2 The current theoretical model

This work considers sound scattering in a lined duct that contains a uniform background flow. Boundary layer and vorticity are omitted to enable the theoretical development in the framework of the Wiener–Hopf method. By assuming much larger time scale and length scale of fluid dynamics than the corresponding scales of acoustic waves, the sound wave propagation and scattering should satisfy the convective wave equation, such that

where $\unicode[STIX]{x1D719}$ is the acoustic potential, $M$ is the Mach number of the uniform background flow and $\unicode[STIX]{x1D709}_{s}=\unicode[STIX]{x1D709}_{s}(x,\unicode[STIX]{x1D703},t)$ is the kinematic displacement of the hypothetical infinitely thin vortex sheets developing along the lining surfaces (Rienstra Reference Rienstra2007). In the following discussion, all variables are non-dimensionalised using the mean flow density $\unicode[STIX]{x1D70C}_{0}$ , the speed of sound $c_{0}$ and the duct radius $R_{0}$ . As shown in figure 2, we have a circular duct with a unit outer radius, a hard wall along $x<0$ and a lined wall for $x>0$ with azimuthally non-uniform impedance $Z(\unicode[STIX]{x1D714},\unicode[STIX]{x1D703})$ (normalised by $\unicode[STIX]{x1D70C}_{0}c_{0}$ ). In our previous work (Liu et al. Reference Liu, Jiang, Huang and Chen2015), $Z(\unicode[STIX]{x1D714},\unicode[STIX]{x1D703})$ is infinite for the rigid splices and is a finite constant for the lined wall. Therefore, the associated distribution along the azimuthal direction was a rectangular Pi function, and it was possible to adopt a manipulation to simplify the relevant matrix kernel factorisation (Liu et al. Reference Liu, Jiang, Huang and Chen2015). In contrast, a more generic set-up with $Z(\unicode[STIX]{x1D714},\unicode[STIX]{x1D703})$ being any arbitrary function of $\unicode[STIX]{x1D703}$ is considered in this paper, which calls for a more rigorous mathematical treatment.

First, we build up the mathematical model and assume the acoustic field with angular frequency $\unicode[STIX]{x1D714}$ has the following relations:

where $(\cdot )_{i}$ and $(\cdot )_{s}$ represent the incident and scattered wave fields, respectively. From the generic solution of duct acoustics, the tonal and axisymmetric incident field inside a cylindrical duct has the form:

where $m$ is the circumferential mode number and $\text{J}_{m}$ is the $m$ th-order Bessel function of the first kind; $\unicode[STIX]{x1D6FC}_{mn}$ is the $n$ th solution of the characteristic equation $\text{J}_{m}^{\prime }(x)=0$ for the hard wall; and $\unicode[STIX]{x1D707}_{mn}^{+}=(\sqrt{1-(1-M^{2})\unicode[STIX]{x1D6FC}_{mn}^{2}/\unicode[STIX]{x1D714}^{2}}-M)/(1-M^{2})$ is the downstream-directing axial wavenumber. Now consider the total sound field $\unicode[STIX]{x1D719}(x,r,\unicode[STIX]{x1D703},t)=\unicode[STIX]{x1D719}_{i}(x,r,\unicode[STIX]{x1D703},t)+\unicode[STIX]{x1D719}_{s}(x,r,\unicode[STIX]{x1D703},t)=\unicode[STIX]{x1D713}(x,r,\unicode[STIX]{x1D703})\text{e}^{\text{i}m\unicode[STIX]{x1D703}-\text{i}\unicode[STIX]{x1D714}t}$ . The scattered field and the corresponding (conceived) vortex sheets from the hard–soft interface (Rienstra Reference Rienstra2007; Jiang & Huang Reference Jiang and Huang2017a ; Jiang, Lau & Huang Reference Jiang, Lau and Huang2017b ) take the harmonic forms:

For brevity, the harmonic factor, $\text{exp}(\text{i}m\unicode[STIX]{x1D703}-\text{i}\unicode[STIX]{x1D714}t)$ , will be dropped in the remainder of this paper when no confusion will arise. In addition, the problem of interest mainly contains the following boundary conditions:

