2. Analytical derivation

To allow analytical progression, we start from a simplified model that is widely used in the literature (Howe Reference Howe1978; Lyu et al. Reference Lyu, Azarpeyvand and Sinayoko2016), i.e. the aerofoil is simplified as a flat plate placed in a uniform flow aligned in the streamwise direction, as shown in figure 1. As mentioned in § 1, the Green's function would be applicable to both trailing-edge and leading-edge scattering because of mathematical symmetry. In this paper, we use the trailing-edge scattering as an example. The flat plate is assumed to be semi-infinite, i.e. the leading edge extends to the upstream infinity and both side edges are also infinitely far away, so only the serrated trailing edge needs to be considered. We restrict our analysis to periodic serrations with a wavelength of $\tilde {\lambda }$. The problem is non-dimensionalized using the serration wavelength $\tilde {\lambda }$, the speed of sound $\tilde {c}$ and the fluid density $\tilde {\rho }$. Note we have used the symbols with a tilde to denote dimensional variables, whereas those without represent non-dimensional variables. We will adhere to this convention throughout this paper unless explicitly noted otherwise. In terms of the non-dimensional variables, the serration has a wavelength $1$ and half-root-to-tip amplitude $h$, and the uniform flow from left to right has a dimensionless velocity $M$, which is just the Mach number.

Figure 1.

Schematic illustration of the Green's function problem. The source represented by the black dot in the diagram is located near the edge at $\boldsymbol {y}$ and the observer is located in the far field at $\boldsymbol {x}$.

A Cartesian coordinate system shown in figure 1 is used in the analysis, where $x_1$, $x_2$ and $x_3$ denote the dimensionless streamwise, spanwise and normal-to-plate coordinates, respectively. In such a coordinate system, the profile of the serration, or the trailing edge of the plate, can be described by the periodic function $x_1 = hF(x_2)$, where $F(x_2)$ obtains a maximum value of $1$ and a minimum value of $-1$. Under the harmonic assumption of $\exp ({-\mathrm {i} \omega t})$, where $\omega$ is the non-dimensionalized angular frequency, the Green's function $G(\boldsymbol {x}; \boldsymbol {y},\omega )$ satisfies the following inhomogeneous convective equation (Amiet Reference Amiet1976a; Lyu et al. Reference Lyu, Azarpeyvand and Sinayoko2016):

(2.1)\begin{equation} \left(\beta^2 \frac{\partial^2 }{\partial x_1^2} + \frac{\partial^2 }{\partial x_2^2} + \frac{\partial^2 }{\partial x_3^2} + 2\mathrm{i} k M \frac{\partial }{\partial x_1} + k^2 \right) G(\boldsymbol{x}; \boldsymbol{y}, \omega) = \delta(\boldsymbol{x}-\boldsymbol{y}), \end{equation}

and the boundary condition

(2.2a,b)\begin{equation} \left.\frac{\partial G}{\partial x_3}\right|_{x_3 ={\pm} 0} = 0, \quad x_1 < hF(x_2), \end{equation}

where $\beta = \sqrt {1 - M^2}$ and $k = \omega /c$, and as shown in figure 1, $\boldsymbol {y}$ denotes the source position, while $\boldsymbol {x}$ denotes the observer position.

Note the observer location $\boldsymbol {x}$ is often in the far field, therefore, a standard technique is to use the reciprocal theorem to calculate the adjoint Green's function $G^a(\,\boldsymbol {y}; \boldsymbol {x}; \omega )\equiv G(\boldsymbol {x}; \boldsymbol {y}, \omega )$ so that the advantage of a plane wave incidence can be taken. However, as we aim to include the mean-flow convection effect in this paper, i.e. $M \ne 0$, (2.1) is no longer self-adjoint. In other words, the adjoint Green's function $G^a(\,\boldsymbol {y}; \boldsymbol {x}, \omega )$ does not satisfy (2.1). Nevertheless, it can be shown that the equation where $G^a(\,\boldsymbol {y}; \boldsymbol {x}, \omega )$ does satisfy differs from (2.1) only by the sign in front of the term $2\mathrm {i} kM ({\partial }/{\partial x_1})$, i.e.

(2.3)\begin{equation} \left(\beta^2 \frac{\partial^2 }{\partial y_1^2} + \frac{\partial^2 }{\partial y_2^2} + \frac{\partial^2 }{\partial y_3^2} - 2\mathrm{i} k M \frac{\partial }{\partial y_1} + k^2 \right) G^a(\,\boldsymbol {y}; \boldsymbol{x}, \omega) = \delta(\,\boldsymbol {y}-\boldsymbol{x}). \end{equation}

Physically, this is equivalent to solving the acoustic pressure at $\boldsymbol {y}$ while the point source is at $\boldsymbol {x}$, assuming a uniform flow of Mach number $M$ travels from right to left. In other words, the problem can be cast as ‘reciprocal’ by reversing the uniform mean flow.

Because $\boldsymbol {x}$ is in the far field, the incidence wave from the source $\boldsymbol {x}$ can be approximated by a plane wave, whose amplitude depends on the distance between $\boldsymbol {x}$ and $\boldsymbol {y}$. Because of linearity, we can start with an incident wave of magnitude $1$, i.e.

(2.4)\begin{equation} p_{in} = \exp({-\mathrm{i} k_1 y_1 / \beta})\exp\left({\mathrm{i}\frac{kM}{\beta^2}y_1}\right) \exp({-\mathrm{i} (k_2 y_2 + k_3 y_3)}), \end{equation}

where $k_1$ and $k_2$ are constants related to the radiation angle, the precise definition of which will be given later, and $k_3 = \sqrt {(k/\beta )^2 - k_1^2 - k_2^2}$. It can be verified that (2.4) satisfies the homogeneous version of (2.3). We decompose the total adjoint pressure field $G^a = p_{in} + p_r + R_s$, where the hypothetically reflected wave $p_r$ off an infinite flat plate is defined as $p_r = p_{in}(\kern0.7pt y_1, y_2, -y_3)$ and $R_s$ is the reflection-removed scattered pressure field. We could also have decomposed the pressure field as $G^a = p_{in} +G_s$, and this approach is shown in Appendix B. Note however no matter which decomposition is used, it should in no way affect the final solution.

