The two-dimensional laminar boundary layer at hypersonic speeds

TECHNISCHE HOGESCHOOL VLIEGTUICTHOUWKUN DE Karaabtrcat 10 - DELfT

M HEI 1953

REPORT No. 67

THE COLLEGE OF AERONAUTICS

CRANFIELD

THE TWO-DIMENSIONAL LAMINAR BOUNDARY

LAYER AT HYPERSONIC SPEEDS

by

T. NONWEILER, B.Sc.

This Report must not be reproduced without the permission of the Principal of the College of Aeronautics.

: 11053

R e p o r t No. 67 J a n u a r y 1955 TECHNISCHE HOGESCHOOL VLIEGTUIGïOUVv'KUNDE Konaolstiaat 10 - DELFT T H E C O L L E G E O F A E R O N A U T I C S C R A N F I E L D

The Tvifo-Dimensional Laminar Boundary Layer a t Hypersonic Speeds

t y

-T. Nonweilers B.Sc. -—oOo

S U M M A R Y

A numerical solution is found for the equations

governing the motion of a two-dimensional laminar houndary layer, in the absence of a pressure gradient,\ïhich would

2 "be valid if the flight Mach Number is very high (i.e. M^>1).

The effects of surface slip, and the finite thickness of the boundary layer are shown to be negligible if the

Reynolds Number (R) exceeds about 10^, and are neglected. Account is taken of the variation of specific heat, Prandtl Number and viscosity, with temperature, although

(for air) only the latter effect is important. Sutherland's formula is used for viscosity variation, and the results

imply that for M <.10, there is little variation of skin friction coefficient (c„) vv'ith Mach Number. For high Mach Number, however, c„ fj^, ^/^Ë:itanö. the heat transfer coefficient k„ = 0.51c„ for air. The surface

temperature has a negligible effect on these quantities if it is sm.all compared with the stagnation temperature. Numerical results are given, and show that skin friction and heat transfer vary as the square root of the surface pressure. The velocity and temperature profiles across

the boundary layer are also deduced! the boundary layer M / R at

high Mach Number, and there is an important interaction betvireen the boundary layer and the external flow.

Some remarks on the stability of a laminar layer are included, and a comparison is made of the above results with those relating to loY/er Mach Nuiiibers of flight. PDF •

-TECHNISCHE HOGESCHOOL VLiEGTUIGBOUVvKUNDE Kanaalstraat 10 — DELfT

1 2

ME! 1953

C O N T E N T S N o t a t i o n .

1. Introduction.

2. The Equations Governing the Boundary Layer in High Speed Plow.

3. Interpretation of the Plow Behaviour near the Surface.

i|.. Behaviour of the Flow near the Outside Edge of the Boundary Layer.

5. Solution of the Equations.

6. Limitations of the Theory at High Altitudes and Speeds.

7. Effect of Surface Shape. 8. Viscous Form Drag.

9. Effect of Atmospheric Turbulence and Surface Roughness on Transition.

10. A Comparison vifith other Results. 11. Conclusions.

References. Figures.

N O T A T I O N

C S u t h e r l a n d ' s C o n s t a n t ( i . e . M-OGSTTQ)

2 Cyj viscous form drag *- ip=u=,

2

C^ friction drag f- ip = u

P shearing stress at surface

i='^Q{hjf)Q)

H Total heat, or enthalpy (={"0 dT)

L length of body surface

M Mach Number of stream outside boundary layer (=u„/a )

M„ Mach Number of free stream (=u„/a^)

el a a

Q heat flux into surface

i=^Qi^)Q)

R Reynolds' Number (=p,u.L/|i-)

Rm Reynolds' Number of transition (=p UgX^A»)

T static temperature of gas

Tu.. t h e r m o m e t e r t e m p e r a t u r e (i. e. t h a t v a l u e of T-. f o r

^^ " w h i c h kjj = 0. ) ^

U' free stream velocity (in ft/sec. )

Y = h/o

Z =

3 T

21^

r,h2

a speed of sound (=Yp/p)

b constant used in equation (3.10)

c = C/T^

c gas specific heat at constant pressure

c gas specific heat at constant volume

f = f(n) = u/u^

g . g ( n ) = - L ^

M ° r>

h 5 h ( n ) = (H-H^)/i-uJ

k T h e r m a l c o n d u c t i v i t y o f g a s

kjj = Q / i p ^ u 3

m m o l e c u l a r w e i g h t o f g a s

n

= ^

^ J 2 M ^ s

Tl dh - o dn

2 -Notation - continued.

p root mean square of atmospheric pressure fluctuations p gas pressure

CL

r air density relative to sea-level conditions (^Pn/pgrN

s = X/L

t time

u,v components of gas velocity parallel to the x- and y-axes respectively.

u^ velocity of slip at surface _ / X 8v I2Rs

w = w(n) = - j ^ - ^ ^3 >j M^

x,y system of orthogonal co-ordinates parallel and perpendicular to surface of plate, with origin at leading-edge of plate.

X' value of x in feet

Xm value of x at transition point.

jp proportional Increase in c at elevated temperatures

% momentum thickness of boundary layer.

a inclination of airflow at outside of boundary layer to free-stream direction

a^ inclination of surface to free-stream direction o

P arbitrary finite constant in definition of n and w Y =(c /c ) (except in equation (5,7)0

0 boundary layer thickness e roughness height

e maximum tolerable roughness height to prevent separation of the flow

- f \ 1

'-1-^ = '-1-^ '-1-^'-1-^) = M It;

0 deflection of stream at outside of boundary layer Ö

A mean free path of molecules at surface la coefficient of viacoaity of gas

p gas density

0 Prandtl Number (= c M-/k)

Notation - continued. df

/T duv

"^ =tdT)

Sviffix: 'SL' denotes conditions in the ambient air at sea level.

'a' denotes conditions in the free stream 'n' denotes differentiation with respect to n 'o' denotes conditions at the surface

'ö' denotes conditions at the outside of the boundary layer (where y = 0)

denotes conditions at tip of small prodection (where y = s)

Primed symbols (e.g. T') denote differentiation with

d-t- df-respect to f (i.e. T' = -r^)

k

-1. Introduction

In this report we shall attempt to examine the

properties of the laminar two-dimensional boundary layer existing in a compressible boundary layer at very high Mach Numbers of flight. To be precise, we shall assume that this Mach Number M is sufficiently large that 1/M can be neglected compared with unity. The resulting flow conditions we describe as 'hypersonic' in a similar way as we can classify as 'incompressible' those flows in Which we can neglect M*^ compared with unity. Like the

results for incompressible flow, those relevant to the hypersonic boundary layer display a certain simplicity

of form, which greatly facilitates their Interpretation and application. This simplicity also enables us to relax many assumptions which are normally made to obtain a numerical solution of the equations involved.

For instance, we do not find it necessary to restrict the discussion to a gas with a Prandtl Number of unity, or to stipulate that the surface temperature is a constant. We shall use Sutherland's formula for the variation of viscosity with temperature, and we shall find it possible to make some allowance for the variation of molecular specific heats v/ith temperature. Such factors greatly enhance the value of the results we can obtain and throw light upon the accuracy of the assumptions more usually made.

Of the assumptions which we do make, the most restrictive is that the pressure over the surface is a constant - although this allows us to consider not only the heat transfer to a flat plate moving parallel to itself, but also to plane inclined surfaces which in supersonic flow are acted upon by a uniform pressure, and to bodies (such as, for instance, the double-wedge wing) composed of several such surfaces. ¥/e shall also make the usual assumptions associated v/ith that of a high Reynolds Number - for example, that the boundary layer is thin, and that the velocity of slip is zero - although we shall examine these in the light of the results to which they lead. In regard to the variation of the molecular specific heats with temperature, we shall be guided by the results of statistical thermodynamics in assuming that they increase to a certain asymptotic value at very high temperatures.

As an extension of the assumption that we may treat (1/M ) as small, we shall also neglect the surface

temperature comr^ared with the thermometer temperature of the boundary layer air. In other words, we shall be dealing only with the boxmdary layer in the presence of high rates of heat transfer. This is easily justifiable

if it is recalled that the thermometer temperature is commensurate with the stagnation temperature, which latter is given by

1 2

(1 + •?• M ) X ambient a i r t e m p e r a t u r e .

