Small Perturbation Theory

Table of contents :
Cover
PREFACE
CONTENTS
1. Introduction
2. The Equation of Sound Propagation
3. The Prandtl-Glauert Equation
4. Application to Wing Theory
5. Application to Other Bodies
6. Pressure-Correction Formulas for Steady Flow
7. The CriticaJ Mach Number
8. Experimental Confirmation
9. The Slender Body Theory of Munk and Jones
10. Rotational Small Perturbation Flow
11. The Stream Function in Rotational Steady Flow
12. Unsteady and Periodic Flow Problems. Fundamental Solutions
13. Cited References

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NUMBER 4 PRINCETON AERONAUTICAL PAPERBACKS

COLEMAN duP. DONALDSON, GENERAL EDITOR PRINCETON AERONAUTICAL PAPERBACKS 1. LIQUID PROPELLANT ROCKETS David Altman, James 1'1. Carter. S. S. Penner, ~1rutin Summerfield. High Temperature Equillbrium, Expansion Processes, Combustion 0£ Liquid Propellant.s, The Liquid Propellant Rocket Engine. 196 pages. $2.95 2. SOLID PROPELLANT ROCKETS Clnyton Huggett, C. E. Bartley nnd t.i ark lvf. ~lilli. Combustion of Solid Propellants, Solld Propellant Rockets. 176 pages. $2.45

SMALL PERTURBATION THEORY

3. GASDYNAl\1IC DISCONTlNUITIES Wallace 0. Hnyes. 76 pages. $1.45 4. S/\1ALL PERTURBATION T HEORY

\V. R. Sears. 72 pages. $1.45

BY W.R. SEARS

5. ffiCHER APPROXIMATIONS IN AERODYNMUC TlIEORY. M. 156

pages.

J.

Ughthill.

$1.95

6. !{)CH SPEED \VINC THEORY Robert T. Jones and Doris Cohen. 248 pages. $2.95 PRINCETON UNIVERSITY PRESS • PRINCETON, N. J. PRlNCETON, NE\V JERSEY PR11'1CETON UNIVERSITY PRESS

1960

'

@ COPYRICRT,

1954, 1960, BY PlllNCl!TON UNtVEllllrt'Y Ptwss

HlC H SPEED AERODYNAMICS L. C. CAruJ

AND JET PROPULSION

60-12051

BO ARD OF EDITORS Reproductioo, translation, publication, use, and disposal by and for the United States Government Md its ollicers, agents, and emplo~ acting within the S(l()pe of their official dlllies, for Government use only, 15 per• mitted. At the expiration of ten yean from the date of

KAaM.l.'1, Chairman HvcH L. DRYI>EN HvcH S. TAYLOR CoLEMAN DvP. DoNALDSON, General Editor, 1956AssoclJ>te Editor, 1955-1956 JOSEPH V. CHAan:, General Editor, 1952Assoc.iato Editor, 1949-1952 t.lARTIN SVMMERFD!LD, General Editor, 1949-1952 RICHARD S. SNEl>EJ"D, for supersonic flow around a corner. In fact, the identity of these methods with those of clll88ical acoustics- to which they are reduced by a simple translation of coordinate ax~ not to have been noticed by aerodynamicista until somewhat late.r. I t is not difficult to see how this would occur, first because of the aeronautical preoccupation with parallel streams and also because, in reality, the idea of tnwll perturbalion• of a pnrallel stream was already well known, having been exploited for somewhat different reasons in the literature of airfoil theory. Pmndtl's famous lifting-line wing theory (1918), although it treated ,vingii in incompressible flow, was a small pert urbation theory, 118 were the thin airfoil theories of Munk (1922), Birnbaum, Gla.uert, and others. Consequently, it wa.s more natural to proceed from this background toward a n approximate theory of compreSBible fluids than to return to the equations of acoustics. To be sure, there aro undoubtedly prior invelltigations in the field of a coustics that anticipated Sjl.Jient features of the Prandtl-Glauert or Acke.r et theories, and could be thought of as linking the acoustic and a.eronautical versions of email perturbation theory. One of these is a 1907 paper by von KJ!.rm"1 (6] concerning standing waves in supersonic jets. Nevertheless, the main course of the idea, so far as aeronautical engineers are concerned, baa clearly been through airfoil theory in the work of Prandtl and GJauert, rather than from Rayleigh directly. l n spit.e of this, our treatment of small perturbation flows in the present section will begin with the acoustical situation of a gaa at res\ and proceed to Prandll'a and Ackeret's pr0blems as subcases. This procedure simplifies and clarifies certain matters, especially pertaining to unsteady flows. After introducing the ba&o ideas in this way, and obtaining the f Wldamental equations in several different forms, we shall devote ourselves in thia ecction principally to their exploitation in subsonic flow situations. I n Sec. D the applications to supel'80nic Bight conditions are treated. This constitutes a report on the vast and rapidly growing body of literature on t he aerodynamics of thin wingii at supersonic speeds, and on

