# Statistical and geometrical properties of thermal plumes in turbulent
Rayleigh-Bénard convection

Quan Zhou1,2 and Ke-Qing Xia1 1Department of Physics, The Chinese University
of Hong Kong, Shatin, Hong Kong, China
2Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University,
Shanghai 200072, China<EMAIL_ADDRESS>
###### Abstract

We present a systematic experimental study of geometric and statistical
properties of thermal plumes in turbulent Rayleigh-Bénard convection using the
thermochromic-liquid-crystal (TLC) technique. The experiments were performed
in three water-filled cylindrical convection cells with aspect ratios 2, 1,
and 0.5 and over the Rayleigh-number range $5\times 10^{7}\leq Ra\leq
10^{11}$. TLC thermal images of horizontal plane cuts at various depths below
the top plate were acquired. Three-dimensional images of thermal plumes were
then reconstructed from the two-dimensional slices of the temperature field.
The results show that the often-called sheetlike plumes are really one-
dimensional structures and may be called rodlike plumes. We find that the
number densities for both sheetlike/rodlike and mushroomlike plumes have
power-law dependence on $Ra$ with scaling exponents of $\sim 0.3$, which is
close to that between the Nusselt number $Nu$ and $Ra$. This result suggests
that it is the plume number that primarily determines the scaling exponent of
the $Nu$-$Ra$ scaling relation. The evolution of the aspect ratio of
sheetlike/rodlike plumes reveals that as $Ra$ increases the plume geometry
changes from more-elongated to less-elongated. Our study of the plume area
fraction (fraction of coverage over the surface of the plate) further reveals
that the increased plume numbers with $Ra$ mainly comes from increased plume
emission, rather than fragmentation of plumes. In addition, the area,
perimeter, and the shape complexity of the two-dimensional horizontal cuts of
sheetlike/rodlike plumes were studied and all are found to obey log-normal
distributions.

##  1 Introduction

As an important class of turbulent flows, turbulent thermal convection is
ubiquitous in nature, ranging from those in the planets and stars, in the
Earth’s mantle and its outer core, and in the atmosphere and oceans to the
convection in heat transport and mixing in engineering applications. Turbulent
Rayleigh-Bénard (RB) convection, a fluid layer sandwiched between two parallel
plates and heated from below, has long been used as a model system to study
natural convections [1, 2, 3]. Two of the important issues in the study of
turbulent RB convection are heat transport and coherent structures such as
thermal plumes. In the first one, one tries to understand how heat transported
upwards across the fluid layer, which is characterized by the Nusselt number
$Nu=J/(\chi\bigtriangleup/H)$, depends on the turbulent intensity, which is
characterized by the Rayleigh number $Ra=\alpha g\Delta H^{3}/\nu\kappa$.
Here, $\Delta$ is the applied temperature difference across the fluid layer,
$g$ is the gravitational acceleration, and $\alpha$, $\nu$, $\chi$ and
$\kappa$ are, respectively, the volume expansion coefficient, kinematic
viscosity, thermal conductivity, and thermal diffusivity of the convecting
fluid. In many theories and experiments, it is often assumed that there is a
simple power law relation between $Nu$ and $Ra$, i.e. $Nu\sim Ra^{\beta}$. In
the second issue one wants to understand the statistical and geometrical
properties of thermal plumes, which is a localized thermal structure and has
been shown to play a key role in many natural phenomena and engineering
applications, such as in mantle convection where mantle plumes just below
Earth’s crust are responsible for the formation of volcanoes (see, for
example, Ref. [4, 5]), in nuclear explosions and stellar convection where
plumes dominate both the dynamics and the energy transport (see, for example,
Ref. [6]). The two issues are not independent of each other. It has been shown
recently that for turbulent RB system most heat is carried and transported by
thermal plumes [7, 8, 9]. Accordingly, it remains a challenge to establish a
quantitative relationship between thermal plumes and heat transport. Although
a theoretical effort [10] has recently focused on this connection, few such
experimental studies have been made.

Thermal plumes generated by a localized hot/cold spot have a well organized
structure, which consists of a mushroom cap with sharp temperature gradient
and a stem that is relatively diffusive [11, 12]. Plumes with such a structure
are often referred to as mushroomlike plumes, which are also observed in
turbulent RBC when viewed from the side [13, 14, 15, 11, 16, 17, 18, 19, 7,
20]. By extracting plumes from temperature time series measured locally near
the cell sidewall, Zhou and Xia [21, 22] showed that the size of mushroomlike
plumes obey log-normal statistics. However, the morphology of the plumes is
totally different when one observes from above (or below): thermal plume is
extended in one horizontal direction but concentrated in the orthogonal
horizontal direction [23, 13, 15, 24, 25, 26, 27, 28, 29, 30, 31]. Plumes with
such a structure are often assumed to have significant vertical extent and
thus are called sheetlike plumes. Puthenveettil and Arakeri [27] studied near-
wall structures in turbulent natural convection driven by concentration
differences across a membrane and found that the plume spacings show a common
log-normal probability density function (PDF). Zhou _et al._ [29] further
revealed that both the area and the heat content of sheetlike plumes are log-
normal distributed. Shishkina and Wagner [32] investigated quantitatively
geometric properties of sheetlike plumes using direct numerical simulations.
However, systematical experimental studies of geometrical structures of
sheetlike plumes are still missing. Recently, Funfschilling _et al._ [30]
suggested that when viewed from above thermal plumes near the two plates
should be better referred to as linelike plumes, as the vertical extent of
these structures do not seem to be established and it appears more likely that
they are one dimensional excitations in the marginally-stable boundary layers.

