3.1. Development of the thermal boundary layer and formation of plumes

In this subsection, we present how the plume formation process is correlated to the development of the thermal boundary layer (TBL). Figure 4(a) shows four snapshots of shadowgraph images near the top surface of the heat source (indicated by the yellow dashed line). The measurement was made at $Pr = 904.7$ and $Ra_{f} = 6.5 \times 10^6$ with continuous heating, so the generated thermal structure was in the starting stage of a plume, i.e. a starting plume. According to the working principle of shadowgraph technique, the TBL is displayed as a dark strip in these images, and the location with the strongest temperature gradient is the edge of the TBL (indicated by the yellow dots). Therefore, we can determine the thickness $\delta$ of the TBL by the vertical distance between the yellow line and the yellow dot. Note that the absolute value of $\delta$ is affected by the inevitable optical distortion in the shadowgraph measurements, so we just examine its change with time here.

Figure 4.

(a) Shadowgraph images at four typical moments during the formation of a starting plume at $Pr = 904.7$ and $Ra_{f} = 6.5 \times 10^6$. The complete process is shown in the supplementary movie available at https://doi.org/10.1017/jfm.2024.266. The pink dashed line highlights the position of the emitted plume. The yellow dashed line and dot represent the top surface of the resistor and the edge of the TBL, respectively, and their vertical distance is used to characterize the thickness $\delta$ of the TBL. (b) Plot of $\delta$ as a function of time after the heating power was turned on, from which four distinct stages separated by three characteristic times can be determined. (c) The three characteristic times as functions of $Ra_f$ at $Pr = 904.7$. Here, all the characteristic times have been normalized by the diffusive time $\tau _{0}=r_{0}^{2} / \kappa$ defined in § 2.1. The solid lines are power-law fits to each data set: $t_{emit}/\tau _{0}\sim Ra_f^{-0.31\pm 0.01},\ t_{recover}/\tau _{0}\sim Ra_f^{-0.52\pm 0.01}$ and $t_{static}/\tau _{0}\sim Ra_f^{-0.56\pm 0.04}$.

The $\delta$ thus determined as a function of time is plotted in figure 4(b). It is seen that the evolution of $\delta$ can be divided into four stages: the growth stage, the emission stage, the recover stage and the quasi-static stage. Specifically, after the heating power is turned on at $t_{start}$, heat is conducted into the fluid and the TBL begins to build up. During the growth stage, the TBL grows in its thickness $\delta$, but remains stable. When $\delta$ reaches a critical value at $t = t_{emit}$, a plume starts to emit due to the instability of the TBL. Because some hot fluid parcels inside the TBL are taken away by the plume, the TBL shrinks, and its thickness $\delta$ decreases during the emission stage. As time goes by, the head of the plume becomes completely detached from the TBL at $t = t_{recover}$ and its tail appears, indicating the formation of a well-defined starting plume. At this critical moment, $\delta$ reaches a local minimum, then the TBL gets into the recover stage. During the recover stage, the TBL grows again gradually thanks to the continuous heating supplied from the heat source, and eventually becomes quasi-static at $t = t_{static}$. In the quasi-static stage, because all the heat supplied is taken away by the convection flow, the TBL becomes stable again and its thickness remains unchanged. It is interesting to note that similar evolution of the TBL thickness (growth–drop–stationary) was observed in some numerical studies of buoyancy-driven flows on a heated horizontal plate (Jiang, Nie & Xu Reference Jiang, Nie and Xu2019a,Reference Jiang, Nie and Xub; Jiang et al. Reference Jiang, Nie, Zhao, Carmeliet and Xu2021). However, these studies did not observe the recover stage as in the present experimental study. Further investigations combined with experiments and simulations are required to find the reason for this difference.

The complete process described above can be found in the supplementary movie, which clearly shows how the plume formation is correlated to the development of the TBL. During this process, three characteristic times are identified, and their $Ra_f$ dependencies are plotted in figure 4(c). It is seen that $t_{emit} / \tau _{0}$, $t_{recover} / \tau _{0}$ and $t_{static} / \tau _{0}$ exhibit distinct $Ra_f$-dependent scaling exponents, being $-0.31$, $-0.52$ and $-0.56$, respectively. The scaling of $t_{recover}/ \tau _{0}$ recalls the study by Moses et al. (Reference Moses, Zocchi and Libchaberii1993), who predicted that the build-up time for starting plumes follows a relation $t_{build\text {-}up} / \tau _{0} \sim Ra_{f}^{-0.5}\,Pr^{0.5}$.

