## 2 Numerical Methods
### Tip-Vortex Regeneration Using Vortex Model  
It is, generally, difficult to capture the vortex structure using the RANS simulation where the vortex tends to be over-dissipated due to numerical viscosity of numerical difference schemes and excessive turbulence viscosity of the turbulence model. To resolve this issue, the tip-vortex structure is regenerated by applying a vortex model to the background flow field obtained from the RANS simulation. The position of the vortex core should be identified in the RANS result before the vortex model is applied. The location of the vortex core is determined using the \(\lambda_{2}\) criterion, which is the second-largest eigenvalue of the matrix. It is the sum of the strain tensor (\(S\)) and vorticity tensor (\(\Omega\)) of the flow, as shown in the following equations for the \(y\)-\(z\) plane. \[\
abla\overrightarrow{u}=\left( \begin{array}{cc}\frac{\partial\varphi}{ \partial y}&\frac{\partial\varphi}{\partial z}\\ \frac{\partial\varphi}{\partial y}&\frac{\partial\varphi}{\partial z}\end{array}  \right)=S+\Omega \tag{1}\]  
\[\lambda_{2}=\frac{\left(\frac{\partial\varphi}{\partial y}\right)^{2}+\left( \frac{\partial\varphi}{\partial z}\right)^{2}}{2}+\frac{\partial\varphi}{ \partial z}\frac{\partial\varphi}{\partial y} \tag{2}\]  
Here, \(y\) and \(z\) denote Cartesian coordinates on the cross-sectional plane perpendicular to the mean flow direction, and \(v\) and \(w\) denote velocity components in the \(y\)- and \(z\)-directions, respectively. The \(\lambda_{2}\) criterion is widely used to identify coherent vortex structure because it can accurately find the coherent vortex core for various flow fields. The region of negative \(\lambda_{2}\) is defined as the vortex core, and the  
Figure 4: (a) Distribution of \(y^{+}\) on NACA16-020 wing surface; (b) grids in wing near zone and vortex zone. minimum corresponds to the central axis of the vortex core. It is possible to define a vortex core on a cross-section plane perpendicular to the tip-vortex vector [34]. In this study, \(\lambda_{2}\) is defined in the two-dimensional plane (\(y\)-\(z\) plane) in Equation (2) because the vortex is formed on a cross-sectional plane normal to the mean flow direction (\(x\)-dir.). The vortex model is applied on each plane normal to the vortex core line determined from the background flow field. [PERSON] et al. [35] generated vortex flow fields using the Rankine vortex model. They determined the vortex properties of the circulation, \(\Gamma\), and vortex core-radius, \(a_{c}\), using the reference length and freestream velocity based on the classical thin wing theory and assuming a fully turbulent boundary layer. However, this method can overestimate the vortex strength. Therefore, in this study, the circulation is estimated using the vortex core-radius and maximum tangential velocity, \(u_{\theta}\), in the region near downstream of the wing-tip, where the vortex begins to form. The vortex flow field is regenerated using the Scully vortex model. The Rankine vortex model, which assumes a solid body rotation due to the viscous effect inside the vortex core and a free (potential) vortex outside, exhibits a discontinuity at the boundary of the vortex core from the fixed vortex to free vortex. The Scully vortex model used an algebraic velocity distribution to overcome this singularity problem. The tangential velocity in the Scully vortex model is defined as a function of \(r\) as follows:  
\[V_{\theta}(r)=\frac{\Gamma}{2\pi}\bigg{(}\frac{r}{a_{c}^{2}+r^{2}}\bigg{)} \tag{3}\]  
Equation (3) is used to determine the circulation from the tangential velocity and the vortex core-radius. Subsequently, the pressure distribution is determined from the three-dimensional steady-state Navier-Stokes equation for the radial direction as follows:  
\[