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Stochastic bifurcation analysis of Rayleigh–Bénard convection

Published online by Cambridge University Press:  06 April 2010

DANIELE VENTURI
Affiliation:
Department of Energy, Nuclear and Environmental Engineering, University of Bologna, viale del Risorgimento 2, 40136 Bologna, Italy
XIAOLIANG WAN
Affiliation:
Department of Mathematics, Louisiana State University, 226 Lockett Hall, Baton Rouge, LA 70803-4918, USA
GEORGE EM KARNIADAKIS*
Affiliation:
Division of Applied Mathematics, Brown University, 182 George Street, Providence, RI 02912, USA
*
Email address for correspondence: gk@dam.brown.edu

Abstract

Stochastic bifurcations and stability of natural convection within two-dimensional square enclosures are investigated by different stochastic modelling approaches. Deterministic stability analysis is carried out first to obtain steady-state solutions and primary bifurcations. It is found that multiple stable steady states coexist, in agreement with recent results, within specific ranges of Rayleigh number. Stochastic simulations are then conducted around bifurcation points and transitional regimes. The influence of random initial flow states on the development of supercritical convection patterns is also investigated. It is found that a multi-element polynomial chaos method captures accurately the onset of convective instability as well as multiple convection patterns corresponding to random initial flow states.

