2.1 The convection cell

The experiments were carried out in four cylindrical convection cells. Each of the convection cells consisted of top and bottom plates and a Plexiglas side wall. Except for the geometry of roughness elements on the top and bottom plates, the constructions of the convection cells were otherwise identical.

The side wall was made of Plexiglas tube with a wall thickness of 4 mm. The convection cells had a diameter $D$ of 19.2 cm and a height $H$ of 20.3 cm measured from the base of the roughness elements. Thus the aspect ratios $\unicode[STIX]{x1D6E4}=D/H$ of the convection cells were approximately unity. The top and bottom plates were made of 12 mm thick copper except for the ones for rough cells with $\unicode[STIX]{x1D706}=1.9$ which were made of aluminium. The thermal conductivity of aluminium is $205~\text{W}~{\text{m}^{-1}\,}^{\circ }\text{C}^{-1}$ at temperature $25\,^{\circ }\text{C}$ , which is approximately half that of copper. The differences in heat transport properties in convection cells with plates made of copper and aluminium, i.e. the effects of the finite conductivity of the plates (Verzicco Reference Verzicco2004; Brown et al. Reference Brown, Nikolaenko, Funfschilling and Ahlers2005), have been shown to be minimal in the $Ra$ and $Pr$ range reported in the present studies (Wei, Ni & Xia Reference Wei, Ni and Xia2012). The pyramid-shaped roughness elements were directly machined on the top and bottom plates that were then plated with nickel to prevent oxidation. The roughness elements were arranged in a square lattice form (see figure 1), which was the same as those used previously (Wei et al. Reference Wei, Chan, Ni, Zhao and Xia2014; Xie et al. Reference Xie, Huang, Funfschilling, Li, Ni and Xia2015). By keeping the height of roughness elements $h$ a constant at 8 mm and changing the base width $w$ of individual roughness elements from 16 mm to 2 mm, rough plates with roughness parameter $\unicode[STIX]{x1D706}$ , defined as the ratio of $h$ and $w$ , at 0.5, 1.0, 1.9 and 4.0 were used in the present study. With the increase of $\unicode[STIX]{x1D706}$ , the number density $n$ (in unit of number of roughness element per $\text{cm}^{2}$ ) also increases, i.e. $n=0.38$ ( $\unicode[STIX]{x1D706}=0.5$ ), $1.5~(\unicode[STIX]{x1D706}=1.0)$ , 6.1 ( $\unicode[STIX]{x1D706}=1.9$ ) and 24.5 ( $\unicode[STIX]{x1D706}=4.0$ ). To clarify whether the number density or the $\unicode[STIX]{x1D706}$ is the key parameter that characterises the system, another set of rough plates with $\unicode[STIX]{x1D706}=4.0$ and a roughness height of 4 mm was used ( $n$ was increased by a factor of 4 in this case). The present study implies that $\unicode[STIX]{x1D706}$ is a more suitable parameter to characterise the geometry of the roughness elements and the response of the system, rather than the number density of the roughness elements.

The top plate was cooled by passing temperature-controlled water from a water circulating bath through a chamber fitted on its top (Poly Science, 9702). The cooling water entered the chamber through two inlets and left the chamber through two outlets. The arrangements of the inlets and outlets were such that the two inlets were along one diameter of the plate and the two outlets were along another diameter perpendicular to the inlets. Temperature stability of the cooling water was better than $0.01\,^{\circ }\text{C}$ . Furthermore, a stirring blade driven by the inlet and outlet flows was installed inside the cooling chamber to ensure temperature uniformity across the top plate. The bottom plate was electrically heated by two rubber heaters connected in series and sandwiched in between the bottom plate and a copper plate with a thickness of 10 mm (see, e.g. figure 1). The heaters were connected to a DC power supply with a long term voltage stability of 99.99 % (Sorensen, XFR300-4). To ensure good thermal contact, a thin layer of thermally conductive paste was used between the heaters and the plates. The largest temperature difference within the same plate was approximately $1.5\,\%\unicode[STIX]{x0394}T$ . It should be noted that in this arrangement, the temperature boundary condition for the top plate was constant temperature and that for the bottom plate was constant heat flux.

To prevent heat leakage and to minimise the influence of the environmental temperature fluctuation, the convection cells were wrapped with Styrofoam with a thickness of 6 cm. A temperature regulated copper basin was placed beneath the convection cell to prevent heat leakage from the bottom plate by maintaining the same temperature at the top and bottom of the basin. The whole set-up was put into a thermostat where the temperature was regulated to match the mean temperature $T_{m}$ of the top and bottom plates. The temperature stability of the thermostat was better than $0.1\,^{\circ }\text{C}$ .

Figure 1.

Photos of the bottom plate of a rough cell with $\unicode[STIX]{x1D706}=1.0$ . (a) Top view; (b) side view. The red heater connection wires can be seen at the top of (a). Plates with $\unicode[STIX]{x1D706}$ values of 0.5, 1.9 and 4.0 have the same design.

2.2 The working fluids

Only approximately one and a half decades of $Ra$ can be achieved in a single experimental set-up using a classical working fluid such as water. To extend the $Ra$ range, two kinds of working fluids, namely deionised water and Flourinert FC770 (FC770 hereafter, 3M Inc.) were used. Water has been widely used in the study of turbulent convection and we will mention here only the properties of FC770. The data listed below are given by the manufacturer (3M Inc.) and are all obtained at the temperature $25~^{\circ }\text{C}$ . The FC770 has $\unicode[STIX]{x1D712}=0.063~\text{W}~{\text{m}^{-1}~}^{\circ }\text{C}^{-1}$ , $\unicode[STIX]{x1D6FC}=0.00148~^{\circ }\text{C}$ , $\unicode[STIX]{x1D708}=7.9\times 10^{-7}~\text{m}^{2}~\text{s}^{-1}$ , the specific heat $c_{p}=1038~\text{J}~{\text{kg}^{-1}~}^{\circ }\text{C}^{-1}$ and the density $\unicode[STIX]{x1D70C}=1793~\text{kg}~\text{m}^{-3}$ .

The working fluid was filled into the convection cell through a hole attached with a stainless steel tube ( ${\sim}8$  mm in diameter) located at the centre of the top plate. The working fluids were degassed in the following way to ensure there was no bubble forming during a single run of the experiment (i.e. a $Ra$ scan of $Nu$ for a fixed $Pr$ ), which typically lasted for three weeks. Deionised water was boiled for a hour and then cooled down to room temperature before filling into the convection cell. For FC770, it was first filled into the convection cell and degassed by keeping the temperature of the top and bottom plate, and thus the whole convection cell, at $50\,^{\circ }\text{C}$ for 24 h. Then the temperature of the convection cell was then lowed to $25\,^{\circ }\text{C}$ which was the $T_{m}$ for all the experiments using FC770, and the bubbles formed were totally removed.

Figure 2.

$Ra$ – $Pr$ phase space explored for different values of $\unicode[STIX]{x1D706}$ . The case for $\unicode[STIX]{x1D706}=0.5$ is only studied for $Pr=23.34$ . All experiments were conducted in convection cells with roughness height $h=8$  mm, except for the ones corresponding to hexagons that were conducted in convection cell with $h=4~\text{mm}$ .

