Bioinspired aerofoil adaptations: the next steps for theoretical models

Abstract

The extended introduction in this paper reviews the theoretical modelling of leading- and trailing-edge noise, various bioinspired aerofoil adaptations to both the leading and trailing edges of blades, and how these adaptations aid in the reduction of aerofoil–turbulence interaction noise. Attention is given to the agreement between current theoretical predictions and experimental measurements, in particular, for turbulent interactions at the trailing edge of an aerofoil. Where there is a poor agreement between theoretical models and experimental data the features neglected from the theoretical models are discussed. Notably, it is known that theoretical predictions for porous trailing-edge adaptations do not agree well with experimental measurements. Previous works propose the reason for this: theoretical models do not account for surface roughness due to the porous material and thus omit a key noise source. The remainder of this paper, therefore, presents an analytical model, based upon the acoustic analogy, to predict the far-field noise due to a rough surface at the trailing edge of an aerofoil. Unlike previous roughness noise models which focus on roughness over an infinite wall, the model presented here includes diffraction by a sharp edge. The new results are seen to be in better agreement with experimental data than previous models which neglect diffraction by an edge. This new model could then be used to improve theoretical predictions for far-field noise generated by turbulent interactions with a (rough) porous trailing edge.

This article is part of the theme issue ‘Frontiers of aeroacoustics research: theory, computation and experiment’.

1. Introduction

Aviation noise from modern passenger airplanes is generated by a vast number of different sources [1]. Of interest in this paper is the large contribution to total aviation noise due to the interaction of turbulence with aerofoils such as the wings or blades within the turbofan.

Turbulence–aerofoil interaction noise can be categorized into two main parts as illustrated in figure 1; leading-edge noise, where turbulence from some upstream source impinges on the front of the aerofoil as is the case in rotor–stator interaction noise within the turbofan; and trailing-edge noise, where the turbulent boundary layer built up on the aerofoil itself scatters off of the sharp trailing edge as is the case in airframe (i.e. wing) noise. This latter source is also a key noise source for wind turbines and UAVs due to the aerodynamic sharp trailing edges of blades used within these machines.

Figure 1.

Illustration of upstream turbulence which generates leading-edge noise and boundary layer turbulence which generates trailing-edge noise.

To reduce turbulence–aerofoil interaction noise, adaptations to the leading and trailing-edges of aerofoils, often inspired by the silent flight of owls [2,3], have become plentiful over recent years. A range of theoretical, numerical and experimental approaches are used to understand how these various adaptations can reduce far-field noise, with designs including leading-edge serrations [4–6] and hooks/combs [7], and trailing-edge serrations [8–10], brushes [11], porosity and/or elasticity [12–14], and surface treatments such as finlets [15]. Here, we discuss some of the popular design adaptations, their mechanisms for noise reduction, and why there may be a discrepancy between theoretical models and experimental data.

(a). Leading-edge adaptations

(i). Theoretical models of leading-edge noise

Leading-edge noise is generated by upstream turbulence impinging on an aerofoil in mean flow. To model this analytically, the turbulence in decomposed into Fourier ‘gust’ components ∼A e^{ik · x−iωt}, and the interaction of a single gust is modelled via the inviscid linearized Euler equations. The equations governing the scattered acoustic field with (non-dimensional) velocity v_a = ∇ϕ e^−iωt, due to this gust interacting with a flat-plate x > 0, y = 0 can be written as

$${\beta }^2\frac{{\mathrm{\partial }}^2\varphi }{\mathrm{\partial }{x}^2}+\frac{{\mathrm{\partial }}^2\varphi }{\mathrm{\partial }{y}^2}+\frac{{\mathrm{\partial }}^2\varphi }{\mathrm{\partial }{z}^2}+2\text{i}kM\frac{\mathrm{\partial }\varphi }{\mathrm{\partial }x}+k^2\varphi =0,$$ 1.1

where β² = 1 − M², with M the upstream Mach number of the steady flow, and k = ω/c₀ with c₀ the speed of sound of the background steady flow. We assume the gust convects with the background flow therefore k = k₁M. The zero normal velocity boundary condition on the plate surface requires

$${\frac{\mathrm{\partial }\varphi }{\mathrm{\partial }y}|}_{y=0}=-A_2{\text{e}}^{\text{i}{k}_{1}x+\text{i}{k}_{3}z}x>0.$$ 1.2

For simplicity we shall normalize A₂ = 1. We also impose continuity of the potential upstream

$${\mathrm{\Delta }\varphi |}_{y=0}=0x<0.$$ 1.3

This mixed boundary condition problem can be solved exactly via the Wiener–Hopf technique [16] to yield

$$\varphi (x,y,z)=\frac{\hspace{0.17em}{\mathrm{e}}^{-i{k}_1{M}^{2}x/{\beta }^2}}{2\pi \sqrt{-\delta -w_n}}\mathrm{sgn}(y){\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\hspace{0.17em}{\mathrm{e}}^{-i\mathrm{\lambda }x/\beta }\frac{{\mathrm{e}}^{-|y|\sqrt{{\mathrm{\lambda }}^2-w^2}}}{\sqrt{\mathrm{\lambda }+w}(\mathrm{\lambda }+\delta )}\hspace{0.17em}\mathrm{d}\mathrm{\lambda },$$ 1.4

where δ = k₁/β and w² = (δM)² − k²₃. The scattered pressure is recovered as

$$p=-\frac{D_0}{Dt}(\varphi \hspace{0.17em}{\mathrm{e}}^{-i\omega t}),$$ 1.5

where D₀/Dt denotes the material derivative with respect to the steady background flow.

