Dynamic inflow-based analysis of pitch changes to hovering rotor

Abstract

There are two parts to solving rotor aerodynamics. First is pertaining to the rotor blades and other to the wake. Dynamic inflow modelling is a computationally cheap framework to estimate the latter. It is essentially an approach to recreate the non-uniform pressure (and by extension, inflow) distribution over the rotor surface with sufficient accuracy. The non-uniform pressure and inflow distributions are obtained using simple harmonic functions or separate functional decomposition in radial and azimuthal states. The field of dynamic inflow is very mature and has been in existence for nearly a century with volumes of improvement and alteration has already been published on the topic. There is a resurgence in their use today to model conventional and coaxial rotors powered by electric drives that has potential for rapid change in operational state. Such a sudden change, for e.g. in rotor frequency or blade pitch, can drastically affect the rotor wake and dynamic inflow modelling is a computationally inexpensive approach to quantify this effect. However, despite the final implementation of the model being simple, dynamic inflow theories, especially the versatile ones like the Peters-He model, are deeply rooted in abstract and complex mathematical domains like spherical harmonics, differential calculus of curvilinear space, functional analysis, numerical analysis and algebraic number theory. They even venture into certain aspects of tensor calculus and differential topology. Given the time elapsed since they were first proposed and the complexity of these theories, it has become increasingly difficult to follow the recent publications on the topic given their apparent choice to avoid stating much of the background mathematical formulations. To further compound the problem, one has to track some extensively used references back to 1950s to find rationale for some derivations and ideas involved. In the current work, two popular dynamic inflow theories - Pitt-Peters and Peters-He, are revisited in their entirety to understand and present a complete and coherent mathematical framework with their necessary assumptions and other supplementary theories. Alternate and more intuitive derivations are proposed. Order of convergence and gain function analyses are performed. Analytical solutions are obtained and state-of-the-art experimental measurements along with CFD results available in literature used to assess the ability of Pitt-Peters and Peters-He models in predicting rotor inflow with prescribed dynamic pitch inputs.