stability condition (3.13) and determines the region of initial perturbations, such that with conditions (3.13) satisfied inequality (3.19) will be fulfilled for all values of S under conditions (3.20). With such a choice of initial conditions, we have throughout the subsequent time of motion as long as (3.20) holds (3.21)

2+

w< w,

by (3.16) and (3.19). Hence, W < W,, and this will hold at least until lqil or Ill exceeds A . But qi(O) and E(O) are smaller than A , and V(O)> .do), and since qi, 1 and change continuously in course of time, lqil and 111 cannot exceed A without first becoming equal to A . And inequalities lqil = A ,

111 = A ,

owing to inequality (3.21) with condition impossible.

V > EE are, quite evidently,

222

V. V . RUMYANTSEV

I t follows from inequality (3.%1),account taken of (3.17), that 171 < L,. On this ground we conclude that

Consequently, if the motion of the system proceeds continuously, so that qi,l,V change continuously in the course of time, then, starting from to, we shall have inequalities

and none will cease to hold as long as the last of them is valid. The theorem has thus been proved. I t is worth noting that for a completely filled cavity the conditions for 1 and V become superfluous; under the conditions of the theorem the following inequalities hold for any 12 t o : /qal < L,,

/%’I < L,,

(221

< L,.

Note 1. As outlined above, the liquid was considered ideal. But the theorem remains valid also for a viscous liquid as in this case the only thing required is to replace equation (3.16) by the inequality

without changing anything in the proof.

Note 2 . Lyapunov 1291 noted that to define the difference between a perturbed shape of a liquid and an unperturbed one it is possible to introduce instead of the separation 1 some other quantities which can vanish only in the case of unperturbed shape. For instance, one may take the deviation V as such a quantity and prove in a way similar to the one above: If for the given steady motion of a solid body with a cavity not filled completely W has minimum Wo in the sense that W - Wo> 0 for all values of (qi( and V which are not simultaneously equal to zero and are smaller than a certain constant bound, then, for sufficiently small perturbations lqil and Jqi’/,V and %, will throughout the motion remain smaller than preassigned constants, however small the latter are. Note 3. If there are theorems of energy and areas for the motion of the solid-liquid body, relative to the centre of the mass of the whole system, theorem 3.1 is also valid for this relative motion. In this respect the theorem is a generalization of Lyapunov’s theorem about the stability of the figure of equilibrium of a homogeneous rotating liquid whose particles attract each other according to Newton’s law.

STABILITY OF MOTION OF SOLID BODIES

223

As a matter of fact, if our system consists of only one gravitating liquid mass, then

and if the minimum of the expression Wjnf occurs for the equilibrium shape of the liquid, then this shape is stable [28, 291. Corollary. If in the equilibrium position of a solid body with a liquidfilled cavity -(KO = 0 ) , the potential energy of the system V has an isolated minimum V,,, then the equilibrium is stable [48]. Note that this result can also be applied in the case of relative equilibrium of the same system. To have something definite in mind, we assume that our system is acted upon, in addition to conservative forces, by nonconservative forces which are the cause of a moment N directed along axis C whose magnitude is such that throughout the motion the angular velocity w of the body about 5 remains constant. In this case we have instead of the integrals of energy (3.1) and areas (3.2) equations

d(T

+ V ) = Nwdt,

dGc = N -

at

151 1

from which it follows that throughout the motion T

+ V - OCC= const.

Introducing again the mobile frame O1(lq,C and bearing in mind formulae (3.3), we can rewrite this integral as (3.22)

1

1+ V--dS=const. 2

The positions of relative equilibrium of our system are again determined by (3.7). By repeating almost literally the proof of Theorem 3.1, it is easy to see that the following theorem is valid: Theorem 3.2. If for the position of relative equilibrium of a solid body with a liquid-filled cavity

w*=v - - w212 s has an isolated minimum, this position is stable. As noted above, condition (3.7) is equivalent to condition (3.10) if (3.4) holds. Owing to this, we are able to compare the state of relative equilibrium of our system at constant angular velocity w with that of steady motion

224

V. V. RUMYANTSEV

possessing an integral of areas (3.2). If for some position of relative equilibrium W , has a minimum, W likewise has a minimum for the corresponding steady motion [51], as can be easily seen. Indeed, let W , be a minimum of expression W,, i.e. I’ -

v, - -21 w2(S - So) > 0,

and assume that W has no minimum in the corresponding steady motion, i.e. there will be points in a sufficiently close vicinity of the position of relative equilibrium, in which

: (f,

-ko2

:)

--- -

It+

v,>o,

By replacing in the latter inequality k , by Sow and adding it to the former, we obtain a contradiction. Consequently, if for constant w the state of relative equilibrium of a solid body with a liquid is stable, the corresponding steady motion is likewise stable provided G: = const. Suppose W has a minimum for some particular value of the parameter, i.e. the steady motion is stable. Now let us change continuously parameter k,, then the roots of equation (3.10) will describe some branch C of the curve of “equilibria”. Also W will change continuously, and for all points of curve C, for which W retains the minimum property, the steady motions are stable. A change in stability can only occur at a point of bifurcation [31]. 3. The Problem of Minimum Consider the region of variables qi in the vicinity of the steady motion (qi = 0) of a solid body with a liquid: (3.23)

lqil ,< H

where H > 0 is a sufficiently small constant. Let qi be the co-ordinates of some fixed point of region (3.23). We wish to find out what shape of the free surface of the liquid produces for fixed qi an extremum of the expression W . To answer this question, we equate the first variation of W a t fixed qi to zero:

where S is the moment of inertia of the system about axis 5 for given q,, and the unknown shape of the free surface of the liquid is to be determined. By carrying out the variations under the integral sign and taking into account the equation of continuity and the boundary conditions for the

STABILITY O F MOTION OF SOLID BODIES

225

liquid, we find that (3.24) is possible if and only if the free surface of the liquid appears in the form (3.25)

For steady motion with q, = 0 equation (3.25) takes the form (3.12). Under rather general assumptions regarding the smoothness of the cavity walls, the Ievel of the liquid and the nature of the body forces acting on the liquid it can be shown [50] that expression W has a minimum if for fixed q, the free surface of the liquid is determined by equation (3.25). We shall not give the proof of this statement. To any given set of values q, belonging to region (3.23), the solid body with liquid in its cavity can be made to correspond to a certain solid body, which we shall call transformed and which consists of the given solid body and the liquid with free surface (3.25) solidified. According to what has been stated in the foregoing paragraph, W for the transformed body will have a minimum compared with W-values that belong to any free surface permissible for the liquid and sufficiently close to surface (3.25). Theorem 3.3. For the existance of a minimum of W in the steady motion of a solid body with a liquid in its cavity the existence of a minimum of W at q’ = 0 for the transformed solid bodies in region (3.23) is necessary and sufficient. Proof. Suppose expression W for all possible transformed solid bodies in region (3.23) has a minimum W,, with q, = 0 ; then W - W,> 0. For a solid body with a liquid, the difference W - W , will be still more positive, by the definition of the transformed body. This proves the sufficiency. To prove necessity, suppose expression W for the solid body with a liquid has a minimum at q, = 0. This means that for every possible set of q,, I , and V , satisfying 1q,1 H , )I1 H , 2 EZ, all the values of W - W , will remain positive, becoming zero only for q, = 1 = = 0. Consequently, this difference will also be positive for values of separation 111 < H that characterize the transition from shape f o of the liquid in the unperturbed motion to shapes f determined by equation (3.25) with any q, from region (3.23). This in turn means that expression W has the minimum W , for transformed bodies with q, = 0. The theorem has thus been proved. Hence, the problem of the minimum of expression W is reduced to a minimum problem for a function of a finite number of variables W for a solid body with a liquid bounded by walls o1 of the cavity and the free surface (3.25). Let us find the change in magnitude W for the transformed solid body when passing from a position corresponding to steady motion of the system at q, = 0 to a perturbed position in region (3.23). This passage can be

<

<

226

V. V.

RUMYANTSEV

realized in two stages: 1) displacement to the perturbed position of the whole system as a single solid body; 2) deformation of the shape lo of the liquid by putting on its free surface a layer of “vanishing volume” tl [51j, which gives it the shape f with free surface (3.25). The corresponding increase in magnitude will appear as

d W = AIW

(3.26)

with obvious meaning of A , and A,. we have

+d2W Neglecting third order terms in q,,

+ 7[2d1S . A 2 S + (A2S)’] + . . . w2

2SO

where ( f 0 denotes that the appropriate quantity is calculated in the unperturbed position of the system, and AIS A,S stands for A S . To calculate A,W it is convenient to introduce a mobile frame xyz rigidly connected to the solid body whose axis z supposedly coincides in the unperturbed position of the system with axis 5. Let us designate by r$(x,y,z,qi) the integrand in the expression for A,W after transformation to variables x,y,z. The equation of the free surface (3.12) of the solidified liquid in variables x,y,z appears as:

+

The magnitude of constant c is determined by the amount of the liquid in the cavity of the body. Suppose equation (3.28) permits a one-valued solution for one of the variables x,y,z, e.g. for z [if (ar$/az), # 0 everywhere on surface (3.28)], and denote by Q the region of plane xy bounded by the projection on this plane of the closed curve in which surface (3.28) and c1 intersect. Surface (3.25) appears now as (3.29)

r$1(x>Y,z>qi)= c1

+

where constant c1 = c dc is determined by the condition that the liquid bodies bounded by (3.28) and (3.29) have the same volume. This is equivalent

STABILITY OF MOTION OF SOLID BODIES

to the vanishing of the volume of this is equivalent to

t l ,or

227

to SZldr= 0. In first approximation

where zo and z1 refer to surface points of (3.28) and (3.29) respectively. Replacing z by the new variable [49] ,n = &x,y,z,q,) - c, we obtain with the same accuracy

(3.30) Neglecting second order terms we have n-

I

(3.31)

4(x,Y,z,,qi) - c

=AC

+ w2 (x' + y2)dS + . . . SO -

since in first approximation functions +(x,y,z,gi) and +l(x,y,z,qi) differ only by w2/S0(x2 y z ) d S . By substituting these values p o and pl in (3.30), we obtain a linear relation between dc with A,S. A similar equation is obtained bv calculating in first approximation

+

Equations (3.30) and (3.32), with the substitutions (3.31), determine dc and d,S as functions of q,. Note that A c and A,S vanish in this approximation, if

Thus we obtain

(3.34)

228

V. V. RUMYANTSEV

According t o (3.26) and (3.279, AW appears as a quadratic form of the variables q l , . . . ,q” - in this approximation. The conditions of positive definiteness for the latter are the conditions for the existence of a minimum of

,

w.

Example 6. Here we consider a solid body with a single fixed point in a homogeneous gravity field; the cavity is partly filled with liquid. Axis C of the fixed frame O [ q [ with origin a t the fixed point is directed vertically upwards. Let the unperturbed motion be a uniform “wheel rotation” of the whole system about axis z coinciding with axis and assume that it is the principal central axis of inertia of the system. The equation of the free surface of the liquid (3.28) is now (3.35)

1

f$(x,y,z,O) = 3 d ( x 2

+ y2) - gz = c.

The potential energy of the system and its moment of inertia about axis [ in the perturbed position equal respectively

Function #( x , v,z,qi) becomes 1

+(x,y,z,Yi) = - . w 2 [ x 2 2 (3.36)

+ y 2 - x 2 ~ I-2 y2y,2 + z2(yI2+ ~ 2 )

- 2XYYlY2 - W

Y 1 +

Y Y 2 ) v1

- Y12 - Yz2I

______

- d Y l + YY2 + v1 - Y12 - Y 2 2 ) , Let us assume that region Q represents a circular ring with radii R, and R, ( R , > R2). In this case (3.33) holds, and dc = d2S = 0. Then pJr,d(XrJ),ZjQi)dt = h(y12 yf2)a, where

+

Thus by (3.26) we find that

+ [(C, - Bo)w2 - Mgzo(0)- a]y,2}

+ ... .

STABILITY OF MOTION OF SOLID BODIES

229

The conditions for the existence of a minimum of W reduce to one inequality

(C, - A,)w2 - Mgz,(O’ - a > 0 (assuming A , 2 B,). The surface of the paraboloid (3.35) a t large o differs little from the surface of a cylinder .r2

(3.37)

+ V 2 = b2

and coincides with the latter in the limit, if 2g/w2+O. Let us consider this extreme case which occurs when the liquid is weightless. Instead of (3.36) we have now

and

(%)o

=d

b,

( ,I

= - w2bz(cos87,

+ sin By2).

Supposing cylinder (3.37) intersects the cavity surface u1 along circles with centres on the axis z a t -7 = h & d . In first approximation the condition of volume preservation becomes now

in the same approximation

hence

AC = A,S = 0. Finally, we find

5

p $dt=zpb2w2 71

3h2

+ d2

3

(YI2

+Y a 2 P

230

V. V. RUMYANTSEV

The minimum condition for W thus reduces to the single inequality [48]

References (Titles of Russian publications are translated) 1. STOKES,G. G., On some cases of fluid motion, Trans. Cambridge Phil. Sot. 8 (1849). -.> STOKES,G. G., On the critical values of sums of periodic series, Trans. Cambridge Phtl. SOC.8 (1849). H. v., Uber Reibung tropfbarer Flussigkeiten, Sitzungsbevichte der 3. HELMHOLTZ, K. Akademie der Wissenschaften zu Wien, 40 (1860). 4. LOBECK,G., Uber den EinfluO, welchen auf die Bewegung eines Pendels mit einem kugelformigen Hohlraume eine in ihm enthaltene reibende Flussigkeit ausubt, Journal f . reine u . a n t . Math, 7 7 (1873). 5. LAMB,H., “Hydrodynamics”, Cambridge, 1932. F., “Hydrodynamische Untersuchungen”, Leipzig, 1883. 8. NEUMANN, N. E., On the motion of a rigid body having cavities filled with a 7. ZHUKOVSKII, homogeneous liquid, Collected Works, vol. 3, Moscow 1938. 8. KELVIN,LORD,Mathematical and Physical Papers, vol. 4, Cambridge, 1882. A. G., On the general motion of a liquid ellipsoid, Proc. Camb. Phtl. 9. GREENHILL. SOC.4 (1880). 10. SLUTSKII,F., De la rotation de la terre suppose6 fluide a son interieur, Bull. de la Socikttt des Naturalistes de Moscou 0 (1895). 11. HOUGH,S., The oscillations of a rotating ellipsoidal shell containing fluid, Phil. Trans. (A) 186 (1895). 12. PO IN CAR^, H., Sur la precession des corps d6formables. Bull. Asfronontique 27 (1910). 13. BASSET,A. B., On the steady motion and stability of liquid in an ellipsoidal vessel, Quart. J . Math. 46 (1914). G. E., ”The Heaving of Ships”. Moscow, 1935. 14. PAVLENKO, S. L., On the motion of a symmetrical top with a liquid-filled cavity, 15. SOBOLEV, in Zb. Prikl. Mekh. Tekh. Fiz., No. 3, 1960. N. G., On stability of the rotating motions of a solid body, whose cavity 16. CHETAEV, is filled with an ideal liquid, Prikl. Mat. Mech. 21 (1957). L. N., Oscillation of a liquid in a moving vessel, Izu. Akad. Nauk 17. SRETENSKII, S S S R , Old. Tekh. Nauk. No. 10, 1951. D. E., On the theory of the motion of a body, having cavities partial 18. OKHOTSIMSKII, filled with liquid, Prikl. Mar. Mech. 20 (1958). 19. KREIN,S. G., and MOISEEV,N. N., About oscillations of a solid body containing liquid with a free surface, Prikl. Mat. Mech. 21 (1957). 20. MOISEEV,N. N.. The problem of the motion of a solid body containing liquid masses having a free surface, Math. Zb. 82 (1953). G. S., About the motion of a solid body having a cavity partially 21. NARIMANOV, filled with liquid, Prikl. Mat. Mech. 20 (1958). S. V.. and TEMCHENKO, M. E., About a method of experimental 22. MALASHENKO, investigation of the stability of motion of a top, inside of which is a cavity filled with liquid, in Zb. F’rikl. Mekh. Tekh. Fiz. No. 3, 1960. A. Yw.,and TEMCHENKO, M. E., Small vibrations of the vertical axis 23. ISHLINSKII, of a top, having a cavity completely filled with an ideal incompressible fluid, i n Zb. Prikl. Mekh. Tekh. Fiz. No. 3, 1980.

