C.P. No.635

AERUNAUKAL RESEAtkW COUNGtL

CUiNWJT PAPIE&

Three-Dimensio,nal’ Turbulent Boundary Layers

bY

1. C. Cooke, j &SC.

C.P. No.635

LONDON: HER MAJESTY’S STATIONERY &FlCE

I%3

PRICE 8s 6d NET

U.&C, NO. 532.526.4 : 53~517.4 : 532.517.3

C.P. No. 635 ame, 1961

J. C, Cooke, D.Sc.

E *

,

This paper reviews the curarcnt state of knowledge of three-dimensional

turbulent boundary layers, mainly from the point of view of making

calculations. The nature of the boundary layer is discussed and the

equations for gcncrul. steady turbulent Slow are given. Next follow

momentum-integral equations, which can then be solved with suitable

assumptions as to the velocity profiles and shear stress components. f%.n

account of these assumptions follows and a few sample results are given.

The paper end s with a short nccount of some matters connected with

transition.

c--m---

ua- A-- - -3-

Replnoes R,A.E, Tech. Note No. Aero. 2762 - A,R,C. 23,259,

T,IST OF CONTENTS -a_ --T.-L --_I

I IWI'RODUCTIOnT

2 G~ENERAI,

2.1 Boundary layer and momentum-integral 2.2 Three-dimensional effects 2.3 The axially symmetric analogy

equations

3 VEr>OCITY E)ROi?ILES AND S,<lX FRICTION COXPONEXTS

3.1 Velocity profiles 3.2 Wall friction values

4 GENERAL KETHODS

4.1 Cooke's method It.2 Becker's method 4.3 Johnston's solution along a line of symmetry 4.4 Braun's solution for the slightly yawed cone

5 TRANSITlON

6 CONCLUDING l!LEX'IhXS

LIST Or' SYMBOLS

LIST Ol? REFtiRENCES

5

5 8 9

II

12 16

17

18 19 22 24

24

25

26

28

Fig.

Laninar cross-flow velocity profiles. External point of inflexion at 5 = 0.5

Oil flow patterns showing rF;edges of turbulence (Brebner and Wyatt)

Experimental arrangement of Johnston

Johnston's form for the cross-flow

Polar plots by Johnston from data of Kuethe, IMcZee and Curry

Cross-flow according to Johnston's model :;zith u/ue = (us) v7 , e = 1.47, A = 0.26

Isometric figure for the velocity distributions of Fig.6

1

6

7

-2-

LIST 0% IXLUSTIUTIOMS CCont'd) S.-x= -

V&e COmi3011entS Wh and W j0 (Blackman and Joubert)

Values of 0,, and 6, from measurements by Brebner and Yyatt

Values of a for measurements by Brebner and Wyatt

Calculated external streamline- (Cooke)

0 on delta wing at zero incidence

Growth of momentum thickness along streamlines of Fig.11 (Cooke)

Deflection of limiting atreamliqes on yawed cone (Braun)

Possible transition front (Cooke)

Development of upper surface showing calculated streamline at -3O incidence and calculated shapes of turbulent wedges for x/o = 0.20, 0.05, 0.02 and 0.01 positions (Gregory)

Spread or' turbulence due to conical excrescences on GO0 swept wing. Prom photOgrapiiS by Gregory

Momentum and displacement tilicknesscs at transition. Prom curves drawn by Kotta from measurements on a flat plate by Schubauer and Klebanoff

&

8

9

IO

II

12

13

-I4

15

Id

17

G

f

-3-

I IFi!RODUCTION --

The study of three-dimensional turbulent boundary layers has been greatly neglected in the literature, Thus in two reviews, one by Sears' in 1954 and one by Moore2 in 1356 which together include about 50 mferences on

r_ the subject of three-dimensional boundary layers, only 4 refer to the turbu- lent problem and these were all concerned with flow over infinite yawed wings and cannot be considered to be quite general. In the review by Cooke and Ea113, written in 1960, there are about 50 more references, and of these only * about 12 refer to genuinely three-dimensional turbulent boundary layers. Again, in Applied. Mechanics Reviews for 1960 there are about 175 references to boundary layers of all kinds (excluding Magnetohydrodynamics) of which only 1 deals with a properly three-dimensional turbulent problem, whilst two others do so in part (and only a small part). Since flying machines of the future are likely to be more f'three-din~ensionalf' in shape than in the past it will certainly be necessary to make a greater study of the subject than has been done up to the present time. The neglect has been due to the difficulties of the subject, which are immense, even in Tao dimensions, and to the fact that the collecting of experimental evidence is so tedious and unrewarding.

This paper is an attempt to review what little that has been done. In it me shall exclude unsteady mean flows and will also exclude in the main axially symmetric flow, since the latter hns usually merely involved a simple extension to work on two-dimensional flow. Attention will be con- centrated almost entirely on theoretical work, and in particular on the description of such calculation methods as have been attempted. These have all assuned small (or zero) cross-flo:nr, that is, small mean velocity components normal to the streamlines in the external flow, and have assumed well-knovnl t::-o-dimensional expressions for th e streamwise velocity and akin friction components. More variety has been found in the assumptions for the cross-f low. The only more unified attempt to write down a general form for the velocity profiles is due to Coles 4, This is an extension of the law of the wall and the law of the v&e; so far only a small amount of experimental evidence exists in supoort of this model. examined by JohnstonS,f;

The subject has been deeply who suggested a form for the cross-flow quite

different from any of the other s and presented considerable experimental evidence in its support.

The flow over infinite yawed :F;ings has considerable literature in laminar flow. This is because there exists an "independence" principle accord- ing to which the c,hordwise components of flow satisfy ordinary two-dimensional boundary layer equations. This principle does not hold for turbulent flow. A simple theoretical proof of this was given by Rott and Crabtree? and experiments of Ashkcnas and Riddell conf-kmad it. A full discussion of the subject was given by Turcottey, who advocated a "line of flow" principle, by which the streamwise wall shear stress is considered to be a function of the distance over which the fluid has aotually flowed, rather than the perpendicular distance from the leading edge. Most methods of calculation so f,ar put forward have used stroamline co-ordinates and they have tacitly followed this principle.

Turbulent sep aration in t,hroe-dincnsions has scarcely been touched upon by workers in the field, except in its purely topological aspects, which apply equally ~11 to lsninar or turbulent boundary layerslOrll, Ve do not consider it here.

In this paper we give the general three-dimensional equations for the mean flow, and then explain the axially symmetric analogy applicable in cases where the moan cross-flow is small. Momentum-integral equations are given, since only by their use h&s any progress been made. Their solution depends on assumptions made about velocity profiles and surface shear stress.

-4-

An account of these assumptions follows. iYext is given a description of the few three-dimensional turbulent boundary layor calculations that have been made, and we conclude with a short account of some matters connected with transition.

2 GENERAL

2.1 Boundary lzr and momentum-intLf;ral esu@&~s_ -n--m

Curvilinear co-ordinates c, 71, Z: are used. The surface over which the boundary layer lies is < = 0 and ;= denotes distance from tho surface measured along a normal. On the surface Z = 0 are two families of co-ordinate curves 5= constant and 7 = constant, orthogonal to one another. i?or this system the element of length is given by

ds2 = h; de2 -+ h; dq2 + dZ2 .

