Modified pressure and vorticity variables using Helmholtz decomposition for solution of the incompressible flow equations

Abstract

Incompressible Navier–Stokes equations are reformulated using the Helmholtz decomposition of a velocity vector into rotational and potential components. By substituting the decomposed velocity in the time derivative term in a momentum equation, the potential component representing a gradient of a pressure-like term is combined with the gradient of the pressure modifying the physical pressure field. The rotational component representing a curl of a vortex-like vector is combined with the vorticity vector, making a non-physical vorticity vector that modifies the fluid viscosity. Thus, the unsteady Navier–Stokes equation is transformed into an explicitly steady-state form in terms of non-primitive variables. The stream function vector is governed by a parabolic equation in time, while the vorticity vector is governed by the Poisson function with a source term function of the convection stretching and time dependency of the physical flow vorticity. Therefore, the resulting system of equations is numerically independent of the cell Reynolds number stability condition that hunted the convection–diffusion Navier–Stokes equation. Numerical results are obtained for the two-dimensional driven cavity problem for Reynolds number of 1000 with 21 × 21, 41 × 41, 81 × 81, and 161 × 161 grid points. The computational grids correspond to cell Reynolds numbers 25, 12.5, 6.25, and 3.125, respectively. The computed results are smooth in all cases and validate the method.