some important aspects of the mathematical techniques found uaeful for these applications. Sec. D clo$es with a consideration of those special situations (transonic and hypersonic) where more careful analysis ie required and nonlinear termll remain, even in the small perturbation equations. Finally, in Sec. E, we complete our study of small perturbation methods by considering their shortcomings and methods that have been proposed to improve them. Actually, these considerations will bring u.s to a much clearer understanding of the first order perturbation theories themselves; COllBl!QuenUy the authors of Sec. C and D preeent these theories without complete justification of the approximations involved, depending upon Sec. E to make up these deficiencies.

where D/ Dt ia the usual "convective" or "substantive" derivative that me&8Ures a rate of change for an indi"idual particle of gas, and 4• denotes the quantity dp/dp for reversible adiabatic compre$8ion, i.e. the square of the speed of sound. Clearly, the replacement of Dp/Dt by a-•Dp/Dt can only be made on the basis stat~ above, that each particle moves about under adiabatic conditions. Both of the statements in Eq. 2-1 are nonlinear, and consequently have been solved only for a few special ~ and under rather severe limitations. In the particular case of sound propagation through a body of gas uniformly at rest, we proceed to make the small perturbation approximation in Eq. 2-1. We assume the velocity and ita derivatives to be 61llall, and the COrre5ponding density and pressure to differ only slightly from the values p,., p,., which pertain to the undisturbed gas. Denoting t hOllO small departures from the steady val ues by p', p', and neglecting second

( 4 }

( 5 )

C,2. The Equation or Sound Propagation. AB in most of the seotion3 of this volume, tho equations of inviscid fluid flow provide the basis for our considerations. These are satisfactory to describe the flow of real gases in regions where neither friction nor heat transfer are important: nrunely, as we have seen (III,A,6), where velocity and t.cmpemture gradients are not large. It has already been noted (III ,A, 7) that, in the ab.scnoe of friction and heat tmlll!fer, each particle of the fluid undergoes reversible adiabatic changes of state. ConsequentJy, if the entropy of the ga$ was initially uniform, a11 in the oaso of a homogeneous gas at rest or flowing in a. uniform stream, it must remain constant and uniformly distributed tluougbout the flow. The equations of the flow are

~~

- - ; grad p;

E£ + p div q Dt

pp-r - const

(2-1 ) =

.!. QJ! + p div q ~ 0 a 1 Dt

C ·SMALL PERTURBATION THEORY

C,3 · THE PRANDTL-GLAUERT EQUATION •

order terms in these and their derivati vee, \Ve have q , = - grad

(f:)

~ V, + p,. div q a..

(2-2)

~)

= 0

( 1 - o... "'"'"

where a! denotes 'YP...fp.,., the square of the speed of sound in the und.i&turbed gas. Eq. 2-2 arc linear. Moreover, in view of the initial condition of uniform rest, the first of Eq. 2-2 implies that q is the gradient of a scalar function, i.e. that a pot.ential function ¢exists. Eq. 2-2 can then be written as follows: (2-3) p,.tf>t = -p' ~.. =

a!,v•~ (24) These e.re the classical equations of sound propa.ga.tion {1, p. 15). (Eq. 2-4 is known as the wave equation.) They a.re useful in a.erodynamice to describe a great variety of small perturbation situations outside of boundary layers and where shock \va.ves ei~her do not exist or are so weak a.s t-0 justify our approximations. They can sometim.es be used to describe the flow behind stronger shock waves, which are nearly uniform in strength, but this case requires special ca.re, as is made clear below (Art. 10). An essential al!Sumption in the derivation of Eq. 2-3 and 2-4 is that the initial condition is one of uni.form rest, which has been used to obtain q as the gradient of a potential function. Clearly, these equations mus~ be discarded whenever there is evidence that the small disturbance hypothesis is violated; we shall see (Art. 4) that this sometimes occurs even when the fl.o,v is due t-0 the motion of a slender body. T h e .Prandtl-Glaue~t Equation. Suppose now that the undisturbed flow is the familiar one of aerodynamics, namely, a uni.form steady stream. This is reduced to the preceding case by steady translation of the reference axes. Let the :i: axis be chosen in the stream direction and the stream speed be called U. The coordinate transformation between tho moving (:i:', y', z', t') sygtom and the stationary (:t, y, z, !) one is, of course, C,3.

