## Abstract

Due to climate change, sea level rise and anthropogenic development, coastal communities have been facing increasing threats from flooding, land loss, and deterioration of water quality, to name just a few. Most of these pressing problems are directly or indirectly associated with the transport of cohesive fine-grained sediments that form porous aggregates of particles, called flocs. Through their complex structures, flocs are vehicles for the transport of organic carbon, nutrients, and contaminants. Most coastal/estuarine models neglect the flocculation process, which poses a considerable limitation of their predictive capability. We describe a set of experimental and numerical tools that represent the state-of-the-art and can, if combined properly, yield answers to many of the aforementioned issues. In particular, we cover floc measurement techniques and strategies for grain-resolving simulations that can be used as an accurate and efficient means to generate highly-resolved data under idealized conditions. These data feed into continuum models in terms of population balance equations to describe the temporal evolution of flocs. The combined approach allows for a comprehensive investigation across the scales of individual particles, turbulence and the bottom boundary layer to gain a better understanding of the fundamental dynamics of flocculation and their impact on fine-grained sediment transport.

### Keywords

- cohesive sediment
- floc measurement
- particle-resolved direct numerical simulation
- continuum model
- population balance equation

## 1. Introduction

Cohesive sediment transport is a tightly coupled system driven by hydrodynamic forcing, resuspension, deposition, and flocculation. Turbulence intensity in the bottom boundary layers and the water column mixed layers controlled by currents (e.g., river outflows, tidal currents), surface waves, and estuarine circulations can vary by several orders of magnitude in terms of turbulent dissipation rate. Hence, the turbulent shear rate defined as

Computer simulation models are commonly the chosen tools that coastal managers use to predict sediment transport rates. In these continuum models, the fluid motion is computed via the continuity equation and the Navier-Stokes equations or spatially and temporally averaged variants thereof, whereas the sediment is represented as a concentration field [8]. This approach allows to simulate large spatial scales covering entire estuaries, because the governing equations are solved on the same Eulerian grid. However, these types of models require closures to account for the unresolved physics of the sediment dynamics, in particular vertical sedimentary distributions [9, 10] and mass fluxes. The latter is the product of the concentration and the settling velocity. Manning and Bass [11] found that mass settling fluxes can vary over four or five orders of magnitude during a tidal cycle in mesotidal and macrotidal estuaries; therefore, a realistic representation of flux variations is crucial to an accurate depositional model.

The specification of the flocculation term within numerical models depends on the sophistication of the model structure. Until recently, even the conceptual relationship between floc size, suspended particulate matter (SPM) concentration and turbulent shear stress proposed by Dyer [12] (see Figure 1a) remained largely unproven. Hence, much more work needs to be done in order to arrive at robust simulation tools with predictive capacity. The present contribution, therefore, provides a review on the state-of-the-art of floc measurements in both the field and laboratory. In addition, we review a newly emerging technique of particle-resolved simulations that can provide a promising alternative avenue to generate data of small-scale sediment dynamics to derive the missing constitutive equations for continuum models. Finally, we provide a summary of current techniques that are used in continuum models to account for cohesive sediment dynamics, where we explicitly point to the components that deserve more research in the future.

## 2. Assessment of temporal floc evolution

### 2.1 Floc measurements

#### 2.1.1 Direct floc size measurements

The presence of large estuarine macroflocs was initially observed in situ using underwater photography [13]. However, floc breakage occurs during sampling in response to the additional shear created by the instrumentation [14]. To overcome this problem, less-invasive techniques for measuring floc properties in situ have been developed. Usually, these can be divided into devices that solely measure floc size (

The strength of video-based floc measurements is that they minimize the number of assumptions used during the data processing and interpretation stages. Devices that only measure the size component require additional gross and often incorrect assumptions regarding the relationship between settling velocity, floc size, and floc density. The settling velocity of a floc is a function of both its size and effective density, and both of these floc components can display variations spanning three to four orders of magnitude within any one floc population [30, 31, 32].

