Abstract
Direct numerical simulation of turbulent convection in a horizontal liquid layer heated from below is performed within the framework of the nonstationary Navier—Stokes equations with the use of the Bubnov—Galerkin method. The main attention is given to calculations for superhigh supercriticalities. Computational burden is reduced by the use of the splitting method at each step of integration. Previously, the smallness of the residual arising from substitution of simulated results into the initial system of equations is demonstrated and the residual’s dependence on the number of reference functions and supercriticality is considered. A good agreement of the results obtained with the use of different numerical implementations of the Bubnov—Galerkin procedure is shown, in particular, for the stochastic processes corresponding to a low supercriticality and appearing with the formation of strange attractors close to a Mobius strip. The calculations were carried out for a wide range of supercriticality (from 1 to 34000). It is shown that simulations and experiment are in good qualitative agreement.
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References
J. B. McLaughlin and P. C. Martin, “Turbulence in a Statically Stressed Fluid System,” Phys. Rev. A 12, 186–203 (1975).
S. Ya. Gertsenshtein and V. M. Shmidt, “Nonlinear Development and Interaction of Finite-Amplitude Disturbances During a Convective Instability of a Rotating Plane Layer,” Dokl. Akad. Nauk SSSR 225(1), 59–62 (1975).
V. N. Gryaznov and V. I. Polezhaev, Numerical Solution of Nonstationary Navier—Stokes Equations for a Turbulent Regime of Natural Convection, Preprint No. 81 (Inst. of Problems of Mechanics, Russian Academy of Sciences, 1977).
F. V. Dolzhanskii, V. I. Klyatskin, A. M. Oboukhov, and A. M. Chusov, Nonlinear Systems of Hydrodynamic Type (Nauka, Moscow, 1974) [in Russian].
E. N. Lorenz, “Deterministic Nonreriodic Flow,” J. Atmos. Sci. 20, 130–141 (1963).
S. Ya. Gertsenshtein and V. N. Popov, “Integral Method of Calculating Turbulent Convection in a Horizontal Layer of a Solution,” Dokl. Akad. Nauk 363, 769–771 (1998).
I. N. Sibgatullin, S. Ya. Gertsenstein, and N. R. Sibgatullin, “Some Properties of Two-Dimensional Stochastic Regimes of Double-Diffusive Convection in Plane Layer,” Chaos 13, 1231–1241 (2003).
E. B. Rodichev and O. V. Rodicheva, “Two-Dimensional Turbulence in the Rayleigh-Benard Problem,” Dokl. Akad. Nauk 359, 486–489 (1998).
S. Gertsenstein and I. Sibgatullin, “Bifurcations, Transition to Turbulence and Development of Chaotic Regimes for Double-Diffusive Convection,” WSEAS Trans. Appl. Theor. Mech. 1(1), 110–114 (2006).
V. I. Arnold and A. Avez, Problemes Ergodique de la Mechanique Classique (Gauthier Villars, Paris, 1967).
Ya. G. Sinai, “On the Concept of Entropy in Smooth Dynamical Systems,” Dokl. Akad. Nauk SSSR 124, 754 (1959).
D. V. Anosov, “Introductory Paper,” in Smooth Dynamical Systems, Ed. by A. N. Kolmogorov and S. P. Novikov (Mir, Moscow, 1977) [in Russian].
F. V. Dolzhanskii, “Mechanical Prototypes of Fundamental Hydrodynamic Invariants and Slow Manifolds,” Usp. Fiz. Nauk 175, 1257–1288 (2005).
S. Smeil, “Differentiable Dynamical Systems,” Usp. Mat. Nauk 25, 113–185 (1970).
V. I. Yudovich, “Stability of Stationary Flows of a Viscous Incompressible Fluid,” Dokl. Akad. Nauk SSSR 5, 1037–1040 (1965).
G. M. Zaslavskii and R. Z. Sagdeev, Introduction into Nonlinear Physics: From Pendulum to Turbulence and Chaos (Nauka, Moscow, 1988) [in Russian].
I. B. Palymskii, “Numerical Simulation of Two-Dimensional Convection at a High Supercriticality,” Usp. Mekh., No. 4, 3–28 (2006).
R. Farhadien and R. S. Tankin, “Interferometric Study of Two-Dimensional Benard Convection Cells,” J. Fluid Mech. 66, 739–752 (1974).
R. A. Denton and I. R. Wood, “Turbulent Convection between Two Horizontal Plates,” Int. J. Heat Mass Transfer 22, 1339–1346 (1979).