The explored $Ra$ and $Pr$ phase space at different values of $\unicode[STIX]{x1D706}$ is shown in figure 2. Except for $\unicode[STIX]{x1D706}=0.5$ which was investigated for a single $Pr$ (23.34) in the present studies, all the other values of $\unicode[STIX]{x1D706}$ were studied for the four values of $Pr$ , i.e. $Pr=3.67$ (water, $T_{m}=50.0\,^{\circ }\text{C}$ ), $Pr=4.34$ (water, $T_{m}=40.0\,^{\circ }\text{C}$ ), $Pr=6$ .14 (water, $T_{m}=25.0\,^{\circ }\text{C}$ ) and $Pr=23.34$ (FC770, $T_{m}=25.0\,^{\circ }\text{C}$ ). The experiments covered $7.50\times 10^{7}\leqslant Ra\leqslant 1.31\times 10^{11}$ . The temperature differences applied across the convection cell using water as the working fluid for different values of $\unicode[STIX]{x1D706}$ were: $0.98\,^{\circ }\text{C}$ to $26.99\,^{\circ }\text{C}$ ( $\unicode[STIX]{x1D706}=4.0$ ); $0.47\,^{\circ }\text{C}$ to $20.16\,^{\circ }\text{C}$ ( $\unicode[STIX]{x1D706}=1.9$ ) and $0.89\,^{\circ }\text{C}$ to $24.75\,^{\circ }\text{C}$ ( $\unicode[STIX]{x1D706}=1.0$ ). The temperature differences applied using FC770 as working fluid at different values of $\unicode[STIX]{x1D706}$ were: $1.40\,^{\circ }\text{C}$ to $22.14\,^{\circ }\text{C}$ ( $\unicode[STIX]{x1D706}=4.0$ ); $0.99\,^{\circ }\text{C}$ to $17.25\,^{\circ }\text{C}$ ( $\unicode[STIX]{x1D706}=1.9$ ); $1.11\,^{\circ }\text{C}$ to $27.63\,^{\circ }\text{C}$ ( $\unicode[STIX]{x1D706}=1.0$ ) and $0.96\,^{\circ }\text{C}$ to $29.66\,^{\circ }\text{C}$ ( $\unicode[STIX]{x1D706}=0.5$ ).

2.3 The temperature measurements

Temperature of the top plate was measured using four thermistors (Omega, Model 44031) embedded at a distance of $D/4$ from the centre of the plate and uniformly in the azimuthal direction. Four thermistors at the mirror location of the top plate embedded in the bottom plate with one additional thermistor at the centre of it were used to measure the temperature of the bottom plate. The thermistors were calibrated individually and separately in a temperature-controlled water bath. The temperature range of calibration was from $5\,^{\circ }\text{C}$ to $70\,^{\circ }\text{C}$ with a temperature accuracy better than $0.01\,^{\circ }\text{C}$ . The resistances of the thermistors were measured using a $6(1/2)$ digital multimeter (Keithley, 2700) and were converted to temperature using the calibration curve. The sampling rate of the temperature measurement was ${\sim}0.7$  Hz.

Local temperatures at the cell centre and ${\sim}1~\text{cm}$ away from the side wall were measured using two small thermistors with a bead diameter of $0.38~\text{mm}$ and a time constant of 30 ms in liquid (Measurement Specialities, Model: G22K7MCD419). These small thermistors had a temperature accuracy of $0.01\,^{\circ }\text{C}$ . They were guided into the convection cell using a stainless steel tube with an outer diameter of 1.1 mm and wall thickness of 0.2 mm (Goodfellow, Stainless Steel – AISI 304 Tube). Each of the small thermistors had an independent temperature measurement system specially designed to measure the temperature fluctuation. Detailed design of the measurement system can be found in Zhou & Xia (Reference Zhou and Xia2001). Briefly, the thermistor acted as one arm of a AC Wheatstone electrical bridge. The resistance fluctuations of the thermistor induced by the temperature fluctuations in the convection cell were converted to the voltage fluctuations of the electrical bridge which were magnified using a lock-in amplifier (Stanford Research System, SR830). The output signal from the lock-in amplifier was then digitised and stored by a dynamic signal analyser (HP, 35670A). The measured voltage was converted to resistance and then to temperature using the calibration curve of the thermistors. The sampling rate of the local temperature varied from 16 Hz to 128 Hz depending on $Ra$ and $Pr$ to ensure that the small-scale temporal temperature fluctuations were fully resolved.

From the temperature measurements, the heat transport across the convection cell, i.e. the Nusselt number $Nu=qH/(\unicode[STIX]{x1D712}\unicode[STIX]{x0394}T\unicode[STIX]{x03C0}(D/2)^{2})$ , local temperatures and the Reynolds number $Re=UH/\unicode[STIX]{x1D708}$ were measured as a function of $Ra$ , $Pr$ and $\unicode[STIX]{x1D706}$ , where $q$ was the input heating power at the bottom plate, $\unicode[STIX]{x1D712}$ was the thermal conductivity of the working fluid and $U$ was the characteristic velocity of the LSC. Each measurement lasted at least 12 h after the system had reached a steady state to ensure that the system had explored all flow states.

2.4 The viscous boundary layer measurements

The dynamics of the viscous boundary layer in the rough cell with $\unicode[STIX]{x1D706}=1.0$ was measured using a commercial PIV system that had been documented elsewhere (Xia, Sun & Zhou Reference Xia, Sun and Zhou2003). The flow was seeded with particles with a diameter of $2~\unicode[STIX]{x03BC}\text{m}$ and density matched with water. A 2-D flow region with a size of $20~\text{mm}\times 28~\text{mm}$ was measured within the LSC’s circulation plane with $x$ denoting the direction along the circulation path of the LSC and $z$ denoting the direction pointing upwards at the centre of bottom plate (see, e.g. figure 7). The origin of the coordinate system was located at the centre of the plate. To minimise optical distortion of the particle images induced by the curved surface of the cell, a jacket filled with water with flat window was adhered to the convection cell. The optical arrangement was such that the measurement plane was within one groove of the rough plate and the images were acquired perpendicular to it. A mask with the shape exactly matching that of the plate geometry was applied to the captured images to exclude the parts blocked by the roughness elements.

The velocity field was obtained by cross-correlating two consecutive images taken with a time separation that matched the velocity of the flow at different $Ra$ . The velocity map consisted of $63\times 79$ vectors in the $x$ - and $z$ - directions respectively. Totally 26 136 vector maps were acquired at a sampling rate of 2.2 Hz, which lasted for ${\sim}3.3$  h. The velocity fluctuation map was obtained by subtracting the mean velocity map from each of the instantaneous velocity map. The measurements were done at $Pr=4.34$ and $3.55\times 10^{8}\leqslant Ra\leqslant 9.62\times 10^{9}$ . To lock the direction of the LSC, the convection cell was tilted with respect to the vertical direction by ${\sim}0.5^{\circ }$ .

