Stochastic bifurcation analysis of Rayleigh-Bénard convection

Journal of Fluid Mechanics

Volumn 650, Issue , 2010, Pages 391-413

  Venturi, Daniele ^a   Wan, Xiaoliang ^b   Karniadakis, George Em ^c  

^a UNIVERSITY OF BOLOGNA   (Italy)

^b LOUISIANA STATE UNIVERSITY   (United States)

^c BROWN UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

BIFURCATION (MATHEMATICS); NATURAL CONVECTION; STOCHASTIC MODELS;

CONVECTION PATTERNS; CONVECTIVE INSTABILITIES; STABLE STEADY STATE; STEADY STATE SOLUTION; STOCHASTIC BIFURCATION; STOCHASTIC SIMULATIONS; SUPERCRITICAL CONVECTION; TRANSITIONAL REGIMES;

STOCHASTIC SYSTEMS;

BIFURCATION; FLOW STABILITY; RAYLEIGH NUMBER; SIMULATION; STABILITY ANALYSIS; STEADY-STATE EQUILIBRIUM; STOCHASTICITY; TWO-DIMENSIONAL FLOW;

EID: 77952420921     PISSN: 00221120     EISSN: 14697645     Source Type: Journal    
DOI: 10.1017/S0022112009993685     Document Type: Article

References (47)

1. 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), 1934-1954.
2. Ahlers, G., Meyer, C. W. & Cannell, D. S. 1989 Deterministic and stochastic effects near the convective onset. J. Stat. Phys. 54 (5/6), 1121-1131.
3. Allgower, E. L. & Georg, K. 1990 Numerical Continuation Methods: An Introduction. Springer.
4. 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, 1185-1195.
5. Asokan, B. W. & Zabaras, N. 2005 Using stochastic analysis to capture unstable equilibrium in natural convection. J. Comput. Phys. 208, 134-153.
6. Bousset, F., Lyubimov, D. V. & Sedel'nikov, G. A. 2008 Three-dimensional convection regimes in a cubical cavity. Fluid Dyn. 43 (1), 1-8.
7. 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), 368-391.
8. 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), 665-672.
9. Chandrasekhar, S. 1981 Hydrodynamic and Hydromagnetic Stability. Dover.
10. Cotta, R. M. 1993 Integral Transforms in Computational Heat and Fluid Flow. CRC Press.
11. Dhooge, A., Govaerts, W. & Kuznetsov, Yu. A. 2003 MATCONT: a Matlab package for numerical bifurcation analysis of ODEs. ACM Trans. Math. Softw. 29, 141-164.
12. Doostan, A., Ghanem, R. G. & Red-Horse, J. 2007 Stochastic model reduction for chaos representation. Comput. Meth. Appl. Mech. Engng 196, 3951-3966.
13. Drazin, P. G. & Reid, W. H. 1961 Hydrodynamic Stability. Cambridge University Press.
14. Figueira, E. & Cotta, R. M. 1996 Benchmark results for internal forced convection through integral transformation. Intl Commun. Heat Mass Transfer 23 (7), 1019-1029.
15. Fishman, G. S. 1996 Monte Carlo: Concepts, Algorithms and Applications. Springer.
16. Ganapathysubramanian, B. & Zabaras, N. 2007 Sparse grid collocation schemes for stochastic natural convection problems. J. Comput. Phys. 225 (1), 652-685.
17. Gelfgat, A. Yu. 1999 Different modes of Rayleigh-Bénard instability in two and three-dimensional rectangular enclosures. J. Comput. Phys. 156, 300-324.
18. 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), 213-241.
19. 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, 315-334.
20. Ghanem, R. G. & Spanos, P. D. 1998 Stochastic Finite Elements: A Spectral Approach. Springer.
21. Hammersley, J. M. & Handscomb, D. C. 1967 Monte Carlo Methods. Fletcher & Son Ltd.
22. Hydon, P. E. 2000 How to construct the discrete symmetries of partial differential equations. Eur. J. Appl. Math. 11, 515-527.
23. Hydon, P. E. 2008 Symmetry Methods for Differential Equations: A Beginner's Guide, 2nd edn. Cambridge University Press.
24. Kelly, R. E. & Pal, D. 1978 Thermal convection with spatially periodic boundary conditions: resonant wavelength excitation. J. Fluid Mech. 86 (3), 433-456.
25. Kirchartz, K. R. & Oertel, H., Jr 1988 Three-dimensional thermal cellular convection in rectangular boxes. J. Fluid Mech. 192, 249-286.
26. Le Mâ?tre, O., Njam, H. N., Ghanem, R. G. & Knio, O. M. 2004 Uncertainty propagation using Wiener-Haar expansions. J. Comput. Phys. 197, 28-57.
27. Le Mâtre, 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), 375-394.
28. 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, 3977-3990.
29. Lucor, D., Xiu, D., Su, C.-H. & Karniadakis, G. E. 2003 Predictability and uncertainty in CFD. Intl J. Numer. Meth. Fluids 43 (5), 485-505.
30. Luijkx, J. M. & Platten, J. K. 1981 On the onset of free convection in a rectangular channel. J. Non-Equilib. Thermodyn. 6, 141.
31. Ma, X. & Zabaras, N. 2009 An adaptive hierarchical sparse grid collocation algorithm for the solution of stochastic differential equations. J. Comput. Phys. 228, 3084-3113.
32. Özişik, M. N. 1985 Heat Transfer: A Basic Approach. McGraw-Hill.
33. Özişik, M. N. 1999 Heat Conduction, 2nd edn. Wiley-Interscience.
34. 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), 208-218.
35. 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), 3233-3247.
36. 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, 753-769.
37. 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), 3639-3654.
38. 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, 393-427.
39. Stork, K. & Müller, U. 1972 Convection in boxes: experiments. J. Fluid Mech. 54 (4), 599-611.
40. Vekstein, G. 2004 Energy principle for the onset of convection. Eur. J. Phys. 25, 667-673.
41. 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, 215-254.
42. 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, 339-367.
43. Wan, X. & Karniadakis, G. E. 2005 An adaptive multi-element generalized polynomial chaos method for stochastic differential equations. J. Comput. Phys. 209 (2), 617-642.
44. Wan, X. & Karniadakis, G. E. 2006a Multi-element generalized polynomial chaos for arbitrary probability measures. SIAM J. Sci. Comput. 28 (3), 901-928.
45. Wan, X. & Karniadakis, G. E. 2006b Stochastic heat transfer enhancement in a grooved channel. J. Fluid Mech. 565, 255-278.
46. Xiu, D. & Karniadakis, G. E. 2002 The Wiener-Askey polynomial chaos for stochastic differential equations. SIAM J. Sci. Comput. 24 (2), 619-644.
47. Xiu, D. & Karniadakis, G. E. 2003 modelling uncertainty in flow simulations via generalized polynomial chaos. J. Comput. Phys. 187, 137-167.