A Numerical Method for the Incompressible Navier-Stokes Equations Based on an Approximate Projection

TLDR

The paper proposes a fractional step discretization for the time‑dependent incompressible Navier–Stokes equations. It employs a projection‑based fractional step that first solves diffusion–convection equations with a second‑order upwind scheme, then approximately projects the intermediate velocity onto a divergence‑free field using a Laplacian discretization solved by multigrid. Numerical tests confirm second‑order convergence for Euler, finite‑Reynolds, and Stokes flows, and demonstrate the algorithm’s behavior on an unstable shear layer.

Abstract

In this method we present a fractional step discretization of the time-dependent incompressible Navier–Stokes equations. The method is based on a projection formulation in which we first solve diffusion–cnvection equations to predict intermediate velocities, which are then projected onto the space of divergence-free vector fields. Our treatment of the diffusion-convection step uses a specialized second-order upwind method for differencing the nonlinear convective terms that provides a robust treatment of these terms at a high Reynolds number. In contrast to conventional projection-type discretization that impose a discrete form of the divergence-free constraint, we only approximately impose the constraint; i.e., the velocity field we compute is not exactly divergence-free. The approximate projection is computed using a conventional discretization of the Laplacian and the resulting linear system is solved using conventional multigrid methods. Numerical examples are presented to validate the second-order convergence of the method for Euler, finite Reynolds number, and Stokes flow. A second example illustrating the behavior of the algorithm on an unstable shear layer is also presented.