Rankine-Hugoniot equation

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, u₁ and u₂, 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

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 (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 ρ₂ / ρ₁ approaches a finite limit of 4 for a monatomic gas (γ = 5/3) and 6 for a diatomic gas (γ = 1.4).