3.1. Particle resuspension

As described in greater detail in § 3.2, dunes form because the horizontal flow along the bottom of the cell, caused by the downward plumes, pushes the settled particles towards the base of the rising plumes, where they accumulate. As these particles maintain contact with the hot bottom wall, they heat up and localise the ascending plumes above the dune. By this process, in spite of their small volume fraction, the dunes are not a mere by-product of the circulation, but they play an important role in structuring the naturally convective flow in the domain. The fluid rising along the sides of the dunes drags the particles along their slopes and places them in the upward plumes detaching from the summit, thus enabling their entrainment and resuspension. Thus it is drag, rather than lift, that is ultimately responsible for the particle resuspension. The effectiveness of this mechanism is illustrated by figure 2, which displays the probability density function (PDF) of the normalised distance of the particle centres $z_c/{d}_p$ above the cell base at the moment of resuspension. Particles are considered resuspended when they lose direct contact with the wall or a dune. A notable feature of this figure is the presence of oscillations with a period of approximately one particle diameter, which reflects the height of the dune-forming particle layers from which the particles detach when they undertake their upward motion. In the smallest cell, with $\Gamma =1$ , the PDF has a tall, narrow peak slightly above ${d}_p/2$ , indicating that some particles start their ascent somewhat above the base. Since the number of these particles is proportional to the area under the peak, this effect is seen to play a minor role. Most particles start their rise for $ {d}_p \lesssim z_c \lesssim 1.8\,{d}_p$ , namely from layers one or two particle diameters thick. As $\Gamma$ increases, the thin peak for $z_c/{d}_p\lt 1$ rapidly decreases while the PDF extends to many particle layers. The peaks for $\Gamma =2$ are sufficiently sharp to permit a quantitative analysis that shows that their period is very close to $(\sqrt {3}/2){d}_p$ , which is the distance between the parallel planes passing through the centres of well-stacked spheres. This suggests that particles detach from the hollow formed by neighbouring spheres. For $\Gamma =4$ , the peaks are more smeared, given the less orderly dune structure that can form with many particles. These results provide a compelling proof of the significance of dunes in the resuspension process.

Figure 2.

The PDF of the normalised particle height $z_c/{d}_p$ above the cell base at the moment when they are resuspended, where $z_c$ is the $z$ coordinate of the particle centre.

Figure 3.

The number of sedimented particles $N_{p,sed}$ , normalised by the total number of particles $N_p$ , versus time (solid lines), for different aspect ratios in increasing order; the dotted lines are the normalised number of particles forming the biggest dune.

The size of the dunes can be quantified by comparing the number of sedimented particles $N_{p,sed}$ , normalised by the total number of particles $N_p$ , in figure 3; the solid lines show the total number of sedimented particles, and the dotted lines show the number of particles in the largest dune. Since $t=0$ is the instant at which the particles kept fixed at random locations in the domain are released, $N_{p,sed}(t=0)=0$ . With time, settling progresses until, in all cases, the lines appear to reach what may be approximately described as a statistically steady state. (Steady state may not have been quite attained for $\Gamma =4$ , but the trend towards a very large dune is nevertheless sufficiently clear from the results in this figure and in figure 1 c.) The initial phase of dune build-up lasts for a considerably shorter time for the smaller aspect ratios: for $\Gamma =1$ , a steady state can be estimated to have been reached at approximately $t^*\sim 50$ , for $\Gamma =2$ at approximately $t^*\sim 70$ , and for $\Gamma =4$ at approximately $t^*\sim 100$ . There are fluctuations of different magnitudes about the asymptotic state. In Chen & Prosperetti (Reference Chen and Prosperetti2024), we have referred to the strong ones for $\Gamma =1$ as ‘breathing’ to indicate the situation in which the dune appears to repeatedly condense and evaporate. The fluctuations are particularly strong in this case because, as explained below, resuspension is very effective for $\Gamma =1$ , and at the same time, flow fluctuations are particularly strong for small aspect ratios, as found in single-phase Rayleigh–Bénard simulations (see e.g. Stevens et al. Reference Stevens, Blass, Zhu, Verzicco and Lohse2018). The dunes for higher aspect ratios are more stable, experiencing smaller fluctuations after the initial transient. It is also evident that, all other parameters being kept constant, the normalised size of the dune is a strongly increasing function of the aspect ratio; an explanation of this fact will be provided later. For $\Gamma =4$ , the dotted line initially fails to track the solid line, but it catches up, showing only small differences. The implication is that with time, a progressively larger dune establishes itself by absorbing isolated smaller dunes to the point that, for $\Gamma =4$ , it ends up consisting of nearly 97 % of the total number of sedimented particles. The difference between the dotted and solid lines in figure 3 is greater for $\Gamma =1$ than for the other aspect ratios, indicating that for the smallest aspect ratio, several other smaller dunes coexist with the largest one. For $\Gamma =2$ and 4, one notices the sudden appearance of nearly vertical dotted lines, which indicates brief time intervals during which a ‘crack’ temporarily splits the largest dune into smaller ones that quickly re-merge.

