3.1. Effect of inclination on flow structures

It is well-known that flow structures in RBC are sensitive to the inclination angle (Wang et al. Reference Wang, Xia, Wang, Sun, Zhou and Wan2018a) and aspect ratio (Wang et al. Reference Wang, Wan, Yan and Sun2018b). The sensitivity here refers to a change in the state of large-scale rolls. Figure 2 shows the instantaneous temperature fields for all five tilt angles and four $Ra$ cases. The effect of the inclination angle is evident in all $Ra$ cases, where flow structures transform from a double-roll state at $\phi =0^{\circ }$ to a thermally stratified state at $\phi =90^{\circ }$. While the former is a consequence of a double aspect ratio, the thermal stratification occurs due to a temperature gradient acting normal to the gravity in the latter. These states have already been identified in previous studies (Shishkina Reference Shishkina2016; Shishkina & Horn Reference Shishkina and Horn2016; Khalilov et al. Reference Khalilov, Kolesnichenko, Pavlinov, Mamykin, Shestakov and Frick2018). Similar transformations of the flow states were observed by Wang et al. (Reference Wang, Xia, Wang, Sun, Zhou and Wan2018a) for the same governing parameters. However, for $\varGamma =1/2$, Wang et al. (Reference Wang, Wan, Yan and Sun2019) observed an unstable triple-roll state at smaller tilt angles (close to the level case), and a single-roll state beyond $\phi \approx 10^{\circ }$.

Figure 2.

Instantaneous temperature field overlaid by streamlines showing the variation in large-scale rolls as the inclination angle increases from $\phi =0^{\circ }$ to $\phi =90^{\circ }$. The $Ra$ values are (a–e) $10^{6}$, (f–j) $10^{7}$, (k–o) $10^{8}$, and (p–t) $10^{9}$. The $\phi$ values are (a,f,k,p) $0^{\circ }$, (b,g,l,q) $15^{\circ }$, (c,h,m,r), $45^{\circ }$ (d,i,n,s) $75^{\circ }$, and (e,j,o,t) $90^{\circ }$. The arrows in (a–e) indicate the direction of gravity. For any specific tilt angle except $\phi =0^{\circ }$, the flow states change with $Ra$.

At $Ra=10^{6}$, it is observed that flow structures attain the shape of the cell at $\phi =15^{\circ }$ (see figure 2b), where the direction of thermal plumes is aligned such that hot plumes move rightwards along the hot isothermal wall, and cold plumes opposite to it. As a consequence, an LSC forms instead of a double-roll state. In the earlier studies, a large-scale roll is termed as LSC in a unit-aspect-ratio cell (Shishkina & Horn Reference Shishkina and Horn2016). Therefore, inclination incites LSC that spans the entire cell, which is consistent with Guo et al. (Reference Guo, Zhou, Qu, Cen and Lu2017), though the flow state changes with further increase in inclination angle. At $\phi =45^{\circ }$, movement of hot and cold fluid is restricted to the right and left parts of the cell, respectively, indicating weak thermal stratification with the central region showing fluid mixing. At $\phi =75^{\circ }$ and $90^{\circ }$, flow structures are nearly the same, and the flow becomes thermally stratified. Flow states at $Ra=10^{7}$ remain the same as for $Ra=10^{6}$, and this is also consistent with previous studies (Wang et al. Reference Wang, Xia, Wang, Sun, Zhou and Wan2018a). However, large-scale structures undergo significant changes beyond $Ra=10^{8}$; see figures 2(p–t), where these effects are more pronounced for $Ra=10^{9}$. It is clearly evident that a double-roll state persists even at $\phi =15^{\circ }$, and the organization of corner rolls is similar to that in the level case. At $\phi =45^{\circ }$, signatures of thermal stratification are absent, and sudden detachment of thermal plumes from the smooth walls is observed at randomly located spots. Unlike the level case, in a tilted set-up, thermal plumes (travelling along the isothermal surface) experience both buoyancy and induced shear. Therefore, it is quite apparent that both these forcings are competing with each other. While travelling along the hot surface, thermal plumes gain heat continuously. Once they attain sufficient energy to overcome the viscous resistance, they detach and move directly into the bulk, which impacts the heat flux to be described in § 3.2. The observed weak thermal stratification suggests that an increase in $Ra$ delays it, where the delay is in terms of inclination angle. This feature becomes more prominent for the highest $Ra$ case, where evidence of relatively higher fluid mixing is seen even at $\phi =75^{\circ }$. In this case, a complex flow state is observed, where several circulation cells are evident in relatively smaller regions of the cell, which restrict the movement of fluid. Notably, we find that the transition to a single large-scale circulation shifts towards a greater inclination angle as $Ra$ increases. This observation is in contrast to the results reported by Wang et al. (Reference Wang, Xia, Wang, Sun, Zhou and Wan2018a). They found that the above-mentioned shift first increases with inclination from $10^{7}$ to $5\times 10^{7}$, and eventually drops beyond $7\times 10^{7}$.

