International Journal of Heat and Mass Transfer 139 (2019) 1056–1064

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International Journal of Heat and Mass Transfer

journal homepage: www.elsevier.com/locate/ijhmt

Effect of surface roughness on heat transfer in Rayleigh-Bénard

convection
Mark J. Tummers ⇑, Martin Steunebrink
Delft University of Technology, Faculty of Mechanical, Maritime and Materials Engineering, Mekelweg 2, 2628 CD Delft, the Netherlands

a r t i c l e i n f o a b s t r a c t

Article history: This paper reports on an experimental study of the effects of surface roughness on the flow and heat
Received 2 January 2019 transfer in cubical Rayleigh-Bénard convection cells for Rayleigh numbers between 107 and 1010. In
Received in revised form 14 May 2019 the rough cells the top and bottom surfaces are equipped with square arrays of copper cubes. In line with
Accepted 21 May 2019
other studies, three different regimes occur in the rough cells, with each regime having a different rela-
Available online 1 June 2019
tion between the Nusselt number, Nu, and the Rayleigh number, Ra. In the first regime the Nu-Ra relation
equals that of the smooth cell, but in the second and third regimes the Nu-Ra relation deviates from that
Keywords:
of the smooth cell with significantly higher Nusselt numbers. To better understand these observations,
Rayleigh-Bénard convection
Natural convection
the flow and temperature fields in both the smooth and rough cells were visualised by using particle
Heat transfer enhancement image velocimetry with suspended thermochromic liquid crystals as flow tracer particles.
Surface roughness Ó 2019 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license (http://
Thermochromic liquid crystals creativecommons.org/licenses/by/4.0/).
Particle image velocimetry

1. Introduction effect of wall roughness on heat transfer in natural convection may

be of interest to practical applications such as electronics cooling
Rayleigh-Bénard (R-B) convection is the fluid flow in the space where heat produced by electronic components must be removed
between two horizontal walls where the fluid is heated from below to prevent the occurrence of hot spots, or in meteorology where
and cooled from above. Many researchers have studied this system surface boundaries are usually not smooth.
using convection cells with different geometries and different Several other studies have addressed the effects of rough upper
working fluids. The case with smooth horizontal walls is well doc- and lower surfaces on R-B convection. It has been reported that
umented in a large number papers reporting on numerical and/or heat transfer increases in rough cells (with respect to smooth cells)
experimental studies. For an overview the reader is referred to when the thermal boundary layer thickness dh becomes less than
Ahlers et al. [1], Lohse et al. [2], and Chillà and Schumacher [3]. the roughness height h. If the Rayleigh number is increased beyond
Some studies focus on the characteristics of the temperature and this point, some researchers have observed a short transition to a
flow fields inside the cell, but the most frequently studied topic regime where the Nusselt number is increased (compared to the
is the relation between the Nusselt number (Nu) and the Rayleigh smooth case) with a constant factor, i.e., the pre-factor a in the
number (Ra), which represent the dimensionless heat transfer and relation Nu ¼ aRac is larger than in the corresponding smooth case,
the dimensionless buoyancy, respectively. Great efforts have been see for instance Shen et al. [7], Du and Tong [6]. Qiu et al. [8] con-
made to theoretically describe this aforementioned relation ducted experiments in a cylindrical cell and reported on an
between the Nusselt number and the Rayleigh number, see Gross- increase in both the pre-factor and the scaling exponent due to
mann and Lohse [5] and Stevens et al. [4]. the rough wall. Others have reported on a change of the value of
In an interesting variation on the standard type of R-B convec- the exponent c only (Roche et al. [9], Tisserand et al. [10]). Ciliberto
tion cells with smooth walls, ordered structures are mounted on and Laroche [11] considered roughness elements in the form of
the upper and/or lower walls, giving rise to so-called rough cells. glass spheres (with relatively low thermal conductivity) which
It has been shown that the heat transfer can be significantly were glued to the surface with thermal conductive paint. They
increased when the roughness elements on the conducting plates measured a lower Nusselt number with respect to the smooth case,
have proper characteristics, see for instance Du and Tong [6]. This although with a higher value of the exponent c. Villermaux [12]
proposed a model that predicts a modest increase (10%) of value
of c for irregular surface roughness while the value of the exponent
⇑ Corresponding author.
remains the same for regular roughness elements.
E-mail address: m.j.tummers@tudelft.nl (M.J. Tummers).

