Wall-to-wall optimal transport in two dimensions

Abstract

Gradient ascent methods are developed to compute incompressible flows that maximize heat transport between two isothermal no-slip parallel walls. Parameterizing the magnitude of velocity fields by a Péclet number Pe proportional to their root-mean-square rate-of-strain, the schemes are applied to compute two-dimensional flows optimizing convective enhancement of diffusive heat transfer, i.e., the Nusselt number Nu up to Pe ≈ 10 5 . The resulting transport exhibits a change of scaling from Nu − 1 ∼ Pe 2 for Pe < 10 in the linear regime to Nu ∼ Pe 0.54 for Pe > 10 3 . Optimal fields are observed to be approximately separable, i.e., products of functions of the wall-parallel and wall-normal coordinates. Analysis employing a separable ansatz yields a conditional upper bound Pe 6/11 = Pe 0.54 as Pe → ∞ similar to the computationally achieved scaling. Implications for heat transfer in buoyancy-driven Rayleigh-Bénard convection are discussed.

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

“…This divergence in results is apparently not because any further 2-D bifurcations have been missed in the wall-to-wall calculations (G. Chini, private communication 2019) but more a reflection of the ‘duality gap’ suggested by Souza et al. (2019) being realised. Figure 3( a , b ) shows the first mode with wavenumber so the flow field contains one pair of convection cells.…”

“…From a different perspective, Souza et al. (2019) have also recently argued that this should happen when exploring the connection between the wall-to-wall approach (a max–min problem) with the associated background method (a min–max problem). A duality gap means that (making the connection and with the variables used by Souza et al.…”

“…Finite-Prandtl-number Rayleigh–Bénard with non-slip smooth walls, however, is most studied partly because of its wide application and partly because the current best bound appears to have the wrong exponent and therefore calls for the most improvement. Of particular interest in recent efforts to lower the bound has been the introduction of the ‘wall-to-wall’ approach by Hassanzadeh, Chini & Doering (2014) (see also Souza (2016) and Souza, Tobasco & Doering (2019)). Here, 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.…”

“…Concurrent work by Souza et al. (2019) has considered how the background method is connected to the wall-to-wall approach and speculated that there could be a ‘duality gap’ between them. Coming from a different perspective (the specific details of solving the variational equations), we share this speculation and confirm it here beyond a certain Rayleigh number.…”

Much faster heat/mass than momentum transport in rotating Couette flows

“…In recent theoretical studies, incompressible steady flows are computed that maximize heat transfer for a given power input (Hassanzadeh, Chini & Doering 2014;Alben 2017;Motoki et al 2018;Souza, Tobasco & Doering 2020). Motoki et al (2018) consider plane Couette flow and show that the optimized flow has a much higher heat transfer for a given power input than ordinary turbulent flow.…”