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i ROYAL, AIRCRAF[' ~iSTA~U~HN~INT

M I N I S T R Y OF A V I A T I O N

A E R O N A U T I C A L R E S E A R C H C O U N C I L

R E P O R T S A N D M E M O R A N D A - , / . .

The Growth of Compressible B o u n d a r y Layers o n Isothermal

.... Adiabatic Walls

B y D . A. SPENCE

Turbulent and

R. & M. No. 3191 (21,144)

A.R.C. Teehni©al Report

. t . . k

~ Z

• k

. , . , k , , .

k

LONDON: HER MAJESTY'S STATIONERY OFFICE

. . . . : . I96I

PRXC~. 6s. 6d~ NET

- a .

SECTION V

REFERENCES

.

Z°

.

.

.

.

.

.

.

I0.

Clauser, F. : The Turbulent Boundary Layer. Advances ,n AppI

Mechanics, Vol. IV, Academic Press, Inc., New York, 19%6.

Von Doenhoff, A.E. and Tetervin, N. : Determination of C, ener/]

Relations for the Behavior of Turbulent Boundary Layers.

NACA Report 77Z, 1943.

Tetervin, N. and Lin, C. C. : A General Inte_gral Form of the Bot

L_~er Equatio___n fo~ressible F1 _o)v with an Application to th

Calculation of the Separation Point of Turbulent Boundar Z ! ~

NACA R e p o r t 1046, 1951.

Maskell~ E . C . : .A_22roximate C a l c u l a t i o n of the T u r b u l e m ~',~Jund

L a y e r in T w o - D i m e n s i o n a l I n c o m p r e s s i b l e F low. B r i t i s h ~' A~ ]~

R e p o r t No. AERO 2443, 1951.

Resho tko , E. and T u c k e r , M. : A p p r o x i m a t e C a l c u / a t i o n of t he C

p r e s s i b l e T u r b u l e n t B o u n d a r y L a y e r w i th Hea t T r a n s f e r and Arbi ,

P r e s s u r e G r a d i e n t . NACA TN 4154, 1957.

McLafferty, G.H. and Barber, R. E. : The Effect of Adverse Pr~

Gradients on the Characteristics of Turbulent Boundary Layers i)

S_~ersonic Streams. (Based on United Aircraft Corporation Repc

R-1285-Ii). J. Aero. Sci., 29, I, pp. 1-18, January 1962.

Ludwieg, H. and Tillmann, W'. : Investigations of the Wall Sheari

Stress in Turbulent Boundary Layers. NACA TN 1285, 1950.

Mager, A. : Transformation of the Compressible Turbulent Boun,

Layer. J. Aero. Sci., Z__~5, 5, pp. 305-311, May 1958.

Coles, D.E. : The Turbulent Boundary Layer in a Compressible

Rand Corp. R. 403-PR, September 1962.

Tetervin, N. : The Application of the Reference-Enthalpy Meth~ -~

Friction Formulas. J. Aero. Sci., Z9, 6, p. 493, June 1962.

Z6

Eckert,

T r a n s f e r to S u r f a c e s in High V e l o c i t y Flow.

8, pp. 585-586, Augus t 1955.

Libby, P . A . , B a r o n t i , P . O . , and Napol i tano , P. :

T u r b u l e n t B o u n d a r y L a y e r w i th P r e s s u r e G r a d i e n t .

E. R. G. : E n g i n e e r i n ~ R e l a t i o n s fo r F r i c t i o n and H e a t

5. Aero. S c i . , Z_~2,

A Study of the

G e n e r a l

App l i ed Sc i ence L a b o r a t o r i e s , I n c . , T e c h n i c a l R e p o r t 333, 1963.

Sanborn , V .A. and Slogan, R. 5. : Study of the M o m e n t u m D i s t r i b u -

t i on of T u r b u l e n t B o u n d a r y L a y e r s i n A d v e r s e P r e s s u r e G r a d i e n t s .

NACA TN 3264, 1955.

Newman, M.A. : Some Contributions to the Study of the Turbulent

Boundary-Layer Near Separation. Aeronautical Research Consultative

Committee {Australia), Report ACA-53, 1951.

Schubaue r , G . B . and Klebanoff ,

the T u r b u l e n t B o u n d a r y L a y e r ?

