Chapter 17.7, Problem 151RP

To determine

The expressions for the ratio of the stagnation pressure after a shock wave to the static pressure before the shock wave as a function of $k$ and the Mach number upstream of the shock wave ${\text{Ma}}_1$.

Answer to Problem 151RP

The expressions for the ratio of the stagnation pressure after a shock wave to the static pressure before the shock wave as a function of $k$ and the Mach number upstream of the shock wave ${\text{Ma}}_1$ is obtained and shown in Equation (VI).

Explanation of Solution

Write the Equation 17-38 as in text book (the relation between the pressures after shock and before shock for an ideal gases).

$\frac{P_2}{P_1}=\frac{1+k{\text{Ma}}_1^2}{1+k{\text{Ma}}_2^2}$ (I)

Here, the specific heat ratio is $k$, the Mach number is $\text{Ma}$, the subscript 1 and 2 indicates the states of before and after shocks.

Write the relation between the stagnation pressure $\left(P_0\right)$ and static pressure $\left(P\right)$ for ideal gas at isentropic flow.

$\frac{P_0}{P}={\left[1+\left(\frac{k-1}{2}\right){\text{Ma}}^2\right]}^{\frac{k}{k-1}}$ (II)

Here, the subscript 0 indicates the stagnation state.

Write the Equation 17-39 as in text book (the expression for Mach number after shock).

${\text{Ma}}_2^2=\frac{{\text{Ma}}_1^2+2/\left(k-1\right)}{2{\text{Ma}}_1^{2}k/\left(k-1\right)-1}$ (III)

Conclusion:

Rearrange the Equation (I) to obtain $P_2$.

$P_2=P_1\left(\frac{1+k{\text{Ma}}_1^2}{1+k{\text{Ma}}_2^2}\right)$

Express the Equation (II) for state 2 i.e. after shock.

$\frac{P_{02}}{P_2}={\left[1+\left(\frac{k-1}{2}\right){\text{Ma}}_2^2\right]}^{\frac{k}{k-1}}$ (IV)

Substitute $P_1\left(\frac{1+k{\text{Ma}}_1^2}{1+k{\text{Ma}}_2^2}\right)$ for $P_2$ in Equation (IV)

$\begin{array}{l}\frac{P_{02}}{P_1\left(\frac{1+k{\text{Ma}}_1^2}{1+k{\text{Ma}}_2^2}\right)}={\left[1+\left(\frac{k-1}{2}\right){\text{Ma}}_2^2\right]}^{\frac{k}{k-1}}\\ \frac{P_{02}}{P_1}=\left(\frac{1+k{\text{Ma}}_1^2}{1+k{\text{Ma}}_2^2}\right){\left[1+\left(\frac{k-1}{2}\right){\text{Ma}}_2^2\right]}^{\frac{k}{k-1}}\end{array}$ (V)

Refer Equation (III).

Substitute $\frac{{\text{Ma}}_1^2+2/\left(k-1\right)}{2{\text{Ma}}_1^{2}k/\left(k-1\right)-1}$ for ${\text{Ma}}_2^2$ in Equation (V).

$\begin{array}{c}\frac{P_{02}}{P_1}=\left(\frac{1+k{\text{Ma}}_1^2}{1+k\left(\frac{{\text{Ma}}_1^2+2/\left(k-1\right)}{2{\text{Ma}}_1^{2}k/\left(k-1\right)-1}\right)}\right){\left[1+\left(\frac{k-1}{2}\right)\left(\frac{{\text{Ma}}_1^2+2/\left(k-1\right)}{2{\text{Ma}}_1^{2}k/\left(k-1\right)-1}\right)\right]}^{\frac{k}{k-1}}\\ =\left(\frac{1+k{\text{Ma}}_1^2}{\frac{\left[2{\text{Ma}}_1^{2}k/\left(k-1\right)-1\right]+\left[k{\text{Ma}}_1^2+k2/\left(k-1\right)\right]}{2{\text{Ma}}_1^{2}k/\left(k-1\right)-1}}\right){\left[1+\frac{\left(k-1\right)\left[{\text{Ma}}_1^2+2/\left(k-1\right)\right]}{2\left(2{\text{Ma}}_1^{2}k/\left(k-1\right)-1\right)}\right]}^{\frac{k}{k-1}}\\ =\left(\frac{\left(1+k{\text{Ma}}_1^2\right)\left(2{\text{Ma}}_1^{2}k/\left(k-1\right)-1\right)}{2{\text{Ma}}_1^{2}k/\left(k-1\right)-1+k{\text{Ma}}_1^2+k2/\left(k-1\right)}\right){\left[1+\frac{\left(k-1\right){\text{Ma}}_1^2+2}{2\left(2{\text{Ma}}_1^{2}k/\left(k-1\right)-1\right)}\right]}^{\frac{k}{k-1}}\\ =\left(\frac{\left(1+k{\text{Ma}}_1^2\right)\left(\frac{1}{k-1}\right)\left(2{\text{Ma}}_1^{2}k-1\left(k-1\right)\right)}{2{\text{Ma}}_1^{2}k/\left(k-1\right)-1+k{\text{Ma}}_1^2+k2/\left(k-1\right)}\right){\left[1+\frac{2\left[\left(k-1\right){\text{Ma}}_1^2/2+1\right]}{2\left(2{\text{Ma}}_1^{2}k/\left(k-1\right)-1\right)}\right]}^{\frac{k}{k-1}}\end{array}$

