1. Introduction
Steep slopes and severe bed changes during floods are the common characteristics of the rivers in Taiwan. Floods often cause channel incision, bank erosion, and meandering migration and endanger the safety of structures or river protection works. Spur dikes are protections commonly used for river engineering to lead water flow and reduce bank erosion. However, the local scour around spur dikes influences the stability of the dikes’ foundation. To reduce the risk of foundation failure, the understanding of local scour phenomenon around hydraulic structures in rivers is important.
One can understand the formation mechanism of scour hole by the flow field and sediment movement behavior of a single cylinder pier.
Figure 1 displays the scouring process around a pier structure to the front of the scour hole. Generally, the sediment entrainment and transport around the scour hole were most related to the turbulence in the approach flow. Through strong vertical flows, the turbulence fluctuations in the water are brought into the sediment layer. The turbulence fluctuation near the sediment layer enhanced the ability of the water to entrain sediment. When scour hole occurs gradually around the hydraulic structure, turbulence fluctuation in the approach flow contributes to the scouring process where there is sediment particles’ pick up, trajectory, deposition, and bed evolution processes with a mixed non-equilibrium sediment transport. Moreover, the characteristics of sediment, bed load with the gravity projection, and the bed geometry of a slope have crucial effects on incipient sediment motion. The sliding effect of scour hole occurs when a local bed slope is higher than the angle of repose. The angle of repose is the steepest angle of descent relative to the horizontal bed to which the bed material can be piled without slumping. At this angle, the bed material on the slope face is on the verge of sliding.
Most mobile-bed models proposed in the past few decades have focused on the sediment transport of an alluvial channel. Jia et al. [
2,
3] developed a two-dimensional (2D) mobile-bed model, known as CCHE2D numerical model, which is a numerical system for modeling 2D non-steady turbulent river flow, sediment transport, and water quality prediction. The model was also designed for use in multiprocess simulation of surface water with complex geometry, such as riverbed geometry; of bank erosion with both uniform and non-uniform sediment; and of meander river migration. The model was validated through many physical experiments and field cases in the development process.
The CCHE2D numerical model can be used for river engineering and design of hydraulic structures, such as grade control structures and dikes. In the review of past research on mobile-bed models, Lai et al. [
4] proposed a 2D model, known as SRH-2D numerical model that was employed to simulate the bed evolution that occurred downstream of the Chi-Chi weir in the Choushui River, Taiwan. Engineering plans pertaining to scour prevention and treatment were proposed and simulated using the model. Liao et al. [
5] developed a 2D mobile-bed model, known as EFA2D numerical model that considers the non-equilibrium suspended sediment concentration profile with a bedrock erosion mechanism for natural rivers.
In the review of the research on the mechanism of modifying sediment transport behavior, Juez et al. [
6,
7,
8] discuss the numerical assessment of bedload formulations for transient flow. The dam break flows over dry/wet initial conditions and erodible channel of experimental cases are simulated by considering the improved model, called Smart CFBS (Smart Combined Friction and Bed Slope model). The study has allowed a detailed analysis of the relative behavior to be performed in 2D situations of different sediment discharge formulas that were derived from 1D laboratory cases, and use of the Smart CFBS formula is suggested, regardless of the hydro-morphodynamic situation. The bedload transport over steep slopes with gravity projection has been studied and embedded in a non-trivial way, not only in the sediment transport rate but also in hydrodynamic equations. It is noted that the gravity effects play an important role on the sediment transport magnitude.
Horvat et al. [
9] developed a 2D model that couples the active layer and multiple size-class approaches for modeling sediment transport by using an enhanced advection algorithm. The focus of their study was improvement of advection computation from the point of view of numerical methods by reducing the limitation of simulation time. The study proposed the advection scheme that allows for larger time steps in simulation. This restriction was overcome by introducing modifications to the characteristic method. The model was assessed and validated using field measurements conducted on the Danube River.
