1

PHAROS UNIVERSITYME 253 FLUID MECHANICS II

Boundary Layer (Two Lectures)

Flow Past Flat Plate• Dimensionless numbers involved Re Ma Fr

Ul U U

c gl

• for external flow: Re>100 dominated by inertia, Re<1 – by viscosity

Boundary Layer

Flows over bodies. Examples include the flows over airfoils, ship hulls, etc.

Boundary layer flow over a flat plate with no external pressure variation.

laminar turbulenttransition

Dye streakU U U

U

Boundary layer characteristicsfor large Reynolds number flow can be dividedinto boundary region where viscous effect are important and outside region where liquid can be treated as inviscid Rex

Ux

The Boundary-Layer Concept

The Boundary-Layer Concept

Boundary Layer Thicknesses

Boundary Layer Thicknesses

• Disturbance Thickness,

Displacement Thickness, *

Momentum Thickness,

Boundary Layer Thickness

Boundary layer thickness is defined as the height above the surface where the velocity reaches 99% of the free stream velocity.

10

Boundary Layer(BL)

Three Thicknesses of a Boundary Layer

d*

1

11

Displacement Thickness *

Volume flux:0

udy Q

Ideal flux: 1Q U 10

udy U

1 0

udy

U

*1 0

(1 )u

dyU

12

Velocity Distribution

U

SolidBoundary

Equivalent Flow Rate

U

Velocity Defect

VelocityDefect

*

Ideal FluidFlow

13

Eqn. for Displacement Thickness

• By equating the flow rate for velocity defect to flow rate for ideal fluid

–

• If density is constant, this simplifies to

–

* would always be smaller than

0

* dyuUU

0

* 1 dyU

u

Displacement Thickness Laminar B.L.

15

Eqn. for Momentum Thickness

• By equating the momentum flux rate for velocity defect to that for ideal fluid

–

• If density is constant, this simplifies to

–

would always be smaller than * and

0

2 uUudyU

01 dyU

u

U

u

Momentum ThicknessThe rate of mass flow across an element of the boundary layer is ( u dy) and the mass has a momentum ( u2 dy ) The same mass outside the boundary layer has the momentum ( u ue dy)

is a measure of the reduction in momentum transport in the B. Layer

Empirical Equations of Laminar B. Layer Parameters

• Boundary Layer Thickness

• Momentum Thickness

• Displacement Thickness

• Skin Friction Coefficient

Skin Friction Coefficient

Boundary layer characteristics• Boundary layer thickness

• Boundary layer displacement thickness: *

0

1u

dyU

• Boundary layer momentum thickness (defined in terms of momentum flux):

2

0 0

1u u

bU b u U u dy bU U

Drag on a Flat Plate

• Drag on a flat plate is related to the momentum deficit within the boundary layer

2

0

2

1

w

u ubU b

U U

db bU

dx x

D

D

• Drag and shear stress can be calculated by assuming velocity profile in the boundary layer

Boundary Layer Definition Boundary layer thickness (d): Where the local velocity reaches to 99% of the free-stream velocity. u(y=d)=0.99U Displacement thickness (δ*): Certain amount of the mass has been displaced (ejected) by the presence of the boundary layer ( mass conservation). Displace the uniform flow away from the solid surface by an amount δ*

0 0* ( ) , or * (1 )

uU w U u wdy dy

U

Amount of fluid being displaced outward

*

U-u

equals

Laminar Flat-PlateBoundary Layer: Exact Solution

• Governing Equations

Laminar Flat-PlateBoundary Layer: Exact Solution

• Boundary Conditions

Laminar Flat-PlateBoundary Layer: Exact Solution

• Results of Numerical Analysis

25

MOMENTUM INTEGRAL EQN

• BOUNDARY LAYER EQUATIONS

• BOUNDARY CONDITIONS y=0, u=v=0 & y = δ , u= U

• Integrating the momentum equation w.r.t y in the interval [0,δ]

