Effects of structural damping on acoustic scattering by flexible plates

Abstract

We investigate the effects of structural damping on the interaction of a turbulent eddy with flexible plates with respect to the efficiency of aerodynamic noise generation. Potential benefits are studied using a model based on a point-reacting compliant semi-infinite plate on a spring-damper foundation. This scattering problem is solved using the Wiener–Hopf technique. We compare results for semi-infinite compliant plates with finite ones. In both cases, plate vibration lead to reductions of sound radiation, especially at resonance; damping tends to reduce such acoustic benefits. We also present a formulation that considers the effect of structural damping on the acoustic properties of finite elastic plates. Numerical results are obtained by applying a boundary element method to solve the Helmholtz equation subject to the boundary conditions imposed by the plate vibration. Under specific conditions, such as high fluid loading factor and low bending-wave Mach number, the acoustic power scattered by an edge tends to be smaller than that which propagates over the plate as bending waves. Results show that structural damping attenuates these waves and may modify the far-field acoustic pressure, mostly by reducing the scattered sound at structural resonances. All models show that large damping coefficients lead to locally over-damped responses. There is thus an ideal range of structural damping to reduce both plate vibration and acoustic scattering.

1. Introduction

For decades, the sound generated by aerodynamic bodies in fluid flow has motivated significant research. Trailing-edge noise is an important and unavoidable source of aerofoil noise; it is generated by turbulence in the boundary layer scattering off the sharp trailing edge of an aerofoil. Hayden [1] states that the introduction of surfaces into fluid flows results in the conversion of incompressible pressure (or velocity) fluctuations which are travelling subsonically into compressible pressure fluctuations which then propagate away from the surface at the sound speed of the medium. Curle [2] was the first to consider the influence of solid boundaries upon aerodynamic sound. Through the extension of Lighthill's general theory of aerodynamic sound [3], it was shown that a rigid boundary could be replaced by a distribution of surface dipoles. This result leads to a dependence law of radiated intensity on the sixth power of the flight speed, for compact bodies and low Mach number flows; this is a quite different behaviour from the well-know u⁸ law found by Lighthill [3] for sound generation by free turbulence. Williams & Hall [4] showed that when a turbulent eddy lies within an acoustic wavelength of a sharp edge on a large surface, the radiated sound has a u⁵ velocity dependence. This results highlights the important role of trailing-edge noise for low Mach number flows, as their sound radiation tends to dominate emissions when compared to compact surfaces or free turbulence. Later, Howe [5] made a critical review on the theory of noise generation by the interaction of a semi-infinite rigid surface with a turbulent low Mach number flow. In addition to the acoustic analogy of Lighthill theories based on the solution of linearized acoustic problems and ad hoc aerodynamic source models were considered, for example, [6,7], for the development of a unified theory that also incorporates the effect of mean motion and of the Kutta condition.

Trailing edge noise is usually estimated by assuming the lifting surface to be rigid [1,8,9]. However, considering an aerofoil as a rigid plate may be an oversimplification if one wishes to evaluate the role of the edge-generated sound in lightweight structures. The first work considering scattering by flexible surfaces was carried out by Crighton & Leppington [10], who studied acoustic scattering by a semi-infinite compliant plate immersed in a turbulent flow. Here, unlike elastic plates that have stiffness and may deform when subjected to stress, compliant refers to a plate with specific mass, which moves as a result of the pressure difference between its two sides. When the plate appears to be relatively limp, they found that the radiated intensity increases with flow velocity according to a u⁶ law, decreasing thus sound radiation for low velocities with respect to the rigid case. Further refinements take into account that bending waves are also excited on the plate [11], carrying elastic energy that will be converted into sound by scattering on possible structural discontinuities [12]. The sound received in the far-field thus consists of the direct radiation from the edge coupled to that portion which is scattered by structural waves. In many practical situations involving high fluid loading, the prevailing source of sound attributable to turbulent flow over a trailing edge will not be the edge, but one or more non-homogeneous or discontinuous positions. For example, Howe [12] presents numerical results for steel plates in water in which the bending wave power exceeds the total sound power generated at the edge by 20–40 dB, which happens when the frequency is smaller than the coincidence value (leading to in vacuo bending waves with phase velocity equal to the sound speed). Thus, the approach based on the rigid surface is unable to make correct estimates even if only a small part of the elastic energy is scattered.

Jaworski & Peake [13], inspired by the silent flight of owls, used the Wiener–Hopf technique [14] to predict the far-field trailing-edge noise from a semi-infinite poroelastic plate. The Rayleigh conductivity was used to model porosity. Through the analysis of limiting cases, namely porous-rigid and impermeable-elastic, they show that for a porous edge, the far-field acoustic power scales like u⁶, and a scaling of u⁷ is found for an elastic edge, to be compared with the previously cited [4] dependence of u⁵ on a rigid impermeable edge; thus, both porosity and elasticity lead to considerable reductions of trailing-edge noise. Acoustic benefits are also obtained for a poroelastic extension of an otherwise rigid impermeable plate [15,16]. Recently, the work of Jaworski & Peake [13] has been extended by Cavalieri et al. [17] to consider the effects of a finite leading edge, and by Pimenta et al. [18] to study three-dimensional plates, with finite span. These works developed a boundary element method (BEM), which rewrites the boundary conditions in terms of the vibration modes of the plate, and allows a direct numerical solution of the acoustic problem. Finite-plate effects are seen to lead to structural resonance, which affects the sound radiation by the plate. When compared to theoretical results for semi-infinite plates, it is seen that finite flexible plates lead to lower sound reductions with respect to the rigid case; the use of lighter, composite materials may then be of interest, as they lead to higher fluid loading and hence to more significant decreases of trailing-edge noise [19].

