A uniform supersonic airstream with Mach number $M_1 = 1.5$ and specific heat ratio $\gamma = 1.4$ expands around a sharp convex corner through a deflection angle $\theta = 15^\circ$. Given the Prandtl-Meyer function values $\nu(1.5) \approx 11.9^\circ$ and $\nu(2.0) \approx 26.4^\circ$, which of the following statements regarding the downstream flow after the expansion fan are **CORRECT**?

(A) The downstream Mach number is approximately $M_2 \approx 2.0$

(B) The static pressure and static temperature both decrease across the expansion fan ($p_2 < p_1$ and $T_2 < T_1$)

(C) The total (stagnation) pressure drops across the expansion fan due to entropy generation ($p_{02} < p_{01}$)

(D) The flow across a centered Prandtl-Meyer expansion fan is isentropic ($s_2 = s_1$)

Answer Key:

Verified in official key

### 1. Strategy & Theoretical Principles For a centered Prandtl-Meyer expansion fan: 1. The expansion is smooth and composed of an infinite family of Mach waves (infinitesimal disturbances). 2. Across Mach waves, entropy change is zero to third order ($ds = 0$). Hence, the entire expansion process is **isentropic** ($s_2 = s_1$). 3. Because the flow is isentropic, the stagnation pressure is strictly conserved: $p_{02} = p_{01}$. 4. The downstream Mach number $M_2$ is determined from the Prandtl-Meyer angle equation: $$\nu(M_2) = \nu(M_1) + \theta$$ ### 2. Analytical Formulation & Calculation Given $\nu(M_1) = 11.9^\circ$ and $\theta = 15.0^\circ$: $$\nu(M_2) = 11.9^\circ + 15.0^\circ = 26.9^\circ$$ From the given table values, $\nu(2.0) = 26.4^\circ$, confirming $M_2 \approx 2.0$. In any expansion fan: - Flow accelerates ($M_2 > M_1$ and $V_2 > V_1$). - Static pressure drops ($p_2 < p_1$). - Static temperature drops ($T_2 < T_1$). - Stagnation pressure remains unchanged ($p_{02} = p_{01}$). ### 3. Statement Evaluation - **Statement A:** **CORRECT.** $\nu(M_2) = 26.9^\circ \approx \nu(2.0) = 26.4^\circ$. - **Statement B:** **CORRECT.** Flow expands; static pressure and temperature drop. - **Statement C:** **INCORRECT.** Total pressure does NOT drop; unlike shock waves, expansion fans are isentropic ($p_{02} = p_{01}$). - **Statement D:** **CORRECT.** Expansion occurs through an infinite series of isentropic Mach lines. ### 4. Exam Trap & Standard Reference - **Trap Alert:** Never confuse oblique shocks (entropy increases, $p_0$ drops) with Prandtl-Meyer expansion fans (isentropic, $p_0$ constant). - **Standard Reference:** Anderson, J.D., *Modern Compressible Flow*, Chapter 4 (Oblique Shock and Expansion Waves).

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