Aspect Ratio Dependence of Heat Transfer in a Cylindrical Rayleigh-B´enard Cell

While the heat transfer and the flow dynamics in a cylindrical Rayleigh-B´enard (RB) cell are rather independent of the aspect ratio Γ (diameter/height) for large Γ a small- Γ cell considerably stabilizes the flow and thus affects the heat transfer. Here, we first theoretically and numerically show that the critical Rayleigh number for the onset of convection at given Γ follows Ra c; Γ ∼ Ra c; ∞ ð 1 þ C Γ − 2 Þ 2 , with C ≲ 1 . 49 for Oberbeck-Boussinesq(OB) conditions.We thenshow that,ina broadaspect ratiorange ð 1 = 32 Þ ≤ Γ ≤ 32 , the rescaling Ra → Ra l ≡ Ra ½ Γ 2 = ð C þ Γ 2 Þ(cid:2) 3 = 2 collapses various OB numerical and almost-OB experimental heat transport data Nu ð Ra ; Γ Þ . Our findings predict the Γ dependence of the onset of the ultimate regime Ra u; Γ ∼ ½ Γ 2 = ð C þ Γ 2 Þ(cid:2) − 3 = 2 in the OB case. This prediction is consistent with almost-OB experimental results (whichonlyexistfor Γ ¼ 1 , 1 = 2 ,and 1 = 3 )forthetransitioninOBRBconvectionandexplainswhy,insmall- Γ cells, much larger Ra (namely, by a factor Γ − 3 ) must be achieved to observe the ultimate regime.

(Received 20 April 2021; accepted 13 January 2022; published 24 February 2022) While the heat transfer and the flow dynamics in a cylindrical Rayleigh-Bénard (RB) cell are rather independent of the aspect ratio Γ (diameter/height) for large Γ, a small-Γ cell considerably stabilizes the flow and thus affects the heat transfer. Here, we first theoretically and numerically show that the critical Rayleigh number for the onset of convection at given Γ follows Ra c;Γ ∼ Ra c;∞ ð1 þ CΓ −2 Þ 2 , with C ≲ 1.49 for Oberbeck-Boussinesq (OB) conditions. We then show that, in a broad aspect ratio range ð1=32Þ ≤ Γ ≤ 32, the rescaling Ra → Ra l ≡ Ra½Γ 2 =ðC þ Γ 2 Þ 3=2 collapses various OB numerical and almost-OB experimental heat transport data NuðRa; ΓÞ. Our findings predict the Γ dependence of the onset of the ultimate regime Ra u;Γ ∼ ½Γ 2 =ðC þ Γ 2 Þ −3=2 in the OB case. This prediction is consistent with almost-OB experimental results (which only exist for Γ ¼ 1, 1=2, and 1=3) for the transition in OB RB convection and explains why, in small-Γ cells, much larger Ra (namely, by a factor Γ −3 ) must be achieved to observe the ultimate regime. Physics is abstraction, often assuming systems of infinite size. In the real world, this is not possible and finite-size effects come into play and thus must be understood. Here we will do so for the Rayleigh-Bénard convection (RBC), which has always been the most paradigmatic system to study buoyancy driven heat transfer in turbulent flow [1][2][3], which is of great importance in geophysical flows and in industry. The dimensionless control parameters are the Rayleigh number, the Prandtl number, and the aspect ratio Γ of the cell, defined, respectively, as where H and D are the height and diameter of the cylindrical cell, α is the isobaric thermal expansion coefficient, ν is the kinematic viscosity, κ is the thermal diffusivity, g is the gravitational acceleration, and Δ ≡ T b − T t is the temperature difference between the hot bottom plate and the cold top plate. The boundary conditions (BCs) are no-slip at all walls and the sidewalls are adiabatic. Within the Oberbeck-Boussinesq (OB) approximation, the flow dynamics for the velocity u, the temperature T, and the kinematic pressure p is given by the continuity equation ∇ · u ¼ 0 and the Navier-Stokes and convection-diffusion equations The key response parameter is the Nusselt number (the dimensionless heat transfer) where h·i z denotes the average in time and over a horizontal cross section at height z from the bottom and h·i is the time and volume average.
