# Numerical Study of the Flow and Blockage Ratio of Cylindrical Pier Local Scour
## 2 Mathematical Model
### Morphodynamic Model  
The present study analyzes sediment transport in the clear-water and live-bed regimes. Despite this, the Rouse number (\(W_{s}/u_{\tau}\), where \(W_{s}\) is the settling velocity) is not high enough for the suspended load to be significant; therefore, only the bedload formulation is considered. Thus, the morphological evolution of the sediment bed is described using Equation (7), also known as the Exner-Polya equation.  
\[(1-\eta)\frac{\partial z_{b}}{\partial t}+\frac{\partial q_{b_{i}}}{\partial x _{i}}=0, \tag{7}\]  
where \(\eta=0.4\) is the porosity of the bed, \(z_{b}\) is the bed elevation, and \(q_{b_{i}}\) (\(i=1,2\)) is the bedload transport rate given by Equation (8), proposed by [PERSON] and [PERSON], 1976 [35].  
\[q_{b_{i}}=\frac{\pi d}{6}p_{{}_{EF}}u_{bi}, \tag{8}\]  
where \(d\) is the particle diameter, \(p_{{}_{EF}}\) is the percentage of particles in motion modeled, and \(u_{b_{i}}\) (\(i=1,2\)) are the mean velocity components of a sediment particle in movement. The value \(p_{{}_{EF}}\) is given by Equation (9).  
\[p_{{}_{EF}}=\left\{\begin{array}{ll}\left[1+\left(\frac{\pi\mu_{d}}{6(\theta -\theta_{c})}\right)^{4}\right]^{-\frac{1}{4}}&,\ \ \theta>\theta_{c}\\ 0&,\ \ \theta\leq\theta_{c}.\end{array}\right. \tag{9}\]  
Here, \(\mu_{d}\) is the dynamic friction coefficient, \(\theta\) is the Shields number, and \(\theta_{c}\) is its critical value for the initiation of motion. The mean velocity is given by the Equation (10).  
\[u_{b_{i}}=a\left(1-0.7\sqrt{\frac{\theta_{c}}{\theta}}\right)u_{\tau}, \tag{10}\]  
where \(a\) is an empirical constant and \(\vec{u}_{\tau}=(u_{\tau_{1}},u_{\tau_{2}})\) is the friction velocity. As implemented in [PERSON] et al., 2020 [20], \(\theta_{c}\) is modelled using Equation (11).  
\[\theta_{c}=\theta_{c_{0}}\left(\cos\beta\sqrt{1-\frac{\sin^{2}\alpha\tan^{2} \beta}{\mu_{s}^{2}}}-\frac{\cos\alpha\sin\beta}{\mu_{s}}\right), \tag{11}\]  
where \(\theta_{c_{0}}\) is the critical Shields number for the initiation of motion in a bed with slope zero, \(\mu_{s}\) is the static friction coefficient, \(\beta\) is the maximum angle of repose, and \(\alpha\) is the difference between the angle of the flow near the bed and the maximum angle of repose \(\beta\). The friction velocity is calculated by solving Equation (12), as proposed by [PERSON], 1950 [38]:  
\[\frac{u}{u_{\tau}}=\kappa^{-1}\ln\left(\frac{z}{z_{0}}\right), \tag{12}\]  
where \(z_{0}\) is governed by the friction Reynolds number \(\text{Re}_{\tau}=u_{\tau}k_{s}/\
u\), with \(k_{s}\) being the so called Nikuradse sand roughness. Based on the value of \(\text{Re}_{\tau}\), three hydraulic regimesare defined to compute \(z_{0}\). The flow is hydraulically smooth for \(\mathrm{Re}_{\tau}\leq 5\), and rough for \(\mathrm{Re}_{\tau}\geq 70\). [PERSON], 1997 ([PERSON], 1997 ([PERSON], 1997), developed a transitional formula in Equation (13) to link both regimes:  
\[z_{0}\;=\;\left\{\begin{array}{ll}\frac{\
u}{9\,u_{\tau}}&,\,\text{for}\;Re_ {\tau}\;\leq\;5\\ \\ \frac{\
u}{9\,u_{\tau}}\;+\;\frac{k_{s}}{30}\biggl{[}1\;-\;\exp\biggl{(}-\; \frac{u_{\tau}k_{s}}{27\,\
u}\biggr{)}\biggr{]}&,\,\text{for}\;5\;<\;Re_{\tau} \;<\;70\\ \\ \frac{k_{s}}{30}&,\,\text{for}\;Re_{\tau}\;\geq\;70\end{array}\right. \tag{13}\]