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On the normalised energy dissipation rate in homogeneous isotropic turbulence

Published online by Cambridge University Press:  28 April 2025

T. Kitamura Open the ORCID record for T. Kitamura [Opens in a new window] ,
K. Nagata Open the ORCID record for K. Nagata [Opens in a new window] ,
K. Shimoyama Open the ORCID record for K. Shimoyama [Opens in a new window]  and
T. Nanri
T. Kitamura*
Affiliation:
Graduate School of Integrated Science and Technology, Nagasaki University, Nagasaki 8528521, Japan
K. Nagata
Affiliation:
Department of Mechanical Engineering and Science, Kyoto University, Kyoto 6158540, Japan
K. Shimoyama
Affiliation:
Department of Mechanical Engineering, Kyushu University, Fukuoka 8190395, Japan
T. Nanri
Affiliation:
Research Institute for Information Technology, Kyushu University, Fukuoka 8190395, Japan
*
Corresponding author: T. Kitamura, [email protected]
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Abstract

The Reynolds number dependence of the normalised energy dissipation rate $C_{\epsilon }=\epsilon L/u^3$ is studied, where $\epsilon$ is the energy dissipation rate, $L$ is the integral length scale and $u$ is the root-mean-square velocity. We present the derivation of the exact relationship between the normalised energy dissipation rate and integrated form of the Kármán–Howarth equation in homogeneous isotropic turbulence. The present mathematical formulation is valid for both forced and decaying turbulence. The discussion of $C_{\epsilon }$ is developed under the assumption that the term resulting from the nonlinear energy transfer appearing in $C_{\epsilon }$ is constant at sufficiently high-Reynolds-number turbulence. The fact that the integrated term originating from nonlinear energy transfer is constant plays the role of a lower bound in $C_{\epsilon }$, implying that the energy dissipation rate is finite in high-Reynolds-number turbulence. Furthermore, the origin of the non-equilibrium dissipation law could be the imbalance between $u$ and ${\rm d}L/{\rm d}t$, the influence of external forces, or both. In decaying turbulence with forced turbulence as the initial condition, the imbalance between $u$ and ${\rm d}L/{\rm d}t$ causes the non-equilibrium dissipation law. The validity of the theoretical analysis is investigated using direct numerical simulations of the forced and decaying turbulence.

JFM classification

Turbulent Flows: Homogeneous turbulence Turbulent Flows: Isotropic turbulence Turbulent Flows: Turbulence theory
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 1010 , 10 May 2025 , A14
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© The Author(s), 2025. Published by Cambridge University Press

