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February  2011, 4(1): 155-168. doi: 10.3934/dcdss.2011.4.155

Two-point closure based large-eddy simulations in turbulence, Part 1: Isotropic turbulence

Marcel Lesieur 1,

1. 

Member of French Academy of Sciences, Laboratory for Geophysical and Industrial Flows, Grenoble

Received  February 2009 Published  October 2010

This is the first of a series of two articles dedicated to Claude-Michel Brauner and Roger Temam on turbulence large-eddy simulations using two-point closures. The present paper deals with applications to isotropic turbulence. First, some personal memories related to my collaboration with Claude-Michel Brauner are given. Then we recall the formalism of large-eddy simulations (LES) of turbulence in physical space for flows of constant density. We consider also a passive scalar, very important for combution applications. Afterwards we study the same problem in Fourier space, on the basis of the Eddy-Damped Quasi-Normal Markovian (EDQNM) theory which is used as subgrid model. This is applied to isotropic turbulence, with particular emphasis put on turbulence decay. We discuss the issue of singularity for Euler equations. We give finally some LES results on pressure statistics.
Keywords: two-point closures., Isotropic turbulence, large-eddy simulations.
Mathematics Subject Classification: Primary: 35Q30, 76F0.
Citation: Marcel Lesieur. Two-point closure based large-eddy simulations in turbulence, Part 1: Isotropic turbulence. Discrete & Continuous Dynamical Systems - S, 2011, 4 (1) : 155-168. doi: 10.3934/dcdss.2011.4.155
References:
[1]

J. C. André and M. Lesieur, Influence of helicity on high Reynolds number isotropic turbulence, J. Fluid Mech., 81 (1977), 187-207. Google Scholar

[2]

G. Balarac, Private communication, 2007. Google Scholar

[3]

G. K. Batchelor and A. A. Townsend, Decay of vorticity in isotropic turbulence, Proc. Roy. Soc. Series A, 191 (1947), 534-550. Google Scholar

[4]

C. Bogey and C. Bailly, Investigation of downstream and sideline subsonic jet noise using large eddy simulation, Theor. Comp. Fluid Dyn., 20 (2006), 23-40. doi: doi:10.1007/s00162-005-0005-7.  Google Scholar

[5]

J. Borue and S. A. Orszag, Spectra in helical three-dimensional homogeneous isotropic turbulence, Phys. Rev. E, 55 (1997), 7005-7009. doi: doi:10.1103/PhysRevE.55.7005.  Google Scholar

[6]

C. M. Brauner, M. Frankel, J. Hulshof, et al, Stability and attractors for the quasi-steady equation of cellular flames, Interfaces and free boundaries, 8 (2006), 301-316. Google Scholar

[7]

J. P. Chollet and M. Lesieur, Parameterization of small scales of three-dimensional isotropic turbulence utilizing spectral closures, J. Atmos. Sci., 38 (1981), 2747-2757. doi: doi:10.1175/1520-0469(1981)038<2747:POSSOT>2.0.CO;2.  Google Scholar

[8]

Y. Dubief and F. Delcayre, On coherent-vortex identification in turbulence, J. Turb., 1 (2000), 11. doi: doi:10.1088/1468-5248/1/1/011.  Google Scholar

[9]

C. Foias and P. Penel, Dissipation totale de l'énergie dans l'équation de Frisch-Kraichnan-Lesieur, C. R. Acad. Sci., Paris, A-B, 280 (1975), A629-A632.  Google Scholar

[10]

U. Frisch, "Turbulence: The Legacy of A. N. Kolmogorov," Cambridge University Press, 1995. Google Scholar

[11]

M. Germano, U. Piomelli, P. Moin and W. Cabot, A dynamic subgrid-scale eddy-viscosity model, Phys. Fluids A., 3 (1991), 1760-1765. doi: doi:10.1063/1.857955.  Google Scholar

[12]

T. Hughes, L. Mazzei, A. Oberai and A. Wray, The multiscale formulation of large-eddy simulation: decay of homogeneous isotropic turbulence, Phys. Fluids, 13 (2001), 505-512. doi: doi:10.1063/1.1332391.  Google Scholar

[13]

