eurm10 msam10

# Aspect ratio dependence of heat transport by turbulent Rayleigh-Bénard
convection in rectangular cells

Quan ZHOU Email address for correspondence<EMAIL_ADDRESS>Bo-Fang LIU
Chun-Mei LI and Bao-Chang ZHONG  Shanghai Institute of Applied Mathematics and
Mechanics, and Shanghai Key Laboratory of Mechanics in Energy Engineering,
Shanghai University, Shanghai 200072, China

(?? and in revised form ??)

###### Abstract

We report high-precision measurements of the Nusselt number $Nu$ as a function
of the Rayleigh number $Ra$ in water-filled rectangular Rayleigh-Bénard
convection cells. The horizontal length $L$ and width $W$ of the cells are
50.0 cm and 15.0 cm, respectively, and the heights $H=49.9$, 25.0, 12.5, 6.9,
3.5, and 2.4 cm, corresponding to the aspect ratios $(\Gamma_{x}\equiv
L/H,\Gamma_{y}\equiv W/H)=(1,0.3)$, $(2,0.6)$, $(4,1.2)$, $(7.3,2.2)$,
$(14.3,4.3)$, and $(20.8,6.3)$. The measurements were carried out over the
Rayleigh number range $6\times 10^{5}\lesssim Ra\lesssim 10^{11}$ and the
Prandtl number range $5.2\lesssim Pr\lesssim 7$. Our results show that for
rectangular geometry turbulent heat transport is independent of the cells’
aspect ratios and hence is insensitive to the nature and structures of the
large-scale mean flows of the system. This is slightly different from the
observations in cylindrical cells where $Nu$ is found to be in general a
decreasing function of $\Gamma$, at least for $\Gamma=1$ and larger. Such a
difference is probably a manifestation of the finite plate conductivity
effect. Corrections for the influence of the finite conductivity of the top
and bottom plates are made to obtain the estimates of $Nu_{\infty}$ for plates
with perfect conductivity. The local scaling exponents $\beta_{l}$ of
$Nu_{\infty}\sim Ra^{\beta_{l}}$ are calculated and found to increase from
0.243 at $Ra\simeq 9\times 10^{5}$ to 0.327 at $Ra\simeq 4\times 10^{10}$.

##  1 Introduction

Convection is ubiquitous in nature and in our everyday life. It can be found
in the stars and planets, in the Earth’s mantle and outer core, in the oceans
and atmosphere, as well as in heat transport and mass mixing in many
engineering applications. The paradigmatic example for natural convection is
the Rayleigh-Bénard (RB) convection of an enclosed fluid layer between the
colder top and the warmer bottom plates (Ahlers, Grossmann $\&$ Lohse 2009 _b_
; Lohse $\&$ Xia 2010). A key issue in the study of turbulent RB convection is
to understand how heat is transported upwards across the fluid layer by
convective flows. The global heat transport by convection is usually expressed
in terms of the Nusselt number, namely,

$Nu=\frac{QH}{\lambda_{f}\Delta},$ (1)

where $Q$ is the heat current density across a fluid layer of thermal
conductivity $\lambda_{f}$ with height $H$ and with an applied temperature
difference $\Delta$. The dynamics of the system is determined by the
geometrical configuration of the convection cell and by two dimensionless
control parameters, i.e. the Rayleigh number and the Prandtl number,

$Ra=\frac{\alpha g\Delta H^{3}}{\nu\kappa}\mbox{\ \ and\ \
}Pr=\frac{\nu}{\kappa}.$ (2)

Here, $g$ is the acceleration due to gravity and $\alpha$, $\nu$, and $\kappa$
are the isobaric thermal expansion coefficient, the kinematic viscosity, and
the thermal diffusivity of the working fluid, respectively. The cell geometry
is usually described in terms of one or more aspect ratios, such as
$\Gamma\equiv D/H$ for a cylindrical cell of inner diameter $D$ and
$(\Gamma_{x}\equiv L/H,\Gamma_{y}\equiv W/H)$ for a rectangular cell of
horizontal length $L$ and width $W$. The $Ra$\- and $Pr$-dependence of $Nu$
for various working fluids and cell geometries have been studied, both
experimentally and numerically, in great detail for many years (Castaing _et
al._ 1989; Kerr 1996; Chavanne _et al._ 1997, 2001; Du $\&$ Tong 2000; Kerr
$\&$ Herring 2000; Niemela _et al._ 2000; Ahlers $\&$ Xu 2001; Roche _et al._
2002, 2005; Xia, Lam $\&$ Zhou 2002; Verzicco $\&$ Camussi 2003; Niemela $\&$
Sreenivasan 2003; Shishkina $\&$ Wagner 2007; Ahlers, Funfschilling $\&$
Bodenschatz 2009 _a_ ; Funfschilling, Bodenschatz $\&$ Ahlers 2009; Song $\&$
Tong 2010; Stevens, Verzicco $\&$ Lohse 2010; Stevens, Lohse $\&$ Verzicco
2011 _a_ ; Silano, Sreenivasan $\&$ Verzicco 2010; He _et al._ 2012). In
addition, various theoretical models have been advanced to predict the
behavior of convective heat transport Castaing et al. (1989); Shraiman &
Siggia (1990); Grossmann & Lohse (2000, 2001, 2003, 2004, 2011); Dubrulle
(2001, 2002). For more detailed elucidation of the problem, we refer
interested readers to the recent review paper by Ahlers et al. (2009b). On the
other hand, there are fewer measurements focusing on the aspect-ratio-
dependence (Xu, Bajaj $\&$ Ahlers 2000; Fleischer $\&$ Goldstein 2002; Cheung
2004; Nikolaenko _et al._ 2005; Funfschilling _et al._ 2005; Sun _et al._ 2005
_a_ ; Niemela $\&$ Sreenivasan 2006; Roche _et al._ 2010), and those
measurements were all performed in containers with a cylindrical geometry. The
objective of the present experimental investigation is to fill this gap by
making high-precision measurements of $Nu$ over a wide range of the aspect
ratio in convection cells with a rectangular geometry.

A lateral sidewall is indispensable for any convection experiment in the
laboratory. The interaction between the sidewall and fluids would change the
velocity and temperature distributions in the cell, and in turn change the
flow structures of the system. Indeed, previous experimental studies for both
cylindrical (du Puits, Resagk $\&$ Thess 2007) and rectangular (Xia, Sun $\&$
Cheung 2008) cells have shown that with increasing the aspect ratio the large-
scale circulation (LSC) departs from a single-roll structure and becomes a
multi-roll pattern. Thus, the $\Gamma$-dependence of $Nu$ may reflect the
influence of flow structures on heat transport characteristics via the
influence on the boundary layers. Measurements in cylindrical samples using
water as working fluid ($Pr\approx 4.3$) revealed that for $\Gamma\lesssim 6$
$Nu$ decreases, albeit only a few percent, with increasing $\Gamma$
Funfschilling et al. (2005); Sun et al. (2005a). This suggests that the global
heat transport properties of the RB system are not very sensitive to the flow
structures for such parameter ranges. However, the situation is very different
for lower $Pr$ and for the two-dimensional (2D) case. Using 3D direct
numerical simulation (DNS), Bailon-Cuba, Emran $\&$ Schumacher (2010) found
for $Pr=0.7$ that the minimum of $Nu$ occurs at the aspect ratio where the LSC
undergoes a transition from a single-roll to a double-roll structure and the
variations in $Nu$ for different $\Gamma$ are significant and can yield
$11.3\%$ for $Ra=10^{8}$. For 2D steady-state calculations, Ching & Tam (2006)
obtained a power-law of $Nu\sim\Gamma^{-1}$ for $\Gamma\leqslant 3$. In 2D
numerical RB flow, van der Poel, Stevens $\&$ Lohse (2011) identified
different turbulent states for both $Pr=0.7$ and 4.3, corresponding to
different roll structures and associated with different overall heat
transfers. Transitions among these states thus lead to jumps and sharp
transitions in $Nu(\Gamma)$. By connecting the structures of $Nu(\Gamma)$ to
the way the flow organizes itself in the sample, van der Poel et al. (2011)
explained why the aspect-ratio dependence of $Nu$ is more pronounced for small
$Pr$. Compared with the 3D cases, we note that the 2D simulations show larger
variations in $Nu(\Gamma)$ for both $Pr=0.7$ and 4.3, which may be explained
by the different flow structures formed in the 2D and 3D cases.

