Stabilization of Explicit Flow Solvers Using a Proper-Orthogonal-Decomposition Technique

A new numerical technique for the stabilization of explicit computational-fluid-dynamic solvers is presented. When using computational-fluid-dynamic codes to solve practical problems one sometimes finds the solution fails to converge due to the presence of a small number of eigenvalues of the flow solver outside the unit circle. In this paper we use the proper orthogonal decomposition technique to estimate the unstable eigenvalues and eigenmodes of the flow solver as the solution diverges for linear problems or exhibits limit-cycle oscillations for nonlinear problems. We use the resulting eigeninformation to construct a preconditioner that repositions the unstable eigenvalues from outside the unit circle to a point inside the unit circle close to the origin resulting in a stable flow solver. In this work the proposed method is applied to a nonlinear steady cascade solver. However, the technique can be easily applied to external flow solvers and to unsteady frequency-domain (time-linearized and harmonic-balance) flow solvers.