STABILITY OF MOTION OF SOLID BODIES

231

24. STEWARTSON, K., On the stability of a spinning top containing liquid, J . of Fluid Mech. 6 , 577 (1959). 25. COOPER,R. M., Dynamics of liquids in moving containers, A H S Journal 80 (1960). 26. RUMYANTSEV, V. V., Stability of motion of a solid body with liquid-filled cavities, in Proceedings of the All-Union Congress on Theoretical and Applied Mechanics, Moscow, 1962. 27. LYAPUNOV, A. M.. “General Problem of the Stability of Motion”. Moscow, 1950. Russian reprint of the original that appeared in 1892. A French translation, dating from 1907, is available as “ProblBme GCn6ral de la Stabilit6 du Mouvement” by M. A. Liapounoff, republished as Number 17 of Annals of Mathematics Studies, Princeton University Press, 1949. 28. LYAPUNOV, A. M., On the stability of ellipsoidal forms of equilibrium of rotating liquids, Collected works, vol. 3, Moscow, 1959. 29. LYAPUNOV, A. M., The problem of minimum in the question on stability of figures of equilibrium of rotating fluid, Collected works, vol. 3, Moscow, 1959. H., Sur 16quilibre d’une masse fluide animee d’un mouvement de rota30. POINCAR&. tion, Acta Math. 7 (1885). 31. CHETAEV,N. G., “The Stability of Motion”. Moscow, 1955, English Translation (same title) published by Pergamon Press 1961. 32. RUMYANTSEV, V. V., Equations of the motion of a solid body, having cavities partially filled with liquid, Prikl. M a t . Mech. 18 (1954). V. V.. On the equations of the motion of a solid body with a liquid33. RUMYANTSEV, filled cavity, Prikl. Mat. Mech. 19 (1955). 34. KOCHIN,N. E., KIBEL, I. A , , and ROSE, N. V., ”Theoretical Hydromechanics”. Moscow, 1955. 35. GORYACHEV, D. N., “Some general integrals in the problem on the motion of a solid body”. Warsaw, 1910. 36. BELETSKII,V. V., Some questions of the motion of a solid body in a Newtonian force field, Prikl. M a t . Mech. 21 (1957). 37. CHETAEV, N. G., On a property of Poincar6’s equations. Prikl. Mat. Mech. 19 (1955). 38. RUMYANTSEV, V. V., On the stability of permanent rotations of a solid body about a fixed point, Prikl. M a t . Mech. 21 (1957). 39. RUMYANTSEV, V. V., About the stability of the motion of gyrostats, Prikl. Mat. Mech. 26 (1961). 40. RUMYANTSEV, V. V., Stability of the rotation of a solid body, having an ellipsoidal cavity filled with liquid, PrikI. M a f . Mech. 21 (1957). 41. ZHAK,S. V., About the stability of some particular cases of motion of a symmetrical gyroscope, containing liquid masses, Prikl. M a t . Mech. 20 (1958). 42. RUMYANTSEV, V. V., About stability of motion with respect to a part of the variables, Vestn. Mosk. U. Ser. Mat. Mekh. Nr. 4, 1957. 43. RUMYANTSEV, V. V.. About the stability of rotational motions of a solid body with liquid content, Prikl. Mat. Mech. 23 (1959). 44. RUMYANTSEV, V. V., A theorem on the stability of motion, Prikl. Mat. Mech. 24 (1960). 45. RUMYANTSEV, V. V., About the stability of the motion of a top having a cavity filled with a viscous fluid, Prikl. Mat. Mech. 24 (1960). 46. KOLESNIKOV, N. N., About the stability of a free solid body having a cavity filled with a n incompressible viscous fluid, Prikl. Mat. Mech. 08 (1962). 47. RUMYANTSEV, V. V., About the stability of equilibrium of a solid body, having cavities filled with a liquid, Dokl. A . N . S S S R 124 (1959). 48. RUMYANTSEV, V. V., About the stability of steady motions of solid bodies with liquid-filled cavities, Prikl. Mat. Mech. 26 (1962).

232

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49. POZHARITSKII, G. K., The problem of the minimum in the problem of stability of

equilibrium of a solid body having a cavity partialy filled with liquid, PrikZ. M a t . Mech. 06 (1962). 50. POZHARITSKII, G. K., and RUMYANTSEV, V. V., The problem of the minimum in the question of stability of the motion of a solid body with a liquid, PrikZ. Mat. Mech. 27 (1963). 51. APPELL,P., Figures d’bquilibre d’une masse liquide homoghe en rotation, Trait6 de Mecanique Rationnelle, vol. 4. Paris, 1932. A brief introduction to Lyapunov’s main idea can be found in LaSalle, J., and Lefschetz, S., “Stability by Liapunov’s Direct Method”. Academic Press, 1961.

Introduction to the Theory of Oscillations of Liquid-Containing Bodies

BY N. N. MOISEEV Computing Centre of the U . S . S . R . Academy of Sciences, Moscow, U . S . S . R . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I. Survey of Special Problems . . . . . . . . . . . . . . . . . . . . . . 1. Oscillations of a Heavy Liquid Enclosed in a Vessel a t Rest . . . . 2. The Stokes-Zhukovskii Problem . . . . . . . . . . . . . . . . . . 3. The Pendulum Problem . . . . . . . . . . . . . . . . . . . . . . 4. Oscillations of a Conservative System with a Liquid Member . . . . 5 . Torsional Oscillations of a Beam with a Liquid-Containing Cavity . . 6. Torsional-Flexural Oscillations of a Beam with a Liquid-Containing Cavity 11. General Properties of the Equations . . . . . . . . . . . . . . . . . 1. Oscillations of a Liquid in a Vessel . . . . . . . . . . . . . . . . 2. Oscillations of a Conservative System with a Liquid Member . . . . 3. Oscillations of a Beam with a Liquid-Containing Cavity . . . . . . . 4. Investigation of Static Stability . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Selected Bibliography . . . . . . . . . . . . . . . . . . . . . . . . .

Page 233 235 . 230 . 243 246 . 252 . 258 266 . 268 . 268 . 270 . 273 . 280 287 288

.

INTRODUCTION The dynamics of a body containing a liquid I as a history that goes back to the classical contributions of nineteenth-century scientists, but only in the last two decades was the case taken up when the liquid enclosed within the body has a free surface on which waves may appear. Almost simultaneously, and independently of one another, several scores of papers, devoted to this problem, were published in various countries. This interest was evoked by a great number of technical problems calling for a theory of motion of a body with a cavity containing liquid under conditions when the cavity is partly filled. These include, above all, the problem of seismic oscillations of structures under water pressure and of fuel reservoirs, the dynamics of rockets, the calculation of an airplane wing, taking into account the fuel which it may contain, etc. The considerable efforts made by scientists and engineers have produced the result that a t present our knowledge about oscillations of a body with a liquid represents a rather comprehensive theory branching out in several directions. It is not possible to expound within this article all the aspects of this theory, and the author is faced with the difficult problem of selecting the 233

234

N . N . MOISEEV

proper material. Here only the linear oscillations of conservative systems are treated. The first chapter deals with the following points in order of ascending complexity: oscillations of a liquid in a fixed vessel, oscillations of systems of solid bodies with a liquid (systems with a liquid “member”), and oscillations of an elastic body containing a cavity with a liquid but subject to the hypothesis of plane cross sections remaining plane. The chapter derives the principal equations and outlines a procedure for the application of Ritz’ method to such problems. The second chapter analyzes the general properties of the resulting equations, proves their solvability, investigates the structure of the spectrum, and provides a justification of Ritz’ method. In addition, a study is made of a remarkable property of these systems. As already shown by Stokes and Zhukovskii, a body enclosing a cavity completely filled with an ideal and incompressible liquid, whose absolute motion is irrotational, is dynamically equivalent to some other solid body (without liquid). If the liquid has a free surface, then there exists no such equivalent solid body since in this case the liquid introduces new degrees of freedom. Nevertheless, for the position of equilibrium of such a body to be stable, it is necessary and sufficient that the equilibrium of some other solid bodies (without liquid) should be stable. Oscillations of conservative systems containing a liquid represent the most completely developed part of the theory, and it is therefore natural that the review should begin with outlining precisely these questions. Apart from this, the problem of determining free oscillations is the first problem facing the engineer who studies the oscillations of systems with a liquid. The questions outlined in the article form a part of the fundamentals of the dynamics of bodies with a liquid, and the study of more complex questions is based on the problem of free oscillations. Most attention is given to the basic questions of the theory. As to effective methods of calculation, we may at present point to just one factor which permits to simplify the calculations very considerably. The fact is that in the applications problems occur frequently in which the energy of oscillations of the liquid is small compared with that of a system in which the free surface has been replaced by a solid “lid”. In this case the methods of the perturbation theory prove to be highly effective. Even in this simplest case calculations call for the use of high-speed computers. For this reason a number of papers have been published in recent years, which substantially develop the methods of numerical calculation of free oscillations. Several ingenious schemes of numerical calculation and of methods of standardizing computations have been published. ,411 this may provide the subject for a separate article. Nearly all the published methods are, however, based on Ritz’ variational method. This is yet another reason why Hamilton’s principle and Ritz’ method have received most attention in this review.

INTRODUCTION TO THE THEORY O F OSCILLATIONS

235

The article does not deal with the two directions in which intensive investigations are being currently conducted. These are non-linear oscillations and the problem of damping. A study of these questions involves difficulties of a fundamental nature. A great many algorithms pertaining to the theory of non-linear oscillations have been published, but all of them are still very clumsy and, most important, no one has so far succeeded in proving their convergence. Moreover, the very question of the existence of periodic solutions of resulting non-linear systems still remains open. Still more complicated is the problem of oscillations of a viscous liquid. The very formulation of the problems comprises here a great number of difficulties. The problems of the dynamics of a body with a liquid when the latter is under conditions of weightlessness have become pertinent most recently, but only the first steps have so far been made in this direction and it is still premature to speak of results. The article supplies references to those papers only which may clarify the essence of the material in question and facilitate the reading of the article. A selected bibliography is given in addition. 1. SURVEY OF SPECIAL PROBLEMS

By applying Hamilton’s principle, this chapter brings a consistent derivation of the equations of oscillations of bodies with a liquid on the assumption that the external forces are conservative. It is also assumed for simplicity that the oscillatory system contains but one “liquid link”. This means that there is but one cavity in the system with an ideal incompressible liquid, but all results apply trivially to the case of an arbitrary number of cavities. The general properties of the equations so obtained are analyzed in the second chapter. 1. Oscillations of

a Heavy Liquid Enclosed i n a Vessel at Rest

1. Consider the linear oscillations of a heavy liquid enclosed in a fixed shell. Z (see Fig. 1 for notations). With zo denoting a unit vector in z-direction, the gravity vector is g = - gz0. It is assumed that the free surface in the position of equilibrium (S in the figure) coincides with plane xOy. The volume occupied by the liquid in the position of equilibrium is denoted by t. The kinetic and potential energy of the liquid in volume t are expressed by formulae [l]

236

N . N. MOISEEV

Here v is the velocity of the liquid particles, z = C(P,t) is the equation of the free surface, p is the density of the liquid, P is the point with coordinates X,Y,Z.

According to Hamilton’s principle, for the actual motion

6L = 0

(1.1)

where 6L is an isochronous variation of the Lagrangian

5

L = .(T-L!)dl. 0

Let us term (1.1) a variational equation.

FIG.1

We now assume that the liquid enclosed in volume z is subject to the following conditions : 1. condition of irrotationality (1 4

v=

vcp;

2, condition of continuity

v v = 0;

(1.3)

3. condition of impermeability of the shell (1.4)

if

v,=O

PEZ;

4. kinematic condition

(1.51

Ct = u,

if

P ES.

The second and the third conditions are determined by the physics of the problem, and the fourth is a corollary of the second for small amplitudes. The condition of irrotationality, as will be seen below, may be disregarded in the linear theory.

INTRODUCTION TO THE THEORY OF OSCILLATIONS

237

The problem is now reduced to finding in the class of functions satisfying conditions (1.2)-(1.5), all possible functions p(P,t), P € 7 and E(P,t), P E S satisfying (1.1). 2 . Using (1.2) rewrite (1.1):

But, by (1.4) and da, = 0,

In addition, it follows from (1.5) that

Therefore, (1.6) may be rewritten in the following way:

Integrating by parts and using the isochronism of the variations, we obtain

,.,.

In virtue of the arbitrariness of the variations, we obtain from this (1.7)

pt

+ gc

= 0,

p E s.

(1.7) is the known condition of constancy of pressure on the free surface [3]. By using (1.7), we can represent L in the following way: t

O

r

S

Let us also introduce Neumann's operator H which sets the function j ( P ) [ P E S , $fdS = 01 in correspondence with the function p(Q),Q E t which 5

is harmonic in

T;

9 satisfies conditions

238

N . N . MOISEEV

We shall write this correspondence as:

91 = H / . H is the integral operator

whose kernel is Green’s function for Neumann’s problem. According to the general theory (see [2] and [8]),the kernel is symmetrical and possesses for P = Q a source-like singularity: log r - l in the two-dimensional and Y - 1 in the three-dimensional problem. Thus H is a completely continuous self-adjoint operator. By using the obvious representation

p can be eliminated from (1.7) : (1.10)

gc

+ Herr

= 0.

Assuming that Q E S in ( l . l O ) , we obtain an integro-differential equation. Such an approach proves to be especially convenient in studying general properties such as the spectrum and the convergence of variational methods. By using (1.9), the Lagrangian may be represented as

(1.11) 3. We study here the question of natural oscillations of the liquid. Let us assume, therefore, that

y ( P , t ) = cos at@(P), (1.12)

C(P,t)= sin at+(P). The natural frequency a is to be determined. Let us write down the Lagrangian L taken as in (1.8) and (1.11) for the case when p or appears as in (1.12). Integrating over t from 0 to 2n/a and omitting a non-essential factor, we can write L as (1.13) T

5

INTRODUCTION TO THE THEORY OF OSCILLATIONS

239

or

H#.#dS -

(1.14)

I

#'dS,

S

s

where I = a2/g. Hence the determination of the natural oscillations of the liquid is reduced to a variational problem for functionals (1.13)and (1.14). Note: According to the above, the solution of the extremal problem should be sought in the class of harmonic functions. I t can, however, be shown that the extremum will coincide with the value obtained if we consider any functions p E L, as admissible functions. ( L , = class of square-summable functions.)