Different geometrical shapes represent different symmetries, and hence may
lead to different features of the flow and heat transfer. This prompts us to
study the $\Gamma$-dependence of $Nu$ in a non-cylindrical system. In the
present study, we choose rectangle as the shape of the cells, which has been
widely used in the past (Xia, Sun $\&$ Zhou 2003; Gasteuil _et al._ 2007;
Maystrenko, Resagk $\&$ Thess 2007; Zhou $\&$ Xia 2008, 2010 _a_ ,_b_ ; Zhou
_et al._ 2007, 2010). It was found that the convective flows in cells with
such geometry share some similar dynamics as those in 2D RB system, such as
the reversals of the LSC Sugiyama et al. (2010).

The remainder of this paper is organized as follows. We give detailed
descriptions of the experimental apparatus and conditions in $\S$2\.
Experimental results are presented and analyzed in $\S$3, which is divided
into two parts. In $\S$3.1, we compare the measured $Nu$ for different aspect
ratios and with those obtained in cylindrical cells. In $\S$3.2 we consider
the finite conductivity corrections of the top and bottom plates and estimate
$Nu_{\infty}$ for perfectly conducting plates. The scaling behaviors of
$Nu_{\infty}$ are also discussed in $\S$3.2. We summarize our findings and
conclude in $\S$4.

##  2 Experimental apparatus and methods

Figure 1: Schematic diagram of front view of a rectangular cell. A: the
Plexiglas cover; B: the copper top plate; Ci and Ci’ ($i=1,2,3,4$): nozzles
connecting the channels to the refrigerated circulators; D: thermistors; E:
the Plexiglas sidewall; F: the copper bottom plate; G: the copper cover for
the bottom plate, H1: nozzle for transferring fluid into the cell; H2: nozzle
for letting air out of the cell.

Figure 1 is a schematic drawing of the front view of our apparatus and the
drawing corresponds to $(\Gamma_{x},\Gamma_{y})=(1,0.3)$. The sidewall of the
cell, indicated as E in the figure, is composed by four transparent Plexiglas
plates of 1.2 cm in thickness. The inner length $L$ and inner width $W$ of the
cell are 50 cm and 15 cm, respectively. Six sidewalls of heights $H=49.9$,
25.0, 12.5, 6.9, 3.5, and 2.4 cm were used in the experiment. The
corresponding aspect ratios are $(\Gamma_{x}\equiv L/H,\Gamma_{y}\equiv
W/H)=(1,0.3)$, $(2,0.6)$, $(4,1.2)$, $(7.3,2.2)$, $(14.3,4.3)$, and
$(20.8,6.3)$, respectively. As $\Gamma_{y}$ is proportional to $\Gamma_{x}$,
in the remainder of the paper we use only $\Gamma_{x}$ to indicate the cell’s
aspect ratio for ease of presentation. The top (B) and bottom (F) plates are
made of pure copper of 56 cm in length and 21 cm in width and their fluid-
contact surfaces are electroplated with a thin layer of nickel to prevent the
oxidation by water. The thickness of the top plate is 3.5 cm and that of the
bottom plate is 1.5 cm. Silicon O-rings are placed between the copper plates
and the sidewall plates to avoid fluid leakage. Eight stainless steel posts
(not shown) hold the top and bottom plates together. They are insulated from
the plates by Teflon sleeves and washers. Four parallel channels (not shown)
of 1.5 cm in width and 2 cm in depth are machined into the top plate and the
separation between adjacent channels is 1.5 cm. The channels start and end,
respectively, at the two diagonal ends of the long edges. A silicon rubber
sheet (not shown) and a Plexiglas plate (A) of 1.4 cm in thickness are fixed
on the top to form the cover and also to prevent interflow between the
adjacent channels. At two ends of the $i^{th}$ channel ($i=1,2,3,4$), there
are two nozzles (Ci and Ci’) located, through which the channel is connected
to a separate refrigerated circulator (Polyscience 9712) that has a
temperature stability of 0.01 ∘C. The channels and the circulators are
connected such that the incoming cooler fluid and the outgoing warmer fluid in
adjacent channels always flow in opposite directions. To provide constant and
uniform heating, two rectangular Kapton film heaters of 25 cm in length and 15
cm in width are sandwiched between two copper plates (F and G) and are
connected in parallel to a d.c. power supply (SGI 330X15D) with $99.99\%$
long-term stability. Therefore, the experiments were conducted under constant
heating of the bottom plate while maintaining a constant temperature at the
top plate. Note that recent high-resolution 2D Johnston & Doering (2009) and
3D Stevens et al. (2011a) simulations have revealed that turbulent thermal
convection with boundary conditions of constant temperature and constant heat
flux display identical heat transport at sufficient high Rayleigh numbers.

Figure 2: The normalized horizontal temperature differences,
$(T_{i}-T_{m})/\Delta$, for the top (_a_ , _c_) and bottom (_b_ , _d_) plates
and for $\Gamma_{x}=1$ (_a_ , _b_) and 7.3 (_c_ , _d_). The index $i$
($=1,2,3,4,5$) of the thermistor is listed in figure 1 (see the top plate B
and the bottom plate F).

Degassed water was used as the convecting fluid and the cell was leveled to
better than $0.1^{\circ}$. During the measurements the entire cell was wrapped
with several layers of Styrofoam. The temperature of each conducting plate was
measured by five thermistors (D), which are embedded uniformly beneath the
fluid-contact surface of the respective plate. When calculating the
temperature difference $\Delta$ between the bottom and top plates, a
correction was made for the temperature change between the thermistor position
and the fluid-contact surface. In each measurement after $Ra$ was changed it
took about 4 to 8 hours for the system to reach the steady state and we waited
for at least 12 hours to start the measurements. A typical measurement lasted
over 12 hours and more than 24 hours for low-$\Delta$ experiments ($\Delta<4$
∘C). No long-term drift of the mean temperature in the plates was observed
over the duration of the measurement and the standard deviations were less
than $0.5\%$ of $\Delta$ for all measurements.

To see the temperature uniformity of each conducting plate, we plot in figure
2 the normalized temperature variation, $(T_{i}-T_{m})/\Delta$, for both the
top and bottom plates. Here, $T_{i}$ ($i=1,2,3,4,5$) is the time-averaged
temperature measured by the $i^{th}$ thermistor in a given plate and $T_{m}$
is the mean value of all the five thermistors in the same plate. It is found
that the $\Gamma_{x}=1$ cell has the largest variation of plate temperature
and we plot the $\Gamma_{x}=1$ results in figures 2(_a_) and (_b_). The
temperature variation for the other five $\Gamma_{x}$ are similar and we
choose $\Gamma_{x}=7.3$ as an example and plot the results in figures 2(_c_)
and (_d_). One sees that $(T_{i}-T_{m})/\Delta$ is at most $4\%$ for
$\Gamma_{x}=1$ and is less than $2\%$ for all $\Gamma_{x}=7.3$ measurements.
We thus estimate that a systematic error of the order of $1\%$ could be
introduced into the measured $Nu$ by this horizontal temperature inhomogeneity
of the conducting plates. Several thermistors were placed around the cell
sidewall and around the bottom plate to monitor the environment temperature,
based on which heat leakages to the environment and conduction by the posts
and Plexiglas sidewall were calculated. The relative leakages to the total
heat current were found to become more significant with decreasing $\Delta$
and thus we kept heat leakages from all sources to be less than $7\%$ of the
total applied heat current by working with sufficiently large $\Delta$. We
found that the largest source of leakage, especially for the small aspect-
ratio cells, is through the cell sidewall which is insulated by multi-layers
of Styrofoam. The errors in calculating the leaks come mainly from
uncertainties in the thermal conductivities of the material involved, which
are estimated to be less than $20\%$. This translates into an uncertainty of
less than $1.5\%$ in the results of $Nu$.

