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Stochastic dynamics of fluid–structure interaction in turbulent thermal convection

Published online by Cambridge University Press:  12 September 2018

Jinzi Mac Huang Open the ORCID record for Jinzi Mac Huang [Opens in a new window] ,
Jin-Qiang Zhong ,
Jun Zhang  and
Laurent Mertz
Jinzi Mac Huang
Affiliation:
Applied Math Lab, Courant Institute, New York University, New York, NY 10012, USA
Jin-Qiang Zhong
Affiliation:
School of Physics Science and Engineering, Tongji University, Shanghai, 200092, China
Jun Zhang
Affiliation:
Applied Math Lab, Courant Institute, New York University, New York, NY 10012, USA NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai, Shanghai, 200062, China Department of Physics, New York University, New York, NY 10003, USA
Laurent Mertz*
Affiliation:
NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai, Shanghai, 200062, China
*
†Email address for correspondence: laurent.mertz@nyu.edu
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Abstract

The motion of a free-moving plate atop turbulent thermal convection exhibits diverse dynamics that displays characteristics of both deterministic and chaotic motions. Early experiments performed by Zhong & Zhang (Phys. Rev. E, vol. 75 (5), 2007, 055301) found an oscillatory and a trapped state existing for a plate floating on convective fluid in a rectangular tank. They proposed a piecewise smooth physical model (ZZ model) that successfully captures this transition of states. However, their model was deterministic and therefore could not describe the stochastic behaviours. In this study, we combine the ZZ model with a novel approach that models the stochastic aspects through a variational inequality structure. With the powerful mathematical tools for stochastic variational inequalities, the properties of the Markov process and corresponding Kolmogorov equations could be studied both numerically and analytically. Moreover, this framework also allows one to compute the transition probabilities. Our present work captures the stochastic aspects of the two aforementioned boundary–fluid coupling states, predicts the stochastic behaviours and shows excellent qualitative and quantitative agreements with the experimental data.

JFM classification

Convection: Convection Turbulent Flows: Turbulent Flows
Type
JFM Rapids
Information
Journal of Fluid Mechanics , Volume 854 , 10 November 2018 , R5
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Copyright
© 2018 Cambridge University Press 