To compare our results with this theoretical prediction, we plot different characteristic times for all the $Pr$ in figure 5. It is seen that $t_{recover}/\tau _{0}$ obeys a scaling relation $Ra_{f}^{-0.48}\,Pr^{0.48}$ for the present parameters explored, which is in good agreement with the theoretical prediction mentioned above. In terms of magnitude, $t_{recover}$ measured directly in the present study is very close to the experimental values of $t_{build\text {-}up}$ reported by Moses et al. (Reference Moses, Zocchi and Libchaberii1993). However, Moses et al. (Reference Moses, Zocchi and Libchaberii1993) did not measure $t_{build\text {-}up}$ directly. Instead, they assumed that the starting plume was originated from a ‘virtual point source $y_{virt}$’ at $t = 0$, and the plume's ascent height could be described by a function $y = V_{starting}(t-t_{build\text {-}up})$, where $V_{starting}$ is the rising velocity of the starting plume at the steady state. Then $t_{build\text {-}up}$ can be evaluated to be $t_{build\text {-}up} = -y_{virt}/V_{starting}$. Although the result yielded from this estimation ($t_{build\text {-}up}/\tau _{0} \sim Ra_{f}^{-0.46}\,Pr^{0.46}$) is consistent with their theory, the obtained $t_{build\text {-}up}$ is physically ambiguous, and the data plotted in figure 5 are relatively scattered. This is in contrast to $t_{recover}$ in the present study, which can be determined precisely thanks to its clear physical meaning as revealed by figure 4.

Figure 5.

Different normalized characteristic times as functions of $Ra_{f}^{-0.5}\,Pr^{0.5}$. The solid symbols are the build-up times for starting plumes measured indirectly by Moses et al. (Reference Moses, Zocchi and Libchaberii1993). The solid lines are the best power-law fits to each data group.

The physical meaning of $t_{emit}$ is also revealed unambiguously by figure 4. This data group can be fitted by a power law $t_{emit}/\tau _0 \sim Ra_{f}^{-0.41}\,Pr^{0.41}$. To understand this scaling relation, we note that at the moment of plume emission, the time duration for heating power being supplied to the fluid (i.e. the heating time $t_{heat}$) is equal to $t_{emit}$. Therefore, we can substitute $t_{emit}$ into the definition of Rayleigh number $Ra = t_{heat}\,Ra_{f}/\tau _0\,Pr$ (see the last paragraph in § 2.1) to quantify the total buoyancy supplied to the fluid at this moment. Specifically, we can substitute $Ra_{f}/Pr = Ra\,\tau _0/ t_{emit}$ into the scaling relation $t_{emit}/\tau _0 \sim Ra_{f}^{-0.41}\,Pr^{0.41}$, which leads immediately to $t_{emit}/\tau _0 \sim Ra^{-0.69}$. This result is in good agreement with a phenomenological model developed by Howard (Reference Howard1966), who predicted that the onset time for the instability of the TBL scales with $Ra$ as $Ra^{-2/3}$. Similar scaling exponents have been reported in some previous experiments (Vest & Lawson Reference Vest and Lawson1972; Davaille & Vatteville Reference Davaille and Vatteville2005). However, these studies determined the onset time by watching the convection patterns instead of measuring the TBL directly, so the correlation between the plume emission and the TBL instability has not been revealed explicitly as in the present study.

