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, <math>u_1<math> and <math>u_2<math>, are eliminated.
It is usual to denote downstream conditions with subscript 1 and upstream conditions with subscript 2. Here, <math>\rho<math> is density, <math>u<math> speed, <math>p<math> pressure. The symbol <math>e<math> means internal energy per unit mass; thus if ideal gases are considered, the equation of state is <math> p=\rho(\gamma-1)e<math>.
The following equations
- <math>\rho_1u_1=\rho_2u_2<math>
- <math>p_1+\rho_1u_1^2=p_2+\rho_2u_2^2<math>
- <math>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)
<math>
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:
- <math>
2\left(h_2-h_1\right)=\left(p_2-p_1\right)\cdot
\left(\frac{1}{\rho_1}+\frac{1}{\rho_2}\right)
<math>
where <math>h=\frac{p}{\rho} + e<math>.
Now if the ideal gas equation of state is used we get
- <math>
\frac{p_1}{p_2}=
\frac{(\gamma+1)-(\gamma-1)\frac{\rho_1}{\rho_2}}
{(\gamma+1)\frac{\rho_1}{\rho_2}-(\gamma-1)}
<math>
Thus, because the pressures are both positive, the density ratio is never greater than <math>(\gamma+1)/(\gamma-1)<math>, or about 6 for air (in which <math>\gamma<math> 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.