To ascertain the weakening of turbulence with increasing inclination, we show time series of vertical velocity fluctuations $v^{\prime }$ obtained from a probe placed near the geometric centre of the cell, $x=1.1$, $y=0.5$, at $Ra=10^{7}$ and $10^{9}$ in figure 3. The vertical velocity is chosen because it signifies the rise of thermal plumes in the bulk region. At $Ra=10^{9}$, in the level case, large fluctuations are evident in figure 3(a). With increase in inclination angle, the fluctuations drop and nearly vanish at $\phi =90^{\circ }$. The sharp fluctuations signify movement of thermal plumes at the geometric centre. As shown in figures 3(c–e), a weak fluctuating signal in intermediate and higher tilt angles shows the absence of vertical movement of thermal plumes in the bulk region. It indicates movement of the flow along the walls, and the diminishing effect of buoyancy in the transverse direction is responsible for such movement. In comparison to $Ra=10^{7}$, evidence of fluctuations in vertical velocity at $Ra=10^{9}$ is seen clearly at $\phi =75^{\circ }$, which confirms better mixing of fluid relative to the lower $Ra$ case.

Figure 3.

Time series of vertical velocity fluctuation $v^{\prime }=v-\langle v \rangle _t$ in the vicinity of the geometric centre of the convection cell, $x=1.1$, $y=0.5$, for $Ra=10^{7}$ and $10^{9}$ at all five inclination angles. With increase in inclination angle, intensity of fluctuations drops in both the $Ra$ cases. Plots are for (a) $\phi = 0^{\circ }$, (b) $\phi = 15^{\circ }$, (c) $\phi = 45^{\circ }$, (d) $\phi = 75^{\circ }$, (e) $\phi = 90^{\circ }$.

3.2. $Nu\unicode{x2013}\phi$ dependence

The Nusselt number from the present study is compared with a few previously reported data in figure 4(a). Our data agree well with Wang et al. (Reference Wang, Xia, Wang, Sun, Zhou and Wan2018a) for $Ra=10^{7}$. A marginal difference is noted for $Ra=10^{6}$ with Shishkina & Horn (Reference Shishkina and Horn2016), who used $\varGamma =1$. At small to moderate $Ra$ ($\leq 10^{8}$), the $Nu(\phi )$ trend remains the same, while it drops monotonically for $Ra=10^{9}$, which is shown in a separate scale. To compare directly the $Nu\unicode{x2013}\phi$ dependence at different $Ra$ values, we show the normalized heat flux in figure 4(b). For $Ra \leq 10^{8}$, a drop in $Nu$ at $\phi =15^{\circ }$ is evident in all the cases. With further increase in $\phi$, it starts rising again to attain a maximum at $75^{\circ }$. Interestingly, the $Nu(\phi )$ trend becomes weaker with increasing $Ra$, and drops monotonically beyond $Ra>10^{8}$. In contrast to the present work, Wang et al. (Reference Wang, Wan, Yan and Sun2019) obtained a similar and robust $Nu(\phi )$ trend up to $Ra=10^{9}$ in the convection cell of $\varGamma =1/2$. This inconsistency can be attributed to the differences in the aspect ratio. For VC, the distance between the hot and cold plates in $\varGamma =1/2$ is twice of that in $\varGamma =2$, which favours thermal stratification and delays the onset of turbulence. Since greater tilt is effective in transporting heat due to laminar characteristics of the flow, Wang et al. (Reference Wang, Wan, Yan and Sun2019) did not observe any variation in the $Nu(\phi )$ trend due to the dominating thermal stratification. They attributed the minimum $Nu$ at smaller inclination angles to the double-roll state at $Ra=10^{6}$, and stable triple-roll state at $Ra=10^{7}$ and $10^{8}$. On the other hand, as seen in figures 2(b,g,l,q), we observe that flow exhibits a single-roll state in the two lower $Ra$ cases, and a double-roll state in the other two higher ones. Contrary to the previous studies where the single-roll state was found responsible for enhancing the heat flux (Wang et al. Reference Wang, Wan, Yan and Sun2019), we find the same state to diminish it for $Ra\leq 10^{7}$. This shows that the $Nu(\phi )$ dependence is similar for $\varGamma =1/2$ and $2$ at lower $Ra$ ($\leq 10^{8}$) despite different flow states.

Figure 4.