Compared to $t_{recover}$ and $t_{emit}$, the data group $t_{static}$ is relatively scattered and does not collapse well for different $Pr$. Nevertheless, a simple power-law fit to this data group yields a relation $t_{static}/\tau _{0} \sim Ra_{f}^{-0.38}\,Pr^{0.38}$. If a least squares method is used to determine the best $Ra_f$ and $Pr$ scaling exponents, then we obtain $t_{static}/\tau _{0} \sim Ra_{f}^{-0.49}\,Pr^{0.33}$ (not shown here). The latter scaling relation might be more reasonable, but we do not have a theory for it at this stage. According to figure 4(b), $t_{static}$ is the moment when the TBL changes from the recover stage to the quasi-static stage. Thus it is more meaningful to check the time duration for the TBL to recover, i.e. $t_{static}-t_{recover}$. However, no clear $Ra_{f}$–$Pr$ dependence is found for this time duration either, so we do not present the data here. It is noteworthy that a similar recover process exists in turbulent convection experiments due to the so-called finite conductivity effect, which could influence the plume emission and even the heat transport properties (Brown et al. Reference Brown, Nikolaenko, Funfschilling and Ahlers2005; Sun & Xia Reference Sun and Xia2007). It would be interesting to quantify this process in turbulent convection experiments and compare with the present results in future studies.

To further examine the correlation between the plume formation and the TBL development, we plot in figure 6 the plume's rising velocity and the corresponding acceleration together with the TBL thickness. Because similar behaviours have been observed for different $Pr$, the data measured at $Pr = 904.7$ are taken as the example for discussion. Note that the acceleration plotted here is calculated by taking the time derivative of the rising velocity. A Gaussian convolution method was adopted to reduce the data noise (Mordant et al. Reference Mordant, Crawford and Bodenschatz2004), so there are no acceleration data at the very beginning of the time series. In addition, the plume has not formed before $t = t_{emit}$, so the velocity and acceleration before this moment are the data inside the TBL. It is seen in figure 6(a) that there is a time delay in the evolution curves between the plume and the TBL. Such a time delay is much clearer in figure 6(b), which seems to be a constant in units of $t_{recover}$. While the TBL thickness becomes unchanged after $t \simeq 1.5t_{recover}$, the rising velocity of the plume does not reach the steady state until $t \simeq 2.5t_{recover}$. The time delay can also be observed in the acceleration of the plume (see figures 6c,d). Noticeably, the normalized accelerations $A/A_{max}$ for different $Ra_f$ almost fall into a single curve once they are plotted against $t/t_{recover}$, just as for the normalized rising velocities $V/V_{max}$ shown in figure 6(b). These results not only provide further information about the correlation between the plume formation and the TBL development, but also suggest that $t_{recover}$ is a crucial time scale during the plume evolution process, which will be discussed further in the following subsections.

Figure 6.

(a) Temporal evolution of the TBL thickness determined from shadowgraph (lines) and the plume's rising velocity obtained from PIV (symbols) for different $Ra_f$ at $Pr = 904.7$. (b) The same data as in (a), but normalized by using the maximum TBL thickness, the maximum rising velocity and $t_{recover}$. Note that these quantities used for normalization have different values for different $Ra_f$. (c,d) The relation between the TBL thickness and the plume's acceleration is examined in a similar way as in (a,b).

3.2. Effects of heating time

In the previous subsection, we have shown that there exist four distinct stages during the plume formation process. In this subsection, we present a quantitative comparison of the properties of thermal structures emitted at different stages, i.e. generated by different heating times $t_{heat}$. To examine the effects of $t_{heat}$ in a better way, $Ra_f$ was fixed at $Ra_f = 6.5 \times 10^6$. We first show the results measured at $Pr=904.7$. The three characteristic times for this $Pr$ at $Ra_f = 6.5 \times 10^6$ are $t_{emit} = 6.6$ s, $t_{recover} = 10.6$ s and $t_{static}= 16.3$ s, so $t_{heat}$ was varied from 3.0 to 16.3 s, corresponding to an $Ra$ ($= t_{heat}\,Ra_{f}/\tau _0\,Pr$) number range $900 \leq Ra \leq 4900$.