For a finite plate, 0 < x < b, y = 0 one can obtain an analytic solution by first supposing there is no trailing edge (as above), then correcting this solution with a second solution which has a finite trailing edge, but no leading edge; the pressure jump across the wake, y = 0, x > b, must be set to zero. The process could be continued indefinitely (with a third solution correcting the pressure jump upstream then fourth the pressure jump across the wake once more, and so on). When including two terms, this results in Amiet's solution [17] for the pressure jump across the aerofoil

$$\mathrm{\Delta }p(x,y,z,t)=2\pi g(x,k_1,k_2)\hspace{0.17em}{\mathrm{e}}^{i\mathbf{k}\cdot x-i\omega t},$$ 1.6

where g is the transfer function detailed in [17].

To go from these single-gust results to a fully turbulent field, one integrates over a turbulent wavenumber spectrum, Φ, which represents the statistical quantities expected from the upstream turbulence. Typically, a Liepmann spectrum well represents experimentally generated grid turbulence, thus the far-field sound pressure level (SPL) at a fixed observer locations, (x, y, x) is given theoretically by

$$\mathrm{SPL}(x,y,k_1)=10{\log }_{10}({\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}|p(x,y,z){|}^2\mathit{\Phi }(k_1,k_3)\hspace{0.17em}\mathrm{d}{k}_3),$$ 1.7

where

$$\mathit{\Phi }(k_1,k_3)=\frac{3\overline{u^2}{L}^2}{4\pi }\frac{L^2(k_1^2+k_3^2)}{{(1+L^2(k_1^2+k_3^3))}^{5/2}},$$ 1.8

with L the integral lengthscale of turbulence and $\overline{u}$ the turbulence intensity.

Theoretical models can also include small amounts, O(ϵ) say, of thickness, camber and angle of attack of the aerofoil with respect to the mean flow by using perturbation methods and expanding the scattered pressure p₀ + ϵp₁, where p₀ is the solution for the flat plate, and p₁ contains the corrections due to geometry [18]. Rapid distortion theory is often used to describe the deformation of the incident gust on approach to the thin (but not flat) aerofoil: more details on Rapid distortion theory can be found throughout this special issue.

Overall theoretical predictions for scattering by semi-infinite plates, finite plates and aerofoils with realistic geometry agree well with experimental measurements.

(ii). Serrated adaptation

Leading-edge serrations are illustrated in figure 2, and are inspired by the frontal comb-like structure of owls wings. The principal difference in modelling the interaction theoretically is now (1.2) and (1.3) are no longer defined for x > 0 and x < 0, respectively, but for x > cF(z) and x < cF(z) respectively, where c varies the sharpness of the serration and F(z) provides its periodic shape (typically non-dimensionalized such that the wavelength of the serration, λ = 1). The inclusion of the spanwise direction into the mixed boundary condition problem creates difficulty for the Wiener–Hopf method, thus modal expansions in the spanwise direction are used in current approaches [19,20]. This results in a solution for the far-field scattered potential derived in [20] as

$$\varphi (x,y,z)=\frac{{\mathrm{e}}^{-i{k}_1{M}^{2}x/{\beta }^2}}{2\pi \sqrt{-\delta -w_n}}\mathrm{sgn}(y)\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\hspace{0.17em}{\mathrm{e}}^{-i\mathrm{\lambda }(x/\beta -cF(z)/\beta )}\frac{E_n(\mathrm{\lambda })\hspace{0.17em}{\mathrm{e}}^{-|y|\sqrt{{\mathrm{\lambda }}^2-w_n^2}}}{\sqrt{\mathrm{\lambda }+w_n}(\mathrm{\lambda }+\delta )}{Z}_n(\mathrm{\lambda },z)\hspace{0.17em}\mathrm{d}\mathrm{\lambda },$$ 1.9

where w²_n = (δM)² − (k₃ + 2nπ)²,

$$Z_n(\mathrm{\lambda },z)={\mathrm{e}}^{-i\mathrm{\lambda }cF(z)}\hspace{0.17em}{\mathrm{e}}^{i{k}_3\zeta +2n\pi iz}$$ 1.10

and

$$E_n(\mathrm{\lambda })={\int }_0^1\hspace{0.17em}{\mathrm{e}}^{i{k}_{1}cF(\zeta )+i{k}_3\zeta }\overline{Z_n(\overline{\mathrm{\lambda }},z)}\hspace{0.17em}\mathrm{d}z.$$ 1.11

By comparing (1.9) with (1.4), we see the effect of the serration geometry is contained simply within the expansion of the incident surface pressure

$${\mathrm{e}}^{i{k}_{1}cF(z)+i{k}_{3}z}=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{E}_n(\mathrm{\lambda })Z_n(\mathrm{\lambda },z).$$ 1.12

This expansion, therefore, provides a natural decomposition of the scattered field into modes with frequencies w_n and amplitudes governed by E_n which aids in the understanding of why serrated edges reduce noise.

Figure 2.

Illustration of a leading-edge serration. (a) Aerofoil surface (flat plate) lies in the region x > cF(z), y = 0. Parameter c measures the sharpness of the serration. (b) Tip (red) and root (blue) sources. (Online version in colour.)

