# An efficient cellular flow model for cohesive particle flocculation in turbulence

###### Abstract

We propose a one-way coupled model that tracks individual primary particles in a conceptually simple cellular flow setup to predict flocculation in turbulence. This computationally efficient model accounts for Stokes drag, lubrication, cohesive and direct contact forces on the primary spherical particles, and allows for a systematic simulation campaign that yields the transient mean floc size as a function of the governing dimensionless parameters. The simulations reproduce the growth of the cohesive flocs with time, and the emergence of a log-normal equilibrium distribution governed by the balance of aggregation and breakage. Flocculation proceeds most rapidly when the Stokes number of the primary particles is O(1). Results from this simple computational model are consistent with experimental observations, thus allowing us to propose a new analytical flocculation model that yields improved agreement with experimental data, especially during the transient stages.

——————————————————————————————————————

## 1 Introduction

Cohesive sediment, commonly defined as particles with diameters , plays a central role in a wide range of environmental and industrial processes. For these small grain sizes, attractive van der Waals forces can outweigh hydrodynamic, buoyancy and collision forces, and trigger the formation of large aggregates via flocculation (Yoshimasa, 2017). Following the pioneering work by Levich (1962), current approaches for modeling the flocculation process often employ population balance equations (Maggi et al., 2007; Verney et al., 2011; Shin et al., 2015) or simplified versions thereof (Winterwerp, 1998; Son and Hsu, 2008, 2009; Lee et al., 2011; Shen et al., 2018). These semi-empirical models, which require calibration with experimental data, usually do not account for the detailed profiles of the various forces governing particle-particle interactions.

The present investigation presents a conceptually simple model to obtain flocculation data via one-way coupled simulations that track individual primary particles and accurately capture the inter-particle forces, based on the recent development of advanced collision models in viscous flows (Biegert et al., 2017a, and references therein), along with strategies for accurately modeling cohesive forces (Vowinckel et al., 2019). Towards this end, we employ the well known initial configuration of cellular Taylor-Green flow as a simple, quasi-steady analytical model of a turbulent flow at the Kolmogorov scale. This flow has previously been used successfully in elucidating fundamental aspects of particle-vortex interactions (Maxey, 1987; Bergougnoux et al., 2014). We will exploit this conceptually simple, computationally efficient scenario to systematically investigate the influence of key physical parameters, and propose a new flocculation model that agrees closely with experimental data.

## 2 Computational model

### 2.1 Particle motion in cellular flow fields

In the spirit of earlier investigations by Maxey (1987) and Bergougnoux et al. (2014), we apply a simple model flow in order to investigate the effects of turbulence on the dynamics of cohesive particles. We consider the one-way coupled motion of small spherical particles in the two-dimensional, steady, spatially periodic cellular flow field commonly employed as initial condition for simulating Taylor-Green vortices (cf. figure 0(a)), with fluid velocity field

(1) |

where and represent the characteristic length and velocity scales of the vortex flow.

Keeping in mind that cohesive sediment grains in nature may be non-spherical, we nevertheless approximate each primary particle as a sphere that moves with the translational velocity and the angular velocity . These are obtained from the linear and angular momentum equations

(2) |

(3) |

where the primary particle moves in response to the Stokes drag force , the gravitational force , and the particle-particle interaction force . Here and indicate the fluid and particle velocities evaluated at the particle center. denotes the particle’s mass, its diameter, its density, and the total number of particles in the flow. We assume all particles to have the same diameter and density. and denote the dynamic viscosity and the density of the fluid, respectively, and is the gravitational acceleration. accounts for the direct contact force in normal and tangential direction, as well as for short-range forces due to lubrication and cohesion , where the subscript indicates the interaction between particles and . denotes the moment of inertia of a particle. represents the torque due to particle-particle interactions, where we distinguish between the direct contact torque and lubrication torque .

Following Biegert et al. (2017a), we represent the direct contact force by means of spring-dashpot functions, while the lubrication force is accounted for based on Cox and Brenner (1967) as implemented in Biegert et al. (2017b). The model for the cohesive force is based on the work of Vowinckel et al. (2019). It assumes a parabolic force profile, distributed over a thin shell surrounding each particle.

### 2.2 Non-dimensionalization

We choose , and as the characteristic length, velocity and time scales. Conceptually, these can be thought of as representing Kolmogorov scales. In this way, we obtain the dimensionless equation of motion for the particles as

(4) |

where dimensionless quantities are denoted by a tilde. The dynamics of the primary particles are characterized by the Stokes number and the settling velocity , where is the Stokes settling velocity of an individual, isolated primary particle. The dimensionless particle mass and density ratio are defined as and , respectively.

