Bouncing behaviour of a particle settling through a density transition layer

To understand the underlying physics of particle bouncing behaviour, we attempt to decompose its hydrodynamic force into different components. We start from the motion equation of a particle settling from the rest in a quiescent homogeneous fluid, which can be written as

The left-hand side is the total inertia force acting upon the particle, where $m_{p}$ is the particle mass. It arises due to the imbalance of forces. At the right-hand side, $G$ and $F_{b}$ are respectively the gravity and buoyancy forces of the particle, $F_{d}$ is the ‘steady-state’ drag force at the considered time instant, $F_{a}$ is the inertia force of added mass, and $F_{h}$ is the history (Basset) force. Here, $F_{d}$ can be evaluated according to (10) and (11). In the limit of potential flow, $F_{a}$ can be calculated as

At the Stokes regime, $F_{h}$ has an analytic solution

For the particle settling in a stratified fluid, we follow the same way of drag force decomposition as in (13), while introducing an extra term $F_{s}$, accounting for the stratification effects. The motion equation becomes

It is reasonable to further decompose $F_{s}$ into two components as

where $F_{sb}$ is the enhanced buoyancy caused by dragging the upper fluid to the lower layer, therefore modifying the density distributions, and $F_{sj}$ is the force caused by the induced flow structure due to the stratification, represented typically by the upward jet flow at the rear of the particle. As we have discussed in figure 8, the observed jet is conjectured to be the dominant flow structure as the particle bounces. The equation of motion for a particle settling in a stratified fluid can therefore be written as

It is possible to evaluate numerically the different force components. The most convenient way is to reorganise (18) as

where $F_{static}$ is the hydrostatic force caused by density stratification and its nonuniform distributions, and $F_{dynamic}$ is the force caused by the non-zero velocity field at uniform density distribution. Excluding the steady-state drag $F_{d}$, the sum of ($F_{a}+F_{h}+F_{sj}$) represents the force caused by unsteady flow, i.e., the flow caused by the stratification effect. For a given density field, the hydrostatic pressure $p_{s}$ can be obtained by solving

$F_{static}$ is the integration of $p_{s}$ over the particle surface:

where $\bm{n}$ is the unit normal at the surface. Since $F_{b}$ can be calculated by considering an undisturbed density profile, we have

The total hydro-force $F_{hydro}$ acting on the particle, i.e., $F_{static}+F_{dynamic}$, can be calculated by solving the equations (4), (6) and (7). Then, we have

If we evaluate the steady-state drag $F_{d}$ using (10) and (11), we can write the drag component contributed by unsteady flow as

For a particle settling through a density interface, if $U_{min}\leq 0$ (keeping in mind that the positive direction of $z-$axis points downwards), it bounces up. At the instant when the first time the particle reaches $U=0$ (see the arrow in figure 10), the particle satisfies

Thus at $U=0$ we have

In figure 11(a), we plot $F_{a}$ and $F_{h}$ predicted by (14) and (15). We note that the solutions given by (14) and (15) can actually not be accurately applied to our Reynolds number ($Re_{u}\approx 200$). However, at least we know that $F_{a}$ and $F_{h}$ are opposite to the acceleration of the particle, i.e, positive at this time instant. Apparently, the necessary condition for (26) to hold is

Whether the jet is necessary for the occurrence of bouncing depends on the magnitude of $F_{sb}$. When $|F_{sb}|\leq|G+F_{b}|$, the contribution of jet is necessary for the bouncing behaviour. Observed from our experiments, as discussed in section 4.2, the particle reverses its motion direction after the wake detaches, so that $F_{sb}$ is relatively small, therefore $F_{sj}$ is determinant for the bouncing behaviour.