4. According to the general theory, the first (smallest) eigenvalue 1, is determined by formula

s (A@)2dt

ill = min '

(1.15)

S@2dS

'

s

The second eigenvalue ,I2 is determined as a solution of the variational problem

J" (VQ2dt 1, = min * J@,dS in the class of functions orthogonal to @, if is a function solving the variational problem (1.15),etc. Orthogonality refers here to the metric defined in L,, the functions themselves being defined in S. 5 . To solve the variational problem (1.1)for the functional (1.13)it is advantageous to apply Ritz' method. The standard procedure reduces to the following: a system of coordinate functions { x , ( P ) } is introduced, and the solution of the problem is sought as a sum iv

Then, proceeding from (1.13) and (l.ll), we arrive at the following system of algebraic equations N

(1.16)

2 am(%,,,- ilp,,,,) 9U-l

= 0,

n = 1,4,.. ., N ,

240

N. N . MOISEEV

where

with anm = am%,

Pnm

Pmn.

The system of equations (1.16) is homogeneous. For non-trivial solutions of this system to exist, it is necessary and sufficient that the eigenvalue I should satisfy equation (1.17)

lanrn -

=0

which we shall call the equation of frequencies, We shall designate its zeroes by I n ( n = 1,2,. . . N ) , Owing to the symmetry of matrices awmand prim, the eigenvalues In are real. Once they are known, the natural frequencies are determined by (1.18)

aw2=

In&

The actual determination of the natural frequencies of an oscillating liquid calls for the use of electronic computers. Any numerical scheme should meet the requirement of simplicity in standardizing a vast number of calculations, and Rit.z’ method satisfies this requirement. The chief difficulty encountered in the practical realization of this method consists in selecting the coordinate functions. No general recommendations are available, but in the course of solving the problem three points should be borne in mind. I. The value of I1 is “only little sensitive” to the selection of functions xu. This means : if we replace O1which produces the minimum of the functional in (1.15) by another 01* (such that ~VO,~’Qr,*dr # 0), then I , will change T

but slightly. 11. The boundary conditions for QI belong to the category of natural conditions, and therefore it need not be required that the functions xn should strictly satisfy all boundary conditions. According to the general theory, the minimizing sequence constructed by Ritz’ method will nevertheless converge to the exact solution. may be selected rather 111. Hence, the system of coordinate functions roughly. I t suffices to provide only for the completeness of the system. It is therefore advisable to select the as eigenfunctions of some volume which contains the given volume but has a simpler shape. For instance, if the liquid oscillates inside a conical tank, we may take as system of coordinate functions the eigenfunctions of the liquid in a cylindrical vessel whose cross section equals the largest cross section of the cone.

INTRODUCTION TO THE THEORY OF OSCILLATIONS

24 1

6. Let us now consider the variational problem (1.15) for two volumes equal free surface S , but with surface Zlenveloping surface Z2 (see Fig. 2). Hence t l > t2,and for any function @

tland t2with

(1.19)

Let A']), A(2),@(l)and T~ and T ~ i.e. ,

@(2)

solve the variational problem (1.15) for volumes

FIG. 2

Since

A(')

is the minimum, we have, on replacing

A'2'

<

@)

by

J (V@(l')2dT J- @ ( 1)2dS S

Using the first equation (1.20), we obtain

+I

But, according to (1.19), the factor attached to Consequently )L(2)

is less than unity.

<

Thus, if we have two vessels with the same area of free surface but such that Zlof the first vessel envelops Z2 of the second vessel, then the corresponding natural frequencies will be greater in the vessel whose volume is larger.

242

N. N. MOISEEV

An example will show the possibilities afforded by this theory for an approximate determination of natural frequencies. Consider a two-dimensional problem for simplicity. For the magnitude of the first natural frequency of oscillations of a liquid occupying the volume sketched in Fig. 3 we find the following estimate, using known results on rectangular shapes: tanh (h, + hl) > 1, > tanh h,. 7. At the beginning of this section we have noted, without supplying any proof, that the condition of irrotationality in problems about linear oscillations may be omitted. Let us prove this proposition.

FIG. 3

Suppose the motion of the liquid is rotational. Let us represent the vector of velocity of liquid particles as a sum v = u v, where

+

(1.21) (1.22)

v,= V O , Ap,=O,

VU=O,

PET;

(1.23)

(1.24)

Designate by E the space of the solenoidal vectors u, defined in

T

by

(1.22)-(1.24), depending on 1 and possessing continuous t-derivatives. Define

the scalar product as (12 5 )

j-

(v1,v2) = v1 v,dt.

Let E, E E be the sub-space of irrotational vectors and Elits complement in E . Then vo E E ~ , u € E l . It can be easily seen that u is orthogonal to v, in the sense of (1.25).

243

INTRODUCTION TO THE THEORY OF OSCILLATIONS

Indeed, vector u is solenoidal and consequently

(u,vo)=

5

pnr&

z+s

But, owing to (1.23) and (1.24), ( u , ) ~ = + ~0. Hence

(u,vo) = 0. We shall express this result as follows:

vo = n0v where I7, is the operator of orthogonal projection from E on E,. Vector v satisfies Euler’s equation (126)

Vf

+ (v,V)v = - v-PP - gzo.

Applying the operator of projection

nowe

obtain

P !% +no(v.v)v =-v at P

-

- gzo.

Denoting by l7, the operator of projection from E on E,, we obtain by projection of (1.26)

ut

(1.27)

+ I7,(V,B)V

= 0.

For the infinitesimal motions under consideration, the linear terms of (1.26) and (1.27) produce

v

(2

-+p++z

)= o

or

3 + p + gz = const, at

111

= 0.

The last equation has a simple physical sense: the vortical component of the velocity vector remains constant at every point of volume t if second order terms are neglected. The pressure and the shape of the free surface is determined with the same degree of accuracy by the irrotational component alone which can be found in this approximation independently of the vortical component as a solution of the problem under investigation in this section. Conversely, the omission of condition (1.2), i.e. of irrotationality, will not affect the firstorder accuracy of the solution. 2. The Stokes-Zhukovskii Problem 1. Before proceeding to a liquid with free surface, we have to consider the classical problem of the motion of a body with a cavity completely filled by an ideal incompressible liquid.

244

N. N. MOISEEV

Let the liquid occupy volume t (but the plane S is no longer a free boundary) and assume in addition that volume t is moving and that this motion is known to us. This means that vector v,, of its instantaneous translational motion and vector w of its instantaneous angular velocity are known. The velocity potential of the absolute motion of the liquid should satisfy condition

_ aP an - v,,

P E Zf

s

where v, = von0

(12 8 )

+ (ox r)no, +

no is the unit vector of the normal to the surface Z S and r is the radiusvector of the points of this surface. Let us denote by voi,wi,uz,xi(i = 1 , 2 , 3 ) the projections of the vectors vo,w,noand r on the axes of the system x1,xz,x3, rigidly connected to the moving volume t. We seek the harmonic function p, the potential of the absolute motion of the liquid, in its dependence on the coordinates of the mobile system. As has already been noted by Stokes, the solution of problem (1.28) should be sought as a linear function of scalars zloi and w,, 3

p=

3

2

voipi*

i=1

+ 2 wipi*+s. 1

=1

Functions pi* are determined only by the geometry of region t and do not depend on its motion. They are harmonic and satisfy the following boundary conditions : --

--

an - az z h2,

an

-a%* - u3xz - Q X , an

(1.29)

= h,,

_ a%*_ -azxl - ulxz an

= h,.

an

an

3

=h

-

.

31

= h,,

- UIX3 - U3Xl

-a%*

--

According to accepted terminology, we call these functions Stokes-Zhukovskii potentials [4, lo]. 2 . The. Stokes-Zhukovskii potentials can likewise be determined by Ritz' method. We bring here the necessary considerations, following in the main reference [2].

INTRODUCTION TO THE THEORY OF OSCILLATIONS

Denote by t,h, functions defined in and satisfying

_ -- h i , an

PEZ+S

t,twice

245

differentiable in this region

for

i=l,21...16,

and assume the St.-Zh. potentials in the form p;* =

*; +

u;.

For the functions ui we thus have the boundary problem

-Au;=A#;=f,

PET;

(1.30)

This boundary value problem is equivalent to a variational problem for functional

!

- Au; * uidt - 2 uifdt, which can be written in terms of pi* and

T

T

#i:

T

where

Since C does not depend on the unknown function, the solution of the variational problem for functional F is reduced to solving the variational problem for functional

Applying Green's formula, we obtain

J

I;* = (Vp;*)'dt S

[p;*h;- $;h;]dS, +z

S +z

246

but

N. N. MOISEEV

s t,bihidS

does not depend on the unknown function and may be

s +r

disregarded. Hence, the problem of determining the Stokes-Zhukovskii potentials is equivalent to finding the minimum of functional r

r

J

J S

(1.31) 7

3. The Pendulum Problem 1. We begin the study of the motion of a body with a liquid with a simple example (see [l] and [7]) and analyze the oscillations of the pendulum shown in Fig. 4. The pendulum bob is an open vessel filled with liquid; waves may appear on its surface. The vessel is connected to an ideal hinge

FIG.4

by a rigid weightless rod. Let us place the fixed system of coordinates (OZY) a t the point of suspension and connect the mobile system (ozy) to the free surface in the state of rest, as shown in the sketch. To analyze the plane oscillations of the pendulum requires, generally speaking, some additional assumptions regarding the shape of the cavity. We begin with constructing an expression for the kinetic energy of the system body liquid

+

T = T , + T,; (1.32)

T,is the kinetic energy of the shell, Jo is its moment of inertia about the hinge 0 ; T , is the kinetic energy of the oscillating liquid, p is its density, and v is the velocity vector of the liquid on the assumption that the free surface has been replaced by a solid wall:

INTRODUCTION TO THE THEORY OF OSCILLATIONS

247

v = wvV* where ~ 1 is * the Stokes-Zhukovskii potential and is determined by the shape of the vessel only, and tp is the potential of wave motion in the vessel. Thus

T=

J

8’2

2

+

+

( V ~ ) ~ d8’p t VVVq*dt

+ pe’2

(Vp*)2dt T

7

The integral in the last term depends on the geometry of the cavity only. For this reason we shall call the quantity

an “adjoint mass” (in this case an adjoint moment of inertia) of the liquid. On writing J = J 0 + m, (1.32) becomes (1.33) T

7

We consider this problem in the linear sense and therefore should retain only the quadratic terms in the expression for the kinetic energy. Consequently, we should carry out the integration in (1.33) over the volume which the liquid occupies in the equilibrium position. Let us now construct in a similar way an expression for the potential energv of the system

where

nois the potential energy of

the pendulum,

Mo is the mass of the pendulum shell, and I, is the distance from its centre of inertia to point 0. In this expression the integration extends over the volume occupied by the liquid,

Here we denote by tothe volume occupied by the liquid in the position of equilibrium, and by tlthe volume enclosed between the free surface z = [(P,t)

248

N. N. MOISEEV

and plane S. The first item is the potential energy possessed by the liquid if the free surface is replaced by a solid lid. With pg f Z d r = pgZ*, where 7.

Z* is the ordinate of the centre of mass of the liquid volume. Further, Z* = - I* cos 8 2 (1*02/2) const. Here I* is the distance from the hinge 0 to the centre of mass of volume to. Let us transform the second integral, noting that

+

Z = (z-L)cosO+ ysin8 where 1 = 00. We have [ ( z - 2 ) cos 8 s

7,

+ y sin 8]dz

o

=

$

C2dS

+ pg8

s

in deriving this expression we have used

5

CydS

+ O ( P );

S

s CdS = 0.

S

Neglecting the higher order terms, we finally obtain (1.34)

n=-2 + Bpg u2e2

1

CydS

5

+p

5

C2dS.

S

Here u2 = loM&

+ pgt$*.

Note: Subsequently we shall need yet another expression for the kinetic energy. Applying Green's formula to (1.33) and taking into account that aP

-= an

Sr,

P ES,

we find that (1.35)

2. Let us now construct the equation of motion, calculating for this purpose 6L where

L=

i

(T-Wdt

0

INTRODUCTION TO T H E THEORY OF OSCILLATIONS

249

while T and 17 are taken as in (1.33) and (1.34). By using Green’s formula and the kinematic relationship (1.5), we obtain

Integrating by parts and taking into consideration the isochronism of the variations, we find

For real motions 6L = 0. Hence, by using the arbitrariness and independence of variations 66 and SC, we come to the following system of intergr-differential equations

(1.36)

In this system there are functions y and C connected by the kinematic relationship which allows to eliminate 5. For this purpose it is necessary to differentiate the second equation (1.36) with respect to t and make the substitution pa= Ct. But this procedure is inconvenient in that it leads to an artificial increase in the order of the system. Okhotsimskii [9] suggested the use of the potential of displacements instead of velocities. It is determined by Qt = p On the solid wall, Q satisfies aQlan = 0, but on the free surface it satisfies a Q p n = 5. Using Q, system (1.36) may be rewritten as follows:

(1.37)

250

N. N. MOISEEV

In theoretical investigations it is more convenient to eliminate the velocity potential p, expressing it through the free boundary p = HCt as in (1.9). We thus obtain

3. Let us now consider the problem of free oscillations. Assuming for this purpose that

0 = X sin at,

(1.39)

cp = @ cos at,

5 = $sin at,

system (1.38) after substitution will appear as

X ( K~ ]a2)

1

+p

* ( g y - cp*a2)dS= 0,

5

(1.40)

+ pg*

X p ( g y - cp*u2)

- p@H* = 0.

Let us designate by K , the resolvent operator for operator :I - a2H where I is a unit operator. Then, from (1.402),

$ = - XR,(gy

- q*a2).

