An implicit and adaptive nonlinear frequency domain approach for periodic viscous flows

An implicit nonlinear Lower-Upper symmetric Gauss-Seidel (LU-SGS) solver has been extended to the adaptive Nonlinear Frequency Domain method (adaptive NLFD) for periodic viscous flows. The discretized equations are linearized in both spatial and temporal directions, yielding an innovative segregate approach, where the effects of the neighboring cells are transferred to the right-hand-side and are updated iteratively. This property of the solver is aligned with the adaptive NLFD concept, in which different cells have different number of modes; hence, should be treated individually. The segregate analysis of the modal equations prevents assembling and inversion of a large left-hand-side matrix, when high number of modes are involved. This is an important characteristic for a selected flow solver of the adaptive NLFD method, where a high modal content may be required in highly unsteady parts of the flow field. The implicit nonlinear LU-SGS solver has demonstrated to be both robust and computationally efficient as the number of modes is increased. The developed solver is thoroughly validated for the laminar vortex shedding behind a stationary cylinder, high angle of attack NACA0012 airfoil, and a plunging NACA0012 airfoil. An order of magnitude improvement in the computational time is observed through the developed implicit approach over the classical modified 5-stage Runge-Kutta method.