1. (i) rigid wall

   (2.5a,b ) $$\begin{eqnarray}\left.\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}_{s}}{\unicode[STIX]{x2202}r}(x,r,\unicode[STIX]{x1D703})\right|_{r=1}=0,\quad \forall x<0;\quad \left.\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}_{i}}{\unicode[STIX]{x2202}r}(x,r,\unicode[STIX]{x1D703})\right|_{r=1}=0,\quad \forall x;\end{eqnarray}$$

2. (ii) kinetic displacement

   (2.6) $$\begin{eqnarray}\left(-\text{i}\unicode[STIX]{x1D714}+M\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x}\right)\unicode[STIX]{x1D709}(x,\unicode[STIX]{x1D703})=\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}{\unicode[STIX]{x2202}r}(x,\unicode[STIX]{x1D709}^{-},\unicode[STIX]{x1D703})=\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}{\unicode[STIX]{x2202}r}(x,1,\unicode[STIX]{x1D703})=\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}_{s}}{\unicode[STIX]{x2202}r}(x,1,\unicode[STIX]{x1D703}),\quad \forall x>0;\end{eqnarray}$$

3. (iii) pressure continuity

   (2.7) $$\begin{eqnarray}\left(-\text{i}\unicode[STIX]{x1D714}+M\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x}\right)[\unicode[STIX]{x1D713}_{s}(x,\unicode[STIX]{x1D709}^{-},\unicode[STIX]{x1D703})+\unicode[STIX]{x1D713}_{i}(x,\unicode[STIX]{x1D709}^{-})]=-\text{i}\unicode[STIX]{x1D714}\unicode[STIX]{x1D713}(x,\unicode[STIX]{x1D709}^{+},\unicode[STIX]{x1D703}),\quad \forall x>0;\end{eqnarray}$$

4. (iv) lined wall

   (2.8) $$\begin{eqnarray}p(x,\unicode[STIX]{x1D709}^{+},\unicode[STIX]{x1D703})\text{e}^{-\text{i}m\unicode[STIX]{x1D703}}=-\text{i}\unicode[STIX]{x1D714}Z(\unicode[STIX]{x1D714},\unicode[STIX]{x1D703})\unicode[STIX]{x1D709}(x,\unicode[STIX]{x1D703}),\quad \forall x>0.\end{eqnarray}$$

Here $(\cdot )^{\pm }$ denotes the upper and lower sides of $(\cdot )$ , respectively. In addition, combining (2.6) and (2.8) together results in the well-known Ingard impedance boundary condition, but the current equations enable the following development of the Wiener–Hopf model in a straightforward way. Further, the description of edge condition is omitted for brevity and a detailed discussion of this topic can be found in Liu et al. (Reference Liu, Jiang, Huang and Chen2015). Next, taking the axial Fourier transform, we have

for the scattered field, where $u$ is the normalised wavenumber in the axial $x$ -direction. Since $\unicode[STIX]{x1D6FD}$ and $\unicode[STIX]{x1D713}_{s}$ are $2\unicode[STIX]{x03C0}$ -periodic in the $\unicode[STIX]{x1D703}$ -direction, both can be expanded into

where $(\cdot )^{\unicode[STIX]{x1D705}}$ denotes the $\unicode[STIX]{x1D705}$ th component, which satisfies the Bessel equation:

where $\unicode[STIX]{x1D706}^{2}=(1-uM)^{2}-u^{2}$ , $\unicode[STIX]{x1D706}=\unicode[STIX]{x1D706}^{+}\unicode[STIX]{x1D706}^{-}$ , and then $\unicode[STIX]{x1D706}^{\pm }=(1-u(M\pm 1))^{1/2}$ . The two branch cuts are two separate paths joining the branch points $u_{0}^{\pm }=1/(M\pm 1)$ to $\pm \infty$ through the two half-planes $R_{\mp }$ as shown in figure 3(a), respectively, where $R_{\pm }$ are defined as: $\pm \text{Im}(u-u_{0}^{\pm })<\mp \text{tan}(\unicode[STIX]{x1D716})\text{Re}(u-u_{0}^{\pm })$ , with $\unicode[STIX]{x1D716}=\text{tan}^{-1}(\text{Im}(\unicode[STIX]{x1D714})/\text{Re}(\unicode[STIX]{x1D714}))$ . The intersection of the two half-planes is a strip ${\mathcal{S}}$ . The solutions of (2.11) should have the form:

where $\text{J}_{m+\unicode[STIX]{x1D705}}$ is the $(m+\unicode[STIX]{x1D705})$ th-order Bessel function of the first kind and $A^{\unicode[STIX]{x1D705}}$ is the associated amplitude. From (2.12), The scattered field might consist of a series of circumferential modes other than the original mode $m$ of the incident wave, due to non-uniform impedance in the $\unicode[STIX]{x1D703}$ -direction.

