module 1 : hydrostatics constant flows surfacewavesnovember 2007 schedule oe4620 module 1 • 13...
TRANSCRIPT
![Page 1: Module 1 : Hydrostatics Constant Flows SurfaceWavesNovember 2007 Schedule OE4620 Module 1 • 13 November –Constant potential flow phenomena • 20 November –Constant real flow](https://reader035.vdocuments.us/reader035/viewer/2022081323/5fffcd32cdd5857bfd68062a/html5/thumbnails/1.jpg)
November 2007
Offshore Hydromechanics
Module 1 : HydrostaticsConstant FlowsSurface Waves
OE4620 Offshore Hydromechanics
Ir. W.E. de Vries
Offshore Engineering
![Page 2: Module 1 : Hydrostatics Constant Flows SurfaceWavesNovember 2007 Schedule OE4620 Module 1 • 13 November –Constant potential flow phenomena • 20 November –Constant real flow](https://reader035.vdocuments.us/reader035/viewer/2022081323/5fffcd32cdd5857bfd68062a/html5/thumbnails/2.jpg)
November 2007
Today
First hour:
• Schedule for remainder of hydromechanics module 1
• Assignment stability
• Basic flow properties
• Potential flow concepts
Second hour:
• Potential flow elements
• Superposition of flow elements
• Cylinder in uniform flow
![Page 3: Module 1 : Hydrostatics Constant Flows SurfaceWavesNovember 2007 Schedule OE4620 Module 1 • 13 November –Constant potential flow phenomena • 20 November –Constant real flow](https://reader035.vdocuments.us/reader035/viewer/2022081323/5fffcd32cdd5857bfd68062a/html5/thumbnails/3.jpg)
November 2007
Schedule OE4620 Module 1
• 13 November – Constant potential flow phenomena
• 20 November – Constant real flow phenomena 1
• 27 November – Constant real flow phenomena 2
• 4 December – Ocean surface waves 1 (!)
• 11 December – Ocean surface waves 2
• 18 December – Extra/Examples
![Page 4: Module 1 : Hydrostatics Constant Flows SurfaceWavesNovember 2007 Schedule OE4620 Module 1 • 13 November –Constant potential flow phenomena • 20 November –Constant real flow](https://reader035.vdocuments.us/reader035/viewer/2022081323/5fffcd32cdd5857bfd68062a/html5/thumbnails/4.jpg)
November 2007
Assignment: Stability
• Make the assignment for next lecture
• Next lecture the assignment will be worked out briefly
• Questions regarding the assignment or the subjects of module 1? Contact:
• Room 2.78
• Tel. 015 278 7568
• E-mail [email protected]
![Page 5: Module 1 : Hydrostatics Constant Flows SurfaceWavesNovember 2007 Schedule OE4620 Module 1 • 13 November –Constant potential flow phenomena • 20 November –Constant real flow](https://reader035.vdocuments.us/reader035/viewer/2022081323/5fffcd32cdd5857bfd68062a/html5/thumbnails/5.jpg)
November 2007
CONSTANT POTENTIAL FLOW PHENOMENA
Flows in this chapter:
• Obey simplified laws of fluid mechanics
• Are subject to limitations
• Give qualitative impressions of flow phenomena
![Page 6: Module 1 : Hydrostatics Constant Flows SurfaceWavesNovember 2007 Schedule OE4620 Module 1 • 13 November –Constant potential flow phenomena • 20 November –Constant real flow](https://reader035.vdocuments.us/reader035/viewer/2022081323/5fffcd32cdd5857bfd68062a/html5/thumbnails/6.jpg)
November 2007
Basic Flow Properties
• Ideal fluid :
• Non – viscous
• Incompressible
• Continuous
• Homogeneous
![Page 7: Module 1 : Hydrostatics Constant Flows SurfaceWavesNovember 2007 Schedule OE4620 Module 1 • 13 November –Constant potential flow phenomena • 20 November –Constant real flow](https://reader035.vdocuments.us/reader035/viewer/2022081323/5fffcd32cdd5857bfd68062a/html5/thumbnails/7.jpg)
