# 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, [itex]u_1[itex] and [itex]u_2[itex], are eliminated.

It is usual to denote upstream conditions with subscript 1 and downstream conditions with subscript 2. Here, [itex]\rho[itex] is density, [itex]u[itex] speed, [itex]p[itex] pressure. The symbol [itex]e[itex] means internal energy per unit mass; thus if ideal gases are considered, the equation of state is [itex] p=\rho(\gamma-1)e[itex].

The following equations

[itex]\rho_1u_1=\rho_2u_2\,[itex]
[itex]p_1+\rho_1u_1^2=p_2+\rho_2u_2^2[itex]
[itex]u_1\left(p_1+\rho_1e_1+\rho_1u_1^2/2\right)=
      u_2\left(p_2+\rho_2e_2+\rho_2u_2^2/2\right)


[itex]

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:

[itex]

2\left(h_2-h_1\right)=\left(p_2-p_1\right)\cdot \left(\frac{1}{\rho_1}+\frac{1}{\rho_2}\right) [itex] where [itex]h=\frac{p}{\rho} + e[itex]. Now if the ideal gas equation of state is used we get

[itex]

\frac{p_1}{p_2}= \frac{(\gamma+1)-(\gamma-1)\frac{\rho_1}{\rho_2}} {(\gamma+1)\frac{\rho_1}{\rho_2}-(\gamma-1)} [itex]

Thus, because the pressures are both positive, the density ratio is never greater than [itex](\gamma+1)/(\gamma-1)[itex], or about 6 for air (in which [itex]\gamma[itex] is about 1.4). As the strength of the shock increases, the downstream gas becomes hotter and hotter, but the density ratio approaches a finite limit.de:Rankine-Hugoniot-Gleichung

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