The reflection-removed scattered wave $R_s$ satisfies

(2.5)\begin{equation} \beta^2 \frac{\partial^2 R_s}{\partial y_1^2} + \frac{\partial^2 R_s}{\partial y_2^2} + \frac{\partial^2 R_s}{\partial y_3^2} - 2 \mathrm{i} k M \frac{\partial R_s}{\partial y_1} + k^2 R_s = 0 \end{equation}

and the following boundary conditions due to the periodicity of the serrations (Ayton Reference Ayton2018):

(2.6a)$$\begin{gather} \left.\frac{\partial R_s}{\partial y_3}\right|_{y_3=0}= 0,\quad y_1 < hF(y_2); \end{gather}$$

(2.6b)$$\begin{gather}\left.R_s\right|_{y_3 = 0} ={-}\exp({-\mathrm{i} k_1 y_1 / \beta}) \exp\left({\mathrm{i}\frac{kM}{\beta^2}y_1}\right)\exp({-\mathrm{i} k_2 y_2}), \quad y_1 > hF(y_2); \end{gather}$$

(2.6c)$$\begin{gather}\left.R_s\right|_{y_2 = 0}= \left.R_s\right|_{y_2 = 1} \mathrm{e}^{\mathrm{i} k_2}; \end{gather}$$

(2.6d)$$\begin{gather}\left.\frac{\partial R_s}{\partial y_2}\right|_{y_2 = 0} = \left.\frac{\partial R_s}{\partial y_2}\right|_{y_2 = 1} \mathrm{e}^{\mathrm{i} k_2}. \end{gather}$$

Eliminating the first-order term in (2.5) by the transformation $R_s = \bar {R}_s \exp ({\mathrm {i} k M y_1 / \beta ^2})$, we obtain

(2.7)\begin{equation} \beta^2 \frac{\partial^2 \bar{R}_s}{\partial y_1^2} + \frac{\partial^2 \bar{R}_s}{\partial y_2^2} + \frac{\partial^2 \bar{R}_s}{\partial y_3^2} + \left(\frac{k}{\beta}\right)^2 \bar{R}_s = 0. \end{equation}

Earlier work (Ayton Reference Ayton2018) often used the non-orthogonal coordinate transformation ${\xi _1 = (y_1 - h F(y_2)) / \beta}$, $\xi _2 = y_2$ and $\xi _3 = y_3$ to enable the use of separation of variables. We show that this coordinate transformation is not necessary and the same Wiener–Hopf equation can be obtained by using the Fourier transform directly. We follow this approach here. Introducing the stretched coordinate $\xi _1 = y_1 / \beta$, $\xi _2= y_2$, $\xi _3 = y_3$, we see that the governing equation reduces to

(2.8)\begin{equation} \frac{\partial^2 \bar{R}_s}{\partial \xi_1^2} + \frac{\partial^2 \bar{R}_s}{\partial \xi_2^2} + \frac{\partial^2 \bar{R}_s}{\partial \xi_3^2} + \bar{k}^2 \bar{R}_s = 0, \end{equation}

where the stretched constants are defined as $\bar {k} = k/\beta$. Now the boundary conditions read

(2.9a)$$\begin{gather} \left.\frac{\partial \bar{R}_s}{\partial \xi_3}\right|_{\xi_3=0}= 0, \quad \xi_1 < \bar{h} F(\xi_2); \end{gather}$$

(2.9b)$$\begin{gather}\bar{R}_s|_{\xi_3 = 0} ={-}\exp({-\mathrm{i} (k_1 \xi_1 + k_2 \xi_2)}),\quad \xi_1 > \bar{h} F(\xi_2); \end{gather}$$

(2.9c)$$\begin{gather}\bar{R}_s|_{\xi_2 = 0}= \bar{R}_s|_{\xi_2 = 1} \mathrm{e}^{\mathrm{i} k_2}; \end{gather}$$

(2.9d)$$\begin{gather}\left.\frac{\partial \bar{R}_s}{\partial \xi_2}\right|_{\xi_2 = 0} = \left.\frac{\partial \bar{R}_s}{\partial \xi_2}\right|_{\xi_2 = 1}\mathrm{e}^{\mathrm{i} k_2}, \end{gather}$$

where $\bar {h}$ is defined as $\bar {h} = h/\beta$. We can now perform the Fourier transform along the $\xi _1$ direction, i.e.

(2.10)\begin{equation} \mathcal{R}(s, \xi_2, \xi_3) = \int_{-\infty}^{\infty} \bar{R}_s(\xi_1, \xi_2, \xi_3) \exp({\mathrm{i} s \xi_1}) \,\mathrm{d} \xi_1. \end{equation}

Function $\mathcal {R}(s, \xi _2, \xi _3)$ can be decomposed into two parts, i.e.

(2.11)\begin{align} \mathcal{R}(s, \xi_2, \xi_3) &= \int_{-\infty}^{\bar{h}F(\xi_2)}\bar{R}_s(\xi_1, \xi_2, \xi_3) \exp({\mathrm{i} s \xi_1}) \,\mathrm{d} \xi_1 \nonumber\\ &\quad + \int_{\bar{h}F(\xi_2)}^\infty \bar{R}_s(\xi_1, \xi_2, \xi_3) \exp({\mathrm{i} s \xi_1}) \,\mathrm{d} \xi_1 \nonumber\\ &= \int_{-\infty}^0\bar{R}_s(\xi_1+\bar{h}F(\xi_2), \xi_2, \xi_3) \exp({\mathrm{i} s (\xi_1+\bar{h}F(\xi_2))}) \,\mathrm{d} \xi_1 \nonumber\\ &\quad + \int_{0}^\infty \bar{R}_s(\xi_1+\bar{h}F(\xi_2), \xi_2, \xi_3) \exp({\mathrm{i} s (\xi_1+\bar{h}F(\xi_2))}) \,\mathrm{d} \xi_1\nonumber\\ &= \exp({\mathrm{i} s \bar{h} F(\xi_2)}) (\mathcal{R}^-(s, \xi_2, \xi_3) + \mathcal{R}^+(s, \xi_2, \xi_3) ) \end{align}

where functions $\mathcal {R}^-$ and $\mathcal {R}^+$ are complex functions that are analytical in the lower and upper half-$s$ planes, respectively. A similar Fourier transform (and decomposition) is applied to the function $\partial \bar {R}_s / \partial \xi _3$, the result of which will be denoted by $\mathcal {R}^\prime$ in the rest of the paper.

