Mx-w-).R’.r No. 876

CONTENTS

suMM.4RY_______________________________________________~N~ODrC~OX -------------------------------------------sYmoM ------------------------------------------------I. PEEU~IIXART eONsIDEE.iTIOXS--------------------------2. 13ALCULATIOXOF THE LIMITS OF STABILITY OF L.WIX.AR

BOUNDARYLAYER IFi AVISCOIJSf20XDCCTIVEGas. _______3. DESTABILIZINGIXFLUEXCEOF VISCOSITY AT T’ERY LARGE

REYNOLDS XUMBERS; EXTENSION OF HEISEXBERG’SCRITERIONTO THE COMPREssIB= FLUID_______________

4. STABILITY OF LXJIXAR BOUXDARYLAYER A-r J.ARGER~Y-NoLrIs NuMB~Rs----------------------------------------

A. Subsonic Free-Stream Veloeity (M,< I)____ . ..-. _.B. Supersonic Free-Stream l’elocity(.llo>l) _________

5. CRm~RIOX- FORTm311NIMU.V CRITICALREYXOLDSNTCTMBER_& PHYSICALSIGAWICANCEOFRESCLTSOFSTABILITY.kALYSIS_

A. General---------------------------------------B. EtTeet of Free-Stream Mach Number and Thermal

Conditions at Sofid Surface on Stability ofLami-nar Boundary Layer --------------------------

C. Results of DetaiIed Stability Calculations for lJXW-Iated and hioninsulated Smfaees----------------

D. Instability of Laminar Boundary Layer and Tram~i-ticmto Turbulent Flow ________________________

7. STABKJTY OF LAMIXAR BOEXD~RY-IiATERFLOW OF .L GASWITH A PRESSUREGRADI~NTIN THE D~RZCTIOXOF rrmFRWZ,STRZ.M_________________ _______________________

33133~332

332

335

343

3453453+6348350350

350

352

353

35-I

CONCLUSIONS-------------------------------------------

APPEXDIX .4.-CALCEL-knOX OFIXTEGR.4LSAPPEARIXGm THEkiTSCID SOLUTIONS--------------------------------------

General PIanof Calculation ---------------------------DetMed CaIctiatiom ---------------------------------Evaluation of$k~nJ-----------------------------------Order of Magnitude of Imaginary parts of Integrals Hs,

M2, and ArS----------------------------------------

APPEXDIX B.—CALCCLATIOX OF l1E.W-I’EL0CIT% AXD MEAWTEMPERAT~EEDISTRIBUTrOXACROSSBOUXDARYLAYER KNDTHE VELOCITY AND TEMPEE*TVRE DERIYATITES AT THESOMDSURFACE________________________________________

31ean VeIocity-Tem~rature Distribution Across Bound-a~ Layer _________________________________________

Calculation of Mean-VeIocity a~d Mean-TemperatureDerivati\-m------------------------------------------

APPEXDIX C.—RAPID APPROXIM~~ION TCJ THE FmCmON(1—2A)u(c) &im T= 311NIMUMCRITICAL REYNOLDS N-UM-BER__________________________________________________

APPEXDIX D.-BEEAYIOROF &()

P ‘: FROM EQUATIONS OF

MEAX 310nox_ ____________________________________–_

.&FP~XDIX E-—CALWIATION OF CRITICAL S1.~CHKUXBm FOR

STABILIZATION OF LAMIXARBOUXD.~RT L.+YER -------------

RFiFEREXCES ----------------------------- _______________

329

Page

355

357357358366

367

368

368

36-9

37I

372

373

574

REPORT NO. 876

THE STABIWI’Y OF THE LAMINAR BOUNDARY’ LAYER ~’ A COMPRESSIBLE FLUID

BF LESTERLEES

SU3131ARY

The present paper is a continuation of a i%eoretical inresti-gation Oj-the stability of the laminar boundary layer in a com-pressible $uid. An approrima~e estimate for the minimumcritical Reynolds number ReCT~~x,or stability limit, is obtained

in terms of the distribution of the kinematic ri.scody and the—

product of the mean density ~ and mean cortieity ‘A* acrow thedy”

boundary layer. With the help of this estimate jor l?o.,~ti

it is shown that withdrawn”ng heat -from the f?uid through thesolid surface increases Rff and stabilizes the $owr as

‘mix

compared with the $OW ocm an ins-dated surface at the sameMach number. Conduction of heat to the jluid through the

solid surface has exactly the opposite e~ect. The ralue of R$w~iz

for the insulated surface decreases as the Mach number in-creases=for the case of a uniform free+trea m re[ocity. Thegegeneral concksion~ are wpplemented by de~ailed calculationsoj the curres of ware number (inrerse ware kngth) againstReynolds number for the neutral disturbance.s for 10 representa-tive cases of insulated and noninsulated surfaces.

So far as laminar stability is concerned, an important difer-ence e.risf~ between the case of a subsonic and supersonic free-strearn reloeity outfi”de the boundary layer. The neutralboundary-layer disturbances that are signi$cant for laminar sta-bility die out exponentially with dis(ance from the solid surface;therefore, the phase relocii!y c* of these disturbances iz subsonicrelatire to the free-s~ream ce~ocity ~—or UO<—6*< p,

7u~where ~ is the local sonic relocity. Then ~=.llo<l

(where 310 is jree-stream Mach number), it ~~llows that()~G*~~* ~aq and any larninar boundary-layer$ow is ultimatelyunstable at s-uj%kit fly high Reynolds numbers because of thedestabilizing action of uiscosity near the solid surface, asexplained by Prandtlfor the incompreseible$uid. When MO> 1,

is large enough negat ire$y, the rate at which energy passes fromthe dis~urbance to the mean $OW, which is proportional to

-C*[+4%)1.=C*, can always be large enough to coun-

terbalance the rate at which energy passes from the mean $OWto the disturbance because of the destabilizing action of viscositynear the solid surface. In that case only damped disturbancesexist and the laminar boundary layer is completely stable a~ all

Reynolds numbers. This condifiion occurs when the rate atwhich heat is withdrawn from the$uid through ~hesolid surfacereaches or exceeds a critical ralue that depends only on theMach number and the propertiie~ of the gas. Calculations.shou~that for MO>3 (approx.) the Zaminar boundary-layer$owfor thermal equilibrium-where the heat conduction through thesolid surface balances the heat radiated from the surface—i~cornp[e~e[y stable at all Reynolds numbers under free-j’ightconditions if the free-strea m relocity is uniform.

The results of the analysis of the stability of the [aminarboundary layer must be applied with care to discussions oftransition; howewr, withdrawing hea~ from the jluid throughthe solid surface, for examp[e, not only increases R8C,~~8but

also decreases i’heinitial rate of amphjication of the se~-ezcited

disturbances, which is roughly proportional to 1 /3~=.

7’hu.Y,the e~ect of the thermal condifiions at the solid swja.ce on

the transition Reynolds number R6,, is similar to the e$ect on

Ren~f.. A comparison betzceen ~his conclusion and experi-

mental in res+igations of the e~ect of surface heating on &ansi-tion at low speeds shows that the results of the present papergire the proper direction of this e~ect.

The extension of the results of the stability analysis to larninarboundary-layer gas $OWS with a pressure gradient in thedirection of the free stream is discussed.

INTRODUCTION

By the Wmreticd studies of 13eiseuberg, Tolbnien,S4dicb.ting, and Lin (references 1 to 5) and the carefulexperimental investigations of Liepmann (reference 6) andH. L. Dryden and his associates (reference 7], it has beendeiiniteIy established that. the flow- in the laminar bounc{arylayer of a viscous homogeneous incompressible fluid is un-stabIe above a certain characteristic critical Reynoldsnumber. When the level of the disturbances in the freestream is low, as in most cases of teckicaI interest, thisinherent instability of the laminar motion at sticientlyhigh Reynolds numbers is responsible for the ultimatetransition to turbulent flow in the boundary layer. Thesteady laminar bouuclary-Iayer flow would always representa possible soIution of the steady equations of motion, butthis steady flow is in a state of unstable dynamic equilibriumabove the critical Reynolds number. Self-excited dis-turbances (Tolkien waves) appear in the flow, and thesedisturba~ces grow large enough eventuality to destroy thelaminar motion.

331

332 REPORT hTO. 876—NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS

‘Ile question naturally arises w to how the phenomenaof kninar instability and transition to turbulent flow aremodified when the fluicl velocities and temperature varia-tions in the boundary Iayer are large enough so that thecompressibility and conductivity of the fluid can no longerbe neglected. The present paper represents the secondpb ase of a theoretical investigation of the stability of thelaminar boumkry-layer flow of a gas, in which the com-pressibility and heat conductivity of the gas as welI as itsviscosity are taken into account. The first pm% of this workwas presented in reference 8, The objects of this investiga-tion are (1) to cletermine how the stabiIity of the laminarboundary layer is affected by the free-stream Illach numberand the thermaI conditions at the solicl boundary and (2) toobtain a better understanding of the physical basis for theinstability of laminar gas flows. In this sense, the presentstudy is an extension of Lhe Tollmien-ScMichting analysis ofthe stability of the laminar flow of an incompressible fluid,but the investigation is also concerned with the generalquestion of bounclary-layer disturbances in a compressiblefluicl and their possible interactions with the main externalflow.

With minorthe same asquantities arecorresponding

SYMBOLS

exceptions the symbok used in this paper arethose introduced in reference 8. Physicaldenoted by a~ asterisk, or star; whereas thenoncIimensional quantities are unstarred.

A bar over a quantity denotes mean value; a prime denotesa fluctuation; the subscript O clenotes free-stream valuesat- the “edge” of the boundary layer; the subscript 1 denotesvalues at the solid surface; and the subscript c denotesvalues at Lhe inner “critical layer,” where the phase velociiyof the disturbance equals the mean ffow velocity, Thefree-shream values are the characteristic measures for allnondimensional quantities. The ch a.racteristic le~gtb meas-ure is the bounds.ry-Iayer thickness 3, except where otherwiseindicated, Note that in order to conform with standardnotation. the symbol 6 for boundary-layer thickness isunstarred; whereas the symbols ~* and 0 are used forboundary-layer displacement thickness and boundary-layermomentum thickness, respectively.** distance along surface

Y* distance normal to surfacet* timeU* component of velocity in x*–direction

~~=~

u~*V* component- of veIocity in y*–direction

*f~=$

+* stream function for mean flowP* density of gas

P* pressure of gasT“ temperature of gas~+ laminar shear stress

ordinary coefficient of viscosity of gaskinematic viscosity of gas (M*/P*)thermal conductivity of gasspecific heat at constant volumespecific heat- at constant-pressuregas constant per gramratio of specific heats (co/c,); 1.405 for aircomplex phase velocity of b oun clary-layer

disturbancewave Iength of boundary-Itiyer clisturbancebouncktry-layer thicknessboundary-layer displacemen~ thiclincss

/ P. \(j ( l—pw)dy*

o )

boudary-layer momentum thickness

(J

.pw(l-w)dy*

o )

~,al-e number of bouncIa.ry-Iayer clisturbfi.nce(27r/A*)

()——pfi*uo*C!

Reynolds number -==-Plo

-. —po”uo*e

R@==PIO*

M.

(-)

Mach number -==~~+A’ * TO*

G- ()7

I?randtl number c, !%~o*

1. PRELIRIINARY CONS1DERATIONS

In the first phase of this investigation (reference 8) Lhcstabdity of the laminar boundary-layer fio~v of a gas isanalyzed by the method’ of small perturbations, whit% wzsalready so successfully utilized for the study of the stubilit,yof the laminar flow of an incompressible Quid. (S-w refer-ence 5.) By this method z nonsteady gas flow is investigatedin which all physical quantities cliffer from their -dues in agiven sieady gas flow by smaII pert.llrbations that are func-tions of the time and space r.oorclinates. This nonslmclyflow rnusi satisfy the complete gas-dynamic equations of mo-tion and the same bounclary conclit.ions as the giwn skadyflow. The question is whether ~he noustcady flow dmnps tothe steady flow, oscillates about it, or diverges from it withtime—that. is, whether the smaII perturbations are damped,neutral, or self-excited disturbances in time, tincl thus whetherthe given steady gas flow is stabIe or unstatk. The analysisis particularly concerned with the conditions for the oxis[enceof neutral clisturbances$ which mark the transition from staMeto unstable flow and define the minimum critical RcynoIdsnumber.

THE STABILITY OF THE LAMINAR BOUKOARY LAYER IN A COMPRESSIBLE FLUID 333

In order to bring out some of the principal features of thestability problem without becoming in-rolved in hopelessmathematical complications, the solid boundary is taken astwo-dimensional and of negligible curvature a~d the boundary-layer flow is regarded as pIane and essentially para~el;that is, the velocity component in the direction normal tothe surface is negligible and the velocity component parallelto the surface is a function mainly of the distance normalto the surface. The smaIl disturbances, which are also tn-o-dirnensional, are analyzed into Fourier components, or normalmodes, periodic in the direction of the free stream; and theampIitude of each one of these partiaI oscillations is a functio~of the distance normal to the solid surface, that is}

ZL*r=@f (V)eiti(z-et)

In the study of the stability of the Iaminar boundary layer,it \\ill be seen that only the local properties of the “pandlel”flow are significant. To include the ~ariatiort of the meanvelocity in the direction of the free stream or the velocitycomponent normal to the solid bounda~ in the probIemwouId lead only to higher order terms in the dii7erentiaIequations go-reining the disturbances, since both of thesefactors are inversely proportional to the local Reynolds num-ber based on the boundary-layer thickness. (See, for ex-ampIe, reference 2.) By a carefuI analysis, Pretsch hasshown that wen -with a pressure ~aclie~t in the direction ofthe free stream the local mean-~ elocity distribution alonedetermines the stability characteristics of the IocaI boundary-Iayer flow at large ReynoMs numbers (reference 9j. Sucha statement appIies only to the stability of the flow withinthe boundary layer. For the interactio~ between the bound-ary Iayer and a main “e~ternal” supersonic flow, for example,it is obviously the variation in bounclary-]ayer thickness andmean velocity along the surface that is significant. (Seereference 10.)

The aforementioned considerations also Iead quite natu-rally to the study of incli-ridual partial oscillations of theform -f(y) ei~(’-”j, for mhich the differential equations ofdisturbance do not contain r and t e.splicitly. These partiaIosc~ations are idea~y suited for the study of i.RstabiIit-y, for

in order to show that a flow is unstable it is tmnecessary toconsider the most general possible disturbance; in fact, thesimplest will sufiice. IL is only necessary to show that aparticular disturbance satisffig the equations of motion andthe boundary conditions is self-excited or, in this case, that

‘ the imaginary part of the complex phase ~elocity c is positive.In reference S the differential equations governing one

normai mode of the disturbances in the Iaminar bounda~layer of a gas -were deri~ed and studied -rerj thoroughly.The complete set. of sohtions of the disturbance equationswas obtained and the ph~sical boundary concLitions that.these soIutions satisfy were in~estigated. lt n-as found that.the final relation between the values of c, a, and R thatdetermines the possible neutral disturbances (limits ofstability) is of the same form in the compressible fluid ZLSinthe incompressible fluid, to a fist approximation. The basis

for this result is the fact that for Reynolds numbers of theorder of those encountered in most. aerodynamic problemsthe temperature disturbances have only a negligible effecton those particular veIocity solutions of the disturbanceequations that depend primarily on the viscosity (viscoussoIutions). To a first approximation, these -riscous solutionstherefore do not depend directly on the heal conductivityand are of the same form as in the incompressible fluid,~xcept that they in-roIve the Re-ynolds number based on thekinematic tiscosity near the soIid boundary (-where the vis-cous forces are import ant.) rather than in the free stream.In this fist approximation, the seconcI viscosity coefficient,-which is a measure of the dependence of the pressure on therate of change of density, does not affect the stability of theIaminar bouncl~ry layer. From these results it was inferredthat aL Iarge Reynolds numbers the infiuence of the viscousforces on the stability is essentiaHy the same as in an incom-pressible fluid. This inference is borne out by the resultsof the present paper,

The influence of the inertial forces on the stability of theIaminar boundary la~er is reflected in the behavior of theasymptotic intiscid solutions of the disturbance equations,which are independent of Reynolds number in first approm-- ‘-mation. The restits obtained in reference S show that thebehavior of the inertial forces is dominated by the dishibu-tion of the produci of the mean density and mean -rorticity

P~—~across the boundary Iayer. (The gradient of this quan-

d dw

()tity, or ~Y PG , which pIays the same roIe as the gradient

of the -vorticit.y in the case of an incompressible fluid, is ameasure of the rate at. which the x-momentum of the &nlayer of fluid near the critical Iayer (where w= c) increases,or decreases, because of the transport of mome~tum by thedisturbance.) In order to clarify the behavior of the inertiaIforces, the Iimi&ing case of an inviscid fluid (R+ ~ ) is studiedin detail in reference S- The following general criterions are

d dw

()P— ranishes forsome~alueobtained: (1) If the quantity ~Y dv

of w> I —+, then neutral and self-excited subsotic disturb-~o

anees e-xist, and the inviscid compressible flow is unstable.

()(2] If the quantity $ P% does not vanish for some value

1of w> 1—~ then W subsonic disturbances of finite -wave

length are damped and the inviscid compressible flow isstable. (Outs;de the boundary .Jayer, the relat i-ve velocitybetween the mean flow and the r-component. of the phasevelocity of a subsonic disturbance is Iess than the mean sonicvelocity. The magnitude of such a disturbance dies out ex-

—.c

ponentiaI1-y with distance from the solid surface.) (3) IngeneraI, a disturbance gains energy from the mean flow

d dw

()if~qj is positive at the critical layer (where w~=c) and

[%31W=C<”-loses energy to the mean flo~ if ~Y

,

334 REPORT NO--- 876—N.4TIONAL ADVISORY COMMITTEE FOR AERONAUTICS

The generaI stability criterions for inviscid compressibleflow give some insight into tlte effect of the inertial forces onthe stability, but they cannot be taken over bodiIy to thereal compressible fluid. Of course, if a flow is unstable inthe limiting case of an infinite Reynolds number, the flow isunstuble for a certain finite range of Reynolds number. Acompressible flow that is stable when R+ co~however, is notnecessarily stable at alI finite ReynoIds numbers when theeffect of viscosity is taken into account. One of the objectsof the present paper is to setlIe this question.

On the basis of the stability criterions obtained in reference 8,some general statements were made concerning the eflect ofthermaI conditions at the solid boundary on the stability oflaminar boundary-layer flow. It is concluded from physicalreasoning ancl a study of thG equations of mean motion

d dw

()that the quantity ~ ‘d~ vanishes for some value of w> O

if bT()— ~ O, that k, if heat is added to the fluid through the.l!i , –

(7solid surface or if the surface is imndated. If k >0by ,

and is sufficiently large, that is, if heat is withdrawn from thefluid through the solid surface at a sufficient rate, the quan-

d dW

() (?

atity ~ p~ never vanishes. Thus, when — s O, thedy ,–

laminar boundary-layer fiow is destabilized by the action

of the inertiaI forces but stabilized through the increase of

kinematic viscosity near the solid surface. When(,7

~ ,>0,

the reverse is true. The question of which of these effects ispredominant can be answered only by further study of thestability problem in a real compressible fluid,

ln the present paper this investigation is continued alongthe following lines:

(1) A study is made of how the general criterions forinstability in an inviscid compressible fluid are modified bythe introduction of a small viscosity (stability at very largeReynolds numbers).

(2) The conditions for the existence of neutraI disturb-ances at large Reynolds numbers are examined (study ofasymptotic form of relation between eigen-values of c, a,ant] R).

(3) A relatively simple expression for the approximatevalue of the minimum critical ReynoIds number is derived;this expression involves the local distribution of meanvelocity ancl mean temperature across the bounclary layer.This approximation will serve as a criterion from which theeffect of the free-stream Mach number and thermaI con-ditions at the solid surface on the st.aMity of kminarbounc~ary-layer now is readily evaIuated. The question of

the relative influence of the kinematic viscosity and the

distribution of ~~~ on stability WOUMthen be settled.

(4) The energy balance for smalI disturbances in the redcompressible fluid is considered in an attempt to ckmify the.physicaI basis for the instability of laminar gas flows.

(5) In order to supplement the investigations outlinedin the four preceding paragraphs, detailed calculations memade of the limits of st.abiIit.y, or the curve of a against 1?for the neutral disturbances for several representative casesof insulated and noninsulated surfaces. The resuIts of thecalculations are presented in figures 1 to 8 and tabhx Ito IV. The method of computnkion of the stabiIity limits isbriefly outlined in reference 8, although the calcuImtionswere not carried out in that paper.

/.o —

.9 /

.8

.7---

.7 - f. l--

/ _ _ _ _ _ _ _ _.6

M! 676

.5.502777

.90 31851./0 3.476

w

/[/

1.30 3.B25

.4

.3

L — — —

..2

./

o / 2 3 4 5 @ 7

rv% JQ-VO*x =

FIGURE1.—Boundary-k+yer vcIocity profiles for insulated surface, Since u is taktu cquaI to

[ ( - )] Z+% W?,unity, the temperature profile is given by T= TI— Ii— l+ZZ: Mz

THE STABILITY OF THE L&ML\’AR BOUNDARY l14YER IN A COMPRESSIBLE FLUID 335

Lo

.9

1

.8 / // /

/!

.7 -0.;-” -. I

i Ji)f ~.8 -.. It [-g. . ---,,-Biasim;~ =/;.4{~=O ,

Wit? /[.2 - .. ..