https://doi.org/10.1016/j.ijheatmasstransfer.2019.05.066
0017-9310/Ó 2019 The Authors. Published by Elsevier Ltd.
This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
 M.J. Tummers, M. Steunebrink / International Journal of Heat and Mass Transfer 139 (2019) 1056–1064 1057

Experiments by Xie and Xia [13] in a cylindrical cell with rough b of the rough cell according
By adjusting the physical height, H,
top and bottom walls (with pyramid-shaped roughness elements) to Eq. (1) the surface-averaged height of the rough cell equals the
indicated the existence of three regimes. A first regime in which height of the smooth cell. As a consequence, the liquid volumes in
the relation between the Nusselt number and the Rayleigh number both cells are equal.
is the same as in the case of smooth walls, a second regime with an The top and bottom plates of each cell are equipped with a
increased exponent, and a third regime also with an increased number of epoxy coated NTC thermistors of type C100 (General
exponent, but its value is generally somewhat lower than in the Electric). The relation between the temperature and resistance in
second regime. Rusaouën et al. [14] carried out experiments in a the range 273.15 K < T < 323.15 K is specified by the manufacturer
cylindrical R-B cell with one smooth and one rough wall (with as:
cubical roughness elements) that confirmed the existence of these
    2   3
three regimes. Rth Rth Rth
T 1 ¼ a1 þ a2 ln þ a3 ln þ a4 ln
Shishkina and Wagner [15] carried out 2D numerical simula- R25 R25 R25
tions for cases with a small number of relatively large rectangular
roughness elements and found an increase of the exponent. The where a1 = 3.354  103 K1, a2 = 2.562  104 K1,
6 1 8 1
DNS study by Wagner and Shishkina [16] showed that the increase a3 = 2.082  10 K , a4 = 7.300  10 K , Rth is the electrical
of the exponent depends on the geometry and spacing of the resistance of the thermistor at temperature T (in K) and R25 is the
roughness elements, and that for increasing Rayleigh numbers electrical resistance of the thermistor at 298.15 K. The value of R25
the value of the exponent becomes similar to that for the smooth differs for each thermistor but is usually somewhere between
walls. 9900 X and 10100 X. The individual thermistors are calibrated by
Zhu et al. [17] carried out a DNS of Taylor-Couette flow with measuring the value of R25 when the thermistors are positioned
grooved walls and found that the power law relation between inside the copper plates. A Pt100 thermometer with an inaccuracy
the dimensionless torque and the Taylor number (which are the of 0.05 K is used as the temperature standard during the calibration.
analogues of dimensionless heat transfer and Rayleigh number, Water from a thermostatic circulator flows through a brass
respectively) also has three characteristic regimes depending on plate that is mounted on the top copper plate (see Fig. 1a) to keep
the whether the boundary layer thickness (which depends on the this top plate at a pre-set constant temperature. The convection
Taylor number) is larger or smaller than the groove height. cells are heated from below with electrical heating foils that are
This study focuses on the effect of (well conducting) ordered attached to the bottom of the lower plate. The heating foils have
surface roughness on the flow field and the heat transfer in cubical dimensions 80  80 mm2, so the small cells are equipped with
R-B convection cells. The objective is to determine the Nu-Ra rela- one foil and the large cells with four foils. The cells are fully cov-
tion for this type of roughness, and to find out if there are specific ered in a 3 cm thick layer of insulating foam, insuring that the heat