November 2007
Continuity Condition
•Increase of mass per unit time:
inm u dydzdtρ= ⋅ ⋅( )mdxdydz
t tρ∂ ∂= − ⋅
∂ ∂
![Page 8: Module 1 : Hydrostatics Constant Flows SurfaceWavesNovember 2007 Schedule OE4620 Module 1 • 13 November –Constant potential flow phenomena • 20 November –Constant real flow](https://reader035.vdocuments.us/reader035/viewer/2022081323/5fffcd32cdd5857bfd68062a/html5/thumbnails/8.jpg)
November 2007
Continuity Condition
•Mass through a plane dxdzduring a unit of time:
inm u dydzdtρ= ⋅ ⋅
inm v dxdzdtρ= ⋅ ⋅
![Page 9: Module 1 : Hydrostatics Constant Flows SurfaceWavesNovember 2007 Schedule OE4620 Module 1 • 13 November –Constant potential flow phenomena • 20 November –Constant real flow](https://reader035.vdocuments.us/reader035/viewer/2022081323/5fffcd32cdd5857bfd68062a/html5/thumbnails/9.jpg)
November 2007
Continuity Condition
•Mass through a plane dxdzduring a unit of time:
inm u dydzdtρ= ⋅ ⋅
( )out
vm v dy dxdzdt
y
ρρ
∂ = + ⋅ ∂
inm v dxdzdtρ= ⋅ ⋅
![Page 10: Module 1 : Hydrostatics Constant Flows SurfaceWavesNovember 2007 Schedule OE4620 Module 1 • 13 November –Constant potential flow phenomena • 20 November –Constant real flow](https://reader035.vdocuments.us/reader035/viewer/2022081323/5fffcd32cdd5857bfd68062a/html5/thumbnails/10.jpg)
November 2007
Continuity Condition
•Mass through a plane dxdzduring a unit of time:
inm u dydzdtρ= ⋅ ⋅
( )out
vm v dy dxdzdt
y
ρρ
∂ = + ⋅ ∂
inm v dxdzdtρ= ⋅ ⋅
( )vmv dy dxdz udxdz
t y
ρρ ρ
∂ ∂ = + ⋅ − ∂ ∂
![Page 11: Module 1 : Hydrostatics Constant Flows SurfaceWavesNovember 2007 Schedule OE4620 Module 1 • 13 November –Constant potential flow phenomena • 20 November –Constant real flow](https://reader035.vdocuments.us/reader035/viewer/2022081323/5fffcd32cdd5857bfd68062a/html5/thumbnails/11.jpg)
November 2007
Continuity Condition
•Mass through a plane dxdzduring a unit of time:
inm u dydzdtρ= ⋅ ⋅
( )out
vm v dy dxdzdt
y
ρρ
∂ = + ⋅ ∂
inm v dxdzdtρ= ⋅ ⋅
( ) ( )v umv dy dxdz udxdz dxdydz
t y y
ρ ρρ ρ
∂ ∂ ∂ = + ⋅ − = ∂ ∂ ∂
![Page 12: Module 1 : Hydrostatics Constant Flows SurfaceWavesNovember 2007 Schedule OE4620 Module 1 • 13 November –Constant potential flow phenomena • 20 November –Constant real flow](https://reader035.vdocuments.us/reader035/viewer/2022081323/5fffcd32cdd5857bfd68062a/html5/thumbnails/12.jpg)
November 2007
Continuity Condition
Combining for x, y and z direction yields the Continuity Equation
( ) ( ) ( )u v wmdxdydz dxdydz
t t x y z
ρ ρ ρρ ∂ ∂ ∂ ∂ ∂= − = + + ∂ ∂ ∂ ∂ ∂
![Page 13: Module 1 : Hydrostatics Constant Flows SurfaceWavesNovember 2007 Schedule OE4620 Module 1 • 13 November –Constant potential flow phenomena • 20 November –Constant real flow](https://reader035.vdocuments.us/reader035/viewer/2022081323/5fffcd32cdd5857bfd68062a/html5/thumbnails/13.jpg)
November 2007
Continuity Condition
Combining for x, y and z direction yields the Continuity Equation
( ) ( ) ( )u v wmdxdydz dxdydz
t t x y z
ρ ρ ρρ ∂ ∂ ∂ ∂ ∂= − = + + ∂ ∂ ∂ ∂ ∂
![Page 14: Module 1 : Hydrostatics Constant Flows SurfaceWavesNovember 2007 Schedule OE4620 Module 1 • 13 November –Constant potential flow phenomena • 20 November –Constant real flow](https://reader035.vdocuments.us/reader035/viewer/2022081323/5fffcd32cdd5857bfd68062a/html5/thumbnails/14.jpg)
November 2007
Continuity Condition
Combining for x, y and z direction yields the Continuity Equation
( ) ( ) ( )u v wmdxdydz dxdydz
