Off-surface integral solutions to an inhomogeneous wave equation based on acoustic analogy could suffer from spurious wave contamination when volume integrals are ignored for computation efficiency and vortical/turbulent gusts are convected across the integration surfaces, leading to erroneous far-field directivity predictions. Vortical gusts often exist in aerodynamic flows and it is inevitable their effects are present on the integration surface. In this work, we propose a new sound extrapolation method for acoustic far-field directivity prediction in the presence of vortical gusts, which overcomes the deficiencies in the existing methods. The Euler equations are rearranged to an alternative form in terms of fluctuation variables that contains the possible acoustical and vortical waves. Then the equations are manipulated to an inhomogeneous wave equation with source terms corresponding to surface and volume integrals. With the new formulation, spurious monopole and dipole noise produced by vortical gusts can be suppressed on account of the solenoidal property of the vortical waves and a simple convection process. It is therefore valid to ignore the volume integrals and preserve the sound properties. The resulting new acoustic inhomogeneous convected wave equations could be solved by means of the Green’s function method. Validation and verification cases are investigated, and the proposed method shows a capacity of accurate sound prediction for these cases. The new method is also applied to the challenging airfoil leading edge noise problems by injecting vortical waves into the computational domain and performing aeroacoustic studies at both subsonic and transonic speeds. In the case of a transonic airfoil leading edge noise problem, shocks are present on the airfoil surface. Good agreements of the directivity patterns are obtained compared with direct computation results.

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In this study, asymptotic analysis of the frequency-domain formulation to compute the tonal noise of the small rotors in the now ubiquitously multi-rotor powered drones is conducted. Simple scaling laws are proposed to evaluate the impacts of the influential parameters such as blade number, flow speed, rotation speed, unsteady motion, thrust and observer angle on the tonal noise. The rate of noise increment with thrust (or rotational speed) is determined by orders of blade passing frequency harmonics and the unsteady motion. The axial mean flow influence can be approximated by quadratic functions. At given thrust, the sound decreases rapidly with the radius and blade number as the surface pressure becomes less intensive. The higher tonal harmonics are significantly increased if unsteady motions, although of small-amplitude, are existed, as indicated by the defined sensitivity function, emphasizing that the unsteady motions should be avoided for quiet rotor designs. The scaling laws are examined by comparing with the full computations of the rotor noise using the frequency-domain method, the implementation of which has been validated by comparing with experiments. Good data collapse is obtained when the proposed scaling laws, which highlights the dominant influence of the design parameters, are incorporated.