2 Model formulation

Similar to the simplification introduced in the earlier work of the authors (Lyu et al. Reference Lyu, Dowling and Naqavi2017), the thickness and camber of the wing are neglected and the wing is therefore treated as a flat plate, as shown in figure 1. The flat plate has a swept trailing edge, with a mean chord $c$ and span $d$ , and is placed near a round jet nozzle. The centreline of the jet nozzle does not necessarily lie in the mid-span plane. However, as we will discuss in more detail in the rest of this paper, the nozzle does have to be sufficiently far away from either side edge of the flat plate, such that the side-edge scattering can be neglected. The bottom surface of the flat plate is $H$ above the jet centreline. The swept trailing edge is $L$ downstream from the jet nozzle exit, measured in the vertical plane containing the jet centreline. A Cartesian coordinate system has its origin fixed on the trailing edge in this vertical plane with $x_{1}$ in the streamwise direction, $x_{2}$ in the spanwise direction and $x_{3}$ in the direction normal to the plate. In this coordinate system, the swept trailing edge is specified by $x_{1}=F(x_{2})$ . Since the trailing edge is linearly swept,

(2.1) $$\begin{eqnarray}\displaystyle F(x_{2})=\unicode[STIX]{x1D70E}x_{2}, & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D70E}\equiv \tan \unicode[STIX]{x1D6FC}$ is a parameter specifying the sweep angle $\unicode[STIX]{x1D6FC}$ . In this coordinate system, the two side edges are located at $x_{3}=0,x_{2}=-l$ , where $0<l<d$ , and $x_{3}=0,x_{2}=d-l$ , respectively.

Figure 1.

A schematic illustration of the simplified model problem.

As discussed in the earlier study, outside the jet there is a region of near-field pressure fluctuations, which are primarily induced by the jet’s hydrodynamic instability waves and decay exponentially in the radial direction (Jordan & Colonius Reference Jordan and Colonius2013). Because of this exponential decay, they contribute very weakly to the far-field sound. However, when a surface with sharp edges is present in the near field of the jet, the jet instability waves’ previously non-radiating pressure field can be efficiently scattered into sound by the edges. This is the primary physical mechanism of the observed noise intensification at low frequencies. Strictly speaking, when a flat plate is placed near the jet, the instability waves would be different from those of an isolated jet. However, as discussed in earlier works (Bychkov & Faranosov Reference Bychkov and Faranosov2014; Lyu et al. Reference Lyu, Dowling and Naqavi2017), the influence of the flat plate on the instability characteristics is not significant provided $H$ is not so small as to significantly change the jet mean flow. We can therefore use the instability waves of an isolated jet as the input for the scattering problem considered in this paper. Note that the convection velocities of the instability waves of an isolated jet are found to be frequency dependent. For example, by analysing both large-eddy simulation (LES) and experimental data (Lyu & Dowling Reference Lyu and Dowling2017; Lyu et al. Reference Lyu, Dowling and Naqavi2017) the convection velocity $U_{c}$ is found to be significantly lower than $0.6U_{j}$ at low frequencies (e.g. $St<0.1$ ), where $U_{j}$ is the jet exit velocity.

We start to develop the model by assuming that the near-field pressure fluctuation is dominated by the evanescent wave of the first few azimuthal modes in the cylindrical coordinate system whose origin is on the jet centreline. As mentioned in § 1, the scattering mainly occurs in the local vicinity of the edge, and we may therefore neglect the axial growth, saturation and decay behaviour of the instability waves in the scattering process. The amplitude modulation may also contribute to the far-field sound directly through the wavenumber leakage (Tam & Viswanathan Reference Tam and Viswanathan2008; Jordan & Colonius Reference Jordan and Colonius2013), but this is much weaker than the scattered sound when the flat plate is placed close to the jet, in which this paper is particularly interested. We can therefore write the instability wave (omitting $\text{e}^{\text{i}\unicode[STIX]{x1D714}t}$ ) near the swept edge as (for more detail see Lyu et al. (Reference Lyu, Dowling and Naqavi2017))

(2.2) $$\begin{eqnarray}\displaystyle p^{\prime }(\unicode[STIX]{x1D714},\boldsymbol{x})=\mathop{\sum }_{m=-N}^{N}\hat{p}(\unicode[STIX]{x1D714},m)K_{m}(\unicode[STIX]{x1D6FE}_{c}r)\text{e}^{-\text{i}k_{1}x_{1}}\text{e}^{\text{i}m\unicode[STIX]{x1D703}}, & & \displaystyle\end{eqnarray}$$

where $\hat{p}(\unicode[STIX]{x1D714},m)$ denotes the amplitude of the pressure fluctuations of mode $m$ and frequency $\unicode[STIX]{x1D714}$ , $K_{m}$ is the $m$ th-order modified Bessel function of the second kind, $k_{1}$ is the streamwise wavenumber of the instability waves and the radius $r$ here is now defined as $\sqrt{x_{2}^{2}+(x_{3}+H)^{2}}$ . Note that in practice the amplitude $\hat{p}(\unicode[STIX]{x1D714},m)$ can be obtained through pressure measurements in the near field of an isolated jet. We will discuss this in detail at the end of this section. The convection velocity $U_{c}$ (and hence  $k_{1}$ ) is a function of $\unicode[STIX]{x1D714}$ and $m$ . Angle $\unicode[STIX]{x1D703}$ is the azimuthal angle (measured from the $x_{3}$ axis in the clockwise direction) and $N$ is a small integer; for example, according to the LES data, we find $N=1$ is sufficient (Lyu & Dowling Reference Lyu and Dowling2016; Lyu et al. Reference Lyu, Dowling and Naqavi2017). The convective radial decay rate $\unicode[STIX]{x1D6FE}_{c}$ , in the presence of ambient mean flow of Mach number $M_{a}$ , is given by (Amiet Reference Amiet1976; Lyu et al. Reference Lyu, Dowling and Naqavi2017)