The dimensionless direct contact force between particles includes the normal component and the tangential component , which are defined as

(5) |

where is the normal surface distance between particles and . We account for the surface roughness of the particles, which is set to . is the tangential spring displacement, which denotes the accumulated relative tangential motion between two particles in contact. and denote the normal and tangential components of the relative velocity of particles and . represents the outward-pointing normal on the particle surface, and points in the direction of the tangential force. We use the parametrization for silicate grains described in Biegert et al. (2017a), so that we chose a standard friction coefficient of and obtain stiffness and and damping and to obtain a specified restitution coefficient as the ratio of impact to rebound velocity () for the normal component and rolling conditions for the tangential component of , respectively.

The dimensionless lubrication force between particles and has the normal and tangential components and , respectively, which are defined as

(6) |

where is the range of the lubrication force, and denote the tangential components of the relative translational velocity and the relative rotational velocity of the particles, respectively. The coefficients and take the values and , respectively (Biegert et al., 2017b).

The dimensionless cohesive force is defined as

(7) |

where represents the range of the cohesive force. The cohesive number indicates the ratio of the maximum cohesive force at to the characteristic inertial force

(8) |

where the Hamaker constant is a function of the particle and fluid properties and the characteristic distance . Vowinckel et al. (2019) provide representative values of for common natural systems.

To summarize, the simulations require as input parameters the dimensionless particle diameter , the number of particles , the density ratio , the settling velocity , the Stokes number and the cohesive number . For convenience, the tilde symbol will be omitted henceforth.

### 2.3 Validation: Aggregation and breakage of two particles

To validate our numerical implementation of the cohesive force model, we consider the interaction of two neutrally buoyant particles with , , that are placed symmetrically to the left and right of the stagnation point at (1,1) in figure 0(a). The particles are at rest initially, at a surface distance of , so that the cohesive force is at its maximum. Figure 2 presents the temporal evolution of the various forces acting on the particle to the left of the stagnation point, for the two scenarios of (a) floc breakage, and (b) floc aggregation. For the smaller value of , the drag force that tries to separate the particles is initially larger than the cohesive force that attracts them to each other (figure 1(a)). As the surface distance between the particles increases, the cohesive force decays and approaches zero. While the lubrication force acts to slow the separation of the particles, the overall net force acting on the particle is always negative, so that the particles gradually move apart. When the surface distance between the particles becomes larger than the range of the cohesive and lubrication forces, the net force equals the drag force.

Figure 1(b), on the other hand, focuses on a case in which the cohesive force initially is larger than the drag force, so that the particles approach each other. This process is slowed down by the lubrication force. The particles asymptotically approach an equilibrium position of near contact in which the separating drag force is balanced by the attractive cohesive force.

## 3 Large ensemble of particles

### 3.1 Computational setup

We now investigate ensembles involving more particles, to obtain insight into the flocculation dynamics of larger systems. We employ a computational domain of size , with periodic boundaries (figure 0(a)). All particles have identical diameters and densities. Initially they are at rest and separated, and randomly distributed throughout the domain. When the distance between two particles is less than , we consider them as part of the same floc. We then track the number of flocs as a function of time, with an individual particle representing the smallest possible floc. To improve the statistics, we repeat each simulation five times for different random initial conditions as the simulation results are statistically independent of the initial particle placement.

A typical floc configuration is shown in figure 0(b). Figure 2(a) presents results for a series of simulations with particles that have a size of 10% of the Kolomogorov length scale, i.e. . Further, the parameters for this scenario were , , , and . Since the particles are dispersed initially, the initial number of flocs . Subsequently decreases rapidly due to flocculation, before leveling off around a constant value that reflects a stable balance between aggregation and breakage. The transient variation of can be fitted by an exponential function of the form

(9) |

where we evaluate as the average number of flocs during the equilibrium stage . The agglomeration rate with the constraint is obtained via a least-square fit. We define the characteristic flocculation time scale as the time it takes for the number of flocs to decrease from its initial value to a characteristic number of flocs . Hence the corresponding characteristic time can be calculated as .

Figure 2(b) displays the statistical floc size distribution during the equilibrium stage , where the ”floc size” denotes the number of particles in a floc. refers to the number of the flocs of the same size. We find that the floc size distribution closely follows a log-normal distribution, consistent with previous experimental observations (Bouyer et al., 2004; Verney et al., 2011; Hill et al., 2011).

