*Article* **Secondary Flow E**ff**ects on Deposition of Cohesive Sediment in a Meandering Reach of Yangtze River**

**Cuicui Qin 1,\*, Xuejun Shao <sup>1</sup> and Yi Xiao <sup>2</sup>**


Received: 27 May 2019; Accepted: 9 July 2019; Published: 12 July 2019

**Abstract:** Few researches focus on secondary flow effects on bed deformation caused by cohesive sediment deposition in meandering channels of field mega scale. A 2D depth-averaged model is improved by incorporating three submodels to consider different effects of secondary flow and a module for cohesive sediment transport. These models are applied to a meandering reach of Yangtze River to investigate secondary flow effects on cohesive sediment deposition, and a preferable submodel is selected based on the flow simulation results. Sediment simulation results indicate that the improved model predictions are in better agreement with the measurements in planar distribution of deposition, as the increased sediment deposits caused by secondary current on the convex bank have been well predicted. Secondary flow effects on the predicted amount of deposition become more obvious during the period when the sediment load is low and velocity redistribution induced by the bed topography is evident. Such effects vary with the settling velocity and critical shear stress for deposition of cohesive sediment. The bed topography effects can be reflected by the secondary flow submodels and play an important role in velocity and sediment deposition predictions.

**Keywords:** secondary flow; cohesive; deposition; 2D depth-averaged model; meandering; Yangtze River

#### **1. Introduction**

Helical flow or secondary flow caused by centrifuge force in meandering rivers plays an important role in flow and sediment transport. It redistributes the main flow and sediment transport, mixes dissolved and suspended matter, causes additional friction losses, and additional bed shear stress, which are responsible for the transverse bed load sediment transport [1–3]. Moreover, the secondary flow may affect lateral evolution of river channels [4–6]. Extensive researches have been conducted about secondary flow effects on flow and sediment transport, especially bed load in a singular bend [7] or meandering channels of laboratory scale [8] and rivers of field scale [9,10]. However, few researches focus on suspended load transport. In China, sediment transport in most rivers is dominated by suspended load, such as Yangtze River and Yellow River. On the Yangtze River, the medium diameter of sediment from upstream is ~0.01 mm [11], which has taken on cohesive properties to some extent [11,12]. More importantly, these cohesive suspended sediments have been extensively deposited in several reaches which have blocked the waterway in Yangtze River [13]. As most of these reaches are meanders with a central bar located in the channel, to what extent the secondary flow has affected the cohesive suspended sediment deposition should be investigated.

Cohesive sediment deposition is controlled by bed shear stress [14], which is determined by the flow field. In order to investigate the secondary flow effects on cohesive sediment transport, its effects on flow field should be considered first. Secondary flow redistributes velocities, which means the high velocity core shifts from the inner bank to the outer bank of the bend [15,16]. Saturation of

secondary flow takes place in sharp bends [17]. Due to the inertia, the development of secondary flow lags behind the curvature called the phase lag effect [18]. All these findings mainly rely on laboratory experiments or small rivers with a width to depth radio less than 30 [19] probably resulting in an exaggeration of secondary flow. When it comes to natural meandering or anabranching rivers, especially large or mega rivers, secondary flow may be absent or limited in a localized portion of the channel width [20–24]. However, those researches are only based on field surveys and mainly focused on influences of bifurcation or confluence of mega rivers with low curvatures and significant bed roughness [23] at the scale of individual hydrological events. On contrary, Nicholas [25] emphasized the role of secondary flow played in generating high sinuosity meanders via simulating a large meandering channel evolution on centennial scale. Maybe it depends on planimetric configurations, such as channel curvature, corresponding flow deflection [26] and temporal scales. Therefore, whether secondary flow exists and has the same effects on the flow field in a meandering mega river as that in laboratory experiments should be further investigated. Besides, the long-term hydrograph should be taken into account.

