p0824 - 1146 CHAPTER 8 ONE-DIMENSIONAL COMPRESSIBLE FLOW...

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1146 CHAPTER 8. ONE-DIMENSIONAL COMPRESSIBLE FLOW 8.24 Chapter 8, Problem 24 Problem: Consider subsonic airflow past an airplane wing. The flow speed far ahead of the wing is U . The flow accelerates over the wing, reaching a maximum speed of 3 2 U . The freestream density and temperature are ρ = 0.00234 slug/ft 3 and T = 519 o R, respectively. Compute M and the percentage change in density at the point where the maximum velocity is achieved for the following. (a) U = 100 mph (b) U = 450 mph Solution: Since total temperature and total density are constant for isentropic flow, we have T t = T } 1 + γ 1 2 M 2 ] = T } 1 + γ 1 2 M 2 ] ρ t = ρ } 1 + γ 1 2 M 2 ] 1 / ( γ 1) = ρ } 1 + γ 1 2 M 2 ] 1 / ( γ 1) Now, since M 2 = U 2 /a 2 = U 2 / ( γ RT ) , there follows T + γ 1 2 U 2 γ R = T + γ 1 2 U 2 γ R = a 2 + γ 1 2 U 2 = a 2 + γ 1 2 U 2 But, we are given U = 3 2 U , so that a 2 = a 2 + γ 1 2 } U 2 9 4 U 2 ] = a 2 5 8 ( γ 1) U 2 = a 2 } 1 5 8 ( γ 1) M 2
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