Rankine–Hugoniot relation

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 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. See also Salas (2006) for some historical background.

In a coordinate system that is moving with the shock, the Rankine–Hugoniot conditions can be expressed as:

where u_s is the shock wave speed, ρ₁ and ρ₂ are the mass density of the fluid behind and inside the shock, u₂ is the particle velocity of the fluid inside the shock, p₁ and p₂ are the pressures in the two regions, and e₁ and e₂ are the specific (with the sense of per unit mass) internal energies in the two regions. These equations can be derived in a straightforward manner from equations (12), (13) and (14) below. Using the Rankine-Hugoniot equations for the conservation of mass and momentum to eliminate u_s and u₂, the equation for the conservation of energy can be expressed as the Hugoniot equation:

where v₁ and v₂ are the uncompressed and compressed specific volumes per unit mass, respectively.

Consider gas in a one-dimensional container (e.g., a long thin tube). Assume that the fluid is inviscid (i.e., it shows no viscosity effects as for example friction with the tube walls). Furthermore, assume that there is no heat transfer by conduction or radiation and that gravitational acceleration can be neglected. Such a system can be described by the following system of conservation laws, known as the 1D Euler equations, that in conservation form is:

where

Assume further that the gas is calorically ideal and that therefore a polytropic equation-of-state of the simple form

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