1 Introduction
Turbulent Rayleigh–Bénard convection is a model system for natural convection. Theoretically, it consists of a horizontal infinite layer of fluid inserted between two plates: a hot one at the bottom and a cold one at the top. The thermal forcing sets the fluid into motion. The intensity of the forcing can be assessed by the Rayleigh number
$$\begin{eqnarray}\displaystyle Ra=\frac{g\unicode[STIX]{x1D6FC}\unicode[STIX]{x0394}TH^{3}}{\unicode[STIX]{x1D708}\unicode[STIX]{x1D705}}, & & \displaystyle\end{eqnarray}$$
where
$\unicode[STIX]{x0394}T$
is the temperature drop across the cell,
$H$
is the distance between the two plates,
$\unicode[STIX]{x1D6FC}$
is the expansion coefficient of the fluid,
$\unicode[STIX]{x1D708}$
its kinematic viscosity,
$\unicode[STIX]{x1D705}$
the thermal diffusivity and
$g$
the acceleration due to gravity. The fluid properties are characterized using the Prandtl number
$$\begin{eqnarray}\displaystyle Pr=\frac{\unicode[STIX]{x1D708}}{\unicode[STIX]{x1D705}}, & & \displaystyle\end{eqnarray}$$
which compares the two diffusion mechanisms that impede convection. In this work, the model experiment is a Rayleigh–Bénard cell where the layer of fluid is inserted into a cylindrical container. The only geometrical parameter is the aspect ratio
$\unicode[STIX]{x1D6E4}=D/H$
, where
$D$
is the diameter of the cell, and
$H$
its height.
The system response is the thermal heat flux,
$Q$
, which is larger than the case without convection. The non-dimensional heat flux is the Nusselt number, which compares the global heat flux to the purely conductive one for similar temperature drop,
$$\begin{eqnarray}\displaystyle Nu=\frac{QH}{\unicode[STIX]{x1D706}\unicode[STIX]{x0394}T}, & & \displaystyle\end{eqnarray}$$
where
$\unicode[STIX]{x1D706}$
is the thermal conduction.
One objective is to be able to predict the thermal heat flux for a given forcing, i.e. to relate the Nusselt number to the control parameters. Several models derive predictions in the form of a scaling law,
$$\begin{eqnarray}\displaystyle Nu={\mathcal{C}}Ra^{a}Pr^{b}. & & \displaystyle\end{eqnarray}$$
One of the first of such models was proposed by Malkus (Reference Malkus1954) and yields
$a=1/3$
, which means that the heat flux does not depend on the distance between the plates. This is a strong indication that the plates can be described independently of one another. Many published experimental data are in fair agreement with this scaling, although there are some deviations. The reader may refer to the review of Chillà & Schumacher (Reference Chillà and Schumacher2012) for more details.
An alternative description was proposed by Grossmann & Lohse (Reference Grossmann and Lohse2000), where the relation is no longer a simple scaling law, but rather a superposition of scaling laws. It accounts well for the evolution of the effective scaling exponent,
$a$
, when the Rayleigh number increases. This is why it is used throughout this work to provide estimates of reference Nusselt numbers in the case of hydrodynamically smooth plates.
For asymptotically large forcings, one may expect the boundary layer to become fully turbulent, which yields
$a=1/2$
(Kraichnan Reference Kraichnan1962; Grossmann & Lohse Reference Grossmann and Lohse2011). This exponent is also a rigorous upper bound (Goluskin & Doering Reference Goluskin and Doering2016), and the corresponding regime is sometimes called ultimate regime of convection, as there could not be a more efficient regime beyond. Several groups have claimed to observe this regime at very large Rayleigh numbers, using cryogenic gaseous helium (Chavanne et al.
Reference Chavanne, Chillà, Castaing, Hébral, Chabaud and Chaussy1997; Roche et al.
Reference Roche, Gauthier, Kaiser and Salort2010), or compressed sulphur hexafluoride (He et al.
Reference He, Funfschilling, Nobach, Bodenschatz and Ahlers2012).
In this paper, we consider the case of a Rayleigh–Bénard cell with rough boundaries. The addition of a controlled roughness on the boundaries produces an enhancement of the heat transfer beyond a critical Rayleigh number,
$Ra_{c}$
, determined by the roughness size. Indeed, below this critical value, the thermal boundary layer is larger than the typical roughness size, and the boundary is hydrodynamically smooth. Enhancement is observed when the boundary layer thickness is the size of the roughness.
In the past, several types of enhancement have been reported. Roche et al. (Reference Roche, Castaing, Chabaud and Hébral2001a
), Qiu, Xia & Tong (Reference Qiu, Xia and Tong2005), Tisserand et al. (Reference Tisserand, Creyssels, Gasteuil, Pabiou, Gibert, Castaing and Chillà2011) and, in some configurations, Wei et al. (Reference Wei, Chan, Ni, Zhao and Xia2014) observe an increase of the scaling law exponent
$a$
: before the transition to Nusselt enhancement,
$a$
is close to
$2/7$
or
$1/3$
then it increases and reaches nearly
$1/2$
. In several other configurations, the exponent
$a$
is unchanged but the prefactor
${\mathcal{C}}$
increases (Du & Tong Reference Du and Tong1998; Wei et al.
Reference Wei, Chan, Ni, Zhao and Xia2014).
