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