.6-

W

.5 1(/ / ‘ .0.?0 – %6

M I.60 2.177.S’0 2.435

I “[.25—3.343”

.4 h //

/’vi

.3 I’ll/

11[/

.2 i!

— — — — — — — — — —

/o [ 2 3 4 5 6 7.

FIGt%E Z.-Boundary-layer reloeity profdes for noktdsted surfwe. X&= O.iO. TI is theratio of surface temperature (deg abs.) to free-stremn temperature (deg abs.). InEectionmore pronounced and farther out into fkdd for T1=125 then for insufaced surface ( 7’1=1,10).k~oM@ction for T,=0.70,0.82,0.90.

IrI the present investigation the work of Heisenberg(reference 1) and Lin (reference 5) on the stability of a realincompressible fluid is naturalJy an indispensable guide. IDfact, the methods utilized in the present study are analogousto those developed for an incompressible fluid.

The present paper is concerned only with the subsonicdisturbances. The ampIitude of the subsonic disturbancedies out rapidly -with distance from the soLid bouncIar-y. Inother worcls, the neutral subsonic disturbance is an “ eigen-oscillation” con.tlned mainI-y to the boundary layer and existsonIy for discrete eigen-dues of c, a, ancl R that determinethe limits of stabdity of Iaminar boundar-y-Iayer flow. Dis-turbances classified in reference 8 as neutral “supersonic,”that is, d~twrbances such that the reIative velocity between

the x-component of the phase velocity of such a clisturbmweand the free-stream ~elocity is greater than the local meansound speed in the free stream, are actuaIly progressive sound-waves that impinge obliqueIy on the bounclary layer andare refIected wi&h change of amplitude. For disturbancesof this type the -wave length a~d phase velocity are obviouslycompIet ely arbitrary (eigen-values are continuous), and thesedisturbances have no significance for boundary-layerstability.

lThen the free-stream velocity is supersonic (.310> 1),thesubsonic boundary-Iayer disturbances must satisfy the re-

quirement that ~–c*<~ or c>l –&O (for Mo<l, cZO).

N’ow-, by anaIogy with the case of an incompressible fluid itis to be mpected that for values of c greater than somecriticaI ~alue of cO,say, alI subsonic disturbances are damped.Thus, when MO> 1, there is the possibility that, for certainmean veIocity-temperature distributions across the boundarylayer, neutraI or seIf-excited disturbances satisfying thedifferential equations of motion, the boundary conditions,

and ako the physical requirement that c> 1—-& cannot be

found. In thak e-rent, the Iaminar boundary flow is stabIeat all Re+ynoIds numbers. This interesting possibfity isinvestigated iu the present paper.

2. CALCULATION OF THE LIMITS OF STABILITY OF LA311NARBOUNDARY LAYER 1X A f’ISCOUS CONDUCTIYE GAS

In order that the complete system of solutions of the differ-ential equations for the propagation of small disturbancesin the laminar boundary layer shall satisfy the physicalboundary concbtions, the phase velocity must depend on thewave Iength, the Reynolds number, and the Mach numberin a manner that is determined entirely by the local distribu-tion o{ mean -relocity and mean temperature across theboundary Iayer. In other words, the only possible subsonicdisturbances in the laminar boundary layer are those forwhich there exists a deiinite relation of the form (reference 8)

c= c(a, R, M02) ._(l)

Since a, R, and .ll~’ are real quantities, the relation ~1) is

equivalent to the two relations

c,= c,fa, R, .V;2) (la)

ci=ci(a, R, .lfoz) (lb)

The cur~e c,(cr, R, MO?=0 (or CK=a(R, M;)) for the neutraIdisturbances gkes the limits of stability of the laminarboundary layer at a given ndue of the Mach number. Fromthis curve can be determined the -ralue of the Reynolds

336 REPORT NO. 876—NATIONAL ADVISORI” COMMITTEE FOR AERONAUTICS

v (c)

.7

.6

Mo=o .5.7.9 /./ 1.3I

/// // ‘/.5

.4 — — — — — — — –

/

.3 / // i H

/

.2

///

,

,1 / /

/ /’A A /

/ “

//

-i (a)

o ./ .2 .3 .4 .5 .6c

(a) Insulated surface.

‘(c)=-~~ [% $ (~)]u~c'whicha~~ears directl~ins&btiity ~~lculation. Thevalue of c=co at which (1-2,) u=O.MO(k is small) is a measure of the stability of a giwnlaminar bouudary.layer flow.

number ?.M1OWwhich disturbances of all wave lengths aredamped and above which self-excited disturbances of certainwa~e lengths appear in a given laminar boundary-layer fiowT.

In reference 8, it is shown that the relation (1) between thephase velocity and the wave length takes the folIowing form:

E(a, c, M?) =F(z) (2)

k equation (2), F’(z) is the Tietjens function (reference 1I)defined by the. rektion

()

aRwC’ 113z= — (?/.-J/JVc (4)

and the quantity HI ,Y(1Jis the ~anke] function of the fkt kind

of order j{. The prime denotes differentiation with respecttoy, The function E(a, c, .M$), which depends only on the

.6

- “1-v”<1,25 :-Jb50-. #.5 , t f / --

.4 i

.3 t I

_ _ _

.2

l-f(c)

./

0 I

\/

\ \.—— .

-. I —

-.2 — —

(b)-.30:: .,

.2 .3 .4 ..5 .6c

(b) Noniusulated surface. .MI=O.70.

FIGUIrE3.—&mdU~ed,

asymptotic inviscid solutions PIand p2 (section 4 of refercnce18)and not on the Reynolds number, is defined as folIows:

Pij=dYJ 1 (6)

~,j=l,2

and VI and Vi are the coordinates of the solid surface and the“edge” of the boundary layer, respectively.

The Tietjens function was carefully recalculated in refer-ence 8, and the reaI and imaginary parts of the function

@(+=1_&2)are plotted in figure 9, (The functio~ @(z) is

found to be more suitable than F(z) for the ac~ual calcula-tion of the stability limits.)

THE STABLLITY OF TEE LAMN.4R BOUNDARY IL4YER IN A COWPRESSIBLE FLUID 337

Om ~~ ~ 800 10JQ 1200 [400 k5D0 DUORe

(@Insulated surfwe. M=O (RIasiwsflow).(b) Insulated mrfm?. Mo=O.W.w) InmMed >mrfwx. 3f#=0 .X3.

FIGmE +.—Wat-enumtw’ a ogoid Reynokds number I?r for neutral stability of Iamimubmmchry-layer flow.

,0 200 +09 ma 6W2 1~ 12C0 [4~ 16C0 18~.Re

(d) Insulated swfwe. M=O.W(e) InmMed mrfsce. M= 1.10.(f) Inssted surhce. .M=l.3%

FIG~E l.-C!ontinnd.

338 RETORT NO. 876—NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS

Cl&

Cfo

o 5 [0 15 20 25 30 35x/03R@

(g) NOfinsul~ted surkce. JM=o.70; T1=o.70; (~),=o >0.

(h) Noninsulated surface. MO=O.70;T’,=O.SO;(%),=0 ‘o

Jo

.08

.06

ffe

.04

.02

0 509 [000 [500 2#Q 2500 5000 3500R@

(i) ifoninsulated surface. MQ=0.70; T, =0.W; (g),=o >0.

FIGCRE4.—Continued.

.20

./6

.[2

&@

.08

.04

u 200 400 600 80D [000 /.200 14Q0 /6cZ7 1800R,

(i)Noninsulated swface. Mo=O.70;TI=1,25;(%),=0 ’04

FIGURE.4+Continued. I FIGURE4.–Contiuued,

THE ST.4BILITT OF THE IAM2WR BOUNDARY LAYER IN A COMPRESSIBLE FLUID 339

MO=0{./5

I 1 I Illt III I I I II I I I ~~,ly ,1 I I I 1111 Ill I I I I II I I II 1/1 I II I I I I Ill I I I I I

,. .r ‘... “. ::

,,. . .“

!“ ‘.. .. . .~.-., ... “. . .,,: “’. ‘“.,. .“ ,,,,,.,,, ,4”Y, ““’.,,””,, ‘,, “. : “ . . ,,

‘.,~,-” . . , ‘,, ., .

!,.

I ./,1,

:,.-.4.,,. i.. ,.

-.,:, . .. ~“’:,, ., I

;1

., :.., r~ ,.., . . .. . . . . . . . . . .

:. . . :... . . . .

,,. . . ::

. .

,. ~“, :.“.

., ... ,

I ,.... :..,

I ,..

‘,: *. ,.

.

I

.,

,,, .,, ,,, ,,,,:. ’... . . . -.

—-,, .../0 /02”” [03””” /@

(k) Insulated surbce. ~f&#o. NIJWbat Q+z#c as R~m for insulated surhw.

(1) ~onimtited surf-. 31,=0.70. Noti that tia.#0 as R-@ when T,=l.M.

FiGC_EK4.—ConeIuded.

340 REPORT NO. 876—NATIONAL ADVISORY COhI~LiTTEE FOR AEROh’AUTICS

60,000 200[-.

\\

\\,\;\

50,000 160 ‘\ \\

:~R* \\

\\ \ .- -Esfimafedfrom

‘<”\ equaTion (71)40,000 /20

R=:} \c \.M.: \

E f ,’ \

.Hb \ \R 4°

30,000 80kldculde d fro m; ?igure 4

%

20,000 40 N

/0, 000 0.4 .8 [.2 i.6

M*

FIGURE5.—Variation of minimum critical Reynolds number with Mach rmmber for le.mirmr

boundary-layer flow. InsuIated surface.

“’(%);RZ*=(%)=225RR’

FFFl\\ ‘iI

~ ColcJJofeb [flq.4),

9000 600~ ~ ~o.7.9 ,

“--- Estimated equafio; [7//);\ J\ /vcc TO’/

—- Es~m~t;d (equation (71)),-

7000 \ t\

400 ‘\

I f!t

I

6002 \ 300 \

,1 h+

\b5000 ; 200

sf I \.

d’ \ ,>

4000

\ i:

/00>+. .

\ ----<

3000\ \, o

\\ \ I I

\

\2000

\ ‘

//000 \ \

.

0 (a)P << --- _

I -- -----— 1.7 .8 .9 Lo /.\ 1.2 /.3

T11

(a) R6,rminagainst Ti.

FIGrRE 6.—Dependence of mti]mum crifical Re~molds number on thermaI conditions atsolid surface. .Mo=O ,70.

11

(b) R:*ermin against TI crdculated from values of R#Cr=t~ tsken from flgl~re 4.

FIGrRE i3.-Concludcd.

9000”-

woo r

--- -h’f~ .a7

cc7000 I

1 — — . — .

[6000 — —+ — — — — — — ‘“ — — — “

II .-M~ =13— — — — —/-,,..

5000 f~E

~iQ T

4000 .—. .I

I

3000 — — –

M0 =.i5,.-

2000 -qQ =0-- “’ I- ..

coI

,*MO =20/000 — —

1: *~ . .

k“Cp

1.Mo=3’0 \,,.tio=kO

,/’

o I 2 3 4 5 611

FIGURE7.—Stabilizingeffect ef supersonicMach numbtrs of wlthdmrvrd of heat from fluid

through solid surface. At each value of .W>i, there is a critical value Of TI- Ti ti ~ueh

that for T, STI _ the Iamkar boundary-km’ flow is stiible at sII vrdursO( the RwnoIds

number. (Curves for 310=0 and 3fn=0.70 ~Clud~d for COmParfsd ~~ti=fm=thrr~~~d

from equation (71). pmfi ff=l.

THE STABILITY OF THE IL4XINAR BOUNDARY LAYER IX A COMPRESSIBLE FLUID

o I 2 3 4 5

Ma

FIGrEE 8.—G_itical temperarrue ratio T~mfor stsbfi-ty of laminw boundwy Iayer against

Mach rmmber.%

Z. 6

24

22

20

[.8

/.6

/.4

‘#+) *

/.2

/.0

8

.20 2?4 28 3.2 36 4.o .&f 4.8z

FIGUBE9,—The furmioms*,(z) ad $, (z),

ss302&5G23

The inviscid solutions p, and p, were obtainedseries in a2 as follows (section 6 of reference 8):

3’41

M power

(7]

(8)

andho=l.O

and for na 1

1s

Y (~—c)~M: dy y12.+ (y;c,Mo2)dy (10)

VI

The lower limit in the integrals is taken at the surfacemerely for convenience. ‘PThen y>yc, the path of integra-tion must be taken below the point Y=Y. in the compIexy-pIane. The power series in az is theu uniformly conver-gent for any finite value of a.

At the surface, the in-riscid soIutions are readiIy evaluated

ql’ = –: (T,–M02C’)J

At the “edge” of the boundary layer, the inviscid solutions

are most conveniently expressed as follows:

p.,z= (1 –c) >0 CY2’H,.[C,M*) 1

Er,.-,(c,ML12)=[ 1

1–M,’(1-C)- -’(1–c)’

h~n’(y~;c,MO’j

K2n(c,M;’) =[ 1

I—wo’(l- c)’ -1(1–c)’

I&,’ (y,;c,MJ)

KO=l.O

(13)

342 REPORT NO. 876—NATIONAL ADVISORY COMM~TEE- FOR AERONAUTICti

With the aid of equations (11), the expression forE (al c, M02) can be rewritten as follows:

E(C?,C,MJ) =*WI‘( 922’+ @qzz)

(14)WI!(9*2’+ p$oy.)+: (@l~’+&olJ

where‘wl’(Yc-Yl)_lx(c)= ~ - (15)

The relation (2) between the phase velocity and the wavelength is broughi into a form more suitable for the calculationof the stabiIity limits by making use of the fact that for real

values of c the imaginary part of E’ (a, c, M02) is contributed

largely by the integral K1 (c, M/). (The procedure to befollowed is identical with that used by Lin in the limitingcase of the incompressible fluid (reference 5, part HI).)Define the function 0(z) by the relation

(16)

I~+~u=y

where

and for nz 3

and

When c is real,

Then,(U+io)@(z) =*l=(l+A) l+~(u+io) -

where

“’’(:$”+%2)u+iv=l+T.

Equation (17) is equivalent to the two real rektions

(1+-X)U . ___@@)=(~+~u)2+k2u*

@,(z) =(l+x)[ 1

U(l+AU)+M?(l+ A?l)’+m’

(17)

(18)

(19)

(20)

The red ancl imaginmy parts of @ (z) are p~otted agains~ z infigure 9.

The clominant term in the imaginary par~ of the right-handside of equation (18), which involves K, (cl M02), is extracted

by means of straightforward aIgebraic transfornmtious. Re-lation (18) becomes

for those values of a and c that occur in the stabiIity calcuktions. (This approximation isimaginary part of the integral K] (c, Mo2) is readily computed. It is fou~id that

I. P. KI(c, MOZ)=–mF3wKw..c

‘-”w%-%)

(22b)

ustified later in appendix A,) Tho

(23)

hTow k(c) is generally quite small, therefore @i(z) can be taken equaI to V(C) and @,(z) can be taken equaI to u m a

zerotlh approximation. From equations (19) and (2o), when c is reaI,

@i(o) (~(o)) =~= —

%%&%)(24)

~(o)=@,(o)(z(o)) (25)

By equation (24), z(o) is reIated to c with the aid of figure 9; and by equation (25), u(o) is aIso related to c. The quan[it.y cik’is connected with c by means of the identi~y

Vc (–)2?011g~R =W.’(l+ky c

(26)

and the corresponding values of a are obtained from equation (21) (slightIy transformed) by a method of successiveapproximations. Thus

(27)

THE ST.ABJIJITY OF THE IAUINAR BOUNTOARI” LAI-ER IN’ A COMPRESSIBLE FLUID 343

where

‘=% ’’+-%3)

{The syml-ds .Ilk and f~t now designate the real parts of the

integrals Mk and N...) The iteration process is be~gun by

taking a suitable initial value of a on the right-baud side of

equation (27). The metbocls adopted for computhg these

in tefywds when the mean veIocity-t emperature profile is

known are described in appenclkes A to C.For greater accuracy, the values of z and a for a given real

value of c are computed by successive approximations. Fromequations (19) and (20),

(28)

The value of L’is always approximated by relation (24).Cur-i-es of wa-re number against Reynolds number for the

neutral disturbance have bee~ caIcnIatecI for 10 representa-tive cases (fig. 4), that is, i.rdated surface at Mach rmmbersof O, 0.50, 0.70, 0.90, 1.10, and 1.30 ancl heat tram~fer acrossthe solid surfare at a llach number of 0.70 with vaJues of theratio of surface temperature to free-stream temperature T1 of0.70, 0.80, 0.90, and 1.25. (It is founcI more desirable to basethe nondimensional wa-re number and the Re.ynokls numberon the momentum thickness 6, which is a clirect measure ofthe skin friction, rather than on the bounclary-la-yer thickness8, which is some-wh%t indefinite.)

In fi.~re 5 the minimum critical Reynolds numberR,=,mf~,or the stability limit., is pIottecl against Mach uum-

ber for the insulated surface; and in figure 6(a) ROCrmiRis

plottedagainstT, for the cooled or heated surface ab a Machnumber of 0.70. The marked stabilizing influence of a with-(kawal of heat from the ffuid is clearl$y evident. Discussionof the pb-ysicaI significance of these numerical results isreser~ed until after general criterions for the stability of thelaminar boundary Iayer ha~e been obt~ined.

3. DES~ABILIZING INFLUENCEOF l’ISCOSITY AT VERY LARGEREYNOLDS NUMBERS ; EXTEXSION OF HEISENBERG’SCRITERION TO THE C031PRESSIBLE FLUID

The numerimd cakuktion of the limits of stability forseveral particular cases gives some indication of the effectsof free-stream lkch number and thermaI conditions at thesolid surface on the stabiIity of the laminar boundary layer.It WOUIC1be very desirabIe, howe~-er, to establish generalcriterions for Iaminar instability. For the incompressiblefluid, Ileise~berg has shomn that the infiuerice of viscosity isgenerally destabilizing at very Iarge R eynolck numbers(refermce 1). His criterion can be stated as follows: If aneutral disturbance of no.nwmish.irg phase velocity andfinite via~e length exists in an inviscid fluid (1?- ~ ) for agiven mean -relocity c{istributio-n, a disturbance of thegarfie ware Zcngtk is unstable, or self-exeit ed, in the real fluid

at ~ery large (but finite) ReynoIds numbers.The same co~clusion can be draw-n from Prandtl’s discussion

of the energy baIance for small disturbances in the laminarboundary layer (reference 12).

Heisenbc&s criterion is established for subsonic clisturb-antes in the Iaminar boundary Iayer of a compressible fluidby an argument quite simiIar t.o thtit which he gave originallyfor the incompressible fluid and which was later supple-mented by Lin (reference 5, part HI). .It ~ery IargeReynolds numbers, the relation (1) between the phasevelocity and the -wa~~elength can be considerably simplified.Khen x is finite and e does not vanish, IzI>>1 at largeReynoIds numbers. The asymptotic beha~ior of the Tietjensfunction F(z) as [zI+ ~ is given by (reference 5, part 1)

(!/1–YC)NZ) ==/~ LY;C

and the relation (I) becomes

w-here i?(ci, c, MOY)is gi~en by equation (14).Suppose that a neutral clisturbance of non-mnishing

the in-riscid fluid (Limit@~ case of an infinite Reynoldsnumber). The phase velocity c is a contkuous funct ion of R,and for a disturbance-of gi-ve~ wave number a, the value of cat very large Re-moIds numbers -w-illdiflw from c: by a smallincrement Le. Both sides of ecluation (31) can be developedin a Taylor’s series in Ac, and an expression for Ac can beobtained as folIows:

‘e; [l~O(Ae)]

‘Ic ci*— C*c<

The boundary condition

9,,’(CY.,c,, M02)+- /3.@, (a,, c*, M,’)= o (33)

must be satisfied for the inviscid neutral disturbance, andthe function li’~(as, c,, Maz) vanishes (equation (14)). Recog-nizing that

?L171 > 1

(-)ac Cs.Usd

E~z — c~

~c~

reduces equation (32) for Ac to the form

From equation (14),

(34)

(35)

344 REPORT NO. 876--h’ATIONAL ADVISORY “COMMLTTEE FOR AERON” 4UTICS

()By equations [12) and the bouncky condition (33), the quantity b%, ~,,~, is evaluated as foIIol\7s:

where the primes now denote differen~iation with respect to c. ()For small values of c, and a,, the quantity ;~~l ~.,d, is given

approximately by the relation

()ZLE’, CS2

[

2–M*2(1-C,)Z—— ~.-f-K1’ (es,3102)be .,,U,=—T

—— ——a,(l —c,)3Jl-Mo~(l —C$) 1

and the expression for Ac is

T,Ac= —

a. Ill —M02(I —cJ*e’C’4

/R ~2-M02(1-cJZ

\ %<cs (1–C,)3+ a, 41 –MQ_’(1 –mK,’(es, M,z)

‘(c’’02)=-fi+&[$ G)lu=.’’n’-i”)+O (’)

(37)

(39)

w?]Since the quanti~Y @ 1’ ,,=,, ~ranishes (reference 8), differentiatio~~ of equation (39) gives

(A{%[EF)I.=C})C.C$ ’l’’C-’+’(’)’)(40).K,’ (c,, M02) ‘a+ bc (W:)3 CZy T

Thus, K*’ (c,, M?) is approximately real and positive for

small values of c$. With c.>1–j&, I. P. Ac must also be

posit ive (equation (38)); therefore, a subsonic disturbance

of wave length As# O; which is neutral in the inviscid com-

pressible fluid, is self-excited in the reaI compressible fluid

a~ very farge (but finite) Reynolds numb em.