3.1 Heat transport measurements

We show in this section that the heat transport in turbulent RBC over rough plates may be classified into three regimes based on the scaling law of $Nu$ versus $Ra$ . In addition, the heat transport scaling in the enhanced regime (Regimes II and III) can be manipulated by changing the roughness geometry. The transition from Regime I to Regime II in a rough cell with $h=4$  mm and $\unicode[STIX]{x1D706}=4.0$ is discussed in section § 3.1.1. The heat transport measurement with changing the geometry of roughness elements, i.e. for different values of $\unicode[STIX]{x1D706}$ , is presented in § 3.1.2 together with the observation of Regime II and Regime III.

3.1.2 Effects of roughness geometry in the heat transport enhanced regime

We present measurements of $Nu$ as a function of $Ra$ in convection cells with a roughness height $h$ of 8 mm and different roughness geometries in this subsection. The roughness height was much larger than $\unicode[STIX]{x1D6FF}_{th}$ for the explored range of $Ra$ . Therefore, Regime I behaviour was not observed here. Instead, we focus on Regime II and Regime III (a new regime to be defined below).

Figure 3.

(a)  $Nu$ as a function of $Ra$ in a rough cell with a roughness elements height of 4 mm. The down-pointing triangles are taken from Funfschilling et al. (Reference Funfschilling, Brown, Nikolaenko and Ahlers2005) ( $\unicode[STIX]{x1D6E4}=1.0,Pr=4.38$ ), the hexagons are taken from Sun et al. (Reference Sun, Ren, Song and Xia2005) ( $\unicode[STIX]{x1D6E4}=2.0,Pr=4.34$ ), the stars are taken from Wei et al. (Reference Wei, Chan, Ni, Zhao and Xia2014) ( $\unicode[STIX]{x1D6E4}=1.0,Pr=4.34$ ). (b) The compensated plot of the same data as in (a).

Figure 4.

(a,d,g,j)  $Nu$ as a function of $Ra$ in rough cells for (a)  $\unicode[STIX]{x1D706}=4.0$ ; (d)  $\unicode[STIX]{x1D706}=1.9$ ; (g)  $\unicode[STIX]{x1D706}=1.0$ and ( $j$ )  $\unicode[STIX]{x1D706}=0.5$ . The down-pointing triangles are from Funfschilling et al. (Reference Funfschilling, Brown, Nikolaenko and Ahlers2005) ( $\unicode[STIX]{x1D6E4}=1.0,Pr=4.38$ ), the hexagons are from Sun et al. (Reference Sun, Ren, Song and Xia2005) ( $\unicode[STIX]{x1D6E4}=2.0,Pr=4.34$ ) and the stars are from Wei et al. (Reference Wei, Chan, Ni, Zhao and Xia2014) ( $\unicode[STIX]{x1D6E4}=1.0,Pr=4.38$ ). Data at $Pr=4.34$ in $(j)$ are from Wei et al. (Reference Wei, Chan, Ni, Zhao and Xia2014);  $Nu$ at $Pr=23.34$ at $\unicode[STIX]{x1D706}=4.0$ and 1.9 are larger than those at $Pr=3.57$ at the same $Ra$ . For a better comparison, they are shifted downwards respectively by 1.5 and 1.34 and displayed using squares with a cross. The solid lines are power-law fits to the data larger and smaller than $Ra^{\ast }\sim 4\times 10^{9}$ . (b,c,e,f,h,i,k,l) The compensated plots of $Nu$ with the fitted $Ra$ scaling for $Ra$ larger and smaller than $Ra\sim 4\times 10^{9}$ from (a,d,g,j).

Figure 4(a,d,g,j) shows $Nu$ as a function of $Ra$ and $Pr$ in rough cells with $\unicode[STIX]{x1D706}=4.0,1.9,1.0$ and 0.5, respectively. The solid lines in the figure are power-law fit to each data set for $Ra$ larger and smaller than a critical Rayleigh number $Ra^{\ast }=4\times 10^{9}$ . The same data from smooth cells as shown in figure 3 are shown in these figures. The general conclusion that one can draw from these figures is that the heat transport is enhanced in rough cells.

We first discuss the case with $\unicode[STIX]{x1D706}=1.0$ (figure 4 g). Within the precision of the experiments, data at different values of $Pr$ but the same $Ra$ collapse onto each other, suggesting that the $Pr$ -dependence in the rough cell with $\unicode[STIX]{x1D706}=1.0$ is very weak in the present parameter range, which is consistent with observation in a smooth cell (Xia, Lam & Zhou Reference Xia, Lam and Zhou2002). Detailed examination of the data shows that there is a transition of the $Nu$ – $Ra$ scaling at $Ra^{\ast }\sim 4\times 10^{9}$ . Below and above $Ra^{\ast }$ , the data can be fitted by different scaling laws, i.e. $Nu\sim Ra^{0.43\pm 0.01}$ and $Nu\sim Ra^{0.35\pm 0.01}$ . The different scaling laws suggest that the heat transport enhancement regime in a rough cell can be further classified into two regimes for the $Pr$ range covered in the present experiments. We term the two regimes before and after $Ra^{\ast }$ as Regime II and Regime III. The data are compensated by the scaling exponents in Regimes II and III and shown in figure 4(h,i). A gradual transition at $Ra^{\ast }\sim 4\times 10^{9}$ is clearly seen.

Data at $\unicode[STIX]{x1D706}=4.0,1.9$ and 0.5 are plotted in figure 4(a,d,j). One interesting feature revealed in figure 4(a,d) is that the $Nu$ at $Pr=23.34$ is larger than that at $Pr=3.57$ when $Ra$ is the same. We note that this $Nu$ -enhancement for FC770 ( $Pr=23.34$ ), as compared to water ( $Pr=3.57$ ), only occurs for the large $\unicode[STIX]{x1D706}$ cases. In fact, for the smallest value of $\unicode[STIX]{x1D706}$ ( $=0.5$ ) there is a slight decrease in $Nu$ . This is quite surprising and unexpected, as in the case of a smooth cells, $Nu$ peaks around $Pr=3\sim 4$ (Xia et al. Reference Xia, Lam and Zhou2002). It is not clear why the heat transport enhancement in rough cells has this peculiar $Pr$ -dependence, and furthermore this behaviour has a strong $\unicode[STIX]{x1D706}$ dependence, i.e. it is 150 % for $\unicode[STIX]{x1D706}=4.0$ , 134 % for $\unicode[STIX]{x1D706}=1.9$ , 4 % for $\unicode[STIX]{x1D706}=1.0$ and $-8\,\%$ for $\unicode[STIX]{x1D706}=0.5$ . One possible reason leading to such a $Pr$ -abnormality is that the lifetime of the thermal plumes at larger $Pr$ is larger compared to the small $Pr$ case. Assuming that the thermal plumes have typical length scale comparable to the thermal BL thickness, the estimated thermal diffusion time scale, i.e. $\unicode[STIX]{x1D6FF}_{th}^{2}/\unicode[STIX]{x1D705}$ , for FC770 is 14.5 s and that for water is 2.6 s at $Ra\approx 10^{10}$ . When normalised by the LSC turnover time, the dimensionless plume lifetime for FC770 is 0.25 and that for water is 0.11 for $\unicode[STIX]{x1D706}=4.0$ . Another likely reason is the stronger plume clustering effect in rough cells with larger $\unicode[STIX]{x1D706}$ as shown by the shadowgraph images in figure 16, which also has the effect of increasing plume lifetime or slowing the decay of plumes. While it is difficult to quantify the combined effect of these two factors, it appears that larger values of $\unicode[STIX]{x1D706}$ together with a larger $Pr$ result in a more efficient enhancement in the heat transport. For small $\unicode[STIX]{x1D706}$ and/or small $Pr$ , thermal plumes lose more heat to the surrounding fluid and therefore $Nu$ will not be enhanced as much.

