2. The acoustic governing equations

Neglecting mass sources, body forces and any heat addition and thermal diffusion, the conservation of mass, momentum and energy for a single-component thermally perfect gas gives

(2.1)\begin{gather} \frac{\partial \rho}{\partial t}+ \boldsymbol{\nabla}\boldsymbol{\cdot} (\rho \boldsymbol{u})=0, \end{gather}

(2.2)\begin{gather}\frac{\partial \boldsymbol{u}}{\partial t} +\boldsymbol{\nabla} B = \boldsymbol{u}\times \boldsymbol{\omega}+ T\boldsymbol{\nabla} s + \frac{1}{\rho} \boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{\tau}, \end{gather}

(2.3)\begin{gather}\rho T \frac{\textrm{D} s}{\textrm{D} t}= \boldsymbol{\tau} : \boldsymbol{\nabla} \boldsymbol{u}, \end{gather}

where $\rho , t, \boldsymbol {u}, T$ and $s$ denote density, time, velocity, temperature and entropy, respectively, $B=h+|\boldsymbol {u}|^2/2$ is the stagnation enthalpy, $h$ the enthalpy, $\boldsymbol {\omega }=\boldsymbol {\nabla } \times \boldsymbol {u}$ the vorticity, $\boldsymbol {\tau }$ the viscous stress tensor, $\textrm {D} /\textrm {D} t = \partial / \partial t +\boldsymbol {u}\boldsymbol {\cdot } \boldsymbol {\nabla }$ the material derivative, and : the double dot product, such that ${\boldsymbol{\mathsf{M}}} :{\boldsymbol{\mathsf{N}}} ={tr}({\boldsymbol{\mathsf{M}}} {\boldsymbol{\mathsf{N}}}^\textrm {T})$. Dividing (2.1) by $\rho$ and taking $\partial /\partial t$, multiplying (2.2) by $\rho$ and taking the divergence, and then combining gives

(2.4)\begin{equation} - \rho \frac{\textrm{D}}{\textrm{D} t} \left( \frac{1}{\rho} \frac{\partial \rho}{\partial t}\right) + \boldsymbol{\nabla} \boldsymbol{\cdot}(\rho \boldsymbol{\nabla} B) = \boldsymbol{\nabla} \boldsymbol{\cdot} (\rho \boldsymbol{u}\times \boldsymbol{\omega}+ \rho T\boldsymbol{\nabla} s + \boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{\tau}). \end{equation}

Taking $\boldsymbol {u} \boldsymbol {\cdot }$ (2.2), adding $(1/\rho )\partial p/\partial t$ to both sides, then using the relation $\textrm {d} B = \textrm {d}h + \textrm {d}|\boldsymbol {u}|^2/2$ and the generic thermodynamic relations $\textrm {d} h = \textrm {d} p/\rho + T \textrm {d} s$ and $\textrm {d} \rho = \textrm {d} p/c^2 - \rho \textrm {d} s/ c_p$ (where $c$ is the sound speed and $c_p$ the heat capacity at constant pressure) gives

(2.5)\begin{equation} \frac{1}{\rho} \frac{\partial \rho}{\partial t} = \frac{1}{c^2}\left(\frac{\textrm{D} B}{\textrm{D} t} - T \frac{\textrm{D} s}{\textrm{D} t} -\frac{\boldsymbol{u}\boldsymbol{\cdot} (\boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{\tau})}{\rho} \right)-\frac{1}{c_p}\frac{\partial s}{\partial t}. \end{equation}

Substituting (2.5) into (2.4) finally gives

(2.6)\begin{align} \left[\rho \frac{\textrm{D}}{\textrm{D} t}\left(\frac{1}{c^2} \frac{\textrm{D}}{\textrm{D}t}\right) - \boldsymbol{\nabla} \boldsymbol{\cdot} (\rho \boldsymbol{\nabla})\right]B &=\rho\frac{\textrm{D}}{\textrm{D}t}\left(\frac{T}{c^2}\frac{\textrm{D}s}{\textrm{D}t} + \frac{\boldsymbol{u}\boldsymbol{\cdot} (\boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{\tau})}{\rho c^2}+\frac{1}{c_p}\frac{\partial s}{\partial t} \right) \nonumber\\ &\quad - \boldsymbol{\nabla} \boldsymbol{\cdot} (\rho \boldsymbol{u}\times \boldsymbol{\omega} + \rho T \boldsymbol{\nabla} s + \boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{\tau}). \end{align}

