The Compressible Euler Equations
The Euler equations form a set of hyperbolic equations governing inviscid and adiabatic flow. The Compressible Euler Equations interface solves the transient compressible version of the Euler equations for ideal gases
(6-12)
(6-13)
(6-14)
where
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ρ is the density (SI unit: kg/m3)
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u is the velocity vector (SI unit: m/s)
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ρu is the momentum vector (SI unit: kg/(m2 s))
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p is the absolute pressure (SI unit: Pa)
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etot is the total energy (SI unit: J/m3)
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F is the volume force vector (SI unit: N/m3)
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Q is the volumetric heat source (SI unit: W/m3)
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g is the acceleration of gravity (SI unit: m/s2)
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I is the identity matrix
The volume force and heat source terms can be added using the Volume Force and Heat Source features, respectively. The gravity terms are added when Include Gravity is selected at the interface level in the Physical Model settings.
The physics interface assumes that the fluid is an ideal gas. This is used in the definition of the Inlet and Outlet fluxes and the positivity-preserving limiter (see Shock Capturing and Positivity-Preserving Limiters). The equation of state for the internal energy per unit volume e is
Where γ is the ratio of specific heats (dimensionless), which is assumed constant. The total energy is expressed as the sum of the internal energy and the kinetic energy
The physics interface solves for the conservative dependent variables, which are the density, momentum, and total energy. In some features, the primitive variables, density, velocity, and pressure, may be used to impose initial or boundary conditions.
The absolute temperature T (SI unit: K) can be obtained from the equation of state for an ideal gas
where Rs is the specific gas constant (SI unit: J/(kg·K)).