The plate model used by Jaworski & Peake [13], Cavalieri et al. [17] and Nilton et al. [19] was developed by Howe [20], it is written in terms of transverse displacement of the plate and the displacement of the fluid through apertures that mimic the porosity. This equation has explicit terms for the plate stiffness and inertia; however, it does not take into account the structural damping. Such damping, inherent of elastic materials, may affect the radiated sound by attenuating flexural waves in the plate, changing thus the fluid–structure interaction near the edge. To the best of our knowledge, the role of structural damping has not been considered in studies of trailing edge noise. The present work seeks thus to evaluate the influences of structural damping on the scattering efficiency of semi-infinite and finite plates in the vicinity of a turbulent source. As a fundamental study, we do not consider here how this damping would be implemented in a real plate, but a recent investigation by our group showed that the addition of viscoelastic layers is a feasible procedure to increase damping in a composite plate [21]. The current study is undertaken by considering problems of different complexity: scattering by a semi-infinite compliant plate, by a finite compliant plate, and finally by a finite elastic plate. In all cases, damping is introduced. For the first two problems, a compliant plate is considered to lie on a mass-spring-damper foundation, whereas for the third case vibrations of a damped plate are considered. The model equations for the scattering of a source located far from a semi-infinite compliant plate, similar to the work of Crighton & Leppington [10] are developed in §2a, whose near-field solution provides the far-field sound due to a source near the plate by reciprocity arguments. Section 2b outlines the model equations for the acoustic scattering problem and the vibration of a finite plate, and we use such equations to analyse a finite compliant plate, similar to the one studied in §2a. The third problem is considered in §2c, where an explicit damping term is added to the vibration equation used in the finite plate problem. Results and discussion are presented in §3, where numerical solutions for semi-infinite and finite compliant plates are presented in §3a and 3b, respectively, and finite elastic damped plates are considered in §3c. The paper is closed by the presentation of conclusions in §4.

2. Acoustic scattering models

(a). Acoustic scattering by a semi-infinite compliant plate

A semi-infinite compliant plate, on a spring-damper foundation, is modelled by the extension of the problem discussed by Crighton & Leppington [10]. According to Lighthill's theory, turbulent eddies are replaced by a volume distribution of quadrupoles. The reciprocal theorem is used to transform the quadrupole scattering problem into one of diffraction of a plane acoustic wave. The model problem at hand is shown schematically in figure 1. A plate of mass m per unit area, stiffness $\mathcal{K}$ per unit area, damping $\mathcal{C}$ per unit area and negligible thickness lies in the half-plane: x≥0, y = 0, −∞ < z <  + ∞. The semi-infinite plate has a trailing edge at x = 0, but extends infinitely, and thus has no leading edge. Quiescent compressible fluid of density ρ_f and speed of sound c₀ surrounds the plate.

Figure 1.

Schematic of a semi-infinite compliant plate; by the reciprocal theorem we solve for the field at a general point x due to the presence of a monopole at a distant point x₀. Z-axis is normal to the plane of the figure.

This problem is solved by the Wiener–Hopf technique, which is particularly well suited to problems of scattering of the near-field of multipole sources by semi-infinite bodies [10]. Using the reciprocity property, we seek for solutions at a general point x due to the presence of a monopole source at a distant point x₀. In this case, acoustic waves reach the plate as incident plane waves. By suitable differentiations of the field, we may generate the field at x₀ due to a multipole of arbitrary order at x. Omitting the variation in span z, the velocity potential of the incident plane wave from the far-field is

$${\varphi }_0(x,y)=\exp [\mathrm{i}k(x\cos {\theta }_0+y\sin {\theta }_0)],$$ 2.1

as $r_0\to \mathrm{\infty }$, with r finite. The scattered field must satisfy the Helmholtz equation

$$({\mathrm{\nabla }}^2+k^2)\varphi (x,y)=0,$$ 2.2

where k = k₀sinψ₀ is the reduced wavenumber.

The boundary conditions to close the problem are: the velocity potential ϕ(x, y) is continuous across y = 0 for x < 0 which is a continuity condition for the potential in free air; ϕ′(x, y) is continuous across y = 0 for all x, where a prime denotes the operation ∂/∂y; finally, the pressure in the scattered field p is related to the transverse plate displacement η through the relation

$$p(x,0^{-})-p(x,0^{+})=-m{\omega }^2\eta (x)+\mathcal{K}\eta (x)-\mathrm{i}\omega \mathcal{C}\eta (x),$$ 2.3

the pressure at the plate surface and the plate and fluid displacements are related by the linearized Euler equation as

$${\rho }_f{\omega }^2\eta ={\frac{\mathrm{\partial }p}{\mathrm{\partial }y}|}_{y=0}.$$ 2.4

Combining equations (2.3) and (2.4) and writing in terms of velocity potential

$$\varphi (x,0^{+})-\varphi (x,0^{-})=\frac{m-\mathcal{K}/{\omega }^2+\mathrm{i}\mathcal{C}/\omega }{{\rho }_f}[{\varphi }^{\prime }(x,0)+\mathrm{i}k\sin {\theta }_0\exp (\mathrm{i}kx\cos {\theta }_0)],\mathrm{for}x>0.$$ 2.5

Equations (2.3) and (2.5) represent a compliant plate lying on a spring-damper foundation. It is convenient to non-dimensionalize the set of governing equations by introducing the natural resonance frequency ${\omega }_N=\sqrt{\mathcal{K}/m}$, which can be related to a natural wavenumber k_0N = ω_N/c₀. We also define the damping coefficient $\xi =\mathcal{C}/(2{\omega }_{N}m)$ and the loading parameter μ₀ = 2ρ_f/m analogous to what is done in Crighton & Leppington [10]. Denoting dimensionless variables by hats, the following scalings are employed:

$$\begin{array}{rl}\eta & =k_{0_N}^{-1}\hat{\eta },p={\rho }_f{\omega }_N^2{k}_{0_N}^{-2}\hat{p},\varphi ={\omega }_N{k}_{0_N}^{-2}\hat{\varphi },\\ \omega & ={\omega }_N\hat{\omega },{\mu }_0=k_{0_N}\widehat{{\mu }_0},(x,y,z)=k_{0_N}^{-1}(\hat{x},\hat{y},\hat{z}),\end{array}$$ 2.6