To determine more accurately the scaling law of $Nu$ versus $Ra$ , the data at $Pr=23.34$ in figure 4(a,d) are shifted downwards by a constant, i.e. 1.50 in (a) and 1.34 in (d), to collapse data at different $Pr$ but the same $Ra$ . The transition of the heat transport scaling around $Ra^{\ast }$ is observed for all $\unicode[STIX]{x1D706}$ studied. The power-law fits of $Nu=ARa^{\unicode[STIX]{x1D6FC}}$ in Regimes II and III for different values of $\unicode[STIX]{x1D706}$ are summarised in table 1. It should be noted that the shifted high $Pr$ data were included in the fitting. Since $Ra$ obtained using water as working fluid at $Pr=3.57$ and that using FC770 as working fluid at $Pr=23.34$ had an overlapping range of less than a half-decade, i.e. from $6\times 10^{9}$ to $9\times 10^{9}$ , a determination of the $Pr$ dependence in Regime III was not possible in the present experiments. It is seen that the heat transport law is distinctively different in Regime II and Regime III. This can be seem more clearly from the compensated plot in the right panel next to figure 4(a,d,j).

The experimentally determined scaling exponent $\unicode[STIX]{x1D6FC}$ , i.e. $Nu\sim Ra^{\unicode[STIX]{x1D6FC}}$ , in both Regime II and Regime III increases significantly with $\unicode[STIX]{x1D706}$ . In Regime II, the $\unicode[STIX]{x1D6FC}$ increases from 0.36 to 0.59 when $\unicode[STIX]{x1D706}$ is increased from 0.5 to 4.0; and in Regime III, it increases from 0.30 to 0.50. These data are plotted in figure 5. If a power-law fit is attempted with these data within the limited range of $\unicode[STIX]{x1D706}$ , we obtain $\unicode[STIX]{x1D6FC}_{II}=0.42\unicode[STIX]{x1D706}^{0.23\pm 0.02}$ and $\unicode[STIX]{x1D6FC}_{III}=0.35\unicode[STIX]{x1D706}^{0.24\pm 0.02}$ which are shown as solid lines in the figure. Within the error bar, the $\unicode[STIX]{x1D706}$ -dependences of $\unicode[STIX]{x1D6FC}_{II}$ and $\unicode[STIX]{x1D6FC}_{III}$ are the same except that the prefactor of $\unicode[STIX]{x1D6FC}_{II}$ is larger than $\unicode[STIX]{x1D6FC}_{III}$ . The results suggest that the scaling law of the heat transport in turbulent RBC over rough plates can be manipulated using $\unicode[STIX]{x1D706}$ . In this sense, the present experiments demonstrate clearly the importance of roughness geometry on turbulent heat transport. Any realistic model for turbulent thermal convection in nature should take into account the effects of wall roughness.

Very recently, it has been found that there is an upper bound for the heat transport in turbulent convection over rough plates, i.e. $Nu$ cannot grow faster than $Nu\sim Ra^{1/2}$ in the limit of $Ra\rightarrow \infty$ (Goluskin & Doering Reference Goluskin and Doering2016). With increasing $Ra$ , Regime II is just a transient and Regime III is the dominant one. Our experiments showed that at $\unicode[STIX]{x1D706}=4.0$ the scaling exponent already reached 0.50, which was consistent with the upper bound derived (Goluskin & Doering Reference Goluskin and Doering2016). One interesting question is whether the scaling exponent will saturate to 0.50 with further increasing $\unicode[STIX]{x1D706}$ which is very challenging to achieve experimentally due to technical difficulties in machining the plates. DNS studies may provide helpful insights into this aspect of the problem.

Table 1.

Measured scaling law of the Nusselt number $Nu$ and normalised temperature fluctuations $\unicode[STIX]{x1D70E}_{T}/\unicode[STIX]{x0394}T$ in Regime III for different values of $\unicode[STIX]{x1D706}$ .

We next study the $\unicode[STIX]{x1D706}$ -dependence of the heat transport enhancement in Regime II and Regime III. Figure 6 shows the heat transport enhancement in rough cells compared with that in a smooth cell. It is seen that the heat transport enhancement in Regime II (open symbols in figure 6) first increases and then decreases with increasing $\unicode[STIX]{x1D706}$ . The situation is very different in Regime III (solid symbols in figure 6) where it is found that the heat transport enhancement increases monotonically with increasing $\unicode[STIX]{x1D706}$ . A maximum heat transport enhancement of 637 % is observed at $\unicode[STIX]{x1D706}=4.0$ .

We note that in a DNS study, Wagner & Shishkina (Reference Wagner and Shishkina2015) show that the heat transport is enhanced more for slender roughness elements, i.e. larger $\unicode[STIX]{x1D706}$ , using only four roughness elements with a roughness height of 0.125H (‘obstacles’ in Wagner & Shishkina Reference Wagner and Shishkina2015) in a quasi-2-D convection cell. The observation in Wagner & Shishkina (Reference Wagner and Shishkina2015) is qualitatively in agreement with what we find here, e.g. heat transport enhancement generally increases with $\unicode[STIX]{x1D706}$ (see e.g. figure 6).

3.2 Transition from Regime II to Regime III: the viscous boundary layer cross-over

To understand what may have happened at $Ra^{\ast }$ , i.e. the transition from Regime II to Regime III, we recall that there are two types of BLs in turbulent RBC, i.e. the thermal BL and the viscous BL. The viscous BL is thicker than the thermal BL for $Pr>1$ . According to the Grossmann–Lohse model for turbulent RBC, BLs play crucial role in determine the global heat transport (Grossmann & Lohse Reference Grossmann and Lohse2000). The transition seen in figure 4 may be understood in terms of the viscous BL cross-over, i.e. the roughness elements not only strongly perturb the thermal BL but also alter the viscous BL in Regime III. To verify this, we directly measured the viscous boundary layer in the rough cell with $\unicode[STIX]{x1D706}=1.0$ .

Figure 5.

The scaling exponent $\unicode[STIX]{x1D6FC}$ from power-law fits to $Nu$ versus $Ra$ , i.e. $Nu\sim Ra^{\unicode[STIX]{x1D6FC}}$ , in Regime II (circles) and Regime III (triangles) as a function of $\unicode[STIX]{x1D706}$ . Also plotted in the figure is the scaling exponent $\unicode[STIX]{x1D6FD}$ from power-law fits to $Re$ versus $Ra$ (squares), i.e. $Re\sim Ra^{\unicode[STIX]{x1D6FD}}$ , as a function of $\unicode[STIX]{x1D706}$ . The solids lines are empirical power-law fits to the respective data sets: $\unicode[STIX]{x1D6FC}_{II}=0.43\unicode[STIX]{x1D706}^{0.24\pm 0.02}$ , $\unicode[STIX]{x1D6FC}_{III}=0.37\unicode[STIX]{x1D706}^{0.23\pm 0.02}$ and $\unicode[STIX]{x1D6FD}=0.49\unicode[STIX]{x1D706}^{0.08\pm 0.01}$ .

