# Aeroacoustics of vortex-wedge interaction

Marios Spiropoulos, Florent Margnat, Vincent Valeau, Peter Jordan

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Marios Spiropoulos, Florent Margnat, Vincent Valeau, Peter Jordan. Aeroacoustics of vortex-wedge interaction. Journal of Fluid Mechanics, 2025, 1019, pp.A43. 10.1017/jfm.2025.10580. hal-05374913

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# Aeroacoustics of vortex-wedge interaction

## Marios I. Spiropoulos¹†, Florent Margnat¹, Vincent Valeau¹ and Peter Jordan¹

¹Institute PPRIME UPR 3346, CNRS - University of Poitiers - ENSMA, 86022, Poitiers, France.

(Received xx; revised xx; accepted xx)

We consider the vortex-wedge interaction problem, taking as departure point Howe's model of a point vortex interacting with a semi-infinite half plane, where the vortex-path is influenced by its image and a closed-form analytical solution is obtained for the sound field. We generalise Howe's model to consider wedges of arbitrary angles and explore the influence of vortex circulation, distance from the edge and the wedge half-angle. The effect of wedge angle on sound emission involves a reduced amplitude of the latter as the former is increased. An extension of the model is proposed to account for convection effects by a non-zero ambient flow. We identify a non-dimensional parameter that characterises the vortex kinematics close to the edge and the associated acoustic effect: high and low values of the parameter correspond, respectively, to high- and low-amplitude sound emission of high- and low-frequency.

## 1. Introduction

Sound generated by the interaction of turbulent flows with edges is prevalent in many industrial applications: side-edge noise of flaps (Hardin 1980; Guo 2011), airfoil-noise (Howe 2001; Roger & Moreau 2005; Moreau & Roger 2009; Lee et al. 2021), jet-flap interaction noise (Cavalieri et al. 2014; Jordan et al. 2018; Huber et al. 2024) and whistling of side-view mirrors (Stoffel et al. 2025), to cite just a few examples. The problem of trailing-edge noise has, in particular, received a lot of attention and there is a substantial number of associated theoretical and experimental studies. Hybrid numerical approaches have been proposed where the flow-field containing the sound sources is obtained by Unsteady Reynolds Averaged Navier Stokes (URANS) simulations, Large Eddy Simulations (LES) or Direct Numerical Simulations (DNS) and the sound in the far-field computed using acoustic analogies and the tailored Green functions (Wang 2000). The first review on trailing edge noise was conducted by Howe (1978) who considered theoretical developments, while detailed reviews on the subject can be found in the works of Doolan & Moreau (2015) for experimental studies with application to wind-turbine noise and Lee et al. (2021) for a more global classification of numerical, experimental & theoretical approaches on trailing-edge noise and its control.

Theoretical models based on Lighthill's acoustic analogy (Lighthill 1952) and the works of Curle (1955), Ffowcs Williams & Hawkings (1969) or the theory of Vortex Sound (Powell 1964; Howe 1975a) can shed light on the mechanism of aerodynamic noise generation for wall-bounded flows and many of these have been used for the problem of airfoil noise. The sound due to the interaction of a flow with geometrical objects is mostly generated at the

<footnote>† Email address for correspondence: marios-ioannis.spiropoulos@ensma.fr</footnote>
edges (Roger et al. 2016). The scattering properties of geometries are often taken into account by considering the edges as semi-infinite half planes or wedges, for which exact (Macdonald 1915; Biot & Tolstoy 1957) and approximate analytical solutions (Menounou & Nikolaou 2017; Menounou et al. 2023) exist, and the flow-structures interacting with the object are frequently accounted for via a sound-source term crafted according to the acoustic analogy considered. Such approaches have led to fundamental results for the aeroacoustics of turbulent flows interacting with edges. For instance, Ffowcs Williams & Hall (1970) applied Lighthill's theory to show that compact quadrupole sources close to the edge of a semi-infinite half plane lead to an acoustic intensity that scales with the fifth power of the turbulent velocity. Crighton & Leppington (1971) calculated scaling laws for acoustically compact quadrupole and dipole sources for soft and hard wedge-like objects by considering that the Green function can be obtained as a solution of the Laplace equation when the observer is many wavelengths away and the object is acoustically compact. Crighton (1972) examined the aerodynamic sound generated by a line vortex with its axis parallel to the edge of a semi-infinite rigid half plane. Howe (Howe 1975a,b), by introducing the concept of the compact Green functions, extended this idea to two-dimensional problems in the time domain and obtained simple closed-form analytical solutions that clarify the physical mechanisms underpinning aerodynamic sound generation (Howe 1998, 2003). Priddin et al. (2018) illustrated how simplified models derived in that framework can be used to investigate methods of passive and active control of the trailing edge noise. The aeroacoustics of the half-plane/vortex system was also studied by Kambe et al. (1985) who performed a theoretical and experimental study, considering that the vortex kinematics are entirely governed by free-stream convection. Their theoretical model was then extended to wedges of arbitrary angles by Chang & Chen (1994) and to porous half-planes by Chen et al. (2022). Despite the fact that previous works have treated the problem of sound generation by vortex-wedge interactions (Crighton & Leppington 1970; Chang & Chen 1994) and resulted in useful scaling laws, to the best of the authors' knowledge, no explicit closed form, analytical formulae are presented that describe the sound field produced when a vortex induced by its image follows a trajectory in the vicinity of a semi-infinite rigid wedge. Furthermore, most of the models related to point vortex-edge interactions consider either that the point vortex is purely convected by a free-stream (Kambe et al. 1985; Chang & Chen 1994; Chen et al. 2022) or its path is solely influenced by its image as in the works of Crighton (1972); Howe (2003). Kambe et al. (1985) and more recently Chen et al. (2022) mentioned that the path of a convected vortex ring alters its trajectory when passing in the vicinity of a half-plane and this path-deviation depends on the vortex circulation. However, a more detailed investigation on the competing mechanisms of convection and edge effects on the vortex path and sound field was not carried out.

In the present work, we propose a generalisation of Howe's vortex half-plane interaction problem, by deriving closed-form analytical solutions in both time and frequency domains, when a point vortex is induced around the edge of a semi-infinite rigid wedge. The model is then extended to include the combined effects of vortex induction and convection by a non-zero ambient flow.

The paper is structured as follows: In Section 2, Howe's model is briefly recalled while its extension to arbitrary wedge-angles is proposed in Section 3 in frequency- and time-domains. The combined effect of convection and induction is considered in Section 4 and in Section 5 we conclude the paper and include a discussion regarding possible applications of the model for aeroacoustic problems such as noise generation by the trailing edges of airfoils.
<figure><img src="image_2.png" /><figcaption>Figure 1: Acoustic waves with wavelength $\lambda$ generated by the two-dimensional interaction between a point vortex (black dot) with circulation $\Gamma$ and a semi-infinite, rigid half plane. The red-dashed line corresponds to the vortex trajectory due to the interaction with its image point and the gray solid lines to the streamlines of the incompressible potential around the edge. $l$ is the minimum distance between the vortex and the edge ($l \ll \lambda$).</figcaption></figure>

## 2. Vortex-edge Interaction: Howe's Half plane problem

A generic turbulent fluctuation in the vicinity of the edge of a semi-infinite rigid plate ($x_1 < 0, x_2 = 0$) is modelled as a point vortex, shown in Fig. 1. It is assumed that the acoustic wavelength $\lambda$ is much larger than the smallest distance $l$ between the edge and the vortex. Any effect of the free-stream flow is omitted and the trajectory of the point vortex is influenced only by its image. Howe's theory of vortex sound is applied to obtain the pressure pulse in the far field,

$$p(\vec{x}, \tau) = -\rho \int_V \int_{-\infty}^{\infty} (\vec{\omega} \times \vec{v})(\vec{y}, t) \vec{\nabla} G(\vec{x}, \vec{y}, t-\tau) dt dy^3, \quad (2.1)$$

where $G$ denotes the tailored Green function for the half-plane, $\rho$ the medium's density (assumed to be constant), $\vec{\nabla} = (\frac{\partial}{\partial y_1}, \frac{\partial}{\partial y_2}, \frac{\partial}{\partial y_3})$ the del-operator and $\vec{\omega}, \vec{v}$ are the vorticity and velocity fields respectively.

