Skip to main content Accessibility help

Stochastic dynamics of fluid–structure interaction in turbulent thermal convection

  • Jinzi Mac Huang (a1), Jin-Qiang Zhong (a2), Jun Zhang (a1) (a3) (a4) and Laurent Mertz (a3)


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.


Corresponding author

Email address for correspondence:


Hide All
Ahlers, G., Grossmann, S. & Lohse, D. 2009 Heat transfer and large scale dynamics in turbulent Rayleigh–Bénard convection. Rev. Mod. Phys. 81 (2), 503537.
Allshouse, M. R., Barad, M. F. & Peacock, T. 2010 Propulsion generated by diffusion-driven flow. Nat. Phys. 6 (7), 516519.
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, 491511.
Bensoussan, A. & Lions, J. L. 1982 Contrôle Impulsionnel et Inéquations Quasi Variationnelles. Gauthier-Villars.
Bensoussan, A. & Mertz, L. 2012 An analytic approach to the ergodic theory of a stochastic variational inequality. C. R. Mathematique 350 (7–8), 365370.
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), 33743396.
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), 27832805.
Bensoussan, A. & Turi, J. 2008 Degenerate Dirichlet problems related to the invariant measure of elasto-plastic oscillators. Appl. Math. Opt. 58 (1), 127.
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. 923. Springer.
Benzi, R. 2005 Flow reversal in a simple dynamical model of turbulence. Phys. Rev. Lett. 95, 024502.
Bernardin, F. 2003 Multivalued stochastic differential equations: convergence of a numerical scheme. Set-Valued Anal. 11 (4), 393415.
Brown, E., Nikolaenko, A. & Ahlers, G. 2005 Reorientation of the large-scale circulation in turbulent Rayleigh–Bénard convection. Phys. Rev. Lett. 95, 084503.
Elder, J. W. 1968 Convection – the key to dynamical geology. Sci. Prog. 56 (221), 133.
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), 4760.
Gurnis, M. 1988 Large-scale mantle convection and the aggregation and dispersal of supercontinents. Nature 332 (6166), 695699.
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), 581584.
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), 123142.
Krishnamurti, R. & Howard, L. N. 1981 Large-scale flow generation in turbulent convection. Proc. Natl Acad. Sci. USA 78 (4), 19811985.
Laurière, M. & Mertz, L. 2015 Penalization of a stochastic variational inequality modeling an elasto-plastic problem with noise. ESAIM: Proc. Surv. 48, 226247.
Lions, P.-L. & Sznitman, A.-S. 1984 Stochastic differential equations with reflecting boundary conditions. Commun. Pure Appl. Maths 37 (4), 511537.
Lowman, J. P. & Jarvis, G. T. 1993 Mantle convection flow reversals due to continental collisions. Geophys. Res. Lett. 20 (19), 20872090.
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), 5368.
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.
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), 13871408.
Mertz, L. & Feau, C. 2012 An empirical study on plastic deformations of an elasto-plastic problem with noise. Prob. Engng Mech. 30, 6069.
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.
Sreenivasan, K. R., Bershadskii, A. & Niemela, J. J. 2002 Mean wind and its reversal in thermal convection. Phys. Rev. E 65, 056306.
Turcotte, D. L. & Schubert, G. 2002 Geodynamics. Cambridge University Press.
Whitehead, J. A. 1972 Moving heaters as a model of continental drift. Phys. Earth Planet. Inter. 5, 199212.
Whitehead, J. A. & Behn, M. D. 2015 The continental drift convection cell. Geophys. Res. Lett. 42 (11), 43014308.
Zhang, J. & Libchaber, A. 2000 Periodic boundary motion in thermal turbulence. Phys. Rev. Lett. 84 (19), 4361.
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.
Zhong, J.-Q. & Zhang, J. 2005 Thermal convection with a freely moving top boundary. Phys. Fluids 17 (11), 115105.
Zhong, J.-Q. & Zhang, J. 2007a Dynamical states of a mobile heat blanket on a thermally convecting fluid. Phys. Rev. E 75 (5), 055301.
Zhong, J.-Q. & Zhang, J. 2007b Modeling the dynamics of a free boundary on turbulent thermal convection. Phys. Rev. E 76 (1), 016307.
MathJax is a JavaScript display engine for mathematics. For more information see

JFM classification


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed