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.
Referenced By
Euler's equation | Euler equations | List of equations | List of mathematical topics (P-R) | Rankine | Shock wave
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