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References

Abramowitz, M. & Stegun, I.A. 1964 Handbook of Mathematical Functions. Courier Corporation.Google Scholar
Alvelius, K. 1999 Random forcing of three-dimensional homogeneous turbulence. Phys. Fluids 11 (7), 1880–1889.CrossRefGoogle Scholar
Antonia, R.A. & Burattini, P. 2006 Approach to the 4/5 law in homogeneous isotropic turbulence. J. Fluid Mech. 550 (−1), 175–184.CrossRefGoogle Scholar
Ayyalasomayajula, S. & Warhaft, Z. 2006 Nonlinear interactions in strained axisymmetric high–Reynolds–number turbulence. J. Fluid Mech. 566, 273–307.CrossRefGoogle Scholar
Batchelor, G.K. 1953 The Theory of Homogeneous Turbulence. Cambridge University Press.Google Scholar
Bos, W.J.T. & Rubinstein, R. 2017 Dissipation in unsteady turbulence. Phys. Rev. Fluids 2 (2), 022601.CrossRefGoogle Scholar
Bos, W.J.T., Shao, L. & Bertoglio, J.P. 2007 Spectral imbalance and the normalized dissipation rate of turbulence. Phys. Fluids 15 (4), 045101.CrossRefGoogle Scholar
Cafiero, G. & Vassilicos, J.C. 2019 Non-equilibrium turbulence scalings and self-similarity in turbulent planar jets. Proc. R. Soc. A 475 (2225), 20190038.CrossRefGoogle ScholarPubMed
Cambon, C. & Rubinstein, R. 2006 Anisotropic developments for homogeneous shear flows. Phys. Fluids 18 (8), 085106.CrossRefGoogle Scholar
Dairay, T., Obligado, M. & Vassilicos, J.C. 2015 Non–equilibrium scaling laws in axisymmetric turbulent wakes. J. Fluid Mech. 781, 166–195.CrossRefGoogle Scholar
Davidson, P.A. 2000 Was Loitsyansky correct? J. Turbul. 1, 14.CrossRefGoogle Scholar
Davidson, P.A. 2011 Long–range interactions in turbulence and the energy decay problem. Phil. Trans. R. Soc. A 369 (1937), 796–810.CrossRefGoogle ScholarPubMed
Davidson, P.A. 2015 Turbulence: An Introduction for Scientists and Engineers. Second edn. Oxford University Press.CrossRefGoogle Scholar
Doering, C.R. & Foias, C. 2002 Energy dissipation in body-forced turbulence. J. Fluid Mech. 467, 289–306.CrossRefGoogle Scholar
Doering, C.R. & Petrov, N.P. 2005 Low-wavenumber forcing and turbulent energy dissipation. In Progress in Turbulence, pp. 11–18. Springer.CrossRefGoogle Scholar
Durbin, P.A. & Reif, B.A.P. 2011 Statistical Theory and Modeling for Turbulent Flows. John Wiley & Sons.Google Scholar
Eswaran, V. & Pope, S.B. 1988 An examination of forcing in direct numerical simulations of turbulence. Comput. Fluids 16 (3), 257–278.CrossRefGoogle Scholar
Ferchichi, M. & Tavoularis, S. 2000 Reynolds number effects on the fine structure of uniformly sheared turbulence. Phys. Fluids 12 (11), 2942–2953.CrossRefGoogle Scholar
Fukayama, D., Oyamada, T., Nakano, T., Gotoh, T. & Yamamoto, K. 2000 Longitudinal structure functions in decaying and forced turbulence. J. Phys. Soc. Japan 69 (3), 701–715.CrossRefGoogle Scholar
Goto, S. & Vassilicos, J.C. 2015 Energy dissipation and flux laws for unsteady turbulence. Phys. Lett. A 379 (16–17), 1144–1148.CrossRefGoogle Scholar
Goto, S. & Vassilicos, J.C. 2016 Unsteady turbulence cascades. Phys. Rev. E 94 (5), 053108.CrossRefGoogle ScholarPubMed
Gotoh, T., Fukayama, D. & Nakano, T. 2002 Velocity field statistics in homogeneous steady turbulence obtained using a high–resolution direct numerical simulation. Phys. Fluids 14 (3), 1065–1081.CrossRefGoogle Scholar
