Air at Mach number \(M=2\) expands around a convex corner to \(M=3\). What is the angle \(\delta\) between the forward and rearward Mach lines of the Prandtl–Meyer expansion fan (in degrees)? Given \(\nu(3)=49.76^\circ\) and \(\nu(2)=26.38^\circ\).
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Centered expansion fan: the spread between its extreme Mach lines equals the turning angle \(\theta=\nu(M_2)-\nu(M_1)\).
Step 1: Prandtl–Meyer relation for an expansion. For an isentropic centered expansion from \(M_1\) to \(M_2\), the flow deflection angle \(\theta\) equals the increase in the Prandtl–Meyer function: \[ \theta \;=\; \nu(M_2)-\nu(M_1). \] Step 2: Evaluate for the given Mach numbers. \[ \theta \;=\; \nu(3)-\nu(2) \;=\; 49.76^\circ-26.38^\circ \;=\; 23.38^\circ. \] Step 3: Connect \(\theta\) to the fan angle \(\delta\). The angle between the first (forward) and last (rearward) Mach lines in a centered fan equals the total turn \(\theta\). Hence \(\delta=\theta=23.38^\circ\). \[ \boxed{\delta=23.38^\circ} \]