In our previous works, we showed that roughness triggers turbulent instabilities in the boundary layers at moderate Rayleigh numbers (Salort et al. Reference Salort, Liot, Rusaouen, Seychelles, Tisserand, Creyssels, Castaing and Chillà2014) and enhance the velocity fluctuations (Liot et al. Reference Liot, Ehlinger, Rusaouen, Coudarchet, Salort and Chillà2017). The objective is to gain insights into the role of turbulence on the thermal transfer, at a given Rayleigh number, hence without the need for non-conventional working fluids.
Roughness can be added to only one of the horizontal plates (Ciliberto & Laroche Reference Ciliberto and Laroche1999; Tisserand et al. Reference Tisserand, Creyssels, Gasteuil, Pabiou, Gibert, Castaing and Chillà2011; Wei et al. Reference Wei, Chan, Ni, Zhao and Xia2014), to both plates (Du & Tong Reference Du and Tong1998, Reference Du and Tong2000; Qiu et al. Reference Qiu, Xia and Tong2005) or even to the entire cell (Roche et al. Reference Roche, Castaing, Chabaud and Hébral2001a ). Several geometries of structure are used such as square pyramids, (Du & Tong Reference Du and Tong1998, Reference Du and Tong2000), pyramidal grooves (Roche et al. Reference Roche, Castaing, Chabaud and Hébral2001a ), spheres (Ciliberto & Laroche Reference Ciliberto and Laroche1999) or square structures (Tisserand et al. Reference Tisserand, Creyssels, Gasteuil, Pabiou, Gibert, Castaing and Chillà2011). Among these experiments, that of Ciliberto and Laroche does not directly compare because the roughness elements are glass spheres coated with copper varnish, so they can be considered as thermally insulating the plate. Although the details of the roughness geometry are of high interest for optimization purposes (García et al. Reference García, Solano, Vicente and Viedma2012), we focus on the effects of roughness-triggered turbulence in general.
Recently, Toppaladoddi, Succi & Wettlaufer (Reference Toppaladoddi, Succi and Wettlaufer2017) and Zhu et al. (Reference Zhu, Stevens, Verzicco and Lohse2017) focused on the influence of the density of roughness structures on the thermal transfer. To do so, they performed several two-dimensional numerical simulations in a Rayleigh–Bénard system with sinusoidal roughness on both plates. Both studies report the dependency of the exponent
$a$
with
$\unicode[STIX]{x1D6EC}=d/H$
, the ‘wavelength’ associated with the roughness horizontal dimensions, and the existence of an optimal wavelength value,
$\unicode[STIX]{x1D6EC}_{opt}$
, at which
$a$
is maximum. Above
$\unicode[STIX]{x1D6EC}_{opt}$
,
$a$
recovers the smooth case value. Horizontal spacing was also identified as an important parameter in studies involving roughness in wind tunnel such as Perry, Schofield & Joubert (Reference Perry, Schofield and Joubert1969).
Recent experiments from Xie & Xia (Reference Xie and Xia2017) also evidence this role of roughness geometry. They have varied the roughness aspect ratio,
$\unicode[STIX]{x1D706}$
, defined as the height of a single roughness element over its base, and found that the asymptotic scaling law exponents increase with
$\unicode[STIX]{x1D706}$
. However, the roughness density also increases when
$\unicode[STIX]{x1D706}$
increases, so it is not yet possible to disentangle the effect of aspect ratio and the effect of roughness density.
They evidence two transitions in the
$Nu$
versus
$Ra$
scaling: the first transition occurs when the thermal boundary layer thickness is the height of the roughness, consistent with past experiments. They call this regime of enhanced heat transfer ‘Regime II’. Then, a second transition occurs when the velocity boundary layer thickness is the height of the roughness, yielding ‘Regime III’. The scaling exponent in Regime III is lower than in Regime II.
In the present paper, new heat transfer measurements are presented in several rough configurations. We observe both the regime of enhanced scaling exponent
$a$
, and the regime of enhanced prefactor
$C$
. Because the roughness aspect ratio is fixed (
$h_{0}/d=0.4$
), we cannot disentangle the role of roughness height from the role of the roughness wavelength.
After a description of the experimental apparatus, in § 2, we will detail some reference results obtained in the classical smooth configuration of the cell, § 3. Then, we will present the new results obtained in the rough cell with larger roughness elements and compare them to other published measurements, § 4. This allows us to explore the thermal behaviour of the cell when the height of the thermal boundary layer is significantly smaller than the height of the elements.
2 Experimental apparatus
2.1 The cell
The experimental apparatus consists of a cylindrical Rayleigh–Bénard cell: see figure 1. The diameter
$D$
is 0.5 m. Two sidewalls can be installed, one of height 20 cm, the second of height 1.0 m. They are made of 3 mm thick stainless steel. Figure 1 sketches the 1 m high cell, ‘tall cell’ (
${\mathcal{T}}{\mathcal{C}}$
), of aspect ratio
$\unicode[STIX]{x1D6E4}=0.5$
. The smaller configuration, ‘small cell’ (
${\mathcal{S}}{\mathcal{C}}$
), has an aspect ratio
$\unicode[STIX]{x1D6E4}=2.5$
.
Figure 1. Sketch of the
$\unicode[STIX]{x1D6E4}=0.5$
Rayleigh–Bénard cell. Two aspect ratios can be used:
$\unicode[STIX]{x1D6E4}=0.5$
corresponding to a diameter
$D=50~\text{cm}$
and a height
$H=1~\text{m}$
, and
$\unicode[STIX]{x1D6E4}=2.5$
with
$D=50~\text{cm}$
and
$H=20~\text{cm}$
.
