**2. Numerical model**

CCHE3D model [3] is a Finite Element Method (FEM) model based on a partially staggered 3D sigma layered mesh system, which consists of multiple layered structured meshes. It was developed for 3D numerical simulation and analysis for free surface turbulent flows in rivers, lakes, reservoirs, and estuaries and associated processes, such as sediment transport [18], morphological changes [17], heat transfer [19], pollutant transport and water quality evaluation [20], etc. Since it can provide more accurate and detailed local flow fields around in-stream structures [18] than 2D depth averaged models, CCHE3D model was selected to evaluate the erosion control plans with multiple weir structures installed in the study channel.

### **2.1 3D RANS model**

The full 3D Reynolds-averaged Navier-Stoker (RANS) equations are solved in CCHE3D model, which are simply listed in the form of index notations for Cartesian coordinates as follows:

$$\frac{\partial u\_i}{\partial \mathbf{x}\_i} = \mathbf{0} \tag{1}$$

$$\frac{\partial u\_i}{\partial t} + u\_j \frac{\partial u\_i}{\partial \mathbf{x}\_j} = -\frac{1}{\rho} \frac{\partial p}{\partial \mathbf{x}\_j} + \frac{\partial}{\partial \mathbf{x}\_j} \left( \nu \frac{\partial u\_i}{\partial \mathbf{x}\_j} - \overline{u\_i' u\_j'} \right) + f\_i \tag{2}$$

where *ui* = Reynolds-averaged velocities defined at *xi (i, j =* 1, 2, 3*)*; *t* = time ; *ρ* is water density; *p* is pressure; *ν* is kinematic viscosity; �*u*<sup>0</sup> *i u*0 *<sup>j</sup>* = Reynolds stress; and, *fi* represents the body force.

For surface flows, the free surface kinematic equation is applied:

$$\frac{\partial \eta}{\partial t} + u\_{\eta} \frac{\partial \eta}{\partial \mathbf{x}} + v\_{\eta} \frac{\partial \eta}{\partial y} - w\_{\eta} = \mathbf{0} \tag{3}$$

where *η* is water surface elevation; and (*uη*,*vη*,*wη*) denotes velocity at water surface.

In CCHE3D model, a partially staggered stencil and a velocity correction algorithm are used for solving the momentum and continuity equation. Several turbulence closure schemes including the zero equation models (parabolic, mixing length, and wind-induced) and the *k* � *ε* models are provided. More details of CCHE3D model can be found in Ref. [3].

#### **2.2 3D sediment transport model**

Sediment transport is one of the most complex and least understood phenomena in nature. In 3D, sediment particles' movements are highly affected by vertical motion of fluid flows in addition to horizontal movements. This is particularly true in the vicinity of hydraulic structures where the flow impacts on the solid walls of the structures (bridge pier and abutment, for instance). If the structures are very large (dam), the fluid flows in an open channel are forced to change their speed and direction near structures in order to pass through them, the sediment transport capacity due to this impact is adjusted significantly causing localized scouring and deposition over the sediment bed. In CCHE3D model, in addition to general sediment transport capabilities, special sediment transport features, such as local scouring around structures and channel head-cut migration have been developed as well.

In CCHE3D model, the 3D convection-diffusion equation for the suspended sediment is solved as follows:

$$\frac{\partial \mathbf{C}}{\partial t} + \boldsymbol{\mu} \frac{\partial \mathbf{C}}{\partial \mathbf{x}} + \boldsymbol{\nu} \frac{\partial \mathbf{C}}{\partial \mathbf{y}} + (\boldsymbol{\nu} - \boldsymbol{\alpha}\_{\mathrm{s}}) \frac{\partial \mathbf{C}}{\partial \mathbf{z}} - \frac{\partial}{\partial \mathbf{x}} \left[ \frac{\nu\_{t}}{\sigma\_{\mathrm{s}}} \frac{\partial \mathbf{C}}{\partial \mathbf{x}} \right] - \frac{\partial}{\partial \mathbf{y}} \left[ \frac{\nu\_{t}}{\sigma\_{\mathrm{s}}} \frac{\partial \mathbf{C}}{\partial \mathbf{y}} \right] - \frac{\partial}{\partial \mathbf{z}} \left[ \frac{\nu\_{t}}{\sigma\_{\mathrm{s}}} \frac{\partial \mathbf{C}}{\partial \mathbf{z}} \right] = \mathbf{S} \mathbf{T} \tag{4}$$

where *C* is the suspended sediment concentration; *u, v*, and *w* are velocity components (m/s); ω*<sup>s</sup>* is the sediment settling velocity; *ν<sup>t</sup>* is the eddy viscosity (m<sup>2</sup> /s); *ST* is the source term. and *σ*<sup>s</sup> is the Schmidt number to convert the turbulence eddy viscosity to eddy diffusivity for suspended sediment.

At the free surface, the vertical sediment flux is zero, so the gravity effects *ωsC*balance the diffusion effects *ε<sup>s</sup> ∂C <sup>∂</sup><sup>z</sup>* , and the following condition is applied:

$$
\rho \mathbf{o}\_t \mathbf{C} + \varepsilon\_t \frac{\partial \mathbf{C}}{\partial \mathbf{z}} = \mathbf{0} \tag{5}
$$

At the bottom, the following condition is applied:

$$
\rho \rho\_t \mathbf{C} + \varepsilon\_t \frac{\partial \mathbf{C}}{\partial \mathbf{z}} = D\_b - E\_b \tag{6}
$$

where *ω<sup>s</sup>* is settling velocity (m/s); *ε<sup>s</sup>* ¼ *νt=σ<sup>c</sup>* is diffusion coefficient for sediment; *Db* and *Eb* (kg/m<sup>2</sup> /s) are deposition rate and erosion (re-suspension) rate at bottom, respectively.

