3.1. The velocity field

Using the PIV velocity measurements, we are able to describe the flow field as a function of tilt angle ${\it\beta}$ . PIV measurements of RBC in a horizontal rectangular cell with ${\it\Gamma}\simeq 1$ have been reported by Xia et al. (Reference Xia, Sun and Zhou2003). In their experiments, the distinct features of the mean flow field were well described in terms of a central low-velocity elliptical core, two high-velocity bands outside of the central core, and two counter-rotating vortices at the cell corners. Among these flow structures, the high-velocity band reflects one of the basic characteristics of the LSC, and is the main focus in the following discussion.

Figure 2.

Coarse-grained vector map of the time-averaged velocity field for an angle ${\it\beta}$ of (a) 0, (b) 0.15, (c) 0.23, (d) 0.64, (e) 0.92, ( f) 1.05, (g) 1.17 and (h) 1.41 rad. At each ${\it\beta}$ , the velocity magnitude is scaled by the different colour bar. The trajectories produced by the Tecplot software are plotted to highlight the flow pattern. At (d)  ${\it\beta}=0.64\ \text{rad}$ and (e)  ${\it\beta}=0.92\ \text{rad}$ , some velocity distortions in the upper left corner manifest due to the weak laser light, and are shaded.

Figure 2 shows the time-averaged velocity maps at eight different tilt angles. Note that here the velocity vectors have been further averaged by using a $3\times 3$  window, so that they can be more clearly exhibited. The velocity magnitude $V$ is defined as $V=\sqrt{U^{2}+W^{2}}$ , where $U$ and $W$ are the time-averaged velocity components along the $x$ and $z$ axes, respectively. In the level case (figure 2 a), a clear clockwise LSC spans the whole cell, with two counter-rotating eddies at the upper left and lower right corners. These flow features are similar to those described by Xia et al. (Reference Xia, Sun and Zhou2003). As ${\it\beta}$ increases from 0 to 0.64 rad (figure 2 a–d), the pattern of the LSC gradually changes from oblique ellipse-like to square-like. Meanwhile, the counter-rotating eddies seem to disappear. In previous studies it was reported that the LSC pattern shifted from an elliptical shape at low Rayleigh numbers to a squarish shape at high Rayleigh numbers in a level cell (Niemela & Sreenivasan Reference Niemela and Sreenivasan2003; Xia et al. Reference Xia, Sun and Zhou2003). The present observations may suggest that the increasing value of ${\it\beta}$ plays the same role as the increasing value of $\mathit{Ra}$ in modifying the LSC pattern. When ${\it\beta}$ is increased further the flow pattern becomes relatively complicated. At ${\it\beta}=0.92\ \text{rad}$ (figure 2 e), the flow along the sidewall first strikes the lower right (or upper left) corner, then bounces towards the bulk region, but finally turns back to the plates. That is, the flow moves in a tortuous way. At ${\it\beta}=1.05\ \text{rad}$ (figure 2 f), the flow pattern is similar to that at ${\it\beta}=0.92\ \text{rad}$ , but the bouncing height seems to be greater. In both cases, a single LSC can still be discernible. At ${\it\beta}=1.17\ \text{rad}$ (figure 2 g), the high-velocity band is still adjoining the sidewall, and two large backflows occur in the right and left halves of the cell. For ${\it\beta}\geqslant 1.41\ \text{rad}$ , the bulk region is full of small chaotic flow structures, which cannot be well distinguished because of the extremely slow flow speed. Generally, figure 2 shows that the high-velocity zone approaches the sidewall and departs from the conductive plates as the tilt angle ${\it\beta}$ increases.