The cold plate is made of copper coated with a thin layer of nickel to prevent chemical attack from the working fluid, deionized and degassed water. It is thermalized by a water circulation on its top which is controlled by a regulated bath. The hot plate is made of aluminium. It is heated by Joule effect using a spiralled resistor of
$13~\unicode[STIX]{x03A9}$
inserted into the plate.
The cell is covered by a thermal insulator, 4 cm thick neoprene foam, and enclosed into a thermal screen made of copper. The mean temperature of the screen is regulated at the bulk temperature by a water bath to prevent interaction between the cell and the environment. The entire apparatus is placed on a table whose temperature is also regulated at the bulk temperature.
2.2 Measurement techniques
Measurements are focused on thermal transfer. To do so, the cell is instrumented with different kinds of thermometers. Six resistance temperature detectors (Pt100 type), three per plate, measure the absolute value of the temperature of each plate. Six thermocouple junctions measure the temperature at mid-height and the temperature of the bottom plate relative to the top plate with high accuracy. The common reference is then inserted into the cold plate which provides the relative zero value in the system (
$T_{c}$
). Another junction is also introduced into this plate at a different radius. Two junctions are placed into the hot plate, and provide the hot temperature
$T_{h}$
. As the thin lateral walls are in stainless steel, they are thermalized at the bulk temperature
$T_{b}$
. Two junctions are then placed onto those walls to access
$T_{b}$
. We also measure the temperature of the thermal screen and the table. The thermocouple junctions are connected to an electronic amplifier with negligible offset. The signal is amplified 2000 times. One measurement consists of averaging over several hours (typically 48 h) of recording.
By varying the bulk temperature of the fluid, we induce variations of the fluid properties and consequently of the Prandtl number. We perform measurements at fixed Prandtl number by keeping the bulk temperature constant. This allows us to check the potential influence of the Prandtl number in a range of values between 2.5 and 6.5.
2.3 Non-Boussinesq and lateral wall corrections
Since our experimental procedure is similar to the one used by Tisserand et al. (Reference Tisserand, Creyssels, Gasteuil, Pabiou, Gibert, Castaing and Chillà2011), we will use the same corrections, as detailed below. First, we take care of the non-Boussinesq (NOB) effects and we show that they are negligible in our case. The Boussinesq approximation assumes that all the physical properties of the fluid are independent of the temperature except the density
$\unicode[STIX]{x1D70C}$
in the buoyancy term, which can be approximated as
$\unicode[STIX]{x1D70C}=\unicode[STIX]{x1D70C}_{0}(1-\unicode[STIX]{x1D6FC}\unicode[STIX]{x0394}T)$
, where
$\unicode[STIX]{x1D70C}_{0}$
is the density of water at the temperature
$T_{b}$
. Experiments performed by Ahlers et al. (Reference Ahlers, Brown, Araujo, Funfschilling, Grossmann and Lohse2006) suggest that, in water, the main sources of NOB effects are the variation of kinematic viscosity
$\unicode[STIX]{x1D708}$
and thermal diffusivity
$\unicode[STIX]{x1D705}$
only. In liquid water, the second one is nearly constant which leaves only the effects of
$\unicode[STIX]{x1D708}$
. Ahlers et al. (Reference Ahlers, Brown, Araujo, Funfschilling, Grossmann and Lohse2006) define a parameter
$\unicode[STIX]{x1D712}$
, as previously done by Wu & Libchaber (Reference Wu and Libchaber1991), corresponding to the dissymmetry of the system,
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D712}=\frac{T_{h}-T_{b}}{T_{b}-T_{c}}=1-c_{2}\unicode[STIX]{x0394}T. & & \displaystyle\end{eqnarray}$$
Tisserand et al. (Reference Tisserand, Creyssels, Gasteuil, Pabiou, Gibert, Castaing and Chillà2011) used a logarithmic dependence of
$c_{2}$
on Prandtl number but no dependence on Rayleigh number.
$$\begin{eqnarray}\displaystyle c_{2}=-0.061Pr^{0.25}\frac{\text{d}\ln (\unicode[STIX]{x1D708})}{\text{d}T}. & & \displaystyle\end{eqnarray}$$
This results in corrective prefactors for the Nusselt number such as
$Nu_{s}^{cor}=(1+c_{2}\unicode[STIX]{x0394}T_{s}/2)Nu_{s}$
. The deviation remains smaller than 1% in all our experimental conditions, and thus can be neglected. This was also suggested by Ahlers et al. (Reference Ahlers, Brown, Araujo, Funfschilling, Grossmann and Lohse2006). In their study, they performed thermal transfer measurements in water, reaching a difference of temperature of nearly 40 K, and found only small deviation of the Nusselt number from the Oberbeck–Boussinesq case. In the experiment presented here, the largest temperature difference is 20 K and the highest Rayleigh number we can attain is
$1.5\times 10^{12}$
. We thus do not expect NOB corrections to be large in these conditions.
The second effect we have to consider is the spurious heat conduction in the sidewalls; see Ahlers (Reference Ahlers2000) and Roche et al. (Reference Roche, Castaing, Chabaud, Hébral and Sommeria2001b
). The thermal conductivity of these walls has to be taken into account. It behaves as if the effective surface,
$S_{eff}$
, of the horizontal plates was larger than the real one
$S$
.
