Investigation of various solution strategies for the time-spectral method for incompressible, viscous flows

Abstract

Time-spectral methods show a huge potential for decreasing computation time of time-periodic flows. While time-spectral methods are often used for compressible flows, applications to incompressible flows are rare. This paper presents an extension of the time-spectral method (TSM) to incompressible, viscous fluid flows using a pressure-correction algorithm in a finite volume flow solver.Several algorithmic treatments of the time-spectral operator in a pressure-correction algorithm have been investigated. Initially the single time instances were solved using the Jacobi method as preconditioner. While the existing fluid code is easily adapted, the solver shows a fast degradation in stability. Thus the solution matrix was reordered with respect to time and a block Gauss–Seidel preconditioner was applied. The single time blocks were directly solved using the Cholesky algorithm. The solver is more robust, but the current implementation is inefficient. To alleviate this problem an approach, coupling all time instances and control volumes, was developed. For the complete time and spatial system two different treatments in the preconditioner were researched.To outline the advantages and disadvantages of the proposed solution strategies the laminar flow around the pitching NACA0012 airfoil was investigated. Moreover, unsteady simulations using first and second order time-stepping techniques were used and the time-spectral results were compared to regular time-stepping approaches. It is shown that the time-spectral implementations solving the whole temporal-spatial system are faster than the regular time-stepping schemes. The efficiency of the time-spectral solver decreases with increasing number of harmonics. Furthermore, with a small number of harmonics the lift coefficient over time is not accurately predicted.