Of note, the LabSFLOC suite of high-resolution, low intrusive, underwater video camera systems for the past 20+ years have been regarded internationally as a benchmark device for the sampling and dynamical simultaneous measurement of sizes and settling velocities for entire populations of flocs and a range of bio-sedimentary particles, and this enables individual floc effective density values to be determined by applying a modified Stokes Law [33]. This also enables the calculation of additional floc properties including: structural composition (porosity, fractal dimension), shape, sedimentary mass flux, and floc population mass-balancing, all from within a wide range of aquatic environments. In conclusion, selection of the most appropriate instrumentation is paramount when attempting to parameterise flocculated cohesive sediments. Manning et al. [34] provide a detailed review of many of these floc measuring systems.

#### 2.1.2 Parameterizing floc data

To aid the interpretation of floc characteristics and for their inclusion in sediment transport models, each floc population can be segregated into various subgroupings according to floc size. Sample-mean floc values can be computed (i.e., a single value per floc population) to show generalized floc property trends.

Dyer et al. [35] reported that a single mean or median settling velocity did not adequately represent an entire floc spectrum, especially in considerations of a flux to the bed. Dyer et al. [35] recommended that the best approach for accurately representing the settling characteristics of a floc population was to split a floc distribution into two or more components, each with their own mean settling velocity. Both Eisma [13] and Manning [32] concur with this finding by suggesting that a more realistic and accurate generalization of floc behavior can be derived from the macrofloc and microfloc fractions. These two floc fractions form part of Krone’s [36] classic order-of-aggregation theory and produce two floc property values per floc population.

Macroflocs are large (typically

The smaller microflocs (typically

In terms of flocculation kinetics [44], the macroflocs tend to control the fate of purely muddy sediments in an estuary [45]; this is because the smaller microflocs generally settle at less than 1 mm

#### 2.1.3 Floc data example

In order to illustrate the spectral variability of floc properties, each floc population can be divided into various size bands. The band divisions can be chosen to best fit the data collected. LabSFLOC data for a mud sample from the Medway Estuary, UK, provide a graphical illustration of size banding and the increasing settling velocities associated with larger flocs (Figure 3). Twelve size bands (SB) have been used to represent the Medway floc data, with SB1 representing microflocs less than

#### 2.1.4 Floc settling modeling approaches

#### 2.1.4.1 Constant settling velocity

Specification of the flocculation term within numerical models depends on the sophistication of the model. The simplest parameterisation is a single floc settling velocity value that remains constant in both time and space (one coefficient). These fixed settling values are usually in the range of 0.5–5 mm

#### 2.1.4.2 Power law settling velocity

The next step is to use gravimetric data provided by field settling-tube experiments to relate floc settling velocity to the instantaneous SPM concentration, using a power law with two coefficients (e.g., [48]; see Figure 4a). Empirical results have shown a generally exponential relationship between the mean, or median, floc settling velocity and SPM concentrations for concentrations

#### 2.1.4.3 The van Leussen parameterisation

More recently, a number of authors have proposed simple theoretical formulae interrelating a number of floc characteristics that can then be calibrated using empirical studies. Such an approach has been used by van Leussen [19], who utilized a formula that modifies the floc settling velocity in still water by a floc growth factor, due to turbulence, and then reduces it by a turbulent floc disruption factor. The reference settling velocity (taken at low turbulent shear conditions),

where

#### 2.1.4.4 The Lick et al. parameterisation

A number of authors have attempted to observe how the floc diameter changes in turbulent environments. In particular, Lick et al. [53] derived an empirical relationship based on laboratory measurements made in a flocculator. They found that the floc diameter varied as a function of the product of the SPM concentration and the turbulence parameter as the turbulent shear rate,

where

#### 2.1.4.5 The Manning and Dyer parameterization

The Manning Floc Settling Velocity (MFSV) algorithm for settling velocity [54] is based entirely on empirical observations made in situ using nonintrusive floc and turbulence data acquisition techniques in a wide range of estuarine conditions. The floc population size and settling velocity spectra were sampled using the video-based INSSEV instrument and LabSFLOC data.

The Manning-Dyer algorithms were generated by a parametric multiple regression statistical analysis of key parameters, which were generated from the raw, spectral floc data. Detailed derivations and preliminary testing of the floc-settling algorithms are described by Manning [26, 55]. Although the resulting empirical formulae are not presented in a fully dimensionless form, these formulae have the merit of being based on a large dataset of accurate, in situ settling velocity measurements (157 individually observed floc populations), acquired from different estuaries (Tamar, Gironde and Dollard) and different estuarine locations, such as the turbidity maximum and the intertidal zone.