and any parameters not shown are scaled using ω_N for inverse time and k_0N for inverse length. Dropping the hats on the dimensionless quantities, the complete set of dimensionless equations for the scattering of an incident plane wave by a semi-infinite compliant plate becomes the following:

$$\begin{array}{rl}& {\varphi }_0(x,y)=\exp [\mathrm{i}k(x\cos {\theta }_0+y\sin {\theta }_0)],\end{array}$$ 2.7

$$\begin{array}{rl}& ({\mathrm{\nabla }}^2+k^2)\varphi (x,y)=0,\end{array}$$ 2.8

$$\begin{array}{rl}& \mathrm{\Delta }p=\frac{2}{{\mu }_0}(-{\omega }^2+1-2\mathrm{i}\xi \omega )\eta ,\end{array}$$ 2.9

$$\begin{array}{rl}& k_0^2\eta ={\frac{\mathrm{\partial }p}{\mathrm{\partial }y}|}_{y=0}\end{array}$$ 2.10

$$\begin{array}{rl}\mathrm{and}& D(x)=\varphi (x,0^{+})-\varphi (x,0^{-})=\frac{2}{{\mu }_0}\left(\frac{{\omega }^2-1+2\mathrm{i}\xi \omega }{{\omega }^2}\right)[{\varphi }^{\prime }(x,0)+\mathrm{i}k\sin {\theta }_0\exp (\mathrm{i}kx\cos {\theta }_0)],\end{array}$$ 2.11

for x > 0. Note that the function D(x) has been defined for convenience. The scattered potential must satisfy a radiation or extinction condition at infinity. This problem is solved through the use of D. S. Jones's technique for Wiener–Hopf problems [10].

(i). The Wiener–Hopf equation

Define half-range Fourier transforms in x according to

$$\begin{array}{rl}\mathit{\Phi }(\alpha ,y)& ={\int }_{-\mathrm{\infty }}^0\varphi (x,y)\hspace{0.17em}{\mathrm{e}}^{\mathrm{i}\alpha x}\hspace{0.17em}\mathrm{d}x+{\int }_0^{\mathrm{\infty }}\varphi (x,y)\hspace{0.17em}{\mathrm{e}}^{\mathrm{i}\alpha x}\hspace{0.17em}\mathrm{d}x,\\ & \equiv {\mathit{\Phi }}_{-}(\alpha ,y)+{\mathit{\Phi }}_{+}(\alpha ,y),\end{array}$$ 2.12

α being regarded as a complex variable. It is convenient to regard the acoustic wavenumber as complex as well, k = k₁ + ik₂, where k₂ > 0 is a small artificial dissipative effect which helps identify the strip of analycity by defining branch cuts in the α-complex plane. Then it can be shown that Φ₊(α, y) is a regular function of α in the upper half-plane R₊(Im(α) > − k₂cosθ₀), and that Φ₋(α, y) is regular in the lower half-plane R₋(Im(α) < − k₂). Thus there is a strip S, in which the full-range transform exists as a regular function of α.

The Helmholtz equation takes the form of

$$(\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{y}^2}-{\gamma }^2)\mathit{\Phi }(\alpha ,y)=0,$$ 2.13

where γ² = α² − k², γ is the branch of (α² − k²)^1/2 which tends to α as $\alpha \to \mathrm{\infty }$. Noble [14] shows that solution of equation (2.13) is given by

$$\begin{array}{rl}\mathit{\Phi }(\alpha ,y)& =A(\alpha )\exp (-\gamma y),\mathrm{for}y>0,\\ & =B(\alpha )\exp (\gamma y),\mathrm{for}y<0,\end{array}\}$$ 2.14

which satisfies either a radiation or an extinction condition as $|y|\to \mathrm{\infty }$.

Through the direct application of the Wiener–Hopf technique, whose details are given in [10], and considering that the μ parameter appearing in the kernel function K(α) = μ + (α² − k²)^1/2, is modified by stiffness and damping to

$$\mu ={\mu }_0\left(\frac{{\omega }^2}{{\omega }^2-1+2\mathrm{i}\xi \omega }\right),$$ 2.15

it is possible to obtain,

$${\mathit{\Phi }}_{-}^{\prime }(\alpha ,0)=-\frac{k\sin {\theta }_0}{\alpha +k\cos {\theta }_0}(1-\frac{K_{-}(\alpha )}{K_{-}(-k\cos {\theta }_0)}).$$ 2.16

It is common for the source in problems of aerodynamic sound generation by turbulent flow in the vicinity of a scattering plate to be positioned near the edge, because, as demonstrated by Williams & Hall [4], the presence of the edge of a plate in a turbulent flow results in a large increase in the noise generated by the flow, mainly at low Mach numbers. However, in the reciprocal problem, this corresponds to the observation of the scattered field at a point x very close to the edge and hence equation (2.16) is evaluated in the limit $|\alpha |\to \mathrm{\infty }$. Provided that a certain criterion is met, the first term of equation (2.16) will dominate the solution when evaluated in an asymptotic way. We shall now consider two distinct cases to identify what the constraints will be. First, the case of a nearly rigid surface, i.e. |μ|≪1, the criterion is that k|x|⪷1 which holds within a wavelength of the edge. On the other hand, for a flexible surface |μ| > 1, the requirement is presumably much more stringent, and we need |μx|⪷1. Typical aeronautical applications involve only small values of μ; however, in this investigation, we also address flexible plates so some care should be taken in the analysis of results. Then, for points sufficiently close to the edge, Crighton & Leppington [10] justify the use of the asymptotic form of equation (2.16):

$${\mathit{\Phi }}_{-}^{\prime }(\alpha ,0)\approx \frac{k\sin {\theta }_0}{K_{-}(-k\cos {\theta }_0)}{\alpha }^{-1/2}.$$ 2.17

With this approximation, [10] obtains the distant field emitted by a quadrupole in the presence of a semi-infinite plate

$$\frac{{\mathrm{\partial }}^2\varphi }{\mathrm{\partial }x\mathrm{\partial }y}=\frac{k_0\sin {\theta }_0\sin {\psi }_0\cos \frac{3}{2}\theta }{2{\pi }^{1/2}{R}^{3/2}{K}_{-}(-k_0\sin {\psi }_0\cos {\theta }_0)}\frac{\exp [\mathrm{i}(k_0{r}_0+k_{0}z\cos {\psi }_0+\frac{1}{4}\pi )]}{r_0},$$ 2.18

where ∂²ϕ/∂x∂y gives the scattered potential at x₀ due to a quadrupole at x with one axis in the +x direction and the other in the +y direction.