The derivation so far makes only the assumptions stated at the start of the section. We now apply further simplifications to the right-hand side. Due to the high Reynolds numbers typical for flows of practical interest, the viscous stress tensor is neglected for the body of the flow. (Its effect can be retained at the sharp separation edge, where it leads to the generation of an irrotational jet flow bounded by an axisymmetric shear layer, the latter idealised as an infinitely thin vortex sheet or a ‘free streamline’ (Batchelor Reference Batchelor1967; Howe Reference Howe1979), as shown in figure 1(a). The analysis therefore assumes the vorticity fluctuations are either confined to a thin sheet or are negligible). Strictly, viscosity also leads to an entropy source in (2.3), but this effect has been shown to be negligible (Morgans, Goh & Dahan Reference Morgans, Goh and Dahan2013; Xia et al. Reference Xia, Duran, Morgans and Han2018), such that $\textrm {D} s/\textrm {D} t =0$. Further assuming the gas to be calorically perfect, i.e. $c_p$ to be constant, then allows the first and third terms in the first bracket on the right-hand side of (2.6) to be neglected, simplifying the equation to

(2.7)\begin{equation} \left[\rho \frac{\textrm{D}}{\textrm{D} t}\left(\frac{1}{c^2} \frac{\textrm{D}}{\textrm{D}t}\right) - \boldsymbol{\nabla} \boldsymbol{\cdot} (\rho \boldsymbol{\nabla})\right]B=-\boldsymbol{\nabla} \boldsymbol{\cdot} (\rho \boldsymbol{u}\times \boldsymbol{\omega} + \rho T \boldsymbol{\nabla} s). \end{equation}

Equation (2.7) agrees with Howe (Reference Howe2010, (2.6)). To obtain the governing equations for the acoustics, it needs to be linearised. We perform this differently to Howe (Reference Howe2010). We use $\overline {[~]}$ and $[~]'$ to denote mean and perturbation values, respectively. The thermodynamic relation $B'=p'/\bar {\rho }+\bar {T}s'+(|\boldsymbol {u}|^2/2)'$ indicates that, away from the (very thin) region where vorticity perturbations exist, $B'$ comprises only acoustic and entropic components. These need to be clearly distinguished for identifying the entropy-related sound sources.

By neglecting viscous terms, the time-average of (2.1)–(2.3) and (2.5) gives

(2.8a–c)\begin{equation} \boldsymbol{\nabla}\boldsymbol{\cdot} (\bar{\rho} \bar{\boldsymbol{u}})=0 , \quad \boldsymbol{\nabla} \bar{B} = \bar{\boldsymbol{u}}\times \bar{\boldsymbol{\omega}}+ \bar{T}\boldsymbol{\nabla} \bar{s}, \quad \bar{\boldsymbol{u}}\boldsymbol{\cdot} \boldsymbol{\nabla} \bar{s} =\bar{\boldsymbol{u}}\boldsymbol{\cdot} \boldsymbol{\nabla} \bar{B}=0, \end{equation}

and, defining $\bar{\textrm {D}}/\bar{\textrm {D}} t = \partial /\partial t + \bar {\boldsymbol {u}}\boldsymbol {\cdot }\boldsymbol {\nabla }$, the linear perturbation of (2.3) gives

(2.9)\begin{equation} \bar{\textrm{D}}s'/\bar{\textrm{D}} t + \boldsymbol{u}' \boldsymbol{\cdot} \boldsymbol{\nabla} \bar{s} =0. \end{equation}

