*4.2. Similitude for Model Scaling*

A physical model should satisfy the similarity theory of geometrical, kinematic, and dynamic conditions. In open-channel flow, flow patterns under a free-surface condition follow the Froude number (*Fr*) similarity. Therefore, the following relationship is involved in the scaling of model dimensions for flow patterns: λ*Fr* = λ*u*/ <sup>λ</sup>*g*λ*<sup>h</sup>* = 1, where λ is the ratio of the model to the prototype; λ*u* is the velocity scale ratio; λ*h* is the vertical length scale ratio; and *g* is the acceleration of gravity, with λ*g* = 1. Hence, the flow velocity ratio of the model to the prototype can be derived as follows: λ*u* = λ1/2 *h* . Note that the flow discharge scale is <sup>λ</sup>*Q* = λ*u*λ*<sup>A</sup>* = λ5/2 *h* , and the time scale is λ*t* = λ1/2 *h* . The subscripts *u*, *Q*, *A*, and *t* denote the velocity dimension, discharge dimension, cross-sectional area dimension, and time dimension, respectively.

## 4.2.1. Initiation of Sediment Movement

Similarities in sediment transport mechanisms engender difficulties in combining coarse and fine materials because of the difference in sediment particle sizes. In this study, criteria for the similitude of sediment movements were required to assess coarse sediment for modeling the permeable barrier and to assess cohesive sediment for the dredged material placed in the replenishment area. For coarse sediment, we could adopt the Shields diagram for the condition involving incipient riverbed particle motion [29].

*Water* **2019**, *11*, 1998

For satisfying the similitude of model scaling, the dimensionless shear parameter in the Shields diagram was employed and can be expressed as follows:

$$\frac{\lambda\_{\tau\_{cr}}}{\lambda\_{(\gamma\_s-\gamma)}\lambda\_{d\_c}} = 1\tag{8}$$

where τ*cr* is the critical shear stress (λτ*cr* = λρλ*g*λ*R*λ*s*), γ*s* is the specific density of sediment, γ is the specific density of water, *dc* is the grain size of coarse sediment, ρ is the density of water, *R* is the hydraulic radius, and *S* is the energy slope. The energy slope ratio λ*s* is equal to 1. If we substitute λτ*cr* with λρλ*g*λ*R*λ*s* and substitute λ*R* with the vertical length scale ratio λ*h*, Equation (8) becomes <sup>λ</sup>*dc* = λρλ*h*<sup>λ</sup>(ρ*s* − ρ) −1, where ρ*s* is the sediment density. If we use the same sediment material in the physical model as that in the field, the density scale would be equal to 1. Then, the scale ratio for the particle size of coarse sediment can be expressed as <sup>λ</sup>*dc*= λ*h*.

As the scouring process acts on the surface of replenished materials, fine sediment with cohesiveness can be washed away when picked up by running flow, which may behave as a wash load [30]. The scour potential with respect to the flow shear strength can be related to the dry density of cohesive deposits in exponential form, and this relationship has been derived by several researchers [31]. Although this relationship is rather an approximate and is site specific, it is useful for estimating the critical shear stress generated by a flood flow to characterize the structures of replenished sediment as deposits with cohesiveness. Based on flume experiments, the regressed relationship between critical velocity *uc* and dry density ρ*d* can be expressed as follows:

$$u\_{\mathfrak{c}} = a \rho\_d^{\mathfrak{b}} \text{ or } \rho\_d = \left(u\_{\mathfrak{c}}/a\right)^{1/b} \tag{9}$$

where the coe fficient *a* and exponent *b* can be determined by conducting a flume experiment [32]. According to the preceding expressions regarding the similarities in the scouring mechanisms of fine sediment and similarities in the Froude number of flows, the scale ratio of dry density can be expressed as follows:

$$
\lambda\_{\mathcal{P}\_{\rm d}} = \lambda\_{u\_{\rm c}}^{1/b} = \lambda\_{h}^{1/2b} \tag{10}
$$

