Computation of Boundary Layers

Computation of Boundary Layers

József Dénes, István Patkó

Budapest Tech

Doberdó út 6, H-1034 Budapest, Hungary, E-mail:

denes.jozsef@nik.bmf.hu patko@bmf.hu

Abstract:This paper is the first part of a series of studies where we examine several methods for the solution of the boundary layer equation of the fluid mechanics. The first of these is the analytical or rather quasi analytical method due to Blasius. This method reduces a system of partial differential equations to a system of ordinary differential equations and these in turn are solved by numerical methods since no exact solution of the Blasius type equations is known. We determind all the Blasius equation neccessary for up to 11-th order approximation. Our further aim to study the finite difference numerical solutions of the boundary layer equation and some of the methods applying weighted residual principles and by comparing these with the ”exact” solutions arrived at by Blasius method develop a quick reliable method for solving the boundary layer equation.

Keywords: Boundary Layer, Blasius Method, Boundary Layer equation

1 Boundary Layer

The motion of a fluid around a solid body according to Prandtl (1904) can be described by the Euler equation of the perfect (that is nonviscous) fluid motion except in a thin layer near the surface of the solid body where the speed of the motion increases from zero to the speed that would be in case if the fluid had no viscosity at all. Outside the boundary layer the fluid may be considered as nonviscous. This is the case when the velocity of the fluid in the direction of the flow around the body increases. When it decreases that is the pressure increases, often the fluid motion unable to follow the bodies’ surface and it gets detached and the space between the surface of the solid and the detached fluid is filled with irregularly moving fluid. Prandt’s theory of boundary layer, more precisly his equations describing the motion within the boundary layer can predict the point(s) of detachment accurately. The detachment begins where the curve of the velocity profile starts out perpendiclar to the surface of the solid. After this point a backward flow develops. The typical values used for describing the boundary layer are:

δ

: Boundary layer thickness is the distance measured from the surface of the solid where the speed of the fluid is within 1% of the speed outside of the boundary layer.

δ

1: Displacement thickness

⁼

^∞

∫

0

1

1 (

δ U U − u )dy

δ

2: Impulseloss thickness

⁼ ∫

^∞

0

2

1

2

(

δ U U − u )udy

*

δ

* : Energyloss thickness

⁼ ∫

^∞

0 3

*

*

1 (

U U

δ − u )

²

udy

Profile parameter

H

2

*

* 2 ,

1

δ

= δ

2 The Eqations of the Boundary Layer Flow

Inside the boundary layer that is in the vicinity of the body the forces due to viscosity are comparable in magnitude with the forces of inertia they can however be neglected outside of it. The pressure in the boundary layer could be taken as constant and its value equal to the pressure belonging to the corresponding perfect fluid flow, that is the pressure outside of the boundary layer. Without going into more details we give the equations of the boundary layer motion in case of two dimensional staionary incompressible fluid flow:

= 0

∂ + ∂

∂

∂

∂ = + ∂

∂

∂

y v x u

dx U dU y v u x u u

2

∂ + ∂

y ν u

u v y = 0

= u U ( x )

where u is the velocity of the fluid in the boundary layer parallel to the tangent of the surface of the solid v is perpendicular to it and U is the velocity outside of the boundary layer. and must also satisfy the boundary conditions: :

v 0

u = 0 y = ∞ =

3 Blasius’ Method for Solving the Boundary Layer Equation

The boundary layer equations can be reduced to an infinite system of ordinary differential equations with the following method due to Blasius. By substituting for the velocity components

∂ y u = ∂ ψ

and

v x

∂

− ∂

= ψ

, where

ψ

is the stream function of Lagrange the second of the two boundary layer equations is automaticly satisfied and the first one becomes:

νψ

yyy

+ Ψ

yy x xy

y

dx

U dU ψ

ψ ψ

ψ − =

.

This equation then can be reduced to a set of ordinary differential equation if for in case of symmetric bodies the following power series expansion is substituted:

+ )

7

(

7

η

+ +

+

= {

_{1 1}

( ) 4

₃ ³ ₃

( ) 6

₅ ⁵ ₅

( ) 8

₇

1

η η

ν η

ψ u xf u x f u x f u x f

u

....

