6.1. Motion of nearly neutrally buoyant particles

To target attractors for the particle motion that are caused mainly by particle–wall interaction, any inertia effects must be minimised. Therefore, we first consider particles whose density is nearly matched to that of the fluid. Nearly neutrally buoyant conditions were realised by adjusting the temperature of the fluid in the cavity experiment, yielding relative density $\varrho =1.0001\pm 0.0001$ for all particles considered in this section, i.e. for particles with sizes $a = 0.0064$, 0.0111, 0.0272, 0.0390, 0.0494, 0.0586 and 0.0703. Their motion is considered for ${\textit {Re}} = 100$.

6.1.1. Limit cycles of period one

For $\varrho = 1.0001$, we find a periodic attractor with period one in very close vicinity of the closed streamline $L_1$ for all particle sizes mentioned above. Figure 5 shows the locations of the limit cycles in the Poincaré plane $y=0$ (here and in the following, the dashed lines indicate the location of the curved walls in this plane). As can be seen, the attractors arise as mirror symmetric pairs and they grow out of the closed streamline (white diamond in figure 5b) as the particle size (coded by colour) increases from $a=0.0111$. With increasing $a$, the attractors move away from the moving wall in the $y=0$ plane, and closer to the symmetry plane $z=0$. The growth of the distance from the moving wall with increasing particle size is also seen in the projections of the limit cycles onto the $(x,y)$ plane shown in figure 6. In addition, the periodic orbits extend slightly further in the positive $y$ direction as $a$ increases. The Poincaré points for the smallest particle with $a=0.0064$ – corresponding to the dark grey crosses in figure 5(b) – near the closed streamline belong to a transient state at $t\in [6700,7200]$ s, which finally converges to a limit cycle for much longer time (see figure 19(b) below).

Figure 5. (a) Poincaré section on $y=0$ of trajectories of nearly neutrally buoyant spherical particles ($\varrho =1.0001$) on their respective attractors (colours). The colour indicates the particle radius $a$: $0.0064$ dark grey, 0.0111 blue, 0.0272 yellow, 0.0390 cyan, 0.0494 green, 0.0586 orange, and 0.0704 red. For comparison, the Poincaré section of KAM tori and closed streamlines are shown as light grey dots. The black dots represent the Poincaré section of the attractors for an inertial particle with $a=0.0119$ and $\varrho =1.052$. The large square indicates the zoom shown in (b), where attractors are shown by crosses and streamlines on KAM tori by light grey dots. The dashed lines indicate the boundary of the domain in the plane shown.

Figure 6. (a) Projection of all trajectories onto the $(x,y)$ plane of nearly neutrally buoyant spherical particles ($\varrho =1.0001$) moving on their respective attractors. The colour indicates the particle radius $a$: $0.0064$ light grey and dark grey, 0.0111 blue, 0.0272 yellow, 0.0390 cyan, 0.0494 green, 0.0586 orange, and 0.0704 red. Black and maroon lines indicate numerically computed closed streamlines of periods one and six, respectively. (b) Zoom, the x and y axes are scaled differently.

As a typical example, we consider in more detail the trajectory of a particle with $a=0.0272$ and $\varrho =1.0001$. Initially, the particle is transferred by the boundary repulsion effect to one of the regions occupied by the two main sets of KAM tori $T_1$ of the flow. There, the particle approximately moves on KAM tori, but slowly spirals into the limit cycle. A three-dimensional view of the resulting limit cycle is shown in figure 7(a), where 40 trajectories have been superimposed. The close agreement of the trajectories demonstrates the reproducibility of the two limit cycles. Figure 7(b) shows all Poincaré points for two realisations. It is seen that during the asymptotic transient motion, the trajectories are characterised by two incommensurate frequencies whose ratio is close to five.

Figure 7. (a) Forty trajectories for $a=0.0272$ ($a_{p}=1.1$ mm) and $\varrho =1.0001$ recorded during $t\in [500,600]$ s. Shown is a three-dimensional view of the two periodic attractors. The arrow indicates the direction of the motion of the lid/cylinder. (b) Poincaré section (black dots) of two such particle trajectories on the plane $y=0$ recorded during $t\in [0,600]$ s. The particles are initially located in different half domains ($z>0$ or $z<0$) of the cavity. Poincaré sections of numerically computed streamlines on KAM tori are shown as light grey dots.

A particle moving on a limit cycle makes a periodic motion in which the components of the trajectory $\boldsymbol X(t)$ are periodic functions of time. A characteristic quantity of this motion is the period $\tau _1$ that is the inverse of the fundamental frequency $f_1$ of the Fourier spectrum of any of the periodic coordinate functions. The amplitude spectrum $\hat X$ of the $x$ coordinate of the limit cycle is presented in figure 8 (solid line). The fundamental frequency of the limit cycle is $f_1=0.165$ Hz, corresponding to the non-dimensional frequency $F_1=13.33$. The spectrum of the limit cycle compares very well with the spectrum of the fluid motion on the closed streamline that is shown by a dashed line for comparison. To determine the asymptotic decay to the limit cycle, the distance functions $d_n$ from the fixed point in the Poincaré plane for all forty realisations are fitted by (5.3). The result is shown in figure 9(b), yielding the decay rate $\sigma =0.743$.

Figure 8. Amplitude spectrum $\hat X(f)$ (solid line) of the trajectory of a particle with $a=0.0272$ ($a_{p}=1.1$ mm) and $\varrho =1.0001$ moving on its periodic attractor with fundamental frequency $f_1=0.165$ Hz ($F_1=13.33$) in comparison with the spectrum $\hat X_{L_1}$ of the numerically determined closed streamline (dashed line).

Figure 9. (a) Poincaré section on $y=0$ of a trajectory of a single particle (black dots and lines) with $a=0.0272$ and $\varrho =1.0001$. The final phase from $\tau = 6.4335$ onwards is shown by red dots. Grey dots indicate the largest contiguous KAM torus; the white diamond marks the closed streamline; and the circle defines the threshold distance $d^*$ for Poincaré points to be included in the fit (5.3). (b) The distance function $d_n$ for 40 realisations (grey plus signs). A fit (solid black line) of the data according to (5.3) yields the attraction rate $\bar \sigma =0.766\pm 0.07$.