Figure 6.

The heat transport enhancements in Regime II (open symbols) and Regime III (solid symbols) as a function of $\unicode[STIX]{x1D706}$ .

Figure 7.

The instantaneous (a) and time-averaged (b) velocity fields measured near the bottom plate within the large-scale circulation plane ( $Ra=4.22\times 10^{9}$ , $Pr=4.34$ and $\unicode[STIX]{x1D706}=1.0$ ). The velocity magnitude (in units of  $\text{mm}~\text{s}^{-1}$ ) is colour coded with the colour bar nearby each figure and is also represented by the length of the vectors. The velocity field is not accessible experimentally below the dashed lines.

Figure 8.

(a) Typical time-averaged horizontal velocity $U$ and root-mean-square horizontal velocity $\unicode[STIX]{x1D70E}_{u}$ profiles ( $Ra=4.22\times 10^{9}$ , $\unicode[STIX]{x1D706}=1.0$ and $Pr=4.34$ ). The vertical dashed line indicates the location of the viscous boundary layer. (b) Normalised viscous boundary layer thickness $\unicode[STIX]{x1D6FF}_{v}/H$ versus $Ra$ . The dashed line marks the height of the roughness elements. The datum at the highest $Ra$ (symbols with cross, $\unicode[STIX]{x0394}T=29.83~\text{K}$ ) shows different behaviours with other data points which we believe is an artefact due to the strong optical distortion of the images used for PIV analysis.

Figure 9.

$Nu$ as a function of $Re$ measured in rough cells with different values of $\unicode[STIX]{x1D706}$ at (a)  $Pr=23.34$ (b)  $Pr=6.14$ (c)  $Pr=4.34$ and (d)  $Pr=3.57$ .

An example of the instantaneous velocity map measured at $Ra=4.22\times 10^{9}$ and $Pr=4.34$ is shown in figure 7(a). The triangles outline the space occupied by the roughness elements schematically. The velocity magnitude is coded by both the colour and the length of the vectors. The instant velocity field shows that the mean flow is going from the right to the left and there are intermittent bursts of the flow velocity. Figure 7(b) depicts the time-averaged velocity field with the velocity magnitude coded also in both colour and the length of the vectors. One may notice that the time-averaged maximum velocity magnitude is significantly reduced when compared with the instantaneous one as revealed by the scale bars. After averaging over time one sees that the mean velocity is uniform with respect to $x$ except very close to the plate, e.g. in between the roughness elements. A tiny counter-rotating vortex may be identified in the region $-8~\text{mm}<x<-5~\text{mm}$ and $0~\text{mm}<z<4~\text{mm}$ . The mean velocity field varied weakly with $x$ (figure 7 b). We thus averaged the mean velocity field along $x$ -direction to obtain the mean velocity and the root-mean-square (r.m.s.) velocity profiles.

Figure 8(a) shows an example of the horizontal mean velocity profile $U(z)$ and the profile of the r.m.s. velocity $\unicode[STIX]{x1D70E}_{U}(z)$ measured at $Ra=4.22\times 10^{9}$ and $Pr=4.34$ . A remarkable feature shown in figure 8(a) is that the r.m.s. velocity is comparable to the mean velocity very close to the plate, and it is even larger than the mean velocity within the viscous BL. The origin of the enhanced velocity fluctuation is not clear but speculatively this should largely correlate with the intermittent bursts of thermal plumes. In contrast, the maximum r.m.s. velocity in a smooth cell is only approximately 30 % of the maximum mean velocity and it is always smaller than the mean velocity within the BL region (Sun, Cheung & Xia Reference Sun, Cheung and Xia2008; Wei & Xia Reference Wei and Xia2013).

The viscous BL thickness was determined using the so-called slope method: the intersection between the extrapolation of the linear part of the velocity profile very close to the plate and the maximum mean horizontal velocity was defined as the thickness of the viscous boundary layer $\unicode[STIX]{x1D6FF}_{v}$ (indicated by the vertical dashed line in figure 8 a) (Xin, Xia & Tong Reference Xin, Xia and Tong1996; Xin & Xia Reference Xin and Xia1997; Qiu & Xia Reference Qiu and Xia1998; Sun et al. Reference Sun, Cheung and Xia2008; Wei & Xia Reference Wei and Xia2013). The $Ra$ -dependence of the viscous BL thickness normalised by the height of the convection cell is plotted in figure 8(b). With increasing $Ra$ , $\unicode[STIX]{x1D6FF}_{u}/H$ becomes thinner and thinner. At $Ra\sim 3\times 10^{9}$ , $\unicode[STIX]{x1D6FF}_{u}/H$ approximately equals to $h/H$ . This roughly corresponds to $Ra^{\ast }$ seen in the measured $Nu$ ( $Ra^{\ast }\sim 4\times 10^{9}$ ). This observation indicates that the transition of the heat transport law between Regimes II and III may be caused by the viscous BL becoming smaller than the height of the roughness elements. After this viscous BL cross-over, the roughness elements strongly perturb both the thermal and the viscous BLs. The thermal plumes emitted from the tip of pyramids can then be directly ejected into the bulk. For $Pr=23.34$ , the estimated viscous BL thickness at the smallest $Ra$ explored in the experiments ( $Ra=4.39\times 10^{9}$ ) was 0.04H (Lam et al. Reference Lam, Shang, Zhou and Xia2002), which was very close to the height of the roughness element, implying that the measurements with $Pr=23.34$ were all in Regime III.

In the asymptotically large $Ra$ limit, the thickness of the BLs is so small such that the roughness elements perturb both the thermal and the viscous BLs. The system is in Regime III in this case wherein $Nu$ is very sensitive to the LSC, or equivalently to the Reynolds number $Re$ associated with the LSC (to be defined in § 3.3). We plot in figure 9  $Nu$ as a function of $Re$ for (a)  $Pr=23.34$ , (b)  $Pr=6.14$ , (c)  $Pr=4.34$ and (d) $Pr=3.57$ . On one hand, figure 9(a) shows that $Nu$ increases with $\unicode[STIX]{x1D706}$ for all $Re$ measured at $Pr=23.34$ , which are in the Regime III. On the other hand, $Nu$ at other $Pr$ where mostly within the Regime II shows weak dependence on $Re$ . This is another evidence supporting that the viscous BL cross-over may also play important role in turbulent thermal convection with roughness elements.