The algorithms are based on the segregation of flocs into macroflocs (

Equations are given for: the settling velocity of the macrofloc fraction (

for

for

An example of this is the implementation of the algorithm in a TELEMAC-3D numerical model of the Thames Estuary, UK [57], in which it was shown that the use of the Manning algorithm greatly improved the reproduction of observed distributions of SPM concentrations compared with the other formulations, both in the vertical and horizontal dimensions.

The Manning settling algorithms have been extended to cater for mixed sediment flocculation settling, including different ratios of mud:sand ([24, 47, 56]; see Figure 5). These algorithms are a major step forward in establishing a reliable estimate of the settling velocity. It has been developed based on a large and reliable dataset, it caters for the spectrum of hydrodynamic conditions that occur during a typical tidal cycle [58] (a feature often lacking in the settling terms of many estuarine sediment transport models) and has been shown to more accurately reproduce the distribution of suspended sediment compared with simpler settling models.

Soulsby et al. [59] has developed a more ‘physics-based’ version of the empirical model based on the Manning-Dyer formulation, called Soulsby-Manning 2013. It should be noted that for flocculation algorithms and models that include turbulence as a contributing variable, it is vital to ensure that the turbulence data are accurate, otherwise it has significant implications for the accuracy of the calculated floc settling characteristics.

#### 2.1.4.6 Complex population approaches

Lee et al. [60] applied a time-evolving two-class population balance equation (PBE) to determine the spatially and temporally changing distribution of fixed-size microflocs and size-varying macroflocs for bimodal floc distributions, with a fractal relationship between floc size and mass to derive the distribution of settling velocities. However, the authors felt that further intensive investigation of the aggregation and breakage kinetics would be required before their model was generally applicable when compared with the simpler approach of Manning and Dyer [54] and, presumably, Soulsby et al. [59].

Verney et al. [61] applied a time-evolving, multi-fraction model to determine the spatially and temporally changing distribution of the numbers of flocs in each size fraction, with a fractal relationship between floc size and mass to derive the distribution of settling velocities.

A relationship between the floc settling velocity and floc properties and fractal dimensions is given by Winterwerp [62]. A fractal approach has been used by Winterwerp [63] to solve a differential equation that simulates the time-varying representative floc diameter, from which floc density is derived from fractal considerations, and settling velocity obtained from a Stokes-like formula. Winterwerp et al. [64] also used a simplified fractal model to relate settling velocity to a turbulent shear parameter, the instantaneous concentration, and water depth.

The state-of-the-art model for floc structure is to assume a fractal structure. Many studies [65, 66, 67] indicate that the most sensitive parameters in a fractal model controlling the resulting settling velocity are the primary particle properties (primary particle diameter, density and their distributions) and the fractal dimension. For organic-rich particles, evidence suggests that the fractal dimension highly depends on stickiness [68].

Floc breakup in the existing size-class-based PBE formulation is modeled by assuming an invariant floc structure (e.g., fractal dimension

A fractal model is widely used to parameterize floc structure. Fractals are a mathematical simple approach, where scientists feel computationally ‘comfortable’ and therefore are happy to ‘shoehorn’ all flocs into this framework, and a fractal dimension of around 2 is often used. However, its applicability for heterogeneous sediments remains to be proved. Moreover, for flocs of high organic content, stickiness can be significantly enhanced due to the presence of extracellular polymeric substances (EPS) and transparent exopolymer particles (TEP). Field observations suggest that the fractal dimension for inorganic particles is larger than 2.0 while for organic rich flocs, it can be smaller than 2.0 [68]. There are also empirical formulations suggesting that the fractal dimension depends on floc size [65, 66].

Unlike Verney et al. [61], who use floc diameter, Maggi et al. [71] describe the floc population based on the number of primary particles in the flocs, which appears to make the incorporation of a variable fractal dimension straightforward. Moreover, Maggi et al. [71] adopt a sophisticated collisional efficiency closure that considers the effects of floc size and permeability.

