Journal of Scientific Computing

, Volume 1, Issue 1, pp 3–51 | Cite as

Renormalization group analysis of turbulence. I. Basic theory

  • Victor Yakhot
  • Steven A. Orszag


We develop the dynamic renormalization group (RNG) method for hydrodynamic turbulence. This procedure, which uses dynamic scaling and invariance together with iterated perturbation methods, allows us to evaluate transport coefficients and transport equations for the large-scale (slow) modes. The RNG theory, which does not include any experimentally adjustable parameters, gives the following numerical values for important constants of turbulent flows: Kolmogorov constant for the inertial-range spectrumCK=1.617; turbulent Prandtl number for high-Reynolds-number heat transferP t =0.7179; Batchelor constantBa=1.161; and skewness factor¯S3=0.4878. A differentialK-\(\bar \varepsilon \) model is derived, which, in the high-Reynolds-number regions of the flow, gives the algebraic relationv=0.0837 K2/\(\bar \varepsilon \), decay of isotropic turbulence asK=O(t−1.3307), and the von Karman constantκ=0.372. A differential transport model, based on differential relations betweenK,\(\bar \varepsilon \), andν, is derived that is not divergent whenK→ 0 and\(\bar \varepsilon \) is finite. This latter model is particularly useful near walls.

Key words

Renormalization group turbulence theory inertial range turbulence transport Reynolds number large-eddy simulation computational fluid dynamics 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Antonia, R. A., Chambers, A. J., and Anselmet, F. (1984).Phys. Chem. Hydrodyn. 5, 368.Google Scholar
  2. Batchelor, G. (1959).J. Fluid Mech. 5, 113.Google Scholar
  3. Bayly, B., and Yakhot, V. (1986).Phys. Rev. A, in press.Google Scholar
  4. Deardorff, J. W. (1970).J. Fluid Mech. 41, 453.Google Scholar
  5. Deardorff, J. W. (1971).J. Comp. Phys. 7, 120.Google Scholar
  6. De Dominicis, C., and Martin, P. C. (1979).Phys. Rev. A 19, 419.Google Scholar
  7. Dorfman, R. (1975). In Cohen, E. G. D. (ed.),Fundamental Problems in Statistical Mechanics, North-Holland, Amsterdam.Google Scholar
  8. Forster, D., Nelson, D., and Stephen, M. (1977).Phys. Rev. A 16, 732.Google Scholar
  9. Fournier, J. P., and Frisch, U. (1978).Phys. Rev. A 17, 747.Google Scholar
  10. Fournier, J. P., and Frisch, U. (1983).Phys. Rev. A 28, 1000.Google Scholar
  11. Frenkiel, F. N., Klebanoff, P., and Huang, T. T. (1979).Phys. Fluids 22, 1606.Google Scholar
  12. Hanjaliĉ, K., and Launder, B. E. (1972).J. Fluid Mech. 52, 609.Google Scholar
  13. Herring, J. R., Orszag, S. A., Kraichnan, R. H., and Foxz, D. G. (1974).J. Fluid Mech. 66, 417.Google Scholar
  14. Hohenberg, P. C., and Halperin, B. I. (1977).Rev. Mod. Phys. 49, 435.Google Scholar
  15. Kolmogorov, A. N. (1941).Dokl. Akad. Nauk SSSR 30, 299.Google Scholar
  16. Kraichnan, R. H. (1959).J. Fluid Mech. 5, 497.Google Scholar
  17. Kraichnan, R. H. (1961).J. Math. Phys. 2, 124.Google Scholar
  18. Kraichnan, R. H. (1971).J. Fluid Mech. 47, 525.Google Scholar
  19. Landau, L., and Lifshitz, E. M. (1982).Fluid Mechanics, Pergamon, New York.Google Scholar
  20. Launder, B. E., and Spalding, D. B. (1972).Mathematical Models of Turbulence, Academic Press, New York.Google Scholar
  21. Launder, B. E., Reece, G. J., and Rodi, W. (1975).J. Fluid Mech. 68, 537.Google Scholar
  22. Leslie, D. C. (1972).Developments in the Theory of Turbulence, Clarendon Press, Oxford.Google Scholar
  23. Ma, S. K., and Mazenko, G. (1975).Phys. Rev. B 11, 4077.Google Scholar
  24. Moin, P., and Kim, J. (1981).J. Fluid Mech. 118, 341.Google Scholar
  25. Monin, A. S., and Yaglom, A. M. (1975).Statistical Fluid Mechanics, Vol. 2, MIT Press, Cambridge, Massachusetts.Google Scholar
  26. Orszag, S., and Patera, A. (1981).Phys. Rev. Lett. 47, 832.Google Scholar
  27. Pao, Y. H. (1965).Phys. Fluids 8, 1063.Google Scholar
  28. Pao, Y. H. (1968).Phys. Fluids 11, 1371.Google Scholar
  29. Reynolds, W. C. (1976).Annu. Rev. Fluid Mech. 8, 183.Google Scholar
  30. Sivashinsky, G., and Yakhot, V. (1985).Phys. Fluids 28, 1040.Google Scholar
  31. Smagorinsky, J. (1963).Monthly Weather Rev. 91, 99.Google Scholar
  32. Tennekes, H., and Lumley, J. L. (1972).A First Course in Turbulence, MIT Press, Cambridge, Massachusetts.Google Scholar
  33. Wilson, K. G. (1971).Phys. Rev. B 4, 3174.Google Scholar
  34. Wilson, K. G., and Kogut, J. (1974).Phys. Rev. 12C, 77.Google Scholar
  35. Wood, W. W. (1975). In Cohen, E. G. D. (ed.),Fundamental Problems in Statistical Mechanics, North-Holland, Amsterdam.Google Scholar
  36. Wyld, H. W. (1961).Ann. Phys. 14, 143.Google Scholar
  37. Wyngaard, J. C., and Tennekes, H. (1970).Phys. Fluids 13, 1962.Google Scholar
  38. Yakhot, V. (1981).Phys. Rev. A 23, 1486.Google Scholar
  39. Yakhot, V., and Sivashinsky, G. (1986).Phys. Rev. A, submitted.Google Scholar

Copyright information

© Plenum Publishing Corporation 1986

Authors and Affiliations

  • Victor Yakhot
    • 1
  • Steven A. Orszag
    • 1
  1. 1.Applied & Computational Mathematics and Mechanical & Aerospace EngineeringPrinceton UniversityPrinceton

Personalised recommendations