In order to quantify the global flow pattern, it is assumed that there is still a single large-scale flow roll in the convection cell. The origin of the roll is defined to be the cell centre; then the position ( $x,z$ ) is transformed into ( $r,{\it\theta}$ ), where $r$ is the polar radius and ${\it\theta}$ is the polar angle. Usually, the velocity profiles along $x=L_{x}/2$ and $z=H/2$ are analysed in order to characterize the LSC (Qiu & Tong Reference Qiu and Tong2001; Xia et al. Reference Xia, Sun and Zhou2003). Here, an alternative strategy is applied using the azimuthal average velocity,  $V_{a}$ , so that the global flow field can be involved. However, as shown in figure 2, the LSC pattern is affected by the square shape of the cell. Thus, when the azimuthal average velocity, $V_{a}$ , is examined, it makes sense to analyse the velocity of the same $r/r_{b}$ , where $r_{b}$ is the distance between the boundary and the cell centre and with the same ${\it\theta}$ as $r$ . Therefore, the azimuthal average velocity $V_{a}$ has the form

(3.1) $$\begin{eqnarray}V_{a}(r/r_{b})=\frac{1}{N_{r/r_{b}}}\mathop{\sum }_{r-{\it\delta}r}^{r+{\it\delta}r}\mathop{\sum }_{{\it\theta}=0}^{2{\rm\pi}}V(r/r_{b},{\it\theta}),\end{eqnarray}$$

where ${\it\delta}r=r_{b}/100$ and $N_{r/r_{b}}$ is the number of velocity vectors along the normalized radius $r/r_{b}$ . Figure 3 shows the circulating velocity $V_{a}$ as a function of the normalized radius $r/r_{b}$ for different tilt angles. It indicates that the $V_{a}$ profile varies with the tilt angle ${\it\beta}$ . Generally, a velocity peak appears in each $V_{a}$ profile, which corresponds to the high-velocity band of the LSC. Therefore, the LSC can be described using the amplitude $V_{ap}$ , position $(r/r_{b})_{p}$ and width $hw/r_{b}$ of the velocity peak.

Figure 3.

The azimuthal average velocity $V_{a}$ as a function of the normalized radius $r/r_{b}$ at ${\it\beta}=0$ , 0.08, 0.15, 0.23, 0.64, 0.92, 1.05, 1.17 and 1.41 rad (as indicated).

Figure 4.

(a) The amplitude ( $V_{ap}$ ), (b) position ( $(r/r_{b})_{p}$ ) and (c) width ( $hw/r_{b}$ ) of the high-velocity band as functions of the tilt angle ( ${\it\beta}$ ). In (c), the three grey lines are least-squares linear fits for $0\leqslant {\it\beta}\leqslant 0.15\ \text{rad}$ , $0.15<{\it\beta}<1.05\ \text{rad}$ and $1.05\leqslant {\it\beta}<{\rm\pi}/2\ \text{rad}$ , with slopes $-2.2$ , $-0.3$ and $-0.17\ \text{rad}^{-1}$ , respectively.

The results obtained for the amplitude $V_{ap}$ are shown in figure 4(a). First, $V_{ap}$ increases with increasing ${\it\beta}$ , before reaching a maximum at ${\it\beta}\simeq 0.15\ \text{rad}$ . After this, $V_{ap}$ gradually decreases as ${\it\beta}$ increases further. The position of the high-velocity band has a different ${\it\beta}$ dependence. As shown in figure 4(b), the high-velocity band concentrates at $(r/r_{b})_{p}\simeq 0.5$ (about a quarter width from the boundary) in the level case ( ${\it\beta}=0\ \text{rad}$ ). When the cell is slightly tilted ( ${\it\beta}=0.04\ \text{rad}$ ), the high-velocity band moves abruptly towards the cell boundary ( $(r/r_{b})_{p}\simeq 0.83$ ). As ${\it\beta}$ increases further to ${\rm\pi}/2\ \text{rad}$ , even though the data are somewhat scattered, the high-velocity band can be seen to progressively approach the boundary to $(r/r_{b})_{p}\simeq 0.93$ . The position of the high-velocity band shows that the LSC has different flow patterns between the level and tilted cases. In the level case, the convection cell is axisymmetric with respect to the cell’s vertical centreline, and so is the buoyancy force produced from the conductive plates. It should not matter whether the LSC flow is clockwise or anticlockwise, accompanied with the large eddies in the corners (figure 2 a). In the tilted case, the buoyancy force of the system is no longer axisymmetric with respect to the cell’s vertical centreline, the LSC is forced into a certain flow direction (Ahlers et al. Reference Ahlers, Brown and Nikolaenko2006) and the eddies inside the corners are strongly suppressed. The LSC pattern is, therefore, very sensitive to the symmetry of the system. The width of the high-velocity band, $hw/r_{b}$ , is defined as the width at half the velocity amplitude, $V_{ap}$ , in the velocity profile (figure 3). Figure 4(c) shows the values obtained for the width $hw/r_{b}$ . These indicate that the width $hw/r_{b}$ decreases with increasing ${\it\beta}$ . It seems that there are two transitions at ${\it\beta}=0.15\ \text{rad}$ and ${\it\beta}=1.05\ \text{rad}$ . The reduction slopes are 2.2, 0.3 and $0.17\ \text{rad}^{-1}$ in the ranges $0\leqslant {\it\beta}\leqslant 0.15\ \text{rad}$ , $0.15<{\it\beta}<1.05\ \text{rad}$ and $1.05\leqslant {\it\beta}<{\rm\pi}/2\ \text{rad}$ , respectively. Note that the width, $hw/r_{b}$ , at ${\it\beta}={\rm\pi}/2\ \text{rad}$ is not included in the analysis as it is beyond the low-velocity measurement limit. Recent studies have shown that the aspect ratio ${\it\Gamma}_{y}$ also has a strong effect on the flow dynamics and heat transport (Huang et al. Reference Huang, Kaczorowski, Ni and Xia2013; Wagner & Shishkina Reference Wagner and Shishkina2013). Therefore, the dependence of the LSC on the tilt angle ${\it\beta}$ might be different for different ${\it\Gamma}_{y}$ .

