Operator-Splitting Schemes for the Navier-Stokes Equations on the Staggered Grid.

Abstract: New implicit finite-difference schemes to solve the time-dependent incompressible Navier-Stokes equations in the Boussinesq approximation using primitive variables and the staggered grid of MAC-type are developed and verified. On the staggered grid the pressure is defined at the center points of cells whereas the velocity components are referred to the corresponding faces of the cell. Special approximations of differential operators are used in order to obtain difference operators which heritage the fundamental properties of the corresponding initial operators. Employed spatial approximations are of the second order. A priori estimate for the discrete solution of the methods are obtained. The estimate is similar to the corresponding one for the solution of the differential problem and guarantees boundedness of the solution. To solve systems of grid equations, the following iterative methods have been employed: the local relaxation technique and preconditioned conjugate gradients methods. It is shown that the derived scheme has a very weak restriction on the time-step size due to nonlinerity of the equations. To examine stability and accuracy of the suggested scheme, the following test problems have been studied: a lid-driven cavity flow, a flow over a backward-facing step and free convection of a heat-generating fluid. Model predictions of a flow in the geometry similar to real reactors are conducted too. There is described a parallel version of the local relaxation method adaptive to MIMD computer architecture.Note: Research direction:Mathematical problems and theory of numerical method