#### 2.1.5 Future floc modeling directions

Real flocs are multi component of different densities; even measuring real fractal dimensions is highly problematic. The emergence theory has been utilized in many disciplines (e.g. [72, 73, 74]) and provides a valid alternative and potentially more realistic approach for representing real multi-component mud floc structures. Both Cranford et al. [75], and Rietkerk and van de Koppel [76] have successfully adapted an emergence approach for application to natural biomaterials and ecosystems (respectively).

By utilizing an emergence framework for flocculation, at one end a simple fractal representation would still operate for basic, geometrically repeating simple floc structures (e.g. flocs composed from a single clay primary particle). Whilst as the flocs before more complex in structure, composition and geometry, and fractal theory become less representative, the emergence would adapt to a more suitable floc representation. It is envisaged that this new emergence approach [77] could cope better and more efficiently and realistically for real flocs at a wide range in resolution scales all using real image data at each scale. This will provide a level of error checks that are not supported by regular fractal approaches. Nonetheless, we are some way off from implementing this approach in a numerical floc model and more fundamental research on floc dynamics, properties and characteristics is required, in particular 2D and 3D floc imaging techniques (e.g. [27, 78, 79, 80, 81, 82]).

### 2.2 Grain-resolving simulations

Grain-resolving simulations are a powerful tool to obtain detailed, high-fidelity data of complex fluid-particle systems. Despite its rather large computational costs, recent advances in computational power have made it possible to perform grain-resolving simulations on scales that become relevant for sediment transport phenomena. The idea is to compute the trajectories of all the individual grains in a flow. Typically, the flow is computed on a Eulerian grid that is fixed in space and time. It can either be approximated by assuming a prescribed background flow or by solving the Navier–Stokes equations. The movement of the particles is then computed by their equations of motion in a Lagrangian sense, i.e. the particle is free to move within the entire computational domain. Hence, this representation is commonly referred to as the Euler–Lagrange approach.

Depending on their fluid-particle coupling, several schemes can be employed. If the particles are driven by a fluid flow but do not modify the flow field, the scheme is considered one-way coupled, whereas the fluid-particle mixture is two-way coupled if the flow is modified by the particle motion as well. In addition, momentum exchange of particles may be accounted for by means of collision and contact or any other particle related force. Using a two-way coupled simulation that accounts for particle-particle interactions is considered a fully-coupled scheme. In the context of the present study, attractive cohesive forces can contribute to the particle-particle interaction by binding grains into aggregates that are much larger than the individual primary particle. This process can be understood as flocculation. In the following, we will first review the governing equations to compute the particle motion in a Lagrangian sense and then proceed to present two examples to model flocculation of cohesive sediment to investigate aggregation and settling processes.

#### 2.2.1 Computing the particle motion

Regardless of the background flow, we prescribe the motion of each primary cohesive particle

where the primary particle

Following Biegert et al. [83, 84], Zhao et al. [85, 86] represent the direct contact force

Here,

#### 2.2.2 Aggregation

The flocculation process is strongly affected by the turbulent nature of the underlying fluid flow. Small-scale eddies modify the collision dynamics of the primary particles and hence the growth rate of the flocs, while turbulent stresses can result in the deformation and breakup of larger cohesive flocs. Hence, the dynamic equilibrium between floc aggregation and breakage is governed by a complex and delicate balance of hydrodynamic and inter-particle forces.

In the spirit of earlier investigations [89, 90], Zhao et al. [85] apply a simple model flow in order to investigate the effects of turbulence on the dynamics of cohesive particles. These authors 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 6a), with fluid velocity field

where

The dynamics of the primary particles are characterized by the Stokes number

where the Hamaker constant

Zhao et al. [85] employ a computational domain with periodic boundaries. All particles have identical diameters