As two different configurations of thermal plumes coexist simultaneously in
turbulent RB system, it is natural to ask how sheetlike and mushroomlike
plumes transform from each other. By using thermochromic-liquid-crystal (TLC)
technique, Zhou _et al._ [29] showed, in a cylindrical cell with unity aspect
ratio ($\Gamma=D/H$ with $D$ and $H$ as the inner diameter and the height of
the convection cell, respectively.), that hot fluids (plumes) move upwards,
imping on the top plate from below, then spread horizontally along the top
plate and form waves or sheetlike plumes. As they travel horizontally along
the plate, these sheetlike plumes collide with each other or with the
sidewall, convolute and form swirls. As these swirls are cooler than the bulk
fluid, they move downwards, merge and cluster together. By the symmetry of the
system, the same process is expected for the morphological evolution of
thermal plumes occurring near the bottom plate. However, in Ref. [29] the
morphological evolution was visualized only at a single Rayleigh number
($Ra=2.0\times 10^{9}$), and it is not clear whether this process is universal
for much higher $Ra$ or for cells with different aspect ratios. The
understanding of this evolution process is of great importance, as the
characterization of coherent structures is essential to the understanding of
turbulent flows in many systems.

In the present paper, we report new experiments of the temperature and
velocity fields measured at varying depths from the top plate over the
Rayleigh number range $5\times 10^{7}\leq Ra\leq 10^{11}$ and the Prandtl
number range $4.1\leq Pr=\nu/\kappa\leq 5.3$ and in three water-filled
cylindrical sapphire cells with aspect ratios 0.5, 1, and 2. We present
quantitative results on the relationship between plume number density and $Nu$
and results that relate the evolution of the plume morphology to the heat
transport. The remainder of the paper is organized as follows. We give a
detailed description about the experimental setup and data analysis method in
section 2. In section 3, we study the $z$\- and $Ra$-dependence of plume
number and discuss the relationship between thermal plumes and the $Nu$-$Ra$
scaling. Section 4 presents study on the geometric properties of sheetlike
plumes, which are mainly based on data from the aspect ratio 1 cell. We
summarize our findings and conclude in section 5.

##  2 Experimental setup and procedures

###  2.1 The convection cell and the experimental parameters

Figure 1: A schematic drawing of the convection cell used in the experiments.
The top and bottom plates are made of sapphire so that the flow in a
horizontal plane can be visualized and captured. A: the cooling chamber cover,
B: the top sapphire plate, C: the Plexiglas sidewall, D: the bottom sapphire
plate, E: the heating chamber cover, F: thermistors, G: nozzles for cooling or
heating water, H: stainless steel rings as part of the heating and cooling
chambers, I: nozzle for filling fluid into the cell, J: nozzle for letting air
out of the cell, K: a square-shaped glass jacket.

The experiments were carried out in three cylindrical sapphire cells [33, 29].
The schematic diagram of the cells is shown in figure 1. To capture and study
the horizontal temperature and velocity fields from top, two sapphire discs
(Almaz Optics, Inc) with diameter $19.5$ cm and thickness $5.0$ mm were chosen
as the top and bottom plates for their good thermal conductivity ($35.1$ W/mK
at 300 K) compared with other transparent materials. Two chambers, constructed
by stainless steel rings $H$ and Plexiglas discs $A$ and $E$, are used to heat
the bottom plate and cool the top plate. Each chamber is connected to a
separate refrigerated circulator by four nozzles, water flows into the chamber
by two nozzles in two opposite direction and leaves from the other two nozzles
perpendicular to the inlets. The sidewall of the cell is a vertical tube made
of Plexiglas ($C$) with inner diameter $D=18.5$ cm and wall thickness $8$ mm,
respectively. The separations between the top and bottom plates $H$ are $9.3$,
$18.5$, and $37$ cm so that the aspect ratios of the cells are respectively
$\Gamma=2$, 1, and 1/2. Draining ($I$) and filling ($J$) tubes are fitted on
the Plexiglas tube at a distance of $1.5$ cm from the top and bottom plates. A
square-shaped jacket ($K$) made of flat glass plates and filled with water is
fitted to the outside of the sidewall [20], which greatly reduces the spatial
variation in the intensity of the white lightsheet caused by the curvature of
the cylindrical sidewall. Four rubber O-rings (not shown in the figure) are
placed between the two sapphire plates and the steel rings and between the
rings and the two Plexiglas discs to avoid fluid leakage. Four thermistors $F$
(model 44031, Omega Engineering Inc.) are used to measure the temperature
difference between the two sapphire plates. To keep good contact between the
plates and the thermistors, the thermistors are wrapped by the heat transfer
compound (HTC10s from Electrolube Limited). It is found that the measured
relative temperature difference between the two thermistors in the same plate
is less than $3\%$ of that across the convection cell for both plates and for
all $Ra$ investigated. This indicates the uniform distribution of the
temperature across the horizontal plates.

###  2.2 Liquid crystal measurements

Figure 2: The optical setup for thermochromatic liquid crystal visualization
and measurements: $S$, halogen lamp; $L1$, concave mirror; $L2$ and $L3$,
condensing lenses; $L4$, diverging cylindrical lens.

