# 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, \[C_{p}=\frac{2}{\left ( 1.4 \right )\left ( 0.52 \right )^{2}}\left ( \frac{1.202}{1.555}-1 \right )=-1.199\]

Coefficient of pressure can also be calculated by using the formula,

\[C_{p}=\frac{2}{\gamma M_{\infty}^{2}}\left [ \left ( \frac{1+\left ( \frac{\gamma -1}{2} \right ){M_{\infty}^{2}}}{1+\left ( \frac{\gamma -1}{2} \right )M^{2}} \right )^{\frac{\gamma }{\gamma -1}} – 1\right ]\]\[\Rightarrow \frac{2}{\left ( 1.4 \right )\left ( 0.52 \right )^{2}}\left [ \left ( \frac{1+\left ( \frac{1.4 -1}{2} \right ){\left ( 0.52 \right )^{2}}}{1+\left ( \frac{1.4 -1}{2} \right )\left ( 0.82 \right )^{2}} \right )^{\frac{1.4 }{1.4 -1}} – 1\right ]=-1.1984 \]