Stability characteristics of the virtual boundary method in three-dimensional applications

The refined stability analysis of the virtual boundary method proposed by Goldstein et al. (1994) and modified by Saiki and Biringen (1996) is carried out for applications to three-dimensional turbulent flows in complex geometry. The precise stability boundaries in the forcing parameter space for various time-advancing schemes are provided under the assumption that the virtual boundary points are densely distributed. From these and the relevant investigation of frequency of the forced system, the optimum gains of the feedback forcing are suggested. Stability regimes of the Runge–Kutta schemes of various order are much wider than those of the Adams–Bashforth schemes. Specially, the third-order Runge–Kutta scheme allows the use of an order-one CFL number in the integration of the feedback forcing, rendering the method applicable to turbulent flows with complex boundaries. The three-dimensional turbulent flow caused by a surface-mounted box was simulated using a spectral method for evaluation, confirming the stability limit proposed by theoretical estimate. The method was then applied to simulations of the flow around an impulsively starting cylinder and of the rough-wall turbulent boundary layer flow.