t t x y z
ρ ρ ρρ ∂ ∂ ∂ ∂ ∂= − = + + ∂ ∂ ∂ ∂ ∂
![Page 15: Module 1 : Hydrostatics Constant Flows SurfaceWavesNovember 2007 Schedule OE4620 Module 1 • 13 November –Constant potential flow phenomena • 20 November –Constant real flow](https://reader035.vdocuments.us/reader035/viewer/2022081323/5fffcd32cdd5857bfd68062a/html5/thumbnails/15.jpg)
November 2007
Continuity Condition
Combining for x, y and z direction yields the Continuity Equation
( ) ( ) ( )0
u v w
t x y z
ρ ρ ρρ ∂ ∂ ∂∂ + + + =∂ ∂ ∂ ∂
![Page 16: Module 1 : Hydrostatics Constant Flows SurfaceWavesNovember 2007 Schedule OE4620 Module 1 • 13 November –Constant potential flow phenomena • 20 November –Constant real flow](https://reader035.vdocuments.us/reader035/viewer/2022081323/5fffcd32cdd5857bfd68062a/html5/thumbnails/16.jpg)
November 2007
Continuity Condition
For an ideal, incompressible fluid:
0u v w
x y z
∂ ∂ ∂+ + =∂ ∂ ∂
![Page 17: Module 1 : Hydrostatics Constant Flows SurfaceWavesNovember 2007 Schedule OE4620 Module 1 • 13 November –Constant potential flow phenomena • 20 November –Constant real flow](https://reader035.vdocuments.us/reader035/viewer/2022081323/5fffcd32cdd5857bfd68062a/html5/thumbnails/17.jpg)
November 2007
Continuity Condition
For an ideal, incompressible fluid:
Or, with :
0u v w
x y z
∂ ∂ ∂+ + =∂ ∂ ∂
x y z
∂ ∂ ∂∇ = + +∂ ∂ ∂ 0V∇ =
uv
![Page 18: Module 1 : Hydrostatics Constant Flows SurfaceWavesNovember 2007 Schedule OE4620 Module 1 • 13 November –Constant potential flow phenomena • 20 November –Constant real flow](https://reader035.vdocuments.us/reader035/viewer/2022081323/5fffcd32cdd5857bfd68062a/html5/thumbnails/18.jpg)
November 2007
Deformation of flow particles
Deformation
tanv
xα α∂ = ≈
∂& & tan
u
yβ β∂ = ≈
∂& &
![Page 19: Module 1 : Hydrostatics Constant Flows SurfaceWavesNovember 2007 Schedule OE4620 Module 1 • 13 November –Constant potential flow phenomena • 20 November –Constant real flow](https://reader035.vdocuments.us/reader035/viewer/2022081323/5fffcd32cdd5857bfd68062a/html5/thumbnails/19.jpg)
November 2007
Deformation of flow particles
Deformation
Deformation velocity (dilatation):
1
2 2
v u
x y
α β + ∂ ∂= + ∂ ∂
&&
tanv
xα α∂ = ≈
∂& & tan
u
yβ β∂ = ≈
∂& &
![Page 20: Module 1 : Hydrostatics Constant Flows SurfaceWavesNovember 2007 Schedule OE4620 Module 1 • 13 November –Constant potential flow phenomena • 20 November –Constant real flow](https://reader035.vdocuments.us/reader035/viewer/2022081323/5fffcd32cdd5857bfd68062a/html5/thumbnails/20.jpg)
November 2007
Deformation of flow particles
Deformation
Deformation velocity (dilatation):
Rotation (in 2 D) :
1
2 2
v u
x y
α β + ∂ ∂= + ∂ ∂
&&
1
2
v u
x yϕ ∂ ∂= − ∂ ∂ &
tanv
xα α∂ = ≈
∂& & tan
u
yβ β∂ = ≈
∂& &
![Page 21: Module 1 : Hydrostatics Constant Flows SurfaceWavesNovember 2007 Schedule OE4620 Module 1 • 13 November –Constant potential flow phenomena • 20 November –Constant real flow](https://reader035.vdocuments.us/reader035/viewer/2022081323/5fffcd32cdd5857bfd68062a/html5/thumbnails/21.jpg)
November 2007
Velocity Potential
A scalar function, Φ
The velocity in any point of the fluid and in any direction is the derivative to that direction of the potential
ux
∂Φ=∂
vy
∂Φ=∂
wz
∂Φ=∂
![Page 22: Module 1 : Hydrostatics Constant Flows SurfaceWavesNovember 2007 Schedule OE4620 Module 1 • 13 November –Constant potential flow phenomena • 20 November –Constant real flow](https://reader035.vdocuments.us/reader035/viewer/2022081323/5fffcd32cdd5857bfd68062a/html5/thumbnails/22.jpg)
