Question

A shock moving into a stationary gas can be transformed to a stationary shock by a change in reference frame, as shown in the figure. Which of the following is/are true relating the flow properties in the two reference frames? 

• A. $T'_1 > T_1,\; T'_{01} > T_{01},\; p'_{01} > p_{01},\; \rho'_2 > \rho'_1$
• B. $T'_1 = T_1,\; T'_2 < T_{01},\; p'_{01} > p_{01},\; \rho'_2 = \rho_2$
• C. $T'_1 < T_1,\; p'_1 > p_1,\; p'_{01} > p_{01},\; \rho'_2 > \rho_1$
• D. $T'_1 = T_1,\; p_2 > p_{01},\; T'_{01} > T_{01},\; p'_{01} > p_{01}$

Hint:

When shifting reference frames in compressible flow, stagnation temperature and stagnation pressure always increase if the velocity in the new frame is higher.

Answer 1

The Correct Option is D

Step 1: Understanding the transformation.
When switching from a moving-shock frame to a stationary-shock frame, the stagnation properties upstream increase because the upstream flow velocity increases in the transformed frame. Thus $T'_{01} > T_{01}$ and $p'_{01} > p_{01}$ must hold.

Step 2: Upstream static temperature.
The upstream gas is stationary in the original frame. When we move to the shock-fixed frame, the upstream velocity becomes nonzero, but static temperature remains same: \[ T'_1 = T_1 \]

Step 3: Pressure relation.
Downstream pressure $p_2$ is always greater than upstream stagnation pressure. Hence $p_2 > p_{01}$ is a correct statement.

Step 4: Eliminating wrong options.
Options (A), (B), (C) all contradict at least one of the stagnation or static temperature rules. Only option **(D)** satisfies all correct shock relations.