Spatial symmetry invariance of three-dimensional Navier–Stokes turbulence

Direct numerical simulation (DNS) of the deterministic Navier–Stokes (NS) equations is widely used to provide benchmark solutions of turbulent flow, but its accuracy and reliability have not been rigorously inspected from a mathematical viewpoint. Here, we prove a spatial symmetry invariance theorem (SSIT) of three-dimensional NS equations whereby an exact solution of deterministic NS equations under periodic boundary conditions maintains the same spatial symmetry as its symmetric initial flow, forever. SSIT provides an absolute benchmark test for computational fluid dynamics (CFD). It is shown herein that traditional DNS of NS turbulence quickly violates SSIT, revealing that the spatiotemporal trajectory of NS turbulence predicted by DNS soon becomes badly polluted by numerical noise. Importantly, it is found that even the statistical results of DNS might be unstable, i.e. exhibit sensitivity dependence on the initial flow conditions. In other words, the loss of spatial symmetry in traditional DNS of NS turbulence is not a physical instability, but instead a numerical artefact caused by the chaotic nature of the flow. Using the SSIT, it is straightforward to deduce several key properties of NS turbulence, including ‘ultra-chaos’, the so-called ‘noise-expansion cascade’, ‘permanent memory of symmetric initial flow conditions’ and so on. Note that permanent memory of symmetric initial flow conditions suggests that NS turbulence governed by deterministic NS equations subject to symmetric initial flow should have no ‘mixing’, corresponding to non-ergodicity, although ergodicity is one of the most fundamental assumptions of turbulence theory. SSIT is therefore capable of greatly deepening our physical insight concerning NS turbulence. In the future, SSIT could provide a completely new theoretical cornerstone and a completely novel perspective regarding turbulence, considered by Feynman to be the most important unsolved problem of classical physics.