Problem 5.30 - Problem *5.30 [Difficulty: 3] Stream...

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Problem *5.30 [Difficulty: 3] Given: Stream function for an incompressible flow field: ψ 5 A x 2A y A2 m s Find: (a) Sketch streamlines ψ = 0 and ψ = 5 (b) Velocity vector at (0, 0) (c) Flow rate between streamlines passing through points (2, 2) and (4, 1) Solution: We will generate the velocity field from the stream function. x v y u Governing Equations: (Definition of stream function) Assumptions: Incompressible flow ( ρ is constant) Flow is only in the x-y plane For ψ = 0: 05 A x y Solving for y: y 5 2 x For ψ = 5: 55 A x y Solving for y: y 5 2 x 5 2 m 2 s s 2m 5 2 x 5 2 m 4 2 0 2 4 10 5 5 10 x (m) y (m) Here is the plot of the two streamlines: ψ =0 is in red; ψ = 5 is in blue Generating the velocity components from the stream function derivatives:
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