Daniel Odens Mora

Martin Obligado
email: martin.obligado@univ-grenoble-alpes.fr

Alberto Aliseda

Alain Cartellier

The effect of Re λ and Rouse numbers on the settling of inertial droplets in homogeneous isotropic turbulence

We present an experimental study on the settling velocity of dense sub-Kolmogorov particles in active-grid-generated turbulence in a wind tunnel. Using phase Doppler interferometry, we observe that the modifications of the settling velocity of inertial particles, under homogeneous isotropic turbulence and dilute conditions (i.e., small liquid fraction φv ≤ O(10) -5 ), is controlled by the Taylor-Reynolds number Re λ of the carrier flow. Meanwhile, we did not find a strong influence of the ratio between the fluid and gravity accelerations on the particle settling behavior. Remarkably, we find that that the degree of hindering experienced by the particles (i.e., the measured particle settling velocity is smaller in magnitude than its respective one in still fluid conditions) increases with Re λ . This observation is contrary to previous works at intermediate values of Re λ that report the opposite effect: settling enhancement. Nonetheless, our trend is observed at all particle sizes investigated, and when previous experimental data is included into the analysis, our data suggest that the particle settling behavior may be non-monotonic with Re λ : inducing enhancement at moderate values of Re λ , and at promoting hindering at higher values of Re λ . Moreover, at the highest Re λ studied, the settling enhancement regime ceases to exist. Finally, we find that the difference between the measured particle settling velocity (Vp) and the particle terminal velocity in still fluid conditions (VT ), normalized by the carrier phase rms fluctuations, (Vp -VT )/u scales linearly with the Rouse number Ro = VT /u (i.e., the ratio between the particles settling velocity and the fluid rms fluctuations). However, such behavior (Vp -VT )/u ∼ -Ro appears only to be valid for moderate values of the Rouse number.

I. INTRODUCTION

Turbulent particle-laden flows have a widespread presence in industrial and natural processes, e.g., coatings, spray combustion, pollen dispersion, planetesimal growth, and clouds' formation [START_REF] Aliseda | Effect of buoyancy on the dynamics of a turbulent boundary layer laden with microbubbles[END_REF][START_REF] Vaillancourt | Review of particle-turbulence interactions and consequences for cloud physics[END_REF][START_REF] Sumbekova | Preferential concentration of inertial sub-Kolmogorov particles: The roles of mass loading of particles, Stokes numbers, and Reynolds numbers[END_REF]. Among the several consequences of particle-turbulence interactions, preferential concentration and particle settling velocity modification have received considerable attention in the last decades [START_REF] Balachandar | Turbulent dispersed multiphase flow[END_REF][START_REF] Aliseda | Preferential concentration and rise velocity reduction of bubbles immersed in a homogeneous and isotropic turbulent flow[END_REF][START_REF] Elghobashi | Direct numerical simulation of turbulent flows laden with droplets or bubbles[END_REF]. Preferential concentration describes the tendency of particles to accumulate in space, forming clusters and voids. In contrast, particle settling modification accounts for the enhanced (resp. hindering) of the particles' settling velocity in the direction of a body force acting on them, for instance, gravity. Several theoretical approaches have suggested mechanisms that relate the topology of the background turbulent flow to the observed preferential concentration, and settling modification. Regarding preferential concentration, classical contributions have suggested that sub-Kolmogorov particles, which have a characteristic scale smaller than the Kolmogorov scale η, tend to concentrate in regions of high strain and low vorticity [START_REF] Balachandar | Turbulent dispersed multiphase flow[END_REF][START_REF] Maxey | The gravitational settling of aerosol particles in homogeneous turbulence and random flow fields[END_REF]. However, recent research has proposed that this classical picture does not include the multiscale nature of turbulence. Under this framework, some studies have proposed that particles accumulate at the different (coarse-grained) scales of high strain and low vorticity [START_REF] Bragg | Mechanisms for the clustering of inertial particles in the inertial range of isotropic turbulence[END_REF]. Alternatively, other studies have shown evidence that particles mimic the clustering of the carrier phase zero acceleration points [START_REF] Coleman | A unified sweep-stick mechanism to explain particle clustering in two-and three-dimensional homogeneous, isotropic turbulence[END_REF][START_REF] Obligado | Preferential concentration of heavy particles in turbulence[END_REF], which exhibits a self-similar behavior [START_REF] Goto | Self-similar clustering of inertial particles and zero-acceleration points in fully developed twodimensional turbulence[END_REF]. Regarding particles' settling modification, previous works have suggested that the modification of particles' settling velocity is due to centrifugal effects, a mechanism called preferential sweeping: inertial particles are expelled of eddies but fast-track into downward eddies, thereby enhancing their settling speed [START_REF] Maxey | The gravitational settling of aerosol particles in homogeneous turbulence and random flow fields[END_REF][START_REF] Ghosh | How turbulence enhances coalescence of settling particles with applications to rain in clouds[END_REF][START_REF] Wang | Settling velocity and concentration distribution of heavy particles in homogeneous isotropic turbulence[END_REF]. The opposite effect has also been observed: particles' settling velocity is reduced instead of being enhanced [START_REF] Nielsen | Turbulence effects on the settling of suspended particles[END_REF]. Some research has conjectured that this phenomenon occurs when particles preferentially sample the upward regions of the flow [START_REF] Nielsen | Turbulence effects on the settling of suspended particles[END_REF][START_REF] Nielsen | On the motion of suspended sand particles[END_REF][START_REF] Kawanisi | Turbulent effects on the settling velocity of suspended sediment[END_REF]. Recent research has also attempted to incorporate ideas from the multiscale nature of turbulence to understand the observed particle settling behavior. Some works argue that the centrifugal effect (and the enhanced settling) depends on the relationship between the particle inertia, and all of the carrier phase length scales, i.e., particles of different inertia are affected by different length scales of the turbulent flow [START_REF] Tom | Multiscale preferential sweeping of particles settling in turbulence[END_REF]. Considering the complex interactions between the turbulent carrier phase, and the discrete particle phase, most studies treat preferential concentration and particle settling independently. Recent works [START_REF] Sumbekova | Enhancement versus hindering: impact of the turbulence characteristics on the settling behavior of heavy sub-Kolmogorov particles in a turbulent flow[END_REF][START_REF] Sumbekova | Clustering and settling of inertial particles in turbulence[END_REF][START_REF] Monchaux | Settling velocity and preferential concentration of heavy particles under two-way coupling effects in homogeneous turbulence[END_REF][START_REF] Baker | Coherent clusters of inertial particles in homogeneous turbulence[END_REF][START_REF] Huck | The role of collective effects on settling velocity enhancement for inertial particles in turbulence[END_REF][START_REF] Petersen | Experimental study of inertial particles clustering and settling in homogeneous turbulence[END_REF], however, have aimed to relate both phenomena. For instance, some studies have reported that the enhanced particle settling is due to the increased local concentration (higher liquid fractions φ v ) [START_REF] Huck | The role of collective effects on settling velocity enhancement for inertial particles in turbulence[END_REF][START_REF] Aliseda | Effect of preferential concentration on the settling velocity of heavy particles in homogeneous isotropic turbulence[END_REF]. In other words, particles in high density regions settle (on average) faster with respect to particles in low density regions [START_REF] Baker | Coherent clusters of inertial particles in homogeneous turbulence[END_REF][START_REF] Huck | The role of collective effects on settling velocity enhancement for inertial particles in turbulence[END_REF][START_REF] Petersen | Experimental study of inertial particles clustering and settling in homogeneous turbulence[END_REF][START_REF] Aliseda | Effect of preferential concentration on the settling velocity of heavy particles in homogeneous isotropic turbulence[END_REF]. Numerical and experimental studies do exhibit similar trends on the behavior of preferential concentration and settling velocity with global flow parameters, such as the Taylor-based Reynolds number Re λ = uλ/ν and the Stokes number St = τ p /τ η , where u stands for the RMS value fluid fluctuating velocity u , λ corresponds to the Taylor microscale, ν is the kinematic viscosity, τ p and τ η stand for the particle relaxation and the Kolmogorov timescales, respectively. Nevertheless, quantitative consensus between experiments and simulations has yet to be reached [START_REF] Tom | Multiscale preferential sweeping of particles settling in turbulence[END_REF][START_REF] Monchaux | Settling velocity and preferential concentration of heavy particles under two-way coupling effects in homogeneous turbulence[END_REF][START_REF] Petersen | Experimental study of inertial particles clustering and settling in homogeneous turbulence[END_REF][START_REF] Bosse | Small particles in homogeneous turbulence: settling velocity enhancement by two-way coupling[END_REF][START_REF] Wittemeier | Explanation of differences in experimental and computational results for the preferential concentration of inertial particles[END_REF][START_REF] Ireland | The effect of reynolds number on inertial particle dynamics in isotropic turbulence. part 1. simulations without gravitational effects[END_REF].

Moreover, the origin of these discrepancies could be multi-fold [START_REF] Monchaux | Settling velocity and preferential concentration of heavy particles under two-way coupling effects in homogeneous turbulence[END_REF][START_REF] Bosse | Small particles in homogeneous turbulence: settling velocity enhancement by two-way coupling[END_REF][START_REF] Rosa | Settling velocity of small inertial particles in homogeneous isotropic turbulence from high-resolution dns[END_REF][START_REF] Good | Settling regimes of inertial particles in isotropic turbulence[END_REF]. For example, the numerical and experimental study of Good et al. [START_REF] Good | Settling regimes of inertial particles in isotropic turbulence[END_REF], at Re λ 200, and liquid fractions φ v = 10 -6 has shown that particle settling hindering effects cannot be captured in DNS simulations that only consider linear drag. Conversely, DNS simulations of Rosa et al. [START_REF] Rosa | Settling velocity of small inertial particles in homogeneous isotropic turbulence from high-resolution dns[END_REF] report no variation in the particle settling velocity with the drag model, i.e., their results were insensitive to the choice of the drag law used (e.g., linear, non-linear). Another source of discrepancy may stem from the mechanical coupling between particle phase and the turbulent carrier phase interaction, ignored in most DNS studies. The need to include these inter-phase mechanical coupling effects was recognized early by Aliseda et al. [START_REF] Aliseda | Effect of preferential concentration on the settling velocity of heavy particles in homogeneous isotropic turbulence[END_REF]. They suggested modifying the carrier phase pressure field to account for the flow regions with high particle density. Most DNS studies ignore this coupling and assume that the particles do not affect the carrier phase, a regime known as 'one-way' coupling. Interestingly, Bosse et al. [START_REF] Bosse | Small particles in homogeneous turbulence: settling velocity enhancement by two-way coupling[END_REF], and Monchaux et al. [START_REF] Monchaux | Settling velocity and preferential concentration of heavy particles under two-way coupling effects in homogeneous turbulence[END_REF] observed a larger particle settling velocity when the mechanical coupling between the phases, a regime known as 'two-way' coupling, was included. Their simulations, however, were run at rather small Reynolds numbers (Re λ ≈ 40). A recent numerical study by Rosa et al. [START_REF] Rosa | Effects of turbulence modulation and gravity on particle collision statistics[END_REF] has arrived at similar conclusions at higher Reynolds numbers (Re λ ≈ 100). These findings hint that including two-way coupling interactions may be necessary to describe accurately the physics of the particles' settling modification.