z - z' - Ut',

11 - y',

t - t'

(3-1)

Thus, Eq. 2-3 and 2-4 become p .. (~,,

+ uq.,,,)

The equations for the special case where the flow is steady when viewed from the moving coordinat.e system are called the Prandtl..Olauert equations [£,SJ: (34) p - p., = -p.. Ut/>.-

(3-2)

0

(3-5)

Eq. 3-5 illustrates clearly the change of mathematic.al character of gas flow in the three major speed regimes, subsonic, transonic, and supersonic. This equation is elliptic, parabolic, or hyperbolic \vhen the coefficient of the first term is positJve, zero, or negative, i.e. when the stream speed falls into the three regimes mentioned. I t is a result of our small perturbation hypothesis, of course, that tho charncter of the equation depends only on the Mach number of the undisturbed stream; this will unavoidably restrict the usefulness of this approximate theory, especially io the t ransonic regime. Eq. 3-5 is also of va.lue because it establishes the significance of incompressible Row theory in relation to the aerodynamics of compressible ga.ees. It is apparent that there will be e. range of Me.ch numbers nee.r zero, at least for slender bodies, for which the compressible and incompressible flo'll' patterns wiJJ be nearly alike. There is another way to establish this fe.ct, 'vbich is better because it is not restrictd to slender bodies, namely, the expansion method of Janzen and Rayleigh, which is treated in E,2. The Janzen-Rayleigh theory confirms that the first departures from incompressible fluid theory, as the l\ta.ch number of steady flow, ilf,., ia increased from a small value, appear in terms of order M!., aa might be inferred from Eq. 3-5. This equation is put into the form of Laplace's celebrated equation by a simple affine transformation. Dropping the primes in Eq. 3-4 and 3-5 and writing /J' for (1 - U•/ a';), which is a. positive number for subsonic stream flow, we put (3-6)

'II = n 11,

with m/n = fl.

Eq. 3-4 and 3-0 are immodiat.ely transformed into P -

P.. •

-

l m p,.U'l'1

(3-7)

and 'PH

= -p' = 'P.. - 'P

+ r'v' + q,.,,,. =

+ IP" +

'{'tr '"'

0

(3-8)

where again p.,,, p,., a,. are values pertaining to the undisturbed stroam, and V" involves deri,•atives with respect to z', y' 1 z'.

where 'I'(~, '11 t) has been "Titten for the function obtained by substituting Eq. 3-6 in -

(4-9)

It is also clear that c,..(.,) is just c(y)/fJ. The evaluation of ai.. follows from Eq. 4.-3, which defines the fictit.i.o\lll wing; we see that ai..(11) - a(y). Finally, according to Prandt1'1" the0ry, W;•• is given by an integration of I"..,(>1), which sums up the effects of the sheet of trailing vortices behind the wing: w,,.('I)

1

= •-

...,..

1•12 r ;_(,;)diiCl' -b/2 '1 -

..!.. O' Jb12

=

w= UC,, and CL =

r(y) = 2fJ Uc.c,.

a -

r ' (f}) diJ

12 r ' (tJ)dtJ] Jb 4!1rU Cl' -•12 y - fi l

c,,...

wu.. = const - U ,...R,..

{4-11)

(4-12)

and for the flat wing of elliptic plan!onn, in particular, (4-13)

>

(4-15)

+ .,_ 0
.-(z',

+o) ... u cos A T.(z' ) }

of this plane flow; i.e.

z', 1/, :'coordinates. T he calculation is a simple one; in view of the in6nite cylindrical character of the wing, we know that all velocity components are independent of y', which eliminateB a number of terms in the equa.tion. The result is (/J2 cos' A + sin' A)q,.... + 41,,,, - 0 (4-18)

\01(~. f) =

Then Eq. 4-21 becomes a rch\tion between the prC?ssure coefficienta of compressible th.ree-dimensional now and the rotated plrine incompressible flow, viz. 1

CI' (:r! r z') = -fj' C'" (, + If>., =

0

(5-2) (5-8)

and the boundary condition at the cylinder surface is, to the first order, ef>,(z, r 1) = UR'(z)

=

--..U: sin(..-~)

< 16)

(5-3)

{ 17)

C · SMALL PERTURBATION THEORY

C,5 · APPLICATION TO OTHER BODIES

If the cylinder is very slender, so that r 1/l is a small number, we may substitute in Eq. 5-8 the a ppropriate Mymptotic forms of the Bessel functions, which are (5-9)

perturbations a.re imagined to be due to e. distribution of eourcee and sinka of strength f(E) along the E axis. The resulting velocity potential is UE -

..!..1' vn distorted dillerently in the transformation. Actually, thia language seems a little ha.rah, for there is no a priori reason why the transformed boundary value problem must be a recogni.zlible "body problem," as is discUBSed here. Nevertheless, G6thert's observation led him to the discovery and correction of a widespread error in I.he application of the Pra.ndtl-Olauert. theory (see p. 81.).