Equation (2.8) then reduces to

(2.12)\begin{equation} \frac{\partial^2 \mathcal{R}}{\partial \xi_2^2} + \frac{\partial^2 \mathcal{R}}{\partial \xi_3^2} + (\bar{k}^2-s^2) \mathcal{R} = 0. \end{equation}

Equation (2.12) is the standard Helmholtz equation and its solution can be found through the usual method of separation of variables. After using the last two boundary conditions shown in (2.9), we show that (2.12) can be solved for $\xi _3>0$ (the corresponding result for $\xi _3<0$ is similar due to antisymmetry) to yield

(2.13)\begin{equation} \mathcal{R}(s, \xi_2, \xi_3) = \sum_{n ={-}\infty}^{\infty} A_n(s) \exp({-\gamma_n \xi_3}) \exp({\mathrm{i} \chi_n \xi_2}), \end{equation}

where $\chi _n = 2n{\rm \pi} - k_2$, $\gamma _n = \sqrt {s^2 - \kappa _n^2}$ and $\kappa _n = \sqrt {\bar {k}^2 - \chi _n^2}$. We see that $\kappa _n$ denotes the wavenumber in the $\xi _1-\xi _3$ plane and when $n=0$, it is equal to $\sqrt {k_1^2 + k_3^2}$. The complex function $A_n(s)$ will need to be determined by making use of the first two boundary conditions shown in (2.9) by using the Wiener–Hopf method, i.e.

(2.14a)\begin{align} \mathcal{R}^\prime(s, \xi_2, 0) &= \exp({\mathrm{i} s \bar{h} F(\xi_2)})\sum_{n={-}\infty}^{\infty} \mathcal{R}_n^{\prime+}(s) \exp({-{\rm i} s \bar{h} F(\xi_2)}) \exp({\mathrm{i} \chi_n \xi_2}); \end{align}

(2.14b)\begin{align} \mathcal{R}(s,\xi_2, 0) &= \exp({\mathrm{i} s \bar{h} F(\xi_2)}) \left(\sum_{n={-}\infty}^{\infty} \mathcal{R}_{n}^-(s) \exp({-{\rm i} s \bar{h} F(\xi_2)}) \exp({\mathrm{i} \chi_n \xi_2}) \right.\nonumber\\ &\quad \left.\vphantom{\sum_{n={-}\infty}^{\infty}} -\frac{\mathrm{i}}{s - k_1}\exp({-\mathrm{i} (k_1 \bar{h} F(\xi_2)+ k_2 \xi_2)})\right), \end{align}

where $\mathcal {R}_n^{\prime +}(s)$ and $\mathcal {R}_n^-(s)$ are the expansion coefficients of functions $\mathcal {R}^{\prime +}(s, \xi _2, 0)$ and $\mathcal {R}^-(s, \xi _2, 0)$ using the basis functions $\exp ({-\mathrm {i} s \bar {h} F(\xi _2)}) \exp ({\mathrm {i} \chi _n \xi _2}),\ n =0, \pm 1, \pm 2 \cdots$, and they are unknown at this stage. The last exponential term in the parenthesis of (2.14b) can also be expanded and the resulting coefficients are denoted by $E_n(s)$. Here, $E_n(s)$ can be found to be

(2.15)\begin{equation} E_n(s) = \int_0^1 \exp({\mathrm{i} (s-k)\bar{h}F(\xi_2)})\exp({-\mathrm{i} 2 n {\rm \pi}\xi_2})\,\mathrm{d} \xi_2. \end{equation}

Note that $E_n(s)$ can be arbitrary because no restriction on $F(\xi _2)$ has been imposed apart from it being periodic. For any arbitrary piecewise linear functions, $E_n(s)$ can be integrated analytically. For example, for the conventional sawtooth serration profile defined by (in one period)

(2.16)\begin{equation} F(\xi_2)= \begin{cases} 4 \xi_2, & -\dfrac{1}{4} < \xi_2 < \dfrac{1}{4}, \\ -4 \xi_2 + 2, & \dfrac{1}{4} < \xi_2 < \dfrac{3}{4}, \end{cases} \end{equation}

$E_n(s)$ can be found as

(2.17)\begin{equation} E_n(s)= \frac{4 (s - k_1)\bar{h} \sin((s-k_1)\bar{h} - n{\rm \pi}/2)} {4 (s-k_1)^2\overline{h}^2 - n^2 {\rm \pi}^2}. \end{equation}

Upon comparing (2.13) and (2.14a) and making use of orthogonality of the basis functions $\exp ({-\mathrm {i} s \bar {h} F(\xi _2)}) \exp ({\mathrm {i} \chi _n \xi _2}),\ n =0, \pm 1, \pm 2 \cdots$, we arrive at the following matching conditions for mode $n$, i.e.