The 3D mobile-bed model has the capability to simulate the local scouring process. Jia and Wang [
10] simulated 3D flow in a preformed scour hole. A detailed comparison of their simulation results and measurements were presented. Nagata et al. [
11] developed a 3D model to simulate flow and bed deformation around river hydraulic structures. The model solved the fully 3D, Reynolds-averaged Navier–Stokes equation that was expressed in a moving boundary-fitted coordinate system to calculate the flow field of water and bed surfaces at various times. A non-linear
k–
ε turbulence model was employed to predict the flow near the structure, where 3D flow was dominant. Burkow and Griebel [
12] conducted a numerical simulation of a fully 3D fluid flow to reproduce the current-driven sediment transport processes. The scouring at a rectangular obstacle was investigated, and the typical sedimentary processes and the sedimentary form of a scour mark were well captured by the simulation.
Castillo and Carrillo [
13] studied the scour produced by spillways and outlets of a dam due to the operation of free surface spillways and half-height outlets by using three complementary procedures: obtaining empirical formulas using models and prototypes, using a semiempirical methodology based on the pressure fluctuation–erodibility index, and employing computational fluid dynamics simulations. Jia et al. [
14] proposed a local scour scheme for a 3D mobile-bed model and successfully applied the model to the case of a bridge pier under non-uniform sediment conditions. The sediment entrainment, transport, and turbulence around a bridge pier are simulated. The turbulence fluctuations in the water are brought into the sediment layer through strong vertical flows. The ability of the flow to entrain sediment is enhanced by the intruding turbulence fluctuation near the sediment layer. This enhancement could be considered in 3D model by using additional shear stress related to the intruding turbulence fluctuation. However, most 3D models are still limited by the computational efficiency of the model in filed cases.
The sediment entrainment and transport around the scour hole are related to the turbulence fluctuation, characteristics of sediment, bed-load with the gravity projection, and the bed geometry of a slope that have crucial effects on incipient sediment motion. Yang et al. [
15] investigated the angle of repose of non-uniform sediment by considering the concept of exposure degree to account for the hidden and exposed mechanism of non-uniform sediment transport. Meng et al. [
16] proposed the angle of repose for uniform and non-uniform sediment. The angle of repose is based on the compacted friction relationship between mud and sand particles. Sediment dynamics holds that, when water flows over sediment particles on a loose surface, intersediment collision resists the fluid shear stress. This is consistent with underwater flow and sediment dynamics. Al-Hashemi and Al-Amoudi [
17] reviewed the angle of repose of granular materials. This angle is an essential parameter for understanding the microbehavior of granular material and then relating this behavior to the material’s macrobehavior.
In general, both an empirical formula and a numerical model can be used to evaluate the equilibrium scour depth around a hydraulic structure. Most of the empirical formulas have limitations to calculate the scouring depth or hole shape changing over time. This phenomenon and mechanism must be analyzed using the 2D or 3D numerical models. Although the empirical formula is relatively convenient to use, the numerical model is often complicated to use in terms of practical applications, especially when it is a 3D mobile-bed model. Therefore, obtaining a balance between these two approaches is crucial.
In this study, an empirical formula for calculating the sediment repose angle [
16] and the bed geometry adjustment mechanism are integrated into the CCHE2D numerical model to improve the simulation of the local scour hole around a hydraulic structure. The model is calibrated and validated by simulating an experimental case [
18] involving multiple spur dikes. Data are collected on flow depth, bed material grain size, scour depth around structures, and scour hole shape. These basic data are used to calibrate and validate the numerical model. By improving the 2D numerical simulations on local scour hole around spur dike, it will provide practical application value in terms of model efficiency and speed.
2. Numerical Model
The CCHE2D numerical model is a general surface water flow model. It simulates dynamic processes of flow and sediment transport in natural rivers. The CCHE2D numerical model has been developed at the National Center for Computational Hydroscience and Engineering (NCCHE), the University of Mississippi, for more than 20 years, and the developers continue to improve the model in response to different real problems.
