# Linear reactive control of jet installation noise

Matteo Mancinelli, Diego Bonkowski de la Sierra Audiffred, Eduardo Martini

Rodrigues da Silva, Peter Jordan, André Cavalieri, Anton Lebedev

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Matteo Mancinelli, Diego Bonkowski de la Sierra Audiffred, Eduardo Martini Rodrigues da Silva, Peter Jordan, André Cavalieri, et al.. Linear reactive control of jet installation noise. Journal of Fluid Mechanics, 2025, 1017, pp.A8. 10.1017/jfm.2025.10480. hal-05374763

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1 Linear reactive control of jet installation noise

2 M. Mancinelli¹†, D. Audiffred², E. Martini³, P. Jordan³, A.V.G. Cavalieri², and A.

3 Lebedev³

4 ¹Department of Civil Engineering, Computer Science and Aeronautical Technologies, Università Roma Tre, Via Vito Volterra 62, 00146 Rome, Italy

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6 ²Department of Aeronautical Engineering, Instituto Tecnológico de Aeronáutica, São José dos Campos, Brazil

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8 ³Department of Fluids, Thermal Science and Combustion, Institut Pprime, CNRS-Université de Poitiers-ENSMA, 11 Boulevard Marie et Pierre Curie, 86360 Chasseneuil-du-Poitou, France

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10 (Received xx; revised xx; accepted xx)

11 This paper presents an experimental application of reactive control to jet installation noise based on destructive interference. The work is motivated by the success of previous studies in applying this control approach to mixing layers (Sasaki et al. 2018b), boundary layers (Brito et al. 2021; Audiffred et al. 2023), flow over a backward-facing step (Martini et al. 2022) and, more recently, to turbulent jets (Maia et al. 2021, 2022; Audiffred et al. 2024b). We exploit the fact that jet-surface interaction noise is underpinned by wavepackets that can be modelled in a linear framework and develop a linear control strategy where piezoelectric actuators situated at the edge of a scattering surface are driven in real time by sensor measurements in the near field of the jet, the objective being to reduce noise radiated in the acoustic field. The control mechanism involves imposition of an anti-dipole at the trailing edge to cancel the scattering dipole that arises due to an incident wavepacket perturbation. We explore two different control strategies: (i) the inverse feed-forward approach, where causality is imposed by truncating the control kernel, and (ii) the Wiener-Hopf approach, where causality is optimally enforced in building the control kernel. We show that the Wiener-Hopf approach has better performance than that obtained using the truncated inverse feed-forward kernel. We also explore different positions of the near-field sensors and show that control performance is better for sensors installed for streamwise positions downstream in the jet plume, where the signature of hydrodynamic wavepacket is better captured by the sensors. Broadband noise reductions of up to 50% are achieved.

30 **Key words:** Authors should not enter keywords on the manuscript, as these must be chosen by the author during the online submission process and will then be added during the typesetting process (see Keyword PDF for the full list). Other classifications will be added at the same time.

34 MSC Codes (Optional) Please enter your MSC Codes here

† Email address for correspondence: matteo.mancinelli@uniroma3.it
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# 1. Introduction

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The turbulent jet exhausting from an aircraft engine is one of the main sources of aircraft noise, especially during take off when engines run at maximum power. This issue imposes constraints on the development of commercial aircraft. In addition to the technological and sustainability aspects, jet noise remains a challenging scientific problem because most of the fluctuation energy in subsonic turbulent jets is not converted into sound energy: jets are populated by mostly silent flow structures. There is now consensus that coherent structures, referred to also as wavepackets, are the most acoustically efficient structures (Cavalieri et al. 2011, 2012; Cavalieri & Agarwal 2014; Sinha et al. 2014). These structures, which are largely underpinned by Kelvin-Helmholtz and Orr instability mechanisms, can be described by linear models (Jordan & Colonius 2013; Cavalieri et al. 2019).

The installation of a jet in the vicinity of a scattering surface, such as a wing, generates an additional noise source referred in the literature as jet-surface interaction (JSI) noise (Brown 2013). As shown by Cavalieri et al. (2014), Piantanida et al. (2016), Nogueira et al. (2017) and Nogueira et al. (2019), the physical mechanism underlying generation of JSI noise is the scattering of coherent structures by the trailing edge (TE) of the surface. In these studies, the authors show, using acoustic-analogy-based models, the effectiveness of linear models for the JSI noise problem. The utility of linearised approaches for the phenomenon was also shown by Faranosov et al. (2019), who used a Wiener-Hopf estimator to predict far-field noise on the basis of near-field pressure fluctuations provided by parabolised stability equations. The fact that JSI noise is underpinned by wavepackets and amenable to modelling in a linear framework, suggests that a linear reactive flow-control strategy that targets coherent structures could be used to reduce JSI noise.