Figure 7(a) plots the temporal evolution curves of the rising velocities of thermal structures generated by different $t_{heat}$ (or equivalently different $Ra$). The data of the starting plume are plotted together for comparison. It is seen that the maximum rising velocities in these curves increase monotonically with increasing $t_{heat}$, and the upper limit is bounded by the case of the starting plume. Based on their variation trends, these curves can be divided into two groups. For thermal structures generated by $t_{heat} \geq t_{emit}$ ($=6.6$ s), their rising velocities first increase quickly in a manner like the starting plume, and then decay slowly like a thermal after reaching the maximum value (demonstrated in § 3.3). Note that these thermal structures can be viewed as detached TBLs, so their almost-collapsed curves in the accelerating stage could be originated from the dynamics inherited from the unstable TBL, which is commonly believed in studies of Rayleigh–Bénard convection (Ahlers et al. Reference Ahlers, Grossmann and Lohse2009). However, for the cases with $t_{heat} < t_{emit}$, their rising velocities show a different trend in the accelerating stage. This is because these thermal structures are not emitted from the TBL due to instability. Rather, they are hot fluid parcels adjacent to the heat source, which go up after the heating power is turned off thanks to their own buoyancy. In this context, their generation process is similar to a sudden release of a hot fluid parcel in some experimental studies of thermals (see e.g. Griffiths Reference Griffiths1986).

Figure 7.

(a,b) Temporal variations of (a) the rising velocity and (b) the corresponding acceleration of thermal structures generated by different heating times (or equivalently different $Ra$) at $Pr = 904.7$ and $Ra_f=6.5\times 10^6$. (c,d) The dimensionless maximum velocity $V_{max}/u_{0}$ and the dimensionless maximum acceleration $A_{max}/a_{0}$ as functions of (c) $Ra$ and (d) the dimensionless heating time $t_{heat}/t_{recover}$. The corresponding values of the starting plume are indicated by the purple solid line (dimensionless maximum velocity) and the blue solid line (dimensionless maximum acceleration), respectively.

The acceleration plotted in figure 7(b) confirms that $t_{emit}$ is the characteristic time that determines whether thermal structures generated by finite $t_{heat}$ will have similar evolution dynamics. It is seen clearly that the accelerations of thermal structures with $t_{heat} < t_{emit}$ are much weaker than in the cases with $t_{heat} \geq t_{emit}$, and exhibit an obvious different trend in the accelerating stage. For thermal structures with $t_{heat} \geq t_{emit}$, their acceleration curves largely follow the trend of the starting plume, except for the decay behaviour. To be specific, the acceleration of the starting plume decreases monotonically from the maximum value to a steady level close to zero. By contrast, the data for thermal structures with $t_{heat} \geq t_{emit}$ first decay to negative values that are well below zero, and then increase gradually to the level of the starting plume. A close look at figures 7(a,b) reveals that the moment when the acceleration drops to zero is the moment when the rising velocity reaches the maximum value. This phenomenon can be found for all the thermal structures generated by finite $t_{heat}$, though it is not that obvious for the cases with $t_{heat} < t_{emit}$. To understand this result, we note that thermal structures generated by finite $t_{heat}$ do not have an energy supply after the heating power is turned off, thus the viscous drag becomes significant in the decelerating stage and leads to negative accelerations. As the viscous drag is larger for larger rising velocity, one can see in figure 7(b) that the drop in acceleration is stronger for larger $t_{heat}$.

To examine quantitatively how $t_{heat}$ affects the properties of thermal structures, we take the maximum velocity and the maximum acceleration as the characteristic quantities, and plot their dimensionless values in figures 7(c,d). It is seen that the maximum velocity increases with $Ra$ (or the dimensionless heating time $t_{heat}/t_{recover}$) and approaches the value of the starting plume gradually. Based on the variation trend of $V_{max}/u_0$ in figure 7(d), the maximum velocity of thermal structures generated by finite $t_{heat}$ will be equal to the value of the starting plume when $t_{heat} \simeq 2.5t_{recover}$. This time scale is consistent with the data shown in figure 6(b), where one can see that the rising velocity of the thermal structure reaches the steady state at approximately $2.5t_{recover}$.