By substituting this in (1.40,) we obtain the following equation for the determination of the natural frequencies a: K~

5

- J a 2 - p R,(gy - q*a2)* (gy - p*a2)dS = 0. S

Since operator K , yields a meromorphic function of a2, the problem is reduced to finding the zeros of some meromorphic function. Thus the pendulum with the liquid possesses an enumerable set of natural frequencies. If we are able to solve the problem of oscillations of a liquid in a vessel, then the above process may be carried out effectively and the equation of frequencies can be explicitly written. Let us designate by o, and t,h, the natural frequencies (normaliLed by condition $ $n2dS = 1) and the principal modes of oscillations in the vessel. S

These quantities satisfy

INTRODUCTION TO THE THEORY OF OSCILLATIONS

We shall seek the solution of the equation

g+ - 02H# = f as

After evident calculations, we obtain lCIn

where

FIG. ?.I

In our case

Consequently, the equation of frequencies may be written as follows (1.41)

251

252

N. N. MOISEEV

Equation (1.41) can most simply be studied graphically (see Fig. 5). The unknown zeroes are the points of intersection of the straight line Fl = ic2 - Ja2 with the curve

Curve F2(a2)is shown schematically in Fig. 5. I t has an enumerable set of poles. Graphic analysis of equation (1.41) reveals a number of interesting properties of the motion and permits the construction of the asymptote of the natural frequencies as n -+ 00 (see [7] and [ l l ] ) . The question of stability of this motion is the central point. As can be seen from the drawing, the presence of the liquid inside the pendulum merely reduces the “reserve stability” since it shifts the roots to the left. Naturally enough, one would like to formulate conditions guaranteeing that all roots of equation (1.41) lie to the right of the straight line a2 = 0. In the present case of the pendulum, the question is solved in an elementary way. Later we shall obtain an answer to it from a general theorem to be discussed in the second chapter. Methods of effectively calculating the spectrum will be considered in the next section. 4. Oscillations of a Conservative System with a Liquid Member 1. Let us first consider the motion of a solid body with a liquid-containing cavity about the position of equilibrium. As in (1.32), but with slightly different notation, we assume the velocity of the liquid particles v decomposed according to

v = vp,+v* It has been shown in Section 2 that 3

(1.42)

v* =

3

2 v,vp,,* + 2 o,vq:+3 i=l

I=

1

where ‘p,* (i = 1, 2 , . . . , 6) are Stokes-Zhukovskii potentials satisfying the boundary value problem (1.29). Hence, function ‘p must satisfy

z = c ( P , t ) is the equation of the free surface in the system of coordinates,

rigidly connected with the body. Now let the motion of the solid body be defined by the generalized coordinates ZI,. . ., Z,. Then, by analogy with (1.42), we write

253

INTRODUCTION TO THE THEORY OF OSCILLATIONS

and assume the expression for the kinetic energy of the system as follows:

Here

m$7. .- mij 0

+p

!

Vpi*Vqi*dt,

and {mt.} is the matrix of the coefficients of the quadratic form representing the kinetic energy of the solid body. Using Zhukovskii’s terminology, we shall call mii the masses of the equivalent solid body.* Let us now consider the small oscillations of a conservative system K which has n degrees of freedom, and let 2, (z’ = 1,2,. . . , n) be the generalized coordinates of this system. Supposing there is a solid body with a liquid cavity among the members of this oscillatory system. Then, without loss of generality, the kinetic energy of system K can be described by (1.43) in which the summation now extends from 1 to n. 2. We now compute the potential energy 17 of system K. If the free surface is covered by a lid (in this case we shall denote it by KO),then (1.44)

*

n = no= 2 ,rbj7ZjZj. 1

i,j = 1

If we consider only the case when the equilibrium of system KOis stable,

nois positively definite. This assumption is very important since, if system KO is unstable, the deviations very rapidly cease to be small and the whole theory makes no sense. If the liquid does not fill the cavity completely, the potential epergy of K is made up of (1.44) and of the potential energy of the oscillating liquid. The latter, in turn, may be represented as a sum of two terms, one of which is the potential energy of the liquid oscillating in a fixed vessel 17,=$pgJc2dS. S

As the liquid participates in the motion of the system through transport, its potential energy also depends on the coordinates 2,; consequently, it

* Cf. the corresponding definition of equioalent body in the preceding article, following Eq. (1.9).

154

N. N . MOISEEV

should contain a term of the form

IZ, = 22,

s f,CdS

where functions f i are

S

determined only by the geometry of the cavity.

Since SCdS = 0 we may S

assume, again without loss of generality, that Sfd.5 = 0. s

Hence, we shall write the potential energy of system K as follows: ?I

(1.45)

1

lI= 5

2 biiZiZj + i=l

ij=1

S

S

3. To simplify further investigations, it is convenient to introduce new variables. As system KO is conservative and its position of equilibrium stable, there exist principal coordinates Y i such that the linear transformation 2, = ZaiiYi simultaneously diagonalizes and To:

no

n

1

T -0-2

2 Y,I2;

?I

IZ0

=

1

2 pi2Yi2. i=l

i=l

,uKare the natural frequencies of the system when the free surface is covered

with a lid. In the new variables we have

Here y,** and v i are linear functions of using Green’s formula, (1.46,) becomes

vi*

and f,.

By eliminating cp and

(1.47) 4. Applying Hamilton’s principle and repeating the considerations used in the construction of the pendulum equations, we obtain the following system of equations for the oscillations of system K :

INTRODUCTION TO THE THEORY OF OSCILLATIONS

255

System (1.36), which determines the oscillations of a pendulum, was a particular case of the system of integro-differential equations (1.48). References [7]and [ll] bring the analysis of a number of specific problems whose motion is reduced to the study of system (1.48). 5 . Let us now consider the problem of free oscillations. Suppose

Y i= X,sin at,

(1.49)

q

= @ cos at,

5 = (G sin at.

1

Construct

L

= f ( T - Z7)dt,

where T and I7 are determined by (1.46)

0

and (1.47), and integrate from 0 to 2n/a. Omitting a non-essential multiplier, we obtain

-

(1.50)

Thus the problem of free oscillations of system K is reduced to determining vector Xi, function 4, and number a which make the variation of functional (1$50) vanish. Supposing xs is a system of functions complete with respect to integration over S, let us assume that 'V

s=1

By constructing equations aLIaXi = 0 and aL/aa, = 0 we obtain the system of algebraic equations 'Y

Y (a2A,, - &)a, (ae- p L 2 ) X + , A (1.51) n

i = I$,.

. .,n;

s

2 ( a z ~-, ~B,,)x, + C r=l

Here

= 0,

s=1

1-1

- As)a,= 0,

(a20(Is

s = 1,2. . .N .

256

N. N. MOISEEV

It can be easily seen that ais = asi by applying Green's formula to the integral. (1.51) is a system of homogeneous linear equations for Xl,. ... X,, al,.... aN. T o have non-trivial solutions, a2 must be a root of

dN(a2)= 0 where A N is the determinant

... ...

...

...

...

...

...

...

...

...

... ...

...

...

The elements symmetrical to the main diagonal are equal, thus the roots of A(a2) = 0 are real. We have noted in the preceding section [after (1.40)] that the problem is simplified if the normalized eigenfunctions of the free-oscillation problem are taken as since in this case

xu,

0

g

Hxn = 7 xn, O n

(Xn.Xm)

=

n#m =m'

System (1.51) will then appear as s (a2- pi2)Xi

+ 2 (a2Ai, - Bis)as= 0,

i = 1 , 2 . . .n ;

s=l U

(1.52)

2 (a2A,,- &,)Xi+ (aza,- PS)us= 0,

.

s = 1,2.. N ;

i =1

as = pglws2,

P, = pg.

w, are the natural frequencies of oscillations of a liquid in a vessel.

INTRODUCTION TO THE THEORY OF OSCILLATIONS

257

6. The effective calculation of the natural frequencies and the forms of oscillations is a very arduous task. There is, however, a class of problems in which the wave motion of the liquid’ “relatively little” affects the oscillation of the system. This means that n of the natural frequencies of system K differ but little from p1,p2,.. .,p,,. A similar situation can occur if the coefficients of the terms accounting for the reciprocal effect of oscillations are small. In this case the procedure of perturbation theory can be applied. When applying Ritz’ method in this problem it is natural to take again the eigenfunctions of the oscillations in a fixed vessel as coordinate functions. This leads to (1.52) which we write with a parameter E in the form N

(a2

- pi2)Xi + &

2

(a2Ais - ~ i s ) = a ~ 0,

s=1

(153) n

e

2 (a2As - B ~ J +X ~(a2as- bs)a,

= 0.

i =1

We seek the solution of (1.55) by expanding m

m

k =O

k =O

m

a2=

2

ake&’.

k =O

For the zero-order quantities Xi,,, aso and no2 we have the system

XiO(UO2- /Liz) = 0, (15 4 )

aso(ao2as- Ps) = 0. System (1.54) has a solution if uo2= pi2 or ao2= Ps/as. Since we wish to determine the frequencies of system K close to those of system KO,we must assume that a, = pi. For definiteness suppose a, = ,ul ; then X, = X, = . . X , = a, = 0. We may assume without loss of generality that X , = 1. For the first-order quantities Xi, and asl we have the equations

.

Xll(ao2- pI2) = a12, X,l(ao2- pi2) = 0,

i # 1,

asl(q,2as- PSI = - (Asp12- &I. The condition for solvability of the first equation of this system is

aI2= 0.

258

N. N. MOISEEV

Therefore we can always assume that X,,

X i , = O ( i > I),

XI, and

a,,

=

= 0.

It is further evident that

Bl, - AlSPl2 - -. pl2aS - P s

will then satisfy equation N

(1.55)

X&02

- p12)= - cZ2- 2 a,i(A irp12 - Bis). s-1

To make this equation solvable, it is necessary and sufficient that the right member of (1.56) be equal to zero. Hence,

Approximately, therefore, we have

and use it for E = 1. This scheme of calculation may be recommended for a great many technical problems. 5. Torsional Oscillations of a Beam with a Liquid-Containing Cavity 1. The problem of oscillations of an elastic body containing a liquid is extremely complicated. A comparatively simple theory can only be developed for the beam. This term implies here an elastic body subject to the wellknown hypothesis of plane cross sections remaining plane (Bernoulli). We shall analyze the problem of torsional oscillations in detail, and then write out the general equations of arbitrary flexural-torsional oscillations. Let us connect the system of coordinates with the beam axis containing the shear centres as shown in Fig. 6 (axis oy) and designate the angle of twist by B(y,t). In conformity with our assumptions, kinetic and potei,tial energy of the beam without liquid are

(1 55’)

INTRODUCTION TO THE THEORY OF OSCILLATIONS

259

Here I ( y ) is the moment of inertia per unit of length, K ( y ) is the torsional stiffness, 1 is the length of the beam, and B is a quantity characteristic for the moment of the external force. Together with (1.55') we consider the ordinary differential equation

and subject the function u ( y ) to boundary conditions that make the operators in question self-adjoint. For instance, it may be assumed that u(0) = u'(Z)= 0 (the beam is rigidly clamped a t one end). We shall call the boundary value

i FIG. 8

problem for u the problem associated with that of torsional oscillations. Now let us designate by u, and p,, the eigenfunctions and eigenvalues of that problem. If we assume the condition of normalization as

then functions u, will possess the following property:

i

Iu,u,dy

= 0,

n f m;

0

(15 6 )

i

[KU,,'U,'

+ Bu,,u,]dV

0

=

#m

0,

)a

p,~,

n = m'

Returning to the problem of beam oscillation we assume that s n=l

260

N. N. MOISEEV 1

construct the Lagrangian L = J ( T - Z7)dt, and integrate from 0 to 2nlo. 0

As result we obtain the expression N

where

or on the basis of (1.56) h’

This implies that the natural frequencies of the beam are equal to those of the associated boundary value problem. Note: Investigation of the eigenfunctions referred to above is particularly convenient if the parameters of the beam are independent of y . If they change slowly, it is advisable to use the asymptotic solution of the associated problem, (1.57) 0

where c is an arbitrary constant defined by the condition of normalization, and I

Formula (1.57) refers e.g. to the problem of oscillations of an airplane wing (a beam fixed a t one and free at the other end; u(o) = u’(l)= 0 ) . 1. Let us now consider the torsional oscillations of a beam with a liquid on the assumption that the cavity is completely filled (see Fig. 7 where the designations are given). Since we consider gravity as the only external force acting on the liquid, we may assume the potential energy of the system [beam liquid] as (1.55). The kinetic energy of the system must be set up as

+

INTRODUCTION TO THE THEORY OF OSCILLATIONS

261

Here q~ is the potential of the absolute motion of the liquid, and t the volume occupied by the liquid. Let us assume 0 = cos wt6(y),

9 = - o sin odrP(x,y,z), and construct the Lagrangian:

The potential satisfies on surface C

aP an 5 21, (0

= (thyo x r)no.

denotes a unit vector). By introducing designations

no = a x 0 + Pyo + yzo,

r = xx0

+ yyo + zzo,

we obtain

where f(x,y,z) = za

- xy.

Function f is determined only by the geometry of the cavity. With the notations introduced,

a@

-= 6f. an

In order to solve the problem by Ritz' method, a system of functions u,, which is complete with respect to the integration interval [O,I], must be chosen, and 6 must be represented as N

6=

2 i-1

262

N . N. MOISEEV

(in many cases it is advisable to select the eigenfunctions of the associated boundary value problem as u,,). In addition let us assume that N

where @, are harmonic functions in Neumann problem

t, determined

as solutions of the

-a@u -u,f

an

or

on = HH,(P)f(P),

PE

2.

The algebraic system which determines a,, and w 2 appears now as

2

am [w2(anm

+ prim) -

rum1 = 0.

m

a,, and ynm have been determined above, and

bum = p

I

V@,V@,dt.

r

For the same problem let us construct yet another equation of oscillations, using Hamilton's principle for this purpose Carrying out

I

s~ = {ze,se, - Ke,se, 0 0

- seseyyat

+

jrvm,f. v ~ ~ e ~ f a ~ a t

OT

by partial integration of the first term (remembering that the variations are isochronous and vanish at y = 0 and y = I) and by applying Green's formula to the second term, we obtain L I

I

INTRODUCTION TO THE THEORY OF OSCILLATIONS

263

From the definition of the operator H ,

a -Hu an

= 'u,

hence the second integral appears as t

Let I , denote the curve of intersection of Z with a plane perpendicular to axis O v ; then the second integral may also be written

or, after integration by parts,

0 YI l g

Here the minimum and maximum values of y on Z are y1 and yz. We can always integrate over y from 0 to 1, since I , = 0 if y < y1 or y > yz. By collecting terms and using the arbitrariness of variations, we arrive at the following equation for the torsional oscillations of a beam containing a cavity filled completely with a liquid: ( 15 9 )

mt+ p l f w 6 t t d l . - (KO,), + Be = 0. 'Y

For p 3 0, (1.50) becomes the well-known equation of torsional oscillations. It is one of the principal results of the theory of motion of a rigid body with a completely liquid-filled cavity that the motion of this system is equivalent to that of another solid body under the action of the same forces.* But (1.59) indicates that it is impossible to introduce a beam without a liquid whose motion would be equivalent to the motion of the beam with a liquid. Indeed, if the value of angle 0 and 8, are prescribed in a cross section, then the acceleration of particles in this cross section is determined uniquely

* Cf.

the preceding article, after Eq. (1.12).