### 3.2 Influence of the governing parameters on the flocculation dynamics

Parameter | ||||||
---|---|---|---|---|---|---|

Range | 0.00015 - 0.025 | 0.01 - 9 | 0.04 - 0.1 | 0.0042 - 0.0916 | 1 - 3 | 0 - 2 |

In order to explore the dependence of the flocculation process on the key governing quantities, we carry out a total of 300 simulations covering the parameter ranges listed in table 1. Figure 3(a) shows that the number of flocs during the equilibrium stage decreases for increasing . Beyond , all of the primary particles aggregate into one large floc, as the cohesive forces overwhelm the hydrodynamic stresses trying to break up the floc. The characteristic flocculation time initially decreases as grows, and then levels off and remains constant. Figure 3(b) indicates that for large Stokes numbers the equilibrium floc number also asymptotically approaches one. Interestingly, we observe that the flocculation time displays a pronounced minimum around , which reflects the well-known optimal coupling between particle and fluid motion when the particle response time is of the same order as the characteristic timescale of the flow (Wang and Maxey, 1993). Under these conditions, particles rapidly accumulate near the edges of the vortices, which facilitates the formation of flocs.

### 3.3 A new flocculation model based on the simulation data

According to Khelifa and Hill (2006a, b), for flocs of fractal dimension the mean floc size is related to the average number of primary particles per floc :

(10) |

Eqn. (9) yields for the average number of particles per floc

(11) |

where the initial number of particles per floc is , and the average number of particles per floc during the equilibrium stage is . Fitting the simulation results over the parameter ranges listed in table 1 yields {subequations}

(12) | |||||

(13) |

(14) |

For the present cellular model flow the values in eqns. (12) and (14) yield optimal agreement with the simulation data with the fitting deviation of (figure 5a). For real turbulent flows, we will determine and by calibrating with experimental data, as will be explained below.

Winterwerp (1998) introduced a population balance model that accounts for aggregation and breakage for low turbulence levels. His model has the form

(15) |

where indicates the shear rate of the turbulence (units ), represents the sediment concentration (), and denotes the yield strength of the flocs (). Winterwerp suggests the values , , and . The empirical coefficients and depend on the physico-chemical properties of the sediment and fluid. While this model has enjoyed wide popularity in the literature (Winterwerp et al., 2006; Son and Hsu, 2009, 2008; Lee et al., 2011; Keyvani and Strom, 2014; Strom and Keyvani, 2016), it has also been pointed out that for large turbulent shear and sediment concentrations the model predicts that the floc size will be larger than the Kolmogorov scale , which is not consistent with experimental observations (Keyvani and Strom, 2014; Kuprenas et al., 2018; Sherwood et al., 2018). To address this issue, Kuprenas et al. (2018) recently suggested the modification . In the following, we will compare predictions by the current model with both of these earlier models.

Tran et al. (2018) measured the floc size in turbulence for constant shear rate and sediment concentration (figure 5b-5f). They determined the empirical coefficients and by calibrating with the experimental data for . In a similar fashion, we will determine the constants and required for our model by calibrating with the same case displayed in figure 5b. Towards this end, we need to convert the experimental data into characteristic length and velocity scales that can be employed in our model, eqns. (11)-(14). We accomplish this by setting and . Furthermore, we assume the Hamaker constant to be (see Vowinckel et al., 2019, p. 37-39), and the fractal dimension , which yields the correction constants and . The mean floc size can then be obtained from eqn. (10). For lower sediment loadings, figure 5c shows that our model yields predictions similar to those of Kuprenas for the equilibrium floc size, while it tends to perform somewhat better than Kuprenas’ model during the transient stage. These differences become more pronounced for higher sediment loadings (figure 5d-5f), and they likely reflect the more realistic modeling of the particle-particle interactions in the present simulations. For these particular flow conditions, Winterwerp’s model yields valid predictions only for sediment loads below approximately 200mg/L. Hence, the results encourage the use of our conceptually simplified cellular flow model as a cost-efficient tool to derive scaling laws in the form of eqns. (10)-(14) for a wide parameter range for flocculation of cohesive particles in turbulent flow conditions.

## 4 Conclusions

We have analyzed the flocculation dynamics of cohesive sediment via one-way coupled simulations in a model turbulent flow field. The computational model accounts for Stokes drag, lubrication, cohesive and direct contact forces, and it yields the time-dependent floc size as a function of the governing dimensionless parameters. The simulations reproduce the transient growth of the cohesive flocs, as well as the emergence of a log-normal equilibrium distribution governed by the balance of aggregation and breakage. By accounting for the detailed physical mechanisms governing particle-particle interactions, the simulations demonstrate that flocculation proceeds most rapidly when the Stokes number of the primary particles is O(1). We employ the computational data in order to propose a new flocculation model. As it is based on a more realistic representation of particle-particle interactions, this new model yields improved agreement with the experimental measurements of Tran et al. (2018), especially during the transient stages.

Acknowledgements

The authors thank D. Tran for providing his experimental data for comparison purposes. EM gratefully acknowledges support through NSF grants CBET-1803380 and OCE-1924655, as well as by the Army Research Office through grant W911NF-18-1-0379. TJH received support through NSF grant OCE-1924532. KZ is supported by the China Scholarship Council, as well as by the China National Fund for Distinguished Young Scientists through grant 51425603. BV gratefully acknowledges support through German Research Foundation (DFG) grant VO2413/2-1. Computational resources for this work used the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by NSF grant TG-CTS150053.