(2.3) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FE}_{c}=\frac{\sqrt{(k_{1}\unicode[STIX]{x1D6FD}^{2}+kM_{a})^{2}-k^{2}}}{\unicode[STIX]{x1D6FD}}, & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D6FD}=\sqrt{1-M_{a}^{2}}$ and $k=\unicode[STIX]{x1D714}/c_{0}$ . We note here that (2.2) can also be written in terms of Hankel functions with an imaginary argument. But it can be shown that these two ways are equivalent to each other with appropriate choice of branch cut.

The jet instability waves will be scattered by the finite plate. However, when both side edges are sufficiently far away from the jet centreline, we can neglect the scattering of the instability waves by them (Lyu et al. Reference Lyu, Dowling and Naqavi2017). This is possible because the jet instability waves decay exponentially in the radial direction, and hence the strength of these waves weakens sufficiently fast. As mentioned in the preceding section, in this study, the jet centreline does not have to be in the mid-span plane of the plate. But we do require both $l$ and $d-l$ to be large enough to justify this assumption. Therefore, in the following derivation, we neglect the contribution of side-edge scattering. A previous study has shown that the leading-edge backscattering of the trailing-edge noise of an aerofoil can be neglected when the frequency is sufficiently high, i.e. when $kc>1$ (Roger & Moreau Reference Roger and Moreau2005), and this is shown to be suitable for installed jet noise as well (Lyu et al. Reference Lyu, Dowling and Naqavi2017). Therefore, in this study, we only take into account the scattering due to the sharp trailing edge of the swept plate.

When the flat plate is directly above the jet, the hypothetical incident pressure that would exist on the lower surface of the plate if the plate were absent is

(2.4) $$\begin{eqnarray}\displaystyle & & \displaystyle p_{h}(\unicode[STIX]{x1D714},x_{1},x_{2})=\mathop{\sum }_{m=-N}^{N}\hat{p}(\unicode[STIX]{x1D714},m)K_{m}(\unicode[STIX]{x1D6FE}_{c}\sqrt{x_{2}^{2}+H^{2}})\text{e}^{-\text{i}k_{1}x_{1}}\nonumber\\ \displaystyle & & \displaystyle \quad \times \left(\mathop{\sum }_{k=0}^{[|m|/2]}C_{|m|}^{2k}(-1)^{k}\frac{H^{|m|-2k}x_{2}^{2k}}{\sqrt{x_{2}^{2}+H^{2}}^{|m|}}+\text{i}\,\text{sgn}(m)\mathop{\sum }_{k=0}^{[(|m|-1)/2]}C_{|m|}^{2k+1}(-1)^{k}\frac{H^{|m|-2k-1}x_{2}^{2k+1}}{\sqrt{x_{2}^{2}+H^{2}}^{|m|}}\right),\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

where $[x]$ means taking the nearest integer that is not larger than $x$ and $C_{m}^{n}$ is the binomial coefficient, which results from the use of de Moivre’s theorem. This hypothetical incident pressure can be decomposed into a superposition of a series of plane waves, namely

(2.5) $$\begin{eqnarray}\displaystyle p_{h}(\unicode[STIX]{x1D714},x_{1},x_{2})=\mathop{\sum }_{m=-N}^{N}\int _{-\infty }^{\infty }\tilde{p}(\unicode[STIX]{x1D714},k_{2},m)\text{e}^{-\text{i}(k_{1}x_{1}+k_{2}x_{2})}\,\text{d}k_{2}, & & \displaystyle\end{eqnarray}$$

where

(2.6) $$\begin{eqnarray}\displaystyle & & \displaystyle \tilde{p}(\unicode[STIX]{x1D714},k_{2},m)=\frac{1}{\sqrt{2\unicode[STIX]{x03C0}}}\hat{p}(\unicode[STIX]{x1D714},m)\nonumber\\ \displaystyle & & \displaystyle \quad \times \left\{\mathop{\sum }_{k=0}^{[|m|/2]}C_{|m|}^{2k}H^{-2k+1/2}\unicode[STIX]{x1D6FE}_{c}^{-|m|}\frac{\text{d}^{2k}}{\text{d}k_{2}^{2k}}[(\unicode[STIX]{x1D6FE}_{c}^{2}+k_{2}^{2})^{1/2|m|-1/4}K_{|m|-1/2}(H\sqrt{\unicode[STIX]{x1D6FE}_{c}^{2}+k_{2}^{2}})]-\text{sgn}(m)\right.\nonumber\\ \displaystyle & & \displaystyle \quad \times \left.\mathop{\sum }_{k=0}^{[(|m|-1)/2]}C_{|m|}^{2k+1}H^{-2k+1/2}\unicode[STIX]{x1D6FE}_{c}^{-|m|}\frac{\text{d}^{2k}}{\text{d}k_{2}^{2k}}[k_{2}(\unicode[STIX]{x1D6FE}_{c}^{2}+k_{2}^{2})^{1/2|m|-3/4}K_{|m|-3/2}(H\sqrt{\unicode[STIX]{x1D6FE}_{c}^{2}+k_{2}^{2}})]\right\}.\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