In the boundary layer h, and h2 are usually assumed to be functions of F; and 71 only. For this to hold it is necessary that the curvature of the surface does not change abruptly and that locally the boundary layer thickness is small compared to the principal radii of curvature of the surface.

Ye denote velocity components by u + u', v -i v' w c w', where u, v and VJ are the mean values. Density p,total enthalpy 15 and temperature T are treated in the same way. Terms involving fluctuations of the coefficient of viscosity p, the coefficient of heat conductivity, the spccifio heat at constant pressure 0 and the Prandtl number 2?r are omitted.

P hlean squares

and products of the turbulent fluctuations in velocity and density are taken to be of order 6, the boundary layer thickness. As in the incompressible two-dimensional cast tj5.s last assumption is probably justified in favourable or slightly unfavourable pressure gradients, but is not so when separation is approached. The usual boundary layer approximation, namely that rates of change of mean flow propertie s in the c and 71 directions are of order unity and those in the ;j direction are of order 6-1, are made. We write

K, = 1 %2 R2 t= 1 ah., - e--e - -I__-*

h, h2 X ’ 11, h2 &-J ' (2)

these are the geodesic ourvaturcs of the curves E = constant TJ = constant respectively, J7e denote the mean of a product f'g' by (fTF). A full acoOur&

of the derivation of the equations of motion is given by Vsglio-Laurin'2. steady mean flow, in the absence of body forces and of heat sources the equations are

(3)

-5-

F’ I

c

1 ap la av = ----i--- P9 a7l p a,r, [ li- at; ----p(W) ) 1

& (Ph2u) + g (ph,v) -I- L !ph, h2 ag t [

w -I- F]! = 0, ,'

O+)

(5)

I a ‘-‘af? = ;zj i-J z+ CL

‘s?- a (CpT)] - p(T-iF)] . I+ a2g (6)

Certain terms on the right hand sides of equations (3) and (4), namely

consist of shear stress terms P(~u/'~C) and p(av/a;=) together with Reynolds

stresses - p(n) and - p(?-??-). We will denote their values by T, and a 2 and their values at the surface of the body by zO, and 1;

02'

It will bc: seen that the first three equations are similaz to laminar equations, except that VT is replaced by w + (l/p)(m) and the terms on the right hand side of the equation s ~(8u/~3~) and p(av/az) are zj and 'Go.

mhich in luminar flow would be The energy equation (6) also resembles

the ianinnr flow energy equation, except that it has an additional term

- p(zF>.

-6-

The values of ap/ac and ap/a?~ are obtained from the flow at the edge of the boundary layer. Denoting tho values of flow quantities at the edge by the subscript 'Ierr we have

ue sue ve due - .- + .I - h, x h2 ac

- K2uevo -I- K,ve2 = - - ' 22 phd ag '

(7)

since p is unchanged in travel.ling through the boundary layer along a normal to the surface.

In d&riving momentum-integral equations in laminar flow it is usual to ciiminate w, and in addition no direct use is .made of the energy equation. Here we oliminatc w + (TV)/p and the momentum-integral equations will be the same as those in laminar flow. These equations arc dcrived in Rcf.3 and so we shall only give the results here. For reasons given in Ref.3 it is usual to use "streamline co-ordinates" which have the property that the curves rl = oonstant are the projections on to the surface of the strw,mlines of the external flow. In these co-ordinates v = 0 by dofinition. e

We write

Finaily we assume the ex-tornal flow to be irrotational, so that a velocity gotential exists, which we may put equal to g. It can be shown that in this case h I = l/ue.

r,

The momentum-integral oquatians then become 3

-7-

%I - y+ R, (sl, - 022) = - e PU

3' e e

(9)

1 2:; ce

e 0 6<, ) " I$ 02,

To2 -I--=------- -t- 22 -I- - = II -- ' 3

m

h2Uc, e PU ee

as shown in Kcf. 3.

2.2 Three-dimcnsionsl tiffects e-"_p_ --w-e --<- -P--aFr-n--cz

An account of the genera!. behaviour of three-dimensiontil boundary lZijE1- S (whcthcr laminar or turbulent) is given in Ref. 3, and we shall not repeat this in detail. Iiowcv~ rrje may mention some of the more important points.

As we traverse the boundary layer along a normal to the surface the direction of flow changes; the velocity vector r0tatc.s usually, but not alv~ays, in one direction as we travel from outside towards the surface. If wo take the direction of the txt-rnal flow as ihe standard direction, any component of velocity normal to this direction will be called "cross-flow".

A physical explanation for the czristence of cross-flovrs may be given as follows. The str~amlincs at the edge of the boundary layer are suc;2osod to be curvc;d, so that there must be ,an inward pressure gadiont bal.ancing the centrifugal force. Scar to the surface oi' the body the fluid has been retarded, whilst the pro,, ~"nure gradient is unohLnged. Hence the inwards pressure gradient is now too great for the centrifugal force and the fluid iS made to move inwards.

This is fairly easy to understand so long as the inwards pressure gl-adicnt incrtiascs as WC travel downstream. !Vhen it decreases and changes sign, as it will do if the exttirnal strcamlincs reduce their curvature and bend the other v&y as at a point of inflexion, the matter is more complicated. ?i:c rcduct:ion,vanishing, and reversal of the pressure Gradient does not cause: an immediate reduction, vanishing and reversal of the cross-flow. This tal.:c,s tim~5. :"he reversal first occurs downstream of the point of inf'lcxion and then only na:+r to the body, so that there are places where the cross- flow is in7;ards (referred to the new curvature) at points near to the body, and outwards at points further away. Only after travelling some distance downstream of the point of inflexion will the fiov< bc reversed at all levels of the boundary layer. Pig. 1 shows some calculated cross-flow components for a laminar boundary layer. Similar cross-flows should apllear if the boundary layer is turbulent.

-8-

In general the cross-flow in a turbulent boundary layer is much less than in one which is laminar. 'This is to be expected since in equations (-7) and (2) the virtual shear stress terms such as

shear stress

illustrated in Fig. 2 taken from Ref. 43. This shows oil-flow patterns over a wing vVith 55O sv:eep. The oil-flow lines lit along the direction of flow close to the surface of the ning. The direction of flov: outside the boundary layer may be taken to be almost the same as that of the undisturbed flow. It will be seen. that there are v:ed,c-shaped areas Trchoi+e the direction of flovv is much "straighter" than over the rest of the wing. These arc places where the flow is turbulent and at these places the cross -flow is much less thpan at places where the flov: is laminar. In sencral one may expect turbulent mixing to attempt to maintain the mainstrc;&m velocity in magnitude and direction deeper down into the boundary layer than in laminar Slow. The fact that this happens to the si&nitudc ot the velocity vector is well-known from -u_.. studies in two dimensions. It mill also happen to the direction of the vector. ---__I

Another major effect of the thzoc-dimensional nature of the flow is more familiar since it occurs in axially symmetric fl.07;. This is the effect of diverging or converging streamlines in the external flow, Converging stream- lines cause a thickening of the boundary layer. This is to be expected, since the boundary layer has less room to "spread itself", so to speak, over the s-urface ir, the converging flow. The opposite effect takes $Lacc when the streamlintis divorge.