Figure 3.

(a) Schematic of the two complex half-planes $R_{\pm }$ and the overlapped strip when $M=0.3$ . The path of integration should be carefully designed to avoid all possible acoustic zeros and poles. (b) Poles ( $\times$ ) and zeros (○) of the matrix kernel $\mathscr{K}$ . The definition of poles and zeros of a matrix kernel can be found in our recent work (Huang Reference Huang2017). Incident wave with $\unicode[STIX]{x1D714}=25$ , mode $(18,1)$ , $M=0.3$ . The corresponding azimuthally non-uniform impedance distribution satisfies (4.1) that is to be defined in the following section, with $Z_{\unicode[STIX]{x1D6FC}}=1+0.5\text{i},Z_{\unicode[STIX]{x1D6FD}}=2-\text{i}$ and $N=1$ .

Similarly, we expand the Fourier transform of the vortex sheets into

By first applying Fourier transform to (2.6) and then making use of (2.12)–(2.13), we will have the following relation: $A^{\unicode[STIX]{x1D705}}(u)=-\text{i}(1-uM)F_{+}^{\unicode[STIX]{x1D705}}(u)/\unicode[STIX]{x1D706}\text{J}_{m+\unicode[STIX]{x1D705}}^{\prime }(\unicode[STIX]{x1D706}\unicode[STIX]{x1D714})$ . Here $(\cdot )_{\pm }$ denotes the regularity in the half-planes $R_{\pm }$ , respectively, and all the functions $(\cdot )_{\pm }$ are regular in the overlapped strip ${\mathcal{S}}$ .

Then, we make one of the most important definitions as follows:

where

where $Z(\unicode[STIX]{x1D714},\unicode[STIX]{x1D703})$ is $2\unicode[STIX]{x03C0}$ -periodic in the $\unicode[STIX]{x1D703}$ -direction and, therefore, can be expanded into

which leads Term I to

Making use of (2.12), we also have

We can further define $G_{+}(u,\unicode[STIX]{x1D703})=\text{Terms I + II}$ and derive its analytical expression,

which is obtained by using (2.7) along with (2.3), and the approximation $\unicode[STIX]{x1D709}^{-}\simeq 1$ . Meanwhile, we define $G_{-}(u,\unicode[STIX]{x1D703})\triangleq \text{Term}\;\text{III}=\sum _{\unicode[STIX]{x1D705}}G_{-}^{\unicode[STIX]{x1D705}}(u)\text{e}^{\text{i}\unicode[STIX]{x1D705}\unicode[STIX]{x1D703}}$ . Throughout this paper, $(\cdot )_{\pm }$ represents analytic functions in the upper and lower half-planes $R_{\mp }$ (as defined before (2.12)), respectively. Finally, by combining (2.17)–(2.19), we obtain the Wiener–Hopf equation:

which leads to a matrix Wiener–Hopf model by equating the terms with the same $\text{exp}(\text{i}\unicode[STIX]{x1D705}\unicode[STIX]{x1D703})$ :

with

Here the constant matrix $\unicode[STIX]{x1D63E}$ is dependent on $Z(\unicode[STIX]{x1D714},\unicode[STIX]{x1D703})$ , and $\unicode[STIX]{x1D646}$ has the diagonal entries:

