The Navier-Stokes equations in periodic domains

TL;DR

This paper critically examines the formulation of the Navier-Stokes equations in periodic domains, revealing fundamental issues with boundary conditions and their impact on turbulence simulations and flow properties.

Contribution

It demonstrates that the primitive variable formulation in periodic domains is ill-posed, affecting the accuracy and interpretation of turbulence simulations.

Findings

Taylor-Green solution suffers from Hadamard-divergence

ABC flow in cubes is non-unique

Turbulence simulations exhibit slow convergence and fluctuations

Abstract

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.