In addition, figure 9(a) shows the Poincaré section of a representative trajectory (black and red dots) in relation to the largest of the inner contiguous KAM tori (grey dots), the closed streamline (diamond) and the threshold condition (circle with radius $d^*$ about $(x^*,z^*)$).

Figure 10 shows the mean attraction rate $\bar \sigma := N^{-1}\sum _{n=1}^N \sigma _n$, where $\sigma _n$ is determined according to (5.3), $n$ numbers the samples, and $N$ is the number of repeated experiments for each particle. Data ($\times$) are collected for all particles with $\varrho =1.0001$. As can be seen, the rate of attraction to the limit cycle increases with the particle size $a$. As will be shown below, the density difference between the particles and the fluid is too small to relate the attraction to inertial effects. Therefore, the attraction rates found are attributed mainly to the wall effect, i.e. the confinement effect of a finite size particle, and will be denoted $\bar \sigma _w$ henceforth. Also included in the graph is a power-law fit (solid line) to the three points given by the origin $(0,0)$ and the data for the two smallest particles. We find the approximation $\bar \sigma _w = ca^b$ with $c=3968$ and $b=2.32$ valid for $a < 0.012$. Other fit functions, e.g. polynomials, have been tested as well. They all lead to similar implications regarding the contribution of the wall effect to the attraction rate of heavier particles with size up to $a=0.0120$ (figure 15 below).

Figure 10. Mean attraction rates $\bar \sigma _w$ to the period-one attractors for nearly neutrally buoyant particles with radii $a=0.0064$, $0.0111$, $0.0272$, $0.0390$, $0.0494$, $0.0586$, $0.0704$, corresponding to $a_{p} = 0.26$, 0.45, 1.10, 1.58, 2.00, 2.37, 2.85 mm. The solid line is a fit $\bar \sigma _w = ca^b$ ($c=3968$, $b=2.32$) to the data for the smallest particle sizes (see text).

6.1.2. Limit cycle of period six

For the smallest particle size $a=0.0064$ and for $\varrho =1.0001$, we find, in addition to the limit cycle of period one near $L_1$, another limit cycle of period six. The limit cycle to which the particle is attracted depends on the initial conditions. The period-six limit cycle is found near the closed streamline $L_6$. Figure 11 shows a three-dimensional view of trajectories on the two attractors (blue) and Poincaré sections on $y=0$ during the time interval $t\in [6700,7200]$ s. At this time, particle trajectories have converged to the period-six attractor up the size of the blue dots shown in figure 11(b). The period-six KAM tori seem to be a robust feature of the flow, because they also exist in the exactly cubical cavity at the same Reynolds number (Romanò et al. Reference Romanò, Türkbay and Kuhlmann2020).

Figure 11. Sixteen trajectories for $a=0.0064$ ($a_{p}=0.26$ mm) and $\varrho =1.0001$ recorded during $t\in [6700,7200]$ s. (a) Three-dimensional view of the period-six attractors (blue) and transient toroidal trajectories that are ultimately attracted to one of the two period-one limit cycles. (b) Poincaré sections on the plane $y=0$ (black, blue) of the particle trajectories shown in (a). Poincaré sections of numerically computed streamlines are shown as light grey dots.

Also shown in figure 11 is a three-dimensional view of trajectories and a Poincaré section of particles that end up on one of the two period-one limit cycles near $L_1$. The attraction is so weak that these particles have not yet reached their periodic orbit at $t=7200$ s (see figure 19b). In comparison, the attraction to the period-six orbit is one order of magnitude faster (table 3). This can be explained by the close proximity to the moving wall of the closed streamline $L_6$ and the period-six limit cycle, which are shown in maroon and light grey, respectively, in figure 6(b). Therefore, a particle near $L_6$ repeatedly experiences a much stronger wall-induced repulsive force than the same particle (dark grey in figure 6b) when it moves near $L_1$. From the spectrum of the period-six attractor shown in figure 12, we find the fundamental frequency $f_1=0.133$ Hz. The subharmonics at $f_1/6$ of order six and its integer multiples clearly indicate the period-six motion on this attractor. Properties of the limit cycles for all nearly neutrally buoyant particles found are collected in table 3. The correlation between the limit cycles for the particles and numerically computed closed streamlines can be seen by comparing tables 3 and 2.

Figure 12. Amplitude spectrum $\hat X$ of a trajectory on the period-six attractor with $f_1=0.133$ Hz ($F_1=10.91$) for $a=0.0064$ ($a_{p}=0.26$ mm), $\varrho =1.0001$, ${\textit {Re}}=100$ and $T=24.3\,^\circ$C. For comparison, the spectrum $\hat X_{L_6}$ of the numerically determined closed streamline is shown as a dashed line.

Table 3. Properties of measured trajectories of nearly neutrally buoyant particles on their periodic attractors for ${\textit {Re}}=100$ and $\varrho =1.0001$. Specified are the type (period), fundamental frequencies $f_1$ (dimensional) and $F_1$ (dimensionless), turnover time $\tau _1 = F_1^{-1}$, initial transient time $\bar \tau _I$ required to approach the attractor up to the distance $d_n \le 0.15$ (in the plane $y=0$), asymptotic attraction rate $\bar \sigma _w$, the closest wall-normal distances from the boundaries $\varDelta _{p}$ (the boundary is indicated by the superscript), and the number of samples $N$ used for averages.

$^a$The attraction rate for the P-6 attractor was evaluated by fitting exponentially the local minima of the $X(\tau )$ coordinate of the trajectory.

Even though period-seven KAM tori exist in the flow, embedded in the period-one KAM system, we have not found a period-seven attractor. This may be related to not having probed a particle size suitable for attraction. Since $L_7$ approaches the moving wall even closer than $L_6$ (table 2), the particles that could be attracted to a limit cycle of period seven are expected to be even smaller than $a=0.0064$. A further difficulty of finding such attractors may arise from the particle size selectivity, which is related to the small diameter of the period-seven KAM structures.