Physically, the poles and zeros (see figure 3 b) of each $K^{\unicode[STIX]{x1D705}}(u)$ correspond to the acoustic modes. The group of zeros of the determinant, det $\mathscr{K}$ (see (2.21)), is related to the dynamic behaviour of the vortex sheet. The equation of $K^{\unicode[STIX]{x1D705}}=0$ corresponds to the dispersion relation for uniformly lined ducts with a single azimuthal mode $m+\unicode[STIX]{x1D705}$ , while the matrix $\unicode[STIX]{x1D63E}$ represents the coupling between different azimuthal scattering waves excited by non-uniform impedance. It can be seen that (2.21) contains two unknowns, $\unicode[STIX]{x1D641}_{+}(u)$ and $\unicode[STIX]{x1D642}_{-}(u)$ . Hence, we utilise the analytic properties and apply the Wiener–Hopf method to solve this equation in the next section. A useful property for the application of the Wiener–Hopf method is the behaviour of the Wiener–Hopf kernel when $|u|\rightarrow \infty$ . Similar to Gabard & Astley (Reference Gabard and Astley2006), we find $\mathscr{K}\sim O(u^{2})$ as $|u|\rightarrow \infty$ in the strip ${\mathcal{S}}$ (as defined before (2.12)).

3 The revised Wiener–Hopf solution

The most essential (and usually difficult) step in the Wiener–Hopf method is the kernel factorisation. For the current matrix kernel $\mathscr{K}$ , we use the pole-removal method (Rawlins Reference Rawlins1980; Abrahams Reference Abrahams1987; Veitch & Peake Reference Veitch and Peake2008) to conduct the matrix factorisation. First, we directly factorise the diagonal matrix as: $\unicode[STIX]{x1D646}=\unicode[STIX]{x1D646}_{-}^{-1}\unicode[STIX]{x1D646}_{+}$ , where $\unicode[STIX]{x1D646}_{+}$ and $\unicode[STIX]{x1D646}_{-}$ are analytic, invertible and show algebraic behaviours at $\pm \infty$ . Then, from (2.21), we have:

The left-hand side of (3.1) is regular and non-zero in the plane $R_{+}$ (as defined before, see (2.12)). The subtraction of $\unicode[STIX]{x1D646}_{-}(\unicode[STIX]{x1D707}_{mn}^{+})\unicode[STIX]{x1D642}_{+}(u)$ on both sides of (3.1) cancels the pole $\unicode[STIX]{x1D707}_{mn}^{+}$ of $\unicode[STIX]{x1D642}_{+}(u)$ . Nevertheless, the right-hand side of (3.1) is still not regular in $R_{-}$ due to $\unicode[STIX]{x1D646}_{-}(u)\unicode[STIX]{x1D63E}\unicode[STIX]{x1D641}_{+}(u)$ . However, we can infer the singularity property of $\unicode[STIX]{x1D641}_{+}(u)$ in $R_{-}$ by studying $\mathscr{K}_{+}^{-1}=\mathscr{K}^{-1}\mathscr{K}_{-}^{-1}$ , where $\mathscr{K}_{-}^{-1}$ is analytic and invertible in $R_{-}$ . Then, apart from $\unicode[STIX]{x1D707}_{mn}^{+}$ , all singularities of $\unicode[STIX]{x1D641}_{+}(u)$ in $R_{-}$ are the same as the singularities of $\mathscr{K}^{-1}$ in $R_{-}$ (the detailed derivations can be found in appendix A). In addition, $\unicode[STIX]{x1D641}_{+}(u)$ only consists of meromorphic functions with a number of simple poles (denoted by $\unicode[STIX]{x1D6FD}_{j}^{-}$ with order $s_{j}$ , $j=0,1,2,\ldots$ ), which are either poles of $\mathscr{K}$ (as shown in figure 3(b) with circular symbols in $R_{-}$ ) or solutions of det $\mathscr{K}=0$ (since $\mathscr{K}^{-1}=\text{adj}\,\mathscr{K}/\text{det}\,\mathscr{K}$ , where $\text{adj}$ denotes the associated adjugate). Then, we first cancel the effect of the singularity $u=\unicode[STIX]{x1D707}_{mn}^{+}$ by introducing a term to both sides of (3.1) as follows:

Next, we apply the general pole-removal method to cancel the remaining poles, $\unicode[STIX]{x1D6FD}_{j}^{-}$ , of $\unicode[STIX]{x1D641}_{+}(u)$ in the half-plane $R_{-}$ ,

where $\mathscr{R}_{j}^{-}$ is a constant matrix, each element of which is the residue of the corresponding element in $[\unicode[STIX]{x1D646}_{-}(u)-\unicode[STIX]{x1D646}_{-}(\unicode[STIX]{x1D707}_{mn}^{+})]\unicode[STIX]{x1D63E}\unicode[STIX]{x1D641}_{+}(u)$ at the pole $u=\unicode[STIX]{x1D6FD}_{j}^{-}$ . Now both sides of (3.3) are analytic in the corresponding half-planes. By applying analytic continuation in complex analysis, (3.3) defines an entire function $\unicode[STIX]{x1D640}(u)$ that is regular in the whole complex $u$ plane. It is also easy to see that $\unicode[STIX]{x1D646}(u)\sim O(u^{2})$ and $\unicode[STIX]{x1D646}_{\pm }(u)\sim O(u^{\pm 1})$ as $|u|\rightarrow \infty$ . For outlet cases with $M>0$ , the full Kutta condition (Gabard & Astley Reference Gabard and Astley2006) should be adopted, so $\unicode[STIX]{x1D641}_{+}(u)\sim O(u^{-5/2})$ , $\unicode[STIX]{x1D642}_{-}(u)\sim O(u^{-1})$ ; while for inlet cases with $M<0$ , we adopt the leading edge condition, resulting in $\unicode[STIX]{x1D641}_{+}(u)\sim O(u^{-3/2})$ . Therefore, we can obtain the entire function $\unicode[STIX]{x1D640}(u)\equiv 0$ for (3.3) as $u\rightarrow \infty$ . Here we wish to mention that it is possible to have a non-zero entire function if a different Kutta condition is adopted (Rienstra Reference Rienstra2007). However, the current governing equation is linear wave equation, and the unmodelled physics, such as nonlinearity and viscosity, should lead to a smooth behaviour of the edge condition that results in $\unicode[STIX]{x1D640}=0$ . A similar discussion can be found in Rienstra (Reference Rienstra2007)(before (32) therein). As a result,

where the infinitely many unknown residues, $\mathscr{R}_{j}^{-}$ , can be obtained by the approach proposed in the following (3.7). Finally, the solution of the total sound pressure fields is:

where the integral path $\unicode[STIX]{x1D6E4}$ is a curve from $-\infty \text{e}^{-\text{i}\unicode[STIX]{x1D716}}$ to $+\infty \text{e}^{-\text{i}\unicode[STIX]{x1D716}}$ in the strip ${\mathcal{S}}$ (Gabard & Astley Reference Gabard and Astley2006).

To enable the above pole removal in (3.3), we need to know the values of the residue. To this end, the values are still unknown. Here we propose an effective and interesting algorithm to obtain these values approximately. First, we rewrite (3.1) to

where $\unicode[STIX]{x1D646}_{-}(\unicode[STIX]{x1D707}_{mn}^{+})\unicode[STIX]{x1D642}_{+}(u)$ on both sides of (3.6) cancels the pole $\unicode[STIX]{x1D707}_{mn}^{+}$ of $\unicode[STIX]{x1D642}_{+}(u)$ . However, the left-hand side of (3.6) is still not regular in $R_{+}$ due to $\unicode[STIX]{x1D646}_{-}(u)$ with poles in $R_{+}$ (these poles in $R_{+}$ are denoted by $\unicode[STIX]{x1D6FD}_{j}^{+},j=0,1,2,\ldots$ ). By using the pole removal technique to further remove these poles from (3.6), we have

where $\mathscr{R}_{j}^{+}$ is the residue for $\unicode[STIX]{x1D646}_{-}(u)\unicode[STIX]{x1D63E}\unicode[STIX]{x1D641}_{+}(u)$ at the corresponding pole $u=\unicode[STIX]{x1D6FD}_{j}^{+}$ in the upper half-plane. Now both sides of (3.7) are analytic in the corresponding half-planes. Once again, it is easy to obtain an entire function $\unicode[STIX]{x1D640}(u)\equiv 0$ for (3.7) as $u\rightarrow \infty$ , resulting in

