On the Optimal Design of Wall‐to‐Wall Heat Transport

Abstract

We consider the problem of optimizing heat transport through an incompressible fluid layer. Modeling passive scalar transport by advection‐diffusion, we maximize the mean rate of total transport by a divergence‐free velocity field. Subject to various boundary conditions and intensity constraints, we prove that the maximal rate of transport scales linearly in the r.m.s. kinetic energy and, up to possible logarithmic corrections, as the one‐third power of the mean enstrophy in the advective regime. This makes rigorous a previous prediction on the near optimality of convection rolls for energy‐constrained transport. On the other hand, optimal designs for enstrophy‐constrained transport are significantly more difficult to describe: we introduce a “branching” flow design with an unbounded number of degrees of freedom and prove it achieves nearly optimal transport. The main technical tool behind these results is a variational principle for evaluating the transport of candidate designs. The principle admits dual formulations for bounding transport from above and below. While the upper bound is closely related to the “background method,” the lower bound reveals a connection between the optimal design problems considered herein and other apparently related model problems from mathematical materials science. These connections serve to motivate designs. © 2019 Wiley Periodicals, Inc.

Wall-to-wall optimal transport in two dimensions

“…Were such an equality true, it would not preclude the possibility that these quantities achieve the same asymptotic scaling as Pe → ∞, a situation suggested for 3D wall-to-wall optimal transport by the recent numerical scaling max Nu ∼ Pe 2/3 reported for a finite range of Pe in Motoki et al (2018a). It would also be consistent with the dimensionindependent logarithmic lower bound max Nu C P e 2/3 /(log Pe) 4/3 proved for all large enough Pe in Tobasco & Doering (2017) and Doering & Tobasco (2019). Let us illustrate the possibility that (2.33) holds by considering how the previous manipulations operate on the polynomial…”

“…Figure 4 shows the same for the ξ field and the vertical velocity field u 3 . The modes with smaller singular values may be viewed as a preliminary manifestation of a branching-like pattern, perhaps similar to the ones constructed in Tobasco & Doering (2017) and Doering & Tobasco (2019). The structures corresponding to the largest singular value are the direct analogue for the wall-to-wall problem of the singlewavenumber solutions for the Howard-Busse-Malkus problem produced in Howard (1963).…”

“…In fact, reinterpreting for the moment angle brackets as space and time averages, we note that the bound (2.21) holds for time-independent flows as well, an observation that was exploited in Tobasco & Doering (2017) and Doering & Tobasco (2019) with "branching" trial functions to prove the scaling max N u ∼ P e 2/3 up to logarithmic corrections as P e → ∞. We return to discuss this asymptotic result in the context of our numerical results much further below.…”

“…After leaving the "linear" regime where Nu ∼ Pe 2 we enter a "fully nonlinear" regime where Nu ∼ Pe 0.54 . In view of the results of Tobasco & Doering (2017) and Doering & Tobasco (2019) this scaling cannot persist for the globally maximal transport at sufficiently large Pe. Nevertheless, it does appear to be optimal for the computed range of Péclet in two spatial dimensions.…”

“…The formula (2.19), which first appeared in Tobasco & Doering (2017) and was further analyzed in Doering & Tobasco (2019), is an exact variational characterization of the Nusselt number. It allows the optimal wall-to-wall transport problem to be stated succinctly as…”