## 2 Numerical and Experimental Methods
### Vortex Particle Simulation (VPM)
To account for the vorticity flux from the solid wall, the strengths of the surface vortex panels need to be solved by cancelling the slip velocity on the surface, which is followed by the vorticity flux to the nearby particles, for example, as follows:  
\[\frac{D\omega}{Dt}=v\
abla^{2}\omega \tag{2}\]0  
To maintain good performance of the simulations, the spacings between particles, \(h\), are set to be equal to the particle of the radii, \(\sigma\), to ensure overlaps between particles. As time elapses, the degree of particle overlaps decreases due to the strain in the flow. Therefore, the particles are redistributed to a regularized grid every 10 steps. As the Reynolds number increases, the concentration of vorticity in smaller regions is anticipated, which implies that smaller flow structures could appear with faster local rotation. As a result, smaller time increments in simulation steps are required to resolve the details of advection and diffusion for higher Reynolds numbers. Here, \(\Delta t=0.05\) s and 0.002 s are applied to low and high Reynolds number simulations, respectively. Near the airfoil, the particle radii are maintained near the order of the viscous length scale, i.e., \(\sigma=1.4\sqrt{v\Delta t}\). Therefore, as the Reynolds number increases, the particles become finer, and the number of particles increases accordingly. To economically use the computational elements, variable-sized particles are implemented in the simulations, since the particles occupying the wake region are of secondary importance. The varying-sized particles are established based on the radial grid in each redistribution step. Figure 2a,b schematically shows the variations of the grid size along the streamwise direction. The entire simulation domain is shown in Figure 2a, which resembles a sector. As can be seen, the maximum length of the domain in the streamwise direction can extend up to slightly more than \(x/C=10\). The center (or the singularity point) of the radial grid is located at \((x,y)=(-2\) m, 0 m) and the number of azimuthal divisions, \(N_{\text{m}}\), varies with the Reynolds number (Table 1). The grid cells are the finest upstream and gradually increase their sizes in the downstream direction. If a particle moves out of the simulation domain, it is simply disregarded in simulation. Figure 2b shows an example of a closed view of particles initialized near the bluff body. The surface of the airfoil is numerically achieved by straight-line constant-strength vortex panels [27], which are used to impose surface vorticity flux to the particles near the body. Near the leading and trailing edges, smaller panel lengths are used, as can be seen from the distribution of panel edge points indicated in \[\frac{D\omega}{Dt}=v\
abla^{2}\omega \tag{2}\]1 The setups of VPM simulations are summarized in \[\frac{D\omega}{Dt}=v\
abla^{2}\omega \tag{2}\]2  
\[\frac{D\omega}{Dt}=v\
abla^{2}\omega \tag{2}\]3 Schematic description of the vortex particles initialized on the varying-sized grid for \(R\epsilon_{C}=500\): (**a**) Entire domain. (**b**) Close view of particles near the airfoil. The numbers of particles used in the simulations are shown in Figure 3a. The rapid growth of the particles can be observed for \(Ut/C\leq 3\), while the growth rate is slowed down by the use of a variable-sized particle grid for \(3\leq Ut/C\leq 10\). The number of particles stabilizes after the particles fill the wake region in the simulation domain (\(Ut/C\leq 10\)). The stabilized numbers of the particles are about O(10\({}^{5}\)) for \(Re_{C}=4000\) and one order higher, i.e., O(10\({}^{6}\)), for \(Re_{C}=60{,}000\). As a result, the simulations of low Reynolds number cases are significantly faster and thus longer. Due to the larger numbers of particles and shorter time increments (see Table 1) that are implemented, the simulation times of high Reynolds number cases are shorter.