Discussion on momentum interpolation method for collocated grids of incompressible flow

Discussions are given of the different momentum interpolation methods to evaluate the interface velocity in the collocated grid system. It is pointed out that the interface velocity is used in three cases in the overall numerical procedure of the solution of Navier-Stokes equations by utilizing a collocated grid: in the continuity equation; in the interface flow rate computation for the determination of the coefficients in discretization equation; and in the mass residual in the coefficient A(P). Analysis shows that it is better to adopt the momentum interpolation method in the three cases. Two new momentum interpolation methods, called MMIM1 and MMIM2, are proposed. Analysis shows that the two new methods can achieve numerical solutions that are independent of both the underrelaxation factor and the time step size. Taking lid-driven cavity flow as an example, numerical computations are conducted for several Reynolds numbers and different mesh sizes using the SIMPLE algorithm, and the results are compared with benchmark solutions. Numerical tests demonstrate that both MMIM1 and MMIM2 can give unique solutions for different underrelaxation factors and time step sizes, solutions from MMIM1 are slightly better than that of the momentum interpolation of Majumdar, and solutions from MMIM2 have an appreciably better accuracy when the mesh is not fine.