Question

Which equation is used to compute the critical Mach number of the airfoil?

(a) (Cp)crit = \(\frac {2}{γM_{crit}^{2}} \bigg [ \frac {1 + \frac {1}{2}(γ – 1)M_{crit}^{2}}{1 + \frac {1}{2}(γ – 1)} \bigg ]^{\frac {γ}{γ – 1}}\) – 1

(b) (Cp)crit = \(\frac {2}{γM_{crit}^{2}} \bigg [ \frac {1 + \frac {1}{2}(γ – 1)M_{crit}^{2}}{1 + \frac {1}{2}(γ – 1)} \bigg ]^{\frac {γ}{γ + 1}}\) + 1

(c) (Cp)crit = γM\(_{crit}^{2} \bigg [ \frac {1 + \frac {1}{2}(γ – 1)}{1 + \frac {1}{2}(γ – 1)} \bigg ]^{\frac {γ}{γ – 1}}\) – 1

(d) (Cp)crit = γM\(_{crit}^{2} \bigg [ \frac {1 + \frac {1}{2}(γ – 1)}{1 + \frac {1}{2}(γ – 1)M_{crit}^{2}} \bigg ]^{\frac {γ}{γ – 1}}\) – 1

The question was asked during an internship interview.

This interesting question is from Critical Mach Number in chapter Linearized and Conical Flows of Aerodynamics

Answer 1

The correct choice is (a) (Cp)crit = \(\frac {2}{γM_{crit}^{2}} \bigg [ \frac {1 + \frac {1}{2}(γ – 1)M_{crit}^{2}}{1 + \frac {1}{2}(γ – 1)} \bigg ]^{\frac {γ}{γ – 1}}\) – 1

Explanation: The formula for the coefficient of pressure for an isentropic flow is given by:

Cp = \(\frac {2}{γM_∞^{2}} \bigg ( \frac {p}{p_∞}  – 1 \bigg )\)

For an isentropic flow, the ratio of pressure at a point to the freestream pressure is given by:

\(\frac {p}{p_∞}  = \bigg [ \frac {1 + \frac {(γ – 1)}{2} M_∞^{2}}{1 + \frac {(γ – 1)}{2} M^{2}} \bigg ]^{\frac {γ}{γ – 1}} \)

Substituting this in the above equation we get

Cp = \(\frac {2}{γM_∞^{2}} \bigg [ \bigg ( \frac {1 + \frac {(γ – 1)}{2} M_∞^{2}}{1 + \frac {(γ – 1)}{2} M^{2}}\bigg ) ^{\frac {γ}{γ – 1}} – 1 \bigg ] \)

At critical Mach number, local Mach number M = 1 and freestream Mach number is equal to the critical Mach number. Substituting these we finally arrive at the relation:

(Cp)crit = \(\frac {2}{γM_{crit}^{2}} \bigg [ \frac {1 + \frac {1}{2}(γ – 1)M_{crit}^{2}}{1 + \frac {1}{2}(γ – 1)} \bigg ]^{\frac {γ}{γ – 1}}\) – 1