On the consistency of the direct forcing method in the fractional step solution of the Navier–Stokes equations

In the present work the application of the direct forcing method to the fractional step solution of the Navier–Stokes equations is analyzed in details. The central topic of the study is the satisfaction of the boundary conditions in the presence of immersed bodies. To this aim, simple two-dimensional problems with steady forcing and sharp-edged surfaces are solved with a standard fractional step method. It is shown that the direct forcing scheme is not able to satisfy the impenetrability condition. This fact appears to be strictly related to the use of fractional step methods. A mathematically based explanation is given, suggesting that improvements can be obtained with the iterative solution of the irrotational part of the flow, when spectral methods are used and local modification of the discrete differential operators are difficult to be implemented. The results are then verified in two-dimensional unsteady problems and three-dimensional flows with moving smooth boundaries.