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Turbulence: the filtering approach

Published online by Cambridge University Press:  26 April 2006

M. Germano
Dipartimento di Ingegneria Aeronautica e Spaziale, Politecnico di Torino. Duca degli Abruzzi 24, 10129 Torino, Italy


Explicit or implicit filtered representations of chaotic fields like spectral cut-offs or numerical discretizations are commonly used in the study of turbulence and particularly in the so-called large-eddy simulations. Peculiar to these representations is that they are produced by different filtering operators at different levels of resolution, and they can be hierarchically organized in terms of a characteristic parameter like a grid length or a spectral truncation mode. Unfortunately, in the case of a general implicit or explicit filtering operator the Reynolds rules of the mean are no longer valid, and the classical analysis of the turbulence in terms of mean values and fluctuations is not so simple.

In this paper a new operatorial approach to the study of turbulence based on the general algebraic properties of the filtered representations of a turbulence field at different levels is presented. The main results of this analysis are the averaging invariance of the filtered Navier—Stokes equations in terms of the generalized central moments, and an algebraic identity that relates the turbulent stresses at different levels. The statistical approach uses the idea of a decomposition in mean values and fluctuations, and the original turbulent field is seen as the sum of different contributions. On the other hand this operatorial approach is based on the comparison of different representations of the turbulent field at different levels, and, in the opinion of the author, it is particularly fitted to study the similarity between the turbulence at different filtering levels. The best field of application of this approach is the numerical large-eddy simulation of turbulent flows where the large scale of the turbulent field is captured and the residual small scale is modelled. It is natural to define and to extract from the resolved field the resolved turbulence and to use the information that it contains to adapt the subgrid model to the real turbulent field. Following these ideas the application of this approach to the large-eddy simulation of the turbulent flow has been produced (Germano et al. 1991). It consists in a dynamic subgrid-scale eddy viscosity model that samples the resolved scale and uses this information to adjust locally the Smagorinsky constant to the local turbulence.

Research Article
© 1992 Cambridge University Press

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Boussinesq, J. 1877 Essai sur la thèorie des eaux courantes. Mem. Savants Etrangers Acad. Sci. Paris 26, 1680.Google Scholar
Davydov, B. I. 1961 On the statistical dynamics of an incompressible turbulent fluid. Dokl. Akad. Nauk. SSSR 136, 4750.Google Scholar
Deardorff, J. W. 1970 A numerical study of three dimensional channel flow at large Reynolds numbers. J. Fluid Mech. 41, 453480.Google Scholar
Germano, M. 1986 A proposal for a redefinition of the turbulent stresses in the filtered Navier—Stokes equations. Phys. Fluids 29, 23232324.Google Scholar
Germano, M. 1987 On the non-Reynolds averages in turbulence. AIAA Paper 87–1297.Google Scholar
Germano, M. 1990 Averaging invariance of the turbulent equations and similar subgrid scale modeling. CTR, Stanford, California, Manuscript 116.
Germano, M., Piomelli, U., Moin, P. & Cabot, W. H. 1991 A dynamic subgrid-scale eddy viscosity model. Phys. Fluids A 3, 17601765.Google Scholar
Kampè de Fèriet, J. 1957 La notion de moyenne dans la thèorie de la turbulence. Rend. Sem. Mat. Fis. Milano 27, 167207.Google Scholar
Kim, J., Moin, P. & Moser, R. 1987 Turbulence statistics in fully developed channel flow at low Reynolds number. J. Fluid Mech. 177, 133166.Google Scholar
Leonard, A. 1974 Energy cascade in large-eddy simulations of turbulent fluid flows. Adv. Geophys. 18A, 237248.Google Scholar
Lilly, D. K. 1966 On the application of the eddy viscosity concept in the inertial sub-range of turbulence. NCAR, Boulder, Colorado, Manuscript 123.
Monin, A. S. & Yaglom, A. M. 1971 Statistical Fluid Mechanics. The MIT Press.
Reynolds, O. 1895 On the dynamical theory of incompressible viscous fluids and the determination of the criterion. Phil. Trans. R. Soc. Lond. A 186, 123164.Google Scholar
Rogallo, R. S. & Moin, P. 1984 Numerical simulation of turbulent flows. Ann. Rev. Fluid Mech. 16, 99137.Google Scholar
Schiestel, R. 1987 Multiple-time-scale modeling of turbulent flows in one point closure. Phys. Fluids 30, 722731.Google Scholar
Schumann, U. 1975 Subgrid scale model for finite difference simulations of turbulent flows in plane channels and annuli. J. Comput. Phys. 18, 376404.Google Scholar
Smagorinsky, J. 1963 General circulation experiments with the primitive equations. I. The basic experiment. Mon. Weather Rev. 91, 99164.Google Scholar
Speziale, C. G. 1985 Galilean invariance of subgrid-scale stress models in the large eddy simulation of turbulence. J. Fluid Mech. 156, 5562.Google Scholar
Tchen, C. M. 1973 Repeated cascade theory of homogeneous turbulence. Phys. Fluids 16, 1330.Google Scholar
Yoshizawa, A. 1989 Subgrid-scale modeling with a variable length scale. Phys. Fluids A 1, 12931295.Google Scholar
Zang, T. A., Gilbert, N. & Kleiser, N. 1990 Direct numerical simulation of the transitional zone. In Instability and transition (ed. M. Y. Hussaini & R. G. Voigt), pp. 283299. Springer.
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