Gotoh, T. & Watanabe, T. 2015 Power and nonpower laws of passive scalar moments convected by isotropic turbulence. Phys. Rev. Lett. 115 (11), 114502.CrossRefGoogle ScholarPubMed
Gradshteyn, I.S. & Ryzhik, I.M. 2014 Table of Integrals, Series, and Products. Academic Press.Google Scholar
Hearst, R.J. & Lavoie, P. 2014 Decay of turbulence generated by a square-fractal-element grid. J. Fluid Mech. 741, 567–584.CrossRefGoogle Scholar
Hearst, R.J. & Lavoie, P. 2015 Velocity derivative skewness in fractal-generated, non-equilibrium grid turbulence. Phys. Fluids 27 (7), 071701.CrossRefGoogle Scholar
Herring, J.R. 1965 Self-consistent-field approach to turbulence theory. Phys. Fluids 8 (12), 2219–2225.CrossRefGoogle Scholar
Ishihara, T., Kaneda, Y., Morishita, K., Yokokawa, M. & Uno, A. 2020 Second-order velocity structure functions in direct numerical simulations of turbulence with R λ up to 2250. Phys. Rev. Fluids 5, 104608.CrossRefGoogle Scholar
Ishihara, T., Morishita, K., Yokokawa, M., Uno, A. & Kaneda, Y. 2017 Energy spectrum in high-resolution direct numerical simulations of turbulence. Phys. Rev. Fluids 88, 154501.Google Scholar
Kaneda, Y., Ishihara, T., Yokokawa, M., Itakura, K. & Uno, A. 2003 Energy dissipation rate and energy spectrum in high resolution direct numerical simulations of turbulence in aperiodic box. Phys. Fluids 15 (2), 21–24.CrossRefGoogle Scholar
Kitamura, T. 2020 Single-time Markovianized spectral closure in fluid turbulence. J. Fluid Mech. 898, A8.CrossRefGoogle Scholar
Kitamura, T. 2021 Constant-energetics control-based forcing methods in isotropic helical turbulence. Phys. Rev. Fluids 6 (4), 044608.CrossRefGoogle Scholar
Kolmogorov, A.N. 1941 a Dissipation of energy in the locally isotropic turbulence. Dokl. Akad. Nauk. SSSR 32, 15–17.Google Scholar
Kolmogorov, A.N. 1941 b The local structure of turbulence in incompressible viscous fluid for very large reynolds numbers. Dokl. Akad. Nauk. SSSR 31, 301–305.Google Scholar
Kolmogorov, A.N. 1941 c On the degeneration of isotropic turbulence in an incompressible viscous fluid. Dokl. Akad. Nauk. SSSR 31, 538–541.Google Scholar
Krogstad, P.Å. & Davidson, P.A. 2010 Is grid turbulence Saffman turbulence? J. Fluid Mech. 642, 373–394.CrossRefGoogle Scholar
Lesieur, M. 2008 Turbulence in Fluids. 4th edn. Springer.CrossRefGoogle Scholar
Lesieur, M. & Ossia, S. 2000 3D isotropic turbulence at very high Reynolds numbers: EDQNM study. J. Turbul. 1, 1–25.CrossRefGoogle Scholar
Lohse, D. 1994 Crossover from high to low Reynolds number turbulence. Phys. Rev. Lett. 73 (24), 3223–3226.CrossRefGoogle ScholarPubMed
Lundgren, T.S. 2003 Linearly forced isotropic turbulence. In Annual Research Briefs, Center for Turbulence Research, pp. 461–473. Stanford University.Google Scholar
McComb, W.D., Berera, A., Salewski, M. & Yoffe, S. 2010 Taylor’s, 1935 dissipation surrogate reinterpreted. Phys. Fluids 22 (6), 061704.CrossRefGoogle Scholar
McComb, W.D., Berera, A., Yoffe, S.R. & Linkmann, M.F. 2015 Energy transfer and dissipation in forced isotropic turbulence. Phys. Rev. E 91 (4), 043013.CrossRefGoogle ScholarPubMed
Moisy, F., Tabeling, P. & Willaime, H. 1999 Kolmogorov equation in a fully developed turbulence experiment. Phys. Rev. Lett. 82, 3394–3397.CrossRefGoogle Scholar
