# Assessment of a Mid-Fidelity Numerical Approach for the Investigation of Tiltrotor Aerodynamics
## 2 Numerical Approach Implemented in DUST  
DUST is an open source software that was built with the object-oriented paradigms of the latest FORTRAN standards aiming to solve complex aerodynamics problems with a flexible and reliable approach. The mathematical formulation of the problem relies on [PERSON]'s decomposition of the velocity field \(\vec{u}=\vec{u}_{\varphi}+\vec{u}_{\varphi}\), with \(\vec{u}_{\varphi}\) and \(\vec{u}_{\varphi}\) being the irrotational and solenoidal contributions, respectively, to recast the aerodynamic problem as a combination of a boundary value problem for the potential part of the velocity and a mixed panels-vortex particles model of the free vorticity in the flow. The solution is advanced in time using a time-stepping algorithm that alternates the solution of a three-dimensional boundary element method for \(\vec{u}_{\varphi}\) and the Lagrangian evolution in time of the rotational part of the velocity \(\vec{u}_{\varphi}\). Only the surface mesh of the model is required and different aerodynamic elements allow for different levels of fidelity in the model, ranging from lifting line elements to zero-thickness lifting surfaces and surface panels. A piecewise-uniform distribution of doublets and sources is associated with surface panels, according to a Morino-like formulation for the velocity potential [33]. Thin lifting bodies can also be modelled as zero-thickness surfaces of vortex lattice elements, for which a velocity-based non-penetration condition is assigned. The mixed potential-velocity formulation of the boundary element problem results in a linear system whose unknown is the intensity of the doublet distribution on the surface panels and the equivalent doublet intensity of the vortex lattice elements.  
One-dimensional lifting line elements are used for a proper modelling of lifting bodies with a high aspect ratio, such as rotor blades. These elements naturally represent viscous effects, since they rely on the tabulated aerodynamic lift, drag, and moment coefficients of two-dimensional sections as functions of the relative velocity direction and magnitude. Each lifting line element is modelled as a vortex ring that is composed by the lifting line segment along with its trailing vortices. The last line vortex is released in the wake aligned with the spanwise direction. The intensity \(\Gamma\) of this vortex ring and, thus, of the lifting line, is determined through a fixed point algorithm solving a nonlinear problem, connecting the intensity of the lifting line elements with the tabulated aerodynamic coefficients of the lifting sections. Both a loosely coupled \(\Gamma\) method [34] and a \(\alpha\) method [35] solver are available in the DUST formulation. An iterative procedure is used to solve the nonlinear problem of computing the loads on lifting lines, while taking into account their mutual interference, using the unsteady formulation of the Kutta-Joukowski theorem to retrieve the circulations of the elements from their lift.  
A more detailed description of the whole numerical approach implemented in DUST can be found in [36; 30]. In the following, a brief description of the core of the code that is represented by the implemented VPM is provided.