A supersonic stream at flows past a convex corner where the wall turns away from the…

PHYSICS · Multi-part · Fermi University Physics · Fluids & Continuum Physics · Waves, acoustics and compressible flow · Waves, acoustics and compressible flow

Signal Speeds in Fluids, Ch Oblique shocks, expansion fans and blast waves

Problem

A supersonic stream at M1=1.8M_1 = 1.8 flows past a convex corner where the wall turns away from the flow by 8∘8^\circ (the wall falls away from the stream, as at the shoulder of a wedge). Someone argues: "The wall turns the stream by θ=8∘\theta = 8^\circ, so I use the θ\theta-β\beta-MM relation with M1=1.8M_1 = 1.8. Solving by bisection gives β≈41.7∘\beta \approx 41.7^\circ. Then M1n=M1sin⁡β=1.8sin⁡41.7∘=1.196M_{1n} = M_1\sin\beta = 1.8\sin 41.7^\circ = 1.196. The normal-shock relations give p2/p1=1+2γγ+1(M1n2−1)=1.50p_2/p_1 = 1 + \dfrac{2\gamma}{\gamma+1}(M_{1n}^2 - 1) = 1.50, M2n=0.845M_{2n} = 0.845, and M2=M2n/sin⁡(β−θ)=1.52M_2 = M_{2n}/\sin(\beta-\theta) = 1.52. So the corner produces a shock that raises the pressure by 50 percent and slows the flow to M2=1.52M_2 = 1.52." ((a)) Identify the error in this reasoning. ((b)) Give the correct downstream Mach number and pressure ratio p2/p1p_2/p_1.

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