Plight at a Mach Number of 10 or more thus involves

thermometer temperatures of at least i+OOO C. , and plainly unless the surface temperature is considerably smaller

than this the problems have no great practical significance.' From the mathematical point of view, if the surface

temperature is comensurate with that of the ambient air, (say, 2 or 3 times its value), then since the thermometer

2

temperature is of the order of M times its value, the 2

above assumption is justified if we allow M -»oo.

For such reasons as this, the form of the asymptotic solution we obtain has rather strange properties, and having written down the correct boundary conditions for

2

the condition 1/M = 0 , we shall examine the compatability of the results for finite, but large. Mach Nurober. V/e find that in the same way as the assiimption of

2

incompressibility (M = 0) leads to quantitative deductions which are qualitatively sound (if slightly exaggerated) when compared with conditions at finite Mach Numbers, so also does the hypersonic solution we find here. It remains, of course, to be shown whether the results have the same power and significance,as an asymptotic solution, as those for incompressible flow.

2, The Equations of the Boundary Layer in High Speed Flow Using the notation defined at the beginning of this note, v/e may v/rite down the equation of continuity in

steady motion as

9l2iil ^ di2ll= 0 ....(2.1)

9x 9y

and if the surface pressure is uniform, the Eulerian equations of motion as modified by Prandtl for a thin

6

-TECHNISCHE HOGESCHOOL

VLIEGTUIGEOUWKUNDE Kanaalstraat 10 - DELPT

boundary layer become

dp = o .... (2.3) In addition we need the energy equation for the thin

boundary layer!

DH 3 /k 9Hv , ,, fdus2 (0 ) , \

Pïït = 97 ^ a?) ^

'^W

....(2.i+)

Finally, from (2.3)s the Gas Lav/ may be written as

d(-f) = 0 ....(2.5)

We now introduce the non-dimensional variables s = x/L, f = u/u^ and h = (H-H^)/4u^

where s, f and h are bounded quantities and in general, finite', and where the subscript '5' refers to conditions outside the boundary layer (v/hich are invariant with x ) . To relate p to the enthalpy H, we assume that v/ithin the boundary layer

mc ^

-I—2__ = _p , a constant .... (2. 6) ^ p^ö

which is, at least, a more elastic assumption than that the specific heats of the air are constant. In fact,

there is some evidence to shov/ that at elevated temperatures the molecule's specific heats reach an asymptotic value

higher than that at normal temperatures and pressures.

Pov/ler and Guggenheim (in their 'Statistical Thermodynamics') deduce that, for large T^ (mc ) is increased by a factor

"/y in a diatomic gas. Since the air is composed mainly of diatomic molecules, we might therefore expect that, in the high-speed boundary layer, v/here as v/e shall show later, T = 0 (M T,)* ^^'e could put in (2.6), the value 'V = %

and that this would apply in general over the boundary layer if M » 1 . Existing data for air (up to T = 3000°C) suggests that this ratio is exceeded without any evident falling off in the increase of specific heats v/ith

temperature. This increase in caused by the higher vibrational energy of the gas at these temperatures, and

is still further increased at even higher temperatures (say, about 20,000*^0) by electronic excitation - though this effect may legitimately be ignored here since, as v/e shall see later such high temperatures are not likely in the boundary layer at any high speeds in v/hich we might be interested. Hov/ever, it is evident that except near the edges of the boundary layer, v/here the temperature is low

compared with ö, the boundary layer thickness), equation (2.6) can provide a reasonable indication of the change in specific heats.

Then, if we define g so that Pö 2

-^ = g M^ P

it follows from (2.5) that 1 p^ 1 m^ T g = - ^ — = - ^ M p M m T^ o o r u s i n g e q u a t i o n ( 2 . 6 ) g =

1

° P 1

fM^ TcpT)^ ^2.7)

By the definition of the function h, we find that

h = r^

C-,Ö.T/iul =

7- (T-T )/iu2 = ^ 4 (| . 1 ) ^ .. (2.8)

JT^ ^ 0 P 0 0 If-' M ^5 Pö 2

Since h is finite, it follov/s that if we treat M >> 1, then

S =

0 ( M 2 )

.... (2.9)

Assuming (as before) that at high temperatures, the specific heat attains a constant value, we deduce that the mean

specific heat "c" is equal in general to the asymptotic value within the boundary layer.

Hence, in equation (2.8)

2 1 T ^ n 2 1 C-n^p |S,

^ =

YTT

::2 ( f r ' ' ^ ^ . "

_M

_{o ^0 M ^""P'^ö}

Y-T Z2 j é ^ . ^^ - T^

M^ ^^-^-^0 •- M'

^^°-^A''°'-^''\ ••^'•'°^

or substituting in the equation (2.7)

2

For the asymptotic solution, where M -i co, evidently

g = (^)h .... (2.11) In the same v/ay as we assume that the molecular

specific heats are constant over the boundary layer, v/e also ass-ume that the Prandtl Number, a = uc./k, is a

p

constant (and equal to "o, sa.y) since this number is known to be related to the specific heats, but again

o i4 o, since the Prandtl Number differs at high temperatures

O

8

-In suggesting a non-dimensional form for the viscosity,

"'T

we use Sutherlands Formula!

Suppose we define T] SO that ü^ = M.;

2

then Tj is a finite parameter for M >\ 1, since using (2.11) and (2.9),

\M ^5/ \ Ö / '• M

-2

or in the limiting condition, M -jco, from (2.10) and (2.6)

l.a. M = (1 . £ j ^ ^ ^ j * h * ....(2.13)

This relation also expresses the fact that r\ varies,

at large temperatures, as the square root of the temperature (or as the square root of the enthalpy, if the specific

heats are constant).

If v/e nov/ suppose that all the non-dimensional

variables so far defined (i.e. f, g, h and r]) are functions of a single independent variable n, involving the space co-ordinates x and y, then it may easily be shown that this variable must depend on y/ -Jx. Thus we write in

non-dimensional form!

^

J 2M^S

v/here j3 is some arbitrary finite constant whose value may later be chosen as a matter of convenience. We also define a non-dimensional variable involving v,

3v 2 R r ^

v/ = — —— — ^

^6 -J M ^

v/here v/ may be shov/n to be also dependent only on n. From the definitions of f, g, h, rj and v/ which are each functions of n, the equations (2.1), (2.2) and

(2.i|.) maybe simplified to the forms!

(n|iwj||=-P^[é;(Si)-mr] ....(2.16)

We may eliminate (w/g) from equations (2.15) and (2,16) using equation (2. lij.). If, in the resulting equations, we change the independent variable from n to f = (u/u.), and we write

df W2 /Ms / .9

m^)

^ = ^dH = TJR" V'9y

which is a non-dimensional form of the shear stress, then we find that (treating a as a constant) *,

f = -3^ ^ T T " (2.17) h" +(1-0) ^ h' + 2o = 0 (2.18)

where the primes denote differentiations with respect to f. These two equations are equivalent to the equations of Crocco for momentum and energy, after the Crocco

transformation has been applied. It should be noted that their derivation is dependent only on the definition of the new variables, and is not influenced by any of the

assumptions we have made concerning the relation of density, specific heats, and viscosity with temperature (i.e. the relation of g and T^ v/ith h). Hov/ever v/e note ti\at, if

M -4<^ then h, and of course also f, are finite parameters, as also are g and T] which are related to h in the simple manner described by equations (2.11) and (2.13). Apart from this particular choice of the form of the variables, equations (2.17) and (2.18) are equivalent to the

expressions of Crocco.

In particular, the boundary conditions are also 2

greatly simplified in the condition M -j 0*=*. At the outside of the boundary layer, u = u. or f = 1. Here H = H. and the shear stress is zero! i.e.

h = 0, T = 0 at f = 1 (2. 19) At the surface, v/here pu = 0 (i.e. f = O) v/e also have

that pv = 0, and y = 0 (i.e. ^ = 0 and n = 0). From (2.15),it then follows that

df dn ^"^^dH^ - df - 0* at f _ 0.