2.3 14 The axiall~anal. u---e- au-

Let us suppose that we arc following a definite external streamline q = constant, along which we have

and so along this streamline

L I a aE;=<S’ (11)

In addition we write H c 8,/O,,, h2 = r and denote p-artisl derivatives with

respect to s by primes.

Ye nov,r assume that the cro sswise component of velocity and its derivatives are small. In this case ne find that equations (9) and (10) reduce to

-P-

ek + fJ, c (3 + 2) $ * i$ + G. = .-ILL, e pe I PU ee

(12)

2u’ e;, + e21 I

Pk 2rt c + -f + -=F=- 1

1

4 + y g+ (e,, 4,) = To2

----ye 03) PU e e

It willbe seen that equation (12) is the standard momentum-integral equation for compresnl. ‘blc flow over an axioymmetric body, whose cross sections have radius r. This is what we mean when we speak of the axially symmetric analogy.

with identical to th e well-known tmro-dimensional equation

04)

Now r'/r (as in axisymz~etric flow) is a measure of the amount the streamlines diverge (or converge if r' is negative). Thus it is seen that converging flow (rt < 0) has the same effect on 0,, as an adverse pressure gradient.

In a calculation it is necessary to dotermine the value of r. It can be shown'&,'5 that if the equation of the suri'ace in Cartesian co-ordinates is z = z(x, y) and if U and V are velocity components parallel to the axes x and y, then r is given by

where

05)

(16)

Subscripts attached to z denote partial derivatives.

In many cases z and z are small, and in this case equations (‘l5) X Y

and (16) simplify to

- IO -

a a U c 77; = ‘ax

~+v--. 3Y 1

Certain conditions are necessary in order that v may be small. If terms of order ~2 are ignored we find from equations (3) and (7) that in streamline co-ordinates

07)

= K2 (peue2 - pu2> + +c i

P $p(77) . . 1 08)

Hence if v and m) and their derivatives are to be small either K2 or 9 2

PU ee - pu2 must be small. Tnc former condition implies that the geodesic curvature of the streamlines must be small, as might be expected. However I K2 need not be small if peue2 -

2 pu is small. According to Vaglia-Laurin'2

this condition will hold at moderate Xsch numbers if the wall is highly cooled. Vaglio-Laurin supposes that peue2 = pu2 and points out that equation (I 8) will

then admit the solution v 3 0, (m) 5 0. He is then able to use the axially symmetric analogy completely and has no cross-flow equation to solve.

3 - VZXCCITY :'I?OFILES MD SKDJ'RICTION C0E.'0~7EXTS

In order to make an attempt to solve the momentum-integral equations (the solution of the full turbulent boundar layer equations cannot be achieved at present, even in two dimensions 3 it is necessary to make some assumptions about the velocity profiles and the values of 7Cj and ~~2. These

must depend on experiments not many of vrhich have been made in genuinely three- dimensional cases. Boundary layer measurement is a tedious and time-consuming affair, and the determination of zO, and 202 in particular, is a matter of d I considerable difficulty. So far as I am aware no thrtie-dimensional floating element experiments have ever been made. Preston or Stanton tubes may be used, of course, provided they can be properly calibrated, but their directional properties are poor, so that they may only be expected to give the total skin friction. Oil-flow patterns can of course be used to give direction of flow, but no systematic tests of this nature have been made, so far as I am awcaree

?

Some measurements of velocity profiles not very numerous.

have,keen reported but these are The experiments of Cruschwitz were made in a curved

- 11 -

channel with an initial straight portion. Ku&he, McKee and Curry'7 investigated flow over an elliptic wing fi7Jep-t at an angle of 25'. In these experiments the cross-flow w ;as quite large in places. Wallace~8 made measurements on a SirJept tapered wing, the mean sweep being 30°, whilst Brebner and Tyatt'3 Johnstonls5, t

investigated wings snept at angles of &5O and 55'. * experiments mere carried out over a flat vrall bounding a jet

flowing at right angles al;ainst a wall. See Fig.3. Blackman and Joubertlg examined profiles near to the trailin, 0‘ edge of a delta wing at incidence.

It is from this rather small body of experiment that vrorkers have attempted to develop calculation methods.

3.1 Velocity profiles --

Calculation methods have so far only been a-i3plieci to the case of' small cro,ss- flow and in this ease it is to be hoped that streamwise velooity profiles may be similczr to those in t:iro dimensions. Experimental evidence confirms this in general. It has therefore been usually assumed that the streamwise component follows a power law of the form

09)

and n = 7 is commonly used both for subsonic and supersonic flows. Iviager*' found that this form fitted some experiments of Gruschwitz16 fairly well.

Cooke2' found that the form

2.. = U

e

fitted some experiments of Iliallacc 18

Becker2* quite well except near to the wall.

also used this form.

Zaat23 su&gcsted for compressible flow

(20) *

(21)

where

To2 a = tan a = - . "01

-12 -

He quoted %allace's I 8 experiments to justify the form for v/ue.

The form, based on experiments by Gruschwitz 16 on ducts and Wallaoe 18

on wings,

( > 2 .L =

U I+

e

2. U

e a, b-2)

with a= tan a,, was used by Nager 20 , Braun 24 and Cooke" , vrhilst Becker 22

suggested that

fitted Gruschwitz's experiments very well if

a = 4 -5 tan a = $.h2o 3 =01

(23)

(24)

17 Equation (22) also fits the experiments of Kuethc eta1 ,quite well, but

(19) fits poorly with n = 7.

Johnston5'6 . made a close stu&y of the velocity profiies of referenoes I 6 and 17 and as a result of this and of a long series of experiments of his own he suggested a form fcr the cross-flow as shown in Fig. 4. It ~S.11 be seen that v/u, is expressed as a function of u/u,, so that v = 0 TJhen u/u = 0 e and also when u/ue = 1. Fig. 5, reproduced from Ref. 5, shovs how Johnston's model fits the data of Kuethe et al. This form of -@ot does not shorn very clearly the way the cross-flo:7 varie s across the bound,ary layer. 'Ye have therefore in Fig. 6 plotted v/ue as a function of US for Johnston's fit of

his model to one of the profiles of Gruschwitz " (station IO along line 3), taking for the streamwisc component the value

1 A L

0 7 =

U 8 ' e

r:hich fits the observations quite well. This particular profile was chosen because in most of the others the peak is too near to the v/ue axis to be oonvenicntly drawn. Profiles such as that in Fig. 6 illustrate the difference

-13 -

between laminar and turbulent boundary layers in this respect. The peak is very much nearer to the wall in the turbulent than in the laminar case, Pig. 7 is an attempt to give a full three-dimensional picture of the same profile. Unfortunately Johnston's made1 differs from others mostly in the region close to the wall where measurements are the most difficult to make and suffer from the most inaccuracy.

Johnston' s form for the velocity profile v may be written

V eu -.. = .- u e U e

near to the wall, and

2.. ue

t A 1 -” ( > e

(25)

(26)

further away from the wall. Tlie change Prom one to the other takes plaoe where the tv?o lines in the u, v plane (Fig. 4) intersect. Johnston suggests that for external streamlines which are initially straight and then become circular A is given by

A = 28,

where /'3 is the angle through which the streamlines have turned. He found, hoip<ever that except perhaps 2 in eq:ation (27). Indeed

for flow in ducts, one cannot rely on the factor a factor 1 gave closer agreement in one case and

this was the number he used in a calculation. The quantity e is equal to tan a as already defined (a is the angle between streamlines and limiting streamlines) and is an unknown to be determined in the course of a calculation.

