Abstract
The problem of two-dimensional, periodic in the horizontal coordinate, convection of an incompressible fluid heated from below between two horizontal planes is considered. The problem is solved in two formulations: with (stress-)free and hard (no-slip) boundary conditions on the horizontal planes. It is shown that at small supercriticalities the two-dimensional convection calculation leads to more correct results with hard than with free boundary conditions. It is established that the difference between the free and hard conditions is most strongly manifested in the pulsations of the vertical velocity component, whereas the dependence of the Nusselt number and the pulsations of the horizontal velocity component on the boundary conditions is more weakly expressed.
Similar content being viewed by others
References
A.V. Getling, Rayleigh-Bénard Convection: Structure and Dynamics (World Scientific, Singapore, 1996).
G.Z. Gershuni and E.M. Zhukhovitskii, Convective Instability of an Incompressible Fluid [in Russian] (Nauka, Moscow, 1972).
H. Yahata, “Onset of Chaos in the Rayleigh-Bénard Convection,” Progr. Theor. Phys. Suppl., No. 79, 26–74 (1984).
S.Ya. Gertsenshtein and V.M. Shmidt, “Nonlinear Development and Interaction of Finite-Amplitude Perturbations in the Convective Instability of a Rotating Plane Layer,” Dokl. Akad. Nauk SSSR 225(1), 59–62 (1975).
S.Ya. Gertsenshtein, E.B. Rodichev, and V.M. Shmidt, “Interaction of Three-Dimensional Waves in a Rotating Horizontal Liquid Layer Heated from Below,” Dokl. Akad. Nauk SSSR 238(3), 545–548 (1978).
S.Ya. Gertsenshtein and E.B. Rodichev, “Models of Transition to Turbulence in Convective Instability,” Modelirovanie v Mekhanike 3(20) (4), 59–65 (1989).
A.Yu. Gelfgat, “Different Modes of Rayleigh-Bénard Instability in Two- and Three-Dimensional Rectangular Enclosures,” J. Comp. Phys. 156(2), 300–324 (1999).
J.H. Curry, J.R. Herring, J. Loncaric, and S.A. Orszag, “Order and Disorder in Two- and Three-Dimensional Bénard Convection,” J. Fluid Mech. 147, 1–38 (1984).
T. Cortese and S. Balachandar, “Vortical Nature of Thermal Plumes in Turbulent Convection,” Phys. Fluids A 5(12), 3226–3232 (1993).
A.V. Malevsky, “Spline-Characteristic Method for Simulation of Convective Turbulence,” J. Comput. Phys. 123(2), 466–475 (1996).
G. Grotzbach, “Direct Numerical Simulation of Laminar and Turbulent Bénard Convection,” J. Fluid Mech. 119, 27–53 (1982).
J.B. McLaughlin and S.A. Orszag, “Transition from Periodic to Chaotic Thermal Convection,” J. Fluid Mech. 122, 123–142 (1982).
R.M. Kerr, “Rayleigh Number Scaling in Numerical Convection,” J. Fluid Mech. 310, 139–179 (1996).
N.V. Nikitin and V.I. Polezhaev, “Three-Dimensional Effects in Transitional and Turbulent Czochralski Thermal Convection Regimes,” Fluid Dynamics 34(6), 843–850 (1999).
V.M. Paskonov, V.I. Polezhaev, and L.A. Chudov, Numerical Simulation of Heat- and Mass Transfer Processes [in Russian] (Nauka, Moscow, 1984).
A.V. Getling, “Nonlinear Evolution of the Continuous Spectrum of Nonlinear Perturbations in the Bénard-Rayleigh Problem,” Dokl. Akad. Nauk SSSR 233(2), 308–311 (1977).
K.I. Babenko and A.I. Rakhmanov,Numerical Investigation of Two-Dimensional Convection [in Russian], Preprint No. 118 (M.V. Keldysh Institute of Applied Mathematics, USSR Academy of Sciences, Moscow, 1988).
O.V. Rodicheva and E.B. Rodichev, “Two-Dimensional Turbulence in the Rayleigh-Bénard Problem,” Dokl. Ros. Akad. Nauk 359(4), 486–489 (1998).
E. Zienicke, N. Seehafer, and F. Feudel, “Bifurcations in Two-Dimensional Rayleigh-Bénard Convection,” Phys. Rev. E 57(1), 428–435 (1998).