Taking the gradient of $B'$ gives

(2.10)\begin{equation} \boldsymbol{\nabla} B' =\boldsymbol{\nabla}\left(\frac{p'}{\bar{\rho}}\right)+ \boldsymbol{\nabla}\bar{T}\boldsymbol{\cdot} s' +\bar{T}\boldsymbol{\nabla} s'+\boldsymbol{\nabla}\left(\frac{|\boldsymbol{u}|^2}{2}\right)'. \end{equation}

By combining (2.8a–c) and (2.10), and $B'=p'/\bar {\rho }+\bar {T}s'+(|\boldsymbol {u}|^2/2)'$, we obtain

(2.11)\begin{align} \rho \frac{\textrm{D}}{\textrm{D} t}\left(\frac{1}{c^2} \frac{\textrm{D}}{\textrm{D}t}\right)B = \bar{\rho} \frac{\bar{\textrm{D}}}{\bar{\textrm{D}} t}\left\{ \frac{1}{\bar{c}^2} \left[ \frac{\bar{\textrm{D}}}{\bar{\textrm{D}} t}\left(\frac{p'}{\bar{\rho}} + \left(\frac{|\boldsymbol{u}|^2}{2}\right)'\right) + \bar{T} \frac{\bar{\textrm{D}} s'}{\bar{\textrm{D}} t} + \boldsymbol{\nabla} \bar{B}\boldsymbol{\cdot} \boldsymbol{u}' + \bar{\boldsymbol{u}} \boldsymbol{\cdot} \boldsymbol{\nabla} {\bar{T}} s' \right] \right\}, \end{align}

(2.12)\begin{equation}\boldsymbol{\nabla} \boldsymbol{\cdot} (\rho \boldsymbol{\nabla} B) = \boldsymbol{\nabla} \boldsymbol{\cdot} \left[ \bar{\rho} \boldsymbol{\nabla} \bar{B} + \bar{\rho} \boldsymbol{\nabla}\left( \frac{p'}{\bar{\rho}} + \left(\frac{|\boldsymbol{u}|^2}{2}\right)'\right) + \bar{\rho}\boldsymbol{\nabla} \bar{T} s' + \bar{\rho}\bar{T}\boldsymbol{\nabla} s' + \rho' \boldsymbol{\nabla} \bar{B} \right], \end{equation}

(2.13)\begin{align} & \boldsymbol{\nabla} \boldsymbol{\cdot} (\rho \boldsymbol{u}\times \boldsymbol{\omega} + \rho T \boldsymbol{\nabla} s)\nonumber\\ &\quad = \boldsymbol{\nabla} \boldsymbol{\cdot} [ \bar{\rho}\bar{\boldsymbol{u}}\times \bar{\boldsymbol{\omega}} + \bar{\rho}\bar{T} \boldsymbol{\nabla}\bar{s}+\rho'(\bar{\boldsymbol{u}}\times \bar{\boldsymbol{\omega}} + \bar{T} \boldsymbol{\nabla} \bar{s}) + \bar{\rho}(\boldsymbol{u}\times \boldsymbol{\omega} + T \boldsymbol{\nabla} s)']. \end{align}

Substituting (2.11)–(2.13) into (2.7), cancelling out mean values and incorporating (2.8a–c), (2.9) as well as the thermodynamic relation $c_p \textrm {d} T = T\textrm {d} s + \textrm {d} p/\rho$, the equation gives