(2.18a)$$\begin{gather} -\gamma_n A_n(s) = \mathcal{R}^{\prime + }_n(s), \end{gather}$$

(2.18b)$$\begin{gather}A_n(s) = \mathcal{R}_{n}^-(s) - \frac{\mathrm{i}}{s -k_1} E_n(s). \end{gather}$$

We can proceed by eliminating $A(s)$ and arrive at the Wiener–Hopf equation:

(2.19)\begin{equation} \gamma_n \left(\mathcal{R}^-_n(s) - \frac{\mathrm{i}}{s-k_1} E_n(s)\right) + \mathcal{R}_n^{\prime +}(s) = 0. \end{equation}

The function $E_n(s)$ causes much difficulty in the kernel decomposition. A recent approach (Ayton Reference Ayton2018) assumes that both $\mathcal {R}_n^-(s)$ and $\mathcal {R}_n^{\prime +}(s)$ contain the factor $E_n(s)$ so that a kernel factorization can proceed. However, we find that this assumption appears not to be true, in particular, this leads to results that do not strictly satisfy the boundary conditions. Moreover, as mentioned above, the results obtained by decomposing the total pressure field as either $G^a = p_{in} + G_s$ or $G^a = p_{in} + p_r + R_s$ should yield no difference to the final solution. However, it can be verified that if the assumption that $E_n(s)$ is a factor in $R_n^-(s)$ and $R_n^{\prime +}(s)$ is used, the two methods would yield different solutions (the two are only equal to each other for mode $n=0$, see Appendix D for details), which signals a potential problem with the underlying assumption. In fact, from (2.9) and (2.14a), we see that $E_n(s)$ represents the variation of the incident pressure on the edge. As $\mathcal {R}_n^-(s)$ denotes the scattered pressure upstream of the trailing edge, if $\mathcal {R}_n^-(s)$ had the same $E_n(s)$ factor as the incident wave, the scattering problem would need to be homogeneous in the spanwise direction. This can only be guaranteed if the trailing edge is a straight (or swept) edge. For serrated edges, the homogeneity condition is not satisfied and the $E_n(s)$ variation in $\mathcal {R}_n^-(s)$ cannot be guaranteed.

However, as the frequency increases, the acoustic wavelength becomes increasingly short, and the scattered pressure variation on the edge is expected to become increasingly localized and dominated by the incident phase variation; the assumption of $E_n(s)$ dependence may be approximately valid in the high-frequency limit. Serrations are known to be more effective as the frequency increases (see for example Howe Reference Howe1991a; Gruber Reference Gruber2012; Lyu et al. Reference Lyu, Azarpeyvand and Sinayoko2016), and more importantly, it is the hydrodynamic wavelength that characterizes the incoming (gust) length scale in TE noise modelling, the localized scattering is more likely to be valid. Therefore, in the following part of this paper, we focus on this high-frequency regime aiming to develop a closed-form analytical Green's function, which can be used to develop a 3-D TE noise model.

The kernel is the standard $\gamma _n = \sqrt {s^2 - \kappa _n^2}$, and once $E_n(s)$ is removed from both $\mathcal {R}_n^-(s)$ and $\mathcal {R}_n^{\prime +}(s)$, it becomes a routine procedure to be factorized as $\sqrt {s - \kappa _n}\sqrt {s + \kappa _n}$. Then $A_n(s)$ can be approximated by

(2.20)\begin{equation} A(s) ={-} \frac{1}{\gamma_n} \mathcal{R}^{\prime+}_n(s) ={-}\frac{\mathrm{i}}{\sqrt{s - \kappa_n}}\frac{\sqrt{k_1 - \kappa_n}}{s-k_1} E_n(s). \end{equation}

Substituting (2.20) into (2.13) and taking the inverse Fourier transform yields

(2.21)\begin{align} \bar{R}_s(\xi_1, \xi_2, \xi_3)) &= \sum_{n ={-}\infty}^{\infty}-\mathrm{i}(\sqrt{k_1 - \kappa_n}) \exp({\mathrm{i} \chi_n \xi_2}) \nonumber\\ &\quad \times\frac{1}{2{\rm \pi}}\int_{-\infty}^{\infty} \frac{E_n(s)}{s-k_1}\frac{1}{\sqrt{s - \kappa_n}} \exp({-\mathrm{i} s \xi_1 - \gamma_n \xi_3})\,\mathrm{d} s. \end{align}

Let $r = \sqrt {(y_1/\beta )^2 + y_3^2}$ and $\cos \theta = y_1 / (\beta r)$, we have finally

(2.22)\begin{align} R_s(r,\theta, y_2) &= \frac{1}{2{\rm \pi}}\exp({\mathrm{i} k M y_1 / \beta^2}) \sum_{n ={-}\infty}^{\infty}-\mathrm{i}(\sqrt{k_1 - \kappa_n}) \exp({\mathrm{i} \chi_n y_2}) \nonumber\\ &\quad \times \left[ \int_{-\infty}^{\infty} \frac{E_n(s)}{s-k_1} \frac{1}{\sqrt{s - \kappa_n}} \exp({({-}i s \cos \theta - \gamma_n \sin\theta) r}) \,\mathrm{d} s \right], \end{align}

where $E_n(s)$ is given by (2.15), and the integral is along the path $P$ shown in figure 2. Note that the integrand in (2.22) has a pole at $s=k_1$ and two branch points at $s=\pm \kappa _n$. The integral path $P$ has to pass above the pole at $s=k_1$ due to the analyticity requirement. It is, however, equivalent to integrating (2.22) along the path $P_0$ shown in figure 2, provided that the residue contribution from the pole is subtracted. We see from (2.17) that ${E_n(k_1) = \delta _{n_1}}$, therefore, it is convenient to calculate the residue, which is precisely the hypothetical reflected wave off an infinite flat plate $p_{r}$. Consequently, the total scattered field $G_s$ can be directly calculated by integrating (2.22) along the path $P_0$ instead, i.e.

(2.23)\begin{align} G_s(r,\theta, y_2) &= \frac{1}{2{\rm \pi}}\exp({\mathrm{i} k M y_1 / \beta^2}) \sum_{n ={-}\infty}^{\infty}-\mathrm{i}(\sqrt{k_1 - \kappa_n})\exp({\mathrm{i} \chi_n y_2}) \nonumber\\ &\quad \times \left[\int_{-\infty}^{\infty} \frac{E_n(s)}{s-k_1}\frac{1}{\sqrt{s - \kappa_n}} \exp({({-}i s \cos \theta - \gamma_n \sin\theta) r}) \,\mathrm{d} s\right], \end{align}

where the integral path in (2.23) is given by $P_0$ as shown in figure 2.

