2.1 Statement of the problem

As shown in figure 1, the problem considered in this work contains an infinite number of periodic and straight (in the $x$ direction) serrations in the $z$ direction. Here, we only consider rigid serrations. For poroelastic edges, the interested reader can refer to recent experimental and theoretical works (Jaworski & Peake Reference Jaworski and Peake2013; Ayton Reference Ayton2016a ; Chong & Dubois Reference Chong and Dubois2016). The profile of the serrations is represented by $\unicode[STIX]{x1D712}(z)$ , which is mainly dependent on two geometrical parameters: the period $\unicode[STIX]{x1D706}$ and the root-to-tip distance $2h_{s}$ . The rest of the semi-infinite surface is assumed to be rigid and infinitely thin. For simplicity, a stationary background flow is assumed for the current work. Nevertheless, the theoretical model developed in this work can be easily extended to cases with uniform background flows by using the Prandtl–Glauert transformation (Huang, Zhong & Stalnov Reference Huang, Zhong and Stalnov2014b ) (a brief introduction is included in appendix A). For the current set-up, an incident wave coming from the far-field region with an incident angle $\unicode[STIX]{x1D703}$ will impinge on the serrated flat plate and further generate scattering waves.

Then, the governing wave equation for the current problem is

where $\unicode[STIX]{x1D719}$ is the acoustic potential. All of the variables are non-dimensionalised using the reference length scale $\unicode[STIX]{x1D706}_{0}^{\ast }$ , which is the wavelength of the incident sound at $340~\text{Hz}$ , the density of air $\unicode[STIX]{x1D70C}_{0}^{\ast }=1.225~\text{kg}~\text{m}^{-3}$ and the speed of sound $c_{0}^{\ast }=340~\text{m}~\text{s}^{-1}$ .

From the acoustic potential $\unicode[STIX]{x1D719}$ , we will be able to obtain the sound pressure $p$ , the particle velocity $\boldsymbol{v}=(u,v,w)$ and the acoustic density $\unicode[STIX]{x1D70C}$ by using the following relations:

where the subscripts $(\cdot )_{i}$ and $(\cdot )_{s}$ denote the incident and scattering waves respectively. Further, the incident wave from the far-field region presumably has a time-harmonic form,

where $A_{i}$ is the normalised amplitude of the incident wave, $k$ is the normalised wavenumber (hence, $k=\unicode[STIX]{x1D714}$ ) and $\unicode[STIX]{x1D703}$ is the incident angle (see figure 1). For a linear wave system, the scattered field will also take the harmonic form

For simplicity, the common factor $\exp (-\text{i}\unicode[STIX]{x1D714}t)$ will be omitted throughout the rest of this paper. Then, $\unicode[STIX]{x1D713}_{s}$ should satisfy the Helmholtz equation,

and further be subject to the following boundary conditions.

1. (1) Rigid plate condition.

   On the rigid plate (where $v=0$ ), the scattered field is equal but opposite to the incident field, resulting in

   (2.6) $$\begin{eqnarray}\left.\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}_{s}\left(x,y,z\right)}{\unicode[STIX]{x2202}y}\right|_{y=0}=-\!\left.\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}_{i}\left(x,y,z\right)}{\unicode[STIX]{x2202}y}\right|_{y=0}=\text{i}k\sin \unicode[STIX]{x1D703}\cdot \text{e}^{-\text{i}kx\cos \unicode[STIX]{x1D703}},\quad \forall x\leqslant \unicode[STIX]{x1D712}(z).\end{eqnarray}$$

2. (2) The scattered field $\unicode[STIX]{x1D713}_{s}$ should be an odd function in $y$ and continuous for $x>\unicode[STIX]{x1D712}(z)$ (Noble Reference Noble1958), hence

   (2.7) $$\begin{eqnarray}\unicode[STIX]{x1D713}_{s}(x,y,z)|_{y=0}=0,\quad \forall x>\unicode[STIX]{x1D712}(z).\end{eqnarray}$$

3. (3) The far-field radiation condition and the edge condition.

   Here, the theoretical model is developed for stationary mean flow cases. The edge condition is imposed at the serrated open ends to ensure a finite velocity potential at the edge and to obtain a unique solution of the problem (Munt Reference Munt1977; Gabard & Astley Reference Gabard and Astley2006). More specifically, $\unicode[STIX]{x1D713}_{s}(x,0,z)\rightarrow \boldsymbol{O}((x-\unicode[STIX]{x1D712}(z))^{1/2})$ as $x\rightarrow \unicode[STIX]{x1D712}(z)^{+}$ for a leading-edge case and $\unicode[STIX]{x1D713}_{s}(x,0,z)\rightarrow \boldsymbol{O}((x-\unicode[STIX]{x1D712}(z))^{3/2})$ as $x\rightarrow \unicode[STIX]{x1D712}(z)^{+}$ for a trailing-edge case. Here, $\unicode[STIX]{x1D712}(z)^{+}$ denotes the positive side of $\unicode[STIX]{x1D712}(z)$ in the $x$ direction. This edge condition will ensure the desired algebraic property for solving the scattered sound field in the axial wavenumber domain. For turbulent gusts and aerofoil interactions, some more subtle studies may be required for the cases with a leading-edge stagnation point (Ayton Reference Ayton2016b ) or a trailing-edge unsteady Kutta condition (Ayton, Gill & Peake Reference Ayton, Gill and Peake2016).

