3.1. Basic features of the particle trajectories

Figure 2 shows around 800 superimposed particle trajectories, with the $z$-axis pointing downwards, as they settle through different two-layer medium scenarios at a particle release frequency $f_p = 4\, {\rm Hz}$. The paths of the particles diverge considerably between the two fluid layers. In the upper fluid layer, the trajectories are oblique, indicative of path instability. This behaviour can be attributed to the substantial Galileo number, $Ga= 976$ (Horowitz & Willizmson Reference Horowitz and Willizmson2010). In contrast, the lower fluid layer had relatively straight trajectories, a result of the comparatively smaller $Ga = 16$ (Jenny et al. Reference Jenny, Dusek and Bouchet2004).

Figure 2.

Superimposed particle trajectories for various upper fluid layer heights $H_1/d =$ (a) 10, (b) 20, (c) 30 and (d) 40, all at a particle release frequency of 4 Hz.

The ensemble-averaged lateral particle displacement from the vertical axis of the release location, $L(z) = [{\sum }_{i=1}^{N} (x_i(z)-x_0)^2+(y_i(z)-y_0)^2 ]/N$, is shown in figure 3. Here, $x_i(z)$ and $y_i(z)$ are the horizontal coordinates of the particles, obtained every 1 mm vertically, $x_0$ and $y_0$ denote the release position and $N$ is the total number of particles. Generally, the lateral displacement increased monotonically in the upper fluid layer and the lower layer to $8d$ below the interface (figure 3a–c). The lateral motion of the particles in the lower layer near the interface demonstrates the span dictated by the influence of viscosity in order to suppress the inertial component. As particles settle further from the interface, the lateral distance remains nearly constant, reflecting the low $Ga$ of the lower fluid medium. The asymptotic lateral distance, $L_f$ (figure 3d), depicts the dependency of this quantity on the thickness of the upper layer, $H_1$, and the release frequency. The value of $L_f$ increased with $H_1$, owing to a longer residence time in the upper fluid layer, where particles encounter significant path instability. In general, $L_f$ decreased as $f_p$ increased, suggesting that the lateral motions of particles were more restrained when particle separations were smaller.

Figure 3.

Mean particle lateral displacement from the release position at (a) $f_p = 4\, {\rm Hz}$, (b) $f_p = 6\, {\rm Hz}$ and (c) $f_p =$ 8 Hz. The grey area includes the region between $z/d=0$ and $z/d=8.$ (d) Ensemble average of the asymptotic lateral distance, $L_f$.

The ensemble average of the particles lateral velocities, $v_L = (u^2+v^2)^{1/2}$, where $u$ and $v$ are the velocity components in the horizontal $x$- and $y$- directions, is illustrated in figure 4 for the various scenarios. The release of particles near the free surface marks the early stage of particle motion, as shown by the grey shaded areas in figures 4 and 5. The analysis focuses primarily on particle behaviour after this initial stage. The value of $v_L$ exhibits a significant dependence on the particle-release frequency before reaching the interface. Note that $v_L$ was lowest at the highest particle-release frequency. Once past the density interface, $v_L$ decreased substantially, with only a minimal dependence on release frequency and with some slight variation near the density interface.

Figure 4.

Mean particle lateral velocity, $v_L$, for $H_1/d =$ (a) 10, (b) 20, (c) 30 and (d) 40 at various particle-release frequencies. Grey regions indicate the early stage partially affected by air entrainment, and black dashed lines denote the density interface between the two fluids.

Figure 5.

Mean particle vertical velocity, $w$, for $H_1/d =$ (a) 10, (b) 20, (c) 30 and (d) 40 at various particle-release frequencies. Minimum values are indicated with circles.

The aggregate vertical particle velocity, $w$, depicted in figure 5, exhibited a consistent increase in the upper fluid layer following the initial transient phase, with the exception noted in the $H_1/d = 10$ case, which presented the minimal thickness of the first layer, as shown in figure 5(a). In this case, the initial adjustment encompassed most of the upper layer. As particles approached the interface, their vertical velocity decreased and exhibited a minimum in the lower fluid medium (highlighted by circles in figure 5). Beyond that point, the vertical velocity increased at a very slow rate. Drawing from prior studies (Srdic-Mitrovic et al. Reference Srdic-Mitrovic, Mohamed and Fernando1999; Camassa et al. Reference Camassa, Falcon, Lin, McLaughlin and Myckins2010; Pierson & Magnaudet Reference Pierson and Magnaudet2018b), this behaviour can be attributed to the fluid entrained from the upper layer. The settling velocity is minimized when the volume of the entrained upper fluid is at its peak. As the entrained fluid detaches from the particle, the drag starts to counterbalance the net weight of the particle, resulting in a settling velocity similar to the terminal velocity in a homogeneous medium.