10

-surface is zero. At the -surface we also have that

T = T , say. If the skin temperature is of comparable

magnitude to the ambient air temperature, then T /T.=0(1)

it III -i

c.\i , so that from (2.8) in this condition

h = 0(-U) at f = 0

M

or in the limiting condition, we have

1^ = 0, h = 0, n = 0, at f = 0 .... (2.20)

The condition h = 0 merely expresses the fact that the

temperature of the air at the surface is negligible

compared with that of the air within the boundary layer

2

if M >> 1, and of course if applied to flows with finite

values of M, is only an approximation (unless T = T ).

V/e may immediately infer that the asymptotic solution of

2'

the hypersonic flow equations (with M >> 1) will be

uninfluenced by variations in the surface temperature.

Equations (2.17) and (2.18), with the approximate

relations (2.11) and (2.13)? and the boundary conditions

(2,19) and (2.20) describe the state of the bo\mdary

layer in a hypersonic flow, which we interpret mathematically

as the flow of infinitely large Mach Number. In a later

paragraph we shall attempt to solve these equations

numerically! equation (2.18) yields the formal solution,

using (2.19), that

i; '"•(J?-' ")•

h = 21 I

1 \\ ^ df]d£

(2.21)

where 0' is the value of f where the total heat is a

maximum and must be chosen to satisfy the condition that

h = 0 when f = 0 in equation (2,20). This however is

a solution of little computational value, but it does

indicate that if v/e define q as a non-dimensional form

of the heat flxixli.e.

^ - ^dK -

TY^TTP

J ^

r9?K\"L-")

t h e n , v/e f i n d t h a t

T | | = -2ÏÏ T^f", T^-^ df . . . . ( 2 . 2 2 ) - P f

Hence at f = 1 (i.e. at the outside of the boundary layer)

v/here

1 = 0

(i.e. the shear stress vanishes) then also

q = 0, - in other v/ords, there is no heat flux from the

boundary layer to the ambient air.

The approximations introduced in the solution for hypersonic flow are, as we have seen, valid in general within the boundary layer! that is, except near the

surface and the outer edge. In viev/ of this fact, it is necessary before proceeding with the solution, to

examine its validity in these bounding regions. It v/ill be shov/n in the next paragraph that although the assumptions

are in error, the solution is still an adequate approximation

if

M S > 1 .

Interpretation of the Behaviour of the Flow near the Surface Let us first consider the behaviour of the hypersonic flow solution, already obtained, near the surface (i.e. in the condition f->0).

We first notice, from equation (2.22), that h' / 0 at f = 0, since the value of T (the non-dimensional form of the shear stress) is finite at the surface. Thus

h 'N.. h'^f as f T>0 .... (3.1)

where h'= h' at f = 0. Again, in (2.17), from (2.11) and (2.13) v/e have that g/r^cdi^ and so

T''.._/ const, (f/h") as f-) 0 i. e from equation (3. 1)

T " .- const, f2 as f -> 0

Prom (2.20), T' = 0 at f = 0, so that

T ' . V const. f^^2 as f •-• 0 (3.2) To relate this limiting behaviour to the independent

variable n, we note that

n = 1^ ? df ....(3.3)

-'o

so that from (2. 13 ) and (3.1)? since T: ^ 0 at f = 0, and

1 1

n '..' const, f^^^ as f ~» 0 .... (3-4) Thus, in (3.1) and (3.2), from (3. i+)

f v... const.n ^-^, h--• const, n'^^ > %'-^ const, n, for n-> 0 ....(3.5) The limiting behaviour is thus singular! • the velocity and temperature gradients f and h are infinite at n = 0,

TECHNISCHE HOGESCHOOC

VLIEGTUIGEOUWKUNDE Kanaaistraat 10 - DELfT

12

-stress and heat flux, since the value of r\ tends to zero at the surface.

It will be evident that such a solution for finite Mach Number would be invalid because (~ TT ] = r| is

non-wX:

zero at the surface, and the velocity ^ and temperature gradients are finite. However, v/e shall attempt to show how the true solution departs from the asymptotic one, and to deduce that the value of T and q at f = 0 in the asymptotic solutions for M-»coare the correct values of the non-dimensional form of the shear stress and heat flux if M is large, but not infinite.

Let us then consider the conditions existing for flows in which M is finite. At the surface, v/here u = v = 0 we have from equations (2.2) and (2.14.), that

9 / 9u\ 9 ƒ, 9T , 9u\ ^

or, performing the differentiations of the products! 9^u ^ _i dii 9T 9u

^ H dT 9y 9y

(3.6)

gfr _ 1, dk/9Tx2 iJ:/'9ü^' gy2 - " k 3T'^9y^ " k'^9y^

Rewriting these expressions in terms of the non-dimensional variables f and h which are functions only of n (i.e. f=f(n), h=h(n)) we find that

^ „ n ( 0 ) = - ^ M \ ( 0 ) f „ ( O ) { ( f t | , | ^ }

^ p ''O

v

(0) = - W K ( 0 ) ] f 2£

'J^vll

(V),^ -2o|„(0)]

.... (3.7)

Where subscript 'n' denotes a differentiation with respect to n, and subscript 'o' refers to values at n = 0.

Now in our asjnnptotic solution for M -* 00, v/e have established that both T and q are finite at n = 0! hence, by definition

If M-^ oc, evidently f„(o) -»QC>, which is in accordance with

the asymptotic behaviour of the hypersonic flov/ solution given in (3.5). Similarly

and both fj^(O) and h (0) are, f or large M, magnitudes commensurate with M. It follows from (3.7) that the second derivatives f„^(0) and h (o) are magnitudes of order M . For large values of M, however, both f and

h are finite within the boundary layer. Thus any definitive relation between the derivatives of f or h must involve only finite constants. Hence, for large M, it follows that as n -^ 0

h »v const, h^

nn n

both sides of the expression being of order M^, and the constant being finite. We may, in fact, deduce its value from (3.7) and find that

or \ ^ ^ -b h^, say

It follows, by integration, that as n -»0,

h(n) - h(0)^ ^ {3b n ^[Hf(öT]^]

^""^

(3.10)j

1

2^

[h„{0)J

(3.11) 2 "^

If we now allow M -» OQ, provided that n M » 1, it follows from (3.9) that

h ( n ) ^ - \ ^ n V 3 f i + o (—)) ....(3.12) 2b '/3 ( M'^ j

since h(0) = 0(—^). This expression (3.12) is identical with that in ^ (3.5) found previrusly for the

asymptotic solution! equation (3.11) implies a more exact description of the conditions near n = 0 if M is

large, but not infinite. It will be seen that the asymptotic (hypersonic flow) solution can give the correct description of conditions (v/ith error of order —s- ) if

M

n = O(^) .... (3.13) 1

and fails if n = 0(-^). M^

In a similar manner we may show that, for n-* 0, for large M

t^

| ^ [ h ( n ) - h(0)]^ i;[^(^) - h(°)] •••• (3.1^)

which is likewise compatible with (3.5) except where n = 0(-^).

M^

14

-the variatinno of h and f as obtained ei-ther by -the

hypersonic flovi equations, or by the exact equations for large (but non-infinite) M.

Moreover", from equations (3.6.) it follov/s that the gradient of the shear stress, i.e. ( T ^ ) * tends to zero as n -A-0 for all values of M (which is again compatible

with (3.5))°, so that from (3.13), provided M is sufficiently large, the vslue of a- found from the solution in hyoersonic

1

flow (which is valid v/here n » -^) will not differ greatly M

from the true value at the surface. Similar argxiraents lead to the deduction that the rate of heat flux at the surface is also correctly given by the hypersonic solution.