In this work, we report experimental measurements of a polydisperse population of inertial particles settling under homogeneous isotropic turbulence downstream of an active grid. Considering that the impact of two-way coupling on the settling modification is yet to be dispelled, we run experiments at the smallest liquid fractions attainable, as previous research in our facility has shown that at liquid fractions φ v = O(10 -5 ) the particles' presence may affect the turbulence characteristics [START_REF] Mora | Experimental estimation of turbulence modification by inertial particles at moderate re λ[END_REF] adding an extra layer of complexity. Thus, under these experimental conditions at Re λ ∈ [230 -650] and φ v < O(10 -5 ), our experimental results suggest that the Taylor-Reynolds number (Re λ ) is the leading contributor to the particles' behavior, influencing all the measurable regimes. For instance, the degree of hindering (i.e., measured particle settling velocity smaller than its respective value for still fluid conditions) increases with the value of Re λ . Moreover, when we plot the settling velocity against the Rouse number, which is the ratio between the particles settling velocity and the fluid rms fluctuations Ro = V T /u, we find that the transition point between hindering and enhancement (i.e., particles falling faster than in a quiescent fluid) regimes shifts to smaller Rouse numbers at increasing values of Re λ .

II. METHODS

A. Experimental setup details

The experiments were performed in a close-circuit wind tunnel 'Lespinard ' in the Laboratoire des Écoulements Géophysiques et Industriels (LEGI) at Université Grenoble Alpes. This facility has been regularly employed to study particle clustering under decaying homogeneous isotropic turbulence (HIT) conditions [START_REF] Sumbekova | Preferential concentration of inertial sub-Kolmogorov particles: The roles of mass loading of particles, Stokes numbers, and Reynolds numbers[END_REF][START_REF] Monchaux | Preferential concentration of heavy particles: A Voronoï analysis[END_REF][START_REF] Obligado | Experimental detection of superclusters of water droplets in homogeneous isotropic turbulence[END_REF][START_REF] Mora | Estimating the integral length scale on turbulent flows from the zero crossings of the longitudinal velocity fluctuation[END_REF]. A sketch of our experimental setup is depicted in figure 1. We briefly explain our experimental setup, and the experimental methods used. First, in the experiments, the turbulent flow was generated utilizing an active grid [START_REF] Mydlarski | A turbulent quarter century of active grids: from makita (1991) to the present[END_REF] in triple random mode. We set our experiment following best practice guidelines. First, the active grid mesh size M is roughly a tenth of the tunnel width. As stated on previous studies on similar active grids and protocols (see [START_REF] Mydlarski | A turbulent quarter century of active grids: from makita (1991) to the present[END_REF] and references therein), within this conditions the mean velocity profile in both the vertical y and the spanwise z directions is expected to be fairly homogeneous. This ensures that at our measuring station, our Eulerian measurement is surrounded by a homogeneous box of size comparable to the integral lengthscale L. As we focus on the homogeneous region of the flow, and in the dilute limit, we expect that particles will not have an impact of the background turbulent flow, FIG. 1: Sketch of our experimental setup (not to scale). The wind-tunnel cross-section is 75x75 cm 2 . Its center line is labeled as χ in the figure. The emitter and receiver components of the PDI are on the same horizontal plane. However, the receiver is positioned at 30 degrees (see α in the figure) with respect to the emitter to maximize the capture of the water droplets refraction [START_REF] Sumbekova | Enhancement versus hindering: impact of the turbulence characteristics on the settling behavior of heavy sub-Kolmogorov particles in a turbulent flow[END_REF][START_REF] Bachalo | Phase/doppler spray analyzer for simultaneous measurements of drop size and velocity distributions[END_REF]. Two holes of approximately 10 cm were carved onto the walls to counteract the water accumulation on them. The measuring station was located at the position labeled as M1 on the wind-tunnel center line, and 3 meters downstream of the droplets injection. TABLE I: Parameters of the unladen flow, measured by means of hot wire anemometry, at the measuring station 3 m downstream of the grid. The parameters are defined as: u = u 1/2 , the turbulence energy dissipation rate ε = 15νu 2 /λ 2 , η = ν 3 /ε 1/4 , and L is the integral length scale computed following [START_REF] Puga | Normalized dissipation rate in a moderate taylor reynolds number flow[END_REF]. The kinematic viscosity of the air is taken as ν = 1.5 × 10 -5 (m 2 s -1 ). Finally, γ = ε 3/4 /(gν 1/4 ) is the acceleration ratio.

Re λ U∞ u/U∞ L ε λ η γ [ms -1 ] [cm] [m 2 s -3 ] [cm] [
particularly rendering negligible any effect that mean shear could have on the particles' behavior. Next, we measured the turbulent unladen velocity through hot-wire anemometry, and computed the turbulent parameters using standard methods and assumptions (e.g., Taylor hypothesis). The most relevant unladen flow parameters are summarized in table I. For detailed explanations on the turbulence characterization, see [START_REF] Mora | Energy cascades in active-gridgenerated turbulent flows[END_REF] and references therein. Figure 3a shows the unidimensional power spectral density spectra at the measuring station (see label 'M1' in figure 1). Second, downstream of the 'grid' section (see figure 1) a rack of 18, or 36 spray nozzles -at smaller concentrations fewer injectors were used, see figure 2a-injected inertial water droplets with diameters D p between 20 and 300 microns , i.e, D p ∈ [20 -300] µm. This polydispersity was characterized by a phase Doppler interferometry [START_REF] Bachalo | Phase/doppler spray analyzer for simultaneous measurements of drop size and velocity distributions[END_REF]. The respective droplet distribution is close to a log-normal distribution (see fit in figure 2b). Droplets were considered as spherical particles as their Weber number parameter was, for most droplets, below unity (see in Sumbekova [START_REF] Sumbekova | Concentration préférentielle de particules inertielles : la structure et la dynamique de clusters[END_REF] section 6.3). However, to have an adequate statistical convergence in our experimental realizations, we binned the measured droplets between 7.5 and 155 µm (see figure 2c). Third, the measuring station was placed 3m downstream of the droplet injection (see figure 1). The measuring volume lay at the center-line of the wind-tunnel. We used a phase Doppler interferometry apparatus (PDI), model Artium-PDI-200, which can measure the settling velocity and the particles' diameter simultaneously [START_REF] Bachalo | Phase/doppler spray analyzer for simultaneous measurements of drop size and velocity distributions[END_REF][START_REF] Schwarzkopf | Multiphase flows with droplets and particles[END_REF]. The PDI setup has two components: the receiver and the laser emitter. The laser emitter was placed perpendicular to the gas flow. The receiver (see figure 1) was on the same horizontal plane but rotated 30 degrees to ensure adequate capture of spherical water droplets in the airflow.

The PDI setup has two different laser beams (before splitting to form the fringes) that allow us to measure the two velocity components of the particles captured, and their respective diameter. Although it is desirable to have for each particle these three characteristics altogether (horizontal/vertical component, diameter), a compromise is required as coincident detection does significantly reduce the validation rate, i.e., the ratio between the number of FIG. 2: a) Injector rack sketch. For the lowest volume fractions half of the injectors (filled markers) were utilized. b) Spray characterization coming from Spraytec data from Sumbekova [START_REF] Sumbekova | Concentration préférentielle de particules inertielles : la structure et la dynamique de clusters[END_REF]. c) Droplet histogram for all experimental datasets reported in this work.

particle events deemed as valid (a credible detection) over the total number of particle events. Hence, we set the PDI in non-coincident mode: the vertical velocity component and the particles' diameter were recorded by the largest wavelength beam (green laser in our case), whereas the streamwise velocity was recorded by the shortest wavelength beam (blue laser in our case). We made this choice to improve the validation rate (by a factor five or even larger). To quantify the effects of the carrier phase turbulence on the particles, we tried to match as close as possible, the particles' mean horizontal velocity ( U p ) to respective unladen mean velocity (U ∞ ), i.e., in our experiments U p ≈ U ∞ . The validation rate reported by the PDI software was above 70% or higher in all experimental realizations. The acquisition rate (particles per second) varied between 400 and 3000 Hz depending on the liquid fraction and bulk velocity, i.e., a higher concentration at a higher bulk velocity yielded a higher acquisition rate. For each experimental condition, we collected data from 5 × 10 5 particles. The vertical (resp. horizontal) velocity component had a resolution of 0.010 m/s (resp. 0.04 m/s) for all experimental conditions. Finally, it is worth nothing that the choice of the measurement position (3m downstream of the injection) was based on previous studies in the same facility. These studies report that at 3m downstream of the injection the particle velocity statistics are almost Gaussian (see in the Appendix A, figures 11a-11b), meaning that the particles' velocities have relaxed to the background fluid fluctuations. Hence, our measurements are able to gauge the effects of the background turbulence on the particle behavior. On the other hand, and as mentioned in the introduction, a recent study in our facility reports that the carrier phase turbulence may change due to the particles presence [START_REF] Mora | Experimental estimation of turbulence modification by inertial particles at moderate re λ[END_REF]. To curb the influence of such turbulence modulation due to the particles' presence, we ran the experiments with the smallest liquid fractions attainable in our facility [START_REF] Sumbekova | Concentration préférentielle de particules inertielles : la structure et la dynamique de clusters[END_REF] (i.e., φ v = [10 -6 , 10 -5 ]). We expect that, at these liquid fractions, the turbulence modulation is strongly reduced [START_REF] Elghobashi | On predicting particle-laden turbulent flows[END_REF]. Altogether, these previous considerations led to the exploration of a parameter space aiming to small concentrations and large Reynolds numbers, exploiting the limits of the experimental facility (see figure 3b).