We have two different solutions for $\unicode[STIX]{x1D641}_{+}(u)$ , i.e. (3.4) and (3.8), with many unknown residue values $\mathscr{R}_{j}^{\pm }$ , $j=0,1,\ldots .$ By carefully comparing (3.4) and (3.8), we find that from (3.4) we can calculate $\mathscr{R}_{j}^{+}$ , while from (3.8) we can calculate $\mathscr{R}_{j}^{-}$ , together these constitute a closed set of equations. For simplicity, we assume a problem set-up with only four residues, $\mathscr{R}_{0}^{\pm }$ and $\mathscr{R}_{1}^{\pm }$ . Then, the corresponding equations are

where

The above four equations are closed with four unknown residues $\mathscr{R}_{0,1}^{\pm }$ at $\unicode[STIX]{x1D6FD}_{0,1}^{\pm }$ . Of course, a practical problem set-up usually contains many poles and zeros and a truncation operation has to be adopted to enable a numerical implementation. Generally, for $2\mathscr{N}$ poles ( $\unicode[STIX]{x1D6FD}_{j}^{\pm }$ , $j=0,1,\ldots ,\mathscr{N}-1$ ), we can obtain the corresponding residue values by the following generic formulation:

where

Physically, the $j=0$ pole corresponds to a modal element with the smallest mode number, $j=1$ corresponds to the second smallest mode number, and so on.

Given a problem set-up, we will first set an appropriate truncation number $\mathscr{N}$ and solve (3.12) to obtain the corresponding residue values. Then, we can use either (3.4) or (3.8) to solve $\unicode[STIX]{x1D641}_{+}(u)$ . Eventually, we solve (3.5) to produce the whole sound wave field. If $\mathscr{N}$ is set to 0, the effect due to those poles is neglected and the current model follows the same simplification strategy adopted in the previous work (Liu et al. Reference Liu, Jiang, Huang and Chen2015).

4 Results and discussion

To demonstrate the proposed model, we prepare a couple of test cases. First, we verify the numerical implementation by examining a simple flow duct case with uniform, semi-infinite axial liners. The current model can exactly recover the same results of the previous model (Rienstra Reference Rienstra2007) because $\unicode[STIX]{x1D63E}=0$ in that case, meaning that there are no poles to remove. The corresponding results are omitted for brevity. Then, we consider azimuthally non-uniform liner cases with two representative impedance distributions: one has a segmental set-up with two different acoustic impedance values, $Z_{\unicode[STIX]{x1D6FC}}$ and $Z_{\unicode[STIX]{x1D6FD}}$ , resulting in the following distribution

where $j=1,2,\ldots ,N$ with $N$ the number of segments; and the other has a sinusoidal distribution

where $Z_{\unicode[STIX]{x1D6FC}}$ and $Z_{\unicode[STIX]{x1D6FD}}$ are two constant parameters to be defined by users. For example, figure 4 shows the associated azimuthal distributions under certain parameters.

Figure 4.

Normalised impedance varying along with the azimuthal angle $\unicode[STIX]{x1D703}$ , where the original impedance values are defined by (4.1) and (4.2), and the reconstructed values are from Fourier series expansion. The truncation issue is discussed in the main text.

Figure 5.

Study of the truncation issue, where (a) the sound pressure level ( $\text{SPL}=20~\text{log}_{10}(p_{rms}/2\times 10^{-5})$ , with $(\cdot )_{rms}$ the root mean square value) results at $(x,r)=(0.1,1)$ , where $(m,n)=(24,1)$ , Mach number $M=0.3$ , $\unicode[STIX]{x1D714}=30$ and impedance $Z_{\unicode[STIX]{x1D6FC}}=1+1\text{i}$ , $Z_{\unicode[STIX]{x1D6FD}}=2+2\text{i}$ and $N=2$ . First, we set $\mathscr{N}=0$ and do not perform pole removal. Then, we gradually remove the associated 8 poles from $K^{0}$ . (b) An enlarged view shows that the pole-removal operation for the current set-up only introduces slight modifications.