Leading-edge serrations are thought to be effective in reducing turbulence–aerofoil interaction noise through two primary mechanisms. First, a serrated edge allows for acoustic sources to be generated at different spatial locations, thus permits a destructive interference of noise in the far field [19,21]. The two most important sources arise at the tip/peak and the root/trough of the serration, which we illustrate in figure 2. Second, the ever-varying oblique edge of the aerofoil scatters each gust component of the incoming turbulence into a range of modes (rather than just one mode as is the case for a straight edge), which is seen clearly in the analytic solution. This modal decomposition has fixed frequencies, w_n, relating to the wavelength of the serration and incident frequency, regardless of the precise shape of the serration. However, the amplitude of the modes, E_n, are dependent on the precise geometry of the serration. By altering the serration geometry (either the overall shape, or the sharpness/amplitude of the serration), one alters the modal amplitudes. Given that high-order (large n) modes are cutoff in the far field, one can effectively shift the acoustic energy from cuton modes to cutoff modes [20,21] thus reduce the total noise propagating to the far field by altering the geometry of the serration. In particular, sharpening a sawtooth serration has this effect [20].

A secondary mechanism for noise reduction prevalent for serrations with very narrow sections or slits (with respect to the overall serration wavelength, λ) is an increase in root source strength [22,23] due to secondary vorticity. By increasing this source strength, the interference of tip and root fields can be enhanced [24]. Current theoretical models based on the linearized Euler equations [19,20] do not account for this secondary mechanism as they do not permit vortical solutions, and thus for leading-edge serrations with narrow sections current theory cannot make accurate predictions for the noise reduction, despite there being very good agreement between theoretical models and experimental data for serrations without narrow sections.

To improve upon current theoretical models for leading edges with slits, it is necessary to include the effects of increased root source strength. A suggested route forward may be via an empirical model for the increased source strength which can be used to supplement existing scattering theory. To generate such a model, a numerical dataset for a range of slitted profiles would be required. Preliminary results may be obtained from [22].

(b). Trailing-edge adaptations

The theoretical modelling of trailing-edge noise can be very similar to that for leading-edge noise. The boundary layer turbulence, often supposed to only be present on the upper surface of the aerofoil, may be decomposed into gust components, and the linearized Euler equations are solved for one general component. They key difference is in the boundary conditions, where for a flat plate y = 0, x < 0 we now impose zero normal velocity for x < 0 and a zero pressure jump across the wake vortex-sheet, x > 0. This system may, as for the leading-edge problem, be solved exactly via the Wiener–Hopf technique. Once again, to recover results for a fully turbulent interaction, the far-field pressure is integrated over a wavenumber spectrum describing the boundary layer turbulence.

An alternative approach to predicting trailing-edge noise is through the acoustic analogy. Lighthill's acoustic analogy [25], as presented by Goldstein [26], provides the solution for acoustic pressure fluctuations p(x, t), radiating from a region of turbulent flow in volume, V , bounded by an impermeable surface, S;

$$\begin{array}{rl}p(\mathbf{x},t)& =-{\int }_{-T}^T{\int }_S{\rho }_0{U}_n\frac{\mathrm{\partial }G}{\mathrm{\partial }\tau }\hspace{0.17em}\mathrm{d}S(\mathbf{y})\hspace{0.17em}\mathrm{d}\tau +{\int }_{-T}^T{\int }_S{f}_i\frac{\mathrm{\partial }G}{\mathrm{\partial }{y}_i}\hspace{0.17em}\mathrm{d}S(\mathbf{y})\hspace{0.17em}\mathrm{d}\tau \\ & +{\int }_{-T}^T{\int }_V{T}_{ij}(\mathbf{y},\tau )\frac{{\mathrm{\partial }}^{2}G}{\mathrm{\partial }{y}_i\mathrm{\partial }{y}_j}\hspace{0.17em}\mathrm{d}V(\mathbf{y})\hspace{0.17em}\mathrm{d}\tau .\end{array}$$ 1.13

Here, U_n is the mean flow velocity normal to the surface, and ρ₀ the mean flow density. The Green's function, G(x, t|y, τ), solves the wave equation in V . The term f_i denotes the force per unit area applied to the fluid by the surface, f_i = p_sn_i − σ_ijn_j, where n is the surface normal, p_s is the unsteady component of pressure being scattered by the surface (namely the boundary layer turbulence) and σ_ij is the viscous stress tensor, which for our purposes shall be set to zero as we assume an inviscid flow. The Lighthill stress tensor is denoted by T_ij.

Theoretical trailing-edge noise predictions thus rely on accurately modelling two things: first, the interaction of turbulence with the trailing edge (such as through gust decomposition or the acoustic analogy); and second, the features of the turbulent boundary layer pressure fluctuations on the suction side of the aerofoil, i.e. p_s. We highlight below a range of problems with at least one of these for current theoretical models of far-field noise from aerofoils with different trailing-edge adaptations.

(i). Serrations

Theoretical models for trailing-edge serrations, inspired now by the zig-zag pattern seen at the rear of owls' wings, predict similar mechanisms for noise reduction as seen at the leading edge; a destructive interference of scattered acoustics, and a redistribution of energy to cutoff modes [10,27]. However, unlike the good agreement seen between theoretical models and experimental data for leading-edge serrations, there is poorer agreement for trailing-edge serrations even when the serration geometry is a simple sawtooth or sinusoid [10,27]. The principal difference between leading-edge and trailing-edge models from a theoretical point of view is that the incoming turbulence can be accurately described for leading-edge noise regardless of the adaptation, and is typically done so with a Von Karman or Liepmann spectrum to represent grid-generated experimental turbulence [20]. However, at the trailing edge, since the turbulent source is intrinsically linked to the edge itself, accurate knowledge of what the input should be to theoretical trailing-edge models can be somewhat of a mystery.