##  3 Results and discussion

###  3.1 $Nu$ vs. $Ra$ for different $\Gamma_{x}$

Figure 3: (color online). Compensated $Nu/Ra^{1/3}$ on a linear scale vs. $Ra$ on a logarithmic scale. $\bigtriangledown$, $\Gamma_{x}=20.8$; $\bigcirc$, $\Gamma_{x}=14.3$; $\bigtriangleup$, $\Gamma_{x}=7.3$; $\square$, $\Gamma_{x}=4$; $\vartriangleleft$, $\Gamma_{x}=2$; $\vartriangleright$, $\Gamma_{x}=1$ and $Pr\approx 7.01$; $\lozenge$, $\Gamma_{x}=1$ and $Pr\approx 5.45$. Note that the data are as measured, without the correction for the finite plate conductivity. Table 1: Experimental parameters and results for the $\Gamma_{x}=1$ cell. Here, $Nu_{\infty}$ is calculated using Eq. (3) and the parameters $a=0.275$ and $b=0.39$ obtained by Brown et al. (2005) in cylindrical cells of two sets of plates of different thermal conductivities (Cu and Al). It is also seen that $\alpha\Delta\lesssim 0.01$ for all cases, which is considered to be sufficiently small. Note that two points are listed per line and the data are listed in chronological order. $\Delta$ (K) | $Ra$ | $Pr$ | $Nu$ | $Nu_{\infty}$ | $\alpha\Delta$ | $|$ |  $\Delta$ (K) | $Ra$ | $Pr$ | $Nu$ | $Nu_{\infty}$ | $\alpha\Delta$
---|---|---|---|---|---|---|---|---|---|---|---|---
|  |  |  |  | ($10^{-3}$) | $|$ |  |  |  |  |  | ($10^{-3}$)
$\Gamma_{x}=1$ |  |  |  |  |  | $|$ |  |  |  |  |  |
$3.44$ | $1.06\times 10^{10}$ | $5.45$ | $128.8$ | $130.3$ | $1.04$ | $|$ | $5.24$ | $1.63\times 10^{10}$ | $5.41$ | $147.4$ | $149.5$ | $1.59$
$15.16$ | $4.73\times 10^{10}$ | $5.40$ | $205.5$ | $210.6$ | $4.62$ | $|$ | $18.33$ | $5.57\times 10^{10}$ | $5.48$ | $216.6$ | $222.4$ | $5.49$
$24.36$ | $7.71\times 10^{10}$ | $5.40$ | $239.8$ | $247.2$ | $7.42$ | $|$ | $9.97$ | $3.03\times 10^{10}$ | $5.47$ | $178.3$ | $181.9$ | $2.99$
$12.65$ | $3.77\times 10^{10}$ | $5.52$ | $191.9$ | $196.2$ | $3.74$ | $|$ | $7.52$ | $2.30\times 10^{10}$ | $5.46$ | $163.5$ | $166.4$ | $2.27$
$6.41$ | $1.96\times 10^{10}$ | $5.46$ | $155.4$ | $157.9$ | $1.93$ | $|$ | $8.74$ | $2.66\times 10^{10}$ | $5.47$ | $170.4$ | $173.6$ | $2.63$
$30.93$ | $9.27\times 10^{10}$ | $5.51$ | $255.3$ | $264.0$ | $9.19$ | $|$ | $5.30$ | $9.25\times 10^{9}$ | $7.01$ | $121.9$ | $123.1$ | $1.09$
$7.02$ | $1.22\times 10^{10}$ | $7.02$ | $133.1$ | $134.6$ | $1.44$ | $|$ | $3.20$ | $5.56\times 10^{9}$ | $7.02$ | $105.3$ | $106.1$ | $0.66$
$11.98$ | $2.11\times 10^{10}$ | $6.98$ | $160.2$ | $162.8$ | $2.49$ | $|$ | $4.23$ | $7.34\times 10^{9}$ | $7.03$ | $114.6$ | $115.6$ | $0.87$
$15.10$ | $2.64\times 10^{10}$ | $7.01$ | $171.6$ | $174.7$ | $3.12$ | $|$ | $2.63$ | $4.60\times 10^{9}$ | $7.00$ | $99.3$ | $100.0$ | $0.54$
$21.96$ | $3.87\times 10^{10}$ | $6.99$ | $193.1$ | $197.3$ | $4.56$ | $|$ | $3.02$ | $5.26\times 10^{9}$ | $7.02$ | $102.5$ | $103.2$ | $0.62$
$31.40$ | $5.55\times 10^{10}$ | $6.98$ | $216.9$ | $222.5$ | $6.54$ | $|$ | $18.52$ | $3.23\times 10^{10}$ | $7.02$ | $182.0$ | $185.6$ | $3.82$
Table 2: Experimental results for the $\Gamma_{x}=2$, 4, 7.3, 14.3, and 20.8 cells. Note that two points are listed per line and the data are listed in chronological order. $\Delta$ (K) | $Ra$ | $Pr$ | $Nu$ | $Nu_{\infty}$ | $\alpha\Delta$ | $|$ |  $\Delta$ (K) | $Ra$ | $Pr$ | $Nu$ | $Nu_{\infty}$ | $\alpha\Delta$
---|---|---|---|---|---|---|---|---|---|---|---|---
|  |  |  |  | ($10^{-3}$) | $|$ |  |  |  |  |  | ($10^{-3}$)
$\Gamma_{x}=2$ |  |  |  |  |  | $|$ |  |  |  |  |  |
$17.18$ | $6.67\times 10^{9}$ | $5.26$ | $111.0$ | $114.1$ | $5.41$ | $|$ | $9.39$ | $3.64\times 10^{9}$ | $5.26$ | $90.8$ | $92.6$ | $2.95$
$21.63$ | $8.46\times 10^{9}$ | $5.24$ | $119.5$ | $123.2$ | $6.84$ | $|$ | $7.12$ | $2.79\times 10^{9}$ | $5.23$ | $82.5$ | $84.0$ | $2.25$
$33.33$ | $1.28\times 10^{10}$ | $5.28$ | $135.6$ | $140.5$ | $11.0$ | $|$ | $6.02$ | $2.40\times 10^{9}$ | $5.19$ | $78.5$ | $79.7$ | $1.92$
$29.86$ | $1.17\times 10^{10}$ | $5.23$ | $130.7$ | $135.3$ | $9.45$ | $|$ | $14.62$ | $5.69\times 10^{9}$ | $5.25$ | $105.3$ | $107.9$ | $4.61$
$5.17$ | $2.01\times 10^{9}$ | $5.25$ | $74.7$ | $75.8$ | $1.63$ | $|$ | $27.15$ | $1.08\times 10^{10}$ | $5.18$ | $129.3$ | $133.8$ | $8.37$
$14.94$ | $5.85\times 10^{9}$ | $5.25$ | $106.1$ | $108.9$ | $4.57$ | $|$ | $3.82$ | $1.56\times 10^{9}$ | $5.23$ | $69.2$ | $70.1$ | $1.21$
$\Gamma_{x}=4$ |  |  |  |  |  | $|$ |  |  |  |  |  |
$17.02$ | $8.90\times 10^{8}$ | $5.25$ | $57.4$ | $59.1$ | $5.36$ | $|$ | $1.60$ | $8.33\times 10^{7}$ | $5.25$ | $29.1$ | $29.4$ | $0.50$