For an observer in the acoustic far-field and a source located in the vicinity of the edge of the half plane ($kl \equiv 2\pi l/\lambda \ll 1$), Howe used an asymptotic expression of the half-plane Green function to compute the scattered sound field (Howe 1975a, 1998). An integration along the span is performed to reduce the problem to two dimensions and the resulting low-frequency, 2D Green function is obtained using the method of stationary phase. The time-domain compact Green function for the diffraction field around a half plane is obtained by taking the inverse Fourier transform of its frequency domain counterpart and reads as [details in the appendix of the work of Howe (1975a)],

$$G(\vec{x}, \vec{y}, t-\tau) \approx \frac{\phi^*(\vec{x}) \phi^*(\vec{y})}{\pi|\vec{x}|} \delta(t-\tau), \quad |\vec{x}| \to \infty, \quad (2.2)$$

where $\vec{x} = (x_1, x_2)$ denotes the coordinates of the observer in the far field, such that $k|\vec{x}| \gg 1$ and $\vec{y} = (y_1, y_2)$ corresponds to the source/vortex coordinates and $\delta(\cdot)$ is the Dirac delta-function. The origin $(0, 0)$ coincides with the edge of the half-plane (Fig. 1) and the position
vector of the vortex is denoted as $\vec{y}_0$, such that $k|\vec{y}_0| \ll 1$. $\tau = t - |\vec{x}|/c_0$ is the retarded time, $c_0$ the speed of sound and $\phi^*(\cdot)$, the incompressible velocity potential of the flow around the edge of the half plane, which is used to compute the low-frequency Green function in the limit of $k|\vec{y}| \ll 1$. Note that $\phi^*(\cdot)$ is not related to the actual flow-field, it is rather a mathematical tool to obtain the Green function, denoted by Howe (2003) as a hypothetical velocity potential.

For a point vortex, the source term can be written as (Howe 1975a):

$$ \vec{\omega} \times \vec{v} = \Gamma \hat{e}_3 \times \frac{d\vec{y}}{dt} \delta(\vec{y} - \vec{y}_0(t)), \quad (2.3) $$

where $\Gamma$ the circulation of the vortex. From here on, the polar coordinate system will be used: i) $\vec{x} = r(\cos\theta, \sin\theta)$ and ii) $\vec{y}_0 = r_0(\cos\theta_0, \sin\theta_0)$ unless stated otherwise. Substituting Equations (2.2), (2.3) into (2.1) the pressure pulse in the far field can be written in the form,

$$ p(\vec{x}, \tau) = \frac{\rho \Gamma \sin(\theta/2)}{\pi \sqrt{r}} \left[ \frac{d\vec{y}_0}{d\tau} \cdot \vec{\nabla} \Psi^* \right] (\vec{y}_0), \quad (2.4) $$

where $\Psi^*$ is the stream-function associated with the velocity potential ($\phi^*(\vec{y}_0)$) near the edge of the half-plane. With the operator $D/D\tau$ corresponding to the material derivative with respect to the vortex trajectory for 2-D flow, we can write (cartesian coordinates for simplicity), (Howe 2003)

$$ \frac{d\vec{y}_0}{d\tau} \cdot \nabla \Psi^* = \frac{dy_{01}}{d\tau} \frac{d\Psi^*}{dy_{01}} + \frac{dy_{02}}{d\tau} \frac{d\Psi^*}{dy_{02}} = \frac{D\Psi^*}{D\tau}(\vec{y}_0). $$

One of the key results of this simplified model is the following physical interpretation it enables regarding the mechanism of sound generation. The far-field sound is generated when the vortex path crosses the streamlines, given by $\Psi^*$. Vortices that follow these streamlines do not produce sound. In particular, a vortex far from the edge, translating parallel to the half plane (and hence the streamlines $\Psi^*$) with an equivalent velocity $v = \Gamma/(8\pi l)$ is silent (Howe 1975a, 1998, 2003).

The material derivative $D\Psi^*/D\tau$ depends on the vortex path, which is obtained by taking into account the vortex and its image, details shown in Howe (2003). This provides an expression for the acoustic pressure as a function of vortex circulation and smallest vortex-edge distance,

$$ p(\vec{x}, \tau, l, \Gamma) = \frac{\rho \Gamma^2 \sin(\theta/2)}{(4\pi l)^2} \sqrt{\frac{l}{r}} \left[ \frac{\Gamma \tau / (8\pi l^2)}{\left[ 1 + \left( \frac{\Gamma \tau}{(8\pi l^2)} \right)^2 \right]^{5/4}} \right]. \quad (2.5) $$

Kambe et al. (1985) studied a similar problem theoretically, where a thin vortex ring interacts with a sharp edge. The problem was treated in three-dimensions by applying the theory of vortex sound and using the inverse-Fourier-transformed asymptotic low-frequency forms of the half-plane Green function. In their analysis the vortex does not interact with its image and is considered to pass by the edge without altering its trajectory. Kambe et al. validated their result experimentally, showing that it is the rate-of-change of a volume flux of a hypothetical potential flow around the edge that leads to sound generation. Chang & Chen (1994) later followed the same methodology and used an asymptotic, low-frequency form of the Green function for a semi-infinite rigid wedge to extend Kambe's model to semi-infinite sharp wedges, including the half plane as a special case.

Notice that in the works of Crighton (1972) and Howe (1975b) the problem of sound generation by vortex half-plane interactions is considered in two dimensions, where Howe
<figure><img src="image_3.png" /><figcaption>Figure 2: Geometry of the wedge problem.</figcaption></figure>

has shown that sound is generated due to the crossing of the vortex through the hypothetical streamlines. The works of Kambe et al. (1985) for half-planes and Chang & Chen (1994) for wedges extend Howe's model to three dimensions and hence they conclude that the sound is generated due to the rate-of-change of the volume flux of the hypothetical potential flow when a vortex ring crosses its streamlines. It is thus seen, that the physical mechanism of sound generation is similar, however both models consider different vortex kinematics. On the one hand (Howe's model) the point vortex is self-induced around the edge of the half-plane, while the models of Kambe and Chang & Chen consider that the vortex ring moves on a straight line and its trajectory is not affected by the edge. In the following sections we are going to focus on the two-dimensional problem (Howe's model) and extend it to consider i) wedges of arbitrary angles and ii) the combined effect of convection and self-induction in the vortex kinematics.

### 3. Extension to the semi-infinite rigid wedge

In what follows, the analysis of Howe for the two-dimensional problem of a point vortex interacting with a half-plane is generalised for any arbitrary wedge with half angle $\Omega$. Explicit formulae for the acoustic pressure are given in both frequency and time domains while extension of the model to account for non-zero convection velocity and the associated edge-effect on vortex kinematics and sound will be considered in a subsequent section.

Figure 2 shows the configuration considered. The wedge parameter $\gamma = 2(\pi - \Omega)/\pi$ is introduced and ranges from $\gamma = 2$, which corresponds to the limiting case of a semi-infinite half-plane, to $\gamma = 1$ (infinite wall). The physical space is restricted from the lower to the upper face of the wedge, $\theta_0 \in [-\gamma\pi/2, \gamma\pi/2]$. Note that for $\gamma < 1$ the problem resembles rather the aerodynamic sound generated by a closed triangular cavity, which is beyond the scope of the present study. Following Howe's methodology (Howe 2003), the complex velocity potential close to the edge is given by,

$$ 
\begin{aligned}
W(z_j) &= -iz_j^{1/\gamma}, \\
&= -ir_j^{1/\gamma} (\cos(\theta_j/\gamma) + i \sin(\theta_j/\gamma)),
\end{aligned} 
\qquad (3.1) 
$$
167 where $z_j = r_j e^{i\theta_j}$ and $r_j, \theta_j$ denote source or receiver coordinates, $\phi^* = \Re[-iW(z_j)]$ and

168 $\Psi^* = \Im[-iW(z_j)]$ (Howe 2003). Combining Eqs.(2.1), (2.2), (2.3) and (3.1) one obtains:

$$
\begin{equation}
\begin{split}
p(r, \theta, \tau, \Gamma, l, \gamma) ={}& \frac{\rho \Gamma \sin(\theta/\gamma)}{\pi r^{\frac{\gamma-1}{\gamma}}} \left[ -\frac{r_0^{1/\gamma-1}(\tau)}{\gamma} \frac{dr_0}{d\tau}(\tau) \cos\left(\frac{\theta_0(\tau)}{\gamma}\right) \right. \\
& \left. \qquad + \frac{r_0^{1/\gamma}(\tau)}{\gamma} \sin\left(\frac{\theta_0(\tau)}{\gamma}\right) \frac{d\theta_0}{d\tau}(\tau) \right],
\end{split}
\tag{3.2}
\end{equation}
$$

170 where the position of the vortex in polar coordinates $r_0(\tau), \theta_0(\tau)$ is given as a function of

171 the retarded time $\tau$.

