Implicit Time-Spectral Method for Unsteady Reynolds-Averaged Navier–Stokes Computations of Turbulent Flows

Abstract

An efficient implicit Fourier time-spectral method is developed for the unsteady Reynolds-averaged Navier–Stokes (URANS) computations of turbulent flows. In addition to the time-spectral URANS equations, the time-spectral Spalart–Allmaras turbulence model equation is also solved by the space–time lower-upper symmetric Gauss–Seidel (LU-SGS) implicit scheme. The efficient LU-SGS sweeps are directly extended to the time domain without introducing additional approximation error, delayed implicit temporal coupling, and additional algorithmic steps. The ability of the algorithm is verified through the test cases of the pitching NACA0012 airfoil and pitching ONERA M6 wing. Dominant Fourier modes of the drag coefficient and its components are identified. Special attention is paid to the nonlinear behavior of the surface pressure and skin friction coefficients in the shock-free area, especially the region downstream the shock where the influence of turbulence is significant. For the test case of a pitching ONERA M6 wing, difference in local flow unsteadiness for the two branches of the -type shock is observed. The phenomenon is further discussed through study of the shock motion range and the upstream Mach number for each shock branch.