Figure 2.

Integral path $P$ in (2.22), which passes around a simple pole at $s=k_1$ and two branch points at $s =\pm \kappa _n$. Note the branch point $\kappa _n$ can be an imaginary number depending on the value of $n$, which however does not affect the analyticity of the integrand along the integral path. The integral along path $P$ is equivalent to that along $P_0$ minus a residue contribution around $s=k_1$.

To obtain a closed-form analytical Green's function, the integral in (2.23) has to be evaluated analytically. Note that $E_n(s)$ is arbitrary, but for all piecewise linear serration profiles, $E_n(s)$ can be evaluated analytically. If the far-field scattered pressure is of interest, i.e. $r\to \infty$, (2.23) can be quickly evaluated asymptotically by the powerful method of the steepest descent, as shown by Ayton (Reference Ayton2018). However, as we seek the Green's function, it is the near-field scattered pressure that is of our interest. The steepest descent method can no longer be used, and the contour integral in (2.23) must be integrated exactly. We show that for all piecewise linear profiles, the above complex contour integral can be integrated exactly to yield closed-form analytical solutions. We use the conventional sawtooth serration as an example in the rest of the paper, whereas the Green's functions for other common piecewise linear functions are given in Appendix C.

We begin by noting that for conventional sawtooth serrations, $E_n(s)$ is given by (2.17), in which the sine functions can be expanded using exponential functions. To facilitate a compact notation, we define two auxiliary local polar coordinate frames, i.e. $(r_t, \theta _t)$ and $(r_r, \theta _r)$ in the stretched $y_1 / \beta$–$y_3$ plane (i.e. $\xi _1$–$\xi _3$ plane), as shown in figure 3. Here the stretch factor $\beta$ is to account for the background uniform flow, and when $M=0$, the stretched plane is just the physical $y_1$–$y_3$ plane. We see that $\theta _t$ and $\theta _r$ represent the geometric angles of the observer with respect to the tip and root of the serration in the stretched $y_1/\beta$–$y_3$ plane, respectively, while $r_t$ and $r_r$ represent their corresponding radial coordinates, respectively. With these definitions, we can obtain

(2.24a)$$\begin{gather} r_t = \sqrt{r^2 + \bar{h}^2 - 2 r \bar{h} \cos\theta}, \end{gather}$$

(2.24b)$$\begin{gather}\theta_t = \arccos [(r\cos\theta - \bar{h})/r_t]. \end{gather}$$

And similarly, we have

(2.25)\begin{equation} \left.\begin{gathered} r_r = \sqrt{r^2 + \bar{h}^2 + 2 r \bar{h} \cos\theta}, \\ \theta_r = \arccos [(r\cos\theta + \bar{h})/r_r]. \end{gathered}\right\} \end{equation}

Expanding the sine functions in (2.17) into exponential functions, we can show that

(2.26)\begin{align} G_s(r, \theta, y) &= \frac{1}{2{\rm \pi}}\exp({\mathrm{i} k M y_1 / \beta^2}) \sum_{n ={-}\infty}^{\infty}-\mathrm{i}(\sqrt{k_1 - \kappa_n})\exp({\mathrm{i} \chi_n y_2})\nonumber\\ &\quad \times\left( \exp\left( -\mathrm{i}\left(k_1 \bar{h}+\frac{n{\rm \pi}}{2}\right)\right) H_n(r_t, \theta_t) - \exp\left( \mathrm{i}\left(k_1\bar{h}+\frac{n{\rm \pi}}{2}\right)\right)H_n(r_r, \theta_r)\right), \end{align}

where

(2.27)\begin{align} H_n(r_i, \theta_i) = \int_{-\infty}^{\infty} \frac{-2\mathrm{i}\bar{h}}{(2(s-k_1)\bar{h})^2-(n{\rm \pi})^2} \frac{1}{\sqrt{s - \kappa_n}} \exp({({-}i s \cos \theta_i - \gamma_n \sin\theta_i)r_i}) \,\mathrm{d} s \end{align}

and $(r_i, \theta _i)$ can take the value of either $(r_t, \theta _t)$ or $(r_r, \theta _r)$.

Figure 3.

Definition of the two geometrical angles $\theta _t$ and $\theta _r$ and their corresponding radial coordinates $r_t$ and $r_i$.

To integrate (2.27), when $n\ne 0$, we may expand the first factor of the integrand as partial fractions, i.e.

(2.28)\begin{equation} \frac{-2\mathrm{i}\bar{h}}{(2(s-k_1)\bar{h})^2-(n{\rm \pi})^2}={-} \frac{\mathrm{i} \bar{h}}{n {\rm \pi}}\left[\frac{1}{2(s-k_1)\bar{h} - n{\rm \pi}} - \frac{1}{2(s-k_1)\bar{h} + n{\rm \pi}}\right]. \end{equation}

Equation (2.27) can then be written as the difference between two integrals, i.e.