In reference 8, it was proved that a neutral subsonicboundary -]ayer disturbance of nonvanishing phase velocityancI finite Jiwve ]ength exists in au inviscid compressible fiuid

Citl)()d P~yonly if the qucmhi~y ~ vanishes for some value. of.1

w>l ----- If this condition is satisfied, then self-excited

subsonic disturbances also exis ~ k the fluicl; ancIthe kminarboundary Iayer is unst tiblc in the limit”mg case of an inilnite

Reynokls number. By the extension of Heisenberg’s

criterion to the compressible fkid, it can be seen thatj far

from st a.bilizing the. flow, the small viscosity in the reaI fluid

has, OD the contrary, a destabilimkg influence at very large

Reynolds numbers. Thus, any laminm boundary-layer

flow in a viscous conductive gas for which the cjuantity

c1 dw( .)& P&- vanishes for some. vaIue of w> 1-- & is unstable aL

-o-.” .

sufficiently high (but finite) Reynokls numbers.

()Unless the condition ~ P~~ =Oforsomcvallle w>+-~~

is satisfied, all subsonic clisturkmnccw of finilc wave Irnglll aredamped in the limiting case of infinite RPynolds mmlberj andthe inviscid fiow is stable< Since the effec~ of ~isco~i~y is

clestabiliz~ng at very large Reynolds nunkrs, hc)we ~cr, alaminar boundary flow that is stable i~nthe limit of infu~itcReynokknumber is not necessarily stal}le at. large R[’ynol(knumbers when the viscosity of the fluid is consiclrrcd.(See fig. 4 (1).) In fact, for thf’ incomprwsible flui(l, IJinhas shown that, rvery laminar boundary-layrr flow is IJIE

stable at. sufficiently high Reynolds numbers, whether or no L

the ~-ortkity gradient’~ vanishes (ref(~rcnce 5, pmi. III).

In order-lo set (le this question for the emnprcssihk fluid ingeneraI terms, the relation (1) bet.ween the complt’x pllascvelocity and the wave kng[h LI1huge Rcynohis mlmh~rs

d du,

()must now ‘be studied for flows in w~ch the quantity Z; p~

does not ~anish for any value of w> 1— ~-.M o

THE ST.+ BILIT%” OF THE L.LXINAR BOUNDARY IAYER IN A COMPRESSIBLE FLUID ..345

4 STABILITY OF LA311NAR BOUNDARY LAYER AT LARGEREYNOLDS XUSIBERS

The neutral subsonic disturbance marks a possible“boundary” between the damped and the self-exciteddisturbance, that is, between stable and unstable flow.Thus, the general conditions under which self-excited dis-turbtinces exist in the laminar boundary Iayer at largeReynolds numbers can be determined from a study of thekharior of the eur-re of a against R for the neutral clisturb-antes. Nlen the mea~ free-stream velocity is subsonic(MO< 1), the physics] situation for the subsonic disturbaricesat large ReyDokk numbers is quite similar to the analogoussituation for the incompressible fluid. The cw~e of aagainst R for the neutral disturbances ccm be expectedto have two cIistinct asymptotic branches that encIose aregion of instability in the al?-plane, regardless of the localdistribution of mean velocity ant{ mean temperature acrossthe boundaly layer. When the mean free-stream -i-eIocityis supersonic (MO> 1j, the sit.uation is somewhat differe~~;under certain conditions (soon to be defined) a neutral or

~ . j Camot~fiSLa self-excited subsonic disturbance c> 1—~

at any value of the Reynolck number. For this reason, itis more convenient to study the case of subsonic and super-sonic free-stream Yelocity separately.

.$.SLIBSOXIC FREFXTEEAM VELOCITY (Mc<l!

The asymptotic behavior at Iarge Reynolds numbers of

the cur=i-e of a against 1? for the neutral disturbances is

determined by the relations (19) to (22) betwee~ CP,R, and cfor real -ralues of c. For small }-aIues of m and c, these rela-tioms are gi~-en approximately by

. . W,’C T:

––[%)].=. ‘“)“(c] ‘@i(z) = – T, (WG’)3 C@

‘lI=@r(z) (42!)

R= 7,1.76

–(): ‘[w,’)’

a(43)

& 1+ m, either z+ cu or z remains finite while both a ancl

c approach 0- These two possibilities correspond to twoasymptotic branches of the cur-re of CYagainst R.

Lower branch.—If z remains kite as R-+~, then c-O;and by equation (41), @i(z)~O. Therefore, z~2.29 while?L+2.29(6g.9). From equations (43) and (44), aIOng theIower branch of the cur-re of a tigainst. R for neutral stability

and cr~O at large ReynolcIs numbers (fig. 4 (1)).

(46)

Upper branch.—~oug the upper branch of the curve of aagainst R for neutral stability, z+a and

whiIe ti~l.O (fig. 9 and equation (42)). If the quantity

(’l$ $, cloes not vanish for any due of w>O, then by

equation (47) c must approach zero as z-a. Mong thisbranch,

(49)

and a~O at large Reynol.k n;,mbers (fig. 4 (1)).

()On the other kind, if ~ ~ ccznishes for some -i-due of

w=c.>0, then by equation (47j c~c, and o-~a, m ~oth z and -- “-”R approach cu. ATOW,

If [$ g)], does not vanish. (see appenclix D), then by.

equations (44) and (47), along the upper branch of the curveof a against R for the neutral clisturbances,

.-. .—.-

R=(U’,’)g

27rzT10z4

Wkll}’+ ‘c:’~ ‘-’51’

r Ia—

‘u;: +1 –’1102(1–c~2(52)

and cAc,# 0, a-a, #O at large Reynolds numbers (t@. 4 (k)

and 4 (1)).E [$($)11

-iani.shes, the relation (51) is

replaced by

R ~2(w,’)’”gT,O,24{[$i$)lx:” “2:’82)2 ’53’

which reduces to the relation obtained by Lin in the limitingcase of an incompressible fluid when Mo~O, the solid bound-ary is insulated, and w“=0 for some value of w=cs>O.(See equation (12.22) of reference 5, part III.)

()If the quantity $ ~ vanishes a.? the solid boundary

346 REPORT NO. 876 —NATIONTAL ADVISORY” COMhIITTEE FOR .4ERONAVlYCS

(that is, for w= O), it can be shown from the equations of

[42P)]motion (appendix D) that ~~ T is always positive—

except in the limiting case of an incompressible fluid. Ford w’

small values of y, the quantities ––[)ciy -T

and ‘~ are bokh

positi~e and increasing. For large values of y, however,w’ {—-+0, physically; therefore ‘— must have a maximum, orT Td W’

()fiT’= O for some value of w>O, and this case is no

clifferent from the general case treated in the precedingparagraph. In the Iimiting case of cm incompressible fluid,

,when w” vanishes at the surface, w. lf=wll~~

-(:1’)2 ‘ince “’”ahvays vanishes in this case. From equation (48) the rela-

tion between a ard R along the upper branch of the neutralstability curve is therefore

R=2(W1’)19 1 1

7r’ (w,”)’ m(54)

which is identic,a~with equation ( 12. 19) in reference 5, part HI.

()Thus, regardless of the behavior of the quantity #y ~ –

regardless of the local distribution of mean veIocity ancl meantemperature across the boundary layer—wlum .MO< 1, thecurve of a against R for the neutral disturbances has twodistinct branches at large Reynolds numbers. From physi-caI considerations, all subsonic disturbances must be cla-rnpedwhen the wave length is sufficiently small (a large) or theReynoIcls number is sufficiently low. Consequently, thetwo branches of the curve of a against R for the. neutraI dis-turbances must join eventually, and the region betweenthem in the d-plane is a region of instability; that is, at a,given value of the Reynolds .uumber, subsonic clisturbanceswith wave lengths lying between two critical values AI End12 (al and aJ are self-excited. Thus, when MO<I, any Zami-nar boundary-layer $OW in a viscous conductive gas is znstaldeat sujlciently fvigh (but Jlnite) Rey?mkk numbers.

TII e lower branch of the curve of a against R for theneutral disturbances is virtually unaffected by the distribu-

()tion of $ $ across the bounckwy Iayer, but for the upper

d w’branch the behavior of the quantity —

0dy Tis clecisive.

()l}7hen $ ~ = O for some value of w.= c,>O, the neutral

subsonic disturbance passes continuously into the charac-teristic inviscid disturbance c= c, ancl a= a. as R-+ co.This result is in accordance with the results obtained inreference 9 for the inviscicl compressible fluid and is in agree-ment with Heisenberg’s criterion. In acklition, all subsonic.

disturbances of finite wave length k> A,=: (and nonvanish-

ing phase velocity O< C,<CJ are self-excited in the limitingcase of infinite Reynolcls number. On the other hand, when

d W’(-)%Tdoes not vanish for any vfilue of w> 0, then except for

. .the “skgu]ar” neutral disturbance of zero phase velocityancl infinite wave length (c==O and a= O}, all disturbancesare clamped in the inviscicl comprcssibk fluid. T!kissingular neutral clisturbanre can bc regarded as the limit i]lgcase of the neutral subsonic disturbance in a real compressi-ble fluid as R-m.

B. SUPERSONIC FREE-STREAM VELOCITY (Afn> 1)

When the velocity of the free stremn is supersonic, thesubsonic boundary-layer disturbances must sa tkfy noL onlythe diffe~.emtial equations and the boundary conditions of ~bc

problem but ako the physical rccluiremen~ that c,> 1–1}0’

The asymptotic behavior at kge Reynolds numbers of (1}~curve of a against R for the neutral subsonic distu rbanrcsis determined by the approximate e relations (41 j to (44), with

‘1 Tthe additional restriction that c> 1–~o. & c~l –jl~O,

~+0 b3T equation (44); therefore, R-+ CDby cqtmtion (43).The corr-esponding value (or vaIucs) of z is dckrmit~ cd byequation (41 ) as follows:

()O,(z) =v(c)=u 1–&

— ?’rWl’——

Pk%)l.=c=% ’55’

d W-’

()Now from physical considerations, ~~ -T <0 for large

Therefore if’- “values of y.(’-)‘dy T

= O (clmngcs sign) for some

1~alue of w=c,> 1—1~, thenj in general,1’0 [Hww=,-+:”

WIC1 @ i(Z) 1 <O (equation (55)). From figure 9 i~ can kc=l–tie”

seen that in this case there is only one vrdue of 2 (zl, SRY)correspond ing to the value of @~(.z)given lIY c3quation (55).From equations (42) to (44) along the Zower branch of thccurve of a against R for t$e neutral disturb tmcw}

f

‘“?3!1?:WIr

7cl= “-i&2”’OJ=Fa ’57)

.,and c~l —lJro at k-ge Reynolds numbers (fig. 4(1<)). The

upper branch of the curve in this c.asc is given by equations

(51) and (52), or by equations (53) and (52) if[$;@lL

vanishes, with c-c,> 1—1Llxoand a-%~#0.

THE STABILITY OF TEE LMIINAR BOTECOARY LAYER ~’ A COMPRESSIBLE FLUID 347

()If ~ @ vanishes for w=l—~~ then Z~CC as R4~dy T o

along the upper brancb of the cur-re of a against R for theWIf

DeutraI disturbances, ancl @i(z)-

/

—. Xow a+41 as~2a;c3

C~I —~=1-in this case ako (equation (57) with ul = 1.0) soII

thak

.Mong the loxer branch of the curve of a against R at Iar.geReynoIds numbers, a, R, MIClc are connected by equations(56) and (57), Itith 2,=2.29 and ti,=2.29. In spite of the

fact that. -$ ~$)=0 for w= 1–&O, a neutraI sonic clisturb-

( )ante c= I —~ of finite wave length does not. exist. in

M

H 1the inviscid fluid unIess K,(c)== ,3” &—AIz cZy is

positive. (See section 10 of. -reference 8.) CMeulationshows that III(c) is almost always negative (equation (40));therefore, in generaI, the sonic disturbance of intlnite wavelength (cK=O) mith constant phase across the boundaryIayer e.tists onIy in the inviscid iluid (R+ m ).

()If -$Y $ does not. vanish for any vaIue of W21 —&, it

is’er’’~th’’[$G)l.=c=I.&<o ‘d by ‘qu’tion ’55)

W)c=l.+l> 0. when ~lm1 <0.!xIO (approx. ), there are two

dues of z (% and z3, say, mith ~3>~2) COrreSpOndkg to thevalue of @i(z) given by equation (55). (See fig. 9.) AIongthe two asymptotic branches of the cum-e of a against R forthe neutral clisturbances, a, R, and c are connected by rela-tions of the form of equations (56) and (57), tith z and wreplaced by 22 and uZ, respectively, aIong the 10UW branchand by z’ ancl u3, respecti~ely, cdong the zppw branch. At~ given~alue of the Slach number, the value of u ~ is con-

IztrolIed by the thermal conditions at the solid surface. (Seesection 6.) V%en these conditions are such that, u I =0.580,

1-Yethen Za= 23, and the two &symptotic branches of thecur-re of CYagainst R for the neutral disturbances coincide.When t> I ~ 0.580 (approx.), it is impossible for a neuirdl–~—

or a self-e~cit ed subsonic disturbance to e-tit in the Iaminarboundary Iayer of a viscous conducti~e gas at any due ofthe ReynoIds nhmber. In other words, if u 1 >0.580

~–m—(approx.), the kminar bounclary layer is stable at alI ~aluesof the ReLynoIds number. (Of course, in any gken case, thecritical conditions beyond which only damped subsonic dis-turbances exist can be calctiated more accurately from therelations (28) and (29). See section 5 on minimum criticalReynolds number.)

The preceding conclusion can ako be deduced, at IeastquaIitati~eIy, from the resuks of a study of the energy

balance for a neutral subsonic disturbance iu the kninarbouudar~ layer. .& neutraI subsonic disturbance can exist.ordy w-hen the destabiIizing effect of ~-iscosity near the solidsurf~ce, the clamping effect of ~iscosity in the ffuid, and theenergy transfer bet-ween mean flow and disturbance in thevicinity of the inner “eriticaI layer” aII balance out to givea zero (average) net rate of change of the energg of the dis-turbance. (See SMichting’s discussion for incompressiblefluid in reference 4.) In reference 8 it is shown that the signand magnitude of the phase shift in u*’ through the inner“critical layer” at w= c is determined by the sign and maomi-

t.ude of the quantity[%($)l.=:

The corresponding

—

*—~ m, ~hicb is zero for 10<Capparent shear stress Tc —in the in-riscid compressible fluid, is given by the followingexpression for W>C (reference 8):

[w)]If the quanti~Y dy T .=. is negative, the meau flow ab-

[d”(w=.ispositive$sorbs energy from the disturbance; if ~ ~

enerbq passes from t’he mean Bow to the disturbance. IB ._.the real compressible fluid, the thickness of the inner criticallayer in which the -riscous forces are important is of the order

1R lP, and the phase shift. in U*[ is actually brought. about

()cY—

Vc

by the effects of viscous diffusion(

dwof the quantity p *

)

through this layer.k shown by Prandtl (reference 12), the destabilizing ef-

fect of viscosity near the solid surface is to shif~ the phase ofthe “frictional” component uf,*’ of the disturbance velocityagainst the phase of the “frictionless” or ‘ ‘iuviscid>’ compo-nent ‘Ui~@*’in a thin layer of ffuid of thickness of the order of

/

1—R. By continuity, the associated normal component

\ac—

n

shown in section 1 of reference 8 that for Iarge values of crRthe “frictional” components of the disturbance_ also satisfy

~uw , &*fthe continuity relation ~ ~ — = O in the compress-ay%

ible fluid.) The corresponding apparent shear stress

rl *=%*- is given by the expression

But from equations (11)

(61)

348 REPORT NO. 876—NATIONAL ADVISORY COhfMITTEE FOR AERONAUTICS

mi

(62)

Since the shear stress associated wiih the destabilizingeff ec L of viscosity near the solid surface tmcl the shear stressnear the criticaI layer act roughly throughout the sameregion of the fluicl, the ratio of the rates of energy trans-

( J

h 7 .,.

ferred approximately ~)

* by the two physicalo,

processes is

E,* T.* TW,’c T:

1 ––I[:G)I.=J”3’2~ =~ ‘j T, (WC’)3

VA ere

=;~v(c) [ 2?(2

,., _ R Gz3G -%,, (W,’)*

(63)

.()If the quantity -$ ,% is negative and sutlciently large when

w=cI, say, then khe rate at which energy is absorbed by themean ff0T17 near the inner “critical layer” plus the rat e atwhich the energy of tlw disturbance is dissipated by viscousaction more than counterbaknees the rate at. which energypasses from the mean flow to the clisturbance because of thedestabilizing effect of viscosity Dear the solicl surface. Con-sequently, a neutral subsonic disturbance with the phasevelocity c >cI does not exist; in fact, all subsonic clisturb-ances for which c 2CI arc dumped. When MO< 1, there is

always a range of values of phase velocity O~ c<co for

which the ratio ~f~ : given by equation (63), is small enough

for ncut,raI (ancl ~elf-excited) subsonic disturbances to existfor ReynoMs numbers greater than a certain critical value.TVhen MO> 1, howe.tier, because of the physical requirement

that c> 1—~~>0, the possibilityy exists that for certain ther--.—,,

[w]mal conditions at the solicl surface the quantity ~y T ~=o

EC*

is always sufficiently large negatively (ancl therefore. — isEl*

sufficiently large) so that onIy cIamped subsonic disturbancesof course if ~ ~ ~Tal*ishes

exist a~ all Reynolds numbers.()‘dy T.

for some vaIue of w> I –&, it is certain that v(c) <0.5s0

for some range of values of the plmse velocity 1—&s cs ?;.

In that case, neutral ancl seIf-excited subsonic disturbtineesalways exist for R> R6,n1~and the flow is ahvays unstable “at

sufficiently high Reynolds numbers, in accordance withHeisenberg’s criterion as extended to the compressible fluid(section 2),

~ discussion of the significance of these results is reservedfor a later sectio~ (section 6) in which the behavior of Lhe

‘-” d W’

()quaniit~ @ Twill bc rektecl directly to the thermal

conditions at the solid surface andnumber.

5. CRITERION FOR THE hlIN1fillJllNUhlBER

The object of tht stability analysis is not only to dctur-mine the general conditions undw- which the Iaminar bound-ary kyer is unstable at sufficiently high Reynolds numbersbut also, if possible, to obtain some simple criterion for (Ilelimi~ of stabi]ity of the flow (minimum critimI RPynolclsnumber) in terms of the Iocal distribution of nmm -relocilyand mean temperature across the boundary layer. For plane~ouette motion (linear velocity profile) and plane l?oiseuillemotion (parabolic velocity profile) in an incomprcssibk fluid,Syngc (reference 13) was able to prove riggrol.@y tht aminimum critical ReynoIck number actually mists ‘DCIUWwhich tht flow is stabk. His proof applies also 10 thelaminar boundary layer in an incompressible fluicl, with onlya slight modification (reference 5, part ?31). Such a proof ismore difficult to give for the laminar boundary layer in aviscous conductive gas; however, the existence, in geucral,of a minimum critical ReynoMs number can ‘m inferredfrom purely physical consiclerations. A study of t-k Cnmgybalance for small clisturban~ccs in the larninar bmmdary lfiycrshows that the ratio of the rate of viscous dissipation 10 therate of energy transfer near the critical layer is l/R for aclisturbance of given wave length whiIe tl.w ene~.gy transferassociated with the destabilizing action of viscosity near the

solicl surface bears the ratio 1/@ to tlw cnerg~ transfer near

the. critical layer. Thus, the cff eck of viscous dissipa Lion

w-ill predominate at sutllciently low Reynolds numbers findall subsonic disturbances will be dampccl. The t}vo distinctasymptotic branches of the curve of a against A’ for theneutral disturbances at. large Reynolds numbers must joineventually (section 4) and the flow is staMc for all Reynoldsnumbers less than a certain criticaI ~aluc.

An estimate of tbe vaIue of RC,~f~, which will servo as a

stabiIity criterion, is obtained by taking the phase velocity cto have the maximum possible valm cOfor a neutral suko nicdisturbance; that is, for C>CO all subsonic disturbances arcdamped. This condition is very nearly equivalent to tl.Mcon-dition that UR be a minimum, which was employed by liJ) forthe case of the incompressible fluid (p. 2s5 of refwcnm 5,part HI). The condition C=COoccurs wheu (1~(~) is a maxi-mum; thzt is, when @~(z) =0.580, zO=3.22j ancl (E,(zo)=1.48(fig. 9). The corresponding vaIue of C=COcan bc rakulatcclfrom the relations (19) to (22). A’eglecting terms in A’(h is usually very small) ancl taking u=l.50 givw

0,(2) = [1–2A(C)]O(C) (64)

THE STABILITY OF THE LA2MLNAR BOUND.4RT LAYER IN A COMPRESSIBLE FLUID .349

and‘W’(?JC-U) _ ~x(c)= ~ (66)

It. is only necessary to plot the qwmtity (1 —2k) 0 against. cfor a given Iaminar bour&r-y-Iayer flow and find the valueof C=COfor which (1 —2x) v= 0.5S0. The corresponding valueof cJi’ k determined from the relation

and this w.lue of aR is very close to the minimum value ofal? . .4 rough estimate of the value of a for C=COis given bythe following relation (equation (27)):

This estimated -ralue of a is, in generaI, too small. The fol-lowing estimate of Rc,~ti is obtained by making an approxi-

mate aHowance for this discrepancy and by taking roundnumbers:

(69)

For zero pressure gradient, the slope of the veIocity pro61e&J-

()a~ the surface —dq ~is given very closely by (appemlk B)

tmd

Therefore

(71)

‘l’he expression (71) is useful as a rough criterion for thedependence of R$C,~imon the local distribution of mean

-relocity and mean temperature across the boundary layer.