B. Velocity measurements and angle correction

There will always be a small deviation angle between the PDI axes, and the wind tunnel coordinate system (see figure 4 ) impacting the vertical velocity measurements. Considering that the particles' horizontal velocity is at least an order of magnitude larger than the vertical one, the horizontal component's projection onto the vertical component in the PDI coordinate system will cause an error in the vertical velocity measurements due to such optical misalignment. We address this problem by subtracting the projected mean droplet horizontal ( U p ) velocity from the vertical velocity in the PDI coordinate system (V p ). We define this angle-corrected velocity as:

V c j = V j -U p sin(β) = V j -V β , V β = U p sin(β) (1)
To calculate V β , we used a different configuration in the wind tunnel. We used a single particle injector, positioned at the grid plane and set the grid completely open, thus minimizing turbulence. We circulated air at 3.5 ms -1 and injected olive oil droplets, with a very narrow distribution of sizes, centered around a mean diameter ≈ 8µm (measured with the PDI). Olive oil droplets were convected downstream of grid by the bulk flow, and we recorded droplets' velocities at the PDI measuring volume (see 1). The velocity statistics collected for 2000 droplets in the PDI frame FIG. 3: a) Example of an unidimensional power spectral density F 11 [START_REF] Pope | Turbulent Flows[END_REF] from one of our hot-wire records at measuring station M1 (see figure 1). b) Parameter space for the experiments conducted. The global liquid fraction was estimated as φ v ≈ Q W /Q A , where Q W , and Q A are the volumetric flux of water, and of air, respectively.

of reference (see figure 4) were U p = (-3.52 ± 0.02)m/s, σ Up = (0.11 ± 0.02)m/s, and V p = (-0.09 ± 0.005)m/s, σ Vp = (0.11 ± 0.005)m/s. Then, we estimated the expected settling velocity of droplets via the Schiller and Nauman [START_REF] Clift | Bubbles, drops, and particles[END_REF] drag coefficient semiempirical formula. Our experimental values, the Schiller and Nauman formula and our resolution yielded a correction angle β = -1.5 • ± 0.3 • (the angle uncertainty is fruit of the PDI velocity resolution), which we used for all experimental realizations. This optical alignment correction for all particles is justified under our turbulent conditions because this correction is smaller than the standard deviation of the carrier phase velocity, i.e., V β /u = sin(β) × U p /u ≈ sin(1.5 We will consider the particles' vertical velocity to be positive towards gravity in agreement with the PDI coordinate system (figure 4). We binned our datasets by the droplet diameters (see figure 2c). The amount of droplets per bin follows the droplet distribution PDF (figure 2b . These bins had a size of 5µm (an operation represented by | D ) and their centers spanned D p ∈ [7.5 -155]µm. This latter consideration is due to the injector droplet size distribution and has some consequences: smaller droplets are less common (see figure 2b for an example of size distribution), and therefore, our first bins have a larger variation. We, nevertheless, collected enough samples to have meaningful statistics. For instance, in figure 2c, we show that each bin had at least 1000 particles. The polydispersity of our droplet injection and our active grid turbulence characteristics (e.g., higher values of ε, see table I) allows to explore a wide range of particle Stokes numbers for the different experimental conditions (see figure 5). Interestingly, for all particle sizes, the particles' velocities decrease with increasing Re λ (i.e., slower settling in our convention). Moreover, our raw velocity measurements show that for a fixed experimental condition particles with larger Stokes (larger diameters in our case) fall -on average-faster (see figure 5), as expected. However, there are two sources of uncertainty in our results for the smallest droplets: first, the accuracy of the optical alignment, and second, the vertical resolution used (0.010 m/s). This vertical resolution results from a trade-off between an adequate acquisition rate and the statistical convergence needed.

In this work, we refrained from analyzing the preferential concentration from 1D particle records, for instance via 1D Voronoï tessellations [START_REF] Ferenc | On the size distribution of poisson voronoi cells[END_REF], as a previous work from the group suggests that these statistics in our experimental setup may present biases that could lead to misleading conclusions see [START_REF] Mora | Characterizing 1d inertial particle clustering[END_REF]. Nonetheless, the presence of clusters (higher density regions with respect to the average concentration) and voids (lower density regions with respect to the average concentration) may be of relevance in this problem. Particularly considering the polydispersity of injected droplets. For instance, while particles belonging to clusters and voids tend to have different settling velocities than the particles not belonging to any of them [START_REF] Huck | The role of collective effects on settling velocity enhancement for inertial particles in turbulence[END_REF], the droplet size distribution within these structures can be significantly different from the size distribution at the droplet injection station. While we cannot measure this phenomenon with our experimental setup, it may play an important role and further research is needed. In our experiments, to curb the influence of clusters and voids, we deliberately ran experiments at the lowest of φ v attainable, aiming to reduce the influence of these collective effects in our measurements.

B. Settling parameters, and non-dimensional numbers

The turbulent carrier phase is usually characterized by the Taylor Reynolds number Re λ = uλ/ν. There is still an open debate (see [START_REF] Sumbekova | Concentration préférentielle de particules inertielles : la structure et la dynamique de clusters[END_REF] and references therein) about which parameters are adequate to describe the dynamics of the discrete phase. For instance, in the literature, the impact of the Stokes, Rouse, Reynolds and Froude numbers (defined below through equations 2 to 6) on the settling velocity have indeed been reported but there is no consensus on the governing scaling laws. Our aim is to disentangle (within the limitations of our experimental setup) the independent influence of these parameters. For example, the role of the Stokes number is well known to control preferential sweeping (or fast track, see Maxey [START_REF] Maxey | The gravitational settling of aerosol particles in homogeneous turbulence and random flow fields[END_REF]). On the other hand, for droplets with a high settling velocity (high Rouse number), the settling speed is expected to be modified by loitering (Nielsen[14]) and, hence, it should be controlled by the ratio of fluid to gravity accelerations (a Froude number). In fact, a global non-dimensional analysis (detailed in section 6.5 of [START_REF] Sumbekova | Concentration préférentielle de particules inertielles : la structure et la dynamique de clusters[END_REF]) yields that the non-dimensional settling velocity is expected to depend on five independent parameters: the local volume fraction, the particle to fluid density ratio, the turbulent Reynolds number, the Stokes number, and the Rouse number.

The Froude number could be used in lieu of the Stokes or of the Rouse number as there is an exact relationship between F r, Re, St and Ro (see below equations 5 and 6). In our experiments, we note that the particle to fluid density ratio is very large so that we can consider that the system is within its asymptotic regime with respect to that parameter. As we kept the volume fraction very low (ideally low enough to avoid turbulence modulation), it can be argued (subject to a posteriori verification) that the system is also in an asymptotic regime with respect to the volume fraction. Therefore, we are left with three 'independent' parameters: Re, St and Ro, any of which can be exchanged by the Froude number. Then, inspection of definitions 5 and 6 suggests that relationships can be constructed for any combination of Rouse, Froude, Reynolds, or Stokes numbers. However, among the possible experimental parameters, below we present results using the combination Ro, Re λ and St. Although we do not argue that this parameter space combination is complete, it is supported by our experiments as well as the experiments we found in the literature. Indeed, for ours and previous experimental data plotted under this set of parameters, a key observation is that the settling velocity at large Rouse number is controlled to first order by the fluid velocity fluctuations (represented by the Reynolds number) and not by fluctuations in the fluid acceleration, as we may have expected in the loitering scenario.

Considering the previous discussion, we proceed to define the most common non-dimensional numbers used to analyze the particles settling velocity. Classical numerical and experimental studies [START_REF] Wang | Settling velocity and concentration distribution of heavy particles in homogeneous isotropic turbulence[END_REF][START_REF] Aliseda | Effect of preferential concentration on the settling velocity of heavy particles in homogeneous isotropic turbulence[END_REF] plot the particles settling velocity against the Stokes number St = τ p /τ η (see figure 5); changes in the turbulence dissipation lead to changes in the Stokes number (figure 5).Other non-dimensional parameters of interest involve the ratio between the particle terminal speed (V T ) and the background turbulence rms fluctuation, known as the Rouse number, Ro = V T /u [START_REF] Monchaux | Settling velocity and preferential concentration of heavy particles under two-way coupling effects in homogeneous turbulence[END_REF][START_REF] Sumbekova | Concentration préférentielle de particules inertielles : la structure et la dynamique de clusters[END_REF] (some authors also refer to this non-dimensional number as the settling parameter Sv [START_REF] Petersen | Experimental study of inertial particles clustering and settling in homogeneous turbulence[END_REF][START_REF] Good | Settling regimes of inertial particles in isotropic turbulence[END_REF]). Algebraic manipulations allow combining both Rouse and Stokes numbers as follows:

St = τ p τ η → St = τ p τ η u u g g → St = Ro u τ η g . (2)
The particle relaxation time includes the non-linear drag from Schiller and Nauman [START_REF] Clift | Bubbles, drops, and particles[END_REF],

τ p = ρ p D 2 p 18µ f (1 + 0.15Re 0.687 p ) , V T = τ p g, (3)
where g stands for earth's gravitational acceleration, µ f is the fluid viscosity, Re p is the particle Reynolds number, and ρ p the particles density (water).

In addition to St and Ro, some research suggest that the ratio between the turbulent acceleration (η/τ 2 η ) and gravity may play a role on the results. Some authors refer to this ratio as γ = η/(gτ 2 η ) [START_REF] Nielsen | Turbulence effects on the settling of suspended particles[END_REF][START_REF] Huck | The role of collective effects on settling velocity enhancement for inertial particles in turbulence[END_REF][START_REF] Good | Settling regimes of inertial particles in isotropic turbulence[END_REF], while others refer to it as a Froude number [START_REF] Tom | Multiscale preferential sweeping of particles settling in turbulence[END_REF][START_REF] Bec | Gravity-driven enhancement of heavy particle clustering in turbulent flow[END_REF] (F r). In this work, we will follow the former notation. From equations (2 -4), and taking into account that λ = √ 15τ η u (small scale isotropy), one gets;

γ = ε 3/4 gν 1/4 = η τ 2 η g , (4
)

St = γ RoRe 1/2 λ 15 1/4 , (5)
Ro = 15 1/4 St γRe 1/2 λ . (6)
Moreover, combinations of these parameters such as RoSt (involving the Rouse and Stokes numbers) have recently received some attention, as they appear to give a better collapse of the data [START_REF] Ghosh | How turbulence enhances coalescence of settling particles with applications to rain in clouds[END_REF][START_REF] Petersen | Experimental study of inertial particles clustering and settling in homogeneous turbulence[END_REF][START_REF] Good | Settling regimes of inertial particles in isotropic turbulence[END_REF]. for instance, for the RoSt, one gets from equations 5 and 6 that:

RoSt = V T u τ p τ η ∼ V T τ p λ , (7)
which seems to take into account the influence of the background turbulence on the particle settling velocity; the ratio between the particle stopping distance to the Taylor microscale λ, which scales with the average distance between velocity stagnation points [START_REF] Mora | Energy cascades in active-gridgenerated turbulent flows[END_REF][START_REF] Liepmann | Counting Methods and Equipment for Mean-Value Measurements in Turbulence Research[END_REF][START_REF] Sreenivasan | Zero-crossings in turbulent signals[END_REF][START_REF] Mazellier | The turbulence dissipation constant is not universal because of its universal dependence on large-scale flow topology[END_REF][START_REF] Vassilicos | Dissipation in turbulent flows[END_REF].