(ii). Factorization of the Wiener–Hopf kernel

As we shall see later, the kernel has a large influence on sound directivity; therefore, the required multiplicative factorization of K(α) = μ + (α² − k²)^1/2 must be carefully evaluated. In this work, this decomposition is carried out numerically, but as this case is analogous to that of the paper by Crighton & Leppington [10], asymptotic analytic solutions are also valid for large or small |μ|, as we shall see.

The function K(α) may be numerically split through the use of Cauchy's factorization [14,22] which gives

$$K_{-}(\alpha )=\exp [-\frac{1}{2\pi \mathrm{i}}{\int }_C\frac{\ln (K(\zeta ))}{\zeta -\alpha }\hspace{0.17em}\mathrm{d}\zeta ],$$ 2.19

which shall be evaluated for α =  − kcosθ₀; the contour C should lie within the regularity strip S, and must be indented above the pole at ζ = α [14]. Analytical factorization valid for limiting values of μ is given in the appendix of Crighton & Leppington's paper [10]. Firstly, for $\mu \to 0$, which can be understood as a perturbation of the basic kernel (α² − k²)^1/2, the rigid-plate kernel. After some manipulation it is possible to obtain

$$K_{-}(-k\cos {\theta }_0)=-\mathrm{i}{(2k)}^{1/2}\cos \left(\frac{{\theta }_0}{2}\right)[1+\frac{\mathrm{i}\mu {\theta }_0}{\pi k\sin {\theta }_0}+\mathcal{O}(\mu )].$$ 2.20

On the other hand, an estimate of K₋( − kcosθ₀) for large values of |μ| leads to

$$K_{-}(-k\cos {\theta }_0)\approx {\mu }^{1/2}\hspace{0.17em}{\mathrm{e}}^{-\mathrm{i}\pi /4}.$$ 2.21

Equations (2.20) and (2.21) may be used for small or large values of |μ|. For intermediate values, such that $|\mu |=\mathcal{O}(1)$ numerical integration of equation (2.19) leads to the required factorization. Integration is carried out following Peake [22].

(b). Acoustic scattering by a finite compliant plate

Unlike the previous section, here, we will not use the principle of reciprocity. Now, the problem of acoustic scattering by a finite compliant plate is modelled using a point quadrupole source S positioned near the trailing edge of a plate, which has finite chord and infinite span. The plate has the leading edge clamped and trailing edge free, but other combinations of structural boundary conditions are also possible. The objective is to compare the results for finite and semi-infinite plates.

Cavalieri et al. [17] obtained a set of non-dimensional equations to solve the present acoustic problem. The scattered sound is obtained by the solution of an inhomogeneous Helmholtz equation,

$${\mathrm{\nabla }}^{2}p+k_0^{2}p=-S,$$ 2.22

where S is the acoustic source function, k₀ is the acoustic wavenumber, and an $\exp (-\mathrm{i}\omega t)$ time dependence is implicitly assumed. Equation (2.22) is subject to boundary conditions (2.9) and (2.10), which relate the pressure and its normal derivative on the plate surface, leading to

$${\frac{\mathrm{\partial }p}{\mathrm{\partial }y}|}_{y=0}=-\frac{\mu }{2}\mathrm{\Delta }p.$$ 2.23

The problem of acoustic scattering is solved by a BEM. Using Green's second identity, one can write the following boundary integral equation [17]:

$$T(\mathbf{x})p(\mathbf{x})={\int }_{\mathit{\Gamma }}[\frac{\mathrm{\partial }p(\mathbf{y})}{\mathrm{\partial }{n}_y}G-\frac{\mathrm{\partial }G}{\mathrm{\partial }{n}_y}p(\mathbf{y})]\mathrm{d}\mathit{\Gamma }-\frac{{\mathrm{\partial }}^{2}G}{\mathrm{\partial }{\mathbf{z}}_{i_m}\mathrm{\partial }{\mathbf{z}}_{i_n}}S({\mathbf{z}}_i),$$ 2.24

where T(x) = 1/2 when x is on a smooth boundary surface Γ, and T(x) = 1 when x is a field point anywhere in the fluid region. The derivatives with respect to the inward normal direction of the boundary surface are represented by ∂/∂n and n is an inward unit normal. The ith source location is z_i and the incident quadrupolar field is computed as the second derivative of the Green's function (G), which is a fundamental solution for the Helmholtz equation in the free space. In the current work, the same numerical parameters of [17] were used. The plate is considered to have a thickness h, and its surface is discretized in 802 elements. Further details can be found in the cited paper.

(c). Acoustic scattering by a finite elastic damped plate

In problems with flexible plates bending waves are excited in the structure. According to the specific characteristics of the problem, such as the fluid loading parameter (ϵ) and bending wave Mach number (Ω), the power of flexural waves exceeds the total sound power generated at the edge. These waves carry elastic energy that will be converted into sound by scattering on possible structural discontinuities. The sound received in the far field consists of the direct radiation from the edge coupled to that portion which is scattered by bending waves. Thus, the addition of structural damping may contribute to the reduction of sound radiation.

