Time spectral method for unsteady viscous flow around an oscillating airfoil

The time spectral method casts a time-periodic flow computation into the solution of N coupled steady computations, where N indicates the number of time intervals. This method solves the periodic state directly without resolving transients, by enforcing periodicity and using a Fourier representation in time. This technique is presented to solve time-periodic unsteady Navier-Stokes equations for unsteady viscous flow over an oscillating airfoil. The Roe’s FDS scheme is used to discrete convective terms, and physical time derivative terms are discretized by a Fourier representation, in which the time derivative is formulated with the combination of flow variables at all time levels, leading to spectral accuracy. A pseudo-time derivative term is added into the original equations and the fully implicit BLU-SGS method in the pseudo-time direction is employed. The solution is obtained by marching towards a steadystate in the pseudotime variable. A subsonic case and a transonic case about the NACA0012 airfoil are calculated. BaldwinLomax turbulence model has been used in viscous flow analysis. The numerical results agree well with the experimental data. The computational results show that the small number of time intervals is sufficient to capture the flow physics phenomena.