As to its effects on bed morphology, secondary flow induced by channel curvature produces a point bar and pool morphology by causing transversal transport of sediment, which in turn drives lateral flow (induced by topography) known as topographic steering [9] which plays an even more significant role in meandering dynamics than that curvature-induced secondary flow [3]. The direction of sediment transport is derived from that of depth-averaged velocity due to the secondary flow effect, which has been accounted for in 2D depth-averaged models and proved to contribute to the formation of local topography [4–6,27], especially bar dynamics [28,29], and even to channel lateral evolution [4,6,30]. Although Kasvi et al. [31] has pointed that the exclusion of secondary flow has a minor impact on the point bar dynamics, temporal scale effects remain to be investigated as the authors argued for only one flood event has been considered in their research and the inundation time may affect the effects of secondary flow [32]. Those researches have enriched our understandings of mutual interactions of secondary flow and bed morphology. However, they mainly focused on bed load sediment transport, whereas the world largest rivers are mostly fine-grained system [21] and are dominated by silt and clay, such as Yangtze River [11,12]. Fine-grained suspended material ratio controls the bar dynamics and morphodynamics in mega rivers [23,33]. As is known, such fine-grained sediment is common in estuarine and coastal areas. However, how they work under the impacts of secondary flow in mega rivers is still up in the air and the temporal effects of secondary flow should be investigated.

Numerical method provides a convenient tool for understanding river evolution in terms of hydrodynamics and morphodynamics in addition to the laboratory experiments and field surveys. The 2D depth-averaged model is preferable because it keeps as much detailed information as possible on the one hand and remains practical for investigation of long-term and large-scale fluvial processes on the other hand. The main shortcoming of the 2D depth-averaged model is that the vertical structure of flow has been lost due to the depth-integration of the flow momentum and suspended sediment transport equations, and thus the secondary flow effects on the flow field and suspended sediment transport are neglected. These effects can be retrieved by incorporating closure correction submodels into the 2D depth-averaged model. In order to account for these effects on the flow field, various correction submodels have been proposed by many researchers [34–38]. The differences among these models are whether or not they consider (1) the feedback effects between main flow and secondary flow and (2) the phase lag effect of the secondary flow caused by inertia. Models neglecting the former one are classified as linear models, in contrast to nonlinear models which consider such effects [1,38]. The nonlinear models [1,39] based on the linear ones are more suitable for flow simulation of sharp bends [1,2]. The phase lag effect, which is obviously pronounced in meandering channels [40], has been thought to be important in sharp bends especially with pronounced curvature variations [2], and proven to influence bar dynamics considerably [29]. Although the performances of those above mentioned models have been extensively tested by laboratory scale bends, their applicability to field

meandering rivers, especially mega rivers, needs to be further investigated. Besides, which model is preferable in flow simulation of meandering channels of field mega scale remains to be answered.

To consider the secondary flow effects on the suspended sediment transport, closure submodels should be coupled to the sediment module of the 2D depth-averaged models in a similar way to the flow module [41]. However, as to the cohesive sediment transport, it is mainly related to the bed shear stress determined by the flow field. Besides, according to field survey of two reaches of Yangtze River by Li et al. [11], cohesive sediment transport is controlled by the depth-averaged velocity. Therefore, only the secondary flow effects on flow field are considered to further analyze their effects on bed morphology here. In addition, the turbulence models should be considered in the 2D depth-averaged model, especially when there are recirculating flows [34]. Based on the previous research work [34,36], the depth-averaged parabolic eddy viscosity model can be applied.

