Renormalization Methods Applied to Turbulence Theory

  • David McComb
Part of the International Centre for Mechanical Sciences book series (CISM, volume 442)


In these notes the turbulence problem is interpreted as one in many-body physics. We consider the many-body problem, along with the concept of renormalization, and show how the methods of renormalized perturbation theory have been applied to turbulence. Then a review and assessment of the two-point, two-time closures which arise from renormalized perturbation theory is given. After that, renormalization group is introduced as a more limited but potentially more rigorous way of applying perturbation methods to turbulence with particular relevance to the sub-grid modelling problem. We conclude with a brief discussion of the possible application of these methods to more realistic problems in shear flows.


Partition Function Dissipation Rate Ising Model Isotropic Turbulence Perturbation Expansion 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Allen, J. (1963). Formulation of the theory of turbulent shear flow. Ph.D. Dissertation, Victoria University of Manchester.Google Scholar
  2. Balescu, R., and Senatorski, A. (1970). A new approach to the theory of fully developed turbulence. Ann.Phys(NY) 58: 587.CrossRefADSGoogle Scholar
  3. Bertoglio, J.-P., and Jeandel, D. (1987). A simplified spectral closure for inhomogeneous turbulence applicable to the boundary layer. In Durst, F., ed., Turbulent Shear Flows 5, 323. Berlin: Springer-Verlag.Google Scholar
  4. Burden, A. D. (1991). Towards an EDQNM-Closure for inhomogeneous turbulence. In Johansson, A. V., and Alfredsson, P. H., eds., Advances in Turbulence 3, 387. Berlin: Springer-Verlag.Google Scholar
  5. Cambon, C., and Scott, J. R (1999). Linear and nonlinear models of anisotropic turbulence. Ann. Rev. Fluid Mech. 31: 1.MathSciNetCrossRefADSGoogle Scholar
  6. Edwards, S., and McComb, W. (1969). Statistical mechanics far from equilibrium. J.Phys.A 2: 157.CrossRefMATHADSGoogle Scholar
  7. Edwards, S. (1964). The statistical dynamics of homogeneous turbulence. J. Fluid Mech. 18: 239.MathSciNetCrossRefADSGoogle Scholar
  8. Eyink, G. (1994). The renormalization group method in statistical hydrodynamics. Phys. Fluids 6 (9): 3063–3078.MathSciNetCrossRefMATHADSGoogle Scholar
  9. Forster, D., Nelson, D., and Stephen, M. (1976). Long-time tails and the large-eddy behaviour of a randomly stirred fluid. Phys. Rev. Lett. 36 (15): 867–869.CrossRefADSGoogle Scholar
  10. Frederiksen, J. S., and Davies, A. G. (2000). Dynamics and spectra of cumulant update closures for two-dimensional turbulence. Geophys. Astrophys. Fluid Dynamics 92: 197.MathSciNetCrossRefADSGoogle Scholar
  11. Frederiksen, J., Davies, A., and Bell, R. (1994). Closure theories with non-gaussian restarts for truncated two dimensional turbulence. Phys. Fluids 6 (9): 3153.CrossRefMATHADSGoogle Scholar
  12. Godeferd, F. S., Cambon, C., and Scott, J. F. (2001). Two-point closures and their applications: report on workshop. J. Fluid Mech. 436: 393.MATHADSGoogle Scholar
  13. Herring, J., and Kraichnan, R. (1972). Comparison of some approximations for isotropic turbulence Lecture Notes in Physics, volume 12. Springer, Berlin. chapter Statistical Models and Turbulence, 148.Google Scholar
  14. Herring, J. (1965). Self-consistent field approach to turbulence theory. Phys. Fluids 8: 2219.MathSciNetCrossRefMATHADSGoogle Scholar
  15. Horner, H., and Lipowsky, R. (1979). On the Theory of Turbulence: A non Eulerian Renormalized Expansion. Z.Phys.B 33: 223.MathSciNetCrossRefADSGoogle Scholar
  16. Johnston, C., and McComb, W. (2000). Renormalized expression for the turbulent energy dissipation rate. Phys. Rev E 63: 015304.Google Scholar
  17. Kaneda, Y. (1981). Renormalized expansions in the theory of turbulence with the use of the Lagrangian position function. J. Fluid Mech. 107: 131–145.CrossRefADSGoogle Scholar
  18. Kida, S., and Gotoh. (1997). A Lagrangian direct-interaction approximation for homogeneous isotropic turbulence. J. Fluid Mech. 345: 307–345.MathSciNetCrossRefMATHADSGoogle Scholar
  19. Kraichnan, R., and Herring, J. (1978). A strain-based Lagrangian-history turbulence theory. J. Fluid Mech. 88: 355.CrossRefMATHADSGoogle Scholar
  20. Kraichnan, R. H. (1959). The structure of isotropic turbulence at very high Reynolds numbers. J. Fluid Mech. 5: 497–543.MathSciNetCrossRefMATHADSGoogle Scholar
