Consider a normal discontinuity at rest in its own frame

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

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

Problem

Consider a normal discontinuity at rest in its own frame, with gas entering at (ρ1,u1,p1)(\rho_1,u_1,p_1) and leaving at (ρ2,u2,p2)(\rho_2,u_2,p_2), and specific enthalpy h^=γγ−1pρ\hat h=\dfrac{\gamma}{\gamma-1}\dfrac{p}{\rho} for an ideal gas. ((a)) Write down the mass, momentum and energy conservation conditions across the discontinuity in this frame, and eliminate the velocities u1,u2u_1,u_2 between them to obtain the Hugoniot relation h^2−h^1=12(p2−p1)(1ρ1+1ρ2).\hat h_2-\hat h_1=\tfrac12(p_2-p_1)\Big(\frac{1}{\rho_1}+\frac{1}{\rho_2}\Big). ((b)) Insert the ideal-gas enthalpy and solve for the density ratio as a function of the pressure ratio alone, obtaining ρ2ρ1=(γ+1)(p2/p1)+(γ−1)(γ+1)+(γ−1)(p2/p1).\frac{\rho_2}{\rho_1}=\frac{(\gamma+1)(p_2/p_1)+(\gamma-1)}{(\gamma+1)+(\gamma-1)(p_2/p_1)}. ((c)) Check this relation against the worked example of a jump entered at 680 m s−1680\,\mathrm{m\,s^{-1}} in air at 293 K293\,\mathrm{K}, for which p2/p1=4.4p_2/p_1=4.4: evaluate ρ2/ρ1\rho_2/\rho_1 and compare with the normal-shock table value 2.672.67 quoted for M1=2M_1=2.

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