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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: takeshi@kyoryu.scphys.kyoto-u.ac.jp
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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

REFERENCES

Biferale, L., Daumont, I., Lacorata, G. & Vulpiani, A. 2001 Fluctuation-response relation in turbulent systems. Phys. Rev. E 65 (1), 016302.CrossRefGoogle ScholarPubMed
Brun, C. & Pumir, A. 2001 Statistics of fourier modes in a turbulent flow. Phys. Rev. E 63 (5), 056313.CrossRefGoogle Scholar
Cardy, J. 1996 Scaling and Renormalization in Statistical Physics. Cambridge University Press.CrossRefGoogle Scholar
Carini, M. & Quadrio, M. 2010 Direct-numerical-simulation-based measurement of the mean impulse response of homogeneous isotropic turbulence. Phys. Rev. E 82, 066301.CrossRefGoogle ScholarPubMed
Cugliandolo, L.F., Kurchan, J. & Parisi, G. 1994 Off equilibrium dynamics and aging in unfrustrated systems. J. Phys. I (France) 4, 1641.CrossRefGoogle Scholar
Davidson, P.A. 2004 Turbulence. Oxford University Press.Google Scholar
Eyink, G.L. & Frisch, U. 2011 Robert H. Kraichnan. In A Voyage Through Turbulence (ed. Y. Kaneda, P.A. Davidson & K.R. Sreenivasan), pp. 329–372. Cambridge University Press.CrossRefGoogle Scholar
Forster, D., Nelson, D.R. & Stephen, M.J. 1977 Large-distance and long-time properties of a randomly stirred fluid. Phys. Rev. A 16 (2), 732.CrossRefGoogle Scholar
Frisch, U. 1996 Turbulence. Cambridge University Press.Google Scholar
Gotoh, T., Rogallo, R.S., Herring, J.R. & Kraichnan, R.H. 1993 Lagrangian velocity correlations in homogeneous isotropic turbulence. Phys. Fluids A 5 (11), 2846.CrossRefGoogle Scholar
Harada, T. & Sasa, S.-I. 2005 Equality connecting energy dissipation with a violation of the fluctuation-response relation. Phys. Rev. Lett. 95, 130602.CrossRefGoogle ScholarPubMed
Harada, T. & Sasa, S.-I. 2006 Energy dissipation and violation of the fluctuation-response relation in nonequilibrium Langevin systems. Phys. Rev. E 73, 026131.CrossRefGoogle ScholarPubMed
He, G., Jin, G. & Yang, Y. 2017 Space-time correlations and dynamics coupling in turbulent flows. Annu. Rev. Fluid Mech. 49, 51–70.CrossRefGoogle Scholar
Kaneda, Y. 1981 Renormalized expansions in the theory of turbulence with the use of the Lagrangian position function. J. Fluid Mech. 107, 131–145.CrossRefGoogle Scholar
Kaneda, Y. 1993 Lagrangian and Eulerian time correlations in turbulence. Phys. Fluids A 5 (11), 2835–2845.CrossRefGoogle Scholar
Kaneda, Y. 2007 Lagrangian renormalized approximation of turbulence. Fluid Dyn. Res. 39 (7), 526–551.CrossRefGoogle Scholar
Kaneda, Y. & Gotoh, T. 1991 Lagrangian velocity autocorrelation in isotropic turbulence. Phys. Fluids A 3 (8), 1924.CrossRefGoogle Scholar
Kaneda, Y., Ishihara, T. & Gotoh, K. 1999 Taylor expansions in powers of time of lagrangian and Eulerian two-point two-time velocity correlations in turbulence. Phys. Fluids 11, 2154–2166.CrossRefGoogle Scholar
Kida, S. & Goto, S. 1997 A Lagrangian direct-interaction approximation for homogeneous isotropic turbulence. J. Fluid Mech. 345, 307–345.CrossRefGoogle Scholar
Kraichnan, R.H. 1959 The structure of isotropic turbulence at very high Reynolds numbers. J. Fluid Mech. 5, 497–543.CrossRefGoogle Scholar
Kraichnan, R.H. 1964 a Decay of isotropic turbulence in the direct-interaction approximation. Phys. Fluids 7, 1030–1048.CrossRefGoogle Scholar
Kraichnan, R.H. 1964 b Kolmogorov's hypotheses and Eulerian turbulence theory. Phys. Fluids 7 (11), 1723.CrossRefGoogle Scholar
Kraichnan, R.H. 1965 Lagrangian–history closure approximation for turbulence. Phys. Fluids 8, 575.CrossRefGoogle Scholar
Kraichnan, R.H. 1966 Isotropic turbulence and inertial-range structure. Phys. Fluids 9, 1728–1752.CrossRefGoogle Scholar
Leslie, D.C. 1973 Developments in the Theory of Turbulence. Clarendon Press.Google Scholar
Luchini, P., Quadrio, M. & Zuccher, S. 2006 The phase-locked mean impulse response of a turbulent channel flow. Phys. Fluids 18, 121702.CrossRefGoogle Scholar
Marconi, U.M.B., Puglisi, A., Rondoni, L. & Vulpiani, A. 2008 Fluctuation–dissipation: response theory in statistical physics. Phys. Rep. 461, 111–195.CrossRefGoogle Scholar
Matsumoto, T., Otsuki, M., Ooshida, T., Goto, S. & Nakahara, A. 2014 Response function of turbulence computed via fluctuation-response relation of a Langevin system with vanishing noise. Phys. Rev. E 89, 061002(R).CrossRefGoogle ScholarPubMed
Miyazaki, K. & Reichman, D.R. 2005 Mode-coupling theory and the fluctuation–dissipation theorem for nonlinear Langevin equations with multiplicative noise. J. Phys. A 38 (20), L343–L355.CrossRefGoogle Scholar
Novikov, E.A. 1965 Functionals and the random-force method in turbulence theory. Sov. Phys. JETP 20, 1290–1294.Google Scholar
Pope, S.B. 2000 Turbulent Flows. Cambridge University Press.CrossRefGoogle Scholar
Puglisi, A., Sarracino, A. & Vulpiani, A. 2017 Temperature in and out of equilibrium: a review of concepts, tools and attempts. Phys. Rep. 709–710, 1–60.CrossRefGoogle Scholar
Reichman, D.R. & Charbonneau, P. 2005 Mode-coupling theory. J. Stat. Mech. 2005 (5), P05013.CrossRefGoogle Scholar
Sagaut, P. & Cambon, C. 2008 Homogeneous Turbulence Dynamics. Cambridge University Press.CrossRefGoogle Scholar
Sain, A., Manu, & Pandit, R. 1998 Turbulence and multiscaling in the randomly forced Navier–Stokes equation. Phys. Rev. Lett. 81 (20), 4377.CrossRefGoogle Scholar
Yeung, P.K. & Pope, S.B. 1989 Lagrangian statistics from direct numerical simulations of isotropic turbulence. J. Fluid Mech. 207, 531–586.CrossRefGoogle Scholar
Zinn-Justin, J. 2002 Quantum Field Theory and Critical Phenomena. Oxford University Press.CrossRefGoogle Scholar