3.1. Numerical methods and computational set-up

The confined orifice flow is analysed using a two-level numerical framework. A steady compressible Reynolds-averaged solution is first computed and adopted as the base flow for the subsequent resolvent analysis. The LES is then employed as a higher-fidelity reference, both to assess the validity of the steady base flow and to evaluate the baseline circular and selected chamfered configurations. This arrangement provides a computationally efficient description of geometry-dependent mean-flow variation while retaining access to the unsteady vortical dynamics responsible for wall-pressure fluctuations. The use of a mean-flow-based resolvent formulation has become standard in modern resolvent analysis and has been shown to provide a physically meaningful description of amplification mechanisms in statistically stationary turbulent flows (McKeon & Sharma Reference Mckeon and Sharma2010; Moarref et al. Reference Moarref, Sharma, Tropp and Mckeon2013; Abreu et al. Reference Abreu, Cavalieri, Schlatter, Vinuesa and Henningson2020).

The governing equations are the compressible Navier–Stokes equations written in conservative form as

(3.1) \begin{equation} \frac{\partial \rho }{\partial t}+\boldsymbol{\nabla }\boldsymbol{\cdot }\left(\rho \boldsymbol{u}\right)=0, \end{equation}

(3.2) \begin{equation} \frac{\partial (\rho \boldsymbol{u})}{\partial t}+\boldsymbol{\nabla }\boldsymbol{\cdot }\left(\rho \boldsymbol{u}\otimes \boldsymbol{u}+p\boldsymbol{I}\right)=\boldsymbol{\nabla }\boldsymbol{\cdot }\boldsymbol{\tau }, \end{equation}

(3.3) \begin{equation} \frac{\partial (\rho E)}{\partial t}+\boldsymbol{\nabla }\boldsymbol{\cdot }\left[\left(\rho E+p\right)\boldsymbol{u}\right]=\boldsymbol{\nabla }\boldsymbol{\cdot }\left(\boldsymbol{\tau }\boldsymbol{\cdot }\boldsymbol{u}\right)-\boldsymbol{\nabla }\boldsymbol{\cdot }\boldsymbol{q}, \end{equation}

where ρ, u , p and E denote the density, velocity vector, pressure and total energy per unit mass, respectively, I is the identity tensor, and ⊗ denotes the tensor product. The viscous stress tensor τ is written as τ = μ[▽ u + (▽ u ) ^T – 2/3 · (▽ · u ) I ], and the heat flux q is modelled using Fourier’s law q = –λ▽T, where μ and λ are the molecular viscosity and thermal conductivity, respectively. The steady base flow is obtained by solving the compressible Reynolds-averaged Navier–Stokes (RANS) equations closed with the shear stress transport (SST) k–ω model (Menter Reference Menter1994). In eddy-viscosity form, the mean momentum equation may be written as

(3.4) \begin{equation} \frac{\partial (\overline{\rho }\,\overline{\boldsymbol{u}})}{\partial t}+\boldsymbol{\nabla }\boldsymbol{\cdot }\left(\overline{\rho }\overline{\boldsymbol{u}}\otimes \overline{\boldsymbol{u}}+\overline{p}\boldsymbol{I}\right)=\boldsymbol{\nabla }\boldsymbol{\cdot }\left[\left(\mu +\mu _{t}\right)\left(\boldsymbol{\nabla }\overline{\boldsymbol{u}}+\left(\boldsymbol{\nabla }\overline{\boldsymbol{u}}\right)^{T}-\frac{2}{3}\left(\boldsymbol{\nabla }\boldsymbol{\cdot }\overline{\boldsymbol{u}}\right)\boldsymbol{I}\right)\right]\!, \end{equation}

where the overbar denotes Reynolds averaging and μ _t is the turbulent eddy viscosity provided by the SST closure. The higher-fidelity LES solves the spatially filtered compressible equations, in which the unresolved scales are represented by the subgrid-scale stress tensor τ ^sgs. In the present work, the subgrid-scale viscosity is modelled using the wall-adapting local eddy-viscosity model (Ducros et al. Reference Ducros, Nicoud and Poinsot1998),

(3.5) \begin{equation} {\mu }_{t}^{\textit{sgs}}=\rho (C_{w}\varDelta)^{2}\frac{\big({S}_{ij}^{d}{S}_{ij}^{d}\big)^{3/2}}{\big(\tilde{S}_{ij}\tilde{S}_{ij}\big)^{5/2}+\big({S}_{ij}^{d}{S}_{ij}^{d}\big)^{5/4}}\!, \end{equation}

where $\varDelta$ is the local filter width, $\tilde{\textit{S}}_{\textit{ij}}$ is the resolved strain-rate tensor and ${\textit{S}}_{\textit{ij}}^{\textit{d}}$ is the traceless symmetric part of the square of the resolved velocity-gradient tensor. The wall-adapting local eddy-viscosity model is adopted because it provides the correct near-wall asymptotic behaviour without the need for an additional damping function.

The computational domain follows the confined circular-orifice configuration introduced in § 2, with an upstream length of 3D and a downstream length of 8D, consistent with figure 2(a). At the inlet, a bulk velocity corresponding to Re _D = 10 000 is imposed. At the outlet, a convective outflow boundary condition is prescribed in the spirit of non-reflecting formulations for compressible viscous flow. No-slip boundary conditions are applied at the wall. Spatial and temporal discretisation are based on a second-order finite-volume formulation, and pressure–velocity coupling is handled through a pressure correction-type predictor–corrector procedure. Figure 3(a) presents a close-up view of the resolved turbulent structures in the orifice region, identified by the Q = 1 × 10⁷ isosurface and coloured by the vorticity magnitude. Grid dependence is assessed against the particle image velocimetry (PIV) measurements reported in figure 21 of Shan et al. (Reference Selvam, Peixinho and Willis2021), using supplementary simulations on three meshes containing 5.4 × 10⁶, 8.7 × 10⁶ and 10.2 × 10⁶ cells, respectively. As shown in figure 3(b), the transverse profile of normalised mean streamwise velocity becomes essentially insensitive to further refinement from the fine mesh to the dense mesh, and the converged numerical result reproduces the PIV measurement with satisfactory agreement. The temporal resolution is examined in figure 3(c). With the adopted time step Δt = 1 × 10⁻⁶ s, the maximum convective Courant–Friedrichs–Lewy number remains below 0.4 throughout the baseline calculation. Additional simulations with Δt = 5 × 10⁻⁶ s and 5 × 10⁻⁷ s show that the mean-flow profile obtained with Δt = 1 × 10⁻⁶ s closely agrees with that obtained with the smaller time step, while the larger time step produces a slightly more visible deviation. The result indicates that the mean-flow profile used for the present validation is only weakly affected by further reduction of the time step. The fine mesh and Δt = 1 × 10⁻⁶ s are therefore adopted in the subsequent LES computations as a compromise between numerical accuracy and computational cost. The principal working conditions, numerical resolution parameters and statistical sampling settings used in the numerical simulations are summarised in table 1. The wall-pressure spectra were estimated from 1 s records sampled at 10 kHz. The one-sided power spectral densities (PSDs) were calculated using Welch’s method with a 4096-point periodic Hann window, 75 % overlap (Welch Reference Wang, Lele and Moin1967). Six overlapping blocks were averaged, giving a frequency-bin spacing of 2.44 Hz, corresponding to ΔSt = 0.02.

Table 1.

Summary of the working conditions, numerical resolution and statistical sampling parameters used in the numerical computations.

Table 1. Long description

The table presents parameters and their values used in numerical computations. It has two columns: Parameter and Value. The parameters include Bulk Reynolds number, inlet Mach number, number of cells for baseline circular orifice, maximum wall resolution, time step for LES calculations, statistical averaging time, total simulated physical time, and number of full-field snapshots. The values are ReD = 10 000, M = 9 x 10^−3, 8.7 x 10^6 cells, y+ 0.8, Δt = 1 x 10^+6 s, 5 s, 10 s (55 flow-through times), and 10 000 snapshots (1 s record duration).

Figure 3.

(a) Close-up view of the flow structures in the orifice region, with the Q = 1 × 10⁷ isosurface coloured by the vorticity magnitude. (b) Grid-resolution sensitivity of the transverse profile of normalised mean streamwise velocity obtained with different mesh resolutions, together with the PIV measurements reported by Shan et al. (Reference Selvam, Peixinho and Willis2021) in their figure 21. (c) Time step sensitivity of the same mean-flow quantity, comparing Δt = 5 × 10⁻⁶ s, 1 × 10⁻⁶ s and 5 × 10⁻⁷ s.

A further mean-flow consistency check is provided in figure 4 through a direct comparison between the steady base flow and the corresponding LES statistics for the baseline orifice. The centreline static-pressure distribution in figure 4(a) shows that the RANS and LES solutions give consistent pressure-drop and downstream-recovery trends for the baseline orifice. The transverse mean streamwise velocity profiles in figure 4(b) further show that the steady solution captures the contraction of the orifice jet, its downstream spreading and the gradual recovery of the pipe flow with acceptable agreement. Overall, the steady Reynolds-averaged base flow retains the principal mean-flow features of the confined-orifice configuration, including the confined jet issuing from the orifice, the downstream adverse-pressure-gradient region and the associated recirculating motion near the wall. These comparisons support the use of the steady solution as the base state for the subsequent resolvent analysis. The relative pressure-drop variation and downstream pressure spectra for representative configurations, together with the overall sound pressure level (OASPL) across the chamfer design space, are assessed experimentally in Appendix A.