3.2. Temperature and velocity oscillations

Compared to the cylindrical cell (Cioni et al. Reference Cioni, Ciliberto and Sommeria1997; Chillà et al. Reference Chillà, Rastello, Chaumat and Castaing2004; Sun et al. Reference Sun, Xi and Xia2005a ; Ahlers et al. Reference Ahlers, Brown and Nikolaenko2006), the rectangular cell simplifies the flow dynamics by lowering the degrees of freedom of the LSC. Twisting and sloshing of the LSC are expected to be suppressed in this quasi-two-dimensional case, possibly influencing the temperature and velocity oscillation. Therefore, it is worth examining the dependence of the oscillation intensity and period on the tilt angle in both temperature and velocity fields. Generally, the coherent oscillation is examined by analysing the autocorrelation function,

(3.2) $$\begin{eqnarray}C_{X}({\it\tau})=\frac{\langle {\it\delta}X_{i}(t){\it\delta}X_{i}(t+{\it\tau})\rangle }{\langle {\it\delta}X_{i}(t)^{2}\rangle },\end{eqnarray}$$

where $X$ can be temperature or velocity, ${\it\delta}X$ is the fluctuation in $X$ , the subscript $i$ denotes the sample position and the angle brackets denote the time average. The intensity ( $I$ ) and period ( ${\it\tau}_{o}$ ) of oscillation are defined as the amplitude and time period of the second-order peak in the autocorrelation function, respectively.

Figure 5(a) shows an example of the temperature autocorrelation functions measured by the six thermistors in the plates at ${\it\beta}=0.38\ \text{rad}$ . An obvious oscillation ( ${\it\tau}_{o}=80.7\pm 1.7\ \text{s}$ ) can be identified in all autocorrelation functions, even though the oscillation intensity is different at different positions. In the bottom plate, the oscillation intensity is strongest at position B1, as shown in figure 1(b), where the cold descending flow arrives. The intensity gets weaker and weaker from right to left (from B1 to B3). Similar observations are found at the top plate, where the intensity is weakened from left to right (from T3 to T1). This result is somewhat similar to that in the cylindrical cell (Xi et al. Reference Xi, Zhou, Zhou, Chan and Xia2009), where the oscillations are also found at the positions of plume impingement and are absent at the positions of plume emission. At other tilt angles, the strongest oscillations are always found at positions B1 and T3, too.

Figure 5.

(a) Autocorrelation functions of the temperature signals at the six thermistor positions in the plates for ${\it\beta}=0.38\ \text{rad}$ . (b) Autocorrelation functions of the velocity component, $w$ , at the eight positions for ${\it\beta}=0.38\ \text{rad}$ . These positions are described in the text.