0

y

v

x

u

2

2

)(y

u

dx

dUU

y

uv

x

uu

26

MOMENTUM INTEGRAL EQN

dyy

udy

dx

dUUdy

y

uvdy

x

uu

0

2

2

000

dyy

udy

dx

dUUdy

y

vuUvdy

x

uu y

0

2

2

00

0

0

)(

000

0

0

)(

y

y y

udy

dx

dUUdy

x

uuUvdy

x

uu

0000

2

y

y

udy

dx

dUUdy

x

uUdy

x

uu

dyy

udy

dx

dUUdy

y

uvdy

x

uu

0

2

2

000

dyy

udy

dx

dUUdy

y

vuUvdy

x

uu y

0

2

2

00

0

0

)(

dyy

udy

dx

dUUdy

y

uvdy

x

uu

0

2

2

000

dyy

udy

dx

dUUdy

y

vuUvdy

x

uu y

0

2

2

00

0

0

)(

dyy

udy

dx

dUUdy

y

uvdy

x

uu

0

2

2

000

dyy

udy

dx

dUUdy

y

vuUvdy

x

uu y

0

2

2

00

0

0

)(

dyy

udy

dx

dUUdy

y

uvdy

x

uu

0

2

2

000

27

MOMENTUM INTEGRAL EQN

dyU

u

dx

dUUdy

U

u

dx

dUdy

U

uU

dx

dUudy

dx

dU

dyU

u

dx

dUUdy

U

u

dx

dUdy

U

uU

dx

ddyu

dx

d

dydx

dUUdyu

dx

dUdyu

dx

d

00

2

00

0

22

0

2

0

22

0

2

0

000

2

2

)( 00 )(

yy

u

dyUuUu )/1(/0

UdyuU /)(*0

DISPLACEMENT THICKNESS

MOMENTUM THICKNESS

28

MOMENTUM INTEGRAL EQN

0

00

2

0

0000

2

0

000

2

0

000

2

0

22

0

2

0

000

2

112

12

121

2

)(

dyU

u

dx

dUUdy

U

u

U

u

dx

dUU

dx

dU

dyU

u

dx

dUUdy

U

u

dx

dUUdy

dx

dUUdy

U

u

U

u

dx

dUU

dx

dU

dydx

dUUdy

U

u

U

u

dx

dUUdy

U

u

U

u

dx

dU

dydx

dUUdy

U

u

dx

dUUdy

U

u

dx

dUdy

U

u

dx

dUUdy

U

u

dx

dU

dydx

dUUdyu

dx

dUdyu

dx

d

/)2( 0

*2 dx

dUU

dx

dU

29

MOMENTUM INTEGRAL EQN

• VON KARMAN MOMENTUM INTEGRAL EQUATION

/)2( 0

*2 dx

dUU

dx

dU

30

KARMAN POHLHAUSEN METHOD FOR FLOW OVER FLAT PLATE

/)2(

0

0

2

1

0*2

2

dx

dUU

dx

dU

dx

dUU

dx

dpdx

dUU

dx

dp

constUp BERNOULLI’S EQUATION

xU

2

21 2

0REVISED KARMAN EQUATION

FOR NO EXTERNAL PRESSURE

NO IMPOSED PRESSURE

VON KARMAN EQUATION

31

KARMAN POHLHAUSEN METHOD FOR FLOW OVER FLATE PLATE

• Dimensionless velocity u/U can be expressed at any location x as a function of the dimensionless distance from the wall y/δ.

• Boundary (at wall) conditions y=0, u=0 ; d2u/dy2 = 0• At outer edge of boundary layer (y=δ) u=U, du/dy=0• Applying boundary conditions a=0,c=0, b=3/2, d= -1/2

32

y

dy

cy

bay

U

u

3

2

1

2

3

yy

U

uVelocity profile

32

KARMAN POHLHAUSEN METHOD FOR FLOW OVER FLATE PLATE

3

2

1

2

3

yy

U

u

xU

2

21 2

0

INSERT INTO THIS EQUATION

33

KARMAN POHLHAUSEN METHOD FOR FLOW OVER FLATE PLATE

xU

dyyyyyy

xU

dyyyyy

xU

dyU

u

U

u

xU

xU

2

0

64322

0

32

0

220

280

39

4

1

2

3

2

1

4

9

2

3

2

1

2

3*

2

1

2

31

1

34

KARMAN POHLHAUSEN METHOD FOR FLOW OVER FLATE PLATE

• At the solid surface, Newton’s Law of Viscosity gives:

xU

U

therefore

U

yU

yU

yy

u

yy

2

0

3

0

0

280

39

2

3

2

3

2

1

2

3

35

KARMAN POHLHAUSEN METHOD FOR FLOW OVER FLATE PLATE

Re

64.4

*13

2*140

0,0,0

13

140

2

13

140

2

x

xUx

Cx

CU

x

xU

eRx

0.5

KARMAN POHLHAUSEN SOLUTION

BLASIUS SOLUTION

36

Boundary Layer Parameters• BOUNDARY LAYER THICKNESS INCREASES AS THE

SQUARE ROOT OF THE DISTANCE x FROM THE LEADING EDGE AND INVERSELY AS SQUARE ROOT OF FREE STREAM VELOCITY.