172 3.1. Self-induction around a semi-infinite rigid wedge

173 A single point vortex is first considered in the vicinity of the corner of a semi-infinite wedge

174 and for zero free-stream velocity. The equations of motion are obtained by the following

175 expression (Howe 2003)

$$
\frac{dz_0^*}{dt} = -i \frac{\Gamma \zeta''(z_0)}{4\pi \zeta'(z_0)} + F'(z_0), \quad (3.3)
$$

177 where $z_0$ is the position of the vortex core in the complex plane, $z_0^*$ is the complex conjugate

178 of $z_0$, $\zeta(z_0)$ is the conformal mapping of $z_0$ in the $\zeta$-plane and $F(z_0)$ is a regular function of

179 $z_0$. The symbols ($'$, ($''$) denote the first and second derivatives with respect to the complex

180 variable $z$. Since the convection of the vortex core is not considered, we may write

$$
F(z_0) = \frac{i\Gamma}{2\pi} \ln(\zeta(z) - \zeta^*(z_0)), \quad (3.4)
$$

which corresponds to the velocity field induced by an image vortex. To account for the
presence of a corner the following conformal transform is applied,

$$
\zeta(z) = iz^{1/\gamma}.
$$

182 Using the polar form of the complex number, $z = re^{i\theta}$, Eq.(3.3) results in a 2 × 2 system of

183 non-linear differential equations of first order. The vortex position at any time instant is then

184 given by

$$
r_0(\tilde{\tau}) = l \sqrt{1 + (2\tilde{\tau}/\gamma)^2},
\quad (3.5)
$$

$$
\theta_0(\tilde{\tau}) = \gamma \tan^{-1}(-2\tilde{\tau}/\gamma),
$$

186 where $\tilde{\tau} = \frac{\Gamma \tau}{8 \pi l^2}$ is a non-dimensional time parameter and $l$ denotes the smallest distance from

187 the vortex to the edge and depends on the initial conditions as shown in the full steps of the

188 derivation in Appendix A. Equation (3.5) implies that the kinematics of the vortex are purely

189 driven by the circulation and the smallest distance to the edge $l$, since convection effects are

190 neglected.

191 Substituting Eq.(3.5) into Eq.(3.2) we obtain an expression for the far-field pressure pulse,

192 written as the product of two parameters, one (defined as C) which is constant and contains

193 the amplitude and directivity and a second (defined as $T_{\text{wedge}}$) that depends on time and the

194 wedge angle.
<figure><img src="image_4.png" /><figcaption>Figure 3: Vortex path (top) and normalised pressure pulses ($T_{\text{wedge}} = p/C$) (bottom), obtained for an observer at ($r = 20l, \theta = \pi/2$) for different wedge angles. The vortex arrives close to the edge with $v = 17$ m/s and the closest distance from the edge is $l = 0.025$ m. The circulation of the vortex is computed as $\Gamma = \frac{v}{8\pi l}$. The black, solid lines correspond to the streamlines of the incompressible potential around the edge in the absence of vortices.</figcaption></figure>

$$p(r, \theta, \tilde{\tau}, \Gamma, l, \gamma) = C(\theta/\gamma, \Gamma, l) T_{\text{wedge}}(r, l, \tilde{\tau}, \gamma),$$

$$C(\theta/\gamma, \Gamma, l) = \frac{\rho\Gamma^2}{8\pi^2 l^2} \sin(\theta/\gamma), \qquad (3.6)$$
$$T_{\text{wedge}}(r, l, \tilde{\tau}, \gamma) = 2 \frac{(rl)^{1/\gamma}}{r} \left(\frac{\gamma-1}{\gamma^2}\right) \left[ \frac{2\tilde{\tau}/\gamma}{\left[1+(2\tilde{\tau}/\gamma)^2\right]^{\frac{3\gamma-1}{2\gamma}}} \right].$$

195

Equation (3.6) is a generalisation of Eq.(2.5) for arbitrary wedge angles, $\gamma \in [2, 0)$. Indeed, for $\gamma = 2$ ($\Omega = 0$) Equation (3.6) reduces to Eq.(2.5). It can be seen that the wedge angle influences the vortex path and thus the far-field pressure. In the limiting case of $\gamma = 1$ ($\Omega = \pi/2$) (infinite wall), the vortex path is parallel to the streamlines of the hypothetical potential flow and Eq.(3.6) yields $p = 0$. The pressure pulse is mainly dependent on the wedge time function $T_{\text{wedge}}(r, l, \tilde{\tau}, \gamma)$, since the amplitude is constant and the directivity function ($\sin(\theta/\gamma)$) takes values from -1 to 1. Figure 3 shows the vortex-path and the hypothetical streamlines (top-plots) while on the bottom plot a comparison of the acoustic pulses for different wedge angles is shown. By examining the behaviour of the wedge time function, it can be shown that the maximum value of the pressure pulse depends explicitly on the wedge angle. The maxima are found by,

$$\frac{\partial T_{\text{wedge}}}{\partial \tilde{\tau}} = 0,$$

196 and the non-dimensional time for peak radiation is given by,

$$\tilde{\tau}_{\text{max}} = (\gamma/2) \sqrt{\frac{\gamma}{2\gamma - 1}}. \qquad (3.7)$$

197

198 The non-dimensional time $\tilde{\tau}_{\text{max}}$ decreases as the wedge angle increases, i.e. for vortices with

199 the same circulation and distance from the edge, the radiation-peak that corresponds to an
<figure><img src="image_5.png" /><figcaption>Figure 4: Maximum values of $T_{\text{wedge}}$ with respect to the wedge angle, obtained by Eq.(3.8), for $l = 0.025$ m and $r = 20l$.</figcaption></figure>

arbitrary wedge angle, for instance $\Omega = 40^{\circ}$, happens sooner than the one of the half plane as shown in Fig. 3.

Substituting Equation (3.7) into Eq.(3.6), it follows that the maxima of the pressure pulse are obtained by,

$$ \max [T_{\text{wedge}}](r, l, \gamma) = 2 \frac{(rl)^{1/\gamma}}{r} \left(\frac{\gamma-1}{\gamma^2}\right) \left[ \frac{\sqrt{\frac{\gamma}{2\gamma-1}}}{\left[1+\frac{\gamma}{2\gamma-1}\right]^{\frac{3\gamma-1}{2\gamma}}} \right]. \quad (3.8) $$

Figure 4 depicts the trends of the maxima of the far field pressure pulses (Eq.(3.8)) for $\Omega \in [0, 90^{\circ}]$. The model thus indicates a decrease of the peak sound-radiation amplitude as the wedge angle increases.

Examining Eq.(3.7) and solving for the dimensional time $\tau_{\max}$ that corresponds associated with the maximum amplitude yields,

$$ \tau_{\max} = \frac{4\pi l^2}{\Gamma} \gamma \sqrt{\frac{\gamma}{2\gamma-1}}. \quad (3.9) $$

The time when the sound pulse reaches its maximum amplitude depends on i) the circulation ii) distance from the edge and iii) the wedge angle. For sources closer to the edge and/or vortices with strong circulation, $\tau_{\max}$ decreases and thus the peak radiation occurs sooner; corresponding to higher-frequency sound emission, while $\tau_{\max}$ decreases as $\Omega$ increases, as discussed above for its non-dimensional counterpart.

The asymptotic behaviour of the far field pressure for short non-dimensional times, that is $2\tilde{\tau}/\gamma \ll 1$ can be found as,

$$ p_{\text{short}} = \frac{(rl)^{1/\gamma}}{r} \frac{\rho \Gamma^2}{2\pi^2 l^2} \sin(\theta/\gamma) \left(\frac{\gamma-1}{\gamma^3}\right) \tilde{\tau}, \quad (3.10) $$

showing a linear dependence on $\tilde{\tau}$.