Following the non-equilibrium transport approach (*qbk* 6¼ *qb*<sup>∗</sup> *<sup>k</sup>*) proposed by Wu [21], the bedload transport rate is governed by

$$\frac{\partial(\delta\overline{c}\_{bk})}{\partial t} + \frac{\partial q\_{bk\mathbf{x}}}{\partial \mathbf{x}} + \frac{\partial q\_{bk\mathbf{x}}}{\partial \mathbf{y}} = -\frac{1}{L\_b} \left( q\_{bk} - q\_{b,k} \right) \tag{7}$$

where *qbk* is the bedload transport rate for the *kth* size class, *qbkx* and *qbky* are the component in *x* and *y* directions, δ is the bedload layer thickness, *cbk* is the bedload concentration and *Lb* is the bedload adaptation length; *qb*<sup>∗</sup> *<sup>k</sup>* is the bedload sediment transport capacity for equilibrium transport conditions, which can be estimated using empirical transport formulas.

### **2.3 Bedrock erosion model**

According to previous studies, the bedrock erodibility (strength), stream power, shear stress, sediment supply, and grain size were identified as important impacting factors on bedrock erosion rate, which lead to two popular bedrock erosion mechanisms, plucking and abrasion, widely used in the numerical models [1, 12, 14, 16]. The first one is corresponding to the so-called stream power-based method that the bedrock erosion rate is considered as a function of the stream power, and the varying shear stress causes the hydraulic scouring on soft bedrock [4–6, 13], and the other one is the abrasion-based method, which emphasizes on the important role of the sediment supply (both bedload and suspended load) by considering the eroding and shielding effects of sediment on bedrock [7, 8, 11].

Whipple *et al*. [4] observed in the field that for well-joined rocks with fractures and bedding planes, the plucking is the dominant erosion process, while the abrasion process dominates for rocks with smooth and polished surfaces but with ripples, flutes, and potholes prominently developed. All the aforementioned bedrock erosion models oversimplified and conceptualized the complicated bedrock erosion processes in nature. For any particular mountainous river, these natural processes cannot be separated and modeled accurately using one method. Practically, however, a certain dominant process has to be selected to represent all erosion mechanisms for the modeling purpose. For the downstream channel of JiJi Weir, since the measured bedrock erodibility is available, it is assumed that the plucking is dominating in this reach, and the stream power method [16] is selected for current study.

*Erosion Control at Downstream of Reservoir Using In-stream Weirs DOI: http://dx.doi.org/10.5772/intechopen.108169*

In the stream power method, the bedrock erosion rate *E* is only related to the rock erodibility index [5] and the flow stream power, which is proportional to bed shear stress, as described in Eqs. (8, 9):

$$E = K\_s U \left(\frac{P}{P\_{cm}} - \mathbf{1}\right)^c = K\_s U \left(\frac{\tau U}{P\_{cm}} - \mathbf{1}\right)^c \tag{8}$$

$$P\_{cm} = aK\_h^b \tag{9}$$

where *Ks* is non-dimensional coefficient; *U* is depth-averaged velocity of flow (m/s); *P* is stream power of flow (kW/m<sup>2</sup> ), *P = τU*, *τ* is shear stress (N/m<sup>2</sup> ); *Pcm* is critical stream power (kW/m<sup>2</sup> ); *Kh* is the bedrock erodibility index defined as the capability of earth materials for resisting erosion, which is correlated empirically to the stream power and obtained based on field and laboratory studies; and, *a*, *b*, and *c* are site-specific calibrated parameters.

The above stream power method is further improved to take into account lateral erosion by considering the local lateral bed slope. Thus, a factor *Sb* representing high slope zones is introduced as follows:

$$\mathbf{S}\_b = \max\left(\frac{\mathbf{S}\_l}{k \cdot \mathbf{S}\_R}, \mathbf{1.0}\right)^r \tag{10}$$

where *Sl* is the local lateral bed slope computed in an element, *SR* is a reference slope of the simulation area; it is currently represented by the average slope of all wet elements in a domain. The power *r* is empirical and needs to be calibrated. In the tests, it is found that *r* = 1.5�5.0. It can be seen that this factor is effective only when the local lateral bed slope is larger than the reference slope. *k* = 1�6 is the coefficient to adjust the reference slope, which filters out small slope area from erosion. With this factor, Eq. (8) is modified to:

$$E\_{bb} = K\_s U \left(\frac{P}{P\_{cm}} - \mathbf{1}\right)^c \mathbf{S}\_b \tag{11}$$

where *Ebb* is the erosion rate applicable to both softrock bed and bank erosion. With *r* = 0, Eq. (11) will convert back to Eq. (8).

#### **2.4 Coupling of bedrock erosion model and sediment transport model**

When the sediment transport is simulated, the boundary condition between the moving sediment particles and the bedrock has to be treated. Sediment particles can deposit over the rock surface and form a deposition layer. A concept of a sediment mixing layer over the softrock surface is adopted. If the thickness of the sediment layer is large, no rock erosion is calculated. If no sediment deposition exists on the bed, the proposed stream power method is used for the rock bed erosion. If the thickness of the sediment layer is within a criterion (mixing layer thickness), the stream-powerinduced erosion would be applied at a reduced rate, proportional to the thickness of the mixing layer. The net change rate of the mixing layer is the combined rates of rock erosion and sediment deposition.

There is insufficient knowledge on the interactive and coupling mechanism between the bedrock erosion and the sediment transport. This simple coupling basically considers the shielding effects of the sediments on bedrock, similar to that of the abrasion model, which is concetually reasonable and has been proved and validated in previous studies [7, 8, 11]. However, the actual interactions between the bedrock erosion process and the sediment transport process are much more complicated in nature.
