A pressure-based velocity decomposition procedure for the incompressible Navier-Stokes equations

An alternative formulation is presented for the incompressible form of the Navier-Stokes equations. This is done through the introduction of a new vector variable which represents the sum of the velocity vector and the pressure gradient vector. The introduction of the new variable modifies the continuity equation to a Poisson equation for pressure, with the forcing function on the right-hand side representing the divergence of the new vector variable. Hence, the proposed formulation utilizes pressure to satisfy the divergence-free condition on the differential level in a simple and straightforward manner. Moreover, with pressure introduced to the continuity equation on the differential level, the present formulation allows direct computation of the pressure and velocity fields from the continuity and momentum equations, respectively, and eliminates the need for computing pressure and velocity corrections to satisfy mass conservation as in SIMPLE-type pressure correction methods. The proposed procedure is numerically tested against several two-dimensional and three-dimensional steady flow problems. Discretization is carried out using a second-order accurate central difference scheme on a staggered grid and the discretized equations are solved in a segregated manner. Numerical results from the present formulation are in good agreement with conventional incompressible flow solvers and with experimentation.