Aspect Ratio Dependence of Heat Transfer in a Cylindrical Rayleigh-Bénard Cell

Abstract

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,∞}(1+CΓ^{-2})^{2}, with C≲1.49 for Oberbeck-Boussinesq (OB) conditions. We then show that, in a broad aspect ratio range (1/32)≤Γ≤32, the rescaling Ra→Ra_{ℓ}≡Ra[Γ^{2}/(C+Γ^{2})]^{3/2} collapses various OB numerical and almost-OB experimental heat transport data Nu(Ra,Γ). Our findings predict the Γ dependence of the onset of the ultimate regime Ra_{u,Γ}∼[Γ^{2}/(C+Γ^{2})]^{-3/2} in the OB case. This prediction is consistent with almost-OB experimental results (which only exist for Γ=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.

Full text

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=ðCþΓ2Þ3=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=ðΔH3ÞhuzTi:
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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PHYSICAL REVIEW LETTERS 128, 084501 (2022)
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