One key question-clearly, since Kraichnan's 1962 prediction of an ultimate regime [13][14][15] (i.e., the asymptotic law of heat transport at fixed Pr and extremely large Ra)-is, what is the Nu(Ra) dependence for very large Ra? However, achieving very large Ra and thus this predicted ultimate regime is challenging, both experimentally, as large-scale setups are required, and computationally, as the number of grid points that can be handled is limited, too. Driven by the aim to nonetheless achieve very large Ra, one is tempted to perform experiments or simulations at as small Γ as possible. For a profound judgement on this, a FIG. 1. Critical Ra c;Γ for the onset of convection: Linear growth rates (colored vertically elongated boxes) from the linearized DNS approach (GOLDFISH) compared to the neutral stability curves (blue lines) from the eigenvalue LSA for (a) 2D box with isothermal sidewalls, (b) 2D box with adiabatic sidewalls, and (c) cylinder with adiabatic sidewall. Black lines show Ra c;Γ ¼ 1708ð1 þ C=Γ 2 Þ 2 with a best-fit C for the linearized DNS data (dashed lines) and with theoretical C for isothermal sidewall (solid line). Pluses in (c) show Ra c;Γ from the nonlinearized DNS data (AFiD) [4]. Temperature contours near the onset of convection are shown for some Γ, as obtained from the linearized DNS. See details in [5][6][7][8] and the Supplemental Material [9]. Most data are for Γ ¼ 1 and 1=2, which form the shape of this dependence. The data for extremely small Γ show no discernible dependence. (b) Compensated Nu vs Ra based on the proper length scale l, for the same data as in (a). In the main plot, the theoretical value of C ¼ 1.49 is taken, while in the inset C ¼ 0.77, which corresponds to the best fit of the critical Ra c;∞ for the onset of convection. Now the data for extremely small Γ follow the general trend. good understanding of the Γ dependence of the flow and the heat transfer for small Γ is mandatory. The Göttingen group [34,[39][40][41]50,53] has built large-scale cylindrical cells with 1 ≥ Γ ≥ 1=3 and heights up to H ¼ 2.24 m, filled with pressurized SF 6 (with low viscosity and nearly constant Pr) and has experimentally studied the onset Ra u;Γ of the ultimate regime in almost-OB RBC. Note that building even larger setups is not prohibitive, but simply extremely costly. The Göttingen group found that the onset occurs at Ra around 10 14 (consistent with the theoretical estimate of Grossmann and Lohse [15]) and revealed a Γ dependence as Ra u;Γ ∝ Γ −3.04 [54]; i.e., smaller Γ require considerably larger Ra to observe the onset. Also Roche et al. [55,56], for 1.14 ≥ Γ ≥ 0.23, found a strong Γ dependence of Ra u;Γ with the same trend. Based on an analysis of different experimental data [39][40][41][42][43]50,55,[57][58][59], they also proposed that for small Γ the onset Ra for the ultimate regime goes approximately as Ra u;Γ ∼ Γ −3 .
In fact, due to the stabilizing effect of the sidewalls in small-Γ cells, it is not surprising at all that flow transitions are shifted toward much larger Ra. This already holds at the onset of convection: While without lateral confinement (i.e., Γ → ∞) this onset occurs at a critical Ra c;∞ ≈ 1708 [60], for small Γ the critical Ra c;Γ is much larger [61][62][63][64][65][66][67][68][69][70][71]. In the limit Γ → 0, Catton and Edwards [63] numerically solved the linearized perturbation equations with approximate wall conditions and proposed the scaling Ra c;Γ ∼ Γ −4 for the onset Ra c;Γ in this limit.