$35.83$ | $1.88\times 10^{9}$ | $5.25$ | $73.8$ | $76.8$ | $11.3$ | $|$ | $2.26$ | $1.18\times 10^{8}$ | $5.25$ | $31.9$ | $32.3$ | $0.71$
$32.61$ | $1.68\times 10^{9}$ | $5.29$ | $70.7$ | $73.4$ | $10.2$ | $|$ | $3.03$ | $1.59\times 10^{8}$ | $5.24$ | $34.9$ | $35.3$ | $0.96$
$29.21$ | $1.52\times 10^{9}$ | $5.26$ | $68.0$ | $70.5$ | $9.18$ | $|$ | $5.10$ | $2.67\times 10^{8}$ | $5.26$ | $40.2$ | $40.9$ | $1.61$
$25.98$ | $1.36\times 10^{9}$ | $5.25$ | $66.0$ | $68.4$ | $8.19$ | $|$ | $6.93$ | $3.62\times 10^{8}$ | $5.25$ | $43.9$ | $44.7$ | $2.18$
$22.96$ | $1.19\times 10^{9}$ | $5.27$ | $63.2$ | $65.3$ | $7.20$ | $|$ | $9.16$ | $4.83\times 10^{8}$ | $5.23$ | $47.8$ | $48.8$ | $2.90$
$19.87$ | $1.04\times 10^{9}$ | $5.24$ | $60.4$ | $62.3$ | $6.28$ | $|$ | $11.66$ | $6.06\times 10^{8}$ | $5.27$ | $51.1$ | $52.3$ | $3.66$
$14.25$ | $7.44\times 10^{8}$ | $5.26$ | $54.5$ | $56.0$ | $4.49$ | $|$ | $3.96$ | $2.07\times 10^{8}$ | $5.25$ | $37.3$ | $37.9$ | $1.25$
$1.87$ | $9.78\times 10^{7}$ | $5.26$ | $30.4$ | $30.7$ | $0.59$ | $|$ | $5.88$ | $3.08\times 10^{8}$ | $5.25$ | $41.8$ | $42.5$ | $1.85$
$3.44$ | $1.79\times 10^{8}$ | $5.26$ | $35.9$ | $36.4$ | $1.08$ |  |  |  |  |  |  |
$\Gamma_{x}=7.3$ |  |  |  |  |  | $|$ |  |  |  |  |  |
$16.16$ | $1.42\times 10^{8}$ | $5.24$ | $34.0$ | $35.1$ | $5.10$ | $|$ | $2.91$ | $2.57\times 10^{7}$ | $5.25$ | $21.1$ | $21.4$ | $0.92$
$13.53$ | $1.20\times 10^{8}$ | $5.24$ | $32.1$ | $33.1$ | $4.27$ | $|$ | $4.63$ | $4.08\times 10^{7}$ | $5.25$ | $23.6$ | $24.1$ | $1.46$
$22.21$ | $1.96\times 10^{8}$ | $5.25$ | $36.6$ | $37.9$ | $7.01$ | $|$ | $2.18$ | $1.92\times 10^{7}$ | $5.25$ | $19.4$ | $19.6$ | $0.69$
$9.87$ | $8.69\times 10^{7}$ | $5.25$ | $29.2$ | $30.0$ | $3.11$ | $|$ | $1.75$ | $1.54\times 10^{7}$ | $5.25$ | $18.6$ | $18.9$ | $0.55$
$6.55$ | $5.74\times 10^{7}$ | $5.26$ | $26.1$ | $26.7$ | $2.06$ | $|$ | $3.74$ | $3.31\times 10^{7}$ | $5.24$ | $22.3$ | $22.7$ | $1.18$
$1.51$ | $1.32\times 10^{7}$ | $5.26$ | $17.8$ | $18.0$ | $0.47$ | $|$ | $25.14$ | $2.21\times 10^{8}$ | $5.26$ | $38.4$ | $39.9$ | $7.91$
$8.36$ | $7.37\times 10^{7}$ | $5.25$ | $27.5$ | $28.2$ | $2.64$ | $|$ | $28.43$ | $2.49\times 10^{8}$ | $5.27$ | $39.5$ | $41.1$ | $8.93$
$2.04$ | $1.80\times 10^{7}$ | $5.25$ | $19.1$ | $19.3$ | $0.64$ | $|$ | $19.31$ | $1.69\times 10^{8}$ | $5.26$ | $35.0$ | $36.1$ | $6.07$
$\Gamma_{x}=14.3$ |  |  |  |  |  |  |  |  |  |  |  |
$12.35$ | $1.42\times 10^{7}$ | $5.25$ | $17.8$ | $18.4$ | $3.89$ | $|$ | $2.65$ | $3.03\times 10^{6}$ | $5.26$ | $11.6$ | $11.8$ | $0.83$
$8.96$ | $1.03\times 10^{7}$ | $5.26$ | $16.3$ | $16.8$ | $2.82$ | $|$ | $1.61$ | $1.84\times 10^{6}$ | $5.26$ | $10.6$ | $10.7$ | $0.51$
$5.98$ | $6.84\times 10^{6}$ | $5.27$ | $14.5$ | $14.9$ | $1.88$ | $|$ | $14.95$ | $1.71\times 10^{7}$ | $5.26$ | $18.6$ | $19.3$ | $4.70$
$2.00$ | $2.29\times 10^{6}$ | $5.26$ | $10.9$ | $11.1$ | $0.63$ | $|$ | $18.96$ | $2.19\times 10^{7}$ | $5.24$ | $19.8$ | $20.6$ | $5.99$
$3.44$ | $3.94\times 10^{6}$ | $5.26$ | $12.4$ | $12.6$ | $1.08$ | $|$ | $23.45$ | $2.66\times 10^{7}$ | $5.28$ | $20.8$ | $21.7$ | $7.34$
$4.28$ | $4.90\times 10^{6}$ | $5.26$ | $13.0$ | $13.3$ | $1.35$ | $|$ | $26.52$ | $3.02\times 10^{7}$ | $5.27$ | $21.5$ | $22.4$ | $8.32$
$1.39$ | $1.59\times 10^{6}$ | $5.26$ | $10.1$ | $10.3$ | $0.44$ |  |  |  |  |  |  |
$\Gamma_{x}=20.8$ |  |  |  |  |  |  |  |  |  |  |  |
$11.75$ | $4.34\times 10^{6}$ | $5.26$ | $12.9$ | $13.4$ | $3.70$ | $|$ | $3.19$ | $1.18\times 10^{6}$ | $5.26$ | $9.1$ | $9.3$ | $1.01$
$16.55$ | $6.12\times 10^{6}$ | $5.26$ | $14.2$ | $14.8$ | $5.21$ | $|$ | $5.59$ | $2.07\times 10^{6}$ | $5.26$ | $10.6$ | $10.9$ | $1.76$
$19.15$ | $7.11\times 10^{6}$ | $5.24$ | $14.8$ | $15.4$ | $6.05$ | $|$ | $1.85$ | $6.82\times 10^{5}$ | $5.27$ | $8.1$ | $8.3$ | $0.58$
$22.00$ | $8.18\times 10^{6}$ | $5.24$ | $15.3$ | $16.0$ | $6.95$ | $|$ | $3.97$ | $1.47\times 10^{6}$ | $5.26$ | $9.6$ | $9.8$ | $1.25$
$14.18$ | $5.24\times 10^{6}$ | $5.26$ | $13.4$ | $13.9$ | $4.46$ | $|$ | $2.48$ | $9.16\times 10^{5}$ | $5.26$ | $8.7$ | $8.8$ | $0.78$
$7.51$ | $2.81\times 10^{6}$ | $5.23$ | $11.3$ | $11.6$ | $2.38$ | $|$ | $1.71$ | $6.30\times 10^{5}$ | $5.26$ | $8.1$ | $8.2$ | $0.54$
$9.59$ | $3.55\times 10^{6}$ | $5.25$ | $12.0$ | $12.3$ | $3.02$ | $|$ | $25.00$ | $9.30\times 10^{6}$ | $5.24$ | $15.7$ | $16.4$ | $7.90$
$1.62$ | $5.96\times 10^{5}$ | $5.26$ | $7.9$ | $8.1$ | $0.51$ |  |  |  |  |  |  |