Considering a turbulent eddy modelled as a vortex of circulation $\Gamma = 2\pi\nu R$, where $R$ the radius and $\nu$ a measure of the velocity fluctuation, the time scale can be approximated as $\tau = l/\nu$. It follows that,

$$ \frac{2\Gamma\tau}{8\pi l^2 \gamma} = \frac{1}{2\gamma} \left(\frac{R}{l}\right). $$

The asymptotic expression of Eq. (3.10) can be reformulated in order to derive the following
221 scaling law

222 $p_{\text{short}} \sim v^2 l^{\frac{1-3\gamma}{\gamma}} R^3$, $R/l \ll 1$. (3.11)

223 Kambe et al. (1985) showed experimentally that vortices passing near a semi-infinite plate, with $R/l \ll 1$ follow a scaling law $v^{2.53}l^{-2.24}$. For $\gamma = 2$ the asymptotic expression of Eq. (3.11) gives $p \sim v^2 l^{-2.5}$. However, in the general case Eq. (3.6) illustrates that the pressure scales as $\sim l^{(1-2\gamma)/\gamma}$, which agrees with the results of the scaling to a factor of $l^{-3/2}$ obtained by Ffowcs Williams & Hall (1970) for the infinite plate, when $\gamma = 2$.

228 3.2. Extension to frequency domain

229 The foregoing analysis can be extended to the frequency domain. The fact that the radiation 230 peak occurs sooner for higher wedge angles indicates, as mentioned, that as $\Omega$ increases the 231 acoustic energy is distributed to higher frequencies. This can be shown by considering the 232 Fourier-transform of Eq. (3.6), which has the following form (detailed derivations are shown 233 in Appendix B):

$$ 
\begin{aligned}
\hat{p}(r, \theta, \tilde{\omega}, l, \gamma) &= \tilde{C}(\theta/\gamma) \hat{T}_{\text{wedge}}(r, l, \tilde{\omega}, \gamma), \\
\tilde{C}(\theta/\gamma) &= \rho \sin(\theta/\gamma), \\
\hat{T}_{\text{wedge}}(r, l, \tilde{\omega}, \gamma) &= -i \frac{(rl)^{1/\gamma}}{r} \frac{4}{\gamma} \left(\frac{\gamma-1}{\gamma^2}\right) \left(\frac{4}{\gamma^2}\right)^{\frac{1}{2\gamma}-3/2} \frac{\sqrt{\pi} \left(\frac{\gamma^2}{2}\right)^{\frac{1}{2\gamma}}}{\tilde{\Gamma}\left(\frac{3}{2}-\frac{1}{2\gamma}\right)} \tilde{\omega}^{1-\frac{1}{2\gamma}} K_{\frac{1}{2\gamma}}\left(\frac{\gamma^2}{4}\tilde{\omega}\right),
\end{aligned}
\quad (3.12) $$

234 where $\tilde{\Gamma}(\cdot)$ is the gamma function,

$$ K_{\nu}(q) = \frac{i\pi}{2} e^{i\frac{\pi}{2}\nu} H_{\nu}^{(1)}(qe^{i\pi/2}), $$

235 is the modified Bessel function, $q$ an arbitrary variable, $H_{\nu}^{(1)}(\cdot)$ the Hankel function of the first kind and,

$$ \tilde{\omega} = \frac{8\pi l^2}{\Gamma} \omega, $$

235 the dimensionless frequency of the sound wave. The sound-wavelength associated with $\tilde{\omega}$ is

236 $\frac{\lambda_{\Gamma}}{l} = \frac{1}{\tilde{\omega}} \frac{16\pi^2 lc_0}{\Gamma}$. (3.13)

237 Equation (3.13) is in agreement with the time-domain analysis of the problem, where high 238 values of $\Gamma$ and/or small values of $l$ lead to acoustic waves with shorter wavelengths (higher 239 frequencies), however the frequency-domain analysis allows the derivation of an explicit 240 relation between the acoustic wavelength and its dependence on the main parameters of the 241 problem. Moreover, the analysis indicates that in a realistic turbulent flow around an edge, 242 flow structures that remain close to the edge radiate at wavelengths ($\lambda_{\Gamma}$) that scale with the 243 circulation and the distance from the edge.

244 Figure 5 shows the spectrum of $\hat{T}_{\text{wedge}}(r, l, \tilde{\omega}, \gamma) = \hat{p}/\tilde{C}$ [Eq. (3.12)] as a function of the 245 wedge angle and the non-dimensional frequency. The sound energy for the half-plane ($\Omega = 0$ 246 or $\gamma = 2$) is mostly concentrated at low non-dimensional frequencies, while with increasing 247 wedge angle a slight broadening of the spectrum to higher frequencies is observed. However, 248 as observed in the time-domain analysis, the amplitude of the acoustic pressure has maximum
<figure><img src="image_6.png" /><figcaption>Figure 5: Amplitude of the acoustic pressure [Eq. (3.12)] generated by the vortex-wedge interaction as a function of the dimensionless frequency and the wedge angle. The results are shown for an observer located at r = 20l and l = 0.025m.</figcaption></figure>

values for small wedge angles (Ω → 0) and the half plane is the most acoustically efficient scatterer of all wedges, which is consistent with similar findings in the literature (Crighton & Leppington 1971; Chang & Chen 1994).

The foregoing is restricted to two-dimensions, low frequencies with turbulent flow-structures represented by point vortices that interact with their images inside the wedge. Despite these simplifications, which are justified since quasi 2D structures have been shown to be the key sound-radiation structures for trailing edges in turbulent flow (Sano et al. 2019; Abreu et al. 2021; Yuan et al. 2024; Demange et al. 2024a), the model provides physical insight regarding the parameters that influence the acoustic-field generated by vorticity in the vicinity of edges with arbitrary angles. It was shown that the circulation (Γ) and the minimum distance (l) between the vortex and the edge influences the wavelength of the radiated sound waves as shown by Eq. (3.13).

The models presented in (Kambe et al. 1985; Chang & Chen 1994) assume that the vortex is sufficiently far from the edge so that its trajectory is not altered in the vicinity of the edge. In that case the corresponding non-dimensional time unit scales as

$$T = \frac{U\tau}{L},$$

where L the smallest distance to the edge (Chang & Chen 1994; Kambe et al. 1985). It is noted that l and L are in general different parameters since the first corresponds to the problem of vortices self-induced by the edge, while the second to the minimum distance for which a purely convected vortex passes near the edge without altering its trajectory. The non-dimensional frequency for the problem of pure-convection is $\tilde{\omega} = L/U\omega$ and hence the wavelengths can be expressed as

$$\lambda_U = \frac{2\pi L}{M} \frac{1}{\tilde{\omega}}, \qquad (3.14)$$

where M is the Mach number. It can be seen that in that case the sound-wavelength depends on different parameters to those of the self-induction problem: convection velocity U and a distance L ≫ l where the vortex is located far enough from the edge, that its trajectory is not affected by the image-vortex.
<figure><img src="image_7.png" /><figcaption>Figure 6: Motion of a vortex close to the edge by taking into account convection effects. The red-dashed line corresponds to the vortex trajectory and $l_0$ is a measure of its initial distance from the edge. The black solid lines correspond to the stream-function related to the hypothetical velocity potential, as shown in sub-section 3.1.</figcaption></figure>

In real flows, for instance, in the vicinity of a trailing edge, turbulent eddies of different length scales interact with the edge resulting in acoustic radiation over a wide range of frequencies (Wang 2000; Colonius & Lele 2004). The distinction between purely edge-induction and convection effects is not clear. In the next section we therefore consider the combination of convection, and edge-effects on the trajectory of a point vortex in the vicinity of a corner and its influence on the sound field in order to provide a single model that contains the effects of convection (Chang & Chen 1994; Kambe et al. 1985) and self-induction around the edge (Crighton & Leppington 1971; Howe 1975a, 2003).

## 4. Convection and self-induction effects on vortex kinematics and sound

Turbulent fluctuations in the vicinity of an edge are influenced both by convection effects associated with the mean flow (here uniform velocity), and by near-edge vorticity. In the simplest case of a vortex-ring passing near the edge of a semi-infinite half-plane, experimental results have shown that its trajectory is influenced by its circulation as discussed in the works of Kambe et al. (1985) and in the appendix of Chen et al. (2022). We thus generalise the previous developments to account for a combination of these effects, both of which may influence the vortex path, hence the sound generation.

