A class of fully third-order accurate projection methods for solving the incompressible Navier-Stokes equations

In this paper, a fully third-order accurate projection method for solving the incompressible Navier-Stokes equations is proposed. To construct the scheme, a continuous projection procedure is firstly presented. We then derive a sufficient condition for the continuous projection equations to be temporally third-order accurate approximations of the original Navier-Stokes equations by means of the localtruncation-error-analysis technique. The continuous projection equations are discretized temporally and spatially to third-order accuracy on the staggered grids, resulting in a fully third-order discrete projection scheme. The possibility to design higher-order projection methods is thus demonstrated in the present paper. A heuristic stability analysis is performed on this projection method showing the probability of its being stable. The stability of the present scheme is further verified through numerical tests. The third-order accuracy of the present projection method is validated by several numerical test cases.

Implementation of the Divergence-Free and Pressure-Oscillation-Free Projection Method for Solving the Incompressible Navier-Stokes Equations on the Collocated Grids

In this paper,a divergence-free and pressure-oscillation-free projection method for solving the incompressible Navier-Stokes equations on the non-staggered grid is presented.The exact discrete projection method is used to compute the velocity field,which ensures the discrete divergence of the velocity field is zero.In order to eliminate the odd-even decoupling in the pressure field,a filtering procedure is proposed and applied to the pressure field.We have shown this filter recovers the grid scale ellipticity in the pressure field and the odd-even decoupling can be removed effectively.The proposed numerical scheme is further verified through numerical experiments.