Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-19T04:58:43.475Z Has data issue: false hasContentIssue false
  • English
  • Français

Numerical simulation of the turbulent Rayleigh–Bénard problem using subgrid modelling

Published online by Cambridge University Press:  20 April 2006

Thomas M. Eidson
Thomas M. Eidson
Affiliation:
Georgia Institute of Technology, School of Mechanical Engineering
Get access Rights & Permissions [Opens in a new window]

Abstract

A numerical simulation of turbulent natural convection (the Rayleigh–Bénard problem) has been conducted using large-eddy-simulation (LES) methods and the results compared with several experiments. The development of the LES equation is outlined and discussed. The modelling of the small-scale turbulent motion (called subgrid modelling) is also discussed. The resulting LES equations are solved and data collected over a short period of time in a similar manner to the direct simulation of the governing conservation equations. An explicit, second-order accurate, finite-difference scheme is used to solve the equations. Various average properties of the resulting flow field are calculated from the data and compared with experimental data in the literature. The use of a subgrid model allows a higher value of Ra to be simulated than was previously possible with a direct simulation. The highest Ra successfully simulated was 2.5 × 106. The problems at higher values of Ra are discussed and suggestions for improvements made.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 158 , September 1985 , pp. 245 - 268
Copyright
© 1985 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Antonopoulos-Domis M. 1979 Aspects of large eddy simulation of homogeneous isotropic turbulence. QMC EP 6038, Department of Nuclear Engineering, Queen Mary College.
Businger J. A. 1973 Turbulent transfer in the atmospheric surface layer. Workshop in Micro-meteorology, p. 67. American Meteorology Society.
Busse F. H. 1981 Transition to turbulence in Rayleigh-Bénard convection. In Hydrodynamic Instabilities and the Transition to Turbulence (ed. H. L. Swinney & J. P. Gollub). Springer.
Carroll J. J. 1976 The thermal structure of turbulent convection. J. Atmos. Sci. 33, 642.Google Scholar
Chandrasekhar S. 1961 Hydrodynamic and Hydromagnetic Stability. Clarendon.
Chu, T. Y. & Goldstein R. J. 1973 Turbulent convection in a horizontal layer of water. J. Fluid Mech. 60, 141.Google Scholar
Clark R. A., Ferziger, J. H. & Reynolds W. C. 1979 Evaluation of subgrid-scale turbulence models using a fully simulated turbulent flow. J. Fluid Mech. 91, 1.Google Scholar
Clever, R. M. & Busse F. H. 1978 Large wavelength convection rolls in low Prandtl number fluids. Z. angew. Math. Phys. 29, 711.Google Scholar
Deardorff J. W. 1970a A numerical study of three-dimensional turbulent channel flow at large Reynolds numbers. J. Fluid Mech. 41, 453.Google Scholar
Deardorff J. W. 1970b Convection velocity and temperature scales for the unstable planetary boundary layer and for Rayleigh convection. J. Atmos. Sci. 27, 1211.Google Scholar
Deardorff J. W. 1971 On the magnitude of the subgrid scale eddy coefficient. J. Comp. Phys. 7, 120.Google Scholar
Deardorff J. W. 1972 Numerical investigation of neutral and unstable planetary boundary layers. J. Atmos. Sci. 29, 91.Google Scholar
Deardorff J. W. 1973 Three-dimensional numerical modeling of the planetary boundary layer. In Workshop in Micrometeorology (ed. D. A. Haugen), p. 271. American Meteorology Society.
Deardorff, J. W. & Willis G. E. 1965 The effect of two-dimensionality on the suppression of thermal turbulence. J. Fluid Mech. 23, 337.Google Scholar
Deardorff, J. W. & Willis G. E. 1967 Investigation of turbulent thermal convection between horizontal plates. J. Fluid Mech. 28, 675.Google Scholar
Denton, R. A. & Wood I. R. 1979 Turbulent convection between two horizontal plates Intl J. Heat Mass Transfer 22, 1339.Google Scholar
Eidson T. M. 1982 Numerical simulation of the turbulent Rayleigh-Bénard problem using subgrid modeling. Ph.D. Thesis, University of Michigan.
Fitzjarrald D. E. 1976 An experimental study of turbulent convection in air. J. Fluid Mech. 73, 693.Google Scholar
Goldstein, R. J. & Chu T. Y. 1969 Thermal convection in a horizontal layer of air. Prog. Heat Transfer 2, 55.Google Scholar
