Turning up the Heat in Turbulent Thermal Convection COMMENTARY Charles R

COMMENTARY

Turning up the heat in turbulent thermal convection COMMENTARY Charles R. Doeringa,b,c,1

Convection is buoyancy-driven flow resulting from the fluid remains at rest and heat flows from the warm unstable density stratification in the presence of a to the cold boundary according Fourier’s law, is stable. gravitational field. Beyond convection’s central role in Beyond a critical threshold, however, steady flows in myriad engineering heat transfer applications, it un- the form of coherent convection rolls set in to enhance derlies many of nature’s dynamical designs on vertical heat flux. Convective turbulence, character- larger-than-human scales. For example, solar heating ized by thin thermal boundary layers and chaotic of Earth’s surface generates buoyancy forces that plume dynamics mixing in the core, appears and per- cause the winds to blow, which in turn drive the sists for larger ΔT (Fig. 1). oceans’ flow. Convection in Earth’s mantle on geolog- More precisely, Rayleigh employed the so-called ical timescales makes the continents drift, and Boussinesq approximation in the Navier–Stokes equa- thermal and compositional density differences induce tions which fixes the fluid density ρ in all but the buoyancy forces that drive a dynamo in Earth’s liquid temperature-dependent buoyancy force term and metal core—the dynamo that generates the magnetic presumes that the fluid’s material properties—its vis- field protecting us from solar wind that would other- cosity ν, specific heat c, and thermal diffusion and wise extinguish life as we know it on the surface. The expansion coefficients κ and α—are constant. Rayleigh structure of the Sun itself relies on convection in the showed that stability of the no-flow conduction state is gαΔTH3 outer layers to transfer heat from the interior to radiate controlled by the dimensionless group νκ , where g away from the surface. is the acceleration of gravity. Today this nondimen- The key feature of convection is transport: Thermal sional combination is called the Rayleigh number Ra, ν convection actively transports the heat that generates leaving the fluid’s Prandtl number Pr = κ and the spa- the density variations that produce the buoyancy tial domain’s aspect ratio Γ (the ratio of the horizontal forces, and determining the rate at which “heat rises” extent to its vertical height) as secondary dimension- in turbulent convection is one of the most important less parameters. Rayleigh–Bénard convection has since open problems in fluid dynamics. In PNAS, Iyer et al. come to serve as a principal paradigm of nonlinear phys- (1) report the results of large-scale computational sim- ics extending from symmetry breaking bifurcation and ulations revealing heat transfer rates in accord with pattern formation to chaos and turbulence (4–7). one of two competing theories for turbulent convec- The primary gauge of intensity for convection is the tion in the strongly nonlinear regime. magnitude of the heat flux Q resulting from an im- The problem that Iyer et al. address is one of posed temperature drop ΔT. Of particular interest is longstanding interest and tremendous influence. In- the relation between Q and ΔT that quantifies the spired by Henri B énard’s (2) turn of the 20th century effective thermal conductivity of the convecting layer, experiments, Lord Rayleigh (3) introduced a minimal analogous to how the electrical conductivity of an ob- mathematical model for buoyancy-driven thermal ject is determined by the current resulting from an im- ρ κ ΔT convection in 1916. His model for what has come to posed voltage drop. The heat flux c h in the no-flow be known as “Rayleigh–Bénard convection” consists conduction state is linearly proportional to the tem- of a layer of fluid between impermeable horizontal perature drop. The natural nondimensional measure boundaries separated vertically by distance H and of convective intensity—the rate at which heat rises in held at fixed temperatures differing by ΔT, the higher Rayleigh–Bénard convection—is the heat transfer en- temperature being on the underside where the direc- hancement factor, the ratio of total to conductive flux ≡ QH tion of gravity distinguishes up from down. For rela- known as the Nusselt number Nu ρcκΔT. In the context tively small ΔT the no-flow conduction state, in which of Rayleigh’s model, the challenge for analysis and

aCenter for the Study of Complex Systems, University of Michigan, Ann Arbor, MI 48109-1042; bDepartment of Mathematics, University of Michigan, Ann Arbor, MI 48109-1043; and cDepartment of Physics, University of Michigan, Ann Arbor, MI 48109-1040 Author contributions: C.R.D. wrote the paper. The author declares no competing interest. Published under the PNAS license. See companion article, “Classical 1/3 scaling of convection holds up to Ra = 1015,” 10.1073/pnas.1922794117. 1Email: [email protected]. First published April 28, 2020.

www.pnas.org/cgi/doi/10.1073/pnas.2004239117 PNAS | May 5, 2020 | vol. 117 | no. 18 | 9671–9673 Downloaded by guest on September 30, 2021 Fig. 1. Snapshots of the temperature field in 2D Rayleigh–B´enard convection simulations. (Top) For suitably weak temperature drops ΔT the fluid remains at rest and heat transfers via conduction. (Middle) Sufficiently large ΔT destabilizes the conduction state and coherent convection rolls actively increase the heat flux. (Bottom) Convective turbulence sets in at larger ΔT.