In our experimental analysis, we discerned the presence of the upper fluid trailing in the wake of the particles as they traversed past the density interface (figure 6a). The influence of the thickness of the upper layer on the vertical velocity of the particles is illustrated in figure 6(b,c). To assess the potential interplay of particle interactions, we initially compared the results with the terminal velocity of a single particle in the oil medium, $w_{T_0} = [(729+3Ga^2)^{0.5}-27]\nu /d$ (Goossens Reference Goossens2020). As shown in figure 6(b), there was a reduction in the local minimum velocity as the thickness of the upper layer increased. This trend suggests that, as particles fall through an increasingly extended upper fluid layer, they are capable of entraining a larger volume of fluid from that layer, thereby enhancing buoyancy.

Figure 6.

(a) Particles passing through the density interface indicated by the white, dashed line for the $H_1/d = 10$ case at $f_p = 8\,{\rm Hz}$. (b) Minimum vertical velocity, (c) $H_1$-dependent drag coefficient in the lower fluid medium.

Within the lower fluid medium, the drag coefficient, $C_d$, was determined using vertical velocities at $z/d = 50$. Figure 6(c) displays these values, normalized by the drag coefficient for an isolated particle in a uniform oil medium, $C_{d_0}$. In the two-layer set-ups, the interplay of particle interactions and entrained fluid produced opposing effects on the drag coefficient. When multiple particles settled concurrently, the trail created by leading particles could potentially influence the others, often resulting in reduced drag. Such a phenomenon has been noted in previous studies regarding multiple particle settling (Fortes, Joseph & Lundgren Reference Fortes, Joseph and Lundgren1987; Uhlmann & Doychev Reference Uhlmann and Doychev2014; Ghosh & Stockie Reference Ghosh and Stockie2015). Concurrently, the entrained fluid can increase drag by inducing supplementary buoyancy prior to its complete detachment from the particle. Figure 6(c) reveals that the normalized drag coefficients predominantly exceed unity. This implies that, even in the presence of inter-particle influences, the aggregate drag coefficient either parallels or surpasses that of a single particle. Thus, even at positions as deep as $50d$ below the interface, the impact of the entrained fluid remained significant in our experimental set-up. The drag coefficient displayed an upward trend with expansion in the thickness of the upper layer, reaching a maximum when $H_1/d=40$. This tendency was less evident at the highest release frequency, where particle interactions had the most pronounced impact.

Figure 7(a) presents a selection of particle trajectories viewed from above, illustrating that the particles underwent abrupt alterations in their drift directions. In figure 7(b), we introduce the pivotal vertical position, $z^*$, at which this drift reorientation occurs, and define the drift vectors preceding and succeeding this point as $\boldsymbol {L_1}$ and $\boldsymbol {L_2}$. The magnitude of the drift is quantified by the angle between these vectors, given by $\theta = \arccos (({\boldsymbol {L_1} {\cdot } \boldsymbol {L_2}})/({|\boldsymbol {L_1}| \ |\boldsymbol {L_2}|})$. Figure 7(c) shows the mean values of $z^*$ and $\theta$ across all examined scenarios. Note that this alteration in drift direction is manifested not at the density interface, but approximately $8d$ to $9d$ beneath it, concurring with the onset of lateral velocity stabilization in the denser medium, as depicted in figure 4. The drift redirection was pronounced, averaging close to 90 degrees, albeit with some variability across different conditions. This phenomenon arises from the interplay between varying buoyant force across the density gradient, viscosity contrasts between layers, particle inertia and localized flow structures such as vortices at the density interface. As particles move from a less dense to a more dense fluid, they encounter a shift in buoyancy, which coupled with the increased drag from higher viscosity in the lower layer, triggers a reorientation of their drift path. Moreover, particles carry momentum that leads to a temporary change in drift direction until a new equilibrium is found. The presence of flow structures, especially at the interface, further influences this behaviour by altering trajectories based on the particle position relative to these structures and their magnitude. Upon navigating the density interface and descending within the denser fluid, particles shed the inertia previously imparted by the diminishing flow vorticity, a phenomenon reminiscent of the history-related force contribution, as discussed by Boussinesq (Reference Boussinesq1885) and Basset (Reference Basset1888).

Figure 7.