Behaviour of the Flow near the Outer Edge of the Boundary Layer At the outer edge of the boundary layer where f -^ 1,

v/e have seen from equation (2.21) that o—1

h'/s., const. T ~ .... (4. 1) and since T - » 0 a s f - » 1 , i f a < 1, it follows that

h'-*ooas f'-:* 1. In particular, since from (2.19)

h = 0 at f = 1, it follows that upon integration, as f -• 1 h ^ const. { I -4—•-\ .... ik' 2)

(i: ;^"-)

In our solution for infinitely large M, using (2.11) and (2.13) in (2.17), we have then that as f -* 1

T T:" /^ const.

which since T - > 0 as_f -* 1 , imi^lies that if a > 0 ,

T'v const. (1-f)3+^ as f ->1 (l)..3) and so in (U. 2) j^^

h ^ const, (l-f)" ~ as f-* 1 ....(4.1+) Near the outside of the boundary la.yer, from (3.3) and

using (2. 13) J

2o-3

|g = a ^ const. (l-f)3+^ ....(I+.5) and it follows that if o > 0, n tends to a finite limit

as f -• 1. In other v/ords, the extent of the boundary

layer is finite. Su-opose (by suitable choice of the constant (3 in the definition of n ) that n = 1 then corresponds with the outside of the boundary layer. We have from (I4.. 5) upon integration, that ,

Thus in (i|.. 3) and (I;.. U ) , ,f rom (.;.. 5 ) ,

(1-f) Ai const. (l-n) -^ , h ru const. (1-n) , 1 /

% Ai const. (1-n) ^'^ as n .* 1 ... . (i|.. 7 ) H e r e a g a i n the b e h a v i o u r o f t h e v a r i a b l e s is s i n g u l a r , a s

it w a s near the surface.

T h i s b e h a v i o u r r e s u l t s from the f a c t that the b o u n d a r y layer is f i n i t e in e x t e n t (as m e a s u r e d b y the independent v a r i a b l e n ) , and i s of course o n l y strictly v a l i d in the l i m i t i n g c o n d i t i o n of infinite M a c h Number. F o r finite M a c h N u m b e r w e k n o w that the b o u n d a r y layer e x t e n d s to infinity! b u t this is quite c o m p a t i b l e w i t h t h e

h y p e r s o n i c f l o w s o l u t i o n f o r , a s w e shall n o w shov/, a t l a r g e d i s t a n c e s f r o m the surface (for g i v e n n ) t h e a i r v e l o c i t y d e c a y s v e r y r a p i d l y w i t h increase of M a c h Nvmiber.

In e q u a t i o n s (2.17) a n d ( 2 . 1 8 ) , w h i c h involve n o a p p r o x i m a t i o n s c o n c e r n i n g the value of M , v/e have u s e d the b o u n d a r y c o n d i t i o n s h = T = 0 a t f = 1 , w h i c h a r e a l s o correct f o r a l l M. H o w e v e r in r e l a t i n g f t o n i n

(I4.. 5 ) w e used the a p p r o x i m a t i o n that rj a h ^ , w h e r e a s at

the outside of the b o u n d a r y layer w e h a v e simply b y d e f i n i t i o n

Ti = ^ at f = 1 {k,8)

Similarly we have usee in obtaining (I4..3) the approximation g ö t h , although strictly

g = J_ at f = 1 (I1..9) M

Strictly, for any value of M, we have from (2.11+) that

and so in (2.15),

w\df / f ^ f ^ \df «2 d / df\ /, ,^v

•^°^ dn * dT Iv approaches zero at the outside of the boundary layer: and so from (i|.. 8) and (1+.9), since f -* 1

as n -* «o, we have that 2 df 2 d f

M^n g =

-&^Mf^

,

dn

We may put 3 = 1 without loss in generality, and after integration we find that

TECHNISCHE HOGESCHOOL VLIEG IUIGBOU WKUNDE Kaaaalstraat iO - DELFT

16

-Thus, for a given value of n, the disturbance to the flow decreases exponentially with increase in M. In the limiting condition of infinite Mach Number, it vanishes for all but a finite range of n as v/e have seen.

It should be noted that n is itself dependent on M, and although it is true that the changes within the boundary layer become more concentrated, the actual thickness of the boundary layer increases as the Mach Number is increased,

as we shall see in the next paragraph, and for M =oo is in fact infinite.

5. Solution of the Equations

If in equations (2.17) and (2.18), we put Y = h / 0 , Z = p / ^ T

then they become, with the use of equations (2.11) and (2.13) Z Z " + - = 0 • " + ( 1 - 0 ) ^ ^ + 2 = 0 w i t h t h e boiindary c o n d i t i o n s y = Z ' = O a t f = 0 (5.1) Y = Z 0 at f = 1.

The equations (5.I) have been solved nvimerically to satisfy these conditions, using a relaxation method, for o = 1.0, 0.8 and 0.6. The results are summarised in the Table I below.

Table I! Summary of Numerical Solutions. f 0.0 .1 .2 .3

.4

.5 .6 .7 .8 .9 1.0 Y a=1 0.00 .09 .16 .21 .24 .25 .24 .21 .16 .09 0.00 0=0.8 0.0000 .0922 .1646 .2172 . 2500 .2630 .2560 .2286 .1799 .1073 0.0000 0=0.6 0.0000 .0949 . 1700 .2255 .2615 .2779 . 2 7 4 4 .2502 . 2 0 3 4 . 1284 0.0000 a= 1 0.7200 .7188 -.7131 .7004 .6783 .6439 .5940 .5232 .4223 .2703 0.0000

z

0=0.8 0.7114 .7100 .7037 . 6904 .6679 .6335 .5837 .5133 .4-136 .2643 0.0000 0=0. 6 0.6966 .6954 .6898 .6772 .6552 .6211 .5717 .5019 .4031 .2563 0.0000

We are particularly interested in the values of the

shear stress and heat flux at the wall! these are usually

quoted as the coefficients c„ and

k„9

v/here we define

c - li"i L(^)l /in u^ - ll£^(r,^) - ' ^

°f -

mo

i^^gyll /sPgU^ - jRMs ^'^dn-'o -,

so that

and where

so that

!% _ limf' ^9Ts,, ,,r9ü^) - l i n i ^ J L ^ l

c^ - n-»o I ^ ^gy'^^ö^ ^gy-'j ~ f^o [

2Ü

df j

i. e. k „ J

^ ^

Y'(0) ....(5.3)

c^ 2

In (5.2), we find from (2.11) and (2.13) that

(5.4)

Fe propose to ignore the change in the molecular weight

of the air within the layer, as appreciable dissociation

is unlikely to occur within the range of temperatures

v/e are concerned with, and as Sutherland's formula (used

in for-ming the connecting of viscosity with temperature)

is unlikely to be valid if appreciable dissociation takes

place. Hence, if we put C/T- = c, in (5*4)

nh^^ 1+c /

2tY

0 2 g 0 2 ^^-^ '

Thus in (5.2)

A i l ^ U _ ^ \

Z(0) ....(5.5)

^ { RMs J Lo(Y-1)J

For air, v/e may take

Y - I r» - i

^ - 5 '

r- 7 .... (5.6)

and taking a mean value of Z(0) for o betv/een 0.7 and

0.8 from Table I, since Z(0) does not change greatly

with a, we find that

c = 1.6 f z ^ ^ .... (5.7)

J o^]

^ ' T^RMs

18

-and it is related to the value of the ratio of specific

heats! for this reason it will change at high temperatures, and to account for this change we may use Eucken's Formula!

° = 9Yr5 ^^®^® "^ = % / % ....(5.8) At normal temperatures Y = •^, but at elevated teiirperatures,

from the results of Fowler and Guggenheim v/e find that Y = ^, so that from (5.8)

^o •- ^5 " T9 " °*'^^'^ , ÏÏ = ^ = 0.782 (5.9) since ïï, the value of a assiimed v/ithin the boundary layer,

refers to the condition of elevated temperature. Then from (5.9) in (5.7)* we have that for airj.

T+c

°f ~ ''•"^ ^ RMs (5.10) If we ommitted to account for the variation of specific

heats with temperature, so that we put J* = 1, and a = 0.737 in (5.5)* then in place of the coefficient 1.7 v/e should have 1.62, so that it v/ill be seen that there is only a

small effect on the skin friction of these variations. The inclusion of dissociation effects v/ould have a more important effect, since we see from (5.4) and (5.2) that c„ will vary as (m/ra~)4j.: if the molecular weight of the gas within the boundary layer were only half that at normal temperatures, then c^ v/ould be reduced by 10°/Q. Evidently,

hov/ever, the neglect of dissociation will not greatly affect the numerical answer, which v/ill err, (if anything) on the pessimistic side in the evaluation of both skin friction

and heat transfer.