It is immediately e-ride~t. that RSC,qiX-+~ when cO+l –&O.

When [(1 –2A) t?] I k 0.5S0, the laminar boundary layer isC.l–=e —

stable at ail mlues of the Reynolds number. (This condition

is an impro-rement. on the stability condition ol_A~ 0.580Me

(approx.) stated in section 4.)In the following tables and in figures 5 and 6 (a) the esti-

mated -ralues of ROC,ma~ti-ren by equation (71) can be com-

pared with the vaIues of R@C,=&taken from the calculated

curves of a$ against R@ for the neutral disturbances. For

the insulated skface, the -raJues are

tc,

o.4M.4QIA&Cc.48s3.5139-5450

1!?5Lmm129m$=2 1--

R@.,-i.

(fig. 4)

lEO136lxi11.5MMw

For the noninsulat ed surface when MO= 0.70, the values are —

<HR8 Rt

TI T(G) == i= Cr” ;.

Q(es.] (fig. 4)

o.n 0. 1S72 0.ii12 5377 .51.50 ..S) 1%3 1$40.93

~$j \ :~g 524 mL 25 L 1449 39 63

The expression (71) for RRm~in gi~-es the correct order of

magnitude and the pro-per -rariation of the stability limitwith Mach number and with surface temperature at a giwml~ach number.

The form of the criterion for the minimum critical Rey-nolds number (equation (71 )) and the results of the detailedstability calculations for se~eraI representati~e cases (@. 3and 4) show that the distribution of the product of the

density and the -rorticity p $ across the boundary layer

Iargely determines the limits of stability of laminar boundary-layer flow. The fact that the “proper” Rqmolds numberthat. appears in the boundary-layer stability calculations isbased on the kinematic viscosity St the inner criticaI layer(where the -iiscous forces are important) rather than in thefree stream also enters the problem, but it amounts only toa numericaI and not a qmlitati~-e change when the w_mIRe~oIds number based on free-stream kinematie viscosi~yis finally computed. Ti%ether the mdue of ReC,~,Z for a

given Iaminar bound~-layer flow is larger or smaller thanthe rahe of R80~iz for the B1asius Ho-w, for exampIe, is ‘

determined entireIy by the distribution of p ~ across the

()boundary layer. If the quantity $ p ~ is-~egative and

large near the solid surface so tha~ the quantity (1 –2?Jv(c)reaches the ~aIue 0.5S0 when the ~ahle of c= co is less than0.4186, the flow is relatively more stable than the Blasius

350 REPORT NO. 876–--NATIONAL ADVISORY COMMITTEE FOR AERON.4UTICS

()flow. If the quantity -$ P ~ is positive near the solid

surface, so that, (1 —2k)o(c) =0.580 when w (or c)> O.4186,the flow is relatively less stable than the Blasius flow. Thusjthe question of the relative influence on R9G.~i~of the kine-

mat ic viscosity EL the inner criticaI layer and the distributiondw

of p — moss the bounclary layer, which remained open indy

the concluding discussions of reference 8, is now settle-d.

The physical basis for the predominant infiuence on

R8,,~i~ of the distribution of p ~ across the boundary layer

is to be found in a study of the energy balance for a subsonich~nclary-layer disturbance (section 4). The distribution of

p ~ determines the maximum possible vaIue of the phase●

velocity co or the maximum possible distance of the innercritical layer from the solicI surface for x neutral subsonicdisturbance. The greater the distance of the inner criticallayer from the solid surface, the greater (relatively) the rateof energy absorbed by the mean flow from the disturbancein the vicinity of the criticaI layer (equations (61) and (62)).I}%en co is large, therefore} the energy baIanc.e for a neutralsubsonic disturbance is achieved only when the destabilizingaction of viscosity near the solid surface is relatively large or,

in other words, wheu ~ s c03/2is large ancl the Reynolds

/\ao:

number Ro, which is very nearly equal to RC,~~fi,is correspondi-

ngly small. On the other hand, when co is srnalI and theinn er critical layer is close to the solid surface, the rate atwhich energy is a?morbecl from the disturbance near thecritical layer is relat iveIy small and the rate at which energypasses Lo the disturbance near the solid surface, which is of

the order of1

-===~ is also relatively small for energy bal-

4

Ra—

Vcante; consequently RC,~iz is Iarge.

6. PHYSICAL SIGNIFICANCE OF RESULTS OF STABHJTYANALYSIS

A. GEh-EIl.4L

From the resdts obtained in the present paper and inreference 8, it is clear that the stability of the laminar bound-ary layer in a compressible fluid is governed by the action ofboth \-iscous and inertia forces. Just, as in the. case of anincompressible fluid, the stability problem cannot be under-sioocl urdess the viscosity of the fluid is taken into account.Thus, wheth,er or not a laminar boundary -Itiyer flow is un-stable in the inviscid compressible fhicl (R-+ cc), that is,whether or not the product. of the density and the vorticity

dwp~

has an extremum for some value of w> 1–&, there is

alN7a37ssome value of the Reynolcls iurnher RC,~i~ below

which the effect of viscous dissipation predominates and theflomT is stable. On the other hand, at very large Reynoldsnumbers the influence of viscosity is- dest.ablizing. If the

free-stre5m velocity is subsonic, any laminar boun{la~y-h]ycr

flow is unstable at sufficiently high (but. finite) R(~ynol(ls

numbersl whether or not the flow is stabIe in the invisci(]

fluid when only the inertia forces arc ccmsidere{I,

The action of the in~rtia force.s is more clccisive for thes.tabiIit.y of the laminar boundary layer if the free-streamvelocity is supersonic.. Because of th~ physicciI rcquircrnrntthat the relative phase velocity (1 —c) of the boundary -]ciycr

disturbances mus~ be subsonic, i~ follows that c> 1–X$->tl.

[J G%ol.=c “and the quantity ~ii can be large enough nega-

tively under certain conditions so that the stabilizing actionof the inertia forces near the inner critical kyer (wherew=c>O) is not overcome by the (Icstabiliziug fiction ofviscosity near the solid surface. III that, case, un{ltimpwldisturbances cannot exist in the fluirl, w-d the flow is sfablcat all values of the. Reynolds number.

Regardless of the free-stream velocity, the clistribution of

the p~oduct of the density ancl the vorticity p @ arross th~dy

boundary layer determines the actuaI limit of stability, or the

minimum critical Reynolds number, for laminar k wldarLy-Iayer flow in a viscous concluctiv? gas (ccluat ion (71]).

Since the distribution of p ~ across the boundmy kycr in

turn is determined by the free-stream Mach mlrnbcr andtk therm.aI conditions at the solid surface, the eflcwt of tlicscphysical parameters on the stability of lanlinar boundmy -Iayer flow is readily evaluated.

B. EFFECT OF FREE-STREAkf AIACH KLIMBER AND THE RKI.AL C:OXDITIC)NS

AT SOLID SURFACE ON STABILITY OF LAMWAR BOUNIJ.4RY LAYER

The clistribution of mean velocity zmcl mean temperature

(

dw

)and therefore of P— across the laminar Ixmncltiry layer in

Cly

a viscous conductive gas is strongly influenced by the facL

that the- viscosity of a gas increases with the kmpcraturc.(For mosL gases, pa T’n (m=O.76 for air) over a fairly widetemperature range.) lVhen brat. is transferred to [hc fluid

through the solid surface, doe temperature and viscosity

near the surface both decrease along the out~rard normfil, and

the fluid near the surface is more retarded by the viscuus

shear than the. fiuicl farther out from the surface-as com-

pared with the isothermal Blasius flow. The veloci~y profiktherefore always possesses a point of infk tion (w’here w”= 0)when heat is adcled to the fluicl through the solid surfaeejprovidecl there is no pressure gradient in ~he clircction of the

-d -() ‘f w’T’main flow. Since ~ p* =WT –-T, , [he c~uantity

dw

()

dw~ P*dy

vanishes and P— has an ext.remum aLsome pointdy

in the ffuid. On the other hand, if heat is withdrawn frombT

the fluicl through the solic{ surfrwe, --- and a~ arc both posi-ag ?ly

tive near the surface. ancl the fiuid near the surface k lessretarded thtin tho fluid farther out-—-as compmed with theBlasius flow. The velocity profik is therefore more conwcnear the surface than the Blasius profiIe.

THE STABILITS OF TEE L& MIN’.\RBOUNllAR1- HYEP. m- A COMPRESSIBLE FLUZO 351

.1s pointed out in section 11 of reference S, the Muence. of

the variable ~tiscosity on the behavior of the procIuct. of the

density and the, rorticity p $V can be seen directly from the

equations of motion for the mean flow. Then there is nopressure gradient in the direction of the main How, the fluidacceleration wmishes at the solid surface, or

and

(“72)

Thus, when heat is added to the fluid throwh the solid sur--.—

)face (T,’<()), (~z, * k positive, and the -reIocity profiIe is

concave near the surface and possesses a point of inflectionfor some value of w>O; when heat is -withdrawn from the

fluid ( ~1’>o), (a~$~)lis Dwtive, and the velocity profile

is more con~ex near the surface than the BIasius profiIe.

The behavior of the quantity ~ey’(+%)=&(p*@

$2~*is paraIIeI to that of —~y*2’ From equation (73), in nondi-

mensional form,

Differential ing the dcynarnic equations once and making useof the energy equation gives the folio-iving expression for

[%011 -- : --(appendix Dj

[$(%)11=

, (T,’)’(wl’)3+2(m+l)%u,~a(m+l)(yl).u/ ~(75)

Thus, for zero pressure gradient,[$(311

is always positi~e;

w)]N“o-iv, if the surface is insuIated, the quantity ~y ~ ,

‘mishes>but [$2($)12° a“dw) and? both ‘-

crease with distance from the solid surface. Since $ ~0

()far from the soIicl surface, $ has a maximum and -$ ~

vznishes for some value of w>O. If heat. is added to the~ f’~>’

)fluid through the soLid surface (T,’< 0), ~(~, is a~ead~

positive at the surface; and since[$($)11

>0, the quan-

(.)tity -$ $ vanishes at a point, in the %uicl mhich is farther

from the surface than for an insulated boundary at the sameMach number (figs. 3 (a) and 3 (b)). ~onsequenti-y, the vaIueof c = COfor which the function

T, [F%%)]...(1–2X)D(C) =–F(l–2L) ~

reaches the value 0.5S0 iskrgerthan the -mluefor the insu~atedsurface. By equation (71), the effect of adcling heat to thefluid through the solid surface is to reduce R~a ~ md to

destabilize the flow, as compared with the flow o;er an in-sulated surface at the same Mach number (fig. 6).

If heat. is -withdrawn from the fluid through the solid sur: _.

[&G)llisn@i~e ‘fa’tif ‘he ‘ateface, T,’>0 and —

of heat tramsfer is suEEciently large$ the quantity

& ~T’) does not vanish within the boundary Iayer (fig. 3 (b)).

The vaIue of C=CO for mhich the function (1–2x)?(c)reaches the value 0.580 is smaller than for an inwdateclsurface at the same JIach number; and by equation (71),the effect of ~~ithd~atig beat from the fluidthrough the solid surfcwe is to increase .RR and to stab-

Crmtiilize the flow, as compared with the flow over an insulatedsurface at the same llach number (fig. 6). IT&m the ~eloe-ity of the free stream at. the “ edge” of the boundary la-ye-ris supersonic, the Iamiaar boundar-y layer is completelystaMIized if the rate at which heat is withdrawn tbou@the solid surface reaches or exceeds a critical value thatdepends only on the Alach number, the Reynolds number,and the properties of the gas. The criticaI rate of heat

d W’

()transfer is that for which the quantity ~ ~ k. sti-

ciently large negati~el~ near t!ue surface (see equation (74))

so that (1 —2k)r(c) =0380 m-hen c=cO=l —~O(sections 4

and 5). -Although det aiIecI stabikty calcuIationy for ~per-sonic flow o-rer a noninsuIated surface ha-re not been carriedout, the function (1 —2x) c(c) has been computecI for non-insuIated surfaces at JIO= 1.30, 1.,50, 2.00, 3.00, and 5.00 bya rapid appro.~ate method (appe~clk (2). The corresponcl-ing estimated values of R&a=,=were calculated. from equa-

tion (71), ancl in figure 7 these values are plotted against 1“1,

the rat io of surface temperat m-e (deg abs. ) t o free-stream tem-

perature (deg abs.). A.t any gi-ren Llach number gTeater .

than uni~y, the ~aIue of R@~~i~increases rapicU-yas CO-I —~;al,

when COcliffers only slightly from I —~ ~ the stability of.11,

the ]aminar bounda~ is extremely sem~itive to thermalconditions at the solid surface. At each walue of .TIO>1,there is a criticaI -mJue of the temperature ratio ~,= for which

R$ -+~ . If T,S TIC,, the laminar boundary layer isCrmin

stable at all R e-ynoIds rmmbers. The difference between thestagnation-t emperat ure rat io and the critical-surface-temperature ratio, which is reIatecl to the heat-transfercoefficient, is plotted ~~ainst Mach number in figure S.~TncIer free-flight. conditions, for Slach numbers greaterthan some criticaI Mach number tlmt depencls largely onthe aItitucle, the vaIue of 7’.– T,= is within the order of

352 REPORT NO. 876—NA.TION.4L .4 DVISORY COMMITTEE FOR AERONAUTICS

ma@tu~e of t]le ~iflerence” hetl!~een stagnation temperature

and surface &mperature that actually exists because of heat

mcliation from the surface (references 14-and 15). In otherwords, the critical rate of heat withclrawcd from the fluid

for laminar stability is within tlw order of magnitude of the

calculated rat e of heat comluction through the solid surface

which balances the heat radiated from the surface under

eq nilibrium conclitions. The calculations in appendix Eshow that this critical Mach number is approximately 3 at

50,000 feet altitude and approximately 2 at 100,000 feet

a~titude. Thus, for Mfl >3 (approx.) at 50,00Ci feet altitudeand MO>2 (approx.) at 100,000 feet altitude, the larniuarboundary-layer flow for thermal equilibrium is completelystable in the absence of an adverse pressure gradient in thefree stream.

VVhen there is actuaIly no heat conduction throughthe soIid surface, the limit of stability of the laminarboundary layer depends only on the free-stream Machnumber, that is, on the extent of the “aerodynamic heating”

( ( ))-’7322of the order of V1- ~ near the solid surface, A goodY

indication of the influence of the. free-stream Mach number

dwon the distribution of p ~ across the boundary layer for cm

Yinsulated surface. is obtained from a rough estimate of the

()location of the point at which $ P * reaches a positive

‘Daxi’’’un’ (or $(’%) “’fishes) ‘ifferen’iating ‘hedynamic equations of mean motion twice and making use ofthe energy and continuity equations yields the followingresult for an immlat ecl surface:

[$+311=-: w

/-”---0

where b=3 — I?rorn equations (75) and (76’> QZ*”

(76)

“the value

of c at whit; ~ “()

~,anishes or g w’dyz T > ()dy 7

reaches a maxi-

mum, is given roughly for air by

in which wl’ = 6(0’~20) (appendix B). In other words, the

d W’

()point in the fluicI at which — —dy 1’ attains a maximum moves

farther out from the surface as t,he Mach n.urnber is in-creased—at least in tho rarige OS MO <4.5 (approx.); t,here-

d w’fore the value of c for which —

()dy ~ vanishes and the value

of C=co for which (1 —21) o(c) reticks the value 0.580 both

increase with the Mach number (fig. 3 (a)). By equation(71), the value of R8,,~iz for the lmnimw boundctry-lqwr flow

over an insulated surface decreases as the Nlach numberincreases ancl the flow is destabilized ~as comparwl with [kBlasius flow (fig. 5).

C. RESULTS OF DETAILED STABILITY C.4LCULATIONS FOR INS ULATCH ASDNONINSULATEDSURFACES

From the results of the detailed stability cakukttions forseveral representative cases (figs. 4-to 6), a quantitative esti-mate cali be made of the effect of free-stream Marl] ILumberand therrncd conditions at, the solid surface on the stability ofIaminar boundary-layer ffOW. For the insulated surface, thevalue of R~,,~t~ k 92 when MO= 1.30 as cornparrtl with a valuo

of 150 for the Bksius flow. For the noninsulatwl surface a~MO= 0.70, the: vahle of RoC, ~~ is 63 117hcn T1 = 1.25 (heat

addccl to fluidj, RoC,~i~=126~when Tl= 1.10 (ksu]vtcci sur-

face), and Rec,mia= 5150 when T1=O.70 (lumt witldrawn

/

_

from flui~). Since Rs *= 2.2511~2(the value of 01

_eJwhichv~*x*

is proportional to the skin-friction co@iicicnt.J differs onlyslightly from the Blasius vaIue of 0.6667), tLte cfl’rut of thetkermaI conclit ions at, the solid surface on D.=”is cvcct morepronoune,e.cl. The value of R.* is 60X 10fiwhen Tl= 0.70 andMO= O.70, _ as compared with a vaIue of 5I X 10s for (hoBlasius fl-o~~’lT, = 1 and MO= O). For the insulaf.(.vl sllrfarcthe value of R=”C, cleclines from the Bl~sius value for

~ in

MO=O to “a value of 19x103 at MO=l.30. The cxtrcuncsensitivity of the limit of stability of f-he laminar boundarylayer to +Lermal conditions at the solic{ surfacw when T,< 1is accoum!ed for by the fact thtit COis snmll WIND TI <1 and~?lo< 1 (or MO is not much great m than unity) ctnd

.?m,n ‘-”R8 =7 (equation (71)). Small rhanges in c,, lhcrcfcwe,

procluce large changes in lloC,~i,. In addition, wheu T1<l,

small changes in the thermal renditions cit the solid surf arc

~J ~~~ (equation (74)) ancl,produce appreciable changes in –-

therefore, in the value of co.Not only is the value of R~C,~I~fiffcctcd by the thlllfd

conditions zt th: solid surface and by the free-stream hlachnumber but. the entire curve of a~ against Re for the uetitialdis~urbances is also affect d. (See figs 4 (k) and 4 (1~.)~%en the surface is insulated (and ~fo# 0), or heat is”addccIto the fluid (l’l= 1.25), a~~cr,# O as Re-+ ~ along the upfwrbranch of the curye of neut raI stability. In other words,there is a finite range of unstable WQVClengths even in LLOlimiting case of an infinite Reynolcls number (inviscid fluid).However, a-+0 as Rg~ Q for the Blasius flow, or when heatis withcfrawm from the fluid. This behavior is in comple kagreement with the resu~ts obtained in section 4 and in.reference_ 8.

THE STABILITY OF THE LA311X’.%R BOU?WARY LAYER IN A COMPRESSIBLE FLUID 353

.% comparison between the curves of a~ against Re forTl= 1.25 ancl T,=O.70 at J10=0.70 shows that withdrawingbeak from the fluid not ordy stabilizes the flow by increasingR, ,r~{n but cdso greatly reduces the range of wtable wa~re

numbers (a~). On the other hand, the addition of heat. to

the fluid through the soIid surface greatly increases the range

of unstable wa-re numbers.It shoulc[ also be noted that for gjren values of ca, c,

aml Rg the time frequencies of the bounclary-layer clisturb-arwes in the high-speed flow of a gas are considerably greaterthan the frequencies of the fcmiliar Tollmien wa~-es obser~edin low-speed flow. The actual time frequency n* expressednondimensionally is as foJ.Iows:

—ii.*vO* _ cffc9

(’~jz – 2zR6

For gi~en ~alues of c, a~, and RB the frequency ticreases Mthe square of the free-stream ~elocity. -

~. lNST.4BILJTYOFLhMx.4REG~&l?~R~L.4~ER.%ADTR~::SITIOk-TOTCTr.B~LENTFLOL~

The wdue of R~,,n,B obtained from the st~biIity amd-jsis

for a given kuninar boundary-la-j-er flow is the value of the

Reynolds number at which self-excited disturbances fh-stappear in the boundary layer. ..% Prcmdtl (reference H)

carefuIIy pointed out, these initicd ckturbamws are notturbulence, in any sense, but s~owIy growhg osctiations.The value of the ReynoIck number at which houndcmy-Iayer disturbances propagated along the surface -wiL1besimplified to a sfieient extent to cause turhdence must beIa.rger tkm R$C,r,ti in any case; for the insukted flat-p~ate

flow at Iow speeds and with no pressure graclient, the tran-sition RecynoIds mlmber RS(, is found to be three to seven

times M large as the ~alue of Rea~tm (references 6 ancl 7).

