# An airfoil is in airflow with $$0.52$$ free stream Mach number. Local Mach number at a given point is $$0.82$$. Find the pressure coefficient at this point.

An airfoil is in airflow with $$0.52$$ free stream Mach number. Local Mach number at a given point is $$0.82$$. Find the pressure coefficient at this point.

From isentropic flow properties table, for $$M_{\infty}=0.52,\frac{p_{0}}{p_{\infty}}=1.202$$
for, $$M_{\infty}=0.82, \frac{p_{0}}{p_{\infty}}=1.555$$
Coefficient of pressure is $C_{p}=\frac{p-p_{\infty}}{q_{\infty}}=\frac{p-p_{\infty}}{\frac{\gamma }{2}p_{\infty}M_{\infty}^{2}}=\frac{2}{\gamma M_{\infty}^{2}}\left ( \frac{p}{p_{\infty}}-1 \right )$$\frac{p}{p_{\infty}}=\frac{\frac{p}{p_{\infty}}}{\frac{p_{0}}{p}}=\frac{1.202}{1.555}$
Therefore, $C_{p}=\frac{2}{\left ( 1.4 \right )\left ( 0.52 \right )^{2}}\left ( \frac{1.202}{1.555}-1 \right )=-1.199$
$C_{p}=\frac{2}{\gamma M_{\infty}^{2}}\left [ \left ( \frac{1+\left ( \frac{\gamma -1}{2} \right ){M_{\infty}^{2}}}{1+\left ( \frac{\gamma -1}{2} \right )M^{2}} \right )^{\frac{\gamma }{\gamma -1}} – 1\right ]$$\Rightarrow \frac{2}{\left ( 1.4 \right )\left ( 0.52 \right )^{2}}\left [ \left ( \frac{1+\left ( \frac{1.4 -1}{2} \right ){\left ( 0.52 \right )^{2}}}{1+\left ( \frac{1.4 -1}{2} \right )\left ( 0.82 \right )^{2}} \right )^{\frac{1.4 }{1.4 -1}} – 1\right ]=-1.1984$