6.2. Forces along the trajectories of density-matched particles

In order to assess major forces acting on the particle during its motion, we used a measured particle trajectory and computed all flow gradients along this trajectory from the unperturbed flow field obtained numerically. The flow velocity at the position of the particle is determined by polynomial interpolation of the numerical velocity field, while the velocity of the particle is calculated by central differences of the measured discrete position data. For small particles, the forces arising in the Maxey–Riley equation (Maxey & Riley Reference Maxey and Riley1983) are of interest.

Results for a particle with $a=0.0111$ and $\varrho =1.0001$ for ${\textit {Re}}=100$ are shown in figure 13(a) during the initial (transient) phase, and in figure 13(b) during the motion on the limit cycle. Figures 13(a i) and 13(b i) show the particle location in the form of $x(\tau )$ (black), $y(\tau )$ (red) and $z(\tau )$ (green). Figures 13(a ii–iv) and 13(b ii–iv) show the $x$, $y$ and $z$ components of the acceleration terms evaluated. For the $x$ component, figures 13(a ii) and 13(b ii) show the pressure force, including the added mass effect $(3/2)R\,\mathrm {D} u/\mathrm {D} \tau$ (black), Faxén's correction to the added mass $(a^2/20)\,\mathrm {D} ({\nabla }^2 u)/\mathrm {D} \tau$ (red), the Stokes drag $-R\,{\textit {St}}^{-1}\,(\dot X - u)$ (green) and Faxén's correction to the Stokes drag $(a^2/6)R\,{\textit {St}}^{-1} {\nabla }^2 u$ (blue). Here, $R=2/(1+2\varrho )$. The lines in the plots for the $y$ and $z$ components have the corresponding meanings.

Figure 13. Trajectory $(x,y,z)$ (plot (i)) of the particle with $a=0.0111$ and $\varrho =1.0001$ for ${\textit {Re}}=100$. Plots (ii)–(iv) show different components of the acceleration in the $x$, $y$ and $z$ directions, respectively. For an explanation of the colour code, see the text. (a) Initial transient phase. (b) Final phase when the particle moves on its attractor.

In addition to the above terms, we show forces which are not contained in the Maxey–Riley equation. They derive from the particle velocity, the unperturbed velocity gradient and the distances to the walls. These forces were computed to estimate the magnitude of the corresponding effect on the particle motion. The additional drag force on a particle viscously settling according to Brenner's correction to the drag coefficient $\lambda$ has been computed as $-R\,{\textit {St}}^{-1}\sum _{i=1}^3(\lambda _i -1) \dot {\boldsymbol X}_{i,\perp }$, where the sum runs over the contributions from the top wall ($i=1$), the curved moving wall ($i=2$) and the bottom wall ($i=3$), $\lambda _i$ is the drag coefficient evaluated corresponding to the distance from the $i$th wall, and $\dot {\boldsymbol X}_{i,\perp }$ is the particle velocity normal to the $i$th wall. The resulting contributions, decomposed into Cartesian components, are shown in orange in figure 13. To estimate a possible migration effect, we also monitor (in the $x$ plot only) Saffman's lift force in the $x$ direction $0.727\varrho ^{-1}\,{\textit {St}}^{-1/2}\,(v-\dot Y)\,|\partial v /\partial x|^{1/2}{\rm sgn}(\partial v /\partial x)$ (cyan; Saffman Reference Saffman1965, Reference Saffman1968). Finally, we also consider the lift force (in the $x$ plot only) for a freely rotating particle near a plane wall according to the formula given by Cherukat & Mclaughlin (Reference Cherukat and Mclaughlin1994) (brown).

Throughout its motion, the largest forces on the particle arise during the closest approach to the moving wall. Of these forces, viscous forces (green, orange) dominate. We first consider the motion on the attractor. From figure 13(b iii), one can see that the passage of the particle near the moving wall experiences a viscous drag (green) that is directed in the negative $y$ direction due to the flow acceleration caused by the wall motion and the particle being slightly heavier than the fluid. In the $x$ direction, viscous forces dominate as well. It is seen that the particle starts lagging behind the flow even upon approaching the moving wall. The positive drag forces indicate that the particle is kept away from the moving wall. This is an important property of the lump wall effect (see also the model of Hofmann & Kuhlmann Reference Hofmann and Kuhlmann2011). The repulsion effect is also signalled by the positive contribution of the correction to the viscous drag according to Brenner's drag coefficient $\lambda$ (orange). On the other hand, Saffman's lift force (cyan) tends to push the particle towards the wall. Since the Saffman lift may not be correct near a moving wall, we also included the lift force according to the formula of Cherukat & Mclaughlin (Reference Cherukat and Mclaughlin1994) for a freely rotating particle (brown). The lift force of Cherukat & Mclaughlin (Reference Cherukat and Mclaughlin1994) is much smaller than Saffman's formula suggests, and particle lift seems to be insignificant overall.

The initial transient phase is shown in figure 13(a). This phase is very important for the attraction rate of the particle attraction to the limit cycle (see e.g. figure 9 for $a=0.0272$). One can see that the forces experienced by the particle upon the interaction with the moving wall are even stronger and more impulsive than during its motion on the attractor (figure 13b). The reason is that during the initial phase, the particle approaches the wall even closer and in more irregular intervals. Therefore, the overall dynamics is clearly dominated by the particle–wall interaction.

Finally, we consider the largest density-matched particle with $a=0.0704$ and $\varrho =1.0001$ for ${\textit {Re}}=100$. The forces acting during the motion on the attractor are displayed in figure 14. For this large particle, the wall interaction is even more dominant. Again, viscous forces (green, orange) are dominant and signal the wall repulsion effect. This is plausible, since the particle approaches the moving wall much closer than its radius (see table 3). For this larger particle, also the pressure force becomes relevant, because of the larger Stokes number. The two negative peaks (black) in the $x$ component tend to keep the particle away from the wall. From these considerations, viscous forces seem to play an important role for the particle dynamics. Since these forces arise predominantly during the passage of the particle past the moving wall, its attraction to the limit cycle is attributed to a lump wall effect. Faxén's forces are insignificant, and also lift forces appear to play a very minor role for the particle motion in the present fully three-dimensional flow.