For a plane-wave-like incident pressure given by $\tilde{p}(\unicode[STIX]{x1D714},k_{2},m)\text{e}^{-\text{i}(k_{1}x_{1}+k_{2}x_{2})}$ , the scattered pressure $p_{s}$ is governed by the convective wave equation (with ambient mean flow of Mach number $M_{a}$ )

(2.7) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FD}^{2}\frac{\unicode[STIX]{x2202}^{2}p_{s}}{\unicode[STIX]{x2202}x_{1}^{2}}+\frac{\unicode[STIX]{x2202}^{2}p_{s}}{\unicode[STIX]{x2202}x_{2}^{2}}+\frac{\unicode[STIX]{x2202}^{2}p_{s}}{\unicode[STIX]{x2202}x_{3}^{2}}-2iM_{a}\frac{\unicode[STIX]{x2202}p_{s}}{\unicode[STIX]{x2202}x_{1}}+k^{2}p_{s}=0, & & \displaystyle\end{eqnarray}$$

subject to boundary conditions

(2.8) $$\begin{eqnarray}\displaystyle & p_{s}=-\tilde{p}\text{e}^{-\text{i}(k_{1}x_{1}+k_{2}x_{2})},\quad x_{1}>F(x_{2}) & \displaystyle\end{eqnarray}$$

(2.9) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}p_{s}}{\unicode[STIX]{x2202}x_{3}}=0,\quad x_{1}<F(x_{2}). & \displaystyle\end{eqnarray}$$

With the coordinate transformation $x_{1}^{\prime }=x_{1}-F(x_{2})$ , $x_{2}^{\prime }=x_{2}$ and $x_{3}^{\prime }=\sqrt{\unicode[STIX]{x1D6FD}^{2}+\unicode[STIX]{x1D70E}^{2}}x_{3}$ and the new variable definition $\unicode[STIX]{x1D719}=p_{s}\text{e}^{\text{i}\bar{k}_{2}x_{2}^{\prime }}\text{e}^{\text{i}(\bar{k}_{2}\unicode[STIX]{x1D70E}-kM_{a})/(\unicode[STIX]{x1D6FD}^{2}+\unicode[STIX]{x1D70E}^{2})x_{1}^{\prime }}$ , where $\bar{k}_{2}=k_{2}+k_{1}\unicode[STIX]{x1D70E}$ , the governing equation reduces to

(2.10) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}^{2}\unicode[STIX]{x1D719}}{\unicode[STIX]{x2202}{x_{1}^{\prime }}^{2}}+\frac{\unicode[STIX]{x2202}^{2}\unicode[STIX]{x1D719}}{\unicode[STIX]{x2202}{x_{3}^{\prime }}^{2}}+K^{2}\unicode[STIX]{x1D719}=0, & & \displaystyle\end{eqnarray}$$

where

(2.11) $$\begin{eqnarray}\displaystyle K=\frac{\sqrt{k^{2}-\bar{k}_{2}^{2}+(\bar{k}_{2}M_{a}-k\unicode[STIX]{x1D70E})^{2}}}{\unicode[STIX]{x1D6FD}^{2}+\unicode[STIX]{x1D70E}^{2}}. & & \displaystyle\end{eqnarray}$$

The boundary conditions now read

(2.12) $$\begin{eqnarray}\displaystyle & \unicode[STIX]{x1D719}=-\tilde{p}\text{e}^{-\text{i}k_{1}x_{1}^{\prime }}\text{e}^{\text{i}(\bar{k}_{2}\unicode[STIX]{x1D70E}-kM_{a})/(\unicode[STIX]{x1D6FD}^{2}+\unicode[STIX]{x1D70E}^{2})x_{1}^{\prime }},\quad x_{1}^{\prime }>0 & \displaystyle\end{eqnarray}$$

(2.13) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}}{\unicode[STIX]{x2202}x_{3}^{\prime }}=0,\quad x_{1}^{\prime }<0. & \displaystyle\end{eqnarray}$$

The solution to (2.10) can be readily found using the Schwarzchild technique (Lyu, Azarpeyvand & Sinayoko Reference Lyu, Azarpeyvand and Sinayoko2016; Lyu et al. Reference Lyu, Dowling and Naqavi2017). By transforming back to the original coordinate system, one can show that the scattered pressure is

