Recent experimental, numerical and theoretical advances in turbulent Rayleigh-Benard convection are presented. Particular emphasis is given to the physics and structure of the thermal and velocity boundary layers which play a key role for the better understanding of the turbulent transport of heat and momentum in convection at high and very high Rayleigh numbers. We also discuss important extensions of Rayleigh-Benard convection such as non-Oberbeck-Boussinesq effects and convection with phase changes.

https://link.springer.com/content/pdf/10.1140%2Fepje%2Fi2012-12058-1.pdf

New perspectives in turbulent Rayleigh-Bénard convection.

The structure of the boundary layers in turbulent Rayleigh‐B´ enard convection is studied by means of three-dimensional direct numerical simulations. We consider convection in a cylindrical cell at aspect ratio one for Rayleigh numbers of RaD 3 10 9 and 3 10 10 at fixed Prandtl number PrD 0:7. Similar to the experimental results in the same setup and for the same Prandtl number, the structure of the laminar boundary layers of the velocity and temperature fields is found to deviate from the prediction of Prandtl‐Blasius‐Pohlhausen theory. Deviations decrease when a dynamical rescaling of the data with an instantaneously defined boundary layer thickness is performed and the analysis plane is aligned with the instantaneous direction of the large-scale circulation in the closed cell. Our numerical results demonstrate that important assumptions of existing classical laminar boundary layer theories for forced and natural convection are violated, such as the strict twodimensionality of the dynamics or the steadiness of the fluid motion. The boundary layer dynamics consists of two essential local dynamical building blocks, a plume detachment and a post-plume phase. The former is associated with larger variations of the instantaneous thickness of velocity and temperature boundary layer and a fully three-dimensional local flow. The post-plume dynamics is connected with the large-scale circulation in the cell that penetrates the boundary region from above. The mean turbulence profiles taken in localized sections of the boundary layer for each dynamical phase are also compared with solutions of perturbation expansions of the boundary layer equations of forced or natural convection towards mixed convection. Our analysis of both boundary layers shows that the near-wall dynamics combines elements of forced Blasius-type and natural convection.

Boundary layer structure in turbulent Rayleigh-Bénard convection

We present planforms of line plumes formed on horizontal surfaces in turbulent convection, along with the length of line plumes measured from these planforms, in a six decade range of Rayleigh numbers (10(5) < Ra < 10(11)) and at three Prandtl numbers (Pr = 0.7, 5.2, 602). Using geometric constraints on the relations for the mean plume spacings, we obtain expressions for the total length of near-wall plumes on horizontal surfaces in turbulent convection. The plume length per unit area (L(p)/A), made dimensionless by the near-wall length scale in turbulent convection (Z(w)), remains constant for a given fluid. The Nusselt number is shown to be directly proportional to L(p)H/A for a given fluid layer of height H. The increase in Pr has a weak influence in decreasing L(p)/A. These expressions match the measurements, thereby showing that the assumption of laminar natural convection boundary layers in turbulent convection is consistent with the observed total length of line plumes. We then show that similar relationships are obtained based on the assumption that the line plumes are the outcome of the instability of laminar natural convection boundary layers on the horizontal surfaces.

/pdf/length-of-near-wall-plumes-in-turbulent-convection-2e5x3xw0ka.pdf

Length of near-wall plumes in turbulent convection

Natural convection arising over vertical and horizontal heated flat surfaces is one of the most ubiquitous flows at a range of spatiotemporal scales. Despite significant developments over more than a century contributing to our fundamental understanding of heat transfer in natural convection boundary layers, certain “hidden” characteristics of these flows have received far less attention. Here, we review scattered progress on less visited fundamental topics that have strong implications to heat and mass transfer control. These topics include the instability characteristics, laminar-to-turbulent transition, and spatial flow structures of vertical natural convection boundary layers and large-scale plumes, dome, and circulating flows over discretely and entirely heated horizontal surfaces. Based on the summarized advancements in fundamental research, we elaborate on the selection of perturbations and provide an outlook on the development of perturbation generators and methods of altering large-scale flow structures as a potential means for heat and mass transfer control where natural convection is dominant.

