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
Evaluating the primitive variables in Equation 6-9, 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 6-9 with a superscript
A are computed as averages of the inside and outside values.
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
Applying a supersonic inlet, the full flow at the inlet is specified using the inlet properties in Equation 6-11. 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.
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.