processes occurring within the rough cells that are responsible for produced in the heating foils per unit time Q equals the heat trans-
the heat transfer enhancement. For this purpose, two smooth and fer rate through the fluid. The heat flux, q, through the cell is calcu-
three rough cubical R-B convection cells were built. The two lated from q = Q/H2. The heat transfer rate Q = IDV is determined by
smooth cells have different heights to increase the Rayleigh num- measuring the total electrical current I and the voltage difference
ber range considered in this study. The three rough cells have dif- DV at the electrical connections near the heating foils. The Nusselt
ferent ratios of roughness height and cell height. number is then computed as Nu = q H/(k DT) where k is the thermal
conductivity of the working fluid and DT is the temperature differ-
ence between the bottom and top walls of the cell. The effect of the
2. Experimental setup
conduction through the vertical side walls of the cell was esti-
mated using the procedure proposed by Roche et al. [18] and found
2.1. Rayleigh-Bénard convection cells
to have negligible effect on the Nusselt number. In this study the
Rayleigh number is computed as Ra = b g DT H3/(ma) where g is
The characteristics of the different cells that are used in this
the gravitational acceleration, b is the thermal expansion coeffi-
study are listed in Table 1. All five cells are (nearly) cubical with
cient, m is the kinematic viscosity, and a is the thermal diffusivity
inner dimension H. Fig. 1a shows a vertical cross section of a
of the working fluid. Finally, the Prandtl number is defined as
smooth cell (left) and a cell with surface roughness (right). Each
Pr = m/a. For the highest Rayleigh numbers considered in each
cell consist of copper top and bottom plates and the side walls
measurement series, the uncertainty of the measured Rayleigh-
are made of 2 mm thick glass. The roughness elements on the
and Nusselt numbers are determined as DRa/Ra = 1.9% and DNu/
top and bottom plates are small copper cubes with dimensions
Nu = 1.5%.
h  h  h. The cubes are arranged as a square array with a pitch
of 2h, as illustrated in Fig. 1b. The distance between the top and
bottom plates in the rough cells, H, b is made slightly larger than 2.2. TLC and PIV measurements
that of the corresponding smooth cells, H, because the cubical
roughness elements protrude into the cell. The relation between The flow tracer particles used in this research are encapsulated
Thermochromic Liquid Crystals (TLCs) with a mean diameter of
these two heights is:
about 25 lm. The encapsulated TLC particles are illuminated by a
b ¼ H þ h=2:
H ð1Þ ‘‘white” light sheet that is constructed as follows. Light from a
Waldman MCXFL3S light emitting diode is focused onto a glass
fibre bundle that is connected to a line light (Schott Fostec) produc-
Table 1 ing a homogeneous light intensity along its length. The light com-
Characteristics of the five R-B cells. The height of the cubical roughness elements is h.
ing from the line light is then focused into a vertical sheet that
Cell description h [mm] H [mm] spans the height of the cell by using a cylindrical lens.
Small smooth 0 77 The encapsulated TLC particles are of type R25C60W and pro-
Small rough 1 mm 1 77 duced by Hallcrest. The red start temperature of 25 °C and the tem-
Small rough 3 mm 3 77 perature bandwidth of 60 °C are both specified by the
Large smooth 0 155
manufacturer, and refer to a situation where the viewing angle
Large rough 3 mm 3 155
(i.e. the angle between the incident white light and the recorded
1058 M.J. Tummers, M. Steunebrink / International Journal of Heat and Mass Transfer 139 (2019) 1056–1064

