Spatial symmetry invariance of solution of Kolmogorov flow

TL;DR

This paper proves that solutions of 2D Kolmogorov flow governed by Navier-Stokes equations preserve initial spatial symmetry over time, providing a mathematical basis to evaluate turbulence simulation reliability.

Contribution

It establishes a rigorous theorem confirming symmetry preservation in 2D and 3D Navier-Stokes solutions, supporting the validity of clean numerical simulations over direct numerical simulations.

Findings

CNS results maintain symmetry, unlike DNS results.

Mathematical theorems confirm symmetry preservation in NS turbulence.

DNS results violate the proven symmetry preservation, indicating numerical noise issues.

Abstract

We prove a mathematical theorem that solution for all $t > 0$ of the two-dimensional (2D) Kolmogorov flow governed by Navier-Stokes (NS) equations with periodic boundary condition keeps the same spatial symmetry as its smooth initial condition. The proof of a similar theorem for the three-dimensional NS equations is given in the appendix. These mathematical theorems can be used to check the correctness and reliability of numerical simulations of NS turbulence. For example, they support the corresponding CNS (clean numerical simulation) results of the 2D and 3D turbulent Kolmogorov flows [1-3] that remain the same spatial symmetry in the whole time interval of simulation, but do not support the corresponding DNS (direct numerical simulation) results that lose the spatial symmetry quickly. In other words, these DNS results violate these mathematical theorems. Thus, these mathematical theorems rigorously confirm that the spatiotemporal trajectories of NS turbulence given by DNS are indeed quickly polluted by numerical noises badly. All of these indicate that CNS can indeed provide helpful enlightenments to deepen our understanding about turbulence and besides approach some mathematical truths about NS equations.