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

Guenter Ahlers,4,1 Eberhard Bodenschatz ,1,3,7,8 Robert Hartmann ,2Xiaozhou He ,6,1 Detlef Lohse ,2,3,1

Philipp Reiter ,1Richard J. A. M. Stevens ,2Roberto Verzicco,5,9,2 Marcel Wedi,1Stephan Weiss ,1,3

Xuan Zhang ,1Lukas Zwirner ,1and Olga Shishkina 1,*

1Max Planck Institute for Dynamics and Self-Organization, 37077 Göttingen, Germany

2Physics of Fluids Group, J. M. Burgers Center for Fluid Dynamics and MESA+ Institute,

University of Twente, 7500 AE Enschede, Netherlands

3Max Planck—University of Twente Center for Complex Fluid Dynamics, 7500 AE Enschede, Netherlands

4Department of Physics, University of California, Santa Barbara, California 93106, USA

5Dipartimento di Ingegneria Industriale, University of Rome “Tor Vergata,”Via del Politecnico 1, Roma 00133, Italy

6School of Mechanical Engineering and Automation, Harbin Institute of Technology, Shenzhen, 518055 China

7Institute for the Dynamics of Complex Systems, Georg-August-University Göttingen, 37073 Göttingen, Germany

8Laboratory of Atomic and Solid-State Physics and Sibley School of Mechanical and Aerospace Engineering,

Cornell University, Ithaca, New York 14853, USA

9Gran Sasso Science Institute—Viale F. Crispi, 767100 L’Aquila, Italy

(Received 20 April 2021; accepted 13 January 2022; published 24 February 2022)

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-Γcell 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 Γfollows Rac;Γ∼Rac;∞ð1þCΓ−2Þ2, 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 →Ral≡Ra½Γ2=ðCþΓ2Þ3=2collapses various OB numerical and almost-OB experimental

heat transport data NuðRa;ΓÞ. Our findings predict the Γdependence of the onset of the ultimate regime

Rau;Γ∼½Γ2=ðCþΓ2Þ−3=2in the OB case. This prediction is consistent with almost-OB experimental results

(which only exist for Γ¼1,1=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.

DOI: 10.1103/PhysRevLett.128.084501

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-B´enard convection (RBC),

which has always been the most paradigmatic system to

study buoyancy driven heat transfer in turbulent flow [1–3],

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

Γof the cell, defined, respectively, as

Ra ≡αgΔH3=ðκνÞ;Pr ≡ν=κ;Γ≡D=H; ð1Þ

where Hand Dare the height and diameter of the

cylindrical cell, αis the isobaric thermal expansion

coefficient, νis the kinematic viscosity, κis the thermal

diffusivity, gis the gravitational acceleration, and Δ≡Tb−

Ttis 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 pis given by the continuity equation

∇·u¼0and the Navier-Stokes and convection-diffusion

equations

∂tuþu·∇uþ∇p¼ν∇2uþαgTez;ð2Þ

∂tTþu·∇T¼κ∇2T: ð3Þ

The key response parameter is the Nusselt number (the

dimensionless heat transfer)

Nu ≡huzTiz−κ∂zhTiz

κΔ=H ¼H

κΔhuzTiþ1;ð4Þ

Published by the American Physical Society under the terms of

the Creative Commons Attribution 4.0 International license.

Further distribution of this work must maintain attribution to

the author(s) and the published article’s title, journal citation,

and DOI. Open access publication funded by the Max Planck

Society.

PHYSICAL REVIEW LETTERS 128, 084501 (2022)

0031-9007=22=128(8)=084501(6) 084501-1 Published by the American Physical Society

where h·izdenotes the average in time and over a horizontal

cross section at height zfrom the bottom and h·iis the time

and volume average.

One key question—clearly, since Kraichnan’s 1962

prediction of an ultimate regime [13–15] (i.e., the asymp-

totic 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 experimen-

tally, 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 Γas possible. For a profound judgement on this, a

FIG. 1. Critical Rac;Γfor 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 Rac;Γ¼1708ð1þC=Γ2Þ2

with a best-fit Cfor the linearized DNS data (dashed lines) and with theoretical Cfor isothermal sidewall (solid line). Pluses in (c) show

Rac;Γfrom the nonlinearized DNS data (

AF

i

D

)[4]. Temperature contours near the onset of convection are shown for some Γ, as obtained

from the linearized DNS. See details in [5–8] and the Supplemental Material [9].

FIG. 2. (a) Compensated Nu vs Ra, as obtained in OB experiments and DNSs of RBC in a cylinder for Pr ≈4.4(water) and different Γ.

Most data are for Γ¼1and 1=2, which form the shape of this dependence. The data for extremely small Γshow 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 C¼1.49 is taken, while in the inset C¼0.77, which corresponds to the best fit of the critical Rac;∞for the onset of convection.

Now the data for extremely small Γfollow the general trend.