Figure 4.

(a) Static-pressure distribution along the pipe centreline for the baseline flow. (b) Transverse profiles of the mean streamwise velocity at selected streamwise locations around the orifice, showing the agreement between the RANS and LES results.

3.2. Resolvent analysis implementation

Figure 5 summarises the present resolvent-based optimisation workflow. The central idea is to retain the full flow dynamics in the resolvent formulation, while introducing the wall-pressure quantity relevant to flow-induced noise only at the output stage. In this way, the optimisation is not performed directly on a reduced wall-only system, but on the wall-pressure manifestation of the dominant amplified response supported by the full flow field. This treatment is particularly suitable for the present confined orifice flow, in which the dominant source activity is associated with wall-pressure fluctuations induced by separated and reattaching motions near the inner orifice and the immediate downstream surface.

Figure 5.

Workflow of the resolvent-based optimisation framework for the chamfered orifice, including geometry parameterisation, base-flow computation, full-state resolvent analysis, wall-pressure extraction and proxy-objective construction, constrained optimisation and postoptimal mechanism analysis.

For each geometry ξ = (c, α), the steady base flow $\overline{\boldsymbol{q}}$ ( x ; ξ) obtained in § 3.1 is taken as the reference state, and the flow variables are decomposed as

(3.6) \begin{equation} \boldsymbol{q}\left(\boldsymbol{x},t;\boldsymbol{\xi }\right)=\overline{\boldsymbol{q}}\left(\boldsymbol{x};\boldsymbol{\xi }\right)+\boldsymbol{q}'\left(\boldsymbol{x},t;\boldsymbol{\xi }\right)\!, \end{equation}

where $\boldsymbol{q}^{\prime}$ denotes the fluctuation about the steady base flow. The fluctuation state is written in primitive variables as $\boldsymbol{q}^{\prime}$ = [ρ $^{\prime}$ , ( u $^{\prime}$ ) ^T , p $^{\prime}$ ] ^T , where ρ $^{\prime}$ , u $^{\prime}$ = [u _x $^{\prime}$ , u _y $^{\prime}$ , u _z $^{\prime}$ ] ^T , and p $^{\prime}$ denote the perturbations in density, the three velocity components and the pressure, respectively. Following the standard resolvent framework of McKeon & Sharma (Reference Mckeon and Sharma2010), the nonlinear terms not retained in the linearisation are treated as an intrinsic forcing. This forcing–response interpretation has since been adopted widely in gain-based resolvent studies of turbulent wall-bounded flows (Moarref et al. Reference Moarref, Sharma, Tropp and Mckeon2013; Karban et al. Reference Karban, Martini, Cavalieri, Lesshafft and Jordan2022) and has also been extended to primitive-variable formulations for pressure fluctuations (Luhar et al. Reference Luhar, Sharma and Mckeon2014). The fluctuation dynamics may therefore be written in input–output form as

(3.7) \begin{equation} \frac{\partial \boldsymbol{q}'}{\partial t}=\boldsymbol{A}_{\overline{\boldsymbol{q}},{\mu _{t}}}(\boldsymbol{\xi })\boldsymbol{q}'+\boldsymbol{Bf}, \boldsymbol{y}=\boldsymbol{Cq}'. \end{equation}

Here, $\boldsymbol{A}_{\overline{\boldsymbol{q}},{\mathit{\mu }_{\textit{t}}}}$ is the linearised compressible Navier–Stokes operator about the steady base flow. Using the compact ordering q ’, its block structure can be written as

(3.8) \begin{equation} \boldsymbol{A}_{\overline{\boldsymbol{q}},{\mu _{t}}}=\left[\begin{array}{lll} A_{\rho \rho } & A_{\rho u} & A_{\rho p}\\[3pt] A_{u\rho } & A_{uu} & A_{up}\\[3pt] A_{p\rho } & A_{pu} & A_{pp} \end{array}\right]\!. \end{equation}

The block A _uu contains mean-flow advection, coupling with the mean velocity gradients, and viscous diffusion based on the effective viscosity μ + μ _t . The remaining blocks describe the compressible coupling between density, velocity and pressure perturbations. In this study, the unresolved nonlinear action is represented by an equivalent volumetric forcing in the momentum equations. With f = [f _ux , f _uy , f _uz ] ^T , the input operator has the block form

(3.9) \begin{equation} \boldsymbol{B}=\left[\begin{array}{l} 0\\[3pt] I_{3}\\[3pt] 0 \end{array}\right]\!, \end{equation}

where the identity block maps the three forcing components to the corresponding momentum equations. Therefore, B is not the identity operator on the full perturbation state. No independent forcing is introduced into the continuity or pressure equation. The operator C defines the response retained for the subsequent singular value decomposition. Assuming harmonic time dependence at angular frequency ω,

(3.10) \begin{equation} \boldsymbol{q}'(\boldsymbol{x},t;\boldsymbol{\xi })=\hat{\boldsymbol{q}}\left(\boldsymbol{x},\omega ;\boldsymbol{\xi }\right)e^{i\omega t},\boldsymbol{f}\left(\boldsymbol{x},t;\boldsymbol{\xi }\right)=\hat{\kern-2.5pt\boldsymbol{f}}\left(\boldsymbol{x},\omega ;\boldsymbol{\xi }\right)e^{i\omega t}, \end{equation}

the governing equation becomes

(3.11) \begin{equation} \left(i\omega \boldsymbol{I}-\boldsymbol{A}_{\overline{\boldsymbol{q}},{\mu _{t}}}\left(\boldsymbol{\xi }\right)\right)\hat{\boldsymbol{q}}=\boldsymbol{B}\hat{\kern-2.5pt\boldsymbol{f}}, \end{equation}

and the corresponding output is

(3.12) \begin{equation} \hat{\boldsymbol{y}}=\boldsymbol{C}\hat{\boldsymbol{q}}=\boldsymbol{C}\big(i\omega \boldsymbol{I}-\boldsymbol{A}_{\overline{\boldsymbol{q}},{\mu _{t}}} (\boldsymbol{\xi })\big)^{-1}\boldsymbol{B}\hat{\kern-2.5pt\boldsymbol{f}}=\mathrm{\mathcal{H}}\left(\omega ;\boldsymbol{\xi }\right)\hat{\kern-2.5pt\boldsymbol{f}}. \end{equation}

The operator,

(3.13) \begin{equation} \mathrm{\mathcal{H}}\big(\omega ;\boldsymbol{\xi }\big)=\boldsymbol{C}\big(i\omega \boldsymbol{I}-\boldsymbol{A}_{\overline{\boldsymbol{q}},{\mu _{t}}} (\boldsymbol{\xi })\big)^{-1}\boldsymbol{B}, \end{equation}

is the resolvent operator from the intrinsic forcing to the chosen output. In the present study, C is selected so that the output corresponds to the wall-pressure response on the orifice surface of interest, rather than to a reduced wall-only dynamical system. This choice is consistent with the primitive-variable resolvent treatment of pressure fluctuations in wall turbulence (Luhar et al. Reference Luhar, Sharma and Mckeon2014) and with later resolvent-based estimation frameworks in which the output operator is tailored to the quantity of interest (Martini et al. Reference Martini, Jung, Cavalieri, Jordan and Towne2022; Ying et al. Reference Ying, Chen, Li and Fu2023). After spatial discretisation, the harmonic resolvent problem for each geometry ξ is written in the generic form,

(3.14) \begin{equation} \boldsymbol{K}_{\boldsymbol{\xi }} (\omega )\hat{\boldsymbol{q}}=\boldsymbol{B}\hat{\kern-2.5pt\boldsymbol{f}}, \boldsymbol{K}_{\boldsymbol{\xi }}(\omega)=\boldsymbol{K}_{\mathit{0,}\boldsymbol{\xi }}-i\omega \boldsymbol{L}_{\boldsymbol{\xi }}, \end{equation}

where K _0,ξ contains the linearised advection, pressure-coupling and viscous terms, and L _ξ collects the temporal part of the discretised system. To distinguish forced amplification from autonomous temporal growth, the homogeneous form of the same discretised operator is also considered for the baseline orifice,

(3.15) \begin{equation} \boldsymbol{K}_{\mathit{0,}{\boldsymbol{\xi }_{0}}}\hat{\boldsymbol{q}}_{\!j}=\lambda _{\!j}\boldsymbol{L}_{{\boldsymbol{\xi }_{0}}}\hat{\boldsymbol{q}}_{\!j}, \end{equation}

where λ _j = λ _r,j + iλ _i,j and the temporal dependence is written as

(3.16) \begin{equation} \boldsymbol{q}_{\!j} (\boldsymbol{x},t)=\hat{\boldsymbol{q}}_{\!j}(\boldsymbol{x})e^{(\lambda _{\!j}t)}. \end{equation}

Both the forced resolvent problem in (3.14) and the homogeneous global eigenvalue problem in (3.15) are solved over the same computational domain as the numerical simulations. Homogeneous conditions are imposed on all perturbation variables at the inlet, so that no external disturbance is prescribed. At the rigid walls, homogeneous no-slip conditions are applied to the velocity perturbations, together with zero-normal-gradient conditions for the pressure and density perturbations. Homogeneous terminal conditions are imposed at the outlet. Smooth sponge layers are placed adjacent to the inlet and outlet to damp the velocity and pressure perturbations before they reach the truncated boundaries and thereby reduce artificial reflections. The resulting operator is sparse, and its dimension N _q is set by the number of flow unknowns retained on the computational mesh for the geometry under consideration. The input operator retains the momentum-block structure defined above. The forcing and response norms are both evaluated using the quadrature-weighted inner product,

(3.17) \begin{equation} \left\langle \boldsymbol{a},\boldsymbol{b}\right\rangle _{W}=\boldsymbol{b}^{*}\boldsymbol{Wa}, \end{equation}

where W denotes the weight matrix appropriate to the quantity being measured. For the response, W incorporates the spatial quadrature weights and the variable weights associated with the compressible perturbation-energy norm, following Jung et al. (Reference Jung, Bhagwat and Towne2025). For the forcing, W applies the spatial quadrature weights to the three momentum-forcing components. The same notation is used for both inner products, with the corresponding form of W adopted for the input or response space.