Reshotko, E. and Tucker, M. :

P. S. : Investigation of Separatio n of

NACA TN 2133, 1950.

Effec t of a D i s c o n t i n u i t y on T u r b u l e n t

Boundary- Layer- Thicknes s Parameter s with Application to Shock-

Induced Separation. NACA TN 3454, 1955.

van Driest, Eo R. : The Problem of Aerodynamic Heating. Aero-

nautical Engineering Review, pp. 26-41, October 1956.

Cresci, R. J,, MacKenzie, D.A. and Libby, P.A. : An Investigation

of Laminar, Transitional, and Turbulent Heat Transfer on Blunt-Nosed

Bodies in H y p e r s o n i c F low.

1960.

J. Aero. Sci., ZT, 6, pp. 401-414, June

Zakkay, V. and Callahan, C.J. : Laminar, Transitional, and Turbulent

Hea t T r a n s f e r to a C o n e - C y l i n d e r - F l a r e Body a t M a c h 8 . 0 .

2__9, 12, pp. 1403-1420, D e c e m b e r 1962.

<.," t t ' l "d4,c l , . t .~

J . Aero. S c i . ,

27

The Growth of Compressible Turbulent Boundary Layers on Isothermal and

Adiabatic Walls By D . A. SPENCE

COMMUNICATED BY THE DEPUTY CONTROLLER AIRCRAFT (RESEARCH AND DEVELOPMENT)

MINISTRY OF SUPPLY

Reports and Memoranda No. 3z9 I*

June, z959

S u m m a r y . - - A recent study of velocity profiles in the turbulent boundary layer on a flat plate at high Mach numbers has suggested the formula

T~ _ Tr - H= ~ n , + y - i

for relating the form parameter H to its incompressible value H~ (T~, Te, Tr are the wall, free-stream and recovery temperatures). This formula is used in conjunction with a generalisation of the Stewartson-Illingworth trans- formation to reduce the left-hand side of the integral momentum equation (including pressure gradients) to incompressible form, for arbitrary values of the fiat-plate recovery factor (or of the turbulent Prandtl number), which is about 0-89 for air. The intermediate temperature formula of Eckert is used to relate the skin friction to its incompressible value, for which a 1/nth power law is used. The momentum equation is then integrable when H~ is given a constant value, which may be chosen to secure agreement in the incompressible case with Maskell's quadrature for the momentum thickness /9, or with the author's modification of the latter. The integration is carried out for the cases of zero heat transfer and of a constant-temperature wall, the details of the Stewartson- type transformation being slightly different in the two cases, but the final forms for

® = P._L~ (P~u, OIJ"~ 0 pm ', l, zm I

the same : ® M B F ( M ) = (const) f M 1~ F ( M ) dx. a r e

The constant B depends in the second case on the ratio TwIT o of wall to total stream temperature, but F ( M ) does not. This is of the same form as the equation obtained by Reshotko and Tucker in the special case of unit recovery factor. A numerical example shows results very close to those obtained by Young's method, which involves two quadratures, in a particular case at zero heat transfer. Some calculations have also been made to illustrate the effect of cooling the wall on the boundary-layer growth in a pressure gradient.

1. I n t r o d u c t i o n . - - A recent s tudy 1 of the veloci ty profiles measu red by Lobb , W i n k l e r a nd Persh ~

i n t u r b u l e n t b o u n d a r y layers on a flat plate at M a c h n u m b e r s up to 8 has shown that it is very

largely possible to represen t the effect of compress ib i l i ty by wr i t ing

u, "~]' (1)

* R.A.E. Report Aero. 2619, received 15th October, 1959.

where

~ Y P f0 ' p ' V = - - d y , A = jo p, ~ d y , (2)

ue and pe being velocity and density at the outer edge of the boundary layer where y = 8. The distributions of shear stress and thence of temperature across the layer can then be inferred, and it is found that provided the turbulent Prandtl number a = EC~/K (~ ---- eddy viscosity, ic = eddy

conductivity) is between about 0-6 and 1, the temperature is very nearly a quadratic function of

the velocity*. A value of about 0.85 for a would be consistent with the observed value of about 0.9 for the recovery factor r, which is defined by

r = ( r , . - r~)/(r 0 - r~) (3)

(Here suffixes 0, e and r refer respectively to an isentropic stagnation point in the free stream, to

the edge of the boundary layer, and to wall conditions at zero heat transfer: • will be used for wall conditions in general.)

