2.1. Vortex sound at low Mach numbers

Consider a rigid semi-infinite porous plate with negligible thickness that lies in the region $-\infty < x_1,y_1\leq 0$, $x_2,y_2=0$, $-\infty < x_3,y_3<\infty$ of the coincident Cartesian coordinate systems {$(x_1,y_1), (x_2,y_2), (x_3,y_3)$}, as shown in figure 1. The porous plate is immersed in an unbounded fluid at rest at infinity. The present work studies the acoustic emission by an acoustically compact vortex ring source passing near the edge of a porous half plane. Source compactness requires $l/\lambda \ll 1$, where $l$ is the length scale of the turbulent source, and $\lambda$ is the wavelength of the emitted sound. The Mach number $M$, defined by $U/c$, is assumed to be much smaller than unity, where $U$ is the characteristic speed of the vortex ring and $c$ is the sound speed.

Figure 1.

Schematic of the porous half-plane and the coincident coordinate systems for the compact vortex ring source $\boldsymbol {y} = (\kern 0.05em y_1, y_2, y_3)$ and the observer at $\boldsymbol {x} = (x_1, x_2, x_3)$.

The sound resulting from the vortex ring interaction with the porous edge may be described by the inhomogeneous wave equation forced by the Lamb vector (Howe Reference Howe1998, p. 188),

(2.1)\begin{equation} \left(\frac{1}{c^2}\,{\frac{\partial^2{}}{\partial{t}^2}}-\nabla^2\right) p = \rho\,{\rm{div}}(\boldsymbol{\omega} \times \boldsymbol{v}), \end{equation}

which admits a general solution in terms of a boundary integral,

(2.2)\begin{equation} p(\boldsymbol{x},t)={-}\rho \iint (\boldsymbol{\omega} \times \boldsymbol{v})(\kern 0.06em \boldsymbol{y},\tau)\boldsymbol{\cdot} {\frac{\partial{G(\boldsymbol{x},\boldsymbol{y};t-\tau)}}{\partial{\boldsymbol{y}}}}\, {\rm d}{\boldsymbol{y}}\,{\rm d}{\tau}. \end{equation}

Here, $p$ is the acoustic pressure, $\rho$ is the mean fluid density, $\boldsymbol {\omega }$ is the vorticity distribution in an ideal fluid neglecting viscous dissipation, $\boldsymbol {v}$ is the vorticity convection velocity, and $G(\boldsymbol {x},\boldsymbol {y};t-\tau )$ is the time-domain Green's function. At sufficiently low Mach numbers, $G(\boldsymbol {x},\boldsymbol {y};t-\tau )$ may be approximated by the compact Green's function (Howe Reference Howe1975), which is described further in the next subsection.

2.2. Green's function for a porous edge

In pursuit of an acoustic pressure prediction using (2.2), a Green's function is now sought that produces the solution at distant point $\boldsymbol {x}=(x_1,x_2,x_3)$ due to an impulsive point source of unit strength at position $\boldsymbol {y}=(\kern 0.05em y_1,y_2,y_3)$ near the edge of a porous half-plane (cf. figure 1). The Green's function also must exhibit outgoing wave behaviour and satisfy (Howe Reference Howe1998, p. 39)

(2.3)\begin{equation} \left(\frac{1}{c^2}\,{\frac{\partial^2{}}{\partial{t}^2}}-\nabla^2\right)G(\boldsymbol{x},\boldsymbol{y};t-\tau) = \delta(\boldsymbol{x}-\boldsymbol{y})\,\delta(t-\tau), \end{equation}

where the right-hand side represents the impulsive point source. The reciprocal theorem (Rayleigh Reference Rayleigh1945) enables the positions of source $\boldsymbol {y}$ and observer $\boldsymbol {x}$ to be interchanged, i.e. the mathematical set-up is identical to finding the sound field observed at a near point $\boldsymbol {y}$ due to a monopole at a distant point $\boldsymbol {x}$. The problem thus converts to the solution for the Green's function $G$ as a function of observer positions $\boldsymbol {y}$ close to the edge, which is a diffraction problem that can be solved in the manner described, for example, by Crighton & Leppington (Reference Crighton and Leppington1970) or Jaworski & Peake (Reference Jaworski and Peake2013). The linearity of (2.3) permits the solution to be written as

(2.4)\begin{equation} G = G_0 + G_s, \end{equation}

where $G_0$ and $G_s$ are the time-domain velocity potentials for the incident field and the scattered field, respectively. The time-domain Green's function $G$ is related to its Fourier transform $\hat {G}$ by

(2.5)\begin{equation} G(\boldsymbol{x},\boldsymbol{y};t)={-}\frac{1}{2{\rm \pi}}\int_{-\infty}^{+\infty} {\hat{G}(\boldsymbol{x},\boldsymbol{y};k)\, \text{e}^{-\text{i}\omega t}\, {\rm d}\omega}, \end{equation}

where $k=\omega /c$ is the wavenumber, and $\omega$ is the angular frequency.