In our experiments, we cannot change the magnitude of the acceleration of gravity (g) or the magnitude of the air kinematic viscosity (ν). As a result, we cannot easily disentangle or individually vary, Ro, St, and γ. Actually, we can only increase the turbulence dissipation ε (and the absolute magnitude of the turbulence RMS fluctuations) by increasing the inlet velocity U ∞ . These constraints yield similar functional behaviors for γ, and Re λ . To overcome these restrictions and acquire a broader understanding of the physical phenomena at play, we compare our results with other experimental datasets taken from different experimental studies.

C. Normalized settling velocity

To quantify the degree of settling enhancement, we compute the velocity difference between the particle settling velocity and its terminal speed ∆V = V | D -V T -V β , where V β includes the misalignment effects. ∆V is usually normalized by the carrier phase fluctuations u, or by the particle terminal speed V T [START_REF] Balachandar | Turbulent dispersed multiphase flow[END_REF][START_REF] Wang | Settling velocity and concentration distribution of heavy particles in homogeneous isotropic turbulence[END_REF][START_REF] Sumbekova | Enhancement versus hindering: impact of the turbulence characteristics on the settling behavior of heavy sub-Kolmogorov particles in a turbulent flow[END_REF][START_REF] Petersen | Experimental study of inertial particles clustering and settling in homogeneous turbulence[END_REF][START_REF] Aliseda | Effect of preferential concentration on the settling velocity of heavy particles in homogeneous isotropic turbulence[END_REF][START_REF] Rosa | Settling velocity of small inertial particles in homogeneous isotropic turbulence from high-resolution dns[END_REF][START_REF] Bec | Gravity-driven enhancement of heavy particle clustering in turbulent flow[END_REF]. Interestingly, previous experiments [START_REF] Sumbekova | Enhancement versus hindering: impact of the turbulence characteristics on the settling behavior of heavy sub-Kolmogorov particles in a turbulent flow[END_REF][START_REF] Good | Settling regimes of inertial particles in isotropic turbulence[END_REF], as well as ours, reveal that the particle velocity is hindered (slowed down with respect to the still fluid terminal velocity) as the Re λ increases above a certain threshold (see figure 6). Other experiments, e.g., Akutina et al. [START_REF] Akutina | Experimental evidence of settling retardation in a turbulence column[END_REF] have also reported hindering for particles falling inside a turbulent column. Although particles with small Rouse and Stokes numbers have settling velocities (magnitudes) that depend strongly on the liquid fraction φ v and Re λ , we observe that for Ro > O(0.1) (after the peak of maximum settling enhancement) the normalized particle settling (∆V /u) seems to have a quasi-linear behavior (see figure 6). To the best of the authors' knowledge, this regime is not predicted by any available analytical models. We also observe that hindering is present at large St number when ∆V /u is plotted against the Stokes number. The latter observations imply each other due to the relationship between Rouse and Stokes numbers (c.f. equation 5).

We note that our data exhibits hindering effects at very small St, and Rouse numbers, in agreement with findings in other experimental facilities, e.g., experiments in grid tanks [START_REF] Jacobs | Flow scales of influence on the settling velocities of particles with varying characteristics[END_REF] and in a turbulence box [START_REF] Petersen | Experimental study of inertial particles clustering and settling in homogeneous turbulence[END_REF]. However, we must note that these conclusions require further research given the difficulty of recovering the 'tracer' behavior in similar experimental measurements, i.e., a particle that almost perfectly follows a fluid parcel. To recover this behavior using laser interferometry (e.g. PDI) and imaging (e.g. PIV, PTV), it is required that the optical alignment is very accurate so that the absolute zero is adequately set. Besides proper alignment, we also need two extra elements: very dilute conditions φ v → 0, and, in our specific case, very small particles St → 0. Thus, it is not surprising that most experiments have reported values of ∆V = 0 for St → 0 [START_REF] Sumbekova | Enhancement versus hindering: impact of the turbulence characteristics on the settling behavior of heavy sub-Kolmogorov particles in a turbulent flow[END_REF][START_REF] Good | Settling regimes of inertial particles in isotropic turbulence[END_REF][START_REF] Good | Intermittency and inertial particle entrainment at a turbulent interface: the effect of the large-scale eddies[END_REF]. It is worth noting that we included some previous experimental data at higher liquid fractions than our experiments φ v > 1 × 10 -5 from the experiment of Aliseda et al. [START_REF] Aliseda | Effect of preferential concentration on the settling velocity of heavy particles in homogeneous isotropic turbulence[END_REF]. Although at these higher concentrations the mechanical coupling between phases (two-way coupling) may start to play a role into the settling velocity, all trends with respect to the Rouse and Reynolds numbers seem to be consistent with our results on the diluted regime. As discussed by Aliseda et al. [START_REF] Aliseda | Effect of preferential concentration on the settling velocity of heavy particles in homogeneous isotropic turbulence[END_REF], for those experimental datasets, the higher concentration seems to induce an upwards shift in the settling velocity . We will revisit the effects of concentration in our results in the next section IV B. Moreover, our measurement resolution could also have an impact on the measurements taken in the low St regime. These resolution limitations can be clearly observed when the velocity ∆V is normalized against V T (in the Appendix C 1 see figures 12a, and 12b and the large error bars for small Rouse). We also note that these conclusions could be biased by a condition that may exist due to the spatial domain where the experiments take place (confinement effects): weak recirculation currents that perturb the settling dynamics of the particles. These perturbations could be of the order of the settling velocity for small inertial particles biasing the results measured. These biases imply that the tracer behavior may not be recovered ∆V /V T = 0 for St → 0, and therefore, measuring the true values of ∆V /V T for Ro 1 or St 1 is not straightforward (see figures 12a and 12b in Appendix C 1).

For instance, Good et al. [START_REF] Good | Intermittency and inertial particle entrainment at a turbulent interface: the effect of the large-scale eddies[END_REF] reports ∆V /V T → O(100) for Ro 1 in wind tunnel experiments, but in a following publication, Good et al. [START_REF] Good | Settling regimes of inertial particles in isotropic turbulence[END_REF] suggest their previous experimental observation (i.e. ∆V /V T ≥ O(10) for Ro 1) was due to a weak mean flow. Likewise, Akutina et al. [START_REF] Akutina | Experimental evidence of settling retardation in a turbulence column[END_REF] reports a similar phenomenon in grid-tank experiments: "The intensity of these mean fluid motions can be of the order of the particle settling velocity and therefore strongly affects the measurements."

Given the difficulty of measuring both phases simultaneously at our values of Re λ , we are unable to asses the impact of these recirculation cells on our results. Future research should address the impact of these weak mean flows on the small Rouse regime. To circumvent these non-zero vertical mean flow effects, we suggest in section V to analyze the droplets' settling in a translating frame of reference. Considering these experimental difficulties found in the double limit of φ v → 0, and St → 0, we will focus our analysis on bulk trends of the moderate Rouse regime, which is less sensitive to these measuring uncertainties. FIG. 6: Particle velocity over the carrier phase fluctuations against Rouse (left) and Stokes numbers (right) . In the figures legend GEA refers to the data of Good et al. [START_REF] Good | Settling regimes of inertial particles in isotropic turbulence[END_REF]. AEA refers to the data of Aliseda et al. [START_REF] Aliseda | Effect of preferential concentration on the settling velocity of heavy particles in homogeneous isotropic turbulence[END_REF], and SBK refers to the data of Sumbekova [START_REF] Sumbekova | Enhancement versus hindering: impact of the turbulence characteristics on the settling behavior of heavy sub-Kolmogorov particles in a turbulent flow[END_REF]. Error-bars denote the resolution uncertainty, and they are masked by the size of the markers on the plot.

IV. MODERATE ROUSE REGIME

A. Global behavior

We focus on the regime Ro > O(0.1), and analyze the particles velocity settling curves against Rouse (see figure 7). The curve is defined by its slope, x-axis intercept (crossover between hindering and enhancement) and its maximum. For those datasets that have not reached hindering, we extrapolated the crossover with a linear fit. First, we consider the scaling of Sumbekova et al. [START_REF] Sumbekova | Enhancement versus hindering: impact of the turbulence characteristics on the settling behavior of heavy sub-Kolmogorov particles in a turbulent flow[END_REF] for a similar range of Rouse numbers (other scalings proposed for this regime are included in Appendix C 2). They propose that the crossover Ro cr , which defines the boundary between hindering and enhancement, increases with γ a = a 1/2 0 γ, where a 0 = 0.13Re 0.64 λ is the Lagrangian acceleration correlation proposed by Sawford [START_REF] Sawford | Reynolds number effects in lagrangian stochastic models of turbulent dispersion[END_REF]. This proposal seems to hold to some extent for previous datasets (see figure 8a) but it does not hold for our data (black markers), which seems to be less affected (if at all) by variations of the fluid acceleration. For the sake of completeness, we also plotted our data using different scalings found in the literature (see appendix C 2). Interestingly, our data reveal that Ro cr (figure 8b) becomes smaller with increasing Re λ , in agreement with [START_REF] Kawanisi | Turbulent effects on the settling velocity of suspended sediment[END_REF][START_REF] Good | Intermittency and inertial particle entrainment at a turbulent interface: the effect of the large-scale eddies[END_REF]. Although the liquid fraction does impact Ro cr , the leading order contribution comes from Re λ . It is then left for future research to assess whether these effects could be facility dependent (e.g., non-zero mean vertical flow [START_REF] Good | Settling regimes of inertial particles in isotropic turbulence[END_REF][START_REF] Sumbekova | Concentration préférentielle de particules inertielles : la structure et la dynamique de clusters[END_REF][START_REF] Akutina | Experimental evidence of settling retardation in a turbulence column[END_REF]). The linear fit y-intercepts (i.e. the limit Ro → 0 in table II) also decrease with increasing Re λ . This trend is consistent with the observed reduced settling at increasing Re λ (figure 7). On the other hand, the fitted linear slopes (figure 8c) are of order 1, i.e., (∆V /u)/Ro = ∆V /V T = O(1), and they seem to become steeper with Re λ . The correlation with Re λ , however, is not conclusive, as multiple factors (e.g., recirculation cells, and volume fraction φ v ) could be influencing the results. Interestingly, this quasi-linear behavior has also been recovered in numerical simulations (see appendix B), where the horizontal motion of the particles was suppressed [START_REF] Rosa | Settling velocity of small inertial particles in homogeneous isotropic turbulence from high-resolution dns[END_REF].