(2.14) $$\begin{eqnarray}\displaystyle p_{s}=\tilde{p}(\unicode[STIX]{x1D714},k_{2},m)\text{e}^{-\text{i}(k_{1}x_{1}+k_{2}x_{2})}[(1+\text{i})E_{0}(-\unicode[STIX]{x1D707}(x_{1}-\unicode[STIX]{x1D70E}x_{2}))-1], & & \displaystyle\end{eqnarray}$$

where

(2.15a,b ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D707}=k_{1}+\frac{(kM_{a}-\bar{k}_{2}\unicode[STIX]{x1D70E})}{\unicode[STIX]{x1D6FD}^{2}+\unicode[STIX]{x1D70E}^{2}}+\frac{\sqrt{k^{2}-\bar{k}_{2}^{2}+(\bar{k}_{2}M_{a}-k\unicode[STIX]{x1D70E})^{2}}}{\unicode[STIX]{x1D6FD}^{2}+\unicode[STIX]{x1D70E}^{2}},\quad E_{0}(x)=\int _{0}^{x}\frac{\text{e}^{-\text{i}t}}{\sqrt{2\unicode[STIX]{x03C0}t}}\,\text{d}t. & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

Equation (2.14) shows that the scattered surface pressure has a phase variation along the spanwise ( $x_{2}$ ) direction even when $k_{2}=0$ due to the $x_{2}$ dependence in the square bracket, which does not occur in the case of a straight trailing edge. To show this more clearly, we have plotted a typical pressure distribution on the surface of a swept flat plate according to the real part of (2.14) in figure 2 for a plane-wave-like incident pressure $\tilde{p}(\unicode[STIX]{x1D714},k_{2},m)\text{e}^{-\text{i}(k_{1}x_{1}+k_{2}x_{2})}$ with $k_{2}=0$ . The phase variation along the trailing edge of the flat plate can be clearly observed. The theory of Curle (Reference Curle1955) shows that the scattered surface pressure acts like a distribution of dipole sources, and the far-field sound can be obtained by calculating the sound generated from this distribution of dipole sources. The substantial phase variation exhibited by figure 2 implies that some of these dipole sources are in anti-phase and would result in destructive (can also be constructive depending on the position of the observer) interference, which, as will be discussed in detail in the rest of this paper, is the primary mechanism of installed jet noise reduction observed in the experiments.

Figure 2.

The scattered pressure on the surface of a swept flat plate induced by a plane-wave-like incident pressure. The contour is obtained from the real part of (2.14) with $\tilde{p}=1$ , $kc=10$ , $k_{2}=0$ , $M_{a}=0$ , $M=0.5$ , $U_{c}=0.6U_{j}$ and $\unicode[STIX]{x1D6FC}=20^{\circ }$ .

Because of linearity, the far-field sound induced by the incident pressure given by (2.5) can be readily calculated according to the theory of Kirchhoff and Curle (Lyu et al. Reference Lyu, Azarpeyvand and Sinayoko2016; Lyu & Dowling Reference Lyu and Dowling2017) to be

(2.16) $$\begin{eqnarray}\displaystyle p_{f}(\unicode[STIX]{x1D714},\boldsymbol{x}) & = & \displaystyle (1+\text{i})\frac{\unicode[STIX]{x1D714}x_{3}}{\unicode[STIX]{x03C0}c_{0}S_{0}^{2}}\text{e}^{\text{i}(k/\unicode[STIX]{x1D6FD}^{2})(M_{a}x_{1}-S_{0})}\nonumber\\ \displaystyle & & \displaystyle \times \mathop{\sum }_{m=-N}^{N}\int _{-\infty }^{\infty }\text{e}^{\text{i}\unicode[STIX]{x1D705}(l-d/2)}\frac{\sin [\unicode[STIX]{x1D705}d/2]}{\unicode[STIX]{x1D705}}\frac{1}{\unicode[STIX]{x1D707}_{A}}\unicode[STIX]{x1D6E4}(c,\unicode[STIX]{x1D707},\unicode[STIX]{x1D707}_{A})\tilde{p}(\unicode[STIX]{x1D714},k_{2},m)\,\text{d}k_{2},\end{eqnarray}$$

where $\unicode[STIX]{x1D705}=\bar{k}_{2}-kx_{2}/S_{0}-(k/\unicode[STIX]{x1D6FD}^{2})(x_{1}/S_{0}-M_{a})\unicode[STIX]{x1D70E}$ , $x_{1}$ , $x_{2}$ and $x_{3}$ now denote the Cartesian coordinates of the far-field observer, and the properly bounded function $\unicode[STIX]{x1D6E4}$ in (2.16) is defined as

(2.17) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6E4}(x,\unicode[STIX]{x1D707},\unicode[STIX]{x1D707}_{A})=\text{e}^{\text{i}\unicode[STIX]{x1D707}_{A}x}E_{0}(\unicode[STIX]{x1D707}x)-\sqrt{\frac{\unicode[STIX]{x1D707}}{\unicode[STIX]{x1D707}-\unicode[STIX]{x1D707}_{A}}}E_{0}[(\unicode[STIX]{x1D707}-\unicode[STIX]{x1D707}_{A})x]+\frac{1}{1+\text{i}}(1-\text{e}^{\text{i}\unicode[STIX]{x1D707}_{A}x}), & & \displaystyle\end{eqnarray}$$

where

(2.18) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\displaystyle S_{0}=\sqrt{x_{1}^{2}+\unicode[STIX]{x1D6FD}^{2}(x_{2}^{2}+x_{3}^{2})},\\ \displaystyle \unicode[STIX]{x1D707}_{A}=k_{1}+\frac{k}{\unicode[STIX]{x1D6FD}^{2}}\left(M_{a}-\frac{x_{1}}{S_{0}}\right).\end{array}\right\} & & \displaystyle\end{eqnarray}$$

In reaching (2.16), we have made a simplification by assuming that the leading edge is also swept by an angle of $\unicode[STIX]{x1D6FC}$ . This is permissible because, as mentioned earlier, the leading-edge backscattering is weak enough to be neglected. Consequently, replacing a straight leading edge with a swept one should cause minimal change to the far-field sound. It should be noted that (2.16) is due to the contribution of the scattered pressure only. In order to take the incident wave contribution into consideration, the term 1 in the last parentheses on the right-hand side of (2.17) defining the function $\unicode[STIX]{x1D6E4}(x,\unicode[STIX]{x1D707},\unicode[STIX]{x1D707}_{A})$ should be omitted (Amiet Reference Amiet1978).