The CCHE2D numerical model comprises flow and sediment transport modules. The governing equations of the flow module are 2D depth-integrated Reynolds equations. The free surface elevation of the flow is calculated using the depth-integrated continuity equation. Two methods for calculating the eddy viscosity exist in the model: the depth-integrated parabolic eddy viscosity formula and depth-integrated mixing length eddy viscosity model [
19,
20]. The sediment transport module comprises the suspended sediment transport and bed load transport simulations for non-uniform sediment. More details of the flow and sediment transport modules can be obtained in the CCHE2D technical report and verification and validation test documentation [
2,
3].
2.1. Flow Module
The governing equations of the flow are 2D depth-integrated Reynolds equations expressed in the Cartesian coordinate system and are as follows:
where
u and
v are the depth-integrated velocity components in the
x and
y directions, respectively;
t is the time;
g is the gravitational acceleration;
η is the water surface elevation;
ρ is the density of water;
h is the local water depth;
fCor is the Coriolis parameter;
τxx,
τxy,
τyx, and
τyy are the depth-integrated Reynolds stresses, shown as
Figure 2; and
τbx and
τby are the shear stresses on the bed and flow interface.
The shear stress terms are not considered at the water surface because the wind shear driven effect is small and not considered in the model. The turbulence Reynolds stresses in the governing equations are approximated according to the Bousinesq’s assumption that are related to the main rate of the strains of the depth-averaged flow field with a coefficient of eddy viscosity:
Free surface elevation of the flow was calculated using the depth-integrated continuity equation:
Two methods for calculating eddy viscosity exist in the model: the depth-integrated parabolic eddy viscosity formula and depth-integrated mixing length eddy viscosity model. First, the eddy viscosity coefficient
is calculated using the depth integrated parabolic eddy viscosity formula:
where
is the integration constant,
is shear velocity,
is the von Karman’s constant (0.41) and
is the relative depth of the flow.
is a coefficient to adjust the value of the eddy viscosity. Its default value is set to 1 and it can be adjusted by users from 1–10. In addition to this approach, a depth integrated mixing length eddy viscosity model is also available:
where the depth integrated velocity gradient along vertical coordinate
is introduced to account for the effect of turbulence generated from the bed surface. The eddy viscosity defined by Equation (7) would be zero in the uniform flow condition without this term. It is determined in the way that eddy viscosity shall be the same as that of the uniform flow in the absence of other terms. Assuming that the flow is of logarithmic profile along the depth of the water, the total velocity
U of the vertical gradient should be:
2.2. Sediment Transport Module
The sediment transport module performs suspended sediment transport and bed load transport simulations for non-uniform sediment. The module is also designed to simulate a channel bed with large slopes and curved channel secondary flow effects. The capability of simulating the non-equilibrium sediment transport is achieved by solving the following equations.
Bed changes:
where
εs is the eddy diffusivity of sediment in the vertical
z direction;
Ck is the concentration of the
k-th size class, and
C*k is the corresponding transport capacity;
αs is the adaptation coefficient for suspended load;
ωsk is the sediment settling velocity;
δb is the thickness of bed layer;
cbk is sediment load of the
k-th size class;
qb*k,
q*k,
qbxk and
qbyk are the bed load transport capacity, the bed load transport rate, and transport rate components in
x and
y directions, respectively;
Lt is the adaptation length for bed load that characterizes the distance for a sediment process adjusting from a non-equilibrium state to an equilibrium state, and the model computes the non-equilibrium sediment transport process including the actual and equilibrium sediment load by considering the adaption length and factors;
p’ is the porosity of bed material, and
Zbk is the bed change.
