The Navier-Stokes equations in periodic domains

In the present technical note, we establish that the setting of the primitive variables of the unsteady incompressible fluid dynamics is ill-formulated in spatially periodic domains as the specification of the boundary velocity is too broad to sidestep time-dependency and approximation errors. As an illustration, we show that the Taylor-Green solution in planes suffers from the Hadamard-divergence, and the ABC flow in cubes is non-unique. In direct numerical simulations of homogeneous turbulence with no corrective precautions on the boundary values, our assertion helps us understand the well-experienced nuisances, such as slow rates of convergence in energy dissipation, fluctuations in the statistics moments, or spontaneous surges in the time-averaged flow quantities. In particular, vorticity dynamics is not described by singular integral equations.