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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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stream function velocity potential
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
Stream Lines
  • Consider 2D incompressible flow
  • Continuity Eqn
  • Vx and Vy are related
  • Can you write a common function for both?
stream function
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 function4
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
rotation

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

rotation6

Similarly

Rotation
  • To Calculate Rotation

Time = t + dt

Dy1

a

Dx

rotation7

a

b

Rotation
  • To Calculate Rotation
  • For very small time and very small element, Dx, Dy and Dt are close to zero
rotation8
Rotation
  • To Calculate Rotation
  • For very small q, (i.e.q ~ 0)
rotation9

Simplifies to

Rotation
  • To Calculate Rotation
slide10

Rotation in terms of Stream Function

  • To write rotation in terms of stream functions
  • That is
  • For irrotational flow (wz=0)
slide11

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
Velocity Potential
  • Assume
  • Then
  • In 3D, similarly it can be shown that
  • f is the velocity potential
velocity potential vs stream function
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
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