Yinally, Coles3 suggested that the velocity vector 2 should be given by

where

In these equations q7; is the magnitude of a friction velocity vector

sT which is taken parallel to the surface shear stress vector zO The

- ILL -

functions f and w are the well-known "law of the wall" ‘and "law of the wake" functions, that is

and w is a function tabulated by Coles. According to Coles rlsT is to be a

vector constant in magnitude and direction at any given station. ThUS %

andsv are b tk o 1 vectors constant in direction but varying in magnitude for

varying 2: according to the variations of the functions f and m. zitself varies both in magnitude and diaection. Coles finds in an example that the direction of s,{ is nearly the same as the direction of the external prcssure

gradient at the point concernud, and makes the tentative suggestion that this should hold universally.

Blackman and Joubert 19 attempted to tsst the theory in an experiment. They divided the vector -9 into components wh in thc 2T direction and wj

perpendicular to this. wh and wj, suitably normalised, should both have the shape of the law of the wake function w(t;/6) if the hypothesis of Coles is correct. The results are shonn in Fig. 8. Blackman and Joubcrt claim that the fit is good, since wj has a fairly good fit. They rightly say that wh

is subject to much more uncertainty than w,; J

indeed the lack of experimental

precision obtainable by present methods makes scatter inevitable in this P component. There was, however, very little Gross-flom in the experiments (a < 5O) and it was not yossible to determine the direction of the pressure gradient, which was in any case small.

The question of the fit of experiments to the models of Coles" and Johnston5 is in a state of some corfusion, Coles found that his model fitted the measurements of Ku&he et al'/. very ;-rell, and Johnston has used Pig. 5 to show how his model also fits the s:Lmc observations. On the other hand Johnston found that his own observations could not be made to fit Coles's model. Johns;Czn was able to fit his model fairly well to the experiments of Gruschwitz ; Colts felt that the experiments concerned were probably subject to errors near to the ~-all. In the discussion following Ref. 6 Coles reported on his at-tcnpts to fit Johnston's mcasure- ments to Coles's model. He found that zq could be made to fit ~~11, and

that the dircotion of-TV KS close to the direction of the pressure gradient,

but that the fit to his form for w(u6) X~S not good. Finally we may note d that the profiles of Blackman and Joubertdy helped to confirm Coles's model

.

but could not be made to fit that of Johnston.

&tixch less is l:nobsn about ~lir,~,:e-diner;slon,zl compressible bouni!ary layer profiles than incompresaiblc ones. One ma;/ 5opc that tno-dimensional fiows will again give us a gil-ide. In tLis connection one may expect tbst the transformation

-15 -

where pm ia evaluated at the temperature Tm of equation (31) below, will nc reduce streamwise compressible profiles to incompressible ones. SpenccLY found this transformation to be moat successful in the two-dimensional case. @ore apeoulative would be to assume that the transformation would also hold for the cross-flow. If it did one would be able to formulate a compressibility oorrclation similar to that for laminar flow.

3.2 sll fxiction values -.-

For the streamwise skin friction in incompressible flow Becker 22 used

the Ludwieg-Tillman forr!!ula26.

%I e % ‘7

= o.,23 x ,,-0.578 -“‘268a

PU c > V

e

Cooke", folloxing Spence 27 used the relation

(29)

(30)

where quantities with suffix m are evaluated at a temperature T, determined from the relation

T, = &be + T,) c 0,22(Tr - Te), (31)

where Tr is the rcoovery temperature given by

Tr = Te (32)

r' is the "recovery factor" and is usually taken as G.89, whilst He is the

Usoh number at the edge of the boundary layer. The assumption

- 16 -

p a T"

was also made, 75th L\) = 8/9.

BraW124. used the relation

(33)

with T m given by equation (3-i). The viscosity temperature relation used

WSS

(34)

K is 0.0225 for a f'lat plate. Braun, ti:o -::as dcaline ;:rith a slightly yawed cone, did not take K = 0.0225 but he multiplied this value by 2/sj, saying that this is usually done fcr an unyawed cone.

,' All these forms aretaken from two-dimensional methods. The only evidence

for this comes from axisymnetric bodies, for which it seems to be true that the same flow mechanism holds locally as in t-

. wo-dimcnc~onal bodies. !

Vaglio-Laurin'2, in a problem with zero cross-f'lmr, after using a transformation which reduce h-is compress iblc problem to an incompressible axially symnotric one with zero pressure gradicn t,uses equation (20) which, on integration of thv equation for Oql in cqaation (8) gives zo, in terms

of 8 I I l

The skin friction com,Ionent 702 in all the methods is taken to be

equal to zo, tan a, where a is to be determined in the course of the

solution. All workers except Becker take

tan u = lim z0 z+Q

(35)

Becker 22 inserts a factor 2 on the right of equation (35). There is no special reason wh;r equation (35) should hold exactly, since the forms for u and v used by all workers Co not apply ne.ar to the wall.

Only two general methods of solution stem to have boon suggested, both of which assume small cross- flow and both of which solve the sWiamwise equa- tions by taking over two-dimensional methods. Braun24, :pho found a solution

- 17 -

l

for a cone at small angles of yaw, also implicitLy assumed small cross-flow and used two-dimensional properties for the streamwise flow. 'Jaglio-LaurinA2 of c0urs0 took the cross-flow equal to zero and then also used two-dimensional methods for the streamwise flow.

4.1 Cooke'- m&hod 21 .m.m-A,-.-*-.,

In this method the solution of the streamwise equation follows 5 closely the method of Spence27 in two dimensions. This method uses the

momentum equation (12) and in incompressible flow the strreamwisc shear stress is given by

TOl e 01, - = O.cD88 --

2 PU e >

-'/5 e V

e

This equation is due to Young 28 .

(36)

and assuming that in the determination of EI1, it is sufficiently accurate to write H = 1.5, we find that equation (12) becomes in the incompressible case

6 k ue4.r'*'j = 0.0106 u:r'02, (37)

where r is found from equation (47).

For the cross-flow the form (22) is assumed, with a = tan a a a, and H is again assumed tobc constantly equal to 1.5. Writing

E = a r u -3’2 . e

t Cooke finds

c. dE 0.0166 zi-rE =

2.187 % -=JE-!TK’

e

This method was applied to a 55' swept wing ttsted by Brebner and vJyattI3, and included the determination of EI by Spence's method29, and SO

%' bj and j3 could all be calculated. Some results are shown in Figs.9 and IO.

- 18 -

The method can readily be extended to compressible flow using a transformation for the streamwise flow due to Spence27. For the flow over infinite yawed wings the equations for 0,, in the case of zero heat transfer and also of constant wall temperature are given in Ref. 30.