The above observations raise key questions that the following sections will address. (i) Why does the system prefer a single, stable dune, and how does it form? (ii) How does the dune size depend on the system parameters? Section 3.2 focuses on the first question, and § 3.3 addresses the second question.

3.2. Dune formation

We now investigate in greater detail the process by which a large dune eventually forms, using the case $\Gamma = 4$ as an example. The snapshots of figure 4 illustrate the time evolution of the sedimented particles, with colour representing their height above the cell bottom. Figure 5 provides a top-down perspective, displaying both the particles and the fluid vertical velocity on a horizontal plane at height $1.25{d}_p$ above the bottom, corresponding to figures 4(a), 4(b) and 4(c). Initially, particles start to settle on the ground, with some giving rise to filaments, as illustrated in figure 6, which shows the vertical velocity on a horizontal plane $1.25{d}_p$ above the bottom wall together with vectors indicating the horizontal velocity of particles located between the plane and the bottom wall.

Figure 4.

Sequence of snapshots documenting the evolution of the sedimented particles for $\Gamma =4$ ; colour indicates the particle height above the cell bottom.

Figure 5.

Top-down view of the fluid vertical velocity on a horizontal plane $1.25{d}_p$ above the bottom wall at three instants of time for the $\Gamma =4$ simulation. The particles within the region between the plane and the bottom wall are also shown, coloured by their temperature.

Figure 6.

Top-down view of the fluid vertical velocity, indicated by colour, on a horizontal plane $1.25{d}_p$ above the bottom wall at $t^*=5.3$ for $\Gamma =4$ . Arrows indicate the horizontal velocity vectors for all the particles within the region between the horizontal plane and the bottom wall. The red, green and purple boxes indicate attraction, saddle and repulsion regions; see text.

The formation of these filaments can be described as follows. When it reaches the proximity of the cell base, the fluid falling in descending plumes (such as the one marked with a purple box in figure 6) turns horizontally and pushes the particles that it encounters radially outwards. In this way, the stagnation points at the bottom of the downward plumes become repulsion points for the particle motion in the horizontal plane. The horizontal flow generated by a descending plume encounters similar flows generated by the neighbouring descending plumes. This interaction creates a series of points which, for reasons explained shortly, we call saddle points, at which the flow velocity along the line joining the stagnation points of two downward plumes vanishes. These saddle points are connected by lines, or filaments that, seen from below, give the impression of an irregularly tessellated plane as shown in figure 5(a). As the fluid approaches these lines, it slows down due to its diverging nature and to the adverse pressure gradient, and heats up as it flows in contact with the warm base. In this way, the upward flow induced by the saddle points and generated along the lines that connect them is actually a thin warm rising plume (figure 7 a). Along the bottom, particles are pushed by the flow towards the saddle points and the filaments, and are pushed along them on either side of the saddle points. This fact justifies our terminology as, insofar as the particle motion in the plane is concerned, they do behave as saddle points; an example can be seen inside the green box in figure 6. In this way, the filaments become pathways for the particle motion on the cell bottom, leading them to accumulate at attraction points where three or more filaments come together (red box in figure 6). Particles delivered to the attraction points jam there, accumulate, and form separate islands, or small dunes. This helps to localise the ascending plumes, further stabilising the attraction points and attracting new particles. This is the main mechanism leading to the rapid growth of the islands during the initial stage, as shown in figure 4(b), where the particles more or less follow the single-phase flow structures near the bottom wall. The repulsion points, saddle points, attraction points and pathways are illustrated in figure 7(b). Aside from the presence of the particles, the flow just described closely resembles the typical one observed in single-phase Rayleigh–Bénard convection (see e.g. Stevens et al. Reference Stevens, Blass, Zhu, Verzicco and Lohse2018). Initially, therefore, the particles do not alter the flow structure as might have been expected a priori given their relatively small amount.