In §2a, we discussed the problem of a compliant semi-infinite plate with local damping, and in §2b we showed that the numerical method to compute the acoustic field scattered by finite perforated elastic plates [17] can be used for finite compliant plates on a spring-damper foundation. We now extend the model in §2b to consider acoustic scattering by finite elastic damped plates. This effect is taken into account by the addition to the plate equation [17] of a damping term proportional to the transverse displacement velocity of the plate, which leads to

$${\mathrm{\nabla }}^4\eta -\frac{k_0^4}{{\mathit{\Omega }}^4}(1+2\mathrm{i}\frac{\xi }{{\mathit{\Omega }}^2})\eta =ϵ\frac{k_0^3}{{\mathit{\Omega }}^6}\mathrm{\Delta }p,$$ 2.25

where Ω and ϵ characterize the fluid–structure interaction; and ξ is the non-dimensional damping coefficient. In this section, damping is normalized by the coincidence frequency (${\omega }_c=c_0^2\sqrt{m/B}$, where B is the bending stiffness of the plate and m is the mass per unit area), since the plate has unlimited natural frequencies; thus, $\xi =\mathcal{C}/(2m{\omega }_c)$ is a non-dimensional damping coefficient that depends solely on the material properties. An additional, convenient parameter $\overline{\xi }$ can be defined, where $\overline{\xi }=\xi /{\mathit{\Omega }}^2$. The numerical procedure to solve this problem is the same as that used in §2b. The plate displacement is

$$\eta =\frac{ϵk_0^3}{{\mathit{\Omega }}^6}\frac{⟨\mathrm{\Delta }p,{\mathit{\Psi }}_j⟩}{{\beta }_j^4-(k_0^4/{\mathit{\Omega }}^4)(1+\mathrm{i}2\overline{\xi })},$$ 2.26

where Ψ_j is an orthonormal basis obtained from the solution of an auxiliary eigenvalue problem subject to the same boundary conditions of equation (2.25), see [17] for details. The eigenvalue β_j is identified as the bending wavenumber of a vibration mode Ψ_j of the plate. Equation (2.23), which relates the pressure difference between the two sides of the plate and the pressure gradient, is modified to

$${\frac{\mathrm{\partial }p}{\mathrm{\partial }y}|}_{y=0}=\frac{ϵk_0^5}{{\mathit{\Omega }}^6}\frac{⟨\mathrm{\Delta }p,{\mathit{\Psi }}_j⟩}{{\beta }_j^4-(k_0^4/{\mathit{\Omega }}^4)(1+\mathrm{i}2\overline{\xi })}{\mathit{\Psi }}_j.$$ 2.27

An important effect of the damping parameter $\overline{\xi }$ is that for non-zero damping the denominator in equation (2.27) is never zero for real-valued k₀ and Ω. For undamped plates, resonances appear when the denominator is zero, leading to amplitudes of vibration going to infinity, as seen in Cavalieri et al. [17]. As will be seen later, the inclusion of damping leads to reduction of such resonances to finite amplitudes, analogue to what happens in forced mass-spring-damper systems.

3. Results

The results presented in this section were obtained using the Wiener–Hopf technique for semi-infinite plates and the BEM for finite plates. A detailed validation of the present BEM code was presented in references [17,18]; moreover, a comparison between finite and semi-infinite plates in this section provides further validation of the current methodology.

(a). Acoustic scattering by a semi-infinite compliant plate

We first study acoustic scattering of a compliant plate considering $\mathcal{C}=0$ and $\mathcal{K}=0$, which means that we are recovering the results of Crighton & Leppinton [10]. Three values were chosen for μ, given by: (0.0, 0.2, 5.0), with k = 1.0 + i0.001 held fixed. We verified that changes of the imaginary part of k around this value have negligible effect on the far-field sound. Using equation (2.18), it is possible to obtain the directivity of the acoustic radiation, which is shown in figure 2. Note that the condition of validity of the asymptotic approximation changes for different values of μ. As mentioned in the §2a, the distance from the observer to the edge of the plate in the xy-plane (R according to figure 1) should be smaller than a wavelength for μ≪1. On the other hand, R⪷1/μ when μ≫1. The first point to highlight in figure 2 is that there is a good agreement between the numerical (with numerical evaluation of the integral of equation (2.19)) and asymptotic analytical results for particular cases in which μ≪1 or μ≫1. Another point worth highlighting is that the radiation from the rigid plate (μ = 0.0) approaches sin(θ/2) while that of the compliant plate (|μ| > 1.0) is shaped like a dipole. We can also see that the overall radiation for the rigid plate is greater than the one when the plate is limp relative to the surrounding fluid. According to Crighton & Leppington [10], the reason is that in the case of the limp plate a much weaker type of scattering occurs, in which the u⁵₀ law predicted by Williams & Hall [4] for total acoustic power is replaced by a dependence upon u⁶₀/ϵ, where ϵ is the fluid loading parameter. This deduction does not require that μ be real positive, so it must apply for large |μ| to the cases we shall see further in which μ is complex-valued.

Figure 2.

Sound directivity profiles obtained for k = 1.0 + i0.001 and μ = (0.0, 0.2, 5.0). The numerical and analytical results are displayed. (Online version in colour.)

The Wiener–Hopf kernel is a function of μ, which in the compliant case of Crighton & Leppington [10] is μ = μ₀ = 2ρ_f/m. For the model in §2a, μ is a function of frequency ω, see equation (2.15). This dependency is shown in figure 3 for a fixed damping coefficient ξ = 1.0. Due to damping, μ is now complex with a negative imaginary part, and the spring induces a resonance frequency ω_N for which the amplitude of μ is maximal and its phase is −π/2.

Figure 3.

Evolution of amplitude and phase of μ/μ₀ as a function of ω, for ξ = 1. (a) Amplitude of μ/μ₀ and (b) phase. (Online version in colour.)