Cooke3' has also applied this method to the flow over a delta wing of 130 thi.&ness-chord ratio at zero lift at a Mach number of 2. This wing has been tested in the R.A.E. 8 foot tunnel and it was found that the pressure measurements agreed quite well with calculations by slender thin wing theory. However the separate velocity components were not measured and so the calculated values of these components were used in the boundary layer oalculations. The streamlines could then be found and are drawn in Fig. 11. Turbulent streamwise calculations using equation (12) and Spencc's transforma- tion27 were then made. These calculations could be simplified because it ras found that the term (H + 2) (upe) + p’/p could be ignored compared with the other terms in the ooc?ficient of 0,,. The result is shown in

Fig. 12. There is considerable convergence of the external streamlines near to the centre line and this causes a thickening of the boundary layer in this region. Everywhere else it was found that the value of 13,~ was

much the same as it would have been if the flow had been over a flat plate of the same planform, It was found also that the additional pressure drag due to the displacement thickness was less than 1,x of the total skin friction drag plus wave dra, 0 and it could therefore be ignored.

4.2 Becker's method 22

Becker assumea the equations (20) and (23) fDr the velocity components. For the streamwise shear stress he assumes the well-known Ludwieg and Tillman relation26

=01 - = 0.123 x lo P”e2

and suggests that a clnse approximation to this is

“04 e e,, - = 0.0125 7

c >

44.

P”e2

.

He deals only with incompressible flow. Unfortunately Becker assumes a rather special external flow correspond-

ing to a special value of r in equation (12) namely r = l/ue. This assumption is equivalent to assuming that arrr$ac = O,,which implies that the external flow is two-dimensional, as may be seen by oonsideration of the equation of continuity for the external flow. The assumption is easily avoided and Becker's streamwise equation may then be written

- 19 -

The croar, -flow equation (-13) becomes according to Becker

%l + 02d

c

Tol 2u; &’ Y3(H)~---=---a + -- .i- -

II p"e2 u = "--

r (40) e

where

Y3(Ij) = t z+f+-,

though Becker's equation has the term 2r'/r ;- 2uCyue missing bc.cause of his special value of r.

T In order to solve these equstiop3 one may assume a constwit value of H as we did previously. y3(d = 0.52,

Becker suggests H = 1.29 in one example. This m&es As an alternstivc Becker fin& Ii by an equation which is

I indcpendcnt of r. He writes

and USGS the equation

B =

Equations (39), (40) and (41) 6, H and a, sinoe

are then three equations for the three unknowns

5-1 = ?qirq H-l 6

' e21 = -zm

H-' a6 l

(42)

Becker does two examples, the first or" which is the case where the streamlines are circles of radius R, and ue = f(R), so that ue is constant

along streamlines; he also supposes the boundCar layer is "very thick", so that the term 704 may be ignored in equation . This equation then reduces to

51 e;, = T (l+H),

remembering that Becker writes r = l/u e' If H and 8,, arc constant, and H is equal to I.29 this by equations (42) reduces to

a = I .5ah

where @ is the amount the external streamlines have turned from the start.

For his second example Elecker considers the decay of a cross-flow which had developed earlier, specified by 0 2, = (t32,)o, and is now subject to uniform external flow with straight streamlines. In this case equation (40) becomes

*;I + 0.0065 e2., = 0,

where

Reg ue 'II

II = a?,

and once more H = 1.29.

- 21 -

This gives, using (38),

e21 = (e2,)o ems”,

where e

( >

l/4 e =

154 % Roell *

4.3 ___ Johnston's solution along a line of ssnmetr 2 ,33 4-_-e %- - - .

Vhen applying the axially symm&ric unalom vqe assumed that C,2, e22 and their derivatives in equation (3) could be ignored. Johnston, though he ignores 02" (it is in fact zero along a lint of symmetry), does not

ignore N,,/hi&%

If we assume oonstant density and write

we find that equation (9) takes the form

1 a e12 c UP

e;1 -i---+0 (H+2)$ + "01 =

r aq II LL1pIz) PU ee

(43)

which we may compwe with equation (*12). Johnston points out that the tc-rm rV8,d/z- rupresonts the effect of oonvi.rging or diverging streamlines

(as we have already explained in Section 2.3) and that the term ae12/r&l represents the effect of boundary layer "s&%i.ng". That this term need not be small was shown by lAoore and Richardsor&. In one el;poriment, for instance,

P they found the values

1 a e12 Tol -T=

0.007, - 0.008, - = o.cQ17, r 2

PU ee

so that in this O&SC the oolltribution of d0,2/raq certainly could not be

ignored.

- 22 -

Using his model [equations (25) Johnston shows that for large Reynolds numbers

Hence we have along a line of symmetry

Now, as disousscd in section 3.1 Johnston assumes that k = /3, where p is the angle the streamlines have turned from their initial strai&t path, and since the curvature K, of the lines g = constant is

we have by equation (2)

5 r' =..- = r

2.. = 5%. (45)

Finally Johnston assumes that r = l/u,, the same assumption that was

YF;ed b+ &dker 22 ) which is tantamount to assuming that the external flow is two-dk&&ional and is probably justified for the experiment of Fig. 3.

Heno'e we have from equations (43), (I+&) and (4.5)

U’

e;, = “01

-2e11 <+--=- 2 PU ee

(46)

along a line of symmetry.

% ‘Thfs equation is inteLrated by using the Ludwieg and Tillman 26 formula 9 for ,T~, Combined with the von Docnhoff and Tetervin35 formula for H.

Alternatively a method due to Rotta 36 is followed.

Eii;her of the methods gives results agreeing well with experiment in the special case considered.

- 23 -

4.4 Braun's solution for the sliphtly y awed oone 24

Braun solves the problem of a oone of semi-angle 0 yawed at a small angle y in a supersonio stream, He uses a streamline co-ordinate system and uses the velooit profiles (19) and (22) and wall shear stress values given flow he USeS ,,!: i3;lbg; by equations (33 (32) for zol ana (35) for 1602s For the external

I 37. Writing X = y/sin8 for the yaw parameter he expands tan a and 6 in the form

!E tana = u,h + a212 f ..*,

6 = ho + S,h * l . . l

He is then able to find a ,, b. and 6, in olosed form. Perhaps the most interesting results are the deflection of the limiting streamlines, that is, the value of a to the first order in X, Fig. 13 is taken from Braun's paper, In this figure Jr is the azimuth angle round the oone measured from the generator with the greatest inclination to the flow at infinity. It will be seen that a is increased by heating, but raising the Mach number reduoes it. The deflection angle is much smcaller than in the laminar adiabatio o&se and it decreases with Mach number in contrast to the laminar case, where it increases with Mach number for values of Mach number greater than about 2.

5 TRANSITION

We shall only make a few isolated remarks in this seotion, some of which are speculative and not completely justified either experimentally or theoretically.

So far no calculations appear to have been made in oases when the flow is partly laminar and partly turbulent. First of all the determination of the transition flfrontfl itself is a difficult matter into which we shall not enter, beyond remarking that two kinds of instability seem to be involved, namely two-dimension&l instability (Tollmien-Sohliohting) and sweep instability, It was suggested by Eiohelbrenner and Miche138 that both of these types exist on a surfaoe; in oertain regions one is more important than the other and in other regions their roles are reversed. Thus, for a slender wing with leading edge separation and reattachment, transition fronts as shown in Fig. 14 may be possiblejg, where CDE is a front due to two-dimen- sional instability and ABC and EFG are fronts due to sweep instability. Fronts of this type have been observed. A third type of instability, that described by G'drtle&O as occurring in flow over concave surfaoes, may also have its effect in plaoes.