Current theoretical models [10,27] use an approximation [28] for the wavenumber spectral density in a turbulent boundary layer proposed by Chase [29]; this spectral density is known to be accurate for a turbulent boundary layer over a smooth infinite wall, however, it is unclear if the model remains accurate when a spanwise variation (such as that imposed by a serrated edge) is present. Indeed recent experimental results for trailing-edge serrations note distinct differences in the boundary layer structure for a serrated trailing edge versus a straight trailing edge [30–33]. To be able to accurately predict trailing-edge noise using a theoretical model, we must first have a greater understanding of how the boundary layer changes for serrated edges in order to input a suitable wavenumber spectrum.

(ii). Finlets, fences and fibres

Finlets, fences and fibres placed at the trailing edge of an aerofoil attempt to mimic the owl's downy upper coat which consists of streamwise oriented fibres forming a canopy above the main body of the wing. It is believed these features aid in pushing the turbulence in the boundary layer further away (in the direction normal to the aerofoil) from the trailing-edge, reducing the pressure exerted on the trailing edge, and thus the diffracted noise [15]. These features are discussed in greater detail along with a theoretical model for the case of elastic fibres later in this Special Issue [34].

(iii). Porosity

Naturally, due to the feather structure of birds, owls' wings are slightly permeable to air at the trailing edge. By replacing a section of the impermeable chord of an aerofoil with a porous material at the trailing edge, one can successfully reduce low-frequency trailing-edge noise by reducing the amplification factor of near-field turbulence to a sixth-power velocity dependence rather than a fifth-power associated with an impermeable edge [12]. Porous trailing edges also have the benefit of delaying vortex shedding (in the case of a blunt edge) and altering the boundary layer near the trailing edge, prior to scattering [35–37]. These boundary layer effects include principally a weakening of the pressure fluctuations near the surface (resulting in weaker scattering at the edge), but also a destruction of the spanwise coherence of the boundary layer turbulent structures which reduces the energy content of large low-frequency structures thus reducing the low-frequency scattered noise.

Theoretical models [12,38] can correctly capture the reduction of noise for low-frequency interactions, however, do not see any impact on the noise generation at high frequencies, which is well documented in experimental results [14]. Here, the theoretical model is inaccurately accounting for all of the sources, in particular, the additional noise that arises due to surface roughness. Linear inviscid theory for the scattering of quadrupole sources near the trailing edge predicts that porosity becomes inefficient at reducing noise for high frequencies [12]; however, experimental measurements suggest instead that noise is increased at high frequencies for porous trailing edges compared to impermeable edges [14].

To improve upon current theoretical models, one must develop a supplementary model for roughness-induced trailing-edge noise at high frequencies. The theory for this already exists, thus we shall illustrate a suitable supplemental model here.

(iv). Surface roughness

As mentioned in the previous subsection, surface roughness can be an important feature for adapted aerofoils, in particular, if porous materials are involved. The effect of surface roughness over a wall is to increase turbulent noise from O(ρ₀U²M⁵) for the smooth case (where ρ₀ is the fluid density, U is the fluid velocity and M≪1 is the flow Mach number), to O(ρ₀U²M³) for the rough case, as discussed in detail in earlier work [39]. Measurements of roughness noise also indicate a similar increase [40,41] and predict that wall roughness noise can dominate over traditional (i.e. smooth) trailing-edge noise in certain circumstances due to dipole sources. However, these predictions [41] are based on Howe's early theoretical model [42], which supposes an infinite wall and are therefore lacking trailing edge, thus missing from that model is the diffraction of roughness-generated noise by the trailing edge.

The effect of trailing-edge diffraction for a rough surface is included in the experimental measurements of Hersh [40], where it was found that far-field noise from a cylindrical pipe is enhanced by roughness. Upon comparison of these experimental measurements with the theoretical approach of Howe [42], we see large disagreements of up to 10 dB for small ωR/v_* (frequency, ω, multiplied by the radius of a typical roughness element, R, divided by surface friction velocity, v_*). This discrepancy corresponds to a disagreement for all frequencies less than around 40 kHz (using parameters from Howe of R = 0.00032 m, v_* = 1.3 m s⁻¹). This range includes the high frequencies which exhibited trailing-edge noise increases in experimental results for partially porous aerofoils [14], and the frequencies which exhibit roughness noise overpowering smooth trailing-edge noise [43]. It is important we rectify this discrepancy to be able to accurately model the acoustic effects of roughness near a sharp trailing edge by including the effects of diffraction by the edge. This is particularly important for improving the theoretical modelling of porous trailing-edge noise, which currently does not agree with experimental measurements at high frequencies and thus these theoretical models incorrectly predict the overall SPL reductions, ΔOSPL, achievable by employing porous trailing-edge adaptations.

The remainder of this paper, therefore, extends the analysis of Glegg & Devenport [44] to develop a theoretical model considering the effects of an arbitrary rough surface on trailing-edge noise generation. In particular, we develop predictions for the additional trailing-edge scattered noise due to roughness elements; this additional contribution shall be referred to as the diffracted dipole roughness noise. In the example case of roughness generated by hemispherical bosses [42], it will be seen that the scattering of the roughness-induced dipole sources by the sharp trailing edge is more dominant than the directly generated roughness noise over an infinite wall for sufficiently low frequencies. This analysis and roughness model may provide a useful tool for predicting roughness-induced trailing-edge noise for porous or partially porous aerofoils where the porous section is built from spherical powder/bead materials, such as Reapor [45] and sintered bronze powder [46], or the impact of structural roughness near the trailing-edge of a wing, such as rivets.

