A Direct Discretization Approach near Wall Boundaries for a Cartesian Grid Method (Consideration of Consistency between Velocity and Pressure Fields)

A new discretization scheme for a Cartesian grid method is proposed. The Navier-Stokes equation is discretized directly even in the boundary cells in order to ensure the momentum conservation. Furthermore, the Navier-Stokes equation and the pressure Poisson equation in the boundary cells are constructed with the properly interpolated flux and pressure gradient. This treatment guarantees the consistency between the velocity and pressure fields in the boundary cells. The validity of the present method is assessed in some fundamental flows. It is found that the present method significantly improves the accuracy orders for the wall shear stress as well as the velocity compared to the voxel method and the conventional direct forcing immersed boundary methods. The results by the present method are also found to be less dependent on the Courant number due to the consideration of the consistency between the velocity and pressure fields in the vicinity of the boundary. The method is also applied to a large eddy simulation of a flow past a circular cylinder at a Reynolds number 3900. The flow fields predicted by the present method are found to be in good agreement with those of the experimental and numerical studies reported in the literatures.