• WALL SHEAR STRESS IS INVERSELY PROPORTIONAL TO THE SQUARE ROOT OF x AND DIRECTLY PROPORTIONAL TO 3/2 POWER OF U

• LOCAL & AVERAGE SKIN FRICTION VARY INVERSELY AS SQUARE ROOT OF BOTH x & U

Use of the Momentum Equation for Flow with Zero Pressure Gradient

• Simplify Momentum Integral Equation(Item 1)

The Momentum Integral Equation becomes

Use of the Momentum Equation for Flow with Zero Pressure Gradient

• Laminar Flow– Example: Assume a Polynomial Velocity Profile

(Item 2)

• The wall shear stress w is then (Item 3)

Use of the Momentum Equation for Flow with Zero Pressure Gradient

• Laminar Flow Results(Polynomial Velocity Profile)

Compare to Exact (Blasius) results!

Use of the Momentum Equation for Flow with Zero Pressure Gradient

• Turbulent Flow– Example: 1/7-Power Law Profile (Item 2)

Use of the Momentum Equation for Flow with Zero Pressure Gradient

• Turbulent Flow Results(1/7-Power Law Profile)

Example

Assume a laminar boundary layer has a velocity profile as u(y)=U(y/ for 0y and u=U for y>, as shown. Determine the shear stress and the boundary layer growth as a function of the distance x measured from the leading edge of the flat plate.

u(y)=U(y/

u=U y

x

2

w 0

0 0

U

For a laminar flow ( ) from the profile.

Substitute into the definition of the momentum thickness:

U y(1 ) (1 ) , since u

.6

w

y

d

dxUu

y

u u y ydy dy

U U

2 2

2 2

x

3 2

1U , U

66 12

Separation of variables: , integrate 12( ) ,U U U

13.46 3.46 , where Re

Re

3.46 ,

0.289 10.289 ,

Re

w

x

w w

x

Ud d

dx dx

d x xx

U x

x U x

xx

U

U U U

x x

Note: In general, the velocity distribution is not a straight line. A laminar flat-plate boundary layer assumes a Blasius profile. The boundary layer thickness and the wall shear stress w behave as:

(9.14). ,Re

332.0.(9.13) ,

Re

0.50.5 2

x

w

x

Ux

xU

Laminar Boundary Layer Development

x( )

x0 0.5 1

0

0.5

1• Boundary layer growth: x• Initial growth is fast• Growth rate d/dx 1/x, decreasing downstream.

w x( )

x0 0.5 1

0

5

10

• Wall shear stress: w 1/x• As the boundary layer grows, the wall shear stress decreases as the velocity gradient at the wall becomes less steep.

45

Momentum Integral Relation for Flat-plate BL

xx

y

hU

CV

Stream line

P

d

Free stream

U=const P=const z b

Steady & incompressible

46

Momentum Integral Relation for Flat-plate BL

:X Outlet2

0b u dy

Inlet2b u h

.2 2

0M bU h b u dy

2 2

0D bU h b u dy

Continuity0

hU udy

0

uh dy

U

0( )D b u U u dy

2bU 2dD d

bUdx dx

47

Meantime

0( ) ( )

x

wD x b x dx

w

dDb

dx

2w

dU

dx

For flat plate boundary layer

U const 0dU

dx

48

2wd

dx U

' '

K a m a n

0(1 )

u udy

U U

2u a by cy

0 0y u y u U 0u

y

2

20, , -

U Ua b c

2

2

2u y y

U 0 ( )y x

49

2 2

2 20

2 2( )(1 )

y y y ydy

y

令 ＝

1 2 2

0(2 )(1 2 )dy

2

15

2

15

d d

dx dx

2d

dx U

2|w y o

u U

y

15d x

U

50

0, 0x

21 15

2x

U

5.5 5.5x

xU Ux

5.5

Rex x

*( ) 1.83

Re

x

x x

( ) 0.74

Re

x

x x

*3 7.5 *

H

Shape factor

51

2

0.73

1 2w

f

x

CU Re

Skin-friction coefficient

0

l

f wX bdx

21 2f

D

XC

U bl Drag coefficient

52

Boundary Layer Equation

Inviscid

53

3 Boundary Layer Equation

54

55

Boundary Layer Equation

2-D,steady,incompressible,neglect body force

0u v

x y

2 2

2 2

1( )

u u p u uu v

x y x x y

2 2

2 2

1( )

v v p v vu v

x y y x y

1[ ]Re

[ ]L

[ ]L

[ ]L

[ ]L

[1][1] [1] [1]

