Investigation of particle laden gravity currents using the light attenuation technique

An extensional use of the light attenuation technique is described to study the dynamics of particle laden flows. The method is validated against an analytical absorption/scattering model. It is then applied to particle laden gravity currents (PLGCs) and provides access to quantitative measurements of the particle mass fraction field. The influence of particle concentration on the turbulent dynamics of the flow and the front velocity are evaluated and discussed. In all experiments heterogeneities in the particle distribution are measured, with the maximum particle concentration in the PLGC front and intermediate particle concentration at the turbulent interface. The results highlight two regimes. In the dilute case, particles have no measurable impact on the turbulent structures observed in the flow nor on the bulk velocity. Particle sedimentation occurs due to mixing between the carrier and surrounding fluids. In the dense case, turbulent structures strongly weaken and the bulk velocity of the PLGC quantitatively decreases as the effective flow viscosity grows.


Introduction
Buoyancy-driven longitudinal currents, known as gravity currents are ubiquitous in environmental flows and also met in industry (Simpson 1982;Sher and Woods 2015). Understanding the dynamics of these flows is a major point of interest for many geophysical phenomenon, such as avalanches, mud flows, dam break, and turbidity currents in the ocean. Particle laden gravity currents (PLGCs) propagate across reservoirs, lakes (Best et al. 2005;Cossu et al. 2015), and oceans (Mulder et al. 2003;Mountjoy et al. 2018). PLGCs are the dominant mechanism for the transport of sediment, nutrients, and pollutants from the continental margin to the deep ocean (Wells and Dorrell 2020).
Lock-release experiments are an usual configuration for the study of turbulent gravity currents (Prastowo et al. 2008;Fragoso et al. 2013;Micard et al. 2016;Hughes and Linden 2016). In this configuration, a lock separates two fluid compartments with different density and composition properties. The horizontal pressure gradient between the two compartements leads to the generation of a gravity current when removing the lock.
In monophasic gravity currents (MGCs), the flow is driven a by a density difference induced by a stratifying agent between two miscible fluids. The two fluids can be of constant density (Shin et al. 2004;Prastowo et al. 2008;Fragoso et al. 2013;Micard et al. 2016;Hughes and Linden 2016) or stratified (Maxworthy et al. 2002;Cortés et al. 2014), which lead to a variety regimes controlled by the Reynolds and Froude numbers. The key role played by mixing effects has been quantified in these laboratory configuration (Prastowo et al. 2008;Fragoso et al. 2013;Micard et al. 2016). The lock-release configuration has been adapted to characterize the flow dynamics including the effects of slope (Odier et al. 2009;Tanimoto et al. 2021) and rotation (Negretti et al. 2021).
A range of techniques are available to understand the dynamics of monophasic gravity currents: shadowgraph (Prastowo et al. 2008) Laser Induced Fluorescence (LIF) (Balasubramanian and Zhong 2018), Ultrasound Velocity Profiling (UVP) (Hitomi et al. 2021), Particle Image Velocimetry (PIV) (Krug et al. 2013) and Light Attenuation Technique (LAT) (Fragoso et al. 2013;Micard et al. 2016). However, the experimental studies performed with this technique are limited to MGCs. The investigation of PLGCs remains an experimental challenge. We can mention the studies of Johnson et al. (2012) where PTV technique was used to assess surface flow velocities in debris flows and Sanvitale and Bowman (2016) where PIV is used to measure granular temperature in saturated debris flows. While providing a description of PLGCs dynamics in various concentration, particle properties and stratification regimes, experimental studies carried on PLGCs do not provide a quantitative measure of particle concentration distribution in the flow (Saha et al. 2013;Sutherland et al. 2018;Davarpanah Jazi and Wells 2020). We propose to extend the use of the LAT to investigate the dynamics of the PLGCs in non-newtonian and turbulent regimes. The experimental setup is presented in Sect. 2. The particle induced light attenuation technique and its validation are introduced in Sect. 3. The technique is then applied to investigate the PLGC dynamics relative to the initial mass fraction in Sect. 4, where the local and bulk flow properties are discussed. Conclusions are drawn in Sect. 5.