November 2007
Properties of Velocity Potentials
• Combining the continuity condition with the velocity potentials results in the Laplace equation :
or
• Potential flow is rotation-free by definition :
2 2 2
2 2 20
x y z
∂ Φ ∂ Φ ∂ Φ+ + =∂ ∂ ∂
2 0∇ Φ =
0v u
x y
∂ ∂− =∂ ∂
0w v
y z
∂ ∂− =∂ ∂
0u w
z x
∂ ∂− =∂ ∂
![Page 23: Module 1 : Hydrostatics Constant Flows SurfaceWavesNovember 2007 Schedule OE4620 Module 1 • 13 November –Constant potential flow phenomena • 20 November –Constant real flow](https://reader035.vdocuments.us/reader035/viewer/2022081323/5fffcd32cdd5857bfd68062a/html5/thumbnails/23.jpg)
November 2007
Properties of Velocity Potentials
• Combining the continuity condition with the velocity potentials results in the Laplace equation :
or2 2 2
2 2 20
x y z
∂ Φ ∂ Φ ∂ Φ+ + =∂ ∂ ∂
2 0∇ Φ =
![Page 24: Module 1 : Hydrostatics Constant Flows SurfaceWavesNovember 2007 Schedule OE4620 Module 1 • 13 November –Constant potential flow phenomena • 20 November –Constant real flow](https://reader035.vdocuments.us/reader035/viewer/2022081323/5fffcd32cdd5857bfd68062a/html5/thumbnails/24.jpg)
November 2007
Euler and Bernoulli
• Euler: Apply Newton’s 2nd law to non viscous and incompressible fluid
Du Du pdm dxdydz dx dydz
Dt Dt xρ ∂= ⋅ ⋅ = − ⋅
∂
![Page 25: Module 1 : Hydrostatics Constant Flows SurfaceWavesNovember 2007 Schedule OE4620 Module 1 • 13 November –Constant potential flow phenomena • 20 November –Constant real flow](https://reader035.vdocuments.us/reader035/viewer/2022081323/5fffcd32cdd5857bfd68062a/html5/thumbnails/25.jpg)
November 2007
Euler and Bernoulli
• Euler: Apply Newton’s 2nd law to non viscous and incompressible fluid
Du Du pdm dxdydz dx dydz
Dt Dt xρ ∂= ⋅ ⋅ = − ⋅
∂
![Page 26: Module 1 : Hydrostatics Constant Flows SurfaceWavesNovember 2007 Schedule OE4620 Module 1 • 13 November –Constant potential flow phenomena • 20 November –Constant real flow](https://reader035.vdocuments.us/reader035/viewer/2022081323/5fffcd32cdd5857bfd68062a/html5/thumbnails/26.jpg)
November 2007
Euler and Bernoulli
• Euler: Apply Newton’s 2nd law to non viscous and incompressible fluid
Mass* acceleration = pressure*area
Du Du pdm dxdydz dx dydz
Dt Dt xρ ∂= ⋅ ⋅ = − ⋅
∂
![Page 27: Module 1 : Hydrostatics Constant Flows SurfaceWavesNovember 2007 Schedule OE4620 Module 1 • 13 November –Constant potential flow phenomena • 20 November –Constant real flow](https://reader035.vdocuments.us/reader035/viewer/2022081323/5fffcd32cdd5857bfd68062a/html5/thumbnails/27.jpg)
November 2007
Euler and Bernoulli
• Euler Equations:
With
1u u u u pu v w
t x y z xρ∂ ∂ ∂ ∂ ∂= + + = −∂ ∂ ∂ ∂ ∂
22122
uu
x x x x x
∂ ∂Φ ∂ Φ ∂ ∂Φ = ⋅ = ⋅ ∂ ∂ ∂ ∂ ∂ 22
12
uv
y y x y x y
∂ ∂Φ ∂ Φ ∂ ∂Φ= ⋅ = ⋅ ∂ ∂ ∂ ∂ ∂ ∂ 22
12
uw
z z x z x z
∂ ∂Φ ∂ Φ ∂ ∂Φ = ⋅ = ⋅ ∂ ∂ ∂ ∂ ∂ ∂
![Page 28: Module 1 : Hydrostatics Constant Flows SurfaceWavesNovember 2007 Schedule OE4620 Module 1 • 13 November –Constant potential flow phenomena • 20 November –Constant real flow](https://reader035.vdocuments.us/reader035/viewer/2022081323/5fffcd32cdd5857bfd68062a/html5/thumbnails/28.jpg)
November 2007
Euler and Bernoulli
• Differentiate w.r.t. x, y or z gives 0:
(similar for y, z)
22 21
02
p
x t x y z ρ
∂ ∂Φ ∂Φ ∂Φ ∂Φ + + + + = ∂ ∂ ∂ ∂ ∂