The vaIue of Rti,f depends not onIy on RBWWizbut also on

the initicd magnitude of the disturbances with the most

“ dangerous” frequencies (those with greatest. amplification),

OD the rate of amplification of these disturbances, aDd on the

physical process (as yet unknown) by which the quasi-

stationary laminar flow is finaIly destroyed by the ampMed

osci.llat iom (See, for example, references 16 and 17.) The

results of the stability anaIysk nevertheless permit certa~ge~eraI statements to be made concerning the effect. offree-stream Mach number and thermal conditions at thesolid surface on tramition. The basis for these statements issummarized as follows:

L1I Ln mariy prob~ems of techoical interest. in aeronauticsthe Ie\-el of free-stream turbulence (magnitude of initialdisturbances) is sufficierdIy Iow so that the origin of transitio~is always to be found in the imtcibility of the Iaminar bound-ary Iayer. In other words, the ~alue of Rffcr~tfiis an absolute

lower limit for transition.V2”IThe effect of the free-stream Mach number and the

thermaI conditions at the solid surface on the stability limit

(lhc, ,) is o-rerwheIming. For exampIe, for 31,=0.70, the

vaIu~%f RtC,~ti whe~ Tl= 0.70 (heat w-ithdrawn from fluid)

is more than SO times as ~geat as the value of ReG%iXwhen

T,= 1.25 (heat adc]ed to fluid).(3) The maximum rate of amplification of the self-excited

bounclary-layer clisturbances propagated along the s@acs _____

~aries roughly as 1/%~Re..~,%. (This. approximation agrees

cIoseIy with the numerical resuIts obtained by Pretsch(reference 18) for the case of an incompressible fluid.) Theeffect of withdrawing heat from the fluid, for exarnpIe, isnot ordy to increase RgC,~~~and stabilize the ffo-ir in that

manner but SJSOto decrease the initial rate of amplification

of the unst abIe disturbances. b other words, for a given

level of free-stream turbulence, the inter~al between the

fist appearance of self-excited cksturba.nces ancl the onset of

\ransition is expected to be much Ionger for a relatively

stabIe fiow, for which ReG,~iXis Iarge, than for a relatively

unstabIe floit~, for which R~.,~ti is small and the initial rate

of amplification is large.On the basis of these obserl-ations, transition is delayed

(Rfl, ~creased) by withdraw~a h~~t from the Huicl throu~~_”~_-

thesoIicIsllrfxce anti is ad~-anced by adding heat !O the fluid.

tlwough the soIid surface, as compared with the insulatedsurface at the same liachnumber. For the insulated surfuce,

transition occurs cm-Iier as the llach number is increased, asco~,p.x~.i with the fiat-plate flow at wry Iow llach numbers.“~t-hen the free-stretim velocity at the c*dge of the boundarylayer is supersonic, trmsitiou never occurs if the rate of heattithdramal from the fluid tIwough tk.solid surface reachesor exceeds a criti~:xl ~alu? that depends onl~ on the llachnumber (section 6B ancl figs. 7 and S). ..=.——

A comparison between the results of the present cmal@sancl measurements of transition is possible only when thefree-stream pressure gradient is zero or is Md fi..ed while th~ -free-stream Mach number or the thermal conditions at thesolid suface are -rariecl. Liepmann and Fila (reference 19)

ha-ye measured the mo~ement of the transition point on a

flab plate at a very low free-stream ~e~ocity when heat i.s

applied to the surffice. They found bY means of the hot-wire

anemometer that R..,, declined from 5X 105for the insuIated

surface to a due of approximately 2X105 for T~= 1.36 when

./(~2the levd of free-stream tmhdence /—

1 (~’was 0.17 percent,

/0,u*r 2

or to a ~alue of 3 XIOS ~hen w~ (UO*)Z=005 Percent and

T,= 1.10. The value of Re,, clecl.ines from 470 (approx.) to300 (approx.) in the firsb case and to 365 in the second.

Frick ant{ Mdldlough (reference 20) observed the varia-tion in the transition ReynolcLs number when hea~ is appliedto the. upper surface of an NT.l~.< 65, %-016 airfoil at the nosesection cdone, at the section just aheacl of the minimum pres-sure station, and for the entire laminar run. T&n heat isappIiecI only to the nose section, the trcmsit iort Reynolds

354- REPORT NO. 876—NATI”ONAL ADVLSOR1- COMMITTEE FOR AERONAUTIC%

number (determined by total-pressure-tube measurements)was prfietically unchanged. Near the nose, h?8<R~C,~,~

and the strong favorable pressure gradient in the region of

the stagnation point stabilizes the laminar boundary layer

to suck an extent that the addition of heat to the fluid has

only a negligible effect. When heat is applied, however, tothe section just ahead of the minimum pressure point, wherethe pressure gradients are moderate, the transition Reynoldsnumbter Ret, declined to a -ralue of 1190 for TI = 1.14, com-

parecl with a value of 1600 for the insulatcd surface. When

heat, is applied to the entire laminar run, Rot, declined to avalue of 1070 for TI =1.14.

It }vould be interesting to investigate experimentally thestabilizing effect of a withdrawal of bea~ from the fluid atsupersonic velocities. M. any rate, on the basis of theresults obtained in the experimental investigations of theeffect of heating on transition at low speeds, the results ofthe stability amdysis give the proper direction of this effect.

7. STABILITY OF LAMINAR BOUNDARY-LAYER FLOW OFA GAS WITH A PRESSURE GRADIENT IN THE DIRECTIONOF THE FREE STREAM

For the case of an incompressible fluid, Pretsch (reference 9)has shown that even with a pressure gradient in thedirection of the free stream, the local mean-velocity distri-bution across the boundary layer completely determines thestability characteristics of the local Iaminm boundary-layerflow at large Reynolds numbers. From physical considera-tions this statement should apply also to the compressiblefluid, provic]ed only the stability of the flow in the boundarylayer is considered and not the possible interaction of theboundary layer ancl the main “external” flow. Furtherstudy is required to settle. this question.

If only the locaI mean velocity-temperature clistributionacross the boundary layer is found to be significant forlaminar stability in a compressible fluid, the criterions ob-

tained in the present paper and in reference. 8 are. then im-mediately applicable to lamimw boundary-kyer gas flows inwhich there is a free-stream pressure gradient. The quanti-tative effect of a pressure gradient on laminar stability couIdbe readily determined by means of the approximate estimate

of Rocr~{z (equation (7o)) in terms of the distribution of

the quantity p‘~ across the bounclary layer. Such calcula-

tions (unpublishecl} have already been carried out by.Dr. (2. C. Lin of Brown lTniversity for the incompressible fluidby means of the approximate estimate of RJ.C,n,~ given in

reference 5, part HI.ln any event, the clualitative effect of a free-stream pres-

dwsure. graclient on the. local distribution of ~— across thedyboundary layer is evidentIy the same in a compressible fluidas in an incompressible fluid. lf the effect of the localpressure graclient alone is considerecl, the velocity distribu-tion across the boundary layer is “fuller” or more convex foraced erated than for uniform flow ancl} con~ersely, less

convex for decelerated flow. Thus, frori the rcsulw of ~hcpresent paper the effect of a negztive pressure grad icnt. onthe laminar boundary-layer flow of gas is stabilizing, so faras the local mean velocity-t. emperaturc distribution is con-cerned, while a positive pressure gradien ~ is des tahil izing.For the” incompressible fluid, this fact. is ~vcll-cstal)lisl~t~{lbythe Rayleigh-ToIImien criterion (reference 3), tl~e work ofHeisenberg (reference 1) and Lin (reference 5), and a mass ofdetailed calculations of stability limits from the curves of aagainst R for the neutral disturbances. These calcuh~tiouswere recently carriecl out by several German investigatorsfor a comprehensive series of pressure gradient, profllcs.(See, for example, references 9 and 21.)

Some’idea of the reIative influence 011]aminar sta]jili(y ofthe thermal conditions at the solid surface and the free-stream

pressure gradient is obfiained from the ec~uations of mean

motion. At the surface,

or

In ti region of small or moderate pressure gradients

tknal conditions at the solicl surf are. For e.x[inlple, tlj[]chordwise position of the point of instability of the huninarbounclary layer on an airfoil with a ffat. pre.ssurc rlistribut ionis expert.ed to be strongly influenced by heat conduc~ionthrough the surface. (See reference 20.) For the insulat,udsurface, the equations of mean motion yield the foIloIrillgrelation (appendix D), which does not involve the pressuregradient explici tIy:

The effect of “aerodynamic heating” at the SUI’f:WCO1)JNW.Sthe effect of a favorable pressure gradient so far as [he ciis-

tribution ‘of p ‘—w across the boundary layer is conmmcddy

(equations (79) and (80)). The relative quantitative il~flu-ence of these two effects on lamimm stabiIity can only besettlecI by act uaI calculations of the Iamina r boundtiry-ktycrflow in a compressible fluid with a free-stream pressuregradient.. .4 methocl for the calculation of such flows o~eran insulated surface is given in reference 22.

RThen the local free-stream veIocity at the edge of the.boundary layer is supersonic, a, negative prcssum gradientcan have a clecisive effect on ktminar stability. The Iocnllaminar boundary-layer flow over an insu]at ed surfacrj forexample, is expected to be completely sttible wh[:n the! mag-nitude of the local negative pressure gradient .macl~es orexceeds a critical value that depends only on the local 3iach

THE STABILITS OF THE MMLNAR BOUNWMW LA1-ER IN’ A COIIPRESSIBLE FLUID 355

number and the properties of the gas. The critical. magni-

tude of the pressure gradient is that which makes the-quantitycl dw

()@ PzysufEeiently large negatively near the surface so that

1‘henc= 1–mn”

It has already been shown in th; present paper than w-hen

M0>3 (approx.) the laminar boundary-layer flow with a

uniform free-stream -re]ocity is completely stable under free-

flight conditions when the solid surface is in thermal equi-

librium, th~t is, when the heat conducted from the fluid to

the surface balances the heat radiated from the surface (sec-

tion 6B). The laminar boundary-layer flow for thermal

equi~brium should be completely stable for MO>J1,, say,

where M$<3 if there is a negat i~e pressure gradient in the

direction of the free stream. Favorable pressure gradients

exist. over the forward part of sharp-nosed airfoils arid bodies

of revolution moving at supersonic velocities, and the limits) of the Iaminar bowndary layer shouldof stability (R@C,~{m,

be calculated in such cases.

CONCLUSIONS

From a stud-y of the stability of the Iaminar boundarylayer in a compressible fluid, the following conclusions werereached:

1. k the compressible fluid as in the incompressible fluid,the influence of viscosity on the kminar boundary-~ayer flowof a gas is destabilizing at. very large Reynolds numbers.If the free-stream velocity is subsonic, any laminar boundary-layer flow of gas is met able at sufTicientIy high Reynoldsnumbers.

2. Regardless of the free-stream Mach number, if theprocluct. of the mean density and the mean vorticit.y has an

“trem”nw’%)“ )vamshes for some value of w>l —*0

(~vhere w is’ &’ra& of mean” velocity component para~el

to the surface to the free-stream veIocity, and where MO is

the free-stream ilach number) the flow is unstable at sufEi-eiently high ReynoIds numbers.

3. The actual Iimit of stability of laminar boundary-layerflow-, or the minimum critical Reynolds number Re,,=im, is

determined largely by the distribution of the product of themean density and the mean vorticity across the boundarylayer. .&n approximate estimate of Re,,~,~ is obtained that

serws as a criterion for the influence of free-stream Machnumber and thermaI conditions at the sofid surface onIaminar stability. For zero pressure gradient, this estimatereads as foIIows:

where T is the ratio of temperature at a point within the

boundary layer to free-stream temperature, T, is the ratioof temperature at. the solid surface to the free-stream tem-

perature, and co is the value of c (the ratio of phase ve-

locity of disturbance to the free-stream velocity] for wfich

(1–2k)t2=0.5L30. The functions V(C) and A(C) are defined asfollows :

where

( *,g)T nondimensional distance from surface y

4. On the basis of the stabiIity criterion in conclusion 3and a study of the equations of mean motion, the effect of -

adding heat to the fluid through the So~d s~face “is to .=

reduce R%=~ti and to destabi.li~e the fiow, as comparec~ ‘~th

the flow over an insulated surface at the same l~ach number. .. . .~I&dra~g heat through the solid surface has eRactIY the

opposite effect. The ~alue of R~~=ti for the laminar bounclarj-

layer flow over an insulated surface decreases as theMach number increases, and the flo~ is destabilized, as

compared with the Bkisius flow at low speeds.

5. V7hen the free-stream velocity is supersonic, the lami-

nar bounclary layer is completely stabilized if the rate at.which heat is withdrawn from the fluid tiough the solid __..

surface reaches or exceeds a certain critical value- The ____

criticaI rate of heat transfer, for which ReC,mk-+ ~, isd ()(in

that. which makes the quantity ~ P ~y sticiently lmge

negati~eIy near the surface so that II –~Mc)l ~’(~)=0.~~0.-when c=co=l—l~. CaIcuIations for several supersonic

Mach numbers bet~veen 1.30 and 5.00 show that for MO>3(approx.) the critical rate of heat withdrawal for laminarstablity is within the order of maadtude of the calc~at@ __rat e of heat conduction through the solid surface that bal-ances the heat radiated from the surface under free-flightconditions. Thus, for .110>3 (appro~.) the laminar boundary-Iayer HO-Wfor thermaI equilibrium is completely stabIeat all Re.ynolck numbers in the absence of a positi~e (adverse)pressure graclient in the clirection of the free stream.

6. Detailed calculations of the cum-es of m-aye number(in~erse -wa~e length) against Reynolds number for theneutraI boundary-layer disturbances for 10 representativecases of insulated and nonindated surfaces show that alsoat subsonic speeds the quantitative effect on stability ofthe thermal conditio~ at the SOfiClsurface is ~fw Iarg$-For example, at a >Iaeh number of 0.70, the value of Rs,, ~,=

is 63 whe~ T1= 1.25 (heat added to fluid), R~c,rnti= 126

when T1= 1.10 (insulated surface), cmd R~c,n~=5150 when

T,=O.iO (heat -withdrawn from fluid). Since R.. = 2.25R/,the effect on R..c,%iR is even greater.

356 REPORT N-O . 876—NATIoNAL ADVISORY COMMITTEE FOR AERONAUTICS

7. The results of the analysis of the stability of Iamiuarboundary-Iayer flow by the linearized method of small per-turbations must be appliecl with care to predictions oftransition, which is a nonlinear phenomenon of a differentorder. Ilrithdrawiug heat from the fluid through the solidsurface, however, not only increases Rg,,m~mbut decreases

the initial rate of amplification of the self-excited disturb-

ances, which is roughly proportional to l/l/Rff,, ni~; addition

of heat to the. fluid through the solid surface has the oppositeeffect. Thus, it can be concluded that (a) transition isdelayecl (Rffi, increased) by withdrawing heat from the fluidand advanced by acIding heat to the ffui~ through the soIiclsurface, as compared with the insulated surface at the samehlach number, (b) for the insulated swrface, transitionoccurs earlier as the Mach number is increased, and (c) whenthe free-stream velocity is supersonic, transition neverOCCU]Sif the rate of hea~ withdrawal from the fluid throughthe soIid surface reaches or exceeds the critical value forwhich R8$r~t~+ ~. (See conclusion 5.)

TJnlike lmninar instability, transition to turbuIent flow inthe boundary layer is not, a purely local phmomenon butdepends on the previous history of the flow. The qutintita-tive effect of thermal conditions at the solid surface ontransit ion clepends on the existing pressure gradient in thedirection of the free stream, on the part of the solicl surfaceto which heat is appIiecl, ancl so forth, as well as on theinitial magnitude of the clisturbanws (level of free-stream

turbulence).

.4 comparison between conclusion 7 (a), based on the

results of’ the stabiIit y analysis, aml experimental investi -

gations of the effect of surface heating on transition at low

speeds shows &at the results of the present paper give the

proper direction of this effect,.

8. The results of the present study of Iaminar stabilitycan be extended to include laminar boundary -Inyer [1OWSof a gas in which there is a.pressure graclient in the directionof the free stream. AIthough further shdy is required, itis presumed that onIy the Iocal mean }’cIocity-tcJ1lI}c’r:l IlurecIistribut-im determines t$e stability of Lhe local boundary-layer flow_ If that shouId be the etise, the cffoct of u pressuregradient on laminar stability COUIC1be easily calculatedthrough its ef!eci on the local clistribution of the productof mean clensity and mean vorticity across the boundarylayer.

I}’hen the free-stream velocity at thu” edge” of tk: bound-ary layer-- is supersonic by analogy with the stabilizingeffect of a withdraw-d of heat from the fluid, it is expcc[ cdthat the lami nar boundary-Iayer fIow is compkt cly stddcat all R.?ynolds numbers when the negative (favorable)pressure gradien~ reachw or exceeds a certain critical valuetllfl~ clepends only on the Mach number am] k praperticsof the gws. The laminar boundary-layer flo~v over a surfacein thermal eclui~ibrium shoulcl be completely S(abk forJMo>Msl say] where Ms<3 if tkrc is a negntivc p rrssurcgradient in the direction of the free stream.

~~ivGLEY llEkIORI.4L .AMRONIiUTICAL LL4B0RAT0RY,

JN~TIONAL ADVISORY CIORiW~TEE FOR AERGN.i UTICS,

LANGLEY FIELD, JTA., September 5, 1946.

●

APPENDIX A

CALCULATION OF INTEGRALS APPEARING IX

h order to ca]culate the limits of stability of the laminar boundary

THE INVISCID SOLUTIONS

layer from relations (21) to (29) between the values

of phase velocity, w-aye number, and Reynolds number, it is fist necessary to calculate the ~alues of the integrals Ki,

Hl, Hz, LV1,Mz, .TZ, and so forth, which appear in the expressions for the in-riscicl solutions p,(g) and pi(y) and theirderivati~’es at the edge of the boundary layer. These integrals are as follows (equatiom (13), (9), ancI (10)):

and so forth.

Terms of higher orcler than a3 in the series expression for m and K, are neglected. When a<l, the error inl-olved is

srmdl because the terms in the series cleeline like ~~. Even for a> 1, however, this appro.ximatio~ is justified, at least

for the l-alues of c that appear in the stability calculations for the 10 representative cases selected in the present. p~per,

&&lk-l “’l’ip’i’[’ byFor example, the ]eading term in R. P. ,Vti~,(c), trhere L=2,3, . . ., is approximately ~

the IeacIing term in R. P. .Vs(cl. The quantity iu the brackets is at most 0.12 in the present. czdculations; for example,

R. P. ~~,(C) =0.06 R. P. Il:,~C]. Jloreoyer, R-. P. Ak(e) = (1—c] R. P. .Yti~,(c). SimiIar approximate relatiom exist

kwtwee~ R. P. .ll~k(c) and R. P. M(c); and, in addition, R. P. .113(c)= (l—c) ~ R. P. ~T3(c)==0.015 R. P. ;~~(c), at most. -

The only integral for which the @aginary part is calcdated is K (c). .lt the end of this appencIix, it is shown thatthe contributions of the irnagimary parts of Hj, 313, and .Tz are ne.gIigible in comparison with the contribution of I. P. K*(c).

GESZRALFLAX OF CALCULATION-

The method of calculat iou adopted must take into account

d dw(jthe fact that the -ralue of ~ p ~ at. the point g=y,,\

where W=C, strongly influences the stability of the laminar

boundary layer. .iccordingly, the integrals are broken into

two parts; for example,

I=li*L(c)+ K-,.(C) –M,~

where yf>y,. The integral K,,(c), which involves

[:(491.=:is calculated very accurately, whereas X&(e)

iis calculated by a more approximate method as foIlows:

I

(Al)I

This integral is evaluated as a power series in c. The velocity

profile w(y) is approximate{ b-j a parabotic arc plus a

straight-line s~~ment for purposes of integration. h the more

complex integrals F7?j M3, and & the indefiuit e integrals

point numerical integration by means of Simpson’s rule.

The dues of to(y) are read from the velocity profiks offigures 1 and 2. The -iaIue of ~j–~~= a is 0.40 in the presentseries of calculations; this value is chosen so that. the pointy= yf is ne~er t00 cIose to the singydarity ai Y=YP Take

(.%2j

sThe integraI k,,(c), or the indefkite integral J ~, dy* ~, (w—c)

that appears in Z, M,, and hr,, is evaluated by expanding

357

358 REPORT NO. ‘876—NATIONAL ADVISORT COhHWITTEE FOR AERONAUTICS

the integrancl in a TayIor’s series-in y—y, and then integrat-ing the. series term by term. The path of integration mustibe taken below the pointi y=y, in the complex g-pkme,

Instead of calculating the values of the velocity and tem-perature clerivatives w,(~j and ?’.(”) dircctIy, it is simpIertio relate these derivatives to their values at the surface byTaylor’s series of the form

The derivatives at the surface WI(,1)and Tl(~l are calculated

from the equations of mean motion (appendix B).The in tegraI KII (c), for example, is finally obtained as a

power series in y,–yl=a and in Yj–y,=a–u, pIus termsinvolving log 0-. The phase velocity c is related to a by

where

#jk=w$

Terms up to the order of a5 are retained in order to include

all terms involving wl” i‘,

DETAILED CALCULATIONS

In order to illustrate the method, the evaluation of K, (c)

is given in some detail, as folIows:

(I) Evaluation of K,(c):

K, (c)=J

‘Y’ T~,m dy —M2

(a) Define

svi TK,,(c)= —

,, (w–c)’ ‘~Now

T --(W:CP=(W,’)’(?H,)’V

where

W=l+k (2/-?/,)+–&;(y–y,)’+ ...

The function $ is cleveloped in a Taylor’s series around the

point w= c as follows:

$=($),=,+($)’T “(Y-W)’+ . . .(y–Ye) +(p), 2!

