p1118 - 24 CHAPTER 11. POTENTIAL FLOW 11.18 Chapter 11,...

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24 CHAPTER 11. POTENTIAL FLOW 11.18 Chapter 11, Problem 18 Problem: The streamfunction for an incompressible flow is ψ ( x, y )= x 2 y 2 y 2 , where ψ , x and y are all dimensionless. Plot the streamlines in the first quadrant for ψ = 2 , 0 , 2 , and find any stagnation points. Be sure to indicate flow direction on the streamlines. Is this flow irrotational? Solution: For ψ ( x, y )= x 2 y 2 y 2 the velocity components are u = ∂ψ y = x 2 4 y, v = ∂ψ x = 2 xy At a stagnation point, we must have u = v =0 .Now , u =0 when x 2 =4 y , while v =0 when either x =0 or y =0 . Thus, there is a single stagnation point at the origin. To determine the streamlines, we set ψ equal to a constant. Then, calling the constant C , x 2 y 2 y 2 = C = 2 y 2 x 2 y + C =0 Therefore, the streamlines are given by y = 1 4 x 2 ± 1 4 x 4 8 C The figure below shows the streamlines in the first quadrant for ψ = C = 2 , 0 , 2 .Th
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This note was uploaded on 02/09/2012 for the course AME 309 taught by Professor Phares during the Spring '06 term at USC.

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