It is convenient to decompose $\hat {G}$ into the linear sum $\hat {G}=\hat {G}_0+\hat {G}_s$. The expressions for $\hat {G}_0$ in (A3) and $\hat {G}_s$ in (A8) are determined in Appendix A using the Wiener–Hopf analysis of Jaworski & Peake (Reference Jaworski and Peake2013). Application of the Fourier inversion formula (2.5) to these results yields the time-domain Green's functions $G_0$ and $G_s$, where $G_0$ may be expanded in a series form by following the procedure of Kambe et al. (Reference Kambe, Minota and Ikushima1985). The essential results from the reciprocal problem after reverting to the original configuration are

(2.6)$$\begin{gather} G_0(\kern 0.06em \boldsymbol{y},\boldsymbol{x};t)= \frac{1}{4{\rm \pi} x} \left(\delta(t_r)+\frac{\boldsymbol{x}\boldsymbol{\cdot} \boldsymbol{y}}{cx}\,\text{D}_t\, \delta(t_r)+\frac{(\boldsymbol{x}\boldsymbol{\cdot}\boldsymbol{y})^2}{2c^2x^2}\,\text{D}_t^2\, \delta(t_r)+\cdots\right), \end{gather}$$

(2.7)$$\begin{gather}G_s(\kern 0.06em \boldsymbol{y},\boldsymbol{x};t) = \frac{\mu^n}{2{\rm \pi}^{{3}/{2}}c^m}\,\varPhi(\boldsymbol{Y})\,\frac{M(\psi,\theta)}{ x}\,\text{D}_t^m\,\delta(t_{{\ast}}), \end{gather}$$

where

(2.8a,b)$$\begin{gather} \text{D}_t^m\,\delta(t) = \frac{1}{2{\rm \pi}}\int_{-\infty}^{+\infty} (-\text{i}\omega)^m\,\text{e}^{-\text{i}\omega t}\, {\rm d}\omega,\quad\text{D}_t={\frac{\partial{}}{\partial{t}}}, \end{gather}$$

(2.9a,b)$$\begin{gather}t_r = t - \frac{x}{c},\quad t_{{\ast}} = t-\frac{1}{c}|{\boldsymbol{x}-y_3\boldsymbol{k}}|, \end{gather}$$

(2.10a–c)$$\begin{gather}x =|{\boldsymbol{x}}|,\quad y =|{\boldsymbol{y}}|,\quad \boldsymbol{k} = (0, 0, 1). \end{gather}$$

Here, $\varPhi (\boldsymbol {Y}) = Y^{{1}/{2}}\sin {({\theta _0}/{2})}$ is the velocity potential about the edge, the projection of $\boldsymbol {y}$ onto the $(\kern 0.05em y_1,y_2)$-plane, and $Y=|{\boldsymbol {Y}}|$. The fractional derivative $\text {D}_{t}^{m}$ is used as a convenient and equivalent means of writing the inverse Fourier transform of the Green's function solution of the scattered sound field. The variables $m$ and $n$ are the exponents of the wavenumber $k$ and the dimensional porosity parameter $\mu = \alpha _H \bar {K}_R/R$. In this parametric group, $\alpha _H$ is the open area fraction of the surface with pores of nominal radius $R$, and $\bar {K}_R = 2 K_R/({\rm \pi} R)$, where $K_R$ is the Rayleigh conductivity of the pore (Howe Reference Howe1998; Jaworski & Peake Reference Jaworski and Peake2013); the conditions that the surface is weakly porous and has a pore feature size that is small relative to the acoustic wavelength require $\alpha _H^2 \ll 1$ and $k R \ll 1$, respectively. The directivity function $M(\psi,\theta )$ depends on the angular position of the observer. Note that both $M(\psi,\theta )$ and the exponents $m$ and $n$ vary on the dimensionless porosity parameter $\mu /k$, and must be determined numerically in general.