Consider a point vortex in a free stream, with velocity $U$, angle $\eta$ with respect to the horizontal axis and which passes in the vicinity of a semi-infinite wedge. We cannot now choose, a priori, the closest distance to the edge as in Eq. (3.5). We therefore introduce the length scale $l_0$, which is the initial distance between the vortex and the edge (Fig. 6). The initial angle is $\theta_0$. Equation (3.3) is used as in Section 3 to obtain the kinematics of the point vortex. Convection of the vortex core is accounted for by adding the free-stream complex velocity potential to the Eq.(3.4) :

$$F(z_0) = \frac{i\Gamma}{2\pi} \ln(\zeta(z) - \zeta^*(z_0)) + Uz_0e^{-i\eta}. \quad (4.1)$$

This results in a flow model with two velocity scales: one associated with the free-stream flow $U$ and another that can be considered as a generic, vortical fluctuation, such that $v \sim \Gamma/l_0$,
298 where $v$ is a characteristic vortical velocity associated with the circulation $\Gamma$. By performing

299 the same calculations as in Section 3.1 and Appendix A we obtain:

$$ \left( \begin{array}{cc} \cos \theta_0 & -r_0 \sin \theta_0 \\ \sin \theta_0 & r_0 \cos \theta_0 \end{array} \right) \frac{d}{d\tau} \left( \begin{array}{c} r_0 \\ \theta_0 \end{array} \right) = \left( \begin{array}{cc} \frac{\Gamma(\gamma-1)}{4\pi\gamma r_0} \sin \theta_0 + \frac{\Gamma}{4\pi\gamma r_0} \frac{\sin(\theta_0 \frac{\gamma-1}{\gamma})}{\cos(\theta_0/\gamma)} + U \cos \eta \\ -\frac{\Gamma(\gamma-1)}{4\pi\gamma r_0} \cos \theta_0 - \frac{\Gamma}{4\pi\gamma r_0} \frac{\cos(\theta_0 \frac{\gamma-1}{\gamma})}{\cos(\theta_0/\gamma)} + U \sin \eta \end{array} \right). \quad (4.2) $$

301 The system of Equation (4.2) is solved with the method of determinants and the trajectory of
302 the vortex is described by the coupled system of differential equations,

$$ \dot{r}_0 = -\frac{\Gamma}{4\pi\gamma r_0} \tan(\theta_0/\gamma) + U \cos(\theta_0 - \eta), \quad (4.3) $$

$$ \dot{\theta}_0 = -\frac{\Gamma}{4\pi r_0^2} - \frac{U}{r_0} \sin(\theta_0 - \eta), $$

304 where (·) denotes the first time derivative. The system of Eqs. (4.3) cannot be solved
305 analytically, however it is possible to gain insight regarding the key parameters of the problem.

306 By introducing the non-dimensional quantities $\bar{r}_0 = r_0/l_0$, $\dot{r}_0 = \dot{r}_0/U$ and $\dot{\theta}_0 = \dot{\theta}_0 l_0/U$, we obtain,

$$ \dot{\bar{r}}_0 = -\frac{\Lambda}{\gamma \bar{r}_0} \tan(\theta_0/\gamma) + \cos(\theta_0 - \eta), \quad (4.4) $$

$$ \dot{\bar{\theta}}_0 = -\frac{\Lambda}{\bar{r}_0^2} - \frac{\sin(\theta_0 - \eta)}{\bar{r}_0}, $$

309 where $\Lambda$ is a non-dimensional parameter, that we refer to as the wedge-deflection effect.

$$ \Lambda \equiv \Gamma/(4\pi U l_0) = v/(4\pi U). \quad (4.5) $$

310 In regions with strong circulation or very close to the edge, the velocity field induced by the
311 vortex is dominant and $\Lambda$ takes large values. On the other hand, far from the edge or in regions
312 where vorticity is weak, convection effects dominate and $\Lambda$ decreases. For instance, in the
313 extreme case of $\Lambda \gg 1$ (strong circulation and/or close to the edge and/or very low speed
314 flow) Eq. (4.4) results asymptotically in the equations of motion obtained in the previous
315 sub-section. In the other extreme case of $\Lambda \ll 1$ (weak circulation and/or far from the edge
316 and/or higher Mach number flow), it yields the equations of motion of a vortex convected by
317 the mean flow, for a constant free-stream velocity $U$ the vortex follows a straight line as in
318 the model of Kambe et al. (1985).
319

320 A similar analysis restricted to the case of the half plane was presented by Hardin (1980)
321 in an attempt to model the noise generated by vortices formed at the side edges of flaps.
322 In Hardin's work it is assumed that the vortex is either carried by the potential flow in the
323 vicinity of the edge or its trajectory is deflected by the influence of the edge. In his analysis
324 the term $Uz_0e^{-i\eta}$ in Eq. (4.1) is replaced by $U\sqrt{z_0}$ to account for the local potential flow
325 around the half-plane. In the absence of the edge-effects on the vortex path the latter would
326 yield a silent vortex since its path would match with the streamlines of the potential flow
327 around the wedge (as given by Eq. (3.1)). However, in the present study we are interested in
328 the convection of the vortex by the free stream velocity.
329

329 The system of Eqs (4.3) is solved numerically by implementing a Runge-Kutta scheme
<figure><img src="image_8.png" /><figcaption>Figure 7: Influence of parameter $\Lambda$ on vortex trajectories (left) and far field pressure signature (right). Different vortex paths are shown on the left and the corresponding pressure signatures on the right. The darker colours correspond to lower values of $\Lambda$, while the lighter colours to higher values. The results are shown for $l_0 = 0.2m, \theta_0 = \gamma\pi/2 - \pi/20, U = 30m/s, \Omega = 20^\circ$, an observer in the far field $r = 5l_0$ and $\Lambda = \{0.1; 0.25; 0.5; 1; 1.5; 2\}$ and $\eta = 0$.</figcaption></figure>

of 4th order and Equation (3.2) is used to compute the pressure field. Figure 7 shows a comparison of the far-field pressure pulse generated by vortices subjected to a varying degree of wedge-effect strengths. At a first step, we consider $\eta \approx 0$ to focus on the influence of the wedge-deflection effect on the kinematics of the vortex and the acoustic field. It is observed that for large values of $\Lambda$ the problem reduces to that of sub-section 3.1. Furthermore, a change of sign in the pressure pulse is observed when the vortex passes the bisector of the wedge. As the wedge-deflection effect decreases, the negative peak of the far-field pressure pulse tends to vanish since the vortex is convected downstream and does not move towards the other face of the wedge, $\theta_0 \in (0, -\gamma\pi/2]$. This can be explained by the fact that the basic mechanism of sound generation is the crossing of the vortex path with the hypothetical streamlines around the wedge. When convection effects are absent the vortex path is symmetric around the edge and it becomes parallel to the streamlines of the hypothetical potential flow at $\theta_0 = 0$. Then it passes to the lower side of the wedge, where $\theta_0 < 0$. When convection effects become important, the vortex trajectory changes such that it does not become parallel to the hypothetical streamlines and thus there is no change of sign. The results can be further investigated by rewriting the far-field pressure signature in the following form.

$$p(r, \theta, \tau) = -\rho \sin\left(\frac{\theta}{\gamma}\right) r^{\frac{1}{\gamma}} \frac{\Gamma U}{l_0^{(1-1/\gamma)}} f(\Lambda, \gamma, \bar{r}_0, \theta_0, \eta), \quad (4.6)$$

where $f(\Lambda, \gamma, \bar{r}_0, \theta_0, \eta)$ corresponds to a function that depends on the vortex trajectory,

$$f(\Lambda, \gamma, \bar{r}_0, \theta_0, \eta) = \bar{r}_0^{\frac{1}{\gamma}} \left[ \frac{\Lambda}{\bar{r}_0^2} \sin(\theta_0/\gamma) \left( \frac{1-\gamma}{\gamma} \right) - \frac{\cos\left(\eta - \theta_0 \frac{\gamma-1}{\gamma}\right)}{\bar{r}_0} \right]. \quad (4.7)$$

The influence of the main parameters of the problem is clearer in Eqs. (4.6) and (4.7). The amplitude depends mainly on the flow velocity, the circulation and the distance from the edge as is indicated by the factor $\frac{\Gamma U}{l_0^{(1-1/\gamma)}}$. Further conclusions can be drawn by examining Eq. (4.7). The function $f$ can be decomposed into two terms, one that corresponds to the
<figure><img src="image_9.png" /><figcaption>Figure 8: Vortex path (top) and normalised time-domain acoustic pressure signature (bottom) for different wedge angles (Ω = 5°, 20°, 45°, 60°), constant wedge-deflection parameter Λ = 0.25, angle of incidence η = −Ω and an observer located at (r, θ) = (20l₀, γπ/2).</figcaption></figure>

influence of the vortex close to the edge and a convection term. We therefore have,

$$f = f_1(\Lambda, \bar{r}_0, \theta_0, \gamma) + f_2(\eta, \bar{r}_0, \theta_0, \gamma).$$

When $\bar{r}_0$ is small, the sound pressure field is mainly influenced by $f_1(\Lambda, \bar{r}_0, \theta_0, \gamma)$. On the contrary as the vortex is convected away from the edge, $f_1$ decreases as $\bar{r}_0^{\frac{1}{\gamma}-2}$ and convection effects take over $(\bar{r}_0^{\frac{1}{\gamma}-1})$.

