# Volumetric Concentration Maximum of Cohesive Sediment in Waters: A Numerical Study

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methods and Model Descriptions

#### 2.1. Flocculation Model

^{3}); ρs = density of primary particle (kg/m

^{3}); D = diameter of floc; d = diameter of primary particle; F = three dimensional fractal dimension of floc; µ = dynamic viscosity of fluid (N∙s/m

^{2}); ${B}_{1}$ = coefficient related to the yield strength of floc (see [38], for more details); G = dissipation parameter (so-called “shear rate”, s

^{−1}); and p and q = empirical coefficients. Winterwerp [9] suggests the values of p = 1.0 and q = 0.5 based on several constraints. The dissipation parameter, G, is calculated by the ratio of turbulent energy dissipation rate to kinematic viscosity ($G=\sqrt{\raisebox{1ex}{$\u03f5$}\!\left/ \!\raisebox{-1ex}{$\nu $}\right.}$). It represents the collision frequency causing the aggregation and breakup of flocs. Khelifa and Hill [11] propose a power-law relationship of fractal dimension based on measurement results (Equation (2)).

^{2}} μm and c = {10°} kg/m

^{3}, respectively (see Figure 1 and Figure 2). These results are consistent with the previous in situ measurements (e.g., [39,40]).

Parameter | ${k}_{A}^{\u2019}$ | ${k}_{B}^{\u2019}$ | p | q | B_{1} |
---|---|---|---|---|---|

Value | 64.4770 | 4.82 × 10^{−3} | 1.0 | 0.5 | 2.63 × 10^{−14} |

**Figure 1.**Vertical profiles of calculated concentrations. The solid curves and the dashed curves represent results of non-cohesive sediment and cohesive sediment, respectively. (

**a**) Mass concentration under a steady current; (

**b**) Volumetric concentration under a steady current; (

**c**) Mass concentration under the oscillatory flow condition; and (

**d**) Volumetric concentration under the oscillatory flow condition.

**Figure 2.**Vertical profiles of volumetric concentration (

**a**); size (

**b**); and density (

**c**) of suspended floc under a steady current. The volumetric concentration has a maximum around 0.2 m above the bed. The maximum floc size is shown at 0.23 m of elevation. The minimum density of floc is also located at the same elevation.

#### 2.2. Sediment Transport Model

_{t}= eddy viscosity. The sum of v and v

_{t}is defined as the effective viscosity [43]. In this study, v

_{t}is determined by the k-ε model for cohesive sediment:

_{ϵ1}, C

_{ϵ2}, C

_{ϵ3}, σ

_{k}and σ

_{ϵ}= numerical coefficients; and σ

_{c}= Schmidt number. The values of these numerical coefficients are summarized in Table 2 (see [34], for more details).

Coefficient | ${C}_{\u03f51}$ | ${C}_{\u03f52}$ | ${C}_{\u03f53}$ | σ_{k} | σ_{ϵ} | σ_{c} |
---|---|---|---|---|---|---|

Value | 1.44 | 1.92 | 0.00 | 1.00 | 1.30 | 0.5 |

_{s}) is determined by Stokes’ law of settling particle:

_{f}) and size (D) of floc are variables in Equation (11) due to the flocculation process. Therefore, the settling velocity (W

_{s}) also changes according to flow condition and sediment concentration.

_{0}(t) = prescribed depth-averaged flow velocity; τ

_{s}= surface shear stress set to be zero in this study; h = flow depth; and T

_{rel}= relaxation time set to be 2 × ∆t (time step of computation) in this study. The bottom shear stress (τ

_{b}) is calculated by the bottom friction velocity (u

_{τ})

_{0}= shear stress at the bottom (see [45] for more details).

_{e}= empirical erosion flux coefficient.

## 3. Results and Discussion

_{avg}= 0.5 m/s and oscillatory flow replicating U-tube experiments. The oscillatory flows are defined by the free-stream flow velocity (U) for simplicity (Equation (17)). The oscillatory flow conditions are symmetric in this study.

_{amp}= orbital amplitude; T = period of oscillation; and t = time.

_{amp}= 0.5 m/s and T = 7.0 s. In simulation results of cohesive sediment under a steady current, it is seen that the mean size and density of flocs are 73.8 μm and 1496 kg/m

^{3}, respectively (refer to Figure 2). In the case of oscillatory flow, those are 148 μm and 1263 kg/m

^{3}. Based on the calculation result of cohesive sediment case, the density and size of non-cohesive sediment are fixed to be 73.8 μm and 1496 kg/m

^{3}under a steady current. Under the condition of oscillatory flow, those are set to be 148 μm and 1263 kg/m

^{3}. The depth-averaged mass concentrations in the cases of current and oscillatory flow are 1.166 kg/m