Figure 7.

(a) Streamlines near the bottom wall coloured by vertical velocity at $t^*=5.3$ for $\Gamma =4$ , red and blue indicating upward and downward plumes, respectively. (b) The two-dimensional streamlines in the region of the horizontal plane contained in the pink box (which is $1.25{d}_p$ above the bottom wall) are plotted with a top-down view. The purple, red and green dots represent the stagnation points of the downward plumes, attraction points and saddle points, respectively. The red streamlines show the pathways connecting the saddle points and the attraction points, and correspond to the thin rising plumes.

At later times, however, particles do have a profound effect on the global flow, but for this to happen, the initial islands have to organise into large dunes. The first step in this evolution is the merging of isolated islands formed as just described. This process is driven by two factors, particle jamming and unequal plume strengths. First, as the islands grow due to continuous particle accumulation, their mutual separation decreases, shortening the particle pathways. Since they lie on the ground, the newly sedimented particles moving along the filaments can rarely, if ever, be resuspended before they reach the islands. If this process is too slow, then they will eventually clog the pathways and build a connection between neighbouring islands. This mechanism explains the linear extension of the dunes after the initial formation of separate islands that can be seen in the transition from figure 4(b) to figure 4(c); other examples of this process are visible in figure 5(b). In this way, the dunes continue to grow and merge into elongated structures.

Once the linear structures have formed, they can be pushed into each other, combining into even larger ones. This process can be caused simply by the ‘wind’ blowing along the cell bottom, or by a mechanism reminiscent of dune motion on the bottom of a river. Even though, unlike dunes on a river bottom, in the present flow the currents on both sides of a dune are directed from the base to the crest, there is a similarity if one of the two currents is stronger than the other, as the stronger stream would be able to push more particles in the direction of the weaker stream. Such mismatched plumes can be seen in figure 5(c), and they cause the two large ridges of particles to merge (figures 4 c,d). The particles deep inside a large dune are exposed to negligible flow, and their mobility is severely reduced, a fact that enhances the stability of the dune. The only mobile particles are those on the dune surface, therefore the size and shape of the dune depend essentially on the effectiveness of the resuspension process. When this is weak, the dune can remain stable for a long time. Since the flow is turbulent, occasional strong bursts can appear and cause the temporary fracture of the dune, as shown in figure 3, but particles readily fall into the cracks, producing a single large dune again. A movie of the whole process of dune formation described here can be found in the supplementary movie here https://doi.org/10.1017/jfm.2025.10451.

The mechanism of dune formation is essentially the same for domains with smaller aspect ratios, including $\Gamma =1$ . What makes the details somewhat different in this case is the higher particle resuspension efficiency explained below. According to figure 3, the largest dunes in this case contain a few dozen particles, which rapidly accumulate into the structure. The stronger resuspension, however, gives the dune a very short lifetime because most of it quickly ‘evaporates’, the particles are resuspended, and the process must start anew (Chen & Prosperetti Reference Chen and Prosperetti2024).

3.3. Boundary-layer thickness and particle diameter

The size that the dunes reach at steady state reflects the statistical balance between the particle settling and resuspension rates. Let us start from a consideration of the latter. It was explained before that a key aspect of the physics of resuspension resides in the effectiveness with which the horizontal flow along the cell base is able to push the particles towards the dunes and up their slopes. This effectiveness must be the greater the more the particles stick out of the viscous boundary layer caused by the no-slip condition on the solid surfaces. On average, the distance the particles travel along the base of the cell is greater than the distance along the dune sides, therefore it makes sense to focus on the boundary layer formed at the cell bottom. It is known from the theory of single-phase Rayleigh–Bénard convection that this boundary layer behaves like a normal laminar boundary layer up to $Ra\sim 10^{14}$ , at which point it turns turbulent, ushering in the so-called ‘ultimate turbulent regime’ (see e.g. Lohse & Shishkina Reference Lohse and Shishkina2024). It can therefore be safely assumed that at the Rayleigh numbers of our simulations, namely $10^7$ and $10^8$ , the boundary layer is essentially laminar. Similarly to those of others (see e.g. du Puits et al. Reference du Puits, Resagk, Tilgner, Busse and Thess2007; Shishkina & Thess Reference Shishkina and Thess2009), our attempts to prove this fact from the numerical results have been inconclusive due primarily to flow unsteadiness, which in our case compounds with the complex geometry induced by the presence of particles. These features make it very difficult to calculate clean, uncontaminated average quantities. However, the single-phase particle image velocimetry data of Zhou et al. (Reference Zhou, Stevens, Sugiyama, Grossmann, Lohse and Xia2010) conclusively prove that the mean single-phase velocity profile agrees well with the theoretical Prandtl–Blasius profile. The situation has been reviewed in Zhou et al. (Reference Zhou, Stevens, Sugiyama, Grossmann, Lohse and Xia2010), who support this result beyond doubt. While all the evidence concerns single-phase flow, we feel justified in appealing to it because on the cell base, the particles are sparse, with a density far lower than average. Furthermore, as will be seen, the consequences that follow from this assumption are in very good agreement with the numerical results.

