An analytic solution for the noise generated by gust-aerofoil interaction for plates with serrated leading edges

This paper presents an analytic solution for the sound generated by an unsteady gust interacting with a semi-infinite flat plate with a serrated leading edge in a background steady uniform flow. Viscous and non-linear effects are neglected. The Wiener-Hopf method is used in conjunction with a non-orthogonal coordinate transformation and separation of variables to permit analytical progress. The solution is obtained in terms of a modal expansion in the spanwise coordinate, however for low- and mid-range incident frequencies only the zeroth order mode is seen to contribute to the far-field acoustics, therefore the far-field noise can be quickly evaluated. The solution gives insight into the potential mechanisms behind the reduction of noise for plates with serrated leading edges compared to those with straight edges, and predicts a logarithmic dependence between the tip-to-root serration height and the decrease of far-field noise. The two mechanisms behind the noise reduction are proposed to be an increased destructive interference in the far field, and a redistribution of acoustic energy from low cuton modes to higher cutoff modes as the tip-to-root serration height is increased. The analytic results show good agreement in comparison with experimental measurements. The results are then compared against numerical predictions for the sound generated by a spanwise invariant line vortex interacting with a flat plate with serrated leading edge. Good agreement is also seen between the analytical and numerical results as frequency and tip-to-root ratio are varied

Applying an iterative method numerically to solve n × n matrix Wiener–Hopf equations with exponential factors

This paper presents a generalization of a recent iterative approach to solving a class of 2 × 2 matrix Wiener–Hopf equations involving exponential factors. We extend the method to square matrices of arbitrary dimension n, as arise in mixed boundary value problems with n junctions. To demonstrate the method, we consider the classical problem of scattering a plane wave by a set of collinear plates. The results are compared to other known methods. We describe an effective implementation using a spectral method to compute the required Cauchy transforms. The approach is ideally suited to obtaining far-field directivity patterns of utility to applications. Convergence in iteration is fastest for large wavenumbers, but remains practical at modest wavenumbers to achieve a high degree of accuracy.This work was supported by EPSRC DTP grant no. EP/N509620/1 (M.J.P.), the Sultan Qaboos Research Fellowship at Corpus Christi College at University of Cambridge (A.V.K.) and by EPSRC early career fellowship grant no. EP/P015980/1 (L.J.A.). The authors thank the Isaac Newton Institute for Mathematical Sciences, Cambridge, for support and hospitality during the WHT programme where some work on this paper was undertaken (EPSRC grant no. EP/R014604/1)

Exact solutions for ground effect

"Ground effect" refers to the enhanced performance enjoyed by fliers or swimmers operating close to the ground. We derive a number of exact solutions for this phenomenon, thereby elucidating the underlying physical mechanisms involved in ground effect. Unlike previous analytic studies, our solutions are not restricted to particular parameter regimes such as "weak" or "extreme" ground effect, and do not even require thin aerofoil theory. Moreover, the solutions are valid for a hitherto intractable range of flow phenomena including point vortices, uniform and straining flows, unsteady motions of the wing, and the Kutta condition. We model the ground effect as the potential flow past a wing inclined above a flat wall. The solution of the model requires two steps: firstly, a coordinate transformation between the physical domain and a concentric annulus, and secondly, the solution of the potential flow problem inside the annulus. We show that both steps can be solved by introducing a new special function which is straightforward to compute. Moreover, the ensuing solutions are simple to express and offer new insight into the mathematical structure of ground effect. In order to identify the missing physics in our potential flow model, we compare our solutions against new experimental data. The experiments show that boundary layer separation on the wing and wall occurs at small angles of attack, and we suggest ways in which our model could be extended to account for these effects.Comment: Main body: 10 pages & 3 figures; supplementary material: 6 pages & 5 figures. Submitted to JFM Rapid

