Exact Periodic Solutions of the Forced Incompressible Navier-Stokes Equations in Arbitrary Dimensions

Abstract

Exact periodic solutions of the unforced incompressible Navier-Stokes equations require the convective field to be a pure gradient, absorbed into the pressure; for the cyclic trigonometric families studied previously this occurs only at isolated phases and, over the range so far classified, only at n = 3 and 4. The forced problem, we show, admits an exact construction for every phase vector and every n >= 3. For an n-parameter family of two-term cyclic fields, solenoidal and a Laplacian eigenfunction, the transverse part U^0 of its convective field serves as the body force, giving a closed-form solution with velocity, pressure and force explicit. We prove closed forms for the pressure and for both magnitudes, in every dimension and with no upper bound on n: the forcing is never trivial, and the L^2 magnitude of U^0 stands to that of the whole convective field in the constant ratio 2*sqrt(2)/3, whatever the dimension, phase or units. The forcing does no net work, so the kinetic energy decays purely viscously at arbitrary Reynolds number, giving an exact benchmark in any dimension. A numerical study of vorticity concentration at zero applied work uses it as a reference, and shows that apparent saturation can be a resolution artefact.

15 pages, 2 figures, 2 tables. Verification and solver scripts included as ancillary files