The visualization technique employing thermochromatic liquid crystal (TLC)
particles has been widely used and documented in fluid visualization
experiments and was used in the present work to visualize the temperature and
velocity fields in horizontal fluid layers of varying depth from the cell’s
top plate. Two types of TLC microspheres (Hallcrest, Ltd.) were used in the
experiments. One type (model R29C4W) was used for low-$\Delta$ experiments
with the mean bulk temperature $T_{0}=30$ ∘C and $Pr=5.3$, and the other type
(model R40C5W) was used for high-$\Delta$ measurements with $T_{0}=42$ ∘C and
$Pr=4.1$. Both these particles have a mean diameter of $50$ $\mu$m and density
of $1.03-1.05$ g/cm3, and were suspended in the convection fluid in very low
concentrations (about $0.01\%$ by weight), at which the influence of TLC
particles on the fluid can be neglected. The peak wavelength of light
scattered by these particles changes from red to green and then to blue within
a temperature window of 4 ∘C from about 29 to 33 ∘C for R29C4W, and of 5 ∘C
from about 40 to 45 ∘C for R40C5W. In the experiments, the mean bulk
temperature was set to 30 ∘C for R29C4W (Pr $=5.3$) and to 42 ∘C for R40C5W
(Pr $=4.1$), so that the background fluid appears blue and the red and green
regions correspond to cold fluid, i.e. cold plumes. Figure 2 shows a schematic
diagram of the optical setup for the experiments. A halogen photo optic lamp
($S$) with a power of 650 W was used as the light source. One concave mirror
$L1$ and two condensing lenses $L2$ and $L3$ were used to collect the light
from $S$ and focus it onto the central section of the cell. A horizontal sheet
of white light, generated by a diverging cylindrical lens $L4$ and then
projected onto an adjustable slit, passed through the cell parallel to the top
plate. The thickness of the lightsheet inside the cell is approximately 3 mm.
A Nikon D1X camera, with a resolution $2000\times 1312$ pixels and 24 bit
dynamic range, was placed on the top of the cell to take photographs of the
TLC microspheres. With short camera exposure time ($0.02$ s), the captured
photographs give the instantaneous temperature field, and with long camera
exposure time (0.77s) they will in addition show the the trajectories of the
particles.

Figure 3: Images of TLC microspheres with the camera exposure time of 0.02 s
taken at $2$ mm from the top plate at (a) $Ra=1.1\times 10^{9}$, (b)
$3.0\times 10^{9}$, and (c) $1.0\times 10^{10}$. (d), (e) and (f), which
correspond to (a), (b) and (c), respectively, are extracted sheetlike plumes
with background in the original images removed.

To study thermal plume properties quantitatively, we counted plume numbers and
extracted sheetlike plumes. For sheetlike plumes, we took $150$ to $300$
consecutive images (at $z=2$ mm and with camera exposure time of $0.02$ s) at
$30$ to $60$ s intervals for a given $Ra$, so that two successive images in
each sequence are statistically independent. A cold sheetlike plume is
extracted by first manually drawing a contour around its perimeter and then
using a software to collect all pixels enclosed by the contour. A program is
then used to calculate the perimeter $P$ and the area $A$ of each extracted
plume. To ensure that plumes are identified correctly the operator uses
knowledge gained from viewing movies of plume motions. A total of $3000$ to
$6000$ plumes are identified from the $150$ to $300$ images, which are not
large numbers but should be indicative of the statistical properties. Figures
3(d)-(f) show three examples of extracted plumes with background removed,
which correspond to images of TLC microsperes in figures 3(a)-(c),
respectively, for three different values of $Ra$ obtained from the unity-
aspect-ratio cell. For mushroomlike plumes, we took sequences of images at
varying depth $z$ from the top plate and with camera exposure time of $0.77$ s
at $30$ s intervals for each $Ra$. Each sequence consists $200$ images for a
given depth. A cold mushroomlike plume was identified as an object with
nonzero vorticity and with temperature (color) much lower than that of the
background fluid [29] (see, e.g., figure 4(b)). As mushroomlike plumes are
partly entangled together, they cannot be separated completely. Nevertheless,
we can identify them individually as swirls with cold temperature and thus
count their numbers. The cold sheetlike plumes were also identified and
counted from these images, as objects with a linelike shape and lower
temperature.

Figure 4: Images of TLC microspheres with the camera exposure time of 0.77 s
taken at (a) $2$ mm and (b) $2$ cm from the top plate at $Ra=1.0\times
10^{9}$.

##  3 Plume number statistics

###  3.1 Morphological evolution of thermal plumes

Figure 5: The process of morphological evolution between sheetlike and
mushroom plumes.

The process of morphological evolution between sheetlike and mushroomlike
plumes has been revealed by Zhou _et al._ [29] in a cylindrical cell. Here,
the same process was also observed for all values of $Ra$ and $\Gamma$
investigated in the experiment. Figure 4(a), taken at 2 mm from the top plate,
shows how this evolution comes about. One sees that near the top plate the
motion of TLC microspheres appear to emanate from certain regions or “sources”
with bluish color, suggesting that hot fluids (plumes) are moving upwards,
impinging on and spreading horizontally along the top plate. Along the
particle traces, the color turns from blue to green and red, implying that the
wave fronts are cooled down gradually by the top plate (and the top thermal
boundary layer) as they spread. As they travel along the plate’s surface,
sheetlike plumes collide with each other or with the sidewall. As different
plumes carry momenta in different directions, they merge, convolute and form
swirls (hence generating vorticity). As these swirls are cooler than the bulk
fluid, they move downwards, merge and cluster together (see, e.g, figure
4(b)). We note that, by symmetry of the system, the same morphological
evolution of thermal plumes should occur near the bottom plate. The physical
picture of this evolution process is illustrated in figure 5.

###  3.2 Depth-dependent properties

Figure 6: Mean numbers of (a), (c), and (e) (cold) sheetlike plumes
$N_{pl}^{sheet}$ and (b), (d), and (f) (cold) mushroomlike plumes
$N_{pl}^{mush}$ as functions of the distance $z$ for all measured $Ra$. (a)
and (b) for $\Gamma=2$, (c) and (d) for $\Gamma=1$, and (e) and (f) for
$\Gamma=0.5$.

