Implicit Vectorial Operator Splitting for Incompressible Navier — Stokes Equations in Primitive Variables

The steady incompressible Navier—Stokes equations in primitive variables are coupled by a Poisson equation for the pressure from which the continuity equation is subtracted. The equivalence to the original N-S problem is proved. Fictitious time is added and vectorial operator-splitting is employed leaving the system coupled at each fractionaltime step which allows satisfying the boundary conditions without introducing artificial condition for the pressure. Conservative second-order approximations for the convective terms are employed on a staggered grid. The lid-driven 2D flow in a rectangular cavity is considered as a featuring example. Laminar flow is obtained in our computations for Reynolds numbers up to Re = 11000 on grids with up to 512× 512 cells. The results obtained on different grids confirm the consistency and convergence of the scheme. The flow characteristics calculated here are in very good quantitative agreement with the available numerical solutions form the literature for Re ≤ 10000.