) ( 12

) (

10

₉ ⁹ ₉

+

₁₁ ¹¹ ₁₁

+ + u x f η u x f η

where

ν

¹

y u

η =

.and

u

) (x

are the coefficients in the power series expansion of

U

, that is:

⋅⋅

⋅⋅

, ⋅ ,

₃

1

u

+ + u

₁₅

x

¹⁵

+

+ + +

+ +

= u

₁

x u

₃

x

³

u

₅

x

⁵

u

₇

x

⁷

u

₉

x

⁹

u

₁₁

x

¹¹

u

₁₃

x

¹³

U

...

19 19 17

17

+ +

+ u x u x

From the last two equations it follows that:

( )

( 10 10 5 ) ( 12 12 12 )

8 8

5 6

4

7 5 9

3 11 1 9

2 5 7 3 9

1

3 7 1 5 2 3 5 1 3 3 1 2

1

+ +

+ +

+

+ +

+ +

+

=

x u u u u u

u x

u u u u u

u u u u x u u u x u u x dx u

U dU ( )

...

11 7 5

+ + x

+ )

9(

8f

η

x

+ )

'

(

9

9

η

+ +

+ +

= { _{1 1}( ) 12 ₃ ² ₃( ) 30 ₅ ⁴ ₅( ) 56 ₇ ⁶ ₇( ) 90 ₉

1

η η

η ν η

ψ

u f u x f u x f u x f u

x u

...}.

) ( 132

₁₁ ¹⁰ ₁₁

+ + u x f η

+ +

+ +

=

_{1 1}^'

( η ) 4

₃ ³ ₃^,'

( η ) 6

₅ ⁵ ₅^'

( η ) 8

₇ ⁷ ₇^'

( η ) 10

₉

ψ x f

+ )

'(

9

8

η

u f

x u f

x u f

x u xf

y

u

...

) ( 12

₁₁ ¹¹ ₁₁^'

+ + u x f η

+ +

+ +

= _{1 1}^'(

η

) 12 ₃ ² ₃^'(

η

) 30 ₅ ⁴ ₅^'(

η

) 56 ₇ ⁶ ₇^'(

η

) 90 ₉

ψ

_xy u f u x f u x f u x f u x f

...

) ( 132

₁₁ ¹⁰ ₁₁^'

+ + u x f η

+ )

''

(

7

7

η

+ +

+

=

¹

{

_{1 1}^''

( η ) 4

₃ ³ ₃^''

( η ) 6

₅ ⁵ ₅^''

( η ) 8

₇

ψ u ν u xf u x f u x f u x f

yy

...

) ( 12

) (

10

₉ ⁹ ₉^''

+

₁₁ ¹¹ ₁₁^''

+ + u x f η u x f η

+ )

''

(

' 7

7

η

+ +

+

=

¹

(

_{1 1}^'''

( η ) 4

₃ ³ ₃^'''

( η ) 6

₅ ⁵ ₅^'''

( η ) 8

₇

ψ ν f

...

)

⁵

2 ' 3 2

3

f x +

x u f

x u f

x u xf

u u

yyy

...) )

( 12

) (

10

₉ ⁹ ₉^'''

+

₁₁ ¹¹ ₁₁^'''

+ + u x f η u x f η

and

) 576

480 144

(

) 180 320

100 (

) 192

64 (

48 36

( 12

11 ' 7 ' 5 7 5 '

9 ' 3 9 3 '

11 ' 1 11 1

9 2 ' 5 2 5 '

7 ' 3 7 3 '

9 ' 1 9 1

7 ' 5 ' 3 5 3 '

7 ' 1 7 1

' 5 ' 1 5 1 3

' 3 ' 1 3 1 2 ' 1 2 1

+ +

+

+ +

+

+ +

+ +

+

=

x f f u u f

f u u f

f u u

x f u f

f u u f

f u u

x f f u u f

f u u

u f

f u u x

f f u u xf

xy

u

y

ψ ψ

...

