Across a centered Prandtl-Meyer expansion fan in steady supersonic flow of an ideal gas, which of the following thermodynamic flow properties remains strictly constant?

(A) Total (stagnation) pressure $p_0$

(B) Static temperature $T$

(C) Static pressure $p$

(D) Flow Mach number $M$

Answer Key:

A

### 5-Stage Derivation & Analytical Solution: 1. **Stage 1: Nature of Expansion Waves:** Unlike shock waves, which generate entropy ($Delta s > 0$), expansion waves consist of an infinite number of infinitesimal acoustic Mach waves. 2. **Stage 2: Thermodynamic Constraint:** Because each wave is reversible and adiabatic, the overall process is isentropic ($Delta s = 0$). 3. **Stage 3: Stagnation Property Conservation:** In an isentropic flow with no work/heat interaction, both total temperature $T_0$ and total pressure $p_0$ are conserved: $p_{02} = p_{01}$. 4. **Stage 4: Distractor Elimination:** Static pressure $p$ decreases, static temperature $T$ decreases, and Mach number $M$ increases. 5. **Stage 5: Reference:** John D. Anderson Jr., *Fundamentals of Aerodynamics*, Chapter 9.

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