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Combination of freeand forced vortex

Pressure Distribution in Rotational flows

Apply Euler Equation, we have,

( ) razpdrd ργ =+−

Acceleration the radial directionis given as

rVar

2

−=

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rV ω=For liquid rotating as a rigid body, i.e.

( ) 2ωργ rzpdrd

=+

Integrating the above equation

( ) Crzp +=+2

22ωργ

Cgrzp

=⎟⎟⎠

⎞⎜⎜⎝

⎛−+

2

22ωγ

The above equation describes the pressure variation in rotating flow

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Find the elevation difference betweenthe liquid at the center and the wall during rotation?

Cgrz =⎟⎟

⎠

⎞⎜⎜⎝

⎛−

2

22ω

Cgrzp

=⎟⎟⎠

⎞⎜⎜⎝

⎛−+

2

22ωγ

0=r 0zC =

⎟⎟⎠

⎞⎜⎜⎝

⎛+=

grzz

2

22

0ω

)2

( dr =at ⎟⎟⎠

⎞⎜⎜⎝

⎛+=

gdzz d 2

)2()(22

02ω

Applying of the liquid surface

at the center will reduces to

⎟⎟⎠

⎞⎜⎜⎝

⎛=−

gdzz d 2

)2()(22

02ω

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Find the new level of water during rotation?

Applying about the line of rotationCgrzp

=⎟⎟⎠

⎞⎜⎜⎝

⎛−+

2

22ωγ

P=0 as atmospheric, then

⎟⎟⎠

⎞⎜⎜⎝

⎛−=⎟⎟

⎠

⎞⎜⎜⎝

⎛−

2)(

2)( 22 ωρ

γωρ

γ rr

ll

rz

rz

( )222

2 rlrl rrg

zz −⎟⎟⎠

⎞⎜⎜⎝

⎛=−

ω

36.0=+ rl zz

cmzcmz rl 9.331.2 ==

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Bernoulli's Equation in Irrotational flow

⎟⎠⎞

⎜⎝⎛ −=

rV

drdVFor Irrotational Flow,

Apply Euler Equation, we have, ( )r

Vzpdrd 2

ργ −=+−

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟

⎠⎞

⎜⎝⎛−=+−

2

2Vdrd

drdVVzp

drd ρργ

02

2

=⎟⎟⎠

⎞⎜⎜⎝

⎛++

Vzpdrd ργ

CVzp =⎟⎟⎠

⎞⎜⎜⎝

⎛++

2

2

ργ

⎟⎠⎞

⎜⎝⎛ +=Ω

rV

drdV

z 21

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Pressure variation in a Tornado

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Point 1 represent the forced vortex at the centrePoint 2 represent the junction between forced vortex and free vortexPoint 3 represent the free vortex where the velocity is zero and pressure is atmospheric.

Apply Bernoulli's equation to a free vortexbetween point 3 and any arbitrary point,we have

=⎟⎟⎠

⎞⎜⎜⎝

⎛++

2

2Vzp ργ ⎟⎟⎠

⎞⎜⎜⎝

⎛++

2

23

33V

zp ργ

3033 ,,0 zzppV ===

2

2

0Vpp ρ−=−

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Applying the equation for pressure variation in rotatingflow between point 2 and point 3, we have

⎟⎟⎠

⎞⎜⎜⎝

⎛++

2)( 2

222

rzp ωργ=⎟⎟

⎠

⎞⎜⎜⎝

⎛++

2)( 2rzp ωργ

2max rV ω= 2zz =

⎟⎟⎠

⎞⎜⎜⎝

⎛++

2

23

33V

zp ργ=⎟⎟⎠

⎞⎜⎜⎝

⎛++

2)( 2

222

Vzp ργ

32033 ,,0 zzppV ===2)( max

2

02V

pp ρ−=−

22)( 22

max2

VVpp ρρ +−=

2

22

max0VVpp ρρ +−=

2)( max

2

02V

pp ρ−=−

)( 0pp =The pressure difference between point (1) i.e. the center of the tornado where and point (3) where)0( =V

2max01 Vpp ρ−=−

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Applying the equation for pressure variation in rotatingflow between point (A) and point (B), we have

⎟⎟⎠

⎞⎜⎜⎝

⎛++

2

2B

BBVzp ργ=⎟⎟

⎠

⎞⎜⎜⎝

⎛++

2)( 2

AAA

Vzp ργ

The pressure difference between point (1)i.e. the center of the tornado and the outeredge of the tornado

2max01 Vpp ρ−=−

Is the given velocity at the junction between forced

vortex and free vortex

2maxV

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Pressure variation Around a Circular Cylinder

For an ideal fluid (nonviscous , incompressible irrotational and steady),the flow pattern as shown below.

Pressure distribution on a cylinderirrotational flow

Irrotational flow past a cylinder

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Separation

Flow of a real fluid pasta circular cylinder

Separation point depends on:

(1) Free stream velocity; (2) Cylinder diameter; (3) Reynolds number;(4) Cylinder roughness; (5) Turbulence in the free stream.

Pressure distribution on a circularcylinder Re = 105

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Separation

Flow past a square rod and a disk and through a sharp-edged orifice

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END OF LECTURE (7)