p1132 - 11.32. CHAPTER 11, PROBLEM 32 41 11.32 Chapter 11,...

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11.32. CHAPTER 11, PROBLEM 32 41 11.32 Chapter 11, Problem 32 Problem: A two-dimensional potential flow has streamfunction ψ ( r, θ )= Jr 4 sin 4 θ where J is a constant of dimensions 1 / ( L 2 T ) . (a) Determine the velocity components u r ( θ ) and u θ ( θ ) . (b) Determine the corresponding velocity potential, φ ( θ ) . (c) Locate any stagnation points. Convert to Cartesian coordinates for the sake of clarity. (d) The body shape is given by ψ =0 . If we confine our interest to the first quadrant excluding the y axis, i.e., 0 θ < π / 2 , what is the body shape? (e) Sketch a few streamlines. Include any stagnation points, body contours and flow direction along the streamlines. HINT: Find the flow direction on the body surface. Solution: (a) For the given streamfunction, ψ = 4 sin 4 θ , the velocity components are u r = 1 r ∂ψ ∂θ =4 3 cos 4 θ u θ = r = 4 3 sin 4 θ (b) Now, by definition of the velocity potential, we know that the velocity components are given by u r = ∂φ / r and u θ =(1 /r ) / .Thu s , r = u r 3 cos 4 θ Integrating over r , φ ( θ 4
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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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p1132 - 11.32. CHAPTER 11, PROBLEM 32 41 11.32 Chapter 11,...

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