Experimental setup and measurement technique
Experiments are conducted in a 150 cm long, 10 cm wide and 35 cm high plexiglas tank. A photograph of the water tank is shown in Fig. 1c. After the tank is filled with freshwater to an height h 1 varying from 24.5 to 29.9 cm , a gate with insulating sides is inserted at L 1 = 20 cm from an end of the tank as shown in Fig. 1a. The fluid heights h 1 and h 2 are adjusted to reach a hydrostatic equilibrium between the two compartments at mid-depth (Marino et al. 2005;Prastowo et al. 2009;Maurer et al. 2010;Hughes and Linden 2016). Salt is dissolved in the lock to reach an homogeneous density 1 which is the density of the salted water only in the range 1030-1040 kg m. −3 . Due to the low concentration of added salt, the viscosity is equivalent to that of pure water, in accordance with the measurements performed by Qasem et al. (2021). The particles used are polystyrene spherical beads with a mean diameter d = 1.08 ± 0.02 mm and a average density p = 1039.5 kg m. −3 . The size below which 50% of all particles are found is d 50 = 1.06 ± 0.02 mm . The particle distribution is shown in Fig. 2.
A prescribed mass of particles is poured in a becher filled with saltwater sampled from the lock. The immersed particles are treated in an ultrasonic wave tank for 30 min to remove air bubbles. The latter can in fact lead to the formation of particle aggregates floating at the surface. After that time most of the air bubbles are detached from the particles which are poured in the lock. The added particle mass in the lock varied from 0 to 3000 g corresponding to initial mass fractions 0 of 0-45% in Exp A-G. Homogeneity is controlled by adjusting the density of the carrier fluid to that of the particles. Then, the mixing is performed by using a stirrer and waiting a few seconds for the flow to settle, while sedimentation remains limited at this timescale, before opening the gate. A particle free experiment (Exp A) is used a reference for the flow dynamics in the Monophasic Gravity Current (MGC) regime. An amount of 5 mL of red food dye is added in the lock as a passive tracer for salinity in Exp A and Exp F. The experimental parameters are listed in Table 1. In all the experiments a typical density difference of = 1 − 2 = 25 kg m. −3 is imposed between the two compartments. The varying parameter in the experiments set is the initial mass fraction 0 .
A white LED panel placed behind the tank is used as a quasi-homogeneous light source. Measurements are performed using a 14 bits DSLR Nikon D500 color camera equipped with a Nikkor 53 mm lens. The Nikon camera is also used to record shadowgraph images in PLGC experiments. The camera is placed at a distance of 350 cm of the tank to limit the effects of parallax. The typical spatial resolution in this configuration is 0.2 mm/pixel. The camera exposure is set to optimize the grayscale level range in the fluid seed with particles. Most measurements are performed with a frame rate of 4 fps.