The maximum acceleration $A_{max}/a_{0}$ also increases with $t_{heat}/t_{recover}$ and reaches the value of the starting plume at $t_{heat} \simeq t_{recover}$. For larger $t_{heat}$, $A_{max}/a_{0}$ remains unchanged. This behaviour could be understood as follow. Although the thermal structure begins to emit at $t_{emit}$, the emission process does not complete until $t_{recover}$. During the emission process, the thermal structure can gain buoyancy energy from the TBL directly, so its acceleration can increase to a larger value if the heating power supply is continued. At the moment of $t_{recover}$, the thermal structure is fully detached from the TBL, and its buoyancy energy is fed by the long tail after then, just as in the case of the starting plume. Therefore, the maximum acceleration reaches the value of the starting plume at $t_{heat} = t_{recover}$. And because the rising velocity does not reach the maximum value until the acceleration becomes zero, the maximum velocity approaches the value of the starting plume at a larger value of $t_{heat}/t_{recover}$.

These asymptotic behaviours can be observed for all the $Pr$ explored in the present study. As shown in figures 8(c,d), while $V_{max}/u_0$ for different $Pr$ seem to approach the upper limit values at different $t_{heat}/t_{recover}$, $A_{max}/a_0$ for different $Pr$ reach different plateaus roughly at the same condition ($t_{heat}/t_{recover} \simeq 1$). Noticeably, if $A_{max}/a_0$ is plotted against $Ra$, then the values almost fall onto a single curve before reaching the plateaus (see figure 8b). A more remarkable finding comes from the data shown in figure 8(a). It is seen that $V_{max}/u_0$ for different $Pr$ collapse well over the present $Ra$ number range. However, these data do not follow a simple scaling law. Instead, they experience a transition from $V_{max}/u_0 \sim Ra$ to $V_{max}/u_0 \sim (Ra\ln Ra)^{0.5}$ at $Ra \simeq 6000$. To understand this scaling transition, we note that thermal structures generated by finite $t_{heat}$ contain finite amounts of buoyancy and thus can be viewed as thermals. However, according to the data shown in figure 6(b), if $t_{heat}$ is larger than 2.5$t_{recover}$, then these thermal structures will get into the quasi-static stage and go up with constant rising velocities like plumes. Since the maximum heating times examined in figure 8 are all smaller than 2.5$t_{recover}$, these data can be understood by the behaviours of thermals. For the scaling relation below $Ra \simeq 6000$, it is consistent with previous studies of thermals at low $Ra$ (Morton Reference Morton1960; Shlien & Thompson Reference Shlien and Thompson1975); for that above $Ra \simeq 6000$, it is in excellent agreement with the model prediction and the numerical results by Whittaker & Lister (Reference Whittaker and Lister2008). Interestingly, if a simple power law is attempted to fit all the data in figure 8(a), then the scaling relation turns out to be $V_{max}/u_0 \sim Ra^{0.78}$ (not shown in the figure for clarity), which is in line with the experimental finding and the model prediction ($V_{max}/u_0 \sim Ra^{0.75}$) by Griffiths (Reference Griffiths1986). Therefore, the present data provide not only experimental evidence for the existence of the $(Ra\ln Ra)^{0.5}$ scaling for thermals at large $Ra$, but also a unifying understanding of the seemingly inconsistent scaling relations observed in previous studies (Morton Reference Morton1960; Shlien & Thompson Reference Shlien and Thompson1975; Griffiths Reference Griffiths1986; Whittaker & Lister Reference Whittaker and Lister2008).

Figure 8.

(a,c) The dimensionless maximum velocity $V_{max}/u_0$ and (b,d) the dimensionless maximum acceleration $A_{max}/a_0$ as functions of (a,b) $Ra$, and (c,d) the dimensionless heating time $t_{heat}/t_{recover}$ for different $Pr$ at $Ra_f=6.5\times 10^6$. The black and blue solid lines in (a) are two different power laws, as labelled.

3.3. Characteristics of thermal structures generated by $t_{heat} = t_{recover}$

Based on the results presented in §§ 3.1 and 3.2, we have found that $t_{recover}$ is a crucial time scale during the formation and evolution of thermal structures. In particular, it is the critical moment when the thermal structure is fully detached from the TBL and the corresponding maximum acceleration reaches an upper limit. Since thermal structures generated by different heating times $t_{heat}$ have similar evolution dynamics once $t_{heat} \ge t_{emit}$, we can take the case with $t_{heat} = t_{recover}$ as an example to examine their evolution dynamics in detail, which is the focus of this subsection. Again, we will present first the results measured at $Pr=904.7$ and then the data for different $Pr$.

