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Mathematical proof about spatial symmetry of solutions of the two-dimensional Kolmogorov flow

Shijun Liao

Abstract

We give a mathematical proof that solution for all t > 0 of the two-dimensional (2D) Kolmogorov flow governed by Navier-Stokes (NS) equations with periodic boundary condition remains the same spatial symmetry as its smooth initial condition at t=0. This mathematical theorem supports the corresponding CNS (clean numerical simulation) results of the 2D turbulent Kolmogorov flow[1,2] that remain the same spatial symmetry, but does not support the corresponding DNS (direct numerical simulation) results that lose the spatial symmetry quickly. In other words, these DNS results violate this mathematical theorem. Thus, this mathematical theorem rigorously confirms that the spatiotemporal trajectories of NS turbulence given by DNS are indeed quickly polluted by numerical noises badly. It also illustrates that CNS can provide helpful enlightenments to deepen our understanding about turbulence and besides approach some mathematical truths about NS equations.

Mathematical proof about spatial symmetry of solutions of the two-dimensional Kolmogorov flow

Abstract

We give a mathematical proof that solution for all t > 0 of the two-dimensional (2D) Kolmogorov flow governed by Navier-Stokes (NS) equations with periodic boundary condition remains the same spatial symmetry as its smooth initial condition at t=0. This mathematical theorem supports the corresponding CNS (clean numerical simulation) results of the 2D turbulent Kolmogorov flow[1,2] that remain the same spatial symmetry, but does not support the corresponding DNS (direct numerical simulation) results that lose the spatial symmetry quickly. In other words, these DNS results violate this mathematical theorem. Thus, this mathematical theorem rigorously confirms that the spatiotemporal trajectories of NS turbulence given by DNS are indeed quickly polluted by numerical noises badly. It also illustrates that CNS can provide helpful enlightenments to deepen our understanding about turbulence and besides approach some mathematical truths about NS equations.
Paper Structure (3 sections, 3 theorems, 46 equations)

This paper contains 3 sections, 3 theorems, 46 equations.

Table of Contents

  1. Introduction
  2. Spatial symmetry of 2D Kolmogorov flow
  3. Concluding remarks and discussions

Key Result

theorem 1

For the two-dimensional Kolmogorov flow governed by the Navier-Stokes equation (eq_psi) (with an even number $n_{K}$) under the periodic boundary condition (boundary_condition) subject to a smooth initial condition with the spatial symmetry of rotation (symmetry_psi:A) and/or translation (symmetry_p

Theorems & Definitions (3)

  • theorem 1
  • theorem 2
  • theorem 3