where

where

For the present cellular model flow the values

#### 2.2.3 Hindered settling

It is well known that the settling behavior of a dense suspension differs substantially from the settling behavior of a single grain. Particles settling in a dense suspension induce a counterflow and experience friction by colliding with other particles. These processes yield the so-called hindered settling, which is substantially slower than the settling of an individual particle and depends on fluid and particle properties. Nevertheless, the Stokes settling velocity of an individual grain is still widely used to quantify the settling speed of sediment in particle-laden turbidity currents (e.g., [84]). Hence, constitutive equations to predict the settling speed as a function of the local flow conditions can enhance existing computational frameworks for the analysis of turbidity currents. To investigate the effects of the settling behavior of flocculating cohesive sediment by means of grain-resolving simulations [88, 93], it is important to not only account for frictional contact between particles in a dense suspension but also for the modifications of the fluid flow that is caused by settling particles displacing the fluid underneath them [94]. In this case, one needs to solve the Navier-Stokes equations for an incompressible Newtonian fluid:

along with the continuity equation

where

While this set of equations represents a fully coupled system, it is important to note that the relevant scales that define the system have changed compared to Section 2.2.2. In the scenario that investigates hindered settling of polydisperse cohesive sediment, the fluid flow is driven by moving particles. Hence, the relevant scale becomes

and a characteristic particle Reynolds number

Particles that are placed in a tank will settle due to gravity thereby displacing the fluid and accumulate at the bottom of this tank. The induced counter flow as well as the particle-particle interaction yields frictional contacts and flocculation due to cohesive forces, which is the desired situation for hindered settling [88]. For small particle sizes, where cohesive forces remain relevant, Vowinckel et al. [88, 93] obtain a faster settling behavior for cohesive sediment as compared to its non-cohesive counterpart (Figure 8). During the settling process, the sediment will transform from a suspended state, where the weight is fully supported by the fluid pressure, to a deposited state, where the weight is supported through contact chains of the deposited sediment that extent all the way to the bottom of the tank [93]. This process is described by the effective stress concept, which states that the total stress, i.e. the submerged weight of sediment, is supported by either the particle pressure or the effective stress due to particle contact [97].

## 3. Coastal modeling

Coastal modeling typically refers to the modeling of regional scale (10–100 km^{2}) coastal and estuarine processes over a timescale of hours to months. Due to the large spatial and temporal scales that need to be covered, a Reynolds-averaged Navier-Stokes (RANS) model is adopted (e.g., [98, 99, 100]). Turbulence dissipation and mixing are parameterized with two-equation closure models via a diffusion process. Moreover, when surface waves are present, the individual wave-phase is often not resolved. The generation, transformation and dissipation of the random wave field are represented by a wave spectrum and solved by using the spectral wave action balance equation [101]. Wave-period-averaged wave statistics are then coupled with the coastal models. Consequently, the wave bottom boundary layer processes cannot be directly resolved and additional parameterizations are needed, such as the apparent roughness [102], i.e. the effect of the wave bottom boundary layer on the current resolved by the coastal model, wave-driven small-scale seabed morphological features (e.g., ripples), and the near-bed sediment transport processes (often called bedload or near-bed load). The suspended load transport for a range of non-cohesive sediment classes above the wave bottom boundary layer can be resolved in the coastal models via the conservation of mass. One of the main challenges for extending these coastal models for simulating cohesive sediment transport is the parameterization of settling velocity due to flocculation.

Since the recognition of flocculation in controlling the settling velocity of cohesive sediment in coastal and marine environments [12, 41], significant progress has been made, particularly in the physical parameters controlling the floc dynamics, floc size distribution and their relationships with settling velocity statistics. To name a few, the role of turbulent shear and particle concentration in determining the aggregation rate and the resulting floc size has been quantified [62, 103, 104, 105] (see also Figure 1a). Particularly, in many tidal boundary layers and laboratory experiments of homogeneous turbulence, the mean floc size is observed to be limited by the Kolmogorov length scale (e.g., [106, 107]). Moreover, the relationship between floc size population and settling velocity (or floc density) in both idealized and realistic conditions has been revealed (e.g., [108, 109]) and the fractal dimension has been applied to model these relationships [65, 66, 69, 71] (see also Figure 3). Finally, researchers have begun to understand complex floc characteristics in estuaries dominated by organic particles (e.g., [110]), high cohesion due to TEP (e.g., [111]) and the presence of sand (e.g., [112, 113], see also Figure 5). A more complete discussion on these observational-based empirical parameterizations is provided in Section 2.1.4.

