Rankine-Hugoniot equation

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

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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

$\rho_1u_1=\rho_2u_2\,$

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

$e_1+\frac{p_1}{\rho_1}+\frac{1}{2}u_{1}^2=e_2+\frac{p_2}{\rho_2}+\frac{1}{2}u_{2}^2$

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

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

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 $ρ 2 / ρ 1$ approaches a finite limit of 4 for a monatomic gas ( $γ$ = 5/3) and 6 for a diatomic gas ( $γ$ = 1.4).

Category: Equations of fluid dynamics