Integration and Analytical Linearization of Viscous Terms in State-Space Vortex Particle Methods

This study extends previous work on analytical linearization of state-space vortex particle methods by incorporating viscous terms. The current framework integrates a panel method for modeling blade surfaces and near-wake, and a viscous vortex particle method (VVPM) to model the far-wake. The code is formulated as a system of ordinary differential equations, resulting in a nonlinear time-periodic (NLTP) system in first-order form. The NLTP dynamics are linearized to yield a linear time-periodic (LTP) representation using two approaches: finite differencing and a novel analytical linearization technique. Harmonic decomposition is then applied to approximate the LTP system as a higher order linear time-invariant model, where the LTP system coefficients become states of the time-invariant dynamics, facilitating time-invariant system analysis techniques. The proposed methodology is implemented in MATLAB® and applied to a generic utility helicopter rotor blade, with validation performed against experimental data and CFD. A study was conducted to assess the impact of including viscous effects on the wake structure when modeling ground effect. The accuracy of the linearized models is assessed through comparisons with the nonlinear system in both time and frequency domains. Results indicate that the linearized models effectively capture wake dynamics, particularly for low-mid frequency forcing inputs. Notably, the analytical linearization approach significantly reduces computational cost compared to finite-difference-based methods, achieving an efficiency improvement of O(n2), where n represents the number of system states. This establishes analytical linearization of VVPM as a viable tool for advancing rotorcraft flight dynamics modeling by balancing fidelity with computational efficiency.