Rankine-Hugoniot equation

The Rankine-Hugoniot equation governs the behaviour of shock waves. It is named after physicists William John Macquorn Rankine and Pierre Henri Hugoniot, French engineer, 1851-1887.

The idea is to consider one-dimensional, steady flow of a fluid subject to the Euler equations and require that mass, momentum, and energy are conserved. This gives three equations from which the two speeds, and , are eliminated.

It is usual to denote downstream conditions with subscript 1 and upstream conditions with subscript 2. Here, is density, speed, pressure. The symbol means internal energy per unit mass; thus if ideal gases are considered, the equation of state is .

The following equations

are equivalent to the conservation of mass, momentum, and energy respectively. Note the three components to the energy flux: mechanical work, internal energy, and kinetic energy. Sometimes, these three conditions are referred to as the Rankine-Hugoniot conditions.

Eliminating the speeds gives the following relationship:

where . Now if the ideal gas equation of state is used we get

Thus, because the pressures are both positive, the density ratio is never greater than , or about 6 for air (in which is about 1.4). This result is rather startling at first sight (what happens with really really high Mach number shocks?) but is nevertheless correct; as the strength of the shock increases, the downstream gas becomes hotter and hotter, but the density ratio approaches a finite limit.