![Page 29: Module 1 : Hydrostatics Constant Flows SurfaceWavesNovember 2007 Schedule OE4620 Module 1 • 13 November –Constant potential flow phenomena • 20 November –Constant real flow](https://reader035.vdocuments.us/reader035/viewer/2022081323/5fffcd32cdd5857bfd68062a/html5/thumbnails/29.jpg)
November 2007
Euler and Bernoulli
• Differentiate w.r.t. x, y or z gives 0:
(similar for y, z)
• This expression is a function of time only, providingthe Bernoulli Equation:
22 21
02
p
x t x y z ρ
∂ ∂Φ ∂Φ ∂Φ ∂Φ + + + + = ∂ ∂ ∂ ∂ ∂
( )21
2
pV gz C t
t ρ∂Φ + + + =∂
![Page 30: Module 1 : Hydrostatics Constant Flows SurfaceWavesNovember 2007 Schedule OE4620 Module 1 • 13 November –Constant potential flow phenomena • 20 November –Constant real flow](https://reader035.vdocuments.us/reader035/viewer/2022081323/5fffcd32cdd5857bfd68062a/html5/thumbnails/30.jpg)
November 2007
Euler and Bernoulli
• For a stationary flow along a streamline :
• This can be written in two alternative ways :
21
2
pV gz C
ρ+ + =
2
2
V pz C
g gρ+ + = 21
2 V p gz Cρ ρ+ + =
![Page 31: Module 1 : Hydrostatics Constant Flows SurfaceWavesNovember 2007 Schedule OE4620 Module 1 • 13 November –Constant potential flow phenomena • 20 November –Constant real flow](https://reader035.vdocuments.us/reader035/viewer/2022081323/5fffcd32cdd5857bfd68062a/html5/thumbnails/31.jpg)
November 2007
Stream Function ΨΨΨΨ
• Ψ= constant along a streamline
• Definition : and
• Stream lines and equipotential lines are orthogonal
• No cross flow, impervious boundary of stream tube
uy
∂Ψ=∂
vx
∂Ψ= −∂
![Page 32: Module 1 : Hydrostatics Constant Flows SurfaceWavesNovember 2007 Schedule OE4620 Module 1 • 13 November –Constant potential flow phenomena • 20 November –Constant real flow](https://reader035.vdocuments.us/reader035/viewer/2022081323/5fffcd32cdd5857bfd68062a/html5/thumbnails/32.jpg)
November 2007
Potential Flow Elements
• Uniform flow
• Source
• Sinks
• Circulation
Superposition is allowed
![Page 33: Module 1 : Hydrostatics Constant Flows SurfaceWavesNovember 2007 Schedule OE4620 Module 1 • 13 November –Constant potential flow phenomena • 20 November –Constant real flow](https://reader035.vdocuments.us/reader035/viewer/2022081323/5fffcd32cdd5857bfd68062a/html5/thumbnails/33.jpg)
November 2007
Uniform Flow
U xΦ = − ⋅U xΦ = + ⋅
U yΨ = + ⋅ U yΨ = − ⋅
![Page 34: Module 1 : Hydrostatics Constant Flows SurfaceWavesNovember 2007 Schedule OE4620 Module 1 • 13 November –Constant potential flow phenomena • 20 November –Constant real flow](https://reader035.vdocuments.us/reader035/viewer/2022081323/5fffcd32cdd5857bfd68062a/html5/thumbnails/34.jpg)
November 2007
Source & Sink
ln2
Qr
πΦ = − ⋅ ln
2
Qr
πΦ = + ⋅
2
Q θπ
Ψ = + ⋅2
Q θπ
Ψ = − ⋅
![Page 35: Module 1 : Hydrostatics Constant Flows SurfaceWavesNovember 2007 Schedule OE4620 Module 1 • 13 November –Constant potential flow phenomena • 20 November –Constant real flow](https://reader035.vdocuments.us/reader035/viewer/2022081323/5fffcd32cdd5857bfd68062a/html5/thumbnails/35.jpg)
November 2007
• Counter clockwise flow:
• Clockwise flow
Circulation (Vortex)
2θ
πΓΦ = − ⋅
2θ
πΓΦ = + ⋅
ln2
rπΓΨ = + ⋅
ln2
rπΓΨ = − ⋅
![Page 36: Module 1 : Hydrostatics Constant Flows SurfaceWavesNovember 2007 Schedule OE4620 Module 1 • 13 November –Constant potential flow phenomena • 20 November –Constant real flow](https://reader035.vdocuments.us/reader035/viewer/2022081323/5fffcd32cdd5857bfd68062a/html5/thumbnails/36.jpg)
November 2007
Superposition - principle
