Numerical methods for the unsteady Navier–Stokes equations using primitive variables and partially staggered grids

New implicit finite-difference schemes for solving the time-dependent incompressible Navier–Stokes equations using primitive variables and partially staggered grids are presented in this paper. In the partially staggered grids the pressure is defined at the center points of cells and the velocity components are all defined at the nodes of the mesh. 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 is obtained. The estimate is similar to the corresponding one for the solution of the differential problem and guarantees finiteness of the solution. A way of pressure discrete problem simplification by means of adding of regularizing terms is suggested. The additional terms are proportional to $O(\tau h^2)$. It is shown that the derived scheme has a very weak restriction on a time-step size. To examine stability and accuracy of the suggested schemes, two test problems have been studied: (a) a lid-driven cavity flow (Reynolds number up to 3200 on thesequence of grids $21\times21$, $41\times41$, $81\times81$ and $161\times161$) and (b) a flow over bakward-facing step ($\mathrm{Re}=800$, grids $181\times41$ and $361\times81$). The suggested methods are compared with methods using the non-staggered grid.