It is shown in Appendix I that equation (1), together with the quadratic temperature-velocity relationship, results in the following expression for the form parameter H = 8*/0:

H = - ~ t & + ~ - 1. (4)

This expression for H and a generalisation of the Stewartson-IUingworth transformation are used in Section 2 to reduce the left-hand side of the integral momentum equation to its incompressible form. The momentum thickness 0 and the velocity ue are replaced by

°:(i;° ) ' (5)

lue __

where k and l depend on the flat-plate recovery factor r, and for r = 1 take the values used by Stewartson 3, namely,

k = 1 + 1/( 7 - 1), l = 1. (6)

The latter values reduce the integral momentum equation of the laminar boundary layer, at zero

heat transfer and unit Prandtl number, to its incompressible form, and, as has been pointed out by Culick and Hill ~, would do the same in the turbulent case if the temperature dependence of H were still given by

T° (H~+ 1 ) - 1. H = ~ (7)

Culick and Hill 4 have integrated the momentum equation with the aid of (5) and (6) on the assump-

tion that equation (7) does in fact hold. But (7) is the equation to which (4) reduces for zero heat transfer (T w = T~) 0nly when the Prandtl number or recovery factor is unity, i.e., when T~ = To, and

this is not found to be the case in air. The present paper contains the generalisation to Prandtl

* In this respect the turbulent boundary layer is considerably different from the laminar layer, in which the shear-stress distribution is such that the temperature-velocity relation can be treated as quadratic only when the Prandtl number is close to 1.

numbers different from unity of the integration by Culick and Hill and of a similar one by Reshotko and Tucker 5.

The intermediate temperature formula of Eckert G is used in Section 3 to relate the skin-friction coefficient to its incompressible value for which a flat-plate expression of the form 2C(Ro) -1In is assumed. The momentum equation is then integrated in Section 4 with H i constant. The justification for this procedure is that, a s pointed out in Section 5, it leads to precisely the same form of equation in the incompressible case as was found by MaskelP from a comprehensive examination of experimental results, and which is known to provide an accurate quadrature for 0 when pressure gradients are present. The solution is given for the two cases in which it is particu- larly simple, those of zero heat transfer and of constant wall temperature.

No attempt has been made to go further and calculate the way in which H rises near separation, and the consequential departures of skin friction and heat transfer from the assumed flat-plates values. To do this one would require an auxiliary equation for dH/dx, the derivation of which from incompressible theories would involve much more speculative assumptions than those required to deal with the momentum equation. Maskell's work has, however, shown that such departures are insufficient to affect at all seriously the values of 0 obtained by ignoring them.

2. Transformation of the Integral Momentum Equation. The momentum equation is (Y0ung 8)

dO ~ - ( H + 2 - ~ 0 du e ~ (8) dx M ) ~ dx =peue ~

where M = ue/a e is the local Mach number at the edge of the boundary layer. By differentiating the relations in (5), and using the isentropic relations

To =cons t= Te+uee/2C~= T~{l+7-~-M 2}

which hold in the free stream, we obtain

(9)

se dO_ dO O d~ dO O du~ d x = d x f d x - d x t-(Y-1) h M ~ - - - ue dx '

and l }1 du e 1 &/e_ I + ~ @ _ I ) M ~

ue dx u e dx'

on substitution of Which (8) becomes

d O + [ H + 2 - M ~ + ( y - 1 ) k M ~] O d~, 7 w dx L ] u, & - peu, "

(lo)

The choice of k, l to simplify this equation is slightly different in the adiabatic and isothermal wall c a s e s :

2.1. Zero Heat Transfer (T w = T~).--In this case (4) becomes

H + ! = (Hi+ 1). (11)

The recovery temperature T r is found from (3) and (9) as

{' } T~ = T~+r(To-T~)= T~ l + ~ ( y - 1 ) M ~ .