This paper aims to investigate the secondary flow effects on cohesive sediment deposition in meandering reach of field mega scale during an annual hydrography. The following questions will be addressed; (1) whether secondary flow effects on the flow field can be reflected by typical secondary flow correction models in such mega meandering rivers as laboratory meandering channels, (2) which model should be given priority to flow simulation in meandering channels of such scale, and (3) what the temporal influence of secondary flow is on bed morphology variations associated with cohesive sediment deposition. The contents of this paper are as follows; three secondary flow submodels referring to the aforementioned different effects have been selected from the literature—Lien et al. [37], Bernard [35], and Blankaert and de Vriend [1] models—to reveal secondary flow impacts and distinguish their performances on flow simulation in meandering channels of this mega scale first, and the preferable model is selected. Then, the corresponding model is applied to investigate secondary flow effects on bed morphology variations related with cohesive sediment deposition during an annual hydrograph. Finally, the correction terms representing secondary flow effects have been analyzed to justify their functionalities and performances of these models in meandering channels of such scale. Besides, the roles of cohesive sediment played in secondary flow effects have been investigated as well. The main contributions of this paper are three-fold: (1) the L model has been found to outperform the other models in flow simulation of the field mega scale meandering reach; (2) the bed topography effects have been identified to be reflected by the secondary flow submodels, and the transverse bed topography plays a more important role than the longitudinal one and results in the great improvements of velocity and sediment deposition predictions of the L model in this reach; and (3) secondary flow effects on cohesive sediment deposition become obvious during the last period of an annual hydrography when the sediment concentration is low and the transverse bed topography has been formed. Such effects on the predicted amount of deposition vary with the cohesive sediment properties.

#### **2. Methods**

A 2D depth-averaged model (Section 2.1, referred to as the N model hereafter) has been improved by considering secondary flow effects and cohesive sediment transport. Secondary flow module (Section 2.2) incorporates three different submodels to reflect its different effects, together with the sediment module (Section 2.3) are described briefly. All the equations are solved in orthogonal curvilinear coordinates.

### *2.1. Flow Equations*

The unsteady 2D depth-averaged flow governing equations are expressed as follows [42]

$$\frac{\partial Z}{\partial t} + \frac{1}{J} \left[ \frac{\partial (h\_2 H U)}{\partial \xi} + \frac{\partial (h\_2 H V)}{\partial \eta} \right] = 0 \tag{1}$$

$$\frac{\partial(\overline{\rm H}\overline{\rm I})}{\partial t} + \frac{1}{\overline{\gamma}} \left[ \frac{\partial(h\_2 \overline{\rm H} \overline{\rm I} \overline{\rm I})}{\partial \zeta} + \frac{\partial(h\_1 \overline{\rm H} \overline{\rm I} \overline{\rm I})}{\partial \eta} \right] - \frac{\rm H \overline{\rm V}}{\overline{\gamma}} \frac{\partial \overline{\rm h}\_2}{\partial \zeta} + \frac{\rm H \overline{\rm I} \overline{\rm H}}{\overline{\gamma}} \frac{\partial \overline{\rm h}\_1}{\partial \eta} + \frac{\rm \frac{\rm H}{\overline{\rm H}}}{\overline{\rm h}\_1} \frac{\partial \overline{\rm Z}}{\partial \zeta} + \frac{\rm H \overline{\rm H} \overline{\rm S}}{(\rm \zeta H)^2} = \frac{\nu\_F H}{h\_1} \frac{\partial \overline{\rm E}}{\partial \zeta} - \frac{\nu\_F H}{h\_2} \frac{\partial \overline{\rm F}}{\partial \eta} - S\_{\overline{\rm S}} \tag{2}$$

*Water* **2019**, *11*, 1444

$$\frac{\partial \langle HV \rangle}{\partial t} + \frac{1}{f} \Big[ \frac{\partial \langle h\_2 H l IV \rangle}{\partial \xi} + \frac{\partial \langle h\_1 H l V V \rangle}{\partial \eta} \Big] + \frac{H l IV}{f} \frac{\partial h\_2}{\partial \xi} - \frac{H l II}{f} \frac{\partial h\_1}{\partial \eta} + \frac{\partial H}{h\_2} \frac{\partial Z}{\partial \eta} + \frac{H l I \partial \xi}{\left( \langle H l \rangle \right)^2} = \frac{\nu\_H H}{h\_1} \frac{\partial E}{\partial \eta} - \frac{\nu\_H H}{h\_2} \frac{\partial F}{\partial \xi} - S\_{\eta} \tag{3}$$

$$E = \frac{1}{J} \left[ \frac{\partial (h\_2 l I)}{\partial \xi} + \frac{\partial (h\_1 V)}{\partial \eta} \right] \\ F = \frac{1}{J} \left[ \frac{\partial (h\_2 V)}{\partial \xi} - \frac{\partial (h\_1 l I)}{\partial \eta} \right] \tag{4}$$

