# Effects of Bridge Piers on Flood Hazards: A Case Study on the Jialing River in China
## 2 Materials and Methods
### Model Description  
The numerical model used in this study was the Mike 21 FM, developed by the Danish Hydraulic Institute (DHI), which uses a finite volume method and unstructured triangular mesh to solve the 2D shallow water equations [31]. The governing equations are given by  
Equation of mass conservation:  
\[\frac{\partial h}{\partial t}+\frac{\partial p}{\partial x}+\frac{\partial q}{ \partial y}=0 \tag{1}\]  
Equation of momentum conservation:  
\[\frac{\partial p}{\partial t}+\frac{\partial}{\partial x}\Big{(}\frac{p^{2}}{h }\Big{)}+\frac{\partial}{\partial y}\Big{(}\frac{pq}{h}\Big{)}+\delta h^{2} \frac{\partial c}{\partial x}+\frac{\partial\sqrt{p^{2}+q^{2}}}{\rm C^{2} \times 2h^{2}}-\frac{1}{\rho_{w}}\Big{[}\frac{\partial}{\partial x}\big{(}h \tau_{xx}\big{)}+\frac{\partial}{\partial y}\Big{(}h\tau_{xy}\big{)}\Big{]}- \Omega_{x}=0 \tag{2}\]  
\[\frac{\partial q}{\partial t}+\frac{\partial}{\partial y}\Big{(}\frac{q^{2}}{h }\Big{)}+\frac{\partial}{\partial x}\Big{(}\frac{pq}{h}\Big{)}+\delta h^{2} \frac{\partial c}{\partial y}+\frac{\partial\sqrt{p^{2}+q^{2}}}{\rm C^{2} \times 2h^{2}}-\frac{1}{\rho_{w}}\Big{[}\frac{\partial}{\partial y}\Big{(}h \tau_{yy}\Big{)}+\frac{\partial}{\partial x}\Big{(}h\tau_{xy}\Big{)}\Big{]}+ \Omega_{y}=0 \tag{3}\]  
Figure 1: Location of project section of the Jialing River, China.  
where, \(h=\zeta+d\) is the total water depth; \(d(x,y)\) is the still water depth; \(\zeta(x,y,t)\) denotes free surface elevation; \(p\) and \(q\) are the fluxes per unit width at \(x\) and \(y\) directions, respectively; \(g\) is gravitational acceleration; \(C\) is the Chezy coefficient which is determined by [PERSON]'s formula; \(\rho_{w}\) denotes density of the water; \(\tau_{xx},\tau_{xy},\tau_{yy}\) are the components of lateral shear stress (viscous friction, turbulent friction, and differential advection); \(\Omega_{x}\) and \(\Omega_{y}\) are the Coriolis forces at two horizontal directions(not covered in this study). \(x,y\), are the spatial coordinates and \(t\) denotes time.  
An approximate Riemann solver (Roe scheme) was used to calculate the convective fluxes at cell boundaries. Second-order spatial accuracy was achieved by employing a linear gradient reconstruction technique. A second-order total variation diminishing (TVD) slope limiter (the van-Leer limiter) was used to avoid numerical oscillations. A second-order Runge-Kutta method was used for time integration. More details about the numerical schemes of the model is found in the scientific documentation of the Mike 21 FM.  
To ensure the stability of the model, the time step had to be selected such that the Courant-Friedrich-Levy (CFL) number was less than unity. For the shallow water equations in Cartesian coordinates, the CFL number is defined as:  
\[CFL_{HD}=\Big{(}\sqrt{gh}+|u|\Big{)}\frac{\Delta t}{\Delta x}+\Big{(}\sqrt{gh} +|v|\Big{)}\frac{\Delta t}{\Delta y} \tag{4}\]  
where, \(h\) is the total water depth at the cell center; \(u\) and \(v\) are the depth averaged velocities in the \(x\) and \(y\) directions, respectively; \(g\) denotes gravitational acceleration; \(\Delta x\) and \(\Delta y\) are the mesh sizes in \(x\) and \(y\) directions, respectively, which are approximately the minimum side length for the triangular meshes; and \(\Delta t\) is the time step.