Boundary Layer Finalppt

7/27/2019 Boundary Layer Finalppt

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Boundary Layer Theory

Presented by:

Sadashiv Jha

A-61

B.Tech(Mechanical)


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Contents

Boundary Layers


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Boundary LayersAs a fluid flows over a body, the no-slip condition ensures

that the fluid next to the boundary is subject to large shear. A

pipe is enclosed, so the fluid is fully bounded, but in an open

flow at what distance away from the boundary can we begin

to ignore this shear?

There are three main definitions of boundary layer thickness:

1. 99% thickness2. Displacement thickness

3. Momentum thickness


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99% ThicknessU

U is the free-stream velocity

(x)

x

y

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


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Displacement thickness

There is a reduction in the flow

rate due to the presence of the

boundary layer

This is equivalent to having atheoretical boundary layer with

zero flow

y

u

y

uU

U

d


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Displacement thicknessThe areas under each curve are defined as being equal:

0

dyuUq and Uq d

0

d dyU

u1

Equating these gives the equation for the displacement

thickness:


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Momentum thicknessIn the boundary layer, the fluid loses momentum, so

imagining an equivalent layer of lost momentum:

0 dyuUum andm

2

Um

0

m dyU

u1

U

u

Equating these gives the equation for the momentum

thickness:


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8

BOUNDARY LAYER ON A FLAT PLATE

Consider the following scenario.

1. Asteady potential flow has constant velocity U in the x direction.

2. An infinitely thin flat plate is placed into this flow so that the plate is

parallel to the potential flow (0 angle of incidence).

Viscosity should retard the flow, thus creating a boundary layer on either side

of the plate. Here only the boundary layer on one side of the plate is

considered. The flow is assumed to be laminar.

Boundary layer theory allows us to calculate the drag on the plate!

xy

U

U

u


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9A STEADY RECTILINEAR POTENTIAL FLOW HAS ZERO

PRESSURE GRADIENT EVERYWHERE

xy

U

U

u

plate

A steady, rectilinear potential flow in the x direction is described by the

relations

According to Bernoullis equation for potential flows, the dynamic pressure of

the potential flow ppd is related to the velocity field as

Between the above two equations, then, for this flow

0y

v,Ux

u,Ux

const)vu(21p 22pd

0y

p

x

p pdpd


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11NOMINAL BOUNDARY LAYER THICKNESS

xy

U

U

u

plate

Until now we have not given a precise definition for boundary layer thickness.

Here we use to denote nominal boundary thickness, which is defined to be

the value of y at which u = 0.99 U, i.e.

U99.0)y,x(uy

x

y

u

U

u = 0.99 U

The choice 0.99 is arbitrary; we could have chosen 0.98 or 0.995 or whatever

we find reasonable.


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12STREAMWISE VARIATION OF BOUNDARY LAYER

THICKNESS

Consider a plate of length L. Based on the estimate of Slide 11 of

BoundaryLayerApprox.ppt, we can estimate as

or thus

where C is a constant. By the same arguments, the nominal boundary thickness

up to any point x L on the plate should be given as

UL,)(~L

2/1ReRe

2/12/1

U

L

CorU

L

~

2/12/1

UxCor

Ux~

xy

U

U

u

plateL


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Laminar boundary layer growth

+ d

dy

x

y

Boundary layer => Inertia is of the same magnitude as Viscosity

a) Inertia Force: a particle entering the b.l. will be slowed from a velocity

U to near zero in time, t. giving force FIU/t. But u=x/t => t l/U

where U is the characteristic velocity and lthe characteristic length in the

x direction.

Hence FIU2/l

b) Viscous force: F/y 2u/y2U/2

since U is the characteristic velocity and the characteristic length in the

y direction

(x)


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Laminar boundary layer growth

Comparing these gives:

U

l

So the boundary layer grows according tol

Alternatively, dividing through by l, the non-dimensionalised

boundary layer growth is given by:

lRl

1

Note the new Reynolds number

characteristic velocity and

characteristic length

U

U llRl

)(U

5 Blasius

lU2/lU/2


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Boundary layer growth


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Length Reynolds Number

UlRl

l

U


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Flow at a pipe entry

l

U

d

If the b.l. meet while the flow is still laminar the flow in the pipe will be laminar

If the b.l. goes turbulent before they meet, then the flow in the pipe will beturbulent


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Boundary layer equations for laminarflowThese may be derived by solving the Navier-Stokes equations

in 2d.

