Iterative Construction of n-Dimensional Navier-Stokes Solutions with Non-Gradient Convection

Abstract

Exact periodic solutions of the incompressible Navier-Stokes equations arise when the convective field is a pure gradient and can be absorbed into the pressure. We construct solutions outside this class, in arbitrary spatial dimension, by an iteration in which each step solves a linear diffusion problem forced by the transverse part of the preceding convective field. The construction is dimension-independent, and for a cyclic family of initial fields the first two iterates are available in closed form. Carried out numerically in three dimensions, the iteration converges: at Re=10 successive iterates contract with an observed factor 0.57 and the limit satisfies the mild equation with relative defect 1.0 x 10^-3. The observed factor exceeds unity by Re=30, which identifies the practical range of the iteration rather than a proved convergence boundary. We also show what goes wrong when the projection is omitted, as it was in an earlier iteration of the author's: the iterates cease to be divergence-free and an apparent growth appears that belongs to the expansion and not to the flow. For an n-dimensional generalisation of the initial field the fraction of the convective field not absorbed by pressure is 2sqrt(2)/3, independently of n over the range examined. Divergence and cell kinetic energy provide simple diagnostics that detect failure at the step where it occurs.