$S_{eff}$
can be related to
$S$
by
$S_{eff}=(1+f(W))S$
, where
$W$
balances the heat conductivity of the sidewalls with the value for water and is close to 0.5, and the corrected Nusselt number is
$$\begin{eqnarray}\displaystyle Nu^{cor}=Nu^{raw}(1+\text{f}(W))^{-1}. & & \displaystyle\end{eqnarray}$$
It yields to corrective prefactors shown in figure 2 for a smooth plate. The corrections for the rough plate are expressed in Tisserand et al. (Reference Tisserand, Creyssels, Gasteuil, Pabiou, Gibert, Castaing and Chillà2011), and we shall use the same here.
Figure 2. Corrective prefactors induced by sidewall effects for a smooth plate (for more details on the definitions of
$Nu_{s}$
and
$Ra_{s}$
see expressions (4.3)). Circles are for the
${\mathcal{T}}{\mathcal{C}}$
cell and triangles are for
${\mathcal{S}}{\mathcal{C}}$
. Red refers to series at mean temperature
$60\,^{\circ }\text{C}$
, green is for
$40\,^{\circ }\text{C}$
and blue for
$30\,^{\circ }\text{C}$
.
3 Reference smooth cell
Before presentation of the results obtained in the asymmetric rough cell, we shall discuss the behaviour of
${\mathcal{T}}{\mathcal{C}}$
in the classical configuration where all boundaries are smooth. Indeed, those results were briefly discussed only in the review of Chillà & Schumacher (Reference Chillà and Schumacher2012), but no detailed presentation is available in the literature. The reference case will be referred to as the ‘
${\mathcal{R}}{\mathcal{S}}{\mathcal{C}}$
’ case for ‘reference smooth cell’ in the following.
The experimental apparatus is the same as that described in § 2 except that the two plates are smooth, made of copper and coated with a thin layer of nickel. Results are presented in figure 3(a,b). Figure 3(a) shows the Nusselt number compensated by the Rayleigh number as
$$\begin{eqnarray}\displaystyle \frac{Nu}{Ra^{1/3}}. & & \displaystyle\end{eqnarray}$$
The
${\mathcal{R}}{\mathcal{S}}{\mathcal{C}}$
points are shown as full black diamonds in this figure. This presentation allows us to evaluate the potential departure from a
$Ra^{1/3}$
behaviour. Other results obtained in other cells are also shown for comparison. The open triangles are for Chavanne et al. (Reference Chavanne, Chillà, Chabaud, Castaing and Hébral2001), a cylindrical cell of gaseous cryogenic helium. Green open diamonds and half-diamonds are for Niemela et al. (Reference Niemela, Skrbek, Sreenivasan and Donnelly2000), in a cylindrical cell filled with gaseous helium. Violet circles are for the smooth/smooth values of Du & Tong (Reference Du and Tong2000), in cylindrical cell filled with water at ambient temperature. Finally, blue stars are for Urban, Musilova & Skrbek (Reference Urban, Musilova and Skrbek2011), in a cylindrical cell filled with gaseous cryogenic helium. The black line is the Grossmann–Lohse model (GL model in the following) fitted for
$Pr=3.7$
. The present evaluation of
$Nu_{GL}$
is performed using the updated prefactors proposed by Stevens et al. (Reference Stevens, van der Poel, Grossmann and Lohse2013), as will be the case in the entire paper. Comparison with other experiments also shows a global collapse of all the cells. Some points from Chavanne’s experiment exhibit a departure from other experiments at
$Ra>10^{12}$
. The present experiment does not exhibit such a departure even at similar Rayleigh and Prandtl numbers.
Figure 3. (a) Compensated Nusselt number as a function of the Rayleigh number. The symbols are the same as in (b). The continuous black line is the Grossmann–Lohse model with
$Pr=3.7$
. The present points are represented by the full black diamond. (b) Prandtl number as a function of Rayleigh number.
Table 1. Values of Nusselt, Rayleigh and Prandtl numbers obtained in the
${\mathcal{R}}{\mathcal{S}}{\mathcal{C}}$
, with smooth boundaries.
Figure 3(b) shows the same experiments in a
$(Ra,Pr)$
phase diagram. Several points of
${\mathcal{R}}{\mathcal{S}}{\mathcal{C}}$
have no overlap with previous measurements in this
$(Ra,Pr)$
plane, and thus extend the explored parameter space. Although not all published data are shown, to our knowledge, the
${\mathcal{R}}{\mathcal{S}}{\mathcal{C}}$
data are the only data which range from
$Ra=10^{10}$
to
$Ra=10^{11}$
and
$Pr>6$
. The corresponding values of the Nusselt, Rayleigh and Prandtl numbers are given in table 1.
As the GL model is in fair agreement with all the smooth experiments, and well captures the changes of behaviour in this range of Rayleigh number, we shall use its evaluated Nusselt number to normalize our results in the following and then allow for comparison.
4 Convection cell with rough boundaries
In this section, we consider a rough cell, where roughness is added to the bottom plate only. The top plate and the lateral walls are smooth. The smooth plate is the same as that used for the
${\mathcal{R}}{\mathcal{S}}{\mathcal{C}}$
case previously mentioned. The symmetry is broken, the thermal impedance at the top and bottom boundaries are no longer identical, even within the Boussinesq approximation. This allows in situ comparison of rough and smooth boundary layers.