( 20)

( 21 )

C,,-=

-2tJ-(t)'

(5-25)

""Y"

C,S ·APPLICATION TO OTHER BODIES

C ·SMALL PERTURBATION THEORY

We e.ssume, furthermore, that it is a "slender body," in the sense that the inclination of the surface F to the stream direction is everywhere small, so that the perturbations that it introduces into an initially uniform steady stream can be expected to be small, except, of course, in the neighborhood of stagnation points. '.!.'hen, if the stream speed is U and its Mach number M ,., the differential equation to be satisfied by the perturbation potential q, is the Prandtl-Glauert equation (3-5)

p•q,"'

+ if>•• + q,•• =

fJ'

0j

E

1 - M~

.(x, y, z) -

~ 'Pt(E, 71, t)

.(x, y, z) = .(E, "!, t ) =

~v'(E, .,,, !')

q,,(x, y, z) = r(E,

~ w'(E, 11,

11, !') =

y = 71,

The boundary condition at the body surface is expressed by the statement tha.t the normal velocity component vanishes there. Since the normal direction to the surface F is (F., F., F,), the statement of this boundary condition is

= :. u'(E, 11, t)

z =

(5-30)

!')

r

•Obviously, a.nother interpretation of Eq. 6-29 is that it states t he condition for the surfaoo !f in a stream of speod U /fl, where ., is the perturbation potentiAI. This choice (like a.ny number of others that can be made at this point) leads to results exact.ly equivalent to tboee obtained here. To a.gree with Hess n.nd Gardner, and a.lso to simplify tho identi6eatioo of these general results with our e&rlier ones, wo prefer to rota.in the stream speed U in the fictitious flow .

Let us state this result verbally: to calculate the velocity components in the compressible flow about any slender body, first calculate the components u', v', and w' in the incompressible Bow, at the same stream speed, about a stretched body whose stream wise ilimensions are fr' times as great . The desired velocity components are then r;- •u', p-'v', and {r"ul at corresponding points of the affine transformation which stretches the space in the same fashion. This is Gothert's "e.xtended Prandtl rule" (8). Previously, the application of the Prandtl-Glauert theory to axia.IW symmetric and three-dimensional bodies had been performed erroneously by a number of writers [7,B,12- 16]. The correct application has no'v been given in several places [16- .eJ). The identity of our general equations (5-30) with our earlier results for wings (Eq. 4-4) is not difficult to establish. The mis.sing factor fr' in Eq. 4-4 is provided by decreasing the ordinates of the fictitious wing in accordance with our general rule. That is, in our general method we should ha.ve stretched the wing in the stream direction without simultaneously increru!ing its ordinates so as to maintain its shape and incidence. This would, of course, require t he additional factor fr' to make the results the same. It can also be established that our general rule coincides with Reissner's results for the corrugated cylinder and ours for slender bodies of revolution. In each of these cases we have preferred to use a slightly different affine transformation, but of course, the flow patterns, described in terms of velocity ratio for example, depend only on the shape of a body and not on the size. The identity of the flows, in this sense, can be worked out by means of a little algebra. Finally, it should be pointed out expl icitly that our general rule holds equally well when the slender body involved is immersed in a bounded stream or when it consists of an arrangement of individual bodies. In fact, the surface F(x, y, z) ~ 0 need not be a simple closed surface, but may ha.ve various u nconnected parts, and tb e same deduction still holds. T hus, the fictitious incompressible Bow is one in which the body or bodies and the boundaries of the stream are all stretched in the stream direction.

( 22)

( 23)

+ q,.)F. + q,,F, + q,,F,

(5-27) 0 on F(x, y, z) = 0 Now, an equivalent statement of the slender body hypothesis to the one stated above is IF.1

-0.1

....c-

u

:::l

:l! ~ a..

0.3

u"

-

....c

M..,=0.70

--·-------

Experimental values Prandtl-Glauert theory

-0.2

Cl>

M.., =0.70

-

u -0.1

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

Cl>

0

8 u ~

Experimental val ues Prandtl-Glauert theory

u

I

Cl>

0 .1 I

:J

....

I

0.2

"'"'.... a..

l I

0. 1

Cl>

0.

