T h e o r y f o r t h e H i g h M a c h N u m b e r I n t e r f a c e s

In some industrial applications involving fluid flow, the flow velocity is large enough to introduce significant changes in the density and temperature of the fluid. This occurs because the thermodynamic properties of the fluid are coupled. Appreciable changes in the fluid properties are encountered as the flow velocity approaches, or exceeds, the speed of sound. As a rule of thumb, velocities greater than 0.3 times the speed of sound are considered to be high Mach number flows.

The High Mach Number Flow Interfaces theory is described in this section:

•Consistent Inlet and Outlet Conditions

•Pseudo Time Stepping for High Mach Number Flow Models

•References for the High Mach Number Flow Interfaces

Consistent Inlet and Outlet Conditions

In order to provide consistent inlet and outlet conditions for high Mach number flow the flow situation need to be monitored at the boundary. Because all flow properties are coupled, the number and combinations of boundary conditions needed for well posedness depend on the flow state—that is, with which speeds the different flow quantities are propagated at the boundary. For a detailed specification on the number of physical boundary conditions needed for well posedness, see Ref. 1.

P L A N E W A V E A N A L Y S I S O F I N V I S C I D F L O W

On inlets a plane wave analysis of the inviscid part of the flow is used in order to apply a consistent amount of boundary conditions. The method used here is described in Ref. 2.

Inviscid flow is governed by Euler’s equations, which provided that the solution is smooth and neglecting the gravity terms, can be written as

Q Fj Q

+ -------- ------- = 0

t Q xj

Considering a small region close to a boundary, the Jacobian matrices can be regarded as constant leading to a system of linear equations

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Q Fj Q

+ -------- ------- = 0

t Q 0 xj

where the subscript 0 denotes a reference state at the boundary. Assuming that the state at the boundary, described by a surface normal vector i (pointing out from the domain), is perturbed by a plane wave, the linear system of equations can be transformed to

and corresponds to the direction normal to the boundary. In the unsteady case, Euler’s equations are known to be hyperbolic in all flow regimes; subsonic, sonic and supersonic (Ref. 3). This implies that A0 has real-valued eigenvalues and corresponding eigenvectors, and can therefore be diagonalized according to

TA0T–1 = ii = 1 2 3 4 5

1 = iui2 = 13 = 1

4 = 1 + cs5 = 1 – cs

where cs is the speed of sound.

Using the primitive variables

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u Q = v

w p

The characteristic variables on the boundary are

Each characteristic can be interpreted to describe a wave transporting some quantity. The first one is an entropy wave while the two next correspond to vorticity waves. The fourth and fifth, in turn, are sound waves.

Evaluating the primitive variables in Equation 11-3, the values are taken either from the outside (specified values) or from the inside (domain values), depending on the sign of the eigenvalue corresponding to that characteristic variable. At inlets a negative eigenvalue implies that the characteristic is pointing into the domain, and to ensure upwinding the outside values are used. Correspondingly, for a positive eigenvalue the inside values are used.

Variables in Equation 11-3 with a superscript A are computed as averages of the inside and outside values.

The characteristic variables are then transformed to consistent face values of the primitive variables on the boundary in the manner of

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Characteristics Based Inlets

Applying this condition implies using the plane wave analysis described in Consistent Inlet and Outlet Conditions. Using this condition a varying flow situation at the inlet can be handled. This means that changes due to prescribed variations at the boundary, due to upstream propagating sound waves, or spurious conditions encountered during the nonlinear solution procedure, can be handled in a consistent manner. The full flow condition at the inlet is specified by the following properties

from which the density is computed using the ideal gas law. The dependent variables defined by Equation 11-5 are applied as the outside values used in Equation 11-3, and the boundary values of the dependent variables are defined from Equation 11-4.

Supersonic Inlets

Applying a Supersonic inlet, the full flow at the inlet is specified using the inlet properties in Equation 11-5. Because the flow is supersonic, all characteristic at the inlet are known to be directed into the domain, and the boundary values of the dependent variables are computed directly from the inlet properties.

Hybrid Outlet

When building a model, it is recommended that it is constructed so that as little as possible happens at the outlet. In the high Mach number flow case this implies keeping the conditions either subsonic or supersonic at the outlet. This is, however, usually not possible. For example, often one boundary adjacent to the outlet consists of a no-slip wall, in which case a boundary layer is present containing a subsonic region. The Hybrid outlet feature adds the following weak expression:

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