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Spectral large-eddy simulation of isotropic and stably stratified turbulence

Published online by Cambridge University Press:  26 April 2006

Olivier Métais
Affiliation:
Institut de Mécanique de Grenoble, Institut National Polytechnique de Grenoble, et Université Joseph Fourier, Grenoble, BP 53 X, 38041 Grenoble-Cedex, France
Marcel Lesieur
Affiliation:
Institut de Mécanique de Grenoble, Institut National Polytechnique de Grenoble, et Université Joseph Fourier, Grenoble, BP 53 X, 38041 Grenoble-Cedex, France

Abstract

We first recall the concepts of spectral eddy viscosity and diffusivity, derived from the two-point closures of turbulence, in the framework of large-eddy simulations in Fourier space. The case of a spectrum which does not decrease as $k^{-\frac{5}{3}}$ at the cutoff is studied. Then, a spectral large-eddy simulation of decaying isotropic turbulence convecting a passive temperature is performed, at a resolution of 1283 collocation points. It is shown that the temperature spectrum tends to follow in the energetic scales a k−1 range, followed by a $k^{-\frac{5}{3}}$ inertial–convective range at higher wavenumbers. This is in agreement with previous independent calculations (Lesieur & Rogallo 1989). When self-similar spectra have developed, the temperature variance and kinetic energy decay respectively like t−1.37 and t−1.85, with identical initial spectra peaking at ki = 20 and ∝ k8 for k → 0. In the k−1 range, the temperature spectrum is found to collapse according to the law ET(k, t) = 0.1η(〈u2〉/ε) k−1, where ε and η are the kinetic energy and temperature variance dissipation rates. The spectral eddy viscosity and diffusivity are recalculated explicitly from the large-eddy simulation: the anomalous ∝ ln k behaviour of the eddy diffusivity in the eddy-viscosity plateau is shown to be associated with the large-scale intermittency of the passive temperature: the p.d.f. of the velocity component u is Gaussian (∼ exp − X2), while the scalar T, the velocity derivatives ∂u/∂x and ∂u/∂z, and the temperature derivative ∂T/∂z are all close to exponential exp - |X| at high |X|. The pressure distribution is exponential at low pressure and Gaussian at high.

For stably stratified Boussinesq turbulence, the coupling between the temperature and the velocity fields leads to the disappearance of the ‘anomalous’ temperature behaviour (k−1 range, logarithmic eddy diffusivity and exponential probability density function for T). These are the highest-resolution calculations ever performed for this problem. We also split the eddy viscous coefficients into a vortex and a wave component. In both cases (unstratified and stratified), comparisons with direct numerical simulations are performed.

Finally we propose a generalization of the spectral eddy viscosity to highly intermittent situations in physical space: in this structure-function model, the spectral eddy viscosity is based upon a kinetic energy spectrum local in space. The latter is calculated with the aid of a local second-order velocity structure function. This structure function model is compared with other models, including Smagorinsky's, for isotropic decaying turbulence, and with high-resolution direct simulations. It is shown that low-pressure regions mark coherent structures of high vorticity. The pressure spectra are shown to follow Batchelor's quasi-normal law: $\alpha C^2_{\rm k}\epsilon^{\frac{4}{3}}k^{-\frac{7}{3}}$ (Ck is Kolmogorov's constant), with α ≈ 1.32.

