HW16solution

HW16solution - 2 2 1 ! 2 1 1 1 1 2 1 1 1 x x p p o...

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AAE 334, Fall 2011, Homework 16 SOLUTION Due Wednesday, September 19, at the beginning of class. We will see shortly that for compressible flow of an ideal gas, the ratio of stagnation (or total) pressure to static pressure is related to Mach number and gas properties in this way: 1 2 2 1 1 M p p o Prove with a series expansion that this reduces to Bernoulli’s equation in the limit of small Mach number. Explain your work to receive credit. To begin, note that this is a function of Mach number squared, not Mach number, so substitute some variable, call it x , for M 2 : 1 2 1 1 x p p o Expand in a power series in x around x=0 . Recall      3 3 2 2 ! 3 2 1 ! 2 1 1 ) 1 ( y a z z z y a z z zay ay z and thus,
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Unformatted text preview: 2 2 1 ! 2 1 1 1 1 2 1 1 1 x x p p o Simplify, 2 8 2 1 x x p p o and as M then x and x x 2 so for small Mach number, 2 2 2 1 2 1 2 1 a V M x p p o Speed of sound we know to be RT a or RT a 2 so substitute and cancel gammas, 2 2 2 1 1 2 1 RT V RT V p p o From Ideal Gas Law we know that p RT and so, 2 2 1 1 p V p p o and now multiply both sides by static pressure, 2 2 1 V p p o QED....
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