(2.14)\begin{align} &\overbrace{\bar{\rho} \frac{\bar{\textrm{D}}}{\bar{\textrm{D}} t}\left[ \frac{1}{\bar{c}^2} \frac{\bar{\textrm{D}}}{\bar{\textrm{D}} t}\left(\frac{p'}{\bar{\rho}} + \left(\frac{|\boldsymbol{u}|^2}{2}\right)'\right)\right] - \boldsymbol{\nabla} \boldsymbol{\cdot} \left[\bar{\rho} \boldsymbol{\nabla}\left(\frac{ p'}{\bar{\rho}} + \left(\frac{|\boldsymbol{u}|^2}{2}\right)'\right)\right]}^{\mathcal{L}_1}\nonumber\\ &\quad + \overbrace{\bar{\rho} \frac{\bar{\textrm{D}}}{\bar{\textrm{D}} t}\left[ \frac{(\bar{\boldsymbol{u}}\times\bar{\boldsymbol{\omega}})\boldsymbol{\cdot} \boldsymbol{u}'}{\bar{c}^2}\right]}^{\mathcal{L}_2} +\overbrace{\boldsymbol{\nabla} \boldsymbol{\cdot}\left( \frac{\boldsymbol{\nabla} \bar{s}}{c_p}p'\right)}^{\mathcal{L}_3} = \overbrace{\boldsymbol{\nabla} \boldsymbol{\cdot} [ \bar{\rho}(\boldsymbol{\omega} \times \boldsymbol{u})']}^{\mathrm{I}} + \overbrace{\boldsymbol{\nabla} \boldsymbol{\cdot} \left(\frac{\boldsymbol{\nabla} \bar{p}}{c_p}s'\right)}^{\mathrm{II}}\nonumber\\ & \quad - \overbrace{\boldsymbol{\nabla} \boldsymbol{\cdot} \left( \bar{\rho}\bar{\boldsymbol{u}} \frac{\bar{\boldsymbol{u}}\boldsymbol{\cdot}\boldsymbol{\nabla}\bar{T}}{\bar{c}^2} s'\right)}^{\mathrm{III}} - \overbrace{\frac{\bar{\rho}\bar{\boldsymbol{u}}\boldsymbol{\cdot} \boldsymbol{\nabla}\bar{T}}{\bar{c}^2}\frac{\partial s'}{\partial t}}^{\mathrm{IV}}. \end{align}

This equation takes the form of an acoustic analogy, with the left-hand side comprising terms which predominantly describe acoustic propagation and the right-hand side comprising acoustic sources due to entropy and vorticity perturbations. It is one of the key results of this paper. Its derivation from the flow conservation equations has assumed only (i) a single-component perfect gas with no external mass, volume force or heat sources, (ii) viscosity negligible in the main body of the fluid and (iii) linear perturbations.

In the absence of entropy and vorticity perturbations, terms $\mathrm {II, III, IV}$ and the $\boldsymbol {\omega }'$-related part of term $\mathrm {I}$ disappear, and the equation collapses to one describing sound propagation. When the mean density and sound speed are constant, $\mathcal {L}_1$ simplifies to the familiar convected wave operator $(\bar{\textrm {D}}^2/\bar{\textrm {D}}t^2/\bar {c}^2-\nabla ^2)$. Terms $\mathcal {L}_2$, $\mathcal {L}_3$ and $\mathrm {I}$ capture the effects of the mean entropy gradient and mean vorticity.

Terms $\mathrm {II, III, IV}$ represent sound sources due to entropy perturbations. With their divergence operator, terms $\mathrm {II}$ and $\mathrm {III}$ correspond to dipole-type sources, with the $\boldsymbol {\nabla } \bar {p}/c_p$ factor in $\mathrm {II}$ consistent with the works of Morfey (Reference Morfey1973, (10)), and Ffowcs Williams & Howe (Reference Ffowcs Williams and Howe1975, (1.4)). Term $\mathrm {IV}$ is a monopole source, which is $\textit {O}(M)$ compared to $\mathrm {II}$, consistent with Ffowcs Williams & Howe (Reference Ffowcs Williams and Howe1975, p. 618). Some new insights are facilitated by this equation: dimensional analysis reveals that term $\mathrm {III}$ is $\textit {O}(M^2)$ smaller than term $\mathrm {II}$, and there is a $He= \omega R_u/ \bar {c}$ factor (where $He$ is the Helmholtz number, $\omega$ is the angular frequency and $R_u$ the radius of the upstream duct) as well as a $\textit {O}(M)$ factor in $\mathrm {IV}$ compared to $\mathrm {II}$. Note that the entropy-related sources, including the leading-order term $\mathrm {II}$, are different to those in Howe (Reference Howe2010, (2.13)). There, the entropy-related source term disappears in the zero-frequency limit, making it impossible to recover the quasi-1-D isentropic model of Marble & Candel (Reference Marble and Candel1977). In contrast, as will be shown in § 5, the current model is able to recover it, highlighting the importance of carefully distinguishing the acoustic perturbation from the entropy perturbation.