To calculate the heat transfer coefficient, we need in (5.3) the value of Y' (O). Prom Table I v/e may calculate the following data!

a 1.0 0.8 0.6

Y'(0) 1 1.021 1.0148

— —1 /11

and Y/ithin the a v a i l a b l e accuracy v/e find t h a t Y' (O) p a ^

thus in (5.3)

k

'f

so t h a t , using ( 5 . 9 ) , for a i r

-ii = 0.510 (5.12)

°f

2 = 1 3 - ^ / ^ ^ . . . . ( 5 . 1 1 )

and from (5.10)

'=H = 0-8sJi5l ....(5.13)

The mean skin friction and heat transfer coefficients may be found from the relations!

%

= ll^

^H ^^ * °f = l | ^ '^f ^^

and performing the integrations, we have that

^ ^ ' • ' ' S >. ....(5.14)

*^f

=

^'^

J T I

The local heat flux to the surface in dimensional terms is, using (5.13),

Q

If we r e f e r c o n d i t i o n s o u t s i d e t h e boundary l a y e r (denoted by s u b s c r i p t ' ö ' ) t o those a t s e a - l e v e l (denoted by s u b s c r i p t

' S L ' ) , we then f i n d , using Sutherlands Formula and the P e r f e c t Gas Law, t h a t

/ ^ \ ^ 2 / r^Sl/f

^SL^SL^SLI

1°SL) V L 'SL

^ J

Q = 0.430

Using I . C.A.N, c o n d i t i o n s for t h e p r o p e r t i e s a t s e a - l e v e l (and p u t t i n g C = 117 K), i n metric \ i n i t s , i f x i s i n m e t r e s . u. i n m e t r e s / s e c . , and p i n atmospheres, t h e n

Q = 4.40 X 10"^ u^ y ^ Kw./sq.m (5.15)

In British aeronautical units, if x = x'ft, u = U'ft/sec. , and p. is in atmospheres, then

^ ' ' ^ ' ' f y

Q = 5.07 X 10""^ ^ J ^ ft. Ib/sq. ft/sec (5.16) In a similar we may calculate the local frictional force

to be ^

F = 1.01 X 10"^ IV \ ^ lb./sq.ft. (5.17) Also of interest is an expression for the thickness of

the boundary layer, since we have already inferred that the boundary layer is finite in thickness in the -oresent

2

asymptotic solution for M -i oO . By definition, from (3.3) and (2.13) if m = m„, the value of n for f = 1 (i. e. at the outside of the boundary layer) is!

n^ = (1+c)

20 -Using (5.4)

3/ 3/.//.I _

Y2

i„^ = [(.„)i(^,V'^]([

TT^

Prom the definition of n it also follows that

— \

I n - öQ.

.v2M^S

SO that eliminating n. froxTi the two expressions

4=2(1+C)(^ÏÏ) (J^^df)2|^ ....(5.18)

Using the data of Table I to evaluate the integral, and using the values of Y and £< from (5.6), we calculate that

4 = 0.030 ïï0-^0[-Ü-^] s ....(5.19)

since, within a reasonable approximation the integral

ƒ'• (YVz)df = 0.494

o

-°"^.

Thus, from (5-9), for air!

^

=0.025 r-^^^^js ....(5.20)

In dimensional terms, if u. = U' ft/sec., x = x' ft. and p. is in atmospheres

Ö = 0. 64 X 10""'' f U* ƒ ~ 1ft. <5. 21) It will be noticed that the boundary layer thickness increases with increase of speed - a trend which has been noticed

before in solutions relating to the boundary layer flow at high Mach Number.

Prom equation (2.14) we have

dn

ï = r n - l . ( C \

g Jo^^^^g'

r^ f

i . e. v/ = nf - g| —dn J o ^ o r , from ( 2 . 1 1 ) ,

-C

w = n f - his T-dn

At the outside of the boundary layer, where f = 1, n = n„

o and h = 0 , i t follov/s t h a t

v/. = n„ 0 5

or using the definitions of v/ and n!

•u,

2x

do

dx

(5.22)

since, from (5.20), ö oc x^.

It follows that Ö, which is in the present asymptotic

solution the finite thickness of the boundary layer, is

what is normally termed the 'displacement thickness' of

an infinitely thick boundary layer. Thus the flow is

tangential to the outer edge of the boundary layer.

The momentum thickness is 8 where

c = 2 ^

°f '^dx

and we find from (5-5) t h a t , a f t e r i n t e g r a t i o n

^ ^ L p . , . r2(iH-c) j ^ r 2t 1 %

yf O

c^ ds

^['^fiifei^^'"^^

By comparison with (5.18) it follows that

a= ^^ _ Jz(o)/r Sdfl 0

^

(Y-1

)aU^ ^

Jo ^ 5

For air, we have that

s=m

. . (5.23)

. . (5.24)

and for large Mach Numbers the momentum thickness becomes

small compared with the displacement thickness.

The velocity and temperature distributions within

the boxindary layer may be calculated quite simply from

the data of Table I. Por, by definition

h = 0 Y(f)

1. e.

H - H^ =

-ha

uf Y(;^)

Ö

_u,

— (5.25)

whilst from (3.3), (2.13) and (5.4) we find as before in

deriving equation (5.18) that

^=[<^-'*(^f"'''ltf-) = ^i

R

2M^s

Hence, from (5.18)

Z(f)

i\:"^"]{i

df

.(5.26)

a relation which connects u/u with y/ö.

22

-In figures 1 and 2, the variation of total heat and

velocity across the boiindary layer is shown as a function of (y/ö) for various values of a. The variation of the shear stress is also given in figure 3- It v/ill be seen from figure 2 that the maximum temperature v/ithin the bo.iindary layer ( for air, with o = 0.78 ) corresponds to

a value of total heat of about one fifth of the stagnation (or reservoir) enthalpy.

Limitations of Theory at High Altitudes and Speeds. There are four assiimptions made in the analysis of the boundary layer flow which become invalid if the flight Mach Number is too high in comparison with the Reynolds Number (both of these being assumed of large,

but not necessarily comparable, magnitude in the analysis)! (i) the neglect of the non-linear terms in

viscosity and heat conductivity in the statement of the equations of energy and momentum, which derive from the third order terms in the Boltzmann equation. These so-called 'Burnett terms' have been shown to be equal to quantities of the order of

|Vi ^ YMf

p^ L - R

times those preserved in the equations and their 2 exclusion is equivalent to the neglect of M /R compared v/ith unity.

(ii) the assumption that the boundary layer is

thin. In the equations of the boundary layer, terms 2 2

of order (ö /L ) are neglected compared with those of order unity. Prom (5.16), such an assumption

2

is seen for M >>1 to be equivalent to the neglect

•2

of M /R compared with unity.

(iii) the neglect of the velocity of slip and a temperature 3\imp at the wall. For instance, if there is a slip-velocity u^ at the wall, and if \ is the mean free path, we have neglected

u, o

o.u„X. • o o y=o / \ ^^ But M,, = const. X p..a .'\, so that

Ö 5 ^ = 0 ( K c , )

and a similar result follows for the temperature 3ump. Using (5.6) it follows that for M^>>1, the assumption of zero slip and temperature jump is equivalent to the neglect of JM/R compared with unity.

(iv) the neglect of the disturbance of the external inviscid flov/ by the presence of the boundary layer. The deflection of the external stream by the boundary

layer causes a modification to the pressure distribution along the plate as calculated for inviscid flow. Thus for a plane surface over which we would expect the

pressure to be uniform (as assiimed) there is in fact a small change in pressure producing a pressure

gradient

dx

_{- ^M "Ti"/ = n " T l ; /}

\ M L s'2 /

where 9 is the inclination of the external stream to the surface (i.e. 8 = "^«./'^A =" dö/dx).

