Incremental unknowns for solving the incompressible Navier–Stokes equations

Abstract

Incremental unknowns, earlier designed for the long-term integration of dissipative evolutionary equations, are introduced here for the incompressible Navier–Stokes equations in primitive variables when multilevel finite-difference discretizations on a staggered grid are used for the spatial discretization; extending to this practical case, the notion of small and large wavelengths that stems naturally from spectral methods when Fourier series expansions are considered. Furthermore, for the temporal discretization, we use the θ-scheme, which allows to decouple the nonlinearity and the incompressibility in these equations; then we have to solve a generalized Stokes equation — we consider here a leading preconditioned outer/inner iteration strategy — and a nonlinear elliptic equation — we linearize its nonlinear terms to get an iterative process. Roughly, at each iterative stage several Poisson equations must be solved, incremental unknowns appear there as an efficient preconditioner. The incremental unknowns methodology appears well suited to capture the turbulent behavior of the flow whose small eddies, even for moderate Reynolds numbers, never go to a steady state; instead they converge to a strange attractor, and keep always bringing imperceptible kinetic energy to the flow. First, they wander around, then they converge to the strange attractor.