State-Space Formulation and Analytical Linearization of a Viscous Vortex Particle Method for Flight Dynamics Applications

Abstract

This dissertation presents a state-space formulation of a grid-free, wake-resolving aerodynamic method, developed to enable mid-fidelity aerodynamics to be incorporated directly into flight-dynamics and control analysis. In the hierarchy of rotorcraft aerodynamic models, the proposed approach occupies an intermediate position between low-order finite-state inflow models, which offer computational efficiency at the cost of physical fidelity, and grid-based computational fluid dynamics methods, which resolve complex flow physics but carry substantially higher computational cost. Conventional wake-resolving methods are typically cast as time-marching procedures, which restricts their compatibility with equilibrium analysis, systematic linearization, and control-oriented simulation frameworks. The present work addresses this limitation by reformulating a coupled panel and vortex particle method into a nonlinear state-space representation, in which wake evolution and induced velocities are treated as a unified set of dynamical states.Within this framework, linearized aerodynamic models are derived using both numerical perturbation and a novel analytical linearization approach, enabling efficient construction of linear models suitable for trim and stability analysis. The formulation is first validated for fixed-wing configurations by comparing results with DUST, an open-source aerodynamic solver, confirming the accuracy of the nonlinear model under steady and unsteady conditions. The linearized models show strong agreement with nonlinear responses to sinusoidal angle-of-attack perturbations in both time and frequency domains. The analytical linearization reduces the cost of constructing the linear model by O(n^2) relative to perturbation-based approaches, where n denotes the number of system states, while maintaining comparable accuracy. The formulation is subsequently extended to rotary-wing configurations, where wake dynamics are coupled with rotor motion. Validation against experimental data confirms that the method captures rotor wake behavior and radial load distribution with good agreement. That linearized models closely match the nonlinear response to collective doublet inputs. To improve scalability for large wake systems, the fast multipole method (FMM) is incorporated, reducing the cost of particle-particle interactions from O(Np^2) to O(Np), where Np denotes the number of vortex particles. Results indicate that octree construction and multipole evaluation introduce non-negligible overhead for small particle counts, while substantial computational savings are achieved for larger wake systems. Viscous diffusion is incorporated via the particle-strength exchange (PSE) scheme, which improves wake realism, reduces thrust overprediction, and preserves the efficiency of analytical linearization. The formulation is further extended to account for rotor operation in ground effect using the Method of Images. Finally, the aerodynamic formulation is coupled with GenMR, an in-house flight-dynamics and control framework, enabling a particle-based wake model to interact with rigid-body motion and control inputs within a unified simulation environment. Coupled simulations for representative configurations - including generic helicopter, tiltrotor, and electric vertical takeoff and landing (eVTOL) - demonstrate the method's capability to capture detailed wake dynamics within a flight-dynamics setting. Collectively, these developments establish a mid-fidelity aerodynamic framework suitable for integration within flight-dynamics simulations, and provide a foundation for future extensions to closed-loop flight simulation, aeroelastic modeling, and distributed-propulsion aircraft.