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The renormalization group, the ɛ-expansion and derivation of turbulence models

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We reformulate the renormalization group (RNG) and theɛ-expansion for derivation of turbulence models. The procedure is developed for the Navier-Stokes equations and the transport equations for the kinetic energyK and energy dissipation rate ℰ. The derivation draws on the works of Yakhot and Orszag (1986) and Smith and Reynolds (1992), and all results are found at low order in the underlying perturbation expansion in powers ofɛ. The sum of the source terms in the ℰ-equation is known to beO(1) due to the balance at leading order ofO(R 1/2T ) terms. Smith and Reynolds (1992) showed the cancellation of some of theO(R 1/2T ) terms generated by the RNG procedure. Here we show that including the random-force contribution to ℰ-production results in the cancellation ofall theO(R 1/2 T ) terms. We find that two of theO(1) terms in the RNG equation for the mean dissipation rate ℰ have the same form as those in the widely used model

-equation. The values of the coefficients of the familiar terms are close to those used in practice. An extra production term is predicted which is small for slow distortions, but important for rapid distortions. Hence, it may be a term that should be added to the

model equation. We believe that the present derivation places the

model equation on a more solid theoretical basis.

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  • Batchelor, G. K., and Townsend, A. A. (1948). Decay of turbulence in the final period,Proc. R. Soc. London A194, 527.

    Google Scholar 

  • Comte-Bellot, G., and Corrsin, S. (1966). The use of contraction to improve the isotropy of grid-generated turbulence,J. Fluid Mech. 25, 657.

    Google Scholar 

  • Dannevik, W. P., Yakhot, V., and Orszag, S. A. (1987). Analytical theories of turbulence and theɛ-expansion,Phys. Fluids 30, 2021.

    Google Scholar 

  • De Dominicis, C., and Martin, P. C. (1979). Energy spectra of certain randomly stirred fluids,Phys. Rev. A 19, 419.

    Google Scholar 

  • Durbin, P. A. (1990). Turbulence closure modeling near rigid boundaries, Annual Research Briefs—1990, Center for Turbulence Research, Vol. 3.

  • Durbin, P. A., and Speziale, C. G. (1991). Local anisotropy in strained turbulence at high Reynolds numbers,ASME J. Fluids Eng. 113, pp. 707–709.

    Google Scholar 

  • Forster, D., Nelson, D. R., and Stephen, M. J. (1977). Large-distance and long-time properties of a randomly stirred fluid,Phys. Rev. A 16, 732.

    Google Scholar 

  • Fournier, D., and Frisch, U. (1983). Remarks on the renormalization group in statistical fluid dynamics,Phys. Rev. A 28, 1000.

    Google Scholar 

  • Launder, B. E., Reece, G. J., and Rodi, W. (1975). Progress in the development of a Reynolds-stress turbulence closure,J Fluid Mech. 68, 537.

    Google Scholar 

  • Lee, M. J., and Reynolds, W. C. (1985). Report TF-24, Mechanical Engineering Department, Thermosciences Division, Stanford University.

  • Majda, A., and Avillaneda, M. (1990). Mathematical models with exact renormalization for turbulent transport,Commun. Math. Phys. 131, 381.

    Google Scholar 

  • Mansour, N. N., and Shih, T.-H. (1989).Forum on Trubulent Shear Flows—1989, FED Vol. 76, Am. Society of Mech. Eng., New York.

    Google Scholar 

  • Migdal, A. A., Orszag, S. A., and Yakhot, V. (1990). Intrinsic stirring force in turbulence and the-expansion, Princeton University preprint.

  • Millionshtchikov, M. D. (1939). Decay of turbulence in wind tunnels,Dokl. Akad. Nauk SSSR 22, 236.

    Google Scholar 

  • Orszag, S. A. (1970). Analytical theories of turbulence,J. Fluid Mech. 41, 363.

    Google Scholar 

  • Panda, R., Sonnad, V., and Clementi, E. (1989). Turbulence in a randomly stirred fluid,Phys. Fluids A 1, 1045.

    Google Scholar 

  • Patel, V. C., Rodi, W., and Scheurer, G. (1985). Turbulence models for near-wall and low Reynolds number flows: A review,AIAA J. 23, 1308.

    Google Scholar 

  • Reynolds, W. C. (1976). Computation of turbulent flows,Ann. Rev. Fluid. Mech. 8, 183.

    Google Scholar 

  • Shih, T.-H., Reynolds, W. C., and Mansour, N. N. (1990). A spectrum model for weakly anisotropic turbulence,Phys. Fluids A 2.

  • Smith, L. M., and Reynolds, W. C. (1992). On the Yakhot-Orszag renormalization group method for deriving turbulence statistics and models,Phys. Fluids A 2, 364.

    Google Scholar 

  • Speziale, C., Gatski, T. B., and Fitzmaurice, N. (1991). An analysis of RNG-based turbulence models for homogeneous shear flow,Phys. Fluids. A 3, 2278.

    Google Scholar 

  • Tan, H. S., and Ling, S. C. (1963). Final stage decay of grid-generated turbulence,Phys. Fluids 6, 1693.

    Google Scholar 

  • Tennekes, H., and Lumley, J. L. (1972).A First Course in Trubulence, MIT Press, Cambridge, MA.

    Google Scholar 

  • Yakhot, V., and Orszag, S. A. (1986). Renormalization group analysis of turbulence. I. Basic theory,J. Sci. Comput. 1, 3.

    Google Scholar 

  • Yakhot, V., and Orszag, S. A. (1990). Analysis of theɛ-expansion in turbulence theory: Approximate renormalization group for diffusion of a passive scalar in a random velocity field, Princeton University preprint.

  • Yakhot, V., Orszag, S., and Panda, R. (1988). Computational test of the renormalization group theory of turbulence,J. Sci. Comput. 3, 139.

    Google Scholar 

  • Yakhot, V., Thangam, S., Gatski, T. B., Orszag, S. A., and Speziale, C. G. (1992). Development of turbulence models for shear flows by a double expansion technique, to appear inPhys. Fluids A,7.

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Yakhot, V., Smith, L.M. The renormalization group, the ɛ-expansion and derivation of turbulence models. J Sci Comput 7, 35–61 (1992).

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