von karman integral method (bsl)

IIT-Madras, Momentum Transfer: July 2005-Dec 2005

0yxVV

x y

2

2

yx xx y

VV VdPV V

x y dx y

Von Karman Integral Method (BSL)

PRANDTL BOUNDARY LAYER EQUATIONS for steady flow are

Continuity

N-S (approx)

1

2

If we solve these, we can get Vx, (and hence .

Alternative: We can integrate this equation and obtain an equation in and shear stress



If we assume a rough velocity profile (for the boundary layer), we can get a fairly accurate relationship

Integration is ‘tolerant’ of changes in shape

For all the above 3 curves, the integration (area under the curve) will provide the same result (more or less), even though the shapes are very different


0yxVV

x y

2

2

yx xx y

VV VdPV V

x y dx y


Prandtl equations for steady flow are

Continuity

N-S (approx)

What is Vy?

0

yx

y

VV dy

x

1

2

VdPV

dx x

Pressure gradient (approx)

3a

3b


2

20

yx x x x

x

V V V VVV dy V

x x y x y

Substitute (3a) and (3b) in (2)


4

0 0 0 00

yx x x x

x

V V V VVV dy dy dy V dy

x y x x y

Integrate (4) with respect to y, from 0 to infinity

5


0x

y

V

y

0

0

x

y

V

y

0

yxV

dyx

F

xdV

dx

F y

xG V

xdV

dy

G y

Integration by Parts. Let


Eqn. 5: On the RHS

0x

y

V

y

Eqn 5: On the LHS, for the marked part



0 0

x xx

V VV dy V dy

x x

0 00

yxV Vx

dy dy F dGy x

0

0

F G G dF

This is for the marked region in LHS of Eqn 5

0

xVF dy

x

xG V V

0

00 0xV

F dyx

0 0 0xG V



000 0

x x xx x

V V V VV V dy V dy V dy

x x x x

1. To equation (6), add and subtract0

x

VV dy

x

Note : and areindepent ofV

V yx

Substituting in equation (5)

6

To write equation (6) in a more meaningful form:


2. Note 2

2x x

x

V VV

x x

xx x

VVV V V V

x x x

0 00

0

0 0

2 xx

x

xx

V VV dy V

VV dy

x

V VV dy V dy

x x

dyx x

7


3. Also

... and multiply both sides by -1


Combining the above two

2x x x xV V V V V V

x x

2 x xx x

V VVV V V

x x x


2 x xx x x x

V VVV V V V V V

x x x x


0

0 0

x x x

VV V V dy V V dy

x x

First term is momentum thickness

Second term is displacement thickness

(Note: The density term is ‘extra’ here)

Note : and areindepent ofV

V yx


Equation (7) becomes

Note: Integral method is not only applied to Boundary Layer. It can be applied for other problems also.


Example

Assume velocity profile

It has to satisfy B.C.0 0xV at y

xV V at y 0xV

at yy

For zero pressure gradient

2

20

0x

y

V P

y x

33 1

2 2x

y yV V

For example, use



sinxV a by

2b

2b sin

2x

yV a

cos 0y

by


Or for example, use

What condition should we impose on a and b?

What is the velocity gradient at y= ?

0x

y

V

y


xV V at y

sin2xV a a


What is the velocity at y= ?

a V

Check for other two Boundary Conditions

For zero pressure gradient

OK

sin2x

yV V

OK

0 0xV at y

2

20

0x

y

V P

y x

No slip condition


2

0 0

( ) sin 1 sin2 2x x

y yV V V V dy

2 2 1

2V

Von Karman equation gives

2 2 1

2 2

VV

x

2 2( )

(4 )d dx

V

0 2x

y

VV

y

0V

x

Now, to substitute in the von Karman Eqn, find shear stress

Also


2 22 2

(4 )

x

V x

2

(4 )x xV

2

(4 R) exx

4.7

Rexx


Calculation for comes out ok

Calculation for Cf also comes out ok

0

212

fxV

C

0

0

2

1

12

fL

L

dx

CL

V

Even if velocity profile is not accurate, prediction is tolerable


Now numerical method are more common

( . ) x

s V

F V n dA V dV

01. . .1.2

x x xx x x x x x x

P PF P P P x

2 2

0 0

.x x x x x top

s

V dy V dy V m

0t

Conservation of mass

Von Karman Method (3W&R)


0t

Conservation of mass

.V n dA dVt

00

s

0 0

. . 0x x xx x topV dy V dy m

0 0

. .x x xtop x xm V dy V dy

Von Karman Method


Substitute , rearrange and divide by x

2

0 0

o x x

dP d dV dy V V dy

dx dx dx

0VdP

Vdx x

Outside B.L.

2dP d dV V V

dx dx dx


0

0 0

x x x

d dV V V V V V dy

dx dx

If is constV

If we assume 2 3xV a by cy dy

0

0

x x

dV V V dy

dx


0a

3

2b V

0c

32

Vd

0 0xV at y

xV V at y

0xVat y

y

2

20

0x

y

V P

y x


0

2

0.6461 Re2

fx

x

CV

1.272

RefL

L

C

4.64

Rexx

140

13

dxd

V

von karman integral method (bsl)

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