Equation (2.16) can be further simplified by noting that normally aircraft wings have a large span-to-chord ratio, and $d$ can be quite large compared to the sound wavelength at the frequency of peak noise enhancement in the low-frequency regime. Therefore we use

(2.19) $$\begin{eqnarray}\displaystyle \lim _{d\rightarrow \infty }\frac{\sin [\unicode[STIX]{x1D705}d/2]}{\unicode[STIX]{x03C0}\unicode[STIX]{x1D705}}=\unicode[STIX]{x1D6FF}(\unicode[STIX]{x1D705}) & & \displaystyle\end{eqnarray}$$

to simplify (2.16) to

(2.20) $$\begin{eqnarray}\displaystyle p_{f}(\unicode[STIX]{x1D714},\boldsymbol{x})=(1+\text{i})\frac{\unicode[STIX]{x1D714}x_{3}}{c_{0}S_{0}^{2}}\text{e}^{\text{i}(k/\unicode[STIX]{x1D6FD}^{2})(M_{a}x_{1}-S_{0})}\mathop{\sum }_{m=-N}^{N}\frac{1}{\unicode[STIX]{x1D707}_{A}}\unicode[STIX]{x1D6E4}(c,\unicode[STIX]{x1D707}_{o},\unicode[STIX]{x1D707}_{A})\tilde{p}(\unicode[STIX]{x1D714},k_{o},m), & & \displaystyle\end{eqnarray}$$

where $k_{o}=-k_{1}\unicode[STIX]{x1D70E}+kx_{2}/S_{0}+(k/\unicode[STIX]{x1D6FD}^{2})(x_{1}/S_{0}-M_{a})\unicode[STIX]{x1D70E}$ and $\unicode[STIX]{x1D707}_{o}=\unicode[STIX]{x1D707}|_{k_{2}=k_{o}}$ . We see that when the span is large, an effective destructive interference along the span is possible and (2.20) shows that only the plane wave with a spanwise wavenumber $k_{o}$ can be heard in the far field. Plane waves with other values of $k_{2}$ do not contribute to the far-field sound due to effective destructive interference (see figure 2 for the example of $k_{2}=0$ ). However, the validity of this result hinges on the assumption of a sufficiently long span, which is used to obtain (2.19) and (2.20). Therefore, it is expected that a longer span is required for it to be valid at lower frequencies. Note that $l$ does not enter (2.20). This implies that, provided side-edge scattering can be neglected, the far-field sound is not sensitive to the spanwise position of the plate relative to the jet nozzle.

The far-field sound power spectrum can be found by formulating the ensemble average to be

(2.21) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6F7}_{N}(\unicode[STIX]{x1D714},\boldsymbol{x}) & = & \displaystyle 2\left[\frac{\unicode[STIX]{x1D714}x_{3}}{c_{0}S_{0}^{2}}\right]^{2}\mathop{\sum }_{m=-N}^{N}\mathop{\sum }_{m^{\prime }=-N}^{N}\left.\frac{\unicode[STIX]{x1D6E4}(c,\unicode[STIX]{x1D707}_{o},\unicode[STIX]{x1D707}_{A})}{\unicode[STIX]{x1D707}_{A}}\right|_{m}\left.\frac{\unicode[STIX]{x1D6E4}^{\ast }(c,\unicode[STIX]{x1D707}_{o},\unicode[STIX]{x1D707}_{A})}{\unicode[STIX]{x1D707}_{A}^{\ast }}\right|_{m^{\prime }}\nonumber\\ \displaystyle & & \displaystyle \times \,\lim _{T\rightarrow \infty }\frac{\unicode[STIX]{x03C0}}{T}\,\overline{\tilde{p}(\unicode[STIX]{x1D714},k_{o},m)\tilde{p}^{\ast }(\unicode[STIX]{x1D714},k_{o},m^{\prime })},\end{eqnarray}$$

where $2T$ (Amiet Reference Amiet1975) is the temporal interval used to perform the Fourier transformation to obtain $p_{f}(\unicode[STIX]{x1D714},\boldsymbol{x})$ and the overbar denotes ensemble average. We note that (2.21) is the result obtained using Amiet’s approach where the interval of Fourier transformation is finite. When the interval is infinite, we have (Dowling & Williams Reference Dowling and Williams1983; Howe Reference Howe1991)

(2.22) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6F7}_{N}(\unicode[STIX]{x1D714},\boldsymbol{x})=\int _{-\infty }^{\infty }\overline{p_{f}(\unicode[STIX]{x1D714},\boldsymbol{x})p_{f}^{\ast }(\unicode[STIX]{x1D714}^{\prime },\boldsymbol{x})}\,\text{d}\unicode[STIX]{x1D714}^{\prime }. & & \displaystyle\end{eqnarray}$$

The two approaches, however, yield identical answers for statistically stationary signals, and we use the former one in this paper.