The measured $Nu$ with corresponding values of $\Delta$, $Ra$, and $Pr$ are
given in tables 1 and 2. The $\Gamma_{x}=1$ measurements were made at mean
temperatures of 20∘C and 30∘C, corresponding to $Pr=7.01$ and 5.45,
respectively, and the measurements for the other five values of $\Gamma_{x}$
were conducted at 31.3∘C, corresponding to $Pr=5.25$. Previous studies have
revealed that with increasing Prandtl number $Nu$ first increases, reaches its
maximum value at around $Pr\approx 4$ (depends on $Ra$), and then decreases
slightly or remains independent of $Pr$ Ahlers & Xu (2001); Grossmann & Lohse
(2001); Xia et al. (2002); Silano et al. (2010); Stevens et al. (2011a).
Therefore, within the present $Pr$ range, $Nu$ is expected to depend very
weakly on $Pr$. Indeed, as shown in figures 3 and 5, no significant
differences are observed between the $Pr=5.45$ (circles) and 7.01 (right-
triangles) results. Another source of uncertainty in the measured $Nu$ could
be the non-Boussinesq effects, as some of our measurements were made at larger
$\Delta$. Funfschilling et al. (2005) argued that the applied temperature
difference $\Delta$ should be limited to $\lesssim 15^{\circ}$C to strictly
conform to the Boussinesq conditions. However, it can be seen from figure 2
and tables 1 and 2 that some of our data have $\Delta$ much larger than
$15^{\circ}$C. As we shall see below, the measured large-$\Delta$ data show
the same trend as those of small $\Delta$. This suggests that some of our data
being not strictly Boussinesq will not change the main conclusions of the
present work. Indeed, Ahlers et al. (2006) have shown that for water as the
working fluid the non-Boussinesq effects could only slightly reduce the
measured $Nu$ by at most $1.4\%$ for $\Delta=40$ K and thus $Nu$ is rather
insensitive against even significant deviations from the Boussinesq
conditions. Hence, for completeness all data are listed in the tables and are
plotted in the figures.

Figure 3 shows $Nu/Ra^{1/3}$ as a function of $Ra$ for all six $\Gamma_{x}$.
It is seen that data points for $\Gamma_{x}=20.8$ (down-triangles) collapse
well on top of those for $\Gamma_{x}=14.3$ (diamonds) (within their overlap
$Ra-$range), which in turn collapse well on top of those for $\Gamma_{x}=7.3$
(up-triangles). This implies that all sets of data can be described by a
single curve over such a wide range of $\Gamma_{x}$, i.e., no significant
$\Gamma_{x}$-dependence is observed. As discussed in $\S$1,
$\Gamma$-dependence of $Nu$ essentially reflects the influence of flow
structures on heat transport characteristics. Indeed, using particle image
velocimetry (PIV), Xia et al. (2008) have shown in rectangular cells that the
number of the convection rolls depends systematically on the aspect ratio of
the system: only one convection roll is observed in the $\Gamma_{x}=1$ and 2
cells and the LSC breaks into (horizontally arranging) multi-roll structure
for $\Gamma_{x}$ larger than or equal to 4. Therefore, our present results
suggest that for rectangular geometry turbulent heat transport is very
insensitive to the nature and structures of the large-scale mean flows of the
system. We note that our present results are different from those obtained in
2D numerical case van der Poel et al. (2011), which were made for both
$Pr=0.7$ and 4.3. For the 2D simulation, the stable states with $n$ rolls are
found to enable larger heat transfer than those with $n+1$ rolls for
vertically arranging LSC rolls. One possible reason for this difference may be
attributed to different alignments of the LSC rolls, i.e. the aspect ratios of
the 2D simulation vary between 0.4 and 1.25 and the LSC rolls are stacked
vertically, whereas for the present geometry we have horizontally stacked
rolls Xia et al. (2008).

In figure 4, we compare the present results with those obtained in cylindrical
cells and at $Pr\approx 4.4$. We mainly consider two recent data sets: one is
from Funfschilling et al. (2005) (referred to as FBNA) and the other is from
Sun et al. (2005a) (referred to as SRSX). As our measured $Nu$ is independent
of the cells’ aspect ratio, here we do not distinguish our data for different
$\Gamma_{x}$. Note that all data in figure 4 are as measured, without the
correction for the finite plate conductivity. For the FBNA data, one sees that
the $\Gamma=6$ data of FBNA are in excellent agreement with ours, but their
small-$\Gamma$ data are a few percent larger. Nevertheless, we note that the
$\Gamma=1$ data of FBNA (down-triangles) show similar trends to ours for
$Ra\lesssim 10^{9}$ and $Ra\gtrsim 10^{10}$ and thus the small difference
between the two sets of data may be attributed to different system errors. The
SRSX data and ours are much closer. One sees that parts of the $\Gamma=5$ and
10 data of SRSX agree well with ours and the others lie slightly below our
measured $Nu$.

Figure 4: (color online). Comparison among $Nu/Ra^{-1/3}$ from the present
work ($\bigcirc$), from Funfschilling et al. (2005) ($\bigtriangledown$,
$\Gamma=1$; $\times$, $\Gamma=2$; $*$, $\Gamma=3$; $\square$, $\Gamma=6$; for
clarity their $\Gamma=1.5$ data are not shown) and from Sun et al. (2005a)
($\bigtriangleup$, $\Gamma=2$; $\vartriangleright$, $\Gamma=5$;
$\vartriangleleft$, $\Gamma=10$; $\bigcirc$, $\Gamma=20$). As our results show
that for rectangular geometry $Nu/Ra^{-1/3}$ is independent of the cells’
aspect ratio, here we do not distinguish our data for different $\Gamma_{x}$.
Note that all data are as measured, without the correction for the finite
plate conductivity.

One noticeable difference between the present results and those of FBNA and
SRSX is worthy of note. For both FBNA and SRSX the larger-$\Gamma$ results lie
consistently below those of smaller ones, i.e. $Nu$ is generally smaller for
larger $\Gamma$. This is true even for the largest-$\Gamma$ (i.e. $\Gamma=10$
and 20) data of SRSX (see figure 4). Whereas, our measured $Nu$ for all six
values of $\Gamma_{x}$ fall into a single curve, i.e. $Nu$ is essentially
independent of $\Gamma_{x}$ for our results. To understand such a difference,
we note that all data plotted in figure 4 have not been corrected for the
influence of the finite conductivity of the top and bottom plates Verzicco
(2004). In a cylindrical cell of $\Gamma=0.5$, Sun, Xi $\&$ Xia (2005 _b_)
argued that because of the finite conductivity and finite heat capacity of the
plates the azimuthal sweeping of the circulation plane of the LSC would make
heat transfer more efficient than the case when the LSC is locked in a
particular orientation. Moreover, Xi & Xia (2008) studied the azimuthal motion
of the LSC in cylindrical cells of $\Gamma=0.5$, 1.0, and 2.3 and their
results showed that the LSC’s azimuthal motion is more confined in
larger-$\Gamma$ cells. If we generalize the above two findings to large
$\Gamma$ and taken them together, a possible scenario can be achieved: the
larger-$\Gamma$ cell confines the azimuthal sweeping motion of the LSC which
in turn reduces the measured $Nu$. This scenario could be valid for
cylindrical cells due to their azimuthal symmetry. But for rectangular cells,
this is not expected to work because the rectangular geometry has already
locked the orientation of the LSC Xia et al. (2003). Therefore, our present
$\Gamma_{x}$-independent results are consistent with the lack of the azimuthal
sweeping motion of the LSC in rectangular cells. What we should stress is that
the large-$\Gamma$ cell contains multi-roll structures of the LSC du Puits et
al. (2007); Xia et al. (2008); Bailon-cuba et al. (2010) and thus the
azimuthal motion of the LSC and its influence on heat transfer should be more
complicated for large $\Gamma$. However, as we shall see in figure 6, when
corrections for the finite conductivity of the plates are made,
$\Gamma$-dependence of $Nu_{\infty}$ becomes weaker for both the FBNA and SRSX
data, which suggests that the finite plate conductivity effect is indeed a
major factor for the observed difference of the behaviors of heat transfer in
rectangular and cylindrical cells.

