AAWSAP DIRD, Detection and High-Resolution Tracking of Vehicles at Hypersonic Velocities, November 2010

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D. Compute the Properties on the Downstream Side of the Bow Shock at the Nose

1. The following functions describe the change in properties across a normal shock.

$$
M_2(M_1) := \sqrt{\frac{1 + \frac{(\gamma - 1)}{2} \cdot M_1^2}{\gamma \cdot M_1^2 - \frac{(\gamma - 1)}{2}}}
$$

$$
\rho_{\text{ratio}}(M_1) := \frac{(\gamma + 1) \cdot M_1^2}{2 + (\gamma - 1) \cdot M_1^2}
$$

$$
p_{\text{ratio}}(M_1) := 1 + \frac{2\gamma}{\gamma + 1} (M_1^2 - 1) \qquad u_{\text{ratio}}(M_1) := \frac{1}{\rho_{\text{ratio}}(M_1)}
$$

$$
T_{\text{ratio}}(M_1) := \left[ 1 + \frac{2\gamma}{\gamma + 1} \cdot (M_1^2 - 1) \right] \cdot \left[ \frac{2 + (\gamma - 1) \cdot M_1^2}{(\gamma + 1) \cdot M_1^2} \right]
$$

a. Stagnation temperature and pressure ratios.

$$
T_{\text{o.ratio}}(M_1) := 1.0
$$

$$
p_{\text{o.ratio}}(M_1) := p_{\text{ratio}}(M_1) \cdot \frac{\left[ 1 + \frac{\gamma - 1}{2} \cdot (M_2(M_1))^2 \right]^{\frac{\gamma}{\gamma - 1}}}{\left( 1 + \frac{\gamma - 1}{2} \cdot M_1^2 \right)^{\frac{\gamma}{\gamma - 1}}}
$$

2. Compute the properties across a normal shock for this problem.

$$
M_2 := M_2(M_1) \qquad M_2 = 0.416
$$

$$
T_2 := T_1 \cdot T_{\text{ratio}}(M_1) \qquad T_2 = 1.263 \times 10^3 \text{ K}
$$

$$
p_2 := p_1 \cdot p_{\text{ratio}}(M_1) \qquad p_2 = 7.447 \times 10^5 \text{ Pa}
$$

$$
p_{\text{o}2} := p_{\text{o}1} \cdot p_{\text{ratio}}(M_1) \qquad p_{\text{o}2} = 3.624 \times 10^8 \text{ Pa}
$$

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