me2135 formula sheet
TRANSCRIPT
ME2135E Fluid Mechanics Formula sheet
A0066078X Lin Shaodun Page 1 of 4
( )
Centrifugal Pump
Power ( )
Head
( )
Capacity
Coeff
Head
Coeff
Power
Coeff
Head
(
)
(
)
(
)
(
)
(
)
(
)
Eff
( ) ( ) ( )
Eff ⁄
( ) ⁄|
⁄
⁄
|
Head
(
)
(
)
(
)
Axial Pump (
)⁄
Power ( ) Head ( ⁄ )
( ) ⁄
Cavitation
(∑
∑ )
( )
Cavitation index:
( )
Hydrostatic Bearing
( )
( ⁄ )
(
) ( )
( ⁄ )
( ⁄ )
(
)
( ⁄ )
( ⁄ )
(
)
(
)
( ⁄ )
( ⁄ )
[ ( ⁄ ) ]
𝑢 : Absolute Blade velocity
𝑤 Flow velocity relative to blade
𝑉 Absolute flow velocity
𝑉𝑛 Normal component of 𝑉
𝑉𝑡 Tangential component of 𝑉
𝛽 𝑡𝑎𝑛 𝑉𝑛
𝑢
𝑎 𝑡𝑎𝑛 𝑉𝑛
𝑉𝑡
𝜶𝟏 𝟗𝟎° 𝑽𝒏𝟏 𝑽𝟏 𝑽𝒕𝟏 𝟎
ur1
ub1
V=Vn 𝛽
Inlet velocity triangle
ur2
ub2
Vn2 𝛼
Vt2
V2
Outlet velocity triangle
𝛽
ME2135E Fluid Mechanics Formula sheet
A0066078X Lin Shaodun Page 2 of 4
Minimum pumping loss:
√
(
)
( ⁄ ) ⁄
Hydrodynamics Bearing-Inclined
( )
( )
( )
( )
[ ( ) ]
( ) ( )
(
) ( )
( ) ( )
( )
[ ( )( )
]
(
)
(
)
⁄
⁄
⁄
∫ ( )
( ) (
)
∫ |
∫
|
(
)
Hydrodynamics Bearing-Rayleigh step
( )
( ⁄
⁄ )
(
)
Flat Plate Boundary Layer Flow Rex=5x105
Re ⁄
⁄ ⁄
⁄
Laminar (Blasius
Solution)
⁄
⁄
⁄
⁄
2.59
⁄
Turbulent (1/7 Power
Lower vel. profile )
⁄
⁄
⁄
⁄
1.29
⁄
For laminar flows H varies between 2 and 3, (normal: 1.5~2)
⁄
⁄
( )
⁄
( ) (
)
(
) ⁄
|
Blasius Solution:
( ) √
⁄ ( ) (
)
( )
Boundary Layer with transition
( )
⁄⁄ ⁄⁄
⁄
√
∫ (
)
∫
(
)
Attached Flow Point of separation Separated flow:
|
|
|
Pressure decreases in the flow direction, accelerate
the lower part of boundary layer, Result in thinner
boundary layer than the case
Pressure increases in the flow direction, decelerate the
lower part of boundary layer, and cause some fluid
particles adjacent to the wall to be brought to rest.
Navier-Stokes Equation
(
)
(
)
(
)
(
)
ME2135E Fluid Mechanics Formula sheet
A0066078X Lin Shaodun Page 3 of 4
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)
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Steady Parallel Flow though a straight channel width=2b
(
)
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Stream function: CCWPositive, CWNegative, Volume flow: Incompressible flow:
(
)
∮ ∯ (
)
Velocity Potential
Differential
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)
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)
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)
( )
(
)
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[ ( ) ]
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[ ( ) ]
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Stream Functions
Flow
Type Flow Pattern Stream Function Velocity
Potential Flow Type Flow Pattern Stream Function Velocity Potential
Parallel
Flow
Source and
Uniform flow
(Half Rankine)
(
)
( )
( )
Stagnation Pt: (
)
Uniform
Oblique
Flow
( )
Sink / Source
pair
( ) ⁄
(
)
Source
or Sink
(
)
Sink / Source
pair in Uniform
flow (Full Rankine)
( )
(
)
Point
Vortex
( )
Doublet
Vortex-
Source
A Doublet in
Uniform flow
(Flow over cylinder)
, (
)
Flow with circulation round a
circular cylinder in uniform
flow
(
)
(
) (
) (
)
1
11
0
01
0 tan2
tan2 xx
yyq
xx
yyqyu
21
2
1
1
2
0
2
0
00
22 yyxx
xxq
yyxx
xxquu
For source and sink at general coordinates (x0,y0) and (x1,y1)
𝛼
ME2135E Fluid Mechanics Formula sheet
A0066078X Lin Shaodun Page 4 of 4
Sample Q&A
1. NS equation: 2 liquid with diff. , same , no mixing, no pressure grad. Plate moving at U horizontal, gap =h+h
ANS:
( )
( )
( )
( )
2. NS question: liquid flow on slant plate, zero shear at interface, with gravity g, liquid thickness=h, flow direction as x axis.
ANS:
|
|
( )
(
)
∫ ( )
3. NS question: liquid flow vertically within 2 fixed plates, under gravity g (direction –z), liquid thickness=h.
ANS:
( ) ( ) ( )
4. NS question: liquid flow within 2 plates, one fixed, another moves upwards in velocity U, under gravity g, liquid thickness=h
ANS:
( ) ( ) ( )
(
)
5. NS question: Couette flow, pressure gradient at x direction. Liquid thickness=h
ANS:
(
)
∫
6. Stream function: Flow past a cylinder with its axis perpendicular to the direction of flow.
ANS:
(
)
(
)
( )
ANS: °
⁄ ∫ [
( )]
∫ (
)
7. Stream function: A vortex with wall.
ANS:
[ ( ) ]
[ ( ) ]
|
( )
( )
( )
( )
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( )[
( )]
8. Boundary layer: Skin friction.
⁄
⁄
(
) ⁄
(
) ⁄
* (
) ⁄
⁄ +
9. Stream function: Two sink Q at x=±1 and one source NQ at 0,0
ANS:
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(
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(
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