Eight positions are selected to examine the oscillation in the velocity signals. Three of them are 15 mm above B1, B2 and B3 and are denoted by Bv1, Bv2 and Bv3, respectively. The other three positions are 12 mm below T1, T2 and T3 and are denoted by Tv1, Tv2 and Tv3, respectively. In addition, we select a position at the mid-height of the cell and approximately 25 mm from the sidewall, denoted by S1. The last position is in the centre, denoted by C1. At each position it is found that the velocity components $u$ and $w$ have more or less the same autocorrelation functions. Therefore, only the velocity $w$ is used for the oscillation analysis. The autocorrelation functions of $w$ at these eight positions for ${\it\beta}=0.38\ \text{rad}$ are shown in figure 5(b). They indicate that an obvious oscillation with a period of ${\sim}80\ \text{s}$ appears, which is consistent with the observations in the temperature signals (figure 5 a). At positions Bv1 and Tv3, the oscillation is found to be strongest, also consistent with the temperature observation. The oscillation is found to be very obvious near the sidewall (at position S1), where the oscillation intensity is nearly as strong as those at positions Bv1 and Tv3. On the other hand, the oscillation is weakest in the cell centre (at position C1). Therefore, the oscillation distributions near the sidewall and in the centre are consistent with the previous observations in the level case (Zhou et al. Reference Zhou, Sun and Xia2007).

As shown in figure 5, the intensity of oscillation is the strongest, and is almost the same at the positions B1 (Bv1) and T3 (Tv3) in temperature (velocity) signals. We can use the average intensity at these two positions to represent the temperature (velocity) oscillation of the whole system. Figure 6(a) shows the oscillation intensity ( $I$ ) as a function of the tilt angle ${\it\beta}$ . Within the experimental errors, the oscillation intensity $I$ follows similar trends for the temperature and velocity signals. It first increases with increasing ${\it\beta}$ , until it reaches its maximum at around ${\it\beta}\simeq 0.48\ \text{rad}$ , before decreasing as ${\it\beta}$ is increased further. For ${\it\beta}>1.05\ \text{rad}$ , the oscillation is not detectable in the temperature autocorrelation function at all thermistor positions. For ${\it\beta}>1.32\ \text{rad}$ , the oscillation cannot be detected in the velocity signals either. Figure 6(b) shows the oscillation period ${\it\tau}_{o}$ as a function of the tilt angle ${\it\beta}$ . The temperature and velocity oscillation periods agree well with each other. With increasing ${\it\beta}$ , ${\it\tau}_{o}$ first decreases, it has a remarkable transition at an angle of approximately ${\it\beta}=0.15\ \text{rad}$ , then it increases gradually as ${\it\beta}$ increases from 0.15 to 1.32 rad.

Figure 6.

(a) Oscillation intensity ( $I$ ) and (b) oscillation period ( ${\it\tau}_{o}$ ) as functions of the tilt angle ${\it\beta}$ from temperature and velocity signals.

3.3. The Reynolds numbers

In this section, the Reynolds number, which is defined as $\mathit{Re}=LV/{\it\nu}$ , is discussed using several characteristic speeds for $V$ . In RBC, the oscillation period is comparable to the complete turnover time of the single LSC, for aspect ratios close to one (Ahlers et al. Reference Ahlers, Brown and Nikolaenko2006; Zhou et al. Reference Zhou, Sun and Xia2007; Weiss & Ahlers Reference Weiss and Ahlers2013). Thus, the Reynolds number $\mathit{Re}_{o}$ , based on the oscillation period ( ${\it\tau}_{o}$ ), can be defined as

(3.3) $$\begin{eqnarray}\mathit{Re}_{o}=\frac{L}{{\it\nu}}\frac{4L}{{\it\tau}_{o}},\end{eqnarray}$$