![Page 37: Module 1 : Hydrostatics Constant Flows SurfaceWavesNovember 2007 Schedule OE4620 Module 1 • 13 November –Constant potential flow phenomena • 20 November –Constant real flow](https://reader035.vdocuments.us/reader035/viewer/2022081323/5fffcd32cdd5857bfd68062a/html5/thumbnails/37.jpg)
November 2007
Superposition - principle
![Page 38: Module 1 : Hydrostatics Constant Flows SurfaceWavesNovember 2007 Schedule OE4620 Module 1 • 13 November –Constant potential flow phenomena • 20 November –Constant real flow](https://reader035.vdocuments.us/reader035/viewer/2022081323/5fffcd32cdd5857bfd68062a/html5/thumbnails/38.jpg)
November 2007
Superposition - principle
![Page 39: Module 1 : Hydrostatics Constant Flows SurfaceWavesNovember 2007 Schedule OE4620 Module 1 • 13 November –Constant potential flow phenomena • 20 November –Constant real flow](https://reader035.vdocuments.us/reader035/viewer/2022081323/5fffcd32cdd5857bfd68062a/html5/thumbnails/39.jpg)
November 2007
Superposition - principle
![Page 40: Module 1 : Hydrostatics Constant Flows SurfaceWavesNovember 2007 Schedule OE4620 Module 1 • 13 November –Constant potential flow phenomena • 20 November –Constant real flow](https://reader035.vdocuments.us/reader035/viewer/2022081323/5fffcd32cdd5857bfd68062a/html5/thumbnails/40.jpg)
November 2007
Uniform flow + sink
arctan2
Q yU y
xπ ∞ Ψ = − ⋅ − ⋅
![Page 41: Module 1 : Hydrostatics Constant Flows SurfaceWavesNovember 2007 Schedule OE4620 Module 1 • 13 November –Constant potential flow phenomena • 20 November –Constant real flow](https://reader035.vdocuments.us/reader035/viewer/2022081323/5fffcd32cdd5857bfd68062a/html5/thumbnails/41.jpg)
November 2007
Separated Source + sink
2 2 2
2arctan
2
Q ys
x y sπ
Ψ = ⋅ + + s
![Page 42: Module 1 : Hydrostatics Constant Flows SurfaceWavesNovember 2007 Schedule OE4620 Module 1 • 13 November –Constant potential flow phenomena • 20 November –Constant real flow](https://reader035.vdocuments.us/reader035/viewer/2022081323/5fffcd32cdd5857bfd68062a/html5/thumbnails/42.jpg)
November 2007
Uniform flow + source + sink
• No flow throughboundary
(= streamline)
• Rankine ship forms
2 2 2
2arctan
2
Q ysU y
x y sπ ∞
Ψ = ⋅ + ⋅ + −
![Page 43: Module 1 : Hydrostatics Constant Flows SurfaceWavesNovember 2007 Schedule OE4620 Module 1 • 13 November –Constant potential flow phenomena • 20 November –Constant real flow](https://reader035.vdocuments.us/reader035/viewer/2022081323/5fffcd32cdd5857bfd68062a/html5/thumbnails/43.jpg)
November 2007
Doublet or Dipole
• A source and sinkpair placed veryclose together
• (s -> 0)
2 2
y
x yµΨ = ⋅
+ 2 2
x
x yµΦ = ⋅
+
![Page 44: Module 1 : Hydrostatics Constant Flows SurfaceWavesNovember 2007 Schedule OE4620 Module 1 • 13 November –Constant potential flow phenomena • 20 November –Constant real flow](https://reader035.vdocuments.us/reader035/viewer/2022081323/5fffcd32cdd5857bfd68062a/html5/thumbnails/44.jpg)
November 2007
Dipole + uniform flow
• Model for cylinder
• X-axis is a streamline
• Boundary at
is also a streamline
RU
µ∞
=
![Page 45: Module 1 : Hydrostatics Constant Flows SurfaceWavesNovember 2007 Schedule OE4620 Module 1 • 13 November –Constant potential flow phenomena • 20 November –Constant real flow](https://reader035.vdocuments.us/reader035/viewer/2022081323/5fffcd32cdd5857bfd68062a/html5/thumbnails/45.jpg)
November 2007
Uniform flow around a cylinder
• Vθ= 0 at stagnation points
• Vθ= 2*U∞ at sides of cylinder
sinsin
r R r R
v U rr r rθ
µ θ θ∞= =
∂Ψ ∂ = − = − − ∂ ∂ 2 sinv Uθ θ∞= −