In most of the early sediment transport models, it is usually assumed that the actual sediment transport rate or a saturated concentration is equal to the capacity of flow carrying sediment at equilibrium conditions. Based on this sediment transport state, a sediment transport model is called an equilibrium transport model. It could be a steady exchange of particles between bed material and the sediment load. However, this equilibrium assumption is not realistic in natural alluvial rivers because of variations in flow conditions and bed material properties. Non-equilibrium sediment transport models adapt sediment transport equations to determine the realistic bed load and suspended load transport rates.
In the processes of vertical integration, the water must be assumed to be shallow relative to channel width and the vertical variation in the flow to be negligible; otherwise, the dispersion terms must be maintained to preserve the effect of vertical sediment profiles on changes in the bed form. In the second case, the source term is nonzero but represents the dispersion terms. Computation of the dispersion term is complicated and requires knowledge of the vertical velocity and suspended sediment profiles in the
x and
y directions. The dispersion terms in suspended load transport equation are usually combined with the diffusion terms. More details can be found in the technical report of the sediment module [
2].
3. Bed Geometry Adjustment Mechanism for a Local Scour Hole
Figure 3 displays the modules and flowchart of the calculation process in this study. First, the flow module calculates the flow conditions, e.g., flow field, velocity, and water depth. After calculating the flow conditions, the sediment transport module calculates the sediment transport process, e.g., sediment transport capacity, bed material sorting, and bed changes. Finally, local scour holes around hydraulic structures are adjusted using the bed geometry adjustment mechanism proposed in this study.
The flow velocity profile in the vertical direction is generally non-uniform. The predicted local scour depth and volume are often underestimated due to the hypothesis of the depth-averaged velocity in the 2D model. In this study, an empirical formula for calculating the sediment repose angle [
16] and an algorithm for modifying the bed geometry in the local scour hole are integrated into the 2D mobile-bed model to improve the model’s ability to simulate local scours around the structures. Experiments were conducted by Meng et al. [
16] by using cohesionless heavy sediment to obtain the angle of repose by using the natural accumulation method and angle of surface friction by using the tilting method. The experimental results revealed that the angle of repose over the water is bigger than that in the submarine condition. Moreover, the angle of repose does not increase monotonously with an increase in the sorting coefficient but increases with a decrease in the asymmetry coefficient
S of non-uniform sediment for the same median diameter
D50. The empirical formula (expressed in Equations (13)–(15)) proposed by Meng et al. [
16] for uniform and non-uniform sediment is employed in this study. The procedure of the bed geometry adjustment mechanism for local scour holes can be explained as follows.
First, the angle of repose is calculated according to the sorting coefficient and bed material (e.g., uniform sand, non-uniform sand, gravel, or cobble). Then, the bed elevation around the hydraulic structures in the stream is calculated and modified using trigonometry with different bed slope trends (e.g., downhill or uphill slope). Finally, the adjusted bed geometry of the local scour hole around the hydraulic structures is obtained.
The angle of repose for uniform and non-uniform sediment proposed by Meng et al. [
16] can be expressed as follows:
where
is the angle of repose at computational node
i and
Di is the mean diameter of the bed material at computational node
i. Moreover,
S0 is the sorting coefficient, in which
D75 and
D25 are the particle sizes at 75% and 25% weight of the bed materials. The average angle of repose between computational nodes
i and
i + 1 can be expressed as follows:
Depending on whether there is a downhill or uphill bed slope around the scour hole (
Figure 4), the adjusted bed geometry can be expressed as follows:
where
is the bed elevation at computational node
i that is modified by the angle of repose and
is the distance between computational nodes
i and
i + 1. In the computational nodes of bed elevation in the 2D model, where the set of
i are the nodes along the longitudinal direction (
x-direction) and the set of
j are those along the lateral direction (
y-direction).
Figure 5 shows a flowchart of the bed geometry adjustment mechanism for a local scour hole. According to the characteristics of sediment classes, the bed geometry of local scour hole could be adjusted in the 2D model by considering the adjustment mechanism mentioned above.