Figure 4. Sketch of the roughness pattern. In the present paper, results are obtained using
$h_{0}=4~\text{mm}$
,
$d=10~\text{mm}$
with a periodicity
$2d=5h_{0}$
. In Tisserand et al. (Reference Tisserand, Creyssels, Gasteuil, Pabiou, Gibert, Castaing and Chillà2011),
$h_{0}=2~\text{mm}$
and
$d=5~\text{mm}$
.
Figure 5. Global heat transfer measurements in asymmetric cells with a rough bottom plate and a smooth top plate. Colours refer to the bulk temperature
$T_{b}$
, blue is
$25\,^{\circ }\text{C}$
or
$30\,^{\circ }\text{C}$
, green is
$40\,^{\circ }\text{C}$
, red
$60\,^{\circ }\text{C}$
and brown
$70\,^{\circ }\text{C}$
. Open symbols refer to the
$h_{0}=2~\text{mm}$
of Tisserand et al. (Reference Tisserand, Creyssels, Gasteuil, Pabiou, Gibert, Castaing and Chillà2011), full symbols to the current
$h_{0}=4~\text{mm}$
elements. Triangles are for
${\mathcal{S}}{\mathcal{C}}$
, circles for
${\mathcal{T}}{\mathcal{C}}$
. Solid line: Grossmann–Lohse model for symmetric Rayleigh–Bénard cells with smooth boundaries.
The roughness elements consists of cubic square studs, arranged in a lattice, as shown in figure 4. The height of the roughness elements is
$h_{0}=4~\text{mm}$
, their width is
$d=10~\text{mm}$
. The periodicity of the pattern is
$2d$
. They are machined directly into the plate to preserve the thermal properties of the material. This configuration is similar to that of Tisserand et al. (Reference Tisserand, Creyssels, Gasteuil, Pabiou, Gibert, Castaing and Chillà2011), but with larger roughness elements. They used elements with
$h_{0}=2~\text{mm}$
and
$d=5~\text{mm}$
arranged in the same way, so there is a scaling factor of 2 between the two roughness sizes. The present results will be compared with those obtained in this previous study.
It is always formally possible to define global Rayleigh and Nusselt numbers in this cell. As shown in figure 5, the global Nusselt number is larger than in the case of smooth boundaries, and the scaling exponent is modified. However, due to the mixed nature of the boundaries, and the broken symmetry, it is hard to draw more precise conclusions from these quantities.
Figure 6. Separation of plates. (a) Asymmetric cell, (b) symmetric cell based on the hot plate, (c) symmetric cell based on the cold plate.
4.1 Separation of plates
Following the approach of Tisserand et al. (Reference Tisserand, Creyssels, Gasteuil, Pabiou, Gibert, Castaing and Chillà2011), the asymmetric cell can be divided into two symmetric half-cells, under the assumption that the plates are independent. This procedure is justified only by the broken symmetry of the cell geometry. Indeed, although this has recently triggered some discussion (Skrbek & Urban Reference Skrbek and Urban2015; Shishkina, Weiss & Bodenschatz Reference Shishkina, Weiss and Bodenschatz2016), its relevance as a means to recover from the non-Boussinesq effect is not discussed in this paper because our working conditions are all chosen in a range where the Boussinesq approximation holds. The bulk temperature is not equal to
$(T_{h}+T_{c})/2$
due to the impedance adaptation of the flow induced by the introduction of roughness on only one plate of the cell. The heat flux is imposed at the hot plate and the cold one is temperature regulated, which allows the bulk and hot temperatures to stabilize at free values corresponding to the stationary state of the operating point. The asymmetric cell with roughness on the bottom and without roughness on the top is sketched in figure 6(a). We measure
$T_{c}$
,
$T_{b}$
and
$T_{h}$
. If we focus on the hot/rough plate and its corresponding half-cell, we can construct the symmetrical part by considering that, with respect to the Boussinesq approximation, the corresponding cold plate should be at the temperature
$T_{h}-2(T_{h}-T_{b})$
: see panel (b). This can be done also for the cold plate, resulting in the case shown in panel (c). We then compute a difference of temperature corresponding to cases (b) and (c),
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x0394}T_{r}=2(T_{h}-T_{b}),\quad \unicode[STIX]{x0394}T_{s}=2(T_{b}-T_{c}), & & \displaystyle\end{eqnarray}$$
and the Rayleigh and Nusselt numbers for the rough half-cell,
$Ra_{r}$
and
$Nu_{r}$
, and for the smooth half-cell,
$Ra_{s}$
and
$Nu_{s}$
,
$$\begin{eqnarray}\displaystyle Ra_{r}=\frac{\unicode[STIX]{x1D6FC}g\unicode[STIX]{x0394}T_{r}H^{3}}{\unicode[STIX]{x1D708}\unicode[STIX]{x1D705}},\quad Nu_{r}=\frac{QH}{\unicode[STIX]{x1D706}\unicode[STIX]{x0394}T_{r}}, & & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle Ra_{s}=\frac{\unicode[STIX]{x1D6FC}g\unicode[STIX]{x0394}T_{s}H^{3}}{\unicode[STIX]{x1D708}\unicode[STIX]{x1D705}},\quad Nu_{s}=\frac{QH}{\unicode[STIX]{x1D706}\unicode[STIX]{x0394}T_{s}}. & & \displaystyle\end{eqnarray}$$
This way, the behaviour of each plates can be characterized separately.
Table 2. Heat transfer data in the asymmetric cells with rough bottom plate and smooth top plate.
