Louisiana Tech University Ruston, LA 71272 Boundary Layer Theory Steven A. Jones BIEN 501 Friday,...

Louisiana Tech UniversityRuston, LA 71272

Boundary Layer Theory

Steven A. Jones

BIEN 501

Friday, April 11 2008


Momentum Balance

Learning Objectives:1. State the motivation for curvilinear coordinates.2. State the meanings of terms in the Transport Theorem3. Differentiate between momentum as a property to be transported

and velocity as the transporting agent.4. Show the relationship between the total time derivative in the

Transport Theorem and Newton’s second law.5. Apply the Transport Theorem to a simple case (Poiseuille flow).6. Identify the types of forces in fluid mechanics.7. Explain the need for a shear stress model in fluid mechanics.

The Stress Tensor.Appendix A.5Show components of the stress tensor in Cartesian and cylindrical

coordinates. Vectors and Geometry


Flow Over a Flat Plate

0U 0U

is small

changes slowy in the direction

changes rapidly in the direction

y

x

x

v

v x

v y

What can we say about this flow?

x

x

y – Boundary layer thickness


Flow Over a Body

x

yU

U0


Continuity & Momentum

Use 2-dimensional equations.

2 2

2 2

2 2

2 2

0 continuity

momentum

( momentum)

yx

x x x xx y

y y y yx y

vv

x y

v v v vPv v x

x y x x y

v v v vPv v y

x y y x y

In contrast to previous derivations, we do not say that terms are zero. We say that they are “small.”


Continuity

Non-dimensionalize so that velocities and lengths are of order 1.

L

UV

y

vV

x

v

L

U

y

Lx

Vy

Ux

o

yx

:conclude

continuity

:directiontheinLengthsticCharacteri

:directiontheinLengthsticCharacteri

:directiontheinVelocitysticCharacteri

:directiontheinVelocitysticCharacteri

0*

*

*

*0

0


Exercise

If: 0Characteristic Velocity in the direction:

Characteristic Velocity in the direction:

Characteristic Length in the direction:

Characteristic Length in the direction:

o

x U

Uy V

Lx L

y

How do you non-dimensionalize:

2 2

2 2, , , ?x x x x

x y

v v v vv v

x y x y


x-Momentum

Non-dimensionalize

2 * * 2 * 2 ** 2* *0 0

* * * 2 2 *2 *2

2 * * 2 *** *0

* * * *20

x x x xx y

2x x x

x y

U v v U v vPv v

L x y L x L x y

U v v vPv v

L x y U L x y

P

P

small

21

21

0

Re

LL

U

L

If inertial terms are important

with respect to viscous terms.


Characteristic Pressure

Non-dimensionalize

20

2

0

ULU

LU

0

P

P 2

important is pressure if 1,orderofis

andsince 21

21

0

Re

LL

U

L


y-momentum

From y-momentum

* * 2 * 2 *2 2 * 2*0 0 0

2 * * * 2 2 2

y y y yx y

v v v vU U UPv v

L x y y L L x y

2 2

2 2

1y y y yx y

v v v vPv v

x y y x y

2 2 * * ***

* *2 * *

1y y yx y

v v vPv v

y L y x y

Small in comparison to P

x


Characteristic Pressure

From y-momentum

21constant

2P x U x

0*

y

P

P is not a function of y, so the pressure in the boundary layer is the pressure in the free stream, which can be determined from a potential flow solution.

Take the derivative with respect to x to get something to plug into the x-momentum equation.

dU xdPU x

dx dx


Final x-momentum, B.C.s

2 * * 2 *** *0

* * * *20

2x x x

x y

U v v vPv v

L x y U L x y

P

2

2x x x

x y

dU xv v vv v U x

x y dx y

0 at 0

at

0 at

x

x

x

v y

v U x y

dvy

dy


Flow Over a Flat Plate

0U 0U

x

x

x

0

dU xdPU x

dx dx

Velocity at large y does not depend on x.