To account for the turbulent mean-flow effects in the linear dynamics, the eddy viscosity μ _t obtained from the SST k–ω base-flow computation is retained in $\boldsymbol{A}_{\overline{\boldsymbol{q}},{\mathit{\mu }_{\textit{t}}}}$ . This treatment follows recent developments in eddy-viscosity-enhanced resolvent analysis, which have shown clear benefits for turbulent boundary layers, channel flows and jets (Fan et al. Reference Fan, Kozul, Li and Sandberg2024; Von Saldern et al. Reference Testud, Auregan, Moussou and Hirschberg2024). In addition, recent work on forcing statistics has shown that the structure of the intrinsic forcing can play a non-negligible role in resolvent predictions, especially for broadband turbulent flows (Nogueira et al. Reference Nogueira, Morra, Martini, Cavalieri and Henningson2020; Fan et al. Reference Fan, Kozul, Sandberg and Li2025). In this study, at each frequency, the singular value decomposition searches over all unit-norm momentum forcings admitted by B and identifies the forcing that produces the maximum response norm. The resulting singular values therefore represent optimal linear amplification gains rather than direct predictions of the fluctuation amplitude generated by the actual turbulent forcing. The nonlinear terms omitted from the linear operator are formally retained in the equivalent forcing f , which is treated here as an independent unit-norm input. The formulation therefore does not predict nonlinear saturation or the finite amplitude of a self-sustained oscillation. In the present confined orifice flow, retaining the eddy viscosity in the linear operator is particularly important because the dominant amplification occurs in a separated turbulent region where molecular viscosity alone would not provide an adequate linear representation of the base-flow-supported dynamics. At each frequency, the forcing-to-output resolvent operator is analysed through a singular value decomposition,

(3.18) \begin{equation} \mathrm{\mathcal{H}}\left(\omega ;\boldsymbol{\xi }\right)=\sum_{j=1}^{N_{m}}\sigma _{\left(j\right)}\left(\omega ;\boldsymbol{\xi }\right)\psi _{\left(j\right)}\left(\omega ;\boldsymbol{\xi }\right){\phi }_{\left(j\right)}^{*}\left(\omega ;\boldsymbol{\xi }\right)\!, \end{equation}

where σ _(j) is the jth singular value, ϕ _(j) is the corresponding forcing mode, ψ _(j) is the corresponding response mode, (^⋅)^∗ denotes the Hermitian transpose and N _m is the number of retained singular values. The leading gain σ ₍₁₎ gives the maximum amplification from intrinsic forcing to wall-pressure-related response at a given frequency. The associated forcing and response modes satisfy

(3.19) \begin{equation} \mathrm{\mathcal{H}}\phi _{\left(j\right)}=\sigma _{\left(j\right)}\psi _{\left(j\right)}, \mathrm{\mathcal{H}}^{*}\psi _{\left(j\right)}=\sigma _{\left(j\right)}\phi _{\left(j\right)}. \end{equation}

For later interpretation, the response mode is partitioned into its pressure and velocity components, ψ _(j) = [ψ _ρ,(j), ψ _u,(j), ψ _p,(j)] ^T , where ψ _u,(j) denotes the velocity part of the response mode and ψ _p,(j) denotes its pressure part. The use of forcing and response modes in this manner has become standard in modern resolvent analysis for coherent-structure interpretation, gain analysis and control-oriented modelling (McKeon & Sharma Reference Mckeon and Sharma2010; Liu et al. Reference Liu, Sun, Yeh, Ukeiley, Cattafesta and Taira2021).

Based on the leading response mode, a wall-pressure-related proxy objective is constructed for each geometry. Let S _ξ denote the selected wall surface in the directional sense used for the proxy objective. Geometrically, it comprises the inner-orifice wall, the orifice-downstream face, the chamfered surface when present and the downstream pipe wall within the prescribed axial range. Only those surface elements with a non-zero axial normal component contribute directly to the axial dipole proxy. Let C _p,s be the extraction operator that maps the full-state response to the wall-pressure response on S _ξ . The acoustic response of interest propagates downstream along the pipe axis. The relevant surface-dipole loading is therefore the component of the pressure force projected onto this direction. The leading wall-pressure response and its axially projected loading are written as

(3.20) \begin{equation} \hat{p}_{s,(1)}\left(\boldsymbol{x}_{\boldsymbol{s}},\omega ;\boldsymbol{\xi }\right)=n_{x}\sigma _{\left(1\right)}\left(\omega ;\boldsymbol{\xi }\right)C_{p,s}\psi _{\left(1\right)}\left(\boldsymbol{x}_{\boldsymbol{s}},\omega ;\boldsymbol{\xi }\right)\!, \end{equation}

where x _s denotes the wall position. Since the dipole-related source term of interest is associated with the wall-pressure time derivative, the harmonic representation is given by

(3.21) \begin{equation} n_{x}{\frac{\partial p}{\partial t}}_{s,(1)}=i\omega \hat{p}_{s,(1)}. \end{equation}

Accordingly, the frequency-resolved proxy objective is defined as

(3.22) \begin{equation} J_{\boldsymbol{\xi }}(\omega)=\frac{1}{\big| S_{\boldsymbol{\xi }}\big| }\int _{{S_{\boldsymbol{\xi }}}}\left| n_{x}{\frac{\partial p}{\partial t}}_{s,(1)}(\boldsymbol{x}_{\boldsymbol{s}},\omega ;\boldsymbol{\xi })\right| ^{2}\mathrm{d}S=\omega ^{2}{\sigma }_{(1)}^{2}(\omega ;\boldsymbol{\xi })\frac{1}{\big| S_{\boldsymbol{\xi }}\big| }\int _{{S_{\boldsymbol{\xi }}}}\big| n_{x}C_{p,(1)}\psi _{(1)}(\boldsymbol{x}_{\boldsymbol{s}},\omega ;\boldsymbol{\xi })\big| ^{2}\mathrm{d}S. \end{equation}

For interpretation, (3.22) can be expressed as

(3.23) \begin{equation} J_{\xi }\left(\omega \right)=\omega ^{2}{\sigma }_{\left(1\right)}^{2}{\Gamma} _{p,x}\left(\omega \right)\!, \end{equation}

where

(3.24) \begin{equation} {\Gamma} _{p,x}\left(\omega \right)=\frac{1}{\big| S_{\boldsymbol{\xi }}\big| }\int _{{S_{\boldsymbol{\xi }}}}\left| n_{x}C_{p,\left(1\right)}\psi _{\left(1\right)}\left(\boldsymbol{x}_{\boldsymbol{s}},\omega ;\boldsymbol{\xi }\right)\right| ^{2}\mathrm{d}S \end{equation}

measures the area-averaged squared wall-pressure response projected onto the pipe-axis direction. The three factors in (3.23) have distinct physical roles. The squared singular value σ ₍₁₎ ² measures the amplification of the leading response under unit-norm momentum forcing. The factor ω ² weights the time variation of the wall-pressure loading. The quantity Γ _p,x measures how strongly the amplified response appears as an axially directed pressure loading over the selected wall surface. The projection factor is essential for the present confined geometry. Surfaces whose normals are perpendicular to the pipe axis satisfy n _x = 0 and do not contribute to the axial dipole-source proxy. The downstream surface has |n _x | = 1, while the chamfered surface contributes according to the axial component of its surface normal. The quantity J _ξ (ω) is therefore interpreted as a dipole-source proxy based on the axially projected unsteady wall loading. It is used to rank candidate geometries according to the magnitude and spatial extent of the source-region response. The full-band objective used in the design maps is written as

(3.25) \begin{equation} J_{\boldsymbol{\xi }}= \sum_{m=1}^{N_{\omega }}J_{\boldsymbol{\xi }}\left(\omega _{m}\right){\unicode[Arial]{x0394}} \omega _{m}, \end{equation}

where ${\{\mathit{\omega }\textit{m}\hspace{0pt}\}}_{\textit{m}=1}^{\textit{N}\mathit{\omega }}$ denotes the sampled frequency set within the full band defined in § 2. For a prescribed Strouhal-number band $\Omega$ _b , the corresponding band-limited objective is

(3.26) \begin{equation} {J}_{\boldsymbol{\xi }}^{\left({\Omega} _{b}\right)}=\sum _{\omega _{m}\in {\Omega} _{b}}J_{\boldsymbol{\xi }}\left(\omega _{m}\right){\unicode[Arial]{x0394}} \omega _{m}. \end{equation}

The design-space maps reported later are expressed in terms of the normalised quantities log₁₀(J _ξ /J _ξ0). The objective J _ξ is therefore a source-side acoustic indicator rather than the propagated acoustic pressure level itself. It measures how strongly the leading amplified response is expressed as ∂p/∂t over the wall surface most relevant to the dipole-related loading. The proxy guides the geometric modification of the source region, and its downstream acoustic consequence is assessed in § 4.5 through the plane-wave pressure component extracted from the LES. In this way, the optimisation is guided by a dipole-source proxy constructed from the wall-pressure response, while the duct propagation environment is introduced separately in the postoptimal acoustic assessment.

