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.

中文翻译：

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对流的最终状态：极高瑞利数实验的统一图

在非常高的瑞利数 (Ra) 下发散热传输测量的长期难题由基于经典边界层众所周知的特性的简单模型解决。Rayleigh-Bénard 单元中对流向“最终状态”的转变被建模为由大规模边界层涡流的不稳定性控制的亚临界转变。这些涡流的大小受侧壁或水平板的限制，具体取决于泡孔纵横比（在圆柱形泡孔中，直径与高度比约为 2 或 3 时会发生交叉）。已知的大范围风穿过对流单元被假定对过渡具有拮抗作用，这取决于它的强度，导致风免疫、风阻或风辅助的路线到达最终状态。与标准模型中的假设相反，特别是假设中等强度的风通过破坏热传递来阻碍转变。这种现象学模型能够协调来自格勒诺布尔、尤金、的里雅斯特、哥廷根和布尔诺的十多个对流单元的观察结果。特别是，它解释了在高 Ra 下无法解释的观察结果，例如 Prandtl 数和纵横比依赖性、对侧壁细节的高度接受性以及实验之间传热效率的差异。自然对流在强热强迫极限下如何传递热量？这个古老的问题在对流界仍然存在争议。除了在渐近极限中解开自然对流特性的学术动机之外，了解这些强烈的流动与各种环境（海洋，......）和大规模工业流动（核反应堆......）有关。事实上，自然对流中的强迫强度与流动垂直延伸的立方成正比，导致大规模流动中的强烈强迫，即使是在适度的温差驱动运动时也是如此。一个模型系统在自然对流的实验室研究中很普遍：在 Boussinesq 条件下运行的 Rayleigh-Bénard 单元 [1]（见图 1(a)）。正如下一节中详述的那样，三个无量纲数传统上分别表征了热力、流体特性和由此产生的热传递：瑞利 (Ra)、普朗特 (Pr) 和努塞尔特 (Nu) 数。本研究仅关注具有中间 Pr 的流体，即具有可比粘性和热分子扩散率的流体，例如空气、氦气和水，而不是一种扩散率显着超过另一种扩散率的流体，例如液态金属或油。我们向读者推荐评论 [2] 和其中的参考资料，以了解湍流瑞利-贝纳德对流的一般介绍。给定 Pr 下的传热相关性 Nu(Ra) 提供了关于流动的间接但易于测量的信息。因此，大多数对流模型都为这种依赖性提供了预测。1962 年，R Kraichnan 预测在非常高的 Ra 下存在渐近流动状态，其特征在于不仅在流动的主体中（如在前面的状态中）而且在其边界层中都存在湍流 [3]。Kraichnan 预测了一种独特的 Nu(Ra) 依赖性（稍后回忆），与之前称为硬湍流状态的流动状态相比，具有显着增强的热传输。90 年代后期，格勒诺布尔报道了根据 Kraichnan 的预测解释的对显着 Nu(Ra) 跃迁的首次观察 [4]。“终极政权”[4] 或等同于“终极状态”的表述 © 2020 作者。由 IOP Publishing Ltd 代表物理研究所和 Deutsche Physikalische Gesellschaft New J. Phys 出版。22 (2020) 073056 PE Roche 图 1. (a) 典型的 Rayleigh-Bénard 电池。两个板块之间的流体层通过自然对流运动。板之间的大部分温降集中在两个热边界层中。(b) 补偿传热 Nu · Ra-1/3 与 Ra 的关系，在纵横比 Γ = 0 的圆柱形电池中达到至少 Ra = 10 的实验。5 除非另有说明，并且对于 Prandtl 数在 0.6 < Pr < 7 内。表中提供了对各种实验的参考。表 1. 本文中讨论的非常高的 Ra 瑞利-贝纳实验。位置 长宽比 Complementary Cell height Fluid 注释和参考文献 Γ = φh 名称 h [cm] Grenoble 0.5 Chavanne 20 He 低温 [4, 15] Grenoble 0.23 43 He 低温 [16] Grenoble 0.5 Vintage 20 He 低温参考 Γ = 0.5 Cell with a 1.3° 或 3.6° 倾斜 [16] Grenoble 0.5 Flange 20 He 冷冻机，侧壁中间高度有法兰 [16] Grenoble 0.5 Paper 20 He 冷冻机，内侧壁上有纸层 [16] Grenoble 0.5 ThickWall 20 He 冷冻机侧壁 x 4.4 厚 [17] 格勒诺布尔 1.14 8.8 He 低温 [16] 芝加哥 0.5 40 He 低温 [18] Eugene 0.5 100 He 低温 [19, 20] Trieste 1 50 He 低温 Eugene 实验搬到的里雅斯特 [21] Trieste 4 12.5 He 低温 [22] 布尔诺 1 30 He 低温 [10, 23] Göttingen 0.33 330 SF6 [24] Göttingen 0.5 I, IIb24 [25, 92] Göttingen 0.5 IIe 224 SF6 板和侧壁之间的密封 [26] Göttingen 1 112 SF6 [27] [5] 是当时创造的，以命名在非常大的 Ra 下观察到的新流动状态。我们选择在本文中使用相同的术语。此后，格勒诺布尔、尤金、的里雅斯特、哥廷根和布尔诺的新实验在非常高的 Ra（例如 Ra > 1013）和中间 Pr 下探索了 Rayleigh-Bénard 对流（参考表 1）。如图 1(b) 所示，大多数 Nu(Ra) 测量值都在 Ra # 1011 以下，而较大 Ra 的情况更令人费解，并且一直是科学争议的燃料[6-14]。确实，在一些实验（芝加哥、尤金、布尔诺）中，补偿传热 Nu · Ra−1/3 降低或与 Ra 持平，而在其他实验（格勒诺布尔、的里雅斯特）中则显着增加，导致传热的两倍差异Ra = 1014 附近的效率。对流文献如何处理高 Ra 数据集之间显着的明显矛盾？多年来，三种不同的方法在格勒诺布尔/里昂（例如参见 [​​4, 16]）、的里雅斯特/布尔诺/布拉格（例如参见 [​​21, 23]）和哥廷根/圣巴巴拉/特温特（例如参见 [​​27, 28]）来解释结果1。这些观点要么基于 Kraichnan 预测的变体，要么基于非 Boussinesq 近似效应。单独来看，各自的数据集和解释当然是公平的，而且通常是自洽的，但还没有提议的模型能够解释所有现有的观察结果。因此，大多数论文提出了一个简单的讨论，但在明显矛盾的情况下无视或贬低先前的结果。1 当然，三个小组的成员之间在判断和重点上存在差异。