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

## Abstract

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.

## Keywords

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

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## Notes

### Acknowledgements

This work was supported by JSPS KAKENHI Grant Number JP17K06143.

## References

- 1.Gotoh, T.: Fundamentals of Turbulence Theory. Asakura, Tokyo (1998)Google Scholar
- 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.Hamba, F.: Statistical investigation of the energy dissipation equation in shear turbulence. J. Phys. Soc. Jpn.
**56**, 3771–3774 (1987)CrossRefGoogle Scholar - 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.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.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.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.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.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.Kida, S., Goto, S.: A Lagrangian direct-interaction approximation for homogeneous isotropic turbulence. J. Fluid Mech.
**345**, 307–345 (1997)MathSciNetCrossRefzbMATHGoogle Scholar - 11.Kraichnan, R.H.: Lagrangian-history closure approximation for turbulence. Phys. Fluids
**8**, 575–598 (1965)MathSciNetCrossRefGoogle Scholar - 12.Kraichnan, R.H.: Inertial-range transfer in two- and three-dimensional turbulence. J. Fluid Mech.
**47**, 525–535 (1971)CrossRefzbMATHGoogle Scholar - 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.Leschziner, M.: Statistical Turbulence Modelling for Fluid Dynamics—Demystified. Imperial College Press, London (2016)zbMATHGoogle Scholar
- 15.Leslie, D.C.: Developments in the Theory of Turbulence. Oxford University Press, Oxford (1973)zbMATHGoogle Scholar
- 16.Lumley, J.L.: Similarity and the turbulent energy spectrum. Phys. Fluids
**10**, 855–858 (1967)CrossRefGoogle Scholar - 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.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.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.Pope, S.B.: Turbulent Flows. Cambridge University Press, Cambridge (2000)CrossRefzbMATHGoogle Scholar
- 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.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.Rubinstein, R., Clark, T.T.: Self-similar turbulence evolution and the dissipation rate transport equation. Phys. Fluids
**17**, 095104 (2005)MathSciNetCrossRefzbMATHGoogle Scholar - 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.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.Sagaut, P., Cambon, C.: Homogeneous Turbulence Dynamics, Chap. 3 and 4. Springer, Berlin (2008)CrossRefzbMATHGoogle Scholar
- 27.Sanada, T.: Comment on the dissipation-range spectrum in turbulent flows. Phys. Fluids A
**4**, 1086–1087 (1992)CrossRefGoogle Scholar - 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.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.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.Tennekes, H., Lumley, J.L.: A First Course in Turbulence. MIT Press, Cambridge (1972)zbMATHGoogle Scholar
- 32.Woodruff, S.L., Rubinstein, R.: Multiple-scale perturbation analysis of slowly evolving turbulence. J. Fluid Mech.
**565**, 95–103 (2006)MathSciNetCrossRefzbMATHGoogle Scholar - 33.Yakhot, V., Orszag, S.A.: Renormalization group analysis of turbulence. I. Basic theory. J. Sci. Comput.
**1**, 3–51 (1986)MathSciNetCrossRefzbMATHGoogle Scholar - 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.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.Yoshizawa, A.: Statistical modeling of a transport equation for the kinetic energy dissipation rate. Phys. Fluids
**30**, 628–631 (1987)CrossRefGoogle Scholar - 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.Yoshizawa, A.: Hydrodynamic and Magnetohydrodynamic Turbulent Flows: Modelling and Statistical Theory. Kluwer, Dordrecht (1998)CrossRefzbMATHGoogle Scholar