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

Abstract

A new 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

Exhausting the background approach for bounding the heat transport in Rayleigh–Bénard convection

“…Recently, Wen et al (2015) have proved that when τ is 1-dimensional, i.e. τ = τ (z), appropriately augmenting the (steady) Euler-Lagrange equations with time derivatives leads to a system where the optimal solution is a unique attracting steady state.…”

“…Section 2 describes the set-up of 2D Boussinesq convection ( §2.1), explains how a bound can be found using the background approach ( §2.2) and then discusses the convexity of the optimization problem for a general temperature background field which ensures a unique optimal ( §2.3). Section 2.4 explains how the numerical computations are performed with a choice having to be made between a branch continuation approach (PK03) and a time stepping method (Wen et al 2013(Wen et al , 2015. Section 3 describes the results of tackling the upper bounding problem with the full heat equation imposed in the presence of the same symmetry as used in Hassanzadeh et al (2014).…”

Heat transport by coherent Rayleigh-Bénard convection

“…Rigorous bounds for free-slip 24 yield Nu − 1 0.2295 Ra 5/12 that conflicts with (5) for any Pr (the bound has been improved to Nu − 1 0.106 Ra 5/12 ). 29 We consider 2D flow with v = u(x, y,t)x + v(x, y,t)ŷ and use h = H/2, V =  g ′ h, τ = h/V , and ∆T/2 as our characteristic length, velocity, time, and temperature scales, respectively. Eliminating p by taking theŷ component of the curl of (1) yields…”

A short review on porous media convection

“…Moreover, rigorous upper-bound analyses of the Darcy-Oberbeck-Boussinesq equations by Doering and Constantin [46] and Otero et al [34], by maximizing the heat transport of piecewise linear 'background' profiles, show that F ≤ cRa with different prefactors c. Following this strategy, [47][48][49] compute the optimal upper bound on F under the Constantin-Doering-Hopf variational scheme. They find a lower prefactor c for the unit optimal scaling exponent in the F -Ra relationship.…”