Consistent Inlet and Outlet Conditions
In order to provide consistent inlet and outlet conditions for high Mach number flow, the flow situation needs 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.
Plane Wave Analysis of Inviscid Flow
On inlets a plane wave analysis of the inviscid part of the flow is used in order to apply a consistent number of boundary conditions. The method used here is described in Ref. 3.
Inviscid flow is governed by Euler’s equations, which, provided that the solution is smooth and neglecting the gravity terms, can be written as
Considering a small region close to a boundary, the Jacobian matrices can be regarded as constant, which leads to a system of linear equations
where the subscript 0 denotes a reference state at the boundary. Assuming that the state at the boundary, described by a surface normal vector αi (pointing out from the domain), is perturbed by a plane wave, the linear system of equations can be transformed to
where
and ξ 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 flow (Ref. 4). This implies that A0 has real-valued eigenvalues and corresponding eigenvectors, and it can therefore be diagonalized according to
The matrix T contains the (left) eigenvectors, and the matrix Λ is a diagonal matrix containing the eigenvalues. The eigenvalues are given exactly by
where cs is the speed of sound. Using the primitive variables
The characteristic variables on the boundary are
(5-8)
Each characteristic can be interpreted to describe a wave transporting some quantity. The first one is an entropy wave while the next two correspond to vorticity waves. The fourth and fifth, in turn, are sound waves.
Evaluating the primitive variables in Equation 5-8, the values are taken 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 hence outside values are used. Correspondingly, for a positive eigenvalue the inside values are used.
Variables in Equation 5-8 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
(5-9)
Characteristics Based Inlets
Applying this condition implies using the plane wave analysis described in Consistent Inlet and Outlet Conditions. With 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
(5-10)
from which the density is computed using the ideal gas law. The dependent variables defined in Equation 5-10 are applied as the outside values used in Equation 5-8, and the boundary values of the dependent variables are obtained from Equation 5-9.
Supersonic Inlets
Applying a supersonic inlet, the full flow at the inlet is specified using the inlet properties in Equation 5-10. 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 containing a subsonic region is present. The hybrid outlet feature adds the following weak expression:
where û is the test function for the velocity vector. This corresponds to a pressure, no viscous stress condition in regions with subsonic flow and a no viscous stress condition in regions with supersonic flow. When the static pressure at the outlet is not known beforehand, it is recommended that it is set to the inlet pressure. When a converged solution has been reached, the solution can be analyzed to find the pressure level just outside the sonic point (Ma = 1) along the boundary. You can then apply this pressure level instead.
Supersonic Outlet
When the outlet condition is known to be fully supersonic, the viscous stress is specified in accordance with the equations and hence no physical condition is applied. This is done by prescribing the boundary stress using the full stress vector:
It is often possible to use the supersonic condition at outlets that are not strictly supersonic but mainly supersonic (the main part of the outlet boundary contains supersonic flow).