Abstract
We revisit the optimal heat transport problem for Rayleigh–Benard convection in which a rigorous upper bound on the Nusselt number, , is sought as a function of the Rayleigh number, . Concentrating on the two-dimensional problem with stress-free boundary conditions, we impose the time-averaged heat equation as a constraint for the bound using a novel two-dimensional background approach thereby complementing the ‘wall-to-wall’ approach of Hassanzadeh et al. (J. Fluid Mech., vol. 751, 2014, pp. 627–662). Imposing the same symmetry on the problem, we find correspondence with their maximal result for but, beyond that, the results from the two approaches diverge. The bound produced by the two-dimensional background field approaches that produced by the one-dimensional background field from below as the length of computational domain . On lifting the imposed symmetry, the optimal two-dimensional temperature background field reverts to being one-dimensional, giving the best bound compared to in the non-slip case. We then show via an inductive bifurcation analysis that introducing two-dimensional temperature and velocity background fields (in an attempt to impose the time-averaged Boussinesq equations) is also unable to lower the bound. This then exhausts the background approach for the two-dimensional (and by extension three-dimensional) Rayleigh–Benard problem with the bound remaining stubbornly while data seem more to scale like for large . Finally, we show that adding a velocity background field to the formulation of Wen et al. (Phys. Rev. E., vol. 92, 2015, 043012), which is able to use an extra vorticity constraint due to the stress-free condition to lower the bound to , also fails to further improve the bound.
Highlights
In this paper we consider the fundamental problem of assessing how the heat flux behaves as a function of the Rayleigh number, Ra, in Rayleigh–Bénard convection where a layer of fluid is heated from below and cooled from above
The key novelty has been to consider background temperature and velocity fields whose dimensional dependence matches that of the physical problem
This situation needs a reformulation in the way the variational equations are solved which has the significant consequence of breaking any link between the optimal fields which emerge and single physical temperature and velocity fields
Summary
In this paper we consider the fundamental problem of assessing how the heat flux behaves as a function of the Rayleigh number, Ra, in Rayleigh–Bénard convection where a layer of fluid is heated from below and cooled from above. The particular focus here is on the use of variational methods which seek an upper bound on the heat flux in the hope that this bound will capture the correct high-Ra scaling for turbulent convection This approach involves constructing an optimisation problem constrained by information gleaned from the governing equations. The steady heat equation has been imposed as a constraint with some incompressible boundary-compliant flow field which, apart from an overall amplitude, is otherwise unconstrained and a maximisation problem is solved This appears to give a much improved (reduced) estimate of maximal flux with Nu ∼ Ra5/12 for stress-free boundary conditions in two-dimensional (2-D) convection with Souza (2016) finding a yet stronger (reduced) bound of Nu ∼ Ra0.371 for non-slip boundary conditions.
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