Using sediment sampled from the Agongdian Reservoir located in Southern Taiwan, Lai (1998) [33] conducted experiments by setting the value of *a* to 1.65 and the value of exponent *b* to 1.96. As shown in Figure 2, the particle size distribution of the sediment sample (with 22% of clay) from the Agongdian Reservoir is highly similar to the particle size distribution of the sediment sample from the Shihmen Reservoir. Thus, the relationship presented in Equation (9) for fine sediment from the Agongdian Reservoir is analogous to relationship for fine sediment from the Shihmen Reservoir. From the field data regarding dredged sediment properties provided by the Northern Region Water Resources O ffice (2010) [20], the water content of dredged material ranges from 39.9% to 45.2% at 1 year after deposition and from 81.8% to 98.5% at 16 days after deposition. Therefore, this study adopted water content levels of 40% and 80% to conduct tests in the physical model for sediment replenishment. Based on dredged sediment sample property, the dry densities corresponding to water content levels of 40% and 80% were 1.2 and 0.84 <sup>t</sup>/m3, respectively.

#### 4.2.2. Sediment Concentration and Transport Time

Flood flow from the reservoir generates a bed shear force that acts on the surface of replenished sediment. The scouring depth over time for the entire amount of replenished sediment can be denoted as ρ*dVS*, where *VS* is the volume of sediment eroded. The total amount of eroded sediment in the water column is denoted as *SV*, where *S* represents the sediment concentration in water of volume *V*. The amount of sediment scoured from the riverbed approximately equals that suspended in the

water column, that is, *SV* = ρ*dVS*. Because of the requirement of dynamic similarity during scouring, the sediment concentration ratio of the model to the prototype λ*s* can be written as follows:

$$
\lambda\_S = \frac{\lambda\_{\rho d} \lambda\_{Vs}}{\lambda\_V} = \frac{\lambda\_{\rho d} \lambda\_L^3}{\lambda\_L^3} = \lambda\_{\rho d} \tag{11}
$$

Equation (11) indicates that the scale ratio of the model to the prototype for the sediment concentration is equal to the scale ratio of dry density.

Bed elevation variations during river scouring can be described by the sediment continuity equation:

$$
\rho \frac{\partial (q\mathbb{S})}{\partial \mathbf{x}} + \rho\_d \frac{\partial z\_b}{\partial t} = 0 \tag{12}
$$

where *q* is the discharge per unit width, *x* is the distance along the river, *zb* is the bed elevation, and *t* is the time required for bed variation. On the basis of model similarity, Equation (13) can be derived from Equation (12) and expressed as follows:

$$\frac{\lambda\_q \lambda\_S}{\lambda\_x} = \lambda\_{\rho\_d} \frac{\lambda\_{z\_b}}{\lambda\_t} \tag{13}$$

where λ*x* = <sup>λ</sup>*zb* = λ*h* and λ*q* = λ*u*λ*<sup>h</sup>* = λ3/2 *h* for the undistorted model. Based on the relationship between Equations (10), (12), and (13), the time scale ratio of bed elevation variations due to scouring is λ*t* = λ1/2 *h* . In particular, the ratio is the same as the time scale of flow similarity derived from the Froude number. This indicates that the scouring time scale of flood flow is equal to that of sediment transport.

According to the physical model ratio of the prototype to the model (64:1), the model scale ratios could be derived from similarity theory (Table 1). The installation of replenished sediment and the filter structure to serve as the permeable barrier in the physical model is illustrated in Figure 9. Experimental results and analysis are described on the prototype scale in the following section.

**Figure 9.** Installation of replenished sediment and filter structure (serving as a permeable barrier): (**a**) Zone 1 in the physical model, (**b**) view from the upstream side, and (**c**) view from the downstream side of the replenishment area.


**Table 1.** Scale ratio of the flow patterns and sediment properties.