For each admissible geometry, the resolvent operator is evaluated over the prescribed full-band Strouhal-number range, and the resulting proxy objective is used to assess the design space and identify the most favourable downstream chamfer. The corresponding forcing and response modes are then examined to interpret the physical mechanisms underlying the amplification characteristics and the subsequent wall-pressure reduction.

4.1. Baseline flow and wall pressure characteristics

To identify the baseline flow structures and source-supporting regions that are most relevant to the subsequent resolvent-guided optimisation, we first examine the LES statistics of the circular orifice. The mean fields in figure 6 exhibit the canonical topology of a finite-thickness orifice flow. As the incoming pipe flow passes through the sharp-edged contraction, it accelerates into a confined high-speed core inside the orifice, accompanied by a pronounced pressure drop across the plate. After discharge into the downstream pipe, the jet undergoes sudden expansion and the separated shear layers shed from the trailing edge enclose a recirculating near wake. The turbulent kinetic energy (TKE) is concentrated not only along the downstream shear layers, but also close to the inner wall near the orifice exit (figure 6 c), indicating that the confined jet remains dynamically active within the finite plate thickness. In a finite-thickness aperture, such a distribution is consistent with a small in-hole separation–reattachment process superposed on the larger downstream recirculating wake (Shan et al. Reference Selvam, Peixinho and Willis2021). Previous studies on thick-plate apertures and finite-length holes have likewise shown that the instability-bearing region can reside within the thickness of the plate, and that the reattachment state inside the hole is an important determinant of the unsteady and acoustic response (Sierra-Ausin et al. Reference Shan, Fujishiro, Tsuneyoshi and Tsuji2022; Brokof et al. Reference Brokof, Guzman-Inigo, Yang and Morgans2023).

Figure 6.

Baseline mean-flow topology and separation features obtained from LES within the orifice and near the orifice-downstream surface, shown by the distributions of (a) static pressure p, (b) mean velocity U and (c) TKE.

To characterise the source-related loading associated with these baseline structures, we introduce dipole- and quadrupole-source measures following the classical acoustic analogy framework (Howe Reference Howe1975; Wang et al. Reference Von Saldern, Schmidt, Jordan and Oberleithner1996). The integrated dipole and quadrupole source moments are written as

(4.1) \begin{equation} D_{i}(t)=\int _{S}n_{i}p(\boldsymbol{x},t)\mathrm{d}S, \end{equation}

(4.2) \begin{equation} Q_{ij}(t)=\int _{V}T_{ij}(\boldsymbol{x},t)\mathrm{d}V, \end{equation}

where n _i is the outward unit normal of the rigid wall, and T _ij is the Lighthill stress tensor. Under the present low Mach-number conditions, T _ij is dominated by the inertial contribution and may therefore be approximated as

(4.3) \begin{equation} T_{ij}=\rho u_{i}u_{\!j}. \end{equation}

For the local wall-based source map, we retain D _s,rms = (∂p/∂t) _rms Δt/0.5ρU ² _in as a local wall-loading proxy based on the time variation of wall pressure. Here, D _s,rms represents the local magnitude of the surface dipole-source activity without resolving its orientation relative to the pipe axis. It is used to identify the wall regions supporting strong unsteady pressure loading. The contribution of this local source activity to the downstream low-frequency radiation is determined separately through the axial projection factor n _x . The distribution of D _s,rms in figure 7 shows that the strongest wall-based source activity is concentrated on the inner orifice wall, with a clear maximum near the trailing edge where the confined jet, the separated shear layer and the local reattachment process interact most intensely. Elevated levels persist over the downstream wall region behind the orifice, although their magnitude decreases progressively with streamwise distance. The downstream-face map in figure 7(b) further shows that the orifice-downstream face itself carries a pronounced annular footprint of dipole activity. The dominant wall-pressure-related source is therefore not confined to a single compact point, but is distributed over the inner wall, the trailing-edge junction and the downstream wall region immediately behind the orifice. This spatial pattern identifies the wall regions where the confined jet and the separated shear layer generate the strongest unsteady pressure loading. Their direct contribution to the downstream axial field, however, is determined by the projection factor n _x .

Figure 7.

(a) Side distribution of the surface dipole-source intensity (Ds _rms ) for the baseline orifice. (b) Distribution of Ds _rms on the downstream orifice surface, together with a schematic showing the side and downstream surfaces.

Figure 8.

The PSD of dp/dt at different wall locations for the baseline circular orifice. The black curve denotes the wall measurement point at the streamwise centre inside the orifice, the blue curve denotes the point on the downstream surface located 0.3D from the pipe centreline, and the red curve denotes the point at the junction between the orifice and the downstream surface. The schematic indicates the three measurement locations. Their coordinates are (x/D, y/D, z/D) = (0.125, 0.25, 0), (0.25, −0.3, 0) and (0.25, 0, 0.25), respectively.

The local spectra of dp/dt in figure 8 further clarify the spatial hierarchy of the wall response. The three wall locations share similar dominant narrow-band components, showing that the pressure fluctuations on the inner wall and on the downstream face are organised by the same global unsteady process. Their amplitudes, however, differ markedly with position. The signal at the streamwise centre inside the orifice is the strongest over most of the resolved range, the junction signal remains slightly weaker, and the point on the downstream surface exhibits the same dominant peaks but at clearly lower amplitude. Combined with the D _s,rms distribution in figure 7, this indicates that the most intense wall-pressure loading is anchored primarily to the orifice interior rather than to the downstream surface alone. An interpretation, also consistent with the mean-flow and TKE distributions in figure 6, is that the finite plate thickness allows a small separation–reattachment process to develop inside the orifice, so that strong unsteady wall loading is produced near the inner-wall reattachment zone. This interpretation is supported by previous studies of thick apertures and finite-length holes, which showed that the recirculation region within the plate thickness and the reattachment state inside the hole play a central role in the instability and acoustic response (Fabre et al. Reference Fabre, Longobardi, Citro and Luchini2019; Sierra-Ausin et al. Reference Shan, Fujishiro, Tsuneyoshi and Tsuji2022). It is also consistent with pressure-decomposition study of confined orifice flow, which showed that the source-relevant wall-pressure content is concentrated near the orifice region rather than being distributed uniformly over the surrounding wall (Zhang et al. Reference Yuan, Kou, Zheng and Zhang2025a ). The baseline source field is therefore spatially localised and spectrally organised, with the dominant narrow-band wall-pressure response anchored inside the orifice and along the downstream part of the inner wall, rather than on the downstream surface alone. This provides the physical basis for selecting the inner-orifice wall together with the downstream wall region behind the orifice as the target surface in § 3.2.

Figure 9.

(a) Ratio of the gains of the first and second resolvent modes σ ₍₁₎/σ ₍₂₎, as a function of St. (b) Relative contribution of the nth modal squared gain and cumulative contribution of the first n modal squared gains, obtained by summing σ _(n) ² over the analysis band. The first 20 resolvent modes are retained in the normalisation.

4.2. Baseline resolvent characteristics and amplification mechanism

The rank characteristics of the baseline resolvent operator are assessed using both the frequency-dependent separation of the leading singular values and the modal contribution integrated over the analysis band. As shown in figure 9(a), the ratio σ ₍₁₎/σ ₍₂₎ varies with St, reflecting a frequency-dependent degree of modal separation. Figure 9(b) evaluates the relative contribution of the first 20 modal gains after summing their squared singular values over the analysis band. The squared singular values are used rather than the singular values because the squared response norm under unit forcing, as well as the modal weighting in the proxy objective, depends on σ _(n) ². A similar squared-singular-value measure has been used to assess the modal contribution and low-rank behaviour of resolvent systems by Xuan & Shen (Reference Xuan and Shen2025). Figure 9(b) shows that the leading mode accounts for the dominant fraction of the band-integrated squared gain, while the cumulative contribution approaches unity rapidly after the first few modes are retained. The system is therefore sufficiently low rank in the frequency-integrated sense relevant to the present optimisation, although it is not uniformly rank-one at every individual frequency. This provides the basis for constructing the proxy objective from the leading response mode.

Figure 10.

(a) Gain of the leading resolvent mode σ _0,0(ω) for the baseline circular orifice as a function of St, and (b) the corresponding proxy-objective function log₁₀(J _0,0(ω)). (c) Cumulative proxy contribution P _r of the first r resolvent modes to the wall-pressure-related proxy objective at St = 1.52, 4.73 and 7.73. The first 20 modes are retained in the normalisation. (d) Non-dimensional temporal growth rates λ _r d/U _in of the autonomous global modes for the baseline orifice.