Advancements have also been made in modeling flocculation processes of cohesive sediments. Following the summary presented in Section 2.1, this section focuses on a more in-depth discussion of the complex population approach. Pioneering work by Winterwerp [62, 94] established a robust single-size class (averaged floc size) flocculation modeling framework. This framework has been refined by Kuprenas et al. [91] to limit the floc size growth by the Kolmogorov length scale. A more sophisticated flocculation model [114] based on the population balance equations (PBE) has been incorporated into the Princeton Ocean Models (POM) by Xu et al. [115] to study the dynamics of Estuarine Turbidity Maximum (ETM). Recently, Sherwood et al. [116] incorporated the PBE flocculation model FLOCMOD by Verney et al. [2] into the Regional Ocean Modeling system (ROMS), which is part of the Coupled Ocean-Atmosphere-Wave-Sediment Transport Modeling System (COAWST, [98]). The model is used to study cohesive sediment transport in an idealized setting and a realistic application in the York River estuary. Through a direct simplification of the PBE type model, a tri-modal flocculation model was recently developed in the coastal model TELEMAC [117]. Last but not least, the empirical parameterization of the floc settling velocity MFSV algorithm (see Section 2.1.4) suggested by Manning and Dyer [54] has been incorporated into TELEMAC-3D [57], while more recently the nondimensional version of MFSV proposed by Soulsby et al. [59] has been incorporated into the Finite Volume Community Ocean Model (FVCOM, [99]). These closure models cover a wide range of flocculation physics incorporated or neglected, and the resulting computational cost also varies from minimal to very significant. It is worth to point out that besides the turbulence-averaged models discussed so far, a PBE formulation for flocculation dynamics has recently been incorporated into a turbulence-resolving large-eddy simulation model by Liu et al. [118] to study floc dynamics in the upper-ocean mixing layer subject to Langmuir turbulence.

### 3.1 Structure of continuum model

The Reynolds-averaged suspended sediment mass concentration

where

### 3.2 Population balance equation

In the population balance formation, the sediment mass concentration

where

As mentioned before, the sediment mass concentration distribution can be described by floc size class [61] or number of primary particles in the floc [66]. Here, we focus on the more popular one using floc size class with each class having a floc diameter of

where

By, furthermore, assuming that the floc structure follows a fractal relationship, we can calculate the mass of a floc as [69]:

Therefore, with the given fractal dimension and primary particle properties,

where

where

It is worth noting that in the population balance formulation, the floc settling velocity of a particular size class is treated as a constant determined by the given fractal dimension, primary particle properties, and fluid properties. Therefore, it is more suitable for typical coastal modeling systems due to their numerical treatment of advection. In typical field conditions, a size-class based population balance formulation is reported to require at least 10–20 size classes [61, 116].

After proposing the appropriate gain and loss terms in Eq. (18), the full dynamics of floc transport, settling and re-distribution of sediment mass among all floc size classes due to flocculation can be modeled. In practice, some models solve a system of partial differential equation for the number concentration

In this paper, the essential model formulations and closures of Verney et al. [61] are reviewed. The purpose here is not to discuss the model details, since they can be found in the cited references. Rather, this section is intended to bridge the discussions in Section 1 and Section 2 by focusing on the key model elements that are sensitive to the model results and hence may require more physical understanding. The governing equation for floc number concentration

in which

and

where

The quantity

and

Essentially, gain of flocs

where

As demonstrated in Verney et al. [61], their PBE-based flocculation model can predict several key features of floc dynamics observed in the field. For instance, the model is capable of reproducing the observed slower aggregation and more rapid fragmentation process (so-called clock-wise hysteresis of aggregation/fragmentation process) during a tidal cycle. As the floc size directly controls the settling velocity, capturing the hysteresis of floc aggregation/fragmentation is essential to further predict the net sediment fluxes during a tidal cycle. Moreover, Verney et al. [61] showed that a bimodal distribution of flocs often observed in the field can be reproduced by the PBE model by including a mix of different breakup distribution functions. Although the PBE-based flocculation models provide a promising modeling framework for cohesive sediment transport, there are limitations that require future investigations.