Figure 9 shows the evolution curves of the rising velocities of thermal structures generated by $t_{heat}=t_{recover}$ for different $Ra_f$ at $Pr = 904.7$. The data have been plotted in different ways to show every detailed feature during the evolution. It is seen in figure 9(a) that the curves for different $Ra_f$ have a similar shape, where the higher $Ra_f$ case has a larger maximum rising velocity $V_{max}$ that occurs at a shorter time. If we take $V_{max}$ and $t_{recover}$ as the referenced units, then the normalized velocity $V/V_{max}$ as a function of $t/t_{recover}$ for different $Ra_f$ collapses onto a universal curve when $t / t_{recover} \le 2.5$ (see figure 9b). The three characteristic times for the $Ra_f=6.5\times 10^6$ case are also highlighted in figure 9(b), indicating that the evolution trends are different before and after $t_{emit}$. However, we note that the data collapse could not be achieved by using $t_{emit}$ as the referenced time. Moreover, the universal behaviour applies only to the data within $t / t_{recover} \simeq 2.5$, after which the data for different $Ra_f$ become diverged (see the inset of figure 9b).

Figure 9.

(a) Temporal variations of the rising velocities of thermal structures generated by $t_{heat}=t_{recover}$ for different $Ra_f$ at $Pr = 904.7$. (b) The same results as in (a), but normalized by the maximum rising velocity $V_{max}$ and the characteristic time $t_{recover}$ for each $Ra_f$. The inset shows that the data for different $Ra_f$ begin to diverge when $t / t_{recover} \gtrsim 2.5$. (c) A similar plot as in (b), but the data are normalized by the diffusive velocity $u_{0}$ ($= r_0/\tau _0$). The solid curves are the best fits to each data set for $t / t_{recover} \gtrsim 2.5$. (d) The same data as in (c), with the fitting curves extended to the stagnant time $t_{stag}$, at which the rising velocity becomes zero and the thermal structure reaches the maximum ascent height $H_{max}$. Inset: the dimensionless stagnant time $t_{stag}/t_{recover}$ and the dimensionless maximum ascent height $H_{max}/r_0$ as functions of $Ra_f$. The solid lines are power-law fits to each data set: $t_{stag}/t_{recover} \sim Ra_f^{0.50}$ and $H_{max}/r_{0} \sim Ra_f^{0.31}$. See text for detailed explanations.

For the time period $t / t_{recover} > 2.5$, the thermal structures have got into the decelerating stage without heat supply and are dominant by the viscous drag. Therefore, we take the diffusion velocity $u_0$ ($=r_0/\tau _0$) as the referenced velocity, and plot $V/u_{0}$ as a function of $t/t_{recover}$ in figure 9(c). Although this dimensionless form does not yield a data collapse, the data in the decelerating stage ($t / t_{recover} > 2.5$) for different $Ra_f$ exhibit a similar shape and can be fitted by a relation $V/u_{0} \sim (t/t_{recover}-1)^{-0.5} - V_{virt}$, which is proposed to describe the evolution behaviour of thermals (Morton Reference Morton1960; Griffiths Reference Griffiths1986). Here, $V_{virt}$ is known as the virtual velocity, as the rising velocity should approach zero as the time goes to infinity.

Actually, thermal structure generated by a finite heating time should have a finite lifetime. With this in mind, we define the stagnant time $t_{stag}$ when the rising velocity decays to zero. At this moment, the thermal structure becomes stagnant and reaches the maximum ascent height $H_{max}$. Figure 9(d) illustrates how this characteristic time is determined from the decay curves of the rising velocity. The dimensionless stagnant time $t_{stag} / t_{recover}$ thus determined is found to scale with $Ra_f$ as $t_{stag}/t_{recover} \sim Ra_f^{0.50}$ (see the inset of figure 9d). As $t_{recover}/\tau _0\sim Ra_f^{-0.48}$, we can infer that $t_{stag}$ is almost independent of $Ra_f$, implying that this time scale is determined by a diffusion process. The corresponding maximum ascent height $H_{max}$ can also be obtained by integrating the rising velocity to the moment $t = t_{stag}$. The data thus obtained are plotted in the inset of figure 9(d), and they follow a scaling relation as $H_{max}/r_{0} \sim Ra_f^{0.31}$. This scaling exponent coincides with the predicted scaling $1/3$ by a theoretical model of turbulent atmospheric plumes (Stothers Reference Stothers1989), despite the present scenario being laminar thermal structures in a homogeneous stationary fluid layer.