264

N. N. MOISEEV

for a beam without liquid, but if there is liquid inside the beam, the acceleration in this cross section also depends on the angular accelerations of all the other cross sections. This effect arises from the term

1

W f e t P ,= p j fW,)

'Y

lv

1

H(py,p)f(p)e,(P,t)dPdP,

z

where P , is on the circumference I,, P on the surface Z, and H(P,Q)is Green's function of Neumann's problem for region z. 3. Let us now consider torsional oscillations of a beam containing a cavity partly filled with liquid. A number of new peculiarities arise in this case. (1.59) indicates that the influence of the completely filled cavity DlRECTlON OF

fi

FIG.7

consists in changing the natural frequencies and forms of the principal oscillations.' When the mass of the liquid tends toward zero, frequency and form of the oscillations approach the corresponding quantities of the beam without liquid. The matter is quite different when the liquid has a free surface. In this case new forms of oscillations arise. Let us connect the system of coordinatesOx,y,z, to the free surface in the state of rest, as shown in Fig. 7. The velocity potential p may again be represented as p = (p(1)+ p'2'

where q$') is the velocity potential which the liquid would have had if surface S had not been free ~ ( l=) Hv,(P),

PEL'+ S,

265

INTRODUCTION TO THE THEORY OF OSCILLATIONS

i,e. ql is the velocity potential of the preceding section. p(2)satisfies conditions

where 5' is the elevation of the free surface above the unperturbed level in the absolute system of coordinates. Hence q ' 2 ' = H*(5't - v,) = H*ut

where u = 5' - S,,, and 6, is the displacement of the points of surface S owing to elastic deformations. Operation H*f produces a function harmonic in z whose normal derivative is equal to zero on the wetted part of 2, and is equal to f on S. Hence the kinetic energy of the liquid is represented bv

5

T*=L [VHV, 2p

+ PH*utl2dt.

T

Let us now calculate the potential energy of the liquid

where tois the volume occupied by the liquid in equilibrium, and tlis the volume bounded by the free surface and the plane S. The second term can be written as

Clearly, the terms

are only determined by the elastic displacements and have the same structure as the potential energy of the external forces. Therefore we can include them in the potential energy of the external forces without loss of generality. Thus we write the kinetic and potential energy of the system beam liquid in the following way

+

266

N. N. MOISEEV

+

[ K e y 2 B02]dy 0

+ 5

S

The integrals in (1.61) and (1.62) depending on 6, are written in the system of coordinates oxlylzl. Accordingly, the equations of motion will appear as 18;;

+p

s

JY

(1.63)

+p

fHfettdb

I

dY

fH*%tdd, - (KO,),

+ Be +p g

+ p ~ * u t+t p g f e + pgu

pHettf

I

f d d , = 0,

dY

=

o

where d , is the segment which lies at the intersection of the region S and the plane of the normal cross section whose coordinate equals y . Thus the oscillations of a beam with a liquid are determined by a system of two integro-differential equations. Note. Ritz' method may again be recommended for calculating the principal oscillations and natural frequencies. If the frequencies of the free oscillations of the liquid in a fixed vessel and of the torsional oscillations of a beam are not close to each other, the spectrum of the combined problem (beam + liquid) will be given approximately by the superposition of the spectra of both problems. We have to select two systems of coordinate functions. In this case it is advisable to choose the system of eigenfunctions of the operator H as one of the systems, and the system of eigenfunctions of the associated boundary value problem as the other system. To calculate the spectrum of the combined problem one may advantageously employ the methods of the perturbation theory outlined in the preceding section. 6. Torsional-Flexural Oscillations of a Beam with a Liquid-Containing Cavity 1. Let us denote the deflection in the plane xoy by X , and in the plane yoz by letter 2. Then the kinetic and potential energy of the beam (without a liquid) is given by the following formulae well-known from the elements of strength of materials (see e.g. [ I S ] ) : I

+ 2B13X8+ B 2 g 2+ 2B,ZB + B,03}dy.

267

INTRODUCTION TO T H E T H E O R Y OF OSCILLATIONS

Here the C, are the flexural and torsional stiffnesses, the B,, characterize the external conservative forces (B,?= B J , the A,, are coefficients characterizing the distribution of masses or moments of inertia (in the preceding section I @ ) ) ,and the A,, characterize the asymmetry of the beam, i.e. the non-coincidence of the shear centre and the centroid of the cross section. If there is a liquid-containing cavity inside the beam and the liquid has a free surface, the kinetic and potential energy of the system are

T

=

To

+ ?!2-

1

(C7HylXt

+ PHy2Zt + C7Hy3Bt+ VH*ut}*dt,

1

( I .65)

ZZ =

“1

+2

u2dS

+ pg

u[Xy,

+ Zy2 + 8y3Jds.

Here y1 = cos (nx),

y2 = cos (nz),

y 3 = z cos (nx)- x cos (n2).

+

+

It is worth recalling that u = ( - 6, where 6, = Xy, zy2 8y3 is the elastic displacement of a point of plane S. By applying Hamilton’s principle, we can construct without difficulty the general equations of oscillations of the system beam liquid.

+

268

N . N . MOISEEV

+ B,,Z + ~,,e + p g

I

Y3udl = 0,

dv

+ + pH*utt + pgy& + PgY38 + Pg* = 0

pHXtty1+ pHZtty2

+ PgY,z

pmty3

We obtain system (1.63) as a particular case of this system, in (1.66) X = 2 = 0. We may regard system (1.48) in the same way as a particular case of this system. For this purpose assume, for instance, that X = Z = 0 ; C, = 0 and consider 8 as an n-dimensional vector.

11. GENERALPROPERTIES OF THE EQUATIONS This chapter is devoted to a study of the general properties of the equations derived in the first chapter. The structure of the spectrum, the existence of natural oscillations, and the possibility of employing Ritz’ method are elucidated for the case of conservative external forces. In t h e last section the basic theorem of this theory is proved: for the stability of the equilibrium of the system body liquid it is necessary and sufficient that some other, considerably simpler system should be stable. The body may be either rigid or elastic (but again subject to the hypothesis of Bernoulli’s beam t heorey).

+

1. Oscillations of a Liquid in a Vessel 1. I t has been shown in Section 1.1 that the motion of the liquid is determined by equation (1.10) g[ Hr,,= 0. To find the natural oscillations, we make the substitution (1.12) and obtain

+

AH# - # = 0, To find the solutions, Ritz’ method was recommended. But certain questions remained open. They concern the solvability of this problem, the structure of the spectrum, the completeness of the system of eigenfunctions, and the convergence of Ritz’ method with N 00. ---+

INTRODUCTION TO THE THEORY OF OSCILLATIONS

269

These questions are resolved in an elementary way on the basis of the general theorems of functional analysis. It suffices to prove that operator H possesses certain properties. I t has already been proved above that H is a bounded, completely continuous, selfadjoint operator. 2. Operator H has been defined on a set of functions satisfying J f ( P ) d S= 0. For the Hilbert space E of functions f ( P ) with scalar product S

( f l , f z ) = SflfPdS, the operator H is therefore defined on a subspace of E S

which is orthogonal to unity. We shall now demonstrate that H is positive on this set. Let us consider the scalar product

(2.2) ss

Here H(P,Q)is Green's function of Neumann's problem for region t. Accordingly the inner integral

s

represents a function harmonic in

t

and satisfying

Consequently,

Hence, according to Green's formula,

5

(Hf,!) = ( V u ) 2 d t . 1

This proves the assertion, since the right member is zero if and only if u G 0 , 0. or, which is the same, if f The physical meaning of this statement is quite evident. The scalar product (2.2) determines the kinetic energy of an oscillating liquid if the velocity of the points of the free surface is determined by f(Q) sin at. But it is well-known that the kinetic energy of liquid in irrotational motion in a simply-connected vessel at rest can vanish if and only if f = 0.

270

N. N. MOISEEV

3. Thus operator H is bounded on E , completely continuous, self-adjoint, and positive. The known properties of such operators (see e.g. [2]) enable us to formulate the following Theorem 1. (a) For the motion of a liquid about the equilibrium position in a finite vessel there exist natural oscillations, i.e. solutions of the form (1.12). (b) The eigenvalues u are positive, have finite multiplicity factors and form a sequence increasing without bound: lim u, = 00. n+m

(c) The eigenfunctions $, of operator If, which describe the principal forms of free oscillations are such that sequence 1&,. .. $,. .. is complete in E . (d) The eigenvalues and eigenfunctions may be determined by Ritz' method. This theorem covers the essence of all the theoretical questions about a liquid in free oscillations of infinitely small amplitude. 2. Oscillations of a Conservative System with a Liquid Member 1. Here we study certain general properties of the system of integrodifferential equations (1.48) which are the subject of analysis in 1131. We shall follow the outline given in that article. First of all let us put system (1.48) in operator form. To do this we introduce the n-dimensional space E, of vectors Y with components Y , (i = 1, 2 , . ... n) and the space E of functions C ( P E S),square-integrable in S. Let us define the scalar products. n

2 Y,Z,

(Y,z),=

in E,,

,=1

(C,d =

1

C(P)V(P)dP

in E.

S

+

We further introduce the space rf = E, E . Element x E 6 is the vector with components Y E E, and C E E , and the scalar product in 6 is defined by (X(l),X(Z))

+ (pp).

= (YCl),YCZ)),

We shall use the following operators : (a) acting in E,: unit operator L, and operator M,,

L, =

... 0 0 1 ... 0 ......... ......... 1 0

p1

0

...

0

0

pz

...

0

0 0

............ ............ 0 0 ... P*

...

1

,

M, =

INTRODUCTION TO THE THEORY OF OSCILLATIONS

27 1

(b) acting in E :

L,,

=H ,

M,,= p g ;

(c) acting from E, to E :

Ll*Y = P(qJ**,Yo),

M,, = ( Y ' Y ) ,

where we denote by q** and v the vectors with projections qi** and v,, respectively (i = 1, 2,. . ., n) ; (d) acting from E to E,:

M,,C

L,,C = y*,

= v*

where y* and v* are the n-dimensional vectors

(I {

. .p

y* = p q,**CtdS,. S

S I

q7n**tdS ,

S

v* = [v,CdS.

.. . jVnCdS,1

S

5

By using these designations, system (1.48) may be written as follows:

+ Lo,tt + M,Y + L,OY'' + L,& + M,,Y + M,,t L,Y"

M0,C = 0,

= 0.

This can be expressed still more briefly if the operators L and M acting in d are introduced:

(1.48) now becomes

(2.3)

Lx"

+ M X= 0

where the prime denotes differentiation with respect to time t , and x E 8. 2. As in the preceding section, we intend to analyze the structure of the spectrum and to prove the convergence of Ritz' method. I t appears that all facts of interest to us simply follow from the general theory of linear operators. To prove this, it suffices to establish certain properties of the operators L and M . To begin with, these operators are selfadjoint i.e. (2.4)

(Lx,x)= (x,Lx),

( M x , x= ) (x,Mx).

272

N. N. MOISEEV

This is established by direct verification; apart from this, in Section 1.4 we established the symmetry of the matrix of algebraic equations obtained by applying Ritz’ method ; this result implies selfadjointness of operators L and M . Next, operator L is completely continuous. This is true for L, since it acts in a finite-dimensional space: also Lol acts in the finite-dimensional space E,. Operator L,, acts from a finite-dimensional space and amounts to multiplying an element of E, by the vector v**; thus it is likewise completely continuous. Lastly, operator L,, = H is completely continuous, as has been shown in the preceding section. Furthermore, L is a positive operator since the functional ( L x ’ , ~ ’ ) represents twice the kinetic energy of system K. Hence L is a completely continuous, selfadjoint, positive operator. We shall regard operator M as positive definite. As ( M x , x ) = 2I7, i.e. twice the potential energy of system K, the assumption so introduced means that we confine ourselves to the case when the potential energy has a minimum in the equilibrium position, i.e. when K is statically stable. Assuming that x = qPJt,

we obtain from (2.4)

( M - u2L)q = 0 . Thus the quantities A

=

l/a2are the eigenvalues of the operator C

=

M-lL:

cq = Aq.

(2.5)

Let us introduce Friedrichs’ metric in space E : (x134F =

(Mx19x2).

As operator L is completely continuous, and M-I is bounded, operator C is completely continuous in both metrics, but in Friedrichs’ metric it will also be selfadjoint : (C%,,X,)F =

(MM-’Lx1,x2)= ( L x , x ) = (x1,Lx2)

= (XI,MM-’Lx,) =

(Mx,,Cx,)

= (x1,Cx,)F.

In addition, ( C X , , X = ~ ) (~L x , , ~ ,> ) 0, if x1 f 0. Hence, proceeding from the general theorems of analysis, it follows (see [2]) that: (a) for operator C in space E there exists a complete system of eigenelements orthogonal in Friedrichs’ metric (b) all un2> 0 and on2 do with ~t-+ 00 (c) the eigenelements q,, and quantities u, can be determined by Ritz’ method which converges.

-

INTRODUCTION TO T H E THEORY OF OSCILLATIONS

273

The sum total of the results so obtained can be formulated in the following.

Theorem 1. If system K consists of a finite number of conservative members and contains a finite number of cavities partly filled with liquid, and if the potential energy of the system has a minimum in the equilibrium position, then (a) in the motion of this system about the equilibrium position there exist principal oscillations, and system (1.48) has a solution of the form y-. - Xie*at ;

[ = Zeid ;

% -

(b) the frequencies of these oscillations are real quantities and on 00 with n .-. 60. This means that the position of equilibrium is stable; (c) any free motion of K may be represented as a superposition of oscillations, i.e. the system of principal oscillations is complete; (d) free oscillations and frequencies can be found by Ritz’ method: In formulating this theorem, the concept of stability was used. I t implies the following : the position of equilibrium is considered stable, provided any principal oscillation is bounded. Thus statement b) is an analogue of Lagrange’s theorem: for the equilibrium position of system K to be stable it suffices t h a t the potential energy of the system should have a minimum. In the present case the inversion of Lagrange’s theorem holds. This theory is brought to conclusion by -f

Theorem 2. If the potential energy is not a minimum in the equilibrium position, then there is a t least one negative quantity among the on2. Proof of this theorem is given in [13]; it is based on the results obtained by L. S.Pontryagin in the theory of hermitian operators in spaces with indefinite metric. 3. Oscillations of a Beam with a Liquid-Containing Cavity 1. Let us now proceed to the general properties of system (1.66). As in the preceding paragraph, we write (1.66) in operator form. For this purpose we introduce the following function spaces : (a) spaces E,, E, and E, of functions u,(y), u,(y) and u3(y) defined and summable on the interval [O,l] with the scalar product

u~,u; E Ei,

(b) space E, of functions u4(P),P scalar product

E

i = 1,2, 3 ;

S, square-summable in S , with the

274

N. N. MOISEEV

4

(c) the direct sum of spaces 8 = Z Ei with the scalar product i=l 4

(x,Y)=

2

(.iVi)i.

i=l

Here element x is the vector with components u1,u2,u3,u4.Functions ui should also satisfy certain boundary conditions which are determined by the way in which the beam is fastened. Without going into details, let us consider them as homogeneous and of such a nature that they ensure selfadjointness of the operators associated with the elastic oscillations of a beam without liquid. We further introduce operators Lij and Mi, acting from E j to Ei:

Lipi = A+,

+p

s

yiHyiujdZ,

i,j = 1 , 2 , 3 ;

I,

j = 1,2,3;

Lpju, = pHiyjuj,

5

Lj4u4= p ylH*u4dl,

j = 1,2,3;

I?.

L,u,

=pH*q;

M33u3 = - C3U3Y)Y -I B33U3, M . . u .- B..u. ,I I ,

i,j = 1 , 2 , 3 ,

M4pj = pgyiu,,

j = 1 , 2 , 3;

I

Mj4u4 = p g y,u4dl,

M44u4 = pgu4;

i#j;

j = 1,2,3.

d?.