/ ~- O,I ,~,FAI 7 3

(12)

Substituting from (11), the numerator of the expression in square brackets in (10) is

H + 2 - M 2 + ( Y - 1 ) k M ~ = ~ ( H ~ + I ) + I + ( 2 k - 7 - 2 1 ) ~ M ~ = ~ ( H i + T ~ 2) (13)

2 if 2k - r. (14)

y - 1

And if l = r the denominator of the same expression is ~ /T e. Thus if

equation (10) reduces to

r 1 k = ~ + g ~ f , l = r

(15)

dO 0 d ~ % dx F ( H i + 2 ) ~ dx - ~-- (16)

Pc Ue 2"

If T w # ~, the term H e + 2 in this equation becomes (Tw/T~)H~ + 2. Integration is therefore possible by the method described in Section 4 whenever Tw/T ~ has a constant value, and is not limited to the zero heat-transfer ease.

2.2. Wall at a Constant Temperature T~.--It is convenient to write (4) in this case as

T~Zm+~-I= 1+ M~ M~. - Yooni + ~ ( r - 1) (17)

Then H+2-M~+( r -1 )kM 2 [~ l -4+2k+r- 1)~M = ~ H i + 2 + [ T ~ 2 \ 7 - 1 2 7 -

(18)

2 if 2k+r - 2. (19)

7 - 1

And if l = 1, the denominator of the expression in square brackets is

Thus if

equation (10) reduces to

I + Z ~ M ~.

T~ = const ] k 1--r+ 1-L-L, / = 1 2 7 - 1 J

(20)

dO /T w \ 0 d~ %, - pouo ' (21)

which is of the same form as (16).

4

3. Expression for the Skin Fr ic t ion . - -The intermediate temperature method enables us to write the wall shearing stress at a distance x from the virtual origin of the turbulence on a flat plate as

rw =Pm u~ 2, (22)

where the suffix m indicates evaluation of physical quantities at an empirically defined reference

temperature T,,,, and (u,/ue) z is the same function of u e x/v m as in incompressible flow. The accepted

definition of the reference temperature, due to Eckert 6, is

T m = 0.5(Tw+ Te) +0-22(T~- Te). (23)

With y = 1.4, r = 0.89, this becomes

T~ = 1 +0.12.8M 2 (24)

in the case of zero heat transfer (T~o = ~), and

Tm= 0.5(1 + 0.078M z) + 0.5 ~ (1 + 0.2M 2) (25) Te

when the wall temperature T~ is constant. I t is shown in Appendix II that the local momentum-thickness Reynolds number which corre-

sponds to u e x/v m is Pe Ue O/tzm" Referred to the density and viscosity at an isentropic stagnation point

this is

p~u~O_ (Te~*--i_~-~o[T,,,~-ou~O (26)

in which the viscosity-temperature relation has been taken as /~oc T% The dependence of (G/ue) ~

on this will be taken as an inverse 1/nth power law, so that from (22)

T,w __ tim C( f l eUe O] -1In T m -1+~ r e -@~_li-oJ)/n "It e O. - lha

With C = 0.0088, n = 5, this agrees within about 5 per cent with the KSrm~n-Schoenherr line

in the range 500 < R o < 105 in the incompressible case (for which T~ = T~ = To).

4. Integration of the Momentum Equat ion. -7The right-hand sides of (16) and (21) are both equal

to ~'rw/peUe z, which may be wri t ten in terms of the transformed variables as

where

- t W '

y - 1 ~+co n

p = l ~° n

k and l being defined by (15) and (20) in the two cases. Then, introducing

\ V o /

(28)

(29)

(30)

both (16) and (21) may be written

dO~B O ~ 1 + ~) _tTo\~ ITmX-P ,

where for zero heat transfer, from (16),

and for a constant temperature wall, from (21),

1 / l

(31)

(32)

- -

I f H i is constant, (31) may be integrated to

1 n" (33)

JtTOI \~l ~2dx. In transforming back to the original variables, it is convenient to introduce also

(34)

In passing it may be noted from (27) that

o 2c to ( ' °("2]'", - = ( 'l ( - -~ ?~i- )i (36)

where c s is the local skin-friction coefficient ?wiPe ue ~. Equation (34) then becomes

