  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.