^{3}and 0.1116 kg/m

^{3}, respectively. The mass concentration profile of non-cohesive sediment continuously decreases in the vertical direction. The mass concentration of cohesive sediment under a steady current has a clear lutocline around 1 m above the bed. The volumetric concentration profile of non-cohesive sediment is linearly proportional to the mass concentration because the density of sediment is constant. It is found in Figure 1b that the profile of ${\mathrm{\varphi}}_{f}$ for cohesive sediment shows a peak value around 0.2 m above the bed under a steady current. ${\mathrm{\varphi}}_{f}$ of cohesive sediment increases in the range of 0 m to 0.2 m from the bed whereas the mass concentration of cohesive sediment shows the typical shape of a Rouse profile (see Figure 1a). The sediments under the oscillatory flow condition are confined near the bed similar to sheet flow. However, the maximum value of ${\mathrm{\varphi}}_{f}$ exists at the elevation of 4 cm above the bed whereas that of the non-cohesive sediment exists at the bed. In Figure 1, it can be seen that the elevated maximum of ${\mathrm{\varphi}}_{f}$ is not caused by the type of flow condition because the elevated maximum is shown under both steady and oscillatory currents. The difference between profiles of mass and ${\mathrm{\varphi}}_{f}$ result from the spatial variation of floc density and size. The density of floc decreases as the size of floc increases (see Equations (2) and (3)). Thus, it is deduced that the size of floc is large and the density is low near the elevation where the maximum value of ${\mathrm{\varphi}}_{f}$ exists. Figure 2 shows the vertical profiles of ${\mathrm{\varphi}}_{f}$, size, and density of floc under the steady current (uavg = 0.5 m/s). As discussed above, the maximum size and the minimum density of floc are calculated near the elevation at which the maximum value of ${\mathrm{\varphi}}_{f}$ is located.

_{avg}= 0.4 m/s and u

_{avg}= 0.35 m/s is relatively small compared to the cases of of u

_{avg}= 0.45 m/s and u

_{avg}= 0.50 m/s.

**Figure 3.**Vertical profiles of floc concentration under the different conditions of current velocity. (

**a**) c under Different Conditions of u

_{avg}; (

**b**) ϕ

_{f}under different conditions of u

_{avg}; (

**c**) D under different condition of u

_{avg}; and (

**d**) ρ

_{f}under different condition of u

_{avg}. (

**a**) The mass concentration shows a lutocline which is the important characteristics of cohesive sediment suspension; (

**b**) The volumetric concentration shows the elevated maximum when the velocity exceeds 0.4 m/s; (

**c**) The maximum size of floc in the cases of u

_{avg}= 0.40 m/s and u

_{avg}= 0.35 m/s is also located at elevations of 0.35 m and 0.03 m (not at the bed); (

**d**) The resulting density of floc shows the maximum values not at the bed but above the bed. In (

**a**) and (

**b**), the solid, dashed, dotted and dash-dotted curves represent results of u

_{avg}= 0.50 m/s, u

_{avg}= 0.45 m/s, u

_{avg}= 0.40 m/s, and u

_{avg}= 0.35 m/s, respectively. In (

**c**) and (

**d**), the solid and dashed curves show the results of u

_{avg}= 0.40 m/s and u

_{avg}= 0.35 m/s.

_{c}are plotted in Figure 4. The solid, dashed, dotted, and dash-dotted curves represent calculation results under the conditions of τ

_{c}= 0.17 kPa, τ

_{c}= 0.15 kPa, τ

_{c}= 0.13 kPa, and τ

_{c}= 0.11 kPa, respectively. When u

_{avg}is set to be 0.4 m/s, the elevated maximum of volumetric concentration is calculated as τ

_{c}is decreased from 0.17 to 0.15 kPa. In the case of u

_{avg}= 0.35 m/s, the elevated maximum is calculated with τ

_{c}= 0.11 kPa. As shown in Figure 4a,b, the mass concentration is increased according to decrease of τ

_{c}. The increased mass concentration enhances the aggregation process (the first term of Equation (1)) resulting in calculating the elevated maximum of volumetric concentration.

**Figure 4.**Vertical profiles of concentrations under different conditions of critical shear stress. (

**a**) The mass concentration under the condition of u

_{avg}= 0.40 m/s; (

**b**) The mass concentration under the condition of u

_{avg}= 0.35 m/s; (

**c**) The volumetric concentration under the condition of u

_{avg}= 0.40 m/s; (

**d**) The volumetric concentration under the condition of u

_{avg}= 0.35 m/s. The solid, dashed, dotted, and dash-dotted curves represent results of τ

_{c}= 0.17 kPa, τ

_{c}= 0.15 kPa, τ

_{c}= 0.13 kPa, and τ

_{c}= 0.11 kPa, respectively.

_{avg}and τ

_{c}are set to be 0.45 m/s and 0.17 kPa. As shown in Figure 5a, the mass concentration is increased in accordance with decrease of h whereas the suspension height is almost constant. Figure 5b represents the profiles of volumetric concentration. In Figure 5b, it is found that the vertical gradient of the volumetric concentration and the elevated maximum become more significant as h decreases. The profiles of mass concentration and volumetric concentration under the condition of h = 3.0 m are similar to those of h = 2.0 m and τ

_{c}= 0.15 kPa (see Figure 4a,b).

