Scale-dependent error growth in Navier-Stokes simulations

Abstract

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

Large-scale patterns set the predictability limit of extreme events in Kolmogorov flow

“…In particular, the sum of the positive Lyapunov exponents, the Kolmogorov-Sinai entropy, quantifies the rate at which different possible future states emerge from the present state of the system (Boffetta et al 2002). However, the Lyapunov exponents are very sensitive to small-scale dynamics (Aurell et al 1997;Budanur & Kantz 2022) and are only useful to characterise short-time predictability (Palmer 1993). Moreover, their relationship to certain features of the flow, such as extreme events, is not straightforward.…”

“…However, the Lyapunov exponents are very sensitive to small-scale dynamics (Aurell et al. 1997; Budanur & Kantz 2022) and are only useful to characterise short-time predictability (Palmer 1993). Moreover, their relationship to certain features of the flow, such as extreme events, is not straightforward.…”