Another interesting fact has been point out by Gregorybl. It is that when transition occurs at an isolated point (say near to the leading edge of a swept wing) the usual wedge of turbulenoe appears, spreading out at the usual angle, but the wedge is curved, its edges being the envelopes of lines making angles t10.6’ with the external streamlines starting from the point oonoerned. Fig.15, taken from Ref. 4.2, shows oalculated wedges of turbulence starting from points at different distances from the leading edge. It will be seen that the nearer the disturbance is to the attaohment line the greater the area covered by the turbulence wedge. In particular, if x/o is equal to 0.01 the whole wing outboard of the point of disturbance is l*oontami:.ated". Fig. 16 is taken from photographs by Gregoryb2, and illustrates this phenomenon very well.

When turbulent calculations are to be made they must start with some known initial conditions. In two dimensions it is usual to assume that the

- 24 -

momentum thickness is unchanged at transition; this means that skin frio- tion and displacement thickness are usually discontinuous. (Actually tran- sition does not take place suddenly but is spread over a region which is fairly short, so that skin friction and displacement thiokness vary rapidly but are not quite discontinuous.) The assumption is that there is no rapid change in the momentum thickness. as follows,

An argument in favour of this might run A discontinuity in momentum thickness implies a sudden change

in momentum and this means that there must be not merely a discontinuity in skin friotion but also can infinite foroe at the point of transition. There is some experimental evidence for this assumption, as may be seen from Fig.17 which shows displaoement thickness and momentum thickness as drawn by Rottak3 from experiments by Schubauer and Klebanoffh.

In three-dimensional flow the problem is more difficult. An obvious first suggestion is to make 0,, continuous. This is not suffioient and a second condition is required. Ore that has been suggested is to make a continuous, that is, the directic;? of the limiting streamlines is to be unchanged, Apart from the fact that this direction is conneoted with skin friction which we expect to be discontinuous, the assumption of an unahanged direction is contrary to the evidence of Fig. 2, a careful examination of whioh seems to show a discontinuity in direction of the oil-flow lines at the edges of the wedges of turbulenoe. A discussion which is not, and can- not be, wholly satisfactory, is given in Appendix 1. It would appear from this that the result depends on the angle a' whioh the normal to the tran- sition front makes with the local external direction of flow. The result is that

51 00s a' - 8 12

sin a' and e22

sinat -0 21

co9 u'

are continuous at transition, If the oross-flow is small,012 is small com- pared with Cl, and 822 is small compared with 821. In this case the condi- tions reduce to a simpler form, namely that 8,, and 821 are continuous at transition, independently of the direction of the transition front. This was pointed out to me by Professor A.D. Young.

6 CONCLUDING FUZMARKS

Although little work on turbulent boundary layers has been done the outlook for calculation methods is not quite so dismal as at first appears. Velocity profiles seem to be of a more "universal" shape than in laminar flow, and momentum-integral equations seem to be adequate in favourable or slightly unfavourable pressure gradients. Energy-integral equations could also be further exploited in this connection.

Even in the laminar case it has been found that streamwise velocity profiles are very like two-dimensional ones and, since turbulent mixing is a three-dimensional phenomenon, it may well be expected that this similarity will be borne out more faithfully in the turbulent case. The great differ- ence is that one can never resort to an exact solution to test any method, and experiment is the only possibility. In any case methods are largely empirioal. This inevitably means that calculations with turbulent boundary layers will be inaccurate, and indeed in two dimensions can aocuracsy of from 5 to 10 per cent is all that is usually expected, Within that range the position is not too hopeless &and one may well anticipate improvements in the future if only more systematio and careful experiments can be made.

i

.

We have made no attempt here to discuss the structure of a three- dimensional turbulent boundary layer. This has an extensive literature in two-dimensions but practically none in three dimensions, In addition, so

- 25 -

far as I am aware, no work has been done in the unsteady case or in connection with separation. The problem of detormining separation is difficult enough and doubtful enough even in laminar flow. However separation seems to t,ake place fairly abruptly in the sense that the bending of the streamlines increases considerably just before sepamtion and methods of solution break down when this happens. One may therefore hope that .a place whore a method breaks down is fairly near to the place of separation. Eurthcr than that one cannot go at present.

&ST OF SYMBOLS

A

A, B,

b

c

C chord measured normslto loading edgc

cP e

fs2 ( > Y

I1 a

hl' h2 I$ , R2

x

h¶

n

P

Pr

4

%

R

Ree 41

faotor in vc-locity profile (26)

factors in equation (41)

factor in velocity profiles (21), (22) and (23)

factor in vtlocity profile (21)

constant in viscosity temperature relation (34)

specific heat at constant pressure

factor in velocity profile (25)

law of the wall function, cc,. (28)

form factor

total enthalpy

coefficients in line element ('t)

geodesic curvatures of curves c = const, q = const. Eq. (2)

constant in skin friction formula (33)

Mach number

power in eq (19)

pressure

Prandtl number

velocity

friction velocity

radius

Reynolds numbcr based on 6,.i

- 26 -

F

r

S

T

r'

U¶ v, w

u.

wh9 wj

WLzb)

XI Y, e

a

at

LIST OF SYMBOLS (Conttd)

recovery factor in eq. (32)

h2

length measured along a streamline

temperature

external temperature over unyawed cone. Fig. 13.

velocity components

reference velocity

wske velocity components in direction of limiting streamlines and normal to them

law of the wake component

Cartesian oo-ordinates

defined by tan a = 'co2/'to,

angle between the normal to the transition front and the local direction of the external flow

angle of turn of external streamlines

angle of yaw of cone in Braun's work

boundary layer thiokness

transformed boundary layer thickness

displacement thickness "components"

arue-3~2

momentum thickness "oomponents"

Fig. 12 and section 4.4. Semi-vertical angle of oone

Section 4.1. e.j, he 8, ,/4"5

ooefficient of viscosity

kinematio viscosity

ourvilinear co-ordinates, 4; along streamlines, c normal to surface, q normal to g and ;:

density

shear stress components

wall shear stress oomponents

azimuth angle of cone

- 27 -

LIST,OF SYidBOLS (Cont'd)

0 powcr of T in viscosity-temperature relation H-l

;; K

Subscript e denote s vrtlues at the edge of thi: boun&ary layer

tt

fl

m denotes values at a temperature Tm given by equation (31)

r aeno-tes rccovwy vaiues

2 tioore, F. K.

c

3

.

4

5

6

7

8 ;

7 9

IO

Cooke, J . c . , Ikll > 41 G 0 I* .

Colts, 1,.

Johnston, J. P.

Johnston, J. P.

iTott, K., Crabtrct:, L.P.

Ashkcntls, II., Riddcll, F. 8.

Turcotte, 13. E.

Maskell, 5. C.

Title, etc.

Boundary layers in three-dimensional flow. Agylied Mechanics Reviews, Vol.7, p.281, 11354-a

Three-dirncnsional boundary layer theory. Chapter in Advances in Applied RPechanios, Vol. 4. Edited by B.T,.Dryden and Th, iron Karmon, Accdemic RXSS, 1756.

