Fundamental Solutions of the Potential Flow Equations for Rotorcraft with Applications

Abstract

In previous work, a complete dynamic inflow model for flow above the rotor disk was developed, and numerical results were presented in both the frequency domain and the time domain. In this paper, the inflow model is extended to all three components of inflow below the disk. The inflow model, which is expressed in terms of a finite number of states, is derived in a mathematically rigorous way. The essence of the extension is that if one computes the costates of the inflow equations (along with the normal states), then one can find the velocity in the hemisphere below the plane of the rotor disk (including the velocity within the wake) with accuracy equal to that of the flow above the rotor plane. The derivation is for the case of general skew angle. To optimize the results from the model, truncating the even terms included in the model is adapted to minimize the error norm for different skew angles. A divergence issue in the calculation of the Legendre functions of the second kind far away from the rotor disk is solved by utilization of a new scheme developed in this paper. Numerical comparisons with exact solutions for the component of flow in axial flow (given for some special cases) illustrate the effectiveness of the new model. The simulations also illustrate that the model is valid for either the frequency domain or the time domain.