2. Theoretical model for edge roughness noise

(a). Acoustic analogy

To provide a model for trailing-edge roughness noise, we choose the approach of the acoustic analogy due to the complexity of the shape of the rigid surface. We specifically consider the scattering of a turbulent boundary layer by the sharp trailing edge of a plate y₂ = ξ(y₁, y₃), y₁ < 0, y₃∈( − ∞, ∞), where ξ(y₁, y₃) denotes a rough surface whose elements protrude a typical distance h≪1 from the flat surface. The roughness is considered to only be along the upper surface of the plate y₂ > 0, with the lower surface, y₂ = 0₋, being smooth. We also suppose that the turbulence exists only on the upper surface of the plate, as would be appropriate for an aerodynamic (lifting) body.

We choose a Green's function to use in (1.13) that satisfies the rigid boundary conditions for a smooth half-plate, namely ∂G/∂n = 0 on y₂ = 0, y₁ < 0, and we impose continuity of G along y₂ = 0, y₁ > 0. This will be given explicitly in the next subsection. Following this choice, the second term in (1.13), which represents dipole noise, does not contribute if the surface S is smooth due to the choice of boundary conditions of G, but will have an impact for the upper, rough surface. The first term in (1.13) is set to zero since we suppose the surface S is rigid, thus U_n = 0. The third term represents quadrupole noise and therefore is O(M²) smaller than the dipole term. Typical testing of porous trailing edges in wind tunnel facilities is at low speeds of M ≈ 0.2; therefore, the quadrupole noise is approximately 0.04 times t dipole noise. The main focus of this section shall, therefore, be on the second term to understand how dipole noise generated by roughness elements diffracts off the trailing edge of a rough plate. We shall compare to the level of dipole noise directly radiated by a rough surface, that is the dipole roughness noise over an infinite rough wall as determined previously [44].

(b). Green's function

We first give the transformed Green's function, $\tilde{G}$ defined via

$$G(\mathbf{x},\mathbf{y},t,\tau )=-\frac{1}{2\pi }{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\tilde{G}(\mathbf{x},\mathbf{y},\omega )\hspace{0.17em}{\mathrm{e}}^{-i\omega (t-\tau )}\hspace{0.17em}\mathrm{d}\omega ,$$ 2.1

where k₀ = ω/c₀ is the acoustic wavenumber, with c₀ the speed of sound and ω the angular frequency.

The Green's function satisfying zero normal derivative on a semi-infinite flat plate y₂ = 0, y₁ < 0 and continuity across y₂ = 0, y₁ > 0, for low Mach number flow is given in [47,48]

$$\tilde{G}(\mathbf{x},\mathbf{y},k_0)=\frac{-i{k}_0}{4{\pi }^2}({\int }_{-\mathrm{\infty }}^{s_1}\frac{K_1^{\ast }(i{k}_0{\overline{r}}_1\sqrt{1+u^2})}{\sqrt{1+u^2}}\hspace{0.17em}\mathrm{d}u+{\int }_{-\mathrm{\infty }}^{s_2}\frac{K_1^{\ast }(i{k}_0{\overline{r}}_2\sqrt{1+u^2})}{\sqrt{1+u^2}}\hspace{0.17em}\mathrm{d}u),$$ 2.2

where K_i is the modified Bessel function of the second kind,* denotes complex conjugation, r₁ = |x − y|, and r₂ = |x − y^#|, with y^# = (y₁, − y₂, y₃) an image source. Finally, s_1,2 are given by

$$s_1=\frac{2\sqrt{r_{0}r}}{r_1}\sqrt{\sin \varphi }\cos \left(\frac{\theta -{\theta }_0}{2}\right)$$ 2.3

and

$$s_2=-\frac{2\sqrt{r_{0}r}}{r_2}\sqrt{\sin \varphi }\cos \left(\frac{\theta +{\theta }_0}{2}\right),$$ 2.4

where θ and θ₀ are the cylindrical polar angles made by x and y, respectively, r = |x| and r₀ = |y|, and ϕ is the angle of projection of the observer in the x₂ − x₃ plane with respect to the x₂ axis. As we are interested in only the far-field sound, we approximate |x|≫|y| to obtain an asymptotic approximation of the integrals, and use (2.1) to obtain

$$\begin{array}{rl}G(\mathbf{x},\mathbf{y},t,\tau )& \approx \frac{1}{8\pi |\mathbf{x}|}[\delta (\frac{|\mathbf{x}-\mathbf{y}|}{c_0}+\tau -t)+\delta (\frac{|\mathbf{x}-{\mathbf{y}}^{\mathrm{\#}}|}{c_0}+\tau -t)]\\ & +\frac{{\mathrm{e}}^{-3\pi i/4}}{4\pi |\mathbf{x}|}\sqrt{\frac{2|\mathbf{y}|}{c_0}}\sqrt{\sin \varphi }[\cos \left(\frac{\theta -{\theta }_0}{2}\right){\delta }^{\frac{1}{2}}(\frac{|\mathbf{x}-\mathbf{y}|}{c_0}+\tau -t)\\ & -\cos \left(\frac{\theta +{\theta }_0}{2}\right){\delta }^{\frac{1}{2}}(\frac{|\mathbf{x}-{\mathbf{y}}^{\mathrm{\#}}|}{c_0}+\tau -t)],\end{array}$$ 2.5

where δ is the Dirac delta function, and δ^1/2 denotes the fractional half-derivative of the delta function. We have written, where appropriate and concise, this function in terms of x and y as we shall be differentiating with respect to the components in y.

