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o source time

Subscript:
ret quantity is evaluated at the retarded time, o = t ? r=c

Introduction

High-speed impulsive (HSI) noise is a particularly intense and annoying noise generated by helicopter rotors in high-speed forward flight. This HSI noise is closely associated with the appearance of shocks and transonic flow around the advancing rotor blades. The quadrupole sources in the Ffowcs Williams{ Hawkings (FW{H) equation1 account for nonlinearities in the vicinity of the rotor blade. These nonlinearities are of two types, which are described by Lighthill.2; 3 First, the local speed of sound is not constant but varies due to particle acceleration. Second, the finite particle velocity near the blade influences the velocity of sound propagation. By inclusion of the quadrupole source, the correct physics is mathematically simulated in the acoustic analogy. The quadrupole source in the FW{H equation was identified by Yu et al.4 as a significant contributor to helicopter HSI noise. Hanson and Fink5 also included the quadrupole source for high-speed propeller noise prediction but found that it was not a significant noise source in that application. Even though this early work demonstrated the importance of the FW{H quadrupole, it has not been routinely included in rotor noise predictions because of the difficulty in predicting the source strength of the Lighthill stress tensor Tij and the lack of a computationally efficient algorithm for computing the quadrupole noise.

In the past few years, the computation of the transonic aerodynamic field around rotor blades has become feasible; hence, renewed interest in prediction of HSI noise has emerged. Yu et al.4 were the first to successfully utilize advances in CFD by approximating the quadrupole source strength and integrating in the direction normal to the rotor plane. The integration in the normal direction of the approximate quadrupole source, which is valid in the far field ahead of the helicopter, effectively transforms the volume integration of the quadrupole into a surface integration. More recently, Schultz and Splettstoesser,6 Schultz et al.,7 and Ianniello and De Bernardis8 have also used this technique with good results. Prieur9 and Prieur et al.10 have developed a frequency domain method for computing the quadrupole noise of hovering rotors that has yielded good results.

Some attempts have been made to numerically integrate the entire volume around the blade,7; 8

but the computations generally require computer resources comparable to those required by unsteady three-dimensional computational fluid dynamics (CFD) calculations|significantly more than that required for thickness and loading noise predictions. Farassat11 and his colleagues12; 13 also tried to reduce the computational effort required in computing HSI noise; they recognized that the appearance of a shock wave coincides with the onset of HSI noise. By assuming that the shock is the dominant contributor of quadrupole noise, the acoustic sources are mathematically confined to the shock surface. When the shock-noise theory was implemented, the conclusion that the shock noise was a dominant component of the quadrupole source was verified.13 Nevertheless, the difficulty in accurately extracting the shock geometry, location, and strength from CFD solutions, together with the fact that the shock noise alone did not sufficiently characterize the total quadrupole source contribution, has postponed the complete implementation of the theory.

The goal of this work is to utilize the far-field approximation to the FW{H quadrupole given by Brentner and Holland14 and extend the formulation to include forward-flight computations. This new formulation yields efficient numerical prediction of HSI noise without resorting to unnecessary or ad hoc simplifications of the FW{H quadrupole source term. The mathematical manipulations used in this approach are rigorous and depend only on the far-field assumption, without approximation of the source strength. Numerical time differentiation of integrals is avoided throught the development of an alternate formulation in which the time differentiation is done analytically. Preliminary calculations with this new formulation demonstrate the potential for efficiency and robustness.

The acoustic analogy approach was chosen because of the substantial knowledge base gained in the development and utilization of thickness and loading noise predictions, based on the FW{H equation. Further, the fundamental far-field assumption, which is described in the next section, leads to integrals of precisely the same form as current thickness and loading noise calculations; hence, the existing numerical algorithms can be used directly. Finally, the identification of individual noise components is a unique advantage of the acoustic analogy approach. This new formulation has been coded and is described in the remainder of this paper. The numerical results are compared with experimental data for both hover and forward-flight conditions.