The most sensitive empirical parameters in FLOCMOD are the collisional efficiency

The sensitivity of the modeled mean floc diameter to the prescribed fractal dimension has also been discussed in Verney et al. [61]. More recently, Penaloza-Giraldo et al. [124] report that the temporal evolution of floc sizes (timescale to reach the equilibrium floc sizes) are sensitive to the fractal dimension

#### 3.2.1 Parameterizing the floc structure with a fractal dimension

To estimate floc mass and floc density in a given floc size, further assumptions on floc structures is needed. The state-of-the-art model for floc structure is to assume a fractal structure, which renders Eqs. (21) and (22) useful. In the PBE models for floc dynamics, fractal dimension directly affects the breakup term via the fragmentation probability function (see Eq. (30)). While most of the existing flocculation models assume a constant fractal dimension, examining the field and laboratory data by relating measured floc settling velocity and floc size (see Eqs. (22) and (23)) suggests that the fractal dimension may not be a constant, especially when considering the PBE equations are very sensitive to the prescribed fractal dimension value. Recent grain-resolving simulations (see Section 2.2.2 and the studies of [85, 86] referenced therein) also confirms that the fractal dimension depends on floc size. Researchers have proposed to model fractal dimension as a function of floc size [65, 71], however, whether it significantly improves the modeled flocculation processes remain to be proven.

#### 3.2.2 More complete descriptions of collisional efficiency and fragmentation rate

As discussed by Hill and Nowell [125], the collision efficiency is practically treated as an empirical tuning parameter that parameterizes three main processes: encounter, contact and sticking. Encounter and contact are physical processes while sticking is associated with chemical-biological processes. From a physical perspective, only sticking efficiency is solely an empirical parameter. Maggi et al. [71] used a more complex collisional efficiency formulation that depends on the size and porosity of two colliding flocs. Through detailed laboratory experiments, Soos et al. [126] proposed a collisional efficiency formulation

While the existing studies mostly treat the fragmentation rate to be a model constant, physically the fragmentation rate

with

#### 3.2.3 Improved physical understanding of breakup distribution functions

In the coastal sediment literature, detailed studies on floc breakup, particularly from the observational perspective, are rare. In the water quality literature, Jarvis et al. [127] provide some insights into the breakup distribution function. First, they discern the fragmentation mechanism, similar to the binary breakup, as the most likely scenario to occur when flocs are subjected to tensile stress acting across the floc. Secondly, the erosion mechanism in floc breakup is likely due to shear stress acting tangentially to the floc surface. Based on this argument, researchers hypothesize that the floc breakage types may depend on the ratio of floc size to the Kolmogorov length scale (smallest turbulent eddy size). However, the results are not conclusive and more comprehensive studies on how turbulent eddies interact with flocs and causing floc breakage are warranted.

## 4. Conclusions

We have presented an overview covering different types of floc analyses based on experimental measurements and grain-resolved simulations. These tools are currently emerging and show a very promising perspective to generate the data needed to account for unresolved cohesive sediment dynamics in continuum models with high fidelity. More work will be needed in the future to cover the different aspects laid out in this chapter. Those are in particular, the effects of biofilms, the settling velocity of different types of flocs, as well as the aggregation and break-up efficiencies governing the exchange between different classes of PBE-type models. The knowledge to be gained can lead to a new generation of continuum models that enable simulations with predictive power for entire estuaries, which will bring inestimable advantages for these attractive settlement areas, both in economic terms and in terms of an increased quality of life.

## Acknowledgments

BV gratefully acknowledges the support through the German Research Foundation (DFG) grant VO2413/2-1. KZ is supported by the National Natural Science Foundation of China through the Basic Science Center Program for Ordered Energy Conversion (51888103). LY, AJM, TJH and EM gratefully acknowledge supports by US National Science Foundation through a collaborative research between the University of Delaware (OCE-1924532) and the University of California Santa Barbara (OCE-1924655). AJM’s contribution towards this research was partly supported by the US National Science Foundation under grants OCE-1736668 and OCE-1924532, TKI-MUSA project 11204950-000-ZKS-0002, and HR Wallingford company research FineScale project (ACK3013_62).

