The numerical solution of certain differential equations occuring in Crocco's theory of the laminar boundary layer

TECHNISCHE HOGESCHOOL VUEGTUIGBÓUWKUNK

2 7 JUNI 1953

REPORT No. 74

THE COLLEGE OF AERONAUTICS

CRANFIELD

THE NUMERICAL SOLUTION OF CERTAIN DIFFERENTIAL

EQUATIONS OCCURRING IN CROCCO'S THEORY OF

THE LAMINAR BOUNDARY LAYER

by

S. KIRKBY, Ph.D., and T. NONWEILER, B.Sc, of the Department of Aerodynannics.

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

REPORT NO. 7 4 FiAY. 1953

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 Niraerical Solution of Certain Differential Eqxiations occurring in Crocco's Theory of the

Laminar Boundary Layer

-by-S. Kirkby, Ph.D., and T.Nonweiler, B.Sc. of the Department of Aerodynamics

rECHNISCHE HOGESCHOOl,

VUEGTUiGBÓUWKUNDE

2 7 JUNI 1953

SUI'iMARY

A numerical method is described for the solution of certain differential equations -which result from the application of Crocco's transformation to the laminar boundaay layer equations appropriate to high supersonic I/Iach numbers. (i.e. at hypersonic speeds).

Solution is obtained by continuD\i3 application of a rapidly convergent relaxation process to a pair of simultaneous

differential eq-uations, for which one of boundary conditions is a first derivative. The Prandtl number occurs as a parameter.

3

-The d i f f e r e n t i a l equations can be expressed approx-imately as f i n i t e difference equations i n the form

Z.6^Z + h V y ~ Y + Zjü^ = 0 (2.5)

6 ^ + (l-ö-).|i6Y,|i5Z/Z + 2 h ^ + A 2 = 0 . . . . (2.6)

where ^ . and £s are difference corrections -vïhich include all but the dominant differences when derivatives are expressed in terms of central differences, and where h is the constant Interval betvreen successive pivotal points.

^.7e denote fxmctional values of Y and Z corresponding to f , f + h, ...,f + nh, by suffices 0,1,.,.,n and express

O ' O O / w ^ 7 7

differences in terms of functional values according to the relations

^5Y^ = i(Y^ -Y_^)

ji6z^ = -Hz^ - z ^)

P , > (2.7)

^ \ = (^1 -2^o^^-l)

6^Z = (Z, - 2Z + Z J

o 1 o ""1 _

Thus equations (2.5) and (2.6) may be -written

Z (Z,-2Z +Z , ) + h^f / . / " Y +Z A . = R^ (2.8) o 1 o -V o " o o 1 1 '

(Y^-2Y^+Y_^)+(l-or) (Y^-Y__^) (Z^-Z__^)/4Z^+2h2+Zi2 = R2

(2.9) where R. and R„ are r e s i d i i a l s .

1 2

In the ensuing solution we obtain a first approximation to the dependent variables by neglecting the difference corrections A . , A 2 and applying the method of rela:xation to obtain zero residiaals R , R . More accurate numerical representation of the dependent vajriables is obtained by differencing the above values and including approximate difference corrections before continuing the relaxation.

Leading terms in the difference corrections are

A ^ = -5^zyi2 + 8^zy90- (2.10)

^ 2 = - 6 \ / l 2 + 6 ^ y 9 0 - (2.11)

(1-0-) ((fi6Y - | i 5 \ / 6 + . . . ) ( | i 6 Z -iibh /6+...)-[i8Y .^6Z ƒ / Z

/ o o 0 0 0 0 ( 0

/ 3 . . . .

4. Starting Values for the Solution

It will be seen that equation (2.2) has a closed analytical solution when cr = 1. For in this special case

Y " + 2 = 0 (4.1)

and so, i n viey/ of t h e boundary c o n d i t i o n s ( 2 . 3 ) and ( 2 . 4 ) , we o b t a i n

Y = f ( l - f ) . ( 4 . 2 )

Approximate v a l u e s of Z a t f = 0 and 0 . 5 majr novir be o b t a i n e d from e q u a t i o n s ( 2 . 4 ) , ( 2 . 8 ) , ( 3 . 5 ) , and ( 4 . 2 ) , f o r n e g l e c t i n g j / \ . and R i n eq-uation ( 2 . 8 ) , we have a p p r o x i m a t e l y

z (z, - 2Z + Z J / Y + h^f = 0 ( 4 . 3 )

o 1 o - V V o o

L e t f_^ = 0 , f^ = 0 , 5 , f-, = ^ so t h a t h = 0.3,J~I^ = 0 . 5 by e q u a t i o n ( 4 - 2 ) , t h e boimdary c o n d i t i o n Z. = O by, ( 2 . 4 ) and

Z = 1+ J2 Z /(1+<j 2 - 1) by e q u a t i o n ( 3 . 5 ) , so t h a t eq-uation (4. 3) niay be -written

^ 4 ƒ2 Z \ . . . ° V 4 ^ 2 -1 0 / 2 4 2

from v/hich we obtain

z = 0.564 o

and hence Z = 0 . 6 8 5 .