Type
Research Article
Copyright
© 1992 Cambridge University Press

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References

André, J. C. & Lesieur, M. 1977 Influence of helicity on the evolution of isotropic turbulence at high Reynolds number. J. Fluid Mech. 81, 187207.Google Scholar
Anselmet, F., Gagne, Y., Hopfinger, E. J. & Antonia, R. A., 1984 High-order velocity structure functions in turbulent shear flows. J. Fluid Mech. 140, 6389.Google Scholar
Antonia, R. A., Chambers, A. J., Van Atta, C. W., Friehe, C. A. & Helland, K. N., 1978 Skewness of temperature derivative in a heated grid flow. Phys. Fluids 21, 509510.Google Scholar
Antonia, R. A., Hopfinger, E. J., Gagne, Y. & Anselmet, F., 1984 Temperature structure functions in turbulent shear flows. Phys. Rev. A 30, 27042707.Google Scholar
Antonia, R. A., Rajagopalan, S., Browne, L. W. B. & Chambers, A. J. 1982 Correlations of squared velocity and temperature derivatives in a turbulent plane jet. Phys. Fluids 25, 11561158.Google Scholar
Antonopoulos-Domis, M.: 1981 Large-eddy simulation of a passive scalar in isotropic turbulence. J. Fluid Mech. 104, 5579.Google Scholar
Batchelor, G. K.: 1951 Pressure fluctuations in isotropic turbulence. Proc. Camb. Phil. Soc. 47, 359374.Google Scholar
Batchelor, G. K.: 1953 The Theory of Homogeneous Turbulence. Cambridge University Press, 197 pp.
Batchelor, G. K.: 1959 Small scale variation of convected quantities like temperature in turbulent fluid. Part 1. General discussion and the case of small conductivity. J. Fluid Mech. 5, 113134.Google Scholar
Batchelor, G. K., Canuto, V. M. & Chasnov, J. R., 1992 Homogeneous buoyancy generated turbulence. J. Fluid Mech. 235, 349378.Google Scholar
Bègue, C., Chacón, T., Ortegón, F. & Pironneau, O. 1987 3D simulation of 2 length scales turbulent flows by homogenization. In Advances in Turbulence 1 (ed. G. Comte-Bellot & J. Mathieu), pp. 135142. Springer.
Brachet, M.: 1990 Géométrie des structures à petite échelle dans le vortex de Taylor–Green. C. R. Acad. Sci. Paris II 311, 775780.Google Scholar
Cambon, C.: 1989 Spectral approach to axisymmetric turbulence in a stratified fluid. In Advances in Turbulence 2 (ed. H. H. Fernholtz & H. E. Fiedler), pp. 162167. Springer.
Cambon, C. & Jacquin, L., 1989 Spectral approach to non-isotropic turbulence subjected to rotation. J. Fluid Mech. 202, 295317.Google Scholar
Castaing, B., Gagne, Y. & Hopfinger, E. J., 1990 Velocity probability density functions of high Reynolds number turbulence. Physica D 46, 177200.Google Scholar
Castaing, B., Gunaratne, G., Heslot, F., Kadanoff, L., Libchaber, A., Thomae, S., Wu, X.-Z., Zaleski, S. & Zanetti, G., 1989 Scaling of hard thermal turbulence in Rayleigh–Bénard convection. J. Fluid Mech. 204, 130.Google Scholar
Chollet, J.-P.: 1984 Turbulence tridimensionnelle isotrope: modélisation statistique des petites échelles et simulation numérique des grandes échelles. Thèse de Doctorat d'État, Grenoble.
Chollet, J.-P.: 1985 Two-point closure used for a sub-grid scale model in large eddy simulations. In Turbulent Shear Flows 4 (ed. L. J. S. Bradbury et al.), pp. 6272. Springer.
Chollet, J.-P. & Lesieur, M. 1981 Parameterization of small scales of three-dimensional isotropic turbulence utilizing spectral closures. J. Atmos. Sci. 38, 27472757.Google Scholar
Comte, P., Lesieur, M. & Fouillet, Y., 1990 Coherent structures of mixing layers in large-eddy simulation. In Topological Fluid Mechanics (ed. H. K. Moffatt & A. Tsinober), pp. 649658. Cambridge University Press.