In addition to being a function of ω, the μ parameter also depends on ξ, according to equation (2.15). To verify the potential influences of these parameters, we will analyse them separately. Let us again consider μ₀ = (0.0, 0.2, 5.0) and fix k_0N = 1.0 + i0.001. At first, we will also fix ξ equal to one and analyse the directivity for some values of ω. From now on, we will present only the results based on numerical kernel factorization, since we have already seen that the numerical and analytical results are similar. The rigid plate case is obtained when μ₀ = μ = 0; for comparison, this case is repeated throughout figure 4, which presents results for the different values of ω. Note that the lowest sound radiation corresponds to the resonance frequency, i.e. ω = 1, and the sound directivity profile increases when the frequency moves away from the resonance value. Following this behaviour will be explored in more detail. Another point that we can highlight is that there is a similarity in the profiles between frequencies ω and 1/ω, by comparison of graphs (a) and (c).

Figure 4.

Sound directivity profiles obtained for various values of ω and ξ = 1. (a) ω = 0.2, (b) ω = 1.0 and (c) ω = 5.0. (Online version in colour.)

Now we consider μ₀ = 1.0/k_0N, and study the frequency-dependence of radiated power for some values of the damping coefficient ξ. Note that k_0N remains fixed at 1.0 + i0.001. The results are shown in figure 5a. The first point that is worth highlighting is the symmetry that the curves present in relation to ω = 1.0, as this graph is in logarithmic scale these results corroborate the similarity between ω and 1/ω. These sweeps indicate that the largest reduction in sound power happens at the resonance frequency, and also that the intensity and extent of this reduction is a function of the damping factor. For a large damping factor, the frequency range in which a reduction of the power level happens is larger than for a small ξ, but the intensity of the reduction is lower, as the behaviour of the plate becomes locally over-damped as ξ gets higher.

Figure 5.

Sound power radiated by semi-infinite compliant plates relative to the rigid case. (a) Sound power radiated by compliant plates relative to the rigid case, as a function of ω and different values of ξ. (b) Sound power radiated by compliant plates relative to the rigid case, as a function of ξ and different values of ω. (Online version in colour.)

In the same way, the acoustic power varies with the damping. Figure 5b shows that, as we could expect from the last paragraph, the sound power will tend to its limit value corresponding to the rigid case as ξ gets higher. On the other hand, it will tend to a limit case depending on the frequency as ξ gets low, since μ will tend to a limit value different from zero and frequency-dependent, i.e. it may reach a minimum for various values of ξ, depending on the frequency, provided that it is not equal to one. In this last case, according to equation (2.15), μ continues to increase as ξ approaches zero.

(b). Acoustic scattering by a finite compliant plate

For finite plate calculations, a point quadrupole source was positioned at the vicinity of the trailing edge. The free edge is located at (x, y) = (1, 0), and the quadrupole source is placed at (x, y) = (1, 0.004). We calculate the acoustic power and present directivity results for observers in the acoustic far-field located 50 chords from the plate trailing edge, the polar angle θ is the counterclockwise angle from the x-axis. Cavalieri et al. [17] show that in the case of an acoustically compact chord, the noise reduction capability of the elastic trailing edges is reduced. Therefore, we will use k₀ = 20, which corresponds to the case of a non-compact plate.

The acoustic scattering problem as formulated in §§2a and 2b is governed by three input parameters: μ₀, which is a parameter that characterizes the fluid–structure interaction; ξ, which characterizes the plate damping and ω. Once we have these parameters we can calculate μ and solve the problem of the semi-infinite plate, and we can also solve the analogous problem of the finite plate. At first we set μ₀ = 1/k_0N, and ω = 1.0, and we choose ξ = [0.1, 1.0, 10] for which directivity results are shown in figure 6. Despite the particularities of each formulation, there is a good agreement between the results. The apparent lobes in figure 6b,c are due to backscattering by the leading edge [23]. These are observed to be weaker for lower ξ.

Figure 6.

Sound directivity profiles obtained for different values of ξ and ω = 1.0. (a) ξ = 0.1, (b) ξ = 1.0 and (c) ξ = 10.0. (Online version in colour.)

Parametric studies in ω and ξ were performed with the same configurations of the previous subsection, whose results are shown in figure 5. The corresponding finite plate results are shown in figure 7. In figure 7a note that the trend is similar to that shown in figure 5a: there is a sharp peak of reduction for ξ = 0.1, and this peak is flattened as ξ is increased. However, for ω < 1 (frequencies lower than the natural one), an elevation that exceeds the rigid case appears. This difference from the semi-infinite case is confirmed in figure 7b.

Figure 7.

Sound power radiated by finite compliant plates relative to the rigid case. (a) Sound power radiated by compliant plates relative to the rigid case, as a function of ω and different values of ξ. (b) Sound power radiated by compliant plates relative to the rigid case, as a function of ξ and different values of ω. (Online version in colour.)

(c). Acoustic scattering by a finite elastic damped plate

In this section, we will present results using the formulation described in §2c. Again, a point quadrupole source was positioned at the vicinity of the trailing edge, in the same position that was reported in the previous subsection. We calculate the relative change in acoustic power level due to effects of structural damping and present directivity results for observers in the acoustic far-field located 50 chords from the plate trailing edge. We use one hundred in vacuo bending modes and 802 boundary elements in the plate discretization for all simulations.

First, for comparison, we reproduce the results found by Cavalieri et al. [17] of the pressure field scattered by rigid and aluminium plates immersed in water; the choice of using water as fluid medium simplifies visualization of the relevant waves in the problem due to its high fluid loading parameter (ϵ = 0.135). The Helmholtz number (k₀ = 20) was chosen to ensure that the surface is non-compact in relation to the acoustic wavelength. The results found in Cavalieri et al. can be seen in figure 8a,b. Then we added sample structural damping values, which are shown in figure 8c,d. We observe that compared to the rigid-plate case in figure 8a, the elastic plate in figure 8b presents significant leading-edge scattering of bending waves, which can be seen as the evanescent pressure waves of small wavelength close to the plate surface. Inclusion of damping leads to attenuation of such bending waves, which in turn reduces leading-edge scattering, and higher damping values further reduce the leading edge role. In figure 8, we also represent the displacement η of the plate in black lines, according to equation (2.26); the plate thickness and the amplitude of displacement are exaggerated in figures for ease of visualization. It is possible to observe that introduction of damping leads to decaying amplitudes as bending waves propagate from the trailing to the leading edge.