) 132

360

336 240

120 12

(

90 224

180 96

10 (

) 56

120 72

8 (

) 30

48 6

(

) 12

4 (

11 '' 1 9 1 11 ''

3 9 3 9

'' 5 7 7 5 ''

7 5 7 5 ''

9 3 9 3 ''

11 1 11 1

9 1 '' 3 7 7 3 ''

5 5 2 5 ''

7 3 7 3 '' 9 1 9 1

7 '' 1 7 7 1 '' 3 5 5 3 ''

5 3 5 3 '' 7 1 7 1

5 '' 1 5 5 1 '' 3 3 2 3 '' 5 1 5 1

3 '' 3 3 3 1 '' 3 1 3 1 ''

1 1 2 1

+ +

+ +

+ +

+ +

+ +

+ +

+ +

+ +

+

+ +

+

=

x f f u u f

f u u

f f u u f

f u u f

f u u f

f u u

u u f

f u u f

f u f

f u u f f u u

x f f u u f

f u u f

f u u f

f u u

x f f u u f

f u f

f u u

x f f u u f f u u x f f

yy u

x

ψ

) ⁹

'' 1

9f x +

f

ψ

finally substituting these into the bondary layer equation:

ψ ψ

ψ

ψ

ψ

Comparing the coefficients of

x

yields:

'' ' 11 2 1 2

1

+ u f

''' 1

'

= 1 + f

x

3

'' ' 3 3

4 u

1

u f

'' '

1 + f

3

= x

5

5 '' 1

5

f f ) =

5 1

u u

'' 1 1 2 1 2 ' 1 2

1

f u f f u

u − =

and from here we get:

' 1 1 2 '

1

f f

f −

From the coefficients of we get:

3 1 3 '' 1 3 1 ''

3 1 3 1 ' 3 ' 1 3

1

( 4 12 ) 4

12 u u f f − u u f f + u u f f = u u +

that is:

3 '' 1 '' 3 1 ' 3 '

1

3

3 f f − f f − f f

As for :

''' 5 5 1 2 3 5 1

1 '' 3 3 2 3 ''

5 1 5 1 2 ' 3 2 3 '

5 ' 1 5 1

6 3 6

30 48

6 ( 48

36

f u u u u u

u u f

f u f

f u u f

u f

f u u

+ +

+ +

− +

Dividing by

6

:

'' ' 5 5 1

2

3

f

u

u +

5 '' 1 '' 3 3 5 1

2 '' 3

5 1 2 ' 3 5 1

2 ' 3

5 '

1

2

1 1 ) 5 8

( 8

6 f f f f u

u u f u f u f

u f u

f + − + + = +

f

5

If we seek in the form

5 5 1

2

3

h

u u

+ u g

₅

h

₅

'' ' 5

5

= 1 + g

5

5

g

f =

we get for and the following differentialequations:

'' 1 '' 5 1 ' 5 '

1

5

6 f g − f g − f g

'' 3 3 5 '' 1 '' 5 1 2 ' 3 ' 5 '

1

8 5 8

6 f h + f − f h − f h − f f

₅^'''

2 1 + h

=

x

7

n

n

x

7

Case . This case is sufficiently complex to demonstrate the method of finding the Blasius type diffrential equations for the general case that is for where is an arbitrary odd integer. By comparing the coefficients of we get:

x

'' 1 7

1

f f ) =

7 1

u u

''' 7 7 1 5 3 7 1

7 ''

3 5 3 5 ''

5 3 5 3 ''

7 1 7 1

' 5 ' 3 5 3 '

7 ' 1 7 1

8 ) 8 8

(

56 120

72 8

(

) 192

64 (

f u u u u u u

u u f

f u u f

f u u f

f u u

f f u u f

f u u

+ +

+ +

+

− +

Dividing by

8

gives:

''' 7 7 1

5 3

'' 3 5 7 1

5 '' 3

5 3 7 1

5 '' 3

7 1 ' 5 ' 3 7 1

5 ' 3

7 ' 1

) 1

(

15 9

( ) 24

8 (

u f u

u u

f u f u

u f u

u f u

u f u

f f u f u

u f u

f

+ +

+ +

+

− +

f

7

'' 1

7

)

7 f f =

Let us seek in the form 7

~

7 1

5 3 7

7

h

u u

u g u

f = +

g

7

. Substituting this into the last equation yields for the following ordinary differential equation:

''' 7 ''

= 1 + g

7

h

~

1 7 '' 7 1 ' 7 '

1

7

8 f g − f g − f f

and for

''' 7

~

7 1

5 3 7 1

5 3

5 5 1

2 3 5 '' 3 7 1

5 '' 3

5 5 1

2 '' 3 5 3 7 1

5 3

'' 7

~ 1 7 1

5 ' 3

5 5 1

2 ' 3 5 ' 3 7 1

5 3 '