Natural convection over vertical and horizontal heated flat surfaces: A review of recent progress focusing on underpinnings and implications for heat transfer and environmental applications

The long-standing puzzle of diverging heat transport measurements at very high Rayleigh numbers (Ra) is addressed by a simple model based on well-known properties of classical boundary layers. The transition to the ‘ultimate state’ of convection in Rayleigh–Bénard cells is modeled as sub-critical transition controlled by the instability of large-scale boundary-layer eddies. These eddies are restricted in size either by the lateral wall or by the horizontal plates depending on the cell aspect ratio (in cylindrical cells, the cross-over occurs for a diameter-to-height ratio around 2 or 3). The large-scale wind known to settle across convection cells is assumed to have antagonist effects on the transition depending on its strength, leading to wind-immune, wind-hindered or wind-assisted routes to the ultimate regime. In particular winds of intermediate strength are assumed to hinder the transition by disrupting heat transfer, contrary to what is assumed in standard models. This phenomenological model is able to reconcile observations from more than a dozen of convection cells from Grenoble, Eugene, Trieste, Göttingen and Brno. In particular, it accounts for unexplained observations at high Ra, such as Prandtl number and aspect ratio dependences, great receptivity to details of the sidewall and differences in heat transfer efficiency between experiments. How does natural convection transport heat in the limit of intense thermal forcing ? This old question is still vividly disputed in the convection community. Beyond the academic motivation of unraveling the properties of natural convection in its asymptotic limit, understanding these intense flows is relevant to various environmental (oceans, . . . ) and large scale industrial flows (nuclear reactors, . . . ). Indeed, the intensity of forcing in natural convection is proportional to the cube of the flow vertical extension, resulting in intense forcing in large-scale flows, even when a moderate temperature difference is driving the motion. A model system is prevalent in laboratory studies of natural convection: the Rayleigh–Bénard cell operated in Boussinesq conditions [1] (see figure 1(a)). As detailed in the next section, three dimensionless numbers traditionally characterize respectively the thermal forcing, the fluid properties and the resulting heat transfer: the Rayleigh (Ra), Prandtl (Pr) and Nusselt (Nu) numbers. This study focuses exclusively on fluids with intermediate Pr, that is fluids with comparable viscous and thermal molecular diffusivities, such as air, helium and water, and not on fluids where one diffusivity significantly exceeds the other one, such as liquid metals or oils. We refer the reader to the review [2], and references within, for a general introduction on turbulent Rayleigh–Bénard convection. The heat transfer dependence Nu(Ra) at given Pr provides an indirect but easily measurable piece of information on the flow. Thus, most models of convection provide predictions for this dependence. In 1962, R Kraichnan predicted the existence of an asymptotic flow regime at very high Ra, characterized by the presence of turbulence not only in the bulk of the flow—as in the preceding regime—but also in its boundary layers [3]. A distinctive Nu(Ra) dependence—recalled later—was predicted by Kraichnan, with a significantly enhanced heat transport compared to the preceding flow regime called hard turbulence state. The first observation of a marked Nu(Ra) transition interpreted following Kraichnan’s prediction was reported in Grenoble in the late 90s [4]. The expressions ‘ultimate regime’ [4] or equivalently ‘ultimate state’ © 2020 The Author(s). Published by IOP Publishing Ltd on behalf of the Institute of Physics and Deutsche Physikalische Gesellschaft New J. Phys. 22 (2020) 073056 P-E Roche Figure 1. (a) Typical Rayleigh–Bénard cell. The fluid layer between two plates is set into motion by natural convection. Most of the temperature drop between the plates is concentrated in two thermal boundary layers. (b) Compensated heat transfer Nu · Ra−1/3 versus Ra for experiments reaching at least Ra = 10 in cylindrical cell of aspect ratio Γ = 0.5 unless otherwise noted, and for Prandtl numbers within 0.6 < Pr < 7. References to the various experiments are provided in the table. Table 1. Very high Ra Rayleigh–Bénard experiments discussed in the present paper. Location Aspect ratio Complementary Cell height Fluid Comment and references Γ = φh designation h [cm] Grenoble 0.5 Chavanne 20 He cryo [4, 15] Grenoble 0.23 43 He cryo [16] Grenoble 0.5 Vintage 20 He cryo Reference Γ = 0.5 cell with a 1.3◦ or 3.6◦ tilt [16] Grenoble 0.5 Flange 20 He cryo With a flange at sidewall mid-height[16] Grenoble 0.5 Paper 20 He cryo With layers of paper on the inner sidewall[16] Grenoble 0.5 ThickWall 20 He cryo With sidewall x 4.4 thicker [17] Grenoble 1.14 8.8 He cryo [16] Chicago 0.5 40 He cryo [18] Eugene 0.5 100 He cryo [19, 20] Trieste 1 50 He cryo Eugene experiment moved to Trieste [21] Trieste 4 12.5 He cryo [22] Brno 1 30 He cryo [10, 23] Göttingen 0.33 330 SF6 [24] Göttingen 0.5 I, IIa, IIb 224 SF6 Unsealed [25, 92] Göttingen 0.5 IIe 224 SF6 Seals between plates and sidewall [26] Göttingen 1 112 SF6 [27] [5] were coined at that time to name the new flow state observed at very large Ra. We choose to use the same terminology in this paper. Since, new experiments have explored Rayleigh–Bénard convection at very high Ra (say Ra > 1013) and for intermediate Pr, in Grenoble, Eugene, Trieste, Göttingen and Brno (see table 1 for references). As illustrated by figure 1(b), most Nu(Ra) measurements agree below Ra # 1011, while the situation at larger Ra is more puzzling and has been a fuel for scientific controversy [6–14]. Indeed, the compensated heat transfer Nu · Ra−1/3 decreases or level out with Ra in some experiments (Chicago, Eugene, Brno) while it increases significantly in others (Grenoble, Trieste), leading up to a two-fold difference in heat transfer efficiency around Ra = 1014. How does the convection literature deal with the striking apparent contradiction between the high Ra datasets ? Over the years, three different approaches have consolidated in Grenoble/Lyon (e.g. see [4, 16]), Trieste/Brno/Prague (e.g. see [21, 23]) and Göttingen/Santa Barbara/Twente (e.g. see [27, 28]) to account for the results1. These views are either based on variants of Kraichnan’s prediction or on non-Boussinesq approximation effects. Taken separately, the respective datasets and interpretations are certainly fair and often self-consistent, but no proposed model is yet able to account for all existing observations. As a consequence, most papers propose a simple discussion, but disregard or depreciate previous results in apparent contradiction. 1 Surely, there are differences in judgment and emphasis between the members of the three subgroups.

