## 2 Numerical and Experimental Methods
### Vortex Particle Simulation (VPM)  
To conveniently obtain the normalized flow quantities in the numerical simulation, the chord lengths, \(C\), the free stream velocity, \(U\), and air density, \(\rho\), are set as unity (i.e., \(U\) = 1 m/s, \(C\) = 1 m, \(\rho\) = 1 kg/m\({}^{3}\)) in the numerical method introduced in this section. Two Reynolds numbers, i.e., \(Re_{C}\) = 4000 and 60,000, are considered in the modeling, which can be easy to setup by changing the kinematic viscosity in the model, i.e., \(v=UC/Re_{C}=Re_{C}{}^{-1}\), if unitary values of basic parameters are utilized. Figure 1: Airfoils studied in this work: (**a**) NACA64 and (**b**) DU40. The dots on the airfoil surface represent the edges of the surface panels. The current vortex particle method (VPM) detailed in [23] is used to directly simulate the unsteady incompressible flow in two dimensions (2D). By taking the curl of the Navier-Stokes equations and noting that the vortex stretching terms vanish in 2D, the advection and diffusion of the vorticity form the governing equation for the vortex simulations, for example, as follows:  
\[\frac{D\omega}{Dt}=v\
abla^{2}\omega \tag{2}\]  
where \(\omega=\omega\)\(\mathbf{e}_{3}\) is the vorticity in the out-of-plane direction, \(\mathbf{e}_{3}\) and \(D/Dt\) represents the material derivative. The current method discretizes the vorticity-intense region by vorticity-carrying particles, for example, as follows:  
\[\omega(\mathbf{x})=\sum_{j=1}^{N}\Gamma_{j}\delta_{\sigma}\big{(}\mathbf{x}- \mathbf{x}_{j}\big{)} \tag{3}\]  
where \(\delta_{\sigma}\) is the Gaussian radial function used to distribute the circulation of the \(j\)-th particle located at \(\mathbf{x}_{j}\) to a nearby location, \(\mathbf{x}\), based on the radius of the, \(\sigma_{j}\), for example, as follows:  
\[\delta_{\sigma}(r)=\frac{1}{2\pi\sigma_{j}^{2}}\exp\left(-\frac{r^{2}}{2\sigma _{j}^{2}}\right) \tag{4}\]  
and \(r=\big{|}\mathbf{x}-\mathbf{x}_{j}\big{|}\). The particles are transported by the local velocity, \(\mathbf{u}(\mathbf{x}_{i})\), evaluated using the Biot-Savart law, for example, as follows:  
\[\mathbf{u}(\mathbf{x}_{i})=\frac{1}{2\pi}\mathbf{e}_{(z)}\times\underset{j=1} {\overset{N}{\sum}}\Gamma_{j}\frac{\mathbf{x}_{i}-\mathbf{x}_{j}}{\big{|} \mathbf{x}_{i}-\mathbf{x}_{j}\big{|}^{2}}f\big{(}\big{|}\mathbf{x}_{i}- \mathbf{x}_{j}\big{|}\big{)} \tag{5}\]  
where \(f\) is the velocity mollification kernel corresponding to the vorticity support \(\delta_{\sigma}\), for example, as follows:  
\[f(r)=1-\exp\left(1-\frac{r^{2}}{2\sigma_{ij}^{2}}\right) \tag{6}\]  
Note that the symmetrized particle radius, i.e., \(\sigma_{ij}^{2}=\left(\sigma_{i}^{2}+\sigma_{j}^{2}\right)/2\), is used to conserve moments of order 0, 1, and 2. To evaluate the vorticity diffusion on the right-hand-side of Equation (2), particle-strength-exchange (PSE) is applied, for example, as follows:  
\[\Gamma_{i}^{(n+1)}=\Gamma_{i}^{(n)}+\frac{vdt}{\sigma_{ij}^{2}}\sum_{j=1}^{N }h^{2}\Big{(}\Gamma_{j}^{(n)}-\Gamma_{i}^{(n)}\Big{)}\eta_{\sigma}\big{(}x_{ j}-x_{i}\big{)}, \tag{7}\]  
The interaction kernel of the PSE, \(\eta_{\sigma}\), corresponding to the vorticity kernel is as follows:  
\[\eta_{\sigma}(r)=\frac{1}{\pi\sigma_{ij}^{2}}\exp\left(-\frac{r^{2}}{2\sigma _{ij}^{2}}\right) \tag{8}\]  
Utilizing the 2 nd-order accurate Adam-Bashforth scheme, the numerical procedure in simulating Equation (2) is split into advection, i.e., Equation (9), and diffusion, i.e., Equation (10), as follows:  
\[\mathbf{x}_{i}^{(n+1)}=\mathbf{x}_{i}^{(n)}+\Delta t\bigg{(}\frac{3}{2} \mathbf{u}_{i}\Big{(}\mathbf{x}^{(n)},\Gamma^{(n)}\Big{)}-\frac{1}{2}\mathbf{ u}_{i}\Big{(}\mathbf{x}^{(n-1)},\Gamma^{(n-1)}\Big{)}\bigg{)} \tag{9}\]  
\[\Gamma_{i}^{(n)}=\Gamma_{i}^{(n)}+\Delta t\left.\frac{d\Gamma}{dt}\right|_{ \text{PSE}}\Big{(}\mathbf{x}^{(n)},\Gamma^{(n)}\Big{)} \tag{10}\]where the superscript \((n)\) denotes the \(n\)-th time step.