Figure 14. (a) Trajectory $(x,y,z)$ of the particle with $a=0.0704$ and $\varrho =1.0001$ for ${\textit {Re}}=100$ moving on its attractor. (b–d) Different components of the acceleration in the $x$, $y$ and $z$ directions, respectively. For an explanation of the colour code, see the text.

From the Stokes drag force shown in figures 13(b) and 14, the slip velocity along the limit cycle seems to remain considerable even between the particles’ visits of the moving wall. The $y$ component $\dot Y-v$ of the slip velocity computed varies almost linearly from 5 % to $-7\,\%$ of the wall velocity during the particle's motion in the bulk (from the departure from to the arrival at the moving wall). We cannot rule out completely that a certain amount of slip is caused by extra drag forces on the particle provoked by the confinement of the system by the stationary and moving walls. However, the computed slip velocity in the bulk was found to be almost identical for the particles with $a=0.0704$, 0.0111 and 0.0064 (not shown). This indicates that the computed data overestimate the slip due to different sources of error. The most obvious source of error, which alone can account for the above residual slip, is the accuracy of $|\Delta \boldsymbol X| = 0.0074$ ($\pm 0.3$ mm) by which the position of the particle can be measured. Other errors arise from the sampling time of $\Delta t\approx 6.1\times 10^{-4}$ (0.05 s) due to which the direction of the particle velocity computed from secants of the trajectory deviates from the true particle velocity in regions of fast motion on curved trajectories. Other sources of error are the limited resolutions of the camera sensors. To illustrate this point, the Stokes drag is the only quantity that is shown unfiltered in figures 13 and 14, and thus appears noisy. Finally, the flow in the experiment can deviate from the numerically computed flow such that also the measured particle velocity $\dot {\boldsymbol X}$ can deviate from the numerical flow velocity at the position but in the absence of the particle. An elimination of these errors and a clarification of the magnitude and character of the true residual drag forces acting in the bulk are very difficult due to the nature of the problem.

6.3. Motion of inertial particles

For particles heavier than the fluid with density difference up to 5 %, a picture emerges that is similar to that for density-matched particles. For all particle sizes investigated, there always exists a period-one attractor near the closed streamline $L_1$. The properties of the attractors are provided in table 4. Among the cases considered, the limit cycle with the largest displacement from $L_1$ in the Poincaré plane arises for $a = 0.0119$ ($a_p = 0.48$ mm) and $\varrho = 1.052$. The Poincaré section of the attractor is shown by black symbols in figure 5. From the Poincaré section, the location of the periodic orbit relative to the closed streamline does not follow the systematic trend exhibited by the neutrally buoyant particles. In the Poincaré plane shown, the inertial particle (black) is displaced from the closed streamline much further and in a different direction as compared to a neutrally buoyant particle of comparable size (blue).

Table 4. Properties of measured trajectories of inertial particles on their attractors for ${\textit {Re}}=100$. Specified are the type of attractor (P means periodic, QP means quasi-periodic), fundamental frequencies $f_1$(dimensional) and $F_1$(dimensionless), turnover time $\tau _1=F_1^{-1}$, initial transient time $\bar \tau _I$ required to approach the attractor up to the distance $d_n \le 0.15$ (in the plane $y=0$), asymptotic attraction rate $\bar \sigma$, the closest wall-normal distances from the boundaries $\varDelta _{p}$ (the boundary is indicated by the superscript), and the number of samples $N$ used for averages.

From (4.1), the attraction/repulsion rate for weakly inertial-buoyant particles is expected to scale $\sim (\varrho -1)\,{\textit {St}}$ (see also Muldoon & Kuhlmann Reference Muldoon and Kuhlmann2016; Wu et al. Reference Wu, Romanó and Kuhlmann2021). However, the attraction rates measured and shown as red dots in figure 15 do not exhibit the expected overall linear dependence. It is striking, however, that the attraction rates for small particles of nearly the same size with $a\in [0.0062, 0.0064, 0.0069]$ as well as the attraction rate for the neutrally buoyant particle with $\varrho =1.0001$ (blue square) exhibit a linear dependence on $(\varrho -1)\,{\textit {St}}$, merely offset from the linear law (shown as a solid line) that passes through zero. Furthermore, the slope of the line connecting the growth rates of the two larger particles of similar size ($a\in [0.0119, 0.0120]$) is almost the same as for the smaller particles. These observations, in particular the offset of the growth rate for the nearly neutrally buoyant particle (blue square), suggest that the attraction rate $\bar \sigma =\bar \sigma _w +\bar \sigma _i$ comprises a superposition of a size effect ($\bar \sigma _w$, due to the wall repulsion) and an inertial effect ($\bar \sigma _i$). A linear superposition is a valid assumption here, because the forces responsible for the attraction are small, which is confirmed by the small particle Reynolds number (table 1), leading to a small attraction rate, which justifies a linearisation of the dynamics. We note that the particle Reynolds number ${\textit {Re}}_p$ and corresponding Stokes drag are possibly smaller, because of the difference due to numerical error and imperfections between the numerically computed velocity at the location of the particle and the unperturbed experimental velocity, respectively, which could not have been measured.

Figure 15. Measured total mean attraction rate $\bar \sigma$ (blue square, red dots) to the period-one limit cycle for ${\textit {Re}}=100$ as function of $(\varrho -1)\,{\textit {St}}$ for six particles with different sizes and densities as indicated. The radii are (from left to right) $a\in [0.0064, 0.0069, 0.0062, 0.0064, 0.0120, 0.0119]$ with corresponding densities $\varrho \in [1.0001, 1.022, 1.042, 1.051, 1.019, 1.052]$. The inertial part $\bar \sigma _i$ of the attraction rate is shown by black crosses. The solid line is a linear regression of $\bar \sigma _i$ (see text).