The frequency dependence of the leading gain and the corresponding proxy objective is shown in figure 10. The gain σ _0,0(ω) exhibits a broadband envelope with several local maxima, and three representative peaks are selected at St = 1.52, 4.73 and 7.73 to characterise the low-, intermediate- and high-frequency amplification behaviours of the baseline orifice. The corresponding gain ratios σ ₍₁₎/σ ₍₂₎ are 5.0, 4.7 and 5.9. These values confirm that the leading forcing–response pair is clearly separated from the second mode at the three principal amplification peaks. Figure 10(b) shows that the proxy objective log₁₀(J _0,0(ω)) does not follow the gain curve in a one-to-one manner. This difference follows directly from (3.22), because the proxy depends not only on the leading gain, but also on the axially projected wall-pressure response Γ _p,x . Strong amplification alone is therefore insufficient to produce a large directional source proxy. The amplified response must also generate pressure loading on a surface with a non-zero axial normal component. To examine whether this source-relevant contribution is sensitive to the higher-order resolvent modes, the cumulative modal contribution is defined at each frequency in terms of the frequency-resolved proxy objective as

(4.4) \begin{equation} P_{r}\left(\omega \right)= \sum_{j=1}^{r}J^{(j)}(\omega)/ \sum_{j=1}^{20}J^{(j)} (\omega). \end{equation}

Here, J ^(j)(ω) denotes the contribution of the jth resolvent mode to the frequency-resolved proxy objective. The quantity P _r therefore represents the fraction of the wall-pressure-related proxy contribution recovered by the first r modes, relative to that recovered by the first 20 modes. Figure 10(c) shows that the leading mode alone accounts for approximately 90 % of the proxy contribution at St = 1.52, 4.73 and 7.73. The contribution approaches unity rapidly after the first few modes are included, although the precise convergence rate differs slightly between the three frequencies. Higher-order modes therefore introduce only a limited correction to the source-relevant wall-pressure contribution at the principal amplification peaks. Combined with the band-integrated squared-gain contribution in figure 9(b), this result supports the use of the leading-mode proxy in the subsequent design-space assessment. The autonomous global-mode growth rates in figure 10(d) provide a separate assessment of whether the narrow-band amplification peaks are associated with self-sustained linear instability. All computed eigenvalues satisfy λ _r d/U _in < 0. The baseline mean-flow operator therefore does not predict an autonomously growing hydrodynamic global mode within the examined range. The peaks in σ and J are consequently interpreted as selective forced amplification of temporally stable dynamics, rather than as linear predictions of a self-sustained limit cycle. Their correspondence with the narrow-band peaks obtained from the nonlinear LES indicates that the intrinsic forcing preferentially excites these amplification branches. It does not imply that the linear resolvent model predicts the finite amplitude of the resulting fluctuations. Figure 10 therefore defines the representative frequencies used below for modal interpretation, consistent with the standard resolvent practice of identifying dominant frequencies from amplification and then interpreting their modal structures (Yeh & Taira Reference Yang and Morgans2019).

The leading forcing modes in figure 11 indicate where the baseline flow is most receptive to harmonic forcing. At St = 1.52, the forcing structure is relatively broad and occupies much of the orifice interior, indicating that the low-frequency amplification process is supported by a spatially extended receptivity distribution. As the frequency increases to St = 4.73 and 7.73, the forcing field becomes progressively more concentrated near the downstream part of the orifice and along the inner shear layers issuing from the downstream edge. This shift indicates that the higher-frequency amplification is tied more closely to the separated flow near the downstream lip and the in-hole recirculation region. The forcing support is consistent with the LES source maps, which locate the strongest loading inside the orifice and near the downstream part of the inner wall.

Figure 11.

Contours of the amplitude of the leading resolvent forcing mode ϕ/||ϕ|| for the baseline circular orifice at St = (a) 1.52, (b) 4.73 and (c) 7.73. The forcing structures are concentrated within the orifice and in the immediate downstream region, indicating that these locations are the most receptive to harmonic forcing and therefore most effective in generating amplified responses.

The corresponding leading response modes in figure 12 show the conversion of this receptive region into source-relevant wall-pressure amplification. At St = 1.52, the pressure response displays a comparatively large-scale structure extending over the downstream region, while the velocity response occupies a broad portion of the postorifice jet and near wake. This mode therefore represents a relatively global low-frequency amplification process, in which the confined jet and the downstream recirculating region participate together. By contrast, the responses at St = 4.73 and 7.73 become much more compact in pressure and more organised in velocity. The pressure field is then strongly concentrated near the downstream lip and the neighbouring target wall region, whereas the velocity response aligns with the separated shear layers and the downstream jet core. The baseline amplification mechanism may therefore be summarised as follows. Intrinsic forcing acting primarily within the orifice and near the downstream edge is selectively amplified by the mean flow, and the resulting wall-pressure response is supported most strongly over the inner-orifice wall and the downstream wall region behind the orifice at representative frequencies. From this perspective, downstream chamfering is not merely a geometric smoothing of the edge. It directly modifies the most amplification-sensitive region of the baseline flow and the wall-pressure response supported by that region, which explains why a resolvent-guided redesign focused on the downstream edge is physically well founded.

Figure 12.

Contours of the leading resolvent response modes of (a) pressure ψ _p and (b) velocity ψ _u for the baseline circular orifice at St = 1.52, 4.73 and 7.73.

Figure 13.

(a) Relative change in pressure drop for the chamfered orifice (Δp _c,α – Δp _0,0)/Δp _0,0 in the design space of chamfer length c and chamfer angle α. The chamfer length varies from 0 to 0.9θ and the chamfer angle varies from 0° to 60°. Under the allowable limit of a 1.5 % increase in relative pressure drop, only the cases with c = 0.9θ fall outside the acceptable range, as marked by crosses. (b) Feasible regions obtained with four pressure-drop tolerances, η = 1.0 %, 1.25 %, 1.5 % and 3.0 %, used to assess the sensitivity of the admissible design space to the hydraulic constraint.

4.3. Design-space assessment and optimal resolvent characteristics

The design assessment begins with the hydraulic admissibility of the downstream chamfer. As shown in figure 13(a), the relative pressure-drop variation increases monotonically with both chamfer length c and chamfer angle α. This trend indicates that a larger downstream chamfer does not simply smooth the local expansion, but also enlarges the geometrically modified wall region and displaces the pressure-recovery process farther downstream, so that the end-to-end loss across the orifice section increases gradually. Figure 13(b) further shows the feasible regions obtained with several values of the pressure-drop tolerance, η = 1.0 %, 1.25 %, 1.5 % and 3.0 %. As expected, the admissible design space expands progressively as η is relaxed. The adopted value η = 1.5 % provides a restrictive but not overly narrow feasible region for the subsequent proxy-objective assessment, while the neighbouring thresholds shown in figure 13(b) provide a sensitivity check on the hydraulic constraint. The blank cells at α = 60° and c ≥ 0.6θ correspond to geometrically inadmissible configurations, because an excessively long chamfer at such a large angle would exceed the available plate thickness and can no longer be defined consistently.

Figure 14.

Full-band proxy-objective function log₁₀(J _c,α /J _0,0) in the design space of chamfer length c and chamfer angle α. The crossed cells denote the configurations excluded by the pressure-drop constraint.

Figure 15.

Proxy-objective function log₁₀(J _c,α /J _0,0) in the design space of chamfer length c and chamfer angle α for two Strouhal-number ranges: (a) St = 1 ∼ 8, and (b) St = 8 ∼ 33. The crossed cells denote the configurations excluded by the pressure-drop constraint.

Figure 14 presents the full-band design map of proxy-objective function log₁₀(J _c,α /J _0,0). Negative values indicate that the chamfered orifice yields a lower proxy objective than the baseline circular orifice, whereas positive values indicate deterioration. Most admissible chamfers lie in the reduction regime, but the magnitude of the reduction is clearly non-uniform across the (c, α) plane. The strongest overall benefit appears in the region c = 0.5θ ∼ 0.7θ and α = 40° ∼ 60°, whereas shorter and weaker modifications, such as c ≤ 0.3θ together with α ≤ 20°, provide only limited improvement. For clarity, figure 15 partitions the proxy objective into the two Strouhal ranges St = 1 ∼ 8 and St = 8 ∼ 33, enabling the dominant low- and intermediate-frequency amplification to be examined separately from the higher-Strouhal response. The band-limited maps reveal that the best full-band designs arise from a selective frequency redistribution rather than from a uniform reduction at all frequencies. In the lower band, the reduction is comparatively broad and coherent, indicating that the low- and intermediate-frequency wall-pressure amplification is particularly sensitive to downstream-edge shaping. In the higher band, by contrast, the response becomes much more heterogeneous. A representative example is the configuration c = 0.6θ, α = 40°, which yields a stronger reduction than c = 0.5θ, α = 60° in the range St = 1 ∼ 8, but shows a local penalty in the range St = 8 ∼ 33. This contrast indicates that the higher-Strouhal content is more sensitive to the detailed matching between the chamfer geometry and the characteristic scales of the separated flow, so that a geometry which is highly effective in the lower band does not necessarily remain optimal at higher Strouhal numbers. By contrast, a milder geometry such as c = 0.3θ, α = 20° only weakly perturbs the baseline amplification mechanism and therefore yields modest benefit in both bands. For design purposes, the more important observation is that the neighbourhood of c = 0.5θ, α = 60° combines strong low- and intermediate-frequency reduction with a more balanced higher-frequency response, which is why it is taken as the centre of the subsequent refinement. This band dependence is consistent with previous observations that geometric changes can modify both the level and the spectral content of ducted acoustic responses (Laffay et al. Reference Laffay, Moreau, Jacob and Regnard2020; Quaroni et al. Reference Price and Smith2022).