(a) Comparison of $Nu(\phi )$ curve obtained in the present work with Shishkina & Horn (Reference Shishkina and Horn2016) ($Ra=10^{6}$, $\varGamma =1$ and $Pr=1$), Wang et al. (Reference Wang, Xia, Wang, Sun, Zhou and Wan2018a) ($Ra=10^{7}$, $\varGamma =2$ and $Pr=0.71$), and Wang et al. (Reference Wang, Wan, Yan and Sun2019) ($Ra=10^{7}$, $\varGamma =1/2$ and $Pr=0.71$). Note that the arrow indicates the vertical axis (red) for the highest $Ra$ case. (b) Variation of normalized heat flux with inclination, where $Nu$ data are normalized by those of the level case $\phi =0^{\circ }$.

Next, we identify the inclination angles that yield maximum heat flux for each $Ra$ case. It is evident that IC with greater tilt ($\phi =75^{\circ }$) transports heat most efficiently until $Ra=10^{8}$, where $\phi =75^{\circ }$ is seen as the optimum inclination angle. However, beyond $Ra=10^{8}$, RBC transports heat more efficiently. To observe the $Nu(Ra)$ trend for the optimum inclination angle, we further present the variation of normalized $Nu$ by including four extra data points in figure 5. Heat flux initially rises with $Ra$ to become maximum at $Ra=10^{7}$, and drops monotonically beyond it. Interestingly, the optimum inclination becomes the most effective one at $Ra=10^{7}$, with an increment of $18\,\%$ in $Nu$. As shown by the shaded region, we observe a range $2.15\times 10^{6} \leq Ra \leq 2.15\times 10^{7}$ where increment in heat flux is more than $15.7\,\%$. Below and above this range, a sudden drop in augmented heat flux is noted.

Figure 5.

Identification of the $Ra$ range where IC transports heat in the most efficient manner for $\phi =75^{\circ }$. The shaded region represents the range $2.15\times 10^{6} \leq Ra \leq 2.15\times 10^{7}$ where significant increment in heat flux is obtained.

We further explore why the normalized heat flux drops with increasing $Ra$ at $\phi =75^{\circ }$. To answer this question, we show the time-averaged velocity field in the direction of IS overlaid by velocity vectors in figure 6. It is noted that with increase in $Ra$, the intensity of the velocity drops near the isothermal walls. At the highest $Ra$, hot and cold fluid movement towards the opposite plates is obstructed due to the formation of several small-scale flow structures. These structures redirect the cold (hot) fluid towards the colder (hotter) region, and restrict the fluid mixing. In other words, weakening of thermal convection in the direction of IS is responsible for the reduced heat flux at higher $Ra$. In figure 7, we quantify the drop in the velocity near the hot plate. Here, $N=10$ equispaced probes are placed at distance $y/H=0.05$ to obtain the mean velocity in the direction of IS ($u_p$) and NIS ($v_p$), which are computed as

(3.1)\begin{gather} u_p=\frac{1}{N}\sum_{i=1}^{N}{|\langle u_i \rangle_{t}|}, \end{gather}

(3.2)\begin{gather} v_p=\frac{1}{N}\sum_{i=1}^{N}{|\langle v_i \rangle_{t}|}, \end{gather}

where $i$ and $\langle \,{\cdot } \,\rangle _t$ represent the probe number and time average. Clearly, $u_p$ drops monotonically until $Ra=10^{8}$, and becomes nearly constant beyond it, which shows weak convection in the direction of IS. However, $v_p$ is insignificant compared to $u_p$, which is expected at such a greater tilt due to thermal stratification.

Figure 6.

For $\phi =75^{\circ }$, component of time-averaged velocity field in the direction of IS overlaid by velocity vector at $Ra$ values (a) $10^{6}$, (b) $10^{7}$, (c) $10^{8}$, and (d) $10^{9}$. With increase in $Ra$, the strength and region of velocity in the direction of IS diminish.

Figure 7.

For $\phi =75^{\circ }$, strength of velocity in the direction of the IS and NIS as a function of $Ra$. The velocities are computed from probes placed at distance $y/H=0.05$ units away from the hot plate.

One reason to consider the three decades of $Ra$ in the present study is to establish a $Nu(Ra)$ dependence for each inclination angle, which is described as follows. Figure 8(a) shows the compensated $Nu(Ra)$ scaling law in order to study the deviation from $Nu\sim Ra^{1/4}$. The greater tilt cases follow strictly the $1/4$ law, whereas a significant deviation is observed as $\phi$ approaches the level case. Further, in figure 8(b), we show the variation of the $Nu(Ra)$ scaling exponent $m$ with $\phi$ and find that a slight tilt does not alter the scaling exponent. On the other hand, it drops sharply as the cell is tilted beyond $\phi =15^{\circ }$. Evidently, the exponent becomes nearly invariant for $\phi \geq 75^{\circ }$. It is evident that $m$ is higher ($m=0.284$) in RBC than in VC ($m=0.254$), due to which the $Nu(\phi )$ curve shifts downwards with increasing $Ra$. Consequently, the most efficient heat transport state turns out to be the level case for $Ra >10^{8}$.