computation is to determine Nu as a function of Ra and Pr and argument. Spiegel (12) computed an a priori prefactor from Γ.* Insofar as Nu varies with Ra, Q depends nonlinearly on ΔT. Malkus’s theory, predicting Nu ≈ 0.07 Ra1=3. The essence of clas- The relevant parameter regime for many geo- and astrophys- sical scaling is that the heat flux is limited by transport across the ical and engineering applications corresponds to very large values emergent thermal boundary layers. of the Rayleigh number, from millions and billions to trillions and The ultimate theory asserts Nu ∼ Pr 1=2Ra1=2, at least for Pr K 1. † more times its value (typically in the thousands) at convective It was proposed by Spiegel (13) in the early 1960s based on γ β onset. Power-law asymptotic behavior of the form Nu ∼ Pr Ra is different physics. Spiegel postulated that turbulent convection often presumed for high-Ra heat transport (7), but two alternative layers are populatedp withffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi plumes transporting heat near the buoy- theories predicting distinct scaling exponents γ and β vie to cap- ancy free-fall speed gαΔTH without regard for the nature or structure the asymptotic behavior of Nu as Ra → ∞. These are the so- ture of boundary layers. This γ = 1=2, β = 1=2 scaling implies “ ” “ ” called classical and ultimate theories, and ruling out one or “anomalous” transport wherein Q and the associated turbulent ki- the other of these theories is a major open problem for the field. netic energy dissipation rate are independent of the microscopic ∼ 1=3 The classical theory asserts Nu Ra . It was simultaneously material transport coefficients ν and κ. The underlying assumption and independently proposed by Priestly (9) and Malkus (10) in is that convective heat flux is limited not at the boundaries, but rather 1954. Priestly argued that Q should be independent of H for tur- by transport across the bulk of the fluid layer. bulent convection, at least for sufficiently large aspect ratio do- Soon thereafter Spiegel’s postdoc mentor speculated that ve- mains, while Malkus formulated an elaborate maximal dissipation locity boundary layers on the rigid plates may transition to shear = theory predicting as well that the 1 3 scaling is uniform in Pr. Ten turbulence at ultrahigh Ra and postulated modification of Ra1=2 years later Howard (11) compellingly reframed this γ = 0, β = 1=3 −3=2 scaling by a ðlog RaÞ factor (15). The moniker ultimate was first scaling in terms of a marginally stable thermal boundary layer used in the mid-1990s (16) referring to logarithmic corrections vary- − = − ing from ðlog RaÞ 3 2 to ðlog RaÞ 3, and it has since been adopted by the community for NuRa relations with predominant 1=2 scaling. *For quantitative considerations it is important to distinguish between Rayleigh’s original model—with thermal energy entering and exiting through the boundaries—and variations thereof, for example with heating and cooling † in the bulk to produce temperature variations across the fluid layer (8). Spiegel’s theory actually predates ref. 13 as testified by Batchelor (14).

9672 | www.pnas.org/cgi/doi/10.1073/pnas.2004239117 Doering Downloaded by guest on September 30, 2021 Rayleigh’s model ostensibly contains precise predictions for in an aspect ratio Γ = 2domainfittedNu ≈ 0.030 × Ra0.337 ± 3.4% ð2σÞ possible high-Ra behaviors of Nu, but exact solution formulas for more than two decades of Ra ∈ ½2 × 1012,5× 1014, indistin- are not available and mathematical analysis has to date yielded guishable from classical scaling with a perhaps not unexpectedly only upper bounds on the heat flux. Indeed, ultimate-like reduced (approximately halved) a priori postulated prefactor (23). Nu ∼ Ra1=2 scaling—albeit uniform in Pr—really is ultimate in the It is natural to wonder, however, how much the restriction to two sense that it is a rigorous upper limit on heat transport in Ray- dimensions matters. leigh’s model for three-dimensional convection between isother- To simulate convection to even higher Ra in three spatial di- mal no-slip boundaries (17, 18). But this neither confirms nor rules mensions, Iyer et al. (1) exploit the fact that Rais proportional to out either scaling theory for Pr K 1. Meanwhile sophisticated H3 andconsiderconvectioninanaspectratioΓ = 1=10 cylindri- 21st-century laboratory experiments designed to test Rayleigh’s cal cell. Their Pr = 1 simulation data fit Nu ≈ 0.053 × Ra0.331 ± 3.2%, model, respecting the assumptions employed therein, have not also consistent with classical 1=3 scaling with an even more theoret- settled the matter. Some reports suggest ultimate behavior (19) ically anticipated prefactor, over five decades of Ra ∈ ½1010,1015. while others are more consistent with classical scaling (20–22). Then one wonders how much the restriction to such a small Direct numerical simulations of the equations of motion are an aspect ratio matters. What is perhaps most remarkable about Iyer alternative approach to study the problem, but they become et al.’s (1) simulation data is their quantitative correspondence with increasingly difficult as convection intensifies into ultrahigh Ra much larger aspect ratio Γ = 1(Pr ≈ 1) experiments in cryogenic 4He regimes of interest. Needs for 1) finer spatial meshes to resolve (21) fitting Nu ≈ 0.051 × Ra1=3 ± 5% for Ra ∈ ½2 × 1011,5× 1013 and ever thinner boundary layers and thermal plumes, 2) smaller time- Γ = 1=2(Pr ≈ 0.8) experiments in near ambient temperature high- 0.336 step increments to resolve faster flow dynamics, and 3) longer runs pressure SF6 gas (22) that fitted Nu ≈ 0.050 × Ra ± 3% for to generate reliable statistics for strongly fluctuating turbulent Ra ∈ ½3 × 1012,1015 (24). All in all, classical 1=3 scaling currently solutions combine to quickly turn the task into a large-scale appears to be winning the competition. high-performance computing problem. One way to achieve high Rayleigh numbers with limited Acknowledgments computational resources is to constrain the dynamics to two C.R.D. is very grateful to Hannah L. Swan for producing the simulation images spatial dimensions. (The images in Fig. 1 were produced by such for Fig. 1. C.R.D.’s research is supported by National Science Foundation two-dimensional [2D] simulations.) Recent 2D simulations with Pr = 1 Award DMS-1813003.

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