To test the superposition principle, we compute the inertial attraction rate $\bar \sigma _i$ by subtracting from the measured total attraction rate $\bar \sigma$ the wall-induced attraction rate $\bar \sigma _w$ found for nearly neutrally buoyant particles. Since the particle sizes in figure 15 range up to $a\approx 0.012$, we use the power-law fit to the three smallest data points (including $(\bar \sigma _w,a)=(0,0)$), which is shown by the solid line in figure 10. As a result, we obtain the inertial attraction rate $\bar \sigma _i$ to the period-one limit cycle, which is shown by crosses in figure 15. As can be seen, the data for $\bar \sigma _i$ are consistent with the expected linear dependence. A linear regression (solid line) yields $\bar \sigma _i = c(\varrho -1)\,{\textit {St}}$ with $c=1.24\times 10^5$. Based on this correlation, the inertial attraction rate for the largest density-matched particle from figure 10 with $a=0.0704$ is $\bar \sigma _i=1.39\times 10^{-2}$. This is only $0.6$ % of the attraction rate $\bar \sigma _w=2.354$ measured. This indeed shows that the attraction to the limit cycle is almost entirely due to the wall effect for the particles with $\varrho =1.0001$ (figure 10).

In a similar way as for nearly neutrally buoyant particles, we also find period-six attractors for heavier and lighter particles than the fluid. An example for a heavy particle is shown in figure 16 for $a=0.0064$ and $\varrho =1.051$. In three out of four realisations, the heavy particle is attracted to one of the period-one limit cycles (black), whereas one realisation shows attraction to a period-six attractor (blue) in the vicinity of one of the period-six KAM tori. The attraction rate to the period-six limit cycle is difficult to determine due to the very short asymptotic phase (the KAM tori are small) such that the time required for the attraction depends sensitively on the initial conditions, i.e. on the initial chaotic motion.

Figure 16. Four trajectories for $a=0.0064$ ($a_{p}=0.26$ mm) and $\varrho =1.051$ recorded during $t\in [5500,6000]$ s. (a) Three-dimensional view of the period-one (black) and period-six (blue) attractors for the particle. (b) Poincaré sections on the plane $y=0$ (black, blue) of the particle trajectories shown in (a). Poincaré sections of numerically computed streamlines on the reconstructible KAM tori are shown as light grey dots.

The attractors for a light particle with $a=0.0062$ and $\varrho =0.959$ are shown in figure 17. We find attractors near $L_1$ (black) and $L_6$ (blue). Now, however, the attractors are not periodic, but quasi-periodic. The reason is the change of the sign of $\varrho -1$ which renders a pure inertial-buoyant limit cycle emerging from the closed streamlines to change its stability. Therefore, unstable limit cycles of period one and six must exist for $\varrho -1 < 0$ if the motion would be purely inertial-buoyant. Obviously, such unstable limit cycles also exits under the present (additional) wall effect. Therefore, the particle–wall interaction cannot prevent the unstable limit cycles. However, the repulsion of the particles from the moving wall stabilises the particle motion on a quasi-periodic orbit in some distance from the respective unstable limit cycle.

Figure 17. Nine trajectories for $a=0.0062$ ($a_{p}=0.25$ mm) and $\varrho =0.959$ recorded during $t\in [12000,14000]$ s. The corresponding non-dimensional time interval is $[140.5 ,163.9]$. (a) Three-dimensional view of the toroidal period-one (black) and period-six (blue) attractors. (b) Poincaré sections on the plane $y=0$ (black, blue) of the particle trajectories shown in (a). Poincaré sections of numerically computed streamlines on the KAM tori are shown as light grey dots.

The attraction dynamics of the same light single particle ($a=0.0062$, $\varrho =0.959$) to the period-one torus is shown in figure 18(a). The mean attraction rate is $\bar \sigma =0.0131$, much smaller than values for all other inertial particles with $\varrho >1$. This can be explained by the quasi-periodic attractor being caused by a balance of opposing weak forces (inertia and wall repulsion). The attraction rate has been computed as described in § 5 using data from seven realisations (figure 18b). To check this result, the evolution of the minima of $X(\tau )$ for a single representative trajectory was considered. For a quasi-periodic orbit, the minima of $X(\tau )$ will vary within a certain range and would not converge. To monitor the range of variation, the maximum and minimum over all minima of $X(\tau )$ satisfying $X < 0.2$ were considered. Within temporal bins of 5000 data points corresponding to $\Delta \tau _{bin}=2.926$, the maximum (upper envelope) and the minimum (lower envelope) of all minima of $X(\tau )$ were obtained. Figure 19(a) shows the evolution of the maximum (circles) and the minimum (squares) of the minima of $X(\tau )$. Fitting the last two-thirds (dotted rectangle) of the envelopes to exponentials (solid red and blue lines), we find slightly different attraction rates $\sigma _{max(min)}=0.0111$ (red) and $\sigma _{min(min)}= 0.0105$ (blue). Both values are within the error bar by which $\bar \sigma$ was determined from the ensemble average (figure 18b). Since the two envelopes extrapolate to different values for $\tau \to \infty$ (horizontal dotted lines), the asymptotic state is indeed quasi-periodic. Near its minimum $x$ coordinate, the attracting torus has diameter in the $x$ direction $D_x = 0.0439$. The slight scatter of the black Poincaré points (instead of a sharp torus) in figure 17(b) can be explained by the particle not yet having reached its asymptotic attractor, which is confirmed by figure 19(a). For comparison, figure 19(b) shows the same analysis for the smallest particle ($a=0.0064$, $\varrho =1.0001$), which is nearly density matched. It can be seen that the data extrapolate to a limit cycle.

Figure 18. (a) Poincaré section on $y=0$ of the trajectory of a single particle (black lines) with $a=0.0062$ and $\varrho =0.959$. The Poincaré points during the last phase $\tau \in [156.9, 163.8]$ are shown as red dots (the total measurement time was 3.9 h). Grey dots indicate the largest numerically reconstructible contiguous KAM torus, and the diamond marks the closed streamline. (b) The distance function $d_n$ for seven realisations ($+$). A fit of the data according to (5.3) (solid black line) yields the attraction rate $\bar \sigma =0.0133\pm 0.0043$.