If the laminar nature of the boundary layer is accepted, then it follows that the horizontal scales of the flow near the cell bottom are proportional to $L$ , so the thickness $\delta _v$ of the boundary layer can be estimated, up to a constant of order one, by the standard relation $\delta _v \sim L/\sqrt {LU_f/\nu _f}$ . It is then straightforward to establish the relation

(3.1) \begin{equation} \frac {\delta _v}{{d}_p}\simeq \sqrt {\Gamma }\frac {H}{{d}_p}\left (\frac {Pr}{Ra}\right )^{1/4} = \sqrt {\Gamma }\frac {H}{{d}_p} \left (\frac {\nu _f^2}{\beta (T_h-T_c)gH^3}\right )^{1/4} . \end{equation}

In single-phase Rayleigh–Bénard convection at $Ra=2\times 10^7$ , it is known that the scale of the largest flow structures at the cell mid-height keeps increasing until $\Gamma \sim 64$ (Stevens et al. Reference Stevens, Blass, Zhu, Verzicco and Lohse2018), but comparable information on the boundary-layer thickness is not available. All we can say on the basis of our experience (described below) is that the consequences of (3.1) are supported up to $\Gamma =4$ .

For the present simulations with $Ra=10^7$ , $H/{d}_p=20$ and $\rho _p/\rho _0=1.1$ , (3.1) becomes $\delta _v/{d}_p\simeq 0.36\, \Gamma ^{1/2}$ and varies, therefore, from $\delta _v/{d}_p\simeq 0.36$ for $\Gamma =1$ , to $\delta _v/{d}_p\simeq 0.71$ for $\Gamma =4$ . In the smaller-aspect-ratio cell, therefore, most of the particle is outside the viscous boundary layer, while the opposite is true for the larger-aspect-ratio cell. One may therefore well expect a much greater resuspension efficiency for the smallest aspect ratio than for the largest one. This fact can be demonstrated by plotting the number of resuspended particles $\Delta N_{res}$ during a time unit $\tau _f=H/U_f$ normalised by the number of settled particles $N_{p,sed}$ . These results are shown in figure 8 for $\Gamma =1,2,4$ . The line for $\Gamma =1$ exhibits strong statistical fluctuations, which are a consequence both of the smaller number of particles and of the greater irregularity of the flow for this aspect ratio mentioned before. Although the line for $\Gamma =4$ settles at a lower level than that for $\Gamma =2$ as steady state is approached, the two nearly superpose until $U_ft/H\simeq 50$ . An explanation of this feature is given below.

Figure 8.

Here, $\Delta N_{res}$ is the number of particles resuspended during one time unit $H/U_f$ . The time evolution of this quantity, normalised by the number of settled particles $N_{p,sed}$ , is shown here for, in descending order, $\Gamma =1,2,4$ .