B /Te~a-Bllg [Tm~-fl = { 1 ~ c[[Te~-BII2{Tm~@ 11+~) u~"dx. (37)

Clearly T O could be replaced on both sides of (37) by a different reference temperature if desired. The equation can also be put in terms of the Mach number M at the edge of the boundary layer, in the form

OMS F(M) = ( I +I) c f MB F(M) dx, (38)

where F ( M ) = t ~ ] t Z ] " (39)

r~/TO is (1 + ~M~) -1, and r~lr~ is given in terms of Mach number by (24) or (25). The constants of integration on the right-hand sides of (37) and (38) are to be found from a laminar

boundary-layer calculation, for which a number of methods are available including that recently published by Luxton and Young 9. This will give a value of Ore, say, for the initial momentum thickness, which may be assumed continuous at the transition pointxT.

Non-dimensionally, (38) may be written in terms of a reference length c and the Reynolds number R = a o clv o based on the viscosity and speed of sound at an isentropic stagnation point in the free stream, as

(~) '+" 'MB+' i 'G(M) = (1+ 1) OR -11" ~/71~ M B .~(M) d(~)--]- K, (40)

where it is found with the aid of (36) that

{T~] cl+lm) k+~B+lm~ a -~ /~ (41) a ( M ) = \T0l

and K is the value of the left-hand side at transition. Although B depends on Tw/T o (by equation (33)) in the constant wall-temperature case, both

F(M) and G(M) are independent of this ratio, for since l -- 1 (equation (20)), the terms involving B disappear f rom the indices in equations (39) and (41).

5. Numerical Values.--We shall take 7 = 1.4, r -- 0.89, ~o -- 8/9 for air, and give two alternative sets of values for the remaining constants, between which there is little to choose.

5.1. 1/5th Power Law for Skin Friction (YoungS).--In order to be consistent with an earlier

report 1° we may take C = 0.0088, n = 5, Hi = 1.5. (42)

(40) then becomes

= 0.0106R-°'2 M B F(M)d(;) +K,

where-B, F(M) and G(M) in the two cases are:

(43)

B.

F(M)

GiM) Tm

To

Zero heat transfer Constant wall temperature

4

2oo! T ~ ) 3 " 7 6 5

1 + 0 . 1 2 8 M ~

1 .8To+2 .2

!

1 {1 + 0.078M2 + ~ (1 + 0.2M~)}

(1 + 0.2M~) -1 in both cases

(44)

5.2.--Maskell's ConstantsL--To make (40) agree with Maskell 's formula in incompressible flow, the constants in which were chosen to secure the best-fitting linear relation between d@/dx and

(@/ue) due/dx for a wide range of experimental data, we should require the values

1 - , = 0.2155, C = 0.00965, H ~ = 1.633. (45) n

The values of c I = 2CRo -x/~ corresponding to (42) and (45) are the same at R o = 1000, and differ by at most a few per cent over the whole range 500 < R o < 10,000; thus both sets of constants lead

to very much the same value of 0. Using the values (45), (43) and (44) must be replaced by

(O)l"2155MB+O'215aG(M) = 0.01173R-O'2185f~[//[ M B F(M) d(;)+ K, (46)

where .

B

F(M)

G(M)

Zero heat transfer Constant wall temperature

4.2

(T TOI

1-985 To + 2"215

x/c

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

TABLE 1

%

+ 0.050 + 0.000 =0.031 - 0-048 --0.058 - 0.062 - 0.060 - 0.057 - 0.060

Young

01"~ × 1 0 4

Present method Young

0"116 0"412 0"703 0"993 1"287 1"578 1-874 2"172 2"466

0"116 0-420 0"712 1"010 1"304 1"605 1"907 2"219 2"512

3"54 2"70 2-33 2"18 2"06 1"97 1-92 1-89 1"84

'7" w

90o U¢o 2

Present method

3"70 2"77

2"43 2"23 2"10 2-01 1 " 9 7

1 " 9 3

1"87

favourable (i.e., falling) pressure gradient, in which the Mach number rises linearly with distance from 2 to 6. Equation (43) was used, with a value of R and the Mach-number gradient such that

dM 0.0106/R °'~ = 0.001,

<-;)

