Tri-periodic fully three-dimensional analytic solutions for the Navier-Stokes equations

In this paper we derive unsteady tri-periodic laminar solutions of the Navier-Stokes equations. In particular, these represent fully three-dimensional (3-D) flows, since all the velocity components depend non-trivially on all three coordinate directions. We show that they belong to the class of Beltrami flows and can be gathered in two distinct solutions characterized by positive and negative helicity. These can be regarded as an extension in three dimensions of the bi-periodic vortex solution by Taylor (Phil. Mag., vol. 46, 1923, pp. 671-674). Their use as benchmarks for checking the accuracy of 3-D numerical codes and/or studying the onset of turbulence is suggested.