Figure 8.

(a) The compensated Nusselt number showing the attainment of a $1/4$ power law as inclination increases from $\phi =0^{\circ }$ to $\phi =90^{\circ }$. (b) Variation of the $Nu(Ra)$ scaling exponent with inclination angle.

3.3. $Re\unicode{x2013}\phi$ dependence

The Reynolds number ($Re$) is another response parameter that depends upon $Ra$ and inclination. It is a measure of turbulent intensity and computed as $Re=\sqrt {Ra/Pr}\,U$, where $U=\sqrt {\langle \bar {u}\,\boldsymbol{\cdot}\,\bar {u}\rangle }$; see (2.4a,b). In figure 9(a), we show a comparison of absolute $Re$ among the present study, Shishkina & Horn (Reference Shishkina and Horn2016) and Wang et al. (Reference Wang, Xia, Wang, Sun, Zhou and Wan2018a, Reference Wang, Wan, Yan and Sun2019). We observe that the trend of $Re$ in the present work agrees well with these studies. It is noted that with increasing $\phi$, $Re$ first remains nearly the same up to $\phi =15^{\circ }$, and then drops monotonically. For a direct comparison of $Re(\phi )$ at different $Ra$, we present the normalized $Re(\phi )/Re(0)$ data in figure 9(b), where the shaded region shows $Re<Re(0^{\circ })$. It is observed that $Re$ remains unaltered until $\phi =15^{\circ }$ for $Ra\leq 10^{8}$, whereas it increases by nearly $10\,\%$ in the highest simulated $Ra$ case. For $Ra=10^{9}$, $Nu(\phi )$ shows a reduction in heat flux for $\phi =15^{\circ }$, which is in contrast to the $Re$ value at the same inclination. Thus in a double-aspect-ratio configuration, despite unaltered or a small increase in turbulent intensity, the drop in $Nu$ indicates that the enhanced turbulent intensity does not necessarily improve the heat flux. A similar observation was made by Jiang et al. (Reference Jiang, Sun and Calzavarini2019), who reported that instead of turbulent intensity, the strength of LSC enhances $Nu$. However, the double-roll state at $\phi =15^{\circ }$ in figure 2(q) seems to be responsible for the enhanced turbulent intensity at $Ra=10^{9}$. With a further increase in inclination angle, $Re$ plummets sharply, and decreases by more than $90\,\%$ at $\phi =90^{\circ }$. The observed trend is different to what Wang et al. (Reference Wang, Wan, Yan and Sun2019) found, where $Re$ first drops and then rises again before following the monotonically decreasing trend. Again, this inconsistency may be attributed to the differences in the aspect ratio, which plays a significant role in determining the turbulent intensity due to the gap between the two isothermal walls. This analysis also suggests that $Re$ drops fastest in the intermediate inclination range and not near the vertical convection.

Figure 9.

(a) Comparison of the $Re(\phi )$ trend obtained in the present work with Shishkina & Horn (Reference Shishkina and Horn2016) ($Ra=10^{8}$, $\varGamma =1$ and $Pr=1$), Wang et al. (Reference Wang, Xia, Wang, Sun, Zhou and Wan2018a) ($Ra=10^{7}$, $\varGamma =2$ and $Pr=0.71$) and Wang et al. (Reference Wang, Wan, Yan and Sun2019) ($Ra=10^{7}$, $\varGamma =1/2$ and $Pr=0.71$). (b) Variation of normalized Reynolds number with inclination. Note that the normalization is carried out with respect to the level case ($\phi =0^{\circ }$). The shaded region highlights the drop in Reynolds number below the level case.

In figure 10, we show the least squares fit of $Re(Ra)$ data. For $\phi =0^{\circ }$, we observe that $Re \sim Ra^{0.59}$, which is in good agreement with the previous studies (Sugiyama et al. Reference Sugiyama, Calzavarini, Grossmann and Lohse2009; Zhang et al. Reference Zhang, Zhou and Sun2017; Yang et al. Reference Yang, Wei, Zhu, Dou and Qian2018; Chand et al. Reference Chand, De and Mishra2021b). In addition, we further study the $Re(Ra)$ dependence for other inclination angles, which have not been reported so far. The level case $\phi =0^{\circ }$, and $\phi =15^{\circ }$, follow nearly the same power law, $Re \sim Ra^{0.59}$. In these two cases, the collapse of $Re(Ra)$ curves indicates the insignificant effect of a smaller inclination angle on turbulent intensity, which is also supported by the obtained flow states. With further increase in $\phi$ ($=45^{\circ }$), we observe the same scaling exponent but with a significant drop in absolute $Re$. The observed drop may be attributed to the single-roll state. In comparison to a double-roll state, thermal plumes, driven by a single roll, traverse a larger distance to reach the opposite plate. As a result, thermal plumes diffuse early, which diminishes the strength of the single large-scale roll. In the two highest $\phi$ cases, the $Re(Ra)$ values nearly follow the same power law and the least scaling exponent $n=0.39$. Figure 10(b) shows the variation of $n$ with $\phi$, where the exponent remains nearly the same until $\phi =45^{\circ }$, and subsequently drops to a significantly lower value for the two highest inclination cases. The variation in the exponent signifies differences in the transport mechanism. While the higher exponent marks the presence of turbulence in the flow, its lower counterpart highlights the signature of laminar characteristics, which can also be seen from the stratified temperature field in figure 2.