Special Case: Flat Plate

2

2x x x

x y

v v vv v

x y y

0 at 0

at

0 at

x

x

x

v y

v U x y

dvy

dy

Use the stream function:

,x yv vy x


Special Case: Flat Plate2 2 3

2 3y x y x y y

Assume a similarity solution:

0,Uy

x y f f y fx

2 20 0 0 0 0

3 3 3 2

1, ,

2 2 2

U U U U Uf y f f y f

y x x x x y x x x

32 2 3 32

0 02 2 3 3

,U Uf f

y x y x



2 2 2

2 2

1, ,

2 2 2

f y f f y f

y y x x x y x x

32 2 2 3 3

2 2 2 3,f f

y y y y

2 20 0 0 0 0

3 3 3 2

1, ,

2 2 2

U U U U Uf y f f y f

y x x x x y x x x

3

2 2 3 320 0

2 2 3 3,

U Uf f

y x y x

0Uyx



32 2 2 2 3

2 2 2 2 3

1

2 2 2

f f y f f f f

y x x y y

2 2 3

2 3y x y x y y

2 2 2

2 2

1, ,

2 2 2

f y f f y f

y y x x x y x x

32 2 2 3 3

2 2 2 3 3,f f

y y y y


Final Differential Equation

3 2

3 20

2

d f f d f

d d

Or, more simply

12 0f ff

This equation is 3rd order and nonlinear. There is no closed form solution, but it can be solved numerically.


Final Differential Equation

Remember that f is the stream function, so you must take derivatives to get velocity after you solve for f.

0

1

2

3

4

5

6

7

0.0 0.5 1.0 1.5

Velocity/Uinfinity

x-velocity for flow over a flat plate.

,x y

f fv v

y x


Integral Momentum Boundary

Instead of seeking an exact solution to the differential equation, we can integrate the equations for continuity and x-momentum.

2

2

2

20 0

x x xx y

x x xx y

dU xv v vv v U x

x y dx y

dU xv v vv v dy U x dy

x y dx y

0

0yx xy

vv vv dy

x y x



The terms in these equations have specific meanings.

2

20 0 0 0

x x xx y

dU xv v vv dy v dy U x dy dy

x y dx y

2

200

0x xx x

y

v vv vdy

y y y y

Becomes zero because velocity becomes constant far from the boundary.

Wall shear stress



The terms in these equations have specific meanings.

2

20 0 0 0

x x xx y

dU xv v vv dy v dy U x dy dy

x y dx y

0

xy

vv dy

y

Integrate by parts with , x

y

vu v dv dy

y

00 0

yxy x y xy

vvv dy v v v dy

y y



vx becomes U far from the boundary, and vy() is obtained from the integrated continuity equation.

0

xx y

vv v U dy

x

00 0

yxy x y xy

vvv dy v v v dy

y y

0 0

0 0 yxy x y x y x

vvv dy v v v v v dy

y y

Both velocities are zero at the wall.



Because by Continuity:

0 0 0

0 0

yx xy x

x xx

vv vv dy U y v dy

y x y

v vU y v dy

x x

yxvv

x y



In summary:

0 0 0

x x xy x

v v vv dy U y v dy

y x x

2

20 0

x x xx y

dU xv v vv v dy U x dy

x y dx y

2

20

xw

vdy

y

So:

02 x x

w x

v v Uv U U dy

x x x



With:

If we knew vx, we would be able to integrate for the wall shear stress. It turns out that the exact form of vx is not as important as one might think, and good results can be obtained with a form that looks reasonable and satisfies the boundary conditions.

02 x x

w x

v v Uv U U dy

x x x

Louisiana Tech University Ruston, LA 71272 Boundary Layer Theory Steven A. Jones BIEN 501 Friday,...

Documents

Transcript of Louisiana Tech University Ruston, LA 71272 Boundary Layer Theory Steven A. Jones BIEN 501 Friday,...