The maximum settling enhancement (figure 8d) also decreases with Re λ in agreement with [START_REF] Sumbekova | Clustering and settling of inertial particles in turbulence[END_REF]. Likewise, the Rouse number corresponding to the peak enhancement of the settling velocity decreases with Re λ (figure 8d). This observation may be a direct consequence of the coupling between u and Re λ in our experiment: they both scale with the inlet velocity U ∞ . Thus, Ro = V T /u decreases with increasing Re λ . These characteristics of the ∆V vs Ro are FIG. 7: Parameters computed for the data in 6 . The different line styles refer to different values of Re λ .

summarized in table II in the Appendix B.

B. Local concentration effects

Some experimental studies report that the increased local concentration has an impact on the particle settling velocity due to preferential concentration [START_REF] Huck | The role of collective effects on settling velocity enhancement for inertial particles in turbulence[END_REF][START_REF] Aliseda | Effect of preferential concentration on the settling velocity of heavy particles in homogeneous isotropic turbulence[END_REF], a mechanism frequently referred to as collective effects. In the same facility, previous research has found, utilizing 2D images, evidence of preferential concentration under similar experimental conditions as those studied here [START_REF] Sumbekova | Preferential concentration of inertial sub-Kolmogorov particles: The roles of mass loading of particles, Stokes numbers, and Reynolds numbers[END_REF]. Based on the approach of [START_REF] Obligado | Study on preferential concentration of inertial particles in homogeneous isotropic turbulence via big-data techniques[END_REF], we decided to normalize ∆V by the cluster velocity V cl ∼ C cl A cl ρ p g/(ρ air ν), where ρ p is the particle density, C cl , and A cl are the clusters concentration, and area, respectively. We estimate the latter quantities as C cl ≈ 4φ v from 2D correlations in the same facility [START_REF] Sumbekova | Preferential concentration of inertial sub-Kolmogorov particles: The roles of mass loading of particles, Stokes numbers, and Reynolds numbers[END_REF][START_REF] Huck | The role of collective effects on settling velocity enhancement for inertial particles in turbulence[END_REF][START_REF] Monchaux | Preferential concentration of heavy particles: A Voronoï analysis[END_REF], and A cl = 2.1 × 10 -5 St -0.25 max Re 4.7 λ φ 1.2 v [START_REF] Sumbekova | Preferential concentration of inertial sub-Kolmogorov particles: The roles of mass loading of particles, Stokes numbers, and Reynolds numbers[END_REF]. The mean concentration range has also been reported for anisotropic turbulence [START_REF] Boddapati | A novel approach for conditional measurement of droplet size distribution within droplet clusters in sprays[END_REF] at mass loadings between 1% to 7%.

The normalization by a single velocity scale fails to account for the different trends observed (figure 9a). Tom and Bragg [START_REF] Tom | Multiscale preferential sweeping of particles settling in turbulence[END_REF] claimed that normalizing the settling velocity results with a single length scale (or velocity scale) may not be adequate due to the multi-scale nature of the turbulence. They advance that the particle settling is affected by the multi-scale phenomenology of turbulent flows, and the resulting particle settling is due to an integrated effect of a range of scales that depend on the particle Stokes number. They argue that some physics may be lost by using a single scale to normalize the particle settling velocity enhancement, and therefore, the better collapse of the data brought by the use of the mixed length scales normalizations (Kolmogorov-scale velocity scaling combined by integral-scale Stokes [START_REF] Good | Settling regimes of inertial particles in isotropic turbulence[END_REF]) is an indication of the multi-scale nature of particle settling. Consistent with their observations, we see a slightly better collapse when using mixed scalings (viscous and integral scales combined) (see figure 9b).

V. ANALYSIS ON A MOVING FRAME OF REFERENCE

As stated in section III C, non-zero mean vertical flow effects could potentially impact the results presented here. To address these biases, we conduct a final analysis considering the particle settling velocity in a frame of reference moving with the particle distribution global average; V | all = V (D p )f (D p )dD p , where f (D p ) is the particle distribution PDF (see figure 2b). In this moving frame of reference, we encounter the following question: which parameter does control the evolution of the particle in the translating frame of reference? After some iteration, we find that the scaling RoSt, combining the Rouse and the Stokes numbers (see equation 7), provides the best collapse of the data (see figure 10a) in the x-coordinate that Ro or St individually. FIG. 8: a) Ro cr cross over between enhancement and hindering against γ a = 0.13Re 0.64 λ F r. The solid lines refer to the proposed scaling in [START_REF] Sumbekova | Concentration préférentielle de particules inertielles : la structure et la dynamique de clusters[END_REF]. b) Ro cr : crossover value between enhancement and hindering against Re λ . c) Slope of the settling velocity against Rouse number (∆V /u)/Ro. d) Maximum settling velocity, and Rouse value for these maxima.

Interestingly, in the regime RoSt > 0.1, the relative particle settling velocity has a slow evolution (see figure 10a):

V | D -V | all V T ≈ C, (8)
with C ∈ [0.4 -0.5], and which after algebraic manipulation gives;

V | D -V | all -V T u ≈ (C -1)Ro. ( 9
)
This expression is consistent with the quasi-linear behavior found in figure 6. Although the datasets present some variability at small Rouse numbers, we observe a power-law dependency for small RoSt 10 -2 . If we were to apply this observed power law, algebraic manipulations would yield: This result suggests that at very small Rouse numbers it would be possible to bound these profiles within the values of parameter C † . The data has a better collapse in this framework when multiplied by the mixed scaling (see figure 10b). The effects of V | all and its relationship with the particle size distribution and the observed particle settling should be further investigated in future experiments. For instance, some experiments have advanced that a bidisperse particle distribution may fall faster than any of the two monodisperse ones [START_REF] Wang | Clustering behaviour and settling velocity of bidisperse inertial particles in turbulent open channel flow[END_REF], an enhancement that cannot be explained by simple linear superposition, i.e., by taking an effective diameter of the bidisperse distribution.

V | D -V T -V | all u ≈ C † 15 1/4 γRe 1/2 λ 1/2 -Ro. (10)

VI. CONCLUDING REMARKS

Using phase Doppler interferometry, we experimentally investigate the behavior of polydispersed inertial sub-Kolmogorov particles under homogeneous isotropic turbulence for turbulent Reynolds numbers up to ≈ 650. Combined with previously available experimental results in the range Re λ ∈ [75 -200] taken in different facilities [START_REF] Sumbekova | Enhancement versus hindering: impact of the turbulence characteristics on the settling behavior of heavy sub-Kolmogorov particles in a turbulent flow[END_REF][START_REF] Aliseda | Effect of preferential concentration on the settling velocity of heavy particles in homogeneous isotropic turbulence[END_REF][START_REF] Good | Settling regimes of inertial particles in isotropic turbulence[END_REF], we find that the average settling velocity of particles is mainly a function of the Rouse number of the particle (Ro) and the overall particle-turbulence interactions are governed by the Taylor Reynolds number (Re λ ). We are unable to recover any strong influence from other parameters such as the ratio between the rms acceleration and gravity (γ) on the particle settling behavior. Our results also suggest that at increasing values of Re λ , the particles settling velocity is more hindered: their measured particle settling velocity is smaller than their respective one in still fluid conditions. This observation is recovered when the droplets' settling velocity is plotted against the Rouse numbers for each experimental condition explored.

A close inspection of the difference between the measured particle settling velocity and their respective one in still fluid conditions reveals that the boundary between the particle settling hindering (∆V /V T < 1) and enhancement (∆V /V T > 1) regimes also depends on Re λ . The onset of such transition point seems to behave as Re -2 λ . In addition, we find that after the peak of enhancement ∆V /u ≈ -κRo decreases almost linearly with the Rouse number. This behavior starts in the enhancement region and goes well into the hindering region for all the Rouse numbers considered Ro < 10. Noteworthy, the κ parameter, which accounts for this linear behavior, seems also to be a function of Re λ for a fixed particle distribution: the larger the Re λ , the steeper the decrease.

Although our concentration range is narrow to reach a definite conclusion, we do not recover a strong influence of the concentration on the results presented. This lack of influence seems to be a consequence of the more dilute conditions of our experiments (φ v ≤ O(10) -5 ) with respect to those conducted in the same facility [START_REF] Sumbekova | Preferential concentration of inertial sub-Kolmogorov particles: The roles of mass loading of particles, Stokes numbers, and Reynolds numbers[END_REF], which report the existence of preferential concentration. Previous studies [START_REF] Huck | The role of collective effects on settling velocity enhancement for inertial particles in turbulence[END_REF][START_REF] Aliseda | Effect of preferential concentration on the settling velocity of heavy particles in homogeneous isotropic turbulence[END_REF] have shown that the existence of preferential concentration leads to enhanced settling velocity for those particles inside a clusters. Sumbekova et al. [START_REF] Sumbekova | Preferential concentration of inertial sub-Kolmogorov particles: The roles of mass loading of particles, Stokes numbers, and Reynolds numbers[END_REF] reports that the degree of clustering, as well as the clusters' characteristic size is an increasing function of Re λ . On the contrary, we conjecture that these collective effects [START_REF] Huck | The role of collective effects on settling velocity enhancement for inertial particles in turbulence[END_REF] become less important at increasing values of Re λ where the hindering effect takes control of this phenomenon. 2D PTV measurements taken in the same facility support such conjecture: [START_REF] Sumbekova | Clustering and settling of inertial particles in turbulence[END_REF] (figure 4) reports that for a fixed droplet distribution, increasing Re λ leads to a global reduction in the measured particle settling velocity for particles inside clusters.

