Chemistry - 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, $u 1$ and $u 2$ , are eliminated.

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

The following equations

ρ 1 u 1 = ρ 2 u 2

$p_1+\rho_1u_1^2=p_2+\rho_2u_2^2$

$u_1\left(p_1+\rho_1e_1+\rho_1u_1^2/2\right)= u_2\left(p_2+\rho_2e_2+\rho_2u_2^2/2\right)$

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:

$2\left(h_2-h_1\right)=\left(p_2-p_1\right)\cdot \left(\frac{1}{\rho_1}+\frac{1}{\rho_2}\right)$

where $h=\frac{p}{\rho} + e$ . Now if the ideal gas equation of state is used we get

$\frac{p_1}{p_2}= \frac{(\gamma+1)-(\gamma-1)\frac{\rho_1}{\rho_2}} {(\gamma+1)\frac{\rho_1}{\rho_2}-(\gamma-1)}$

Thus, because the pressures are both positive, the density ratio is never greater than $(γ + 1) / (γ - 1)$ , or about 6 for air (in which $γ$ 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.

Categories: Fluid dynamics | Equations