**Figure 5.**Vertical profiles of concentration under different conditions of water depth (h). (

**a**) The mass concentration profiles; and (

**b**) The volumetric concentration profiles. The solid, dashed, dotted, and dash-dotted curves represent results of h = 1.5 m, h = 2.0 m, h = 2.5 m, and h = 3.0 m, respectively.

_{avg}at the elevations of 0.30 m, 0.25 m, 0.20 m, and 0.15 m under the condition of Figure 5. It is found in Figure 6 that the velocity gradient near the bed decreases as the water depth increases. The fluid shear stress is considered to be proportional to the velocity gradient under the assumption of Newtonian shear stress relationship. Based on Newtonian shear stress relationship, it can be seen in Figure 6 that the fluid shear is also decreased as the water depth increases. Therefore, the sediment suspension is affected by the water depth under the constant conditions of u

_{avg}and τ

_{c}. However, the water depth is not considered as a dominant factor to determine the occurrence of elevated maximum because the elevated maximum of volumetric concentration is found in all cases of elevations tested in this study. This is also found in the study of Fox et al. [31]. In Figure 4 of Fox et al. [31], the elevated maximum of volumetric concentration is found at many different elevations. However, it is also difficult to make a generalized conclusion with Fox et al. [31] because of the hydrodynamics changes at different locations of measurement. In addition, it is impossible to replicate the measurements of Fox et al. [31] using a numerical model due to limited information on the hydrodynamic conditions of the measurements.

_{f}= volume of a floc; V

_{s}= volume of primary particles within a floc; and ${\mathrm{\varphi}}_{sf}$ = ratio of the total volume of solids in a floc to the volume of a floc. Figure 7 shows the vertical profiles of ${\mathrm{\varphi}}_{sf}^{-1}$, ${\mathrm{\varphi}}_{f}$, D, and G when the elevated maximum of ${\mathrm{\varphi}}_{f}$ occurs. It is found in Figure 7 that ${\mathrm{\varphi}}_{sf}^{-1}$ and D have the same shape and elevation of maximum values. The elevation of maximum ${\mathrm{\varphi}}_{sf}^{-1}$ and D is slightly higher than that of ${\mathrm{\varphi}}_{f}$. The elevated maximum of ${\mathrm{\varphi}}_{f}$ means that larger flocs of which ${\mathrm{\varphi}}_{sf}$ is low exist near the location of the concentration maximum (see Figure 7a,b). From the low values of ${\mathrm{\varphi}}_{sf}^{-1}$ and D near the bed, it is also known that small and dense flocs, which have a large yield stress (see [38]), exist near the bed. A floc has a high density and a small size when the turbulence intensity is strong. Therefore, the intensity of turbulence near the elevation of the maximum value of ${\mathrm{\varphi}}_{f}$ is expected to be low whereas the turbulence intensity near the bed has a high value. Figure 7d shows the profile of G, which is a measure of turbulence intensity. G near the bed is about 90 s

^{−1}whereas G near the elevation of the maximum value of ${\mathrm{\varphi}}_{f}$ is about 5 s

^{−1}. The strong turbulence near the bed enhances the breakup process of flow resulting in a small and dense floc. The low intensity of turbulence near the elevation of maximum ${\mathrm{\varphi}}_{f}$ causes growth of flocs having a low density and a large size. Mikkelsen et al. [46] measure the volumetric concentration at four different locations by estimating a proxy for current stress based on the squared current velocity. It is found in Mikkelsen et al. [46] that the volumetric concentration increases at the location where the current stress is high. It is also concluded in Mikkelsen et al. [46] find that the strong turbulence causes the erosion of micro-flocs (not primary particles) from the bed. This conclusion is consistent with the findings of this study. The floc size is calculated to be small near the bed due to the high intensity of turbulence (see Figure 7).

**Figure 6.**Variation of velocity gradients under different conditions of water depth. The asterisk, diamond, circle, and square symbols represent results at elevations of 0.3 m, 0.25 m, 0.2 m, and 0.15 m, respectively. The experimental condition is consistent with Figure 5.

**Figure 7.**Vertical profiles of ϕ

_{f}, D, ${\mathrm{\varphi}}_{sf}^{-1}$, and G. In Figure 7c, the inverse of ϕ

_{sf}is plotted for convenience. (

**a**) Volumetric concentration; (

**b**) Floc size; (

**c**) Solid volume concentration within a floc; and (

**d**) Shear rate.

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Byun, J.; Son, M.; Yang, J.-S.; Jung, T.-H.
Volumetric Concentration Maximum of Cohesive Sediment in Waters: A Numerical Study. *Water* **2015**, *7*, 81-98.
https://doi.org/10.3390/w7010081

**AMA Style**

Byun J, Son M, Yang J-S, Jung T-H.
Volumetric Concentration Maximum of Cohesive Sediment in Waters: A Numerical Study. *Water*. 2015; 7(1):81-98.
https://doi.org/10.3390/w7010081

**Chicago/Turabian Style**

Byun, Jisun, Minwoo Son, Jeong-Seok Yang, and Tae-Hwa Jung.
2015. "Volumetric Concentration Maximum of Cohesive Sediment in Waters: A Numerical Study" *Water* 7, no. 1: 81-98.
https://doi.org/10.3390/w7010081