(a) Particle trajectories from top view for $H_1/d = 20$ at $f_p = 8\,{\rm Hz}$, and (b) a closer view of one trajectory. (c) Average value of the pivotal vertical position $z^*$, at which the direction of particle drift changed, with the inset figure showing the average value of change in angle in particle drift direction.

3.2. On the acceleration of particles in proximity to the density interface

The averaged particle acceleration across all release frequencies normalized with respect to gravitational acceleration, $g$, is depicted for the lateral and vertical components in figure 8, highlighting the variations near the density interface.

Figure 8.

Average particle acceleration across all release frequencies near the density interface in (a) lateral and (b) vertical directions for varying thicknesses of the upper fluid layer. The shaded areas illustrate the variation in acceleration among cases with different release frequencies.

The case with $H_1/d = 10$ showcases a unique trend in particle acceleration, distinct from other scenarios. Here, the density interface acts as a barrier, constraining particle motions in the lateral and vertical directions, as is evident from the negative particle acceleration near the interface. Lateral acceleration $a^*_L$ in figure 8(a) shows reduced deceleration with increasing $H_1$ due to the lower particle velocity in the upper fluid layer near the density interface, resulting in smaller changes before and after the interface (see figure 4). The deceleration in the lateral direction persisted up to approximately $8d$ below the density interface.

Trends in vertical acceleration $a^*_z$ (figure 8b) reveal a decrease in particle velocity prior to crossing the density interface, a phenomenon also noted by Camassa et al. (Reference Camassa, Falcon, Lin, McLaughlin and Myckins2010), where the interface was likened to a rigid barrier slowing the settling velocity. Peak deceleration occurred at the interface ($z/d = 0$), indicating significant resistance to the vertical motion of the particles, with the magnitude increasing with larger $H_1$. This deceleration extended into the lower medium, persisting up to roughly $10d$ below the interface, a location indicating the observed minimum in vertical velocity (figure 5).

The spatial distribution of particle acceleration uncovers distinctive patterns. Specifically, figure 9(a) displays the distribution of vertical particle acceleration at three vertical positions: above the interface ($z/d = -5$), at the interface ($z/d = 0$) and below the interface ($z/d = 15$), for a set-up with $H_1/d = 40$ and a release frequency of $f_p = 4\, {\rm Hz}$. At each location, the distributions appear approximately normal, with the greatest variance observed at the density interface. The standard deviation of particle acceleration normalized by $g$ ($\sigma ^*_{a_L}$ and $\sigma ^*_{a_z}$), as depicted in figures 9(b) and 9(c), consistently reaches a maximum at the density interface. This variance gradually diminishes and approaches a negligible level in the lower medium, approximately $15d$ from the interface.

Figure 9.

(a) Histogram of vertical particle acceleration at varying positions relative to the interface: above ($z/d=-5$), at ($z/d=0$) and below ($z/d=15$) for a case with $H_1/d = 40$ and $f_p = 4\, {\rm Hz}$. (b) Standard deviation of lateral particle acceleration, and (c) corresponding vertical particle acceleration. Colours in (b,c) indicate particle-release frequencies: $f_p = 4\, {\rm Hz}$ (black), 6 Hz (blue) and 8 Hz (red). Symbols represent the thickness of the first layer: $H_1/d = 20$ ($\circ$), 30 ($\times$) and 40 ($\triangle$).

3.3. Particle pair dispersion and the role of the interface

The particle pair dispersion, $R^2 = \langle (r(t)-r_0)^2 \rangle$, was characterized for various initial separations, $r_0$, in all two-layer fluid configurations (figures 11 and 12). Here, $\langle \cdots \rangle$ denotes the averaging operator, and $r(t)$ represents the distance between two particles over time. Within the two-layer fluid environment, the initial separations allow us to classify particle pairs into two categories. The first category, where $r_0$ is less than $H_1$, encompasses pairs where both particles begin in the upper fluid layer. Conversely, in the second category, where $r_0$ exceeds $H_1$, one particle is in the upper fluid layer and the other is in the lower layer. For pairs in the first category, we noted three distinct stages of motion. In the initial stage, both particles are located within the upper fluid layer. The subsequent stage witnesses the leading particle transitioning into the lower fluid layer, while the trailing particle remains in the upper fluid. In the third stage, both particles reside within the lower fluid layer. However, in the second category particle pairs experience only the latter two stages.