Reactive control is a strategy where the actuation signal is driven in real time on the basis of flow measurements. In this framework, Linear Quadratic Gaussian (LQG) controllers have been widely adopted for flow control to design optimal control tools, e.g., for separated boundary layers (Åkervik et al. 2007), flows over an open cavity (Barbagallo et al. 2009) and a backward-facing step (Barbagallo et al. 2012). The drawback of LQG controllers is that their implementation requires the solution of two algebraic Riccati equations for the estimation of the system state and the control-kernel calculation. Due to the high dimensionality of fluid systems, the use of a reduced-order model (ROM) is required. ROMs may be obtained using, for instance, either balanced POD truncation (Bagheri et al. 2009) or linear stability theory (Sasaki et al. 2017). An alternative, simpler approach for flow systems underpinned by convective instability mechanisms involves the elimination of disturbances via a destructive superposition of waves: wave cancellation. This strategy has recently been used, for instance, for the control of free shear flow (Sasaki et al. 2018b) and a boundary layer (Sasaki et al. 2018a), where it also gave evidence of comparable performances with respect to the more computationally demanding LQG controller. Wave-cancelling approaches have the further advantage that they are based on transfer functions (TFs) that can be educed from experiments in a system-identification approach, as shown by Brito et al. (2021) and Audiffred et al. (2023) for the control of Tollmien-Schlichting waves. Concerning jets, wave-cancellation schemes were applied to control: (i) harmonic disturbances in free jets using an open-loop strategy (Kopiev et al. 2013); (ii) stochastic disturbances in forced and unforced laminar and turbulent free jets (Maia et al. 2021, 2022; Audiffred et al. 2024b); harmonic disturbances in excited and unexcited installed jets (Kopiev et al. 2020; Bychkov et al. 2019). With reference to the last work, plasma actuators at the nozzle exit were driven by near-field sensors via a feedback control scheme to reduce the wavepacket amplitude, thus reducing the noise scattered by the TE of the surface.

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The above discussion motivates the present work, where we perform linear reactive control
of JSI noise for an unexcited subsonic jet over a broadband frequency range. We design an experiment comprising a subsonic turbulent jet in the vicinity of a parallel flat plate to mimic jet-wing interaction. Pressure sensors in the near field of the jet are used to drive downstream actuators installed on the TE of the surface to reduce the noise radiated in the acoustic field. The potentialities of this set up to effectively control jet installation noise have been explored in the model designed by Karban et al. (2023), where the jet dynamics were represented by a stochastic wavepacket modelled using the Ginzburg-Landau equation and the actuation was mimicked by a point dipole placed at the TE of a semi-infinite flat plate. The near-field sensors and the acoustic observer consist, respectively, of a 6-microphone azimuthal array to measure the axisymmetric pressure disturbances and two microphones located in the sound field on the shielded and unshielded sides of the jet-plate system. Transfer functions between sensors, observers and actuators used to build the control kernel are directly obtained from the experiments. Consistent with the results of Mancinelli et al. (2024), where the sensor-observer coherence was shown to vary strongly with the streamwise position of the near-field sensors, we explore control for two axial positions of the near-field sensor array where the sensor-observer coherence showed high amplitude, that is, at the nozzle exhaust and in the vicinity of the TE. Two control strategies are tested: (i) the Inverse Feed-Forward Control (IFFC) (Sasaki et al. 2018a; Brito et al. 2021) and (ii) the Wiener-Hopf (W-H) control (Noble & Weiss 1959; Martini et al. 2022). Consistent with the results obtained by Audiffred et al. (2023) and Audiffred et al. (2024b) for the control of Tollmien-Schlichting waves in a boundary layer over a wing profile and the velocity fluctuations in a turbulent jet, respectively, we show that the W-H formalism yields to a better performance than that achieved using the IFFC approach.

The remainder of the paper is organised as follows. §2 provides the theoretical framework of linear reactive control, §3 describes the experimental setup and §4 is devoted to the presentation of the main results. Conclusions and future developments of the present activity are finally discussed in §5.