Fig. 1a. R-B cells with smooth walls (left) and rough walls (right). A = hollow brass plate, B = top copper plate, C = glass side wall, D = copper bottom plate, E = electrical
heating foil, F = thermistor (6); G = bottom insulation plate.

of the fluid, respectively. The term inside the brackets stems from
the added mass of a spherical particle [20]. The density of the encap-
sulated liquid crystals is close to that of water, but typically a bit
higher, i.e., qp varies between 1.00  103 kg/m3 and 1.02  103 kg/
m3 [21]. For the larger particles in the liquid crystal samples
(d = 50 lm) with the highest density (qp = 1.02  103 kg/m3) the
response time is th = 0.22 ms, which is sufficiently small to follow
the velocity fluctuations in the flow considered in this study. The set-
tling velocity of a 50 lm particle in water, as determined from
 
2
V s ¼ qp  qf gd =18l, equals V s = 0.028 mm/s which is negligible

Fig. 1b. Top view of the square array with cubical roughness elements. compared to the typical velocities found in the convection cells in
this study. The thermal response time of the encapsulated liquid
crystals cannot be determined easily, but for chiral-nematic liquid
reflected light) is 180°. In the present experiment the light crystals the response time is relatively small, i.e., in the order of sev-
reflected by the TLC particles is collected in a direction normal to eral milliseconds (Dabiri [21], Abdullah et al. [22]). Günther and
the light sheet so that the viewing angle is 90°. For that viewing Rudolf von Rohr [23] have reported a thermal response time of
angle the red start temperature and bandwidth are reduced to 1 ms for Hallcrest encapsulated liquid crystals of type R25C20W
about 19.7 °C and 8.6 °C, respectively. The scattered light from with a mean diameter of 19 lm which is similar to the encapsulated
encapsulated TLC particles was recorded by a PCO SensiCam SVGA liquid crystals used in the present study.
camera with a 2/300 CCD sensor with 1280  1024 pixels. The cam-
era was equipped with a Nikon lens with 55 mm focal length and 2.3. Working fluids
numerical aperture of 2.8. A background image was created by
determining, for each pixel, the minimum intensity value in the Four different working fluids (water, methanol, ethanol and
series of 25,000 images. This background image was subsequently acetone) were used to extend the Rayleigh number range that
subtracted from each individual image. The commercial software can be obtained for a particular cell. All experiments were con-
DaVis was used for analyzing the image pairs and vector calcula- ducted with the bulk temperature of the fluid nearly equal to room
tion. The image pairs were processed in two consecutive steps with temperature to minimize the driving force for heat transfer
decreasing interrogation area size. In the first step, interrogation between the R-B cell and its surroundings. The thermo-physical
areas of 16  16 pixels were cross-correlated. The resulting particle properties of the four working fluids were determined in the tem-
displacements were used as window displacements for a cross cor- perature range between 293 K and 298 K (by using the Dortmund
relation with interrogation areas of 8  8 pixels corresponding to Data Bank) to enable accurate calculation of the Rayleigh, Prandtl
0.67  0.67 mm2 in physical space. A local median filter was and Nusselt numbers at the actual bulk temperature. The actual
employed to remove spurious vectors during the iterative process- room temperature varied slightly from day to day, but at a mean
ing and the resulting empty spaces were filled by interpolated vec- value of 295 K the Prandtl numbers of water, methanol, ethanol
tors. Based on the guidelines described in Adrian and Westerweel and acetone were determined as 6.7, 7.6, 17.0 and 4.2, respectively.
[19], the uncertainty of the velocity vectors was approximately
3% of the maximum velocity in the convection cells.
2.4. Experimental program
In this study the encapsulated TLC particles are used as both
temperature indicators and flow tracers. The hydrodynamic
The study considered three types of surfaces (smooth, 1 mm
response time th of a spherical particle can be determined from:
roughness and 3 mm roughness), two cell sizes (77 mm height
 
2 1 (small cell) and 155 mm height (large cell)) and four working flu-
t h ¼ qp d 1 þ qf =qp =18l ids. In total twelve different experiments were conducted, and
2
these are listed in Table 2. In the following sections different exper-
where d and qp are the diameter and the density of the particle, iments will be referred to as, for example, ‘‘small/smooth/water”
respectively, and l and qf are the dynamic viscosity and the density meaning that the experiment is conducted in the small R-B cell
 M.J. Tummers, M. Steunebrink / International Journal of Heat and Mass Transfer 139 (2019) 1056–1064 1059

Table 2
List of experiments.

Exp.# Cell description Fluid

1 Small smooth Water
2 Small smooth Ethanol
3 Large smooth Water
4 Large smooth Methanol
5 Small rough 1 mm Water
6 Small rough 1 mm Ethanol
7 Small rough 1 mm Methanol
8 Small rough 3 mm Water
9 Small rough 3 mm Methanol
10 Small rough 3 mm Acetone
11 Large rough 3 mm Water
12 Large rough 3 mm Methanol

with 77 mm height and smooth walls while the working fluid is

water.

3. Results Fig. 2b. NuRa0:294 as a function of Ra for experiments 1, 2, 3 and 4.

3.1. Experiments in the smooth R-B cells

Experiments 1–4 were conducted in the smooth cells, see

Table 2 for details. As an illustration, Fig. 2a shows the Nusselt
number Nu as a function of the Rayleigh number Ra for experi-
ments in the small smooth cell for both water and ethanol. For
each experiment the data points were least-squares fitted to the
functional form Nu ¼ aRac . The resulting lines have nearly identical
slope and the corresponding value of the exponent was (for all four
experiments in the smooth cell) determined as c = 0.294. This
value compares very well to the value of c = 0.297 that was
reported by Xia et al. [24] for measurements in a (cylindrical)
water filled cell at a similar range of Rayleigh numbers. Fig. 2b
shows NuRa0:294 as a function of the Rayleigh number, and the
results indicate that small jumps occur between the four groups
of data points. These jumps are mainly due to the differences in
Prandtl number of the working fluids, and Fig. 2c shows that this
mild Prandtl number dependence can be absorbed by the inclusion
of the term Pr 0:05 , consistent with the findings of Xia et al. [24].
The Prandtl-number-corrected Nusselt number, defined as
Nu ¼ NuPr 0:05 , is then determined as Fig. 2c. NuRa0:294 Pr 0:05 as a function of Ra for data shown in Fig. 2b.

