Scale-dependent Error Growth in Navier--Stokes Simulations

TL;DR

This paper investigates how error growth rates in Navier--Stokes simulations depend on scale and resolution, revealing that finer scales exhibit faster error growth, which limits predictability regardless of Reynolds number.

Contribution

It quantifies the scale-dependent error growth in Navier--Stokes simulations and shows it diverges as grid spacing decreases, establishing a fundamental limit on prediction horizon.

Findings

Lyapunov exponent diverges as inverse power of grid spacing

Error growth rate is independent of Reynolds number when nondimensionalized

Finer scales exhibit much faster error growth, limiting predictability

Abstract

We estimate the maximal Lyapunov exponent at different resolutions and Reynolds numbers in large eddy (LES) and direct numerical simulations (DNS) of sinusoidally-driven Navier--Stokes equations in three dimensions. Independent of the Reynolds number when nondimensionalized by Kolmogorov units, the LES Lyapunov exponent diverges as an inverse power of the effective grid spacing showing that the fine scale structures exhibit much faster error growth rates than the larger ones. Effectively, i.e., ignoring the cut-off of this phenomenon at the Kolmogorov scale, this behavior introduces an upper bound to the prediction horizon that can be achieved by improving the precision of initial conditions through refining of the measurement grid.