Stream Function & Velocity Potential

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# Stream Function & Velocity Potential - PowerPoint PPT Presentation

Stream Function & Velocity Potential. Stream lines/ Stream Function ( Y ) Concept Relevant Formulas Examples Rotation, vorticity Velocity Potential( f ) Concept Relevant Formulas Examples Relationship between stream function and velocity potential Complex velocity potential.

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## PowerPoint Slideshow about 'Stream Function & Velocity Potential' - zelda

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Presentation Transcript
Stream Function & Velocity Potential
• Stream lines/ Stream Function (Y)
• Concept
• Relevant Formulas
• Examples
• Rotation, vorticity
• Velocity Potential(f)
• Concept
• Relevant Formulas
• Examples
• Relationship between stream function and velocity potential
• Complex velocity potential
Stream Lines
• Consider 2D incompressible flow
• Continuity Eqn
• Vx and Vy are related
• Can you write a common function for both?
Stream Function
• Assume
• Then
• Instead of two functions, Vx and Vy, we need to solve for only one function y - Stream Function
• Order of differential eqn increased by one
Stream Function
• What does Stream Function y mean?
• Equation for streamlines in 2D are given by

y = constant

• Streamlines may exist in 3D also, but stream function does not
• Why? (When we work with velocity potential, we may get a perspective)
• In 3D, streamlines follow the equation

b

Time = t + dt

a

Rotation
• Definition of rotation

Time=t

y

Dy

Dx

x

Assume Vy|x< Vy|x+Dx

and Vx|y> Vx|y+Dy

Similarly

Rotation
• To Calculate Rotation

Time = t + dt

Dy1

a

Dx

a

b

Rotation
• To Calculate Rotation
• For very small time and very small element, Dx, Dy and Dt are close to zero
Rotation
• To Calculate Rotation
• For very small q, (i.e.q ~ 0)

Simplifies to

Rotation
• To Calculate Rotation

Rotation in terms of Stream Function

• To write rotation in terms of stream functions
• That is
• For irrotational flow (wz=0)

Rotation and Potential

• For irrotational flow (wz=0)
• This equation is “similar” to continuity equation
• Vx and Vy are related
• Can we find a common function to relate both Vx and Vy ?
Velocity Potential
• Assume
• Then
• In 3D, similarly it can be shown that
• f is the velocity potential
Velocity Potential vs Stream Function
• In 2D inviscid flow (incompressible flow OR steady state compressible flow), both functions exist
• What is the relationship between them?
Stream Function- Physical meaning
• Statement: In 2D (viscous or inviscid) flow (incompressible flow OR steady state compressible flow), y = constant represents the streamline.
• Proof
• If y = constant, then dy=0

Vy

• If y = constant, then

Vx