We should emphasise that the primary focus of this model is the prediction of scattered wave fields from leading-edge serrations. The high-frequency self-noise, which is presumably due to the vortex shedding from serrations (Chong et al. Reference Chong, Vathylakis, Joseph and Gruber2013; Vathylakis, Chong & Joseph Reference Vathylakis, Chong and Joseph2015), is not considered in this work.

2.2 The Wiener–Hopf model

The essential concept behind this model is the incorporation of Fourier series expansions into the Wiener–Hopf method. The same concept has been successfully applied in one of our recent papers for sound propagation inside a cylindrical flow duct with rigid splices (Liu et al. Reference Liu, Jiang, Huang and Chen2016). The new difficulty for the current theoretical development comes from the representation of the serrated flat-plate problem in the framework of a Wiener–Hopf model with the desired analytic properties (especially the complex analytical properties on the upper and lower halves of the complex plane for the transformed scattered field from serrations).

The Wiener–Hopf method will enable us to obtain rigorous solutions for the wave equations with a pair of idealised but mixed boundary conditions. Typically, the first step of the Wiener–Hopf method is to transform the wave equations into Wiener–Hopf equations via Fourier transform. The next step is to decompose the associated Wiener–Hopf kernel by using suitable factorisation methods, which should yield decomposition results that are analytic and bounded on the upper- and lower-half complex planes respectively. Then, a closed-form solution will be available by applying Liouville’s theorem. More detailed developments for the current problem are given below.

First, $\unicode[STIX]{x1D713}_{s}$ is periodic in the $z$ direction and, therefore, can be represented as follows by using the Fourier series:

where $(\cdot )^{\unicode[STIX]{x1D705}}$ denotes the $\unicode[STIX]{x1D705}$ th component.

Substitution of (2.8) into (2.5) yields

Then, the conventional Fourier transform is applied (in the $x$ direction) to the scattered acoustic field as follows:

where $\unicode[STIX]{x1D6FC}$ is the axial wavenumber. It is easy to see that $\unicode[STIX]{x1D6FD}(\unicode[STIX]{x1D6FC},y,z)$ is also periodic in the $z$ direction and, therefore, can be represented by the following Fourier series:

The latter relation assumes that the integration and the summation are commutative.

From (2.11), equation (2.9) can be transformed to

To simplify the following formulations, we define ${\unicode[STIX]{x1D6FE}^{\unicode[STIX]{x1D705}}}^{2}\triangleq \unicode[STIX]{x1D6FC}^{2}-\hat{k^{\unicode[STIX]{x1D705}}}^{2}$ , with $\hat{k^{\unicode[STIX]{x1D705}}}^{2}\triangleq k^{2}-(\unicode[STIX]{x1D705}2\unicode[STIX]{x03C0}/\unicode[STIX]{x1D706})^{2}$ . To satisfy the radiation boundary condition, the solutions of (2.12) should have the following generic form:

where $A^{\unicode[STIX]{x1D705}}(\unicode[STIX]{x1D6FC})$ is the corresponding amplitude of the $\unicode[STIX]{x1D705}$ th component and $\unicode[STIX]{x1D6FE}^{\unicode[STIX]{x1D705}}$ is defined as a fractional power that will lead to a number of branch cuts joining the branch points $\pm \hat{k^{\unicode[STIX]{x1D705}}}$ to infinity. It is worthwhile to mention that $k$ is usually complex (with an additional small imaginary part, which will be set to zero in the final solution), so that $+\hat{k^{\unicode[STIX]{x1D705}}}$ and $-\hat{k^{\unicode[STIX]{x1D705}}}$ lie in the first and third quadrants of the complex $\unicode[STIX]{x1D6FC}$ -plane respectively (Munt Reference Munt1977). The setting of a small imaginary part of $k$ helps to ensure the causality of the final solutions, and the same technique has been adopted in many previous works (Rienstra Reference Rienstra1984; Gabard & Astley Reference Gabard and Astley2006; Liu et al. Reference Liu, Jiang, Huang and Chen2016).

From (2.13), we further have the following relation at $y=0^{+}$ :

where $(\cdot )^{\prime }$ represents $\text{d}/\text{d}y$ .