The sound sources due to vorticity perturbations are term $\mathrm {I}$ as well as the vortical parts of $(|\boldsymbol {u}|^2/2)'$ in term $\mathcal {L}_1$ and $\boldsymbol {u}'$ in term $\mathcal {L}_2$. The vortical parts will be non-zero in the region where $\boldsymbol {\omega }'\neq \boldsymbol {0}$. Howe (Reference Howe1975) showed that term $\mathrm {I}$ is dominant at low Mach numbers, the other terms being $\textit {O}(M^2)$ less efficient sound generators. This justifies our leaving them on the left-hand side, rather than attempting to separate them out as acoustic sources.

In order to formally derive a form for (2.14) which is both simpler and has a stricter separation of propagation and source terms, the source terms are simplified. This is done by restricting our attention to only low-Mach-number flows at low frequencies ($He\le \textit {O}(10^{-1})$). It is worth noting that (2.14) does not require a low-Mach-number assumption. The new insights discussed above then allow us to neglect the entropy source terms $\mathrm {III}$ and $\mathrm {IV}$. Representing $\boldsymbol {u}'$ as the sum of acoustic, $[~]'_a$, and vortical components, $[~]'_v$, the vortical parts of $(|\boldsymbol {u}|^2/2)'$ in term $\mathcal {L}_1$ and $\boldsymbol {u}'$ in term $\mathcal {L}_2$ can also be neglected. Equation (2.14) then simplifies to

(2.15)\begin{align} &\overbrace{\bar{\rho} \frac{\bar{\textrm{D}}}{\bar{\textrm{D}} t}\left[ \frac{1}{\bar{c}^2} \frac{\bar{\textrm{D}}}{\bar{\textrm{D}} t}\left(\frac{p'}{\bar{\rho}} + \left(\frac{|\boldsymbol{u}|^2}{2}\right)_a'\right)\right] - \boldsymbol{\nabla} \boldsymbol{\cdot} \left[\bar{\rho} \boldsymbol{\nabla}\left(\frac{ p'}{\bar{\rho}} + \left(\frac{|\boldsymbol{u}|^2}{2}\right)_a'\right)\right]}^{\mathcal{L}_1} + \overbrace{\bar{\rho} \frac{\bar{\textrm{D}}}{\bar{\textrm{D}} t}\left[ \frac{(\bar{\boldsymbol{u}}\times\bar{\boldsymbol{\omega}})\boldsymbol{\cdot} \boldsymbol{u}_a'}{\bar{c}^2} \right]}^{\mathcal{L}_2} \nonumber\\ &\quad + \overbrace{\boldsymbol{\nabla} \boldsymbol{\cdot}\left( \frac{\boldsymbol{\nabla} \bar{s}}{c_p}p'\right)}^{\mathcal{L}_3} - \overbrace{\boldsymbol{\nabla} \boldsymbol{\cdot} [ \bar{\rho}(\bar{\boldsymbol{\omega}} \times \boldsymbol{u}_a')]}^{{\mathcal{L}_4}} = \overbrace{\boldsymbol{\nabla} \boldsymbol{\cdot} [ \bar{\rho}(\bar{\boldsymbol{\omega}} \times \boldsymbol{u}_v' +\boldsymbol{\omega}' \times \bar{\boldsymbol{u}})]}^{\mathrm{I}} + \overbrace{\boldsymbol{\nabla} \boldsymbol{\cdot} \left(\frac{\boldsymbol{\nabla} \bar{p}}{c_p}s'\right)}^{\mathrm{II}}. \end{align}

The left-hand side is now purely an equation governing sound propagation, with the right-hand side being the simplified sound sources. The left-hand side gives the full equation governing how sound propagates after it is generated within the considered domain or inserted at a boundary: $\mathcal {L}_1$ simplifies to $(\bar{\textrm {D}}^2/\bar{\textrm {D}}t^2/\bar {c}^2-\nabla ^2)$ when the effect of spatial gradients of the mean density and mean sound speed on the sound propagation are neglected, $\mathcal {L}_2$ and $\mathcal {L}_4$ capture the effect of the mean vorticity on the sound propagation, and $\mathcal {L}_3$ captures the effect of the mean entropy gradient on the sound propagation. The right-hand side gives the low-Mach-number sound source terms within the domain – $\mathrm {I}$ and $\mathrm {II}$ are due to vorticity and entropy perturbations, respectively.