Finally, we cannot rule out that our wind tunnel experiments might be affected by a non-zero mean vertical velocity, as proposed by previous research [START_REF] Good | Settling regimes of inertial particles in isotropic turbulence[END_REF][START_REF] Sumbekova | Concentration préférentielle de particules inertielles : la structure et la dynamique de clusters[END_REF]. To address this potential bias, we have plotted our data in a translating frame of reference moving at the mean vertical velocity of our particle distribution. Previous experimental data as well as ours seem to collapse better in this frame, and it aids in explaining the quasi-linear behavior in the absolute (laboratory) frame of reference past the peak enhancement. The darker the color the larger Re λ . In the figures, the normal distribution is plotted as a dashed line ( ). The scaling of Sumbekova et al. [START_REF] Sumbekova | Enhancement versus hindering: impact of the turbulence characteristics on the settling behavior of heavy sub-Kolmogorov particles in a turbulent flow[END_REF] (figure 13a) does not show a better collapse when compared to those of include in the main text. In the figure, some of the curves look closer, but this could be an effect of the y scale used. On the other hand, when large and small fluid scales are are combined with the cluster falling velocity the curves come close together to some extent (figure 13b). This highlights again the including multiple scales may be necessary to understand the underlying physics of the particle settling modification by the turbulent carrier phase.

Rosa et al. [START_REF] Rosa | Settling velocity of small inertial particles in homogeneous isotropic turbulence from high-resolution dns[END_REF] also found a linear hindering behavior, consistent with our findings of section IV A , with a slope close to -0.3, when the lateral movement of the particles was suppressed in direct numerical simulations (figure 13c). [START_REF] Rosa | Settling velocity of small inertial particles in homogeneous isotropic turbulence from high-resolution dns[END_REF]. In the legends, GEA the data of Good et al. [START_REF] Good | Settling regimes of inertial particles in isotropic turbulence[END_REF], AEA refers to the data of Aliseda et al. [START_REF] Aliseda | Effect of preferential concentration on the settling velocity of heavy particles in homogeneous isotropic turbulence[END_REF], and SBK refers to the data of Sumbekova [START_REF] Sumbekova | Enhancement versus hindering: impact of the turbulence characteristics on the settling behavior of heavy sub-Kolmogorov particles in a turbulent flow[END_REF].

FIG. 5 :

 5 FIG. 5: Particle vertical velocity measurements binned by diameter size against the binned Stokes number. Error bars have a size ±5 × 10 -3 ms -1 (half of the PDI resolution).

FIG. 9 :

 9 FIG.9: Settling velocity normalized by different scales including the estimated settling velocity V cl following the approach of Obligado et al.[START_REF] Obligado | Study on preferential concentration of inertial particles in homogeneous isotropic turbulence via big-data techniques[END_REF]. The vertical axis in figures a) and b) is negative log, i.e., -1 × log. The black markers follow the legend found in figure8

FIG. 11 :

 11 FIG. 11: PDFs of the particles velocity for the different records. a) Horizontal component. b) Vertical component.The darker the color the larger Re λ . In the figures, the normal distribution is plotted as a dashed line ( ).

FIG. 12 :FIG. 13

 1213 FIG. 12: Particle velocity over the particle terminal speed. a) Against Rouse. b) Against stokes. The markers follow the legend of figure 8.

TABLE II :

 II Summary of the parameters extracted from figures 8a to 8d.

2. Sumbekova et al.

[START_REF] Sumbekova | Enhancement versus hindering: impact of the turbulence characteristics on the settling behavior of heavy sub-Kolmogorov particles in a turbulent flow[END_REF]

VII. ACKNOWLEDGEMENTS

Our work has been partially supported by the LabEx Tec21 (Investissements d'Avenir -Grant Agreement # ANR-11-LABX-0030), and by the ANR project ANR-15-IDEX-02. We also thank Laure Vignal and Vincent Govart for their help with the experiments.

If the particles datasets were to be normalized by the respective terminal velocity, we obtain the results found in figures 12a and 12b.


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Garry

 et al., 2008;Blondel et al., 2010;Carrière et al., 2013;Gaillardet et al., 2018;Jourde et al., 2018;Ollivier et al., 2019;Barbel-Périneau et al., 2019Long term observation

• Assure long term data acquisition provided to SNO LSBB multi-physics instrumentation

Environment and resources characterization

Instrumentation and metrology development

Unique facility and potential for experimentation

Measurements of waves and radiation

Jumps of magnetic field in time coincidence with "rainbow clouds" observed at different locations before Sichuan earthquake Maufroy et al., 2012;Bereš et al., 2013;Barruol et al., 2017;Matonti et al., 2017;Frau et al., 2018;Godeau et al., 2018;Tendil et al., 2018 The The low background noise interdisciplinary ground and underground based research laboratory (LSBB, CNRS Mixed Service Unit) contributes to the development of knowledge and know-how at national, European and international levels. As an interdisciplinary research facility, the LSBB is able to (i) accommodate large instruments, (ii) host scientific platforms, (iii) provide real-time measurements to national observatories, (iv) develop and to welcome ultra-sensitive multiphysics instrumentations of both academic and industrial collaborations in a low-anthropic-noise environment with particular interest in geological, electromagnetic and hydrogeological fields.

  infrastructure de recherche interdisciplinaire en région PACA pour la recherche académique et industrielle en environnement bas bruit Stéphane Gaffet To cite this version: Stéphane Gaffet. LSBB : LABORATOIRE SOUTERRAIN A BAS BRUIT : Une infrastructure de recherche interdisciplinaire en région PACA pour la recherche académique et industrielle en environnement bas bruit. 4èmes Rencontres Scientifiques et Techniques Résif, Nov 2019, Biarritz, France. , 2019. hal-03383131

•

  Development and characterization of high sensitivity technology • Interdisciplinary test site for benchmarks • Synergy between academics and industry LSBB know-how • Development, implementation & characterization of sensors • Interactions between research teams • Data quality and reliability management • Coupling between instrumentation and environment Research infrastructure • Accommodation of large experiments in controlled environments • Underground operation The LSBB offers • Potential for multi-scales & multi-depths projects • Permanent staff to ensure the experiment survey • Fine monitoring of environmental parameters • Boreholes at different depths and locations • Easy access for materials and water, electricity and network availability • Surface and underground facilities in a low-background noise environment • Underground vaults, clean room, electromagnetically shielded spaces Addressing societal challenges • Increasing stress on water resources due to climate change and anthropogenic impact • Seismic hazard • Optimization of oil bearing in carbonate reservoirs Interest of LSBB's location • Access to underground water • Karstic environment • Within a main seismogenic zone • Access to carbonated platform

  LSBB counts with 54ha of surface and 4 km of underground galleries with depths ranging from 0 to 518 m Easy access to the shallow and deep unsaturated zones of the Fontaine-de-Vaucluse aquifer THE LSBB UNDERGROUND RESEARCH LABORATORY: A UNIQUE FACILITY FOR FUNDAMENTAL & APPLIED LOW BACKGROUND INTER-DISCIPLINARY GROUND AND UNDERGROUND SCIENCE & TECHNOLOGY www.lsbb.eu direction@lsbb.eu Access to Urgonian facies analog to oil fields of the Middle East


Fernando Peruani

Gustavo Sibona

Reaction processes among self-propelled particles

We study a system of self-propelled disks that perform run-and-tumble motion, where particles can adopt more than one internal state. One of those internal states can be transmitted to another particle if the particle carrying this state maintains physical contact with another particle for a finite period of time. We refer to this process as a reaction process and to the different internal states as particle species making an analogy to chemical reactions. The studied system may fall into an absorbing phase, where due to the disappearance of one of the particle species no further reaction can occur or remain in an active phase where particles constantly react. Combining individualbased simulations and mean-field arguments, we study the dependence of the equilibrium densities of particle species with motility parameters, specifically the active speed v 0 and tumbling frequency λ. We find that the equilibrium densities of particle species exhibit two very distinct, non-trivial scaling regimes with v 0 and λ depending on whether the system is in the so-called ballistic or diffusive regime. Our mean-field estimates lead to an effective renormalization of reaction rates that allow building the phase-diagram v 0 -λ that separates the absorbing and active phase. We find an excellent agreement between numerical simulations and estimates. This study is a necessary step to an understanding of phase transitions into an absorbing state in active systems and sheds light on the spreading of information/signaling among moving elements.

1 FIG. 1. The figure shows two scenarios [(a) aand (b)] where a red agent "activates" a blue one agent after an encounter/collision among the two agents. In (a) particles move in straight trajectories (ballistic regime) in between collisions, while in (b) particles perform several turns (diffusive regime). Note that in an over-damped dynamics, a collision event lasts a finite time. Importantly, the steady state densities of particles species (e.g. density of red and blue agents) depend on the motility parameters of the active particles, namely active speed v 0 and tumbfling frequency λ. For movies, see SI.

I. INTRODUCTION

Systems of self-propelled particles are found in biology across scales, from bacteria [START_REF] Zhang | Proc. Natl. Acad. Sci. USA[END_REF][START_REF] Cisneros | [END_REF][3] to animal groups [START_REF] Ballerini | Proc. Natl. Acad. Sci. USA[END_REF][START_REF] Ginelli | Proc. Nat. Acad. Sci[END_REF]. There exist also man-made self-propelled systems such as chemically driven particles [START_REF] Howse | [END_REF]7], vibration-driven disks [8] and various types of active rollers [9,10] among many other examples. Requiring particles to convert energy into work to self-propel in dissipative media, self-propelled particle systems are intrinsically nonequilibrium systems [11][12][13]. Given the nonequilibrium nature of these systems, self-propelled particle systems display a large variety of phenomena that cannot be found in equilibrium systems as for instance the spontaneous, self-organized emergence of long-range order in the form of large-scale collective motion in two-dimensions that was initially found in models [14][15][16] and later confirmed to exist in real-world systems [8,9,17]. It is worth noting that it was recently observed that the presence of spatial heterogeneities such as imperfections on the substrate where particles move -typically present in real systems -prevents the emergence of large-scale collective motion [18,19]. Not surprisingly, most experimental self-propelled systems do not display global collective motion and particle motion remains diffusive at large time scales [START_REF] Zhang | Proc. Natl. Acad. Sci. USA[END_REF][START_REF] Cisneros | [END_REF][3][START_REF] Howse | [END_REF]. However, even in the absence of global order, the self-propulsion of active particles induces remarkable nonequilibrium features such as non-equilibrium clustering [3,[20][21][22], nonequilibrium phase-separation [23][24][25], and enhanced sedimentation [26].

