Theoretical and Computational Fluid Dynamics

, Volume 33, Issue 2, pp 181–196 | Cite as

Analysis of destruction term in transport equation for turbulent energy dissipation rate

  • Fujihiro HambaEmail author
  • Kouta Kanamoto
Original Article


The K\(\varepsilon \) model used in turbulence simulations involves the transport equations for the turbulent kinetic energy and its dissipation rate. In contrast to the turbulent energy equation derived from the Navier–Stokes equation, the transport equation for the dissipation rate has been modeled empirically and has less of a theoretical grounding. An analysis of the dependence of terms in the exact transport equation on the Reynolds number \(\textit{Re}\) has suggested that the two dominant \(O(\textit{Re}^{1/2})\) terms cancel out at the leading order and their O(1) difference yields the terms in the model equation. The two-scale direct interaction approximation (TSDIA) can be used to derive the model equation for inhomogeneous turbulence. In this study, as a first step toward deriving all the terms in the model equation, the two dominant terms in the exact transport equation were investigated theoretically and numerically for the case of homogeneous isotropic turbulence, in order to derive the destruction term. The two terms were first analyzed by using the TSDIA under the assumption of a simple energy spectrum form. However, the finite-width effect of the inertial range of the energy spectrum did not give the expected \(O(\textit{Re}^{-1/2})\) corrections to the leading-order terms. Then, the model equations of the Markovianized Lagrangian renormalized approximation (MLRA) were numerically solved to obtain the accurate energy spectrum profile of a decaying homogeneous isotropic turbulence. Thereby, it was shown that the deviation of the energy spectrum from the \(-5/3\) law was responsible for the \(O(\textit{Re}^{-1/2})\) corrections. By assuming the energy spectrum suggested by the MLRA results, the two dominant terms were analyzed again by using the TSDIA to successfully derive the destruction term in the model equation.


Turbulence model K\(\varepsilon \) model Energy dissipation rate 


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This work was supported by JSPS KAKENHI Grant Number JP17K06143.