Figure 19. Maxima (circles) and minima (squares) of the relative minima of the $x$ coordinate of a single trajectory for ${\textit {Re}}=100$. Each symbol circle (square) is the maximum (minimum) value of the relative minima of $X(\tau )$ within a temporal bin of width $\Delta \tau _{bin}=2.926$. The red and blue solid curves are exponential fits to the binned maxima and minima, respectively. The dotted horizontal lines indicate asymptotic values for $\tau \to \infty$. (a) Plot of 56 bins for an inertial particle with $a=0.0062$ ($a_{p}=0.25$ mm) and $\varrho =0.959$, yielding $\max \{\min [X(\tau )]\} = -0.0251+0.159\,\mathrm {e}^{-0.0116 \tau }$ (red curve) and $\min \{\min [X(\tau )]\} = -0.0690-0.211\,\mathrm {e}^{-0.0105 \tau }$ (blue curve). (b) Plot of 28 bins for a neutrally buoyant particle with $a=0.0064$ ($a_{p}=0.26$ mm) and $\varrho =1.0001$, yielding $\max \{\min [X(\tau )]\} = -0.0740 + 0.220\,\mathrm {e}^{-0.0277 \tau }$ (red curve) and $\min \{\min [X(\tau )]\} = -0.0765 - 0.301\,\mathrm {e}^{-0.0368 \tau }$ (blue curve).

7.1. Motion of nearly neutrally buoyant particles

The behaviour of nearly neutrally buoyant particles in the flow with ${\textit {Re}}=200$ is similar to that for ${\textit {Re}}=100$. However, the shape and multitude of the periodic attractors differ due to the more intricate structure of the KAM tori for ${\textit {Re}}=200$. Furthermore, the attraction rates are larger than for ${\textit {Re}}=100$ due to the higher flow velocities and the closer proximity of the KAM structures to the moving wall, from which a stronger wall effect results.

Figure 20(a) shows an overview on the attracting sets in the Poincaré plane $y=0$, with a zoom into the lower left in figure 20(b). For all particle sizes, we find a period-one attractor near $L_1$, best seen in figure 20(b). As $a$ increases, the limit cycles and their mirror-symmetric counterparts near $L_1$ are displaced in the negative $x$ direction, away from the moving wall and towards the symmetry plane $z=0$, similarly as for ${\textit {Re}}=100$. Moreover, as $a$ increases, the projections of all the period-one limit cycles onto the $(x,y)$ plane (figure 21) are pushed away from the moving wall due to the hindered particle motion caused by its size, and extend further into the positive $y$ direction. This trend was observed also for ${\textit {Re}}=100$, albeit not as clearly.

Figure 20. (a) Poincaré sections on $y=0$ of trajectories of nearly neutrally buoyant spherical particles ($\varrho =1.0001$) moving on their respective periodic attractor for ${\textit {Re}}=200$. The colour indicates the particle radius: $a = 0.0064$ ($a_p=0.26$ mm, P-10 black, P-7 magenta, P-4 turquoise, P-1 dark grey), $a=0.0111$ ($a_p=0.45$ mm, P-4 violet, P-1 blue), $a=0.0272$ ($a_p=1.10$ mm, P-1 yellow), $a=0.0390$ ($a_p=1.58$ mm, P-1 cyan), $a=0.0494$ ($a_p=2.00$ mm, P-1 green), $a=0.0586$ ($a_p=2.37$ mm, P-1 orange), and $a=0.0704$ ($a_p=2.85$ mm, P-1 red). For comparison, the Poincaré sections of KAM tori and of closed streamlines are shown as light grey dots. Diamonds indicate closed streamlines. (b) Zoom into the lower left of (a).

Figure 21. (a) Projection onto the $(x,y)$ plane of trajectories of nearly neutrally buoyant spherical particles ($\varrho =1.0001$) moving on their limit cycles for ${\textit {Re}}=200$. The arrow indicates the moving wall. (b) Zoom into (a) with the x and y axes scaled differently. The non-dimensional particle radius $a$ and the periodicity of the orbit is given in the legend in (b).

In addition to the period-one limit cycle, we also find limit cycles of periods four (turquoise), seven (magenta) and ten (black) for the smallest particle size with $a=0.0064$. Note that due to its shape, the period-ten attractor returns only eight times to the Poincaré plane $y=0$. From the total of three full periods visible for the period-ten attractor, the particle returned to the Poincaré plane only during one full period for the ninth return (arrows in figure 20a) and never for the tenth. This effect is caused mainly by the location of the Poinicaré plane (see also figure 21a). Also, particles with the next higher particle size of $a=0.0112$ can be attracted to different limit cycles: in addition to the period-one limit cycle (blue), we find a limit cycle of period four (violet dots).

Characteristic properties of all limit cycles found for ${\textit {Re}}=200$ are collected in table 5. All limit cycles found form near corresponding KAM tori and associated closed streamlines that are indicated by open diamonds. The limit cycles are created because the corresponding KAM tori approach the moving wall sufficiently closely (see table 5). This property increases the probability for particles to be transferred from the chaotic sea to one of the KAM tori. Once a neutrally buoyant particle is caught in a KAM torus sufficiently close to the moving wall, it is attracted to a limit cycle by the interaction with the moving wall as described by Hofmann & Kuhlmann (Reference Hofmann and Kuhlmann2011).

Table 5. Properties of measured trajectories on the attractor for nearly neutrally buoyant particles with $\varrho =1.0001$ as functions of the particle radius $a$ at ${\textit {Re}}=200$. Specified are the type of attractor (P means periodic, QP means quasi-periodic), fundamental frequencies $f_1$(dimensional) and $F_1$ (dimensionless), turnover time $\tau _1=F_1^{-1}$, initial transient time $\bar \tau _I$, asymptotic attraction rate $\bar \sigma _w$, the closest wall-normal distances from the boundaries $\varDelta _{p}$ (the boundary is indicated by the superscript), and the number of samples $N$ used for averages.