We note that the first two terms in equation (2.5) are the same as those for the infinite wall of Glegg & Devenport [44] (but with an additional scaling factor of 1/2). We can, therefore, attribute those terms to the direct noise that will be generated by the rough surface, while the second two terms in equation (2.5) relate to the diffraction of roughness-induced dipole sources by the trailing edge. As the first two terms have been previously calculated, we shall only present further calculations for the second two terms, relating to the diffraction.

(c). Far-field scattered pressure

We must evaluate the derivatives of the Green's function on the surface y₂ = ξ(y₁, y₃) to approximate the far-field scattered noise. Since ξ is small, we may take a Taylor series of the relevant derivatives, as done by Glegg & Devenport [44];

$$\begin{array}{r}{\frac{\mathrm{\partial }G}{\mathrm{\partial }{y}_1}|}_{y_2=\xi }\approx {\frac{\mathrm{\partial }G}{\mathrm{\partial }{y}_1}|}_{y_2=0}+O(\xi ),\end{array}$$ 2.6

$$\begin{array}{r}{\frac{\mathrm{\partial }G}{\mathrm{\partial }{y}_2}|}_{y_2=\xi }\approx \xi {\frac{{\mathrm{\partial }}^{2}G}{\mathrm{\partial }{y}_2^2}|}_{y_2=0}+O({\xi }^2)\end{array}$$ 2.7

$$\begin{array}{r}\text{and}{\frac{\mathrm{\partial }G}{\mathrm{\partial }{y}_3}|}_{y_2=\xi }\approx {\frac{\mathrm{\partial }G}{\mathrm{\partial }{y}_3}|}_{y_2=0}+O(\xi ).\end{array}$$ 2.8

Note the rough upper surface of the plate corresponds to θ₀ ≈ π.

Using these approximations, along with the transform (2.1) yields an expression for the transform of the diffracted dipole contribution (last two terms in (2.5)) to the far-field pressure, ${\tilde{p}}^{dd}$, given by

$$\begin{array}{rl}{\tilde{p}}^{dd}(\mathbf{x},k_0)& \approx \frac{-i\sqrt{k_0}\hspace{0.17em}{\mathrm{e}}^{i{k}_0|\mathbf{x}|}\sin \frac{\theta }{2}\sqrt{\sin \varphi }}{\sqrt{2}\pi |\mathbf{x}|}{\int }_{\mathit{\Sigma }}\frac{{\mathrm{e}}^{-i{k}_0(x_1{y}_1+x_3{y}_3)/|\mathbf{x}|}}{{(y_1^2+y_3^2)}^{3/4}}{p}_s(\mathbf{y},k_0)\\ & \times \{\frac{\mathrm{\partial }\xi }{\mathrm{\partial }{y}_1}(\frac{1}{2}{y}_1+\frac{x_1}{|\mathbf{x}|}i{k}_0(y_1^2+y_3^2))+\frac{\mathrm{\partial }\xi }{\mathrm{\partial }{y}_3}(\frac{1}{2}{y}_3+\frac{x_3}{|\mathbf{x}|}i{k}_0(y_1^2+y_3^2))\\ & +\frac{\xi }{4y_1^2}((y_1^2-y_3^2)+\frac{y_1{x}_2}{|\mathbf{x}|}(y_1^2+y_3^2)(i{k}_0)[\frac{4y_1{x}_2}{|\mathbf{x}|}(i{k}_0)-\cot \frac{\theta }{2}])\}\mathrm{d}\mathit{\Sigma }(\mathbf{y}),\end{array}$$ 2.9

where we recall f_i = p_sn_i with p_s(y₁, y₃, k₀) the pressure of the turbulent source on the surface, and Σ denotes the area of the (y₁, y₃)-plane over which the roughness extends. We note equation (2.9) has a factor of sinθ/2 which represents the typical trailing-edge cardioid shape reaffirming that this component of the scattered field arises due to diffraction by the edge.

Following Glegg & Devenport [44], we now write the surface pressure in terms of the wavenumber spectrum,

$$p_s(y_1,y_3,k_0)={\int }_{{\kappa }_1,{\kappa }_2}{p}_s({\kappa }_1,{\kappa }_2,k_0)\hspace{0.17em}{\mathrm{e}}^{-i{\kappa }_1{y}_1-i{\kappa }_3{y}_3}\hspace{0.17em}\mathrm{d}{\kappa }_1\mathrm{d}{\kappa }_2,$$ 2.10

and define the power scattered spectral density as

$${\mathit{\Phi }}_{pp}(\mathbf{x},k_0)=\frac{\pi }{T}\mathrm{Ex}\left(|p{|}^2\right),$$ 2.11

where Ex( · ) denotes expectation. By defining Φ^s_pp(κ₁, κ₃, k₀) as the surface pressure spectrum, we obtain

$${\mathit{\Phi }}_{pp}^{dd}(\mathbf{x},k_0)=\frac{h{k}_0\mathit{\Sigma }|\sin \frac{\theta }{2}{|}^2\sin \varphi }{2{\pi }^2|\mathbf{x}{|}^2}{\mathit{\Phi }}_{pp}^s(k_0){\int }_{{\kappa }_{1,3}}{\mathit{\Psi }}_{pp}^s({\kappa }_1,{\kappa }_3,k_0)\mathit{\Gamma }({\kappa }_1,{\kappa }_3,k_0)\hspace{0.17em}\mathrm{d}{\kappa }_1\mathrm{d}{\kappa }_3,$$ 2.12