Boundary layers in three-dimensions. , Chctp. in Progrtiss in AeronauJiical Sciencea, Vol. 2. Pergsmon Press, 1962.

The law of the make in the turbulent boundary layer. J. Fluid. &oh., voi.1, p.171,1956.

The three-dimensional turbulent boundary layer. E.I.T. Gns Turbine Laboratory Report No. 39. 1957.

On the three-dimensional turbulent boundary layer gcneralzd by secondnxy flow, J. of Bo,sic .&q+neering, SeriLa D, Trans. ASME, vol.. 82, y.243, 1960.

Simplified luminar boundqy-lsycr calculations for bodies of revolution ‘and for yawed wings. J. heron. Soi., Vol, 13, p.553, 1952.

Invostigatiori of the turbulent boundary layer on a yavci flat plate. XACX, TE Do, 3383, 1955.

On incomprcssiblc turbulent boundary layer theory rzpplied to infinite yawed bodies. Cornell University Cruduste School of Aeronnuticcl Engintexirq;, 1955.

#low separation in three dimensions. ARC 18,063, 1955.

LIST OF REFERE~~CES_ (Cont'd)

Author(s)

Cooke, J. C., Brebner, G. G.

12 Vaglio-Laurin, Ii.

13 Brebner, G. G., Wyatt, L. A.

14 Cooke, J. c.

15 Zaat, J. A.

16 Gruschwitz, E.

17

18

19

20

Kuethe, A.M., McKee, P.B., curry, vu'. II.

Wallace, R. E.

Blackman, D. R., Tne three-dimensional turbulent boundary layer. Joubert, P.N. J. Roy. heron. SOC,, vol. 64, p.692, 1960.

Magcr, A.

2-l Cooke, J. C.

22 Becker, E,

23 Zaat, J. A.

Title a --,.A

The nature of separation and its prevention by geometric design in a wholly subsonic flow. Chapter in "Boundary Layer and Flow Control". Vol. 1: Edited by G,V. Lachmann, Fergamon Press, London, 1961.

.

Turbulent heat transfer in blunt-nosed bodies in two-dimensional and general three-dimensional hypersonic flow. TADC 'I'I‘I 58-301, 1958.

Boundary layer measurements at low speed on two wings of' 45’ and 55’ sweep. ARC C.F. 55it. August, 1960.

An axially symmetric analogue for general three- dimensional boundary layers. ARC R. and M. Ju'o. 3200, June, 1959.

A simplified method for the calculation of thrce- dimensional laminar boundary layers. NLL Report El&, 1956.

Turbulente Rcibungsschichten mit Sckund%str6'mung. Ing. -Arch. Bd. 6, p.355, 1935.

I

hIessurements in the boundary layer of a yawed wing. NACA Tr< 1946, 191+9.

The experimental investigation of a swept-wing research model boundary layer. Municipal University of Wichitn Aerodynamics Report PTo. 092, 1953.

Gcnerslization of boundary-layer momentum-integral equations to three-dimensional flows including those of 5 rotating system. T~ACA licp0r-t NO. : 067, 1952.

A calculation m&hod for three-dimensional turbulent boundary layers. + ARC R. and M. TTo. 3199, October, 1958.

Bercchnung von Reibungsschichten mit schwscher Sekundarstr%mung nach dem Impulsverfchren.

z. Elugwiss., ~01. 7, ~~63, 1959.

I

Beitrggc zur Theorie der dreidimensionnlen Grenzschichten. HLL Report MP 190, 1960,

- 29 -

No acs,& Author(s)

24 Brcun, '1%. H.

25 Spenoe, D. A.

26 Ludwieg, H., Tillmann, !"v‘.

27 Spence, D. A.

28 Young, A. D.

29 Thwaites, B. (Ed.

30 Cooke, J. C.

31 Cooke, J. C.

32 Kchl, A

33

34

35

36

Johnston, J. P.

Moore, R. 1'. Richardson, D. L.

Doenhoff, A. E. von Determination of general relations for the behaviour Tetervin, N. of turbulent boundary layers‘

NACA Report 772, 1943.

Rotta, J.

LIST OF REFERENCES (Cont'd)

Title, etc. --

Turbulent boundary layer on a yawed cone in a supersonia stream. NASA TR R-7. 1959.

Velocity and enthalpy distributions in the compressible turbulent boundary layer on a flat plate. J, Fluid Moth. Vol. 8, p.368, 1960.

Untersuchungcn iibor die Wandschubspannung in turbulenten Reibungsschichten. Ing.-Arch. Vol. 17, p.288, 1949. Translated as NACA TM I 285, 1952, ARC 14800.

The growth of compressible turbulent boundary layers on isothermal and adiabatic walls. ARC R. and M. No. 3191, June, 1359.

The calculation of the profile drag of aerOfOilS and bodies of revolution at supersonic speeds. Coil, Aeron. Rep. MO. 73, ARC 15970, 1953.

Incompressible Aerodynamics. Oxford University Press, 1960.

Boundary layers over infinite yawed wings. Aeron. Quart. Vol.11, p.333, 1960.

Turbulent boundary layers over delta wings at zero lift. Unpublished MOA note, 196'i.

Uber konvergcnte und divergente turbulente Reibungsschiohten. Ing.-Arch., Vole 13, p.293, ly43.

T'nc turbulent bound,ary layer at a plane of symmetry in a three-dimensional flow. J of Basic Engineering, Series D, Trans. ASME, Vo1.82, p.622, 1960.

Skewed boundary-layer flow near the end vralls of a compressor cascade. Trans. ASME, Vo1.73, p.1789, qY57.

N&\erungsvcrfahren zur Berechnung turbulenter Grenaschichten unter Benutzung des Energiesatzes. Mitteilungen aus dem Max-Planck-Institut f% Stramungsforsohung. Nr. 8,G%tingen, 1953.

37 Kopal, 2. Tables of supersonic flow around yawing cones. M.I.T. Dept. Elec. Eng, Tech. Rep. No.3, 1947.

- 30 -

LIST 07 -RE3E,",TinTSES (Cont'd)

Title, etc. & Author(s)

38 Eichelbrenncr, E. A. Michel, R.

39 Cooke, J. C.

40 G'drtler, II.

41 Gregory, N.

42 Gregory, IT.

43 Rotts, J. C.

@I- Sohubauer, G. B., Klebanoff', P. S.

Observations sur la transition laminaire- turbulent en trois dimensions. La Recherche AEronnutique, No.65, p.3, 1958.

Boundary layer flow between the attcohment lines on r, flat plate delta ;ving at incidence. kro. Quart., Vol. 13, F.1, 1962.

%er eine dreidimensionale InstabilitXt lamincarerGrenzschiohtcn an korikaven WEnden. mm, vol. 21, p.250, 194-q.

Pansition and the spread of turbulence on a 600 swept-back wing. J. Roy. Aeron. soo. vol. 64., p.562, ISO.

Experiments on transition and on the spread of a turbulence wedge on a 60° swept-back wing. J. Roy. Aeron. Sot. Vol. 64, p.562, 1960.

Turbulent boundary layers in incompressible flow. Chapter ir, "Progress in Aeronautical Sciences" Vol. 2. Porgamon Press, 1962.