3.3 Reynolds number measurements

The self-organised large-scale circulation (or ‘wind of turbulence’) is one of the fascinating features of turbulent RBC. The flow dynamics of the LSC is characterised by the associated Reynolds number $Re$ . We study in this section how the roughness geometry alters the dynamics of the LSC by examining the $Re$ dependence on both $Ra$ and $Pr$ . It has been well known that there exists a well-defined low-frequency oscillation in turbulent RBC. The time scale of this oscillation is interpreted as the turnover time $T_{f}$ of the LSC (see, e.g. Xie, Wei & Xia (Reference Xie, Wei and Xia2013) and references therein). By choosing a typical length scale of the LSC, e.g. $4H$ , the Reynolds number is calculated as:

(3.1) $$\begin{eqnarray}Re=UH/\unicode[STIX]{x1D708}=4H^{2}/(T_{f}\unicode[STIX]{x1D708}).\end{eqnarray}$$

Time-resolved temperature signals measured ${\sim}1~\text{cm}$ away from the side wall were used to determine $T_{f}$ and thus the $Re$ as a function of $Ra$ and $Pr$ when changing the roughness geometry, i.e. at different values of $\unicode[STIX]{x1D706}$ . The turnover time of the LSC was determined by locating the time position of the first peak of the temperature autocorrelation function $C(\unicode[STIX]{x1D70F})$ , which is defined as:

(3.2) $$\begin{eqnarray}C(\unicode[STIX]{x1D70F})=\frac{\langle [T(t+\unicode[STIX]{x1D70F})-\langle T(t)\rangle _{t}][T(t)-\langle T(t)\rangle _{t}]\rangle _{t}}{\unicode[STIX]{x1D70E}_{T}^{2}},\end{eqnarray}$$

where $\langle \cdots \rangle _{t}$ denotes time averaging and $\unicode[STIX]{x1D70E}_{T}$ is the r.m.s. temperature. An example of the autocorrelation function measured in a convection cell with $\unicode[STIX]{x1D706}=4.0$ at $Ra=1.01\times 10^{11}$ and $Pr=23.34$ is depicted in figure 10. Clear oscillations can be seen from $C(\unicode[STIX]{x1D70F})$ . Fitting the data points nearby the first peak of $C(\unicode[STIX]{x1D70F})$ with a sixth-order polynomial, the time location of the first peak was determined and denoted as $T_{f}$ . The uncertainty of $T_{f}$ was from $0.06$ s to $0.008$ s with increasing $Ra$ .

Figure 10.

Autocorrelation function of the temperature measured ${\sim}1$ cm away from side wall in a rough cell with $\unicode[STIX]{x1D706}=4.0$ at $Ra=1.01\times 10^{11}$ and $Pr=23.34$ . Here $T_{f}$ is the turnover time of the LSC.

Figure 11.

Reynolds number $Re$ as a function of $Ra$ and $Pr$ in rough cells for (a)  $\unicode[STIX]{x1D706}=4.0$ , (c) $\unicode[STIX]{x1D706}=1.9$ , (e)  $\unicode[STIX]{x1D706}=1.0$ and (g)  $\unicode[STIX]{x1D706}=0.5$ . To determine more precisely the scaling exponent of $Re$ with respect to $Ra$ , data measured at $Pr=3.53,4.34$ and $6.14$ in (a,c,e,g) are multiplied by a $Pr$ -dependent constant $f(Pr)$ to collapse with data at $Pr=23.34$ . The shifted data are shown in the main plot in the right panels (b,d,f,g) with power-law fit $Re\,f(Pr)\sim Ra^{\unicode[STIX]{x1D6FD}}$ shown as solid lines. The insets in (b,d,f,g) show the compensated plot $ReRa^{-\unicode[STIX]{x1D6FD}}$ with respect to $Pr$ with the solid lines representing power-law fits to the data. See table 1 for the fitted power laws.

Figure 11 displays $Re$ as a function of $Ra$ and $Pr$ in rough cells with $\unicode[STIX]{x1D706}=4.0$ (a,b), $\unicode[STIX]{x1D706}=1.9$ (c,d), $\unicode[STIX]{x1D706}=1.0$ (e,f) and $\unicode[STIX]{x1D706}=0.5$ (g,h). Let us first discuss the case for $\unicode[STIX]{x1D706}=4.0$ . The $Re$ as a function of $Ra$ for $Pr=3.57,4.34,6.14$ and $23.34$ are plotted in figure 11(a) on a log–log scale. The $Re$ exhibits power-law dependence on  $Ra$ at different $Pr$ , i.e. $Re\sim Ra^{\unicode[STIX]{x1D6FD}}$ . To determine more precisely $\unicode[STIX]{x1D6FD}$ , data at $Pr=3.57$ , 4.34 and 6.14 were multiplied by a $Pr$ -dependent constant to collapse them with data at $Pr=23.34$ onto a single line. The shifted data are shown in the main plot of figure 11(b). A power-law fit to the data over almost three decades of $Ra$ yielded: $Re\sim Ra^{0.551\pm 0.001}$ . The $Ra$ -dependence of $Re$ was then divided out by compensating the data in figure 11(a) with the so-determined power law, e.g. the inset of figure 11(b). Within a relatively limited range of $Pr$ , the data can also be described by a power law, i.e. $Re/Ra^{0.551}\sim Pr^{-0.66\pm 0.02}$ . The so-determined $Re(Ra,Pr)$ is:

(3.3a,b ) $$\begin{eqnarray}Re=0.088Ra^{0.551}Pr^{-0.66}\quad (\unicode[STIX]{x1D706}=4.0).\end{eqnarray}$$

Applying the same procedure to data obtained in rough cells with $\unicode[STIX]{x1D706}=1.9$ (figure 11 c), $\unicode[STIX]{x1D706}=1.0$ (figure 11 e) and $\unicode[STIX]{x1D706}=0.5$ (figure 11 g), we determined the $Ra$ and $Pr$ dependence of $Re$ for each values of $\unicode[STIX]{x1D706}$ , respectively. The scaling laws of $Re=BRe^{\unicode[STIX]{x1D6FD}}Pr^{\unicode[STIX]{x1D716}}$ for different values of $\unicode[STIX]{x1D706}$ are summarised in table 1.

On one hand, with $\unicode[STIX]{x1D706}$ increasing from 0.5 to 4.0 the scaling exponent $\unicode[STIX]{x1D6FD}$ , i.e. $Re\sim Ra^{\unicode[STIX]{x1D6FD}}$ , increases from 0.472 to 0.551, which is plotted as a function of $\unicode[STIX]{x1D706}$ in figure 5. The data may be described by an empirical power law within a very limited range of $\unicode[STIX]{x1D706}$ , i.e. $\unicode[STIX]{x1D6FD}=0.49\unicode[STIX]{x1D706}^{0.08\pm 0.01}$ . On the other hand, the scaling exponent $\unicode[STIX]{x1D716}$ , i.e. $Re\sim Pr^{\unicode[STIX]{x1D716}}$ , remains at the same value around $-0.70$ within experimental uncertainty except for $\unicode[STIX]{x1D706}=4.0$ . Whether there is a change of the $Pr$ -dependence of $Re$ in the rough cell with $\unicode[STIX]{x1D706}=4.0$ requires future investigation with a wider range of $Pr$ . The change of $\unicode[STIX]{x1D6FD}$ with $\unicode[STIX]{x1D706}$ suggests a change of the LSC dynamics when the geometry of the roughness elements is changed. It is thus interesting to study torsional and sloshing oscillations of the LSC dynamics in rough cells with different values of $\unicode[STIX]{x1D706}$ as compared to those oscillations in a smooth cell (Funfschilling & Ahlers Reference Funfschilling and Ahlers2004; Xi et al. Reference Xi, Zhou, Zhou, Chan and Xia2009; Xie et al. Reference Xie, Wei and Xia2013). This is beyond the scope of the present studies and will be investigated in the future. Interestingly, the transition of the heat transport scaling law from Regime II to Regime III does not present in the Reynolds number measurement.