Figure 10.

(a) Least squares fit of the $Re(Ra)$ data for all the inclination angles. (b) Variation of the $Re(Ra)$ scaling exponent with inclination angle.

3.4. Effect of inclination on flow structures in the rough cells

In figures 11(a–c), we present time-averaged temperature fields overlaid by streamlines for R$_1$, R$_2$, and R$_3$ configurations, respectively, to identify the changes in flow state with Rayleigh number and tilt angle. The temperature fields are shown for increasing $Ra$ and $\phi$. In the R$_1$ case, flow features change from a double-roll state in the level case to the stratified flow in the VC ($\phi =90^{\circ }$) for all $Ra$ cases, where only the central part of the cell shows a homogeneous mixing of hot and cold fluid. On the other hand, flow structures in the inclined cases $\phi =15^{\circ }$, $45^{\circ }$ and $75^{\circ }$ show transformation from the double-roll state to a single-roll state (except at $\phi =75^{\circ }$ for the highest $Ra$), spanning the entire cell. In the lowest $Ra$ case, the effect of inclination is significant as it brings the double–single roll state transformation at $\phi =15^{\circ }$ and incites thermal stratification at $\phi =45^{\circ }$. With increasing $Ra$, the effect of inclination becomes weaker as the flow structures appear to be the same at $\phi =0^{\circ }$ and $15^{\circ }$ for $Ra=10^{8}$. Since thermal plumes shrink with increasing $Ra$, smaller-sized plumes, owing to their greater mobility at higher $Ra$, do not align themselves in a single-roll state at $\phi =15^{\circ }$. Further, at $\phi =75^{\circ }$, with respect to the smooth case, a weaker thermal stratification and relatively better fluid mixing is evident in the cell. As $\phi$ approaches $90^{\circ }$, thermal stratification is seen clearly where nearly still, cold and hot fluids are present in the vicinity of the cold and hot isothermal walls of the cell, respectively.

Figure 11.

For configurations (a) R$_1$, (b) R$_2$ and (c) R$_3$, time-averaged temperature fields superimposed with streamlines for $Ra=10^{6}$ (i–v), $10^{7}$ (vi–x), $10^{8}$ (xi–xv), and $10^{9}$ (xvi–xx). The inclination angle varies from left to right in the order: $\phi =0^{\circ }$, $15^{\circ }$, $45^{\circ }$, $75^{\circ }$ and $90^{\circ }$. The arrows in the first row of each roughness case indicate the direction of gravity.

In the intermediate roughness configuration (R$_2$), a large-scale roll is attained at $\phi =15^{\circ }$ for the lowest $Ra$ case. For $Ra=10^{7}$, we observe that the flow state remains the same for $\phi =0^{\circ }$ and $15^{\circ }$. Such a response of flow structures towards inclination is also observed in R$_1$, but at a higher $Ra=10^{8}$. For $Ra=10^{8}$, a triple-roll state is evident at $\phi =15^{\circ }$, while for the highest $Ra$ case, a triple-roll state is found for all the inclinations in the range $15^{\circ }\leq \phi \leq 75^{\circ }$. At $\phi =75^{\circ }$, it appears that thermal stratification has vanished due to the presence of a multiple-roll state. On the other hand, it is still evident in VC, although weaker.