Now, we turn to the key part of this theoretical development. First, we decompose $\unicode[STIX]{x1D6FD}$ into the sum of the following two parts which are analytic on the two (yet unknown) half-planes:

In the whole theoretical development, the above equation is the most essential step, which ensures that the regularity of $\unicode[STIX]{x1D6FD}_{\pm }$ is independent of $\unicode[STIX]{x1D712}(z)$ and thus is consistent with the previous definitions in the classical flat-plate problem (Noble Reference Noble1958). In addition, the above definition actually suggests that $\unicode[STIX]{x1D712}(z)$ is a single-valued function, which should be satisfied for most representative set-ups.

Figure 2.

Argand diagram of the two complex half-planes $R_{\pm }$ and the overlapped strip. The symbol $\times$ denotes the pole at $k\cos \unicode[STIX]{x1D703}$ . For each $\unicode[STIX]{x1D705}$ , two branch cuts join the branch points $\pm \hat{k^{\unicode[STIX]{x1D705}}}$ to infinity through the two corresponding half-planes. Since $k$ is usually complex (Munt Reference Munt1977),  $\pm \hat{k^{\unicode[STIX]{x1D705}}}$ and the corresponding branch cuts lie in the first and third quadrants of the complex plane (for clarity of this figure, the branch cuts are not shown here). The imaginary part is set to zero in the final solution and, hence, the integral path moves back to the horizontal axis with $\unicode[STIX]{x1D700}\rightarrow 0$ , except for a slight distortion (Rienstra Reference Rienstra1984; Gabard & Astley Reference Gabard and Astley2006) around the origin to avoid any possible singular point.

The decomposed $\unicode[STIX]{x1D6FD}_{\pm }$ are regular on the two overlapped half-planes, which are shown in figure 2. It is easy to see that $\exp (\text{i}\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D712})$ , $\unicode[STIX]{x1D6FD}_{+}$ and $\unicode[STIX]{x1D6FD}_{-}$ in (2.15) are periodic in the $z$ direction with the same period $\unicode[STIX]{x1D706}$ . By expanding $\exp (\text{i}\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D712})$ and $\unicode[STIX]{x1D6FD}_{\pm }$ into Fourier series, we immediately have

where

We should mention that $\unicode[STIX]{x1D6FC}$ in (2.17) may contain a small but non-zero imaginary part to ensure causality (details can be found in Gabard & Astley (Reference Gabard and Astley2006), Appendix A. Numerical methods). However, a Fourier series expansion is still available as long as $\unicode[STIX]{x1D712}(z)$ is periodic.

From (2.18), we can further define the partial derivatives with respect to $y$ for the $\unicode[STIX]{x1D705}$ th component:

From the hard-wall boundary condition (2.6), we can analytically represent $\unicode[STIX]{x1D6FD}_{-}^{\prime }(\unicode[STIX]{x1D6FC},0^{+},z)$ as follows:

with the Fourier series coefficients

which mathematically represents the effect of different profile shapes of the serrations on the scattering of the incident waves. By comparing (2.19) and (2.20), it is easy to see that

Next, by equating the terms of the same exponential orders in (2.16) and (2.11), we have the following relation:

On substituting the boundary conditions at $y=0^{+}$ , i.e. equations (2.7) and (2.22), into (2.23), we have

where $\unicode[STIX]{x1D6FD}_{-}(\unicode[STIX]{x1D6FC},0^{+})$ is always zero due to (2.7) and is thus omitted in (2.24). The above two equations should illuminate the mechanisms of the serrations in noise control. The right-hand side of (2.25) is actually a convolution of $\unicode[STIX]{x1D701}$ , which is only dependent on $\unicode[STIX]{x1D712}(z)$ . Mathematically, in the complex $\unicode[STIX]{x1D6FC}$ -plane, the scattered field is a convolution from the serrations. For brevity, the input variables $(\unicode[STIX]{x1D6FC},0^{+})$ will be omitted in the rest of this paper when no confusion will arise.

Substitution of (2.24) and (2.25) into (2.14) results in

By repeating (2.26) for all possible values of $\unicode[STIX]{x1D705}$ , we have a system of infinite linear equations which can be represented using the following matrix form:

where

where $(\cdot )^{\text{T}}$ denotes the transpose operation, $\unicode[STIX]{x1D701}^{\unicode[STIX]{x1D705}}$ is defined in (2.17) and $\unicode[STIX]{x1D710}^{\unicode[STIX]{x1D705}}$ is defined in (2.21). In a practical implementation, the above infinite Wiener–Hopf matrix system is truncated to a finite dimension (Veitch & Peake Reference Veitch and Peake2008). The author has conducted convergence tests by examining the order of Fourier series $\unicode[STIX]{x1D705}$ in between $\pm 21$ , and found that values smaller than 21 could be sufficient as long as the corresponding finite summation of $\unicode[STIX]{x1D712}(z)$ converges satisfactorily. Hence, all of the results in the next section are calculated using $\unicode[STIX]{x1D705}\in [-10,10]$ .

