Time-stepping approach for solving upper-bound problems: Application to two-dimensional Rayleigh-Bénard convection

Baole Wen, Greg Chini, Richard Kerswell, Charles Doering

Research output: Contribution to journalArticle (Academic Journal)peer-review

19 Citations (Scopus)
244 Downloads (Pure)


An alternative computational procedure for numerically solving a class of variational problems arising from rigorous upper-bound analysis of forced-dissipative infinite-dimensional nonlinear dynamical systems, including the Navier-Stokes and Oberbeck-Boussinesq equations, is analyzed and applied to Rayleigh-Bénard convection. A proof that the only steady state to which this numerical algorithm can converge is the required global optimal of the relevant variational problem is given for three canonical flow configurations. In contrast with most other numerical schemes for computing the optimal bounds on transported quantities (e.g., heat or momentum) within the "background field" variational framework, which employ variants of Newton's method and hence require very accurate initial iterates, the new computational method is easy to implement and, crucially, does not require numerical continuation. The algorithm is used to determine the optimal background-method bound on the heat transport enhancement factor, i.e., the Nusselt number (Nu), as a function of the Rayleigh number (Ra), Prandtl number (Pr), and domain aspect ratio L in two-dimensional Rayleigh-Bénard convection between stress-free isothermal boundaries (Rayleigh's original 1916 model of convection). The result of the computation is significant because analyses, laboratory experiments, and numerical simulations have suggested a range of exponents α and β in the presumed Nu∼PrαRaβ scaling relation. The computations clearly show that for Ra≤1010 at fixed L=2√2,Nu≤0.106Pr0Ra5/12, which indicates that molecular transport cannot generally be neglected in the "ultimate" high-Ra regime.

Original languageEnglish
Article number043012
Number of pages13
JournalPhysical Review E: Statistical, Nonlinear, and Soft Matter Physics
Issue number4
Publication statusPublished - 16 Oct 2015


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