## 2 Numerical Methods
### Eulerian Approach Based on Continuum Properties
#### 2.1.2 Vortex Core Identification and Vortex Model
The vortex core region is defined as the region where \(\lambda_{2}\) is negative (\(\lambda_{2}<0\)), and the vortex core center is defined as the location where \(\lambda_{2}\) is a minimum. The tangential velocity and pressure fields of vortex structure regenerated by using the Scully vortex model are written in the forms, respectively, [27]  
\[V_{\theta}(r)=\frac{\Gamma}{2\pi}\bigg{(}\frac{r}{a_{c}^{2}+r^{2}}\bigg{)}. \tag{7}\]  
\[P(r)=P_{\infty}-\frac{\rho\Gamma^{2}}{8\pi^{2}}\bigg{(}\frac{1}{a_{c}^{2}+r^{2 }}\bigg{)} \tag{8}\]  
By separating the tangential velocity (\(V_{\theta}\)) to \(x\)-, \(y\)- and \(z\)- velocity components, the velocity field in the Cartesian coordinate can be obtained. Pressure and velocity fields in the regenerated vortex are a function of distance (\(r\)) from the vortex center with parameters of initial vortex core radius (\(a_{c}\)) and circulation (\(\Gamma\)). The vortex flow field can be divided into two areas according to its flow characteristics: one is the inner region of the vortex core, where a forced rotating flow forms, and the other is the outer region of the vortex core where a free vortex flow occurs. Some vortex models are available to define the vortex structure, such as the Rankine vortex, Scully vortex, Vatistas vortex, and Lamb-Oseen vortex. The Rankine vortex is a simple mathematical model of a vortex in a viscous fluid. However, this model has a singularity presented by a sharp velocity profile at the interface between two regions. This singularity can be overcome by using the algebraic velocity profile as Equations (7) and (8) of the Scully vortex model. The Vatistas vortex is also of the algebraic swirl-velocity profile but is based on the stationary vortex assumption. The Lamb-Oseen vortex model is an analytic solution of the one-dimensional Navier-Stokes equation, but with the assumption that the flow is laminar and all velocity components except the azimuthal velocity are zero [32]. Therefore, in this study, the Scully vortex model is used to define the vortex structure due to its merits over the others. However, since the vortex structure is physically dissipated by fluid viscosity during its convection along the downstream, the physical dissipation effects need to be considered in the reconstructed vortex field. [PERSON] and [PERSON] [33] predicted the variation of tip vortex core radius along with the downstream by considering the vortex dissipation due to viscous effect, and [PERSON] et al. [21] applied this method in the vortex modeling process, named the dissipation vortex model (DVM). The formula of DVM is written as  
\[a_{c}^{*}=\frac{\Delta a_{c}}{D}=2.92\left(\frac{x}{D}\right)^{0.5}\frac{1}{ Re_{D}^{0.5}}. \tag{9}\]  
\[a_{c}(x)=a_{c,init}+\Delta a_{c} \tag{10}\]  
Here, \(a_{c}^{*}\) denotes a non-dimensional vortex core radius, and \(a_{c}\) and \(\Delta a_{c}\) are the vortex core radius and its variation along with the downstream distance, \(x\), respectively. \(D\) and \(Re_{D}\) denote the chord length and the chord-based Reynolds number, respectively. The vortex core radius at the location downstream is then the sum of the initial vortex core radius (\(a_{c,init}\)) and the change in vortex core radius as Equation (10). The initial vortex core radius can be determined from the RANS simulation result.