56

For BL, 1L

2

2

1( )

u u p uu v

x y x y

0u v

x y

0p

y

( )P P x

dp

dx

External flow U U 0V (Inviscid Flow)

57

Euler Equation1dU dP

Udx dX

2

2

u u dU uu v U

x y dx y

2

2( )

u u

y y y

1( )

u

y y

1

y

u

y

____

' 'uu v

y

0, 0y u v , ( )y u U x

Laminar flow

Turbulent flow

58

Blasius 1908

u

U ( )

y y

x x U

'( ) ( )u

fU

( ) ( )f d '' '1

( ( ) ) 02

d

''' ''1( ) 0

2f f f

0, 0y '0, ( ) 0u f

, by ', 1u U f

59

5.0

xx Re

* 1.721

xx Re

0.664

xx Re

|w o

u U

y x U

''| (0)o

f Uf

x U

2

0.66412

wf

x

CReU

1.328

ReD

L

C

60

U =10m/s, =17x10-6 m2/s

DISPLACEMENT AND MOMENTUN THICKNESS

• Typical distribution of , * and

Influence of Adverse Pressure GradientAdverse Pressure Gradient dp/dx>0 can cause flow separation

Pressure Gradients in Boundary-Layer Flow

Boundary Layer and separation

gradient pressure

favorable ,0

x

P

gradient no ,0

x

P

0, adverse

pressure gradient

P

x

Flow accelerates Flow decelerates

Constant flow

Flow reversalfree shear layerhighly unstable

Separation point

Flow Separation

SeparationBoundary layer

Wake

Stagnation point

21/ 2

P PCp

U

Inviscid curve 21 4sinCp

Turbulent

Laminar

1.0

0

-1.0

-2.0

-3.0

Drag Coefficient: CD

Supercritical flowturbulent B.L.

Stokes’ Flow, Re<1

Relatively constant CD

Drag

• Drag Coefficient

with

or

Influence of Adverse Pressure GradientAdverse Pressure Gradient dp/dx>0 can cause flow separation

WHY DOES BOUNDARY LAYER SEPARATE?• Adverse pressure gradient interacting with velocity profile through B.L.• High speed flow near upper edge of B.L. has enough speed to keep moving

through adverse pressure gradient• Lower speed fluid (which has been retarded by friction) is exposed to same

adverse pressure gradient is stopped and direction of flow can be reversed• This reversal of flow direction causes flow to separate

– Turbulent B.L. more resistance to flow separation than laminar B.L. because of fuller velocity profile

– To help prevent flow separation we desire a turbulent B.L.

69

EXAMPLE OF FLOW SEPARATION

• Velocity profiles in a boundary layer subjected to a pressure rise– (a) start of pressure rise– (b) after a small pressure rise– (c) after separation

• Flow separation from a surface– (a) smooth body– (b) salient edge

70

BOUNDARY LAYER SEPARATION

• Separation takes place due to excessive momentum loss near the wall in a boundary layer trying to move downstream against increasing pressure, i.e., which is called adverse pressure gradient.

dx

dp

y

u

dx

dp

dx

dUU

y

u

y

ydx

dUU

y

uv

x

uu

wall

wallwall

1

1)(

2

2

2

2

MOMENTUM EQUATION

AT WALL v=u=0

71

BOUNDARY LAYER SEPARATION

• In adverse gradient, the second derivative of velocity is positive at the wall, yet it must be negative at the outer layer (y=δ) to merge smoothly with the mainstream flow U(x).

• It follows that the second derivative must pass through zero somewhere in between, at the point of inflexion, and any boundary layer profile in an adverse must exhibit a characteristic S –shape.

72

BOUNDARY LAYER SEPARATION

• In FAVOURABLE GRADIENT, profile is rounded, no point of inflexion, no separation.

• In ZERO PRESSURE GRADIENT, point of inflexion is at the wall itself. No separation.

• In ADVERSE GRADIENT, point of inflexion (PI) occurs in the boundary layer, its distance from the wall increasing with the strength of the adverse gradient

• CRITICAL CONDITION is reached where the wall shear is exactly zero (∂u/∂y =0). This is defined as separation point

0w SHEAR STRESS AT WALL IS ZERO

73

BOUNDARY LAYER SEPARATION

74

BOUNDARY LAYER SEPARATION• The mathematical explanation of flow-separation :

– The point of separation may be defined as the limit between forward and reverse flow in the layer very close to the wall, i.e., at the point of separation

– This means that the shear stress at the wall, .But at this point, the adverse pressure continues to exist and at the downstream of this point the flow acts in a reverse direction resulting in a back flow.

EXAMPLE: SLATS AND FLAPS