First, we use the case set-up with an impedance distribution such as (4.1) to examine the truncation issue. Given a problem set-up (the spinning mode number $(m,n)=(24,1)$ , Mach number $M=0.3$ , $\unicode[STIX]{x1D714}=30$ and impedance $Z_{\unicode[STIX]{x1D6FC}}=1+1\text{i}$ , $Z_{\unicode[STIX]{x1D6FD}}=2+2\text{i}$ and $N=2$ ). To check the convergence, we remove those poles one by one, by using either (3.4) or (3.8), and compare the corresponding solutions. The sound pressure results shown in figure 5 shows that the pole-removal technique introduces little difference to the final predictions. Here we only remove the poles of $K^{0}$ but the same observations have been found when we further remove the poles of $K^{\pm 2}$ . Overall, figure 5 suggests that the effect due to those poles from the matrix kernel is negligible. Therefore, the simplified strategy adopted in the previous work (Liu et al. Reference Liu, Jiang, Huang and Chen2015) is acceptable, even for the current more generic case set-ups. As a result, we also apply the same strategy to study the set-ups with azimuthally non-uniform liner in this work.

Figure 6.

The SPL results at $(x,r)=(0.1,1),(0.3,1)$ and $(0.5,1)$ calculated by the proposed Wiener–Hopf analytical model and the numerical solver, respectively. The horizontal axis represents $\unicode[STIX]{x1D703}$ (deg.). The set-up is $(m,n)=(24,1)$ , Mach number $M=0.3$ , $\unicode[STIX]{x1D714}=30$ and impedance $Z_{\unicode[STIX]{x1D6FC}}=1+1\text{i}$ , $Z_{\unicode[STIX]{x1D6FD}}=2+2\text{i}$ , for (a) $N=2$ and (b) $N=4$ .

Figure 7.

Instantaneous sound pressure field, with incident spinning mode $(18,1)$ , $\unicode[STIX]{x1D714}=25$ , $M=0.3$ , $N=-1$ and $Z$ is given by (4.2), where $Z_{\unicode[STIX]{x1D6FC}}=1+0.5\text{i}$ , $Z_{\unicode[STIX]{x1D6FD}}=2-1\text{i}$ .

Figure 8.

Instantaneous sound pressure field solved (a) by the proposed theoretical model and (b) by the commercial software ACTRAN^®, where the incident spinning mode $(18,1)$ , $\unicode[STIX]{x1D714}=25$ , $M=0.3$ , $N=-1$ and $Z$ is given by (4.2), where $Z_{\unicode[STIX]{x1D6FC}}=1+0.5\text{i}$ , $Z_{\unicode[STIX]{x1D6FD}}=2-1\text{i}$ .

Figures 6 and 7 compare the model predictions to the numerical results from the commercial finite element software, ACTRAN^®. Both the piecewise segmental impedance values, (4.1), and the sinusoidal impedance values, (4.2), are examined. It can be seen that the analytical and the numerical results agree very well in most regions, with the largest difference less than 0.5 dB. The wiggles of the sound pressure level (SPL) results around the different interfaces between two lined segments are well captured by the theoretical model. In addition, the SPL results along the duct wall ( $r=1$ ) share the same periodicity with the segments, which indicates that there exist new circumferential modes other than the modes of incident waves, and the new modes should be directly related to the parameter $N$ which is defined previously to represent the distribution of liner impedance $Z(\unicode[STIX]{x1D703})$ . Finally, figure 8 shows the overall surface sound pressure field for the azimuthally sinusoidal distribution. Once again, it can be seen that the agreement between the theoretical prediction and the computational result is very good.

Figure 9.

Instantaneous sound pressure fields in the $(r,\unicode[STIX]{x1D703})$ cross-sections at $x=0.1,0.3$ and $0.5$ . (a,c,e) Show results for the piecewise segmental liner with (4.1). (b,d,f) Show results for the sinusoidal liner with (4.2). Other configurations are the same as figure 6(b) and figure 8, respectively.

Figure 10.

Modal spectrum of instantaneous sound pressure field at $x=0.1$ are shown here with respect to circumferential and radial modes. The parameters are same as figure 6, besides (a) $N=2$ and (b) $N=4$ .