Substituting (2.6) into the above equation yields

(2.23) $$\begin{eqnarray}\displaystyle & & \displaystyle \unicode[STIX]{x1D6F7}_{N}(\unicode[STIX]{x1D714},\boldsymbol{x})=\frac{1}{\unicode[STIX]{x03C0}}\left[\frac{\unicode[STIX]{x1D714}x_{3}}{c_{0}S_{0}^{2}}\right]^{2}\mathop{\sum }_{m=-N}^{N}\left|\frac{\unicode[STIX]{x1D6E4}(c,\unicode[STIX]{x1D707}_{o},\unicode[STIX]{x1D707}_{A})}{\unicode[STIX]{x1D707}_{A}}\right|^{2}\unicode[STIX]{x1D6F1}(\unicode[STIX]{x1D714},m)\nonumber\\ \displaystyle & & \displaystyle \quad \times \left\{\mathop{\sum }_{k=0}^{[|m|/2]}C_{|m|}^{2k}H^{-2k+1/2}\unicode[STIX]{x1D6FE}_{c}^{-|m|}\frac{\text{d}^{2k}}{\text{d}k_{o}^{2k}}[(\unicode[STIX]{x1D6FE}_{c}^{2}+k_{o}^{2})^{1/2|m|-1/4}K_{|m|-1/2}(H\sqrt{\unicode[STIX]{x1D6FE}_{c}^{2}+k_{o}^{2}})]-\text{sgn}(m)\right.\nonumber\\ \displaystyle & & \displaystyle \quad \times \left.\mathop{\sum }_{k=0}^{[(|m|-1)/2]}C_{|m|}^{2k+1}H^{-2k+1/2}\unicode[STIX]{x1D6FE}_{c}^{-|m|}\frac{\text{d}^{2k}}{\text{d}k_{o}^{2k}}[k_{o}(\unicode[STIX]{x1D6FE}_{c}^{2}+k_{o}^{2})^{1/2|m|-3/4}K_{|m|-3/2}(H\sqrt{\unicode[STIX]{x1D6FE}_{c}^{2}+k_{o}^{2}})]\right\}^{2},\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D6F1}(\unicode[STIX]{x1D714},m)$ denotes the power spectrum of $m$ th-order near-field evanescent waves and is given by $\unicode[STIX]{x1D6F1}(\unicode[STIX]{x1D714},m)=\lim _{T\rightarrow \infty }(\unicode[STIX]{x03C0}/T)\overline{\hat{p}(\unicode[STIX]{x1D714},m)\hat{p}^{\ast }(\unicode[STIX]{x1D714},m)}$ with $2T$ being the temporal interval of the Fourier transformation used to obtain $\hat{p}(\unicode[STIX]{x1D714},m)$ . Similar to (2.22), $\unicode[STIX]{x1D6F1}(\unicode[STIX]{x1D714},m)$ can also be formulated as $\unicode[STIX]{x1D6F1}(\unicode[STIX]{x1D714},m)=\int _{-\infty }^{\infty }\overline{\hat{p}(\unicode[STIX]{x1D714},m)\hat{p}^{\ast }(\unicode[STIX]{x1D714}^{\prime },m)}\,\text{d}\unicode[STIX]{x1D714}^{\prime }$ . Equation (2.23) is the generic form of near-field scattering model. However, further simplifications can be made in practical cases. First, if we assume that the fluctuation is symmetric with respect to $m$ , i.e. $\unicode[STIX]{x1D6F1}(\unicode[STIX]{x1D714},m)=\unicode[STIX]{x1D6F1}(\unicode[STIX]{x1D714},-m)$ , equation (2.23) can be further simplified to

(2.24) $$\begin{eqnarray}\displaystyle & & \displaystyle \unicode[STIX]{x1D6F7}_{N}(\unicode[STIX]{x1D714},\boldsymbol{x})=\frac{1}{\unicode[STIX]{x03C0}}\left[\frac{\unicode[STIX]{x1D714}x_{3}}{c_{0}S_{0}^{2}}\right]^{2}\mathop{\sum }_{m=0}^{N}\left|\frac{\unicode[STIX]{x1D6E4}(c,\unicode[STIX]{x1D707}_{o},\unicode[STIX]{x1D707}_{A})}{\unicode[STIX]{x1D707}_{A}}\right|^{2}\unicode[STIX]{x1D6F1}_{s}(\unicode[STIX]{x1D714},m)\nonumber\\ \displaystyle & & \displaystyle \quad \times \left\{\!\left(\mathop{\sum }_{k=0}^{[|m|/2]}C_{|m|}^{2k}H^{-2k+1/2}\unicode[STIX]{x1D6FE}_{c}^{-|m|}\frac{\text{d}^{2k}}{\text{d}k_{o}^{2k}}[(\unicode[STIX]{x1D6FE}_{c}^{2}+k_{o}^{2})^{1/2|m|-1/4}K_{|m|-1/2}(H\sqrt{\unicode[STIX]{x1D6FE}_{c}^{2}+k_{o}^{2}})]\right)^{2}\right.\nonumber\\ \displaystyle & & \displaystyle \quad +\left.\left(\mathop{\sum }_{k=0}^{[|m|-1/2]}C_{|m|}^{2k+1}H^{-2k+1/2}\unicode[STIX]{x1D6FE}_{c}^{-|m|}\frac{\text{d}^{2k}}{\text{d}k_{o}^{2k}}[k_{o}(\unicode[STIX]{x1D6FE}_{c}^{2}+k_{o}^{2})^{1/2|m|-3/4}K_{|m|-3/2}(H\sqrt{\unicode[STIX]{x1D6FE}_{c}^{2}+k_{o}^{2}})]\right)^{2}\right\},\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D6F1}_{s}(\unicode[STIX]{x1D714},m)$ is $m$ th single-sided modal PSD, i.e. $\unicode[STIX]{x1D6F1}_{s}(\unicode[STIX]{x1D714},m)=\unicode[STIX]{x1D6F1}(\unicode[STIX]{x1D714},m)+\unicode[STIX]{x1D6F1}(\unicode[STIX]{x1D714},-m)$ for $m\neq 0$ . If we assume that only the $0$ and $1(-1)$ modes are significant, we can show the far-field sound spectral density in a simplified form as