The rates of attraction to the limit cycles have been determined as for ${\textit {Re}}=100$. Figure 22 presents the mean attraction rate as a function of the particle size $a$. The attraction rates are generally larger than for ${\textit {Re}}=100$, but follow a similar trend with respect to a variation of the particle size. As for ${\textit {Re}}=100$, the mean attraction rate is approximated by a power law $\bar \sigma _w=ca^b$, with $c=5.75\times 10^4$ and $b=2.63$, obtained by fitting to the data for the two smallest particles and the origin.

Figure 22. Mean attraction rate $\bar \sigma _w$ to the period-one limit cycle as a function of the particle radius for ${\textit {Re}}=200$ and nearly neutrally buoyant particles with $\varrho =1.0001$ and radii $a=0.0064$, $0.0111$, $0.0272$, $0.039$, $0.0494$, $0.0586$, $0.0704$, corresponding to $a_{p} = 0.26$, 0.45, 1.10, 1.58, 2.00, 2.37, 2.85 mm. The solid curve represents a power-law fit $\bar \sigma _w=ca^b$ ($c=5.75\times 10^4$, $b=2.63$) to the data for small $a$.

7.2. Motion of inertial particles

As the particle density becomes larger, inertial forces cooperate with the forces from the wall in attracting the particle to a limit cycle. Therefore, the attraction rate increases. For all heavy particles, we find the period-one attractor near $L_1$, as for ${\textit {Re}}=100$. However, owing to the different fine structure of the KAM template for ${\textit {Re}}=200$, we find other attractors corresponding to the secondary KAM tori. For particles with small inertia, i.e. for $a=0.0069$, $\varrho =1.022$, and $a=0.0120$, $\varrho =1.019$, we find an additional period-twelve limit cycle. An example is shown in blue in figure 23 for $a=0.0069$ and $\varrho =1.022$. The period-twelve attractor forms in the close vicinity of the period-four KAM tori of the flow field. For the heavier particle with $a=0.0130$ and $\varrho =1.061$, the period-twelve attractor is absent. Instead, an attracting period-three limit cycle is found. It cannot be associated directly with a particular KAM structure, but it seems to reflect the shape of the primary KAM tori around $L_1$. The period-three limit cycle is shown in blue in figure 24.

Figure 23. Nine trajectories for $a=0.0069$ ($a_{p}=0.28$ mm) and $\varrho =1.022$ recorded during $t\in [3000,3600]$ s. (a) Three-dimensional view of the period-one (black) and period-twelve (blue) attractors. (b) Poincaré sections on the plane $y=0$ (black, blue) of the particle trajectories shown in (a). Poincaré sections of numerically computed streamlines on the corresponding largest reconstructible KAM tori are shown as light grey dots.

Figure 24. Seventeen trajectories for $a=0.0130$ ($a_{p}=0.525$ mm) and $\varrho =1.061$ recorded during $t\in [400,500]$ s. (a) Three-dimensional view of the period-one (black) and period-three (blue) limit cycles. (b) Poincaré sections on the plane $y=0$ (black, blue) of the particle trajectories shown in (a). Poincaré sections of numerically computed streamlines on the corresponding largest reconstructible KAM tori are shown as light grey dots.

In figure 25, the mean attraction rates for the three heavy particles to their main period-one limit cycle for ${\textit {Re}}=200$ are shown by red dots as functions of $(\varrho -1){\textit {St}}$. Numerical data are provided in table 6. After subtracting the attraction rate $\sigma _w$ due to the wall effect found for the neutrally buoyant particles (solid line in figure 22), the inertial part $\sigma _i$ of the attraction rate is almost a linear function of $(\varrho -1)\,{\textit {St}}$ (solid line in figure 25).

Figure 25. Mean rates of attraction to the period-one limit cycle for inertial particles with $\varrho > 1$ as functions of $(\varrho -1)\,{\textit {St}}$. Shown are the total attraction rates $\bar \sigma$ (red dots, blue square) measured and the inertial attraction rate $\sigma _i$ (black crosses) after eliminating the wall effect. Three particles with different radii $a\in [0.0069, 0.0120, 0.0130]$ and corresponding densities $\varrho \in [1.022, 1.019, 1.061]$ were tested. The solid line represents the linear regression $\bar \sigma _i = c (\varrho -1)\,{\textit {St}}$ with $c = 2.23\times 10^5$. The blue square shows $\bar \sigma$ for a particle with $a=0.0064$, $\varrho =0.0001$.

Table 6. Properties of measured trajectories of inertial particles on their attractors for ${\textit {Re}}=200$. Specified are the type of attractor (P means periodic, QP means quasi-periodic), fundamental frequencies $f_1$(dimensional) and $F_1$(dimensionless), turnover time $\tau _1=F_1^{-1}$, initial transient time $\bar \tau _I$, asymptotic attraction rate $\bar \sigma$, the closest wall-normal distances from the boundaries $\varDelta _{p}$ (the boundary is indicated by the superscript), and the number of samples $N$ used for averages.

Finally, we investigate the behaviour of a light particle with $a=0.0112$, $\varrho =0.940$. As expected, the periodic attractor near $L_1$ for heavy particles has become an unstable limit cycle for light particles, which cannot be observed directly. Obviously, the repulsion is strong enough to prevent a wall-induced stable limit cycle that would exist near $L_1$ in the absence of inertia. Only at a certain distance from $L_1$ can the stabilising wall effect balance the destabilising inertia effect leading to a toroidal equilibrium trajectory QP-1 with period one (figure 26a).

Figure 26. Thirty-seven trajectories for $a=0.0111$ ($a_{p}=0.45$ mm) and $\varrho =0.940$ recorded during $t\in [1000,1180]$ s. (a) Three-dimensional view of the toroidal period-one attractors QP-1 (black) and the stable period-four limit cycles P-4 (blue). (b) Poincaré sections on the plane $y=0$ (black, blue) of the particle trajectories shown in (a). Poincaré sections of numerically computed streamlines of characteristic KAM tori are shown as light grey dots.