Features of the flow structures that appeared in the R$_2$ case become more pronounced in the tallest roughness configuration (R$_3$), as shown in figure 11(c). The tendency of transformation of the flow state in the level case to higher tilt angles is found to be the same for the lower $Ra$ cases (up to $Ra=10^{7}$) in the R$_2$ and R$_3$ configurations. However, beyond $Ra=10^{7}$, significant differences are observed in the R$_3$ case. Increasing roughness height delays the emergence of a single large-scale roll that depends upon the inclination. In particular, this transition appears at $\phi =75^{\circ }$ and $90^{\circ }$ for $Ra=10^{8}$ and $10^{9}$, respectively. For the highest $Ra$, multiple smaller rolls persist at $\phi =45^{\circ }$ and $75^{\circ }$, respectively. As observed for the R$_2$ case, these rolls improve mixing of fluid in the cell. The most interesting feature appears in VC, where we observe that thermal stratification is nearly absent. It is evident that roughness delays thermal stratification due to its ability to break the large-scale rolls and emit thermal plumes directly into the bulk region. It can also be viewed as early onset of turbulence in greater tilt cases. Since increase in roughness height reduces the distance between isothermal walls, increase in turbulent intensity is obvious, which is reflected in delayed thermal stratification. This result elucidates why Wang et al. (Reference Wang, Wan, Yan and Sun2019) observed greater heat flux at greater inclination even at $Ra=10^{9}$.

3.5. Effect of inclination on $Nu$ in the presence of roughness

Figure 12 shows a comparison of $Nu(\phi )/Nu(0^{\circ })$ among the smooth, R$_1$, R$_2$ and R$_3$ cases for the four $Ra$ values that cover the entire simulated $Ra$ range. For the lowest $Ra$, the $Nu(\phi )/Nu(0^{\circ })$ trend is same for all three roughness cases which is similar to the smooth case. It happens because of a weaker thermal forcing, due to which fluid remains inside the cavities (Zhang et al. Reference Zhang, Sun, Bao and Zhou2018; Chand et al. Reference Chand, Sharma and De2021a). Also, all the roughness elements are buried under a thick thermal boundary layer. The stagnant fluid and the thicker boundary layer suppress the effect of roughness at lower $Ra$, yielding a $Nu(\phi )$ trend similar to that of the smooth case. We observe that $Nu(\phi )$ drops until $\phi =15^{\circ }$, and rises with further increase in inclination angle. In the smooth, R$_1$ and R$_2$ cases, the maximum heat is transported at $\phi =75^{\circ }$. On the other hand, VC ($\phi =90^{\circ }$) yields maximum heat flux in R$_3$. For any particular tilt angle, heat flux decreases as the configuration changes from the smooth to the R$_3$ case. Since the cavity space increases from R$_1$ to R$_3$, the volume of stagnant fluid inside it also increases. In such a scenario, heat is transferred only through conduction. Therefore, for lower $Ra$, this study suggests that the taller the roughness height, the lower the heat flux, which answers why the smooth case yields the highest heat flux, and R$_3$ the least.

Figure 12.

Dependence of heat flux on inclination angle for $Ra$ values (a) $10^{6}$, (b) $10^{7}$, (c) $10^{8}$, and (d) $10^{9}$, in R$_1$, R$_2$ and R$_3$ configurations. The $Nu(\phi )$ curve does not change with roughness configurations at the lowest and highest $Ra$.

For $Ra=10^{7}$ (see figure 12b), the normalized $Nu(\phi )$ curves in R$_1$ and R$_2$ are nearly similar except for $\phi =15^{\circ }$, whereas a completely different trend is observed for the R$_3$ case. Contrary to the smooth case, $Nu$ increases monotonically in the R$_1$ and R$_2$ cases, to become maximum at $90^{\circ }$. This suggests that small asperities on the isothermal walls make the vertical convection the most efficient heat transport system. In R$_3$, $Nu$ is either higher or comparable to R$_1$ and R$_2$. This happens because roughness elements in R$_3$ become active at this $Ra$ ($=10^{7}$) and perturb the thermal boundary layer to modify the flow structures by directly injecting thermal plumes in the bulk (Chand et al. Reference Chand, Sharma and De2021a). On the other hand, owing to their relatively shorter height with respect to the thermal boundary layer, roughness elements in the other two cases remain buried inside it. Here, active roughness refers to those elements that actively participate in emitting thermal plumes and are not buried inside the thermal boundary layer. The idea of activation is seen clearly from the nearly the same $Nu(\phi )$ trends in the two smaller roughness cases.

For $Ra=10^{8}$, the trend in all three configurations reverses completely due to differences in active roughness peaks. In this case, flow states in R$_1$ and R$_2$ are the same beyond $\phi =15^{\circ }$, which is also reflected by a similar $Nu(\phi )$ trend in figure 12(c). On the other hand, in R$_3$, the trend becomes nearly the same beyond this $Ra$. This similarity is attributed to the early onset of transition to a multiple-roll state in R$_3$ through taller roughness peaks. Also, beyond this $Ra$, R$_3$ yields higher heat flux than the smooth, R$_1$ and R$_2$ cases. The reason for the highest heat flux in R$_3$ is again multiple rolls and boundary layer penetrating peaks. In Chand et al. (Reference Chand, Sharma and De2021a), it was already noted for the level case that onset of the enhanced heat flux regime occurs at $Ra=2.15\times 10^{8}$, $5.50\times 10^{7}$ and $10^{7}$ in the R$_1$, R$_2$ and R$_3$ cases, respectively, which supports the above arguments.