The ultimate state of convection: a unifying picture of very high Rayleigh numbers experiments

We present a systematic experimental study of geometric and statistical properties of thermal plumes in turbulent Rayleigh-B\'{e}nard convection using the thermochromic-liquid-crystal (TLC) technique. The experiments were performed in three water-filled cylindrical convection cells with aspect ratios 2, 1, and 0.5 and over the Rayleigh-number range $5\times10^7 \leq Ra \leq 10^{11}$. TLC thermal images of horizontal plane cuts at various depths below the top plate were acquired. Three-dimensional images of thermal plumes were then reconstructed from the two-dimensional slices of the temperature field. The results show that the often-called sheetlike plumes are really one-dimensional structures and may be called rodlike plumes. We find that the number densities for both sheetlike/rodlike and mushroomlike plumes have power-law dependence on $Ra$ with scaling exponents of $\sim 0.3$, which is close to that between the Nusselt number $Nu$ and $Ra$. This result suggests that it is the plume number that primarily d ermines the scaling exponent of the $Nu$-$Ra$ scaling relation. The evolution of the aspect ratio of sheetlike/rodlike plumes reveals that as $Ra$ increases the plume geometry changes from more-elongated to less-elongated. Our study of the plume area fraction (fraction of coverage over the surface of the plate) further reveals that the increased plume numbers with $Ra$ mainly comes from increased plume emission, rather than fragmentation of plumes. In addition, the area, perimeter, and the shape complexity of the two-dimensional horizontal cuts of sheetlike/rodlike plumes were studied and all are found to obey log-normal distributions.

Ke-Qing Xia

Papers

Statistical and geometrical properties of thermal plumes in turbulent Rayleigh-B\'{e}nard convection