PHYSICAL REVIEW LETTERS 128, 084501 (2022)

084501-2

good understanding of the Γdependence of the flow and the

heat transfer for small Γis mandatory. The Göttingen group

[34,39–41,50,53] has built large-scale cylindrical cells with

1≥Γ≥1=3and heights up to H¼2.24 m, filled with

pressurized SF6(with low viscosity and nearly constant Pr)

and has experimentally studied the onset Rau;Γof 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 1014 (consistent with the theoretical estimate of

Grossmann and Lohse [15]) and revealed a Γdependence as

Rau;Γ∝Γ−3.04 [54]; i.e., smaller Γrequire considerably

larger Ra to observe the onset. Also Roche et al. [55,56], for

1.14 ≥Γ≥0.23, found a strong Γdependence of Rau;Γwith

the same trend. Based on an analysis of different experi-

mental data [39–43,50,55,57–59], they also proposed that

for small Γthe onset Ra for the ultimate regime goes

approximately as Rau;Γ∼Γ−3.

In fact, due to the stabilizing effect of the sidewalls in

small-Γcells, 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 Rac;∞≈1708

[60], for small Γthe critical Rac;Γis much larger [61–71].In

the limit Γ→0, Catton and Edwards [63] numerically

solved the linearized perturbation equations with approxi-

mate wall conditions and proposed the scaling Rac;Γ∼Γ−4

for the onset Rac;Γin this limit.

In this Letter, we will derive the scaling relation Rac;Γ∼

Γ−4for Γ→0and, 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 Rac;Γon

Γ. Our result can be cast in the form that the relevant length

scale in RBC is

l∼D= ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

Γ2þC

p¼H= ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

1þC=Γ2

q;ð5Þ

with a constant Cthat 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;ΓÞfrom

OB experiments and direct numerical simulations (DNSs)

for various 1=32 ≤Γ≤32 onto one universal curve.

FIG. 3. (a) Compensated Nu vs Ra for OB RBC in a cylinder for Pr near 0.8 and in a 3D cell with periodic BCs for Pr ¼1and different

Γ. Vertical lines indicate the onset of the transition at high Ra, observed in Göttingen experiments (the onset moves to higher Ra with

decreasing Γ). (b) Compensated Nu vs Ra based on the proper length scale l, for the same data as in (a). Now the transition happens at

the same location for all Γ[the vertical lines from (a) merge into one line]. The here presented experimental data from Chavanne et al.

[42,43] hold δρ=ρ<0.2for the density variation and δκ=κ<0.2for the thermal diffusivity variation, as well as αΔ<0.2and

0.68 ≤Pr ≤1, i.e., similar almost-OB conditions as in [34,39–41,50] (however, in [34,39–41,50] the upper bounds for the fluid

parameter variations are even slightly stricter). Data for Pr ¼0.74 (gas N2) and Pr ¼0.84 (gas SF6) 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 [10–12]).

PHYSICAL REVIEW LETTERS 128, 084501 (2022)

084501-3

Theoretical background.—We first recall that the mean

kinetic energy dissipation rate ϵuand the thermal dissipa-

tion rate ϵθfulfil the exact relations [72,73]

ϵu≡νhð∇uÞ2i¼αghuzTi¼ν3

H4ðNu −1ÞRa

Pr2;ð6Þ

ϵθ≡κhð∇TÞ2i¼ðκΔ2=H2ÞNu:ð7Þ

Decomposing the temperature field as

T≡Tlþθ;T

lðzÞ≡Tb−ðz=HÞΔ;ð8Þ

and taking into account huziz¼0for any z, one obtains

huzTiz¼huzθizand, hence,

huzTi¼huzθi:ð9Þ

From (4) and (7)–(9), we get

huzθi¼ðκH=ΔÞhð∇θÞ2ið10Þ

and then with (6) and (1) we obtain

hð∇uÞ2i¼Ra½κ=ðΔH3ÞhuzTi:

From this, applying successively (9), the Cauchy-Schwarz

inequality, and relation (10), we derive

Ra ¼ΔH3

κhð∇uÞ2i

huzTi¼ΔH3

κhð∇uÞ2ihuzθi

huzθi2

≥

ΔH3

κhð∇uÞ2ihuzθi

hu2

zihθ2i≥H4hð∇uÞ2ihð∇θÞ2i

hu2ihθ2i:ð11Þ

For a slightly supercritical Ra ≳Rac;Γthe flow is sym-

metric so that hui¼0and hθi¼0holds. Therefore, we

can apply the Poincar´e-Friedrichs inequality to the right-

hand side of (11) to obtain

Rac;Γ≳H4hð∇uÞ2ihð∇θÞ2i

hu2ihθ2i≳Λ2;ð12Þ

where Λis 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,

Λ¼m2π2þ4α2

nkΓ−2∼1þCΓ−2:ð13Þ

For Dirichlet or Neumann boundary conditions, αnk are the

first relevant roots of the Bessel function Jnor of its

derivative, respectively. Under the assumption that the

relevant eigenvalues admit positive as well as negative

values of θand uin both horizontal and vertical directions,

we obtain an estimate of the smallest relevant value of Λfor

m¼2,n¼k¼1, 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, rela-

tions (13) and (12), under assumption that Γand Rac;∞are

independent parameters, we obtain

Rac;Γ∼Rac;∞ð1þCΓ−2Þ2ð14Þ

as estimate for the critical Rac;Γfor the onset of convection

in a container with finite aspect ratio Γ.