## 2. Linear reactive control

The principle of linear reactive control for installed jet noise consists of using actuators installed at the TE of the surface to generate an anti-dipole that would cancel the scattering dipole that arises due to an incident perturbation field associated with convecting wavepackets. A simplified sketch of the control strategy just outlined is shown in figure 1, where we use lower- and upper-case symbols to denote time- and frequency-domain quantities, respectively.

The actuation signal in the frequency domain is (Brito et al. 2021),

$$U(f) = K(f) Y(f), \qquad (2.1)$$

where $U$ is the actuation signal, $Y$ is the sensor reading and $K$ is the control kernel. In the time domain, $u$ is given by the convolution

$$u(t) = \int_{-\infty}^{+\infty} k(\tau) \nu(t-\tau) \, d\tau, \qquad (2.2)$$

leading to a solution where, in general, the actuation signal is non-causal, that is, it depends on negative values of $\tau$. In real applications this is not feasible, given that the actuation would depend on sensor readings in the future. For the control to be causal,

$$k(\tau < 0) = 0 \qquad (2.3)$$
<figure><img src="image_2.png" /><figcaption>Figure 1: Schematic representation of the feed-forward wave-cancellation strategy for the JSI noise control. $v$ (Y) denotes the near-field sensor, $u$ (U) the actuator and $\zeta$ (Z) the observer in the acoustic field. Lower-case symbols denote quantities in the time domain, upper-case symbols denote their counterparts in the frequency domain.</figcaption></figure>

needs to be imposed. We address this causality issue using two approaches: (i) the IFFC, where causality is imposed by the truncation of the control kernel to positive $\tau$ and (ii) the Wiener-Hopf control, where causality is imposed a priori.

## 2.1. Inverse Feed-Forward Control

The control law design is based on that presented by Sasaki et al. (2018a) and Maia et al. (2021) and the corresponding block diagram is shown in figure 2. We seek to eliminate fluctuations at the observer $\zeta$, which in the frequency domain is given by

$$Z(f) = \mathcal{Y}(f) H_{v\zeta}(f) + U(f) H_{u\zeta}(f), \qquad (2.4)$$

where $H_{v\zeta}$ and $H_{u\zeta}$ are the sensor-observer and actuator-observer transfer functions, respectively. The actuation can be written as follows

$$U(f) = K(f) (\mathcal{Y}(f) + H_{u\nu}(f) U(f)), \qquad (2.5)$$

where $H_{u\nu}$ is the actuator-sensor transfer function, accounting for potential contaminations of the sensor measurements by the actuators and $\mathcal{Y}(f)$ is, consequently, the uncontaminated sensor reading. As pointed out by Maia et al. (2021), this should not be confused with a classic feedback configuration, where actuation is modified by a downstream sensor, and use of the contamination transfer function is exploited to avoid undesirable noise in the actuation signal. Combining (2.4) and (2.5) and setting $Z(f) = 0$, we find the control kernel,

$$K(f) = - \frac{H_{v\zeta}}{H_{u\zeta} - H_{u\nu} H_{v\zeta}}. \qquad (2.6)$$

To regularise the problem and avoid divisions by zero (or values close to zero), (2.6) is recast as,
<figure><img src="image_3.png" /><figcaption>Figure 2: Block diagram of the control scheme implemented herein.</figcaption></figure>

$$K(f) = \frac{(H_{u\zeta} - H_{u\nu}H_{\nu\zeta})^* H_{\nu\zeta}}{(H_{u\zeta} - H_{u\nu}H_{\nu\zeta})^* (H_{u\zeta} - H_{u\nu}H_{\nu\zeta}) + R'} \quad (2.7)$$

where the superscript * denotes the complex conjugate and R is a penalisation factor typically 148 frequency dependent, analogous to the actuator penalty in the LQG and Wiener-Hopf 149 frameworks (see §2.2). In this work we consider R as a constant in the frequency domain, 150 that is, corresponding to a white noise signal in time, and explore different values with a 151 view to maximising the control performance. The control law in the time domain is, 152

$$u(t) = \int_{0}^{\infty} k(\tau) \nu(t-\tau) \, d\tau, \quad (2.8)$$

where the control kernel is truncated to positive τ to impose the causality constraint a 154 posteriori. The transfer functions are obtained from the experiments and estimated as the 155 ratio between the cross-spectral densities (CSD) of inputs and outputs and the power spectral 156 densities of inputs (Bendat & Piersol 2011): 157