Figure 8.

Scattered pressure fields for an aluminium plate immersed in water (ϵ = 0.135), with k₀ = 20 and Ω = 0.3: (a) rigid (b) elastic plate without damping, (c,d) elastic plate with damping. Figures are plotted using the same contour levels for acoustic pressure. (a) Rigid, (b) $\overline{\xi }=0.0$, (c) $\overline{\xi }=0.1$ and (d) $\overline{\xi }=0.2$. (Online version in colour.)

The overall radiated sound is also reduced, as can be seen in more detail in figure 9, which shows the directivity results of the sound scattered by the aluminium plate with and without the effect of damping, for observers in the acoustic far-field located 50 chords from the plate trailing edge.

Figure 9.

Amplitude of scattered far-field pressure fluctuation from a aluminium plate in water with k₀ = 20 and Ω = 0.3. The rigid case is also shown for comparison. (a) Rigid, (b) $\overline{\xi }=0.0$ and (c) $\overline{\xi }=0.1$. (Online version in colour.)

(i). Parametric study on Ω with fixed k₀ and ξ

To represent compact and non-compact plates in the parametric studies, three Helmholtz number values were chosen: k₀ = 0.1 is representative of the compact limit; k₀ = 10 moves towards the non-compact limit; and k₀ = 1 was chosen as an intermediate point. For each of these values of k₀, we performed a sweep in Ω with different values of structural damping. Decreasing Ω with fixed k₀ corresponds to lowering the bending stiffness of the plate, for instance, by reducing its thickness. We focus on vacuum bending wave Mach numbers less than one (Ω < 1) since it is in this region that the effects of elasticity are observed. To evaluate the effect of damping in acoustic scattering, figures 10–12 show the change of sound power level relative to the rigid case as a function of Ω, considering an aluminium plate immersed in air (ϵ = 0.0021).

Figure 11.

Change in power level radiated by elastic plates relative to the rigid case as a function of bending wave Mach number Ω for k₀ = 1.0 and different values of damping. (Online version in colour.)

Figure 10.

Change in power level radiated by elastic plates relative to the rigid case as a function of bending wave Mach number Ω for k₀ = 0.1 and different values of damping. (Online version in colour.)

Figure 12.

Change in power level radiated by elastic plates relative to the rigid case as a function of bending wave Mach number Ω for k₀ = 10.0 and different values of damping. (Online version in colour.)

The first point to be observed in figures 10–12 is the presence of sharp peaks, which correspond to the resonance points of the fluid-loaded plates without damping. As Ω is reduced for a fixed k₀, the bending wavenumber is increased k_B = k₀/Ω, and one can observe that the acoustic excitation frequency crosses successive resonance conditions. Unlike the preceding cases of plates mounted on spring-damper foundations, for which only decreases in trailing-edge scattering are observed, elastic resonances may cause both increases and decreases of radiated sound. These resonances occur for slightly lower k_B than the corresponding in vacuo resonance wavenumber β_i, a feature expected theoretically [24], and shown by numerical simulations [17]. With the addition of the structural damping coefficient $\overline{\xi }$, it is possible to observe that the resonance peaks are attenuated and as the damping increases the curves begin to present a smoother behaviour. Thus, resonance peaks that would lead to increases of radiated sound (e.g. the peak at Ω ≈ 0.022 in figure 10) now present decreases of PWL compared to the rigid case, since damping eliminates the sharp increase of acoustic radiation associated with structural resonance and leading-edge scattering of bending waves. In particular, for the non-compact surface limit, where k₀ = 10, it is possible to observe that in addition to the attenuation effect, damping contributes to the reduction of the sound power level.

(ii). Parametric study on ξ with fixed plate thickness h

In this subsection, we present results for a plate with thickness h equals to 0.2% of its chord, which contrasts with the preceding parametric study, where changing Ω with fixed k₀ amounts to modifying h. Here, with a fixed thickness the parameters k₀ and Ω are related by Ω = [k₀(h/l)/(ω_ch/c₀)]^1/2 [19]. When considering elasticity, we refer to an aluminium plate immersed in the air, and under these conditions, the fluid loading parameter ϵ = 0.0021 and ω_ch/c₀ = 0.22, according to the literature [20].

We consider first the scattering of an acoustic quadrupole in the vicinity of the trailing edge of a rigid plate, which can be seen in figure 13; this case is obtained when the transverse pressure gradient evaluated at the plate surface is null, ∂p/∂y|_y=0 = 0. Figure 13 also shows results for an elastic undamped plate. Again, one can observe the presence of sharp peaks, which correspond to resonances of the fluid-loaded plates. In what follows some sample values of the acoustic wavenumber will be chosen from the graph of figure 13, and we will analyse what happens when we gradually add damping.

Figure 13.

Sound power radiated by rigid and non-damped elastic plates as a function of k₀. (Online version in colour.)

