Kolmogorov flow — a sinusoidal forcing on a two-dimensional Navier-Stokes fluid with periodic boundary conditions — has been a workhorse model for studying turbulence since the 1960s. Its simplicity makes it computationally tractable; its richness makes it physically relevant.
Liao proves a theorem: solutions of 2D Kolmogorov flow preserve the spatial symmetry of their initial conditions for all time. If the initial velocity field has a specific reflection or rotation symmetry, that symmetry persists through the entire evolution, however turbulent the flow becomes. This is an exact, rigorous result about the Navier-Stokes equations.
The theorem becomes a diagnostic tool. Standard direct numerical simulations (DNS) lose this symmetry rapidly — within short integration times, the computed solution breaks symmetries that the mathematics guarantees must be preserved. The symmetry violation is not physical; it is numerical noise contaminating the trajectory. The flow has not actually broken symmetry; the simulation has.
Clean numerical simulations (CNS), which explicitly control numerical error to within specified bounds, preserve the symmetry throughout, consistent with the theorem.
The structural point: a mathematical property of the equations becomes a thermometer for the quality of their numerical solutions. The theorem tells you what the simulation must preserve, and any deviation is a direct measurement of computational infidelity. Physics provides the test that no benchmark suite can — an exact, problem-specific, time-unlimited diagnostic of whether the numbers you are computing bear any relation to the equations you wrote down.