Because of the collision and convolution of sheetlike plumes and the
transformation from sheetlike to mushroomlike ones, it is expected that the
number of sheetlike plumes should decrease from the top plate, while that of
mushroomlike ones should increase, and this is indeed the case as shown in
figure 6. Figures 6(a), (c), and (e) show the mean numbers of (cold) sheetlike
plumes $N_{pl}^{sheet}$ versus the normalized height $z/\delta_{th}$ obtained
from the cells with $\Gamma=2$, 1, and 0.5, respectively. (Here, $\delta_{th}$
is the thickness of thermal boundary layer, which was measured in [34, 35].)
One sees that $N_{pl}^{sheet}$ decays quickly away from the top plate and the
behavior is similar for all $Ra$ studied. Beyond the depth $z\approx
15\delta_{th}$, one can hardly identify any sheetlike plumes from the acquired
images. Note that, in addition to transforming into mushroomlike plumes, the
rapid reduction of $N_{pl}^{sheet}$ with depth could also be attributed to the
limited vertical extent of sheetlike plumes themselves, i.e. plumes, whose
vertical extents are shorter than a certain height $h$, could not be captured
by the images that are taken farther than $h$ from the top plate. We further
note that the $z$-dependence of plume numbers $N_{pl}^{sheet}$ may be well
described by a decreasing exponential function, i.e.
$N_{pl}^{sheet}=N_{0}e^{-b_{r}z/\delta_{th}}$ with fitting parameters $N_{0}$
and $b_{r}$ for nearly all measured $Ra$. Hence $z_{e}=1/b_{r}$ may be used as
a typical height of vertical extent of sheetlike plumes. Figure 7(a) shows the
scaled height $z_{e}/\delta_{th}$ as a function of $Ra$. It is seen that
$z_{e}/\delta_{th}$ increases with $Ra$ for the $\Gamma=2$ and 0.5 cells,
while keeps nearly constant for the $\Gamma=1$ cell. We can not judge the
significance of the different behaviors of $z_{e}/\delta_{th}$ varying with
$Ra$ in cells with different $\Gamma$. However, $z_{e}$ varies only in the
range of a few times the thickness of thermal boundary layer
($2\delta_{th}<z_{e}<5\delta_{th}$). This suggests that the vertical extent of
the so-called sheetlike plumes does not have enough spatial extension in the
vertical direction to form sheets.

Figure 7: (a) $z_{e}/\delta_{th}$ and (b) $z_{C}/\delta_{th}$ vs $Ra$ for
$\Gamma=2$ (up-triangles), 1 (circles), and 0.5 (down-triangles). The dashed
line in (b) marks the mean value of $z_{c}$ for $Ra>3\times 10^{8}$.

For mushroomlike plumes, the mean numbers of (cold) mushroomlike plumes
$N_{pl}^{mush}$ as a function of the scaled depth $z/H$ is shown in figures
6(b), (d), and (f). With increasing $z$, $N_{pl}^{mush}$ for all $Ra$ first
increases rapidly, then decreases and finally remains approximately constant
for positions $z>0.2H$. When $N_{pl}^{mush}$ first increases with $z$, it
crosses over with $N_{pl}^{sheet}$ at some crossover-depth $z_{C}$. The
normalized crossover depths $z_{C}$ are shown in figure 7(b). It is seen that
$z_{c}\approx 2.5\delta_{th}$ when $Ra\leq 3\times 10^{8}$. For $Ra>3\times
10^{8}$, although the data points seem to be somewhat scattered, they exhibit
no clear dependence on $Ra$ and have a mean value of $\sim 4.84\delta_{th}$.
This implies that the transformation occurs within a region, whose height is
only associated with the thermal boundary layer thickness. If one takes this
value as the typical vertical extent of sheetlike plumes, these results again
suggest that the vertical extent of what is called sheetlike plumes is only
the order of several $\delta_{th}$, not much larger than it, and thus not
large enough to have a sheetlike shape. This will be further discussed in
section 4.4. After $N_{pl}^{mush}$ attains its maximum value around $z=z_{p}$,
it then drops sharply beyond the depth $z\simeq 0.2H$, because of the mixing,
merging, and clustering of mushroomlike plumes. As found by Zhou _et al._
[29], this region corresponds to the region of full width at half maximum of
vertical vorticity profile and can be used as a quantitative definition and
measure of the mixing zone [36, 21]. It is further found that the peak
position in the profiles of $N_{pl}^{mush}$, $z_{p}$, has no obvious
$Ra$-dependence for each $\Gamma$, but decreases slightly from $0.08H$ for
$\Gamma=2$ to $0.03H$ for $\Gamma=0.5$.

###  3.3 The Rayleigh number dependency

Table 1: The fitted parameters $\alpha_{s}$, $\beta_{s}$, $\alpha_{m}$, and $\beta_{m}$ for all three cells. $\Gamma$ | $\alpha_{s}$ | $\beta_{s}$ | $\alpha_{m}$ | $\beta_{m}$
---|---|---|---|---
0.5 | 1.2 | $0.29\pm 0.03$ | 0.45 | $0.31\pm 0.03$
1 | 1.5 | $0.28\pm 0.03$ | 0.37 | $0.33\pm 0.03$
2 | 1.6 | $0.30\pm 0.03$ | 1.86 | $0.28\pm 0.03$
Figure 8: $Ra$-dependency of sheetlike plume number density
$N_{pl}^{sheet}/(\pi D^{2}/4)$ (dark-green symbols), counted just outside the
thermal boundary layer, and mushroomlike plume numbers $N_{pl}^{mush}/(\pi
D^{2}/4)$ (red symbols), counted at the peak of the height-profile of
$N_{pl}^{mush}$, for the $\Gamma=2$ (up-triangles), 1 (circles), and 0.5
(down-triangles) cells. For both $N_{pl}^{sheet}$ and $N_{pl}^{mush}$, the
$\Gamma=2$ (0.5) data have been shifted down (up), so that they agree with
corresponding data for $\Gamma=1$ in their overlap regions of $Ra$. Solids
lines are power-law fittings to the data shown in the same color. The fitted
scaling exponents are $0.29\pm 0.03$ and $0.32\pm 0.03$ for $N_{pl}^{sheet}$
and $N_{pl}^{mush}$, respectively.