Calibration
The objective of the measurements is to provide high resolution data on the particle and/or stratified flow dynamics in the gravity current. The classical light attenuation technique has been used to infer local, cross-averaged density from light absorption in a dyed fluid (Allgayer and Hunt 1991;Hacker et al. 1996;Sutherland et al. 2012). In the present experiments an extended application of the LAT technique is used to infer cross-averaged local particle mass fractions in PLGGs. The acronym PILAT (Particle Induced Light Attenuation Technique) is adopted in the remainder of the manuscript to refer to the latter technique. Due to the combined effects of particle scattering and absorption the light issued from the panel is attenuated when traversing the tank. In the calibration step, the light attenuation is calibrated against particle concentration in a shorter tank with the same width as the experimental tank. The objective of the PILAT technique is to measure a quasicontinuous particle concentration field averaged in the crosswise direction and at the pixel scale in the spanwise direction as shown in Fig. 1b. Here the pixel size is five times smaller than the particle diameter. To a sufficient sensitivity for the technique, the condition that must be met is that the particle diameter must be much smaller than the tank width. In the present geometry, the maximum number of particles in the crosswise dimension is N p = l∕d 50 = 95 which is sufficient to extract concentration heterogeneities in the PLGC dynamics (Figs. 3,4).
The calibration tank is filled with water at the same density as the particles. A reference image with an intensity distribution I 0 (x, z) is taken before adding particles, with x and z the horizontal and vertical coordinates. A controlled mass of particles is added to the tank before each calibration measurement. The mixture is thoroughly mixed with a hand stirrer before an image with an intensity distribution I n (x, z) is recorded, where n is the calibration step. The equivalent absorption A n (x, z, t) = log(I 0 (x, z)∕I n (x, z)) is calculated and the integrated added mass m n is converted into a mass fraction n . The calibration is shown in Fig. 3a. A linear increase in absorption analogous to a Beer-Lambert regime is observed for mass fraction up to 3% . For larger mass fractions the absorption increases at a slower but measurable rate up to the packing limit = 60% . The camera sensor allows to measure absorption differences at the largest mass fraction values. The absorption dynamics closely resembles the one of a blue food dye used in previous light attenuation technique experiments (Fig. 3b), for which a Beer-Lambert and a  Table 1 Experimental parameters used for gravity current experiments The initial density and height of the carrier fluid and the surrounding fluid are ( 1 , h 1 ) and ( 2 , h 2 ), respectively. s  saturated regimes are also observed. In particular, the maximum absorption values reached in the calibration for the particle-fluid and the dye are in the same range A = O(1) . This shows that the dynamics for measurement of particle mass fraction is potentially as large as the dynamics using dyes in monophasic experiments. The calibration curve is fitted by a spline interpolation function using the Curve Fitting Toolbox in Matlab. The calibration function is then used to convert grayscale fields to mass fraction fields in PLGC experiments. An example barrier affects the light distribution in this picture, as seen by the apparent higher concentrations in the vicinity of the barrier. However the barrier is absent in the course of the experiments which allows for homogeneous background in the experimental field of view. In order to limit reflections, a black sheet is placed under the transparent base of the tank. Limited reflection effects are nonetheless visible in the concentration signal in a bottom layer of typical thickness 1 cm. The region where these effects occur is hidden by a gray rectangle in Figs. 5,6,7,8,9,and 10. (1) (1 + ( + ) 2 − 2 ( + ) cos( )) 3∕2 (5) k = max (k a,w , k a,p ) + k s,p (6) = − + l j−1 − of conversion from grayscale to mass fraction before the start of an experiment is displayed in Fig. 4. The average mass fraction in the left compartment inferred from the added particle mass is 2.95 ± 0.05 % . After the calibration is applied to the grayscale image, one obtains a heterogeneous mass fraction field induced by the particle distribution. The average mass fraction and standard deviation in the left compartment are 3.05% and 0.62% . The analyzed average mass fraction matches correctly the inferred value confirming the robustness of the PILAT technique. The

