Operator-splitting methods for the incompressible Navier-Stokes equations on non-staggered grids. Part 1: First-order schemes

New implicit finite difference schemes for solving the time-dependent incompressible Navier-Stokes equations using primitive variables and non-staggered grids are presented in this paper. A priori estimates for the discrete solution of the methods are obtained. Employing the operator approach, some requirements on the difference operators of the scheme are formulated in order to derive a scheme which is essentially consistent with the initial differential equations. The operators of the scheme inherit the fundamental properties of the corresponding differential operators and this allows a priori estimates for the discrete solution to be obtained. The estimate is similar to the corresponding one for the solution of the differential problem and guarantees boundedness of the solution. To derive the consistent scheme, special approximations for convective terms and div and grad operators are employed. Two variants of time discretization by the operator-splitting technique are considered and compared. It is shown that the derived scheme has a very weak restriction on the time step size. A lid-driven cavity flow has been predicted to examine the stability and accuracy of the schemes for Reynolds number up to 3200 on the sequence of grids with 21 × 21, 41 × 41, 81 × 81 and 161 × 161 grid points.