G. Veronis, “Large-Amplitude Bénard Convection,” J. Fluid Mech. 26(Pt 1), 49–68 (1966).
D.R. Moore and N.O. Weiss, “Two-Dimensional Rayleigh-Bénard Convection,” J. Fluid Mech. 58(Pt 2), 289–312 (1973).
I. Palymskii, “Direct Numerical Simulation of Turbulent Convection,” in Progress in Computational Heat and Mass Transfer (Proc. 4th Intern. Conf. on Comput. Heat and Mass Transfer ICCHMT2005), Ed. by R. Bennacer (Lavoisier, Paris, 2005), Vol. 1, 101–106.
I.B. Palymskii, “Numerical Investigation of Stochastic Convection of Chemically Equilibrium Gas,” in Proc. 2nd Intern. Conf. Applied Mech. and Mater. ICAMM 2003 (Durban, South Africa, 2003), 145–150.
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(9), 2105–2115 (1991).
I. Goldhirsch, R.B. Pelz, and S.A. Orszag, “Numerical Simulation of Thermal Convection in a Two-Dimensional Finite Box,” J. Fluid Mech. 199, 1–28 (1989).
E.E. DeLuca, J. Werne, R. Rosner, and F. Cattaneo, “Numerical Simulation of Soft and Hard Turbulence: Preliminary Results for Two-Dimensional Convection,” Phys. Rev. Letters 64(20), 2370–2373 (1990).
J. Werne, “Structure of Hard-Turbulent Convection in Two Dimensions: Numerical Evidence,” Phys. Rev. E 48(2), 1020–1035 (1993).
T.B. Lennie, D.P. McKenzie, D.R. Moore, and N.O. Weiss, “The Breakdown of Steady Convection,” J. Fluid Mech. 188, 47–85 (1988).
I.B. Palymskii, “A Method of Numerical Simulation of Convective Flows,” Vychisl. Tekhnologii 5(6), 53–61 (2000).
I.B. Palymskii, “Linear and Nonlinear Analysis of a Numerical Method for Calculating Convective Flows,” Sib. Zh. Vychisl. Matematiki 7(2), 143–163 (2004).
R. Farhadieh and R.S. Tankin, “Interferometric Study of Two-Dimensional Bénard Convection Cells,” J. Fluid Mech. 66(Pt 4), 739–752 (1974).
R.A. Denton and I.R. Wood, “Turbulent Convection between Two Horizontal Plates,” Intern. J. Heat and Mass Transfer 22(10), 1339–1346 (1979).
H.T. Rossby, “A Study of Bénard Convection with and without Rotation,” J. Fluid Mech. 36(Pt 2), 309–335 (1969).
D.E. Fitzjarrald, “An Experimental Study of Turbulent Convection in Air,” J. Fluid Mech. 73(Pt 4), 693–719 (1976).
A.M. Garon and R.J. Goldstein, “Velocity and Heat Transfer Measurements in Thermal Convection,” Phys. Fluids 16(11), 1818–1825 (1973).
T.Y. Chu and R. J. Goldstein, “Turbulent Convection in a Horizontal Layer of Water,” J. Fluid Mech. 60(Pt 1), 141–159 (1973).
W. V. R. Malkus, “Discrete Transitions in Turbulent Convection,” Proc. Roy. Soc. London, Ser. A 225(1161), 185–195 (1954).
G. Neumann, “Three-Dimensional Numerical Simulation of Buoyancy-Driven Convection in Vertical Cylinders Heated from Below,” J. Fluid Mech. 214, 559–578 (1990).
J.W. Deardorff and G.E. Willis, “Investigation of Turbulent Thermal Convection between Horizontal Plates,” J. Fluid Mech. 28(Pt 4), 675–704 (1967).
Additional information
Original Russian Text © I.B. Palymskii, 2007, published in Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, 2007, Vol. 42, No. 4, pp. 50–60.
Rights and permissions
About this article
Cite this article
Palymskii, I.B. Numerical simulation of two-dimensional convection: Role of boundary conditions. Fluid Dyn 42, 550–559 (2007). https://doi.org/10.1134/S0015462807040059
Received:
Issue Date:
DOI: https://doi.org/10.1134/S0015462807040059