Figure 12.

Reynolds number $Re$ as a function of $RaNu$ for different values of $\unicode[STIX]{x1D706}$ with the solid line representing a power-law fit to the data irrespective of $\unicode[STIX]{x1D706}$ . (a)  $Pr=23.34$ and $Re=0.076\times (RaNu)^{0.36\pm 0.01}$ ; (b)  $Pr=6.14$ and $0.266\times (RaNu)^{0.35\pm 0.01}$ ; (c)  $Pr=4.34$ and $0.281\times (RaNu)^{0.36\pm 0.01}$ ; (d)  $Pr=3.57$ and $Re=0.340\times (RaNu)^{0.36\pm 0.01}$ .

It is found very recently that $Re$ is insensitive to the plate morphology for a given input heat flux for $\unicode[STIX]{x1D706}=0$ and 0.5 (Wei et al. Reference Wei, Chan, Ni, Zhao and Xia2014), i.e. when $Re$ is plotted against the flux Rayleigh number $RaNu$ , data at different values of $\unicode[STIX]{x1D706}$ collapse. We plot $Re$ as a function of $RaNu$ for different values of $\unicode[STIX]{x1D706}$ and $Pr$ in figure 12. The data at the same $Pr$ but different $\unicode[STIX]{x1D706}$ indeed overlap which is in agreement with the observation in Wei et al. (Reference Wei, Chan, Ni, Zhao and Xia2014), but now for a wider range of $\unicode[STIX]{x1D706}$ and $Pr$ . In addition, data at the same $Pr$ can be fitted by a power law, i.e. $Re=0.076\times (RaNu)^{0.36\pm 0.01}$ $(Pr=23.34)$ , $Re=0.266\times (RaNu)^{0.35\pm 0.01}$ $(Pr=6.14)$ , $Re=0.281\times (RaNu)^{0.36\pm 0.01}(Pr=4.34)~$ and $Re=0.340\times (RaNu)^{0.36\pm 0.01}$ $(Pr=3.57)$ . The scaling exponents of these power-law fits are the same within experimental uncertainty but the magnitudes decrease with $Pr$ .

3.4 Local temperature fluctuations

Since Regime II is a transitional state, we focus on temperature fluctuations measured in Regime III only. To simplify the analysis, we do not take into account the effects of $Pr$ and limit the analysis to measurements done in FC770 only ( $Pr=23.34$ ). The results may shed light upon the mechanism leading to the change of the heat transport scaling in rough cells with different roughness geometries in Regime III.

Figure 13.

Probability density functions (PDFs) of the standardised temperature measured (a) at the cell centre and (b) near the side wall for different values of $\unicode[STIX]{x1D706}$ ( $Ra=2.1\times 10^{10}$ and $Pr=23.34$ ). The r.m.s. temperatures as a function of $\unicode[STIX]{x1D706}$ at the cell centre and side wall are plotted in (c) and (d), respectively.

The probability density functions (PDFs) of the standardised temperature $(T-\langle T\rangle )/\unicode[STIX]{x1D70E}_{T}$ measured at $Ra=2.1\times 10^{11}$ at four different values of $\unicode[STIX]{x1D706}$ at the cell centre and side wall are displayed in figures 13(a) and 13(b) respectively, where $\unicode[STIX]{x1D70E}_{T}$ is the r.m.s. temperature that is shown as a function of $\unicode[STIX]{x1D706}$ in figure 13(c,d) at the cell centre and side wall. It is seen that the PDFs at different $\unicode[STIX]{x1D706}$ overlap very nicely at the cell centre, suggesting that the bulk turbulence shares similar dynamics at different values of $\unicode[STIX]{x1D706}$ . A notable feature seen in figure 13(a) is that the PDFs are significantly skewed toward the negative side, i.e. more cold plumes are detected in the cell centre than hot ones. This feature has not been observed either by us or reported by others in smooth cells with the same boundary conditions, i.e. constant temperature at the top boundary and constant heat flux at the bottom boundary. Thus, the reason for this phenomenon remains unknown to us at present. At the side wall, the shapes of the PDFs remain approximately invariant with $\unicode[STIX]{x1D706}$ except the case with $\unicode[STIX]{x1D706}=0.5$ . For measurements with $\unicode[STIX]{x1D706}=0.5$ the thermistor was positioned such that the upwelling hot plumes of the LSC were detected, while for the other cases the downwelling cold plumes of the LSC were detected. The positive tails of the PDFs overlap well onto each other while the negative tails, especially those larger than $3\unicode[STIX]{x1D70E}_{T}$ , change with $\unicode[STIX]{x1D706}$ . As the large temperature fluctuations are caused by thermal plumes, this observation suggests a change of the plume dynamics with increasing $\unicode[STIX]{x1D706}$ . The r.m.s. temperature at the cell centre increases considerably with $\unicode[STIX]{x1D706}$ as can be seen from figure 13(c), and that at the side wall first remains almost the same for $\unicode[STIX]{x1D706}=0.5$ and 1.0 and then increases with $\unicode[STIX]{x1D706}$ as well.

Figure 14.

$\unicode[STIX]{x1D706}$ -dependence of the normalised r.m.s. temperature as a function of $Ra$ at (a) the cell centre and (b) the side wall. The solid lines are power-law fits to the individual data set with the scaling exponents displayed nearby each data set. The data are measured at $Pr=23.34$ .

The Rayleigh number dependence of normalised r.m.s. temperatures $\unicode[STIX]{x1D70E}_{T}/\unicode[STIX]{x0394}T$ for different values of $\unicode[STIX]{x1D706}$ measured at the cell centre and near the side wall are shown in figures 14(a) and 14(b), respectively. With increasing $\unicode[STIX]{x1D706}$ , $\unicode[STIX]{x1D70E}_{T}/\unicode[STIX]{x0394}T$ enhances considerably. The changes lie not only in the magnitudes but also the scaling exponents with $Ra$ . The scaling exponents of the power-law fits, e.g. $\unicode[STIX]{x1D70E}_{T}/\unicode[STIX]{x0394}T=C\times Ra^{\unicode[STIX]{x1D6FE}}$ , are displayed nearby each data set in figure 14(a,b) and the scaling laws of each data set are summarised in table 1. The scaling exponent increases from $-0.16$ to $-0.09$ with $\unicode[STIX]{x1D706}$ increased from 0.5 to 4.0 in the cell centre. For comparison, Wei & Ahlers (Reference Wei and Ahlers2016) obtained an exponent of $-0.17$ in the centre of a smooth cell for a comparable $Pr$ ( $Pr=12.3$ ). With a possible $Pr$ -dependence of the temperature fluctuation unknown at the present, the experimental results suggest that the $Ra$ -dependence of the temperature fluctuation at the cell centre becomes stronger with increasing $\unicode[STIX]{x1D706}$ . Near the side wall, the scaling exponent increases from $-0.23$ to $-0.08$ in the $\unicode[STIX]{x1D706}$ range covered in the experiments, suggesting a weakened dependence of temperature fluctuation on $\unicode[STIX]{x1D706}$ at the side wall. To the best of our knowledge, there has been no report on the smooth cell value for this exponent in the similar $Pr$ range near the side wall.