Mons, V., Cambon, C. & Sagaut, P. 2016 A spectral model for homogeneous shear–driven anisotropic turbulence in terms of spherically averaged descriptors. J. Fluid Mech. 788, 147–182.CrossRefGoogle Scholar
Nedić, J., Tavoularis, S. & Marusic, I. 2017 Dissipation scaling in constant–pressure turbulent boundary layers. Phys. Rev. Fluids 2, 032601.CrossRefGoogle Scholar
Nedić, J., Vassilicos, J.C. & Ganapathisubramani, B. 2013 Axisymmetric turbulent wakes with new nonequilibrium similarity scalings. Phys. Rev. Lett. 111 (14), 144503.CrossRefGoogle ScholarPubMed
Nie, Q. & Tanveer, S. 1999 A note on third–order structure functions in turbulence. Proc. R. Soc. Lond. Ser. A 455, 1615–1635.CrossRefGoogle Scholar
Novikov, E.A. 1965 Functionals and the random-force method in turbulence theory. Sov. Phys. JETP 20, 1290–1294.Google Scholar
Onsager, L. 1949 Statistical hydrodynamics. Nuovo Cimento 6, 279–287.CrossRefGoogle Scholar
Ortiz-Tarin, J.L., Nidhan, S. & Sarkar, S. 2021 High-Reynolds-number wake of a slender body. J. Fluid Mech. 918, A30.CrossRefGoogle Scholar
Pearson, B.R., Krogstad, P.Å. & van de Water, W. 2002 Measurements of the turbulent energy dissipation rate. Phys. Fluids 14, 1288–1290.CrossRefGoogle Scholar
Piquet, J. 2001 Turbulent Flows: Models and Physics. Springer.Google Scholar
Saddoughi, S.G. & Veeravalli, S.V. 1994 Local isotropy in turbulent boundary layers at high Reynolds number. J. Fluid Mech. 268, 333–372.CrossRefGoogle Scholar
Saffman, P.G. 1968 Lectures on homogeneous turbulence. In Topics in Nonlinear Physics (ed. N.J. Zabusky), pp. 485–614. New York: Springer.CrossRefGoogle Scholar
Sagaut, P. & Cambon, C. 2018 Homogeneous Turbulence Dynamics. 2nd edn. Springer.CrossRefGoogle Scholar
Sreenivasan, K.R. 1984 On the scaling of the turbulence energy dissipation rate. Phys. Fluids 27 (5), 1048–1051.CrossRefGoogle Scholar
Sreenivasan, K.R. 1998 An update on the energy dissipation rate in isotropic turbulence. Phys. Fluids 10 (2), 528–529.CrossRefGoogle Scholar
Taylor, G.I. 1935 Statistical theory of turbulence. Proc. R. Soc. Lond. Ser. A 151 (873), 421–444.CrossRefGoogle Scholar
Tennekes, H. & Lumley, J.L. 1972 A First Course in Turbulence. MIT Press.CrossRefGoogle Scholar
Townsend, A.A. 1976 The Structure of Turbulent Shear Flow. Cambridge Univ. Press.Google Scholar
Valente, P.C., Onishi, R. & Silva, C.B. 2014 The origin of the imbalance between energy cascade and dissipation in turbulence. Phys. Rev. E 90 (2), 023003.CrossRefGoogle ScholarPubMed
Valente, P.C. & Vasslicos, J.C. 2012 Universal dissipation scaling for nonequilibrium turbulence. Phys. Rev. Lett. 108 (21), 214503.CrossRefGoogle ScholarPubMed
Vassilicos, J.C. 2015 Dissipation in turbulent flows. Annu. Rev. Fluid Mech. 47 (1), 95–114.CrossRefGoogle Scholar
Vassilicos, J.C. 2023 Beyond scale–by–scale equilibrium. Atomosphere 14 (4), 8.Google Scholar
Waclawczyk, M., Nowak, J.L., Siebert, H. & Malinowski, S. 2022 Detecting nonequilibrium states in atmospheric turbulence. J. Atmos. Sci. 79 (10), 2757–2772.CrossRefGoogle Scholar
Wilcox, D.C. 1998 Turbulence modeling for CFD. In DCW Industries La.Google Scholar
Yang, P.-F., Pumir, A. & Xu, H. 2018 Generalized self-similar spectrum and the effect of large-scale in decaying homogeneous isotropic turbulence. New J. Phys. 20 (10), 103035.CrossRefGoogle Scholar