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# p1108 - 10 CHAPTER 11 POTENTIAL FLOW 11.8 Chapter 11...

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10 CHAPTER 11. POTENTIAL FLOW 11.8 Chapter 11, Problem 8 Problem: For steady, compressible, two-dimensional flows with (variable) density ρ , we can define a “streamfunction” according to u = 1 ρ ˜ ψ y ,v = 1 ρ ˜ ψ x (a) What are the dimensions of ˜ ψ ? (b) Verify that conservation of mass is satisfied. (c) If the flow is irrotational, verify that 2 ˜ ψ = u × ρ where ˜ ψ ˜ ψ k . Solution: (a) To determine the dimensions of ˜ ψ , note that from its definition, [ u ]= 1 [ ρ ] ^ ˜ ψ y ± = L T = 1 M/L 3 [ ˜ ψ ] L = L 2 M [ ˜ ψ ] Therefore, we conclude that [ ˜ ψ M LT (b) Substituting into the continuity equation, we find x ( ρ u )+ y ( ρ v )= x X ˜ ψ y ~ + y X ˜ ψ x ~ = 2 ˜ ψ x y 2 ˜ ψ y x =0 (c) If the flow is irrotational, we have v x u y = x ^ 1 ρ ˜ ψ x ± y ^ 1 ρ ˜ ψ y ± Performing the indicated differentiations and multiplying through by ( 1) , 1 ρ 2

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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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p1108 - 10 CHAPTER 11 POTENTIAL FLOW 11.8 Chapter 11...

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