The first value chosen for the acoustic wavenumber is k₀ = 0.1, representative of the compact limit. In order to evaluate the damping effect, a sweep in the damping coefficient was performed, and the results are shown in figure 14a. For this value of k₀, it is possible to observe that the damping does not bring any additional acoustic benefit. This is explained by the fact that k₀ = 0.1 does not correspond to a resonance point, as can be seen in figure 13. For k₀≪1, it is seen that the damping acts exclusively by attenuating the peaks corresponding to the resonance conditions. Later, we consider k₀ = 0.2, which is in the region of the first resonance peak shown in figure 13. Results for the damped case are shown in figure 14b. Let us analyse the graph starting from the left; at low values of ξ, we observe that the sound radiation surpasses the rigid case, as we are dealing with a structural resonance. As we increase ξ, the acoustic scattering reduces to values that are lower than the rigid case until reaching a minimum value, and as we continue to increase ξ, the behaviour of the curve becomes locally over-damped. Finally, to illustrate the non-compact limit, consider k₀ = 10, a value that does not correspond to a region of resonance and for which the results are shown in figure 14c. We observe a range of damping values for which the acoustic scattering is slightly lower than that of the purely elastic plate, even though it is not a resonance condition. Within this range, there is a damping value for which the reduction of the sound radiation is maximum. Therefore, besides the acoustic benefit given by the elasticity, there is a small contribution given exclusively by the damping. We examine directivities for the three values of k₀ in figure 15. As the Helmholtz number is increased, the directivity shape moves from that of a compact dipole to a cardioid; backscattering by the leading edge [23,25] is related to additional lobes in the directivity in figure 15c.

Figure 14.

Acoustic power radiated by damped elastic plates as a function of ξ. (a) k₀ = 0.1, compact limit, (b) k₀ = 0.2, resonance condition and (c) k₀ = 10.0, non-compact limit. (Online version in colour.)

Figure 15.

(|p′|) of the scattered sound by damped plates, compared to the reference non-damped plate and to the rigid case. (a) k₀ = 0.1, compact limit. In the damped case, ξ = 9.1 × 10⁻⁵. (b) k₀ = 0.2, resonance condition. In the damped case, ξ = 4.5 × 10⁻⁴. (c) k₀ = 10.0, non-compact limit. In the damped case, ξ = 3.3 × 10⁻². (Online version in colour.)

Another feature that can be observed by analysing the graphs of figure 14a–c is that the behaviour becomes locally over-damped for values of ξ≥1. To understand why this happens, let us rewrite equation (2.27) for an impermeable plate using ξ,

$${\frac{\mathrm{\partial }p}{\mathrm{\partial }y}|}_{y=0}=\frac{ϵk_0^5}{{\mathit{\Omega }}^6}\frac{⟨\mathrm{\Delta }p,{\mathit{\Psi }}_j⟩}{{\beta }_j^4-(k_0^4/{\mathit{\Omega }}^4)(1+2\mathrm{i}(\xi /{\mathit{\Omega }}^2))}{\mathit{\Psi }}_j.$$ 3.1

Analysing only the term in parentheses (1 + 2i(ξ/Ω²)), we observe that if ξ≫Ω² the absolute value of the denominator in the equation (2.27) becomes large, and therefore ∂p/∂y at plate surface approaches zero, which corresponds exactly to the rigid case.

4. Conclusion

We present a detailed analysis of the effect of damping on the problem of acoustic scattering by flexible plates. We restrict the analysis to a two-dimensional acoustic problem involving plates with semi-infinite and finite chord; both formulations show, as expected, that structural damping influence the radiation of sound.

A model based on the point-reacting compliant semi-infinite plate was developed, with modifications done to include damping and stiffness terms corresponding to a spring-damper foundation. This problem was solved by the Wiener–Hopf technique. The multiplicative factorization required in this type of problem was done numerically and validated by comparison with asymptotic analytical results. Results showed that damping is more efficient in reducing noise radiation at frequencies close to the resonant frequency. These benefits are found for values of the non-dimensional damping coefficient such that ξ ≤ 1. For values of ξ≫1, the behaviour becomes locally over-damped, whereas for values of ξ≪1 the radiation stabilizes in a plateau below the rigid case, which is frequency-dependent.

The comparison of directivity results between finite and semi-infinite compliant plates presented a good agreement. In some cases, there is a small difference in terms of directivity, due to the backscattering of the acoustic waves by the finite-plate leading edge that generate lobes in the figures. The absence of lobes in other cases indicates that the leading edge is not as important for acoustic scattering, and the approach to the finite plate is close enough to that for the semi-infinite plate.

Finally, we have extended the BEM formulation, which solved the problem of acoustic scattering by finite elastic undamped plates, to plates with structural damping. We have observed that damping attenuates the bending waves in the plate; consequently, as bending waves attain the leading edge with amplitudes significantly reduced, there is a reduction in the importance of the leading edge as a secondary source of sound; furthermore, the overall radiated sound also reduces. There is a range of damping coefficients, capable of reducing peaks in the acoustic spectra associated with structural resonance, while maintaining the reduction of scattered sound due to elasticity. This is the main effect for k₀≪1. However, when analysing the other extreme, k₀≫1, we observe that in addition to the reduction of acoustic scattering due to elasticity there is also a slight reduction that is due exclusively to damping. Another conclusion that has been observed for both formulations is that the introduction of damping above a certain threshold (ξ≫Ω²) can cause the behaviour of the elastic plate to approximate that observed for a rigid plate, in terms of sound radiation.

To avoid excessive vibration induced by trailing-edge noise, and also to reduce radiation peaks associated with structural resonance, it is important to consider in aeronautical projects the use of intrinsically damped materials, such as viscoelastic materials. However, prior to that, it is necessary to be able to predict the real benefits capabilities of these solutions, so for this purpose formulations like those presented in this paper are relevant.

Data accessibility

All PWL results presented in this work are accessible as electronic supplementary material of this article.

Authors' contributions

M.M.N. derived the mathematical models for finite plates and generated the corresponding numerical results. A.S.M. derived the mathematical model for semi-infinite plates and generated the corresponding results. A.V.G.C., M.V.D. and W.R.W. revised the models. All authors worked on the development of the numerical code, on the writing of the manuscript and on the analysis of results.

Competing interests

The authors have no competing interests.

Funding

This work was supported by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) (grant no. 88882.180835/2018-01); the national research council CNPq (grant nos 310523/2017-6, 301053/2016-2 and 305277/2015-4).