Figure 16 shows the refined search around this favourable region. Once the coarse map has identified the most promising zone, the parameter range is narrowed to 0.45θ ≤ c ≤ 0.6θ and 50° ≤ α ≤ 60°. The refined map confirms that the lowest values of the full-band objective are clustered within a small pocket of this region, and the optimisation history identifies the final optimum at c = 0.55θ and α = 50°. This refinement step is essential because the coarse scan is sufficient to reveal the favourable part of the design space, but not to determine the best geometry with adequate resolution. The final design is therefore selected through a systematic local refinement inside the hydraulically feasible region, rather than by choosing an isolated low-valued cell from the coarse map.

Figure 16.

(a) Full-band proxy-objective function log₁₀(J _c,α /J _0,0) in the refined design space, where the chamfer length varies from 0.45θ to 0.6θ and the chamfer angle varies from 50° to 60°. (b) Optimisation history from the baseline circular orifice to the chamfered orifice in the refined design space, identifying the optimal configuration.

The optimal resolvent characteristics are presented in figure 17, together with a direct comparison of the baseline and chamfered velocity response modes. The leading-gain spectrum of the selected chamfered orifice still exhibits a pronounced low-frequency peak at St = 1.52, while the intermediate peak is shifted from St = 4.73 in the baseline case to St = 5.65, and the higher-frequency branch remains weaker and decays more rapidly. The response modes reveal a clear spatial reorganisation. At St = 1.52, the pressure response extends less strongly into the orifice-exit region, and the velocity response forms a narrower downstream wave packet. At the intermediate and higher representative frequencies, the response support moves away from the downstream lip and develops farther downstream. This change is shown directly in figure 17(c), where the upper and lower halves compare the baseline and chamfered velocity response modes for the corresponding amplification branches. The red contours of ψ _u /||ψ _u || = 0.5 identify the strong velocity-response region and show that its upstream boundary is displaced downstream after chamfering. Using the first streamwise location where ψ _u /||ψ _u || > 0.5, the onset locations for the baseline orifice are x/D = 0.6, 0.3 and −0.3 at St = 1.52, 4.73 and 7.73, respectively. The corresponding locations for the chamfered orifice are x/D = 1.5, 1.1 and 0.4 at St = 1.52, 5.65 and 7.72. The strong response support is therefore shifted downstream by approximately 0.7D to 0.9D over the representative branches. The selected chamfer therefore changes both the spectral amplification and the location of the dominant response, reducing its overlap with the wall region associated with the strongest baseline dipole loading.

Figure 17.

(a) Gain of the leading resolvent mode σ _opt (ω) for the resolvent-predicted optimal chamfered orifice as a function of St. (b) Contours of the corresponding leading resolvent response modes of pressure ψ _p at St = 1.52, 5.65 and 7.72. (c) Comparison of the velocity response modes ψ _u for the corresponding amplification branches. The upper-half of each panel shows the baseline circular orifice, while the lower-half shows the selected chamfered orifice. The red contours indicate ψ _u /||ψ _u || = 0.5.

4.4. Wall-pressure reduction and source redistribution

To assess whether the resolvent-predicted optimum is realised in the actual wall-pressure field, figure 18 compares the LES spectra of dp/dt for the optimised chamfered orifice. Figure 18(a) uses the temporal signals at three individual wall locations. For figure 18(b), the instantaneous wall signal is first averaged using the original surface-element areas over a region that begins at the orifice inlet, follows the inner-orifice wall and the chamfered surface, includes the annular downstream face and extends along the downstream pipe wall to x = θ + 2D. In the present low Mach-number rigid-walled duct, this quantity is taken as a monotonic source proxy for the dominant transmitted dipole-type acoustic output. At the lowest Strouhal numbers, the three spectra remain relatively close, indicating that the large-scale low-frequency motion still projects onto the inner wall, the junction and the downstream surface. As St increases, the spectrum at the orifice centre becomes the dominant one over most of the resolved range. At higher Strouhal numbers, however, the junction and downstream spectra collapse towards a comparable level, while both remain distinctly below the orifice-centre spectrum. This ordering differs from the baseline behaviour in figure 8 and indicates that the compact high-frequency source region is displaced by chamfering. The junction no longer acts as a distinct high-frequency loading site. Instead, the junction and downstream positions sample the same developing shear layer, while the residual jet–wall interaction inside the orifice keeps the centre spectrum above them. The spectrum of the surface-area-averaged signal in figure 18(b) supports this interpretation, since the chamfered configuration remains below the baseline across almost the entire full band below the higher-order duct-mode cut-on frequency and shows particularly strong attenuation around the dominant narrow-band peaks.

Figure 18.

(a) The PSD of dp/dt at different wall locations for the optimised chamfered orifice. The black curve denotes the wall measurement point at the streamwise centre inside the orifice, the blue curve denotes the point on the downstream surface located 0.45D from the pipe centreline and the red curve denotes the point at the junction between the orifice and the downstream surface. The schematic indicates the three measurement locations. Their coordinates are (x/D, y/D, z/D) = (0.125, 0.265, 0), (0.25, −0.45, 0) and (0.25, 0, 0.414), respectively. (b) Area-weighted PSD of dp/dt over the inner-orifice wall and the downstream wall region behind the orifice (from x = 0 to x = θ + 2D), comparing the baseline configuration and the resolvent-predicted optimal chamfered orifice.

Figure 19.

Mean-flow topology and separation features of the resolvent-predicted optimal chamfered orifice obtained from LES in the orifice and near-wake regions, shown by the distributions of (a) static pressure p, (b) mean velocity U and (c) TKE.

Figure 20.

Contours from LES of the (a) mean streamwise velocity with streamlines and (b) Reynolds shear stress for the resolvent-predicted optimal chamfered orifice and the baseline circular orifice.

Figures 19 and 20 relate this spectral attenuation to a reorganisation of the separated mean flow. The overall topology of a finite-thickness orifice flow is preserved, but the jet issues through the orifice in a smoother manner and remains compact over a longer downstream extent before substantial lateral spreading develops, as shown in figure 19(b). The highest TKE no longer lies immediately at the orifice exit. Instead, it is displaced downstream and away from the inner-orifice region, as shown in figure 19(c). This shift is important because, in separated and reattaching flows, strong wall-pressure fluctuations are controlled primarily by the development of vortical structures in the separated shear layer rather than by the geometric discontinuity alone. Ji & Wang (Reference Ji and Wang2012) showed that, once the step-induced separation is sufficiently strong, the separated shear layer becomes the dominant pressure source and the maximum surface-pressure level scales with the local Reynolds-stress peak. Related smooth-ramp studies by Simmons et al. (Reference Sierra-Ausin, Fabre, Citro and Giannetti2024) further showed that replacing a sharp step by a rounded geometry changes the downstream evolution from a geometry-fixed separation process to one more strongly governed by pressure-gradient development and subsequent shear-layer growth. The comparison in figure 20 makes the present mechanism more explicit. The mean streamlines show that the shear layer separated from the inlet edge of the baseline square-edged orifice reattaches to the inner wall before leaving the finite-thickness passage. In the selected chamfered configuration, the jet remains detached from the modified wall throughout the passage and enters the downstream pipe without an in-hole reattachment. The optimisation therefore crosses the transition between the reattaching and fully detached flow states, rather than perturbing only a single fixed flow topology. Relative to the baseline circular orifice, the near-wall recirculating motion immediately behind the plate is weakened in the near field and its core is displaced farther downstream. The Reynolds shear stress shows the same redistribution. Its largest magnitude is no longer concentrated directly behind the orifice near the wall, but is shifted downstream and spread over a broader part of the developing shear layer. The effect of the chamfer is therefore not to suppress separation altogether. It changes where the separated shear layer amplifies, where the primary recirculation region interacts with the wall, and where the strongest wall-pressure loading is produced. This downstream transfer explains the spectral change in figure 18(a). Once the junction is released from the sharp-corner-fixed separation pattern, it no longer behaves as a distinct high-frequency loading site, and its spectrum approaches that of the downstream wall, which is governed by the same developing shear layer and downstream recovery process. The strong link between reattachment-zone motion and the unsteady behaviour of the primary recirculation region has likewise been documented in backward-facing-step flow by Schäfer, Breuer & Durst (Reference Roumen2009).

Figure 21.

(a) Side distribution of the surface dipole-source intensity Ds _rms for the resolvent-predicted optimal chamfered orifice. (b) Distribution of Ds _rms on the downstream orifice surface.

