Numerical simulations of unsteady viscous incompressible flows using general pressure equation

In fluid dynamics, an important problem is linked to the knowledge of the fluid pressure. Recently, another approach to study incompressible fluid flow was suggested. It consists in using a general pressure equation (GPE) derived from compressible Navier-Stokes equation. In this paper, GPE is considered and compared with the Chorin's artificial compressibility method (ACM) and the Entropically damped artificial compressibility (EDAC) method. The three methods are discretized in a staggered grid system with second order centered schemes in space and a third order Runge-Kutta scheme in time. Three test cases are realized: two-dimensional Taylor-Green vortex flow, the traveling wave and the doubly periodic shear layers. It is demonstrated that GPE is accurate and efficient to capture the correct behavior for unsteady incompressible flows. The numerical results obtained by GPE are in excellent agreement with those obtained by ACM, EDAC and a classical finite volume method with a Poisson equation. Furthermore, GPE convergence is better than ACM convergence. The proposed general pressure equation (GPE) allows to solve general, time-accurate, incompressible Navier-Stokes flows. Finally, the parametric study realized in terms of Mach and Prandtl numbers shows that the velocity divergence can be limited by an arbitrary maximum and that acoustic waves can be damped.