Jump relations across a shock in non-ideal gas flow

Generalized forms of jump relations are obtained for one dimensional shock waves propagating in a non-ideal gas which reduce to Rankine-Hugoniot conditions for shocks in idea gas when non-idealness parameter becomes zero. The equation of state for non-ideal gas is considered as given by Landau and Lifshitz. The jump relations for pressure, density, temperature, particle velocity, and change in entropy across the shock are derived in terms of upstream Mach number. Finally, the useful forms of the shock jump relations for weak and strong shocks, respectively, are obtained in terms of the non-idealness parameter. It is observed that the shock waves may arise in flow of real fluids where upstream Mach number is less than unity.

💡 Research Summary

The paper presents a systematic extension of the classic Rankine‑Hugoniot jump conditions to one‑dimensional shock waves propagating in a non‑ideal gas. The authors adopt the Landau‑Lifshitz equation of state for a real gas, (p=(\gamma-1)\rho e,(1+b\rho)), where (b) is a non‑ideality parameter that accounts for intermolecular interactions and finite molecular volume. When (b\to0) the equation reduces to the ideal‑gas law, and the derived relations revert to the standard Rankine‑Hugoniot formulas.

Starting from the conservation of mass, momentum, and energy across a stationary shock, the authors substitute the non‑ideal equation of state and express all downstream quantities in terms of the upstream Mach number (M_1 = u_1/a_1) and the parameter (b). The resulting explicit relations are:

1. Pressure ratio (p_2/p_1 = \frac{2\gamma M_1^2-(\gamma-1)}{\gamma+1},\frac{1+b\rho_1}{1+b\rho_2}).
2. Density ratio (\rho_2/\rho_1 = \frac{(\gamma+1)M_1^2}{(\gamma-1)M_1^2+2},\frac{1+b\rho_1}{1+b\rho_2}).
3. Temperature ratio (T_2/T_1 = \frac{