We next focus on the $Ra$-dependency of plume number density. For sheetlike
plumes, the number was counted just outside the thermal boundary layer and the
counted number, $N_{pl}^{sheet}$, for each cell can be well described by a
power law $N_{pl}^{sheet}/(\pi
D^{2}/4)=\alpha_{s}(\Gamma)Ra^{\beta_{s}(\Gamma)}$, with two fitting
parameters $\alpha_{s}$ and $\beta_{s}$ as functions of $\Gamma$. Here, the
fitted values of $\alpha_{s}$ and $\beta_{s}$ for all three aspect ratios are
listed in table 1. It is seen that $\alpha_{s}(\Gamma)$ varies with the cell’s
aspect ratio $\Gamma$ while $\beta_{s}(\Gamma)$ is approximately the same for
all three cells. As the range of $Ra$ is limited for each aspect ratio, we
shifted the plume number densities for the $\Gamma=2$ and 0.5 cells to agree
with data for the $\Gamma=1$ cell in their respective overlapping range of
$Ra$. We then fitted a single power-law to the combined data set from the
three cells that span almost three decades of $Ra$. Figure 8 shows the shifted
numbers, normalized by the area of the top plate $\pi D^{2}/4$, as well as the
plume number from the $\Gamma=1$ cell as dark-green symbols. It is seen that a
single power law

$N_{pl}^{sheet}/(\pi D^{2}/4)=\alpha_{s}Ra^{\beta_{s}}\mbox{\ \ with\ \
}\alpha_{s}=1.4\mbox{\ \ and\ \ }\beta_{s}=0.29\pm 0.03$ (1)

can be used to well describe the data from all three cells.

For mushroomlike plumes, one sees from figures 6(b), (d), and (f) that there
is a peak on the height-profile of mushroomlike plume number $N_{pl}^{mush}$
for all three cells and for all measured $Ra$. We use the value at the peak of
the height-profile of mushroomlike plume number to examine the $Ra$-dependence
of $N_{pl}^{mush}$. It is found that a power law relation $N_{pl}^{mush}/(\pi
D^{2}/4)=\alpha_{m}(\Gamma)Ra^{\beta_{m}(\Gamma)}$ can also be used to
describe the $Ra$-dependence of $N_{pl}^{mush}$. Again, the fitted
$\beta_{m}(\Gamma)$ is nearly $\Gamma$-independent (see table 1). Figure 8
shows the $Ra$-dependence of $N_{pl}^{mush}/(\pi D^{2}/4)$ (red symbols).
Here, the $\Gamma=2$ and 0.5 data have also been shifted to agree with the
$\Gamma=1$ in their respective overlap regions of $Ra$. It is seen that
$N_{pl}^{mush}$ shows similar trend with $Ra$ as $N_{pl}^{sheet}$. Again, the
relationship between $N_{pl}^{mush}$ and $Ra$ can be well represented by
power-law fits

$N_{pl}^{mush}/(\pi D^{2}/4)=\alpha_{m}Ra^{\beta_{m}}\mbox{\ \ with\ \
}\alpha_{m}=0.41\mbox{\ \ and\ \ }\beta_{m}=0.32\pm 0.03.$ (2)

To make a quantitative connection between thermal plumes and the heat flux, we
use the previous experimental finding from both Eularian and Lagrangian
measurements that the heat flux are mainly carried by thermal plumes [7, 9].
As mushroomlike plumes are evolved morphologically from sheetlike plumes [29],
the total heat flux carried by either type should be the same, i.e.,

$Nu\simeq N_{pl}^{sheet}F_{pl}^{sheet}\simeq N_{pl}^{mush}F_{pl}^{mush}\sim
Ra^{\beta}.$ (3)

Here, $F_{pl}^{sheet}$ and $F_{pl}^{mush}$ are the mean heat flux carried by
individual sheetlike and mushroomlike plumes, respectively, and $\beta\simeq
0.3\pm 0.02$ in the $Ra$ and $Pr$ ranges of the experiment [37, 38]. Recall
that the $Ra$-dependence of $N_{pl}^{sheet}$ [Eq. (1)] and
$(N_{pl}^{mush})_{peak}$ [Eq. (2)] discussed above, one sees that the scaling
exponents of the $Nu$-$Ra$ relation and of the $Ra$-dependence of plume
numbers are the same within experimental uncertainty. This indicates that the
heat flux transported by individual plumes is nearly independent of the
turbulent intensity, i.e. $Ra$, and hence the $Nu$-$Ra$ scaling relation is
determined primarily by the number of thermal plumes.

##  4 Geometric properties of sheetlike plumes

###  4.1 Aspect ratio of sheetlike plumes

Figure 9: $Ra$-dependence of the mean aspect ratio of sheetlike plumes
$\langle\gamma_{pl}^{sheet}\rangle$ vs. $Ra$ in a log-log plot. Figure 10:
$Ra$-dependence of (a) the mean normalized area $\langle 4A/\pi D^{2}\rangle$
and (b) the mean normalized perimeter $\langle P/D\rangle$ of sheetlike
plumes.