As the wedge angle increases the vortex structures (here point vortices) move parallel to the face of the wedge before shed in the wake. Therefore, we now consider an incidence angle equal to the half-wedge angle Ω and investigate the effect of the wedge angle on the vortex path and the acoustic field. The results are shown in Fig. 8 for a constant value of the wedge-deflection parameter (Λ = 0.25) and different wedge angles and angles of incidence η = −Ω. It should be noted that in the absence of convection effects, the increase of the wedge angle results in slightly higher frequency radiation (Section 3.2). However, when both convection and edge-induction effects on the vortex path are present, vortices near sharp wedges tend to be more strongly influenced by the edge, while as the wedge angle increases the relative effect of edge induction is reduced, resulting in longer pulses with lower amplitudes. This illustrates the importance of considering both convection and edge-infuction effects in modelling the aeroacoustics of sources in the vicinity of an edge.

The directivity of the acoustic field depends solely on the parameter sin (θ/γ) for both models developed. Directivity plots for different wedge half-angles are depicted in Fig. 9 which shows the typical cardiod shape, characteristic of low-frequency diffraction by semi-infinite half-planes and rigid corners (Howe 2001; Roger et al. 2016). Note that the directivity is shown at the non-dimensional time when the vortex reaches its minimum radial distance from the edge.

The previous theoretical investigation highlighted that the interplay between convection and self-induction due to near-edge vorticity affects the kinematics of a point vortex interacting with a rigid corner and the corresponding sound-field. In actual flows, the idealised point-vortex hypothesis is not valid, however, the proposed analysis allows a qualitative
<figure><img src="image_10.png" /><figcaption>Figure 9: Directivity plots obtained for different wedge angles. The acoustic-field is shown for $\Lambda = 0.25$, $\eta = -\Omega$ and $r = 20l_0$, at the time instant when the vortex reaches its minimum radial distance from the edge. Note that for each figure the radius-axis of the polar plots, corresponding to the amplitude of the acoustic pressure, are adjusted accordingly to enhance the visualisation of directivity pattern. The results are consistent with decreasing amplitude as $\Omega$ increases.</figcaption></figure>

374 description of several aeroacoustic problems. Particularly, the wedge deflection parameter 375 allows to extend the modeling approaches by relaxing the frozen gust assumption for a 376 point vortex (turbulent fluctuation in the vicinity of the edge). The latter has been proven 377 successful in the classical trailing-edge noise modelling (Amiet 1975; Howe 1978), while 378 fails to produce accurate results when applied to serrated trailing-edges (Moreau & Doolan 379 2013; Avallone et al. 2018; Tiomkin & Jaworski 2024), where the hydrodynamic field is 380 strongly affected by the presence of the edge. In the limiting case of small values of $\Lambda$ the 381 proposed analysis is in agreement with the frozen gust assumption, since the point vortex 382 passes near the edge without altering its trajectory. On the contrary, as $\Lambda$ increases, the more 383 we deviate from this hypothesis. Provided that the noise generated by a serrated trailing 384 edge consists of a more complex three-dimensional flow field, the proposed model could be 385 adapted to explore the sound field produced due to the streamwise vorticity on the serration 386 (see Figure 6 of Avallone et al. (2016)) and also provide a simplified description for the 387 interaction between the main vortices and the trailing edge by appropriately informing the 388 wedge-deflection parameter. A more detailed classification and comparison of the vortex 389 structures near serrated trailing edges is however out of the scope of the present work.

390 **5. Conclusions and discussion on extensions to applied problems**

391 We have considered the kinematics and aeroacoustics of point vortices in the vicinity of a 392 wedge-shaped edge. In a first step, the point vortices are considered close to the edge (solely 393 influenced by the image vortex), neglecting convection effects. Closed-form solutions are 394 obtained for the radiated sound in both time and frequency domains, and these show how the
<table><caption>Table 1: Comparison of vortex-edge interaction models. All approaches assume long wavelengths relative to vortex-edge distance ($kl \ll 1$).</caption><thead><tr><th>Model</th><th>Geometry</th><th>Assumptions</th><th>Contribution</th></tr></thead><tbody><tr><td>(Crighton 1972)</td><td>half-plane</td><td>2D, no convection</td><td>analytical expressions using singular perturbation method</td></tr><tr><td>(Howe 2003)</td><td>half-plane</td><td>2D compact Green function; no convection</td><td>vortex-sound model; analytical expressions; enhanced physical interpretation</td></tr><tr><td>(Kambe et al. 1985)</td><td>half-plane</td><td>no edge effects on vortex path</td><td>3D; studies convection effects</td></tr><tr><td>(Chang & Chen 1994)</td><td>wedge</td><td>same as (Kambe et al. 1985)</td><td>extends to wedge geometry</td></tr><tr><td>(Chen et al. 2022)</td><td>half-plane</td><td>same as (Kambe et al. 1985)</td><td>porous edge extension of (Kambe et al. 1985)</td></tr><tr><td>Present (Sec. 3)</td><td>wedge</td><td>2D compact Green function; no convection</td><td>extends (Howe 2003) to wedge geometry</td></tr><tr><td>Present (Sec. 4)</td><td>wedge</td><td>2D compact Green function</td><td>convection & edge effects; wedge deflection parameter</td></tr></tbody></table>

scattered sound depends only on the wedge angle, circulation and the shortest distance from the edge. An extension of the model accounts for convection effects and shows that there exists a non-dimensional *wedge-deflection* parameter that dominates the kinematics of the vortex close to the edge. Large values of the *wedge-deflection* parameter correspond to vortex trajectories dominated by self-induction around the edge that radiate at high frequencies, while low values of the parameter indicate trajectories dominated by convection, leading to lower-frequency radiation. Finally, it was shown that for a constant value of the *wedge-deflection*, as the wedge angle increases the convection effect becomes dominant, leading to lower-frequency radiation with lower amplitude.

To conclude, the proposed model incorporates features from various independent studies found in the literature such as diffraction by semi-infinite wedges (Crighton & Leppington 1970; Chang & Chen 1994) and the sensitivity of the vortex path on edge-induction (Crighton 1972; Howe 1975a) and convection effects (Kambe et al. 1985; Chang & Chen 1994) to examine the sound-field of vortex-wedge interactions. The foregoing analysis thus provides a more general and complete physical insight in the competing flow-mechanisms (edge-induction and convection) that drive the motion of the vortex near an edge of arbitrary angle and dominate the sound field. Table 1 summarises the existing models and shows the contributions of the one proposed in this study.

In addition to the theoretical contributions in edge-vortex interaction noise, the proposed model could further assist in identifying sound-generation mechanisms of more complex flows. A detailed analysis is out of the scope of the present paper and could be provided in a subsequent study, however we give an example of its potential use. Recent approaches aiming to explore bluff-body noise (Prinja et al. 2025) and more specifically airfoil noise
(Sano et al. 2019; Yuan et al. 2024; Demange et al. 2024a,b) adapted the strategy of describing the aeroacoustic source in terms of coherent flow structures (motions), which has been successfully applied in jet-noise modelling (Jordan & Colonius 2013). The extraction of those coherent structures is achieved either by post-processing techniques on experimental or simulation data (Sano et al. 2019) or by physics-based models such as resolvent analysis (Demange et al. 2024b). Both approaches decompose the flow field in modes that contribute to the total flow or acoustic energy. These acoustically important structures can be used to inform the present model and provide an in-depth exploration of the sound generation mechanism due to edge vortex interactions, expressed in terms of a non-dimensional parameter. Similar approaches could be adapted to investigate the aeroacoustics of vorticity-edge interactions for larger wedge angles (Ω ≈ 45°), such as the noise generated by side-view mirrors (Stoffel et al. 2025), or side edges of flaps (Roger et al. 2016).

**Funding.** This work was supported by the DGAC (Direction Générale de l'Aviation Civile), by the PNRR (Plan National de Relance et de Résilience Français) and by NextGeneration EU via the project MAMBO (Méthodes Avancées pour la Modélisation du Bruit moteur et aviOn).