For the highest $Ra$ ($=10^{9}$), maximum heat flux is produced by the tallest case (R$_3$), and the minimum by the smooth case. Interestingly, the trend of increasing heat flux with roughness heights at the lowest $Ra$ value reverses at the highest value. The reason behind this change is the presence of multiple rolls (five large-scale rolls) in R$_3$, whereas maxima of three and two rolls are present in the R$_2$ and R$_1$ cases, respectively. Basically, a larger number of rolls improves the plume ejection spots, which enhance $Nu$ (Wang et al. Reference Wang, Verzicco, Lohse and Shishkina2020). Another physical argument that can be made here is widening of the cavities with increasing roughness height. Wider cavities have a higher tendency to get washed out more efficiently to enhance the emission of thermal plumes (Toppaladoddi et al. Reference Toppaladoddi, Succi and Wettlaufer2017; Zhang et al. Reference Zhang, Sun, Bao and Zhou2018).

For each inclination angle, we further show the variation of normalized heat flux with $Ra$ in figure 13. Based on $Nu/Nu(0)$, we define enhanced, $Nu>Nu(0)$, and reduced, $Nu<Nu(0)$, heat flux regimes. Note that the $Ra$ value beyond which the enhanced heat flux regime changes to the reduced one is termed the critical Rayleigh number ($Ra_c$). For the R$_1$, R$_2$ and R$_3$ cases, $Ra_c$ is found as $10^{8}$, $10^{7}$ and $3\times 10^{6}$, respectively. The width of the enhanced heat flux regime diminishes with increasing roughness height. In this regime, $Nu/Nu(0)$ increases with inclination, as shown by upward arrows. However, a contrasting effect of $\phi$, represented by the downward arrows, is seen in the reduced heat flux regime. The transition in the $Nu(\phi )$ trend reflects the onset of turbulence, which again supports our observation that RBC transports heat more efficiently than IC at larger $Ra$. From R$_1$ to R$_3$, the decrease in $Ra_c$ signifies early onset to turbulence due to a surge in plume emission (Chand et al. Reference Chand, De and Mishra2021b). In particular, we observe maximum improvements of $18\,\%$, $24\,\%$ and $25\,\%$ in heat flux for the R$_1$, R$_2$, and R$_3$ cases, respectively. On the other hand, following the same order, maximum reductions of $32\,\%$, $25\,\%$ and $17\,\%$ in heat flux are observed in the reduced heat flux regime.

Figure 13.

Variation of the normalized heat flux $Nu(\phi )/Nu(0)$ with $Ra$ for (a) R$_1$, (b) R$_2$, and (c) R$_3$ configurations at four different inclinations. While the vertical dashed line demarcates the enhanced and reduced heat flux regimes, the horizontal dash-dotted line represents $Nu(\phi )/Nu(0^{\circ })=1$. The upward (downward) arrows indicate increasing (decreasing) $Nu$ with $\phi$.

Next, we study the effect of tilt on $Nu(Ra)$ dependence by showing the least squares fit of the $Nu$ data in figures 14(a–e), and $m(\phi )$ behaviour in figure 14(f). In the R$_1$ case, note that deviation from the fitting at the highest $Ra$ is observed up to $\phi =45^{\circ }$, which indicates the late onset of turbulence (higher $Ra_c$). The deviation is also evident from the rising error bar in figures 14(a–c). However, as $\phi >45^{\circ }$, the error bar nearly vanishes, which signifies the laminar heat transport throughout the simulated $Ra$ range. This is apparent from the similar $m(\phi )$ trend in R$_1$ and the smooth case beyond $\phi =45^{\circ }$. For these two cases, the exponent drops to $m\approx 1/4$ in VC, which is consistent with previous results (Churchill & Chu Reference Churchill and Chu1975; Shishkina Reference Shishkina2016). On the other hand, the early onset of turbulence is responsible for the higher exponent in the two taller roughness configurations. A direct connection between the turbulent intensity and scaling exponent is evident as the configuration changes from R$_1$ to R$_3$ in figure 14(f). It is interesting to note that despite significant differences in roughness heights, the $m(\phi )$ trend remains nearly the same in the three cases. Here, the effect of decreasing $m(\phi )$ corresponds to the drop in heat flux as $Ra$ increases beyond $Ra_c$.

Figure 14.