The analysis above suggests three characteristic quantities for further examination with different $Pr$: the maximum rising velocity $V_{max}$ during the evolution, the stagnant time $t_{stag}$, and the maximum ascent height $H_{max}$. We first check the $Pr$ dependencies of $t_{stag}$ and $H_{max}$. Figure 10(a) shows the dimensionless stagnant time $t_{stag}/t_{recover}$ as a function of $Pr$ for different $Ra_f$. It is seen that these data sets can all be described by power-law fittings, with the scaling exponents ranging from 0.06 to 0.12. Because the $Ra_f$-dependent scalings of $t_{stag}/t_{recover}$ are close to 0.50 for all $Pr$ (not shown here), we compensate the data in figure 10(a) by ${Ra_f}^{0.50}$ to collapse them into a single line (see figure 10b); then a power-law fit to all the data yields a $Pr$-dependent scaling 0.08, i.e. $t_{stag}/t_{recover}\sim {Ra_f}^{0.50}\,Pr^{0.08}$, or equivalently, $t_{stag}/\tau _0\sim {Ra_f}^{0.02}\,Pr^{0.56}$. Through a similar procedure, we obtain the $Ra_f$–$Pr$ dependence of the maximum ascent height as $H_{max}/r_0\sim {Ra_f}^{1/3}\,Pr^{0.16}$ (see figures 10c,d). To the best of our knowledge, no theory has studied the stagnant time before. However, the maximum ascent height has received much of interest, as it is relevant to a range of natural phenomena, such as cloud rise and volcanic eruption (Wilson et al. Reference Wilson, Sparks, Huang and Watkins1978; Sparks Reference Sparks1986; Stothers Reference Stothers1989; Hansen et al. Reference Hansen, Esposito, Stewart, Colwell, Hendrix, Pryor and West2006; Glaze et al. Reference Glaze, Self, Schmidt and Hunter2017). The scaling relations obtained here, especially the less-explored $Pr$ dependencies, might be helpful to understand the relevant processes in nature.

Figure 10.

(a) The dimensionless stagnant time $t_{stag}/t_{recover}$ as a function of $Pr$ for different $Ra_f$. The solid lines are power-law fits to each data set. (b) Plots of $t_{stag}/t_{recover}$ compensated by ${Ra_f}^{0.50}$ as a function of $Pr$. The solid line is the best power-law fit to all the data points: $t_{stag}/t_{recover}\sim {Ra_f}^{0.50}\,Pr^{0.08}$. (c) The dimensionless maximum ascent height $H_{max}/r_0$ as a function of $Pr$ for different $Ra_f$. The solid lines are power-law fits to each data set. (d) Plots of $H_{max}/r_0$ compensated by ${Ra_f}^{1/3}$ as a function of $Pr$. The solid line is the best power-law fit to all the data points: $H_{max}/r_0\sim {Ra_f}^{1/3}\,Pr^{0.16}$.