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References

Ahlers, G., Grossmann, S. & Lohse, D. 2009 Heat transfer and large scale dynamics in turbulent Rayleigh–Bénard convection. Rev. Mod. Phys. 81 (2), 503–537.Google Scholar
Allshouse, M. R., Barad, M. F. & Peacock, T. 2010 Propulsion generated by diffusion-driven flow. Nat. Phys. 6 (7), 516–519.Google Scholar
Bakhuis, D., Ostilla-Mnico, R., van der Poel, E. P., Verzicco, R. & Lohse, D. 2018 Mixed insulating and conducting thermal boundary conditions in Rayleigh–Bénard convection. J. Fluid Mech. 835, 491–511.Google Scholar
Bensoussan, A. & Lions, J. L. 1982 Contrôle Impulsionnel et Inéquations Quasi Variationnelles. Gauthier-Villars.Google Scholar
Bensoussan, A. & Mertz, L. 2012 An analytic approach to the ergodic theory of a stochastic variational inequality. C. R. Mathematique 350 (7–8), 365–370.Google Scholar
Bensoussan, A., Mertz, L., Pironneau, O. & Turi, J. 2009 An ultra weak finite element method as an alternative to a Monte Carlo method for an elasto-plastic problem with noise. SIAM J. Numer. Anal. 47 (5), 3374–3396.Google Scholar
Bensoussan, A., Mertz, L. & Yam, S. C. P. 2016 Nonlocal boundary value problems of a stochastic variational inequality modeling an elasto-plastic oscillator excited by a filtered noise. SIAM J. Math. Anal. 48 (4), 2783–2805.Google Scholar
Bensoussan, A. & Turi, J. 2008 Degenerate Dirichlet problems related to the invariant measure of elasto-plastic oscillators. Appl. Math. Opt. 58 (1), 1–27.Google Scholar
Bensoussan, A. & Turi, J. 2010 On a class of partial differential equations with nonlocal Dirichlet boundary conditions. In Applied and Numerical Partial Differential Equations, pp. 9–23. Springer.Google Scholar
Benzi, R. 2005 Flow reversal in a simple dynamical model of turbulence. Phys. Rev. Lett. 95, 024502.Google Scholar
Bernardin, F. 2003 Multivalued stochastic differential equations: convergence of a numerical scheme. Set-Valued Anal. 11 (4), 393–415.Google Scholar
Brown, E., Nikolaenko, A. & Ahlers, G. 2005 Reorientation of the large-scale circulation in turbulent Rayleigh–Bénard convection. Phys. Rev. Lett. 95, 084503.Google Scholar
Elder, J. W. 1968 Convection – the key to dynamical geology. Sci. Prog. 56 (221), 1–33.Google Scholar
Feau, C., Laurière, M. & Mertz, L. 2018 Asymptotic formulae for the risk of failure related to an elasto-plastic problem with noise. Asymptotic Anal. 106 (1), 47–60.Google Scholar
Gurnis, M. 1988 Large-scale mantle convection and the aggregation and dispersal of supercontinents. Nature 332 (6166), 695–699.Google Scholar
Gurnis, M. & Zhong, S. 1991 Generation of long wavelength heterogeneity in the mantle by the dynamic interaction between plates and convection. Geophys. Res. Lett. 18 (4), 581–584.Google Scholar
Howard, L. N., Malkus, W. V. R. & Whitehead, J. A. 1970 Self-convection of floating heat sources: a model for continental drift. Geophys. Astrophys. Fluid Dyn. 1 (1–2), 123–142.Google Scholar
Krishnamurti, R. & Howard, L. N. 1981 Large-scale flow generation in turbulent convection. Proc. Natl Acad. Sci. USA 78 (4), 1981–1985.Google Scholar
Laurière, M. & Mertz, L. 2015 Penalization of a stochastic variational inequality modeling an elasto-plastic problem with noise. ESAIM: Proc. Surv. 48, 226–247.Google Scholar
Lions, P.-L. & Sznitman, A.-S. 1984 Stochastic differential equations with reflecting boundary conditions. Commun. Pure Appl. Maths 37 (4), 511–537.Google Scholar
Lowman, J. P. & Jarvis, G. T. 1993 Mantle convection flow reversals due to continental collisions. Geophys. Res. Lett. 20 (19), 2087–2090.Google Scholar
Lowman, J. P. & Jarvis, G. T. 1995 Mantle convection models of continental collision and breakup incorporating finite thickness plates. Phys. Earth Planet. Inter. 88 (1), 53–68.Google Scholar
Mercier, M. J., Ardekani, A. M., Allshouse, M. R., Doyle, B. & Peacock, T. 2014 Self-propulsion of immersed objects via natural convection. Phys. Rev. Lett. 112 (20), 204501.Google Scholar
Mertz, L. & Bensoussan, A. 2015 Degenerate Dirichlet problems related to the ergodic property of an elasto-plastic oscillator excited by a filtered white noise. IMA J. Appl. Maths 80 (5), 1387–1408.Google Scholar
Mertz, L. & Feau, C. 2012 An empirical study on plastic deformations of an elasto-plastic problem with noise. Prob. Engng Mech. 30, 60–69.Google Scholar
Mertz, L., Stadler, G. & Wylie, J.2017 A backward Kolmogorov equation approach to compute means, moments and correlations of non-smooth stochastic dynamical systems. Preprint, arXiv:1704.02170.Google Scholar
Sreenivasan, K. R., Bershadskii, A. & Niemela, J. J. 2002 Mean wind and its reversal in thermal convection. Phys. Rev. E 65, 056306.Google Scholar
Turcotte, D. L. & Schubert, G. 2002 Geodynamics. Cambridge University Press.Google Scholar
Whitehead, J. A. 1972 Moving heaters as a model of continental drift. Phys. Earth Planet. Inter. 5, 199–212.Google Scholar
Whitehead, J. A. & Behn, M. D. 2015 The continental drift convection cell. Geophys. Res. Lett. 42 (11), 4301–4308.Google Scholar
Zhang, J. & Libchaber, A. 2000 Periodic boundary motion in thermal turbulence. Phys. Rev. Lett. 84 (19), 4361.Google Scholar
Zhong, J.-Q., Sterl, S. & Li, H.-M. 2015 Dynamics of the large-scale circulation in turbulent Rayleigh–Bénard convection with modulated rotation. J. Fluid Mech. 778, R4.Google Scholar
Zhong, J.-Q. & Zhang, J. 2005 Thermal convection with a freely moving top boundary. Phys. Fluids 17 (11), 115105.Google Scholar
Zhong, J.-Q. & Zhang, J. 2007a Dynamical states of a mobile heat blanket on a thermally convecting fluid. Phys. Rev. E 75 (5), 055301.Google Scholar
Zhong, J.-Q. & Zhang, J. 2007b Modeling the dynamics of a free boundary on turbulent thermal convection. Phys. Rev. E 76 (1), 016307.Google Scholar