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

Published online by Cambridge University Press:  13 December 2023

Jin Ge Open the ORCID record for Jin Ge [Opens in a new window] ,
Joran Rolland Open the ORCID record for Joran Rolland [Opens in a new window]  and
John Christos Vassilicos Open the ORCID record for John Christos Vassilicos [Opens in a new window]
Jin Ge*
Affiliation:
Univ. Lille, CNRS, ONERA, Arts et Métiers ParisTech, Centrale Lille, UMR 9014 – LMFL – Laboratoire de Mécanique des Fluides de Lille – Kampé de Feriet, F-59000 Lille, France
Joran Rolland*
Affiliation:
Univ. Lille, CNRS, ONERA, Arts et Métiers ParisTech, Centrale Lille, UMR 9014 – LMFL – Laboratoire de Mécanique des Fluides de Lille – Kampé de Feriet, F-59000 Lille, France
John Christos Vassilicos*
Affiliation:
Univ. Lille, CNRS, ONERA, Arts et Métiers ParisTech, Centrale Lille, UMR 9014 – LMFL – Laboratoire de Mécanique des Fluides de Lille – Kampé de Feriet, F-59000 Lille, France
*
†Email addresses for correspondence: jin.ge@centrale.centralelille.fr; joran.rolland@centralelille.fr; john-christos.vassilicos@centralelille.fr
†Email addresses for correspondence: jin.ge@centrale.centralelille.fr; joran.rolland@centralelille.fr; john-christos.vassilicos@centralelille.fr
†Email addresses for correspondence: jin.ge@centrale.centralelille.fr; joran.rolland@centralelille.fr; john-christos.vassilicos@centralelille.fr
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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.

JFM classification

Turbulent Flows: Isotropic turbulence Nonlinear Dynamical Systems: Chaos Turbulent Flows: Turbulence theory
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 977 , 25 December 2023 , A17
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Copyright
© The Author(s), 2023. Published by Cambridge University Press

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