4.1.1 Smooth plate case
Let us first consider the smooth plate. In our previous work (Tisserand et al. Reference Tisserand, Creyssels, Gasteuil, Pabiou, Gibert, Castaing and Chillà2011), the thermal transfer of the smooth plate was not modified by the presence of the roughness on the hot plate. To verify that this still holds in the case of larger roughness elements, the Nusselt number of the smooth half-cell is plotted in figure 7.
Four sets of points are presented, obtained for the tall and small cells, and each with two roughness sizes. These four experimental configurations are compared to the
${\mathcal{R}}{\mathcal{S}}{\mathcal{C}}$
configuration previously discussed in § 3 (black open diamonds on figure 7). We then can discuss four different experimental configurations compared to the reference points. Open triangles and circles correspond to Tisserand et al. (Reference Tisserand, Creyssels, Gasteuil, Pabiou, Gibert, Castaing and Chillà2011), for aspect ratios
$\unicode[STIX]{x1D6E4}=2.5$
and
$\unicode[STIX]{x1D6E4}=1/2$
respectively, colours are Prandtl number series: blue for
$T_{b}=25\,^{\circ }\text{C}$
, green for
$T_{b}=40\,^{\circ }\text{C}$
and brown for
$T_{b}=70\,^{\circ }\text{C}$
corresponding to Prandtl numbers of 6.1, 4.3 and 2.5. Full symbols are the new results presented here, blue is for
$T_{b}=30\,^{\circ }\text{C}$
, red for
$T_{b}=60\,^{\circ }\text{C}$
. The ordinate has been extended in order to range the same values as those of figure 8 for comparison. The Nusselt numbers of the smooth half-cells and those of the reference smooth cell collapse on a single horizontal line when compensated by the prediction from the GL model. The dispersion around the horizontal line is less than 10 %.
Figure 7. Thermal transfer of the smooth plate:
$Nu_{s}$
normalized by the Grossmann–Lohse model computed with respect to the experimental Prandtl number as a function of
$Ra_{s}$
. Colours refer to the bulk temperature
$T_{b}$
, blue is
$25\,^{\circ }\text{C}$
or
$30\,^{\circ }\text{C}$
, green is
$40\,^{\circ }\text{C}$
, red
$60\,^{\circ }\text{C}$
and brown
$70\,^{\circ }\text{C}$
. Open symbols refer to the
$h_{0}=2~\text{mm}$
case of Tisserand et al. (Reference Tisserand, Creyssels, Gasteuil, Pabiou, Gibert, Castaing and Chillà2011), full symbols to the current
$h_{0}=4~\text{mm}$
elements. Triangles are for
${\mathcal{S}}{\mathcal{C}}$
, circles for
${\mathcal{T}}{\mathcal{C}}$
. Black diamonds are for the reference smooth/smooth cell presented in § 3.
Figure 8. Thermal transfer of the rough plate:
$Nu_{r}$
normalized by the Grossmann–Lohse model computed with respect to the experimental Prandtl number as a function of
$Ra_{r}$
. Symbol and colour choices are the same as figure 7. Lines stand for expression (4.5): blue lines are
${\mathcal{S}}{\mathcal{C}}$
, black lines are
${\mathcal{T}}{\mathcal{C}}$
, full lines are for
$h_{0}=4~\text{mm}$
and dashed lines for
$h_{0}=2~\text{mm}$
.
The thermal efficiency of the smooth/cold plate is not modified by the presence of roughness on the opposite hot plate and this is in good agreement with the reference results obtained in the fully smooth cell. This backs up the independence of the plates. Similar observations were also made by Wei et al. (Reference Wei, Chan, Ni, Zhao and Xia2014) in an asymmetrical cell. The behaviour of the smooth plate was not changed by the introduction of roughness elements on the opposite plate. This was tested with roughness either on the bottom or the top plate and suggests that the independence of the plates is a robust result in the range of Rayleigh numbers explored here (
$Ra$
larger than
$10^{8}$
). However, some recent particle image velocimetry (PIV) measurements (Liot et al.
Reference Liot, Ehlinger, Rusaouen, Coudarchet, Salort and Chillà2017) performed in an asymmetric rough cell seem to show a major increase of the root mean square of the velocity close to the smooth plate at nearly constant Rayleigh number. Although the introduction of roughness on one plate yields larger fluctuations in the bulk, it does not change the efficiency of the smooth plate.
4.1.2 Rough plate case
The heat transfer measurements of the rough half-cell are presented in figure 8. The four sets of points are clearly disjoined and correspond to the
${\mathcal{S}}{\mathcal{C}}$
(low Rayleigh numbers) and
${\mathcal{T}}{\mathcal{C}}$
(large Rayleigh numbers) and to the two sizes of roughness elements: the small one used by Tisserand et al. (Reference Tisserand, Creyssels, Gasteuil, Pabiou, Gibert, Castaing and Chillà2011) and the large one presented here. The comparison between those two plots gives another argument for the independence of the plates: the scaling law behaviour of one plate may significantly differ from the other.
The open symbols are the previous points from Tisserand et al. (Reference Tisserand, Creyssels, Gasteuil, Pabiou, Gibert, Castaing and Chillà2011), with a roughness size
$h_{0}=2~\text{mm}$
. The full symbols are the new measurements obtained with
$h_{0}=4~\text{mm}$
. Both exhibit a regime of enhanced heat transfer, with a scaling exponent
$a$
higher than the smooth case which starts when the height of the thermal boundary layer,
$\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D703}}$
,
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D703}}=\frac{H}{2Nu_{r}}, & & \displaystyle\end{eqnarray}$$
gets smaller than the roughness height,
$h_{0}$
, i.e. when the plate is hydrodynamically rough. That is why the Rayleigh number threshold differs for the four values of
$h_{0}/H$
(0.02, 0.01, 0.004, 0.002 from left to right in figure 8).