If $G_0$ in (2.6) is substituted into (2.2), then the first two terms on the right-hand side of (2.6) contribute nothing to the sound field due to the vanishing integral around the half-plane surface, as has been shown previously by Powell (Reference Powell1964). Therefore, the total Green's function $G = G_0 + G_s$ may be approximated to leading order by $G_s$ only,

(2.11)\begin{equation} G(\boldsymbol{x},\boldsymbol{y};t) \approx \frac{\mu^n}{2{\rm \pi}^{{3}/{2}}c^m}\,\varPhi(\boldsymbol{Y})\,\frac{M(\psi,\theta)}{x}\, \text{D}_t^m\,\delta(t_{{\ast}}), \end{equation}

provided that the third term of $G_0$ in (2.6) remains subdominant. The form of the Green's function in (2.11) obviates that $\partial G/\partial y_3$ is indeed subdominant to the magnitudes of the derivatives in other two directions, i.e. $\partial G/\partial y_1$ and $\partial G/\partial y_2$. Therefore, the gradient of the Green's function with respect to $\boldsymbol {y}$, $\partial {G(\boldsymbol {x},\boldsymbol {y};t)}/\partial {\boldsymbol {y}}$, can be approximated by the two-dimensional vector

(2.12)\begin{equation} {\frac{\partial{G(\boldsymbol{x},\boldsymbol{y};t)}}{\partial{\boldsymbol{y}}}} = \frac{\mu^n}{2{\rm \pi}^{{3}/{2}}c^m}\,\frac{M(\psi,\theta)}{ x}\,\text{D}_t\, \delta(t_*)\left({\frac{\partial{}}{\partial{y_1}}}\varPhi(\boldsymbol{Y}), {\frac{\partial{}}{\partial{y_2}}}\varPhi(\boldsymbol{Y}),0\right). \end{equation}

Equation (2.12) may be reworked into a more useful form by introducing the stream function $\varPsi (\boldsymbol {Y})$,

(2.13)\begin{equation} \varPsi(\boldsymbol{Y}) ={-} Y^{{1}/{2}}\cos{\frac{\theta_0}{2}}, \end{equation}

which is related to the velocity potential $\varPhi (\boldsymbol {Y})$ by the Cauchy–Riemann equations

(2.14a,b)\begin{equation} {\frac{\partial{}}{\partial{y_1}}}\varPhi(\boldsymbol{Y}) = {\frac{\partial{}}{\partial{y_2}}}\varPsi(\boldsymbol{Y}), \quad {\frac{\partial{}}{\partial{y_2}}}\varPhi(\boldsymbol{Y}) ={-}{\frac{\partial{}}{\partial{y_1}}}\varPsi(\boldsymbol{Y}). \end{equation}

Therefore, (2.12) can be rewritten as

(2.15)\begin{equation} {\frac{\partial{G(\boldsymbol{x},\boldsymbol{y};t)}}{\partial{\boldsymbol{y}}}} ={\frac{\partial{}}{\partial{\boldsymbol{y}}}} \times [\varPsi(\boldsymbol{Y})\,\boldsymbol{k}]\,\frac{\mu^n}{2{\rm \pi}^{{3}/{2}}c^m}\, \frac{M(\psi,\theta)}{x}\,\text{D}_t\,\delta(t_*). \end{equation}

The substitution of (2.15) into (2.2), the application of the incompressible, inviscid vorticity equation (Howe Reference Howe2003, p. 86),

(2.16)\begin{equation} {\frac{\partial{\boldsymbol{\omega}}}{\partial{t}}}+ \nabla \times (\boldsymbol{\omega} \times \boldsymbol{v})=\boldsymbol{0}, \end{equation}

and an integration by parts determine the final expression of the acoustic pressure in the far field:

(2.17)\begin{align} p(\boldsymbol{x},t) &= \rho \text{D}_t{\iint \omega_3(\kern 0.06em \boldsymbol{y},\tau)}\,F(\boldsymbol{x},\boldsymbol{Y})\,\text{D}_t^m\, \delta(t_* - \tau)\, {\rm d} \boldsymbol{y}\, {\rm d} \tau\nonumber\\ &= \rho \text{D}_t^{m+1} \int \omega_3(\kern 0.06em \boldsymbol{y},t_r)\,F(\boldsymbol{x},\boldsymbol{Y})\, {\rm d} \boldsymbol{y}, \end{align}

where

(2.18)\begin{equation} F(\boldsymbol{x},\boldsymbol{Y}) = \frac{\mu^n}{2{\rm \pi}^{{3}/{2}}c^m}\,\frac{M(\psi,\theta)}{ x}\,\varPsi(\boldsymbol{Y}). \end{equation}

Note that $t_{\ast }$ has been replaced by $t_r = t -x/c$ in (2.17) for a compact turbulence source. Specific details on the compact vortex ring source and its trajectory are described in the next subsection.

