## 2 Panel/Viscous Vortex Particle Aerodynamic Model
### Vortex Particle Method  
It is well established that the wake has a large influence on the aerodynamics of rotorcrafts [3]. As such, an accurate wake model is essential for the rotorcraft aerodynamic model. In the present work, the wakes are modeled using the viscous vortex particle method, which is a Lagrangian formulation of the incompressible Navier-Stokes equation:  
\[\frac{\mathrm{d}\mathbf{\omega}}{\mathrm{d}t}=\mathbf{\omega}\cdot\
abla\mathbf{u}+v \
abla^{2}\mathbf{\omega}, \tag{7}\]  
where \(\mathbf{\omega}\) is the vorticity, \(\frac{\mathrm{d}\mathbf{\omega}}{\mathrm{d}t}\) is the material derivative, and \(v\) is the kinematic viscosity. The vorticity field \(\mathbf{\omega}\) is represented by \(N\) vortex particles with a vector-valued vorticity strength and by using moving Gaussian basis functions \(\xi_{\sigma}\) as follows:  
\[\mathbf{\omega}(\mathbf{x},t)=\sum_{j=1}^{N}\xi_{\sigma}(\mathbf{x}-\mathbf{x}_{j})\mathbf{\alpha }_{j}=\sum_{j=1}^{N}\xi_{\sigma}(\mathbf{x}-\mathbf{x}_{j})\mathbf{\omega}_{j}V_{j}, \tag{8}\]where \(\mathbf{\alpha}_{j}\) is the total vorticity inside the vortex particle, \(\mathbf{x}_{j}\) is the location of the \(j\) th particle, \(V_{j}\) is the particle volume, and \(\sigma_{j}\) is the particle core radius. Here, \(\xi_{\sigma}(\mathbf{x})=\xi(|\mathbf{x}|)/\sigma^{3}\), and \(\xi(|\mathbf{x}|)\) is the Gaussian basis function that is used to smooth the vorticity field and deprecate singularity and is defined by  
\[\xi(|\mathbf{x}|)=\frac{1}{(2\pi)^{3/2}}e^{-|\mathbf{x}|^{2}/2}. \tag{9}\]  
With the discretized vortex particles, the viscous vorticity dynamic equation can be rewritten as follows:  
\[\frac{\mathrm{d}\mathbf{x}}{\mathrm{d}t}=\mathbf{u}, \tag{10}\]  
\[\frac{\mathrm{d}\mathbf{\alpha}}{\mathrm{d}t}=\mathbf{\alpha}\cdot\
abla\mathbf{u}+v \
abla^{2}\mathbf{\alpha}. \tag{11}\]  
Equation (10) describes the vortex particles convection; in Equation (11), the first term on the right hand-side is the stretching-effect term, which stands for vortex stretching and rotation due to the velocity gradient, while the second term on the right-hand side is the viscous diffusion term, which represents the vorticity diffusion caused by viscous effects. The velocity of particles can be calculated by the well-known Biot-Savart law, which is the solution to the unbounded Poisson equation:  
\[\mathbf{u}(\mathbf{x},t)=\sum_{j=1}^{N}\frac{1}{\sigma_{j}^{3}}K(\rho)(\mathbf{x}-\mathbf{x}_ {j})\times\mathbf{\alpha}_{j}, \tag{12}\]  
where \(\rho=\big{|}\mathbf{x}-\mathbf{x}_{j}\big{|}/\sigma_{j}\) is the nondimensional distance parameter, \(K(\rho)=\frac{G(\rho)-\xi(\rho)}{\rho^{2}}\) is the regularized Biot-Savart kernel for velocity evaluation, and \(G(\rho)\) is the Green's function for vector potential and can be expressed as follows:  
\[G(\rho)=\frac{1}{4\pi\rho}\mathrm{erf}\bigg{(}\frac{\rho}{\sqrt{2}}\bigg{)}, \tag{13}\]  
where  
\[\mathrm{erf}(s)=\frac{2}{\sqrt{\pi}}\int_{0}^{s}e^{-s^{2}}\mathrm{d}v \tag{14}\]  
is the error function. The vortex particle method was shown to convergence as the number of particles increases provided that the smoothing core radius is larger than the distance between neighbor particles. The stretching term is handled by using the so-called direct scheme, which is the dot product between the velocity gradient and the particle vorticity; the velocity gradient is obtained by analytically differentiating the velocity field, formulated as  
\[\mathbf{\alpha}_{i}\cdot\
abla\mathbf{u}(\mathbf{x}_{i},t)=\sum_{j=1}^{N}\frac{1}{\sigma _{ij}^{3}}[\mathbf{\alpha}_{j}\times]\cdot\
abla[K(\rho)(\mathbf{x}-\mathbf{x}_{j})]\cdot \mathbf{\alpha}_{i}, \tag{15}\]  
where \(\mathbf{\alpha}_{j}\times\) is the skew symmetric matrix built from vector \(\mathbf{\alpha}_{j}\).