Analytical absorption/scattering model
In order to compare the experimental calibration against a theoretical approach, an analytical model based on Monte Carlo method is used. For this purpose, a model of light propagation in the medium is established. It is based on the estimation of the transmittivity T within the tank by the Monte Carlo method, a reference method in the field of radiative transfers (Howell et al. 2020). The transmitivity is related to the mesured absorption A = − log(T) as T = I(x, z)∕I 0 (x, z).
In the studied case, the medium is considered as a participating medium with heterogeneous radiative properties and the light source emits Lambertian radiation. In order to represent as accurately as possible the heterogeneities of the medium, we use a null-collision Monte Carlo algorithm (Galtier et al. 2013) whose extinction coefficient is presented in Eq. 5. The representations of the interactions between matter and radiation are all probabilized : the Beer-Lambert law is simply an expression of the probability of photons traveling a given distance before being absorbed or scattered while the Henyey-Greensein phase function corresponds to the probabilities that a photon initially in a direction will scatter or be reflected in a direction + and the nature of the medium is defined as a propensity depending on the volume fraction. The transmitivity expression is then expressed in an iterative form as presented in the Eqs. 1 and 2. The scattering phenomena are represented by a Henyey-Greenstein phase function (Eq. 4) whose asymmetry factor value = 0.92 comes from the work of Shahin et al. (2019). The absorption/scattering optical path stops when absorbed by water or polystyrene. In order to identify the absorption and scattering coefficients of the polystyrene used in the experiment, the transmitivity estimation model is combined with a particle swarm optimisation (PSO) code (Eberhart and Kennedy 1995). This results in a very good agreement between the experimental measurements and the absorption/scattering model as shown in Fig 3a.

Quasi-inertial regime ( ˚0 < 10%)
The reference experiment A shows the development of a monophasic gravity current with turbulent features commonly observed in lock-release experiments (Fig. 5).
The current front where large density and velocity gradients, along with a weak mixing occur precedes the head region. In the latter the formation of Kelvin-Helmholtz billows and shear instabilities contribute to an intense mixing between the carrier and the surrounding fluids. Mixing processes weaken in the body and tail regions of the gravity current, as extensively described in (Balasubramanian and Zhong 2018).
When opening the gate, the development of a turbulent gravity current similar to exp A is observed in exp B, C, D. The front exhibits a similar shape and vortices associated with Kelvin-Helmoltz billows are observed in Fig. 4 Conversion from grayscale to absorption, using the calibration function. Validation of the absorption to mass conversion in in the left compartment Page 7 of 12 23 all experiments. The periodicity of the billows is clearly identified as shown in Fig. 7d.
Important horizontal and vertical gradients in the particle distribution in the gravity current are measured in exps C (Fig. 7) and D (Fig. 8). The largest concentrations are located near the current front where mixing is limited, while they are weaker in the interfacial region.
The progressive particle sedimentation can be explained by mixing effects between the carrier and surrounding fluids. In fact, as the current advances, diapycnal mixing contributes to a gradual decrease in the density of the carrier fluid that become less dense than neighbouring particles. Hence, buoyancy forces are no longer able to sustain particles, which result in their sedimentation. The processes by which turbulence enhances particle settling have been described using Direct Numerical Simulations by Henniger et al. (2010). It was shown that the intensity of mixing between the carrier and surrounding fluids play a more important role than the particle inertia on particle settling. On the other hand, the coherent Kelvin-Helmholtz billows act as a particle supply from the mixed region toward the front of the PLGC. This phenomenon leads in particular to a large increase in the particle concentration up to a factor of three in the current front with respect to the upper mixing region as shown in Figs. 7e and 8e.
This sedimentation process also occurs in exp B, although less visible as mass fraction measurements are limited in the interfacial region due to low particle concentration (Fig. 6d).
After reflection on the right end wall, the gravity current propagates leftwards. Turbulent structures in the MGC have weakened in the course of the propagation and interaction with the end wall, which results in a decrease in the maximum current height, from O(15 cm) to O(10 cm) . In the PLGC case instead a dramatic collapse of the particle flow is observed in Figs. 6f 7f and 8f, with a particle current maximum height decreasing to O(5 cm) . The collapse leads to an increase in the particle concentration with mass fractions reaching values larger than 20% in most of the gravity current below the interface.