Figure 5: (color online). (_a_) Compensated $Nu_{\infty}/Ra^{1/3}$ on a linear
scale vs. $Ra$ on a logarithmic scale: symbols as figure 3. Here,
$Nu_{\infty}$ is calculated using Eq. (3) and the parameters $a=0.275$ and
$b=0.39$ obtained by Brown et al. (2005) in cylindrical cells. For comparison,
the uncorrected data $Nu/Ra^{1/3}$ of figure 3 are replotted in (_b_).

###  3.2 $Nu_{\infty}$ vs. $Ra$ for different $\Gamma_{x}$

Brown et al. (2005) suggested an empirical correction factor
$f(X)=1-exp[-(aX)^{b}]$, namely

$Nu=f(X)Nu_{\infty}=\\{1-exp[-(aX)^{b}]\\}Nu_{\infty}(Ra,Pr),$ (3)

to obtain estimates of the ideal Nusselt number $Nu_{\infty}$ for plates with
perfect conductivity from the measured $Nu$. Here, $X=R_{f}/R_{p}$ is the
ratio of the effective thermal resistance of the working fluid,
$R_{f}=H/(\lambda_{f}Nu)$, to the thermal resistance of the plates,
$R_{p}=e/\lambda_{p}$, $\lambda_{p}$ ($=401$ W/m K) is the conductivity of
plates (Cu), $\lambda_{f}$ ($\lambda_{f}=0.614$ W/m K for $Pr=5.25$ and
$\lambda_{f}=0.589$ W/m K for $Pr=7$) is the conductivity of water, and $e$
($=1.5$ cm) is the mean thickness of the top (the part of the top plate below
the cooling channels is used here) and bottom plates Verzicco (2004). To apply
the relation (3) to our measured $Nu$, one needs to determine the values of
$a$ and $b$. The best way to do this is to use plates of different thermal
conductivities, as was done by Brown et al. (2005), who used two sets of
plates made of Cu and Al. However, the lack of aluminum-plate measurements in
the present study prevents us to determine the values of $a$ and $b$.
Alternatively, to estimate $Nu_{\infty}$, we use the parameters $a=0.275$ and
$b=0.39$ obtained by Brown et al. (2005) in cylindrical samples of 50 cm in
diameter.

Figure 5(_a_) shows the calculated $Nu_{\infty}/Ra^{1/3}$ as a function of
$Ra$. For comparison, the uncorrected data $Nu/Ra^{1/3}$ of figure 3 are
replotted in figure 5(_b_). The corrected data in figure 5(_a_) seem to
display a small aspect ratio dependence, e.g., near $Ra=2\times 10^{8}$ the
$\Gamma_{x}=7.3$ data lie slightly above the $\Gamma_{x}=4$ data and near
$Ra=2\times 10^{9}$ the $\Gamma_{x}=4$ data lie slightly above the
$\Gamma_{x}=2$ data. We note that this dependence is consistent with the FBNA
data (see figure 6, where data points for $\Gamma=6$ lie slightly above those
for $\Gamma=3$, which in turn lie slightly above those for $\Gamma=2$).
Nevertheless, the differences of $Nu_{\infty}$ for the two
adjacent-$\Gamma_{x}$ sets of data are only a few percent, which is smaller
than or comparable with the experimental errors. Hence, we may also conclude
that no significant $\Gamma_{x}$-dependence of heat transfer is observed.

We want to emphasize that here we adopted the parameters of $a=0.275$ and
$b=0.39$ determined by Brown et al. (2005) for cylindrical samples with a
diameter of 50 cm. Brown et al. (2005) have shown experimentally that $a$ and
$b$ are independent of the aspect ratio, but vary with the diameter of the
sample. In the present study, we chose the rectangle as the geometry of the
convection cells, which can possibly have an influence on the finite plate
correction, as we have discussed at the end of $\S$ 3.1. However, without the
measurement performed with aluminum plates it is difficult to assess whether
the geometry would (significantly) influence the values of $a$ and $b$. By
using different sets of $a$ and $b$ to perform the finite conductivity
correction, we estimate that the uncertainties of $a$ and $b$ may yield a few
percent uncertainty on the obtained $Nu_{\infty}$, which is the same order of
the difference between $Nu$ and $Nu_{\infty}$. We should also stress that the
best way to estimate $Nu_{\infty}$ is to use plates with different thermal
conductivities, as was done by Brown et al. (2005) for cylindrical samples.
Therefore, new measurements with Al plates are essential for concretely
settling the problem and this will be the objective of future studies.

Figure 6: (color online). Comparison among $Nu_{\infty}/Ra^{-1/3}$ from the
present work, from Funfschilling et al. (2005), and from Sun et al. (2005a).
Note that the data have been corrected for the finite plate conductivity.

In figure 6 we directly compare our $Nu_{\infty}$ and those of FBNA and SRSX.
It is seen that our data display the similar $Ra$-dependence as the $\Gamma=1$
data of FBNA, and an excellent agreement between our data and those of SRSX
can be found for nearly all the overlap $Ra$ range. As the three sets of data
were taken independently from the samples with very different geometries, this
agreement is just remarkable.

Table 3: Fitted parameters from equations (4). $\Gamma_{x}$ |  | 1 |  | 2 |  | 4 |  | 7.3 |  | 14.3 |  | 20.8
---|---|---|---|---|---|---|---|---|---|---|---|---
$A_{(}\Gamma_{x})$ |  | 0.074 |  | 0.059 |  | 0.108 |  | 0.173 |  | 0.212 |  | 0.242
$\beta(\Gamma_{x})$ |  | 0.324 |  | 0.335 |  | 0.307 |  | 0.282 |  | 0.271 |  | 0.262

Figure 7: (color online). Local scaling exponent $\beta_{l}$ of
$Nu_{\infty}(Ra)$, determined from a power-law fit over a sliding window of
half a decade, as a function of $Ra$ for the present data (solid circles) and
for the $\Gamma=1$ data of Funfschilling et al. (2005) (open triangles).
Dashed line marks $\beta_{l}=1/3$ for reference. Note that both sets of
$\beta_{l}$ were obtained from the corrected data.