![Page 46: Module 1 : Hydrostatics Constant Flows SurfaceWavesNovember 2007 Schedule OE4620 Module 1 • 13 November –Constant potential flow phenomena • 20 November –Constant real flow](https://reader035.vdocuments.us/reader035/viewer/2022081323/5fffcd32cdd5857bfd68062a/html5/thumbnails/46.jpg)
November 2007
Uniform flow + circulation
Pressure asymmetry creates a lift force
![Page 47: Module 1 : Hydrostatics Constant Flows SurfaceWavesNovember 2007 Schedule OE4620 Module 1 • 13 November –Constant potential flow phenomena • 20 November –Constant real flow](https://reader035.vdocuments.us/reader035/viewer/2022081323/5fffcd32cdd5857bfd68062a/html5/thumbnails/47.jpg)
November 2007
Continuity Condition
Continuity Equation, general case
Ideal fluid, incompressible
( ) ( ) ( )0
u v w
t x y z
ρ ρ ρρ ∂ ∂ ∂∂ + + + =∂ ∂ ∂ ∂
0
u v w
x y z
∂ ∂ ∂+ + =∂ ∂ ∂
![Page 48: Module 1 : Hydrostatics Constant Flows SurfaceWavesNovember 2007 Schedule OE4620 Module 1 • 13 November –Constant potential flow phenomena • 20 November –Constant real flow](https://reader035.vdocuments.us/reader035/viewer/2022081323/5fffcd32cdd5857bfd68062a/html5/thumbnails/48.jpg)
November 2007
Deformation of flow particles
Deformation velocity or dilatation:
Rotation (in 2 D) :
1
2 2
v u
x y
α β + ∂ ∂= + ∂ ∂
&&
1
2
v u
x yϕ ∂ ∂= − ∂ ∂ &
![Page 49: Module 1 : Hydrostatics Constant Flows SurfaceWavesNovember 2007 Schedule OE4620 Module 1 • 13 November –Constant potential flow phenomena • 20 November –Constant real flow](https://reader035.vdocuments.us/reader035/viewer/2022081323/5fffcd32cdd5857bfd68062a/html5/thumbnails/49.jpg)
November 2007
Velocity Potential
A scalar function, Φ
The velocity in any point of the fluid and in any direction is the derivative to that direction of the potential
u = ∂Φ/∂x etc
![Page 50: Module 1 : Hydrostatics Constant Flows SurfaceWavesNovember 2007 Schedule OE4620 Module 1 • 13 November –Constant potential flow phenomena • 20 November –Constant real flow](https://reader035.vdocuments.us/reader035/viewer/2022081323/5fffcd32cdd5857bfd68062a/html5/thumbnails/50.jpg)
November 2007
Properties of Velocity Potentials
• The continuity condition results in the Laplaceequation :
• Potential flow is rotation-free by definition :
![Page 51: Module 1 : Hydrostatics Constant Flows SurfaceWavesNovember 2007 Schedule OE4620 Module 1 • 13 November –Constant potential flow phenomena • 20 November –Constant real flow](https://reader035.vdocuments.us/reader035/viewer/2022081323/5fffcd32cdd5857bfd68062a/html5/thumbnails/51.jpg)
November 2007
Euler and Bernoulli
Force = mass * acceleration : Euler
non viscous and incompressible fluid
Energy conservation along a flowline : Bernoulli
non viscous and incompressible fluid
stationary flow :
![Page 52: Module 1 : Hydrostatics Constant Flows SurfaceWavesNovember 2007 Schedule OE4620 Module 1 • 13 November –Constant potential flow phenomena • 20 November –Constant real flow](https://reader035.vdocuments.us/reader035/viewer/2022081323/5fffcd32cdd5857bfd68062a/html5/thumbnails/52.jpg)
November 2007
Stream Function ΨΨΨΨ
• Companion of the Velocity Potential
• Definition :
• Stream lines and equipotential lines are orthogonal
• No cross flow, impervious boundary of stream tube
![Page 53: Module 1 : Hydrostatics Constant Flows SurfaceWavesNovember 2007 Schedule OE4620 Module 1 • 13 November –Constant potential flow phenomena • 20 November –Constant real flow](https://reader035.vdocuments.us/reader035/viewer/2022081323/5fffcd32cdd5857bfd68062a/html5/thumbnails/53.jpg)