Table 3. Values of the parameter
$\unicode[STIX]{x1D70E}$
.
The effective scaling exponent is close to
$1/2$
and the prefactor agrees fairly well with the roughness-induced turbulent structure of the boundary layer described by Salort et al. (Reference Salort, Liot, Rusaouen, Seychelles, Tisserand, Creyssels, Castaing and Chillà2014). The lines shown in figure 8 are estimates from equation (23) of Salort et al. (Reference Salort, Liot, Rusaouen, Seychelles, Tisserand, Creyssels, Castaing and Chillà2014),
$$\begin{eqnarray}\displaystyle Nu=\frac{(2\unicode[STIX]{x1D70E})^{3/2}}{2}\left(\frac{h_{0}}{H}\right)^{1/2}Ra^{1/2}, & & \displaystyle\end{eqnarray}$$
where
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D70E}=\frac{Nu_{c}}{Ra_{c}^{1/3}}, & & \displaystyle\end{eqnarray}$$
and
$Nu_{c}$
and
$Ra_{c}$
are the critical values of
$Nu_{r}$
and
$Ra_{r}$
at the transition. Because this transition is controlled by the height of the thermal boundary layer, one can write
$$\begin{eqnarray}\displaystyle Nu_{c}=\frac{H}{2h_{0}}. & & \displaystyle\end{eqnarray}$$
The range of Rayleigh numbers is wider than was considered by Salort et al. (Reference Salort, Liot, Rusaouen, Seychelles, Tisserand, Creyssels, Castaing and Chillà2014), therefore it is not possible to choose one single value for
$\unicode[STIX]{x1D70E}$
: typically
$\unicode[STIX]{x1D70E}=0.06$
for
$Ra>10^{11}$
where the scaling exponent is
$1/3$
. For lower Rayleigh numbers, the effective value of
$\unicode[STIX]{x1D70E}$
is larger. One way to estimate its value is to use the GL model which is well suited to yield estimates of the Nusselt number as long as the plate is hydrodynamically smooth.
Figure 9. (a) Rough half-cell heat transfer measurements with
$Ra^{1/3}$
compensation. Circles and triangles are data from the present work with the same conventions as in figure 7. (b) Rough half-cell heat transfer enhancement
$Nu_{r}/Nu_{GL}$
versus compensated Rayleigh number
$Ra_{r}/Ra_{c}$
. The collapse is better because the use of GL model accounts for variation of the effective exponent. Black solid line: roughness-triggered turbulent model from (4.5). (c) Compilation of several cell geometries. Black symbols are from the cylindrical asymmetric cell from Wei et al. (Reference Wei, Chan, Ni, Zhao and Xia2014). Stars are
$h_{0}=8~\text{mm}$
R/S,
$+$
-circles are
$h_{0}=8~\text{mm}$
S/R and
$\times$
-circles are
$h_{0}=3~\text{mm}$
R/S. Squares are from the rectangular Rayleigh–Bénard cell from Salort et al. (Reference Salort, Liot, Rusaouen, Seychelles, Tisserand, Creyssels, Castaing and Chillà2014).
Let
$f_{GL}$
be the function that gives the Nusselt number for a given Rayleigh number in the GL model, i.e.
$$\begin{eqnarray}\displaystyle Nu_{GL}=f_{GL}(Ra). & & \displaystyle\end{eqnarray}$$
Then the critical Rayleigh number,
$Ra_{c}$
, beyond which the plate gets hydrodynamically rough can be estimated as
$$\begin{eqnarray}\displaystyle Ra_{c}=f_{GL}^{-1}\left(\frac{H}{2h_{0}}\right), & & \displaystyle\end{eqnarray}$$
and therefore
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D70E}=\frac{H}{2h_{0}\,f_{GL}^{-1}(H/(2h_{0}))}. & & \displaystyle\end{eqnarray}$$
The values of
$\unicode[STIX]{x1D70E}$
are shown in table 3. The prediction (4.5) has no free parameters: it is fully determined by the two geometrical parameters,
$h_{0}$
and
$H$
. As shown in figure 9, this estimate collapses the 4 datasets onto one master curve. There is however still some dispersion left. This could be caused by the effect of the Prandtl number, which is not taken into account in this description. Recent experiments of Xie & Xia (Reference Xie and Xia2017) also suggest that the heat transfer efficiency in the case of rough boundaries gets larger when the Prandtl number is increased. They evidence such increase only for larger Prandtl numbers than we do, but the geometry of the cells and roughness significantly differ. Our results are consistent with theirs in the sense that the Nusselt number tends to increase with the Prandtl number in the rough configuration only. As long as the boundary is hydrodynamically smooth, the Nusselt number does not significantly depend on the Prandtl number. However, it is possible that the velocity boundary layer thickness also plays a role near the threshold, and therefore the critical Rayleigh number may depend on the Prandtl number.