To investigate the effect of liner non-uniformity, we first calculate the modal spectrum (shown in figure 10) at $x=0.1$ cross-section for results shown in figure 9. For a set-up of dimensionless frequency $\unicode[STIX]{x1D714}=30$ and background flow $M=0.3$ , modes of circumferential order greater than 28 or less than $-$ 28 are cutoff (the sign of $m$ denotes if the mode is spinning clockwise or counterclockwise). It can be seen that the single mode injected is scattered, yielding many new azimuthal modes. As is expected, the modal spectrum has gaps in azimuthal mode axis and the width of each gap is equal to $N$ , which is the number of periods of $Z(\unicode[STIX]{x1D703})$ in $2\unicode[STIX]{x03C0}$ . Furthermore, the patterns in figure 9 suggest that the single incident mode is scattered into many new azimuthal modes.

In this work, the proposed theoretical model is implemented with MATLAB^®. The computational cost on a desktop computer with Intel^® Core E5 processor @ 3.30 GHz is of the order of $O(10^{2})\;\text{s}$ . As a comparison, the same set-up is solved by a commercial numerical solver, ACTRAN^®, on the same computer. Each three-dimensional unstructured mesh used in the calculation has $3\times 10^{6}$ grid points and the geometrical domain of $-2\leqslant x\leqslant 2$ , $0\leqslant r\leqslant 1$ and $0\leqslant \unicode[STIX]{x1D703}\leqslant 2\unicode[STIX]{x03C0}$ is considered. The computational cost is of the order of $O(10^{4})\;\text{s}$ , which is two orders of magnitude greater than that of the proposed model. Moreover, when a computational solver such as ACTRAN^® is used for more complicated set-ups with more irregular and azimuthally non-uniform impedance values, the preparation of the computational meshes and other initial numerical settings will become even more tedious. In contrast, the proposed theoretical model can rapidly predict sound fields for various azimuthal distributions by a simple change of the corresponding mathematical formula. Therefore, the proposed model is more computationally efficient and convenient to use compared to three-dimensional numerical solvers.

Overall, figures 8 and 9 suggest two possible new noise control strategies by using azimuthally non-uniform liner designs. First, when the number of segments is much larger than the azimuthal mode of incident waves, it is possible to make all the scattered modes cutoff. As a result, each azimuthal segment can be designed to especially attenuate a particular incident wave at a certain frequency, which will enable a broadband liner design. Second, the azimuthal distribution can be carefully designed to re-direct sound propagation to certain angles and thus minimise the interference to other regions.

5 Summary

In this work, we have developed an analytical model for the modelling of flow duct liners with azimuthally non-uniform impedance. To enable the theoretical development, we have to assume an axial symmetrical duct enclosing uniform mean steady flow. By adopting some modifications, it may be possible to apply the proposed modelling method to a slightly curved duct with non-uniform steady flow. However, we believe the resultant model will become less (mathematically) delicate and a direct numerical simulation may be more appropriate for such practical set-ups.

For the current idealised set-up with straightened ducts, we have incorporated Fourier series expansion into the Wiener–Hopf method to investigate spinning modal sound wave scattering from the axial hard–soft interface and azimuthally non-uniform lined walls. The key achievement of this work is the development of the mathematical model and the interesting technique applied in the pole-removal process. In particular, two sets of Wiener–Hopf equations are organised to infer the residues in the upper and lower half-planes, which enable the subsequent matrix kernel factorisation. Another key observation is that the effect due to the removed poles are usually negligible, at least for the current segmental and sinusoidal impedance distributions in the azimuthal direction. As a result, we are also able to validate the simplified strategy adopted in matrix kernel factorisation in a previous work (Liu et al. Reference Liu, Jiang, Huang and Chen2015). Moreover, the proposed model has a distinctive feature that allows it to shed light on the associated physical mechanism of the scattering process, i.e. new modes directly depend on the azimuthal distribution will be scattered.

Although the rapid development of new materials makes it possible to design and employ liners with azimuthally non-uniform impedance inside engine duct to reduce noise emission more effectively, three-dimensional numerical solvers for non-uniform liners are usually computationally expensive for liner optimisation which often requires thousands of runs (Jiang et al. Reference Jiang, Lau and Huang2017b ). The efficiency of the proposed analytical model is at least two orders of magnitude higher than that of three-dimensional numerical solvers. Therefore, the current analytical model should be very useful for a host of new aeroengine (and even airframe (Boorsma, Zhang & Molin Reference Boorsma, Zhang and Molin2010)) liner design and optimisation problems.