This mean-flow reorganisation is reflected directly in the surface dipole-source distribution shown in figure 21. In the optimised chamfered orifice, the inner-orifice wall no longer carries the strong loading observed in the baseline case. Instead, the residual maximum of Ds _rms is located on the downstream wall behind the orifice, while the orifice-downstream face exhibits only a weak annular footprint. The dipole-source field is therefore not only reduced in magnitude but also redistributed in space. This redistribution is physically consistent with the LES results in figures 19 and 20. The chamfer weakens the near-wall recirculating motion immediately behind the plate and shifts the region of strongest Reynolds shear stress farther downstream, so the wall no longer experiences the same compact loading just outside the orifice exit. In separated and reattaching flows, wall-pressure fluctuations are known to be weaker inside the separation bubble and stronger downstream of reattachment, where the Reynolds stresses reach their maximum, as shown by Na & Moin (Reference Na and Moin1998). The present dipole-source map exhibits a similar pattern. As the most energetic shear-layer development is displaced away from the immediate postorifice wall, the strong, localised surface dipole-source activity along the downstream portion of the inner wall is alleviated. The remaining wall-based source contribution is shifted farther downstream and becomes more broadly distributed and less concentrated. A related shift from geometry-fixed separation to downstream shear-layer development has also been documented in smooth-ramp separated flows by Simmons et al. (Reference Sierra-Ausin, Fabre, Citro and Giannetti2024).

Figure 22.

Frequency dependence of the (a) surface-pressure contribution Π _D and (b) volume Reynolds-stress contribution Π _Q to the downstream-propagating plane-wave pressure for the baseline and chamfered orifices. The source region extends over 0 < x < 3D. The vertical markers indicate the representative amplification frequencies of the baseline orifice, St = 1.52, 4.73 and 7.73, selected from figure 10.

To compare the surface and volume source pathways under the present duct confinement, the postoptimal acoustic assessment is based on the one-dimensional Green function for plane-wave propagation in an infinite rigid tube (Hirschberg Reference Hirschberg2009). For frequencies below the first higher-order duct-mode cut-on, the downstream-propagating pressure at the axial locations x _m is decomposed as

(4.5) \begin{equation} \hat{p}^{+} (x_{m},\omega )={\hat{p}}_{D}^{+} (x_{m},\omega )+{\hat{p}}_{Q}^{+} (x_{m},\omega), \end{equation}

(4.6) \begin{equation} {\hat{p}}_{D}^{+} (x_{m},\omega )=-\frac{1}{2A_{p}}\int _{S}n_{x}(\boldsymbol{y})\hat{p} (\boldsymbol{y},\omega)e^{[-i{k^{+}}({x_{m}}-{y_{x}})]}\,\text{d}S, \end{equation}

(4.7) \begin{equation} {\hat{p}}_{Q}^{+}(x_{m},\omega)=\frac{i\omega }{2A_{p}c_{0}}\int _{V}\hat{T}_{xx} (\boldsymbol{y},\omega)e^{[-i{k^{+}}({x_{m}}-{y_{x}})]}\,\text{d}V, \end{equation}

where A _p = πD ²/4 is the pipe cross-sectional area, y denotes the source position and y _x is its axial coordinate. The downstream-propagating wavenumber is k ⁺ = ω/(c ₀ + U _b ), consistent with the plane-wave decomposition introduced in § 4.5. The surface S and volume V are the source domains introduced in (4.1) and (4.2), respectively. For the calculations in figures 22 and 23, both domains are restricted to 0 < x< 3D. Under the present low Mach-number conditions, the axial component of the Lighthill stress tensor is approximated as T _xx = ρu _x u _x . The corresponding dimensionless surface and volume contributions are defined as

(4.8) \begin{equation} \varPi _{D}=\frac{| {\hat{p}}_{D}^{+}(x_{m},\omega )| ^{2}}{(0.5\rho {U}_{\textit{in}}^{2})^{2}}, \varPi _{Q}=\frac{| {\hat{p}}_{Q}^{+}(x_{m},\omega )| ^{2}}{(0.5\rho {U}_{\textit{in}}^{2})^{2}}. \end{equation}

Figure 23.

Comparison of the source contributions for the resolvent-predicted optimal chamfered orifice and the baseline circular orifice. (a) Full-band surface-pressure contribution Π _D and volume Reynolds-stress contribution Π _Q to the downstream plane-wave pressure. (b) Cumulative streamwise source-support measures for the surface and volume terms.

These definitions place the surface-pressure and volume Reynolds-stress contributions on the same downstream plane-wave pressure scale. The one-dimensional duct formulation contains no spherical-spreading factor, and the propagation direction is fixed by the pipe axis. Consequently, only the axial component of the surface pressure force contributes to ${\hat{\textit{p}}}_{\textit{D}}^{+}$ . Surfaces parallel to the pipe axis satisfy n _x = 0 and make no direct contribution, whereas the orifice end faces and the chamfered surface contribute according to their axial normal components. In the present lossless one-dimensional formulation, changing the downstream location x _m changes only the common propagation phase. The magnitudes of Π _D and Π _Q are therefore independent of the downstream microphone position selected. The duct-based contributions in figure 22 show that the selected chamfer affects the two source pathways differently. The surface-pressure contribution Π _D remains below the baseline over nearly the entire analysed frequency range, and the narrow-band peaks associated with the baseline orifice are strongly attenuated. The volume Reynolds-stress contribution Π _Q is also reduced over most of the frequency range, although the difference between the two configurations is smaller at the lowest Strouhal numbers. A residual low-frequency volume contribution is therefore retained, while the intermediate- and high-frequency peaks are substantially weakened. Since Π _D and Π _Q are evaluated using the same duct Green function and the same downstream propagation direction, their magnitudes provide a direct comparison of the two contributions to the downstream plane-wave pressure. The selected chamfer acts most strongly on the surface-pressure pathway targeted by the resolvent proxy, while the distributed volume contribution is more readily sustained by the downstream shear-layer motion. The marked frequencies in figure 22 connect the duct-based source contributions with the representative resolvent branches identified in figure 10. At the same frequencies, figure 12 shows that the baseline pressure and velocity responses are supported by the downstream lip, the downstream part of the inner-orifice wall and the neighbouring shear layer. These locations coincide with the wall region where figures 7 and 8 identify the strongest dipole loading and the dominant narrow-band wall-pressure response. After chamfering, the response support is displaced downstream, as quantified above from the ψ _u /||ψ _u || > 0.5 onset locations, and figure 18(b) shows that the area-weighted wall-pressure spectrum loses the sharp peaks associated with the baseline response. The source indicators in figure 22 follow the same trend. At St = 1.52, the surface-pressure contribution is already reduced by the chamfer, whereas the two volume contributions remain of comparable order. At St = 4.73, both pathways are attenuated, with a more pronounced reduction in the surface-pressure contribution. At St = 7.73, the strong baseline peaks in both Π _D and Π _Q are largely removed. After chamfering, the strongest velocity-response support moves away from the downstream lip and develops farther downstream. The resulting wall-pressure loading is less efficiently projected onto the axial surface-pressure contribution, while the remaining volume activity is associated with the displaced shear layer. The duct-based results therefore support the source-reduction mechanism inferred from the resolvent modes and the LES wall-pressure spectra.

Figure 23 provides a full-band comparison of the two duct-based source pathways. The full-band quantities shown in figure 23(a) are obtained by integrating Π _D and Π _Q over 1 < St < St _c . To examine their spatial support in figure 23(b), the surface and volume source domains are partitioned into streamwise slabs, and the non-negative magnitudes of the corresponding local source terms are accumulated from the orifice to the specified streamwise position. These cumulative curves are used to identify where each source pathway is supported. The surface-pressure contribution decreases from 3.22 for the baseline orifice to 3.92 × 10^–4 for the selected chamfer, corresponding to a reduction by approximately four orders of magnitude. The volume Reynolds-stress contribution decreases from 6.98 × 10^–3 to 7.67 × 10^–5, corresponding to a reduction of two orders of magnitude. The surface-pressure contribution therefore remains larger than the volume contribution for both configurations, although the difference between the two pathways becomes smaller after chamfering. The ratio Π _D /Π _Q decreases from approximately 461 in the baseline case to 5.1 after chamfering, showing that the chamfer suppresses the surface-pressure pathway much more strongly than the volume pathway, although the surface-pressure contribution remains dominant in absolute magnitude. The cumulative curves in figure 23(b) reveal different spatial responses of the two pathways. The surface-pressure contributions for both configurations reach their plateaus within the immediate orifice region. This behaviour follows from the axial projection in (4.6), because the direct surface contribution is confined to the orifice end faces and the chamfered surface for which n _x ≠ 0. The principal effect of the chamfer on this pathway is therefore a strong reduction in magnitude rather than a downstream displacement of the directly radiating surface contribution. The volume contribution exhibits a clearer downstream redistribution. The half-contribution location moves downstream, from x/D = 0.7 to a broader region centred near x/D = 1.8. The selected chamfer therefore suppresses both source pathways, while the downstream transfer of the residual source support occurs primarily through the volume Reynolds-stress contribution.