(k-l-l )

*C(’)= (kw~I) W,’Then

– “-’”’(y-yc~[($).+($I( y-’,)+JKil (c)= (w:/)’ ,,-,. (y–$/J

T “(y-y,)’

0~c ~! +“”1

(),~y!f,

‘rzpc [(yf–lJc)J–(?J,-7JC)’]+ . . ,

T (k+l)

()+(EJ+l!) ~ . [(Yj–’Yc)k-(yl-?/c) ’]+< . ,}

whereyl—y, =lyl —y,le-tr

y,—y,= (?J,-yJ— (yc—yJ =a—c

U=yc—y[

(?(k)

The coefficients —@

are expressed in terms of derivatives

of T and w at y=~l as follows:

Define

fo(!/)=-&

[(-)1_fk(Yc)= (W,9’ (L)m ; ,,

fk’’tw)=~k(Yl) +.fk’(Yl) (YC–Y1) + -jr (Ye–w)’+ . . .

(The method acloptecl for the ccdcuktion of fkn(y,) from (he

velocity and temperature derivatives wlc~j and T, ‘~~ is givcu

at the encl of this appenclix.)From the expression for K,, (c),.—

1, P. KI, (c)=l. P. K,(c)

=~fl (tic)

[‘z fl(yl) +afl’ (?/l) + . . . +: -fIv(YI)

1

and

R. P. ~ll(C) +#--=Co+C,@+C@2+ . . . +c~d+

[& fo(YI)+ufo’(YIJ+ . . . +

w}]ere

a=yc—yI

8.= af2(y1)+ a2j3(y1)+cz3f4(yJ + a4f5(yJ -+-a5fa(yl) + . . ,

THE STABILITI- OF THE LAMIA’AR BOUND.4Rlr LAy13R IN’ A COMPRESSIBLE FLUID 359

s3=~[af,’” (w)+a73’’’(yl)+ . . . ]–

+ [3aj,’’(yl)+ . . . ]–[4a~,’(yl)+ . . . ]+

[5af,(yl) + . . . ]

.4+1 &d~=–* (j-+ I)!

(CZ*=l.0)

((l= o.40)

(b) Detlne

J

‘uh’,,(c) = ~, dy

,, [w–c)

r1.0 T. —, d(y–yl). n.w (W—c,)

=~a,(l+l)c’

w-here

The velocity profiIe w(y) is approximated by a parabolic

are in the interval 0.40 ~y—y~ = &I—~l and by a straight

line (w= Constant. =w(yJ) in the interml ys—yl =y–yl <1.0.

The ~-aIue of y? is determined by imposing the conclition

that the area uncler the parabolic-arc straight-line

segment equals the area under the actual ~elocity profile

W(Y) in the interval 0.40 =Y—LJ1 = 1.0. The paraboIic

arc w= 1+ rn(Y—YI) + n (Y—VJZ is determined by the

following conditions: n-hen Y=v4< 1,

W’=o

when y=yj and ~j—yl=o-~o;

where ~~~(yj) is read off the velocity profile of figures 1 and 2.

The value of Y4 is chosen so that the parabolic arc fits the

velocity curve w(y) closely over the widest possible range.

l-or G= 1,

and

lk is e-raIuated b-y approximating w(y) by a parabolic arcas follows:

[

m+2n(y—yl)

1

r3-91+1*=– (&l [1+m(y–y,)+n(y–yJ’]’-l mm

where .4=m2—41n.As a control in the calculation of the series expression

~ ak(k+ l)c’ for Ar,z(c), use is made of the fact that, fromk=othe definition of 1~ and J~,

Lim (1.+ J,)=1

k—m

[1

~ ‘W’(~j)— [W(gf)]kW Q/j)

and therefore

()

a~~l lklim — –-—

k—- a~ —W(.yj) k+l

The remainder after ATterins in the series for A“IZ(c) is gi-renapproximately b-y

~(~1’+ 1) term]

[’-%1

The real part. of K,(c) is obtained by combining the r~snlts

of (a) and (b); that is,

(2) Evacuation of H,(c):

The integrand of this integral is free of singularities in theregion of the complex y-pIane boumled by y=yl and y=~;therefore, 271(c) is e-raluat ed by pureIy rmmerical int egrat ion.The actual procedure employed for the calculation of in-tegrak of this type is as folIom-s: (The integra~ HI(c) servesas an illustration.)(a) Define

360 REPORT h’O. 876—hrATIOhTAL ADVISORY COMMITTEE FOR AERONAUTICS

where,—

and,—_—

(b) ~TTitll tlle3pproximation tllattl leviscosiiy}- ariesIinearIJ~

with the absolute tempcrzturej the veIoc.ity w is the same

function of the nonclime.nsionaI. _stream function f as in the

Blasius flow; that is,

W=w(f-) =wB(f) .

where f is defined by the rektion d~=pw dq (appendix B).

From these relat ious .

SiIICC. (i{=wB dqB. Jforeoyer,

(iq=---&=T(wB)d~B

wh e~e

[ 1.T(W.)=T, – (T1–l)–~Mo’ w.–% Alo’w.’

for Cr=l.

(c) Finally, from the relations given in (b),

(J J13,(c)=; : WB’CL?B-2C :0 w. CL?.+c’

)

l—~where bOis the value of 8 /’_~ for the Blasius flow. For

\ ~(,,x:

the insulated surfaces, /10,which is somewhat arbitrary, was

(:hosen zs 5.60; whereas for the noninsulated surfaces,

bO=6.00. (The value of wB at ~B=5.60 is 0.9950; when

V~=6.00, WB= 0.997-5, The value of b for the insuIatedsurfaces is the vaIue of ~1at which w= O.9950; ~ihereas b forthe noninsuIated surfaces is the value of .T for whichw=0,9975. ) The advantage of this procedure is that the

“’b@

integralsJ

wBndTB are calculated once and for all and theo

value of ~1 (c) depends only upon the values of b and c.

k fact,

J

bo

( *,/~)B-(~\/-)B=bo-239~~WB2d~B= ba— 6

0

since

(’*@B=’.’30

and!— 1—1

AIso,

sbo

WBdqB=bO—1.~30.0

and

SJb= : dq= “b’TdVB

o

=bO+l,73(T,–1) +0.6667 y~~lIo’

[ 1‘—1 Ml)z +2.3967 ~;-l M02=bo+l.73 (T,–1)–~

See appendix B. (Incidentally, the lasL rclatiol~ shows the

eflect of free-stream illach number and thermal conditions

at the solic] surface on the “thickness” of the. lx-lllndarylayer.)

(3) IZvaluation of ~,(c):

H,(c) =J

Vz T— ,~fo2(w—c) 2

-J

Y(u>—--c)~

(W–C)2dy -7,=- dy

n $’1

Define

(a) The integral ~,,(c) is evaluated by methwls similar to

those aLre&dy outlined for the evaluation of 111(c). TII us

where

exprcssiori for H22(c) are evaluated

using Simpson)s ruk.

@# dy

JH,,, (c) = ‘i T ,dy

Ju @#C&

,, (w–c) fi,

JH,,,(c) = ;: ~w-:aj dy~: ~)-’ dy

1

THE STABILITY OF THE LAMINAR BOUNDARY LAYER IAT A COMPRESSIBLE FLUID 361

The integd Hzlz(c) is evaluated as follows:

@# dy

Define

wk?re

and

[ 1T= T, – (T,– 1) –T~~I/ U)– q.ll#w’

The integral ~(c) is e-raluatecl by numerical integration using

Simpson’s rule: the required values of w are read directly

off the velocity protles of figures 1 and 2. I?hally,

H,,,(c) =K,,(c) H,(c) –P(c)

The integral H2,1(c) is e-raluat ed in exactly the same way zs

KI, (c) -where[W—C)2—= (WC’)2(Y–YC)2 ($)

T’and

4(!/) =l+W+,(M +“&k?/c)’+ “ - “

R. P. [HZ,,(C)]= (boay+coa~+doa~+noa~ +

h~+j(’’;-j2-

T1’ ,1” ,‘3–2A’ z– T1 ‘2

,1’ 2

(-)1,1

362 REPORT NO. 876—NATIOhTAL ADVISORY COMMITTEE FO-R AERON~UTIC~

.-lk=w~

At’’ =Ak+1’—A2AA2—Ak’Ak (2~k~5)

Ak’’’=Ak+l° –A2.4h’’–2Ak’A2 ”Ak2”Ak (2 s ks4)

.Clki’=lik+l’ ”-. $J4 A~’’’—3A~’3A~’– 3A~’A2~ff A~’ffA~(/?=2, 3)

At’’= A~~1’’—A2Aa4A~4 A2’6A2’’—6A2’’At”-

4A2’’’AA2~AAk’Ak (k=.2)

B7(m)=Ai+1(m) —Ai(m+l)

B7=A3’

BT’= 2AaA3’

B7’’=2(A3’)2+ 2A3A3°

B8=.Q4,

B~’= A~’AA+ A~A~{

Bg= AL’

B,0=A3A5

l?inally,

R. l?. 172(c) =R. P. FZ211(C) +~21z&-Mo2&z(c)

(4) Evaluation of M,(c): .

.lx~(c)=J

V2(w_c)2u~ dy ; &

)S—M02 dy ‘@#dy

VI ih=.uj, (c) –-M,’M3,(C)

where

and

4rw’d’”1:1’TwB’’Bd’~--2c3(flW.dq”~:TqBdqB+

J3f..l’ )Tw, d~, dq, +

JJC’ bodq~ b’TqB d~B

(1 lln

where b. has the same meaning as in the evaluation of Ilzz (c)and where

[7—1 ‘

1T= T(w.) =T, – (T,–lj –7 MO w.–% ilf #wJ

The integrals in M32(c) are evaluated by numericaI intqgw-

tion using Simpson’s ruIe. Yalues of w~ are taken from ~he

table in appendix B.

(b) For em.venience, the int egrd .1131(c) is transformed asfollows:

M3,(C)=M311(C)+313,2(C)–M3,,(C)where

.313,,(C)=J J

‘i W dy “ -~s

‘J (W—C)2Y1

~dyy (w—c) $fl T-dy

s J“ ‘~g~ dy ‘2~

J~ (w—c)~11312(c) =

lil ,, (w–c) 2‘lY ,, ---F’- ‘~

m,(c) =J,J

“ *T@ dy “ –—,, (W:c)’ dy,[ ‘%2 ‘~

It is recognized that

J

L!(W—C)2~ dy=H, (c)

gl

J

u,, &2 @~: ‘K;cr @=~~2,2(c)

ThercfomMg12(c) =~1 (cj_Hz12(c)

By additional transformations, the. following equations areobtained:

M3,,(c) =H, (c)P(c) – Q(c)

BOUND.ARY LAYER IN .4 COMPRESSIBLE FLUID 363THE ST.LBILI’IT OF THE ILMIHT.A.R

where

Q(c) =J J

““ @#’ dy ; & dyJ” @# dyYi f

or

The integral Q(c) is eduated by numericaI integrationusing Simpson>s ruIe; P(c) is e-duakd in the calculationof H22(c).

The integraI M~,I (cj k obtained in exactly the same -wayas KIL(c) and H211(c); that. is,

are

‘2J)JWJN

The integrak in .iV~A(c)are evaluated by mmnerieal integration

in a manner similar to that usecl in the evaluation of M3Z(C),

and so forth. Most. of the integrals wiI1 already have been

evaluated in the calculation of ~l(c), lZZZ(c), and M&(c).

(bj For convenience, Na2(cj is broken clown as foIIows:

J

~’ T

s

r-(w-- e)~1?’32(c)= — —

~, (w–c)’ ‘y ~, T “’-y)dy

J

9i.

,, &#yJ?# @-@d~+

s~ dy

J

f (w—c)’;’ (w–c) ,, T ‘~’–y)d~

LetN3j (c} =.N321 (c) + 11522(c)— 41T323(c)

where

1’

Y,llr3zI(C)=

=2 dg ~ (W–C)2~ k-y)dy._ w L1

J

1!

J

~%fw—G~2ti322 (C)= ~j +2 dy y, ~ (Yz–y)dy

N3,3(c)=f: a dyf’ %#’ (yz–u)dy

NTOW,

N323(c)=l“ ~, dyJ~ : (’w-c)- ,“ ‘;< [(Y2–Y1) – (y–yJldy

Since y,–y,= 1.0, and

JP(c) = ,: &, dysy’ @# dy

Y

it. is found that

N323(c) =P(c) —PI (c)

where

JP, (c)= ‘*—

s“ q (?4-Y1) dy

Fi (W:c)’‘y ,

sb T

—, G, (q;c)dn‘; o.,~(w–c)

and‘b ~2

G, (?j’;C) =

J J

“b ? dq,Tqdq–2c b?

J, ~qdq+c2 —

,T

PI(c} is evaluated by numerical integration using Simpson’srule. Define

J

h.i\$22(c)= ,, &zdY~Q~@ ~(YZ-YI)-(Y-YI)] dy

364 REPORT hTO. .876—NATIONAL ADVISORY COM.MLTTEE

Since

J

v?~i & dy= K-,2(C)

and

\

‘W (W)-C)2~ dy=H, (c]

. uit is recognized that”

N,,,(c) =K,,(c) [H,(c) –,~ + (+iq

FCIR AERONAUTICS

J

V2(W—C)24’1~ (y–u)dy=; J :“w-c’’d’lqd’=~[r’(w’-’cw)d’l””l””l

J

‘M (w—c)~—T— (mkh=i. f/] (lbdwB:d”lB ‘d’’-’’rw’””r ‘d’”+c2J7d’’rTd’”)

The integral(‘“% (Y–w) is evaluated hY l~umerical integration in ~~act~y the same W~Y aS MA(c).

. 11The integral ~S21(c) is tnmsformed as follows:

J

#iN321(c)== ,, ##YJ:9 [(Y2-YC)-(WWC)] d,

But

J

>Yi

s

v (W—cy

,, (W:c)’ ‘y ,1 T ‘ ‘~= H2’’(c)and

‘Y2-?/c= (W-yl)-(gc-w) =1–0

so thatN,,,(e)= (1–U)E?2,1(C)–JR,(C)

where

& dys

Y (W—c)f=J2,,(4=J; (w–c) W—T (g–y.)dy

Theintegral J,,, (c) is evaluated in the same way as .&,( c).

Thus

[<+”’(-$’”)l+’’ f=)[*;#a’+”’(-E,m’” 4 {m IM,l)lz f)l+/2a’+’4c”-

4 jl(y,) +~ f,’ (?/1)_jl @M1’ (YJ 1R. P. ~,,1 (C)=+r ~

[ 1(–~2–4a300+a4(5Do+ @ +a2 :+6a2co– 10a3~0–4a3d,

)(+ U3 ~–4aCo+ 10a’D,+ 6u20, -1-asll$+ a

)

[a.ff_4a Ql-5a DD +UC’ –: po+~ ~o-(gl)

1 5(-:”) ___whe~e

lflOA.co=g–~fo(vl)

{1 $1’ (w) _J (?/I)fo’ (m)

cl=+–~” jo(ylJL?O(YI)12 }

4 yl(Y1) ] 1 fz(Yl)Do=&% jO(y,)

— ‘0 Zofo(yl)

R. P. ~32(c) =R . P. N321(c)+~T322(c) ‘AT32s(C)

J?inally

THE STABILITI- OF THE LAMINAR BOUNI).4RY LA?lER IN A C031PRESSIBLE FLUID 365

After severaI transformations, the int egraI .LVS,(c) k brought into a more con-renienfi form

The integral

been performed.

The integral

Li,,,(c) is e~aluated by numerical integration using Simpson’s rule. Some of tie iutegrat ions have already

The integrzal is. given as

N,,,(c) is e-raIuated in exactI-y the same way as KI~(c); that is,

366 REPORT NO. 876—NATIONAL ADVISORY COM&fITTEE FOR AERONAUTICS

F=-p,fI (y,)+ gofo(w) –; fdw)

EVALU.4TION OF fkcd

The functions -fk@’j(y,), which appear repeatedly in theevaluation of the integrals Kl(c), H1 (c], and so forkh, areevaluated in terms of Wl(k) and Tit@ as follows:

[()1fk’’(?h) ~’+ . . .T ‘ ,,=fk(YI)+fk’(3h) a+~“f’~’=(wc’)’:– l)J! ~

fo(~)(YJ = – (Tgo)vl(’@

[nz(m—-l)T1(nz-2)g2+...+=–T1(’@go+mT1@-l)gl+ ~1

~!

(m–r)!r! 1TI@-’lg,-f- . . . + Tlg., (Os?nsq

where

g“=&’

g~= — 2g@4j

g,= - 2(goA,’ +91Z42)

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

9?n=-2[90~2(m-1)+ (m—l)gIx42@–2)+. . .+

~~!~j !r! 9T~2(m~’-’) +; . .+9m_~AJ

( )~I(m)(YJ = T’go+~ g,T ‘m)= (T’gO)vl@)+; (g, T) ,IOZJ

q

(0s?72s5)

f2(m)(y,)=+ (T’go) ,l(m~’)+; (Tgo&) u,(m) (os?ns4)

where

; (~90s2) VI“n)=--; [jok’1) ~zl,, ‘m)and

&@)=: B2(k~—; &@} (osks4)

( )f3(n)(Y1)=~ T“’go+: T“gl (w+* (T’90S,)U @J_

VI 1

& lYo(!A)s31,1(~) (os7ns3)

TEE STABILITY OF T3E LA31LX’AR BOUNDARY LAYER IA’ A COMPRESSIBLE FLUID 367

~6m=~8w —+ ~i(t+l)

P,=B,

C,= A,A,A,

m~)=w+ (72@’fi~

D3@)=; (zc’,(~~+ c.(~~— ca(~+~))

(E}+ (&(a_~z(w))D4C’)=$ (2C,

~,(k)=+ (~c,(~)+ ~,(~1– p,(~+u)

E2(~)=D3@)_+ ~2(k+I)

~z(k)=l 3~5(k)+~4(k)_~3 (E+I))~ (’

Fzc~)=Eat’j –~ ~(~+)

ORDER OF M.4G>TTKIDE OF 1MAGINAR% PARTS OF ISTEGRAIS H?,.11:,AN’D Xz

In the detailed stability calculations the contributions ofthe imaginary parts of the integrak .HZ,M~, AT~,and so forth,to the function u(c) are considered to be negligible in com-

parison with the contribution of the imaginary part of

K’, (c). A. calcuktion of the orders of ruaetit ude of I. P. n,(c),

I. P. J12(c), and I. P. ~Wa(c) from the general expressions

gi-ren in the preceding pages shows that this step is justitled,

at ~east for the values of phase velocity e that appear in the

stability calculations.

For example,

I. ~. ~,(c) =T. P. ~211(C) =zA(wC’)~, (gc)

where

A=––*- —32-’. (%:’) ’+0(’4)+ ---

Therefore

I. P. H,(C)2

= ‘~~1 (~c)T,(u,,,)

The contribution of 1. P. ~z(c) to u(cj is approximately equal

‘oo’[f&)l’wh’re r”=%l-p-K’(” ‘Lequanti’yhthe brackets is of the order of O.O3,at most, in the caIcuktionsof the present paper. (In the approximate calculations ofRg~,minfor Mach numbers very much greater than unit~, c

becomes large because c> 1—-$0; however, a is smaIl when

c is not much greater than 1—& and the results of the

calculations of Re=,~t~ based on the approximation

~(c) = T* I. P. K,(c) are qualitatively correct (fig. 7).)I

From the expressio~ for i$’z(c),

1.P. .iV,(c) = c’ I.P. K,,(c)2 (25,’)’

so thd the eontribut io~ of I.P. .i17s(c) to o(c) is approsi-

[1

~’2c2mately equal to U. — - The quantity in brackets is of

2(.W,’)’the order of 0.06 ab the most.

The imaginary part of .lf~(cj is considerably smaller. In ftict,

c’1.P. ~~,(C) =9 Tl,(w,~)~ r.P. K,(c)

and the contribution of I.P. Ma(c) to u(c) is approximate eIy

‘qualtO’&%:)21-The quantity in brackets is of the

order of 0.001 at. maximum c.

APPENDIX B

CALCULATION OF MEAN-VELOCITY AND MEAN-TEMPERATURE DISTRIBUTION ACROSS_ BOUNDARY LAYER ANI) TIWVELOCITY AND TEMPERATURE DERIVATIVES AT THE SOLID SURFACE

The mean-veIocity ancl mean-t empe.rature profiles for tlIeseveral representative cases of insulated and noninsuhited

surfaces are calculatec~ by a rzpid approximate methoc{ that

gives the slope. of the velocity protiies at the surface with a

m~ximum error of abou~ 4 percent in the extreme case, for

\vbich ~1= 0.70 and JWO=0.70. Tkw surface \ralues of the

]]igher ~’elocity derivatives and t]] e t emperat,ure ckwivatives

required in the stability calcul.ations are obt~inecl directly

from the equations of mean nlotion in terms of the calculated

vctlue of the slope of the velocity profile. The l’randtl

number is taken as unity.

MEAN VELOCITY-TEhlPERATURE DISTRIBUTION ACROSS BOUNDARY

LAYER

In a seminar held at the California Institute_ of Tech-nology in 1942, the present author has shown that a goodfirst a.pproxima tion to tke mean velocity distribution acrossthe boundary lciyer is obtained by assuming that the vis-cosity varies linearly with the absolute temperature. TViththis assumption, the velociby w(f) is the same func Lion of

@the nondimensional stream fun-ction f= ~_— as in the~vo*~*~*

Blasius case, and the corresponding distance from the

surface q=y* /–~ is obtained by a simple qu adrat.ureJ T;*

when o-= 1. Actucd]y, the approximation w(~) =w~(r) isthe first stage of an iteration process appIiecl to the differ-ent ial equations of mean motion in the Iaminar boundaryIayer, in ~vhich Pm T~-’ (e is a small parameter equcd to 0.24for air), and w(f) =w~(f) + WJI({)+ e%o(f)+ . . . CaIcu-Iation of WI(() for T,= 1.50 and TI = 2.00 for MO~O showedthat the iteration process is rapidly convergent; the con-tribution of the second term to the slope of the veloc.i typrofile at the surftce is 5 percent for TI = 1.50 ancl S percenifor TI =2.00. In the present calcuktions the maximumerror in the slope introduced by taking w(~)= WB(f) is ~bout4 percent in the extreme case. (See reference 15, in whichthe authors make use of a linear l“iscosity-temperaturerelation, See also reference 23.)