The corresponding variation of Z with f, as given in equation (3.4) , is

Z = 0.685 0 - f^/^) (4.4) v/hich may be used as an interpolation formula to estimate Z in

the range O^f^CI for the special case cr = 1.

This special solution, for cr = 1, has limited physical significance since for real fluids the Prandtl nijmber cr is less than unity, say about 0.7.

However, to obtain numerical solutions for real fluids, it is convenient to use values given by the special solution, eqi;iations (4.2) and (4.4), ^-s initial val-ues for the relaxation method described below.

/ 5. ...

* On the assumption that Z varies parabolically -with f, one obtains Z = 0.685 + 0.202f - 0.888f^.

TCHNISCHE HOGESCHOOL VLiEG I U.GüOUWKÜNDE

-7-dependent variables.

6. The Relaxation Pattern

Since the given differential equations (2.1) and (2.2) are non-linear, special attention has to be paid to the relaxation pattern which is as follcfws.

When relaxing equation (2,8) to obtain values of Z corresponding to given values of Y we note that if we vary Z by e then we miist change the residual R. at f , f , and f by

and

(R^)^ = e(Z^ - 4Z^ + Z_^ - 2e)

respectively.

Similarly, when relaxing equation (2.9) to obtain values of Y, if we vary Y by e the corresponding changes in the residual R„ at f ., f , and f, are

2 -1' o' 1 and (^2)-1

(Vo

e J1 + (l-a-)(Z^ -

Z_^) / hZ_A

U

- (l-o-)(Z2 - Z^) / 4 z J

Relaxation could be effected simultaneously in both variables but the above procedure is preferred since, for real fliiids, the value of Y only differs slightly from the values calculated for the special solution (4.2).

7. Computational Procedure

In practice it has proved satisfactory to use six or eleven equally spaced pivotal points in the range O ^ f $1.

Values of Y = f ( l - f ) and Z = 0,685(1-f^'^'^^) are c a l c u l a t e d from equations (4-2) and (4.4) of the s p e c i a l s o l u t i o n f o r rr = ^.

Residuals R., R„ a r e then c a l c i i l a t e d frcan equations 1 2

(2.8) and ( 2 . 9 ) , -with difference c o r r e c t i o n s neglected, for each p i v o t a l p o i n t except f = 0 and f = 1.

-9-Ti'^LE 1

Six points. Three decimals

f 0 . 0 . 2 . 4 . 6 . 8 1.0 Y 0- = 1 0 . 0 0 . 1 6 . 2 4 . 2 4 . 1 6 0.00 0 . 8 0.000 .163 .247 .252 .175 0 . 0 0 0 0 . 6 0.000 .167 .255 . 2 6 5 .192 0 . 0 0 0

z

Eleven points. Fo-ur decimals

f 0 . 0 .1 . 2 . 3 . 4 . 5 . 6 . 7 . 8 . 9 1.0 Y cr = 1 0 . 0 0 . 0 9 . 1 6 .21 . 2 4 . 2 5 . 2 4 .21 . 1 6 .09 0 . 0 0 ,. cr = 0 . 8 0.0000 .0922 .1646 .2172 .2500 .2630 .2560 .2286 .1799 .1073 0.0000 cr = 0. 6 0.0000 .0949 .1700 .2255 .2615 .2779 . 2 7 4 4 .25^2 . 2 0 3 4 . 1 2 8 4 0.0000 Z cr = 1 0.7200 .7188 .7131 . 7 0 0 4 .6783 .6439 .5940 .5232 .4223 .2703 0.0000 cr = 0 , 8 0 . 7 1 1 4 .7100 .7037 . 6 9 0 4 .6679 .6335 .5837 .5133 .4136 .2643 0.0000 cr = 0 . 6 0.6966 . 6 9 5 4 .6898 .6772 .6552 ,6211 .5717 .5019 .4031 .2563 0.0000