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Related Experiment Video

Updated: Mar 30, 2026

Evolution of Staircase Structures in Diffusive Convection
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Time-stepping approach for solving upper-bound problems: Application to two-dimensional Rayleigh-Bénard convection.

Baole Wen1,2,3, Gregory P Chini1,2,4, Rich R Kerswell5

  • 1Program in Integrated Applied Mathematics, University of New Hampshire, Durham, New Hampshire 03824, USA.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|November 14, 2015
PubMed
Summary

A new computational method simplifies solving complex fluid dynamics problems. It accurately determines heat transport bounds in Rayleigh-Bénard convection, revealing key scaling relationships.

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Area of Science:

  • Fluid dynamics
  • Nonlinear dynamical systems
  • Computational mathematics

Background:

  • Variational problems arise in analyzing forced-dissipative infinite-dimensional nonlinear dynamical systems.
  • Rigorous upper-bound analysis is crucial for understanding phenomena like convection.
  • Existing numerical methods often require accurate initial conditions and numerical continuation.

Purpose of the Study:

  • To introduce and analyze an alternative computational procedure for solving variational problems.
  • To apply this method to Rayleigh-Bénard convection and determine optimal bounds on heat transport.
  • To prove the algorithm's convergence to the global optimum for specific flow configurations.

Main Methods:

  • Development of an alternative computational procedure for variational problems.
  • Application to Rayleigh-Bénard convection with stress-free isothermal boundaries.
  • Analysis of convergence to the global optimum for three canonical flow configurations.

Main Results:

  • The algorithm converges to the global optimum of the variational problem.
  • Optimal bounds for heat transport (Nusselt number, Nu) were determined as a function of Rayleigh number (Ra), Prandtl number (Pr), and aspect ratio (L).
  • For Ra≤10(10) and fixed L=2√[2], Nu≤0.106Pr(0)Ra(5/12), indicating molecular transport is significant in the high-Ra regime.

Conclusions:

  • The novel computational method is easy to implement and avoids numerical continuation.
  • The findings provide a precise scaling relation for heat transport in Rayleigh-Bénard convection.
  • Molecular transport cannot be neglected in the ultimate high-Rayleigh number regime.