(2.25) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6F7}_{N}(\unicode[STIX]{x1D714},\boldsymbol{x})\approx {\mathcal{H}}_{0}(\unicode[STIX]{x1D714},\boldsymbol{x})\unicode[STIX]{x1D6F1}_{s}(\unicode[STIX]{x1D714},0)+{\mathcal{H}}_{1}(\unicode[STIX]{x1D714},\boldsymbol{x})\unicode[STIX]{x1D6F1}_{s}(\unicode[STIX]{x1D714},1), & & \displaystyle\end{eqnarray}$$

where

(2.26) $$\begin{eqnarray}\displaystyle & \displaystyle {\mathcal{H}}_{0}(\unicode[STIX]{x1D714},\boldsymbol{x})=\left[\frac{\unicode[STIX]{x1D714}x_{3}}{c_{0}S_{0}^{2}}\right]^{2}\left\{\left|\frac{\unicode[STIX]{x1D6E4}(c,\unicode[STIX]{x1D707}_{o},\unicode[STIX]{x1D707}_{A})}{\unicode[STIX]{x1D707}_{A}}\right|^{2}\frac{\text{e}^{-2H\sqrt{\unicode[STIX]{x1D6FE}_{c}^{2}+k_{o}^{2}}}}{2(\unicode[STIX]{x1D6FE}_{c}^{2}+k_{o}^{2})}\right\}_{m=0}, & \displaystyle\end{eqnarray}$$

(2.27) $$\begin{eqnarray}\displaystyle & \displaystyle {\mathcal{H}}_{1}(\unicode[STIX]{x1D714},\boldsymbol{x})=\left[\frac{\unicode[STIX]{x1D714}x_{3}}{c_{0}S_{0}^{2}}\right]^{2}\left\{\left|\frac{\unicode[STIX]{x1D6E4}(c,\unicode[STIX]{x1D707}_{o},\unicode[STIX]{x1D707}_{A})}{\unicode[STIX]{x1D707}_{A}}\right|^{2}\frac{\text{e}^{-2H\sqrt{\unicode[STIX]{x1D6FE}_{c}^{2}+k_{o}^{2}}}}{2\unicode[STIX]{x1D6FE}_{c}^{2}}\left(1+\frac{k_{o}^{2}}{k_{o}^{2}+\unicode[STIX]{x1D6FE}_{c}^{2}}\right)\right\}_{m=1}. & \displaystyle\end{eqnarray}$$

In the above equations, the near-field pressure spectra $\unicode[STIX]{x1D6F1}_{s}(\unicode[STIX]{x1D714},m)$ need to be known in order to predict the far-field sound PSD. These spectra could be obtained from simple models validated by experiments or LES simulations. Let $\unicode[STIX]{x1D6F1}_{0}(\unicode[STIX]{x1D714},0)$ and $\unicode[STIX]{x1D6F1}_{0}(\unicode[STIX]{x1D714},1)$ denote the zeroth- and first-order single-sided modal PSDs measured at a location of $r=r_{0}$ in the near field of the jet. Then it follows that $\unicode[STIX]{x1D6F1}_{s}(\unicode[STIX]{x1D714},0)=\unicode[STIX]{x1D6F1}_{0}(\unicode[STIX]{x1D714},0)/K_{0}^{2}(\unicode[STIX]{x1D6FE}_{c}r_{0})$ and $\unicode[STIX]{x1D6F1}_{s}(\unicode[STIX]{x1D714},1)=\unicode[STIX]{x1D6F1}_{0}(\unicode[STIX]{x1D714},1)/K_{1}^{2}(\unicode[STIX]{x1D6FE}_{c}r_{0})$ . Therefore, the input of this model can be found by measuring the near-field pressure fluctuations using at least two microphones (to resolve mode 0 and mode 1) at suitable radial locations. To ensure that acoustic fluctuations are negligible, one must choose $r_{0}$ to be small such that the microphones are sufficiently close to the jet. Since near-field pressure spectra vary with axial position, it is desirable to put the microphones at the position where the trailing edge of the flat plate would be (in the mid-span plane) if a swept flat plate were to be present, i.e. the place where the incident wave is to be scattered most strongly.