As H is an integral operator whose kernel is Green’s function, it is necessary to impose some more limitations on the functions A,,, Bii and Ci so that operators L,, and M j j act from E, to Ei. Lire shall assume once and for all that these functions satisfy all the required conditions.

INTRODUCTION TO THE THEORY OF OSCILLATIONS

275

With these notations it is possible to write (1.66) as follows

In matrix notation

L

= llLijlll

M

=

llMiill

(2.6) may be written

Lx*t

(2.7)

+ M x = 0.

2. Let us elucidate some properties of operator L . This operator is again bounded and it is selfadjoint i.e. it satisfies condition

( L X , Y ) = (X,LY).

(2.8)

Let us write (2.8) in greater detail: 4

i,i = 1

4

i,j = I

The validity of this equality is established by simple verification. Let us calculate, for instance,

= (ulL14v4)1.

The rest of the relationships proving the validity of (2.9) are established in a similar way. Unlike operator L of the preceding section (see (2.3)),the present operator L is not completely continuous. The difference is caused by the fact that, previously, the space Eo was finite-dimensional; but now the part of E , is played by E , E, E, which is infinite-dimensional.

+ +

276

N. N . MOISEEV

3. Proceeding now to M , this operator is unbounded and selfadjoint. The latter is an elementary corollary of the symmetry of matrix {B;,} and the selfadjointness of the boundary conditions for ui. Let us also calculate the scalar product

(2.10)

By comparing (2.10) and (1.65), we convince ourselves that

( M x , x ) = 2z7. It appears that only those problems are of practical interest, in which the equilibrium position is statically stable, i.e. I? in the position of equilibrium is a minimum. We shall therefore consider operator M as positive definite. Hence, the difference between equations (2.7) and (2.3) consists in the fact that operator L in (2.7) is not completely continuous. This prevents us from applying directly the general theorems about linear operators. Note. If operator M - l had been completely continuous, the above difficulty would have been of no significance, since operator M - l L owing to the boundedness of L would have been completely continuous. 4. Instead of (1.66), let us construct a system of equations equivalent to it. For this purpose we calculate the variation of the Lagrangian I

J ( T - n ) d t and substitute in the resulting equation 0

(2.11)

lc

=

- y l x - y2z -

y3e.

If we now set up the equations for the variables X , Z, 8, and v , and omit the clumsy calculations (for more details see [19]), we arrive a t the following system of integro-differential equations :

+ ~ +~~~~e~~ z + ~ (clxyy)vv ~ + B;X + B : ~ Z + B:,e + y1[ H l x I I y l+ H ~ + HlettY3 z ~ + ~~ * v ~~ ~=1~0,d s

A,,X~~A

1

lv-dv

A

~

+~A x~ +~~~~~e~~ z~ +~ (czz,,),, ~ + ~ 2 :+x ~2*zz + + B,*,e + Y z [ H I x I t y +l HIZttYz+ HletIY3+ H*vttids

1

1 ,--J 3

Y

= 0,

INTRODUCTION TO T H E THEORY OF OSCILLATIONS

277

+ &&f + A d t t - (c3e,),+ ~3+ ~ $ 2 + B,*,e + + w t t y 2+ H ~ + H~ * ~~M S =~ 0, Y

~ 3 1 ~ t t

5

Y3[H1Xttyl

1Y- d Y

p [HIXttyl $- H1ztty,

+

H1eltY3

f

H*vlt

+ gvl

= O*

In these equation H , = H - H*, Bii* = B,i - pli, and p 11. = p"7% are functions of variable y , depending only on the configuration of volume z (the expressions are given in greater detail in the next section). The main difference between the transformed system (2.12) and (1.66) is that its first three equations do not include terms containing (they include only derivatives of v with respect to t ) . If we introduce the space E,' = El E, E3 and let w E E,' where w is the vector with projections X,Y,O; then (2.12) may be written as

+ +

(2.13)

4 1 '

(2.14)

43'

I,' = L21' L22r L,' L31'

Mll'

(2.15)

42'

M'

=

L32'

M12'

'33'

Ml3'

M21' MZ2' M23' M31r

M32'

Y1

r = Y2 Y3

M33'

~

278

and

N. N. MOISEEV

I" is the line matrix

If operators A and N are introduced in space d

D

L'

A=l/PHII'.

= E,'

+ E,

by

N = l / Il l M'

PH*ilJ

Pg

then (2.13) appears as

LIZ,

(2.16)

+ NZ= 0.

It is easy to establish that A , in the same way as L , is a selfadjoint operator. One has only to check the validity of (Az,z,) = (Lx,x,),

(z,Az,) = (x,Lx,),

and for this purpose it suffices, in turn, to spell out the corresponding scalar products and to make in the resulting expressions the substitutions

H

+ H*,

= Hl

v=u

+ y , x + 722 + y3e.

As to operator M', it is like operator M unbounded and positive definite, but, unlike the latter, its inverse M'-l is completely continuous. This can be easily demonstrated. Consider the equation

M ' s =f.

(2.17)

In the particular case of torsional oscillations, (2.17) appears as (2.18)

-

where 0 satisfies boundary conditions ensuring the selfadjointness of M . By using Green's function of the boundary-value problem for (2.18), we obtain I

e m = ] G ~ P . Q ) ~ ( w Q= c,f. 0

As Green's function of the homogeneous differential equation corresponding to (2.18) is continuous, the kernel of the integral operator G is square integrable. Consequently, (see [2]), G is completely continuous. Similarly we can convince ourselves that also in the general case operator G = M'-l is completely continuous

INTRODUCTION TO THE THEORY OF OSCILLATIONS

279

Thus M has been split into M‘ acting in E,’ and an operator that multiplies by a constant, the split being carried out in such a way that M’-1 becomes completely continuous. And this was the purpose of the transformation above. 5. Let us assume that in system (2.13)

(2.19)

w=

w cos ot,

v=

v cos ot.

Then (2.13) appears as

(2.20)

Now substitute

and rewrite (2.19):

Since operator L’ is bounded, M ’ - 1/2L’M‘-112 is completely continuous (since M’-l is). Operators HI and H* are completely continuous, the kernels having only a weak singularity, and therefore operator D is also completely continuous. Hence, operator

M’- 1/2L’M’- 112

R=

1 MI- ll2D VPT

P H,r‘M’-

__

VPg

1/2

P H*

VPg

1 ~

VPg

is completely continuous. The symmetry of operator R is established by a simple verification. Hence the problem is reduced to the eigenvalue problem

280

N . N . MOISEEV

for a completely continuous selfadjoint operator. This reduction finishes the study: proceeding from well-known theorems, the eigenvalues A, of operator R form a denumerable sequence of positive quantities such that A,, + O if n -r w, the eigenelements are orthogonal and have finite multiplicity factors; they can be found by Ritz’ method. The results so obtained are formulated in the following

+

Theorem. If the system [beam liquid] is statically stable, i.e. if form 17 is positive definite, the system possesses principal oscillations of the form (2.19) where the natural frequencies w, are positive numbers and form a denumerable sequence such that cu,, -r 00 if n -> co. A finite number of forms of principal oscillations corresponds to every eigenvalue, and these can be found by Ritz’ method. This theorem again contains as a corollary the Lagrangian theorem about the minimum of the potential energy: for the boundedness of oscillations it is sufficient that the potential energy be a minimum in the equilibrium position. In systems which have apart from the liquid member only a finite number of degrees of freedom, it is possible to prove the inverse theorem. In the general case of a beam, the inverse theorem has not been proved. The system of principal oscillations is complete in E in the sense of Friedrichs’ norm. To see this we carry out the replacement (2.19) in (2.7), rewrite it so that it reads M-1L.z

= AX.

and introduce Friedrichs’ norm

To prove the completeness it suffices to show that condition M-’Lh = 0 implies h 0. Let us consider

=

( M - l L h , h )= ~ (Lh,h) where the scalar product to the right is the kinetic energy. Consequently, unless h z 0, this expression cannot equal zero. 4. Investigation of Static Stability

+

1. Let us call the system body liquid statically stable if the potential energy of body + liquid has a minimum in the position of equilibrium. It was assumed in the preceding sections that the system under investigation is statically stable since this condition is sufficient for dynamic stability (in a finite-dimensional case it is also necessary). In this section we shall make a more detailed study of the condition of static stability.

281

INTRODUCTION TO T H E THEORY OF OSCILLATIONS

Let us spell out functional I7:

1

+

+

17 = 2 ( C I X i y+ CzZfy C3OY2 Bl,X2

+ 2B12XZ + Bz2Z2

0

The substitution (2.11), which we already used in the preceding section, will again be applied, but the calculations will now be made in greater detail. In the new variables, I7 appears as

where

5

I

IT* = { C I X t , + CzZ;, 2 1

+ C30y2+ B l l X 2 + 2B1,XZ + 2B13X0

0

(2.22)

+ BzzZ2+ 2B&O + B,02}dy - pg

1

s’

y2y3ZBdS -

I

I

5

5

- pg yly2XZdS - pg yly3XOdS

‘I

J

y12X2dS- - pg yz2Z2dS- pg y 3 V S . 2 5 S

This expression can be given a more “telling” form that clarifies the meaning of a number of terms. As y1 = cos (nx) and yz = cos (nz) are constant on S (the normal to S is always vertical) and as y3 = zyl - xy2, we can write

S

0

282

N

1

Here 1 , and J , denote the static moments of the interval d , ( J , = Jxdl,. . . ), dY

while Jz,, J,,,JlX are its moments of inertia. Thus the quantities p , , = pi, depend only on the geometry of the cavitv. If we introduce the designations

R? - B.. - plj, I1 I1 functional Z7* will appear as 1

0

(2.23)

+ 2B;XO + B;Z2 + 2BGZO + B3*,02}dy.

2. Hence by the transformation (2.11), functional 17 appears as the sum of two functionals, one of which, svzds, is always positive definite, S

while the other does not depend on v . Therefore we have

+

Theorem 1. For the system beam liquid to be statically stable, i.e. for functional 17 to be positive definite, it is necessary and sufficient that functional 17* should be positive definite. This theorem has a simple mechanical significance. For static stability of a beam with a liquid it is necessary and sufficient that some beam without liquid, possessing the same elastic characteristics but under the action of other external forces, should be stable. It has been shown above that there exists no beam without liquid which would be dynamically equivalent to a beam with liquid. For the stability,

INTRODUCTION TO THE THEORY OF OSCILLATIONS

283

however, it is necessary and sufficient that some equivalent beam without liquid should be stable. 3. The theorem so proved contains the following result.

Theorem 2. For the stability of the equilibrium position of a rigid body containing a cavity partly filled with a liquid it is necessary and sufficient that the equilibrium of some other rigid body should be stable. Consider I7 as given by (1.46). In this case the transformation (2.11) takes the form

t=v-

1 -2ViYi, Pg

and the potential energy may be written

n=II1+n* where

b,, is Kronecker’s symbol, and aii = (l/pg)Sv,v,dS. S

Hence, for positive definitiveness of form 17 of an infinite number of variables it is necessary and sufficient that form II* of a finite number of variables should be positive definite. But since for the stability of the equilibrium position of the system [rigid bodies liquid] the positive definitiveness of 17 is necessary and sufficient we arrive at the following result: for the stability of an equilibrium position of system [rigid bodies liquid] the positive definitiveness of the form Z7* of a finite number of variables is necessary and sufficient. If there is but one solid body in the system, form 11* depends on six variables and describes the potential energy of some solid body, which we have a right-tocall equivalent as far as stability is concerned. 4. Now let us consider an example. In the first chapter we have dealt with a liquid-containing pendulum. For this particular problem, 11* appears as follows:

+

+

But

284

N. N. MOISEEV

where J s is the moment of inertia of the free surface. Thus, the condition of static stability may be represented as (2.24)

K'

> pgJ.s

In other words, for the stability of a pendulum with liquid it is necessary and sufficient that some pendulum without liquid should be stable. Its reduced length is determined from condition

I,

=

MI1

-pJs

M

where I , is the given length of the pendulum whose free surface is covered by a lid, and M is the mass of the whole system. Hence, the presence of the liquid can only weaken the stability of the pendulum. 5. Analysis of the regularity of functional in the general case is not trivial, and there exist no criteria which would not only be sufficient but also necessary, We shall only point here to a few conditions that are sufficient for static stability. Functional (2.23) can be written as:

n*

n*= TIl* + n,* where

If the coefficients C, are everywhere positive (a case that is typical for many applications), then only U2*must be positive definte. Therefore Sylvester's inequalities should hold :

> 0. Conditions (2.25) impose limitations on the geometrical characteristics and can be expressed more directly. In the case of torsional oscillations of a beam, (2.25) reduces to p:> 0 or

INTRODUCTION TO THE THEORY OF OSCILLATIONS

If the axis of the beam is horizontal, y1

=

(2.27)

= 0,

B33 - P g J x x

285

0, y, = 1, and (2.26) becomes

i.e. we have arrived at a condition equivalent to (2.24),but in the case of the pendulum this condition was also necessary. In the case of the beam it is only sufficient. If the axis of the beam is vertical, y1 = y, = 0, and condition (2.27) appears as

B3, > 0. In other words, it is sufficient for the stability of the beam that the external forces should be of the nature of a “restoring” force. In this case the liquid does not affect a t all the stability of the beam. 6. Let us consider now the bending oscillations of the beam in two mutually perpendicular directions. IT,* will appear as 1

“d Conditions (2.25) yield two inequalities: (2.28)

If we return to the old designations, the second condition (2.28) results in (2.29) (2.28) also implies B,, > pu, B,, > p,,. The region of stability is shown in Fig. 8. It is sufficient for static stability that the point with coordinates B,, and B,, lies above the hyperbola B,, = pzz ((I$, - p g y , y ~ ) ~ / ( B -,pll)). ~ The dotted curve in Fig. 8 shows the hyperbola which limits the region of stability in the case when the liquid inside the beam has no free surface. 7. The conditions we have investigated form a natural analogue to the stability conditions for systems with a finite number of degrees of freedom for which they are both necessary and sufficient. They are not indispensable for the beam problem. Indeed they require, for example, that the external forces should be of the nature of restoring forces and should compensate the destabilizing effect of the liquid. I t appears a t the same time that, owing to the rigidity, the beam is capable of retaining stability even in the case when the external forces are of a “tipping” nature. It is difficult to conduct such a study in a general way. For the particular case of torsional oscillations of a beam fastened at one end, we shall point here to one of the possible ways of developing criteria which do not involve the “restoring” nature of the external forces.

+

286

N. N. MOISEEV

For torsional oscillations

5

I

1

n*= 2

(2.30)

I I

C3By2dy

+

1 2 BS+S02dy. 0

0

Let us assume that there exists such a 6 > 0 that

BG2 -4

Y E

[W],

giving the estimate (2.31)

FIG.I)

where

1

Since 8 = SO,,dy, it follows from Schwarz' inequality that 0 I

1

By integrating this inequality, we obtain

INTRODUCTION TO THE THEORY OF OSCILLATIONS

287

Therefore estimate (2.31) may be replaced by

Hence it is sufficient for the positive definiteness of functional function C, should satisfy inequality

LI* that

1

This condition imposes a lower bound on C,. I t will be satisfied if, for instance, function C, is larger than some constant q, which depends on the external forces and the geometry of cavity.

