Advances in Applied Mechanics, Volume 8

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ADVANCES IN APPLIED MECHANICS

VOLUME 8

This Page Intentionally Left Blank

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


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


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.


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


(1.12)

q=U+v

and

H=H,+h

and

J h ( 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.


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


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


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. 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


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)


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


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


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


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


sLowKd

WAVES

0-

-

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

FIG.2.6.

a, = V Z A , .

(The dimen-


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 i,

-.


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


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.


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


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 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


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


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:


(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)


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


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,.


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


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


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)


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


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


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).


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) 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


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


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 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


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


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 ] . )


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).


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 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)


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.

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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.


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-


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


(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

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€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)


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 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

194

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


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

196

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.

197


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

198

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


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.

<
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.


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;,


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

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


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 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,

€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.

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

.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.


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

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


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}

+ ... .


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.


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

V. V. RUMYANTSEV

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


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.


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'

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


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. ---+


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, =


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.


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,


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


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


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,


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


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


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).


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)


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

Advances in Applied Mechanics, Volume 8

Advances in applied mechanics

Advances in Applied Mechanics