(2.29)\begin{equation} H_n(r_i, \theta_i) ={-}\frac{\mathrm{i}}{2n{\rm \pi}}[D_n^+(r_k, \theta_i) - D_n^-(r_k, \theta_i)], \end{equation}

where

(2.30)\begin{equation} D_n^\pm(r_k, \theta_i) =\int_{-\infty}^{\infty} \frac{1}{s-(k_1 \pm n{\rm \pi}/2 \bar{h})}\frac{1}{\sqrt{s - \kappa_n}} \exp({({-}i s \cos \theta_i - \gamma_n \sin\theta_i)r_i}) \,\mathrm{d} s. \end{equation}

We see from (2.30) that the presence of the serration introduces a modulated streamwise wavenumber of $k_1\pm n{\rm \pi} /2\bar {h}$ in the solution. Physically, this can be understood as follows. The presence of the periodic serration modulates the wavenumber of the incoming plane wave. The incoming plane wave has a spanwise wavenumber of $k_2$, consequently the $n$th mode of the scattered pressure has a spanwise wavenumber of $k_2 - 2 n {\rm \pi}$, where $n$ is an integer. Because the serration extends in both $y_1$ and $y_2$ directions, the scattered pressure along the edge varies in both $y_1$ and $y_2$ directions. Therefore, the streamwise wavenumber must be modulated simultaneously in a similar way as the spanwise wavenumber. Because the half-wavelength (one tooth) and root-to-tip amplitude of the serration are $1/2$ and $2h$, respectively, the $n$th-order plane wave would have corresponding modulated streamwise wavenumbers of $k_1+ n{\rm \pi} /2h$ and $k_1- n{\rm \pi} /2h$, due to the presence of the left and right teeth, respectively. Therefore, each spanwise mode ($n$th for example) corresponds to two equally weighted plane waves, one with a streamwise wavenumber of $k_1+n{\rm \pi} /2h$ and the other with $k_1-n{\rm \pi} /2h$. We can define two geometrical angles representing the effective incident angles in the $y_1/\beta - y_3$ plane for the two plane waves, i.e.

(2.31)\begin{equation} \varTheta_n^\pm{=} \arccos \frac{k_1 \pm n {\rm \pi}/2\bar{h}}{\kappa_n}, \end{equation}

where $n$ is an integer. Clearly, for $n=0$, both $\varTheta _0^+$ and $\varTheta _0^-$ reduce to $\varTheta _0 \equiv \arccos k_1/\kappa _0$ representing the incident angle of $p_{in}$.

To evaluate (2.30), we deformed the integration path $P_0$ shown in figure 2 to the curve $P_1+P_2+P_3$ shown in figure 4. The path $P_2$ is described by $s= -\kappa _n \cos (\theta _i + \mathrm {i} t)$, where the real number $t$ varies from $+\infty$ to $-\infty$. Figure 4 shows that $s\to \infty$ in the second quadrant as $t \to +\infty$, whereas as $s\to \infty$ in the third quadrant, $t \to -\infty$. It can be shown that integration along $P_1$ and $P_3$ approaches $0$ as $|s|\to \infty$. Therefore, the integral in (2.30) can be evaluated along $P_2$ instead provided $P_2$ passes the simple pole from below. Because the deformed path $P_2$ intersects with the real axis at $-\kappa _n \cos \theta _i$, such a condition is met when $-\kappa _n \cos \theta _i < k_1$, i.e. when $\theta _i > {\rm \pi}-\arccos (k_1/\kappa _n)$. When $0<\theta _i < {\rm \pi}- \arccos (k_1/\kappa _n)$, we can show that a residue contribution must be added. However, this pole contribution is exactly cancelled by the jump in the resulting integral and the final solution takes the same form as that for $\theta _i > {\rm \pi}- \arccos (k_1 /\kappa _n)$ (see Chapter 2 of Noble (Reference Noble1958) for details). Therefore, in the rest of the paper, we choose not to distinguish the two cases. By deforming the integral along $P_2$ and making use of the definition of (2.31), (2.30) reduces to

(2.32)\begin{equation} D_n^\pm(r_i, \theta_i)={-}\sqrt{\frac{2}{\kappa_n}} \int_{-\infty}^{\infty}\frac{\sin\dfrac{1}{2}(\theta_i + \mathrm{i} t)} {\cos(\theta_i+\mathrm{i} t) + \cos\varTheta_n^{{\pm}}} \exp({\mathrm{i} \kappa_n r_i \cosh t}) \,\mathrm{d} t.\end{equation}

Equation (2.32) can be integrated analytically to yield (see Appendix A for more details)

(2.33)\begin{equation} D_n^\pm(r_i, \theta_i) ={-}{\rm \pi}\sqrt{\frac{2}{\kappa_n}} \frac{I(\kappa_n r_i, \theta_i; \varTheta_n^\pm)}{\sin \dfrac{1}{2}\varTheta_n^\pm}, \end{equation}

where function $I(kr, \theta ; \varTheta )$ is the classical Fresnel solution denoting the pressure field scattered by a straight trailing edge (Noble Reference Noble1958), i.e.

(2.34)\begin{align} I(kr, \theta; \varTheta) &= \frac{\exp\left({-\mathrm{i} \dfrac{\rm \pi}{4}}\right)}{\sqrt{\rm \pi}} \left[\exp({-\mathrm{i} kr \cos(\varTheta+\theta)}) F\left(\sqrt{2kr} \cos \frac{\varTheta+\theta}{2}\right) \right. \nonumber\\ &\quad \left.-\exp({-\mathrm{i} kr \cos(\varTheta-\theta)}) F\left(\sqrt{2kr} \cos\frac{\varTheta-\theta}{2}\right)\right]. \end{align}

The Fresnel integral $F(x)$ in (2.34) is defined as

(2.35)\begin{equation} F(x) = \int_x^\infty \mathrm{e}^{\mathrm{i} u^2} \,\mathrm{d} u \end{equation}

and can be conveniently computed using the standard error function.

Figure 4.

Deformed path $P_1+P_2+P_3$, where $P_2$ is described by the $s = -\kappa _n \cos (\theta _i+\mathrm {i} t)$ as $t$ varies from $+\infty$ to $-\infty$. When $t=0$, the path $P_2$ intersects with the real axis at $-\kappa _n\cos \theta _i$. When $-\kappa _n\cos \theta _i > k_1$ as shown above, the simple pole is crossed, and a residue contribution must be included. The case when $\kappa _n$ is imaginary is similar.

