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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.

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Stream Function & Velocity Potential

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  1. 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

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

  3. 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

  4. 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

  5. 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

  6. Similarly Rotation • To Calculate Rotation Time = t + dt Dy1 a Dx

  7. a b Rotation • To Calculate Rotation • For very small time and very small element, Dx, Dy and Dt are close to zero

  8. Rotation • To Calculate Rotation • For very small q, (i.e.q ~ 0)

  9. Simplifies to Rotation • To Calculate Rotation

  10. Rotation in terms of Stream Function • To write rotation in terms of stream functions • That is • For irrotational flow (wz=0)

  11. 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 ?

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

  13. 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?

  14. 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

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