Robustness and efficiency of encapsulated selective frequency damping using different operator-splitting schemes: Application to laminar cylinder flow and transonic buffet

A modified encapsulated selective frequency damping (MESFD) method based on the Strang splitting scheme is developed in this study. As a comparison the widely-used encapsulated form of selective frequency damping (ESFD) method constructed from the sequential splitting scheme is also introduced. Both methods are implemented into an implicit-time stepping Unsteady Navier-Stokes Equation (UNSE) solver and applied to solve the unstable steady solutions of the laminar cylinder flow and transonic buffet. It turns out that ESFD performs better than MESFD in convergence to the steady solution of the cylinder flow while MESFD does better in calculating the unstable steady solution of transonic buffet than ESFD. The traditional perspective of the global modes or the stability region fails to explain such discrepancies given that the same parameter combination of the control gain $\chi$ and the filtered width $\Delta$ is chosen in both methods. However, the viewpoint of local splitting errors established from the Lie operator formalism that relates the structure of the local splitting errors to the spatial accuracy and the parameter set $\left(\chi ,\Delta \right)$ gives new insights into the differences between MESFD and ESFD. How the local splitting errors contribute to the calculation and how to choose proper $\left(\chi ,\Delta \right)$ accordingly are discussed in these two flow problems. For the laminar cylinder flow problem, both ESFD and MESFD are independent of the mesh size. For the turbulent transonic buffet problem, although both ESFD and MESFD are highly sensitive to the mesh set up, MESFD is more robust and efficient for convergence to the unstable steady solution compared to ESFD.