Having obtained the analytical result of $D_n^\pm (r_i, \theta _i)$, it follows that

(2.36)\begin{equation} H_n(r_i, \theta_i) = \frac{\mathrm{i}}{\sqrt{2\kappa_n}n} \left(\frac{I(\kappa_n r_i, \theta_i; \varTheta_n^+)}{\sin\dfrac{1}{2}\varTheta_n^+} -\frac{I(\kappa_n r_i, \theta_i; \varTheta_n^-)}{\sin\dfrac{1}{2}\varTheta_n^-}\right), \end{equation}

where, as mentioned above, the subscript $i$ takes the value of either $t$ or $r$. Note that in (2.36), $H_n(r_i, \theta _i)$ decays at least as fast as $n^{-3 / 2}$ as $n\to \infty$, and because $n$ appears in the denominator, (2.36) works only for $n\ne 0$. However, if treating $n$ as a real variable, we may obtain the result for $n=0$ by taking the limit as $n\to 0$. To facilitate practical computations, we also derive an explicit formula for $H_0(r_i, \theta _i)$ from (2.27). This can be found in Appendix B.

Substituting (2.36) into (2.26), the total scattered pressure $G_s$ can be readily evaluated. The important fact is that (2.26) is an exact evaluation of (2.23), and therefore is not only valid in the far field, but also in the near field. When $r\to \infty$, (2.26) would recover the far-field approximation obtained using the steepest descent method by Ayton (Reference Ayton2018). Note again that (2.36) consists of the standard Fresnel solution describing the scattered field by a straight edge. This suggests that the pressure field scattered by the sawtooth edge is equivalent to the sum of the Floquet modes scattered by two imagined semi-infinite flat plates with their straight trailing edges located at the tip and root of the serration, respectively. This appears to be somewhat consistent with a number of previous findings showing that noise generation by serrated edges is dominated by the root or tip regions (Kim, Haeri & Joseph Reference Kim, Haeri and Joseph2016; Turner & Kim Reference Turner and Kim2017; Avallone et al. Reference Avallone, van der Velden, Ragni and Casalino2018). This view, however, results from the use of the $E_n(s)$ assumption and is therefore not exact. Because (2.17) is used in the derivation, (2.26) is therefore only valid for sawtooth serrations; however, as mentioned earlier, we can easily obtain analytical Green's functions for any arbitrary piecewise linear serration profiles. Appendix C contains the Green's functions for other serration profiles, such as the square shapes.

When a point source is located at $\boldsymbol {x}$, i.e. $(x_1, x_2, x_3)$, the incident plane wave near the serration has an amplitude of

(2.37)\begin{equation} A(\boldsymbol{x}) ={-}\frac{1}{\beta} \frac{1}{4 {\rm \pi}R} \exp({\mathrm{i} k R / \beta}) \exp\left({-\mathrm{i}\frac{kM}{\beta^2}x_1}\right), \end{equation}

where $R = \sqrt {(x_1/\beta )^2 + x_2^2 + x_3^2}$. Furthermore, the value of $k_1$ and $k_2$ in the definition of $p_{in}$ can be found to be

(2.38a,b)\begin{equation} k_1 = \frac{k}{\beta} \frac{x_1 /\beta}{R},\quad k_2 = \frac{k}{\beta} \frac{x_2}{R}. \end{equation}

By linearity, the Green's function can be readily obtained as

(2.39)\begin{equation} G(\boldsymbol{x}; \boldsymbol{y},\omega)= G^a(\,\boldsymbol {y}; \boldsymbol{x}, \omega) = A(\boldsymbol{x}) \left(p_{in} + G_s\right), \end{equation}

where $p_{in}$ is shown in (2.4) and $G_s$ is given by (2.26). Equation (2.39) is the fundamental equation of this paper. It can be seen that the Green's function consists of two terms; the first term $A(\boldsymbol {x}) p_{in}$ represents the sound propagating directly from the source $\boldsymbol {y}$ to the observer $\boldsymbol {x}$, while the second term $A(\boldsymbol {x})G_s$ represents the scattered pressure off the serrated plate then propagating to the observer $\boldsymbol {x}$. The direct propagating sound is trivial and most importantly does not depend on the serration profiles; therefore, it is the scattered part that we are interested in.

3. Validation

We see from § 2 that to obtain the analytical Green's function, considerable algebra is involved. Therefore, it is necessary to validate the result before the Green's function is used to study the scattering characteristics. In this section, we choose to validate the Green's function using two approaches. The first is to numerically integrate (2.23) so as to ensure that the complex analytical evaluation of the integral is correct. The second approach is to make use of the FEM technique to compute the scattered pressure under the incident wave shown in (2.4) using COMSOL so as to examine to what extent the assumption regarding $E_n(s)$ serves as a good approximation.

Figure 5 shows a comparison between the scattered pressure $|G_s|$ obtained by the numerical integration and from (2.26). As can be seen from figure 5(a), the scattered pressures obtained using the two approaches completely collapse along the line of $y_2 = 0$ in the plane of the flat plate. Similarly, the scattered pressure obtained by numerical integration along the line of $y_1=0$ in the flat plate plane is identical to that from (2.26). Pressure values at other locations show exactly the same agreement. This excellent agreement shows that the analytical evaluation of the contour integral in the complex $s$ domain is indeed correct and exact.

Figure 5.

Comparison of the scattered pressure field $|G_s|$ on (a) $y_2=0$, $y_3=0$ and (b) $y_1=0$, $y_3 = 0$. The serration amplitude is $h=10$, the wavenumber is $k=1$, the Mach number $M=0$ and the observer angle $\varTheta _0=\frac {3}{4}{\rm \pi}$.

Although figure 5 shows that the analytical derivation from (2.23) to (2.36) is correct, it cannot show to what extent (2.39) approximates the exact solution to (2.1). This is because (2.23) is based on the assumption of $E_n(s)$, and to examine its validity, we need to use FEM to numerically calculate the scattered pressure so that a direct comparison between the numerical and analytical Green's functions can be made. We again choose to compare the near-field $|G_s|$ under the incident wave shown in (2.4).