After this truncation, it is straightforward to define a matrix inverse and the poles of $\unicode[STIX]{x1D729}$ . First, to achieve the Wiener–Hopf equation, equation (2.27) can be further simplified to

As suggested by (2.16), the truncated matrices $\unicode[STIX]{x1D729}$ and  $\unicode[STIX]{x1D729}^{-1}$ may contain entries that are polynomials in $\unicode[STIX]{x1D6FC}$ . Mathematically, the poles of a polynomial matrix are the roots of $\det \mathfrak{D}(\unicode[STIX]{x1D6FC})=0$ , where $\mathfrak{D}$ is the denominator of the coprime matrix fraction descriptions of the polynomial matrix. The poles can also be simply calculated as the union of the poles for each entry of a matrix. This simple treatment has been adopted in some commercial software such as MATLAB^® and is also used in this work. On the other hand, for either sawtooth- or sinusoidal-shaped serrations, it is easy to prove that each entry of the corresponding $\unicode[STIX]{x1D729}$ (or $\unicode[STIX]{x1D729}^{-1}$ ) is analytic because the real and imaginary parts of $\unicode[STIX]{x1D729}$ (and $\unicode[STIX]{x1D729}^{-1}$ ) satisfy the Cauchy–Riemann equations. For other possible profiles of the serrations, the corresponding $\unicode[STIX]{x1D729}$ may still have a number of unknown poles and zeros (i.e. poles for $\unicode[STIX]{x1D729}^{-1}$ ) at $\unicode[STIX]{x1D6FC}_{j}$ , and, therefore, neither $\unicode[STIX]{x1D729}$ nor $\unicode[STIX]{x1D729}^{-1}$ satisfies the analytic condition, which is most essential in the Wiener–Hopf method. To cancel out these poles, the term of $[\unicode[STIX]{x1D6F1}(\unicode[STIX]{x1D6FC}-\unicode[STIX]{x1D6FC}_{j})]^{-1}\unicode[STIX]{x1D6F1}(\unicode[STIX]{x1D6FC}-\unicode[STIX]{x1D6FC}_{j})$ is multiplied to both sides of (2.33), resulting in

where $\unicode[STIX]{x1D6F1}$ represents the product of a sequence of $(\unicode[STIX]{x1D6FC}-\unicode[STIX]{x1D6FC}_{j})$ . By the above multiplication, the poles of $\unicode[STIX]{x1D729}$ and $\unicode[STIX]{x1D729}^{-1}$ can be cancelled. Then, $\hat{\unicode[STIX]{x1D729}}$ and $\hat{\unicode[STIX]{x1D729}}^{-1}$ in (2.34) become analytic. It will become clear that the above multiplication operation is only for the rigour of the whole theoretical development and is not required in the actual implementation, since the term $[\unicode[STIX]{x1D6F1}(\unicode[STIX]{x1D6FC}-\unicode[STIX]{x1D6FC}_{j})]^{-1}\unicode[STIX]{x1D6F1}(\unicode[STIX]{x1D6FC}-\unicode[STIX]{x1D6FC}_{j})$ itself equals 1.

It can be seen that (2.34) still contains two unknowns, $\unicode[STIX]{x1D6F7}_{+}^{\prime }$ and $\unicode[STIX]{x1D6F7}_{-}$ . To solve this equation, the Wiener–Hopf method should be considered. First, $\unicode[STIX]{x1D6E4}$ is a diagonal matrix and its multiplicative factorisation satisfying $\unicode[STIX]{x1D6E4}=\unicode[STIX]{x1D6E4}_{+}\unicode[STIX]{x1D6E4}_{-}$ can be easily obtained (see (2.28)). Then,

By rearranging terms and cancelling out the pole at $k\cos \unicode[STIX]{x1D703}$ , we finally achieve the following Wiener–Hopf equation:

2.3 The analytical solutions

The left- and right-hand sides of (2.36) should be regular on the positive and negative half-planes $R_{\pm }$ respectively, where $R_{+}$ is the positive half-plane above all possible $-\hat{k^{\unicode[STIX]{x1D705}}}$ , and $R_{-}$ is the negative half-plane below all possible $\hat{k^{\unicode[STIX]{x1D705}}}$ and $k\cos \unicode[STIX]{x1D703}$ (see figure 2). Both $R_{+}$ and $R_{-}$ will share an overlapped region. As a result, according to analytic continuation in the complex analysis theory, $\boldsymbol{E}(\unicode[STIX]{x1D6FC})$ is an entire function that is regular on the whole complex $\unicode[STIX]{x1D6FC}$ -plane. By considering the classical case with a flat rigid plate, it can be shown that $\boldsymbol{E}(\unicode[STIX]{x1D6FC})\equiv 0$ as $|\unicode[STIX]{x1D6FC}|\rightarrow \infty$ . In addition, it is very easy to see that the edge condition will also force $\boldsymbol{E}(\unicode[STIX]{x1D6FC})\equiv 0$ as $|\unicode[STIX]{x1D6FC}|\rightarrow \infty$ . Otherwise, the finiteness of the scattering wave at the edge (where $\unicode[STIX]{x1D712}(z)=0$ ) will be violated. However, the problem becomes more complicated for serrated ends with $\unicode[STIX]{x1D712}(z)\neq 0$ . With no loss of generality, we assume $\unicode[STIX]{x1D712}(z)\geqslant 0$ . Then, it is easy to prove that each element in $\hat{\unicode[STIX]{x1D729}}$ ,

