Solving harmonic linear problems in unsteady turbomachinery flows using a preconditioned GMRES solver

Solving harmonic linear problems in unsteady turbomachinery flows using a preconditioned GMRES solver

Stridh, M.L.O.
Eriksson, L.E.

2006-09-06

The linear harmonic approach has been applied for solving problems involving for example blade vibrations or unsteady blade row interaction effects in multistage axial compressors and turbines. The flow is decomposed into a time averaged flow and an unsteady flow perturbation which are governed by the Reynolds Averaged Navier Stokes (RANS) and Linearized Navier-Stoke (LNS) equations respectively. The LNS equations are traditionally solved by a time stepping procedure, which unfortunately has its limitation in cases with large separations in the averaged flow field, which often are the most important cases to compute. Hence the need for a numerically more robust solution algorithm. As far as the authors are aware this is one of the first applications of a really robust preconditioned GMRES algorithm to solve the LNS equation in turbomachinery problems. GMRES by itself is a method which gives an approximate solution based on minimizing the residual in a Krylov subspace of given dimension. To be able to use the GMRES in an efficient way some intelligent pre-conditioning procedure is needed. Here a preconditioning based on the basic time-stepping scheme is constructed such that the effective spectrum of the linear system is transformed into a shape which is ideal for resolving on a Krylov subspace. We have used this technique on a 3.5 stage compressor, where large separation occurs in the average flow field. The traditional time stepping procedure fails to reach to a converged solution while the GMRES with the above mentioned pre-conditioning is able to supply the desired unsteady effects in the compressor.

fluid dynamics
GMRES
linearized Navier-Stokes
harmonic
preconditioned

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conference paper

(c) 2006 Stridh, M.L.O.; Eriksson, L.E.