For Rayleigh numbers lower than
$Ra_{c}$
, the plate is hydrodynamically smooth, and the scaling exponent is similar to that of a smooth plate. However, several Nusselt numbers are lower than predicted by the GL model. This indicates that the thermal transfer is less efficient than in the classical smooth case. The interpretation proposed by Tisserand et al. (Reference Tisserand, Creyssels, Gasteuil, Pabiou, Gibert, Castaing and Chillà2011) was that it is caused by the additional thermal impedance of the fluid which fills the space around the roughness elements and locally thickens the boundary layer.
In the high Rayleigh number limit, the data exhibit a second transition, both for the tall and small cells. In this range, the heat transfer is still larger than the smooth case, and larger than effective surface increase (40 %), but the scaling exponent
$a$
is less than
$1/2$
. It may be related to other published works which have reported an increase of the prefactor
$C$
caused by enhanced plume emissions, rather than an increase of the exponent
$a$
due to a change in the structure of the boundary layer. This seems consistent, in particular, with the recent results of Xie & Xia (Reference Xie and Xia2017) in their symmetric rough cell where both plates have pyramid-shaped roughness elements.
To the best of our knowledge, there are only two other published datasets in asymmetric Rayleigh–Bénard cells: Wei et al. (Reference Wei, Chan, Ni, Zhao and Xia2014) and Salort et al. (Reference Salort, Liot, Rusaouen, Seychelles, Tisserand, Creyssels, Castaing and Chillà2014). The former performed experiments in a cylinder of aspect ratio close to 1. They used pyramidal elements as roughness, sketched in figure 10. The base length
$d$
is equal to
$2h_{0}$
. The surface increase induced was 41 %. The latter uses similar square-stud roughness but within a rectangular cell with vertical aspect ratio
$\text{width}/\text{height}=1$
and horizontal aspect ratio
$\text{depth}/\text{width}=0.25$
.
Figure 10. Sketch of the pyramidal roughness used by Wei et al. (Reference Wei, Chan, Ni, Zhao and Xia2014).
These datasets are shown in figure 9. As could be expected, the points do not exactly collapse. Indeed, the details of the geometry may change prefactors in the Nusselt numbers. However, the observations of the three regimes hold, the Rayleigh number thresholds are all consistent and the enhancement goes beyond the increase solely yielded by the increase of the effective surface area.
5 Discussion
Thermal transfer measurements have been carried out in a cylindrical Rayleigh–Bénard cell with square roughness on the bottom plate for two different cell aspect ratios. The results have been compared to previous studies, with different roughness shapes and dimensions. The Grossmann–Lohse model is used to estimate the Nusselt numbers when the boundaries are smooth.
We have shown that both plates are independent, at least when the thermal impedance is considered. The impedance of the smooth boundary is fully determined by its temperature difference to the bulk, and the classical scaling law is fully recovered when the Rayleigh number of the smooth half-cell is considered. Beyond a critical Rayleigh number corresponding to a thermal boundary layer smaller than the typical roughness size, the thermal impedance of the rough boundary is smaller than the smooth case, and is well described by a turbulent destabilization of the boundary layer.
Such destabilization was confirmed directly by the work of Liot et al. (Reference Liot, Salort, Kaiser, du Puits and Chillà2016) where PIV measurements have exhibited a turbulent velocity profile in the boundary layer. However, other enhancement mechanisms were also proposed in the past. In particular, Du & Tong (Reference Du and Tong2000) used thermochromic liquid crystals to measure temperature fields close to a rough plate. Tips of roughness elements seemed to be preferential points of nucleation of thermal plumes. Following this hypothesis, we can assume that a cube can induce a higher increase of the thermal transfer than a pyramid since a cube is formed of four singularities (i.e. corners) interacting with the fluid whereas a pyramid only exhibits one singularity. The details of the shape of the roughness elements, such as sharp edges and vertical surfaces, surely bear some importance and may be investigated further. As can be seen in figure 9, the curves obtained with cubic roughness elements are above those obtained with pyramids, which backs up the idea that more plumes are induced by cubes than pyramids.
At lower Rayleigh numbers,
$Nu_{r}/Nu_{GL}$
is lower than 1 for several experiments (Tisserand et al.
Reference Tisserand, Creyssels, Gasteuil, Pabiou, Gibert, Castaing and Chillà2011; Wei et al.
Reference Wei, Chan, Ni, Zhao and Xia2014). This could be caused by an additional thermal resistance induced by motionless fluid between the roughness elements, as was suggested by PIV measurements (Liot et al.
Reference Liot, Salort, Kaiser, du Puits and Chillà2016).
Though the actual Nusselt numbers depend on the details of the geometry, several observations hold for all known set-ups, regardless of the roughness or cell geometry. Like Xie & Xia (Reference Xie and Xia2017), three regimes can be consistently exhibited: (i) below
$Ra_{c}$
,
$Nu_{r}/f_{GL}(Ra_{r})$
are horizontal lines, meaning that the behaviour is similar to a smooth plate, and consistent with the GL model within
$\pm 20\,\%$
; (ii) a regime of increased scaling exponent occurs beyond
$Ra_{c}$
, fairly compatible with
$a=1/2$
, and more precisely with (4.5), which suggests a roughness-triggered turbulent boundary layer structure; (iii) a third regime, where the heat transfer is enhanced, more than the increase of effective surface area, but the scaling exponent is lower than in the second regime, which suggests that turbulent destabilization of the boundary layer may no longer be the dominant enhancement mechanism.
Acknowledgements
We thank M. Moulin for technical assistance. We also want to thank A. Sergent and B. Podvin for useful discussions on this subject. This work has been supported by the Région Rhône-Alpes (Cible 2011, no. 2770).