3. Sound generation at the sudden expansion

The theory developed in § 2 is now applied to the sudden area expansion problem of figure 1. For this problem, the mean flow is predominantly in the $x$-direction, and the frequencies of interest are low. This means that (2.15) can be further simplified.

At low frequencies, any mean flow non-uniformity has much smaller spatial extent than the acoustic wavelength. The effect on the sound propagation is small, such that for the propagation terms on the left-hand side, $\bar {\rho }$, $\bar {c}$ and the mean flow velocity can be taken as uniform ($\bar {\boldsymbol {u}}=\bar {u}\boldsymbol {{i}}$ where $\boldsymbol {{i}}$ is the unit vector in the axial direction) in $\mathcal {L}_1$. While $\mathcal {L}_2$ and $\mathcal {L}_4$ exist where $\bar {\boldsymbol {\omega }}\neq \boldsymbol {0}$, dimensional analysis shows that they are $\textit {O}(M)$ compared to the first and second terms in $\mathcal {L}_1$, respectively, and can be neglected. Term $\mathcal {L}_3$ is approximately $\textit {O}(M^2)$ compared to the second term in $\mathcal {L}_1$ and is also neglected. Defining the acoustic component of the stagnation enthalpy fluctuation as $B'_a= p'/\bar {\rho }+(|\boldsymbol {u}|^2/2)'_a$ and using the ansatz $[~]'=\widetilde {[~]}\,\mathrm {e}^{\mathrm {i}\omega t}$ where $\widetilde {[~]}$ denotes the Fourier amplitude, (2.15) then reduces to

(3.1)\begin{equation} \left[\frac{1}{\bar{c}^2}\left(\mathrm{i}\omega+\bar{u}\frac{\partial }{\partial x}\right)^2 - \nabla^2 \right] \tilde{B}_a = \boldsymbol{\nabla} \boldsymbol{\cdot} \left(\tilde{\boldsymbol{\omega}} \times \bar{\boldsymbol{u}}_c \right) + \frac{1}{\bar{\rho}}\boldsymbol{\nabla} \boldsymbol{\cdot} \left(\frac{\boldsymbol{\nabla} \bar{p}}{c_p}\tilde{s}\right). \end{equation}

For the first term on the right-hand side, $\bar {\boldsymbol {u}}_c$ is the convection velocity of the shed vorticity, which is typically the average velocity in the resulting shear layer (Howe Reference Howe1979; Dupere & Dowling Reference Dupere and Dowling2001), and $\bar {\boldsymbol {\omega }}\times \tilde {\boldsymbol {u}}_v$ is neglected based on Howe (Reference Howe1979), Dupere & Dowling (Reference Dupere and Dowling2001), Yang & Morgans (Reference Yang and Morgans2016). When the frequency tends to zero, the vorticity phase variation along the vortex sheet near the expansion becomes negligible, and term $\mathrm {I}$ in (2.15) can also be approximated by a quasi-steady approach as in Howe (Reference Howe2010, § 4.3).