4.5. Reynolds number robustness and in-duct acoustic assessment

Figure 24 places the optimal chamfer in a Reynolds number context. The quantity log₁₀(J _0,0/J _5.5,5) remains positive at Re/Re ₀ = 0.6, 0.8, 1.0 and 1.2, reaches its largest value near the design condition Re/Re ₀ = 1, and becomes slightly negative at Re/Re ₀ = 1.4. In the current plot, the corresponding values are approximately 0.5, 5.7, 8.0, 2.7 and −1.7, respectively. These results indicate that the selected chamfer remains beneficial over a moderate Reynolds number neighbourhood around the design condition, but that the benefit is not symmetric with respect to Reynolds number variation. The largest reduction appears at Re/Re ₀ = 1, which is expected because the chamfer geometry is selected at this condition and is therefore best matched to the corresponding baseline amplification and wall-pressure source support. At Re/Re ₀ = 0.6, the wall-adjacent shear and vortical activity are weaker, but the baseline amplification and source region are also less developed. The same fixed chamfer therefore has less intense source activity to suppress, and the relative benefit is smaller than at the design condition. To examine the deterioration on the high-Reynolds-number side, figure 24(b) compares two near-wall base-flow indicators for the selected chamfer. The integration band follows the wall-adjacent region where the wall-pressure source is most sensitive to the separated shear layer. It has a thickness of 0.05D, starts from the orifice inlet at x = 0, covers the inner-orifice wall and the chamfered surface, extends downstream to x = θ + 2D, and includes an additional thin band on the downstream end face near the lip. The first indicator, I _s = ∫∫|∂U _x /∂y|²dA, measures the strength of the near-wall mean shear, while the second indicator, I _ω = ∫∫|U _y /∂x – U _x /∂y|² dA, measures the near-wall mean-vorticity activity. Both quantities increase monotonically with Re/Re ₀, showing that the selected chamfer is exposed to progressively stronger wall-adjacent shear and vortical motion as the Reynolds number is raised. This trend helps explain why the proxy reduction decreases rapidly for Re/Re ₀ > 1. The chamfer selected at the design condition weakens the source proxy by moving the strongest amplification-supporting structures and wall-pressure loading away from the inner-orifice wall and the downstream lip. At higher Reynolds number, however, the strengthened shear layer increases the near-wall shear and vorticity levels in the same source-relevant region. The fixed geometry then becomes less well matched to the high-Reynolds-number shear-layer dynamics, and the wall-pressure manifestation of the amplified response is no longer suppressed as effectively. This interpretation is consistent with gradually expanding pipe flow, where increasing Reynolds number can lengthen the recirculation region, trigger symmetry breaking and induce shear-layer oscillations (Selvam et al. Reference Schafer, Breuer and Durst2015), and with separated turbulent flows in which the wall-pressure field is closely linked to Reynolds-stress levels and large-scale separated-flow structures (Abe Reference Abe2017). From this perspective, the present optimal chamfer should be regarded as Reynolds-robust over a moderate neighbourhood around the design condition, rather than as effective only at a single Reynolds number.

Figure 24.

Reynolds-number dependence of the selected chamfered orifice. (a) Proxy-objective ratio between the baseline circular orifice and the selected chamfered orifice, shown as log₁₀(J _0,0/J _5.5,5). (b) Near-wall mean-shear and vorticity indicators, I _s and I _ω , evaluated in a 0.05D-thick wall-adjacent band extending from the orifice inlet through the inner wall and chamfered surface to x = θ + 2D, with the thin end-face band near the downstream lip also included.

Figure 25.

In-duct transmitted acoustic response extracted from the LES. (a) Schematic of the measurement points in the downstream straight duct and the plane-wave decomposition into downstream-propagating and upstream-propagating components. (b) Spectrum of the downstream-propagating plane-wave pressure Lp+ for the baseline circular orifice and the selected chamfered orifice. (c) Frequency-resolved attenuation ΔL_{p +} of the downstream-propagating pressure after chamfering. (d) Ratio R_p = | $\hat{p}$ ^–|/| $\hat{p}^+$ | between the upstream- and downstream-propagating plane-wave amplitudes for the two configurations. The horizontal line R_p = 1 denotes equal amplitudes of the two components.

Figure 25 examines whether the source-side reduction indicated by the wall-pressure proxy is reflected in the downstream transmitted pressure extracted a posteriori from the LES. Since the analysed frequency range lies below the cut-on frequency of the first higher-order duct mode, the downstream acoustic field is assumed to be dominated by plane-wave propagation. This treatment follows the standard in-duct acoustic decomposition used in transfer function and multimicrophone methods (Allam & Åbom Reference Allam and Åbom2006). Three virtual microphones are placed in the downstream straight pipe at x/D = 4, 6 and 8, as shown in figure 25(a). These locations are consistent with the downstream reference positions introduced in the duct-based source assessment in § 4.4. The complex pressure signal is decomposed into a downstream-propagating plane-wave component $\hat{\textit{p}}$ ⁺ and an upstream-propagating component $\hat{\textit{p}}$ ^–. With the harmonic convention e ^iωt , the pressure field at these locations is written as

(4.9) \begin{equation} \hat{p}\left(x_{m},\omega \right)=\hat{p}^{+}\left(\omega \right)e^{-i{k^{+}}({x_{m}}-{x_{0}})}+\hat{p}^{-}\left(\omega \right)e^{i{k^{-}}({x_{m}}-{x_{0}})}, \end{equation}

where $\hat{\textit{p}}$ (x _m , ω) is the Fourier-transformed pressure fluctuation at the mth microphone and x ₀ is the reference axial location, taken as the first microphone position. The downstream-propagating wavenumber k ⁺ is defined in § 4.4, while the upstream-propagating wavenumber is k ^– = ω/(c ₀ – U _b ). When the mean-flow correction is neglected, k ⁺ = k ^– = ω/c ₀, and the expression reduces to the usual two-wave form $\hat{\textit{p}} = \hat{\textit{p}}$ ⁺ e ^–ikx + $\hat{\textit{p}}$ ^– e ^ikx . In the present postprocessing, the three microphone signals form an overdetermined system, and the complex amplitudes p ⁺ and p ^– are obtained by a least-squares fit to the two-wave model. The relative strength of the counter-propagating plane-wave component is quantified by R _p = | $\hat{\textit{p}}$ ^–|/| $\hat{\textit{p}}$ ⁺|. A smaller value of R _p indicates that the downstream-propagating component remains dominant at the corresponding frequency. The downstream-propagating pressure level shown in figure 25(b) is defined from the PSD of p ⁺ as

(4.10) \begin{equation} L_{{p^{+}}}\left(f\right)=10\log _{10}\left[\frac{S_{{p^{+}}{p^{+}}}(f)}{{p}_{\textit{ref}}^{2}}\right]\!, \end{equation}

where S _p+p+ is the PSD of the downstream-propagating pressure component and p _ref is the reference pressure used consistently in the spectral normalisation. The spectrum shows that the chamfered configuration remains below the baseline circular orifice over most of the analysed Strouhal-number range. More importantly, the dominant peaks in the downstream-propagating pressure are consistent with the amplification frequencies identified by the resolvent analysis. In the baseline case, pronounced acoustic-pressure peaks appear near the representative intermediate- and high-frequency branches, St = 4.73 and 7.73, selected from the baseline resolvent spectrum in figure 10. After chamfering, the intermediate branch shifts to a weaker peak near St = 5.65, consistent with the shift of the leading-gain peak in figure 17(a), while the high-frequency branch is strongly attenuated. This agreement between the resolvent-predicted spectral branches and the downstream-propagating pressure spectrum supports the connection between the wall-pressure source reduction and the in-duct transmitted acoustic response. The trend is also consistent with the area-weighted wall-pressure spectrum in figure 18(b) and with the duct-based surface-pressure reduction shown in figures 22 and 23. Negative values of ΔL _p+ = L _p+(chamfer) – L _p+(baseline) therefore indicate that the downstream-propagating pressure is reduced by chamfering at that frequency. The value of ΔL _p+ remains negative over almost the entire analysed band, with stronger attenuation in the intermediate- and high-Strouhal-number ranges. A band-integrated evaluation gives the same conclusion. The downstream-propagating pressure level is reduced by 45.3 dB over 1 < St < St _c , by 45.1 dB over 1 < St < 8 and by 55.8 dB over 8 < St < St _c . These values confirm that the downstream chamfer not only reduces the wall-pressure-related source indicators, but also lowers the downstream-propagating plane-wave pressure extracted from the LES. The amplitude ratio in figure 25(d) remains below unity for both configurations over the analysed frequency range. In particular, the principal narrow-band branches are not accompanied by an upstream-propagating plane-wave component of comparable amplitude to the downstream component. The present decomposition therefore does not show the strong counter-propagating signature expected for a pronounced standing-wave response at these frequencies. The negative temporal growth rates in figure 10(d) further support interpreting the narrow-band peaks as forced hydrodynamic amplification under the present computational conditions. This assessment is limited to the acoustic field established by the inlet and outlet treatments used here. Alenius, Åbom & Fuchs (Reference Alenius, Åbom and Fuchs2015) showed that the interaction between incident sound and the shear layer inside an orifice is sensitive to the symmetry and disturbance content of the inflow. Sovardi et al. (Reference Simmons, Thomas, Corke and Gluzman2016) treated the noise generated by turbulence and the acoustic scattering matrix as separate properties of a duct singularity. This study does not determine the scattering matrix or the acoustic power production of the selected chamfer under independently imposed incident waves. Its whistling potentiality under other duct terminations would require a separate acoustic scattering or power-balance analysis.