In this Letter, we will derive the scaling relation Ra c;Γ ∼ Γ −4 for Γ → 0 and, in fact, generalize it to any Γ, be it large or small. We will then show that our numerically performed linear stability analysis (LSA) is consistent with the suggested generalized functional dependence of Ra c;Γ on Γ. Our result can be cast in the form that the relevant length scale in RBC is with a constant C that depends on the shape of the cell. We then apply this insight to the fully turbulent case and are able to collapse various heat transfer data NuðRa; ΓÞ from OB experiments and direct numerical simulations (DNSs) for various 1=32 ≤ Γ ≤ 32 onto one universal curve.  [42,43] hold δρ=ρ < 0.2 for the density variation and δκ=κ < 0.2 for the thermal diffusivity variation, as well as αΔ < 0.2 and 0.68 ≤ Pr ≤ 1, i.e., similar almost-OB conditions as in [34,[39][40][41]50] (however, in [34,[39][40][41]50] the upper bounds for the fluid parameter variations are even slightly stricter). Data for Pr ¼ 0.74 (gas N 2 ) and Pr ¼ 0.84 (gas SF 6 ) were taken using the same apparatus as in [47] but were not published there. The inset shows an enlargement at the highest Ra in normal representation for both axes (see also Supplemental Material [9], which includes [10][11][12]).
Theoretical background.-We first recall that the mean kinetic energy dissipation rate ϵ u and the thermal dissipation rate ϵ θ fulfil the exact relations [72,73] Decomposing the temperature field as and taking into account hu z i z ¼ 0 for any z, one obtains hu z Ti z ¼ hu z θi z and, hence, From (4) and (7)- (9), we get and then with (6) and (1) we obtain From this, applying successively (9), the Cauchy-Schwarz inequality, and relation (10), we derive For a slightly supercritical Ra ≳ Ra c;Γ the flow is symmetric so that hui ¼ 0 and hθi ¼ 0 holds. Therefore, we can apply the Poincaré-Friedrichs inequality to the righthand side of (11) to obtain where Λ is the smallest relevant eigenvalue of the Laplacian in a cylindrical domain with a unit height and aspect ratio Γ, for certain integers m, n, and k, For Dirichlet or Neumann boundary conditions, α nk are the first relevant roots of the Bessel function J n or of its derivative, respectively. Under the assumption that the relevant eigenvalues admit positive as well as negative values of θ and u in both horizontal and vertical directions, we obtain an estimate of the smallest relevant value of Λ for m ¼ 2, n ¼ k ¼ 1, leading to C ≈ 1.49.
For an infinite fluid layer (or for a cell with an infinite diameter D, i.e., Γ → ∞) Ra c;∞ ≈ 1708. Using this, relations (13) and (12), under assumption that Γ and Ra c;∞ are independent parameters, we obtain as estimate for the critical Ra c;Γ for the onset of convection in a container with finite aspect ratio Γ. Similarly, we estimate the growth of Nu near Ra c;Γ from (11), the Poincaré-Friedrichs inequality, and hθ 2 i ≤ Δ 2 , Ra ≥ ΛH 2 hð∇θÞ 2 i=hθ 2 i ≥ ΛH 2 Δ −2 hð∇θÞ 2 i: ð15Þ From (8), (7), and (15) we finally obtain Ra ≥ ΛðNu − 1Þ, which, when combined with (13), implies that close to the onset of convection, the Nusselt number behaves as From this and the fact that, in the classical turbulent regime (for not too small Pr and not extremely high Ra), Nu roughly grows as ∼Ra 1=3 , one can expect a collapse of the OB numerical and experimental data for various Γ, if these are plotted as f ≡ ðNu − 1ÞRa −1=3 against (for fixed Pr). Close to the onset of convection, this dependence reduces to f ∼ Ra 2=3 l , while in the developed, statistically steady convective flow f ∼ Ra 0 l ∼ const. The variable Ra l is nothing else but a Rayleigh number not based on the cell height H, but on the proper length scale l, Eq. (5). In the limit Γ → ∞, the length scale l equals H, while for Γ → 0, it is D.