Finally, we examine the maximum rising velocity $V_{max}$ of thermal structures generated by $t_{heat} = t_{recover}$ in figure 11. Also plotted in the figure are the results of starting plumes obtained in two previous studies (Moses et al. Reference Moses, Zocchi and Libchaberii1993; Kaminski & Jaupart Reference Kaminski and Jaupart2003) as well as in the present work. Note that for starting plumes, their maximum rising velocities are the values at the steady state (see figure 6b). According to Moses et al. (Reference Moses, Zocchi and Libchaberii1993), the dimensionless maximum rising velocities of starting plumes $V_{max}/u_0$ obey a scaling law as $V_{max}/u_0 \sim Ra_{f}^{0.5}\,Pr^{-0.5}$. This scaling law is supported by their experimental data obtained by the shadowgraph technique, which can be described by a relation $Ra_{f}^{0.46}\,Pr^{-0.46}$ approximately. However, by conducting new measurements with the differential interferometry technique, Kaminski & Jaupart (Reference Kaminski and Jaupart2003) found that starting plumes can rise much faster than the results reported by Moses et al. (Reference Moses, Zocchi and Libchaberii1993), though the scaling exponents in these two studies are basically consistent (see black and blue lines in figure 11). As discussed by Kaminski & Jaupart (Reference Kaminski and Jaupart2003), the discrepancy in magnitude could be due to some experimental artefacts, such as sidewall effects, free-surface effects and thermal leakage. Our present results obtained by PIV measurements over a different $Pr$ range are in good agreement with the data by Kaminski & Jaupart (Reference Kaminski and Jaupart2003), including both the scaling exponent and the magnitude. This good agreement provides support for the study by Kaminski & Jaupart (Reference Kaminski and Jaupart2003) on the one hand, and verifies the reliability of the present experiments on the other.

Figure 11.

The dimensionless maximum rising velocities of thermal structures as functions of $Ra_{f}^{0.5}\,Pr^{-0.5}$ for different parameter settings. The results obtained in previous studies by Moses et al. (Reference Moses, Zocchi and Libchaberii1993) and Kaminski & Jaupart (Reference Kaminski and Jaupart2003) are plotted for comparison. See text for a detailed explanation.

In contrast to the behaviour of starting plumes, $V_{max}/u_0$ of thermal structures generated by $t_{heat} = t_{recover}$ exhibit a different $Ra_f$–$Pr$ dependence. A simple power-law fit to these data yields a scaling relation $V_{max}/u_0 \sim Ra_{f}^{0.37}\,Pr^{-0.37}$. If a least squares method is used to fit this data group, then the $Ra_f$ and $Pr$ scaling exponents change little, being 0.41 and $-$0.37, respectively (not shown here). To understand this scaling relation, we define a Reynolds number $Re$ as $Re = V_{max}r_0/\nu$ based on this data group. By using the definitions of $\tau _0$, $u_0$, $Ra$ and $Pr$, namely $\tau _0 = r_0^2 / \kappa$, $u_0 = r_0/\tau _0 = r_0\,Pr/\nu$ and $Ra = t_{heat}\,Ra_{f}/\tau _0\,Pr$, and noticing that $t_{heat} = t_{recover}$ for this data group where $t_{recover}/\tau _0\sim Ra_f^{-0.48}\,Pr^{0.48}$, we obtain $Re \sim Ra^{0.71}\,Pr^{-1}$. This result recalls the scaling relation $Re \sim Ra^{2/3}\,Pr^{-1}$ predicted by a theoretical model for large-$Pr$ natural thermal convection (Shishkina et al. Reference Shishkina, Emran, Grossmann and Lohse2017). It is also worth mentioning that in previous studies of turbulent Rayleigh–Bénard convection with large $Pr$ (Lam et al. Reference Lam, Shang, Zhou and Xia2002; Silano, Sreenivasan & Verzicco Reference Silano, Sreenivasan and Verzicco2010; Li et al. Reference Li, He, Tian, Hao and Huang2021a), the observed $Re$–$Ra$ scaling exponents (ranging from 0.65 to 0.74) are compatible with the present value. However, it should be stressed that the convection systems examined in these studies (Lam et al. Reference Lam, Shang, Zhou and Xia2002; Silano et al. Reference Silano, Sreenivasan and Verzicco2010; Shishkina et al. Reference Shishkina, Emran, Grossmann and Lohse2017; Li et al. Reference Li, He, Tian, Hao and Huang2021a) are turbulent flows, while the present study addresses laminar thermal structures. Thus it is interesting to see the good match of the scaling exponents between different studies. One possible explanation is that large-$Pr$ thermal convection is dominant by slender thermal structures, and no large-scale mean flow exists in the system (Li et al. Reference Li, He, Tian, Hao and Huang2021a). As a result, the flow velocity first increases with growing distance away from the boundary, and then decays slowly to zero after reaching the maximum, which is similar to the present situation. These similar flow features might be the origin of the similar scaling relations observed in different studies.