Since inertial forces dominate in the bulk, and repulsive forces from the moving wall decay from the wall within a distance of the order of $a$, the quasi-periodic attractor should approach the moving wall up to a distance of the order of $a$. In fact, the minimum distance of the quasi-periodic attractor QP-1 from the moving wall is $\approx 3a$ (table 6). The Poincaré section of a particle trajectory on QP-1 is shown by black dots in figure 26(b). In addition, a period-four attractor is found (blue in figure 26), similarly as for the neutrally buoyant particle with a comparable size (violet in figure 20). Even though this attractor is expected to be quasi-periodic, just like the toroidal attractor of period six for ${\textit {Re}}=100$ for the light particle (blue in figure 17), the measured time-asymptotic trajectories cannot be distinguished from a limit cycle. This behaviour might be caused by a dominating wall effect on the particle that overcomes the expected destabilising inertia effect. This interpretation is supported by the much closer approach to the moving wall of the period-four orbit P-4 as compared to the toroidal attractor QP-1 (table 6). The locus of P-4 in the Poincaré plane is somewhat displaced from the closed streamline $L_4$ of period four.

It is interesting to note that the light particle has almost the same attractors as a heavy particle with the same size and the same magnitude of the density mismatch, but with the direction of the wall motion reversed, i.e. ${\textit {Re}}\to -{\textit {Re}}$. This is demonstrated by the side-by-side comparison in figure 27 of the Poincaré sections for the light particle with $a=0.0111$, $\varrho =0.94$ and ${\textit {Re}}=200$ (on the left) and the ones for the heavy particle with $a=0.0130$ and $\varrho =1.061$ and ${\textit {Re}}=-200$ (reversed wall motion, on the right). This means, in both cases there exists an unstable limit cycle (periodic repeller) surrounded by the observed QP-1 type of attractors (black).

Figure 27. Symmetry of the attractors QP-1 and P-4 with respect to the symmetry $[{\textit {Re}},(\varrho -1)]\to - [{\textit {Re}},(\varrho -1)]$. Shown are Poincaré sections on the plane $y=0$ of particle trajectories attracted to a period-four limit cycle (blue) and a quasi-periodic orbit (black). For $A$ (to the left of the red line), $a=0.0112$, $\varrho =0.94$ and ${\textit {Re}}=200$. For $B$ (to the right of the red line), $a=0.0130$, $\varrho =1.061$ and ${\textit {Re}}=-200$. Poincaré sections of numerically computed KAM tori are shown as light grey dots.

This may appear surprising, because the sign of the density mismatch ($\varrho -1$) determines the character of the limit cycle, being stable or unstable. To better understand the situation, we consider the inertial equation (4.1) in components:

(7.1a)$$\begin{align} \dot{X} &= u -(\varrho-1)\, {\textit{St}} [u\,\partial_x u + v\,\partial_y u + w\,\partial_z u ], \end{align}$$

(7.1b)$$\begin{align}\dot{Y} &= v -(\varrho-1)\, {\textit{St}} [u\,\partial_x v + v\,\partial_y v + w\,\partial_z v + {\textit{Fr}}^{{-}2}], \end{align}$$

(7.1c)$$\begin{align}\dot{Z} &= w -(\varrho-1)\, {\textit{St}} [u\,\partial_x w + v\,\partial_y w + w\,\partial_z w]. \end{align}$$

The inertial equations (7.1) are invariant under a rotation by ${\rm \pi}$ about the $x$ axis, corresponding to a reversal of the Reynolds number (${\textit {Re}}\to -{\textit {Re}}$)

(7.2) \begin{equation} \left(\begin{array}{@{}c@{}} u \\ v \\ w \end{array}\right) (x,y,z) \longrightarrow \left(\begin{array}{@{}c@{}} u \\ -v \\ -w \end{array}\right) (x, -y, -z), \end{equation}

provided that gravity is reversed as well: ${\textit {Fr}}^{-2}\to -{\textit {Fr}}^{-2}$. This is just a change of the coordinate system. The symmetry shown in figure 27 suggests that (7.1) is also invariant under (7.2) (${\textit {Re}}\to -{\textit {Re}}$) combined with $(\varrho -1)\to -(\varrho -1)$, keeping the orientation of the gravity vector (${\textit {Fr}}$) constant, as in the experiment. But obviously, (7.1) is not invariant under $[{\textit {Re}},(\varrho -1)]\to - [{\textit {Re}},(\varrho -1)]$.

The fact that the QP-1 attractors are almost equal and encircle unstable limit cycles leads us to conclude the following. The inertial term $\boldsymbol u\boldsymbol {\cdot }\boldsymbol {\nabla } \boldsymbol u$ in (7.1) is not of critical importance for the stability of the limit cycle, because its components change their signs relative to the components of $\dot {\boldsymbol X}$ upon $[{\textit {Re}},(\varrho -1)]\to - [{\textit {Re}},(\varrho -1)]$. Thus the inertial terms tend to predict a stable limit cycle for $-[{\textit {Re}},(\varrho -1)]$ (right-hand side of figure 27) rather than the observed repeller (surrounded by the QP-1 type of attractor). Only the buoyancy term $(\varrho -1)\,{\textit {St}}\,{\textit {Fr}}^{-2}$ keeps its sign (relative to $\dot Y$) upon $[{\textit {Re}},(\varrho -1)]\to - [{\textit {Re}},(\varrho -1)]$, and thus favours an unstable limit cycle. We conclude that the buoyancy term is of crucial importance for the stability property of the limit cycles emerging from the closed streamline $L_1$ in the present experiment, and must be dominant. Interestingly, this term alone cannot create limit cycles, because it is conservative. But it does so in combination with the inertial term $\boldsymbol u\boldsymbol {\cdot }\boldsymbol {\nabla }\boldsymbol u$. A similar conclusion was arrived at by Wu et al. (Reference Wu, Romanó and Kuhlmann2021) for attractors and repellers in the two-sided lid-driven cavity.