J. Hunt, A. Wray and P. Moin, Eddies, stream, and convergence zones in turbulent flows, Center for Turbulence Research Rep., CTR-S88 (1988), 193. Google Scholar

[14]

J. Jimenez, Turbulence, in "Perspectives in Fluid Mechanics" (G. K. Batchelor, H. K. Moffatt and M. G. Woster eds), Cambridge University Press, (2000), 231-283. Google Scholar

[15]

R. H. Kraichnan, Eddy viscosity in two and three dimensions, J. Atmos. Sci., 33 (1976), 1521-1536. doi: doi:10.1175/1520-0469(1976)033<1521:EVITAT>2.0.CO;2.  Google Scholar

[16]

E. Lamballais, O. Métais and M. Lesieur, Spectral-dynamic model for large-eddy simulations of turbulent rotating channel flow, Theor. Comp. Fluid Dyn., 12 (1997), 149-177. doi: doi:10.1007/s001620050104.  Google Scholar

[17]

J. Leray, Sur le mouvement d'un fluide visqueux emplissant l'espace, (French), J. Acta Math., 63 (1934), 193-248. doi: doi:10.1007/BF02547354.  Google Scholar

[18]

M. Lesieur and D. Schertzer, Amortissement auto similaire d'une turbulence à grand nombre de Reynolds, (French), Journal de Mécanique, 17 (1978), 609-646.  Google Scholar

[19]

M. Lesieur, S. Ossia and O. Métais, Infrared pressure spectra in two- and three-dimensional isotropic incompressible turbulence, Phys. Fluids, 11 (1999), 1535-1543. doi: doi:10.1063/1.870016.  Google Scholar

[20]

M. Lesieur, O. Métais and P. Comte, "Large-Eddy Simulations of Turbulence," With a preface by James J. Riley, Cambridge University Press, New York, 2005.  Google Scholar

[21]

M. Lesieur, "Turbulence in Fluids," 4th edition, Springer, 2008. doi: doi:10.1007/978-1-4020-6435-7.  Google Scholar

[22]

M. Lesieur and O. Métais, Large-eddy simulations for geophysical fluid dynamics, in "Handbook of Numerical Analysis" (R. Temam, J. Tribbia and P. Ciarlet, eds.), vol XIV, North-Holland, 2008. Google Scholar

[23]

O. Métais and M. Lesieur, Spectral large-eddy simulations of isotropic and stably-stratified turbulence, J. Fluid Mech., 239 (1992), 157-194. Google Scholar

[24]

S. A. Orszag, Analytical theories of turbulence, J. Fluid Mech., 41 (1970), 363-386. doi: doi:10.1017/S0022112070000642.  Google Scholar

[25]

J. Smagorinsky, General circulation experiments with the primitive equations, Mon. Weath. Rev., 91 (1963), 99-164. Google Scholar

[26]

A. Wray, Private communication, 1988. Google Scholar

show all references

References:
[1]

J. C. André and M. Lesieur, Influence of helicity on high Reynolds number isotropic turbulence, J. Fluid Mech., 81 (1977), 187-207. Google Scholar

[2]

G. Balarac, Private communication, 2007. Google Scholar

[3]

G. K. Batchelor and A. A. Townsend, Decay of vorticity in isotropic turbulence, Proc. Roy. Soc. Series A, 191 (1947), 534-550. Google Scholar

[4]

C. Bogey and C. Bailly, Investigation of downstream and sideline subsonic jet noise using large eddy simulation, Theor. Comp. Fluid Dyn., 20 (2006), 23-40. doi: doi:10.1007/s00162-005-0005-7.  Google Scholar

[5]

J. Borue and S. A. Orszag, Spectra in helical three-dimensional homogeneous isotropic turbulence, Phys. Rev. E, 55 (1997), 7005-7009. doi: doi:10.1103/PhysRevE.55.7005.  Google Scholar

[6]

C. M. Brauner, M. Frankel, J. Hulshof, et al, Stability and attractors for the quasi-steady equation of cellular flames, Interfaces and free boundaries, 8 (2006), 301-316. Google Scholar

[7]