## Nomenclature

αij=α | collisional efficiency |

βi=β | fragmentation rate |

Γp,i | fluid-particle interface of particle i |

δi,j | Kronecker delta |

ζ0 | characteristic distance |

ζn | gap size in between particles |

ζmin | limiting gap size in between particles |

λ | cohesive force range |

μ | dynamic fluid viscosity |

ν | kinematic fluid viscosity |

νt | turbulent viscosity |

Πki | breakup distribution function |

ρf | fluid density |

ρkf | floc density of class k |

ρp,i | density of particle i |

ρs | density ratio |

ρs | primary particle density |

ρw | density of water |

τ | turbulent shear stress |

τ | hydrodynamic stress tensor |

τs | nondimensional reference time |

ϕ | particle volume fraction |

ωp,i | Angular velocity vector of particle with index i |

Aij | Shear-driven binary collision probability function |

Ah | Hamaker constant |

a1 | model constant |

a2 | model constant |

b | agglomeration rate |

Bi | fragmentation probability function |

C | SPM concentration |

c | empirical constant |

ck | sediment mass concentration of class k |

Co | cohesive number |

D | floc size |

D50 | median grain size |

D¯f | mean floc size |

Df,k | floc diameter of class k |

Dp | primary particle diameter |

Dp,i | diameter of particle i |

E | empirical constant |

Fc,i | particle-interaction force vector of particle with index i |

Fcoh,ij | cohesive force vector of particle i interacting with particle j |

Fcon,ij | normal contact force vector of particle i interacting with particle j |

Fh,i | hydrodynamic force vector of particle with index i |

Fg,i | gravitational force vector of particle with index i |

fIBM | artificial volume forced introduced by the IBM i |

Flub,ij | lubrication force vector of particle i interacting with particle j |

Fy | floc breakage force |

G | turbulence parameter / shear rate |

G1,k | mass gain due to aggregation of flocs |

Ga | gain of flocs in a size class due to aggregation |

Lbs | gain of flocs in a size class due to shear driven break-up |

g | gravitational acceleration |

g′ | specific gravity |

hco | thickness of cohesive shell |

Ip,i | moment of inertia of particle i |

k | empirical constant |

k | index of class for sediment mass concentration |

kA′ | agglomeration constant |

kB′ | breakup constant |

L0 | characteristic length |

L1,k | mass loss due to breakup of flocs |

La | loss of flocs in a size class due to aggregation |

Lbs | loss of flocs in a size class due to shear driven break-up |

m | empirical constant |

mk | mass of an individual floc in class k |

N | number of classes for sediment mass concentration |

nf | fractal dimension |

Nf | total number of flocs in the flow |

nk | number concentration of class k |

Np | total number of particles in the flow |

N¯p,local | average number of primary particles per floc |

m50 | mass of the median grain size |

mp,i | mass of particle i |

mk | mass of a floc |

r | radial position vector |

Re | Reynolds number |

St | Stokes number |

t | time |

Tc,i | Torque vector of particle with index i due to particle-interaction |

Tcon,ij | collision torque vector of particle i interacting with particle j |

Th,i | hydrodynamic torque vector of particle with index i |

Tlub,ij | lubrication torque vector of particle i interacting with particle j |

U0 | characteristic velocity |

uf,i=u | fluid velocity vector |

up,i | translational velocity vector of particle with index i |

us | buoyancy velocity |

vs | Stokes settling velocity |

Ws | settling velocity of microflocs |

ws,k | settling velocity of of class k |

Ws,macro | settling velocity of macroflocs |

## Abbreviations

ADV | acoustic doppler velocimeter |

COAWST | Coupled Ocean-Atmosphere-Wave-Sediment Transport Modeling System |

DNS | direct numerical simulation |

EPS | extracellular polymeric substance |

ETM | estuarine turbidity maximum |

FVCOM | finite volume community ocean model |

IBM | immersed boundary method |

INSSEV | IN-Situ SEttling Velocity instrument |

LabSFLOC | laboratory spectral flocculation charactersitics |

MFSV | manning floc settling velocity |

MSF | mass settling flux |

OBS | optical backscatter |

PBE | population balance equation |

PICS | particle imaging camera system |

POM | Princeton ocean models |

RANS | Reynolds-averaged Navier-Stokes |

ROMS | regional ocean modeling systems |

SB | size band |

SPM | suspended particulate matter |

TEP | transparent exopolymer particles |

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