7 ' ~ 1 7 1

5 3

) (

15 ) (

9

) (

24 8

u h u

u u u u

u u

u h u g u u f u

u h u

u u g u u f u

u u

h u f u

u h u

u u g u u f u

u h u

u f u

u u

+

− +

− +

−

− +

+

'' 7 1

7 h

~

f =

Substituting for 7

h

~ ₇

7 1

2

3

k

u u

+ u h

₇

''' 7

7

= 1 + h

k

7

'''

''

k k

f =

7 7

~

h

h =

yields for

'' 1 5 '' 3 '' 5 3 '' 7 1 ' 5 ' 3 '

7 '

1

24 9 15 7

8 f h + f g − f h − f g − f g − f h

and for

'' ''

'' ' ' '

'

24 9 15 7

8 f k + f h − f k − f h − f h −

f

n

n f

_n

f

n

With the same method we can arrive at the equations for all the Blasius functions. With increasing has to be broken down into more and more terms. Here the forms of and the corresponding differential equations are given up to the order of 11.

5 5 1

2

3

h

u u + u

5

5

g

f =

7 7 2 1 2

3

k

u u + u

7 7 1

5 3 7

7

h

u u

u g u

f = +

9 9 3 1 4

3

q

u u + u

9 9 2 1

5 2 3 9 9 1

2 5 9 9 1

7 3 9

9

j

u u

u k u

u u h u u u

u g u

f = + + +

11 11 4 1 5 3

1 11 11 2 1

2 5 3 11 11 2 1

7 2 3 11 11 1

7 5 11 11 1

9 3 11 11

u n u

u

u q u u u

u j u

u u

u k u

u u

u h u

u u

u g u

f = + + + + +

₁₁

11 3

5 3

3

m

u

u +

1 3 5 5 7

k

7

g

₉

m

''' 1

'

= 1 + f

'' '

1 + f

3

=

'' ' 5

5

= 1 + g

The differential equatios that the functions , , , , , , , , , , , , , , , , , , have to satisfy are:

f

11

f

11

g n

11

h

g

₇

h

h k

9 ₉

j

₉

q

₉

g

₁₁

h

₁₁

k

₁₁

j

₁₁

q

' 1 1 2 '

1

f f

f −

3 '' 1 '' 3 1 ' 3 '

1

3

3 f f − f f − f f

'' 1 '' 5 1 ' 5 '

1

5

6 f g − f g − f g

'''

2

5

1 + h

=

''' 7 ''

= 1 + g

''' 7

7

= 1 + h

''' 7 7 ''

1

k k

f =

''' 9

9

= 1 + g

'' 3 3 5 '' 1 '' 5 1 2 ' 3 ' 5 '

1

8 5 8

6 f h + f − f h − f h − f f

1 7 '' 7 1 ' 7 '

1

7

8 f g − f g − f f

'' 1 5 '' 3 ''

5 3 '' 7 1 ' 5 ' 3 '

7 '

1

24 9 15 7

8 f h + f g − f h − f g − f g − f h

5 '' 3 '' 5 3 '' 7 1 ' 5 ' 3 '

7 '

1

24 9 15 7

8 f k + f h − f k − f h − f h −

'' 1 '' 9 1 ' 9 '

1

9

10 f g − f g − f g

''' 9

9

= 1 + h

'' 1 7 '' 3 ''

7 3 ''

9 1 ' 7 ' 3 '

9 '

1

32 9 , 6 22 , 4 9

10 f h + f g − f h − f g − f g − f h

'''

2

9

1 + k

=

5 ''

18 g

5

h −

7 ''

4

3

, f k −

'''

1 + g

11

=

'''

1 + h

11

=

'''

1 + k

11

=

7 ''

28 h

5

g −

7 ''

28 g

5

h −

'' 7

20 h

5

h −

9 ''

30 f

3

q −

,

(x ) U ( x )

9 '' 1 '' 5 5 ''

9 1 2 ' 5 '

9 '

1

18 18 9

10 f k + g − f k − g g − f k

''' 9 9 '' 1 7 '' 3

'' 5 5 ''

7 3 ''

9 1 ' 5 ' 5 '

7 ' 3 '

9 ' 1

9 4

, 22

18 6

, 9 36

32 10

j j f h f

h g h

f j

f h g h

f j

f

=

−

−

−

−

− +

+

''' 9 9 '' 1

'' 5 5 '' 7 3 ''

9 1 2 ' 5 '

