Topic 1 Surface Resistance Part II

7/27/2019 Topic 1 Surface Resistance Part II

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Surface Resistance

Part II


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Fully Developed Flow between Parallel

Plates Using the Navier-Stokes Equations


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Navier-Stokes Equations without Body Forces

( ) ( ) ( )Continuity: 0

u v w

t x y z

+ + + =

( ) ( ) ( ) ( )2 1x-momentum:

Re

xyxx xzuu uv uw p

t x y z x x y z

+ + + = + + +

( ) ( ) ( ) ( )2 1y-momentum:

Re

xy yy yzvv uv vw p

t x y z y x y z

+ + + = + + +

( ) ( ) ( ) ( )2 1z-momentum:

Re

yzxz zzww uw vw p

t x y z z x y z

+ + + = + + +

( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

Energy:

1

RePr

1

Re

T T T T

yx z

xx xy xz xy yy yz xz yz zz

E uE vE wE up vp wp

t x y z x y z

qq q

x y z

u v w u v w u v wx y z

+ + + =

+ + +

+ + + + + + + + +


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Fully Developed Flow between Parallel Plates Using

the Navier-Stokes Equations

The governing equation for laminar flow over a surface can also be

derived using the Navier-Stokes equations.

Introduce the new variables xis the coordinate in the direction of

flo

w (same as s)and y perpendicular to the plates.

x


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The flow field is fully developed, so the derivatives

Also the flow is steady, so

The components of the gravity force are

The continuity equation is

Therefore

Continuity equation is automatically satisfied!

0u

x

=

0

v

x

=

0v

t

=

-direction sinx g

u

x

0

v

y

+ =

0 constant 0v

v vy

= = =

0u

t

=

-direction cosy g


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Momentum equation in the y direction is

As v is zero everywhere, there is no acceleration of the fluid in

the y direction and equation reduces to

By integrating last equation we can get

where py=0(x) is the pressure distribution along the lower wall.We can see that pressure is decreasing with the elevation in

the duct.

We can rewrite this equation as

where

v

t

vux

vvy

2

2

p v(y x

2

2

v

y

) g cos

cosgyp =

)(cos 0 xpygp y=+=

)(0 xpzp y==+

zy =cos


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The gradient of pressure in the x direction is

Momentum equation in the x direction is

For steady, fully developed flow all the terms in the left-handside of the equation equal zero and the equation becomes

As u is a function ofy and is function of x only we can

rewrite this equation in this form

As the slope is

We can rewrite the last equation as

u

t

uu

x

+

v+

2

2(

u p u

y x x

= +

2

2) sin

ug

y

+ +

dx

dp

dx

dp

x

p y==

=0

2

2

siny

upg

x

p

=

xp /

2

2

sindy

udgdxdp =

dx

dz=sin

2

2

)(dy

udzp

dx

d =+


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2

2

d d up zdx dy

)(12

2

zpds

d

dy

ud

+=From SimpleForce Balance

From N-S

Equations


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Surface Resistance

Solution of Laminar Boundary

Layer by Blasius


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Surface Resistance

Turbulent Velocity Profile


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Turbulent

Boundary

Layer


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Surface Resistance

Shear Stress at Turbulent Region


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Boundary Layer


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Boundary Layer

Topic 1 Surface Resistance Part II

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