$\begin{array}{c}=\left(\frac{\left(1+k{\text{Ma}}_1^2\right)\left(2{\text{Ma}}_1^{2}k-k+1\right)}{\left(k-1\right)\left[2{\text{Ma}}_1^{2}k/\left(k-1\right)-1+k{\text{Ma}}_1^2+k2/\left(k-1\right)\right]}\right){\left[1+\frac{\left(k-1\right){\text{Ma}}_1^2/2+1}{2{\text{Ma}}_1^{2}k/\left(k-1\right)-1}\right]}^{\frac{k}{k-1}}\\ =\left(\frac{\left(1+k{\text{Ma}}_1^2\right)\left(2{\text{Ma}}_1^{2}k-k+1\right)}{2{\text{Ma}}_1^{2}k-1\left(k-1\right)+\left(k-1\right)k{\text{Ma}}_1^2+2k}\right){\left[1+\frac{\left(k-1\right){\text{Ma}}_1^2/2+1}{2{\text{Ma}}_1^{2}k/\left(k-1\right)-1}\right]}^{\frac{k}{k-1}}\\ =\left(\frac{\left(1+k{\text{Ma}}_1^2\right)\left(2{\text{Ma}}_1^{2}k-k+1\right)}{2{\text{Ma}}_1^{2}k-k+1+k^2{\text{Ma}}_1^2-k{\text{Ma}}_1^2+2k}\right){\left[1+\frac{\left(k-1\right){\text{Ma}}_1^2/2+1}{2{\text{Ma}}_1^{2}k/\left(k-1\right)-1}\right]}^{\frac{k}{k-1}}\\ =\left(\frac{\left(1+k{\text{Ma}}_1^2\right)\left(2{\text{Ma}}_1^{2}k-k+1\right)}{2{\text{Ma}}_1^{2}k+k^2{\text{Ma}}_1^2-k{\text{Ma}}_1^2-k+1+2k}\right){\left[1+\frac{\left(k-1\right){\text{Ma}}_1^2/2+1}{2{\text{Ma}}_1^{2}k/\left(k-1\right)-1}\right]}^{\frac{k}{k-1}}\end{array}$

$\begin{array}{c}=\left(\frac{\left(1+k{\text{Ma}}_1^2\right)\left(2{\text{Ma}}_1^{2}k-k+1\right)}{k{\text{Ma}}_1^2\left(2+k-1\right)+k+1}\right){\left[1+\frac{\left(k-1\right){\text{Ma}}_1^2/2+1}{2{\text{Ma}}_1^{2}k/\left(k-1\right)-1}\right]}^{\frac{k}{k-1}}\\ =\left(\frac{\left(1+k{\text{Ma}}_1^2\right)\left(2{\text{Ma}}_1^{2}k-k+1\right)}{k{\text{Ma}}_1^2\left(k+1\right)+k+1}\right){\left[1+\frac{\left(k-1\right){\text{Ma}}_1^2/2+1}{2{\text{Ma}}_1^{2}k/\left(k-1\right)-1}\right]}^{\frac{k}{k-1}}\end{array}$ (VI)

Thus, the expressions for the ratio of the stagnation pressure after a shock wave to the static pressure before the shock wave as a function of $k$ and the Mach number upstream of the shock wave ${\text{Ma}}_1$ is obtained and shown in Equation (VI).