References (Titles of Russian publications are translated) 1. MOISEEV,N. N., About two liquid-filled pendulums, P M M 16 (1952). 2. MIKHLIN, S. G., “Variational methods in mathematical physics”. GIFML, Moscow, 1957. A German translation is available : S. G. Michlin, “Variationsmethoden der mathematischen Physik”, Akademie-Verlag, Berlin, 1962 ; an English

translation is in preparation at Pergamon Press. 3. KOCHIN. N. E., KIBEL’,I. A., ROZE,N. V., “Theoretical hydromechanics”. GITTL, Moscow, 1948. 4. LAMB,H., “Hydrodynamics”. Cambridge, 1932. 5. MOISEEV, N. N., On the theory of elastic bodies with liquid cavities, P M M 28 (19.59).

6 . MOISEEV,N. N., About the oscillations of an ideal incompressible liquid in a container, Dokl. AN SSSR 88 (1952). 7. MOISEEV,N. N., The problem of the motion of a rigid body containing liquid masses with a free surface, Mat. Sbornik 88 (1953). 8. GYUNTER, N. M., “Potential theory and its application to the fundamental problems of mathematical physics”. GIFML, Moscow, 1953. The original French edition is in Collection Borel, Paris, 1934; a German edition appeared a t Teubner, Leipzig, 1957 (Giinter, Potentialtheorie. . .). 9. OKHOTSIMSKII, D. E., On the theory of motion of a body with cavities partly filled with liquid, P M M 81 (1956). 10. ZHUKOVSKII, N. E., About motions of a rigid body with cavities filled with homogeneous liquid, in Sobr. Soch. (= coll. works), vol. 1, GITTL, Moscow, 1949. 11. MOISEEV,N. N., “Investigations about the motion of a rigid body containing liquid masses with a free surface”. Inst. im. Steklova, 1955. 12. MOISEEV,N. N., The motion of a rigid body with a cavity partly filled with ideal liquid, Dokl. A N SSSR 86 (1952). N. N., About the oscillations of a rigid body containing 13. KREIN,S. G., and MOISEEV, liquid with a free surface, P M M 2 1 (1957).

288

N . N . WOISEEV

14. MIKHLIN,S. G., “The minimum problem for a quadratic functional”. GIFML, Moscow, 1951. 15. RIESZ, F., et SZ.-NAGY.B., “Leqons d’analyse fonctionelle”, 3d ed., Acad. Sci. de Hongrie, Budapest, 1955. 16. SRETENSKII. L. N., “Theory of wave motions of a liquid”. GITTL, Moscow, 1936. 17. PONTRYAGIN, L. S., Hermitian operators in a space with indefinite metric, Izv. A N S S S R , ser. mat. 8 (1944). 18. TIMOSHENKO, S., “Strength of Materials”. Van Nostrand, New York, 1930. 19. MOISEEV,N . N., Variational problems in the theory of oscillations of a liquid and of a body with liquid, i n “Variational methods in problems of oscillations of a liquid and of a body with liquid”. Vychisl. Tsentr A N S S S R , Moscow, 1962.

Selected Bibliography 1 . ABRAMSON,H. N., CHUWEN-HWA,RANSLEBEN, G. E., J R . , Representation of fuel sloshing in cylindrical tanks by a n equivalent mechanical model, A R S Jouvnal 11, 1697-1705 (1961). 2. BUBLIK,B. N., and MERKULOV, V. 1.. Dynamic stability of thin elastic shells, reinforced by rigid ribs and filled with liquid. i n 29. 119-179. 3. COOPER,R. M.. Dynamics of liquids in moving containers, A R S Journal 80, 725-729 ( 1960). 4. KIRILLOV,V. V., investigation of the oscillations of a liquid in a n immovable container with consideration of drainage. T r . Mosk. fiz.-tekhn. instituta, 1960, Nr. 5, 62-72. 5 . LAWRENCE, H. K.,WANG,C. J . and HEDDY,H . R., Variational solution of fuel sloshing modes, Jet Propulsion 28. 729-736 (1958). ti. MILES,J. W., On the sloshing of liquid in a flexible tank, J . A p p l . Mech. 26, 277-283 (1958). 7. MILES, J . W., Note on the damping of free-surface oscillations due t o drainage, J . Flrtrd Mech. 12. 438-440 (1962). 8 . MOISEEV,N. N., The problem of small oscillations of a n open container with liquid under the action of a n elastic force, Llhuazn. rnatern. zh. 4, 168-173 (1952). 9. MOISEEV. N. N., Dynamics of a ship with liquid loads, Izv. A N S S S R . Old. tekhn. n. 1954, Nr. 7. 10. RIoIsEEv, N. N., On a problem in the theory of waves on the surface of a limited volume of liquid, P M M 19, 342-347 (1955). 11. MOISEEV,N. N., On the boundary-value problems for the linearized Navier-Stokes equations in the case of small viscosity, Z h V M i M F 1 (1961). 1‘7. MOISEEV, N . N., On mathematical methods for the investigation of nonlinear oscillations of liquids, in Proceedings of the international symposium on nonlinear oscillations, Kiev, 1961. 13. MOISEEV,N. N., Some mathematical questions concerning the motion of a satellite. Lecture given in the international symposium on the dynamics of satellites, Paris, 1962. 14. NARIMANOV, G. S.. On the motion of a rigid body whose cavity is filled with liquid, P M M 20, 21-38 (1956). 15. NARIMANOV, G. S., On the oscillations of liquid in movable cavities, I z v . A N S S S H , Otd. tekhn. n. 1957, Nr. 10, 71-74. 16. NARIMANOV, G. S., On the motion of a vessel partially filled with liquid, the motion of the latter not being considered as small, P M M 21, 513-524 (1957). 17. PENNEY, W. G. and PRICE,A. T., Some gravity-wave problems in the motion of perfect liquid, Philos. Trans. Roy. SOC.London A244, Nr. 882. 254 (1952).

INTRODUCTION TO THE THEORY OF OSCILLATIONS

289

18. PETROV,A. A , , Oscillations of a liquid in an annular cylindrical container with

horizontal generators, Z h V M z M F 1, 741-746 (1961). 19. PETROV, A. A,, Oscillations of liquid in cylindrical containers with horizontal generators, in 29, 179-203. 80. PETROV, ‘4.A,, Approximate solution of the problem concerning oscillations of a liquid in a cylindrical container with horizontal generators, in 39, 203-213. 21. PIERSON.J . D., Water waves, A p p l . Mech. Reus. 14, 1-3 (1961). 22. RILEY, J . IJ. and TREMBATH, N. W., Sloshing of liquids in spherical tanks, J. Aevospace Sci. 3s. 245-246 (1961). 23. KUMYANTSEV,V. V.. On the stability of rotation of a top containing a cavity filled with viscous liquid, P M M 24, 603-609 (1960). 24. SEKERZH-ZENKOVICH, YA.I., Three-dimensional standing waves of finite amplitude on the surface of heavy, infinitely deep liquid, Tr. morsk. gidrof. institufaA N S S S R , 1959 Nr. 18, 3-29. 25. SRETENSKII, L. N., Oscillations of a liquid in a movable container, f z v . ..IN S S S H . Otd. iekltn. R. 1951, Kr. 10. 26. SRETENSKII, I.. S., The three-dimensional problem of steady waves with finite amplitude, Vestn. mosk. un. 1954, Xr. 5 , 3-12. 27. TROECH, B. A , , Free oscillations of a fluid in a container, in “Boundary Problems in Differential Equations”, R . E. Langer, Ed., University of Wisconsin Press, 1960, pp. 279-299. 28. URSELL,F., DEAN,R. G. and Yu, G. S., Forced small-amplitude water waves: a comparison of theory and experiment, J . Fluid Mech. i , 33-52 (1960). 29. “Variational methods for problems of oscillations of a liquid and of a liquidcontaining body”. { A collection of 7 articles.) Vychisl. tsentr. An SSSR. Moscow, 196’2. Abbreviations:

PM llil

=

,4 N

= Akad. nauk

Zh L’M z ,VIF GIFICfL GI T T L

=

= =

Prikladnaya matematika i mekhanika Zhurn. vychisl. matem. i matem. fiz. Gos. izd. fiz.-mat. lit. (= Fizmatgiz) Gos. izd. tekh.-teor. lit. ( = Gostekhizdat)

This Page Intentionally Left Blank

Author Index Numben in parentheses are reference numbers and are included to assist in locating references in which authors' names are not mentioned in the text. Numbers in italics refer to pages on which the references are listed.

A

Corcos, C. M., 118(36), 139, 179 Corrsin, S., 136(53). 137(53), 180 Cowling, T. G., 52, 53(65), 54(67), 67 Csanady, G. T., 139, 182 Cumberbath. E., 10(8), 37, 64, 67 Curle, N., 112, 779

Akhiezer. A. I., 40(60), 67 Alfvhn, H., 10, 64 Appell, P., 224(51), 226(51). 232

B Barratt, M. J., 119, 120, 121, 122(40), 123(40), 126(40), 129(40), 150(40), 151(40), 171(40), 172(40), 179 Basset, A. B., 184, 230 Batchelor, G. K., 149(71), 166(71), 187 Bazer, J., 38(55, 56), 67 Beletskii, V. V., 191(36), 231 Benninghoff, J. M., 139(57), 180 Blokhintsev, D. I., 118(37), 179 Boltzmann, L., 90, 101 Boynton, F. P., 136(94),'782 Broch, J. T., 140(66), 180 Brown, D. R., 85, 101

C Cabannes, H., 38(58), 40, 67 Carrier, G. F., 18, 65 Chapman, S., 54(67), 67 Chester, W., 12(17, 18), 65 Chetaev, N. G., 184, 185(31), 190, 191(16, 37), 192(37), 194, 195(16, 31), 196(16), 207(31), 208(31), 217(31), 224(31), 230, 231 Chu, C. K., 35, 67 Clarkson, B. L., 105(90), 181 Clauser, F. H.,11, 65 Coleman, B. D., 70(2), 73(6), 76(10), 77(10, l l ) , 78(10), 79(11, 15, 17), 81(10, l l ) , 82(10. 17), 83(10), 87(23), 88(23), 89(5), W 2 , 51, 91(2, 5 ) , 92(6), 96(15, 30), 99(10), 100. 701 Contursi, G., 68(72), 68 Cooper, R. M., 184(25), 231

D Davies, P. 0. A. L., 119, 120, 121, 122(40), 123(40), 126(40), 129(40), 150(40), 151(40), 171(40), 172(40), 179 De Witt, T. W., 79(14), 81(19), 93, 701 Dumas, R., 150(72), 181 Dyer, I., 141(67), 181

E Eisenberg, H., 79(13), 100 Elsasser, W. M., 8(2), 64 Elyash, L. J., 79(14). 101 Ericksen, J. L., 70, 71, 74, 100 Ericson. W. B., 38(55, 56), 67 Etkin, B., 105(20), 179

F Favre, A., 150(72), 181 Ferry, J. D.. 95, 96(29), 701 Feshbach, H., 162(84). 781 Fisher, M. J., 119, 120, 121, 122(40), 123(40), 126(40), 129(40). 150(40), 151(40), 171(40), 172(40), 179 Fishman, F. J., 64(71), 68 Ford, G. W., 118(35), 179 Fowell, L. R., 105(16), 178 Franken, P. A., 105(22), 159(22), 179 Franz, C. J., 148(70), 181 Frei, E. H., 79(13), 700 Friedrichs, K. O., 30, 31, 66

291

292

AUTHOR INDEX

G

L

Gaviglio, J., 150(72), 187 Geffen, N., 40, 41, 67 Germain, P., 40(59), 67 Gerrard, J. H., 133(49), 180 Goryachev, D. N., 190(35), 231 Gottlieb, P., 162(80), 781 Gourdine, M. C.. 19, 65 Greatrex, F. B., 139f59. 60), 140(59), 180 Greenhill, A. G., 184, 190(9), 230 Greenspan, H. P., 18, 65 Gross, B.. 93, 94(28), 101 Gyunter, N. M., 238(8), 287

Lamb, H., 8(1), 64, 97(33), 701, 111(31), 179, 184, 186(5), 187(5), 188f5),197(5), 205(5), 230, 244(4), 287 Lary, E. C., 20, 66 Lassiter, L. W., 129, 135, 136(47), 137(47), 138(47), 171(87), 177, 780, 781 Laurence, J . C., 119, 120, 121, 133, 135, 137(57), 163(86), 107, 171(39), 779,

H Hardcastle, D., 138(50), 780 Hasimoto, H., 11, 13, 65 Heitkotter, R. H., 177, 187 Helfer, H. L., 38(53), 67 Helmholtz, H. v., 184, 230 Herlofsen, N., 30(48), 66 Hough, S., 184, 190(11), 203(11), 230 Howes. W. L., 133(51), 130(51), 780 Hubbard, H. H., 129, 135, 130(47), 137(47), 138(47), 171(87), 780, 787

I Ishlinskii, A. Yu., 184, 230

J Johnson, W. R., 162(85), 787

K Kantrowitz, A., 54(66), 67 Kantrowitz, A. R., 04(71), 68 Kelvin, Lord, 184, 230 Kibel, I. A., 197(34), 205(34), 237, 237(3), 287

Kochin, N. E., 197(34), 205(34), 231, 237(3), 287 Korbacher, G. K., 105(16). 178 Kraichnan, R. H., 147, 175(08), 181 Kranzer, H., 30(47), 31, 66 Krein, S. G., 184, 230. 270(13), 273(13), 287

Kryter, K. D., 140(64, 05), 180

180, 187

Leaderman, H., 91, 94(25), 707 Lee, Robert, 131, 133(50), 140, 141, 180 Lewellen, W. S., 18(28), 65 Lighthill, M. J., 105(3. 30), 100(3), 113(1, 3), 114, ll6(1, 2, 3), 119, 124(3), 127, 137, 138(2, 3). 139(3), 149(1), 150. 151(3), 152(1), 153, 154, 178, 179 Lilley, G. M., 105(4), 120, 124(4), 125(4), 139, 147, 150, 151, 153(4), 156(4), 163, 165, l09(4), 170(4), 171(4), 778, 180 Lin, S. C., 54(00), 67 Liubarskii, G. I., 40(60), 67 Ludford, G. S. S., lO(5, 0), 12(15, 10, 19), 21, 64, 65, 66 Liibeck, G., 184, 230 Lust, R., 38(54), 67 Lyamshev, L. M., 118(92, 93). 171(92), 781, 182

Lyapunov, A.M., 184, 185, 180, 190, 192, 193(27),208(27), 210(28), 213(28), 214, 215(29), 218, 219(29), 220(29), 222. 223(28, 29). 231 Lykoudis, P. S., 18(20), 65 Lynn, Y. M., 35(51), 67

M McCune, J. E.. ll(12, 13), 29(45), 35, 36(45), 43(13). 65, 66 Maczynski, J. F. J., 157(75), 781 Malashenko, S. V., 184, 230 Markovitz, H., 79(14), 85(21), 86(2l), 91, 707

Matschat, K. R., 140(62), 780 Mawardi, 0. K.; 147, 175(09), 187 Mayes, W. H., 130(52), 180 Meacham, W. C., 118(35), 779