A striking feature that one can observe from figure 3 is that, as the Rayleigh
number is increased, the morphology or geometry of sheetlike plumes change
from more elongated shape to less elongated and more fragmented. The extracted
horizontal cuts of sheetlike plumes may be characterized by a typical length
$l$ and a typical width $w$. With increasing $Ra$, the length of sheetlike
plumes seems to decrease, while the plume’s width seems to increase. To
describe these geometric properties quantitatively, we define the aspect ratio
of sheetlike plumes, $\gamma_{pl}^{sheet}$, as

$\gamma_{pl}^{sheet}=l/w\mbox{,\ with\
}\left\\{\begin{array}[]{ll}P=2(l+w),\\\\[8.0pt] A=lw,\\\\[8.0pt] l\geq
w.\end{array}\right.$ (4)

Figure 9 shows the $Ra$-dependency of the mean aspect ratio of sheetlike
plumes $\langle\gamma_{pl}^{sheet}\rangle$ in a log-log plot. It is seen that
$\langle\gamma_{pl}^{sheet}\rangle$ decreases with increasing $Ra$ and the
relation can be described by a power-law with a scaling exponent $-0.23$. We
note that $\langle\gamma_{pl}^{sheet}\rangle$ can also be described by a
logarithmic function of $Ra$ and we cannot definitively conclude which one,
between power-law and logarithm, is a better choice. Figures 10(a) and (b)
show $Ra$-dependence of (a) the mean normalized area $\langle 4A/\pi
D^{2}\rangle$ and (b) the mean normalized perimeter $\langle P/D\rangle$ of
sheetlike plumes. Again, these two quantities are found to decrease with
increasing $Ra$. All these behaviors are well illustrated in figure 3, and
suggest that when the flow becomes more turbulent, the increased mixing,
collision, and convolution can more easily fragment large-sized sheetlike
plumes into smaller ones.

Figure 11: The area fraction $f_{pl}$ of sheetlike plumes (area coverage of
plumes over the top plate) as a function of $Ra$ in a log-log plot. The solid
line represents a power-law fit to the data.

Figure 11 shows the plume area fraction $f_{pl}$ as a function of the Rayleigh
number in a log-log plot. It is seen that $f_{pl}$ increases with increasing
$Ra$ and a power-law function with a scaling exponent 0.23 can be used to
describe well the relationship between $f_{pl}$ and $Ra$. Using the
shadowgraph technique, [30] studied $Ra$-dependence of the plume area
fraction. Although the scaling range is limited, they found a power-law
scaling of $f_{pl}$ with $Ra$ for $f_{pl}<0.1$ and the obtained scaling
exponent is around 2.

Figure 12: (a) Scatter plot of the normalized perimeter $P/D$ vs the
normalized size $(4A/\pi D^{2})^{1/2}$ of sheetlike plumes at $Ra=3.0\times
10^{9}$. (b) The conditional average $\langle P|A\rangle/D$ on the normalized
size $(4A/\pi D^{2})^{1/2}$ for the same data as (a). Inset is the plot of
fractal dimensions of sheetlike plumes’ boundary $d_{A}$ vs Ra. Solid lines in
both (a) and (b) are power-law fits.

Next we examine the relationship between plume perimeter and area. Figure
12(a) shows a scatter plot of the normalized perimeter $P/D$ and the
normalized size $(4A/\pi D^{2})^{1/2}$ of sheetlike plumes, which contains a
total of $6071$ sheetlike plumes, extracted from a sequence of $260$ images at
Ra $=3.0\times 10^{9}$. Here, the perimeter and area are normalized by the
diameter and the area of the conducting plate, respectively. Although the data
look somewhat scatter, one sees that the perimeter increases with area and all
data points can be fitted by a power-low function,

$(P/D)\sim(\sqrt{4A/\pi D^{2}})^{d_{A}}\mbox{\ \ with\ \ }d_{A}=1.50.$ (5)

To better explore this feature, the conditional average $\langle P|A\rangle/D$
on the normalized size $(4A/\pi D^{2})^{1/2}$ is shown in figure 12(b). In the
figure, a good scaling range can be seen and the solid line is a power-law fit
$(P/D)=28.1\times(4A/\pi D^{2})^{1.50\pm 0.02}$. We find that such a scaling
behavior exists for all $Ra$ investigated. The inset of figure 12(b) shows the
$Ra$-dependence of $d_{A}$. One sees that these $d_{A}$ have a mean value of
$1.50$, but decrease from $1.53$ to $1.46$ when $Ra$ increases from $6.7\times
10^{8}$ to $6.3\times 10^{9}$. These measured values of $d_{A}$ are further
found to be larger than the value of 1.29 found for isoconcentration contours
of passive scalars measured in the same system [40].

Mathematically, as introduced by Mandelbrot [41] and Lovejoy [42], $d_{A}$ is
the fractal dimension of the boundary of sheetlike plumes and satisfies $1\leq
d_{A}<2$. Fractal dimensions in turbulence have been widely studied (see, for
example, Ref. [43]). In the present case, our results seem to suggest that the
boundary of sheetlike plumes is fractal with the dimension of around $1.50$
and the slight decrease of $d_{A}$ may suggest that the shape of sheetlike
plumes becomes smoother when the flow becomes more turbulent, as a result of
the increased mixing. However, the extracted sheetlike plumes, as shown in
figure 3, do not look like fractal objects. To understand this and to see if
applying the machinery of fractal analysis to the geometric properties of
plume can shed some light on the problem, we study the shape complexity of the
plumes.