The previous interpretation is strengthened by a comparison of the normalised number of resuspended particles for $Ra=10^7$ and $10^8$ with $\Gamma =2$ . As already explained, the simulation at $Ra=10^8$ was carried out by increasing the cell height, keeping all other dimensional quantities at the same values as before. In particular, for this case, $H/{d}_p=43.2$ . For these conditions, (3.1) becomes $\delta _v/{d}_p\simeq 0.24(H/{d}_p)^{1/4}$ and therefore changes from $\delta _v/{d}_p\simeq 0.50$ for $Ra=10^7$ and $H/{d}_p=20$ to $\delta _v/{d}_p\approx 0.61$ for $Ra=10^8$ and $H/{d}_p=43.2$ . The difference between the two situations is smaller than before and, indeed, the two normalised resuspensions are closer, as shown in figure 9. Remarkably, in spite of the increase of the Rayleigh number, which would lead one to expect a more vigorous resuspension and therefore a smaller fraction of sedimented particles, the opposite is found, in agreement with the prediction of (3.1). In this case, as can be seen in figure 10, the dune is indeed larger than that formed for $\Gamma =2$ at $Ra=10^7$ (figure 1 b), and it has at least seven particle layers.

Figure 9.

Temporal evolution of the number of particles resuspended during one time unit $H/U_f$ , normalised by the number of settled particles $N_{p,sed}$ , for $Ra=10^7$ ,with $H/{d}_p=20$ , and $Ra=10^8$ with $H/{d}_p=43.2$ .

Figure 10.

Particle configuration of sedimented particles for $Ra=10^8$ and $\Gamma =2$ at $t^*=90.1$ . Particles are coloured by their height above the cell base. Suspended particles are neglected in the plot.

We now turn to the smaller-particle simulation for $Ra=10^7$ and $\Gamma =1$ . In this case, (3.1) gives $\delta _v/{d}_p\simeq 0.018 (H/{d}_p)$ , namely $\delta _v/{d}_p\simeq 0.36$ and 0.509 for $H/{d}_p=20$ and 28.3, respectively. The line for the former case is the same as that shown in figure 8 for $\Gamma =1$ . The difference between the two values of $\delta _v/{d}_p$ is comparable to that encountered earlier in the simulations for different aspect ratios, and as can be seen from figure 11, indeed there is a significant difference between the two simulations. The resuspension of the smaller particles is significantly weakened compared to that of the larger ones, with the consequence that the fluctuations of the dune size are much more moderate. As figure 12 shows, the dune is more stable and also contains a much greater fraction of particles than for $H/{d}_p=20$ . The greater mass fraction of the particles in this case may influence these results, but as shown in § 3.4, the effect is very minor.

Figure 11.

Temporal evolution of the number of particles resuspended during one time unit $H/U_f$ , normalised by the number of settled particles $N_{p,sed}$ , for normal-sized particles with $H/{d}_p=20$ , and smaller particles with $H/{d}_p=28.3$ . Here, $\Gamma =1$ and $Ra=10^7$ .

Figure 12.

The number of sedimented particles normalised by the total number of particles, for smaller and normal-sized particles for $\Gamma =1$ .

Figure 13.

Number of suspended particles settling during one time unit $H/U_f$ normalised by the number of suspended particles versus time for aspect ratios $\Gamma =2,4$ . The vertical red dash-dotted line at $t^* = 45$ indicates the time at which the two elongated dunes visible at time $U_ft/H=34.0$ in figure 4(c) begin to merge into a single dune spanning the diagonal of the domain for $\Gamma =4$ .

Figure 14.

$(a)$ Warm particles with $(T_p-T_0)/(T_h-T_c)\gt 0.125$ at $t^*=30.13$ . $(b)$ Streamlines coloured by the vertical velocity. $(c)$ Mid-height vertical velocity, time-averaged over $23.8 \leqslant t^* \leqslant 36.5$ . $(d)$ Warm particles with $(T_p-T_0)/(T_h-T_c)\gt 0.125$ at $t^*=116.2$ . $(e)$ Streamlines coloured by the vertical velocity. $(f)$ Mid-height vertical velocity, time-averaged over $109.9 \leqslant t^* \leqslant 122.5$ . Here, $\Gamma =4$ .