Here, we aim at studying the spreading of information/signaling among actively moving units that do not display large-scale order. With this goal in mind, we analyze a system of self-propelled disks that perform run-and-tumble motion, where particles can adopt more than one internal state, see Fig. 1. One of those internal states can be transmitted to other disk if the disk carrying this state maintains physical contact with another disk for a finite period of time. Making an analogy to chemical reaction, we refer to this process as a reaction process and say that particles in the same internal state belong to the same particle species. The objective of current study is to understand how the steady state densities of particle species depend on the motility parameters of the self-propelled disks -shedding light on the way information spreads in active systems -in the context of reaction processes with an absorbing state. It is worth recalling that phase transitions into absorbing states is one of the fundamental problems in non-equilibrium statistical physics [27][START_REF] Krapivsky | A kinetic view of statistical physics[END_REF][START_REF] Marro | Nonequilibrium phase transition in lattice models[END_REF]. Here we intend to make a first step to understand how activity in the form of self-propulsion affects this classical problem of nonequilibrium statistical physics. It is important to stress that the understanding of the role of particle motion in reaction processes is key for a large number of applications beyond the context of active systems. For instance, it is well-known that the average concentration of (non self-propelled) chemical elements depends on how the system is stirred [START_REF] Taylor | [END_REF]31]. In ecology, the mobility of individuals affects the level of biodiversity [32,33]. In epidemics, it has been shown that the motion of individuals impacts the statistics of disease outbreaks [34][35][36][START_REF] Hufnagel | Proc. Natl. Acad. Sci. USA[END_REF][START_REF] Gonzalez | [END_REF][39]. In microbiology, the spreading of pathogens [40,41] and the spatial distribution of gene expressions and cell types [32,[START_REF] Dworking | Myxobacteria II[END_REF] are known to be correlated to cell motility. Finally, in the context of social dynamics, it has been shown that moving-agent models can be used as a proxy to mimic realistic social dynamics [START_REF] Gonzalez | [END_REF], which paved the way to study opinion dynamics in moving-agent systems [44,45]. In summary, studying how the spreading of information/signaling is affected by the motility parameters of active particles is a fundamental problem in active matter -and in the broad context of nonequilirbium statistical mechanics -which may have important implications for a large number of applications.

II. MODEL DEFINITION A. Particle motion

We consider, as in [46], a two-dimensional system of N self-propelled disks moving in a box of linear size L with periodic boundary conditions. The equation of motion of the i-th disk is given by:

ẋi (t) = v 0 v(θ i ) - j =i ∇U(x i (t), x j (t)) , (1)
where x i (t) is the position of the particle, v 0 is the active speed, v(θ i ) = (cos(θ i ), sin(θ i ))

with θ i an angle that denotes the propulsion direction. The propulsion direction θ i obeys a classical Poisson process: at rate λ -which we refer to as tumbling frequency -a new angle is selected from the interval [0, 2π). This defines a classical run-and-tumble process [START_REF] Berg | Random walks in biology[END_REF][START_REF] Cates | [END_REF],

whose distribution p(θ i , t) obeys ∂ t p(θ i , t) = -λp(θ i , t) + dθ ′ T (θ ′ → θ i )p(θ ′ , t),
where

T (θ ′ → θ i ) = λ 2π is the transition probability from θ ′ → θ i . Note that p(θ i , t → ∞) = 1
2π and the particle stays in a given direction a characteristic time λ -1 . In between turnings, particles interact through a soft-core potential U, which penalizes particle overlapping. Specifically, we implement a two-body repulsive potential, which depends on the distance between the center of mass of the two disks as follows:

U(x, x ′ ) =              a(v 0 ) |x-x ′ | 2r -b -1 if |x -x ′ | < 2r 0 if |x -x ′ | ≥ 2r ( 2
)
where r is the radius of the disks, b is a constant, and a(v 0 ) is a linear function of v 0 , a(v 0 ) = c 1 +c 0 v 0 such that the maximum overlapping area between two agents is independent of v 0 . By appropriately choosing the units of a, we have absorbed the mobility constant in such that ∇U has units of speed instead of force. Note that b controls how soft/hard is the potential U, becoming increasingly harder as b is increased. In the following we fix the parameters r = 1, c 0 = 1.62 (in units of distance), c 1 = 10 -4 (in units of speed squared), b = 1, and the density to d 0 = N/L 2 = 0.045 ensuring that for the explored range of v 0 and λ the system remains in the gas phase. Under these conditions, the motion of the self-propelled disks can be approximated by Fürth's formula:

x 2 (t) ≃ 2 v 2 0 λ 2 (λt -1 + e -λt ) , (3)
and thus, for t << λ -1 we can consider that particles move ballistically at speed v 0 , while for t >> λ -1 their motion is characterized by a diffusion coefficient D = v 2 0 λ -1 .

B. Reaction processes

Our intention is to make a necessary step towards an understanding of phase transitions into absorbing states by studying the dependencies of the equilibrium densities of particles species with motility parameters. Given our goal, any reaction process with an absorbing state serves to our purpose. Prototypical examples of such reaction processes are -using the terminology of epidemics [START_REF] Murray | Mathematical Biology[END_REF] -the Susceptible-Infected-Susceptible (SIS) reaction, which defines the so-called contact process in physics [27][START_REF] Krapivsky | A kinetic view of statistical physics[END_REF][START_REF] Marro | Nonequilibrium phase transition in lattice models[END_REF] or the Susceptible-Infected-Recovered-Susceptible (SIRS) dynamics, which defines a simple spatially extended excitable system, e.g. the Forest-Fire model [27,[START_REF] Krapivsky | A kinetic view of statistical physics[END_REF]. Note that though here we use terminology of epidemics, many physical systems fall in the same universality class as indicated in [27]. Given that for dilute systems, we expect the SIRS model to behave as the SIS model for fast R → S transitions, and to recover the SIR model in the absence of such transition. Thus, to remain as general as possible, we choose the SIRS model, whose dynamics is defined by:

S + I α → 2I, I β → R, R γ → S , (4)
where α, β, and γ are (constant) transition rates. Note that the reaction S + I → 2I to occur requires a particle of the species "S" and an particle of the species "I" to maintain physical contact for a finite time. We say that two particles are in physical contact whenever the center of mass of these two particles, e.g. x and x ′ are such that U(x, x ′ ) > 0. Note that since we are considering active particles that obey an over-damped dynamics, see Eq. ( 1), collisions are not instantaneous and last a finite time; and more importantly, during a collision particles maintain physical contact. In the following, without loss of generality we fix β -1 = 200 and γ -1 = 500 (in arbitrary time units) and vary the reaction rate α since it is the only rate connected to a transition that is affected by particle motion.

III. MEAN-FIELD

At higher densities and active speeds, a system of self-propelled disks can undergo a phase separation [23,24]. However, here we use parameters that ensure that the system remains Symbols correspond to simulations, while the solid line is a power-law fit, see Eq. ( 13). The inset shows that p(ω) is exponential distributed. The different curves correspond to various values of v 0 .

in a gas-like phase, where clusters are small, collisions are mainly binary, and the system is well-mixed. Under these conditions, the temporal evolution of the densities of particle species S, I and R can be described by a mean-field approach of the form:

ρS = γ(1 -ρ S -ρ I ) -ψ(v 0 , λ, α)R(d 0 , v 0 , λ)ρ I ρ S , (5) ρI
= ψ(v 0 , λ, α)R(d 0 , v 0 , λ)ρ I ρ S -βρ I , (6)
where the dot denotes time derivative and ρ S , ρ I , and ρ R are normalized densities such that ρ S + ρ I + ρ R = 1; the actual densities correspond to ρ k d 0 , with k ∈ {S, I, R} and d 0 = N/L 2 the (global) density of particles. Note that in deriving Eqs. ( 5) and ( 6) we have used that due to particle conservation ρ R = 1 -(ρ S + ρ I ). For details on the derivation of population mean-field models, see e.g. [START_REF] Murray | Mathematical Biology[END_REF]. It is important to understand that in a standard all-to-all mean-field scheme the expected term in front of ρ I ρ S in Eqs. ( 5) and ( 6) is α times a constant (i.e. N for N ≫ 1). However, here due to the active motion of the disks, particles are rarely in contact and when they do it, it is only for a short period of time. The question we pose is then how to effectively renormalize the transition rate I → S and how such an effective transition rate depends on the motility parameters. To compute an effective transition rate, we need to estimate the rate at which particles meet -let us call this rate the collision frequency and denote it by R(d 0 , v 0 , λ) -and the probability that a reaction I → S occurs for a random encounter between a particle S and a particle I; let us use the symbol ψ(v 0 , λ, α) to refer to this probability. In summary, R(d 0 , v 0 , λ)ψ(v 0 , λ, α) defines the effective transition rate I → S. Now, let us assume we know R and ψ (we estimate both of them below). It is straightforward to verify that Eqs. ( 5) and ( 6) have two steady states.

One of these states, often referred to as the absorbing phase, is given by ρ S (t → ∞) = 1 and ρ I (t → ∞) = 0. The other one, called the active phase, is given by:

ρ S (t → ∞) = ρ * S = β ψ(v 0 , λ, α) R(d 0 , v 0 , λ) , (7)
ρ I (t → ∞) = ρ * I = γ γ + β (1 -ρ S (t → ∞)) . (8)
Strictly speaking, Eqs. ( 7) and ( 8) correspond to the (active) steady states of an infinite system of density d 0 . Finite size fluctuations, which we have neglected here, may lead to deviation of mean-field predictions. The linear stability analysis around the absorbing state (ρ S = 1, ρ I = 0) -obtained by inserting ρ S = 1 -δ(t) and ρ I = δ(t) into Eq. ( 8) and keeping linear terms in δ -provides the following condition (within the mean-field approximation)

for the existence of the active phase:

ψ(v 0 , λ, α)R(d 0 , v 0 , λ) > β . (9)
In the following, we provide estimates for R(d 0 , v 0 , λ) and ψ(v 0 , λ, α).

