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Correlation function and linear response function of homogeneous isotropic turbulence in the Eulerian and Lagrangian coordinates

Published online by Cambridge University Press:  25 May 2021

Takeshi Matsumoto Open the ORCID record for Takeshi Matsumoto [Opens in a new window] ,
Michio Otsuki Open the ORCID record for Michio Otsuki [Opens in a new window] ,
Takeshi Ooshida Open the ORCID record for Takeshi Ooshida [Opens in a new window]  and
Susumu Goto Open the ORCID record for Susumu Goto [Opens in a new window]
Takeshi Matsumoto*
Affiliation:
Division of Physics and Astronomy, Graduate School of Science, Kyoto University, Kyoto606-8502, Japan
Michio Otsuki
Affiliation:
Graduate School of Engineering Science, Osaka University, Toyonaka560-8531, Japan
Takeshi Ooshida
Affiliation:
Department of Mechanical and Aerospace Engineering, Tottori University, Tottori680-8552, Japan
Susumu Goto
Affiliation:
Graduate School of Engineering Science, Osaka University, Toyonaka560-8531, Japan
*
†Email address for correspondence: [email protected]
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Abstract

We study the correlation function and mean linear response function of the velocity Fourier mode of statistically steady-state, homogeneous and isotropic turbulence in Eulerian and Lagrangian coordinates through direct numerical simulation (DNS). As the Lagrangian velocity, we here adopt Kraichnan's Lagrangian-history framework where Lagrangian particles are labelled with current positions and their velocities are measured at some time before. This Lagrangian velocity is numerically calculated with a method known as the passive vector method. Our first goal is to study the relation between the correlation function and the mean linear response function in Eulerian and Lagrangian coordinates. Such a relation is known to be important in analysing the closed set of equations for the two functions, which are obtained by direct-interaction-approximation-type closures. We demonstrate numerically that the fluctuation–dissipation theorem (proportionality between the two functions) does not hold. The relation is further investigated with general analytical expressions of the mean linear response function under stochastic settings, which are known as the fluctuation-response relations in non-equilibrium statistical mechanics. Our second goal is to identify characteristic times associated with the two functions and to compare the times between the Eulerian and Lagrangian coordinates. Our DNS result supports the common view that the Eulerian characteristic times have the sweeping-time scaling ($\propto k^{-1}$, where $k$ is the wavenumber) for both functions and the Lagrangian characteristic times in the inertial range have the Kolmogorov-time scaling ($\propto k^{-2/3}$) for both functions.

JFM classification

Turbulent Flows: Turbulence theory Turbulent Flows: Turbulence simulation
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 919 , 25 July 2021 , A9
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

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