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3.1 Relationships for Small Mach Number

$\displaystyle {P_0\over P} = 1 + {(k -1) M^2 \over 4} + {k M^4\over 8} + {2(2-k)M^6 \over 48} \cdots$ (17)

$\displaystyle {\rho_0\over \rho} = 1 + {(k -1) M^2 \over 4} + {k M^4\over 8} + {2(2-k)M^6 \over 48} \cdots$ (18)

$\displaystyle {P_0 - P \over {1 \over 2 } \rho U^2} = 1 + \overbrace{{ M^2 \over 4} + {(2-k) M^4\over 24} + \cdots} ^{compressibility\; correction}$ (19)

$\displaystyle M^{*} = {U \over c^{*} } = \sqrt{k+1 \over 2} M \left( 1 - {k -1 \over 4} M^2 + \cdots \right)$ (20)

$\displaystyle {P_0 -P \over P} = {kM^2 \over 2} \left( 1 + {M^2 \over 4} + \cdots \right)$ (21)

$\displaystyle {\rho_0 -\rho \over \rho} = {M^2 \over 2} \left( 1 - {kM^2 \over 4} + \cdots \right)$ (22)

$\displaystyle {\dot{m} \over A} = \sqrt{k {P_0}^2 M^2 \over RT_0} \left( 1 + {k-1 \over 4}M^2 + \cdots \right)$ (23)

The ratio of the area to star area is

$\displaystyle {A \over A^{*}} = \left(2 \over k +1 \right)^{k +1 \over 2 (k-1)} \left( {1\over M} + {k+1 \over 4}M + {(3-k) (k+1)\over 32 } M^3 + \cdots \right)$ (24)

$\displaystyle {A \over A^{*}} = { 1 \over M} \left( { 1 + {k -1 \over 2} M^{2} \over {k +1\over 2}} \right) ^ {k+ 1 \over 2 (k -1 )}$ (25)


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Next: 3.2 Isentropic Isothermal Flow Up: 3 Isentropic Flow Previous: 3 Isentropic Flow
Genick Bar-Meir ||| www.potto.org
copyright Dec , 2006

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