5 Comparison with previous experimental results

The model can now be compared against the experiment carried out by Piantanida et al. (Reference Piantanida, Jaunet, Huber, Wolf, Jordan and Cavalieri2016). In the experiment, the jet diameter $D$ was 0.05 m and the jet Mach number was 0.4. Four flat plates were used, and their trailing edges were tilted by $0^{\circ }$ , $15^{\circ }$ , $30^{\circ }$ and $45^{\circ }$ , respectively. All these plates shared the same average chord $c=9D$ and span $d=15D$ . During the experiment, these plates were placed near the jet at $H=1.5D$ and $L=4D$ and the jet centreline was aligned with the middle span of these plates. The azimuthal directivity at $St=0.2$ was measured at a distance of $14.3D$ from the jet centreline, at the axial location $x_{1}=0$ .

To compare our model with experimental results, the contribution from the Lighthill quadrupole sources is first obtained according to the results shown in the earlier work by Lyu et al. (Reference Lyu, Dowling and Naqavi2017). The validity of doing so has been discussed in § 4. In fact, the scattering of the quadrupole sources by a semi-infinite flat plate with a swept trailing edge can be modelled analytically. But having recognized that it is virtually the same as that for the straight edge in the frequency range of interest, and that the primary purpose of this paper is to model the noise due to the scattering of jet instability waves, such an endeavour seems to be unnecessary. Secondly, the contribution due to the scattering of the near-field instability waves is obtained from (2.25). The near-field spectrum at only $St=0.2$ (one point on the full spectrum) is needed. Because we do not yet have an experimental database for such a spectrum for a jet of Mach number 0.4, the amplitude of such a spectrum at $St=0.2$ is chosen such that the far-field sound for the straight edge matches the corresponding experimental results. This amplitude is subsequently used to predict the results for swept edges.

Figure 9.

Comparison of the azimuthal noise directivity (PSD in dB with a reference value $4\times 10^{-10}~\text{Pa}^{2}~\text{Hz}^{-1}$ ) for a fixed polar angle of $90^{\circ }$ between the model and experiment when $\unicode[STIX]{x1D6FC}=15^{\circ }$ ; blue solid: prediction for straight edge; red dashed: prediction for swept edge; yellow solid with markers: measurement for straight edge; purple dashed with markers: prediction for swept edge.

Results when the trailing edge is tilted by $15^{\circ }$ are shown in figure 9. The experimental results are shown only on the shielded side of the plate. This is because the scattered sound on this side is not affected by the refraction effect of the jet plume. As we can see from the figure, the predicted directivity shape for the straight edge agrees well with the experimental result. When the straight edge is replaced with a swept edge, the experiment reveals that a noticeable noise reduction can be achieved. In particular, greater reduction occurs at large azimuthal angles (close to $180^{\circ }$ ). At low azimuthal angles (close to $0^{\circ }$ ), a slight noise increase occurs. This difference in the noise reduction effectively changes the directivity pattern from a symmetric to titled shape. Such a change is well captured by the model, especially at large azimuthal angles. The slight disagreement at low azimuthal angles, however, could be caused by the side edges of the plates and/or the slow variation of jet instability spectra along the axial direction (because the trailing edge is swept, so strictly speaking the scattering occurs locally at different axial locations, but the slow spectra variation because of this is ignored in the model).

Figure 10.

Comparison of the azimuthal noise directivity (PSD in dB with a reference value $4\times 10^{-10}~\text{Pa}^{2}~\text{Hz}^{-1}$ ) for a fixed polar angle of $90^{\circ }$ between the model and experiment when $\unicode[STIX]{x1D6FC}=30^{\circ }$ ; blue solid: prediction for straight edge; red dashed: prediction for swept edge; yellow solid with markers: measurement for straight edge; purple dashed with markers: prediction for swept edge.

When the sweep angle increases to $30^{\circ }$ , as shown in figure 10, the experimental noise reduction is more pronounced. The change to the directivity from a symmetric to tilted pattern remains the same, and at low azimuthal angles a slight noise increase still occurs. The model predicts both behaviours very well. In particular, the predicted noise spectrum in the large azimuthal angle range has a nearly perfect agreement with the experimental result. The agreement at low azimuthal angles is not quite as good as that at high angles, and this might have been for the same reasons as described in the preceding paragraph.

Figure 11.

Comparison of the azimuthal noise directivity (PSD in dB with a reference value $4\times 10^{-10}~\text{Pa}^{2}~\text{Hz}^{-1}$ ) for a fixed polar angle of $90^{\circ }$ between the model and experiment when $\unicode[STIX]{x1D6FC}=45^{\circ }$ ; blue solid: prediction for straight edge; red dashed: prediction for swept edge; yellow solid with markers: measurement for straight edge; purple dashed with markers: prediction for swept edge.

Figure 11 shows the comparison for the largest sweep angle, i.e. $45^{\circ }$ . The maximum noise reduction, up to 8 dB, occurs at around the azimuthal angle $130^{\circ }$ . The agreement between the experimental results and the model predictions continues to be good. We note again that deviation occurs mostly at low azimuthal angles, with possible causes discussed above. The good agreement shown in this section suggests that the current model can capture the scattering physics very well, and also demonstrates that the destructive interference is the main mechanism of sound reduction by using aircraft wings with swept trailing edges.