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### Transcript of Boundary Layer Theory_2

• CE F312 Hydraulics Engineering 1

STANDARD BOUNDARY LAYER THICKNESS

That distance from the boundary surface (or

plate) upto which the velocity reaches 99% of the

velocity of the main stream.

y = where u = 0.99U

Somewhat an easy to understand

but arbitrary definition as the limit

of boundary layer is not easily

defined13

99% THICKNESS

U

U is the free-stream velocity

(x)

x

y

(x) is the boundary layer thickness when u(y) ==0.99U

14

• CE F312 Hydraulics Engineering 2

For greater accuracy the boundary layer

thickness is defined in terms of certain

mathematical expressions which are the

measures of the effect of boundary layer on the

flow.

Displacement thickness (*)

Momentum thickness ()

Energy thickness (E)

THICKNESS OF BOUNDARY LAYER

15

DISPLACEMENT THICKNESS (*)

The boundary layer retards the fluid, so that the mass flux is less

than it would be in the absence of the boundary layer.

Thus the displacement thickness is the distance by which the

boundary surface would have to be moved so that the actual

discharge deficit would be same as that of an ideal (or

frictionless) fluid past the displaced boundary.

Amount of fluid

being displaced

outward

*

Equal

Areas

U-u

16

• CE F312 Hydraulics Engineering 3

DISPLACEMENT THICKNESS (*)

=

0

*dy

U

u1

Uq *=

The areas under each curve are defined as being equal:

( )

=0

dyuUq

and

Equating these gives the equation for the displacement

thickness:

The loss due to boundary layer

17

MOMENTUM THICKNESS ()

The distance from the actual boundary surface

such that the momentum flux corresponding to

the main stream velocity V through this distance

is equal to the deficiency or loss in momentum

due to the boundary layer formation.

18

• CE F312 Hydraulics Engineering 4

MOMENTUM THICKNESS ()

Um 2=

=

0

dyU

u1

U

u

In the boundary layer, the fluid loses momentum, so imagining an

equivalent layer of lost momentum:

( )

=0

dyuUum and

Equating these gives the equation for the momentum thickness:

19

ENERGY THICKNESS (E)

The distance from the actual boundary surface

such that the energy flux corresponding to the

main stream velocity U through this distance (E)is equal to the deficiency or loss of energy due to

the boundary layer formation

20

• CE F312 Hydraulics Engineering 5

ENERGY THICKNESS (E)

3 2 2

0

1 1( )

2 2EU U u udy

=

2

2

0

1Eu u

dyU U

=

21

SOLVE! Air flowing into a 2-ft-square duct with a uniform velocity of 10 ft/s

forms a boundary layer on the walls as shown in Fig. The fluid withinthe core region (outside the boundary layers) flows as if it were inviscid.From advanced calculations it is determined that for this flow theboundary layer displacement thickness is given by

where * and x are in feet. Determine the velocity U = U(x) of the air within the duct but outside of

the boundary layer.

( ) 2/1* 0070.0 x=

22

• CE F312 Hydraulics Engineering 6

SOLUTION

The volume flow rate across any section of the duct is equal to

that at the entrance (i.e., Q1=Q2). That is

According to the definition of the displacement thickness, the

flowrate across section (2) is the same as that for a uniform flow

with velocity U through a duct whose walls have been moved

inward by *

( ) ===)(

3 /ft ft ft/s 2

2

1140210 udAsAU

( ) ( )

( )ft/s

.

.*/ft

/

/

)(

3

221

2212

2

007001

10

00700142240

xU

xUUudAs

=

===

23

PROBLEM

Assuming that the shear stress distribution in a

laminar boundary layer is such that

Calculate the displacement and momentum

thickness of this boundary layer in terms of .

=

y

10

24

• CE F312 Hydraulics Engineering 7

BOUNDARY LAYER EQUATIONS

Equations of continuity and motion for the

steady flow of an incompressible, inviscid fluid in

2-D without body forces are:

1v v pu vx y y

+ =

CONTINUITY EQUATION

EULERS EQUATION IN X DIR.

EULERS EQUATION IN Y DIR.

0u v

x y

+ =

1u u pu vx y x

+ =

Chapter 6, Sec 6.6

Chapter 7, Sec 7.3

25

If a viscous fluid is considered, eqn. of continuity

remains unchanged

Additional terms due to viscous stresses will be

introduced in the eqns. of motion

BOUNDARY LAYER EQUATIONS

+ d

y

x

y

x

Edge of

boundary layer

( ) x y xy

+ = +

Net force = x yy

26

• CE F312 Hydraulics Engineering 8

1 1u u pu vx y x y

+ = +

BOUNDARY LAYER EQUATIONS

u

y

=

Simplification.. Since the boundary layer is thin, it is expected that the component

of velocity normal to the plate is much smaller than the parallel tothe plate and that the rate of change of any parameter across theboundary layer should be much greater than that along the flowdirection. That is

Again2

2

1u u p uu vx y x y

+ = +

v