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08/12/201008/12/2010 11
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
−=
08/12/201008/12/2010 22
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
08/12/201008/12/2010 33
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
08/12/201008/12/2010 44
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 ==
08/12/201008/12/2010 55
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
08/12/201008/12/2010 66
Pressure variation in a Tornado
08/12/201008/12/2010 77
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−=−
08/12/201008/12/2010 88
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−=−
08/12/201008/12/2010 99
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
08/12/201008/12/2010 1010
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
08/12/201008/12/2010 1111
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
08/12/201008/12/2010 1212
Separation
Flow past a square rod and a disk and through a sharp-edged orifice
08/12/201008/12/2010 1313
END OF LECTURE (7)