Least-squares fits of $Nu(Ra)$ at different inclination angles: (a) $\phi =0^{\circ }$, (b) $\phi =15^{\circ }$, (c) $\phi =45^{\circ }$, (d) $\phi =75^{\circ }$, (e) $\phi =90^{\circ }$. (f) The $Nu(Ra)$ scaling exponent $m$ as a function of inclination angle for the R$_1$, R$_2$ and R$_3$ cases. The dashed line in the last frame represents $m=0.287$ obtained for the smooth case at $\phi =0^{\circ }$. Note that the $Nu(Ra)$ data are presented with error bars in the R$_1$ case to show deviation from the least squares fit.

3.6. Effect of inclination on $Re$ in the presence of roughness

Figure 15 shows the variation of Reynolds number with inclination at four $Ra$ values that cover the entire simulated $Ra$ range. Similar to the smooth case, the $Re(\phi )$ curve shows a consistent trend where it remains nearly identical or changes slightly at small inclination ($\phi =15^{\circ }$), and drops monotonically with further increase in $\phi$ in all four $Ra$ cases. However, roughness impacts the turbulent intensity, which is reflected in the deviation from the smooth case in $Re(\phi )$. At lower $Ra$ ($\leq 10^{7}$), roughness height suppresses the turbulent intensity, i.e. the taller the roughness height, the lower the turbulent intensity. This happens due to the stagnant fluid residing in the cavities at lower $Ra$. It is also reflected from the nearly the same $Re$ values in the smooth and R$_1$ cases, where the flow structures are similar. Beyond $Ra=10^{8}$, as expected, differences in $Re$ in the three configurations reduce due to movement of fluid in the cavities (secondary vortices), which washes them out. The role of roughness heights is remarkable for a greater tilt at $Ra=10^{9}$. Here, while $Re$ between $\phi =75^{\circ }$ and $90^{\circ }$ changes marginally in the smooth case, a significant difference is evident as the roughness configuration changes from R$_1$ to R$_3$. This indicates clearly weaker thermal stratification and greater mixing of fluid at these two inclinations in taller roughness configurations, as observed in the flow structures in § 3.4. Contrary to lower $Ra$ cases, we observe that the $Re(\phi )$ trend does not show notable changes until $\phi =45^{\circ }$ for the highest $Ra$. This can be attributed to the similar flow structures that show a greater turbulent intensity at this inclination range for $Ra=10^{9}$. On the other hand, evidence of thermally stratified flow at $\phi =45^{\circ }$ for lower $Ra$ cases is responsible for the sudden drop in $Re$.

Figure 15.

$Re\unicode{x2013}\phi$ dependence for three roughness cases and the smooth case at four $Ra$ values: (a) $10^{6}$, (b) $10^{7}$, (c) $10^{8}$, and (d) $10^{9}$. The $Re\unicode{x2013}\phi$ curves in the roughness cases are similar to the smooth case. Note that while $Re$ decreases from R$_1$ to R$_3$ at lower $Ra$, it becomes comparable in all three configurations at the highest $Ra$.

We study further the effect of inclination on $Re(Ra)$ dependence for different inclination angles in figure 16. The least squares fit of $Re(Ra)$ data clearly shows that the scaling laws $Re\sim Ra^{n}$ for $\phi =0^{\circ }$ and $15^{\circ }$ are nearly the same, where the scaling exponents for R$_1$, R$_2$ and R$_3$ are $0.61$, $0.64$ and $0.68$, respectively. The nearly the same exponents for the $\phi =0^{\circ }$ and $15^{\circ }$ cases indicates a similar transport mechanism, which is also reflected in their flow structures. Note the effect of roughness on the scaling exponent, which is nearly same for the smooth and tiny roughness heights. However, as roughness changes from R$_1$ to R$_3$, the exponent increases with respect to the smooth case. For $\phi =45^{\circ }$, although turbulent intensity decreases, the scaling exponent remains nearly the same as in the two smaller inclinations, which is reflected by the $n(\phi )$ trend in figure 16(d). However, beyond $\phi =45^{\circ }$, the exponent as well as turbulent intensity drops remarkably due to settling down of the cold and hot fluid near the the cold and hot walls of the cell, respectively (see the time-averaged temperature field in figure 11). For $\phi =75^{\circ }$ and $90^{\circ }$, we observe that the exponent in the smooth and R$_1$ cases does not change appreciably, which indicates a similar flow state. On the other hand, flow structures in R$_2$ and R$_3$ change significantly at these inclinations, which is also reflected in their $n(\phi )$ trend. In R$_3$, greater turbulent intensity (weaker thermal stratification) as compared to the other smaller roughness set-ups at $\phi =75^{\circ }$ is responsible for a higher scaling exponent.

Figure 16.

For five inclinations, least squares fits of $Re$ data for (a) R$_1$, (b) R$_2$, and (c) R$_3$. (d) The $n\unicode{x2013}\phi$ trend for the three roughness cases and the smooth case. The scaling exponent remains nearly the same until $\phi =45^{\circ }$, and drops beyond this range.