Efficiency of two approaches to computing the flow around an airfoil with flaps in the case of flow separation

Model equations describing convective transport are used to analyze the approximation errors of an explicit numerical scheme and various implicit schemes with the same approximation of spatial derivatives. It is shown that, under time step constraints determined by the Courant–Friedrichs–Lewy condition, the implicit scheme is inferior in accuracy to the explicit one and, with a further increase in the time step, the accuracy of simulated convective processes degrades substantially. Two methods for implementing the marching procedure in time are considered, namely, a fractional step in the case of an explicit scheme and a dual step in the case of an implicit scheme. It is shown that the fractional step method is efficient only on grids with a scatter of cell sizes of 100–1000. For the numerical solution of problems with no-slip conditions on solid walls (scatter of cell sizes of 10$^4$–10$^5$), two approaches are proposed: an implicit scheme with a dual step in all cells and an zonal approach, in which a dual step is used in a thin near-wall domain (about 3% of the thickness of a developed turbulent boundary layer), while a fractional step is applied in the rest of the domain. These two approaches are used to compute the flow over an airfoil with flaps. Numerical and experimental data are compared. The accuracy of the numerical results is estimated. The causes of error formation are examined. The domain of efficient application is determined for each of the indicated approaches.