2.3. Vorticity distribution and vortex ring motion

Consider a thin-cored vortex ring, whose centre moves in a plane perpendicular to the $y_3$-axis, as shown in figure 2. Here, $O$ is the coordinate origin located at the half-plane edge, and $L$ is the nearest distance of the vortex ring path to the edge. It is assumed that $L$ is larger than the vortex radius $a$ such that the vortex does not collide with the edge. The unit vector normal to the plane of the vortex ring lies in the$(\kern 0.05em y_1, y_2)$-plane, and the $\xi$-axis denotes the vortex path direction. The $\eta$-axis is taken to be perpendicular to the $\xi$- and $y_3$-axes as illustrated, and the origin of the $(\xi, \eta, y_3)$ coordinate system is located at the vortex ring centre. The vorticity of the vortex ring is assumed to be concentrated on a circle of radius $a$ with a small vortex core of radius $\sigma$, and $\sigma / a \ll 1$. The vorticity components in the $(\xi, \eta, y_3)$ system are therefore

(2.19)\begin{equation} (0, -\varGamma\,\delta(\xi)\,\delta(\zeta-a) \sin{\phi}, \varGamma\, \delta(\xi)\,\delta(\zeta-a) \cos{\phi}), \end{equation}

where $\zeta =\sqrt {\eta ^2 + y_3^2}$, $\varGamma$ is the fixed strength of the vortex ring, $\phi$ is the azimuthal angle of the vortex centre from the $y_3$-axis, and $\zeta$ the radial coordinate in the $(\eta, y_3)$-plane. This model set-up is the same as in Kambe et al. (Reference Kambe, Minota and Ikushima1985), which in this work permits parametric comparisons against their analysis for an impermeable edge.

Equation (2.17) determines the sound produced by the vortex ring passing near an edge provided that its strength and trajectory are known. Here, it is assumed that the vortex ring follows the rectilinear path shown in figure 2, which is justified on physical grounds in Appendix C. Direct substitution of (2.19) into (2.17) yields

(2.20)\begin{equation} p\left(\boldsymbol{x}, t+\frac{x}{c}\right) = \rho \varGamma \text{D}_t^{m+1} \int_0^{2{\rm \pi}}{F(\zeta=a,\xi=0;\boldsymbol{C}(t))\,a\cos{\phi}\,{\rm d} \phi}, \end{equation}

where $\boldsymbol {C}(t)$ represents the position of the vortex centre at time $t$, and the dependence of $F$ on $x$ has been suppressed here for simplicity. Following the procedure by Kambe et al. (Reference Kambe, Minota and Ikushima1985), $F$ may be approximated by the two-term Taylor expansion with respect to $\eta$:

(2.21)\begin{equation} F(\zeta=a, \xi=0; \boldsymbol{C}) \approx F(\boldsymbol{C}) + {\frac{\partial{}}{\partial{\eta}}}F(\boldsymbol{C})\,a\cos{\phi}. \end{equation}

Using (2.21), the integration in (2.20) produces

(2.22)\begin{equation} \int_0^{2{\rm \pi}} {Fa\cos{\phi}\,{\rm d} \phi} = {\rm \pi}a^2\,{\frac{\partial{}}{\partial{\eta}}} F(\boldsymbol{C}) = \frac{{\rm \pi} a^2 \mu^n}{2{\rm \pi}^{{3}/{2}}c^m}\,\frac{M(\psi,\theta)}{x}\,v_{\xi} (\boldsymbol{C}), \end{equation}

where $v_{\xi }$ is the complex velocity of the flow in the $\xi$-direction:

(2.23)\begin{equation} v_{\xi} = {\frac{\partial{}}{\partial{\eta}}}\varPsi(\boldsymbol{C}) = {\frac{\partial{}}{\partial{\xi}}}\varPhi(\boldsymbol{C}). \end{equation}

Thus the acoustic pressure expression is

(2.24)\begin{equation} p\left(\boldsymbol{x},t+\frac{x}{c}\right) = \frac{\rho \varGamma \mu^n}{2{\rm \pi}^{{3}/{2}}c^m}\,\frac{M(\psi,\theta)}{ x}\, \text{D}_t^{m+1}[{\rm \pi} a^2\,v_{\xi}(\boldsymbol{C})]. \end{equation}

