Find the pressure, temperature , Mach number, total pressure and total temperature behind the oblique shock wave.
An oblique shock wave is generated with a wave angle of \({30^ \circ }\) in a Mach \(4\) flow, at an altitude of \(9000\,m\). Find the pressure, temperature , Mach number, total pressure and total temperature behind the oblique shock wave.
Normal component of the upstream Mach number to the oblique shock wave,
\[M_{n1}=M_{1}Sin\beta =4\left ( Sin30^{\circ} \right )=2\]
From Normal shock wave properties table, for \(M_{n1}=2\)
\(\frac{p_{2}}{p_{1}}=4.5\), \(\frac{T_{2}}{T_{1}}=1.687\), \(\frac{p_{02}}{p_{01}}=0.7209\), \(M_{n2}=0.5774\)
At an altitude of \(9000\, m \) , \(p_{1}=3.08\times 10^{4} N/m^{2} \), \(T_{1}=229.74K\)
Therefore,
\[P_{2}=\left ( \frac{P_{2}}{P_{1}} \right )P_{1}=\left ( 4.5 \right )\left ( 3.08\times 10^{4} \right)= 13.86\times 10^{4} N/m^{2}\]
\[T_{2}=\left ( \frac{T_{2}}{T_{1}} \right )T_{1}=\left ( 1.687 \right )229.74=387.571K\]
From the \(\theta -\beta -M\) diagram , for, wave angle of \(\beta =30^{\circ}\) and Mach number of \(4\), deflection angle, \(\theta =17.7^{\circ}\).
Therefore,
\[M_{2}=\frac{M_{n2}}{Sin\left ( \beta -\theta \right )}=\frac{0.5774}{Sin\left ( 30^{\circ} -17.7^{\circ}\right )}=2.710\]
From the table of isentropic flow properties , for
\[M_{1}=4,\frac{p_{01}}{p_{1}}=151.8,\frac{T_{01}}{T_{1}}=4.2\]
Total pressure,
\[p_{02}=\left ( \frac{p_{02}}{p_{01}} \right )\left ( \frac{p_{01}}{p_{1}} \right )p_{1}=\left ( 0.0709 \right )\left ( 151.8 \right )\left ( 3.08\times 10^{4} \right )=331488.696N/m^{2}\]
Total temperature,
\[T_{02}=T_{01}=\left ( \frac{T_{01}}{T_{1}} \right )T_{1}=4.2\times 229.74K=964.908K\]