It was noted in connection with figure 8 that the normalised numbers of resuspended particles for $Ra=10^7$ and $\Gamma =2,4$ track each other until $U_ft/H \simeq 50$ , at which point they deviate, with the resuspension for $\Gamma =4$ becoming appreciably smaller than that for $\Gamma =2$ . This evolution can also be looked at from a different angle by considering the normalised number of particles deposited during a time unit $\tau _f$ , which is shown in figure 13. Here, we see that the curves for the two aspect ratios are very close to each other up to $U_ft/H \sim 50$ , but then start diverging. An explanation of these features suggests itself if we look at the particle and flow arrangements before and after this time. Figure 14(a) shows, for $\Gamma =4$ , an instantaneous image of the arrangement of particles with $(T_p-T_0)/(T_h-T_c) \geqslant 0.125$ at $U_ft/H=30.1$ . Being relatively warm, the majority of these particles rest on the cell bottom. Figure 14(b) shows the fluid streamlines averaged over the time interval $23.8 \leqslant U_ft/H \leqslant 36.5$ , coloured by the local vertical velocity. It can be seen that the settled particles are mostly arranged into two dunes, and the flow is correspondingly arranged in two recirculating flow structures separated by a distance approximately equal to half the cell, i.e. $2H$ . In this sense, the situation is similar to the $\Gamma =2$ case, as can be seen from figures 15(a) and 15(b), and indeed, the entrainment and deposition rates for the two aspect ratios are very close. Note that what we see in the $\Gamma =4$ cell is not the periodic repetition of the $\Gamma =2$ one, but the break-up of the flow into two approximately rectangular cells, with length $4H$ in the direction parallel to the dunes, and $\sim 2H$ in the transverse direction. In this situation, the average length of the particle path along the cell base towards the dunes is comparable to that of the particles in the $\Gamma =2$ case. It is only later that the two dunes of the $\Gamma =4$ simulation start to merge (figures 4 d,e), and at this point, the average horizontal particle path doubles and, according to (3.1), the boundary layer thickens and the resuspension becomes less effective.

Figure 15.

$(a)$ Warm particles with $(T_p-T_0)/(T_h-T_c)\gt 0.125$ at $t^*=75.2$ . $(b)$ Streamlines coloured by the vertical velocity, time-averaged over $68.9 \leqslant t^* \leqslant 81.5$ . $(c)$ Time-averaged mid-height vertical velocity. Here, $\Gamma =2$ .

Figure 13 shows that near $U_ft/H\sim 50$ , when the two dunes of the $\Gamma =4$ simulation merge, the number of settling relative to suspended particles increases so that the necessary balance with the decreased fraction of resuspended particles shown in figure 8 is ensured. (Note that this statement concerns the relative, not absolute, numbers of both types of particles, which actually decrease as the dune size increases.) This increase of the fraction of settling particles can be imputed to a decreased coherence of the flow structures, which is evident when comparing the streamlines in figure 14. A similar point was made by Marsh & Maxey (Reference Marsh and Maxey1985) and Martin & Nokes (Reference Martin and Nokes1989). By means of a simple model of particles settling in steady, cellular convection, the former authors found that a large proportion of particles could remain suspended indefinitely, and argued that the addition of turbulent fluctuations to such a flow would reduce the proportion of particles that could remain suspended indefinitely. Martin & Nokes (Reference Martin and Nokes1989) also argued that all particles would eventually deposit on the bottom in an incoherent flow.

The streamlines of figure 14 are not the only indication of the eventual loss of coherence of the flow for $\Gamma =4$ . This fact is confirmed and strengthened by a consideration of the correlation coefficients between the mid-height vertical velocity and the temperature near the cell bottom for $\Gamma =2,4$ , shown in figure 16. A large value of this coefficient would indicate that plumes rise/sink fairly rectilinearly above positions of high/low bottom temperature, but both these coefficients start at a low value at early times. This result indicates that these early-time plumes (which figures 4(a) and 5(a) show to be irregularly distributed with hardly any dunes at $U_ft/H= 5.4$ ) are relatively weak and easily deviated in their ascent/descent. The correlation increases at $U_ft/H= 13.7$ , at which point a number of well-formed dunes can be seen in figures 4(b) and 5(b), and reaches a maximum at approximately $U_ft/H= 34$ , a time at which figures 4(c) and 5(c) show two large dunes. The relatively large value of the correlation coefficient reached at these times suggests that the plumes are strong and well developed. Past $U_ft/H \sim 50$ , however, the coefficient for $\Gamma =4$ starts deviating from that for $\Gamma =2$ , decreasing to a much lower value. Whether this is due to a weakening of the plume caused by the greater height of the dune, which makes its top cooler due to the relative inefficiency of conduction, or by the onset of horizontal convection in the cell, which strongly tilts and disrupts the plumes in their ascent/descent, it is not possible to say with assurance. In any event, in either case, the flow is disrupted and becomes more irregular, with the consequence that particle settling is favoured.