is analytic but unbounded in the half-plane $R_{-}$ (when the imaginary part of $\unicode[STIX]{x1D6FC}$ approaches $-\infty$ ). To apply Liouville’s theorem (for an analytic and bounded function), we should modify (2.34) to

which results in

It should be straightforward to prove that the left- and right-hand sides of (2.39) are analytic and bounded in $R_{-}$ and $R_{+}$ respectively. Then, we can achieve the final solution by the following steps. First, by applying analytic continuation, a single function $\boldsymbol{E}(\unicode[STIX]{x1D6FC})$ that is regular on the whole complex $\unicode[STIX]{x1D6FC}$ -plane can be defined from (2.39). Second, by Liouville’s theorem, this bounded entire function must be a constant. Then, by studying the behaviour of $\boldsymbol{E}(\unicode[STIX]{x1D6FC})$ as $|\unicode[STIX]{x1D6FC}|$ approaches $\infty$ , we have $\boldsymbol{E}(\unicode[STIX]{x1D6FC})\equiv 0$ . Finally, by substituting $\boldsymbol{E}(\unicode[STIX]{x1D6FC})\equiv 0$ into (2.39), we have

All of the right-hand side terms of (2.40) are known, provided that the profiles of the serrations and other parameters related to the incident waves are already known. Therefore, we immediately have the solution of $\unicode[STIX]{x1D6F7}_{-}(\unicode[STIX]{x1D6FC})$ , and the corresponding solutions for $\unicode[STIX]{x1D6FD}_{-}^{\unicode[STIX]{x1D705}}(\unicode[STIX]{x1D6FC},0^{+})$ using (2.30), $\unicode[STIX]{x1D6FD}^{\unicode[STIX]{x1D705}}(\unicode[STIX]{x1D6FC},0^{+})$ using (2.24) and $\unicode[STIX]{x1D6FD}^{\unicode[STIX]{x1D705}}(\unicode[STIX]{x1D6FC},y)$ using (2.13).

Finally, we have the analytical solution of the scattering waves as follows:

It is worthwhile to mention that the theoretical model requires Fourier series of infinite order and the numerical implementation can only use truncated orders. For the current validation cases (see figures 3 to 7), a study of the convergence of Fourier series orders has been conducted, and the results showed that a truncation with $\unicode[STIX]{x1D705}$ in between $\pm 10$ is already sufficient (at least for the two types of serrations considered in the current work). In addition, we found that it is appropriate to use an approximation with a 1024-point discretised $\unicode[STIX]{x1D6FC}\in [-4\unicode[STIX]{x1D714},4\unicode[STIX]{x1D714}]$ in the calculation of (2.41). We can obtain all possible poles and zeros of $\unicode[STIX]{x1D6F7}_{-}(\unicode[STIX]{x1D6FC})$ from (2.40), such as $k\cos (\unicode[STIX]{x1D703})$ and $\hat{k}^{\unicode[STIX]{x1D705}}$ (with $-k\unicode[STIX]{x1D706}/2\unicode[STIX]{x03C0}<\unicode[STIX]{x1D705}<k\unicode[STIX]{x1D706}/2\unicode[STIX]{x03C0}$ ). Accordingly, the actual integral path in (2.41) is slightly deformed around the origin to avoid passing through these poles and zeros (see figure 2). More details of this technique can be found in Rienstra (Reference Rienstra1984), Gabard & Astley (Reference Gabard and Astley2006) and Liu et al. (Reference Liu, Jiang, Huang and Chen2016).

The above model is seemingly elusive but is actually a straight extension from the classical Wiener–Hopf model for a 2D flat plate (Noble Reference Noble1958; Wright Reference Wright2008). The additional scattering effects due to the serrations are described in (2.26) in a form of convolution. As a summary, the whole model can be straightforwardly implemented in the following steps.

1. (1) Prepare the integral path of $\unicode[STIX]{x1D6FC}$ slightly distorted around the origin.

2. (2) Factorise the kernel across $\unicode[STIX]{x1D6FC}$ by using (2.28) and obtain the resultant $\unicode[STIX]{x1D6E4}_{\pm }$ .