where ${\it\tau}_{o}$ is the average oscillation period from the temperature and velocity signals at the same tilt angle ${\it\beta}$ . In addition, the maximum velocity of the whole velocity field, $V_{m}$ , is selected, which corresponds to the Reynolds number $\mathit{Re}_{m}$ . As mentioned in § 3.1, $V_{ap}$ is the velocity amplitude of the LSC, with the corresponding Reynolds number being denoted by $\mathit{Re}_{ap}$ . The results are shown in figure 7. It is interesting that $\mathit{Re}_{o}$ and $\mathit{Re}_{m}$ are more or less the same over the ${\it\beta}$ range between 0 and 1.05 rad, and their deviation appears and gets larger as ${\it\beta}$ increases. This is probably caused by complicated flow patterns in the bulk region, such as the backflows shown in figure 2(g). Compared to $\mathit{Re}_{o}$ , $\mathit{Re}_{ap}$ has a smaller magnitude, which should be due to the different definitions of the characteristic speed. The various Reynolds numbers $\mathit{Re}_{o}$ , $\mathit{Re}_{m}$ and $\mathit{Re}_{ap}$ have a common trend. With increasing ${\it\beta}$ , the $\mathit{Re}$ number first increases until it reaches its maximum at approximately ${\it\beta}=0.15\ \text{rad}$ , after which it gradually decreases.

Figure 7.

The Reynolds numbers, $\mathit{Re}_{o}$ , $\mathit{Re}_{m}$ and $\mathit{Re}_{ap}$ , as functions of the tilt angle ${\it\beta}$ . Equation (3.4) is plotted as a grey dashed line.

For small values of ${\it\beta}$ , Chillà et al. (Reference Chillà, Rastello, Chaumat and Castaing2004) and Ahlers et al. (Reference Ahlers, Brown and Nikolaenko2006) also found that cell tilting leads to an enhancement of the Reynolds number of the LSC. Chillà et al. (Reference Chillà, Rastello, Chaumat and Castaing2004) modelled this by considering the equilibrium of the buoyancy and shear forces for the top and bottom boundary layer. Ahlers et al. (Reference Ahlers, Brown and Nikolaenko2006) deduced that the Reynolds number is of the form

(3.4) $$\begin{eqnarray}\mathit{Re}=\mathit{Re}(0)\left(1+\frac{\mathit{Ra}}{8\,\mathit{Re}(0)\mathit{Pr}\,\mathit{Nu}^{2}}{\it\beta}\right),\end{eqnarray}$$

where $\mathit{Re}(0)$ is the Reynolds number at ${\it\beta}=0$ , and the Reynolds number is based on the oscillation period. By substituting the present experimental parameters, we can obtain $\mathit{Re}=\mathit{Re}(0)(1+3.56{\it\beta})$ . This model is shown as the grey dashed line in figure 7; it agrees with the data for small angles ( ${\it\beta}<0.15\ \text{rad}$ ). As the model predicts that the Reynolds number increases with increasing angle ${\it\beta}$ , however, it cannot explain the observation of a decreasing trend at larger values of ${\it\beta}$ .

3.4. Heat transport

Lastly, the heat transport of thermal convection is discussed. The calculation of $\mathit{Nu}$ needs the values of the heat flux $F_{T}$ and the temperature difference ${\rm\Delta}T$ . Considering the variances of these quantities, the main measurement error comes from the temperature inhomogeneity of the plates, which is less than 2 % for all tilt angles. When the cell is level, the value of $\mathit{Nu}$ is calculated to be 97.8 at $\mathit{Ra}=4.42\times 10^{9}$ . In a recent experiment with a rectangular cell with ${\it\Gamma}_{x}=1$ and ${\it\Gamma}_{y}=0.3$ , $\mathit{Nu}=99.3$ has been obtained at $\mathit{Ra}=4.6\times 10^{9}$ and $\mathit{Pr}=7$ (Zhou et al. Reference Zhou, Liu, Li and Zhong2012). This is close to the present measurement. As the thermal boundary condition at each ${\it\beta}$ is almost the same, e.g. the temperatures of the plates are kept constant, additional heat transport corrections, such as heat leakage (Xia, Lam & Zhou Reference Xia, Lam and Zhou2002), sidewall coupling (Ahlers & Xu Reference Ahlers and Xu2001) and finite plate conductivity (Nikolaenko et al. Reference Nikolaenko, Brown, Funfschilling and Ahlers2005), have not been applied to avoid new artificial errors.