Nu ¼ 0:132Ra0:294 ð2Þ

and shown as a solid line in Fig. 2d together with all data points
from the four experiments in the smooth cell. The Nu-Ra relation
given by Eq. (2) will serve as a reference in the comparison with
the results for the experiments in the rough cells.

3.2. Effects of the wall roughness

Experiments 5–12 are conducted in the cells with rough sur-

faces, see Table 2. The results of these experiments are presented
in Figs. 3a, 3b and 3c. The data in Figs. 3a and 3b pertain to equal
cell height, but different roughness heights, while the data in
Figs. 3b and 3c pertain to equal roughness height, but different cell
heights. The solid line represents the correlation for the Nusselt
number determined for the smooth wall, Eq. (2). It can be observed
in Figs. 3a, 3b and 3c that roughness elements in the form of copper
cubes have a very significant effect on the heat transfer in the cell.
The graphs indicate that the Nu* values for the rough cells start to
deviate from those of the smooth cells at different Rayleigh num-
Fig. 2a. Nu as a function of Ra for experiments in the small smooth cell for both bers. This is usually explained (see Du and Tong [6] and Shen
water and ethanol (experiments 1 and 2). et al. [7]) by assuming that the roughness starts to have an effect
1060 M.J. Tummers, M. Steunebrink / International Journal of Heat and Mass Transfer 139 (2019) 1056–1064

Fig. 2d. Nu* as a function of Ra for all experiments together with the correlation. Fig. 3b. Nu* as a function of Ra for the small cell with 3 mm roughness elements.
Nu ¼ 0:132Ra0:294 : The vertical dotted line denotes Ra = Ralow according to Eq. (3) for Pr = 6.7 (water).

Fig. 3a. Nu* as a function of Ra for the small cell with 1 mm roughness elements. Fig. 3c. Nu* as a function of Ra for the large cell with 3 mm roughness elements. The
The vertical dotted line denotes Ra = Ralow according to Eq. (3) for Pr = 6.7 (water). vertical dotted line denotes Ra = Ralow according to Eq. (3) for Pr = 6.7 (water).

when the thermal boundary layer thickness dh is approximately for the smooth case but with a pre-factor equal to 2.7 a instead
equal to the height of the surface roughness h, i.e., when dh ¼ h. of a. The value of 2.7 in the pre-factor is interesting in the sense
The corresponding Rayleigh number, Ralow , is calculated by using that it is larger than the ratio of the surface areas of the rough
the relation dh ¼ H=ð2NuÞ (see Belmonte et al. [25]) together with and smooth walls, which in the present study equals Arough/
the Nu-Ra relation given by Eq. (2) as: Asmooth = 2.
 1c The Nu-Ra relation for rough surfaces is qualitatively sketched
H in Fig. 3d where the Rayleigh number range is divided into three
Ralow ¼ : ð3Þ
2ahPr0:05 regimes. For the present experiments, the first regime (smooth
regime) and the third regime both have the same exponent
Note that the results in Fig. 3a do not show evidence of a low-
c1 = c3 = c = 0.294 with pre-factors a1 = a = 0.132 and
ering of the Nusselt number just before the transition at Ra = Ralow
a3 = 2.7a = 0.356. The second regime has exponent c2  0.54 and
as was observed in experiments by Tisserand et al. [10] and in the
results of numerical simulations by Stringano et al. [26]. pre-factor a2 ¼ aðRalow Þcc2 . The Rayleigh number Rahigh that marks
Figs. 3b and 3c also indicate that at higher Rayleigh numbers, the border between the second and third regimes can then be
the Nu-Ra relations for both the small and large cells with 3 mm determined as:.
roughness, tend to approach an asymptote that has the same (or 1
Rahigh ¼ Ralow 2:7c2 c : ð4Þ
very similar) slope as that found for the smooth cells, but with a
higher pre-factor. The Nu-Ra relation for the rough cells apparently The transition from the first regime (smooth regime) to a
equals that for the smooth case for low Rayleigh numbers regime with enhanced heat transfer when the boundary layer
(Ra < Ralow) and after a transition to a region with a higher expo- thickness is equal to the roughness height (at Ra = Ralow) has some
nent (c = c2  0.54), the Nu-Ra relation becomes parallel to that degree of universality. It is confirmed in a large number of exper-
 M.J. Tummers, M. Steunebrink / International Journal of Heat and Mass Transfer 139 (2019) 1056–1064 1061