0 yv

xu

dtdu

yu

xu

xp

1 2

2

2

2

Continuity MomentumU

Assume:

1. The b.l. is very thin compared to the length

2. Steady state


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Boundary layer equations for laminarflow

y

uv

x

uu

y

u

x

p

12

2

This gives Prandtls b.l. equation:

rate of change of u with

x is small compared to y

Blasius produced a perfect solution of these equations valid

for 0


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Blasius Solution

0

5

0 1

u/U

y'

y' f' (or u/U) f''

0 0 0.332

1 0.330 0.332

2 0.630 0.3233 0.846 0.267

4 0.956 0.161

5 0.992 0.064

6 0.999 0.002

7 1.000 0.000

l

Uy'y


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Laminar skin frictionThe shear stress at the surface can be found by evaluatingthe velocity gradient at the surface

00 y

u

The friction drag force along the surface is then found by

integrating over the length

dxy

ubF

0y0

f

l

where b is the breadth of the surface


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Laminar skin frictionFrom the Balsius solution, the gradient of the velocityprofile at y=0 yields the result:

0.5

x0 Rx

U

0.332

The shear force can be obtained by integration along the surface

0.5

0

0f R0.664UbdxbF l

l

The frictional drag coefficient can then be calculated

21

R33.1AU

FC

2

21

ff

l


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Force and momentum in fluid

mechanics - refresherNewtons laws still apply. Consider a stream

tube:

u1,A1

q1=u1A1

u2,A2

q2=u2A2

mass entering in time, t, is u1A1tmomentum entering in time, t, is m1 = (u1A1t)u1

momentum leaving in time, t, is m2 = (u2A2t)u2

Impulse = momentum change, F = (m2m1)/ t = (u22

A2-u12

A1)


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The von Karman Integral Equation(VKI)

A

B

C

D

Flow enters on AB and BC, and leaves on CD

1 2

2 -1

x

U

u1(y)u2(y)


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VKIThe momentum change between entering and leaving the control volume

is equal to the shear force on the surface:

122

0

2

1

0

2

20 Udyudyux12

(CD) (AB) (BC)

By conservation of fluid mass, any fluid entering the control volume must

also leave, therefore

12

0

1

0

212 )(

dyudyuU

12

0

1

2

1

0

2

2

20 dyUuudyUuux

Force onfluid


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VKI

0

2

0 dyUuuddx

As x 0, the two integrals on the right become closer and

the equation may be written as a differential:

0

2

0 dyU

u1

U

u

dx

dU

The integral is the definition of the momentum thickness, so

dx

dU m20

dx

dUUd if U(x)


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Turbulent boundary layers

The assumption is made that the flat plate approximates tothe behaviour in a pipe. The free stream velocity, U,

corresponds to the velocity at the centre, and the boundary

layer thickness, , corresponds to the radius, R.

1/7 Power Law

From experiments, one possibility for the shape of the

boundary layer profile is71

y

U

u

and measurements of the shear profile give

41

U

U0.0225 20


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Turbulent boundary layers

Putting the expression for the 1/7 power law into the

equations for displacement and momentum thickness

72

7

,8

md =99%

d

m


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Turbulent boundary layersdx

dU m20 becomes

dx

dU

72

7 20

Equating this to the experimental value of shear stress:

41

U

0.0225

d

d

72

7

x

Integrating gives:

5

1

Ux0.37x

The turbulent boundary grows as x4/5, faster than the

laminar boundary layer.


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Turbulent boundary layersMomentum thickness

51

Ux0.036x

72

7m

To find the total force, first find the shear stress

dx

dU m20

then integrate over the plate length

m

2

0

m2

0

0 Udx

dx

dUdxF

ll

f

For a plate of length, l, and width b,

51

UbU0.036F 2

l

lf51

0.074RC lf )10R10*5(

75 l


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Logarithmic boundary layerFrom the mixing length hypothesis it can be shown that

the profile is logarithmic, but the experimental values

are different from those in a pipe

yVln85.556.5

V

u *

*

and the friction coefficient llf R

A

Rlog

455.0

C 58.2

(A is a correction constant if part of the b.l. is laminar)

)10R0( 9 l ritrit

58.2

rit

RR

1.328

Rlog

455.0c

cc

A


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Quadratic approximation to the

laminar boundary layer

2

y

y2U

u

Remember - boundary layer theory is only applicable inside

the boundary layer.

This is sometimes written with =y/ and F()=u/U as

2

2F It provides a good approximation to the shape of the

laminar boundary layer and to the shear stress at the surface


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Turbulent Boundary Layer


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Laminar Sub-Layer


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