where ξ and η=longitudinal and transverse direction in orthogonal curvilinear coordinates, respectively; *h*<sup>1</sup> and *h*<sup>2</sup> = metric coefficients in ξ and η directions, respectively; *J* = *h*1*h*2; *g* = acceleration gravity, m/s2; *u* = (*U*, *V*) depth-averaged resultant velocity vector and (*U*, *V*) = depth-averaged velocity in ξ and η directions, separately; *H* = water depth; *Z* = water surface elevation; *C* = Chezy factor; ν*<sup>e</sup>* = eddy viscosity; and *S*<sup>ξ</sup> and *S*<sup>η</sup> = correction terms related to the vertical nonuniform distribution of velocity.

#### *2.2. Secondary Flow Equations*

In order to consider different effects of secondary flow on flow, three secondary flow models are selected from literature to calculate the dispersion terms (*S*ξ, *S*η) in Equations (2) and (3). Among them, the Lien et al. [37] (L) model has been widely applied, which ignores the secondary flow phase lag effect and is suitable for fully developed flows. As secondary flow lags behind the driving curvature due to inertia [2], it will take a certain distance for secondary flow to fully develop, especially in meandering channels. There are several models using a depth-averaged transport equation to consider these phase lag effect, such as the Delft-3D [43] model, Hosoda et al. [44] model, and Bernard [35] model. The Delft-3D model has two correction coefficients to calibrate and Hosoda model is complex to use. In addition, both of them focus on flow simulation in channels with a single bend. In contrast, Bernard (B) model is simple, practicable and has been validated by several meandering channels. Moreover, the sidewall boundary conditions considered by B model is more reasonable, that is, the production of secondary flow approaches zero on the sidewalls [16]. Therefore, the B model is selected as another representative model. Because the above mentioned two models are linear models which are theoretically only applicable to mildly curved bends, a simple nonlinear (NL) model [1] is selected as a typical model to reflect the saturation effect of secondary flow [17] in sharply curved bends. All of the three models can reflect the velocity redistribution phenomenon caused by secondary flow at different levels. These models serve as submodels coupled to the 2D hydrodynamic model to account for different effects of secondary flow on flow field. The major differences of them are summarized in Table 1, while L and B models can refer to the authors [45] for more details. Only NL model are briefly described as follows.


Based on linear models, the NL model is able to consider the feedback effects between secondary flow and main flow to reflect the saturation effect through a bend parameter β [1] (Equation (10)). However, the NL model proposed by Blanckaert and de Vriend [1] is limited to the centerline of the channel. Ottevanger [46] extended the model to the whole channel width through an empirical power law (*fw*, Equation (9)). This method is as follows

$$\begin{array}{l} S\_{\xi} = \frac{1}{\gamma} \frac{\partial}{\partial \xi} (h\_2 T\_{11}) + \frac{1}{\gamma} \frac{\partial}{\partial \eta} (h\_1 T\_{12}) + \frac{1}{\gamma} \frac{\partial h\_1}{\partial \eta} T\_{12} - \frac{1}{\gamma} \frac{\partial h\_2}{\partial \xi} T\_{22} \\ S\_{\eta} = \frac{1}{\gamma} \frac{\partial}{\partial \xi} (h\_2 T\_{12}) + \frac{1}{\gamma} \frac{\partial}{\partial \eta} (h\_1 T\_{22}) - \frac{1}{\gamma} \frac{\partial h\_1}{\partial \eta} T\_{11} + \frac{1}{\gamma} \frac{\partial h\_2}{\partial \xi} T\_{12} \end{array} \tag{5}$$