Figure 15.

Probability density function of the temperature deviation from the mean $T-\langle T\rangle$ normalised by the applied temperature between the top and bottom plate $\unicode[STIX]{x0394}T$ at (a) the cell centre and (b) the side wall ( $Ra=2.1\times 10^{10}$ , $Pr=23.34$ ).

To gain further insight into the dynamics of temperature fluctuations with increasing $\unicode[STIX]{x1D706}$ , we examine the PDFs of the normalised temperature deviation from the mean $(T-\langle T\rangle )/\unicode[STIX]{x0394}T$ for different values of $\unicode[STIX]{x1D706}$ at $Ra=2.1\times 10^{10}$ at the cell centre and the side wall respectively, as displayed in figure 15(a,b). The figure reveals that large events are becoming more probable with increasing $\unicode[STIX]{x1D706}$ at the same $Ra$ , i.e. hotter and colder plumes, suggesting that the effects of changing roughness geometry are to modify the dynamics of thermal plumes and make them more coherent and energetic (therefore decay slower). This is also supported by the shadowgraph visualisation of the flow field, such as those presented in figure 16. These images are taken at $Ra=1.22\times 10^{11}$ and $Pr=23.34$ . It is seen that, with increasing $\unicode[STIX]{x1D706}$ , both the density and the morphology of the thermal plumes change drastically. In the smooth cell there is a well-defined LSC with few thermal plumes passing through the cell centre (figure 16 a). In the rough cell with $\unicode[STIX]{x1D706}=1.0$ , more thermal plumes are passing through the cell centre (figure 16 b). In the rough cell with $\unicode[STIX]{x1D706}=4.0$ , thermal plumes occupy almost all the space in the convection cell (figure 16 c). Furthermore, the plumes are now packed (or clustered) much more densely. This will result in their slower decay through thermal diffusion, which explains why it is now much more probable to find large temperature excursions in the cell centre (figure 15 a).

Figure 16.

Shadowgraph images of the instantaneous flow fields in a rough cell with (a)  $\unicode[STIX]{x1D706}=0.0$ , (b) $\unicode[STIX]{x1D706}=1.0$ and (c)  $\unicode[STIX]{x1D706}=4.0$ . The images were taken at $Ra=1.22\times 10^{11}$ and $Pr=23.34$ .

Figure 17.

(a) The time-averaged plume amplitude $\langle A\rangle$ normalised by $\unicode[STIX]{x0394}T$ and (b) the time-averaged plume width $\langle w\rangle$ normalised by the large-scale circulation turnover time $T_{f}$ as a function of $Ra$ extracted from the temperature time series measured at the side wall at $Pr=23.34$ . The solid lines are power-law fits to individual data sets with the scaling exponents displayed nearby each data set.

To quantitatively understand the change of the dynamics of thermal plume, we extract time series of thermal plume amplitude $A$ and width $w$ using the time-resolved temperature data measured near the side wall. The detailed plume extraction method can be found in Zhou et al. (Reference Zhou, Xie, Sun and Xia2016). The normalised time-averaged plume amplitude $\langle A\rangle /\unicode[STIX]{x0394}T$ and normalised time-averaged plume width $\langle w\rangle /\unicode[STIX]{x1D70F}_{0}$ are shown in figures 17(a) and 17(b) respectively. Figure 17(a) reveals that with the increase of $\unicode[STIX]{x1D706}$ , the time-averaged plume amplitude $\langle A\rangle /\unicode[STIX]{x0394}T$ increases. The solid lines are power-law fits to the data at different values of $\unicode[STIX]{x1D706}$ with the scaling exponents listed nearby each data set. The scaling exponent of time-averaged plume amplitude increases from $-0.386$ to $-0.266$ with $\unicode[STIX]{x1D706}$ increasing from 0.5 to 4.0. In contrast, the $Ra$ -dependence of the time-averaged plume width remains almost the same with a scaling exponent around 0.25 for $\unicode[STIX]{x1D706}$ smaller than 4.0. It should be noted that the scaling exponent of plume width for $\unicode[STIX]{x1D706}=4.0$ increases considerably (0.341) when compared with other values of $\unicode[STIX]{x1D706}$ , which may be correlated with the drastically changed plume dynamics and morphology in the rough cell with $\unicode[STIX]{x1D706}=4.0$ . As the thermal plumes are main heat carriers in turbulent RBC, the experiments suggest that the change of plume dynamics may lead to the change of the scaling law of heat transport for different values of $\unicode[STIX]{x1D706}$ .

3.5 A proper parameter characterising the system: $\unicode[STIX]{x1D706}$ versus number density of roughness elements

When varying $\unicode[STIX]{x1D706}$ the number density $n$ of roughness elements also varies, i.e. $n\simeq \unicode[STIX]{x03C0}/4\unicode[STIX]{x1D6E4}^{2}\unicode[STIX]{x1D706}^{2}(H/h)^{2}$ . In fact, the two parameters have one-to-one correspondence provided that the ratio of the roughness element height $h$ to the cell height $H$ and $\unicode[STIX]{x1D6E4}$ are fixed, which is the case for the present study. To figure out whether $\unicode[STIX]{x1D706}$ is the most relevant parameter that characterises the system, we carried out another experiment, in which $\unicode[STIX]{x1D706}$ was kept constant at 4.0 and the height of the roughness elements was halved. Thus the number density of the roughness elements was increased by a factor of 4.

We show in figure 18(a) $Nu$ as a function of $Ra$ for measurements for $\unicode[STIX]{x1D706}=4.0$ rough cells with $h=8~\text{mm}$ and $h=4~\text{mm}$ together with the same smooth cell data which have be shown in figures 3 and 4. The estimated viscous boundary layer thickness at the highest $Ra$ reached in the experiment is 4.8 mm. Thus, the measurements at $Pr=23.34$ and $h=4~\text{mm}$ are in Regime II. A scaling exponent $\unicode[STIX]{x1D6FC}=0.60$ is observed in rough cell with $h=4~\text{mm}$ in Regime II, which is consistent with 0.59 observed in Regime II of rough cell with $h=8$  mm. This can be seen from the two parallel lines in figure 18(a). The similar $Ra$ -dependence of $Nu$ in Regime II in rough cells with $h=4$  mm and 8 mm can be seen more clearly from the compensated plot in figure 18(b). These observations suggest that $\unicode[STIX]{x1D706}$ is not only a suitable parameter describing the geometry of the pyramid-shaped roughness elements, but also a suitable parameter to characterise the response of turbulent RBC over rough plates with different roughness geometries.

A noticeable feature from figure 18(a) is that the measured $Nu$ in the rough cell with $h=8~\text{mm}$ is larger than that in the rough cell with $h=4~\text{mm}$ in Regime II despite the fact that they have similar $Ra$ -dependence. This may be explained by the measured Reynolds number based on the circulation time of the LSC as plotted in figure 19. It is seen that $Re$ in the rough cell with $h=8~\text{mm}$ is larger than that in the rough cell with $h=4~\text{mm}$ , which is true for both $Pr=4.34$ and $Pr=23.34$ . The result suggests that a faster LSC can transport heat more efficiently across the convection cell.