J. P. Chollet and M. Lesieur, Parameterization of small scales of three-dimensional isotropic turbulence utilizing spectral closures, J. Atmos. Sci., 38 (1981), 2747-2757. doi: doi:10.1175/1520-0469(1981)038<2747:POSSOT>2.0.CO;2.  Google Scholar

[8]

Y. Dubief and F. Delcayre, On coherent-vortex identification in turbulence, J. Turb., 1 (2000), 11. doi: doi:10.1088/1468-5248/1/1/011.  Google Scholar

[9]

C. Foias and P. Penel, Dissipation totale de l'énergie dans l'équation de Frisch-Kraichnan-Lesieur, C. R. Acad. Sci., Paris, A-B, 280 (1975), A629-A632.  Google Scholar

[10]

U. Frisch, "Turbulence: The Legacy of A. N. Kolmogorov," Cambridge University Press, 1995. Google Scholar

[11]

M. Germano, U. Piomelli, P. Moin and W. Cabot, A dynamic subgrid-scale eddy-viscosity model, Phys. Fluids A., 3 (1991), 1760-1765. doi: doi:10.1063/1.857955.  Google Scholar

[12]

T. Hughes, L. Mazzei, A. Oberai and A. Wray, The multiscale formulation of large-eddy simulation: decay of homogeneous isotropic turbulence, Phys. Fluids, 13 (2001), 505-512. doi: doi:10.1063/1.1332391.  Google Scholar

[13]

J. Hunt, A. Wray and P. Moin, Eddies, stream, and convergence zones in turbulent flows, Center for Turbulence Research Rep., CTR-S88 (1988), 193. Google Scholar

[14]

J. Jimenez, Turbulence, in "Perspectives in Fluid Mechanics" (G. K. Batchelor, H. K. Moffatt and M. G. Woster eds), Cambridge University Press, (2000), 231-283. Google Scholar

[15]

R. H. Kraichnan, Eddy viscosity in two and three dimensions, J. Atmos. Sci., 33 (1976), 1521-1536. doi: doi:10.1175/1520-0469(1976)033<1521:EVITAT>2.0.CO;2.  Google Scholar

[16]

E. Lamballais, O. Métais and M. Lesieur, Spectral-dynamic model for large-eddy simulations of turbulent rotating channel flow, Theor. Comp. Fluid Dyn., 12 (1997), 149-177. doi: doi:10.1007/s001620050104.  Google Scholar

[17]

J. Leray, Sur le mouvement d'un fluide visqueux emplissant l'espace, (French), J. Acta Math., 63 (1934), 193-248. doi: doi:10.1007/BF02547354.  Google Scholar

[18]

M. Lesieur and D. Schertzer, Amortissement auto similaire d'une turbulence à grand nombre de Reynolds, (French), Journal de Mécanique, 17 (1978), 609-646.  Google Scholar

[19]

M. Lesieur, S. Ossia and O. Métais, Infrared pressure spectra in two- and three-dimensional isotropic incompressible turbulence, Phys. Fluids, 11 (1999), 1535-1543. doi: doi:10.1063/1.870016.  Google Scholar

[20]

M. Lesieur, O. Métais and P. Comte, "Large-Eddy Simulations of Turbulence," With a preface by James J. Riley, Cambridge University Press, New York, 2005.  Google Scholar

[21]

M. Lesieur, "Turbulence in Fluids," 4th edition, Springer, 2008. doi: doi:10.1007/978-1-4020-6435-7.  Google Scholar

[22]

M. Lesieur and O. Métais, Large-eddy simulations for geophysical fluid dynamics, in "Handbook of Numerical Analysis" (R. Temam, J. Tribbia and P. Ciarlet, eds.), vol XIV, North-Holland, 2008. Google Scholar

[23]

O. Métais and M. Lesieur, Spectral large-eddy simulations of isotropic and stably-stratified turbulence, J. Fluid Mech., 239 (1992), 157-194. Google Scholar

[24]

S. A. Orszag, Analytical theories of turbulence, J. Fluid Mech., 41 (1970), 363-386. doi: doi:10.1017/S0022112070000642.  Google Scholar

[25]

J. Smagorinsky, General circulation experiments with the primitive equations, Mon. Weath. Rev., 91 (1963), 99-164. Google Scholar

[26]

A. Wray, Private communication, 1988. Google Scholar

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