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*Ti,j* (*i*, *j* = 1,2) is called dispersion terms [37]. When the L model is adopted as the linear model, *Ti,j* is expressed as

$$\begin{aligned} T\_{11} &= H l l l l \left(\frac{\sqrt{\mathfrak{S}}}{\kappa \mathbb{C}}\right)^2 \quad T\_{12} = T\_{21} = H l l V \left(\frac{\sqrt{\mathfrak{S}}}{\kappa \mathbb{C}}\right)^2 + f\_{\text{sn}}(\boldsymbol{\beta}) f\_{\text{w}} \frac{(Hl l)^2}{\kappa^2 r} \frac{\sqrt{\mathfrak{S}}}{\kappa \mathbb{C}} F \mathbf{F} \mathbf{1} \\\ T\_{22} &= H l V l \left(\frac{\sqrt{\mathfrak{S}}}{\kappa \mathbb{C}}\right)^2 + f\_{\text{sn}}(\boldsymbol{\beta}) f\_{\text{w}} \frac{2 H l l H l V}{\kappa^2 r} \frac{\sqrt{\mathfrak{S}}}{\kappa \mathbb{C}} F \mathbf{F} \mathbf{1} + f\_{\text{mu}}(\boldsymbol{\beta}) f\_{\text{w}}^2 \frac{(Hl l)^2 H}{\kappa^4 r^2} \mathbf{F} \mathbf{F} \mathbf{2} \end{aligned} \tag{6}$$

where κ = the Von Karman constant, 0.4; *r* = the channel centerline, m; *fw* = the empirical power law equation over the channel width; *FF*1, *FF*2 = the shape coefficients related to the vertical profiles of velocity which can refer to Lien et al. [37] for details; and *fsn*(β) and *fnn*(β) are the nonlinear correction coefficients expressed as Equations (7) and (8) [47], which directly reflect the saturation effect of secondary flow [17].

$$f\_{\rm Sh}(\beta) = 1 - \exp\left(-\frac{0.4}{\beta(\beta^3 + 0.25)}\right) \tag{7}$$

$$f\_{\rm fin}(\beta) = 1.0 - \exp\left(-\frac{0.4}{1.05\beta^3 - 0.89\beta^2 + 0.5\beta}\right) \tag{8}$$

$$f\_{\rm tr} = \left[1 - \left(\frac{2n}{w}\right)^{2n\_p}\right] \tag{9}$$

$$\beta = \mathbb{C}\_f^{-0.275} (H/R)^{0.5} (1+a)^{0.25} \tag{10}$$

$$\alpha\_s = \left[ w \partial \mathcal{U}\_s / \partial n / \mathcal{U}\_s \right]\_{\mathcal{U}\_c} \tag{11}$$

β = the bend parameter which is a control parameter distinguishing the linear and nonlinear models; α*<sup>s</sup>* = the normalized transversal gradient of the longitudinal velocity *U* at the centerline; and *nc* = the position of channel centerline.

The phase lag effect of secondary flow is considered with the following transport equations [46].

$$\frac{1}{\lambda} \left[ \frac{\partial (h\_2 H I I Y)}{\partial \xi} + \frac{\partial (h\_1 H V Y)}{\partial \eta} \right] = \frac{h \left| \overleftrightarrow{u} \right|}{\lambda} (Y\_\varepsilon - Y) \tag{12}$$

λ = the adaption length described by Johannesson and Parker [18]. *Y* = the terms referring to *fsn*, *fnn* in Equation (6), *Ye* = the fully developed value of *Y*.

As L, B, and NL models serve as closure submodels in hydrodynamic Equations (1)–(4), the correction terms (*S*<sup>ξ</sup> and *S*η) are associated with the computed mean flow field, and the information on the relative variables of correction terms is available when solving these submodels. This is similar to the way to solve turbulence submodels. Detailed procedure for solving the NL model is shown in Figure 1. Equations (1)–(4) are solved first without considering the correction terms (*S*<sup>ξ</sup> and *S*η) for water depth and depth-averaged velocity. The nonlinear parameters in Equations (7)–(11) have been calculated next. Afterwards, the transport Equation (12) is solved for evaluating dispersion terms (*Ti,j*, Equation (6)) and (*S*<sup>ξ</sup> and *S*η) (Equation (5)). The correction terms (*S*<sup>ξ</sup> and *S*η) are then included in Equations (1)–(4), which are solved again to get new information on the mean flow field. The procedure continues until no significant variations in the magnitude of depth, velocity, and other variables in the model (Figure 1).

**Figure 1.** Solution procedure.