0.2 0.3

M..,=0.90

-0.2

-0.2

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

-0.1

M..,=0.90

- 0.I

--------------I

0. I

0.1

0.

0.2

10

20

30

40

50

I

10

20

30

40

50

Per cent dist. from nose

Per cent dist. from nose Fig. C,Se. Comparison of experimental prellllure dlstributions with predictions of the Prsndtl-Glauert theory for an ellipsoid of revolution (prolate spheroid) in a cloeed-throat circular tunnel at zero yaw. Thicknees rst.io 1/6. (From [!11).)

I

Comparison of experimental pressure distributions with predictions of the Prandtl-Glauert theory for an ellipsoid of revolution (prolate spheroid) in a cl-.1.-thro&.t circular tunnel at Jero yaw. Thickness ratio 1/10. (From [ill).) Fi.g. 0,81.

{ 37 )

C · S!tlALL PERTURBATI01V THEOR Y

C,9 · THEORY OF MUNK AND JONES

theoretical pressure coefficient with Mach number is very well predicted

Now, in the compressible llow, the perturbation potential satisfies

by this simple rule. We might conclude, therefore, tha.t more elaborate theoretical methods are of doubtful value in predicting compressibility effects on slender bodies if the incompressible flow pressures are known. This conclusion seems inescapable in view of the comparison bet ween a.ny of these theoretical results and the measured ones; as suggested previowily, this agreement can only be considered fair. \Vhereas, as has just been mentioned, the theoretical compressibility correction is additive, and in fact is coll!ltant for each M., for a spheroid, the experimental curves for M .,, = 0.90 do not exhibit this behavior at all. It should be mentioned explicitly that (21) includes considerably more experimental and theoretical data than have been specifically referred to here. Matt hews gives both measured and calculated results for ellipsoids of revolution at angles of attack, and also considers two nonelliptic bodies, one being a "typical transonic body" and the other an ellipsoid with an annular bump. Two other experimental investigations of slender bodies of revolution in subsonic flow should be mentioned; these \vere both carried out at t he Technische fiochschule in Ztirich, and are reported in [26,t7). The Slen der Body Theory of J\l unk and J ones. I t has been known for some time that there are certain incompressible fto,vs about elongated bodies in which the conditions near the body can be approximated by neglecting the first term in the pot.entia.l equation C,9.

(9-1)

where t he x direction is the streaJJ1 direction and the direction of elongation of the body. This was first discovered by J\llunk and given in his famous paper on the cross-force distribution on airships at yaw [SS]. I t has been extended to compressible fto,vs and to the calculation of pressure distributions more recently, especially by R. T. Jones [34). Natu rally, this approximation depends primarily on the assumption that, for a very long slender body whose cross section varies slowly with x, the first term in Eq. 9-1 is negligible compared to t he others. Let us first assume that t he Munk approximation is valid for a certain body in incompressible flow, and demonstrate, on the basis of the PrandtlGlaucrt theory, that the analogous approximation is valid for compressible How. Let the body surface be given by F(x, y, z) = 0 ; it is our hypothesis that, for incompressible flow about this body,

+ .. ~ 0

According to our extended Prandtl rule (Art. 5), a determination of ¢can be made by calculating the flow about an elongated body whose surface is F'(fJE, 11, r) "' S'(~, 11, ?") = 0, in incompressible Row in the~. TJ, r space, and then putting¢ equal to l/{J times the perturbation potential of that flow. Let this potential be called x(t ,,, I) ; t hen ¢(x, y, z) = ti- 1x(~, " ' t). But the body S'(~, T/, I) ~ 0 is even 1nore slender than the original one, and since it is considered for incompressible flow, there is no doubt that the Munk-Jones theory a pplies to it as well as to t he original, i.e. and x .. + xu ~ 0 Transforming these back to the original variables, we have, XH

fJ 3¢.,

a' .. UaR'(x) sin w

r

..

ri

Finally, the two terms given by Eq. 5-23 and 9-9 are to be combined, giving the total perturbation potential for the yawed body. Let u, 11, and w denote the total fluid velocity component.a in the x, r, and.., directions, and let 11> (x, r) denote the symmetric flow perturbation potential of Eq. 5-23; then the formulas for the velocity components are

-=

Fig. C,9. Slender body of revolution at incidence or yaw.

typical aerodynamic problem. Examp~: Body f revolution at yaw. Consider the body of revolution sketched in Fig. C,9 in an unconfined stream at a small angle of attack, a. The boundary condition at the body surface is approximately

r

q,~z> = U etR•(x) co~ w r

u(x, r, w)