Type
Papers
Copyright
Copyright © Cambridge University Press 2010

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References

REFERENCES

Acharjee, S. & Zabaras, N. 2006 A concurrent model reduction approach on spatial and random domains for the solution of stochastic PDEs. Intl J. Numer. Meth. Engng. 66 (12), 19341954.Google Scholar
Ahlers, G., Meyer, C. W. & Cannell, D. S. 1989 Deterministic and stochastic effects near the convective onset. J. Stat. Phys. 54 (5/6), 11211131.Google Scholar
Allgower, E. L. & Georg, K. 1990 Numerical Continuation Methods: An Introduction. Springer.Google Scholar
Alves, L. S. De, Cotta, R. M. & Pontes, J. 2002 Stability analysis of natural convection in porous cavities though integral transforms. Intl J. Heat Mass Transfer 45, 11851195.Google Scholar
Asokan, B. W. & Zabaras, N. 2005 Using stochastic analysis to capture unstable equilibrium in natural convection. J. Comput. Phys. 208, 134153.Google Scholar
Bousset, F., Lyubimov, D. V. & Sedel’nikov, G. A. 2008 Three-dimensional convection regimes in a cubical cavity. Fluid Dyn. 43 (1), 18.Google Scholar
Burkardt, J., Gunsberger, M. & Webster, C. 2007 Reduced order modelling of some nonlinear stochastic partial differential equations. Intl J. Numer. Anal. Mod. 4 (3–4), 368391.Google Scholar
Catton, I. 1972 The effect of insulating vertical walls on the onset of motion in a fluid heated from below. Intl J. Heat Mass Transfer 15 (4), 665672.Google Scholar
Chandrasekhar, S. 1981 Hydrodynamic and Hydromagnetic Stability. Dover.Google Scholar
Cotta, R. M. 1993 Integral Transforms in Computational Heat and Fluid Flow. CRC Press.Google Scholar
Dhooge, A., Govaerts, W. & Kuznetsov, Yu. A. 2003 MATCONT: a Matlab package for numerical bifurcation analysis of ODEs. ACM Trans. Math. Softw. 29, 141164.Google Scholar
Doostan, A., Ghanem, R. G. & Red-Horse, J. 2007 Stochastic model reduction for chaos representation. Comput. Meth. Appl. Mech. Engng 196, 39513966.Google Scholar
Drazin, P. G. & Reid, W. H. 1961 Hydrodynamic Stability. Cambridge University Press.Google Scholar
Figueira, E. & Cotta, R. M. 1996 Benchmark results for internal forced convection through integral transformation. Intl Commun. Heat Mass Transfer 23 (7), 10191029.Google Scholar
Fishman, G. S. 1996 Monte Carlo: Concepts, Algorithms and Applications. Springer.Google Scholar
Ganapathysubramanian, B. & Zabaras, N. 2007 Sparse grid collocation schemes for stochastic natural convection problems. J. Comput. Phys. 225 (1), 652685.Google Scholar
Gelfgat, A. Yu. 1999 Different modes of Rayleigh–Bénard instability in two and three-dimensional rectangular enclosures. J. Comput. Phys. 156, 300324.Google Scholar
Gelfgat, A. Yu. & Bar-Yoseph, P. Z. 2004 Multiple solutions and stability of confined convective and swirling fows – a continuing challenge. Intl J. Numer. Meth. Heat Fluid Flow 14 (2), 213241.Google Scholar
Gelfgat, A. Yu., Bar-Yoseph, P. Z. & Yarin, A. L. 1999 Stability of multiple steady states of convection in laterally heated cavities. J. Fluid Mech. 338, 315334.Google Scholar
Ghanem, R. G. & Spanos, P. D. 1998 Stochastic Finite Elements: A Spectral Approach. Springer.Google Scholar
Hammersley, J. M. & Handscomb, D. C. 1967 Monte Carlo Methods. Fletcher & Son Ltd.Google Scholar
Hydon, P. E. 2000 How to construct the discrete symmetries of partial differential equations. Eur. J. Appl. Math. 11, 515527.Google Scholar
Hydon, P. E. 2008 Symmetry Methods for Differential Equations: A Beginner's Guide, 2nd edn. Cambridge University Press.Google Scholar
Kelly, R. E. & Pal, D. 1978 Thermal convection with spatially periodic boundary conditions: resonant wavelength excitation. J. Fluid Mech. 86 (3), 433456.Google Scholar
Kirchartz, K. R. & Oertel, H. Jr 1988 Three-dimensional thermal cellular convection in rectangular boxes. J. Fluid Mech. 192, 249286.Google Scholar
Le Maître, O., Njam, H. N., Ghanem, R. G. & Knio, O. M. 2004 Uncertainty propagation using Wiener–Haar expansions. J. Comput. Phys. 197, 2857.Google Scholar
Le Maître, O., Reagan, M. T., Debusschere, B., Najm, H. N., Ghanem, R. G. & Knio, O. M. 2005 Natural convection in a closed cavity under stochastic non-Boussinesq conditions. SIAM J. Sci. Comput. 26 (2), 375394.Google Scholar
Leal, M. A., Machado, H. A. & Cotta, R. M. 2000 Integral transform solutions of transient natural convection in enclosures with variable fluid properties. Intl J. Heat Mass Transfer 43, 39773990.Google Scholar
Lucor, D., Xiu, D., Su, C.-H. & Karniadakis, G. E. 2003 Predictability and uncertainty in CFD. Intl J. Numer. Meth. Fluids 43 (5), 485505.Google Scholar
Luijkx, J. M. & Platten, J. K. 1981 On the onset of free convection in a rectangular channel. J. Non-Equilib. Thermodyn. 6, 141.Google Scholar
Ma, X. & Zabaras, N. 2009 An adaptive hierarchical sparse grid collocation algorithm for the solution of stochastic differential equations. J. Comput. Phys. 228, 30843113.Google Scholar
Özişik, M. N. 1985 Heat Transfer: A Basic Approach. McGraw-Hill.Google Scholar
Özişik, M. N. 1999 Heat Conduction, 2nd edn. Wiley-Interscience.Google Scholar
Pallares, J., Arroyo, M. P., Grau, F. X. & Giralt, F. 2001 Experimental laminar Rayleigh–Bénard convection in a cubical cavity at moderate Rayleigh and Prandtl numbers. Exp. Fluids 31 (2), 208218.Google Scholar
Pallares, J., Cuesta, I., Grau, F. X. & Giralt, F. 1996 Natural convection in a cubical cavity heated from below at low Rayleigh numbers. Intl J. Heat Mass Transfer 39 (15), 32333247.Google Scholar
Pallares, J., Grau, F. X. & Giralt, F. 1999 Flow transitions in laminar Rayleigh–Bénard convection in a cubical cavity at moderate Rayleigh numbers. Intl J. Heat Mass Transfer 42, 753769.Google Scholar
Puigjaner, D., Herrero, J., Giralt, F. & Simó, C. 2004 Stability analysis of the flow in a cubical cavity heated from below. Phys. Fluids 16 (10), 36393654.Google Scholar
Puigjaner, D., Herrero, J., Simó, C. & Giralt, F. 2008 Bifurcation analysis of steady Rayleigh–Bénard convection in a cubical cavity with conducting sidewalls. J. Fluid Mech. 598, 393427.Google Scholar
Stork, K. & Müller, U. 1972 Convection in boxes: experiments. J. Fluid Mech. 54 (4), 599611.Google Scholar
Vekstein, G. 2004 Energy principle for the onset of convection. Eur. J. Phys. 25, 667673.Google Scholar
Venturi, D. 2006 On proper orthogonal decomposition of randomly perturbed fields with applications to flow past a cylinder and natural convection over a horizontal plate. J. Fluid Mech. 559, 215254.Google Scholar
Venturi, D., Wan, X. & Karniadakis, G. E. 2008 Stochastic low-dimensional modelling of a random laminar wake past a circular cylinder. J. Fluid Mech. 606, 339367.Google Scholar
Wan, X. & Karniadakis, G. E. 2005 An adaptive multi-element generalized polynomial chaos method for stochastic differential equations. J. Comput. Phys. 209 (2), 617642.Google Scholar
Wan, X. & Karniadakis, G. E. 2006 a Multi-element generalized polynomial chaos for arbitrary probability measures. SIAM J. Sci. Comput. 28 (3), 901928.Google Scholar
Wan, X. & Karniadakis, G. E. 2006 b Stochastic heat transfer enhancement in a grooved channel. J. Fluid Mech. 565, 255278.Google Scholar
Xiu, D. & Karniadakis, G. E. 2002 The Wiener–Askey polynomial chaos for stochastic differential equations. SIAM J. Sci. Comput. 24 (2), 619644.Google Scholar
Xiu, D. & Karniadakis, G. E. 2003 modelling uncertainty in flow simulations via generalized polynomial chaos. J. Comput. Phys. 187, 137167.Google Scholar