Spatial symmetry invariance of solution of Kolmogorov flow
Provides a rigorous mathematical check for numerical simulations of turbulence, validating CNS as a reliable method for studying Navier-Stokes equations.
The paper proves that solutions of the 2D and 3D Navier-Stokes equations for Kolmogorov flow preserve spatial symmetry for all time if the initial condition is symmetric. This theorem is used to validate CNS simulations and invalidate DNS results that lose symmetry, showing DNS trajectories are polluted by numerical noise.
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.