3. (3) Given the profile of serrations, prepare the matrices of $\unicode[STIX]{x1D730}$ and $\unicode[STIX]{x1D729}$ by solving (2.32) and (2.29) respectively.

4. (4) Calculate $\unicode[STIX]{x1D6FD}^{\unicode[STIX]{x1D705}}(\unicode[STIX]{x1D6FC},0^{+})$ from $\unicode[STIX]{x1D6F7}_{-}(\unicode[STIX]{x1D6FC})$ after solving (2.40).

5. (5) Finally, obtain the whole scattered field by summing up $\unicode[STIX]{x1D6FD}^{\unicode[STIX]{x1D705}}(\unicode[STIX]{x1D6FC},0^{+})$ and calculating its inverse Fourier transform by using (2.41).

3.1 Validations

The theoretical development of the abovementioned model has been the most challenging and time-consuming part of this work. The implementation itself is actually straightforward. The whole model has been implemented by using MATLAB^®. Although the model is seemingly complicated, the very fundamental part is still similar to the classical rigid flat-plate problem (Noble Reference Noble1958; Wright Reference Wright2008). The main complexity in the implementation comes from the Fourier decomposition and the matrix Wiener–Hopf factorisation. If the reader is already familiar with the implementation of the rigid flat-plate scattering model, it should be quite straightforward to extend that implementation to the current one based on the abovementioned formulations.

Next, we turn to the validation of the proposed theoretical model. The in-house high-order computational aeroacoustic solver (Huang et al. Reference Huang, Chen, Ma and Zhang2008b ; Peers & Huang Reference Peers and Huang2013) requires a considerable time to construct 3D structured grids for the serrated plates. Hence, the finite element method (FEM) solver of the commercial software COMSOL^® 5.1 is used for numerical simulations. The dimensional computational domain is $7~\text{m}\times 7~\text{m}\times 0.5~\text{m}$ , enclosing a complete serration in the $z$ direction. An infinite span is imposed by using periodic boundary conditions on the two lateral $x$ – $y$ faces of the computational domain. The inlet boundary condition allows the introduction of the incident wave into the computation domain, which is intentionally designed in such a way (as in figure 3) to simplify the set-up of the incident wave and minimise any possible reflection from the computational domain. Furthermore, the plane wave radiation boundary condition is used to reduce possible sound wave reflections from the computational domain. Different refinement levels of the mesh have been tested to ensure a proper convergence of the simulation. The whole computation was performed on a desktop computer with an Intel core i7-4790 @ $3.60~\text{GHz}$ processor and $24~\text{GB}$ memory. The computational set-ups are first verified by using the 2D cases with a straight edge. The corresponding theoretical predictions are from the well-known 2D diffraction model (Noble Reference Noble1958; Wright Reference Wright2008). Figure 4 shows the sound pressure level (SPL, $20\log _{10}\overline{p(t)}/p_{0}$ , where $\overline{(\cdot )}$ is the time root mean square and $p_{0}=20~\unicode[STIX]{x03BC}\text{Pa}$ ) results at a radial distance $r\triangleq \sqrt{x^{2}+y^{2}}=1~\text{m}$ with $f=500$ , $1000$ and $1500~\text{Hz}$ . It can be seen that the computational results and the theoretical predictions are almost identical, which verifies the computational set-up. For 3D computations of the cases with serrated edges, the maximum number of grid cells currently allowed on our desktop computer is approximately $2.5\times 10^{6}$ , which ensures a satisfactory points-per-wavelength value only for incident sound waves with frequencies $f<1000~\text{Hz}$ . The generalised minimal residual method (GMRES) solver with an iterative method is adopted and the corresponding computation time for each set-up is approximately $5000~\text{s}$ .

Figure 3.

The computational domain and the boundary conditions used in the FEM commercial solver.

Figure 4.

Comparisons of the SPL results for the total sound field at $r=1~\text{m}$ for the straight edge set-up. The results are from (– –) the numerical FEM solver and (–) the theoretical model (Noble Reference Noble1958; Wright Reference Wright2008) at (a) $500~\text{Hz}$ ( $k=9.24$ ), (b) $1000~\text{Hz}$ ( $k=18.48$ ) and (c) $1500~\text{Hz}$ ( $k=27.72$ ).

Figure 5.

Two representative profiles $\unicode[STIX]{x1D712}(z)$ are considered in this work. One is sinusoidal-shaped and the other is sawtooth-shaped. The circles denote the three representative cross-sections passing through the tip, medial and root positions of the sinusoidal-shaped serrations.

Figure 6.

The near-field results from the theoretical model, where the serrations are sinusoidal-shaped with $\unicode[STIX]{x1D706}=0.5$ and $h_{s}=0.3$ . The incident wave of $f=1000~\text{Hz}$ is at $\unicode[STIX]{x1D703}=45^{\circ }$ . The instantaneous sound pressure (a,c,e) and the SPL results (b,d,f) are collected at (a,b) the tip cross-section, (c,d) the medial cross-section and (e,f) the root cross-section. The black straight line in each panel denotes the silhouette of the serrated flat plate at the corresponding cross-section.

