A normal shock wave stands in a supersonic flow of air ($\gamma = 1.4$) where upstream Mach number is $M_1 = 2.0$. Which of the following statements regarding the flow changes across this normal shock are **CORRECT**?

(A) The downstream Mach number is strictly subsonic and equals $M_2 = \sqrt{\frac{2 + (\gamma-1)M_1^2}{2\gamma M_1^2 - (\gamma-1)}} \approx 0.577$

(B) The static pressure and static temperature both increase across the shock ($p_2 > p_1$ and $T_2 > T_1$)

(C) The stagnation temperature drops across the shock wave ($T_{02} < T_{01}$) because shock compression causes thermal dissipation

(D) The stagnation pressure drops across the shock wave ($p_{02} < p_{01}$) due to non-zero entropy production across the shock

Answer Key:

Verified in official key

### 1. Strategy & Theoretical Principles Across a normal shock wave in an ideal gas: 1. **Energy Conservation:** No external heat is added or removed ($q = 0$), so the total enthalpy is constant: $h_{01} = h_{02}$. For a calorically perfect gas ($h_0 = c_p T_0$), the **stagnation temperature is constant**: $T_{01} = T_{02}$. 2. **Thermodynamic Irreversibility:** The shock is an irreversible process governed by the second law of thermodynamics ($\Delta s > 0$). The entropy jump causes a **drop in stagnation pressure**: $$\frac{p_{02}}{p_{01}} = \exp\left(-\frac{\Delta s}{R}\right) < 1$$ 3. **Downstream Mach Number:** From 1D normal shock relations: $$M_2^2 = \frac{2 + (\gamma - 1) M_1^2}{2\gamma M_1^2 - (\gamma - 1)}$$ For $M_1 = 2.0$ and $\gamma = 1.4$: $$M_2^2 = \frac{2 + 0.4(4)}{2(1.4)(4) - 0.4} = \frac{3.6}{11.2 - 0.4} = \frac{3.6}{10.8} = \frac{1}{3} \implies M_2 = \frac{1}{\sqrt{3}} \approx 0.5774$$ ### 2. Statement Evaluation - **Statement A:** **CORRECT.** $M_2 = 1/\sqrt{3} \approx 0.577$. - **Statement B:** **CORRECT.** Pressure and temperature jump across the compression shock. - **Statement C:** **INCORRECT.** Stagnation temperature is strictly conserved ($T_{02} = T_{01}$). - **Statement D:** **CORRECT.** Stagnation pressure drops ($p_{02} < p_{01}$) due to entropy generation. ### 3. Exam Trap & Standard Reference - **Trap Alert:** Across a shock wave, stagnation pressure drops ($p_{02} < p_{01}$), but stagnation temperature remains strictly unchanged ($T_{02} = T_{01}$). - **Standard Reference:** Anderson, J.D., *Modern Compressible Flow*, Chapter 3.

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