Corrsin, S.: 1964 The isotropic turbulent mixer: Part II. Arbitrary Schmidt number. AIChE J. 18, 417423.Google Scholar
Craya, A.: 1958 Contribution à l'Analyse de la Turbulence Associée à des Vitesses Moyennes. P. S. T. Ministère de l'Air, 345 pp.
Domaradzki, J. A., Metcalfe, R. W., Rogallo, R. S. & Riley, J. J., 1987 Analysis of subgridscale eddy viscosity with the use of results from direct numerical simulations. Phys. Rev. Lett. 58, 547550.Google Scholar
Eaton, J. K. & Johnston, J. P., 1980 Stanford Rep. MD 39.
Fournier, J.-D.: 1977 Quelques méthodes systématiques de développement en turbulence homogène. Thèse, Université de Nice.
Fung, J. C. H., Hunt, J. C. R., Malik, N. A. & Perkins, R. J., 1992 Kinematic simulation of homogeneous turbulence by unsteady random Fourier modes. J. Fluid Mech. 236, 281317.Google Scholar
Gagne, Y.: 1987 Etude expérimentale de l'intermittence et des singularités dans le plan complexe en turbulence développée. Thèse, Université de Grenoble.
Gargett, A. E.: 1985 Evolution of scalar spectra with the decay of turbulence in a stratified fluid. J. Fluid Mech. 159, 379407.Google Scholar
Georges, W. K., Beuther, P. D. & Arndt, R. E. A. 1984 Pressure spectra in turbulent free shear flows. J. Fluid Mech. 148, 155191.Google Scholar
Gradshteyn, I. S. & Ryzhik, I. M., 1965 Table of Integrals Series and Products. Academic.
Herbert, T.: 1988 Secondary instability in boundary layers. Ann. Rev. Fluid Mech. 20, 487526.Google Scholar
Herring, J. R.: 1974 Approach of axisymmetric turbulence to isotropy. Phys. Fluids 17, 859872.Google Scholar
Herring, J. R.: 1990 Comparison of closure to the spectral-based large eddy simulations. Phys. Fluids 2, 979983.Google Scholar
Herring, J. R. & Kerr, R. M., 1982 Comparison of direct numerical simulations with predictions of two-point closures for isotropic turbulence convecting a passive scalar. J. Fluid Mech. 118, 205219.Google Scholar
Herring, J. R., Schertzer, D., Lesieur, M., Newmann, G. R., Chollett, J.-P. & Larchevêque, M. 1982 A comparative assessment of spectral closures as applied to passive scalar diffusion. J. Fluid Mech. 124, 411437.Google Scholar
Hunt, J. C. R. & Vassilicos, J. C. 1991 Kolmogorov's contributions to the physical and geometrical understanding of small-scale turbulence and recent developments. Proc. R. Soc. Lond. A 434, 183210.Google Scholar
Kaimal, J. C., Wyngaard, J. C., Izumi, Y. & Coté, O. R. 1972 Spectral characteristics of surface-layer turbulence. Q. J. R. Met. Soc. 98, 563589.Google Scholar
Kerr, R. M.: 1985 Higher-order derivative correlations and the alignment of small-scale structures in isotropic numerical turbulence. J. Fluid Mech. 153, 3158.Google Scholar
Kolmogorov, A. N.: 1941 The local structure of turbulence in incompressible viscous fluid for very large Reynolds number. Dokl. Akad. Nauk SSSR 30, 301305.Google Scholar
Kolmogorov, A. N.: 1962 A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds number. J. Fluid Mech. 12, 8285.Google Scholar
Kraichnan, R. H.: 1966 Isotropic turbulence and inertial-range structure. Phys. Fluids 9, 17281752.Google Scholar
Kraichnan, R. H.: 1968 Small-scale structure of a scalar field convected by turbulence. Phys. Fluids 11, 945953.Google Scholar
Kraichnan, R. H.: 1976 Eddy viscosity in two and three dimensions. J. Atmos. Sci. 33, 15211536.Google Scholar
Kraichnan, R. H.: 1990 Models of intermittency in hydrodynamic turbulence. Phys. Rev. Lett. 65, 575578.Google Scholar