Figure 7.

The near-field results from the theoretical model, where the serrations are sawtooth-shaped. Other set-ups are the same as those in figure 6.

Figure 5 shows the two representative serrations with either a sinusoidal- or a sawtooth-shaped profile in the numerical validations, for which various test conditions have been investigated and similar conclusions have been reached. For brevity, we primarily show numerical results for an incident wave of $1000~\text{Hz}$ at $\unicode[STIX]{x1D703}=45^{\circ }$ below. In this numerical set-up, the two profiles have $h_{s}/\unicode[STIX]{x1D706}=0.6$ and $\unicode[STIX]{x1D706}=0.5$ . In addition, the infinitely thin flat plate in the theoretical model has to be replaced by a very thin flat plate with a small thickness of $0.01$ in the numerical set-ups. Figure 5 also shows three cross-sections, where results from the theoretical model and the numerical solver will be extensively examined in the following. For simplicity, these three cross-sections will be referred to as the tip, medial and root cross-sections respectively.

First, to demonstrate the capability of the proposed theoretical model, figures 6 and 7 show the near-field results for the instantaneous sound pressure fields $p(t)$ and SPLs at the three representative $x$ – $y$ cross-sections. The SPL results in figure 6 clearly show that the serrated open ends affect the local sound pressure fields to some extent. In particular, the SPL results at the tip and root cross-sections are quite different from the SPL result at the medial cross-section, although the corresponding instantaneous sound pressure fields are still seemingly similar. The same observation can be made from the corresponding theoretical results for the sawtooth-shaped serrations (see figure 7).

Figure 8.

The near-field results from the commercial FEM solver. The serrations are sawtooth-shaped. Other set-ups are the same as those in figure 7.

The instantaneous pressure fields and the SPL fields in figures 6 and 7 are very similar, which suggests that the noise control performance of the sinusoidal- and sawtooth-shaped serrations is almost the same for the current geometrical set-ups at the frequency $f=1000~\text{Hz}$ . The following parametric study will show that the noise control performance becomes more different at higher frequencies (figure 13) and various dimensions (see figure 14). To ensure a satisfactory points-per-wavelength value, the current computation still adopts an incident sound wave frequency of $f=1000~\text{Hz}$ . To validate the proposed model, figure 8 shows the computational results from the FEM solver. The sawtooth-shaped serrations are used in the computation and other test set-ups are the same as those in figure 7. It can be seen that the theoretical predictions shown in figure 7 and the numerical results shown in figure 8 have almost identical patterns at the three representative cross-sections. The theoretical model, however, is quite efficient and can produce the above results in seconds.

Figure 9.

Comparisons of the SPL results at $r=1$ , where (– –) denotes the results from the numerical solver and (–) denotes the theoretical predictions. (a,c,e) Results for the set-up with the sinusoidal-shaped serrations and (b,d,f) for sawtooth-shaped serrations, at (a,b) the tip cross-section, (c,d) the middle cross-section and (e,f) the root cross-section. Other set-ups are the same as those in figures 6 and 7.

To further quantitate the differences, figure 9 shows the SPL directivity patterns from the theoretical model and the FEM solver along the observation circle with $r\triangleq \sqrt{x^{2}+y^{2}}=1$ . The agreement between the two sets of results is quite good in most angular regions, with the largest difference of less than $1~\text{dB}$ in the angular region with $\unicode[STIX]{x1D703}$ between $180^{\circ }$ and $240^{\circ }$ in the tip and root cross-sections. However, the difference is relatively large (up to approximately $3~\text{dB}$ in the angular region close to $180^{\circ }$ ) for the middle cross-section. We believe that two numerical issues cause the slight differences. First, the commercial software adopts an FEM solver that requires a large points-per-wavelength value to ensure satisfactory simulations of sound wave propagation and scattering. Second, for the convenience of generating computational meshes, the flat plat has a very small but finite thickness (set to 0.01) in the current numerical validation cases, which might affect the relatively weak scattered sound fields in the third quadrant. A better computer with more memory could easily address the above two issues.

Figure 10.

The scattering cross-sections at (a) the tip, (b) the medial and (c) the root cross-sections, (–) without serrations, (– –) with sawtooth-shaped serrations and (○) with sinusoidal-shaped serrations. Other set-ups are the same as those used in figures 6 and 7.

3.3 Parametric studies

The overall computational cost for the theoretical model on a normal desktop computer with an Intel^® Core i7 processor is significantly lower than the cost of the corresponding numerical validation cases run by the commercial finite element software. Specifically, our model is able to generate scattering cross-sections for various set-ups in seconds. In contrast, the same case set-up solved by the commercial finite element software with the unstructured meshes of $2.5\times 10^{6}$ gridpoints usually requires a few dozens of minutes. Overall, the proposed theoretical model is so efficient that we can use it to rapidly perform parametric studies. A couple of test cases are considered below to demonstrate this capability.