We shall now illustrate the Munk-Jones theory by applying it to a

dx

v(x, r, w) =

. dR'(x) cos w U cos a+ ~1l(x, r) - Ua d X

~"(x, r) + Ua [R:~x)

w(x, r, w) = Ua [

1+ R:~x)] sin

- l] cos

(9-11)

w

w

In particular, at the body surface, these become u(x, R, w) = U cos a

+ 'l'(:r:, R)

- 2UaR'(x) cos"' (9-12)

v(x, R, w) = UR'(x)

w(x, R, w) - 2Ua sin w

We recall (cf. Eq. 5-24) that q,~11 (:i:, R) is of order.,-• ln .,., for a slender body whose thickness ratio is .,., while the other perturbation term in u(x, R, w) is of order ar, which we shall assume to be the same order aa a• or .,.•. Clearly, v and w in Eq. 9-12 are of the first order, i.e. a or "· As before (Art. 5), it is advisable to carry certain second order terms in calculating the surface pressure coefficient, viz. Cp(x, R) = a• - 2

q,~ocu

R)

+ 4etR'(x) cos w -

[R'(:i:)]2 -

(9-9)

T

4a2

sin' w (9-13)

the work of Ward (48), at leut in the supersonic aa.se. For example, for symmetric flow problems Wa.rd is able I identify g(z) by compa>-ll>g .,(z; 11, •) with an appropriate solution of Eq. 9-4. Ward's work is discussed at some length in E,7 and [49).

T he cross-force loading at any station is found by integration of C,,(x, R)(pU•/2) cos"' around the body contour at that station; it is evident that the only contribution comes from the third term of Eq. 9-13. The result is (pU 2/2)2ir[R•(:i:)]'a, in agreement 'vith Munk's classical result [SS), which was calculated directly from the rate of change of cross-wind moment\llXl. This slender body theory of Munk and Jones baa been applied with

( 40 )

( 4I )

As stated above, the requirement that ¢1!' be antisymmetric with respect to the horizontal plane tells us that g(x) is at most a constant.

C ·SMALL PERTURBATION THEORY

C,10 ·ROTATIONAL FLOW

considerable success to highly sweptback wings, wing-body combinations, and the like, as discussed in Sec. D and in VII,A, B, and C. There is considerable experimental verification, as is mentioned there; in particular, regarding the pressure distribution on yawed bodies of revolution, good agreement was found by Ackeret, Degen, and Rott [27].

vorticity. For maximum generality it is easiest to begin with flow nearly at rest relative to our coordinate axes. Repeating the analysis of Art. 2, w·e arrive again at the approximate relations

C,10. Rotational Small Perturbation Flow. In our derivation (Art. 2) of the equation of sound propagation, we assumed explicitly that the entropy of the initial state was uniform. Nevertheless, tbe sound propagation equation (2-4) derived there and exploited in the succeeding articles requires no modification if the undisturbed state of the gas is one of rest involving small variations of entropy. T his might be the case, for example, in a room filled with air at rest but ·with small differences between densities at various points, due to variations of temperature. The last of Eq. 2-1 is still correct and the small perturbation approximations lead again to Eq. 2-2, 2-3, and 2-4, 'vhere now P~ and a~ denote average values for the nonuniform initial state. Our conclusion is that any small perturbation flow produced from a state of initi&l rest must be irrotational, to the first order, regardless of first order nonunifonnitie.s of the undisturbed gas. Obviously, by the same coordinate t ransformation as in Art. 3, the equation of sound propagation leads again to the equations of that article for unsteady and steady perturbations of a stream, even though the undistu rbed stream carries first order variations of density and entropy, provided its velocity is uniform. Now, although t he class of flows described in the preceding paragraph is of importance in acoustics and also includes interesting aeronautical cases of bodies moving through a stratified atmosphere, it does not include steady flows with uniform stagnation enthalpy and small variations of entropy and vorticity. T hese are especially important, since they are the steady flows downstream of shock waves. Although the changes of velocity, density, and the other properties of the gas through a stationary shock all depend upon the strength of the wave, the stagnation enthalpy is always conserved through the shock (cf. III,B). Consequently, a steady flow that is produced from a uniform stream and contains shock waves is generally one of uniform stagnation enthalpy (i.e. isoenergic) and nonuniform entropy and vorticity (III,A,11), at least outside of boundary layers and similar viscous regions. In particular, if the shock waves are weak or are nearly straight, the variations of entropy and vorticity will be small; this type of flow occurs frequently in practical applications (cf. (44]). Our derivation in Art. 2 fails for these cases because, relative to the acoustical frame of reference, the initial state is not one of uniform rest, but involves small rotational velocities. We are led, therefore, to consider flows having, initially, first order