Finally, we studied the power-law relation between $Nu_{\infty}$ and $Ra$,
namely,

$Nu_{\infty}=A(\Gamma_{x})Ra^{\beta(\Gamma_{x})}.$ (4)

Table 3 displays the fitted results for the power-law relation (4) at fixed
aspect ratios. One sees that as $\Gamma_{x}$ increases in general the
prefactor $A$ increases and the scaling exponent $\beta$ decreases. Here,
$\Gamma_{x}$-dependences of $A$ and $\beta$ essentially reflect their
$Ra$-dependence as our results do not reveal significant aspect-ratio
dependence of $Nu_{\infty}$. We thus do not distinguish the data for different
$\Gamma_{x}$ and take them together to study the scaling behaviors of
$Nu_{\infty}(Ra)$. The local scaling exponent $\beta_{l}$ is obtained by a
power-law fit, $Nu_{\infty}\sim Ra^{\beta_{l}}$, to the data for
$Nu_{\infty}(Ra)$ within a sliding window that covers half a decade of $Ra$.
Figure 7 shows the results for $\beta_{l}$ as a function of $Ra$. It is seen
that $\beta_{l}$ roughly increases linearly with $\log{Ra}$ from
$\beta_{l}=0.243$ at $Ra\approx 9\times 10^{5}$ to $\beta_{l}=0.342$ at
$Ra\approx 3.6\times 10^{9}$. At small $Ra$, our results differ very much from
the DNS results for cylindrical samples of unit aspect ratio by Wagner,
Shishkina $\&$ Wagner (2012), who found that $\beta_{l}$ increases again as
$Ra$ decreases below $2\times 10^{7}$ and grows to about 0.30 at $Ra=10^{6}$
[see figure 2 of Wagner _et al._ (2012)]. The apparent differences in
$\beta_{l}$ may be explained by the different Prandtl number in the two
studies, i.e., Wagner _et al._ (2012) carried out their simulations at
$Pr=0.786$, while our measurements were made at $Pr\approx 5.45$. For higher
$Ra$, $\beta_{l}$ drops slightly and fluctuates around 0.32. The FBNA results
for $\beta_{l}$ are also displayed together with ours in figure 7 for
comparison. For $Ra\geq 10^{8}$ the two data sets both increase with $Ra$ with
ours being a little scatter. A source of uncertainty for our data could be the
non-Boussinesq effect, i.e. the FBNA data were obtained in the strictly
Boussinesq range, while some of our data are beyond the Boussinesq range (i.e.
$\Delta>15^{\circ}$C). However, as have discussed in $\S$3.1, some of our data
being not strictly Boussinesq will not change our main conclusions. What is
worthy of note is that around $Ra\approx 10^{10}$ our measured $\beta_{l}$ has
a value that is close to the value of 1/3. The exponent $\beta\approx 1/3$ was
obtained before in cylindrical cells of $\Gamma=1$ by FBNA at $Ra\approx
7\times 10^{10}$ (see open triangles in figure 7) and of $\Gamma=4$ by Niemela
& Sreenivasan (2006) for $Ra>10^{10}$. Here, our results in rectangular cells
seem to be qualitatively consistent with these findings

##  4 Conclusion

In conclusion, our high-precision measurements of $Nu$ in rectangular cells
with $(\Gamma_{x},\Gamma_{y})$ varying from $(1,0.3)$ to $(20.8,6.3)$ show
that $Nu$ is independent of the aspect ratio. This is slightly different from
the observations by both FBNA and SRSX in cylindrical cells where $Nu$ is
found to be in general a decreasing function of $\Gamma$, at least for
$\Gamma\sim 1$ and larger. Such a difference may be attributed to different
azimuthal dynamics of the large-scale circulation (LSC) and is probably a
manifestation of the finite plate conductivity effect. To make finite
conductivity corrections, an empirical correction factor
$f(X)=1-exp[-(aX)^{b}]$, together with the parameters $a=0.275$ and $b=0.39$
obtained by Brown et al. (2005) in cylindrical samples, were adopted to
estimate $Nu_{\infty}$ for plates with perfect conductivity from the measured
$Nu$. The obtaiend $Nu_{\infty}$ were found to be consistent with the FBNA and
SRSX data measured in cylindrical samples to only a few percent. The scaling
behaviors between $Nu_{\infty}$ and $Ra$ were studied for all six aspect
ratios. The local scaling exponents $\beta_{l}$ of $Nu_{\infty}\sim
Ra^{\beta_{l}}$ were calculated and found to increase with increasing $Ra$.
Around $Ra\approx 10^{10}$ our measured $\beta_{l}$ has a value that is close
to the value of $1/3$.

###### Acknowledgements.

This work was supported by the Natural Science Foundation of China (Nos.
11161160554, 11002085, and 11032007) and Shanghai Program for Innovative
Research Team in Universities.

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#  QuasinormalModes.jl: A Julia package for computing discrete eigenvalues of
second order ODEs

Lucas Timotheo Sanches

††margin:  DOI:
https://doi.org/10.21105/joss.04077https://doi.org10.21105/joss.04077 Software
• https://github.com/openjournals/joss-
reviews/issues/4077https://doi.orgReview • https://github.com/lucass-
carneiro/QuasinormalModes.jl/https://doi.orgRepository •
https://doi.org/10.5281/zenodo.6478503https://doi.orgArchive Editor:
http://pdebuyl.be/https://doi.orgPierre de Buyl
Reviewers: • https://github.com/JamieBamberhttps://doi.org@JamieBamber •
https://github.com/cescalarahttps://doi.org@cescalara Submitted: 22 September
2021
Published: 25 May 2022 License
Authors of papers retain copyright and release the work under a Creative
Commons Attribution 4.0 International License
(http://creativecommons.org/licenses/by/4.0/https://doi.orgCC BY 4.0).

## Summary

In General Relativity, when perturbing a black hole with an external field, or
particle, the system relaxes by emitting damped gravitational waves, known as
_quasinormal modes_. These are the characteristic frequencies of the black
hole and gain the _quasi_ prefix due to the fact that they have a real
frequency, which represents the oscillation frequency of the response, and an
imaginary frequency that represents the decay rate of said oscillations. In
many cases, such perturbations can be described by a second order homogeneous
ordinary differential equation (ODE) with discrete complex eigenvalues.

Determining these characteristic frequencies quickly and accurately for a
large range of models is important for many practical reasons. It has been
shown that the gravitational wave signal emitted at the final stage of the
coalescence of two compact objects is well described by quasinormal modes
(Buonanno et al., 2007; Seidel, 2004). This means that if one has access to a
database of quasinormal modes and of gravitational wave signals from
astrophysical collision events, it is possible to characterize the remnant
object using its quasinormal frequencies. Since there are many different
models that aim to describe remnants, being able to compute the quasinormal
frequencies for such models in a reliable way is paramount for confirming or
discarding them.

## Statement of need

QuasinormalModes.jl is a Julia package for computing the quasinormal modes of
any General Relativity model whose perturbation equation can be expressed as
second order homogeneous ODE. Not only that, the package can be used to
compute the discrete eigenvalues of _any_ second order homogeneous ODE (such
as the energy eigenstates of the time independent Schrödinger equation)
provided that these eigenvalues actually exist. The package features a
flexible and user friendly API where the user simply needs to provide the
coefficients of the problem ODE after incorporating boundary and asymptotic
conditions on it. The user can also choose to use machine or arbitrary
precision arithmetic for the underlying floating point operations involved and
whether or not to do computations sequentially or in parallel using threads.
The API also tries not to force any particular workflow on the users so that
they can incorporate and adapt the existing functionality on their research
pipelines without unwanted intrusions. Often user friendliness, flexibility
and performance are treated as mutually exclusive, particularly in scientific
applications. By using Julia as an implementation language, the package can
have all of theses features simultaneously.

Another important motivation for using Julia and writing this package was the
lack of generalist, free (both in the financial and license-wise sense) open
source tools that serve the same purpose. More precisely, there are tools
which are free and open source, but run on top of a proprietary paid and
expensive software framework such as the ones developped by Jansen (2017) and
Fortuna & Vega (2020), which are both excellent packages that aim to perform
the same task as QuasinormalModes.jl and can be obtained and modified freely
but, unfortunately, require the user to own a license of the proprietary
Wolfram Mathematica CAS. Furthermore, their implementations are limited to
solve problems where the eigenvalues must appear in the ODE as a polynomials
of order $p$. While this is not prohibitively restrictive to most astrophysics
problems, it can be an important limitation in other areas. There are also
packages that are free and run on top of Mathematica but are not aimed at
being general eigenvalue solvers at all, such as the one by O’Toole et al.
(2019), that can only compute modes of Schwarzschild and Kerr black holes.
Finally, the Python package by Stein (2019) is open source and free but can
only compute Kerr quasinormal modes.

