## 2 Methodology
### Hydrodynamic Model
The \(k-\omega\) model uses the specific dissipation rate, \(\omega=\varepsilon/k\) instead of \(\varepsilon\) to model \(\
u_{t}\) defined as:  
\[\
u_{t}=\frac{k}{\omega} \tag{7}\]To resolve Equation (7), the additional transport equations for \(k\) and \(\omega\) are presented as:  
\[\frac{\partial k}{\partial t}+\frac{\partial(U_{i}k)}{\partial x_{i}}=\frac{ \partial}{\partial x_{j}}\bigg{[}\bigg{(}\frac{v_{t}}{\sigma_{k}}+v\bigg{)} \frac{\partial k}{\partial x_{j}}\bigg{]}+G_{k}-\beta^{*}k\omega \tag{8}\]  
\[\frac{\partial\omega}{\partial t}+\frac{\partial(U_{i}\omega)}{\partial x_{i} }=\frac{\partial}{\partial x_{j}}\bigg{[}\bigg{(}\frac{v_{t}}{\sigma_{\omega} }+v\bigg{)}\frac{\partial\omega}{\partial x_{j}}\bigg{]}+\alpha G_{\omega} \frac{\omega}{k}-\beta\omega^{2} \tag{9}\]  
where \(\alpha_{\omega}\), \(\alpha\), \(\beta\), and \(\beta^{*}\) are the empirical constants; and \(G_{\omega}\) is the production of the dissipation rate. Here, the empirical constants are suggested as \(\alpha_{k}=2\), \(\alpha_{\omega}=2\), \(\alpha=0.55\), \(\beta=0.075\), and \(\beta^{*}=0.09\)[22]. The \(k-\omega\) model is superior to the \(k-\varepsilon\) model for simulating the boundary layer flows and streamwise pressure gradients. Nevertheless, the \(k-\omega\) model has a disadvantage in that the simulation results are sensitive to the free-stream boundary conditions [25]. The SST \(k-\omega\) model compensates for the shortcomings of these two turbulence models by implementing the \(k-\omega\) model in the boundary layers and the \(k-\varepsilon\) model in the free-shear layers. Thus, the SST \(k-\omega\) model is advantageous to reproduce the wall-bounded flows under low Reynolds conditions without extra damping functions. In the SST \(k-\omega\) model, \(v_{t}\) is modeled as:  
\[v_{t}=\frac{a_{1}k}{\max(a_{1}\omega,\;SF_{2})} \tag{10}\]  
where \(a_{1}\) is the closure coefficient; \(S\) is the absolute value of the vorticity; and \(F_{2}\) is the blending function that gives unity for the boundary layers and zero for the free-shear layers. The SST \(k-\omega\) model adopts the reformulated \(\omega\) equation as follows:  
\[\frac{\partial U_{i}}{\partial t}+U_{j}\frac{\partial U_{i}}{\partial x_{j}}= -\frac{1}{\rho}\frac{\partial P}{\partial x_{i}}+\frac{\partial}{\partial x_ {j}}\bigg{[}v\bigg{(}\frac{\partial U_{i}}{\partial x_{j}}+\frac{\partial U_{ j}}{\partial x_{i}}\bigg{)}\bigg{]}-\frac{\partial}{\partial x_{j}}\overline{u_{i}u_{j}} \tag{2}\]0  
where \(F_{2}\) is the auxiliary relation; and \(\sigma_{\omega 2}\) is an empirical constant. The SST \(k-\omega\) model exhibits improved simulation results for adverse pressure gradient flows, thereby accurately reproducing the pressure-induced flow separation [21]. [PERSON] et al. [48] simulated flow and sediment transport in curved channels using the SST \(k-\omega\) model, and this turbulence model was found to adequately predict changes in bed elevation by well reproducing 3D flow features in the channel bends. Moreover, [PERSON] et al. [49] successfully applied the SST \(k-\omega\) model to capture flow characteristics of a natural-like meandering stream located in the St. Anthony Falls Laboratory Outdoor StreamLab, University of Minnesota. They compared simulation results of the \(k-\omega\) and SST \(k-\omega\) models with LES and field observation data. As a result, the SST \(k-\omega\) model shows close agreement with the validation data sets, while the \(k-\omega\) model overestimates turbulent kinetic energy. Furthermore, [PERSON] and [PERSON] [50] demonstrated that the SST \(k-\omega\) model accurately simulates 3D free flows past hydraulic structures and over complex topography of a meandering channel.