Grotzbach G. 1980 Numerical simulation of turbulent temperature fluctuations in liquid metals. Intl J. Heat Mass Transfer 24, 475.Google Scholar
Grotzbach G. 1982 Direct numerical simulation of laminar and turbulent Bénard convection. J. Fluid Mech. 119, 27.Google Scholar
Grotzbach G. 1983 Spatial resolution for direct numerical simulation of the Rayleigh-Bénard convection. J. Comp. Phys. 49, 241.Google Scholar
Grotzbach, G. & Schumann U. 1979 Direct numerical simulation of turbulent velocity-, pressure-and temperature-fields in channel flows. In Turbulent shear Flows I (ed. F. Durst et al.), p. 370. Springer.
Hathaway D. H. 1982 Nonlinear simulations of solar rotation effects in supergranules. Solar Phys. 77, 341.Google Scholar
Herring J. R. 1979 Subgrid scale modeling - an introduction and overview. In Turbulent Shear Flows I (ed. F. Durst et al.), p. 347. Springer.
Krishnamurti R. 1970a On the transition to turbulent convection. Part 1. The transition from two- to three-dimensional flow. J. Fluid Mech. 42, 295.Google Scholar
Krishnamurti R. 1970b On the transition to turbulent convection. Part 2. The transition to time-dependent flow. J. Fluid Mech. 42, 309.Google Scholar
Kwak D. 1975 Three-dimensional time dependent computation of turbulent flow. Ph.D. Thesis, Stanford University.
Lilly D. K. 1962 On the numerical simulation of buoyant convection. Tellus 15, 148.Google Scholar
Lilly D. K. 1967 The representation of small-scale turbulence in numerical simulation experiments. Proc. of the IBM Scientific Computer Symposium on Environmental Sciences, p. 195. IBM Form No. 320–1951.
Lipps F. B. 1976 Numerical simulation of three-dimensional Bénard convection in air. J. Fluid Mech. 75, 113.Google Scholar
Lipps, F. B. & Somerville C. J. 1971 Dynamics of variable wave-length in finite-amplitude Bénard convection. Phys. Fluids 14, 759.Google Scholar
Long R. R. 1976 Relation between Nusselt number and Rayleigh number in turbulent thermal convection. J. Fluid Mech. 73, 445.Google Scholar
Long R. R. 1977 Some aspects of turbulence in geophysical systems. Adv. Appl. Mech. 17, 1.Google Scholar
Mcmillan, O. J. & Ferziger J. H. 1979 Direct testing of subgrid-scale models. AIAA J. 17, 1340.Google Scholar
Mansour N. N., Moin P., Reynolds, W. C. & Ferziger J. H. 1979 Improved methods for large eddy simulations of turbulence. In Turbulent Shear Flows I (ed. F. Durst et al.), p. 386. Springer.
Mihaljan J. M. 1962 A rigorous exposition of the Boussinesq approximation. Astr. J. 136, 1126.Google Scholar
Moin, P. & Kim J. 1982 Numerical investigation of turbulent channel flow. J. Fluid Mech. 118, 341.Google Scholar
Moin P., Mansour N. N., Reynolds, W. C. & Ferziger J. H. 1978 Large eddy simulation of turbulent shear flow. Lecture Notes in Physics, Vol. 90. Springer.
Piacsek, S. A. & Williams G. P. 1970 Conservation properties of convection difference schemes. J. Comp. Phys. 6, 392.Google Scholar
Reynolds A. J. 1975 The prediction of turbulent Prandtl and Schmidt numbers. Intl J. Heat Mass Transfer 18, 1055.Google Scholar
Reynolds W. C. 1976 Computation of turbulent flows. Ann. Rev. Fluid Mech. 8, 183.Google Scholar
Robert A. J. 1966 The integration of a low order spectral form of the primitive meteorological equations. J. Met. Soc. Japan 44, 237.Google Scholar
Rogallo, R. S. & Moin P. 1984 Numerical simulation of turbulent flows. Ann. Rev. Fluid Mech. 16, 99.Google Scholar
Schumann W. 1973 Ein Verfahren zur direckten numerischen Simulation turbulenter Stroemungen in Platten- und Ringspaltkanaelen und ueber seine Anwendung zur Untersuchung von Turbulenzmodellen. Thesis, Univesitaet Karlsruhe (NASA Technical Translation, NASA TT F 15, p. 391).
Schumann U., Grotzbach, G. & Kleiser L. 1980 Direct numerical simulation of turbulence. In Prediction Methods for Turbulent Flows (ed. W. Kollman), p. 124. Hemisphere.
Shaanan S. 1975 Numerical simulation of turbulence in the presence of shear. Ph.D. thesis, Stanford University.
Smagorinsky J. 1963 General circulation experiments with the primitive equations. Mon. Weather Rev. 91, 99.Google Scholar
Spiegel, E. A. & Veronis G. 1960 On the Boussinesq approximation for a compressible fluid. Astrophys. J. 131, 442.Google Scholar
Williams G. P. 1969 Numerical integration of the three-dimensional Navier-Stokes equations for incompressible flow J. Fluid Mech. 37, 727.Google Scholar