Scale-dependent error growth in Navier-Stokes simulations.

Abstract

We estimate the maximal Lyapunov exponent at different resolutions and Reynolds numbers in large eddy simulations (LES) and direct numerical simulations 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 cutoff 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.