The decomposed force components as the functions of vertical position is plotted in figure 11(b), where ($G+F_{b}$) is the reduced gravity. Since $F_{a}$ and $F_{h}$ cannot be accurately calculated, we plot ($F_{sj}+F_{a}+F_{h}$), which is the force due to the unsteady flow, and also gives the upper limit of $F_{sj}$. Note that ($F_{a}+F_{h}$) is always positive during the settling process. Apparently, before entering the transition layer, the balance between the reduced gravity ($G+F_{b}$) and the drag force $F_{d}$ makes the particle reach a constant settling velocity. As the particle enters the transition layer, the force components $m_{p}dU/dt$, $F_{sb}$ and $F_{sj}+F_{a}+F_{h}$ arise from their nearly zero values. We also present the time histories of different force components in figure 11(c), with the aforementioned four stages (section 4.2) separated by three blue dashed lines. The wake buoyancy force $F_{sb}$ correlates to the volume of attached upper light fluid. It reaches a maximum at the end of the first stage, after the dragging of the wake (see figure 9(b) for $t=1.2$s), and before the wake detaches from the particle. The sharp rise of ($F_{sj}+F_{a}+F_{h}$) (the green line) is dominated by $(F_{a}+F_{h})$ at the first two stages due to the sudden deceleration of the particle. As approaching the third stage, $F_{sb}$ becomes small since only a thin layer of light fluid left in the attached wake (see figure 9(e) for $t=3.0$s). In the mean time, ($F_{sj}+F_{a}+F_{h}$) appears to be the dominant drag force (to balance the reduced gravity ($G+F_{b}$)), arresting further settling of the particle. At the instant of bouncing occurrence, the settling velocity $U$ approaches zero (marked by the black dashed line in figure 11 and see also figure 9 between $t=5.4$s and $t=6.0$s). At this time, $|F_{sb}|\ll|G+F_{b}|$, and ($F_{sj}+F_{a}+F_{h}$) plays a dominant role in balancing the reduced gravity. Excluding ($F_{a}+F_{h}$), the contribution from $F_{sj}$ takes the primary part of the drag force. Therefore we conjecture that the jet flow is a necessary condition for bouncing the particle up. In conclude, we prefer to the interpretation that $F_{sb}$ is the primary factor for decelerating the particle as it enters the transition layer, while $F_{sj}$ induced by the jet flow further decelerates the particle till reverses its motion as it leaves the transition layer.

To further investigate the relationship between the minimal velocity and the lower Reynolds number, the experimental data from series 1, where the upper Reynolds number and Froude number are $Re_{u}=252\pm 16$ and $Fr=2.6\pm 0.2$ respectively, are plotted over the lower Reynolds number $Re_{l}$ in figure 13(a). The numerical results at $Re_{u}=258$ and $Re_{u}=349$ with the fixed Froude number $Fr=2.6$ are also plotted for comparison. From the trend, the experimental and numerical results are consistent, both indicating that $U_{min}/U_{u}$ increases linearly with $Re_{l}$. We note that for $Re_{u}\sim 252$, the values of $U_{min}/U_{u}$ in the experiments are uniformly higher than that of the simulations, which is possibly caused by the incomplete development of $U_{l}$ in the experiments. After passing through the interface, a long distance is required for the particle to reach a steady state velocity. Such that the velocity measured at the instant when the particle leaves the viewing window in our experiments might be smaller than the fully developed $U_{l}$ in the simulations, resulting a smaller $Re_{l}$. We can fit the data shown in figure 13(a) by the lines expresses as

The linear regression gives $c_{1}=0.0049$ and $c_{2}=-0.1181$ for the experiments ($Re_{u}=252\pm 16$), and $c_{1}=0.0046$ and $c_{2}=-0.1720$ for the numerical results ($Re_{u}=258$). Critical lower Reynolds numbers $Re^{\ast}_{l}=23.9$ and $37.5$ are identified respectively for these two lines, i.e., the position intersected with the horizontal axis, or when $U_{min}=0$. It is easy to understand that the undetermined parameters $c_{1}$ and $c_{2}$ are dependent on $Re_{u}$ and $Fr$. Evidenced from 13(a), with a higher upper Reynolds number $Re_{u}=349$, the fitting line gives a smaller slope compared with both the experiments and simulations with the smaller Reynolds numbers. However, it is interesting to find that the critical lower Reynolds numbers $Re^{\ast}_{l}$ obtained from the numerical simulations with different $Re_{u}$ are very close.

In our experiments, since the terminal Reynolds numbers are not known $a$ $priori$, we practically adjust the fluid density to get varied Reynolds numbers. Previous works show that the bouncing behaviour is related to the fluid density and density ratio (Camassa & Vaidya, 2022; Doostmohammadi & Ardekani, 2014a). Thus the relationship between the minimal velocity and the density ratio should also be investigated. Here, we define the density ratio as $\Delta\rho_{l}=(\rho_{p}-\rho_{l})/\rho_{l}$. Since $Re_{l}$ is correlated to the density difference, the relationship between $U_{min}$ and $\Delta\rho_{l}$ can be derived from (28). For a particle settling to a steady-state velocity in a homogeneous fluid, the drag force balances the reduced gravity, satisfying the following expression:

Using the definition of $F_{d}$ in equation (10) and substituting $U_{l}=\nu_{l}Re_{l}/D$ into (29), we get the following expression between density ratio and Reynolds number:

where $C_{dl}$ is the steady-state drag coefficient in the lower layer. Equations (28) and (30) yield

We find that the power law fitting in the form

is appropriate to describe the dependence of $U_{min}/U_{u}$ on $\Delta\rho_{l}$, as shown in figure 13(b) (recompiling the data from figure 13(a)).