**Declaration of interests.** The authors report no conflict of interest.

**Author ORCIDs.** M. I. Spriopoulos, https://orcid.org/0009-0004-6462-9585; P. Jordan, https://orcid.org/0000-0001-8576-5587; F. Margnat, https://orcid.org/0000-0001-6870-2779

## Appendix A. Derivation steps of Eqs. (3.5),(3.6)

In this appendix the derivation of the vortex path and the far field pressure signature of Eqs. (3.5), (3.6) is presented. The velocity, obtained by the complex velocity potential is written as

$$ \frac{d\bar{z}_0}{d\tau} = -i \frac{\Gamma \zeta''(z_0)}{4\pi \zeta'(z_0)} + \frac{d}{dz_0} \left[ \frac{i\Gamma}{2\pi} \ln(\zeta(z) - \zeta(\bar{z}_0)) \right] \quad (A.1) $$

Substituting

$$ \zeta(z_0) = iz_0^{1/\gamma} = ir_0^{1/\gamma} e^{i\theta_0/\gamma} $$

it yields:

$$ \frac{\Gamma}{4\pi\gamma r_0} \left[ (\gamma - 1) \sin\theta + \frac{\sin\left(\theta_0 \frac{\gamma-1}{\gamma}\right)}{\cos\left(\frac{\theta_0}{\gamma}\right)} \right] + \frac{i\Gamma}{4\pi\gamma r_0} \left[ (\gamma - 1) \cos\theta + \frac{\cos\left(\theta_0 \frac{\gamma-1}{\gamma}\right)}{\cos\left(\frac{\theta_0}{\gamma}\right)} \right] = \frac{d}{d\tau}\{r_0 \cos\theta_0 - ir_0 \sin\theta_0\} \quad (A.2) $$

By taking the derivative in time and separating the real and imaginary parts of the complex number, the following system of Equations is obtained:

$$ \begin{pmatrix} \cos \theta_0 & -r_0 \sin \theta_0 \\ \sin \theta_0 & r_0 \cos \theta_0 \end{pmatrix} \frac{d}{d\tau} \begin{pmatrix} r_0 \\ \theta_0 \end{pmatrix} = \begin{pmatrix} \frac{\Gamma(\gamma-1)}{4\pi\gamma r_0} \sin \theta_0 + \frac{\Gamma}{4\pi\gamma r_0} \frac{\sin(\theta_0 \frac{\gamma-1}{\gamma})}{\cos(\theta_0/\gamma)} \\ -\frac{\Gamma(\gamma-1)}{4\pi\gamma r_0} \cos \theta_0 - \frac{\Gamma}{4\pi\gamma r_0} \frac{\cos(\theta_0 \frac{\gamma-1}{\gamma})}{\cos(\theta_0/\gamma)} \end{pmatrix} \quad (A.3) $$

The solution of Eq. (A.3) is obtained by the method of determinants and the following trigonometric relations are used

$$ \cos x \sin y - \sin x \cos y = -\sin(x - y) $$
$$ \cos x \cos y + \sin x \sin y = -\cos(x - y) $$

446 It follows

$$ \frac{dr_0}{d\tau} = -\frac{\Gamma}{4\pi\gamma r_0} \tan(\theta/\gamma) \qquad (A.4) $$

447

448

$$ \frac{d\theta_0}{d\tau} = -\frac{\Gamma}{4\pi r_0^2} \qquad (A.5) $$

449

450 From (A.5), (A.4) it follows

$$ r_0 \frac{d\theta_0}{dr_0} = \gamma \cot \left( \frac{\theta_0}{\gamma} \right) \qquad (A.6) $$

451

452 which leads to

$$ \int_{\theta_{0,i}}^{\theta_0} \gamma^{-1} \tan(\xi/\gamma) d\xi = \int_{r_{0,i}}^{r_0} x^{-1} dx \Rightarrow r_0 = l \sec(\theta_0/\gamma), \qquad (A.7) $$

453

454 where $r_{0,i}$, $\theta_{0,i}$ the initial radial and angular coordinates of the vortex and $l$ is the smallest
455 distance between the vortex and the edge, defined as $l = r_{0,i} \cos(\theta_{0,i})$. Then, substituting
456 Eqs. (A.7) into (A.5) and following Howe's notation it yields

$$ r_0(\tau) = l \sqrt{1 + (2\tilde{\tau}/\gamma)^2} \qquad (A.8) $$

457

$$ \theta_0(\tau) = \gamma \tan^{-1}(-2\tilde{\tau}/\gamma), $$

where $\tilde{\tau} = (\tau\Gamma)/(8\pi l^2)$. Furthermore by combining Eqs. (A.8) and Eq.(3.2) and using the identities

$$ \cos[\tan^{-1} x] = (x^2 + 1)^{-1/2} $$

$$ \sin[\tan^{-1}(-x)] = -\frac{x}{\sqrt{x^2 + 1}} $$

458 we result in Eq. (3.6).

## 459 Appendix B. Frequency domain representation

460 Starting from Equation (3.6), the acoustic pulse generated by a vortex which is self-induced
461 close to a semi-infinite rigid wedge is given by

$$ p(r, \theta, \tilde{\tau}, \Gamma, l, \gamma) = C(\theta/\gamma, \Gamma, l) T_{\text{wedge}}(r, l, \tilde{\tau}, \gamma), $$

$$ C(\theta/\gamma, \Gamma, l) = \frac{\rho \Gamma^2}{8\pi^2 l^2} \sin(\theta/\gamma), \qquad (B.1) $$

462

$$ T_{\text{wedge}}(r, l, \tilde{\tau}, \gamma) = 2 \frac{(rl)^{1/\gamma}}{r} \left( \frac{\gamma-1}{\gamma^2} \right) \left[ \frac{2\tilde{\tau}/\gamma}{\left[1 + (2\tilde{\tau}/\gamma)^2\right]^{\frac{3\gamma-1}{2\gamma}}} \right]. $$
463 Its frequency domain counterpart is obtained as follows. We write the Fourier-transformed quantity as

$$ \hat{p}(r, \theta, \tilde{\omega}, \Gamma, l, \gamma) = C(\theta/\gamma, \Gamma, l) \hat{T}_{\text{wedge}}(r, l, \tilde{\omega}, \gamma), $$

$$ C(\theta/\gamma, \Gamma, l) = \frac{\rho\Gamma^2}{8\pi^2 l^2} \sin(\theta/\gamma), \qquad (B.2) $$

$$ \hat{T}_{\text{wedge}}(r, l, \tilde{\tau}, \gamma) = 2 \frac{(rl)^{1/\gamma}}{r} \left(\frac{\gamma-1}{\gamma^2}\right) \int_{-\infty}^{\infty} \frac{2\tilde{\tau}/\gamma}{\left[1 + (2\tilde{\tau}/\gamma)^2\right]^{\frac{3\gamma-1}{2\gamma}}} e^{-i\omega\tau} d\tau, $$

466 where $\tilde{\omega}$ is the non-dimensional frequency, that will be obtained in the following calculations

467 by changing the integration variable. The dependence on the frequency and the wedge angle

468 is captured by $\hat{T}_{\text{wedge}}$, while $C$ shows a dependence on the amplitude and the directivity.