where

$$\begin{array}{rl}& {\mathit{\Psi }}_{pp}^s({\kappa }_1,{\kappa }_3,k_0)={\mathit{\Phi }}_{pp}^s\frac{({\kappa }_1,{\kappa }_3,k_0)}{{\mathit{\Phi }}_{pp}^s}(k_0),\end{array}$$ 2.13

$$\begin{array}{rl}& {\mathit{\Phi }}_{pp}^s(k_0)={\int }_{{\kappa }_{1,3}}{\mathit{\Phi }}_{pp}^s({\kappa }_1,{\kappa }_3,k_0)\hspace{0.17em}\mathrm{d}{\kappa }_1\mathrm{d}{\kappa }_3\end{array}$$ 2.14

and

$$\begin{array}{rl}\mathit{\Gamma }({\kappa }_1,{\kappa }_3,k_0)& =\frac{1}{\mathit{\Sigma }h}|{\int }_{\mathit{\Sigma }}\frac{{\mathrm{e}}^{i\mathbf{k}\cdot \mathbf{y}}}{{(y_1^2+y_3^2)}^{3/4}}\{\frac{\mathrm{\partial }\xi }{\mathrm{\partial }{y}_1}(\frac{1}{2}{y}_1+\frac{x_1}{|\mathbf{x}|}i{k}_0(y_1^2+y_3^2))\\ & +\frac{\mathrm{\partial }\xi }{\mathrm{\partial }{y}_3}(\frac{1}{2}{y}_3+\frac{x_3}{|\mathbf{x}|}i{k}_0(y_1^2+y_3^2))\\ & +\frac{\xi }{4y_1^2}((y_1^2-y_3^2)+\frac{y_1{x}_2}{|\mathbf{x}|}(y_1^2+y_3^2)(i{k}_0)[\frac{4y_1{x}_2}{|\mathbf{x}|}(i{k}_0)-\cot \frac{\theta }{2}])\}\mathrm{d}\mathit{\Sigma }(\mathbf{y}){|}^2,\end{array}$$ 2.15

where k = − (κ₁ + k₀x₁/|x|, 0, κ₃ + k₀x₃/|x|). This holds for any rough surface described by y₂ = ξ(y₁, y₃) whose elements typically project from y₂ = 0 by O(h) where h is much smaller than the boundary layer thickness, such that the bosses lie solely within the turbulent boundary layer.

3. Example: hemispherical bosses

Here, we evaluate (2.15) explicitly for the case of roughness composed of N randomly located hemispherical bosses, each of radius R_n with centre at Yⁿ, within some finite area Σ. We let $\overline{R}$ be the mean radius of the bosses, and assume small variance of $O({\overline{R}}^2)$. We further assume the roughness is acoustically compact, $|\mathbf{k}\overline{R}|\ll 1$. In this case, equation (2.15) may be approximated by

$$\mathit{\Gamma }({\kappa }_1,{\kappa }_3,k_0)\approx \frac{1}{4\mathit{\Sigma }h}{|{\int }_{\mathit{\Sigma }}\frac{{\mathrm{e}}^{i\mathbf{k}\cdot \mathbf{y}}}{{(y_1^2+y_3^2)}^{3/4}}\{\frac{\mathrm{\partial }\xi }{\mathrm{\partial }{y}_1}{y}_1+\frac{\mathrm{\partial }\xi }{\mathrm{\partial }{y}_3}{y}_3+\frac{\xi }{2y_1^2}(y_1^2-y_3^2)\}\mathrm{d}\mathit{\Sigma }(\mathbf{y})|}^2.$$ 3.1

To evaluate this, we integrate the first two terms by parts and keep only leading order terms

$$\mathit{\Gamma }({\kappa }_1,{\kappa }_3,k_0)\approx \frac{1}{4\mathit{\Sigma }h}{|{\int }_{\mathit{\Sigma }}\frac{{\mathrm{e}}^{i\mathbf{k}\cdot \mathbf{y}}\xi }{2{(y_1^2+y_3^2)}^{3/4}}\{1+\frac{1}{y_1^2}(y_1^2-y_3^2)\}\mathrm{d}\mathit{\Sigma }(\mathbf{y})|}^2,$$ 3.2

then write the integral as a sum over each hemispherical component, with the parameterization (yⁿ₁, yⁿ₃) = (Yⁿ₁, Yⁿ₃) + r_n(cosϕⁿ, sinϕⁿ)

$$\mathit{\Gamma }({\kappa }_1,{\kappa }_3,k_0)\approx \frac{1}{16\mathit{\Sigma }h}{|\sum _{n=1}^N\hspace{0.17em}{\mathrm{e}}^{i\mathbf{k}\cdot {\mathbf{Y}}^n}\frac{1}{|{\mathbf{Y}}^n{|}^{3/2}}(1+\frac{Y_1^{n\hspace{0.17em}2}-Y_3^{n\hspace{0.17em}2}}{Y_1^{n\hspace{0.17em}2}})\frac{2\pi R_n^3}{3}|}^2.$$ 3.3