and the starting value of O/c is 0.00177 (for which M4"~G(M)(O/c) 1~ = 0.001 when M = 2). The results are plotted in Fig. 3, and it is seen that the lower the wall to stagnation-temperature ratio,

the more rapid is the boundary-layer growth. 7. Concluding Remarks.--7.1. Accuracy of Results.--There is at present insufficient data to test

the accuracy of the value of 0 found from the present calculation in a pressure gradient at high Mach

number. However, the assumptions on which it rests are well accepted, and it is unlikely to be

seriously wrong. The final form of equation (40) is similar to that found by Reshotko'and Tucker 5,

but the present derivation is considerably more straightforward. The close agreemenic with Young's

method shown in the example is reassuring, but does not necessarily mean the answers are correct. 7.2. Axisymmetric Flow.--Professor Cooke has pointed out to the author that, with the same

assumptions as used by Young in dealing with axisymmetric flow about a body of radius ro(X), namely, that the skin friction and form parameter are the same as in two dimensions, the equation

(34) may be replaced by

= c { I - ( 5 1 ) ~3~ Brol+~/, ~ 1 T e ~ % - ~ " n l J ° ~To] \To] u°~

7.3. Steps in Calculation of 0 and c].--The procedure for carrying out the calculation may briefly

be recapitulated: G i v e n a distribution of Mach number, and the boundary-layer momentum thickness at the

transition point, one finds F(M) and G(M) from (44), then

by quadrature and hence (O/c) z'2 from (43); to find c I also, O/c is found from (38), and by (36)

0 c~ = 0.0176~. (52)

9

a

B £

C

c 1

F ( M ) , G ( M )

H

h, l

K

M

n

R

t

T

U

U r

X

Y

7 3, 31, 0

A,~ @

/~, v, p

"/ 'w

5o

K

Suffixes o

i

'w

co

T

N O T A T I O N

Velocity of sound Defined by (32) or (33) Reference length Constant in skin-friction law, equation (27) Local skin-friction coefficient %/½pjte 2

Specific heat at constant pressure Defined by (39), (41) Form parameter 31/0 Indices in generalised Stewartson transformation (see equation (5)) Constant of integration Local Mach number at the edge of the boundary layer Index in skin-friction law, equation (27) Recovery factor defined by (3) Reynolds number a o c/v o

~/A used in Appendix I Temperature (for suffixes see below) Velocity Friction velocity defined by (22) Distance measured along the surface Distance measured normal to the surface Defined by (29). = is also used for the turbulent Prandtl number Ratio of specific heats Boundary-layer thickness, displacement thickness, momentum thickness (for definitions see Appendix I) Defined by (2) Defined by (35) Viscosity, kinematic viscosity, density (TdT0) k

Wall shearing stress Defined by relation/~ocT ~ Eddy conductivity refers to a stagnation point in the free stream the outer edge of the boundary layer to the intermediate temperature, defined by (23) to the recovery (zero heat transfer) temperature to the value in incompressible flow to the wall to the undisturbed stream to the transition point

10

No. Author

1 D . A . Spence . . . .

L I S T

2 R . K . Lobb, E. M. Winkler and J. Persh

3 K. Stewartson . . . . . . . .

4 F . E . C . Culick and J. A. F. Hill . .

5 E. Reshotko and M. Tucker . . . .

6 E . R . G . Eckert . . . . . . . .

7 E .C . Maskell . . °

8 A . D . Young . . . . . -. ..

9 R .E . Luxton and A. D. Young ..

10 D . A . Spence . . . . . . . .

O F R E F E R E N C E S

Title, etc.

A note on mean flow in the compressible turbulent boundary layer.

(To be published in J . Fluid Mech. (1960).)

Experimental investigation of turbulent boundary layers in hypersonic flow.

J . .de . Sci. 22 (1). pp. 1 to 9. January, 1955.

Correlated incompressible and compressible boundary layers. Proc. Roy. Soc. A. 200. p. 84. 1949.

A turbulent analogue of the Stewartson-Illingworth trans- formation.

J. Ae. Sci. 25. pp. 259 to 262. April, 1958.

Approximate calculation of the compressible turbulent boundary layer with heat transfer and arbitrary pressure gradient.

N.A.C.A. Tech. Note 4154. December, 1957.

