# Assessment of a Mid-Fidelity Numerical Approach for the Investigation of Tiltrotor Aerodynamics
## 2 Numerical Approach Implemented in DUST
### Vortex Particle Method  
The wake shed from the trailing edges of lifting bodies is modeled as a panel wake, which shares the same spatial discretisation that is used to model the lifting bodies and the same formulation as vortex lattice elements in terms of geometry and singularity distribution. When advected downstream, the panel wake is converted into vortex particles in order to obtain a more robust wake formulation that is suitable for the representation of the interactional aerodynamics of both rotorcraft and complex aircraft configurations. The vortex particles method (VPM) [23; 24] is a Lagrangian grid-free method describing the wake evolution through the rotational component of the velocity field \(\vec{u}_{\psi}\) by means of the material vortex particles used to obtain the approximated vorticity field--namely:  
\[\vec{\omega}^{h}(\vec{r},t)=\sum_{p=1}^{N_{p}}\vec{\alpha}_{p}(t)\zeta\big{(} \vec{r}-\vec{r}_{p}(t);R_{p}\big{)}\, \tag{1}\]  
where \(\vec{r}_{p}(t)\) is the position, \(\vec{\alpha}_{p}(t)\) is the intensity, and \(R_{p}\) is the radius of the \(p\)-th vortex particle, while \(\zeta(r)\) is the cut-off function while considering the vorticity distribution that is induced by each particle. By substituting (1) in the equation of the dynamics of vorticity,  
\[\frac{D\vec{\omega}}{Dt}=\vec{\omega}\cdot\
abla\vec{u}+\
u\
abla^{2}\vec{ \omega}\, \tag{2}\]  
the dynamical equations for the intensity \(\vec{\alpha}_{p}(t)\) and position \(\vec{r}_{p}(t)\) of all the material vortex particles to be integrated in time can be obtained, as follows:  
\[ \begin{cases}\frac{d\vec{\alpha}_{p}}{dt}=\vec{\alpha}_{p}\cdot\
abla\vec{u}( \vec{r}_{p}(t),t)+\
u\,\"\
abla^{2}\vec{\alpha}_{p}\,\"\\ \frac{d\vec{r}_{p}}{dt}=\vec{u}(\vec{r}_{p}(t),t)\.\end{cases}  \tag{3}\]  
The viscosity diffusion term \"\(\
abla^{2}\vec{\alpha}_{p}\,\"\) is calculated while using the particle strength exchange (PSE) method, which approximates the Laplacian operator acting on the vorticity field with an integral operator, as reported in [24]. The mathematical formulation used in the solver relies on the Helmholtz decomposition of the velocity field \(\vec{u}=\vec{u}_{\varphi}+\vec{u}_{\psi}\). The irrotational velocity \(\vec{u}_{\varphi}\) is induced by the free stream velocity, by the singularity distributions of sources and doublets on the body surface, and by the wake panels, while the rotational velocity \(\vec{u}_{\psi}\) is induced by the vortex particles. Moreover, the solenoidal constraint on the rotational velocity, \(\
abla\cdot\vec{u}_{\psi}=0\), is used in order to define the vector potential \(\vec{\psi}\), s.t. \(\vec{u}_{\psi}=\vec{\
abla}\times\vec{\psi}\). Consequently, [PERSON]'s equation is obtained for \(\vec{\psi}\):  
\[-\
abla^{2}\vec{\psi}=\vec{\omega}\;, \tag{4}\]  
where we consider the gauge condition \(\
abla\cdot\vec{\psi}=0\), the vorticity field definition \(\vec{\omega}=\
abla\times\vec{u}\), and the vector identity \(\
abla\times\vec{u}_{\varphi}=\
abla\times\
abla\varphi=0\). [PERSON]'s equation solution (4) reads:  
\[\vec{\psi}(\vec{r},t) =\int_{V_{0}}G(\vec{r},\vec{r}_{0})\,\vec{\omega}(\vec{r}_{0},t) dV_{0}\;, \tag{5}\] \[\vec{u}_{\psi}(\vec{r},t) =\int_{V_{0}}\vec{K}(\vec{r},\vec{r}_{0})\times\vec{\omega}(\vec {r}_{0},t)dV_{0} \tag{6}\]  
where \(G(\vec{r},\vec{r}_{0})\) is [PERSON]'s function of the Laplace equation and \(\vec{K}(\vec{r},\vec{r}_{0})\) represents its gradient with respect to the first argument. Substituting the definition of the discretised vorticity field of the particles (1) into Equation (6), the contribution of velocity that is induced by the particles can be obtained, as follows:  
\[\vec{u}_{\psi}^{h}(\vec{r},t)=\sum_{p=1}^{N_{p}}\vec{K}^{h}(\vec{r}-\vec{r}_{ p}(t))\times\vec{\alpha}_{p}(t)\;. \tag{7}\]  
The discrete kernel \(\vec{K}^{h}(\vec{r}-\vec{r}_{p}(t))\) must be consistent with the selected cutoff function \(\zeta\). The cutoff function in the singular vortex particle method is a Dirac delta function, and the Biot-Savart kernel is retrieved.