A. Time in between collisions -estimating R

To estimate R(d 0 , v 0 , λ), we consider that there are two clearly distinct regimes that we can identify by constructing a dimensionless quantity (l ball /l ρ ) that results from the ratio between two characteristic length scales in the system: the characteristic distance l ρ in between particles -neglecting clustering effects and assuming a homogeneous distribution of particles -which is given by l ρ = 1/ √ d 0 , and the typical distance l ball that particles move in straight line (i.e. the typical distance in between two tumbling events). Since the typical time in between tumbling events is λ -1 , then l ball = v 0 λ -1 . Now, we are in condition of defining the ballistic and diffusive spreading regimes. We call ballistic spreading, the regime in which we can ensure that in between collisions, active particles move ballistically, i.e. in straight lines. The condition for this regime is given by: Under this condition, we can make use of kinetic gas theory [START_REF] Reif | Fundamentals of statistical and thermal physics[END_REF] and approximate R(d 0 , v 0 , λ) as:

l ball /l ρ = v 0 λ -1 d 0 ≫ 1 . (10)
R(ρ, v 0 , λ) ∼ v 0 σ 0 d 0 , (11)
where σ 0 = 4r is the scattering cross section of particles (v 0 and d 0 were defined above). The diffusive spreading regime corresponds to the opposite limite: l ball /l ρ ≪ 1. In this regime, active particles perform several tumbling events in between collisions. This implies that the time in between collisions is much larger than λ -1 , which according to Eq. ( 3) means that the active particles are deep inside the diffusive regime. In other words, the time in between collisions is dictated by a diffusive process characterized by a diffusion constant D, whose expression was given above after Eq. (3). To estimate the time in between collisions -let us refer to it as τ -we consider that the centers of mass of two neighboring self-propelled disks are separated an average distance 1/ √ d 0 and that the average distance the disks have to travel to collide is 1/ √ d 0 -2r, since the disks have a radius r. Then, under the assumption we are in the diffusive regime, we expect τ to be such that 1/ √ d 0 -2r 2 ≃ Dτ . Thus, the collision frequency R, which is the inverse of τ , takes the form:

R(d 0 , v 0 , λ) ∼ v 2 0 λ -1 1 √ d 0 -2r -2 . ( 12
)
Note that the above expression is only an approximation. A rigorous calculation of the collision frequency would require to solve a first passage time problem; for details see [START_REF] Krapivsky | A kinetic view of statistical physics[END_REF][START_REF] Redner | A guide to first-passage processes[END_REF].

B. Collision duration -estimating ψ

A collision between two particles is a relatively slow process in which particles stay in contact -understanding by this that the potential energy U between the two particles is larger than 0 -for a finite time ω. By looking at histograms of ω in simulations, we learn that ω is exponentially distributed -meaning that p(ω) = exp(-ω/ ω )/ ω -, see inset in Fig. 2. Furthermore, by performing a systematic study of ω vs. v 0 , Fig. 2, we find that ω follows a power-law with v 0 :

ω = k v -ξ 0 , (13)
with k = 1.18 and ξ = 0.957 for λ -1 larger than r/v 0 [? ]. Knowing the statistics of ω, now we can focus on obtaining an estimate for the probability that, during a collision event of duration ω between a particle I and a particle S, a reaction S + I → 2I occurs. We make use of the fact that the reaction is given by a simple Poissonian process [START_REF] Van Kampen | Stochastic processes in physics and chemistry[END_REF], which let us compute the probability that a reaction takes place in the time interval ω as 1 -e -ωα , under

the assumption that the initial particle I does not transition to R in this time interval. The next step to estimate ψ is to make an average over all possible collision durations ω -that we know is distributed exponentially as indicated above -to express ψ as:

ψ(v 0 , λ, α) = ∞ 0 dω p(ω) 1 -e -ωα = α ω 1 + α ω . (14)
Thus, inserting Eq. ( 13) into Eq. ( 14), we find that

ψ = (1 + k -1 α -1 v ξ 0 ) -1 .

IV. COMPARISON WITH SIMULATIONS

Let us start by analyzing an infinitely fast reaction rate α, which let us take the limit α → ∞ in Eq. ( 14) and express ψ = 1. The prediction is that for l ball /l ρ ≫ 1, by inserting Eq. ( 11) into Eq. ( 7), we find ρ * S ∼ v -1 0 , solid curve in Fig. 3(a), while by inserting Eq. ( 12) into Eq. ( 7), we obtain ρ * S ∼ λ v -2 0 , dashed curves in Fig. 3(a). In summary, there are at least two scalings of ρ * S with v 0 , and a crossover between these two scalings at l ball /l ρ ≃ 1. Fixing v 0 , the set of mean-field approximations detailed above predicts that at large enough tumbling frequencies λ such that l ball /l ρ ≪ 1, ρ * S ∝ λ, dashed curves in Fig. 3(b). As λ is decreased, the system should cross l ball /l ρ = 1 and ρ * S becomes independent of λ, as confirmed by the horizontal solid lines in Fig. 3(b). Now, we move to finite reaction rates α, where the approximations given above suggest that the spreading dynamics is strongly affected by the average collision duration ω through Eq. ( 14). The predictions are now ρ * S ∼ v ξ-1 0 for the ballistic regime, that consists of inserting Eq. ( 11) and Eq. ( 14) into Eq. ( 7), and ρ * S ∼ λ v ξ-2 0 for the diffusive regime, obtained by inserting Eq. ( 12) and Eq. ( 14) into Eq. (7) [? ]. Fig. 4(a) shows a direct comparison between the prediction for finite (dashed curves) and infinitely fast reaction rates α (solid curves). In order to get a direct understanding of the role of the probability ψ, we plot in Fig. 4(b)

H = ρ * S (α)/ρ * S (α → ∞) (let us recall that ρ * S (α → ∞) ∼ v -1 0 and ρ * S (α → ∞) ∼ λ v -2
0 for the ballistic and diffusive regime, respectively). Thus, according to the approximations developed in the previous sections, for both regimes, i.e. ballistic and diffusive, H ∼ (1 + k -1 α -1 v ξ 0 ), see solid curve in Fig. 4(b). In the following, we look for the critical active speed v c (and tumbling rate λ c ) above (below) which the active phase should be observed. We are particularly interested in knowing, given a set of parameters α, β, γ, and d 0 , the behavior of v c with the tumbling rate λ. Inserting Eq. ( 11) and Eq. ( 14) into Eq. ( 7), and by requesting ρ * S = 1, we derive a condition for v c which does not depend on λ; see horizontal lines in Fig. 5. Similarly, by inserting Eq. ( 12) and Eq. ( 14) into Eq. ( 7), and under the same condition, we find that v c is solution of v 2 c -qλǫv ξ c = qλ, where q = β(1/ √ d 0 -2r) 2 and ǫ = 1/α. In order to obtain a close expression for v c , we assume that ǫ is small and ξ = 1 + δ, with |δ| ≪ 1, which leads to:

v c = ǫqλ 2 + qλ 1 + ǫ 2 qλ 4 -1/2 (15)
where terms proportional to ǫδ have been neglected. For infinitely fast reactions, ǫ → 0, and v c ∝ λ 1/2 . Note that this result is independent of δ. For large, but finite reaction rates, Eq. ( 15) provides a good estimate of v c as shown in Fig. 5. Using Eq. ( 10) and the fact that the ballistic and diffusive approximation for ρ * S coincide at the crossover point, we find v cross (λ c ), the solid curve in Fig. 5.

V. CONCLUSIONS

We have studied, combining individual-based simulations and mean-field arguments, the dependency of the equilibrium densities of particle species with motility parameters in an active system consisting of self-propelled disks. Importantly, we consider a reaction process with an absorbing and active state. For a finite reaction rate α, we found that there are two distinct regimes in the active phase with the active speed v 0 and tumbling frequency λ that lead to steady state densities ρ * S ∝ v ξ-1 0 and ρ * S ∝ λv ξ-2 0 for the ballistic and diffusive regime, respectively. For a given set of reaction rates α, β, and γ, and density d 0 , we have been able to compute the phase-diagram in terms of motility parameters, namely v c and λ c , that separates the absorbing and active phase. These results were obtained from a combination of stochastic simulations of the microscopic dynamics and mean-field arguments. It is important to stress that the developed mean-field arguments include information on the spatial dynamics of the self-propelled disks. Moreover, the results here derived cannot be obtained by a classical reaction-diffusion process as done in [START_REF] Murray | Mathematical Biology[END_REF], since these approaches decouple particle motility and effective reaction rates and assume that particle transport is always diffusive. A word of warning: the here derived arguments neglect spatial correlations among the particle species. At higher densities, lower dimensions, and/or close to the critical point, deviations between the derived mean-field arguments and simulations are expected, and correlation and fluctuations should be taken into account in order to obtain an accurate description of the system dynamics along the lines explained in [START_REF] Peruani | [END_REF] that makes used of the formalism developed in [54].

Here, we have shown that the (transient) behavior of the active particles in between collisions plays a key role to understand the equilibrium densities of particle species. While here we considered only ballistic and diffusive regimes, we can imagine a more general scenario where x 2 ∝ t χ with 1 ≤ χ ≤ 2 that we expect to lead to different scalings of the equilibrium densities with active speed v 0 and tumbling frequency λ. Importantly, the presence of different transport regimes in systems of active particles suggests that phase transitions into an absorbing state do not necessarily fall into the directed percolation class or related universality classes for active systems [27,[START_REF] Marro | Nonequilibrium phase transition in lattice models[END_REF]. Understanding in which universality class fall "reactive" active systems with an absorbing state remains a fundamental, challenging question that can only be addressed by combining large-scale simulations and renormalization group techniques, which we hope will be the subject of future works.

The simple, but fundamental results reported in this study represent a necessary first step to a better understanding on the spreading of information/signaling among actively moving units, an issue of key importance in the context of reaction processes and diseases [32,[40][41][START_REF] Dworking | Myxobacteria II[END_REF],

synchronization among moving oscillators [55][56][57][58][59], organization of self-propelled particles into collective motion [11,12,[14][15][16]60], as well as several technological and biomedical applications involving moving entities [61,62].

FIG. 2 .

 2 FIG. 2. Collision statistics. Average collision duration time ω as function of the active speed v 0 .

FIG. 3 .

 3 FIG. 3. Instantaneous transmission. (a) ρ * S vs. v 0 for various values of λ. (b) ρ * S vs. λ -1 for various values of v 0 . In (a) and (b), symbols correspond to simulations, while solid lines to the ballistic and dashed line to the diffusive approximation. The absorbing state is indicated by the horizontal dotted line.

FIG. 4 .

 4 FIG. 4. Finite transmission -(a) ρ S (t → ∞) vs. v 0 for various λ values. Open symbols correspond to simulations with α → ∞, while solid symbols to α = 1. The solid and dashed curve correspond to the mean-field approximation for α → ∞ and α = 1, respectively, in both ballistic and diffusive approximation. (b) H = ρ * S (α)/ρ * S (∞) vs. v 0 for various λ values. Simulations (symbols) followthe mean-field prediction (solid curve) as soon as the system moves into the active phase.

FIG. 5 .

 5 FIG. 5. Phase diagram v vs λ -1 . Symbols correspond to the absorbing-active boundary as found in simulations. Dashed and dotted lines correspond to the critical point (λ -1 c , v c ) predicted by the mean-field approximation for α → ∞ and α = 1, respectively. The solid line, v cross , indicates the crossover between ballistic and diffusive spreading.

ACKNOWLEDGEMENTS F. P. was supported by the Agence Nationale de la Recherche via project BactPhys, Grant No. ANR-15-CE30-0002-01.