Figure 16.

Time evolution of the correlation coefficient between the mid-height vertical velocity field $u_{z,m}$ and the temperature field $T_b$ on a horizontal plane near $1.25{d}_p$ above the cell bottom, denoted as $(u_{z,m},T_{b})$ , for $\Gamma = 2,4$ . The red dash-dotted line at $t^* = 45$ indicates when the two elongated dunes merge into a single dune spanning the diagonal of the domain for $\Gamma =4$ .

Figure 17.

(a) The Nusselt number $Nu$ and (b) the Reynolds number $Re$ , versus aspect ratio, both with and without particles. The single-phase results are shown in black. Red and blue lines represent particulate flows with $Ra=10^7$ and $H/{d}_p=20.0$ , and $Ra=10^7$ and $H/{d}_p=28.3$ , respectively. In (b), solid lines with triangles represent the vertical Reynolds number, and dash-dotted lines with diamonds represent the horizontal Reynolds number.

3.4. Nusselt and Reynolds numbers

Although not the focus of this study, it is also of interest to present results for the Nusselt and Reynolds numbers of the flow in the cell. Figure 17(a) collects the present results for the Nusselt number for single-phase flow (circles) and for the two-phase simulations with $Ra=10^7$ . The single-phase results exhibit the same decreases with increasing aspect ratio found by other investigators (see e.g. Stevens et al. Reference Stevens, Blass, Zhu, Verzicco and Lohse2018). The larger particles, with $H/{d}_p=20$ , cause a slight increase for $\Gamma =1$ , a slight decrease for $\Gamma =4$ , and a negligible difference for $\Gamma =2$ . The $\Gamma =1$ result reproduces that of our earlier study (see e.g. Chen & Prosperetti Reference Chen and Prosperetti2024), which was imputed to the fact that, unlike a fluid parcel that is broken up by the surrounding fluid in its ascent, particles maintain their integrity and are therefore able to carry their thermal energy all the way to the top of the cell. The smaller particles with $H/{d}_p=28.3$ , on the other hand, are seen to decrease the Nusselt number somewhat with respect to the single-phase result, in spite of the fact that they have the same terminal Reynolds number and occupy the same fraction of the total volume. This negative effect is probably due to the particle mass fraction, which is approximately 16 % greater than for the larger particles. It is true that, according to figure 12, the fraction of small suspended particles is approximately five times smaller than that of the larger particles, but their total number is three times as large, so even counting only the suspended particles, a net mass fraction increase still results.

Different definitions are possible for the Reynolds number in the cell. Here, we define a horizontal Reynolds number ${{Re}}_h= (\sqrt {\langle u_x^2\rangle +\langle u_y^2 \rangle }H )/\nu _f$ and a vertical Reynolds number ${{Re}}_v= (\sqrt {\langle u_z^2\rangle }\,H )/\nu _f$ , with the angle brackets indicating a volume average. Our computed results for single- and two-phase Rayleigh–Bénard convection for the larger particles with $H/{d}_p=20$ at $Ra=10^7$ are shown in figure 17(b), with diamonds and triangles indicating the horizontal and vertical Reynolds numbers. The trends are the same as those found in the literature for the single-phase case, with ${{Re}}_h$ an increasing function of $\Gamma$ , and ${{Re}}_v$ nearly independent of $\Gamma$ up to $\Gamma =2$ , and decreasing for $\Gamma =4$ . The contrasting trends of ${{Re}}_h$ and ${{Re}}_v$ suggest that the vertical ascent/descent of the plumes is disrupted by an increasing horizontal flow, which agrees with the remarks made earlier in connection with figure 16, and the decline of the Nusselt number for $\Gamma =4$ seen in figure 17(a). For the smaller particles with $H/{d}_p=28.3$ , the flow is slowed down for the reason explained before, and both Reynolds numbers are found to be smaller.

The results for our single simulation with $Ra=10^8$ cannot be shown in figure 17 without an undesirable compression of the vertical scale, and we prefer to quote numerical results. The single- and two-phase Nusselt numbers are approximately 31.3 and 30.7. Since $\Gamma =2$ in this simulation, the situation is therefore seen to be very similar to that shown in figure 17(a) for this aspect ratio. The horizontal Reynolds numbers are approximately 1446 and 1480 for single- and two-phase flow, respectively, while the vertical ones are approximately 1213 and 1285.