Compressible Flow for All Mach Numbers
The High Mach Number Flow interfaces solve the following equations
(6-3)
(6-4)
(6-5)
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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p is the pressure (SI unit: Pa)
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τ is the viscous stress tensor (SI unit: Pa)
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F is the volume force vector (SI unit: N/m3)
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Cp is the specific heat capacity at constant pressure (SI unit: J/(kg·K))
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T is the absolute temperature (SI unit: K)
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q is the heat flux vector (SI unit: W/m2)
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Q contains the heat sources (SI unit: W/m3)
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S is the strain-rate tensor:
These are the fully compressible Navier-Stokes equations for a simple compressible fluid. As can be seen, the same set of equations can be assembled using, for example, a Nonisothermal Flow interface or by manually coupling a Single-Phase Flow interface with a Heat Transfer interface. The difference is that the High Mach Number Flow interface can handle flow of any Mach numbers, while the other physics interfaces are subject to The Mach Number Limit. The Mach number is defined as
where a is the speed of sound. Equation 6-3 is hyperbolic whereas Equation 6-4 and Equation 6-5 are parabolic for time-dependent flow and elliptic for stationary flow. If diffusive effects can be neglected, as is usually the case for high-speed flows, the entire system of equations becomes hyperbolic. When the Mach number passes through unity, the direction of the characteristics associated with the hyperbolic system changes. This means that new phenomena not observed for incompressible flows, such as shock waves and expansion fans, can occur (Ref. 2). The stabilization and boundary conditions must be adapted to the change in direction of the characteristics.
Note that the diffusive effects do not disappear entirely unless these terms are explicitly excluded from the equations. Instead, they are confined to either boundary layers or to “shock-waves”, which are really thin regions with steep gradients. In the High Mach Number Flow interfaces these thin regions are assumed to be underresolved, and the stabilization takes this into account. If the details of these regions are of physical interest they must be adequately resolved.
The physics interface assumes that the fluid is an ideal gas. This is necessary for the formulation of the Consistent Inlet and Outlet Conditions. The ideal gas law relates density and specific heats to the pressure and temperature. The viscosity and thermal conductivity of an ideal gas can be accurately approximated using Sutherland’s Law, which is included as an option in the High Mach Number Flow interface.
The Mach Number Limit