The Rankine–Hugoniot conditions, also referred to as Rankine–Hugoniot jump conditions or Rankine–Hugoniot relations, describe the relationship between the states on both sides of a shock wave or a combustion wave (deflagration or detonation) in a one-dimensional flow in fluids or a one-dimensional deformation in solids. They are named in recognition of the work carried out by Scottish engineer and physicist William John Macquorn Rankine and French engineer Pierre Henri Hugoniot. The basic idea of the jump conditions is to consider what happens to a fluid when it undergoes a rapid change. Consider, for example, driving a piston into a tube filled with non-reacting gas. A disturbance is propagated through the fluid somewhat faster than the speed of sound. Because the disturbance propagates supersonically, it is a shock wave, and the fluid downstream of the shock has no advance information of it. In a frame of reference moving with the wave, atoms or molecules in front of the wave slam into the wave supersonically. On a microscopic level, they undergo collisions on the scale of the mean free path length until they come to rest in the post-shock flow (but moving in the frame of reference of the wave or of the tube). The bulk transfer of kinetic energy heats the post-shock flow. Because the mean free path length is assumed to be negligible in comparison to all other length scales in a hydrodynamic treatment, the shock front is essentially a hydrodynamic discontinuity. The jump conditions then establish the transition between the pre- and post-shock flow, based solely upon the conservation of mass, momentum, and energy. The conditions are correct even though the shock actually has a positive thickness. This non-reacting example of a shock wave also generalizes to reacting flows, where a combustion front (either a detonation or a deflagration) can be modeled as a discontinuity in a first approximation.
Governing equations In a coordinate system that is moving with the discontinuity, the Rankine–Hugoniot conditions can be expressed as:
where m is the mass flow rate per unit area, ρ1 and ρ2 are the mass density of the fluid upstream and downstream of the wave, u1 and u2 are the fluid velocity upstream and downstream of the wave, p1 and p2 are the pressures in the two regions, and h1 and h2 are the specific (with the sense of per unit mass) enthalpies in the two regions. If in addition, the flow is reactive, then the species conservation equations demands that
ω i , 1 = ω i , 2 = 0 , i = 1 , 2 , 3 , … , N , Conservation of species {\displaystyle \omega _{i,1}=\omega _{i,2}=0,\quad i=1,2,3,\dots ,N,\qquad {\text{Conservation of species}}}
to vanish both upstream and downstream of the discontinuity. Here, ω {\displaystyle \omega } is the mass production rate of the i-th species of total N species involved in the reaction. Combining conservation of mass and momentum gives us
p 2 − p 1 1 / ρ 2 − 1 / ρ 1 = − m 2 {\displaystyle {\frac {p_{2}-p_{1}}{1/\rho _{2}-1/\rho _{1}}}=-m^{2}}
which defines a straight line known as the Michelson–Rayleigh line, named after the Russian physicist Vladimir A. Mikhelson (usually anglicized as Michelson) and Lord Rayleigh, that has a negative slope (since m 2 {\displaystyle m^{2}} is always positive) in the p − ρ − 1 {\displaystyle p-\rho ^{-1}} plane. Using the Rankine–Hugoniot equations for the conservation of mass and momentum to eliminate u1 and u2, the equation for the conservation of energy can be expressed as the Hugoniot equation:
h 2 − h 1 = 1 2 ( 1 ρ 2 + 1 ρ 1 ) ( p 2 − p 1 ) . {\displaystyle h_{2}-h_{1}={\frac {1}{2}}\,\left({\frac {1}{\rho _{2}}}+{\frac {1}{\rho _{1}}}\right)\,(p_{2}-p_{1}).}
The inverse of the density can also be expressed as the specific volume, v = 1 / ρ {\displaystyle v=1/\rho } . Along with these, one has to specify the relation between the upstream and downstream equation of state
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