7 ' 3 '

9 ' 1

9

22 18

6 , 9 18

32 10

q q f

h h k

f q

f h k

f q

f

=

−

−

−

− +

+

11 '' 1 ''

11 1 ' 11 '

1

11

12 f g − f g − f g

11 '' 1 9 '' 3 ''

9 3 ''

11 1 ' 9 ' 3 '

11 '

1

40 10 30 11

12 f h + f g − f h − f g − f g − f h

11 '' 1 7 '' 5 ''

7 5 ''

11 1 ' 7 5 '

11 '

1

48 20 28 11

12 f k + g g − f k − g g − g g − f k

''' 11 11 '' 1 9 '' 3

'' 7 5 ''

9 3 ''

11 1 ' 7 ' 5 '

9 ' 3 '

11 ' 1

11 30

20 10

48 40

12

j j f h f

g h h

f j

f g h h

f j

f

=

−

−

−

−

− +

+

''' 11 11 '' 1 9 '' 3

'' 7 5 ''

9 3 ''

11 1 ' 7 ' 5 '

9 ' 3 '

11 ' 1

11 30

20 10

48 40

12

q q f k f

h g k

f q

f h g k

f q

f

=

−

−

−

−

− +

+

''' 11 11 '' 1 9 '' 3 7

'' 5 7

'' 5 ''

7 5

'' 9 3 ''

11 1 ' 7 ' 5 '

7 ' 5 '

9 ' 3 '

11 ' 1

11 30

28 28

20

10 48

48 40

12

m m f j f h

h k

g k

g

j f m

f h g k

h j

f m

f

=

−

−

−

−

−

−

− +

+ +

''' 11 11 '' 1

7 '' 5 ''

7 5 ''

9 3 ''

11 1 ' 9 ' 3 '

11 ' 1

11

28 20

10 40

12

n n f

k h k

h q

f n

f q f n

f

=

−

−

−

−

− +

The

u y ) = 0

at

y = 0

and

lim u ( x , y

y

=

∞

→

h5

g

₇

h

₇

k

₇

g

₉

boundary conditios in terms of , , , , , , , , , , , , , , ,

, , , become

1

f

3

m

11

n f

11

g

5 11

9

k

h

₉

j

₉

q

₉

g

₁₁

h

₁₁

k

₁₁

= 0

j

11

q

at

η

;

'

0

11

=

= k

;

'

0

9

=

= j q

₉

;

'

0

11

=

= j j

9

j

11

;

'

0

9

=

= q g

₁₁

;

'

0

11

11

= q = m

q

;

'

0

11

=

= g h

₁₁

;

'

0

11

11

= m = n

;

'

0

11

=

= h k

₁₁

;

'

0

11

11

= n =

∞ η =

;

= 1

'

f

1

;

4

'

1

3

=

f ;

6

'

1

5

=

g h

₅^'

= 0 ; ; 8

'

1

7

=

g h

₇^'

= 0 ; k

₇^'

= 0 ;

;

= 0 k

₉^'

= 0 ; j

'

h

9 ₉^'

= 0 ; q

₉^'

= 0 ; ; 12

'

1

11

=

g h

₁₁^'

= 0 ; k

₁₁^'

= 0 ; j

₁₁^'

= 0 ;

;

= 0

₁₁^'

= 0 ;

'

q

11

n

₁₁^'

= 0 ;

and at

m

10 ;

'

1

9

=

g

Conclusion

We have derived the ordinary differential equations necessary to carry out numerical approximation to the solution of the boundary layer equation by 11-th order Blasius method.

References

[1] L. Prandtl, Über Flüssigkeitsbewegung bei sehr kleiner Reibung.

Proceedings 3rd Intern. Math. Congr. Heidelberg 1904, 484-491. Reprinted in Coll. Works II 575-584.

[2] H. Blasius, Grenzschichten in Flüssigkeiten mit kleiner Reibung. Z. Math.

u. Phys. 56, 1-37 (1908).

[3] L. Prandtl, Über Flüssigkeitsbewegung bei sehr kleiner Reibung.

Proceedings 3rd Intern. Math. Congr. Heidelberg 1904, 484-491. Reprinted in Coll. Works II 575-584.

[4] H. Blasius, Grenzschichten in Flüssigkeiten mit kleiner Reibung. Z. Math.

u. Phys. 56, 1-37 (1908).