This equation is now solved using the Green's function method. The Green's function, $\tilde {G}(\boldsymbol {x},\boldsymbol {y}; \omega )$, denotes the acoustic response at the location of an observer, $\boldsymbol {x}$, due to a Dirac delta function, $\delta$, at the source location, $\boldsymbol {y}$:

(3.2)\begin{equation} \left[\frac{1}{\bar{c}^2}\left(\mathrm{i}\omega+\bar{u}\frac{\partial }{\partial x}\right)^2 - \nabla^2 \right] \tilde{G} = \delta(\boldsymbol{x}-\boldsymbol{y}). \end{equation}

For a given space $V$, bounded by surfaces $\boldsymbol {s}$ (being positive in the outwards direction), the solution for $\tilde {B}_a$ can be obtained by convolution of the Green's function with the boundary and volume acoustic source terms. Defining $k=\omega /\bar {c}$, this gives

(3.3)\begin{equation} \tilde{B}_a = \int_{\partial V} \left(-2\mathrm{i}k\bar{M}\tilde{G}\tilde{B}_a \boldsymbol{{i}} + \tilde{G}\boldsymbol{\nabla}\tilde{B}_a - \tilde{B}_a\boldsymbol{\nabla}\tilde{G} \right) \boldsymbol{\cdot} \mathrm{d} \boldsymbol{s} - \int_{V} \boldsymbol{\nabla}\tilde{G} \boldsymbol{\cdot} \left( \tilde{\boldsymbol{\omega}} \times \bar{\boldsymbol{u}}_c + \frac{\boldsymbol{\nabla} \bar{p}}{\bar{\rho} c_p}\tilde{s}\right) \mathrm{d}v. \end{equation}

The solution follows the method of Yang & Morgans (Reference Yang and Morgans2016, § 2), dividing the problem into a region upstream of the expansion and a region downstream. For each, the Green's function is expressed as a Fourier–Bessel expansion for the cylindrical geometry and the relevant boundary conditions applied, before substituting into (3.3).

For the acoustic source terms, upstream of the expansion, the entropy disturbance advects within a uniform flow and neither vortical nor entropic acoustic sources exist. Downstream, flow separation causes a thin cylindrical sheet of vorticity at the radius of the upstream duct. When the entropy wave passes through the expansion, it decelerates, setting up an acoustic field. This contains many radial modes in the vicinity of the expansion, even though all other than the plane wave mode will be evanescent at low frequencies. This acoustic field in turn leads to unsteady vorticity to be shed at the separation edge. Thus, both entropic and vortical source terms exist.

The velocity oscillation at the interface is expanded as the sum of a series of Bessel functions to capture the large number of radial modes – the strength of each ‘Bessel mode’ needs to be solved for. The strength of the unsteady vorticity follows from applying a Kutta condition at the separation edge (e.g. Howe Reference Howe1979), and again needs to be solved for. The perturbation $\tilde {B}_a$ just upstream of the expansion interface is represented in terms of the interface velocity oscillation (Yang & Morgans Reference Yang and Morgans2016, (15)). The perturbation $\tilde {B}_a$ just downstream of the expansion is represented in terms of the interface velocity oscillation and the unsteady vorticity (Yang & Morgans Reference Yang and Morgans2016, (32)). An important difference in the present work is the additional presence of an entropic source term downstream of the expansion, with this depending on the mean pressure gradient in both the axial and radial directions:

(3.4)\begin{equation} \tilde{B}_a(x=0^+, r_x\leq R_d)|_s = - \int_{V} \left( \frac{\partial \tilde{G}}{\partial y} \frac{\partial \bar{p}}{\partial y} + \frac{\partial \tilde{G}}{\partial r_y} \frac{\partial \bar{p}}{\partial r_y}\right) \frac{\tilde{s}}{\bar{\rho} c_p}\, \mathrm{d}v. \end{equation}

The mean pressure and entropy wave profiles are obtained from simulations, as described in § 4. Finally, by applying continuity of $\tilde {B}_a$ across the expansion interface and the Kutta condition at the separation edge, both the velocity oscillation and unsteady vorticity can be resolved. Then $\tilde {B}_a$ and the reflected and transmitted acoustic waves can be obtained.

5. Simplified models

The physical mechanism causing the low-frequency acoustic reflection coefficient to differ substantially from that predicted by Marble & Candel (Reference Marble and Candel1977) remains unclear. Two simplified models are proposed in this section to further analyse this difference.

