A Rigorous Solution for Finite-State Inflow throughout the Flowfield

Abstract

In previous work, a complete dynamic inflow model for flow above the rotor disk has been developed; and numerical results have been presented in both the frequency domain and the time domain. In this paper, we extend the inflow model 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 co-states 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. Numerical comparisons with exact solutions for the z-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. Nomenclature n m a : real part for cosine induced inflow expansion coefficients n m b : imaginary part for cosine induced inflow expansion coefficients [ ] D :