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Governing equations

The compressible Euler equations can be formulated as a system of conservation laws for a fluid in absence of diffusive phenomena due to viscous stress and thermal exchanges. The set of differential equations in one dimension for the conserved quantities (density, momentum and energy) is given by :

where

In the above formulae, is the density, is the momentum, is the energy, is the pressure and is the dynamic enthalpy. This last variable is related to the other quantities by the following relation :

This system of three differential equations in four independent variables (, , and ) is closed by a fourth relation coming from thermodynamics, the equation of state. If we assume that the gas is calorically perfect and polytropic, pressure is related to the other variables by the relation:

where is the ratio of specific heats and takes (for air) the value of .