Engineering relations for friction and heat transfer to surfaces in high velocity flow.

J. Ae. Sci. 22 (8). pp. 585 to 586. August, 1955.

Approximate calculation of the turbulent boundary layer in two-dimensional incompressible flow.

A.R.C. 14,654. November, 1951.

The calculation of the profile drag of aerofoils and bodies of revolution at supersonic speeds.

A.R.C. 15,970. April, 1953.

Skin friction in the compressible laminar-boundary layer with heat transfer and pressure gradient.

A.R.C. 20,336. July, 1958.

Aerofoil theory. I I - - T h e flow in turbulent boundary layers. A.R.C. 18,261. March, 1956.

11

where

APPENDIX I

Variation of H with Stream Temperature

Using equations (6) and (7) of Section 1, the displacement thickness is obtained as

~ = 1 - pu ,ty = ~ - d . = ~ - c ~ A , (53)

C 1 = f2f(t) dr.

Similarly the momentum thickness is

-) o = jo [~ p~u~P'~ (1-g~ dy = (c~-c~)~, (54)

where Ce = f : {f(t)} 2 dt.

Thus H = ~ = (8/A - C~)/(C~- Cz). (55)

Now ~ = dy = joC ~ p~p &7 = ~ d~. (56)

If now we use the quadratic temperature-velocity relationship

T = T ~ + ( T ~ - T ~ ) " - - - ( T ~ . - T ~ ) - - , (57) Ue U e

which can be shown to hold very closely for turbulent boundary layers, with'out restriction to unit Prandtl number, we obtain from (56)

~ = f j [ ~ + ( ~ - - T ~ ) f ( t ) - ( ~ - l ) { . f ( t ) } 2 ] d t ~ + ~ C 1 - ( ~ - 1) C 2 . (58)

On substitution of S/A in (55,), there results

(C1-C~)H = (1 -C1)~+(C1-C~ , ) (~ - I). (59)

But in incompressible flow S = A ; thus from (55)

H i = (1 - C~)/(CI- C2) (60)

Tw and (59) becomes H = ~-e H , + ~ - 1, (61)

the value quoted in equation (8) of Section 1.

12

A P P E N D I X II

Intermediate Temperature Formula for Sk in Friction

The intermediate temperature formula is usually expressed in terms of the Reynolds number based on x; thus if in incompressible flow the skin friction is given by

r w = pue2f~--~) , (62)

p and v = t~/p being constant everywhere, then in compressible flow r w is obtained from the above

expression by evaluating density and viscosity at the reference temperature T m given by equation

(19); thus

. / u e x \ rw = pmue J ~ , ~ ) . (63)

The integral momentum equation in flat plate flow is then

dO _ pm f ue x (64)

i.e., - f (65)

whence on integration, and putting 0 = 0 when x = 0 it is seen that p~u~ O/tzm is a function of

u~x/vm, and thus by (63) that

u 2 [peu~O~ % = P~ ~ g ~ - m - J ' (66)

where g is the function which expresses the dependence of %/pue ~ on pu e 0/t~ in incompressible flow.

13

3"0 ,00~

2 - 0

1.2-

I 'O

/ //7

/

/

J/ / /

YOUNG.

P~ESENT METHOD.

I 0 " 2 0 -4 0 " ~ . 0 -8 I ' 0

x / c

FIG. 1. M o m e n t u m t h i c k n e s s as ca l cu la t ed b y two m e t h o d s ,

in a p a r t i c u l a r c a s e at ze ro h e a t t r ans fe r .

"003

"00~

4 T P.~ u,.. a

- - YOUNG

PRESENT

1 METNOb

oo, I O -2 O-4 0 '6 O- 15 I" 0

x / c

FIG. 2. S k i n f r i c t i o n as c a l cu l a t ed b y two m e t h o d s , in a p a r t i c u l a r

case a t ze ro h e a t t r ans fe r .

t ~

"05

5"

7 o

B/c

,O~

"03

"02

"01

o

To

/ T_._W_ ±

f

3 4 5 G PI

FIC. 3. Calculated g rowth of m o m e n t u m th ickness in a favourab le pressure grad ien t at different wall t empera tu res .

HEAT ('rw --Tr)

TRANSFER.

J ,

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