Lee, T. D.: 1952 On some statistical properties of hydrodynamical and magneto hydrodynamical fields. Q. Appl. Maths 10, 6974.Google Scholar
Leonard, A.: 1974 Energy cascade in large-eddy simulations of turbulent fluid flows. Adv. Geophys. A 18, 237248.Google Scholar
Lesieur, M.: 1987 Turbulence in Fluids. Nijhoff. 286 pp.
Lesieur, M.: 1990 Turbulence in Fluids (revised edn). Kluwer. 412 pp.
Lesieur, M., Meatais, O. & Rogallo, R. S., 1989 Etude de la diffusion turbulente par simulation des grandes échelles. C. R. Acad. Sci. Paris II 308, 13951400.Google Scholar
Lesieur, M., Montmory, C. & Chollet, J.-P. 1987 The decay of kinetic energy and temperature variance in three-dimensional isotropic turbulence. Phys. Fluids 30, 12781286.Google Scholar
Lesieur, M. & Rogallo, R. S., 1989 Large-eddy simulation of passive scalar diffusion in isotropic turbulence. Physics Fluids A 1, 718722.Google Scholar
Lesieur, M. & Schertzer, D., 1978 Amortissement auto similaire d'une turbulence à grand nombre de Reynolds. J. Méc. 17, 609646.Google Scholar
Leslie, D. C. & Quarini, G. L., 1979 The application of turbulence theory to the formulation of subgrid modelling procedures. J. Fluid Mech. 91, 6591.Google Scholar
Lighthill, J.: 1978 Waves in Fluids. Cambridge University Press, 504 pp.
Lilly, D. K.: 1967 The representation of small-scale turbulence in numerical simulation experiments. Proc. IBM Sci. Comput. Symp. Environ. Sci., IBM Data Process. Div., White Plains, NY, pp. 195210.Google Scholar
Lilly, D. K.: 1983 Stratified turbulence and the mesoscale variability of the atmosphere. J. Atmos. Sci. 40, 749761.Google Scholar
Mcwilliams, J. C.: 1989 Statistical properties of decaying geostrophic turbulence. J. Fluid Mech. 198, 199230.Google Scholar
Mestayer, P.: 1982 Local isotropy in a high-Reynolds-number turbulent boundary layer. J. Fluid Mech. 125, 475503.Google Scholar
Métais, O. & Chollet, J.-P. 1989 Turbulence submitted to stable density stratification: large-eddy simulations and statistical theory. Turbulent Shear Flows 6 (ed. J.-C. André et al.), pp. 398415. Springer.
Métais, O. & Herring, J. R. 1989 Numerical simulations of freely evolving turbulence in stably stratified fluids. J. Fluid Mech. 202, 117148.Google Scholar
Métais, O. & Lesieur, M. 1986 Statistical predictability of decaying turbulence. J. Atmos. Sci. 43, 857870.Google Scholar
Métais, O. & Lesieur, M. 1989 Large eddy simulations of isotropic and stably-stratified turbulence. In Advances in Turbulence 2 (ed. H. H. Fernholz & H. E. Fiedler), pp. 371376. Springer.
Monin, A. S. & Yaglom, A. M., 1975 Statistical Fluid Mechanics, Vol. 2, MIT Press.
Normand, X. & Lesieur, M., 1991 Numerical experiments on transition in the compressible boundary layer over an insulated plate. Theor. Comp. Fluid Dyn. (in press).Google Scholar
Oboukhov, A. M.: 1949 Pressure fluctuations in a turbulent flow. Dokl. Akad. Nauk SSSR 66, 1720.Google Scholar
Orszag, S. A.: 1977 Lectures on the statistical theory of turbulence. In Fluid Dynamics (ed. R. Balian & J.-L. Peube), Les Houches, Juillet 1973, pp. 235374. Gordon and Breach.
Orszag, S. A. & Patterson, G. S., 1972 Numerical simulation of turbulence. In Statistical Models and Turbulence (ed. M. Rosenblatt & C. Van Atta). Lecture Notes in Physics, Vol. 12, pp. 127147. Springer.
Piomelli, U., Moin, P. & Ferziger, J., 1988 Model consistency in large eddy simulation of turbulent channel flows. Phys. Fluids 31, 18841891.Google Scholar