First, all of the above test cases were prepared for an incident wave at $\unicode[STIX]{x1D703}=45^{\circ }$ . Now, let us consider the possible noise control effect at different incident angles. Here, we modify the incident angle and examine the corresponding so-called total scattering cross-section:

where $S$ is the far-field surface of interest (e.g. the contour surface in figure 11). As shown in figures 10 and 11, the serrations mainly affect the scattering cross-sections in the radiation angular region between $0^{\circ }$ and $90^{\circ }$ . Hence, here we choose $S$ with the related $\unicode[STIX]{x1D703}\in [0,90]^{\circ }$ .

Figure 13.

The total scattering cross-sections with respect to (a,b) different incident angles with (a) $f=1000~\text{Hz}$ and (b) $f=3500~\text{Hz}$ , and (c) a range of incident frequencies (with $\unicode[STIX]{x1D703}=45^{\circ }$ ), ( $--$ ) with a straight edge, (–) with sinusoidal-shaped serrations and ( $\Box$ ) with sawtooth-shaped serrations. Other set-ups are the same as those used in figures 6 and 7.

Figure 14.

The contours of the total scattering cross-sections with respect to different geometrical values of $\unicode[STIX]{x1D706}$ and $h_{s}$ , (a,c) with sinusoidal-shaped serrations and (b,d) with sawtooth-shaped serrations. (a,b) Results at $f=1000~\text{Hz}$ and (c,d) results at $f=3500~\text{Hz}$ . Other set-ups are the same as those used in figures 6 and 7.

Figure 13(a) shows the theoretical variations of $\unicode[STIX]{x1D70E}_{total}(\unicode[STIX]{x1D703})$ with respect to $\unicode[STIX]{x1D703}$ , with all other parameters being the same as the previous test cases, i.e. $h_{s}=0.3,\unicode[STIX]{x1D706}=0.5,f=1000~\text{Hz}$ . It becomes immediately clear that the serrations, either sawtooth- or sinusoidal-shaped, are effective only for some incident waves with incident angles in between a certain range. In particular, for the current set-up, this effective range of incident angles is between $30^{\circ }$ and $72^{\circ }$ . The maximum reduction relative to the plate with no serrations is approximately $10~\text{dB}$ when $\unicode[STIX]{x1D703}$ is around $53^{\circ }$ . Another interesting finding from figure 13(a) is that the sinusoidal-shaped serrations slightly outperform the sawtooth-shaped serrations. This observation seems to contradict a previous work by Howe (Reference Howe1991b ), whose conclusion, however, was made for high-frequency cases. Hence, we study a new case at $f=3500~\text{Hz}$ in figure 13(b), which then shows the same conclusion as Howe’s work (Howe Reference Howe1991b ), i.e. sawtooth-shaped serrations generally perform better than sinusoidal-shaped serrations, and the noise control performance improves along with an increase in wave frequency. If we further examine figure 13(c), it becomes clear that sawtooth-shaped serrations are usually more effective than sinusoidal-shaped serrations when the incident frequency is higher than $1600~\text{Hz}$ . The corresponding wavelength is smaller than $h_{s}$ (currently 0.3), which suggests two points. First, a serration works more effectively when its characteristic length is larger than the wavelength of the incident wave. Second, it should be more appropriate to take the half root-to-tip distance $h_{s}$ than $\unicode[STIX]{x1D706}$ to be the characteristic length. In addition, figure 13(c) shows that for certain set-ups (e.g. $f\in [2000,2400]~\text{Hz}$ and $[4200,4600]~\text{Hz}$ with $\unicode[STIX]{x1D703}=45^{\circ }$ ), instead of a noise reduction, the two serrations actually excite more noise scattering than that from a straight edge. Physically, the corresponding cases must contain constructive scattering waves instead of destructive cancellations.

Finally, we examine the total scattering cross-sections with respect to different possible values of the geometrical parameters $h_{s}$ and $\unicode[STIX]{x1D706}$ at $f=1000~\text{Hz}$ and $3500~\text{Hz}$ . The other parameters are the same as those in figures 6 and 7. This study can give the optimal geometrical design of the serrations at the given working conditions. Figure 14 shows the normalised results (by the maximum value of $\unicode[STIX]{x1D70E}_{total}(\unicode[STIX]{x1D703})$ from the cases with sinusoidal-shaped serrations), which are consistent with all of the abovementioned observations: (1) the sawtooth-shaped serrations are generally more effective in terms of the suppression of the scattering cross-sections, (2) $h_{s}$ is the key geometrical parameter and (3) the serrations are more effective when their characteristic length is larger than the wavelength of the incident waves.