Note that ${\rm \pi} a^2 v_{\xi }$ is the volume flux of the irrotational flow around the edge passing through the vortex ring (Howe Reference Howe2003). It is now clear that the temporal profile of the acoustic pressure signal is proportional to a fractional rate of change of the volume flux through the vortex ring that depends on $m$. The fractional acceleration of the flow through the vortex ring $\text {D}_{\bar {t}}^{m+1}[{\rm \pi} a^2\, v_{\xi }(\boldsymbol {C})]$ must now be evaluated to furnish scaling estimates of the acoustic emission.

Figure 2.

(a) Side view of the interaction between a semi-infinite porous plate and a vortex ring convecting on a rectilinear path. (b) The local coordinate system of the vortex ring.

Recall that $\text {D}_{\bar {t}}^{m+1}\,v_{\xi }(\boldsymbol {C}) = \text {D}_{\bar {t}}^m[\text {D}_{\bar {t}}\,v_{\xi }(\boldsymbol {C})]$, and $\text {D}_{\bar {t}}\,v_{\xi }(\boldsymbol {C})$ may be adapted directly from Kambe et al. (Reference Kambe, Minota and Ikushima1985):

(2.25)\begin{equation} \text{D}_{\bar{t}}\,v_{\xi}(\boldsymbol{C}) = \tfrac{1}{4}UL^{-{3}/{2}}\text{D}_{\bar{t}}[g(\bar{t})], \end{equation}

where $\bar {t}=Ut/L$ is the dimensionless time, and

(2.26)$$\begin{gather} g(\bar{t}) = \bar{Y}^{-{3}/{2}}\sin{\left(\tfrac{3}{2}\varTheta-2\alpha\right)}, \end{gather}$$

(2.27a,b)$$\begin{gather}\bar{Y} = \frac{Y}{L} = (\bar{t}^2+1)^{{1}/{2}},\quad \varTheta = \tan^{{-}1}\left(\frac{\bar{t}\sin{\alpha}\mp \cos{\alpha}}{\bar{t}\cos{\alpha}\pm\sin{\alpha}}\right). \end{gather}$$

It is clear from (2.25) that $g(\bar {t})$ is proportional to the volume flux through the vortex ring, and its first time derivative is proportional to the acceleration of fluid through the vortex ring. Here, the time origin $\bar {t}=0$ denotes the time instant when the vortex ring is at its nearest distance $L$ to the edge.

Therefore, $\text {D}_{\bar {t}}^{m+1}\,v_{\xi }(\boldsymbol {C})$ in (2.24) may now be expressed as

(2.28)\begin{equation} \text{D}_{\bar{t}}^{m+1}[v_{\xi}(\boldsymbol{C})] = \tfrac{1}{4}U^{m+1}L^{({-}m-{3}/{2})}\text{D}_{\bar{t}}^m [g(\bar{t})], \end{equation}

and the substitution of (2.28), (2.26) and (2.27a,b) into (2.24) yields the acoustic pressure:

(2.29)\begin{equation} p\left(\boldsymbol{x},{\bar{t}}+\frac{x}{c}\right) = \frac{\rho \varGamma a^2 \mu^n U^{m+1}}{8{\rm \pi}^{{1}/{2}}c^m L^{m + {3}/{2}}}\,\frac{M(\psi,\theta)}{ x}\, \text{D}_{\bar{t}}^m[g(\bar{t})]. \end{equation}

The undetermined exponents may be shown to be coupled on dimensional grounds, where $n$ is eliminated here using $n + m = \frac {1}{2}$. Also, the velocity $U$ of a vortex ring is proportional to its circulation $\varGamma$ (Lamb Reference Lamb1924, p. 224):

(2.30)\begin{equation} U = \frac{\varGamma}{4{\rm \pi} a}\left(\ln{\frac{8a}{\sigma}}-\frac{1}{4}\right). \end{equation}

From (2.29), the far-field acoustic pressure radiated by a vortex ring passing near a porous edge scales on

(2.31)\begin{equation} p\sim U^{m+2}L^{-{3}/{2}-m}, \end{equation}

and the corresponding acoustic intensity $\varPi$ is

(2.32)\begin{equation} \varPi = \frac{p^2}{\rho c} \propto U^{\gamma} L^{-\upsilon}, \end{equation}

where $\gamma = 2m+4$ and $\upsilon = 2m+3$.