The value of $\mathit{Nu}$ as a function of the tilt angle ${\it\beta}$ is shown in figure 8. This indicates that cell tilting causes a pronounced reduction of $\mathit{Nu}$ . In the range of $0\leqslant {\it\beta}\leqslant 0.15\ \text{rad}$ , the reduction of $\mathit{Nu}$ is approximately 1.4 % and the decay slope is $8.57\times 10^{-2}\ \text{rad}^{-1}$ . This result is significantly different from those reported by Chillà et al. (Reference Chillà, Rastello, Chaumat and Castaing2004) and Weiss & Ahlers (Reference Weiss and Ahlers2013) in cylindrical cells with ${\it\Gamma}=0.5$ . Chillà et al. (Reference Chillà, Rastello, Chaumat and Castaing2004) found that $\mathit{Nu}$ decreased by 5 % for tilt angles as small as ${\it\beta}=0.026\ \text{rad}$ and the decay slope was $2\ \text{rad}^{-1}$ , while Weiss & Ahlers (Reference Weiss and Ahlers2013) found that a small value of ${\it\beta}$ could slightly enhance the heat transport. Comparatively, the dependence of $\mathit{Nu}$ is more or less similar to that in a cylindrical cell of ${\it\Gamma}=1$ , where the decay slope is smaller ( $(3.1\pm 0.1)\times 10^{-2}\ \text{rad}^{-1}$ ) (Ahlers et al. Reference Ahlers, Brown and Nikolaenko2006). In the range of $0.15\leqslant {\it\beta}\leqslant 1.05\ \text{rad}$ , the decay slope of $\mathit{Nu}$ becomes smaller, and it is fitted as $3.27\times 10^{-2}\ \text{rad}^{-1}$ . Coincidentally, this decay slope is close to that in the cylindrical cell of ${\it\Gamma}=1$ over a smaller ${\it\beta}$ range (Ahlers et al. Reference Ahlers, Brown and Nikolaenko2006). When ${\it\beta}$ further increases ( $1.05\leqslant {\it\beta}\leqslant {\rm\pi}/2\ \text{rad}$ ), $\mathit{Nu}$ decays quickly with the slope of $0.24\ \text{rad}^{-1}$ , which is approximately eight times larger than that for small ${\it\beta}\ ({\leqslant}1.05\ \text{rad})$ . Compared to the level case, $\mathit{Nu}$ is weakened only by approximately 5 % at ${\it\beta}=1.05\ \text{rad}$ , but the reduction reaches approximately 18 % at ${\it\beta}={\rm\pi}/2\ \text{rad}$ .

Figure 8.

The normalized Nusselt number $\mathit{Nu}({\it\beta})/\mathit{Nu}(0)$ as a function of the tilt angle ${\it\beta}$ . The three grey lines are least-squares linear functions for $0\leqslant {\it\beta}\leqslant 0.15\ \text{rad}$ , $0.15\leqslant {\it\beta}\leqslant 1.05\ \text{rad}$ and $1.05\leqslant {\it\beta}\leqslant {\rm\pi}/2\ \text{rad}$ , with slopes $-8.57\times 10^{-2}$ , $-3.27\times 10^{-2}$ and $-0.24\ \text{rad}^{-1}$ , respectively.

Combined with the variance of flow dynamics at different values of ${\it\beta}$ , the transition at ${\it\beta}=0.15\ \text{rad}$ can be observed in the dependences of both $\mathit{Re}$ and $\mathit{Nu}$ on ${\it\beta}$ . However, the heat transport of thermal convection is still reduced, even though the LSC speed is highest, which suggests that the evolution of thermal structures must be considered in understanding the heat transport. The transition at ${\it\beta}=1.05\ \text{rad}$ cannot be discerned in the dependence of $\mathit{Re}$ on ${\it\beta}$ , but seems to be obvious in the dependence of $\mathit{Nu}$ on ${\it\beta}$ . As shown in figure 2, there are totally different flow patterns when the cell is at ${\it\beta}<1.05\ \text{rad}$ and ${\it\beta}\geqslant 1.05\ \text{rad}$ , which may be responsible for the variance of  $\mathit{Nu}$ . Further work, such as an instantaneous $\mathit{Nu}$ measurement, is needed in order to understand the dependence of $\mathit{Nu}$ on ${\it\beta}$ .