That w(~) =w~(() for a Iinear variation of viscosity withabsolute temperature is seen directly from the actuations ofmean motion in the Iaminar boundary Iayer. The equtitionof continuity is aut omatica}ly satisfied. by taking

F - a**= ‘“”=~Po*

368

and

The st~eam function ~* and the (distance along [he. surfticcZ* are selectecl as inclependcni variables folkwing the I]ro-cedure of Von- Mkes, and the dynamic equation d m(’a.nmotion becomes for zero pressure gradim~

Define the nondimensional stream fllnction ~ by the relatiorl

r=~+!l;->.The dynamic ecfuation takes Lhc following

form:—

. .

Since p= ~ in the bounclry layer, if p= T} the dynamic cqu a-

tion in this form is identical with the cquat ion fur thcisothernlal BIasius flow, tha.~ is w(~) =w~(~), or the va]~c of

the velocity ratio w is equaI to the Blasius value mt tbo same._—

value of_~. The cor~esponding value of ~=Y* J$; ‘1”

nondimensional ~‘dist ante” from the surfzce} is Obttlined asfollows:

or

If c= l,--the energy and dynamic equations have a. uniqueintegral gad

as shown by Crocco. Therefore,

But-w(r) = w~(f), cind

TEE STABILITY OF TEE IANIA7.%R BOU3TDAR1- J1411ZRIN A CONPREWIBLE FLUID ..369. -_

J

~B

[

*8The integraIs WEd~B and u’~: dqB are gi_ren in the fol-

lowing table, ;;]d the mezm-;~locity and mean-temperatureprofiles can be calculated rapidly by this method. (The

)dues of (+% ~ are used in the approximate calculation of

R~C,~ix(appendix C).)

r=0.02.23...:

I:%L 20L 40L 601.w3.002.212. #2. m2. S#3. m3.2U3.403. m3.25

I ;[

1-

5. m5. m6.03

l.?x

o.Ixoo-0s64, 13=

. 19s9

.2647

.3236

. 3%s

. 4x13

.516.s

. 574S

. 6LX!$

. W3

.m

.7i25

.Sm

.W

.zY61

. W18

.9233

. w]

.95.5.5

.9759

. KS

.s%2

.%

.S?W1

0.m.mm;g22

.01s9

.036i

.m30

.0903~1~~

%%.3054.*64S.mm. io34.WLL -. %!s71.1478lJi1451..x24L FES22.04192.4222.8X13.2LS03.6L67

(%%0.33m.3319.3314.337x!.3~*.3$23Q.3165.3379. 2SS7-~~.2563.24s3.222Q.2iM.1s25.L61S.L40S.mu.0W6.0ss.Cwo

Yi’ith the approximation that Y wmies linearIy with the

absolute tempe~ature, the sIope of the ~eIocit~ profile at thesolid surface is simp~y reIatecI to the slope of the B]asiusprofile. Thus

andau”

()

_o.332~lx

Or

(h ,_ O.332 ~~=w’– T,

where b is the value of ~ at the “edge” of the boundary Iayer{when u’ reaches an arbitrarily prescribed due close tounity). It is seen that the shear stress at the surface (orthe skin friction) has the same ~aIue as in the Blasius case

The reliability of this approximation can be judged from thecalculations of the skin-friction coefficient in reference 2-4,in which p K P ’76. From fi=gyre 2 of reference 24, the vsdueof the skin-friction coefficient for an instdated surface at. aJlach number of 3.0 (T, =2.823) is only 12 percent loWer

than the B1asius -ralue anc~ onIy 2 percent lower at a 31ach

muaber of 2.0 (~1=1.81). For the Doniusulated surface,

-with TI=O.25, the value of the skin-friction coefficient zt..

~10= O is oity 7 percent greater than the Blasius value and12 percent, greaterat a 31ach number of 3.00.

Since the shear stress at the surface is unchanged in fistapproximation, the bounda~-layer momentum thickness hasthe same value as for the Blasius flow

,-6 /%— 0.6667~ g,*-

The, expression for the clispIacement thickness 3* gi~es ameasure of the effect. of the therma~ conditions at the solidsurface and the free-stream Mach number on the thicknessof the boundary layer. By detition,

=1.73 +( TI—1)1.73+ ~+ M,’ (0.6667)

4 MO’ (0.6667)=1.732’1+72

For the BIasius flow

(1 )

[~8* i~$ =1.730

B

The “thiclmess” of the bomdary lwjer b is gi~en by

b=5.60+ (~,–1)1.73+7~filo’(0 .666F)

8*and the form parameter E?=T is

H= 2.50 T, +7~Mo2

For the iusukted surface,

‘=2-50+3-50 r+’’”)

CALCEZ.4TIONOFME~S-~ELOCITYA~D ME~h--TEllPER~TCREDERW~~rVES

Because of the sensititit.y of the stability characteristicsof the Iaminar boundary Iayer to the behavior of the quantityd dw

-( )— , the ~alues of the required velocity and tempera-

dy p dyt.ure derivat i-res at the surface are calculated directly fromthe equations of mean motion, tith P= ~~ (TA=O-~~ for air).

370 REPORT NO. 876–--NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS _.

d~ ,LTOW—= f = bpw so that the dynamic equation is

dy

–b~w?=(pnwt)~. Since ~(0) =~’(0) =0,

w~//_– -~1 T,’w,’

where T1’= WI![

y—l—2—MO*- (TI—l)

1j if u=l. h other words,

the value of wl” is readily computed from the value of W1’.

In general, w, W) is determined from the relation

b—j (( W’)Z–2= (plwi)k-1

or

[

mTlf (k–l)! (T~)I’’w,(, _,)+ +wl~~)= — (k—l)Twl@)+(k–3)!2! T’lm ‘ ‘ ‘

(k– 1)! (;~)l($) T~)l(.k-l)

1+( Tm w,’ –(k–1–~s! Tin “(’-’)+ “““ ,

[~+fi r,(k-”’)w,’+ (k-2) ~l@3)w,”+ . . . +

(k–2)!(k–2–?’j !r! f

(k-2 -r)wJ+l)+, . . +(k–3~(k–2)r,/twl (k-3)

1

where

{l(P) =~(pw)l(P-l)

[=b ~@-~).w+ (~–l)pl(p-z)w,’+. . .+

_(p–l)! (H-l) wl(g)+ . . .+ P1W1(2PV

(P–!z-_~)k~ ‘1 1p=l,2, ...5

and

pl=—;,T1’

Q1’=–v

*[15( T,I)(T1’1)2+10(T,’) ’T,’”I+m(~—l) lOT,~/y,/ff+5T1,T1’v]+~ ;:

T,’

(T1’)4TI”+%=(m, ~) ~+15(m, 4) T,$

(m, 3)~ [45( T~’)2(T1’’)2+20 (Tl’)3Tl’”]+

‘m’ 2) ~60T,’T1’’T,’’’+ 15()2 Tl]v+15(TI()3]+) 3]+T,’

‘(m-l) (j T///’T,, [1 ( , ) +15 T1’’T,’’+6TT, T]+m2~2~-’1

(m,2)=m(m-l)(m-2)

(m, n)=m(m–l)(m-2) , . . (m–n)

m=O.76

(ml l)=–0.1824-

‘ (m,2)=0.226176

(m, 3)= –0.5066S4

(m,4)=l .641495

(m, 5) =–6.959939

T,’=awL’

~=-(—— 1~ M02-(T1-1)

TI’’=aw,’’– l) Jf#(wl()2’)2

T1’’’=aW3(y’—3 (y —I)Mwl°l’w1°

T1[v=awl’v– (7–l)Jfo2 [3(wI’’)2+4wL’wI’”]

Tlv=a-u~lv—5(T— l) MO~(2w1’’w1’’’+ti*l[~l[~

T,v’=awlv[– (~– l).?tIj[lO(wI’’’ )2+ 15wI’’w11’+ 6W1’WIV]

Each velociby derivative is determined from tl~c knowledgeof all the preceding derivatives.

APPENDIX C

RAPID APPROXIMATION TO THE FUNCTION (1– 2x) C(C) AND THE

In section 5, a criterion -was deri~ed for the dependence of

the minim~un critical Re-ynoIds number R@C,~~%on the local

distribution of mean veIocity ancl mean temperature acrossthe boundary layer. It was found that

-where COis the value of e for which (1 —2k)u(c) =0.580 and

=C

A rapid method for the calculation of the function (1–2k)r(c)and the minimum critical Reynolds number is developed bymaking use of the approximation that the viscosity varieslinearly with the absolute temperature (appendix 13).

(Since the eflect of ~ariable viscosity on the mean-veloeiiy

profile is overestimated in this approximatioDj the dues of

R!j.,~t~ (fig. 6(a)) cakdated by ttis method are lower than

the ~aIues calculated for P= TO-75when heat. is added to thefluid through the solid surface and higher when heat isw-ithckawn from the fluid.)

For p= T, the dynamic equation (appendix B) is

But

so that.

%(%)= -[i+ww&

3fIN131U&I CRITICAL REYNOLDS NUMBER

Finally,

={#yi+N$’U\~(c) = T12

ll?,H=c

()The required ~alues of w~, ~~~ ~? and f are obtained from

the table in appendix B.The small correction to the sIope k(c) is easfly calculated

once the mean veIocity profile has been obtained (appendix B).Thus

The quantity (1 —2x)u(c] has been calculated as a function

of c for various values of TI at .31. =0,0.70, 1.30, 1.50, 2.00,3.00, and 5.00, and the results of these calculations aregiven in the foIlowing table. The decisi~e stabfiiz~g

influence of withdrawing heat from the Huid at. supersonic

-velocities is frustrated in figure 7.

i O,%

110.1945 3550.?’69.5 10s2

I.3435 gj~

I: E .54.35 671.m .62:0 33

t [ i

~-3f$=2.m; C>O.5W

1.63 0.5074 6ilL.65 .543S XJ71.m .6155 751.75 .6749 40I. 8L .7275 251.s5 .7612 19

I M=5.CO; C>o.sm

0.70 a. 1670 I 3u5.30 .23!23 2H0.W ~~&- 613

1.25 .5425 74L xl .6265 3s

.\fl= 1.FD;C>O.332

I. 3a o.34% m1.35 .%%5 2751.lJ ..5505 ‘w1.45?6 .’226 49L60 . i732 16

f I

X*=3.53; C>O.W

2-+s 0.67’W\ ,=z 322.62 :%3 242.72 .Sllxl 142.ii .S295 102.*5 .%m 9

APPENDIX D

()BEHAVIOR OF ~ Pdx FROMdy dy

EQUATIONS OF hfEAN MOTION

In order to determine the effec&_of free-stream Jlachnllmber, thermal conditions at the solid surface, orfree-stream pressure gradient on krninar stability,it is necessary to know7 the reIation between thesephysical parameters and the clistribution of the qusmtity

solid surffice is obtained clirectly from the dynamic equation

‘()(equations (74) and (79)). The value of -$ p% at the

surface} which is ako useful in the discussion of la.minar sfia-

bility, is obtained from the dynamic and energy equations

as f OIIOWS:

wlf!’ 2WIJ’TI’ wl’Tl” +2wI’(T1’)’‘~– T,, -- “-T,’ ~~-

Diffwentiating the dynamic equation once yieIcls the result

At the soIid surfme the rate of change of temperature

d ‘~~ and the mte at which the work is done by pressure

— b? both Vanishj and tbe rate at which a fIuidgradient U* —dx*

element loses heat by conduction equals the rate at whichmechanical energy is transformed into heat by viscousdissipation. The energy equation becomes

or(T,’)’T,’’= –a(7– I) Jf,,’(W1’)2-m ~<0

Utilizing the expression for WI”’ and Tl” gives

[HalI=-2’m+’W$G)ll+

(W[’yu(l+m) (Y—l)Jl&z ~

where

[W)11=-3P1’”1’-+5%

From ibis expression for[$G)I1

the following con-

clusions, which are utilized in the stability analysis, arcreac~~ed:

‘hen[$($)ll ‘anis”es ‘heq’’an’ity[$G)lLis”iupositive.

When the free-strewn ~elocity is uniform,

[$ G)ll-

(W,’)’ (T,’)’—a(l+?n)(y-l)flloz 7Ff7+2(l+7n)2 ~ I’M’

‘ha’is[$G)lli’al’vays~’ osi’”” ~When the surface is insukted,

‘nd[$(al,is alwzys positive, regardless of the pressllrc

gradient.

372

APPENDIX E

CALCULATION OF CRITIC.4L 31ACH NUMBER FOR STABILIZATION OF LA?rflNAR BOUNDARY LAYER

For thermaI equilibrium the rate of heat. conduction fromthe gas to the solid surface baIances the rate at which heatk radiated from the surface. If the rate at which heat isw-ithdran-n from the fluid reaehes or exceeds a certain criticalvalue at a given local supersonic lIach number! the laminarboundary-layer flow k stable at all Reynolds numbers.(See section 6B.1 The purpose of the following brief calcu-lation is to determine the equilibrium surface temperatures at.severaI Mach numbers and compare these temperatures withthe critical temperatures for Iaminar stability. (See fig. 8.)

Rlen the solid surface is in thermal equilibrium,

where Eis the emissitity, ~ is the BoItzmann constantj andthe other symbols ha>-e already been defined. (See refer-ences 14 and 15.i Consider the case in which the free streamis uniform ancl the temperature is constant along the surface.For cr=l,

-=ho’.w-here stagnation temperature T. equals 1~ ~1-Y

r=UT 0.332.~ d the approximation p= 7’ is.&lso (-$)1=3 ~ =X. ,

empIoyed. (See appendix B.) Since~=c,2~l=c.~Tl,

Ti%en the integrations in equation (El) are carried out, thefoIlo~ing relatio~ is obtained for the determination of theequilibrium surface temperature:

u-here

The equilibrium surface temperature under free-fightconditions is affected principa.IIy by the variation in density~ with altitucle h. The results of calculations carried out

for altitudes of 50,000 and 100,000 feet are given in the

following table:

E~~ * 7=” “

In this table,

“-T’”=(”’+‘“zwhere

Q heat withdrawn from fluid per second per unit width of ._.

surface

C1 skin-friction coefficient for one side of surface

R=–’%

L l~~gth of surface

T,* free-stream temperature

k~ heat-conduction coefllcient of gas at free-stream tem-perature

In these calculations the foHoi~kg data are used:e=o.~(l

L=2 ft.

~=400° F abs.~=~.~o X 10–13B~u/sec/ftZ/(OF abs.)~

c~=7.73 Btujslug/YF abs.

~=3.02 X 10-7 slugs/f t-see

~= 9S0 ft/sec

~=3.61 X 10-g s@s/ft~ at 50,000 ft=3.31 X10-5 slugs/ftx at 100,000 ft

K=3.35 XIO-’ at 50,000 ft=3.66 X10-S at 100,000 ft

Since T,– T,c@,r> T.—TIC, for MO =3 at 50,000 feet

altitude and for 310=2 at 100,000 feet altitude, the Iaminar

boundary layer is completely stable under these conditions.

It should be noted th%t under wiud-turmel-test conditions

in which the model is stationary, these radiation-conduction

effects are absent, not only because of reradiation from

the walls of the wind tunnel but also because the surface_ ___

temperatures are lore-generally of the order of room

temperatm-e.

373

374 REPORT h’O. 876—NATION-AL ADVISORY COMMITTEE FOR AERONAUTICS

REFERENCES

1. Heisenberg, Werner: ~ber EMabilitat und Tulbulenz ~-on F1iissig-

keitsstrtimen. Ann. d. Phys., Vierte Folge, Bd. 74, No. 15,

1924,pp.577–627.2+ T’o]lmien,TV.:Ulxx die Entstehung der Turbulenz. Nachr. d.

Ges. d. Wiss. zu Gottingen, NIath.-Ws. IQ., lgZ9,PP. zl-~4.(.&~aiIable as NACA TM No. 609, 1931.)

3. Tollmien, W.: Einallgemeines I1riterium der Instabilitat lanlinarerGeschwindigkeitsverteilungen. Nachr. d. Ges. d. Wiss. zuG6ttingcn, Math. -l%ys. Ill., Neue Folge, Bd. 1, Xr. 5, 1935,

pp. 79-114. (Available as NACATMNo. 792,1936.)4. Schlkhting, H.: Amplitudent”erteilung und Energiebilanz der

kleinen Storungen bei der Plattenstr@mng. NTachr.d. Ges. d,~ri~~. ~u Got,tingen, ~~at~l..p~lYs. K1., Neue Folge, Bd, 1, Nr. 4,

1935,pp.47–78.5. Lin, C. C.: On the Stability of Two-DirnensionaI Parallel F1OWS.

Part 1, Quarterly Appl. lMath., vol. 111, no. 2, July 1945,pp. 117–142;Part 11, VO1. III, no. 3, Oct. 1945, pp. 218–234;and Part 111, vol. III, no. 4, Jan. 1946, pp. 277–301.

6. Liepmann, Hans W,: Investigations on Laminar Boundary-LayerStability and Transition on Curvecl Boundaries. NACA. ACRh“O. 3H30, 1943.

7. Schubauer, G. B., and Slirarnstad, H._ K.: Laminar-Boundary -Layer Oscillations and Transition on a Flat Plate. NACA .ACR,April 1943.

8. Lees, Lester, and-Lin, Ch]a Chiao: lnwstigation of the Stability

of the Laminar Boundary Layer in a Compressible Fluid. NACATX ifO. 1115, 1946.

9. Pretsch, J.: Die Stahilitiit der Laminarstromlll~g bei Druekgefalle.und Druckanstieg. Forsehungsbericht Xr. 1343, Deutsche

Luftfahrtforschung (Gottingen), 1941.10. Oswatitsch, K.} and Wieghardt., K.: Theoretische Untersuc.hungen

tlber stationiire PotentiaIstromungen und Grenzschichten beihohen Gesehwindigkeiten. Bericht SI 3i 1. .Teil der Lilient bal-Geselkhaft fur Luftfahrtforschung, 1942, pp. 20-22. (AvaiI-able as A-ACA TM No. 1189, 1948.)

11. Tietjens, 0.: BeitrZge zur Entstehung der Turbulenz. Z. f. a. M. M.jBrt. 5, Heft 3, June 1925, pp. 200-217.

12. Prandtl, L.: The Mechanics of Viscous Fluids. The Developmentof Turbulence. Vol. 111 of Aerodynamic Theory, div. G. sec.26, W. l?. Durand, cd., Julius Springer (Berlin), 1935, pp.

178-190.13. Synge, J. L.: Hydrodynamical Stability. Semicentennial Publica-

tions, American Math, Sot.; vol. 11: Semicentennial Addresses,

1938, pP. 255-263.14. Kiebel, I. A.: Boundary Layer in Compressible Liquid ~rith

Allowance for Radiation. Comptes Rendus Acad. Sci. USSR,

vol. XXV, no. 4, 1939, pp. 275–279.15. Hantzsche, W., and Wendt, H.: Die laminare Grenzschicht der

ebenen Platte mit und ohne W7armeubergang unter 13erucksich-tigung der Kompressibilitat. Jahrb. 1942 der cleutschen

Luftfahrtforschung, R. Oldenbourg (Munich), pp. I 40-1 50.16. Lieprnann, H. W.: Investigation of Boundary Layer Transition on

Concave Wails. NACA ACR ~TO. 4J28, 1945, pp. 14–20.17. Landau, L,: On the Problem of Twbul.ence. Comptes Rendus

Acad. Sei. USSR, vol. XLIV, no. 8, 1944, pp. 311-314.l& Pretsch, J.: Die Anfachung instabiler Sttirungen in einer laminaren

Reibungsschicht. Jahrb. 1942 der deutschen Luftfahrtforschung,R. Oldenbourg (Munich), pp. I 54–I 71.

19. Liepmann, Hans W., and Fila, Gertrude H: Investigztiom ofEffects of Surface Temperature and Single Roughness Elementson Boundary-Layer Transition. XACA Reporf, No. 8!)0, 1047,

20. Frick, Charles W., Jr., and Mc(!ulloughl George B.: Tests of aHeated Low-Drag Airfoil. NilC.% .ACR, DCC. 1942.

21. Schlichting, H., and ~-lrich, A.: Zur Ilereclmung des Umschlageslaminar/turblllent. Jahrb. 1942 der dcutschw Luftfahrt-forschung, R. Oldenbo,urg (Munich), PP. I 8-I 35.

22. Dorodnitzyn, A.: Laminar Bounclary Layer in Compressible 1,’IuicLComptes RencIus Acad. Sci. USSR, vol. XXX IV, no. 8, 1942,pP, 213–219.

23. Brainerrl, J. G., and Emmons, H. W.: Effect uf Variable Vkeosity

on Boundary Layers, -with a Discllssion of Drag Mcasurcliicnts.Jour. Appl. Mech.. vol. 9, no. 1, March 1942, pp. ,\--1-A-6.

24. Von K4rm&, Th,, and Tsien, H. S.: Boundary Layer in Compres-sible Fluids. Jour. Aero. Sci., vol. 5, no. 6, April 1938, ~)p.227-232.