  1. 1.
    Gotoh, T.: Fundamentals of Turbulence Theory. Asakura, Tokyo (1998)Google Scholar
  2. 2.
    Gotoh, T., Kaneda, Y., Bekki, N.: Numerical integration of the Lagrangian renormalized approximation. J. Phys. Soc. Jpn. 57, 866–880 (1988)CrossRefGoogle Scholar
  3. 3.
    Hamba, F.: Statistical investigation of the energy dissipation equation in shear turbulence. J. Phys. Soc. Jpn. 56, 3771–3774 (1987)CrossRefGoogle Scholar
  4. 4.
    Hanjalić, K., Launder, B.: Modelling Turbulence in Engineering and the Environment: Second-Moment Routes to Closure. Cambridge University Press, Cambridge (2011)CrossRefzbMATHGoogle Scholar
  5. 5.
    Hanjalić, K., Launder, B.E.: Contribution towards a Reynolds-stress closure for low-Reynolds-number turbulence. J. Fluid Mech. 74, 593–610 (1976)CrossRefzbMATHGoogle Scholar
  6. 6.
    Horiuti, K., Tamaki, T.: Nonequilibrium energy spectrum in the subgrid-scale one-equation model in large-eddy simulation. Phys. Fluids 25, 125104 (2013)CrossRefGoogle Scholar
  7. 7.
    Ishihara, T., Kaneda, Y., Yokokawa, M., Itakura, K., Uno, A.: Energy spectrum in the near dissipation range of high resolution direct numerical simulation of turbulence. J. Phys. Soc. Jpn. 74, 1464–1471 (2005)CrossRefzbMATHGoogle Scholar
  8. 8.
    Kanamoto, K., Hamba, F.: Destruction term in transport equation for turbulent energy dissipation rate. In: Meeting Abstracts of the Physical Society of Japan, vol. 72, p. 2831 (2017)Google Scholar
  9. 9.
    Kaneda, Y.: Renormalized expansions in the theory of turbulence with the use of the Lagrangian position function. J. Fluid Mech. 107, 131–145 (1981)CrossRefzbMATHGoogle Scholar
  10. 10.
    Kida, S., Goto, S.: A Lagrangian direct-interaction approximation for homogeneous isotropic turbulence. J. Fluid Mech. 345, 307–345 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Kraichnan, R.H.: Lagrangian-history closure approximation for turbulence. Phys. Fluids 8, 575–598 (1965)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Kraichnan, R.H.: Inertial-range transfer in two- and three-dimensional turbulence. J. Fluid Mech. 47, 525–535 (1971)CrossRefzbMATHGoogle Scholar
  13. 13.
    Launder, B.E., Spalding, D.B.: The numerical computation of turbulent flows. Comput. Methods Appl. Mech. Eng. 3, 269–289 (1974)CrossRefzbMATHGoogle Scholar
  14. 14.
    Leschziner, M.: Statistical Turbulence Modelling for Fluid Dynamics—Demystified. Imperial College Press, London (2016)zbMATHGoogle Scholar
  15. 15.
    Leslie, D.C.: Developments in the Theory of Turbulence. Oxford University Press, Oxford (1973)zbMATHGoogle Scholar
  16. 16.
    Lumley, J.L.: Similarity and the turbulent energy spectrum. Phys. Fluids 10, 855–858 (1967)CrossRefGoogle Scholar
  17. 17.
    Mansour, N.N., Kim, J., Moin, P.: Reynolds-stress and dissipation-rate budgets in a turbulent channel flow. J. Fluid Mech. 194, 15–44 (1988)CrossRefGoogle Scholar
  18. 18.
    Martínez, D.O., Chen, S., Doolen, G.D., Kraichnan, R.H., Wang, L.P., Zhou, Y.: Energy spectrum in the dissipation range of fluid turbulence. J. Plasma Phys. 57, 195–201 (1997)CrossRefGoogle Scholar
  19. 19.
    Meldi, M., Sagaut, P.: Further insights into self-similarity and self-preservation in freely decaying isotropic turbulence. J. Turbul. 14, 24–53 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Pope, S.B.: Turbulent Flows. Cambridge University Press, Cambridge (2000)CrossRefzbMATHGoogle Scholar
  21. 21.
    Ristorcelli, J.R., Livescu, D.: Decay of isotropic turbulence: Fixed points and solutions for nonconstant \({G}\sim {R}_{\lambda }\) palinstrophy. Phys. Fluids 16, 3487–3490 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Rodi, W., Mansour, N.N.: Low Reynolds number \(k\)\(\epsilon \) modelling with the aid of direct simulation data. J. Fluid Mech. 250, 509–529 (1993)CrossRefzbMATHGoogle Scholar
  23. 23.
    Rubinstein, R., Clark, T.T.: Self-similar turbulence evolution and the dissipation rate transport equation. Phys. Fluids 17, 095104 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Rubinstein, R., Zhou, Y.: Analytical theory of the destruction terms in dissipation rate transport equations. Phys. Fluids 8, 3172–3178 (1996)CrossRefzbMATHGoogle Scholar
  25. 25.
    Saddoughi, S.G., Veeravalli, S.V.: Local isotropy in turbulent boundary layers at high Reynolds number. J. Fluid Mech. 268, 333–372 (1994)CrossRefGoogle Scholar
  26. 26.
    Sagaut, P., Cambon, C.: Homogeneous Turbulence Dynamics, Chap. 3 and 4. Springer, Berlin (2008)CrossRefzbMATHGoogle Scholar
  27. 27.
    Sanada, T.: Comment on the dissipation-range spectrum in turbulent flows. Phys. Fluids A 4, 1086–1087 (1992)CrossRefGoogle Scholar
  28. 28.
    Smith, L.M., Reynolds, W.C.: The dissipation-range spectrum and the velocity-derivative skewness in turbulent flows. Phys. Fluids A 3, 992–994 (1991)CrossRefGoogle Scholar
  29. 29.
    Smith, L.M., Reynolds, W.C.: On the Yakhot–Orszag renormalization group method for deriving turbulence statistics and models. Phys. Fluids A 4, 364–390 (1992)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Speziale, C.G., Bernard, P.S.: The energy decay in self-preserving isotropic turbulence revisited. J. Fluid Mech. 241, 645–667 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Tennekes, H., Lumley, J.L.: A First Course in Turbulence. MIT Press, Cambridge (1972)zbMATHGoogle Scholar
  32. 32.
    Woodruff, S.L., Rubinstein, R.: Multiple-scale perturbation analysis of slowly evolving turbulence. J. Fluid Mech. 565, 95–103 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Yakhot, V., Orszag, S.A.: Renormalization group analysis of turbulence. I. Basic theory. J. Sci. Comput. 1, 3–51 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Yakhot, V., Smith, L.M.: The renormalization group, the \(\varepsilon \)-expansion and derivation of turbulence models. J. Sci. Comput. 7, 35–61 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Yoshizawa, A.: A statistical investigation of transport equation for energy dissipation in shear turbulence. J. Phys. Soc. Jpn. 51, 1983–1991 (1982)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Yoshizawa, A.: Statistical modeling of a transport equation for the kinetic energy dissipation rate. Phys. Fluids 30, 628–631 (1987)CrossRefGoogle Scholar
  37. 37.
    Yoshizawa, A.: Nonequilibrium effect of the turbulent-energy-production process on the inertial-range energy spectrum. Phys. Rev. E 49, 4065–4071 (1994)CrossRefGoogle Scholar
  38. 38.
    Yoshizawa, A.: Hydrodynamic and Magnetohydrodynamic Turbulent Flows: Modelling and Statistical Theory. Kluwer, Dordrecht (1998)CrossRefzbMATHGoogle Scholar

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© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Institute of Industrial ScienceThe University of TokyoTokyoJapan

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