Riley, J. J., Metcalfe, R. W. & Weissman, M. A., 1981 Direct numerical simulations of homogeneous turbulence in density stratified fluids. In Proc. AIP Conf. on Nonlinear Properties of Internal Waves (ed. B. J. West), pp. 79112.
Rogallo, R. S. & Moin, P., 1984 Numerical simulation of turbulent flows. Ann. Rev. Fluid Mech. 16, 99137.Google Scholar
She, Z.-S., Jackson, E. & Orszag, A., 1990 Intermittent vortex structures in homogeneous isotropic turbulence. Nature 344, 226228.Google Scholar
Siggia, E. D.: 1981 Numerical study of small-scale intermittency in three-dimensional. J. Fluid Mech. 107, 375406.Google Scholar
Siggia, E. D. & Patterson, G. S., 1978 Intermittency effects in a numerical simulation of stationary three-dimensional turbulence. J. Fluid Mech. 86, 567592.Google Scholar
Neto, A. Silveira, Grand, D., Métais, O. & Lesieur, M. 1991 Large-eddy simulation of the turbulent flow in the downstream region of a backward-facing step. Phys. Rev. Lett. 66, 23202323.Google Scholar
Smagorinsky, J. S.: 1963 General circulation experiments with the primitive equations. I: the basic experiment. Mon. Weath. Rev. 91, 99163.Google Scholar
Sreenivasan, K. R. & Antonia, R. A., 1977 Skewness of temperature derivatives in turbulent shear flows. Phys. Fluids 20, 19861988.Google Scholar
Sreenivasan, K. R. & Tavoularis, S., 1980 On the skewness of the temperature derivative in turbulent flows. J. Fluid Mech. 101, 783795.Google Scholar
Sreenivasan, K. R., Tavoularis, S., Henry, R. & Corrsin, S., 1980 Temperature fluctuations and scales in grid-generated turbulence. J. Fluid Mech. 100, 597621.Google Scholar
Staquet, C. & Riley, J. J., 1989 On the velocity field associated with potential vorticity. Dyn. Atmos. Oceans 14, 93123.Google Scholar
Tavoularis, S., Bennett, J. C. & Corrsin, S., 1978 Velocity-derivative skewness in small Reynolds number, nearly isotropic turbulence. J. Fluid Mech. 88, 6369.Google Scholar
Van Atta, C. W. 1971 Influence of fluctuations in local dissipation rates on turbulent scalar characteristics in the inertial subrange. Phys. Fluids 14, 18031804; and erratum 16 (1973), 574.Google Scholar
Vincent, A. & Meneguzzi, M., 1991 The spatial structure of homogeneous turbulence at Reynolds numbers around 1000. In Turbulence and Coherent Structures (ed. O. Métais & M. Lesieur), pp. 191201. Kluwer.
Voke, P. R. & Collins, M. W., 1983 Large-eddy simulation: retrospect and prospect. Physico-Chem. Hydrodyn. 4, 119161.Google Scholar
Warhaft, Z. & Lumley, J. L., 1978 An experimental study of the decay of temperature fluctuations in grid-generated turbulence. J. Fluid Mech. 88, 659684.Google Scholar
Williams, R. M. & Paulson, C. A., 1977 Microscale temperature and velocity spectra in the atmospheric boundary layer. J. Fluid Mech. 83, 547567.Google Scholar
Wray, A. A. & Hunt, J. C. R. 1990 Algorithms for classification of turbulent structures. In Topological Fluid Mechanics (ed. H. K. Moffatt & A. Tsinober), pp. 95104. Cambridge University Press.
Yakhot, V. & Orszag, S. A., 1986 Renormalization group (RNG) methods for turbulence closure. J. Sci. Comput. 1, 352.Google Scholar
Yamamoto, K. & Hosokawa, I., 1988 A decaying isotropic turbulence pursued by the spectral method. J. Phys. Soc. Japan 57, 15321535.Google Scholar
Yeh, T. T. & Van Atta, C. W. 1973 Spectral transfer of scalar and velocity fields in heated-grid turbulence. J. Fluid Mech. 58, 233261.Google Scholar
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