The production of uncertainty in three-dimensional Navier-Stokes turbulence

TL;DR

This paper derives an evolution equation for uncertainty energy in 3D Navier-Stokes turbulence, revealing how strain rate dynamics influence uncertainty growth, with DNS simulations showing self-similar spectra and intermittent extreme events.

Contribution

It introduces a new evolution equation for uncertainty energy in turbulence and demonstrates its behavior through DNS, highlighting the role of strain rates and intermittency in uncertainty growth.

Findings

Uncertainty energy grows exponentially in a similarity regime.

Uncertainty production is highly intermittent and skewed towards extreme compression.

Self-similar evolution of the uncertainty spectrum is observed when normalized.

Abstract

We derive the evolution equation of the average uncertainty energy for periodic/homogeneous incompressible Navier-Stokes turbulence and show that uncertainty is increased by strain rate compression and decreased by strain rate stretching. We use three different direct numerical simulations (DNS) of non-decaying periodic turbulence and identify a similarity regime where (a) the production and dissipation rates of uncertainty grow together in time, (b) the parts of the uncertainty production rate accountable to average strain rate properties on the one hand and fluctuating strain rate properties on the other also grow together in time, (c) the average uncertainty energies along the three different strain rate principal axes remain constant as a ratio of the total average uncertainty energy, (d) the uncertainty energy spectrum's evolution is self-similar if normalised by the uncertainty's average uncertainty energy and characteristic length and (e) the uncertainty production rate is extremely intermittent and skewed towards extreme compression events even though the most likely uncertainty production rate is zero. Properties (a), (b) and (c) imply that the average uncertainty energy grows exponentially in this similarity time range. The Lyapunov exponent depends on both the Kolmogorov time scale and the smallest Eulerian time scale, indicating a dependence on random large-scale sweeping of dissipative eddies. In the two DNS cases of statistically stationary turbulence, this exponential growth is followed by an exponential of exponential growth, which is in turn followed by a linear growth in the one DNS case where the Navier-Stokes forcing also produces uncertainty.