< 42

)

'I• =

-

grad

(J~)

(10-1)

where p,., a,. are average values, say, for the initial flow, and p', q include both periurbations and small initial departures from uniformity. We cannot, in this case, conclude that q is the gradient of a function, but only that it has the form q (:i:, y, z, t) = grad q,(x, y, z, t)

+ q i(x, y, z)

(10-2)

'vhere q, denotes - f (p'/p,.)dt and q 1 (x, y, z} represents the initial, first order, velocity distribution, wruch is, in general, rotational. Eq. 10-2 states that the velocity at any time is m.a de up of an irrotational perturbation plus the initial velocity at the same point. Thus, it can be said that an initial small vorticity is not disturbed by acoustic perturbations, but remains fixed in the space. The differential equation for the perturbation potential follows inunedint.ely from Eq. 10-1: "' v

-

.

-1

a~

ef>u = - div q 1 p' =

(10-3)

-p,.q,,

The corresponding steady flow equations are (1 - M'!.,)r1.P -

+ q,,,,,,, + ¢,.,., = P~

= p' =

-(v 1),,,

-

(w 1).-

(10-4)

-p,.Uq,.,

The right-hand side of the first of these equations has the form assumed by - div q, (x, y, z) if the flo'v is steady with respect to tbe x', y', z', t' system of coordinates defined in Eq. 3-1, as is postulated here. q,,.,, Herc the velocity components at any point are U + u 1 (y', z') v1{11', z') + q,.,., and w1(y', z') + q,,.. The irrotational perturbations q,,.,, q,,,,, and q,,., vanish far upstream. J u this case we say that first order vorticity is transported by the main parallel stream. The PrandtlG laucrt affine transformation puts Eq. 10-4 into the form of Poisson's equation, familiar in potential theory. Its solution can be carried out by standard techniques. Equatitm3 flYf pressure perturbation. An equivalent method of handling these problems is to return to Eq. 10-1, differentiate the second of t.hese with respect to t, and substitute for q ,; t he result is

+

( 43 }

'

C ·SMALL PERTURBATION THEORY

..

..!._ 11. - V'p' a• "

C,10 ·ROTATIONAL FLO W

(10-.5)

= 0

Thus the variable part of the pressure (which includes the initial variations as well as time-J!.

I p)., e·-·· \Pi ••

{11-6)

wheres denotes the specific entropy. We also have the energy equation for steady isoenergic flow:

!2 (u' + v') + 'Y -'Y l pp =

h0 = const {11-7)

- !2 u• + 'Y -.., I ~ Pi It will be noted that at this point we have specifically assumed that the stream is one of uniform stagnation enthalpy. It should also be emphasized that the uniform reference conditions U, p,, p,, and 81 may be .fictitious; nevertheless, they are convenient reference conditions, and a.re completely defined, for given h 0 , by the equations above. Whel! Eq. 11-6 and 11-7 are combined and second order terms are neglected, the result is p p' u' 8 - 81 - .,, I+ - ... 1 - M, - (11-8) P1

Pl

/t,.

+ Y..v.

= -(1

+ (-y -

l )MU O(y) = CO(y)

(11-20)

and Eq. 11-5 as 'Ut

+ p'...! Us P•

~

>/I,

(11-21)

v, = -Y... An appropriate solution of Eq. 11-20, obtained from classical potent ial theory with the aid of the Prandtl-Glauert transformation as de•Thia a.impliJi•&tioD occurs pot only at 2: - 90° but also over a range of :& near 90° where U, ia of the S&llle order of magnitude as the perturbation velocities.

where the symbol Cl' denotes Cauchy's principal value. Crocco's theorem, Eq. 11-10, gives us the vorticity in terms of the entropy, 82(y), of which the variable pa.rt is tabulated in Eq. 11-19; we find (11-26)

where (u 1), is they derivative of the function u 1 immediately before the shock. Combining Eq. 11-23, 11-24, and 11-26 with the boundary values of Eq. 11-19, we find our problem solved, for Eq. 11-23, in the limit :i;-+ 0, has the form !-g(y)

=

const u 1

+ [integral involving (u ).J 1

Thus g(y) can be considered known in terms of the given upstream disturbance, and consequently u 2 and v2 can be ca.lcu.lated anywhere in the downstream region. In particular, all of the interesting quantities at the shock, including the shock angle f,.x)

(12-7)

+ v(.- + a,.(jit + M,.x) )