###  4.2 Shape complexity of sheetlike plumes

Figure 13: (a) Scatter plot of the shape complexity $\Omega_{2}$ vs the
normalized size $(4A/\pi D^{2})^{1/2}$ of sheetlike plumes at $Ra=3.0\times
10^{9}$. (b) The conditional average $\langle\Omega_{2}|A\rangle$ on the
normalized size $(4A/\pi D^{2})^{1/2}$ with a solid line as a power-law fit.
Inset: $d_{\Omega}$ vs $Ra$. (same data as in figure 9)

The geometric complexity of an object can be characterized by its shape
complexity, which is a dimensionless ratio between its area and volume. The
shape complexity was previously used to study the shape of isocontours for
passive scalar fields in turbulence [39]. We introduce it here to describe the
shape of sheetlike plumes. For a two-dimensional (2D) closed contour, the
shape complexity is defined as follows,

$\Omega_{2}=\frac{P}{2\sqrt{\pi A}},$ (6)

where $P$ and $A$ are the perimeter of the contour and area inclosed by the
contour, respectively, and the subscript 2 refers to 2D. As a circle has the
minimum perimeter $2\sqrt{\pi A}$ among all 2D objects with the same area $A$,
the shape complexity $\Omega_{2}$ satisfies $1\leq\Omega_{2}<\infty$ and can
be used to describe the departure of an object from the shape of a circle. For
a fractal object, a larger $\Omega_{2}$ implies a rougher contour (or surface)
the object. For a non-fractal object, a larger $\Omega_{2}$ signifies that the
object is more different from a circle (e.g. it is more elongated).

Figure 13(a) shows a scatter plot of the shape complexity $\Omega_{2}$ and the
normalized dimension (or size) $(4A/\pi D^{2})^{1/2}$ of sheetlike plumes,
obtained from the same data set as those in figure 12. It is seen that the
mean trend of the relation between the size and shape complexity can be
captured by an increasing function, i.e., the plume with larger size has a
higher probability of possessing a larger $\Omega_{2}$. To illustrate this
mean trend more clearly, the conditional average $\langle\Omega_{2}|A\rangle$
on the normalized size $(4A/\pi D^{2})^{1/2}$ is plotted in figure 13(b). The
solid line in the figure is a power-law fit, i.e.,

$\Omega_{2}\sim(\sqrt{4A/\pi D^{2}})^{d_{\Omega}}\mbox{\ \ with\ \
}d_{\Omega}=0.50\pm 0.01$ (7)

and the reasonable scaling range can be found. Here, $d_{\Omega}\simeq
d_{A}-1$ is an immediate consequence of the definition of shape complexity
[Eq. (6)] together with the power-law relation [Eq. (5)]. The inset of figure
13(b) shows the $Ra$-dependence of $d_{\Omega}$. One sees that $d_{\Omega}$
decreases slightly with increasing $Ra$, which may be understood from the
decrease of $d_{A}$.

Figure 14: (a) $Ra$-dependence of the mean shape complexity
$\langle\Omega_{2}\rangle$ of sheetlike plumes. (b) The ratio between
$\langle\Omega_{2}\rangle$ and
$\frac{1}{\sqrt{\pi}}(\sqrt{\langle\gamma_{pl}^{sheet}\rangle}+\frac{1}{\sqrt{\langle\gamma_{pl}^{sheet}\rangle}})$,
$C_{R}$, vs $Ra$.

Figure 14(a) shows the $Ra$-dependence of the mean shape complexity
$\langle\Omega_{2}\rangle$. It is seen that $\langle\Omega_{2}\rangle$
decreases with increasing $Ra$. This suggests that, because of the increased
mixing, collision and convolution of sheetlike plumes, the shape of sheetlike
plumes become closer to the shape of a circle when the flow becomes more
turbulent. This also implies the decreases of plume’s aspect ratio with
increasing $Ra$. Indeed, with the definitions of $\gamma_{pl}^{sheet}$ [Eq.
(4)] and $\Omega_{2}$ [Eq. (7)], one can obtain the relation between of
$\Omega_{2}$ and $\gamma_{pl}^{sheet}$, i.e.,

$\Omega_{2}=\frac{1}{\sqrt{\pi}}(\sqrt{\gamma_{pl}^{sheet}}+\frac{1}{\sqrt{\gamma_{pl}^{sheet}}}),$
(8)

which implies that $\Omega_{2}$ decreases with decreasing
$\gamma_{pl}^{sheet}$ when $\gamma_{pl}^{sheet}\geq 1$. Furthermore,
$\Omega_{2}$ is proportional to $\gamma_{pl}^{sheet}$ when
$\gamma_{pl}^{sheet}\gg 1$. Accordingly, the decrease of
$\langle\Omega_{2}\rangle$, shown in figure 14(a), may be understood as the
result of the change in aspect ratio of the sheetlike plumes, i.e. the
decrease of $\langle\gamma_{pl}^{sheet}\rangle$ (see figure 9), as
$\gamma_{pl}^{sheet}$ is much larger than 1 for most of situations. To see
this more clearly, we study the ratio $C_{R}$ between
$\langle\Omega_{2}\rangle$ and
$\frac{1}{\sqrt{\pi}}(\sqrt{\langle\gamma_{pl}^{sheet}\rangle}+\frac{1}{\sqrt{\langle\gamma_{pl}^{sheet}\rangle}})$.
Note that Eq. (8) is valid only for an individual plume, while $C_{R}$ is in
an average sense. Figure 14(b) shows $C_{R}$ as a function of $Ra$. It is seen
that $C_{R}\simeq 1$ for nearly all $Ra$ investigated. This result
demonstrates clearly that the decrease of $\Omega_{2}$ is caused by the
decrease of the plume’s aspect ratio, rather than by a decreased roughness in
the case of a fractal object which the sheetlike plumes are not.

###  4.3 Distributions of geometric measures of sheetlike plumes

Figure 15: PDFs of (a) normalized area $4A/\pi D^{2}$, (b) normalized
perimeter $P/D$, (c) shape complexity $\Omega_{2}$, (d) normalized length
$l/D$, (e) normalized width $w/\delta_{th}$, and (f) aspect ratio
$\gamma_{pl}^{rod}$ sheetlike plumes, all obtained at $Ra=3.0\times 10^{9}$.
Solid curves are fittings of log-normal function to the respective data.

