title: Time-stepping approach for solving upper-bound problems: Application to two-dimensional Rayleigh-Bénard convection.
abstract: 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∼Pr(α)Ra(β) scaling relation. The computations clearly show that for Ra≤10(10) at fixed L=2√[2],Nu≤0.106Pr(0)Ra(5/12), which indicates that molecular transport cannot generally be neglected in the "ultimate" high-Ra regime.
authors.affiliation: Program in Integrated Applied Mathematics, University of New Hampshire, Durham, New Hampshire 03824, USA.|Center for Fluid Physics, University of New Hampshire, Durham, New Hampshire 03824, USA.|The Institute for Computational Engineering and Sciences, The University of Texas at Austin, Austin, Texas 78712, USA.
authors.forename: Baole
authors.initials: B
authors.lastname: Wen
authors.affiliation: Program in Integrated Applied Mathematics, University of New Hampshire, Durham, New Hampshire 03824, USA.|Center for Fluid Physics, University of New Hampshire, Durham, New Hampshire 03824, USA.|Department of Mechanical Engineering, University of New Hampshire, Durham, New Hampshire 03824, USA.
authors.forename: Gregory P
authors.initials: GP
authors.lastname: Chini
authors.affiliation: School of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, United Kingdom.
authors.forename: Rich R
authors.initials: RR
authors.lastname: Kerswell
authors.affiliation: Department of Physics, University of Michigan, Ann Arbor, Michigan 48109-1040, USA.|Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109-1043, USA.|Center for the Study of Complex Systems, University of Michigan, Ann Arbor, Michigan 48109-1107, USA.
authors.forename: Charles R
authors.initials: CR
authors.lastname: Doering