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

Abstract

We revisit the optimal heat transport problem for Rayleigh-Bénard convection in which a rigorous upper bound on the Nusselt number, N u, is sought as a function of the Rayleigh number Ra. Concentrating on the 2-dimensional problem with stress-free boundary conditions, we impose the full heat equation as a constraint for the bound using a novel 2-dimensional background approach thereby complementing the 'wall-to-wall' approach of Hassanzadeh et al. (J. Fluid Mech. 751, 627-662, 2014). Imposing the same symmetry on the problem, we find correspondence with their result for Ra Ra c := 4468.8 but, beyond that, the optimal fields complexify to produce a higher bound. This bound approaches that by a 1-dimensional background field as the length of computational domain L → ∞. On lifting the imposed symmetry, the optimal 2-dimensional temperature background field reverts back to being 1-dimensional giving the best bound N u 0.055Ra 1/2 compared to N u 0.026Ra 1/2 in the non-slip case. We then show via an inductive bifurcation analysis that imposing the full time-averaged Boussinesq equations as constraints (by introducing 2-dimensional temperature and velocity background fields) is also unable to lower this bound. This then exhausts the background approach for the 2-dimensional (and by extension 3-dimensional) Rayleigh-Benard problem with the bound remaining stubbornly Ra 1/2 while data seems more to scale like Ra 1/3 for large Ra. Finally, we show that adding a velocity background field to the formulation of Wen et al. (Phys. Rev. E. 92, 043012, 2015), which is able to use an extra vorticity constraint due to the stress-free condition to lower the bound to N u O(Ra 5/12 ), also fails to improve the bound.

Coherent heat transport in two-dimensional penetrative Rayleigh–Bénard convection

“…It is interesting that the power exponent of the bound on heat flux in penetrative convection is the same as that in the classical Rayleigh–Bénard convection (; Plasting & Kerswell 2003). However, as discussed in Ding & Kerswell (2020), the upper bound is not observable because there is no field satisfying the momentum and energy equations. This does not mean that the scaling (usually referred to as the ultimate scaling in classical Rayleigh–Bénard convection) is wrong at very large , but their analysis suggested that any true scaling should be lower than the bound.…”

“…It is noted that the upper bound on heat flux is at least an order higher than the numerical and experimental data at high Rayleigh numbers. Recent study by Ding & Kerswell (2020) demonstrated that the upper bound derived from the background method, however, is not observable, because there is no steady state satisfying the momentum equation.…”

Stabilising pipe flow by a baffle designed using energy stability

“…Here the subscript marks the th critical mode of the energy stability problem. In general, the energy stability eigenvalue problem will have multiple critical eigenfunctions which need to be kept marginal – or ‘pinned’ (Ding & Kerswell 2020) – as the baffle is adjusted. Keeping track of new critical modes which emerge as is increased is a crucial part of solving this problem and considerable experience of handling this has been built up in the complementary problem of bounding the energy dissipation rate in turbulent flow (Plasting & Kerswell 2003, 2005; Ding & Kerswell 2020).…”

“…. The same argument can be applied to subsequent bifurcations (Ding & Kerswell 2020) assuming that is generically non-zero at each bifurcation. Since is always 1-D for as , we assume is 1-D for as well.…”

The background method: Theory and computations

“…Rayleigh-Bénard convection [10,[13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32] [ 20,21,30,31,[33][34][35][36][37][38], [39] † Bénard-Marangoni convection [40,41] [42] Porous-media convection [43] [ [44][45][46] Internally heated convection [26,[47][48][49][50][51] [50, 51] Double-diffusive convection [52] none Horizontal convection [53] none Parallel shear flows [7,8,11,15,…”