Fig. 3d. Model for Nusselt-Rayleigh relation for rough walls. Fig. 4a. Snapshot from TLC recording in the (water-filled, small) smooth cell at
Ra = 6.9  107.

imental and numerical studies for different types of roughness. The

occurrence of the third regime (for Ra > Rahigh) with the increase of
the Nusselt number by a constant factor (compared to the Nusselt
number for the smooth case) appears to be less universal. For
example Tisserand et al. [10], Roche et al. [9] and Salort et al.
[27] did not observe a saturation effect in their experiments. How-
ever, this may have been the result of considering a too small range
of Rayleigh numbers since the second regime is quite wide in terms
of Rayleigh number. For example, Rahigh/Ralow = 2.74.07  57 in the
present experiment. Xie and Xia [13] reported on the three differ-
ent regimes and showed that the value of the exponent in the sec-
ond regime depends on the geometry, or more specifically, on the
aspect ratio k which is defined as the ratio of the height and the
base width of the roughness element. They have reported a value
of 0.43 for the case with k = 1. Rusaouën et al. [14] take the expo-
nent in the second regime as 0.5 and assume that the pre-factor
varies with the ratio h/H. In several other experiments (e.g. Roche
et al. [9] and Tisserand et al. [10]) it is also observed that the value
of the exponent in the second region is close to 0.50, which is char-
acteristic for the ultimate regime predicted by Kraichnan [28].
However, the results of the 2D numerical simulations reported
by Zhu et al. [30] showed that an exponent with a value of 0.5 does Fig. 4b. Snapshot from TLC recording in the (water-filled, small) cell with 3 mm
roughness at Ra = 6.9  107.
not indicate the onset of the ultimate regime because the value of
the exponent decreases with a further increase of the Rayleigh
number resulting in the third regime. ture fields in the two cells are quite different. First, individual
plumes that erupt from the horizontal boundary layers in the
3.3. Results from the suspended TLC recordings smooth cell are generally unable to traverse the core of the cell.
These plumes diffuse rather quickly in the core. The plumes in
The specifications of the useful temperature range of the TLCs the smooth cell reach the opposing horizontal wall mainly by mov-
effectively set the temperatures of the top and bottom plates ing with the large scale circulation (LSC) along the periphery of the
(see Section 2.2). The red start is at 19.7 °C and the upper clearing walls while the core is relatively quiescent and well-mixed in
point is around 28.3 °C. These limits result in a Rayleigh number of agreement with earlier results reported by e.g. Xia et al. [29].
Ra = 6.9  107 for experiments in the small cells. The correspond- Plumes (or groups of plumes) that move with the LSC are also
ing Prandtl-number-corrected Nusselt numbers are Nu* = 27.9 observed in the rough cell. The most important difference between
and Nu* = 52.1 for the smooth cell and the rough cell, respectively. the smooth and rough cells is that in the latter the plumes are
The recordings of the TLC tracer images consist of a series of much stronger. As a result, these individual plumes are able to tra-
2.5  104 sequential snapshots acquired at a frame rate of 7.9 Hz verse the core and then impinge on the opposing horizontal wall,
for each cell. The measurement time of 3165 s corresponds to as illustrated in Fig. 4b. This phenomenon is thought to be a main
1558 free fall time scales Tff computed as Tff = (H/(a g reason for the increased heat transfer in the rough cell. Close
DT))1/2 = 2.03 s. inspection of the TLC recordings near, say, the bottom wall shows
Figs. 4a and 4b show snapshots from TLC recordings in the that the increased strength of the plumes is directly related to the
smooth and rough cells, respectively. The instantaneous tempera- prolonged accumulation of hot fluid in the relatively quiescent
1062 M.J. Tummers, M. Steunebrink / International Journal of Heat and Mass Transfer 139 (2019) 1056–1064

spaces in between the roughness elements. As a consequence, a

relatively large amount of hot fluid is released when a plume
erupts from the thermal boundary layer on a rough wall. Secondly,
it is observed that the plume behaviour in the bottom left corner
and in the top right corner is different in both cells. This will be dis-
cussed in the next section.