## References

- The motion of solid spherical particles falling in a cellular flow field at low Stokes number. Phys. Fluids 26 (9), pp. 093302. Cited by: §1, §2.1.
- A collision model for grain-resolving simulations of flows over dense, mobile, polydisperse granular sediment beds. J. Comput. Phys. 340 (), pp. 105–127. Cited by: §1, §2.1, §2.2.
- High-resolution simulations of turbidity currents. Prog. Earth Planet. Sci. 4 (1), pp. 33. Cited by: §2.1, §2.2.
- Experimental analysis of floc size distribution under different hydrodynamics in a mixing tank. AIChE 50 (), pp. 2064–2081. Cited by: §3.1.
- The slow motion of a sphere through a viscous fluid towards a plane surface. Small gap widths, including inertial effects. Chem. Eng. Sci. 22 (), pp. 1753–1777. Cited by: §2.1.
- Observations of the sensitivity of beam attenuation to particle size in a coastal bottom boundary layer. J. Geophys. Res.:Oceans 116 (C02023), pp. . Cited by: §3.1.
- Influence of cycles of high and low turbulent shear on the growth rate and equilibrium size of mud flocs. Mar. Geol. 354 (), pp. 1–14. Cited by: §3.3.
- Kinematic assessment of floc formation using a Monte Carlo model. J. Hydraul. Res. 44 (4), pp. 548–559. Cited by: §3.3.
- Models for effective density and settling velocity of flocs. J. Hydraul. Res. 44 (3), pp. 390–401. Cited by: §3.3.
- A shear-limited flocculation model for dynamically predicting average floc size. J. Geophys. Res.:Oceans 123 (), pp. 6736–6752. Cited by: Figure 5, §3.3.
- A two-class population balance equation yielding bimodal flocculation of marine or estuarine sediments. Water Res. 45 (), pp. 2131–2145. Cited by: §1, §3.3.
- Physicochemical hydrodynamics. Prentice Hall, Inc. (), pp. . Cited by: §1.
- Effect of variable fractal dimension on the floc size distribution of suspended cohesive sediment. J. Hydrology 343 (), pp. 43–55. Cited by: §1.
- The motion of small spherical particles in a cellular flow field. Phys. Fluids 30 (), pp. 1915–1928. Cited by: §1, §2.1.
- A tri-modal flocculation model coupled with TELEMAC for estuarine muds both in the laboratory and in the field. Water Res. 145 (), pp. 473–486. Cited by: §1.
- Cohesive and mixed sediment in the regional ocean modeling system implemented in the coupled ocean atmosphere wave sediment-transport modeling system. Geosci. Model Dev. 11 (), pp. 1849–1871. Cited by: §3.3.
- Stochastic flocculation model for cohesive sediment suspended in water. Water 7 (), pp. 2527–2541. Cited by: §1.
- Flocculation model of cohesive sediment using variable fractal dimension. Environ. Fluid Mech. 8 (1), pp. 55–71. Cited by: §1, §3.3.
- The effect of variable yield strength and variable fractal dimension on flocculation of cohesive sediment. Water Res. 43 (14), pp. 3582–3592. Cited by: §1, §3.3.
- Flocculation in a decaying shear field and its implications for mud removal in near-field river mouth discharges. J. Geophys. Res.:Oceans 121 (), pp. 2142–2162. Cited by: §3.3.
- How do changes in suspended sediment concentration alone influence the size of mud flocs under steady turbulent shearing?. Cont. Shelf Res. 158 (), pp. 1–14. Cited by: §3.3, §4.
- Behaviour of floc population during a tidal cycle: laboratory experiments and numerical modeling. Cont. Shelf Res. 31 (10), pp. 64–83. Cited by: §1, §3.1.
- Settling of cohesive sediment: particle-resolved simulations. J. Fluid Mech. 858 (), pp. 5–44. Cited by: §1, §2.1, §2.2, §3.3.
- Settling velocity and concentration distribution of heavy particles in homogeneous isotropic turbulence. J. Fluid Mech. 256 (), pp. 27–68. Cited by: §3.2.
- A heuristic formula for turbulence-induced flocculation of cohesive sediment. Estuar. Coast. Shelf Sci. 68 (), pp. 195–207. Cited by: §3.3.
- A simple model for turbulence induced flocculation of cohesive sediment. J. Hydraul. Res. 36 (3), pp. 309–326. Cited by: §1, Figure 5, §3.3.
- Flocculation and me. Water Res. 114 (), pp. 88–103. Cited by: §1.