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Rankine-hugoniot EquationThe 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, and , are eliminated. It is usual to denote upstream conditions with subscript 1 and downstream conditions with subscript 2. Here, is density, speed, pressure. The symbol means internal energy per unit mass; thus if ideal gases are considered, the equation of state is . The following equations -
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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 . 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 , or about 6 for air (in which 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.
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