QuasinormalModes.jl fills the existing gap for free, open source tools that
are able to compute discrete eigenvalues (and in particular, quasinormal
modes) efficiently for a broad class of models and problems. The package was
developed during the author’s PhD research where it is actively used for
producing novel results that shall appear in the author’s thesis. It is also
actively used in a collaborative research effort (of which the author is one
of the members) for computing quasinormal modes produced by perturbations with
integer (but different than 0) spins and semi-integer spins. These results are
being contrasted with those obtained by other methods and so far show
excellent agreement with each other and with literature results.

## Underlying algorithm

QuasinormalModes.jl internally uses a relativity new numerical method called
the Asymptotic Iteration Method (AIM). The method was introduced by Ciftci et
al. (2003) but the actual implementation used in this package is based on the
revision performed by Cho et al. (2012). The main purpose of the (AIM) is to
solve the following general linear homogeneous second order ODE:

$y^{\prime\prime}(x)-\lambda_{0}(x)y^{\prime}(x)-s_{0}(x)y(x)=0,$ (1)

where primes denote derivatives with respect to to the variable $x$ (that is
defined over some interval that is not necessarily bounded),
$\lambda_{0}(x)\neq 0$ and $s_{0}(x)\in C_{\infty}$. The method is based upon
the following theorem: let $\lambda_{0}$ and $s_{0}$ be functions of the
variable $x\in(a,b)$ that are $C_{\infty}$ on the same interval, the solution
of the differential equation, Eq. (1), has the form

$y(x)=\exp\left(-\int\alpha\mathrm{d}t\right)\times\left[C_{2}+C_{1}\int^{x}\exp\left(\int^{t}(\lambda_{0}(\tau)+2\alpha(\tau))\mathrm{d}\tau\right)\mathrm{d}t\right]$
(2)

if for some $n>0$ the condition

$\delta\equiv s_{n}\lambda_{n-1}-\lambda_{n}s_{n-1}=0$ (3)

is satisfied, where

$\displaystyle\lambda_{k}(x)\equiv$
$\displaystyle\lambda^{\prime}_{k-1}(x)+s_{k-1}(x)+\lambda_{0}(x)\lambda_{k-1}(x)$
(4) $\displaystyle s_{k}(x)\equiv$ $\displaystyle
s^{\prime}_{k-1}(x)+s_{0}\lambda_{k-1}(x)$ (5)

where $k$ is an integer that ranges from $1$ to $n$.

Provided that the theorem is satisfied we can find both the eigenvalues and
eigenvectors of the second order ODE using, respectively, Eq. (3) and Eq. (2).
Due to the recursive nature of Eq.(4) and Eq.(5), to compute the quantization
condition, Eq.(3), using $n$ iterations the $n$-th derivatives of
$\lambda_{0}$ and $s_{0}$ must be computed multiple times. To address this
issue, Cho et al. (2012) proposed the use of a Taylor expansion of both
$\lambda$ and $s$ around a point $\xi$ where the AIM is to be performed. This
improved version is implemented in QuasinormalModes.jl

## Benchmark

To show QuasinormalModes.jl in action, this section provides a simple
benchmark where the fundamental quasinormal mode ($n=\ell=m=s=0$) of a
Schwarzschild black hole was computed using 16 threads on a Intel(R) Core(TM)
i9-7900X @ 3.30GHz CPU with 256 bit precision floating point numbers.

Figure 1: Error convergence for the fundamental Schwarzschild quasinormal mode
as a function of the number of AIM iterations. Figure 2: Time taken to compute
the fundamental Schwarzschild quasinormal mode as a function of the number of
AIM iterations on a logarithmic scale.

In order to quantify the rate at which the method converges to the correct
results, the error measure $\varepsilon$ was defined as follows: given the
computed real and imaginary quasinormal frequencies, denoted respectively by
$\omega_{R}$ and $\omega_{I}$, and the reference frequencies given by Berti et
al. (2009), denoted respectively as $\overline{\omega}_{R}$ and
$\overline{\omega}_{I}$ we have that
$|\varepsilon|=|\omega_{R,I}-\overline{\omega}_{R,I}|$. In Figure 1
$\varepsilon$ is plotted as a function of the number of iterations for both
the real and imaginary frequencies. We can see that, as the number of
iterations increases, the error in the computed values rapidly decreases.

In Figure 2, the time required to perform a certain number of iterations is
plotted in logarithmic scale. Each data point is the arithmetic mean time
obtained after 10 runs of the algorithm for a given number of iterations. The
time measurement takes care to exclude the overhead induced at “startup” due
to Julia’s JIT compilation. Even though the time taken to perform a certain
number of iterations increases with a power law, the time scale required to
achieve highly accurate results is still around 10s. This time would be even
smaller if one choose to use built-in floating point types instead of
arbitrary precision numbers.

## Acknowledgements

I would like to thank my PhD advisor, Dr. Maurício Richartz, for introducing
me to the AIM and providing helpful discussions and comments about the
method’s inner workings as well as helpful revision and comments about the
paper’s structure. I would also like to thank Dr. Iara Ota for the helpful
comments, discussions and revision of this paper. I would also like to thank
Dr. Erik Schnetter, Soham Mukherjee and Stamatis Vretinaris for the help and
discussions regarding root finding methods and to Dr. Schnetter for directly
contributing documentation typo corrections and suggestions for improving the
package’s overall presentation and documentation. This research was supported
by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES,
Brazil) - Finance Code 001.

## References

Berti, E., Cardoso, V., & Starinets, A. O. (2009). Quasinormal modes of black
holes and black branes. _Class. Quant. Grav._ , _26_ , 163001\.
https://doi.org/10.1088/0264-9381/26/16/163001

Buonanno, A., Cook, G. B., & Pretorius, F. (2007). Inspiral, merger and ring-
down of equal-mass black-hole binaries. _Phys. Rev. D_ , _75_ , 124018.
https://doi.org/10.1103/PhysRevD.75.124018

Cho, H. T., Cornell, A. S., Doukas, J., Huang, T. R., & Naylor, W. (2012). A
New Approach to Black Hole Quasinormal Modes: A Review of the Asymptotic
Iteration Method. _Adv. Math. Phys._ , _2012_ , 281705\.
https://doi.org/10.1155/2012/281705

Ciftci, H., Hall, R. L., & Saad, N. (2003). Asymptotic iteration method for
eigenvalue problems. _Journal of Physics A: Mathematical and General_ ,
_36_(47), 11807–11816. https://doi.org/10.1088/0305-4470/36/47/008

Fortuna, S., & Vega, I. (2020). _Bernstein spectral method for quasinormal
modes and other eigenvalue problems_. http://arxiv.org/abs/2003.06232

Jansen, A. (2017). Overdamped modes in Schwarzschild-de Sitter and a
Mathematica package for the numerical computation of quasinormal modes. _Eur.
Phys. J. Plus_ , _132_(12), 546. https://doi.org/10.1140/epjp/i2017-11825-9

O’Toole, C., Macedo, R., Stratton, T., & Wardell, B. (2019). QuasiNormalModes.
In _GitHub repository_. GitHub.
https://github.com/BlackHolePerturbationToolkit/QuasiNormalModes

Seidel, E. (2004). Nonlinear impact of perturbation theory on numerical
relativity. _Classical and Quantum Gravity_ , _21_(3), S339–S349.
https://doi.org/10.1088/0264-9381/21/3/021

Stein, L. C. (2019). qnm: A Python package for calculating Kerr quasinormal
modes, separation constants, and spherical-spheroidal mixing coefficients.
_Journal of Open Source Software_ , _4_(42), 1683\.
https://doi.org/10.21105/joss.01683