In a previous study, Zhou _et al._ [29] found that the area of sheetlike
plumes obeys log-normal statistics. Here, we further investigate distributions
of other geometric measures of sheetlike plumes. Figures 15(a)-(f) show the
measured PDFs of normalized geometric measures, i.e. $4A/\pi D^{2}$, $P/D$,
$\Omega_{2}$, $l/D$, $w/\delta_{th}$, and $\gamma_{pl}^{sheet}$, of sheetlike
plumes. It is seen that all these quantities have a same distribution, i.e.
the log-normal distribution. The same distributions were also found for all of
these quantities at other $Ra$ investigated in our experiments. Together with
the log-normal distributions found for mushroomlike plumes [21], these
findings suggest that the log-normal distribution is universal for thermal
plumes and log-normal statistics may be used to model them, at least in
turbulent RBC. In addition, this log-normal statistics of thermal plumes is
different from that of passive scalars measured in the same system, which is
found to obey a log-Poisson statistics [40].

###  4.4 Three-dimensional structures of sheetlike plumes

Figure 16: (a) Three-dimensional sheetlike/rodlike thermal plumes obtained
near the top plate by tomographic reconstruction technique ($Ra=2.0\times
10^{9}$, $\Gamma=1$). (b) An enlarged region of (a), showing the morphological
transformation of rodlike plumes into mushroomlike ones through
convolution/spiraling. (c) An individual rodlike plume.

Finally, we study the 3D structures of sheetlike plumes. Results obtained in
section 3.2 have suggested that the vertical extent of what is called
sheetlike plumes is only of the order of the thermal boundary layer thickness,
and hence not large enough to form a sheet. To see this more clearly , we used
a tomographic reconstruction technique to construct the 3D image of thermal
plumes from sequences of 2D images acquired near the top plate of the cell. To
achieve this, the convection cell was placed on a translational stage and, as
the cell traverses continuously at a speed of $\sim 1cm/sec$, a series of
photographs of TLC microspheres were recorded by a Nikon D3 camera (3CCD, with
a resolution of $2397\times 1591$ pixels) operating at 11 frames/s. As the
speed of the the horizontal motion of the sheetlike plumes are about
$0.4cm/sec$, the sequences of the horizontal slices of the plumes may be
regarded as taken at approximately the same time. In post-experiment analysis,
2D horizontal cuts of cold plumes were extracted from the images taken in each
run and a MATLAB script was used to reconstruct 3D thermal plumes from these
extracted 2D cuts. Figure 16(a) shows an example of the reconstruction of 3D
thermal plumes near the top plate. From the figure, one can see how mushroom-
like plumes are formed by the convolution (or spiraling) of sheetlike plumes.
Figure 16(b) shows this process more clearly, which is an enlarged region of
the image. Another noteworthy feature is that most plumes have a one-
dimensional structure, rather than a sheetlike shape. This geometric feature
may be illustrated more clearly by figure 16(c), which shows the
reconstruction of an enlarged individual 3D plume. It is seen that both the
height and the width of this plume extend only to a few millimeters, while its
length extends to a few centimeters. This is the most direct evidence thus far
that shows thermal plumes near the conducting plates are only one-dimensional
structures with horizontal length being much larger than their horizontal
width and vertical extent, which has been suggested previously [30].
Therefore, we may hereafter call thermal structures near the conducting plates
rod-like plumes. Future investigations will be focused on the geometric
properties of 3D structures of both rod-like and mushroom-like thermal plumes.

##  5 Conclusions

In this paper, we have presented a detailed experimental study of the
temperature and velocity fields in turbulent Rayleigh-Bénard convection using
the thermochromic-liquid-crystal technique. The number statistics and
geometric properties of sheetlike/rodlike thermal plumes were investigated.
Major findings are summarized as follows:

(i) When observed from above, the previous-called sheetlike thermal plumes
near the top plate seem to be only one-dimensional structures and should be
called rodlike plumes hereafter. These plumes evolve morphologically, i.e.
convolute or spiral, to mushroomlike plumes. The width ($\ell$) of the region
near the plates within which these rodlike plumes exist and evolve
morphologically is only associated with the thermal boundary thickness (i.e.
$\ell\sim$ several $\delta_{th}$).

(ii) The numbers of sheetlike/rodlike and mushroomlike plumes,
$N_{pl}^{sheet}$ and $N_{pl}^{mush}$, are found to both scale as $Ra^{0.3}$.
This finding suggests that the total amount of heat flow is dominated only by
the number of thermal (rodlike or mushroomlike) plumes and the normalized mean
heat content carried by each type of plumes is approximately independent on
$Ra$.

(iii) As the turbulent intensity is increased, large-sized sheetlike/rodlike
plumes are more easily to be fragmented into smaller ones and hence their
aspect ratios decrease.

(iv) Although sheetlike/rodlike plumes do not look like fractal objects, ,
power-law relations can be used to characterize the relationship between their
perimeter and area and between their size and shape complexity.

(v) Geometric measures of thermal plumes are found to have a universal
distribution, i.e. the log-normal distribution. Thus, log-normal statistics
may be used to model thermal plumes. In contrast, for the mixing of passive
scalars measured in the same system, it is found that the same quantities obey
log-Poisson statistics [40].

(vi) The Ra-dependency of the plume area density ($f_{r}\sim
Ra^{0.23}$)suggests that the plume number increase with $Ra$ ($N\sim
Ra^{0.3}$) can be largely attributed to the increased emission of plumes (as
$Ra$ is increased), rather than as a result of fragmentation of plumes.

## Acknowledgments

We gratefully acknowledge support of this work by the Research Grants Council
of Hong Kong SAR (Nos. CUHK403806 and 403807) (K.Q.X.) and by the Natural
Science Foundation of Shanghai (No. 09ZR1411200), “Chen Guang” project (No.
09CG41), and RFDP of Ministry of Education of China (No. 20093108120007)
(Q.Z.).

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