H. T. Rossby, “A Study of Benard Convection with and Without Rotation,” J. Fluid Mech. 36, 309–335 (1969).
D. E. Fitzjarrald, “An Experimental Study of Turbulent Convection in Air,” J. Fluid Mech. 73, 693–719 (1976).
A. V. Malevsky and D. A. Yuen, “Characteristics-Based Methods Applied to Infinite Prandtl Number Thermal Convection in the Hard Turbulent Regime,” Phys. Fluids. A 3, 2105–2115 (1991).
A. M. Garon and R. J. Goldstein, “Velocity and Heat Transfer Measurements in Thermal Convection,” Phys. Fluids 16, 1818–1825 (1973).
T. Y. Chu and R. J. Goldstein, “Turbulent Convection in a Horizontal Layer of Water,” J. Fluid Mech. 60, 141–159 (1973).
J. W. Deardorff and G. E. Willis, “Investigation of Turbulent Thermal Convection between Horizontal Plates,” J. Fluid Mech. 28, 675–704 (1967).
S. Globe and D. Dropkin, “Natural-Convection Heat Transfer in Liquids Confined by Two Horizontal Plates and Heated Below,” J. Heat Transfer 81, 24–28 (1959).
W. V. R. Malkus, “Discrete Transitions in Turbulent Convection,” Proc. R. Soc. Ser. A, London 225(1161), 185–195 (1954).
G. Z. Gershuni and E. M. Zhukhovitskii, Convective Stability of an Incompressible Fluid (Nauka, Moscow, 1972) [in Russian].
G. Neumann, “Three-Dimensional Numerical Simulations of Buoyancy-Driven Convection in Vertical Cylinders Heated from Below,” J. Fluid Mech. 214, 559 (1990).
X.-Z. Wu, L. Kananoff, A. Libchaber, and M. Sano, “Frequency Power Spectrum of Temperature Fluctuations in Free Convection,” Phys. Rev. Lett. 64, 2140–2143 (1990).
L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics, Vol. 2: Hydrodynamics (Nauka, Moscow, 1988; Pergamon, Oxford, 1975).
R. M. Kerr, “Rayleigh Number Scaling in Numerical Convection,” J. Fluid Mech. 310, 139–179 (1996).
S. Cioni, S. Ciliberto, and J. Sommeria, “Temperature Structure Functions in Turbulent Convection at Low Prandtl Number,” Europhys. Lett. 32, 413–418 (1995).
P. G. Frik, Turbulence: Approaches and Models (Inst. of Computer Studies, Moscow, 2003) [in Russian].
S. Ashkenazi and V. Steinberg, “Spectra and Statistics of Velocity and Temperature Fluctuations in Turbulent Convection,” Phys. Rev. Lett. 83, 4760–4763 (1999).
X.-D. Shang and K.-Q. Xia, “Scaling of the Velocity Power Spectra in Turbulent Thermal Convection,” Phys. Rev. E 64, 065301-1-4 (2001).
A. V. Malevsky, “Spline-Characteristic Method for Simulation of Convective Turbulence,” J. Comput. Phys. 123, 466–475 (1996).
R. A. Brazhe and O. N. Kudelin, “Experimental Implementation of the Lorenz Model of a Convective Instability of a Fluid in a Vertical Toroidal Cell,” Izv. Vyssh. Uchebn. Zaved., Probl. Nonlin. Dyn., 14(6), 88–98 (2006).
J. J. Niemela, L. Skrbek, K. R. Sreenivasan, et al., “Turbulent Convection at Very High Rayleigh Numbers,” Nature 404(20), 837–840 (2000).
R. Verzicco and R. Camussi, “Numerical Experiments on Strongly Turbulent Thermal Convection in a Slender Cylindrical Cell,” J. Fluid Mech. 477, 19–49 (2003).
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Original Russian Text © S.Ya. Gertsenshtein, I.A. Palymskii, I.N. Sibgatullin, 2008, published in Izvestiya AN. Fizika Atmosfery i Okeana, 2008, Vol. 44, No. 1, pp. 75–85.
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Gertsenshtein, S.Y., Palymskii, I.A. & Sibgatullin, I.N. Intense turbulent convection in a horizontal plane liquid layer. Izv. Atmos. Ocean. Phys. 44, 72–82 (2008). https://doi.org/10.1134/S0001433808010088
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DOI: https://doi.org/10.1134/S0001433808010088