3.4. Results from the PIV measurements

The TLC recordings were also used to determine the flow fields
inside the smooth and rough cells. Figs. 5a and 5b depict the mean
velocity fields for the smooth cell and the rough cell, respectively.
In both cells the highest mean velocities are concentrated along the
walls, whilst the centre region (core) is characterized by much
lower mean velocities thus indicating the existence of a LSC. The
mean flow in the smooth cell is quite symmetric with an oval-
shaped LSC very similar to what has been reported by Xia et al.
[29]. The oval-shaped LSC in the rough cell is slightly flatter and
its core is somewhat displaced to the upper left corner.
In each cell there are two secondary rolls in the mean flow field,
one in the bottom left corner and one in the top right corner. The
secondary rolls are counter-rotating with respect to the direction
of the LSC. The area-averaged mean velocity magnitude in the
rough cell is approximately 12% larger than in the smooth cell. Fig. 5b. The mean velocity field in the (water-filled, small) cell with 3 mm
The higher velocity may increase the heat transfer, but this rather roughness at Ra = 6.9  107.
small difference in mean velocity cannot explain the large increase
in the measured Nusselt number. Interestingly, the dynamical
behaviour of the velocity fields in both cells is very different. This
is illustrated in Figs. 6a and 6b which show the turbulence kinetic
 
energy, defined here as k ¼ 1=2 u0 2 þ w0 2 , for the smooth and
rough cell, respectively. In the expression for k, the symbols u0
and w0 represent the deviation from the mean velocity components
 
u and w. Clearly, the intensity of the turbulent velocity fluctuations
is much higher (approximately 45%) in the rough cell. The sec-
ondary rolls play an important role in the increase of the turbu-
lence kinetic energy. Consistent with observations from the TLC
recordings, the rolls are formed in the corners where the LSC
attaches to the top/bottom plates. Plumes that erupt from the por-

Fig. 6a. The turbulence kinetic energy in the (water-filled, small) smooth cell at
Ra = 6.9  107.

tion of the thermal boundary layer in between the (unsteady)

attachment point and the corner regularly tend to flow against
the direction of the LSC. This mechanism is present in both cells,
but the temporary flow of (groups of) plumes against the LSC is
much stronger in the rough cell to such an extent that these
plumes are able to reach the opposing horizontal plate. When a
group of, say, hot plumes from the secondary roll in the lower left
corner of the cell move against the LSC towards the top wall, the
downward motion of the cold plumes along the vertical side wall
is interrupted. This stimulates the release of large cold plumes
Fig. 5a. The mean velocity field in the (water-filled, small) smooth cell at from the top plate (instead of moving with the LSC along the top
Ra = 6.9  107. plate followed by a descent along the vertical side wall) which sub-
 M.J. Tummers, M. Steunebrink / International Journal of Heat and Mass Transfer 139 (2019) 1056–1064 1063

significantly higher (0.356). The boundary between the smooth

regime and the second regime occurs where the height of the cubes
h is equal to the thermal boundary layer thickness dh . This corre-
  1=c
sponds to a Rayleigh number Ralow = H= 2ahPr0:05 , where
Pr is the Prandtl number of the working fluid.
Recordings of encapsulated thermochromic liquid crystal tracer
particles were used to visualise the temperature distribution and
determine the velocity fields (by using particle image velocimetry)
in the smooth and rough cells for Ra = 6.9  107 (in the second
regime with enhanced heat transfer). In both cells the flow fields
exhibit a large scale circulation along the periphery of the cell with
the velocity magnitude in the rough cell about 12% higher than in
the smooth cell. The turbulence kinetic energy level in the rough
cell is about 45% higher than in the smooth cell. The recordings
of the thermochromic liquid crystals showed that hot fluid accu-
mulated in the quiescent regions in between the cubical roughness
elements, which resulted in the formation of strong plumes that
can traverse the core of the cell and directly impinge on the oppos-
ing horizontal plate. Strong plumes originating from the secondary
rolls in the lower left and upper right corners of the cell were also
able to reach the opposing horizontal plates. These impingement
phenomena are thought to be responsible for the enhanced heat
transfer in the rough cell.
Fig. 6b. The turbulence kinetic energy in the (water-filled, small) cell with 3 mm
roughness at Ra = 6.9  107.
Declaration of Competing Interest

sequently traverse the core of the cell and then impinge on the bot- The authors declared that there is no conflict of interest.
tom wall. The size and strength of the secondary rolls in the rough
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