469 Furthermore, we write $\hat{T}_{\text{wedge}}$ as

$$ \hat{T}_{\text{wedge}}(r, l, \tilde{\omega}, \gamma) = 2 \frac{(rl)^{1/\gamma}}{r} \left(\frac{\gamma-1}{\gamma^2}\right) \times \int_{-\infty}^{\infty} 2\tilde{\tau}/\gamma \left[1 + (2\tilde{\tau}/\gamma)^2\right]^{\frac{1-3\gamma}{2\gamma}} e^{-i\omega\tau} d\tau. $$

471 Then we factorise the terms in the brackets as

$$ \begin{aligned} \left[1 + (2\tilde{\tau}/\gamma)^2\right]^{\frac{1-3\gamma}{2\gamma}} &= \left[4/\gamma^2 \left(\gamma^2/4 + \tilde{\tau}^2\right)\right]^{\frac{1-3\gamma}{2\gamma}} \\ &= \left(\frac{4}{\gamma^2}\right)^{\frac{1}{2\gamma} - \frac{3}{2}} \left[\left(\gamma^2/4 + \tilde{\tau}^2\right)\right]^{\frac{1}{2\gamma} - \frac{3}{2}} \end{aligned} $$

473 As a result, Equation (B) becomes

$$ \begin{aligned} \hat{T}_{\text{wedge}}(r, l, \tilde{\omega}, \gamma) = & \frac{(rl)^{1/\gamma}}{r} \frac{4}{\gamma} \left(\frac{\gamma-1}{\gamma^2}\right) \left(\frac{4}{\gamma^2}\right)^{\frac{1}{2\gamma}-\frac{3}{2}} \\ & \times \int_{-\infty}^{\infty} \tilde{\tau} \left[\frac{\gamma^2}{4} + \tilde{\tau}^2\right]^{\frac{1}{2\gamma}-\frac{3}{2}} e^{-i\omega\tau} d\tau. \end{aligned} \qquad (B.3) $$

475 Since we are working with the non-dimensional units $\tilde{\tau}$, we do the following change of variable

$$ \tilde{\tau} = \frac{\Gamma}{8\pi l^2} \tau \Rightarrow \tau = \frac{8\pi l^2}{\Gamma} \tilde{\tau} \qquad (B.4) $$

$$ d\tau = \frac{8\pi l^2}{\Gamma} d\tilde{\tau}, $$

478 and combining Eqs (B.3), (B.4) we obtain

$$ \begin{aligned} \hat{T}_{\text{wedge}}(r, l, \tilde{\omega}, \gamma) = & \frac{8\pi l^2}{\Gamma} \frac{(rl)^{1/\gamma}}{r} \frac{4}{\gamma} \left(\frac{\gamma-1}{\gamma^2}\right) \left(\frac{4}{\gamma^2}\right)^{\frac{1}{2\gamma}-\frac{3}{2}} \\ & \times \int_{-\infty}^{\infty} \tilde{\tau} \left[\frac{\gamma^2}{4} + \tilde{\tau}^2\right]^{\frac{1}{2\gamma}-\frac{3}{2}} e^{-i\tilde{\omega}\tilde{\tau}} d\tilde{\tau}, \end{aligned} \qquad (B.5) $$

479
where

$$\tilde{\omega} = \frac{8\pi l^2}{\Gamma} \omega,$$

480 which corresponds to wavelengths

$$\frac{\lambda}{l} = \frac{1}{\tilde{\omega}} \frac{16\pi^2 l c_0}{\Gamma}. \qquad (B.6)$$

481

Moreover, we write the exponential as

$$e^{-i\tilde{\omega}\tilde{\tau}} = \cos(\tilde{\omega}\tilde{\tau}) - i\sin(\tilde{\omega}\tilde{\tau}),$$

482 and the integrals of Equation (B.5) becomes,

$$\int_{-\infty}^{\infty} \tilde{\tau} \left[ \frac{\gamma^2}{4} + \tilde{\tau}^2 \right]^{\frac{1}{2\gamma} - 3/2} e^{-i\tilde{\omega}\tilde{\tau}} d\tilde{\tau} = \int_{-\infty}^{\infty} F(\tilde{\tau}) d\tilde{\tau} \\ -i \int_{-\infty}^{\infty} G(\tilde{\tau}) d\tilde{\tau}, \qquad (B.7)$$

483

484 where

$$F(\tilde{\tau}) = \tilde{\tau} \left[ \frac{\gamma^2}{4} + \tilde{\tau}^2 \right]^{\frac{1}{2\gamma} - 3/2} \cos(\tilde{\omega}\tilde{\tau}), \text{ where } F(-\tilde{\tau}) = -F(\tilde{\tau})$$

485

$$G(\tilde{\tau}) = \tilde{\tau} \left[ \frac{\gamma^2}{4} + \tilde{\tau}^2 \right]^{\frac{1}{2\gamma} - 3/2} \sin(\tilde{\omega}\tilde{\tau}), \text{ where } G(-\tilde{\tau}) = G(\tilde{\tau})$$

486 Since $F(\cdot)$ is an odd function, the integration over $\tilde{\tau} \in (-\infty, \infty)$ yields zero, and then only

487 the second integral of the right-hand side of Eq. (B.7) needs to be calculated. Since $G$ is an

488 odd function we may write

$$\int_{-\infty}^{\infty} \tilde{\tau} \left[ \frac{\gamma^2}{4} + \tilde{\tau}^2 \right]^{\frac{1}{2\gamma} - 3/2} e^{-i\tilde{\omega}\tilde{\tau}} d\tilde{\tau} = -2i \int_{0}^{\infty} G(\tilde{\tau}) d\tilde{\tau}, \qquad (B.8)$$

489

490 The integral of Eq. (B.8) is computed analytically for $\gamma \in [2, 1]$ by using the formula 2.10,

491 p.121 in Oberhettinger (2012)

$$\int_{0}^{\infty} x \left(a^2 + x^2\right)^{\nu - 3/2} \sin(yx) dx = \frac{\sqrt{\pi} (2a)^{\nu}}{2\Gamma\left(\frac{3}{2} - \nu\right)} y^{1-\nu} K_{\nu}(ay), \quad \Re[v] < 1,$$

492

where $\Gamma(\cdot)$ is the gamma function and

$$K_{\nu}(z) = \frac{i\pi}{2} e^{i\frac{\pi}{2}\nu} H_{\nu}^{(1)}(ze^{i\pi/2})$$

493 is the modified Bessel function and $H_{\nu}^{(1)}$ the Hankel function of the first kind. It follows,

$$\int_{-\infty}^{\infty} \tilde{\tau} \left[ \frac{\gamma^2}{4} + \tilde{\tau}^2 \right]^{\frac{1}{2\gamma} - 3/2} e^{-i\tilde{\omega}\tilde{\tau}} d\tilde{\tau} \\ = -2i \int_{0}^{\infty} G(\tilde{\tau}) d\tilde{\tau} \qquad (B.9) \\ = -2i \frac{\sqrt{\pi} \left(\frac{\gamma^2}{2}\right)^{\frac{1}{2\gamma}}}{2\Gamma\left(\frac{3}{2} - \frac{1}{2\gamma}\right)} \tilde{\omega}^{1-\frac{1}{2\gamma}} K_{\frac{1}{2\gamma}}\left(\frac{\gamma^2}{4}\tilde{\omega}\right),$$

494
495 From Equations (B.5), (B.9) it follows

$$
\begin{equation}
\begin{aligned}
\hat{T}_{\text{wedge}}(r, l, \tilde{\tau}, \gamma) = & -i \frac{8\pi l^2 (rl)^{1/\gamma}}{\Gamma r \gamma} \frac{4}{\gamma} \left(\frac{\gamma-1}{\gamma^2}\right) \left(\frac{4}{\gamma^2}\right)^{\frac{1}{2\gamma}-3/2} \\
& \times \frac{\sqrt{\pi} \left(\frac{\gamma^2}{2}\right)^{\frac{1}{2\gamma}}}{\Gamma \left(\frac{3}{2}-\frac{1}{2\gamma}\right)} \tilde{\omega}^{1-\frac{1}{2\gamma}} K_{\frac{1}{2\gamma}} \left(\frac{\gamma^2}{4} \tilde{\omega}\right).
\end{aligned}
\tag{B.10}
\end{equation}
$$

496

497 Finally the Fourier-transformed expression is given as

$$
\hat{p}(r, \theta, \tilde{\omega}, \Gamma, l, \gamma) = \tilde{C}(\theta/\gamma) \hat{T}_{\text{wedge}}(r, l, \tilde{\omega}, \gamma),
$$

$$
\tilde{C}(\theta/\gamma) = \rho \sin(\theta/\gamma),
$$

498

$$
\begin{equation}
\begin{aligned}
\hat{T}_{\text{wedge}}(r, l, \tilde{\omega}, \gamma) = & -i \frac{(rl)^{1/\gamma}}{r} \frac{4}{\gamma} \left(\frac{\gamma-1}{\gamma^2}\right) \left(\frac{4}{\gamma^2}\right)^{\frac{1}{2\gamma}-3/2} \\
& \times \frac{\sqrt{\pi} \left(\frac{\gamma^2}{2}\right)^{\frac{1}{2\gamma}}}{\Gamma \left(\frac{3}{2}-\frac{1}{2\gamma}\right)} \tilde{\omega}^{1-\frac{1}{2\gamma}} K_{\frac{1}{2\gamma}} \left(\frac{\gamma^2}{4} \tilde{\omega}\right).
\end{aligned}
\tag{B.11}
\end{equation}
$$

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