Analytic solutions for reduced leading-edge noise aerofoils

This paper presents an analytic solution for the sound generated by an unsteady gust interacting with a semi-infinite at plate with a piecewise linear periodic leading edge. The Wiener-Hopf method is used in conjunction with a non-orthogonal coordinate transformation and separation of variables to allow analytical progress. A fully analytic solution is obtained in terms of a modal expansion for the far-field noise which is obtained by summing only a finite number of cuton modes, allowing very quick evaluation. The analytic solution is compared to experimental results for five test case leading-edge geometries. Good agreement is seen indicating the analytic model is capturing the key features of the interaction such as the destructive interference from the tip and root. In four of the five test cases the serrated edges show large reductions of noise compared to the straight edge at mid and high frequencies, however the square wave geometry is seen to be ineffective at noise reduction for high frequencies.</p

An analytical and experimental investigation of aerofoil-turbulence interaction noise for plates with spanwise-varying leading edges

This paper presents an analytic solution for gust-aerofoil interaction noise for flat plates with spanwise-varying periodic leading edges in uniform mean flow. The solution is obtained by solving the linear inviscid equations via separation of variables and the Wiener-Hopf technique, and is suitable for calculating the far-field noise generated by any leading-edge with a single-valued piecewise linear periodic spanwise geometry. Acoustic results for homogeneous isotropic turbulent flow are calculated by integrating the single-gust solution over a wavenumber spectrum. The far-sound pressure level is calculated for five test case geometries; sawtooth serration, slitted v-root, slitted u-root, chopped peak, and square wave, and compared to experimental measurements. Good agreement is seen over a range of frequencies and tip-to-root ratios (varying the sharpness of the serration). The analytic solution is then used to calculate the propagating pressure along the leading-edge of the serration for fixed spanwise wavenumbers, i.e. only the contribution to the surface pressure which propagates to the far field. Using these results, two primary mechanisms for noise reduction are discussed; tip and root interference, and a redistribution of energy from cuton modes to cut off modes. A secondary noise-reduction mechanism due to non-linear features is also discussed and seen to be particularly important for leading edges with very narrow slits

Analytical and experimental investigation into the effects of leading-edge radius on gust–aerofoil interaction noise

This paper investigates the effects of local leading-edge geometry on unsteady aerofoil interaction noise. Analytical results are obtained by extending previous work for parabolic leading edges to leading edges of the form xm for 0 &lt; m &lt; 1. Rapid distortion theory governs the interaction of an unsteady vortical perturbation with a rigid aerofoil in compressible steady mean flow that is uniform far upstream. For high-frequency gusts interacting with aerofoils of small total thickness this allows a matched asymptotic solution to be obtained. This paper mainly focusses on obtaining the analytic solution in the leading-edge inner region, which is the dominant term in determining the total far-field acoustic directivity, and contains the effects of the local leading-edge geometry. Experimental measurements for the noise generated by aerofoils with different leading-edge nose radii in uniform flow with approximate homogeneous, isotropic turbulence are also presented. Both experimental and analytic results predict that a larger nose radius generates less overall noise in low-Mach-number flow. By considering individual terms in the analytic solution, this paper is able to propose reasons behind this result

On the superior performance of leading edge slits over serrations for the reduction of aerofoil interaction noise

Aerofoils operating in a turbulent flow are an efficient source of noise radiation by scattering vorticity into sound at the leading edge. Much work has now been undertaken demonstrating the effectiveness by which serrations, or undulations, introduced onto the leading edge, can substantially reduce broadband leading edge interaction noise. However, all of this work is focused on sinusoidal leading edge serration profiles. In this paper, a family of alternative serration profiles are proposed that are capable of providing significantly greater noise reductions than single-wavelength serrations at optimal conditions. This new family of profiles will be shown to reduce interaction noise through a fundamentally different noise reduction mechanism than conventional single-wavelength profiles. Unlike single-wavelength profiles, which produce a single compact dominant source region per serration wavelength, these new profiles are designed to produce two dominant compact sources per serration wavelength of the same source strength, that are separated in the streamwise direction. Since these sources are arranged to be closer together than the turbulence length-scale, they are highly coherent and therefore radiate with a difference in phase. A frequency therefore exists at which the sources are exactly 180° out of phase leading to very high levels of noise reduction in the far field.</p