This has been neglected in writing down the equation of motion which includes terms of like order of

magnitude to

fy(-i)=°(i)=°'

Hence the neglect of this induced disturbance is equivalent to the neglect of a term of order

compared with unity.

To sum up, in the mathematical analysis we have assumed 1 1

that both —K- and ^ are infinitesimals. For the M

assumptions inherent in the structure of the equations to be valid and if the surface is assumed plane, the quantities M /R, U^/R, Ju/R, and M J M / R

must also be infinitesimals. The last mentioned is the most stringent requirement, that

R » M ^ (6.1)

If the neglect of this term - due to the disturbance of the external flow by the boimdary layer - is to involve no errors of larger magnitude than the assumption that

2

terms of order 1/M may be neglected compared with unity, we must have that

TECHNISCHE HOGESCHOOL VLIEGTUIGEOUWKUNDE Kanaalsbaat 10 - DELFT 24

-1 = 0 ( 4 ) , . . . . ( 6 . 2 )

^ M^

Thus, if M is ntimerically about 10, the hypersonic o, solution here presented will involve errors of about 1 /Q

9 only, provided that the Reynolds Nimiber exceeds about 10 . It will be seen therefore that a satisfactory treatment of the hypersonic laminar boundary layer should include a correction for this effect, since such a high Reynolds Number is unlikely to be achieved in flight even at high Mach Nximber.

On the other hand, if we interpret our initial assumption that the pressure gradient over the sxirface is uniform as being strictly true, - so that the surface shape will not be plane, as we would expect if the flow could be treated as inviscid, - it then appears that the error in our calculations v/ill be no greater than that

2

involved by assuming M » 1 provided that the Reynolds Number satisfies the relation

I

= o(-^) ....(6.3)

" M^

Thus, for example, if the Mach Number is about 10, then the Reynolds Number should exceed 10-^*, below this

Reynolds Number both the assumptions regarding the absence of surface slip and the thinness of the boundary layer introduce significant errors of greater magnitude than

2

the assumption that (1/M ) is small compared with unity. A value of R > 10-^ for M = 10, implies a value of the relative air density not less than 10"^, if the

length of the body is about 10 ft., say. This is reached at altitudes belov/ about 200,000 ft. , and flight at a

higher altitude than this (at a Mach Number of ten) would

involve an indicated air-speed of less than about 100 ft./sec. Provided then that we interpret our ass\amption of

constant surface pressure literally, the results are valid

_

In this event, the co-ordinates x and y are curvilinear, and the statement of the equations of energy and motion also involve the neglect of terms of order (Kö) compared with unity, where K is the v/all-curvature. Since K is

2

of order ö/L , (as may be deduced from an argument on the lines of that appearing in para. 6) their neglect is

p 2

over a wide range of flight conditions. However, this assumption does not imply that the surface is plane without involving a significant error in most flight conditions. Effect of Surface Shat)e.

We have based our analysis on the assumption that M >> 1, but it should be noticed that this is the Mach Number of the flow at the outside of the boundary layer.

(i.e. more explicitly v/e should put M- in place of M. ) If we put M„ equal to the flight Mach Number, then for M_ » 1, the static temperature behind a shock wave producing a finite deflection is large - of the order of

2

M T - and so it follows that the local speed of soxind behind the shock is of the same order as the flight speed*,

in other words M» = 0(1) and is not large. If also the Mach Number M^ » 1, then the stream deflection must be

0 small! viz.

a = 0(1/M^)

where a is the angle of the stream to the direction of motion at the outside ofthe boundary layer.

There is not the same restriction in accelerated flow, since an expansion produces an increase in local Mach Nximber, though for M „ » 1 it is possible to expand the flow through only a small angle a, - again of order 1/M_,- before the air pressure is theoretically zero. Also it should be noted that the analysis is not applicable to flows in which M„ is finite although - due to expansion - M. » 1 ' ,

a 0

for we have assumed in the analysis that T is comparable with the skin temperature, and after such an expansion T. would be small.

Hence, we may observe that our analysis is applicable only to surfaces (over which the pressiire is a constant) moving at a high Mach Number, M , and inclined at a small angle to the direction of motion! i.e.

\al

=0^^^

....(7.1)

It may then be quite of flight,

and p. ^a

simply shown that,

>

i

if U is the speed

26

-Thus in the expressions derived in the previous

paragraph we may everyT,vhere replace u» b y U, the speed of flight, without affecting the accuracy of the solution. It follows from (5.16) and (5.17) that the skin friction and heat transfer to the surface are influenced b y

surface shape only in that they are proportional to

(7.3)

a=0

The true surface slope (measured in relation to the free-stream direction) is simply a ( x ) , say, if

a^{x) + e = a

where 9 is the inclination of the flov/ relative to the surface at the outside of the boundary layer. T h u s ,

dö/dx since 6 = v_/u. 0 o do or, ^O^""' = ^ - d i from (5.20),

%i^)

f. 0.316 — ^ t (1+c)M-^ Rs (7.4) .... (7.5) As shov/n also in para. 6, since from (7. 1 ) , aJVI is finite,

it follows that the displacement effect of the boxindary g

layer is only negligible if R is of magnitude M or more. 8. Viscous Form Drag.

Combined v/ith the thickening of the boundary layer at high Mach Number already noted, there v/ill be an

increasing displacement of the external flow from its form calculable for inviscid flow. This, as already observed, modifies the pressure distribution from its inviscid form, and gives rise in general to an additional form drag. We define this as a viscous form drag, and it is equal to the difference betv/een the pressure drag calculated for an inviscid flov/ about the body and the pressure drag in a viscid flow, v/ith the boundary layer.

The latter is simply given, by

w ( p. sin a dx ^0 o a (x). dx o ^ ' o 1+0(-^)" M .... (8.1) since in the present conditions, p. is a constant, and

a r 2 1 sin a^ = a^ + ^^ +. ... = a^ [^+0(a^)]

where a is, in general, of order 1/M. Prom (7.4) it follows that pressure drag in viscid flow = («•I'-öj „„-r )p.. We may only make an estimate of the pressure drag for an

inviscid flow about the body in the general case if we assume that the condition (6.2) is satisfied! i.e.

1/R = 0(1/M^). In this event it follows from (7.5) that

a = . ( ^ = i | | = 0 ( 4 ) ....(8.2)

a • a ' a dx ^j,2' '• •'

in general*, and then if p "denotes the pressure at the surface as calculated for a purely inviscid flow (in the absence of the boundary layer), we have from the linearised theory of supersonic flow that, using (8.2),

^ ^ = Y(a^-a)M [n-0(9M)] = - Y M ^ fl+0 (-^l . . (8. 3) The drag in inviscid flow is then

^L

p sin a dx

« o 0

uo

so t h a t from (8.1) the difference between the drag i n v i s c i d and i n v i s c i d flow, v/hich i s t h e viscous form d r a g ,

i s , using (8.2) and (8. 3 ) j

(

(P^-PQ)

sin a^ dx = f (p^-p^) a dx n+0(-l)]

CL

= YMa\ J l dx

^ o ^ ^

[l.0(^)] =rp,aM5|^^,[l.0(j^)j

If v/e quote t h i s as a drag c o e f f i c i e n t , C^., based on the l e n g t h L,

E v i d e n t l y , t h e n , f o r M-» oQ, from (5.20)

CD

=

° * 3 1 6

(a.M)Jüpr

so t h a t from (5.14)

Viscous form drag S ^ ^^^^^ ^^^^ ....(8.3) total skin friction f

er This relation only applies if 1/R = 0(1/M°)! for small Reynolds Numbers, the difference between a and a is

o

larger, as well as the difference between p and p,. The o o linearised theory used above overestimates this difference

28

-between p and p^ if it is large. Also we have o o

replaced a by a which is larger than a„. Thus, in

•^ o o

general, the expression (8.3) will overestimate the viscous form drag, particularly at the lower Reynolds Niombers.

9. Effect of Atmospheric Turbulence and Surface Roughness on Transition.

Our calculations have been restricted to the consideration of a laminar boiindary layer, and it is therefore pertinent to enquire under what conditions laminar flow might exist,

One of the main causes of transition is the existence of an adverse pressure gradient over the surface, which is hardly likely to exist at high supersonic speeds in flow over aerofoils! in the present discussion we have considered the pressure to

be uniform, and this is probably the most adverse condition likely to be met in practice.

