A superconsistent collocation method for high Reynolds number flows

Abstract

We use a novel implicit second-order projection method together with a superconsistent collocation scheme (Funaro, 1993; Fatone et al., 2005; De l’Isle and Owens, 2021) for the solution of the primitive variable formulation of the Navier–Stokes equations at very high Reynolds numbers. In particular, we apply a superconsistent collocation scheme to the convection–diffusion equation arising from one step of the projection method and this represents the first time that superconsistent collocation methods have been employed for the solution of a convection–diffusion equation having unsteady convection velocity. In order to evidence the second-order (in space and time) convergence of our scheme for both the velocity and pressure fields we choose to solve the two-dimensional unsteady Taylor–Green vortex problem (Taylor and Green, 1937). The numerical results presented for the solution of the two-dimensional square lid-driven cavity problem are in excellent agreement over the whole range of Reynolds numbers considered (5000≤Re≤1000) with some others in the literature (Ghia et al., 1982; Wang and Liu, 2019; Bruneau and Saad, 2006; Auteri et al., 2002; Pan and Glowinski, 2000).