(12-9)

where.- denotes v'x• + {32 (y' + z'). These solutions have singularities of the type l/lf, i.e. they may be thought of as sources and sinks of variable strength. T he potentials for moving doublets are simply obtained by differentiation. Sinusoidal 08cillation8. Let us now assume that the motion to be described is that produced. by a body vibrating harmonically while moving through the gas at constant speed U. The potential must have the form (x, y, z, t) = e*•F(x, y, z) (12-10) According to Eq. 12-8, the function F must satisfy the equation - w•F

rr = a:O(rl>rr + rl>YT + rl>~z) (12-5) while the solutions given in Eq. 12-3 are immediately transformed to the new coordinate system by noting t hat

(12-8)

a:,V'

and the solutions Eq. 12-7 become

The differential equation, as has been mentioned, is invariant:

T

+ 2U .p,, + U••• =

or, writing G for

+ 2iwUF. + UF,. =

a:,v•F

(12-11)

Fe-wU•l•i.~'

tJ'G..

1 w2

+ G•• + G., + p"-i-. G = a~

O

{12-12)

Tws equation is obviously amenable to treatment by the familiar Prandtl-Glauert transformation, which reduces it to the form of the equation for harmonic sound vibrations in a gas at rest, viz., from Eq. 12-1 , (12-13) V'l/ + kf = 0, k = const It admits solutions of the form

.

G(x y ' z) =

..

const

e±c"'•1··~·

(12-14)

where.- is the same q1iantity introduced in Eq. 12-9. Tws solution could

( 59 }

C · SMALL PERTURBATION THEORY

C,13 · CI TED REFERENCES

also have been deduced immedia.tely by comparison of Eq. 12-9 and Eq. 12-10. Plane flow. The plane flow counterpart of Eq. 12-1 may be written in the form 1 a.

(12-15)

. . 4>11 - = - •• = 0

An important elementary solution is .._ = .,,

v

Const for a!,t 2 a!,t• - x• - 11•

~o

> x' + y•

for a!,t•

(12-16)

< zt + y•

This represents the potential due to an impulse which occurs at the origin at t = 0 and whose effects spread out in a widening circle a.bout the origin. A distribution of these singularities over an interval of the time variable can be used to represent the effects of time-dependent small disturbances in a fluid otherwise at rest, i.e.

to the iotmll (12-3) in three dimensiona and .P(x, t) - f(x - a.,t)

+ g(x + a.,t)

(12-21)

which is the well-known solution of the one-dimensional equation .Pu = a!.4»~ (12-22) In general, the two-dimensional solutions do not have this form. Th.e result is that two-dimensional sound waves do not preserve their shape, and the wave produced by a square impulse propagates with a "tail" of gradually diminishing strength. Unsteady flow problems in both subsonic and supersonic flow are treated further in Sec. D of this volume, where solutions are constructed by use of the elementary solutions derived here and others. Practical problems of airfoils in unsteady motion are treated in more detail in VII,F. C,13.

Cited R efuences.

where 0 and D are complex constants. In this limiting form, we see for the first time a solution of t he two-dimensional equation ( 12-15) analogous

1. Rayleigh, Lord. TM Theory f Sound (1894). Dover, 1945. 2. Ackeret, J . Uber Luftkrifte bei Sehr grossen Geschwindigkeiten insbesondere bei ebenen Str6mungen. Relu. Phys. Acta 1, 301-322 (1928). 3. Glauert, H. The effect of compressibilit y on the lift of an a.erofoil. Pre. &Ji. Soc• .A.118, 113-119 (1928). 4. Aekeret, J. Luftkrtlfte auf Flllgel, die mit gr6™lrer als Sehlillgeeehwindigkeit bewegt werden, Z. Flugtech. u. Mtrluf1$cloiff11hrt 16, 72- 74 (1925). Tran.slated in NACA Technfool M•morandum SI7. 1925. · 5. von K&rmt\n, Th. Uber lltationlire Wellen in Gaastrahleo. Phys. Zeita. 8, 209-211 (1907). 6. Gothert, B. Ebeno und r¨iche Stromung bci hoben Unterschallgesohwindigkeiten. Jali.rb. du114, Por•cliungabuidit 1869. 1942. Translated in NACA Technical Memorandum.1115. 1947. 25. Herriot, J. G. BlockAge ocrreotJons for threo-dimensional-J!ow closed-throat wind tunnels, with conaider&tion ol the effect oI compresaibillty. NACA Reaearci> ilfemorandum A7Bt8. 194.7. 26. Van Orieet, E. R. Die lincarisiert.e Theoric der drcidimensionalen kompressiblen UnterschallatrOmung und die experimentelle Untenuchung 'on RotatioWlkorpcrn in eincm g011