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Modelling Lagrangian velocity and acceleration in turbulent flows as infinitely differentiable stochastic processes

Published online by Cambridge University Press:  11 August 2020

Bianca Viggiano Open the ORCID record for Bianca Viggiano [Opens in a new window] ,
Jan Friedrich ,
Romain Volk ,
Mickael Bourgoin ,
Raúl Bayoán Cal  and
Laurent Chevillard Open the ORCID record for Laurent Chevillard [Opens in a new window]
Bianca Viggiano
Affiliation:
Laboratoire de Physique, Univ Lyon, Ens de Lyon, Univ Claude Bernard, CNRS, 46 allée d'Italie, F-69342Lyon, France Department of Mechanical and Materials Engineering, Portland State University, Portland, OR97201, USA
Jan Friedrich
Affiliation:
Laboratoire de Physique, Univ Lyon, Ens de Lyon, Univ Claude Bernard, CNRS, 46 allée d'Italie, F-69342Lyon, France
Romain Volk
Affiliation:
Laboratoire de Physique, Univ Lyon, Ens de Lyon, Univ Claude Bernard, CNRS, 46 allée d'Italie, F-69342Lyon, France
Mickael Bourgoin
Affiliation:
Laboratoire de Physique, Univ Lyon, Ens de Lyon, Univ Claude Bernard, CNRS, 46 allée d'Italie, F-69342Lyon, France
Raúl Bayoán Cal
Affiliation:
Laboratoire de Physique, Univ Lyon, Ens de Lyon, Univ Claude Bernard, CNRS, 46 allée d'Italie, F-69342Lyon, France Department of Mechanical and Materials Engineering, Portland State University, Portland, OR97201, USA
Laurent Chevillard*
Affiliation:
Laboratoire de Physique, Univ Lyon, Ens de Lyon, Univ Claude Bernard, CNRS, 46 allée d'Italie, F-69342Lyon, France
*
†Email address for correspondence: laurent.chevillard@ens-lyon.fr
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Abstract

We develop a stochastic model for Lagrangian velocity as it is observed in experimental and numerical fully developed turbulent flows. We define it as the unique statistically stationary solution of a causal dynamics, given by a stochastic differential equation. In comparison with previously proposed stochastic models, the obtained process is infinitely differentiable at a given finite Reynolds number, and its second-order statistical properties converge to those of an Ornstein–Uhlenbeck process in the infinite Reynolds number limit. In this limit, it exhibits furthermore intermittent scaling properties, as they can be quantified using higher-order statistics. To achieve this, we begin with generalizing the two-layered embedded stochastic process of Sawford (Phys. Fluids A, vol. 3 (6), 1991, pp. 1577–1586) by considering an infinite number of layers. We then study, both theoretically and numerically, the convergence towards a smooth (i.e. infinitely differentiable) Gaussian process. To include intermittent corrections, we follow similar considerations as for the multifractal random walk of Bacry et al. (Phys. Rev. E, vol. 64, 2001, 026103). We derive in an exact manner the statistical properties of this process, and compare them with those estimated from Lagrangian trajectories extracted from numerically simulated turbulent flows. Key predictions of the multifractal formalism regarding the acceleration correlation function and high-order structure functions are also derived. Through these predictions, we understand phenomenologically peculiar behaviours of the fluctuations in the dissipative range, that are not reproduced by our stochastic process. The proposed theoretical method regarding the modelling of infinitely differentiability opens the route to the full stochastic modelling of velocity, including the peculiar action of viscosity on the very fine scales.

JFM classification

Turbulent Flows: Homogeneous turbulence Turbulent Flows: Isotropic turbulence Turbulent Flows: Turbulence theory
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 900 , 10 October 2020 , A27
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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References

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