  21. Kraichnan, R. H. (1964a). Decay of isotropic turbulence in the Direct-Interaction Approximation. Phys. Fluids 7(7):1030–1048.MathSciNetCrossRefADSGoogle Scholar
  22. Kraichnan, R. (1964b). Direct-interaction approximation for shear and thermally driven turbulence. Phys. Fluids 7 (7): 1048.MathSciNetCrossRefADSGoogle Scholar
  23. Kraichnan, R. H. (1965). Lagrangian-history closure approximation for turbulence. Phys. Fluids 8 (4): 575–598.MathSciNetCrossRefADSGoogle Scholar
  24. Leslie, D. (1973). Developments in the theory of modern turbulence. Clarendon Press, Oxford.Google Scholar
  25. Machiels, L. (1997). Predictability of small-scale motion in isotropic fluid turbulence. Phys. Rev. Lett. 79 (18): 3411–3414.CrossRefADSGoogle Scholar
  26. McComb, W. D., and Johnston, C. (1999). Conditional mode elimination with asymptotic freedom for isotropic turbulence at large Reynolds numbers. In Vassilicos, J., ed., Turbulence Structure and Vortex Dynamics, Proceedings of the Symposium at the Isaac Newton Institute Cambridge. CUP.Google Scholar
  27. McComb, W. D., and Johnston, C. (2000). Elimination of turbulence modes using a conditional average with asymptotic freedom. J. Phys. A:Math. Gen. 33: L15 - L20.CrossRefMATHADSGoogle Scholar
  28. McComb, W. D., and Johnston, C. (2001). Conditional mode elimination and scale-invariant dissipation in isotropic turbulence. Physica A 292: 346.MathSciNetCrossRefMATHADSGoogle Scholar
  29. McComb, W. D., and Shanmugasundaram, V. (1984). Numerical calculations of decaying isotropic turbulence using the LET theory. J. Fluid Mech. 143: 95–123.CrossRefMATHADSGoogle Scholar
  30. McComb, W., Yang, T.-J., Young, A., and Machiels, L. (1997). Investigation of renormalization group methods for the numerical simulation of isotropic turbulence. In Proc. Eleventh Symposium on Turbulent Shear Flows.Google Scholar
  31. McComb, W. D., Hunter, A., and Johnston, C. (2001). Conditional mode-elimination and the subgridmodelling problem for isotropic turbulence. Physics of Fluids 13: 20–30.CrossRefGoogle Scholar
  32. McComb, W. (1978). A theory of time dependent, isotropic turbulence. J.Phys.A:Math.Gen. 11 (3): 613.CrossRefMATHADSGoogle Scholar
  33. McComb, W. D. (1982). Reformulation of the statistical equations for turbulent shear flow. Phys. Rev. A 26 (2): 1078–1094.CrossRefADSGoogle Scholar
  34. McComb, W. (1990). The Physics of Fluid Turbulence. Oxford University Press.Google Scholar
  35. McComb, W. (1995). Theory of turbulence. Rep. Prog. Phys. 58: 1117–1206.CrossRefADSGoogle Scholar
  36. Nakano, T. (1988). Direct interaction approximation of turbulence in the wave packet representation. Phys. Fluids 31: 1420.CrossRefMATHADSGoogle Scholar
  37. Oberlack, M., McComb, W., and Quinn, A. (2001). Solution of functional equations and reduction of dimension in the local energy transfer theory of incompressible, three-dimensional turbulence. Phys. Rev. E 63: 026308–1.Google Scholar
  38. Phythian, R. (1969). Self-consistent perturbation series for stationary homogeneous turbulence. J. Phys. A 2: 181.CrossRefADSGoogle Scholar
  39. Qian, J. (1983). Variational approach to the closure problem of turbulence theory. Phys. Fluids 26: 2098.MathSciNetCrossRefMATHADSGoogle Scholar
  40. Quinn, A. P. (2000). Local Energy Transfer theory in forced and decaying isotropic turbulence. Ph.D. Dissertation, University of Edinburgh.Google Scholar
  41. Reichl, L. (1980). A modern course in statistical physics. Edward Arnold, London.Google Scholar
  42. Rose, H. (1977). Eddy diffusivity, eddy noise and subgrid-scale modelling. J. Fluid Mech. 81 (4): 719–734.MathSciNetCrossRefMATHADSGoogle Scholar
  43. Sreenivasan, K. R. (1995). On the universality of the Kolmogorov constant. Phys. Fluids 7: 27–78.MathSciNetGoogle Scholar
  44. van Dyke, M. (1975). Perturbation Methods in Fluid Mechanics. Stanford, CA: Parabolic Press.MATHGoogle Scholar
  45. Wilson, K. G. (1983). The renormalization group and critical phenomena. Rev. Mod. Phys. 55 (3): 583–600.CrossRefADSGoogle Scholar
  46. Yakhot, V., and Orszag, S. (1986). Renormalization Group analysis of turbulence. I. Basic theory. J. Sci. Comp. 1 (1): 3–51.MathSciNetCrossRefMATHGoogle Scholar
  47. Yeung, P. K., and Zhou, Y. (1997). Universality of the Kolmogorov constant in numerical simulations of turbulence. Phys. Rev. E 56: 1746.CrossRefADSGoogle Scholar

Copyright information

© Springer-Verlag Wien 2002

Authors and Affiliations

  • David McComb
    • 1
  1. 1.Department of Physics and AstronomyThe University of EdinburghEdinburghScotland

Personalised recommendations