Numerical LSA.-We have verified the estimate (14) for the Γ dependence of the critical Ra c;Γ for the onset of convection with linearized DNSs for the 2D and 3D cases and with the eigenspectrum LSA for the 2D case. The growth rates obtained with both methods are in a very good agreement, see Figs. 1(a) and 1(b). The numerically obtained Ra c;Γ as function of Γ [Eq. (14)] for the isothermal sidewalls are in excellent agreement with the analytical estimates. Equation (14) captures the trend and reflects well also the shape of the neutral curve for the case of adiabatic sidewalls. The best-fit constants C (C ≈ 0.52 for the 2D domain and C ≈ 0.77 for the cylinder) are, however, smaller than the theoretical predictions for the isothermal sidewalls, see Figs. 1(b) and 1(c). Isosurfaces of the temperature of the flow fields near the onset of convection are shown for some Γ in Fig. 1 as well. The azimuthalmode transition found for the cylinder between Γ ¼ 1 and 2 is consistent with the experiments [68].
Comparison with heat transfer data from OB experiments and DNS.-Our above theoretical analysis has PHYSICAL REVIEW LETTERS 128, 084501 (2022) 084501-4 suggested the rescaling Ra → Ra l as a central step to collapse the heat transfer data NuðRa; ΓÞ for given Γ, see Eq. (17). This rescaling reflects that the relevant length scale in RBC for general Γ is l ∼ D= ffiffiffiffiffiffiffiffiffiffiffiffiffiffi , see Eq. (5), and not simply the height H. For large Γ one recovers l ¼ H, but for small Γ one has l ¼ D. We will now check whether this collapse holds and plot the compensated Nusselt number f ≡ ðNu − 1Þ=Ra 1=3 from OB experiments and well-resolved DNSs [74] for various Γ, both vs Ra and vs Ra l (with C ¼ 1.49). We do so for two different Pr, namely, for water (Pr ≈4.4, Fig. 2) and for gas (Pr ≈0.8, Fig. 3) at room temperature. While in Figs. 2(a) and 3(a) [fðRaÞ], the data for small Γ show no trend and seem to scatter, in Figs. 2(b) and 3(b) [fðRa l Þ], they nicely collapse on one curve and on the theoretical curve of the unifying theory for turbulent thermal convection [28][29][30]. A comparison with non-OB data for cryogenic gaseous helium [42,43,57,75,76] is given in the Supplemental Material [9]. As the derivation of the scaling relations is for OB conditions, we do not expect non-OB data to fulfil these relations, and indeed, in general, they do not (see [34,77,78] and Supplemental Material [9]).
Let us now estimate the Γ dependence of the onset of the ultimate regime of thermal convection, i.e., Ra u;Γ . (The other aspects of the ultimate regime are beyond the scope of this Letter.) The Γ dependence of Ra u;Γ has been observed in the Göttingen data [34,[39][40][41]50], with increasing Ra u;Γ for decreasing 1 ≥ Γ ≥ 1=3; see the vertical lines for large Ra in Fig. 3(a). However, as suggested by our theory, in the rescaled Fig. 3(b), these vertical lines collapse at the same Ra l;u ≈ 2.4 × 10 13 . This implies that the Γ dependence of Ra u;Γ in the OB case is which for Γ ≪ 1 simplifies to the estimate Ra u;Γ ∼ Γ −3 , in agreement with the experimental data [54]. Note that in Fig. 3 the agreement between the derived relation (18) and measurements is demonstrated for all available almost-OB experimental data, that is, for Γ ¼ 1, 1=2, and 1=3. Figure 3 and Eq. (18) also show that the presented DNS for small Γ by far do not have large enough Ra to see the expected onset of the ultimate regime.
In conclusion, we have developed a theory to account for the Γ dependence of the heat transfer in buoyancy driven convection under OB conditions in cylindrical cells. In particular, we find the Γ dependence of the onset of convection Ra c;Γ [Eq. (14), consistent with the LSA] and of the onset of the ultimate regime Ra u;Γ [Eq. (18), consistent with the Göttingen experiments]. Both equations reflect that the relevant length scale in OB RBC is , which only in the limiting cases Γ → ∞ or Γ → 0 become the cell height H or the cell diameter D, respectively. Speaking more generally, our results show how strongly finite-size effects affect scaling relations and that small-Γ OB DNSs or (almost) OB experiments require much large Ra to achieve the ultimate regime.