The trajectories and velocity profiles of particle $P1$ from the experiment series 1 are plotted in figure 14. It is clearly shown in figure 14(a) that the sedimentation time is dramatically prolonged by the bouncing behaviour. Although the particle enters the interface with the same velocity, the slight variation of $Re_{l}$ significantly alters the behaviour. For a relatively large lower layer Reynolds number, e.g. $Re_{l}=59$, though a minimum velocity is observed in figure 14(b), the particle keeps descending unidirectionally after it passes through the transition layer. We emphasise that all particles have larger density than the fluid in the tank at all attitudes. As $Re_{l}$ decreases, crossing a critical value, i.e. $Re^{\ast}_{l}$, as we have discussed above, the particle reverses its direction of motion and ascends for a transient time scale (see figure 14(a) for $Re_{l}=26$ and $Re_{l}=1$). This bouncing phenomenon is represented by a much deeper and negative minimum velocity shown in the depth vs velocity plot of figure 14(b). Here, $Re_{l}=1$ refers to a very extreme case, when the particle density ($\rho_{s}=1123.69$ kg/m³) is near the density of the bottom layer ($\rho_{l}=1123.66$ kg/m³). In this case, the particle experiences an extraordinarily long transient time scale to reach the terminal velocity of the bottom layer. An explanation for this long transient, given by the previous study (Abaid & McLaughlin, 2004), is that once the plume of upper layer fluid has shed, there still exists around the particle a small boundary layer of upper fluid which diffuses exceptionally slowly due to the very long diffusion of time of salt in water and the absence of a strong turbulence diffusion in this low-speed flow state. This is evidenced by our experimental results in figure 5(p$\sim$t), which show clearly that a few light fluid remains at the particle surface after the bouncing.

It is worthy mentioning some previous studies, where the bouncing behaviour was not observed. Srdić-Mitrović & Fernando (1999) presented the time trajectories of particles obtained by a series of experiments (see their figure 9). They reported the noticeable decrease of velocity within the transition layer, however without finding reverse motion of the particle. It can be possibly explained from two aspects. First, in their experiments, the upper fluid was a mixture of ethyl alcohol and water, which diffuses faster than salty water. Thus the lighter upper fluid dragged by the particle could adjust to the surrounding fluid immediately and weaken the deceleration of the particle. Second, and more importantly, their examined upper Reynolds numbers fall in the range $0.7\leq Re_{u}\leq 23$, much lower than the current studies. As we will discuss later, the upper Reynolds number is also one of the influencing factors for the occurrence of bouncing behaviour. Actually, the second explanation can also be related to a deeper physical mechanism, the rear buoyant jet, which disappears when $Re_{u}$ is low. As evidenced from the work by Yick et al. (2009), there was no sign of such a jet as $Re_{u}\sim 1$.

We now address the effects of Froude number. Referring to the experiment series 3 presented in table 1, we fix both the upper and lower layer fluid densities, varying only the transition layer thickness. This is achieved by releasing the particles in the same tank of fluid with a time interval of 24 hours. The diffusion of salt along such a long time scale gives the variations of transition layer thickness in the range $L=$3.15 cm - 13.06 cm, resulting in various Froude numbers.

We plot the non-dimensional minimum velocities for different particles over the Froude number in figure 15(a). Note that the particle density increases from P1 to P5, therefore both the upper and lower Reynolds numbers increase from P1 to P5. The general trend is a nearly monotonous increase in the non-dimensional minimal velocity along with the increasing Froude number, i.e., as the transition layer becomes thicker. Comparing between the tests of different particles, we can see an overall increase in the minimum velocity when the particle density increases, or equivalently when the Reynolds number increases. Another trend is the less role of Froude number playing in the increase of minimum velocity when the particle density is large (see P5 in figure 15(a)). In contrast, for the lighter particles, such as P1 and P2, the influence of Froude number is remarkable. For example, the behaviour of particle P2 is altered from bouncing to unidirectional settling as the Froude number increases from $Fr=2.6$ to $5.1$. Note that in the tests of P2, the lower Reynolds number ($Re_{l}=28)$ is very close to the critical Reynolds number $Re_{l}^{\ast}$. In figure 15, for P3, P4 and P5, no bouncing phenomenon is recorded.