The first model, which we denote SimpM$1$, simplifies the entropy-related source term in (3.1). Figure 2 indicates that the source term is negligible inside the annular separation bubble and the mean pressure gradient is mainly in the axial direction (the radial gradient of $\bar {p}$ in (3.4) is negligible). At low frequencies, the entropy perturbation remains quasi-1-D. We thus assume a quasi-1-D mean flow profile within the given separation streamline (i.e. a nozzle geometry that follows the separation streamline). Additionally, this mean flow is assumed isentropic so that we can use the classical solutions for an isentropic 1-D nozzle. The entropy perturbation is then obtained from the 1-D advection equation $(\boldsymbol {i}\omega + \bar {u}(x) \textrm {d}/\textrm {d}\kern 0.05em x)\tilde {s}=0$. This greatly simplifies the entropy-related source term.

The second model, which we denote SimpM$2$, applies the simplifications of SimpM$1$, and further simplifies the Green's function in (3.4). An acoustic field is generated where there is flow deceleration, hence where the cross-sectional area occupied by the central jet varies. If the main jet expansion, and hence flow deceleration, occurs substantially beyond the area-increase interface, the generated higher-order acoustic modes decay before they reach the interface. We can then neglect all higher-order terms in the Green's function Bessel expansion and retain only the plane acoustic waves generated by the entropy-related source. This assumption is supported by the simulated pressure field in figure 2(d).

One would expect SimpM$1$ to recover Marble & Candel's compact quasi-1-D isentropic model if the frequency is low. However, we now show this to be true only if the axial extent of the recirculation zone is negligible. In order to artificially consider recirculation zones of different axial extents, an error function is used to model different separation streamlines, as shown in figure 4(a). A small standard deviation, $\epsilon =0.0001$, ensures a steep expansion at $y_{{mid}}$. Figure 4(b) shows that SimpM$1$ predictions tend to those from the Marble & Candel (Reference Marble and Candel1977) model only when $y_{{mid}}\lesssim 10^{-2}$. Beyond this value, the reflected acoustic wave amplitude falls off until it converges, at approximately $y_{{mid}}=1$, to the prediction from SimpM$2$. Thus, if the expansion of the central jet occurs an (upstream) radius or more beyond the expansion interface, the sound generated is very different to that predicted by the compact quasi-1-D isentropic model, even in the low-frequency (compact) limit. The sound is then accurately predicted by the simplest model, SimpM$2$.

Figure 4. Effect of the axial extent of the recirculation zone on the acoustic response. (a) Green dashed line denotes a prescribed separation streamline for the central jet, $r_y=1.35+0.35{\rm erf}[(y-y_{mid})/(\sqrt {2}\epsilon )]$ where $\textrm{erf}$ is the error function, and $\tilde {p}^{\pm }, \tilde {p}^{\pm }_{{mid}}$ are plane acoustic waves. (b) Reflected acoustic wave amplitude at $S_t=0.01$ against axial location of streamline expansion, $y_{mid}$: ‘noV’ denotes that the vortical sound source is neglected.

For real flows across significant area expansions, the recirculation zone usually extends far enough beyond the expansion plane that SimpM$2$ can be expected to provide good sound prediction. This is confirmed for the sudden expansion, now using the separation streamline from the RANS mean flow. Figure 5 shows that SimpM$1$ and SimpM$2$ predict very similar low-frequency acoustic reflection amplitudes, these being much closer to the LNSE results than those from the Marble & Candel model. They both also capture the higher-frequency fall-off, although with less accurate slope prediction than from the full model of figure 3. This suggests that, for the entropy-related source, 2-D effects become important at higher frequencies, consistent with visualisations in figure 2(a).

Figure 5. Reflected acoustic wave amplitude against Strouhal number using the RANS separation streamline: ‘noV’ and ‘V’ denote neglecting and accounting for vortical sound source, respectively.

Finally, both SimpM$1$ and SimpM$2$ can either neglect or account for the vorticity-related source term in (3.1). Figure 5 shows that accounting for it improves the predictions towards the LNSE results, but only slightly. This confirms that the entropy-related source is dominating over vorticity-related source, and that the size of the recirculation zone rather than flow non-isentropicity is the main factor causing different sound to that predicted by the quasi-1-D isentropic models.