If we suppose the boss nearest to the trailing edge is a distance L_T from the edge, and that the wall pressure spectrum is dominated at the convective ridge with κ₁∼ω/U_c, κ₃∼0, we can obtain an overall scaling for the far-field scattered noise

$${\mathit{\Phi }}_{pp}^{dd}(\mathbf{x},k_0)\propto (k_0\overline{R})\frac{{\overline{R}}^5}{L_T^3|\mathbf{x}{|}^2}{\mathit{\Phi }}_{pp}^s(\frac{\omega }{U_c},0,k_0).$$ 3.4

We compare to the noise generated by wall roughness over an infinite plane [44]

$${\mathit{\Phi }}_{pp}^w(\mathbf{x},k_0)\propto {(k_0\overline{R})}^4\frac{{\overline{R}}^2}{|\mathbf{x}{|}^2}{\mathit{\Phi }}_{pp}^s(\frac{\omega }{U_c},0,k_0),$$ 3.5

and see clearly if k₀L_T ≪1 the diffracted dipole noise is much larger than the wall roughness noise. Such a case would occur for a porous material consisting of spherical powder/beads, where $L_T\sim \overline{R}$. Furthermore, the directivity pattern for an infinite plate is cos²θ, while for trailing-edge scattering is sin²θ/2, thus in the far-field close to θ = π/2 the discrepancy between diffracted dipole noise, and infinite wall roughness noise will be very significant.

We briefly mention this analysis does not permit L_T = 0, i.e. a patch of roughness located at the trailing edge tip. Such a case would invalidate the assumption that the trailing edge is sharp, leading to aerodynamic inefficiencies and additional vortex shedding noise which is beyond the scope of this paper.

Figure 3 illustrates the effect of the additional dipole source noise diffracted by the trailing edge. We see the direct roughness noise over an infinite wall decays much more rapidly for small ωR/v* than the experimental measurements of Hersh [40]. Allowing for the correction due to diffraction at the trailing edge increases this lower frequency noise to be aligned well with the experimental data. In both theoretical cases, the surface spectrum has been taken as

$${\mathit{\Phi }}_{pp}^s(\frac{\omega }{U_c},0,\omega )=\frac{{\mathrm{e}}^{-2\omega R/U_c}}{(1+{\mathrm{e}}^{-2\omega R/U_c})(1+[1+4\mu \sigma \omega R/U_c]\hspace{0.17em}{\mathrm{e}}^{-2\omega R/U_c})},$$ 3.6

as given by Howe [42], where σ is the surface roughness density, and μ = (1 + (1/4)σ)⁻¹. Recall also k₀ = ω/c₀. For clarity, overall multiplicative constants have been removed from this definition as they affect all cases in figure 3 in an identical manner.

Figure 3.

Comparison of roughness noise spectra when $L_T=\overline{R}$. Dots show Hersh's experimental data (grit size 50); blue line assumes an infinite wall; yellow line includes wall generated turbulence and trailing-edge diffraction (dipole only). Theoretical curve heights adjusted to fit experimental spectra (both adjusted identically). (Online version in colour.)

Figure 3 takes the overall scaling (3.4). We illustrate this is an appropriate approximation to the numerically calculated expectation value for 50 uniformly distributed roughness elements over an region (yⁿ₁, yⁿ₃)∈[L_T, 100L_T] × [ − 100L_T, 100L_T] in figure 4. We see the agreement between the experimental results and the numerically predicted results are very slightly improved over the purely scaled results of figure 3.

Figure 4.

Comparison of roughness noise spectra when $L_T=\overline{R}$. Dots show Hersh's experimental data (grit size 50); yellow line includes wall generated turbulence and trailing-edge diffraction calculated by numerically integrating the expectation value for uniformly distributed roughness. Curve height is adjusted to fit experimental spectra. (Online version in colour.)

4. Conclusion

This paper has first reviewed a range of bioinspired aerofoil adaptations which aid the reduction of aerofoil–turbulence interaction noise. Reasons arising for discrepancies between theoretical models and experimental data are discussed for a range of adaptations, but particular attention is given to the porous trailing-edge adaptation. Here, the effect of surface roughness is neglected in current theoretical models, thus this paper also presents a theoretical prediction for this additional roughness-induced noise through the use of the acoustic analogy and perturbation expansions.

We observe that if the roughness is close to the trailing edge (a distance of similar magnitude to the roughness elements), the diffraction of the dipole sources by the trailing edge increases low-frequency far-field noise. By including the effects of this diffraction, the final result for the noise generated by a rough trailing edge is seen to be in very good agreement with experimental measurements; in particular, the discrepancy between previous theoretical and experimental results at low frequencies using infinite wall models (i.e. not accounting for the trailing-edge diffraction) is corrected.

The frequency range of interest for trailing-edge roughness noise is typically ≳5 kHz; below this frequency traditional quadrupole trailing-edge noise (for smooth walls) dominates the scattered frequency spectra [43] (although for large roughness elements the affected frequency range shifts towards the low frequencies). The results illustrated in this paper are within this range of interest and therefore show that a complete picture of roughness noise should include diffractive effects of the trailing edge. Of particular importance may be the ability to predict roughness noise generated by porous trailing-edge adaptations to aerofoils, and this paper provides support for the hypothesis that porous trailing edges increase high-frequency noise not due directly to their porosity, but due to their surface roughness. This conclusion bridges the gap between current theoretical predictions for porous trailing-edge noise, and experimental measurements and illustrates how supplementary new theoretical models can be used to improve existing theoretical predictions.

Data accessibility

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Competing interests

I declare I have no competing interests.

Funding

The work in this paper was supported by EPSRC Early Career Fellowship (grant no. EP/P015980/1).