Specifically, we present the velocity profiles of particle P2 at different Froude numbers in figure 15(b). In these tests, the Froude number significantly alters the evolving profile of particle settling velocity. We emphasise that the bouncing motion occurs after the particle leaves the transition layer (see $Fr$=2.6 and 3.7 in figure 15(b)), while the minimum velocity is reached within the transition layer for those tests without bouncing (see $Fr$=4.5, 4.9 and 5.1 in figure 15(b)). There is an interesting phenomenon we want to report. In figure 15(b) we observe that the particle restores to a higher settling velocity for the lower Froude number tests than that with high Froude numbers. It might be caused by the more pronounced diffusion induced by the jet flow. Clearly, the jet flow can enhance the turbulence diffusion and diffuse the remaining upper layer fluid attached on the particle to its ambient lower layer fluid.

We next investigate the effects of upper Reynolds number $Re_{u}$ by numerical simulations. In figure 16(a), the non-dimensional minimal velocity versus $Re_{u}$ at five different values of $Re_{l}$ are presented, at a fixed Froude number $Fr=2.6$. The trend is clear: the minimal velocity increases with the increased lower Reynolds number $Re_{l}$, while at a fixed $Re_{l}$ the minimal velocity decreases with the increase of $Re_{u}$ except for $Re_{l}=13$, where all minimal velocities are negative, indicating the occurrence of bouncing for all cases. We examine a special case, $Re_{l}=37$, when the minimum velocity becomes negative as the upper Reynolds number rises to $Re_{u}=292$. In figure 16(b), we plot the variations of settling velocity with depth for this special case. Despite the distinct differences in entering velocities of the particle among different tests, their minimal velocities are considerably resembling. Also, they reach minimal velocities at nearly the same depth. Note that the thickness of transition layer varies little among different tests. From figure 12 we understand that $Re_{l}=37$ is near the critical Reynolds number for the occurrence of bouncing, therefore this special case is very sensitive to the parameters, such as the upper Reynolds number $Re_{u}$. As demonstrated in figure 16(b), the particle moves unidirectionally for $Re_{u}=127,190$ and 244, while reverses its motion direction for $Re_{u}=292$ and 337.

We thus stress that the upper Reynolds number $Re_{u}$ plays a non-negligible role only when the bottom layer Reynolds number $Re_{l}$ is close to its critical value for bouncing. However, we note that the currently studied upper Reynolds numbers, in both experiments and numerical simulations, are kept at a certain order of $Re_{u}\sim O(100)$, while a much lower $Re_{u}$, e.g. in the order $\sim O(1)$ will give no sign of bouncing, as we have discussed in section 4.5.1.

Since the bouncing behaviour is primarily determined by the lower Reynolds number $Re_{l}$, while the other two parameters, $Re_{u}$ and $Fr$ can also play their roles, as we have discussed in section 4.5.2 and 4.5.3, it is possible to give a better visualisation by examining their combined effects. For example, in figure 17, we draw two maps in the parametric space of $Re_{u}$ and $Fr$ for two selected lower Reynolds numbers $Re_{l}=13$ and $Re_{l}=37$, with the contours representing their minimal velocities. Clearly, the bouncing phenomenon occurs at the higher $Re_{u}$ and lower $Fr$, i.e. the upper-left regimes of the maps. The direct comparison between figure 17(a) and (b) indicates that the bouncing regime shrinks for the higher value of $Re_{l}$, demonstrating again that $Re_{l}$ is the dominant parameter. When $Re_{l}$ is low, the particles are more prone to bounce after passing through the transition layer.

Simulating in the same ($Re_{u}$, $Fr$) parametric space for different values of $Re_{l}$, with a lower limit of $Re_{l}=13$, we summarise the critical lower Reynolds numbers $Re^{\ast}_{l}$ in table 4. We note that those with $Re_{l}<13$ are not presented in this table. We see that $Re^{\ast}_{l}$ varies from $14.4$ to $46.2$ depending on different combinations of $Re_{u}$ and $Fr$. We should point out that the variations of $Re^{\ast}_{l}$ with $Fr$ at a fixed $Re_{u}$ are not always monotonous, particularly when $Re_{u}$ is high.