The entropy jump across a normal shock in an ideal gas is s R = 1 -1 ! [ p_2 p_1 ( _1…

PHYSICS · Multi-part · Fermi University Physics · Fluids & Continuum Physics · Thermal, Statistical & Complex Systems · The laws of thermodynamics

Signal Speeds in Fluids, Ch Steepening and the normal shock wave

Problem

The entropy jump across a normal shock in an ideal gas is ΔsR=1γ−1ln⁡ ⁣[p2p1(ρ1ρ2)γ]=−ln⁡p02p01,\frac{\Delta s}{R}=\frac{1}{\gamma-1}\ln\!\left[\frac{p_2}{p_1}\left(\frac{\rho_1}{\rho_2}\right)^\gamma\right]=-\ln\frac{p_{02}}{p_{01}}, with p2p1=1+2γγ+1(M12−1),ρ2ρ1=(γ+1)M12(γ−1)M12+2,\frac{p_2}{p_1}=1+\frac{2\gamma}{\gamma+1}(M_1^2-1),\qquad \frac{\rho_2}{\rho_1}=\frac{(\gamma+1)M_1^2}{(\gamma-1)M_1^2+2}, and for a weak shock its leading term is Δs/R≈2γε3/3(γ+1)2\Delta s/R\approx 2\gamma\varepsilon^3/3(\gamma+1)^2, with ε=M12−1\varepsilon=M_1^2-1. Take air, γ=1.4\gamma=1.4. ((a)) At M1=1.1M_1=1.1, evaluate the leading term and the exact Δs/R\Delta s/R, and give the ratio of the two. ((b)) Find the upstream Mach number at which a normal shock costs 11 per cent of its stagnation pressure, first from the leading term and then from the exact expression by bisection. What does the difference say about how far the cubic law can be trusted?

Browse the problem library