November 2007
Potential Flow Elements
• Uniform flow
• Sources and sinks
• Circulating flows
Superposition is
allowed
![Page 54: Module 1 : Hydrostatics Constant Flows SurfaceWavesNovember 2007 Schedule OE4620 Module 1 • 13 November –Constant potential flow phenomena • 20 November –Constant real flow](https://reader035.vdocuments.us/reader035/viewer/2022081323/5fffcd32cdd5857bfd68062a/html5/thumbnails/54.jpg)
November 2007
Polar Coordinate Description
Basic definitions :
Source :
(Sink = - source)
Vortex :
![Page 55: Module 1 : Hydrostatics Constant Flows SurfaceWavesNovember 2007 Schedule OE4620 Module 1 • 13 November –Constant potential flow phenomena • 20 November –Constant real flow](https://reader035.vdocuments.us/reader035/viewer/2022081323/5fffcd32cdd5857bfd68062a/html5/thumbnails/55.jpg)
November 2007
Superposition - principle
![Page 56: Module 1 : Hydrostatics Constant Flows SurfaceWavesNovember 2007 Schedule OE4620 Module 1 • 13 November –Constant potential flow phenomena • 20 November –Constant real flow](https://reader035.vdocuments.us/reader035/viewer/2022081323/5fffcd32cdd5857bfd68062a/html5/thumbnails/56.jpg)
November 2007
Uniform flow + sink
![Page 57: Module 1 : Hydrostatics Constant Flows SurfaceWavesNovember 2007 Schedule OE4620 Module 1 • 13 November –Constant potential flow phenomena • 20 November –Constant real flow](https://reader035.vdocuments.us/reader035/viewer/2022081323/5fffcd32cdd5857bfd68062a/html5/thumbnails/57.jpg)
November 2007
Source + sink
![Page 58: Module 1 : Hydrostatics Constant Flows SurfaceWavesNovember 2007 Schedule OE4620 Module 1 • 13 November –Constant potential flow phenomena • 20 November –Constant real flow](https://reader035.vdocuments.us/reader035/viewer/2022081323/5fffcd32cdd5857bfd68062a/html5/thumbnails/58.jpg)
November 2007
Uniform flow + source + sink
![Page 59: Module 1 : Hydrostatics Constant Flows SurfaceWavesNovember 2007 Schedule OE4620 Module 1 • 13 November –Constant potential flow phenomena • 20 November –Constant real flow](https://reader035.vdocuments.us/reader035/viewer/2022081323/5fffcd32cdd5857bfd68062a/html5/thumbnails/59.jpg)
November 2007
DipoleStrength µ = Qs/π
![Page 60: Module 1 : Hydrostatics Constant Flows SurfaceWavesNovember 2007 Schedule OE4620 Module 1 • 13 November –Constant potential flow phenomena • 20 November –Constant real flow](https://reader035.vdocuments.us/reader035/viewer/2022081323/5fffcd32cdd5857bfd68062a/html5/thumbnails/60.jpg)
November 2007
Dipole + uniform flow
![Page 61: Module 1 : Hydrostatics Constant Flows SurfaceWavesNovember 2007 Schedule OE4620 Module 1 • 13 November –Constant potential flow phenomena • 20 November –Constant real flow](https://reader035.vdocuments.us/reader035/viewer/2022081323/5fffcd32cdd5857bfd68062a/html5/thumbnails/61.jpg)
November 2007
Uniform flow around a cylinder
Stagnation points on the X axis
Vmax = 2 U at Y axis
![Page 62: Module 1 : Hydrostatics Constant Flows SurfaceWavesNovember 2007 Schedule OE4620 Module 1 • 13 November –Constant potential flow phenomena • 20 November –Constant real flow](https://reader035.vdocuments.us/reader035/viewer/2022081323/5fffcd32cdd5857bfd68062a/html5/thumbnails/62.jpg)
November 2007
Uniform flow + circulation
Pressure asymmetry creates a lift force