Similarly, we estimate the growth of Nu near Rac;Γfrom

(11), the Poincar´e-Friedrichs inequality, and hθ2i≤Δ2,

Ra ≥ΛH2hð∇θÞ2i=hθ2i≥ΛH2Δ−2hð∇θÞ2i: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

Nu −1∼ð1þCΓ−2Þ−1Ra:ð16Þ

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 ∼Ra1=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=3against

Ral≡Rað1þCΓ−2Þ−3=2ð17Þ

(for fixed Pr). Close to the onset of convection, this

dependence reduces to f∼Ra2=3

l, while in the developed,

statistically steady convective flow f∼Ra0

l∼const. The

variable Ralis 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 lequals H,

while for Γ→0,itisD.

Numerical LSA.—We have verified the estimate (14) for

the Γdependence of the critical Rac;Γfor 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 Rac;Γas 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 Γin Fig. 1as well. The azimuthal-

mode transition found for the cylinder between Γ¼1and 2

is consistent with the experiments [68].

Comparison with heat transfer data from OB experi-

ments and DNS.—Our above theoretical analysis has

PHYSICAL REVIEW LETTERS 128, 084501 (2022)

084501-4

suggested the rescaling Ra →Ralas a central step to

collapse the heat transfer data NuðRa;ΓÞfor given Γ, see

Eq. (17). This rescaling reflects that the relevant length

scale in RBC for general Γis l∼D= ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

Γ2þC

p¼

H= ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

1þC=Γ2

p, see Eq. (5), and not simply the height

H. For large Γone recovers l¼H, but for small Γone has

l¼D. We will now check whether this collapse holds and

plot the compensated Nusselt number f≡ðNu −1Þ=Ra1=3

from OB experiments and well-resolved DNSs [74] for

various Γ, both vs Ra and vs Ral(with C¼1.49). 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 Γshow no

trend and seem to scatter, in Figs. 2(b) and 3(b) [fðRalÞ],

they nicely collapse on one curve and on the theoretical

curve of the unifying theory for turbulent thermal con-

vection [28–30]. 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 Γdependence of the onset of the

ultimate regime of thermal convection, i.e., Rau;Γ. (The

other aspects of the ultimate regime are beyond the scope of

this Letter.) The Γdependence of Rau;Γhas been observed

in the Göttingen data [34,39–41,50], with increasing Rau;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

Ral;u ≈2.4×1013. This implies that the Γdependence of

Rau;Γin the OB case is

Rau;Γ≈Ral;u½Γ2=ðCþΓ2Þ−3=2;ð18Þ

which for Γ≪1simplifies to the estimate Rau;Γ∼Γ−3,in

agreement with the experimental data [54]. Note that in

Fig. 3the agreement between the derived relation (18) and

measurements is demonstrated for all available almost-OB

experimental data, that is, for Γ¼1,1=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 Γdependence of the heat transfer in buoyancy driven

convection under OB conditions in cylindrical cells. In

particular, we find the Γdependence of the onset of

convection Rac;Γ[Eq. (14), consistent with the LSA] and

of the onset of the ultimate regime Rau;Γ[Eq. (18),

consistent with the Göttingen experiments]. Both equations

reflect that the relevant length scale in OB RBC is

l¼D= ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

Γ2þC

p¼H= ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

1þC=Γ2

p, which only in the

limiting cases Γ→∞or Γ→0become the cell height

Hor the cell diameter D, respectively. Speaking more

generally, our results show how strongly finite-size effects

affect scaling relations and that small-ΓOB DNSs or

(almost) OB experiments require much large Ra to achieve

the ultimate regime.

The authors acknowledge the Deutsche Forschu-

ngsgemeinschaft (SPP1881 “Turbulent Superstructures”

and Grants No. Sh405/7, No. Sh405/8, and No. Sh405/

10), the Twente Max-Planck Center, the European

Research Council (ERC Starting Grant No. 804283

UltimateRB), the National Natural Science Foundation

of China (Grant No. 91952101), PRACE (Projects

No. 2020235589 and No. 2020225335), and the Gauss

Centre for Supercomputing e.V. for providing computing

time in the GCS Supercomputer SuperMUC at Leibniz

Supercomputing Centre.

*Olga.Shishkina@ds.mpg.de

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