We must also however take into account atmospheric turbulence as a contributary factor, and as a result of v/hich there will be small variations in pressure on the

surface. A dimensional consideration shov/s that if p is the root-mean-square of these pressure fluctuations, then the Reynolds Number of Transition is R„, where

- as is suggested, for example, in 'Modern Developments of Fluid Dynamics' p.328. Since (ö/F) varies along the surface as x, it follov/s that v/e may put

F dx ~\ Px/p dx *>Fx '' M dCx/Z)

v/here £ = ijL./p«^a is a unit of length characteristic only of the altitude of flight. Hence

r öPg 1 "^CP/PÖ^ 1

Rrp = g[(FF-)M

"dWI) \

....(9.1)

or substituting from (5.17) and (5.21), for M ^ » 1 ,

fi '^(P/Pa)]

[ M d(x/^) J

R^ = function of | ^ a^x/i>^ I .... (9.2) The corresponding result for the incompressible boundary

2

layer, i.e. M « 1 , interpreting 5 as a displacement thickness, is simply that

Rm = function of

M 3 d(x/f) ... (9.3) Hence, since an increase in p at a certain altitude would

certainly be accompanied by a decrease in R_, (i. e a forward movement of transition), it follows that an increase in Mach Number (at constant height) would always increase Rrp, most particularly at low M.

Of the other factors affecting transition, surface roughness is an important contributary factor. At the tip of a small projection of height s, u = u , say where

"'s ^9y^o [x^

^^^^' p u s

\ = - ^ = (^) ' ^ F e ^ if no.T^

If it is supposed that the flow behind a projection closes up if Rg does not ejiceed a certain critical value, then

it follows from (5.10) that the tolerable roughness height 2

s^„^ for M >3 1 is given by a relation of the type max.

<£(^y^°-5 (Mf)^

max _^| u \a)-t-u. :) /ma\ n. /q • \

The corresponding expression for incompressible flow,

i.e. M ^ « 1 , is e

max

< ^ ( | j f ••••(5.5)

Hence an increase in speed at constant height always reduces the tolerable roughness limit, though less

rapidly for high M. On the other hand, an increase in M for a constant R, increases the tolerable roughness limit, particularly at high M. Heat transfer to the surface also serves to increase the tolerable roughness.

It will be seen that no definite conclusions can be drawn about the stability of the laminar boundary layer at high speeds. However, a comparison of (9.2) and

(9.3), and of (9.4) and (9.5), suggests that the high-speed laminar boundary layer is characteristically less sensitive to atmospheric turbulence and roughness effects than at low speeds. In this sense it is more stable than the incompressible layer.

30 -10. A Comparison with Other Results.

Crocco has suggested from his work on the compressible boundary layer that

which may be interpreted as meaning that ^H _ 1 "^th'^o

c ~ TF~ T -T °f 2a''6 ^th Ö For M^ » 1 , it follows that

.... (10.1)

^ = V + of-^1 (10.2)

f 20*^ \M /

which may be compared with the solution (5.11) given by the present work. The variation of this ratio with 0 at high speeds appears to be less rapid than that suggested by Crocco.

2

Again, Young has siiggested by an extension of Crocco's work and guided by other numerical results that if (XC^T'^,

S2R2c^ = O . 6 6 4 [ O . 4 5 + O . 5 5 T ^ +0.09 (Y-1 ) M ^ Ö 2 ] - ^ ^ - ^ ) / 2

5

(10.3) 2

Taking M >^ 1, and putting 00=0.5 as would be appropriate if v/e have to include the effects of variation of p, at the very high temperatures occurring inside the boundary

layer, evidently for air,

n^^

[^ + 0(^)1 ....(10.4)

which apart from a difference in the numerical constant and in the variation with a is the same as (5.7).

In figure 4> the results of the present work in relation to the variation of c„ with M are compared with the values of c« obtained from (10.3) for various values of 0) and T /T (and using a value of a equal to 0.74- ) The present solution gives consistently higher values of the

skin friction ( and so also of the heat transfer ) coefficient than indicated by (10.3) if v/e suppose that v/e must lower

the value of w to O.5 when dealing v/ith flow at high Mach Numbers as in (10.4). At high Mach Number the difference amounts to about 25°/^. About 5 ^ of this difference

we have adjusted the values of the air specific heats and Prandtl Number, but the bulk of the difference lies in the fact that v/e have used Sutherland's Formula for the variation of viscosity instead of a simple power lav/

variation. This has the effect of increasing the values Q-*

1 + ^ compared v;ith that

oDtamea were v/e to assimie merely that M-OCT^, and so brings the results (at least, for Mach Numbers between 10 and 20) more into line with those of equation (10.3) with a)=0.8-0.7.

(This is a mean figure for w across the boundary layer, but with an emphasis towards its value at normal temperatures) There seems no doubt that the lav/ of the variation of

viscosity with temperature is of the greatest importance in influencing the quantitative evaluation of the boundary layer characteristics.

The present analysis, using Sutherland's formula, implies that there is relatively little change in the skin friction coefficient with Mach Number below M=10, and so endorses the applicability of much of the work on the

compressible boundary layer with ÜJ=1 (i.e. viscosity varying linearly with the temperature) in relation to such speeds.

Figure 4 also shows from equation (10.3) that the variation of surface temperature is not important in influencing the skin friction at high M! and so confirms the qualitative deduction of the present v/ork concerning this effect. The figure shows two sets of results for

(T / T ) equal to 1 and to 4s which would cover a range • of T from say about 250°K to 1000°K.

o

11. Conclusions.

(i) It appears that the high-speed boundary-layer equations may be solved for the condition of laminar flow, in the absence of a pressure gradient, by a niimerical process. The results of this report are relevant

" 2

if the assumptions are made that M >> 1 (where M is the Mach Number of the flov/ outside the boundary layer) but that the surface temperature is not high (i.e. that ': it is not commensurate vith the thermometer temperature),

and that the air molecular specific heats reach an asymptotic value at high temperature (v/hich may be different from that at ordinary temperatures. ) The numerical results apply to a flow with Prandtl Numbers between 1.0 and 0.6.

32

-(ii) Prom the solution it appears that the velocity and temperature vary near the surface in proportion to y ' , where y is the distance away from the surface. This suggests that the velocity and temperature

gradient are infinite at the surface. In the limiting condition (as M-i'OO) this is shown to be compatible with the existence of gradients whose actual magnitude

depends on and increases with M. The solution enables these gradients to be calculated, and the corresponding rate of heat transfer and skin friction assessed.

(iil)The temperature and velocity distributions within the boundary layer are shown in figures 1 and 2. These

demonstrate the trends observed in previous analyses! the velocity distribution is at high speeds more nearly linear than at low speeds, and the enthalpy reaches

•1

its maximum value (equal to about -r of the stagnation value for air) at about a third of the way out into the boundary layer. The effect of a decrease in Prandtl Niimber is to cause a reduction in temperature within the boundary layer, but only a slight increase in velocity.

Figure 3 shows that the shear stress within the flow decreases most rapidly near the outside of the layer, since here both the temperature (and so the viscosity) and the velocity gradient are decreasing.

(iv) The asymptotic solution yields a finite boundary layer thickness, which corresponds v/ith the quantity which is usually termed the 'displacement thickness'. The flow is, accordingly, tangential to the outside of the boundary layer. The compatibility of this result with that for finite Mach Nximber (where the disturbance due to the boundary layer extends to infinity) is discussed in para. 4»

(v) The boundary layer thickness varies as JIT, v/here x is measured along the surface*, it varies asJu^/R

and so increases in proportion to the increase of forward speed (as compared with its behaviour at lower Mach Numbers where it decreases with increase of speed). For air, the thickness is given by equations (5.20) and (5.21)! the latter states that

where U' is the free-stream velocity (in f t / s e c ) , p is the surface pressure (in atmospheres), and x' is measured (in ft. ) along the surface.