Contributions on the mechanics of boundary- layer transition. NACA Report No. 1289, 1956.

ATTACHED:

Appendix I

- 31 -

APPENDIX 1

TRANSITION

We shall consider two-dimensional transition as an introduction to the K method to be used in the more general case, and will give a "proof1 of the

continuity of 0 from elementary oonsiderations. Such a proof cannot be rigorous but it is suggestive,

We assume that the same momentum equation holds on either side of tran- sition. This equation in steady inoompressible flow is

aue z

W g+(H+2)8- = - ue ax P 'e* '

(Al >

where H = S*/0, 'Go is the wall skin friotion and Ue the external velocity.

In turbulent flow there is an additional term

00 a

I’ ii ‘2 ‘2 -5

E u2 aY, 0 e

which we shall ignore as is usually done. Thus the same equation applies throughout.

For small dx equation (Al) may be written, if there are no disoon- tinuities, by the mean value theorem

where [z), denotes a "mean" value, that is the value z takes for some value

of x in the range oonsidered, which is taken to include the transition point. Primes denote values at the end of the interval dx.

We now let dx tend to zero; Ul -c Ue and although H and aw may ohange rapidly or even be discontinuous in the limit their "mean" values are finite and so

that is, 0 is oontinuous at transition. The result is simply any analytical expression of the fact that there must not be an infinitely large skin friotion at transition.

In three dimensions we use equations (9) and (IO), If zo, and 202 are

not to be infinite we find that 0,, and e2, must be continuous at transition.

- 32 -

We have arrived at this result on the implicit assumption that the transition front is at right angles to the direction of flow of the external streamlines. IT, however, the normal to the front makes an angle a,' with the direction of E increasing the result must be modified. We change the co-ordinate system to g1 and r)' with E' in the direction normal to the tr=an- sition front. We now use equations (13) and (14) of Bef.3. They are more complicated but they still lead to the oonolusion that 8' ,, and 0,& are con- tinuous; here primes denote values in the new oo-ordinate system, that is

6 6 q, = _1_

Pe ‘iii bet - u’> pu’ dC, 04, = A-

Pe '* ‘vex - vo pu’ a,

0 eO

where Ue (= ue) is the resultant external velooity.

Now we have

ut = u 00s a’ -vsinu’, U' e = ue 00s a' )

U’ = u sin ut + v 00s at , VI e = ue sina' ,

from whioh we may deduoe

% 2 = 00s a1 8,1 - sin a1 cos a1 @,2 f e2,) + sin* U' e2*

es, = sin CL~ 00s ~1 e,, - sin* U' 8 12 + OOS* d e2, - sin a' 00s CL* 8 22 '

and it is these whioh must be continuous at transition. They reduce to the previous values when at = 0 as they should. A neater way of writing the result is found by multiplying these equations respectively by 00s CL' and sin a* and adding, and by sin ut and 00s a' and subtracting. We find that

0 11

009 d - 8 12

sin u' and e22 sin d - 8 21 00s ut

must be oontinuous at transition,

- 33 - WT.2078 C.P. 635 X3 - Printed in England

FIG. I. LAMINAR CROSS-FLOW VELOCITY PROFILES. EXTERNAL POINT OF INFLEXION

AT t =O*S.

FIG.3. EXPERIMENTAL ARRANGEMENT OF JOHNSTON.

0 he

We

FIG.4. JOHNSTON> FORM FOR THE CROSS-FLOW.

0.1 -

Jo 0

0

0

(1

0 quo

0

0

FIG

0 O-2 0-4 0.6 0.8 I.0

u/u0

5. POLAR PLOTS BY JOHN ‘FROM DATA OF KUETHE ET

STON AL .

FIG. 6. CROSS-FLOW ACCORDING To JOHfWmN’S MODEL WITH due - (S/h L e=O-26, A--I*47

0 0.1

FIG& WAKE COMPONENTS Wh AND Wj (BLACKMAN AND JOUBERT)

CALCULA7ED CALCULA7ED MEASURED MEASURED

MOMENTUM THICKNESS MOMENTUM THIcKN~~sS

FlG.g.VALUES OF e,, AND 6, FROM MEASUREMEN-i-S BY BREBNER & WYATT

6:

o(

k

I I

CALCULATED s MEASUREB

4O

i

.

FIGJO. VALUES OF ti FROM MEASUREMENTS BY BREBNER AND WYATT

FIG. I I. CAKULATED EXTERNAL STREAMLH’ ON DELTA WING AT ZERO INCIDENCE (COO

ES <El

0.0004

0~000l

0

WING

---------- FLAT PLATE

I

0.6 \*o

STREAMUYE I

2

3

4

s

FIG. 12. GROWTH OF MOMENTUM THICKNESS ALONG STREAMLINES OF FIG. II. (COOKE)

o-4

TANOC S\N@ y SIN g

0*2

PERATURE R

I 2 3 4 5 MACH NUMBER AHEAD Of SHOCK.

Tw -= I

\

MACH NUMBER AHEAD OF SHOCK.

FIG. 13. DEFLECTION OF LIMITING STREAMLINES 0N YP~WED CONE (BRAUN)

,

0

A 6

FIG. 14. POSSIBLE TRANSITION FRONT (COOKE)

ATTACHMENT LINE. PARALLEL TO LEAONG EOGE

-g.. / / / I

FIG. IS. DEVELOPMENT OF UPPER SURFACE SHOWING CALCULATEO STREAMLINE AT -3O INCIDENCE AND CALCULATED SHAPES

OF TURBULENT WEDGES ORIGINATING FROM X/C= 0.20, 0.05, 0*02 AND O-01 POSITIONS (GREGORY.)

* * 9 -,

.

MA\N FLOW

NICZAL

FRQM PHOTQGRAPH

O-t0

9, &s

O-08

O-06

0 -04

0.02.

LAMINAR

I

FIG.17 MOMENTUM AND DISPLACEMENT THICKNESS AT TRANSITION. FROM CURVES DRAWN BY ROTTA FROM MEASUREMENTS ON A

FLAT PLATE BY SCHUBAUER AND KLEBANOFF:

THREE-DIMENSIONAL TURBULENT BCUNDARY LAYERS Cooke, J. c. June, 1961.

This paper reviews the current state of knowledge of three- dimensional turbulent boundary layers, mainly from the point or view of making calculations. The nature of the boundary layer is discussed and the equations for general steady tur%lent flow are given. Next follow momentum-integral equations, which can then Se solved with suittile assumptions as to the velocity profiles and shear stress components. An account of these assumptions follows and a few sample results are given. The paper ends with a short account Or some matters connected with transition.

A,R,C. C,F, i’c. 635 532.526.4 :

THREE-DIMENSIONAL TURBULENT BOUNDARY LAYERS Cooke, J. c. June, 1961.

Thfs paper reviews the current state of knowledge of three- dimensional turbulent boundary layers, mainly rmm the point or view of making calculatf ens. The nature Of the boundary layer is discussed and the equations for general steady turbulent flow are given. Next follow momentum”integral equations, which con then be solved with suitable assumptions as to the velocity ProfIles and shear stress components. An account of these assumptions follows and a few sample results are given. The paper ends with a short account Of some matters connected with transition.

I -

Q crown Copyright 1963

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