Non-newtonian regime ( ˚0 > 20%)
For increasing mass fraction 0 , effects of particle-particle interactions lead to an increase in the effective viscosity of the flow. As a result, turbulent structures at the interface progressively weaken from Exp E (not shown), to Exp F (Fig. 9) then Exp G (Fig. 10). Experiment F has been performed by adding a red dye as a tracer of the carrier fluid. The dye fluorescence properties allow to carry out PILAT and density measurements (after calibration) with a single camera.
The absorption signal contains two different quantities depending on the sign. Positive absorption values corresponds to light absorption/dispersion by the particles and hence relate to the particle concentration field (PILAT). Negative absorption (hence positive transmission) values are induced by the red dye fluorescence and is scaled against density using the classical LAT technique. The limit of the technique is that particle absorption/dispersion largely overcome red dye emission, so density cannot be assessed in regions where particles are present, even scarcely.
As in the case of the inertial regime, the front, head, body and tail of the PLGC are observed in Exp F (Fig. 9c).
In the initial course of the current propagation, one observes that the carrier fluid hardly separates from the particles in the front and at the interface between the body and the surrounding fluid, as illustrated in Fig. 9c. It is only due to breaking of the K-H billows in the current tail that flow separation is initiated (Fig. 9d). It is further  Fig. 9e. In the final stage of the experiment, the dense particle region is much thinner than the dense fluid layer.
In the Experiment G the current height has greatly decreased compared to the inertial regime. While billows are still visible up to heights greater than 15 cm in Exp A-D, the development of such billows are strongly limited in Exp G (instabilities are still visible in Fig. 10d) and the maximum current height is O(10) cm.

Particle influence on the front velocity
The front velocity is a bulk property that enables to characterize the overall impact of the particles on the PLGC dynamics. The impact of 0 on the gravity current velocity is analyzed here through the temporal evolution of the front position shown in Fig. 11.
H is plotted against the relative lock viscosity assessed from the Quemada's relation for viscosity r = (1 − ∕ M ) −2 (Quemada 1977), with M the volume fraction at close packing as shown in Fig. 12a, b. The velocity during the first transit for the four experiments in the quasi-inertial regime has a typical value of 0.13 m s −1 , in agreement with the analytical value predicted by Simpson (1982) The velocity slighty decreases with the mass fraction for quasi-inertial experiments. No significant difference is observed between the MGC and PLGC. The particle effect on the current viscosity is limited and the nondimensional velocity matches the theoretical value of 1/2. For larger initial mass fractions concentrations ( 0 > 20% ) the front velocity decreases quantitatively with 0 . This significant decrease occurs when the effective viscosity departs strongly from the slowly varying linear behavior Fig. 12a. In particular the front velocity has decreased by 25% with respect to the MGC in exp G. These results confirm the influence of particle concentration on the turbulent dynamics and the bulk velocity of dense PLGCs.

Conclusion
In this study, the dynamics of Particle Laden Gravity Currents has been investigated using high-resolution light attenuation measurements, enabling to highlight two flow regimes. The choice of the optical properties of the particles has allowed to follow the mass distribution in both diluted and concentrated regimes. The density of the particles, comparable to that of pure and salty water, allowed to describe the role of the mixing on the dynamics of the particle and the carrier fluid, including the effects of sedimentation. PILAT measurements indicate that the particle concentration, and hence the relative importance of turbulent mixing and effective viscous properties of the flow are heterogeneous in the structure of PLGCs. In the quasi-inertial regime, the PLGC dynamics is hardly impacted by the presence of particles, with vigorous shear instabilities at the interface and a front velocity that is comparable to the case of the monophasic gravity current. However, a progressive particle sedimentation to the unmixed heavy fluid region is measured. In the non-newtonian regime, the presence of the particle impacts the turbulent dynamics of the flow via particle-fluid, particle-wall and particle-particle interactions, which inhibits the instabilities at the interface between the two fluids. The front velocity decreases quantitatively as the apparent viscosity increases.
The developement of the PILAT technique, potentially combined with LAT in the carrier fluid, will be applied to granular collapse experiments. Recent experiments by Bougouin and Lacaze (2018) have highlighted four regime for granular collapse in a fluid with constant density. In future works, the combination of LAT/PILAT and PIV measurements in the surrounding fluid will enable to study the dynamics of this collapse for varying density of the carrier and surrounding fluids.