(vi) Associated with the thickening of the boundary layer at high speeds, there is also an important interaction between the boxindary layer and the external flow,

causing a significant modification to the stream

deflection and pressure distribution near the surface as calculated on the assumption of nurely inviscid flow. At a Mach Nximber of about 10, it is shown that these effects are imiaortant at Reynolds Numbers

9

less than about 10 , - which includes most likely

flight conditions. Thus the assumption of a constant pressure over the surface in the present argiiment

cannot be interpreted as implying a plane surface, since the displacement effect of the boundary layer on the external flow must be taken into acco\int. (vii)It is shown that such assumptions as the neglect of

surface slip, and the finite thickness of the boundary layer, v/hich are implicit to the present solution of

the boundary layer equations, are justifiable within the accuracy of the solution provided that the Reynolds Number R is a large number of the magnitude of M-^ or more. Thus at a Mach Number of 10 or more, the results will have an accuracy v/ithin a few per cent, provided

that R exceeds 10-^, say! this includes most flight conditions unless the altitude is so high that the indicated airspeed is much less than 100 ft/sec,

(viii) There is no necessity for assuming that the (constant) surface pressure p- is the same as that of the free-stream, p„. Hov/ever, the assiimption that the local

a

Mach Number of the flow outside the bovmdary layer, M, is large, implies that the slope of the surface to the free stream direction must be small. If a is the inclination of the streaia at the outside of the boundary layer, then a must be small so that

v/here M„ is the Mach Number of flight.

(ix) As remarked above in para, (vi), the deflection of the external flow by boundary layer is generally

34

-important, and a (the slope of the surface to the direction of motion) is in general different from a. From (7.5) we have that

(•

- ^ ) = O U R

)

(x) The displacement effect of the boundary layer, in modifying the pressure distribution, will also

modify the pressure drag as calculated on the assumption of inviscid flow. It is shown in para. 8 that for

large R, the additional drag involved by this effect (which we call the viscous form drag) is commensurate with the skin friction drag', we have from (8.3) that

if a is the stream deflection outside the boundary layer

viscous form drag ^ ^^^^^ ^^^^

total skin friction

although this relation probably overestimates the additional drag at lower Reynolds Numbers. The effect will be seen to be particularly important at high Mach Numbers.

(xi) The heat transfer to the surface and the skin friction are shown to be independent of the surface temperature, at least to the first order of approximation if terms

2

of order (1/M ) are neglected in comparison with unity. This fact is borne out by a comparison with existing results relating to the compressible boundary layer flow. It will be recalled that we have ass\amed that T , the surface temperature, is comparable with that

outside the boundary layer so that (T /T., ) is small. (xii)To account for the variation of viscosity (11) with

temperature, Sutherland's Formula, M-CCCT / T + C ) , is used, with no allov/ance for dissociation effects at high temperatures. In this formula, the larger the value of C, the more rapid v/ill be the increase of

viscosity with temperature. From the present analysis, it appears that the skin friction, heat transfer and

boundary layer thickness all increase as C increases. (xili) A comparison of the present results with others

flow, which have used a formula for viscosity of the type .u^cT^, shows that even at Mach Numbers between 10 and 20 a value of w between 0.7 and 0.8

is appropriate to bring the results into line with those using Sutherlands Formula, It seems that it is the variation of viscosity v/ith temperature near the surface which is most important in

choosing a representative value of co* at the high temperatures existing within the boundary layer a value of 0) = 0.5 would be more appropriate. The present results also imply that for M < 1 0 , there is little change in skin friction coefficient with Mach Number, which again indicates a high value of w, and

endorses the applicability of the methods using a value of 0) equal to unity (i.e. assuming uiCT). (xiv)It is assiimed in the analysis that at high

temperatures the specific heat of the gas at constant pressure increasesby a factor |7*. The skin friction and heat transfer are then found to vary as [^ '7;,

and the boundary layer thickness as X*" . For air a value ofJp = 9/7 is chosen which makes little

difference to the numerical results for the value of skin friction or heat transfer.

(xv) Connected v/ith the increase in the gas specific heats there v/ill also be an increase in Prandtl Number (o) of the gas at high temperatures and a mean value of o = 0.78 has been used in preference to the more usual value of a = 0.74. Here again this makes little

difference to the niimerical results. The skin friction coefficient is found to vary as a" ^^, and the ratio of the heat transfer to the skin friction coefficient

' —1 /11

(k^r/c^), varies approximately as o" '^ . These are rather less rapid variations than are predicted for conditions at lower Mach Number. The boundary layer

0 35 thickness varies in üroportion to a * ^•^.

(xvi)For air, it is shov/n (from equation (5.17)) that the shear force at the surface is

F = 1.01 X 10"^ U' jp^/x' lb./sq.ft. where U' is the free-stream velocity (ft/sec), p

36

-along the surface (in ft.) Such a variation corresponds to a skin friction coefficient given by equation (5.10), where it will be seen that

c^ cP 1//RM^, and decreases with increase of Mach

Number - an effect well established qualitatively in previous solutions.

(xvii) The momentum thickness (!^ ) of the bo\mdary layer becomes appreciably less than the displacement

thickness at high Mach Number. Prom (5.24), a = (10.7/M^)ö.

(xvLii) The ratio of the heat transfer to skin friction coeff: 'icients (k„/c.p) is found to be equal to

1 /"11 n I -z

) ) 9 i f k i ° >>aoo'^ /->•" f^'^ II-'T ^ -Oni

H air, kjT = 0. 510 c^.

(1/20 /^ ) , if kj^ is based on (tp.ugL). For

•om

(xix)It follows that k„ also varies as Jl/RM , and frc

(5.16) the local heat flux to the surface i s given as

Q = 5.07 X 10~^'^/p^/x"'" f t . l b . / s q . f t . / s e c .

(xx) Some remarks are made in para. 9 concerning the stability of a laminar layer, and it is concluded that as far as the effects of atmospheric turbulence and surface roughness may affect the transition to turbulence, the laminar boundary layer at high Mach Nixmber is less sensitive to these effects than at

loY/ Mach Number, if the Reynolds Nximber is the same. It does not follow that an increase of speed alone

(at constant height) is stabilising.

(xxi)The results of this work have been derived elsewhere^ by the author, using the approximate method of

momentum and energy integrals. Whilst in qualitative agreement, these results give values of the skin friction and heat transfer coefficients half as large again as those deduced here. Perhaps this serves to emphasise the difficulty of attempting to simulate ab initio the variations in temperature and density within a compressible boundary layer, as is necessary if this method is to succeed.

(xxiO Particularly in view of the strong interaction between the boundary layer and the external flow, the greatest

need in developing a theory of the boundary layer at high speeds, is for one v/hich will take into account variations in pressure over the surface.

References. 1. Crocco, L. 2. Young, A.D. 3. Tsien, H. Nonv/eiler, T. 5. Kirkby, S. and Nonweiler, T.

Lo Strato Limite Lanimare nei Gas. Monografie Scientifiche di Aeronautica No. 3 s 1946.

Skin Friction in the Laminar Boundary Layer in Compressible Flow,

Aero. Quart. Vol.1., Pt. II. August 1949. Superaerodynamics. The Mechanics of Rarifled Gases.

J.Aero. Sci. Vol.13. No. 12.

Aerodynamic Heating at High Speeds. J.Brit. Interplanetary Soc. Vol.10,No.4. July 1951.

The Numerical Solution of Certain Differential Equations occurring in Crocco's Theory of the Laminar Boxmdary Layer.

COLLEGE OF AERONAUTICS REPORT No. 67 l-O

(-D-)

VELOCITY RELATIVE TO O-Ö THAT IN FREE STREAM 0 - 6 0-4 0-2 / 0-( 0 - É /

f

.

.A

^"/Ty^

^ ^ <r=i.o ^ 0-2 0-4 0-6 0-8 l-O

(Vs)

DISTANCE FROM SURFACE IN TERMS OF THICKNESS OF BOUNDARY LAYER.

FIG I VARIATION HM VELOCITY WITHIN BOUNDARY LAYER WITH VALUE OF THE PRANDTL NUMBER or AT HIGH