RankineHugoniot

Rankine–Hugoniot conditions describe the relationship between physical states on opposite sides of a shock wave or other discontinuity in a compressible fluid. They arise from the conservation laws of mass, momentum, and energy applied in an integral form across the moving discontinuity. The conditions are named after William Rankine and Hugo Hugoniot and are fundamental in the analysis of shock waves, detonation fronts, and other abrupt changes in flow properties.

In one-dimensional, inviscid flow, the Rankine–Hugoniot conditions express conservation across the discontinuity in terms of the conserved variables. Denoting the states upstream and downstream by 1 and 2, respectively, with velocity components u1 and u2 in the shock frame and density rho, these conditions are:

- Mass: rho1 u1 = rho2 u2

- Momentum: p1 + rho1 u1^2 = p2 + rho2 u2^2

- Energy: rho1 E1 u1 + p1 u1 = rho2 E2 u2 + p2 u2, where E = e + 0.5 u^2 is the specific total energy (e is the specific internal energy).

Equivalently, the energy form can be written using the conserved total energy density rho E and the flux rho E u + p u.

For an ideal gas with ratio of specific heats gamma, the normal-shock relations can be expressed in terms of the upstream Mach number M1. The density and pressure ratios across the shock are:

- rho2/rho1 = (gamma + 1) M1^2 / ((gamma - 1) M1^2 + 2)

- p2/p1 = 1 + 2 gamma / (gamma + 1) (M1^2 - 1)

The downstream Mach number M2 is given by M2^2 = [1 + ((gamma - 1)/2) M1^2] / [gamma M1^2 - (gamma - 1)/2].

Key properties include entropy increase across shocks and the fact that these relations determine the post-shock state from the pre-shock state. RH conditions also apply to other discontinuities in conservation laws, though contact discontinuities permit pressure and velocity continuity with density jumps.