In figure 10, the lateral pair dispersion, $R^2_{L}$, exhibited unique patterns depending on the initial separation and corresponding stages. For particle pairs where $r_0< H_1$ (represented by black ‘+’ symbols in figure 10c,d), the dispersion in the initial stage followed a ballistic scaling $R^2_{L} \sim t^2$. As the lead particle crossed the density interface, there was a noticeable deceleration in the dispersion rate, transitioning to a Taylor scaling $R^2_{L} \sim t$. Once both particles were sufficiently distant from the density interface, the trend in dispersion rate flattened, suggesting a minimal relative velocity between the particle pair.

Figure 10.

Lateral pair dispersion, $R^2_{L}$, for various initial separations, $r_0$ for (a) $H_1 = 10d$, (b) $H_1 = 20d$, (c) $H_1 = 30d$ and (d) $H_1 = 40d$ at a particle-release frequency of $f_p = 4\, {\rm Hz}$.

The pair dispersion behaviour and magnitude were reasonably consistent, irrespective of the initial separations for $r_0>H_1$ (as depicted by every symbol in figure 10(a,b) and any symbol other than the black ‘+’ in figure 10c,d). Initially, $R^2_{L}$ exhibited a ballistic scaling at a magnitude distinctly lower than the dispersion for $r_0< H_1$. Following this phase, as the trailing particle transitioned through the density interface, the dispersion rate increased to a Richardson scaling $R^2_{L} \sim t^3$. As both particles continued their descent away from the density interface, $R^2_{L}$ plateaued.

Figure 11 examines the impact of the thickness of the upper fluid layer on lateral pair dispersion. These plots show $R^2_{L}$ across all two-layer experiments, focusing on the smallest initial separation where the particles were in separate fluid layers, essentially the minimum $r_0>H_1$. Across all particle-release frequencies, $R^2_{L}$ tended towards a higher value with a larger upper fluid layer. The different cases plotted in figure 11 highlight that the critical time, $t^*$, at which the dispersion converged, increased with greater $H_1$. This indicates that the trailing particle could disperse more from the leading particle as it settled through the longer upper fluid layer, where it showed considerable lateral movement.

Figure 11.

Lateral pair dispersion, $R^2_{L}$, for the minimum initial separation greater than the first layer thickness $H_1$ at the particle-release frequencies of (a) $f_p = 4\,{\rm Hz}$, (b) $f_p = 6\,{\rm Hz}$ and (c) $f_p = 8\,{\rm Hz}$. (d) Critical time $t^*$ indicating $R^2_{L}$ convergence.

The vertical pair dispersion, $R^2_{z}$, also showed dependencies on the initial separations and various stages of particle motion. As indicated in figure 12, when $r_0< H_1$, $R^2_{z}$ showed a ballistic regime (black ‘+’ symbols in figure 12c,d). After this phase, as the particles transitioned into the lower fluid layer, the dispersion rate increased significantly, exhibiting a $R^2_{z} \sim t^{5.5}$ scaling. This relationship shows a distinct or super-diffusive dispersion regime, diverging from the classical $t^3$ Richardson scaling known for turbulent pair dispersion. This observation indicates the presence of strong dynamic interactions, or enhanced inertial effects, within the stratified medium. When both particles were sufficiently far away from the density interface, $R^2_{z}$ plateaued similar to the behaviour of $R^2_{L}$. For $r_0>H_1$, $R^2_{z}$ showed a ballistic regime and then converged to a constant value (every symbol in figure 12(a,b) and any symbol other than black ‘+’ symbols in figure 12c,d). In contrast to $R^2_{L}$, the value of $R^2_{z}$ for $r_0> H_1$ was larger than that for $r_0 < H_1$. Figure 13 illustrates $R^2_{z}$ for the minimum initial separation of the particle pair residing in different fluid layers. Similar to the patterns observed in $R^2_{L}$ (figure 11), the final value of $R^2_{z}$ and the critical time $t^*$ increased with increasing $H_1$, indicating that particles can disperse more from each other with larger upper layer thickness in two-layer fluid scenarios.

Figure 12.

Vertical pair dispersion $R^2_{z}$ for various initial separations, $r_0$, indicated in different symbols for (a) $H_1 = 10d$, (b) $H_1 = 20d$, (c) $H_1 = 30d$ and (d) $H_1 = 40d$ at a particle-release frequency of $f_p = 4\, {\rm Hz}$.

Figure 13.

Vertical pair dispersion $R^2_{z}$ for the minimum initial separation greater than the first layer thickness $H_1$ at the particle-release frequencies of (a) $f_p = 4\,{\rm Hz}$, (b) $f_p = 6\,{\rm Hz}$ and (c) $f_p = 8\,{\rm Hz}$. (d) Critical time $t^*$ indicating $R^2_{z}$ convergence.