No CrossRef data available.
Article contents
Enstrophy Cascade and Smagorinsky Model of 2D Turbulent Flows
Published online by Cambridge University Press: 05 May 2011
Abstract
Direct numerical simulations of 2D turbulent flows, freely decaying as well as forced, are performed to examine the mechanism of the enstrophy cascade and serve as a template of developing LES models. The stretching effect on the 2D vorticity gradients is emphasized on the analogy of the stretching effect on 3D vorticity. The enstrophy cascade rate, the Reynolds stresses and the associated eddy viscosity for 2D turbulence are correspondingly derived and investigated. Proposed herein is that the enstrophy cascade rate to be modeled in a large-eddy simulation can be and should be calculated using the only available large-eddy information, especially when the Reynolds number is not very large or when the flow is not stationary.
The simulation results suggest all Kolmogorov's, Kraichnan's, and Saffman's similarity spectra. The Kolmogorov's spectrum appears in front of forced wave numbers and creates a subrange of a zero enstrophy cascade rate and a constant energy cascade rate. The Saffman's spectrum is the dissipation spectrum at large wave numbers. Kraichnan's spectrum shows up at intermediate wave numbers when the Reynolds number is sufficiently high. When the Smagorinsky model is employed for a large eddy simulation, its inability of capturing the significant reverse cascade phenomenon as observed in the DNS data becomes a fatal defect. Nonetheless, if only the mean cascade rate is concerned, the required Smagorinsky constant is evaluated using the DNS data and compared with the theoretical prediction of the Kraichnan's spectrum.
- Type
- Articles
- Information
- Copyright
- Copyright © The Society of Theoretical and Applied Mechanics, R.O.C. 2001
References
REFERENCES
1Speziale, C. G., “Analytical Methods for the Development of Reynolds-Stress Closures in Turbulence,” Annu. Rev. Fluid Mech., 23, p. 107 (1991).CrossRefGoogle Scholar
2Lesieur, M. and Metais, O., “New Trends in Large-Eddy Simulations of Turbulence,” Ann. Rev. Fluid Mech., 28, p. 45 (1996).CrossRefGoogle Scholar
3Mansoour, N. N. and Wray, A. A., “Direct Numerical Simulation Data: A Tool for Turbulence Modeling,” 4th International Symposium on Computational Fluid Dynamics, UCSD (1991).Google Scholar
4Batchelor, G. K., “Computation of the Energy Spectrum in Homogeneous Two-Dimensional Turbulence,” Phys. Fluids, 12, p. II-233 (1969).CrossRefGoogle Scholar
5Leith, C. E., “Diffusion Approximation to Inertial Energy Transfer in Isotropic Turbulence,” Phys. Fluids, 10, p. 1409 (1967).CrossRefGoogle Scholar
6Leith, C. E., “Diffusion Approximation for Two-Dimensional Turbulence,” Phys. Fluids, 11, p. 671 (1968).CrossRefGoogle Scholar
7McWilliams, J. C., “The Emergence of Isolated Coherent Vortices in Turbulent Flow,” J. Fluid Mech., 146, p. 21 (1984).CrossRefGoogle Scholar
8McWilliams, J. C., “The Vortices of Two-Dimensional Turbulence,” J. Fluid Mech., 219, p. 361 (1990).CrossRefGoogle Scholar
9Rhines, P. B., “Waves and Turbulence on a Beta-Plane,” J. Fluid Mech., 69, p. 417 (1975).CrossRefGoogle Scholar
10McWilliams, J. C., “Statistical Properties of Decaying Geostrophic Turbulence,” J. Fluid Mech., 198, p. 199 (1989).CrossRefGoogle Scholar
11Forster, D., Nelson, D. R. and Stephen, M. J., “Large-Distance and Long-Time Properties of a Randomly Stirred Fluid,” Phys. Rev. A, 16, p. 732 (1977).CrossRefGoogle Scholar
12Lesieur, M., Turbulence in Fluids: Stochastic and Numerical Modelling, Kluwer Academic Publishers (1990).CrossRefGoogle Scholar
13Holland, W. R., “The Role of Mesoscale Eddies in the General Circulation of the Ocean — Numerical Experiments Using a Wind-Driven Quasi-Geostrophic Model,” J. Physical Oceanography, 8, p. 363 (1978).2.0.CO;2>CrossRefGoogle Scholar
14Sadourny, R. and Basdevant, C., “Parametrization of Subgrid Scale Barotropic and Baroclinic Eddies in Quasi-Geostrophic Models: Anticipated Potential Vorticity Method,” J. Atmospheric Sciences, 42, p. 1353 (1985).2.0.CO;2>CrossRefGoogle Scholar
15Smagorinsky, J., “General Circulation Experiments with the Primitive Equations, I. the Basic Experiment,” Mon. Weather Rev., 91, p. 99 (1963).2.3.CO;2>CrossRefGoogle Scholar
16Saffman, P. G., “On the Spectrum and Decay of Random Two-Dimensional Vorticity Distributions at Large Reynolds Number,” Studies in Applied Mathematics, L, p. 377 (1971).CrossRefGoogle Scholar
17Kraichnan, R. H., “Inertial Ranges in Two-Dimensional Turbulence,” Phys. Fluids, 10, p. 1417 (1967).CrossRefGoogle Scholar
18Kraichnan, R. H., “Inertial-Range Transfer in Two-and Three-Dimensional Turbulence,” J. Fluid Mech., 47, p. 525 (1971).CrossRefGoogle Scholar
19Kolmogorov, A. N., “The Local Structure of Turbulence in Incompressible Viscous Fluid for very Large Reynolds Numbers,” C. R. Acad. Sci., U.R.S.S., 30, p. 301 (1941).Google Scholar
20Kolmogorov, A. N., “On Degeneration of Isotropic Turbulence in an Incompressible Viscous Liquid,” C. R. Acad. Sci., U.R.S.S., 31, p. 538 (1941).Google Scholar
21Kolmogorov, A. N., “Dissipation of Energy in Locally Isotropic Turbulence,” C. R. Acad. Sci., U.R.S.S., 32, p. 16 (1941).Google Scholar
22Kraichnan, R. H., “Isotropic Turbulence and Inertial-Range Structure,” Phys. Fluids, 9, p. 1728 (1966).CrossRefGoogle Scholar
23Rogallo, R. S., “Numerical Experiments of Homogeneous Turbulence,” NASA Technical Memorandum, 81315 (1981).Google Scholar
24Orszag, S. A., “Numerical Simulation of Incompressible Flows within Simple Boundaries, I. Galerkin (Spectral) Respresentations,” Studies in Applied Mathematics, L, p. 293 (1971).CrossRefGoogle Scholar
25Boris, J. P., Grinstein, F. F., Oran, E. S. and Kolbe, R. L., New Insights into Large Eddy Simulation,” Fluid Dynamics Research, 10, p. 199 (1992).CrossRefGoogle Scholar