T.4BLE 1.—.4UXILIARY FUNCTIONS FOR CALCULATING TEE STABILITY03?TEE L.4MLVAR BOUNDARY LAYER FOR INSULATED SURF/i~I?

0.0372.0744”.1115. 14s6.1857.222625%L298G

.3323

.3682

.4037

.4143

.?fo =0

o.owl.@ol.omi3.0006.0012.GWl.0033.0354.WilCQ138

.0131

.0142

0.0362e.0723.1085.1446.1806.2166.2525.2S82.32.37.358S.3936.4280..4306.4362

–O.owl–.cml

.Ocdl

.OC83

.lm7

.0014

.0)23(B36CmiC0i6

.0103

.0137

.0140

.0146

0.522n.47’4s.4303.3887.34N.3139.2S682W5

.2232

.1987

.1770

.171I

o.2W7.2740.2.7X.2433.227s.Zlxl.1958. lifli.1639.1487.1350.1312

r–gc&

.tm29

.0107

.0254

.0492

.0846

.1342

.2Q1O

.2s82

.4m

.5407

.5526

.5794

-~. 0148–. 023+–.0244–. 0LL9–: p!);

.0627

.1103

.1695

.2412

.3261

.4247,4327.4501

0.5122

:::;;.3347.3474.312.7.2s07.2513,22!6.2335.1790.1602.1ss9.1560

0.2!222.2127.2319. lr414. li&9. 1W2.1530.1393.1’247.1104.0W30s2s

.Q81.G

.0732

O.orw.OI?JM.0524.04s5.ofo3.GM!.owl.U3.z6.0217.0180.013’3.Oln

I0.04.43.0401.03%.0316.02s2.0210.0217.Q1S8.Q13S012s

.mMm;

.CLlm

0.Zm.30!51.3124,31G1.3211.3236.32U.3174.30s4.2935;:;2:

.

0.19272Q813

,7,~g3.22$$.23G6,2*Z3.242S.24(IG.2333,m79.1914.1414.1307,12G2

I 1. I I I i I

Mo=o.70-0.0353.0705.1058.1410.1762.2114,2464.2813.3161.3505.3847.4185.4352.4452.4559

–0. Oocfl -o. rxlo9–.oml -: IM24-.oo#o -. em5

Klol .0006.0C04 .CQ90.0008 .0248.G016 .OML.@326 0S72.0039 .13s9

. .OX@ .26s203s1 .2985

.0109 .4137

.0126 .4821

.0137 .5270

.0149 . .37W

-0.0321–.05s43—.0791–.0914–. 0951–. 0SIJ6–.07!1–. 047s–.W.M

.0412

.10(37

.1886

.2363

.2674

.3027.

0.9)31.4599.41913W8344s

.3113

. !2.?Q2

.2516

.2255

.231s

.180G

.1619

.1534

.14S6

.1436

0.133ti. 17s3.1721.1652.1569.147S.137Q.1272.1167.1042;:-2:

.07(70

.0733

.0709

\

0.0321.0300.0279.0257.Q233.mx.0M7.0166.0142.0118.KJ85.6m2.W36.031G

–. Q332.

0.1484.1052181u

.19s12[2s

.2250

.2358

.2436

.2-KM

.!Ml?

.2272

. 19s7

.17s7

. 1Q18

.1575

THE STABILITY OF THE MMINAR BOUNDARY L4YER IN A COM.FRESSD3LE FLUID 375

TAEILE IL—AUXIMARl” FUXCTIOXS FOR CALCULATIXC? TEE STABIL~TYOF THE LAWNAR BOU>-DARY L.kYER FOR XONB”STJLA.TED SDRFACE

T.4BLE L—.!GXILIARl- FUN?TIONS FOE CALCULATING THE STABELITY0 F THE LAM IhT.&R BOUXDARY LAYER FOR INSULATED SERFACE—C,mcluded

cI

LI

nI

LI

H,I

H,I

M%I

.N%1

.?@=0.70;2’1=0.70

.m.=o.w

I0.022 0.Lm6 0.0.s2-5

10.C635

.0521 .0U2 .1645 . 6W20.6102 \ o.s$m.5i35 .3157

.07i7 .0165 .2466 .1134 . :s367 .3045

. Io30 .0z2C .323i . lKQ .5022 .2936

. 12u .L~$ .4L46 . lfj32 -4i!13 .2s2%

.1523 .032i ~M9J . 1s04 .43%5 .2724

.1701 .0365 .5s61 .2134 .4L91 .26.51

.1726 .0370 .57s: .2163 .4162 ~ .264!

0.2743.Xml..%331.3233.3333.3519.3610.3623

0.0334 0.m.w: –. lnxuml –. WYY2

:1335 –. m.1669 –. m3L2m2 . (Km

.2335 .00?5

.2566 . mm2$% . Iw22

.3326 .0034

.Mm .Ocm3m6 .Ooiz

.4236 . CmS

.4612 .0130%36 .0132

.4s12 .0153

-0. C015—.CQ4:_, ~~–. 0102–. m–. 0323

. (km

.0312

.~~

. mss

.1697

.24%3

.S51s

.WE

.4313

. 57ss

–o. 0503–. 0%2–. 13s9–. 1746–. :~~–.223—.2357

–.2441–. 2:07–. 22s1–.-w—.lno–.1302–. 0764–. 0744–. &421

O..R16.4?2.4049.3626.3365.3054. m-

:;%.2U26. 1s23.1041. 14%.1340.1330.1261

0.1303 0.0K30. Lxs : ml:. Ml.L2.n .01s2. L213 .0175. L163 .0166. U03 .0157. Kilo .01.s3.0947 .0128. IR55 .0110. mm .Cml.&m .CKm.0560 . OT21.0464 –. 0336.0463 –. 0240.0418 —.0K6

0.0933.1133.1366.1594-1325.2055.3252.2439.~~.2733.-%4.2515.21s5.1431.1373. KM

M.i=o.70; !rL=o.so

o.W33 o.04s6.0366 .0%5.CK199 .1443-0132 . 19%.Om .2417.01% .X26.0220 .3457.0263 . .ioIi.022 . .!614.0331 .~z~.03<9 .5592.0359 . 5’3$)1

0.0237.04i2.0735.0927. LLE8.1397-l&~. 1ss1.Z3Zi-.22%.2404_~~i~

o.Q2i9.0374. 04s0.04s2.05.50.0649.Om3.0%2. L236.1562.1754. lwi

0-5954 0.2s11.5620 .27H.5W --.4934 :?%.4:01 . m 4. 442? .2439.4152 .23M.Swi .Zt3i.3654 .2210.3424 .2133.3313 .2%?4.324S . ail

o.C-493.0!75.0457.0437.WLr.0397.037s.0359.03s9.0321.0310.0302

IM$=LIO

o.G933 –o. m132U –. (Km

:1650 –. c#05.1954 –. ihm.2302 –. QX12.2638 . lm2.2%5 . (lxM.3292 .ma.3616 .0u31.393s .WE!.&~ .mw.4572 .m?,.4S36 .0L26.5164 .oI&l

–O. 2337 0.4026–. Z&l .36s2–.3166 .335s–. 36!0 ; g.–. ‘m49–. 4396 .2506–. w% .2X3–. .mF3 .m4C–.54M . 1s37–. 5239 .1655—.55L6 . 1:9s–. ~i~ .13.50–. 6112 .1245—.65% .1151

–o. 0140–. Om–. O=—.o~~~–. 0232—.0125

.Lu372

.0362

.0s29

.1442

.2247

.3W

. 4-%7

.=Tgg

–o. 0561–. 0505–. 0364–. 0117

.o~owl

. Km

.2449

.3652

. 3s93

.mii

0.Lw3 o.C4m. 05.’s% . IN3s. LE3 .005L.0657 .Oom-13&32 .~. Ozsl . Cm2.0516 .0358.0431 . m!:.0333 .W3L.021s .0035

–. 13332–:%% –. lK.s7–. 020s –. 01s7–. 0360 –. 0z?J3

o.0s33. m-s.1319. 159s. 1%4.2101.2293.24L6-~<~.2310. 1s34.0764

—.oi37–. 2366

3L=0.70; 2-,=0.93

0.0433 0.W36 0.0517.W3 . lxm . 102s. lzzl .0108 . 156s.1714 .0145 .21Z3.2L35 .01% .2%5.2555 .0227 .3745.2%3 .&~4 .4s435.3166 . Gw2 .642!2.326s .0312 .5762

0.ITJ51 o. m–. LIMi . .fgsg–-OLL1 ..$414–.ali9 .3930.Cms .34s5.0462.30s0.1071.ZiM.1’!s9 .2547.1776 .2466

0.Zqo.234-4.2191.mi4. IWL.1325. 169s.1637. U51xi

ao435

:%%.0337.0304.0272.0240.0224.0217

0-1422. K&s. LSS6.2032.322.2339.246!2.2517.mu

.31*= L30o.22im.2440.2644. 2i42.27LM.22X.5. 11s4

–. Om–. 4943—.5ST1

–I. 5!$/3.

0,2541 –o. m–. cm5

:% .OwJ134s!3 .m

.3s33 .LK121411L W3i

.441s . iK15i

.4721 Cm3

.5ml .0114

.5072 .OL%I

.5416 .0167

K (E244 o.WJ3.0233 C816.01s3 . ln)14.0109. LD19 –: ~6

–. mQ3 –.GWS–. 02S6 –. m–. CM&l –. 0169–. 0622 –. 0-234–. 067L –. 0324–. US-34 –. 0549

–O. 59S2–. m–. 69S7—.7430–. 7S3–.3303–. 3S34–. m

—L 09H–L 1334—L 30:4

0.2<s7.2255.WI.13$5.1667.1507.1366. E42. U36.1119. mm

–: O& –O. 0476–.cmi

–. W3s –. 1013–. 0SS6 –. L132—.1021 —.1155–. 10s5 –. 1031—.1055 –. 0912–-0917 –. M45–. 04w –.0231–. 02A13 .0Li9

.~y .0734

. 12s6 . 13X4

.2414 .ma

.3s59 . !2ii0

.51s4 .32?.2

.aT9 .3349

0.0324 0.1462.0310 .163+.02%2 .1794.02i2 .1955.0251 .21!33.022s .323s.020s .2W2. ~~~ .240j-0161 .2409.0139 .22sFi.0113 . 2m9.00s3 .1616.IxM2 . CbS16

–. Cm2 –. CwL-. Mm –. 2262–.LW91 –. ‘w2s

o.m-%.0s92.10!0. 13s9. Ins

:%. 2is9.313s-34s5.3ss1.4Li4.45L2.4.%6.5042. 51$X3

–O. 0016–. ms2–. ms–.~’–. IXK6–. 03si–. am–. 0101–. 0103–. 0103–.0392–.Q379–. cm9–. ml–. m

.Lm6

0.1731. IiLo.1661.1600.1522.1443.1354. L249. llss. Ilm.03io.072s.05.s2.~~.0314.0269

t

i

376 REPORT h’O. 876—NATIOhTAL ADVISOP.1” COMMITTEE FOR AEROhTAUTICS

TAfiLE1ll.-PHASE VELOCITY,lv.4\' Eh-USIBER,.kND REYNOLDS h~U&lBER FOR NEUTRAL SUBSOL’lC DISTTRBAh-CIZ (STABILITY LIhl ITS)

FOR I&TSUL.kTED SURFACE

I

C~Ol:l~,\R,llCl~/R-”l” UBIR4Jro=o &o.xl

0.0372 0.0321 y ~’xgg.0744 .0685

O.(W, I 3,03=4 00354 “:u:::; 111,W31MH3 o.@312;g;;

121690,lm178,(DO .0667

.1115 .11035,m, 0)0 .0028 679,O&l

‘278:COO 33,103 .lm.14S6

.0421.1585

I,030,coo .0348S.3,m .0189 9,8$0

117,ON.1335

.1857....(M32 2W,Wo

.2146.0672 33,m

32,6CWI .0255.2226

3,w .1$359.2808

.-JS85 106,NO .0101M,800

12,104.0334 1,iE4)

.2594.2002 J186

.35W46,COO .0135 5,240

7,700 .0427 917 .2335 “45402%30 .4535

“22%6004,420

.0175.0540

2,570526 . !2666

.3323‘. ~w

.p;12,100

2,760.0223

.0679 3291,380

.2997 .2459 7,020.3682

;(l% 3001,860 .cE62 220

.4037.3326

:95’s9.3053 4,320 492

1,3!?8 .1142 162.4143

.3652 .. a777t. 0770

2,820 .0430 3211,280 .12$2 M3

.4143.3976 ~. 4638 1,950 .052$ 222

L.2730 1,530 ..1515 182 ,‘f~~.4037

.373;L 2940

1,410 .0653 1611,880 .1540 223

.3682.4612

1.1960..i’2eJI 1,090 .0827 123

3,530 .1424 421 .4636.3323 1.0103

..7396 1,0s0.6>710

.0343.1238 799

.2980.4s12 .8810

.87281,010 .1024 ;:

13,300 .1039 1,580.2594

..4812 1.0130 1,230 .1154 1fo.7177 27,600 .0s5{ 3,270 .4636 1.0120 1,740 .1153 199

.4612 1.Q070 1,820 .1148 207

.4296 .0027 3,1s0 .1029 363.}fc,=o .50 .3976 ..7823 5,w .0892 637

.3652 .6642 9,940 .0757 I, 130

\ 0.0029 4,270,1M0.3326 I S523 18,5@3

O.0?62 0.0251 36,ibO,COO.C.329 2,103

.0723 .0538. ,:

2,1-30,@lo .cm3 2’$s,m.1086 .Osl!a 392,C@ .0101 46,7001446 .1250 116,000 .0146

M=l.lo13,500

:1806 .1695 -$44,500 .0198 5,193.2166 .2216 20,200 .0258 2,301.2525 .2829

0.0693 0.03S6 6,730,(I)Ol!, 400

0.0039.0330

618,@xl1,210

28s2.1322

.3556.0268 769,M .3029 82,W

?, 850 .0414 682.3237

.1633 .046s.4442 3,570 .0518

224,603 . C050 24,IM

.3538. 16s0 .0707

.55’49 2,33085,WJ .Mm

.06479,lfm~i;

,3936 .6993.Z309 .0991

1,62038,303

.0815.0107

1894,130

.2638.42S0 .9301

.13291,230

19,300 .0143.1084 144

2,0s0.2965 .T727

.4306 .955810,603

.1,220.0186

.1114 1421,140

.4362.3292 .22AX

1.01406,203 ..0237 675

1,lSQ .1182 139 .3616 .~55,4362

3,923 .0297 4231.1880 .1,410 .1384

.4306,3938

1.2150.3417 2,610 .036s

;%291

1,580 .1416 .4246,4280

,41591.2150

1,850 .04$8 lW1,664 .1416 .194 .4572 ..5193

.39361,350 .0560 M

1.1240 ~,0s0 .1310 35935Ss

.4836.97s8

,.6268 ~ 1,1Q3 .0676 1100,670 . 1L41 661

.3237 .8272.5104 .8010 091 .0864 107

10,SW .0964.2882

1,260 .5104 .9165.6S69

1,220 .OW 13121,100 . 0.s00 ~ 460 4s36 .S927 2,060 .0902

.4572 .8023 3,320 .0865 ;2

.4246M= 0.70

.6785 5,!Xlo .0732 639.393s “‘. 5766 10,400 ,~zj 1,120

0.0353 0.0191 53,.ZT1030 0.0022 G,100,033.0705 O*M

Mo= 1.303,0#3,mo .m47 349,W!3

.1058 .0677 555,030 .0377 63,403

.1410 ;Wf 161,000 .0112 18,4tK 0.2541 0.0451 63.SJIO) 0.0047

.1762 61,100 .015+6,630

6,980.2114 .1766

.2a58 .0818 24,800 .0035 f%27,300 .Om 3,120

.2464.3173

.2268>.1222 12,300 .0125

13,300 .0259 i, 5s0.2S13 .2s57

.34s8 .1636 6,993 .0170.7,630 .0326

‘ 7208?2

.3161 .3570 ;% / ““:% 4,280 .0222 4.454,.5W1 ; 040s :20

.35052,w .Oml 291

.4433 2,WI .0506 331.3847

.3377 I, 930 .0351 Ml.5515 ~,$$0 .0630 224 .4721

.4185 .6951..41OG 1,420 .of33 147

,0794 162 .5ozo.4352

-m.si%

1,110 .0532 115.0%?4 141

.4452 :%;.5072 .5316 1,Oio .0552

1,160” .0989lli

132 ,.4559 .9704 ‘i,llo

,5416 886.1108

.0788 !!2 ~127

.4559.5416

1.1230%; .0928

1,330 .1233 152.4452

.5Q721.1424 1,650 .1304 !

;% .0s00 h?189

.4352.5020

1,12”30.7592 2,554

1,980.0739 265 ~

.12.W ! ~zi..41.S5

.4721. ...645s1.0720 Z,670

4,ml . ISG71.m!25 305

.3847.44i8

.93s1i 5401

4,8107,9s0

.1072.0561

550:$

.3W5 .7%55 8,8S0 .0910 8 1,010 . .;.

.3161 .6659 16,700 .0761 1,Tlo

THE STABILITY OF THE LAMIN.!.R BOUATDARY LAYER LW .4 COMPRESSIBLE FLUID 377

TABLE II-.-PHASE VELOCITY, WAVE NUMBER, LLQ REY5’OLDS NIMBER FOR IJEI?TRAL SUBSCIMC DISTURB4XC?E (ST.iBILITI” LIMITS}

FOR h-OXINSUIATED SURFACES

c a R a#I

R,l]-ciaqtirl?,I—

3&=o.70;r*=o.io Ma=0.7U2-1=0.90

a0262 0.03w 8~m, cmI

O.w 2,wJ,GK& a w33 a 036s.Omi

17,100;w.0734

0.0W2 I,930>ml5,SW,W .cw.s .US63 .EM 1,MO,LEO .03s2

.onHam

.11% l!&o,Gl& ~ .0143 133;003 -L29L .1552 202,cw .0153.1030 .170s -02U5 44,Mlo

Z&m.m~ .m Q5C.3 -~>:~

.1281 -Z3a3 161:m ~7,070

.Om 19,ml .2135 .%5 zJ5& .0314.1529 .3030

Zwl

%% I :%10,ml .2551 .3iX .0422

.1701 .36701,40

6,S70 .~gg !: m .am<ml

7s9.1726 .3i77 .044 6,540 :% .5s14 .m.s 624.1726 .49S6 69,OXI .05s9 S.mi .3.%.s .6347 $; .Om 5$5.1701 .49Z i3, m .CL5SS & Sio .32m .7817 .CK3S4 735. 1S23 .4732 L21,m

,14,m .3166 . iiol 7,9m .OST1

:?E I ;!!J

395.12.$1 .4Li5 3%ml .3963 11,6LXJ .Cr32i. 10M .3460

1,310.2551 :?% 25,m .ma

.0ii7 .2620 2,m: w 3%= .2L352,S53

-5123.0.521

65,W0 .0581. In3 14,WI, ml .0N6 1,m, m . H14 .,~~ 170,ml .W49 It%

.1291 .2S5.s 61i,W3 .G1323 .69,m

.0S53 .1753 3,7*O,m .0233Mt= 0.70;Tl=o.so

@cm

0.0237M= 0.70;2-,=1.25

0.0237 15; ~ ~ o.CX323 1s,3m,ml.0472 .OX-4 . IXJ59 I, 150,ml.Om .@304 1:970:cm .cm4 230,Wo 0.0346 0.0164.0937 . LL33

~m,ml 0.0316 S,om,coo633,cm .01X3 T3,iMl . @592 .0216 * 3%3,cm .LW36

.1168 .1529 ZKJg .01764S3,@xJ

30,m .10$0 . Ki64. 139i

770,~ . 03ss.1923 .0224 15,mQ

79,033-~qgg . 03L9 217,cco .Om 22,!203

. 162j .23.s2 70,m .Om S,m . 173s . lMI m,w .0115

. 1s51 . .Z?KS .0339 .20S.s . l~fi$ ;g

3+ cm.2m5

.Om?.3510 $%% .cMM kE .2439 . lSW 16,5111 .0195

.22$s .42371:7M

I%W3 .0424 :;~ . Zisc! .2103 ;MJ -021j !337.2m9 . .S363 15,WI .Wi.i .3128 .3022 52Q.2475 .4%52 u, 5W .057S 1:6$0 ~3.f~5 .3i2Z 3,110 :E 319.2475 .EL?us .0T33 2,lea .3Ss1 .42* 2,Om .o-m

?! z207

.2J09 .6233 .0726 2,m .~Ji4 .5568 1,w .05S2 IQ

.22$$ .mm 27,230 .0m6 #ii] I -4S12 .m61 I, m .oi22 m

.2075 .5W9 44%%1~, ~m .0654 .4s46 .93137 iEO .OJJ31 78

.1s51 .5062::% \ LJ$J

. 5fr32 1.mm 643.1625 .4465

. mlMl: cm . 5L92 L 44S0 615 . 11% z

.13W .3SZ Zx_Jg .2446 . mm L WJl 6!0

.1168 .3164.1630

.0369 73:m -~~ L 723 w . Uio :.0937 . zag 1,6$0:IxwI .m 167,m .4346 1.5370 1,390 .1577.0705 . 1s22 5,8S0,WI

I.azlz 68&m3 .4512 1.!2.wl $;; .1291 %

.4L74 L03M .Xk31 5.53

..—. .—. ..==———.=——=—.-—.———.