The commercial software COMSOL is used to conduct the numerical simulation, the computational domain of which is shown in figure 6. We can see that a half-cylindrical domain consisting of one serration wavelength is used. The semi-infinite plate is placed on the left-hand side of the bottom surface, as shown in figure 6. Periodic boundary conditions are used between the front and back surfaces. Perfectly matched layers (PMLs) are attached to the outer side of the domain to absorb the scattered pressure due to a plane wave incidence prescribed by (2.4). The PMLs work well for absorbing sound scattered off a finite object, but start to become less accurate to simulate a flat plate that is semi-infinitely long. To improve the accuracy of the PMLs, a small imaginary part of $k$ (in this paper, $\arg {k}\approx -0.02$) is used so that the scattered pressure decays gradually as it propagates. When compared against analytical results, the same $k$ is used in (2.26). This is permissible and can be shown conveniently by analytical continuation. A free tetrahedral mesh is used and the resulting case has up to 4 millions degrees of freedom at the highest dimensionless frequency $k$. Grid independence is examined by using increasingly fine meshes that result in little change in the calculated pressure field.

Figure 6.

Scattered pressure field $G_s$ (only real part is shown) from FEM simulations, where $M=0$, $h=2$, $k=10$ and $\varTheta ={\rm \pi} /4$. A small imaginary part of $k$ is used to improve the perfectly matched layers accuracy.

The scattered near-field pressure is evaluated along two semicircles shown in blue in figure 7. The two semicircles have a dimensionless radius of $1$ and are located in the $y_2=0$ and $y_2 =0.75$ planes, respectively. In the rest of this paper, they are referred to as the SC1 ($y_2=0$) and SC2 ($y_2 = 0.75$), respectively. In the FEM computation, the serration amplitude $h$, the frequency $k$ and the incident angle $\varTheta$ can all be varied. To facilitate comparison, the scattered pressure by a straight trailing edge is also computed and evaluated on the same semicircles.

Figure 7.

Two semicircles of radius $1$ denote the probe locations on which two scattered pressure are compared between the FEM and the analytical formula. The front semicircle is at $y_2=0$ while the back one is at ${y_2 = 0.75}$.

Figure 8 shows the comparison of the scattered pressure calculated by the FEM technique and the analytical Green's function at $M=0$ and $\varTheta ={\rm \pi} /4$. The scattered pressure values from both the straight (blue) and serrated (red) edges at various non-dimensional frequencies are shown. Figure 8(a,b) shows the results when $k=2$, from which we can see that the computed pressure distribution for straight edges agrees excellently with the analytical prediction. This ensures that the PMLs work satisfactorily and the grid is sufficiently fine to resolve the pressure field. However, the computed scattered pressure for serrated edges is significantly smaller than the analytical prediction. This is expected, as the assumption about $E_n(s)$ dependence is not expected to be valid at this frequency. As mentioned in § 2, as the frequency increases, the scattering becomes increasingly localized and the assumption is more likely to be valid. This is indeed the case, as shown in figure 8(c,d). We can see that the baseline results continue to agree excellently with analytical predictions, but the serrated results are now in better agreement with the analytical prediction with slight deviation at large angles. In particular, figure 8(d) shows a significant change in the near-field directivity shape on SC2 due to the use of serrations, and the Green's function can capture this change well apart from the small magnitude deviation at large observer angles. As the frequency increases to $k=50$, figure 8(e, f) shows that the scattered pressures obtained using the two methods agree well with each other both in terms of the shape and amplitude of the directivity patterns.

Figure 8.

Comparison of the analytical and FEM-calculated Green's function at $M = 0$ and $\varTheta ={\rm \pi} / 4$ for both baseline and serrated ($h=1$) trailing edges on the first (SC1) and second (SC2) semicircles at various frequencies: (a) $k=2$, SC1; (b) $k=2$, SC2; (c) $k=10$, SC1; (d) $k=10$, SC2; (e) $k=50$, SC1; ( f) $k=50$, SC2.

Figure 9 shows the comparison between the scattered pressure when the incident angle $\varTheta =3/4{\rm \pi}$ for both the baseline and serrated trailing edges. When the incident angle ${\varTheta =3/4{\rm \pi}}$, the directivity patterns of the scattered pressure are significantly different from those at $\varTheta ={\rm \pi} / 4$. Nevertheless, figure 9(a,b) still clearly shows the discrepancy between the FEM and analytical results for serrated edges, while figure 9(c,d) shows that the agreement is fairly good. Again at the highest frequency ${k=50}$, the two lines virtually collapse, as shown in figure 9(e, f), indicating good agreement between the analytical and calculated Green's functions.

Figure 9.

Comparison of the analytical and FEM-calculated Green's function at $M = 0$ and $\varTheta =3{\rm \pi} /4$ for both baseline and serrated ($h=1$) trailing edges on the first (SC1) and second (SC2) semicircles at various frequencies. Legends are the same as those shown in figure 8. (a) $k=2$, SC1; (b) $k=2$, SC2; (c) $k=10$, SC1; (d) $k=10$, SC2; (e) $k=50$, SC1; ( f) $k=50$, SC2.

In summary, we see that the analytical Green's function approximates the exact Green's function reasonably well at high frequencies. A rule of thumb for the valid regime may be taken as $kh > 10$. It is, however, worth noting that the incident plane wave is by no means limited to propagative waves. Because of analytical continuation, the Green's function must also work for evanescent waves, such as the plane-wave gusts used in TE noise modelling using Amiet's approach. In such cases, the convective Mach number of the gust is typically low and therefore it is the hydrodynamic wavenumber $k_1h$ that determines how localized is the scattering. Considering that this hydrodynamic wavenumber is often much larger than the acoustic wavenumber, especially for low-Mach-number applications, the $E_n(s)$ assumption would be more likely to hold so that (2.26) may be used to develop a 3-D TE noise prediction model. Note that although (2.26) appears complex, it can be simplified considerably when used to model TE noise because both $\varTheta$ and $\theta _i$ are equal to ${\rm \pi} / 2$ in the scattered surface pressure calculation. Therefore, it can be expected the resulting model can be cast into a relatively compact form that facilitates efficient evaluation.