293

AUTHOR INDEX

Mikhlin, S. G., 238(2), 244(2), 270(2), 272(2), 278(2), 287, 288 Miles, J. W., 128(46), 162(46), 180 Moiseev, N. N., 184(19), 185(20), 230, 235(1), 246(1, 7), 252(7,11), 255(7, 11). 270(13). 273(13), 276(19), 287, 288 Mollo-Christensen, E., 119(41,42),175(41), 179, 180 Moretti, G., 162(81), 181 Mori, Y., 18(27), 65 Morse, P. M., 162(84), 187 Miiller, E. A,, 140(62), 780 Murray. J. D., 12(16), 65

N Napolitano, I,. G., 68(72), 68 Narasimha, R., 119(41), 175(41), 179 Narimanov, G. S., 184, 230 Neumann, F., 184, 230 Noll, W., 69, 70(2), 73(6), 74, 76(10), 77(10, 1l ) ,78(10),79(11, 17),81 (10, 11). 82(10, 17), 83(10), 85(20), 86(20), 87(23), 88(23), 89(5), 90(2, 5), 91(2, 5). 92(5), 96(31), 99(10), 100, 107

0 Okhotsimskii, D. E., 184, 230, 249(9), 287

P Padden, F. J . , 81(19), 101 Padden, F. J., Jr., 79(14), 101 Pai, Shih-I., 68(73), 68, 81(18), 93, 107 Pavlenko, G. E., 184, 230 Petschek, H. E., 64(71), 68 Phillips, 0. M., 105(5), 127(5), 136, 175(5), 176(5). 177, 778’ Pietrasanta, A. C., 130, 181 Poincark, H., 184, 185, 197, 198, 230, 231 Polovin, R. V., 40(60). 67 Pontryagin, L. S., 288 Powell. A , , 105(6, 15, 19, 24, 26, 27). 106(2), 116, 124(6, 21), 125(6), 129(15), 135, 159(15, 76). 162(76), 163(15). 171(6), 175(6, 26). 178, 179, 781 Pozharitskii, G. K., 213(49), 214(49, 50), 225(50), 227(49), 232

Pridmore-Brown, D. C.. 162(78, 79), 181 Proudman, I., 105(7), 124(7), 144, 149, 150(7), 778

R Ram, G. S., 105(29), 179 Rayleigh, Lord, 128(44), 780 Reiner, M., 74, 100 Resler, E. L., 54(66), 67 Resler, E. L., Jr., lO(4). 11(4), 29(45), 35, 36(45),43(13), 56(68),62(70),64, 66.67 Ribner, H. S., 105(8, 9, 10, 11. 20, 28, 29), 117(9, 10, 11). 118(9, 10, l l ) , 119(11, 38). 124(8), 127(11), 128(45), 129(9, 10, 11). 130(11), 132(11), 137(10, 11). 138(1l ) , 142(1l ) , 144(1l ) , 145(11). 147(9, 10, l l ) , 148(11), 151(11), 153(10, 11). 154(11), 155(11), 156(10, I l ) , 157(4, 74). 159(9, 10, 11). 160(11), 161(77), 162(45), 163(9, 10, l l ) , 171(8), 175(10, 11). 176(9, 10, 11). 177, 178, 179, 181 Richards, E. J., 105(14, 17), 140(63), 178, 180 Riesz, F., 288 Rivlin, R. S., 70, 71(4), 74, 79(16), 82(16), 85(16), 100 Rollin, V. G., 137(54). 180 Rose, N. V., 197(34), 205(34), 237 Rossow, V. J., 12, 18(21, 22), 6 5 Roze, N . V., 237(3), 287 Rumyantsev, V. V., 184(26), 185(47), 186(32, 33), 194(38), 195(38, 39), 196(40), 197, 198(33), 203(42, 43, 44), 206(32), 208/44), 210(45), 212(46), 214(32, 50), 223(48), 225(50), 230(48), 231. 232

S Sanders, N. D., 163(86), 181 Sarason, L., lO(8). 64 Savic, P., 116(34), 179 Sears, W. R., l0(4), 11(4, 12, 13), l6(25), 18(27), 25(41), 27(42), 31(49), 40(63, 64), 64, 65, 66, 67 Seebass, A. R., 27(44), 66

294

AUTHOR INDEX

Shercliff, J. A., 38(57), 67 Singh, M. P., 21, 66 Slutskii, F., 184, 196(16), 230 Slutsky, S.. 162(81, 82, 83). 781 Smith, T. J. B., 105(24), 779 Sobolev, S. L., 184, 212, 230 Sonnerup, B., 58, 67 Sperry. W. C., 105(23), 113(23), 148(23), 179

Squire, H. B., 157(73), 187 Sretenskii, L. N.. 184, 230, 288 Stewartson, K., 10(9), 12. 21, 64, 65, 66, 184, 185(24), 237 Stokes, G. G.,96(32), 107, 184. 230 Sung, K. S., lO(7). 64 Sz-Nagy, B., 288

T Tamada, KO.,12(20), 22, 65, 66 Tamagno, J., 162(82), 181 Taniuti, T., 27(43), 40(61), 66, 67 Temchenko, M. E., 184122). 230 Timoschenko, S., 266(18), 288 Toupin, R. A., 74(7), 100 Townsend, A. A., 120(43), 121, 123(43), 180

Trouncer, J., 157(73), 781 Truesdell, C., 74, 86(20), 86(20), 100, 101 Tsuda. A., 79(12), 700

U Uberoi, M. S., 136(53), 137(53), 780

V Van de Hulst, H. C., 30, 66 von Gierke, H. E., 105(18), 178

W Weitzmer, H., 10(8), 33(50). 64, 67 Westervelt, P. J., 105, 106(25), 179 Westley, R., 139, 180 Williams, J . E., 105(12, 13), 127, 128, 135(13), 152(13), 154(13), 156(13). 159, 178 Withington, H. W., 139(61), 140(81), 180

z Zhak, S. V., 203(41), 231 Zhukovskii, N. E., 184, 187, 18817). 189(7), 190. 192(7), 195(7), 197, 199(7), 212, 230, 244(10), 287

Subject Index A

Crossed-fields flow, 34, 36 Current layer (MHD), 15, 28, 35 f. Cyclotron frequency, 60f.. 56, 61

Aligned-fields flow, 12, 23, 55 .41fvdn number, 13 Xlfvdn radiation, 10 Alfvdn waves, damping of, 11 diffusion of, 1 1 AlfvCn-wave speed, 26. 33 Ampere's law, 6 f. Autocorrelation, 146 Axial flow between concentric pipes, 82 f. Aeolian tones, 112

D

B Beam with liquid-containing cavity, 273 Boltzmann's theory of linear viscoelasticity. 75, 90 Boundary conditions (MHD). 5 Boundary layer, inviscid, in MHD. 16 f., 42 f.

Density effects (turb.), 136 Developed jet, 121, 123, 124, 125, 170 f., 173 Deviation (stab.), 218 Direction of boundary-layer growth, reversal of, 14 Dilatation, 117, 122, 143 f., 153, 176 Dipole, 111 sources, 116 Directional sound patterns, 158 Directivity (turb.), 129, 130 f., 153 f., 162, 168 Disturbance-shape diagram, 61 Doppler shift, 132, 133, 174 Dynamic shear viscosity, 94

C

E

Characteristic ion Larmor radius, 57 Chetaev's method, 200 Completely continuous operator, 272. 279 Compressible flow (MHD), 23 Conditional stability, 207 Cone and plate flow, 85 Conservative oscillations with a liquid member, 252. 270 Constant stretch history, 96 Continuum flow approximations in MHD, 4 Convection effects, 126, 151 Convection factor, 127 f., 131 f., 133 ff., 152, 154. 171, 174f. Coordinate functions, choice of, 240 Correlation, 122, 144 f., 151, 153, 157, 165, 169, 172 Couette flow, 71, 79 f., 82, 92 f., 93 Cross-spectral density, 147

Eddy Mach waves, 176 Effective calculation of the natural frequencies, 257 Efficiency (turb.), 133 f. Electron pressure gradient, 53 Elliptic regimes (MHD), 26 Elliptic-su bsonic, 27 Equations of motion (stab.), 203 Equivalent solid body, 189, 253 Extra-stress, 75

F Faraday's law, 7 Fast waves, 31 Fading memory, 70, 73, 74 Flexural-torsional oscillations, 258 Flow over a n airfoil (MHD), 29 over a plate, 21

295

296

S U R J E C T INDEX

Flow with strong magnetic field, 21 Friedrich’s wave-speed diagram, 32, 58 affected by resistivity, 62 Force source, 111 Forward-facing waves, 58 Frbchet functional derivatives, 74 Frozen field lines, 12 Fully ionized gas, 54

H Hall effect, 5. 50, 54, 55, .i8 Hamilton’s principle, 234, 236, 254, 262, 267 Hasimoto’s theory, 14 flow, 16 Heavy liquid in a vessel, 235 Helical flow, 79, 82 Helicoidal motion, 191 High-density plasma, 58 History function, 73 Histories with small norm, 90 Hyperbolic regimes, 26 Hyperbolic-supersonic, 27 Hypercritical, 27

I Ideal conductor, 12, 25 Inconipressible simple fluid, 69 Incompressible flow (MHD), 6 Infinitesimal strain tensor, 90 Influence function, 73 Intermediate waves, 33 Inviscid perfect conductor, 9, 15 Ion slip, 53 Irrotational motion (stab.), 186 in linearized problems, 242 Isochronism of the variations, 249 Isolated minimum, 219 Isotropic turbulence, 149 f., 163, 166 Isotropic functional, 72

L Leading-edge stagnation-point flow (MHD), 18 Lighthill parameter (turb.), 124 Linear theory (stab.), 185 h r e n t z force, 50

Lyapunov’s methods, 185 function, 192 Lyapunov’s theorem, modification of, 208

M Mach number redefined (MHD), 26 Mach wave, 127, 176 emission, 134 Magnetic lines “frozen”, 57 Magnetic poles, absence of, 7 Magnetic Prandtl number (def.), 8 Magnetic Reynolds number (def.). 8 Magnetosonic “Mach cone”, 37 waves, 31 Maievskii’s condition, 195 Material function, 78, 87, 89 Material objectivity, principle of, 72 Memory functional, 74, 76 Mixing region, 119 ff., 123, 125, 169 ff., 174 Monopole, 109 Moving-axes (turb.), 152 If., 166 Moving jets, 157 f.

N Keumann’s operator, 237 Newtonian fluid, 70, 74, 76, 78 Norm, 73 Normal stress, 81, 95, ion effects, 71. 75, 78 measurements, 86 Normal stress coefficients, 92 Not completely continuous operator, 176

0 Ohm’s law for ioized gases, 51, 52, 5!b Oscillations of a liquid in a vessel, 268

P Particle motions (MHD), 50 Pedal curve, 31 Pendulum problem (osc.), 246. 283 Perfect fluid (MHD). 74 Plasma frequency, 57 Plasma, collisionless, 56 cold, 62 f. Plasma, composition of, 52 Poiseuille flow, 79

297

SUBJECT INDEX

Power (turb.), 133, 136, 175 total, 134 Precursor, 14, 18, 2 0 f . Propagation speed of small disturbances, 26 Pseudosound, 118, 143 f., 145, 157, 175 f. Problem of the minimum (stab.), 224

Q Quadrupole, 112, 143, 149, 153, 17.7 sources, 115

R Kate of shear, 77 Real history, 74 Refraction (turb.). 126, I28 f., 131, 159, 168

Reiner-Rivlin fluid, 74, 76 Relative deformation function, 71 1. Relative equilibrium (stab.), 223 Relaxation spectrum, 95 Reserve stability, 252 Resistivity effects, 11 Retardation function, 76 Reynolds stresses, 115, 122, 149 f. Ritx’ method, 234, 239. 257, 261, 266. 271, 273

Rivlin-Ericksen fluids, 71. 74, 75 tensors, 70. 74 Rocket data, experimental, 136

S Scale of turbulence, 1.70, 154 Scale anisotropy, 154 Second-order fluid, 70, 76 Self-noise, 125 f., 133, 149 f., 164, 166 ff. Separation (stab.), 218 Shear compliance, 96 Shear-dependent viscosity, 75 Shear loss modulus, 94 relaxation modulus, 71, 91 f. storage modulus, 94 Shear-noise, 125 f., 133, 164, 168 ff. Shearing flow, steady simple, 77 nonsteady simple, 96 Shock waves (MHD), 38 Simple fluid, 69, 72 f., 78. 81, 85 f., 87

Simple shearing motion, oscillatory, 92 Simple source, 109, 117 Sinusoidal oscillations, 98 solutions, 43, 57 Sinusoidal stress and strain, 93 Slightly ionized gas, 53 Slow waves (MHD), 31 Slug flow (MHD). 21 Small-perturbation flow, 8, 19, 23, 41, 55 Small-Rm approximation, 12 Sound power emission, 124 total, 174 Source distribution, 170 f. Special problems (osc.), 23: Spectrum (turb.). 124 f., 131 ff., 146, 1.55, 168 f., 172 ff. Stability, simplest cases of, 186 Stability of motion, definition of, 2l!1 Stability theorem, 21 7 Stability with respect to a part of the variables, 206 Stability of shock waves (MHL)), 40 Standing .4lfvCn-wave, 37 Standing waves inclined forward, 48 diffusion of, 22 Static stability, 280 Steady motion (stab.), 214 Steady extension, 87, 88 Stokes-Zhukovskii potentials (osc.), 244, 247

problem, 243 Stokes effect, 113 Storage modulus, 7 1 Strain measure, 89 Stress source (turb.), 112 Stress relaxation, 75, 90 Sub-AlfvCnic, 13, 47 Sublayer, viscous (MHD), 1 6 f. Substantially stagnant motions, 96 Super-Alfvbnic, 13 Surface-current layers, 6 Surface presslire (MHD), 18, 28 f.

T Temperature effects (turb.), 136 Tensor conductivity, 52, 54 Torsional flow, 84

298

SUBJECT INDEX

Torsional-flexural oscillations with liquidcontaining cavity, 266 Torsional oscillations with liquidcontaining cavity, 258 Total pressure (MHD), 13, 18 Total sound power, 124 Transport properties in MHD, 54 Transverse scale (turb.), 121 Trailing-edge stagnation-point flow (MHD), 18 Turbojet noise reduction, 138 Turbulence, homogeneous, 165 isotropic, 165 Turbulent jet, structure of, 119 Turbulent scales, 122

U Uniform rotation (stab.), 192 Uniform vortex motion (stab.), 196 Units in MHD, 6 Upstream wake (MHD), 14

V Viscoelasticity, linear, 71, 89, 93 second order, 71, 89 Viscometric flow, 79 Viscosity effects (MHD), 11 Viscosity function, 78 Vortex layer (MHD), 6, 15, 28, 36 Vortex stretching, 116

W Waves, damping of (MHD), 45 Waves, forward-facing, (MHD), 49 Wave-propagation (MHD), 30 Wave-speed diagram, 62 f. generalized, 58 Wake, inviscid, (MHD), 20

z Zhukovskii's theorem, 190