$$H_{v\zeta} = \frac{S_{v\zeta}}{S_{vv}}, \quad (2.9a)$$

$$H_{u\zeta} = \frac{S_{u\zeta}}{S_{uu}}, \quad (2.9b)$$

$$H_{uv} = \frac{S_{uv}}{S_{uu}}, \quad (2.9c)$$

where $S_{v\zeta}$ is the sensor-observer CSD, $S_{u\zeta}$ is the actuator-observer CSD, $S_{uv}$ is the actuator-sensor CSD, $S_{vv}$ and $S_{uu}$ are the sensor and actuator signal auto-spectra, respectively. 162 163 Spectral densities are estimated using the Welch's method (Welch 1967), using a Hanning 164 window of 1024 samples with a total length of 60 s and a sampling frequency of 10 kHz, 165 thus leading to a frequency resolution of ≈ 10 Hz, and 50% overlap. 166

## 2.2. Wiener-Hopf control

The Wiener-Hopf control results in an optimal control where a quadratic cost functional is 168 minimised. The cost functional is given by, 169

$$J = \int_{-\infty}^{+\infty} \left( \langle \zeta_c^* Q \zeta_c + u_+^* R u_+ \rangle + k_+ \lambda_- + k_+^* \lambda_-^* \right) dt, \quad (2.10)$$

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where $\zeta_c$ is the controlled observer signal, $\lambda_-$ is a Lagrange multiplier, where $\lambda_-(t > 0) = 0$ to impose causality, that is, $k_+(t < 0) = 0$, $\langle \cdot \rangle$ denotes an ensemble average, $Q$ is a weighting parameter of state deviation from original condition and $R$ a control penalisation of the same kind described above (cf. (2.7)). From (2.10), the functional is, up to a multiplicative constant, a function of $R/Q$. We, thus, use, without loss of generality, $Q = 1$. The subscripts + and - imply that the associated functions are regular in the upper and lower half-planes of the complex frequency plane, respectively, such that plus and minus functions are zero for $t < 0$ and $t > 0$, respectively, when transformed to the time domain. Recalling that, according to (2.2), $u = k * v$, where * denotes a convolution, the controlled signal can be expressed as a linear combination of the uncontrolled signal $\zeta_{unc}$ and the actuation response, as formalised in what follows,

$$ \zeta_c = \zeta_{unc} + h_{u\zeta} * u = \zeta_{unc} + h_{u\zeta} * k * v, \quad (2.11) $$

where $h_{u\zeta}$ is the actuation-observer transfer function in the time domain. The minimisation of the functional in (2.10) leads to the following Wiener-Hopf equation in the frequency domain (for more details on the solution of the Wiener-Hopf problem see §A):

$$ H_l K_+ S_{v\nu} + \lambda_-^* = H_r S_{v\zeta}, \quad (2.12) $$

where $H_l$ and $H_r$ are defined as follows,

$$ H_l(f) = H_{u\zeta}^*(f) Q H_{u\zeta}(f) + R \quad (2.13a) $$

$$ H_r(f) = -H_{u\zeta}^*(f) Q. \quad (2.13b) $$

Applying the additive and multiplicative factorisations described in §A, the following expression for the control kernel is obtained,

$$ K_+ = H_{l+}^{-1} (H_{l-}^{-1} H_r S_{v\zeta} S_{v\nu-}^{-1})_+ S_{v\nu+}^{-1}, \quad (2.14) $$

where the frequency dependence was omitted for notational simplicity.

To account for the feedback effect, that is, the influence of the actuator $u$ on the sensor $v$, the final expression for the control kernel is (Martini et al. 2022):

$$ K'_+(f) = (I + K_+ H_{uv})^{-1} K_+, \quad (2.15) $$

which is then inverse-Fourier transformed to obtain a causal control kernel in the time domain.

## 3. Experimental setup

Experiments were performed in the Bruit & Vent anechoic facility of Institut Pprime in Poitiers. The anechoic chamber is 12.6 × 10.6 × 7.85 m³ in size and is equipped with absorbing acoustic foam leading to a cut-off frequency of 200 Hz. The feed line consists of a compressed dry-air duct supplied by a screw compressor delivering a continuous mass flow rate of up to 1 kgs⁻¹ at 3.5 bar. An electrically driven valve downstream of the compressor allows regulation of the jet velocity by controlling the incoming mass flow. A cooling system is installed downstream of the regulating valve to provide a jet flow in isothermal conditions with respect to the ambient temperature. The stagnation temperature and pressure in the plenum chamber are monitored through a thermocouple and a pressure transducer, respectively. The jet issues from a straight nozzle of diameter $D = 0.05$ m. The boundary
210 layer inside the nozzle is tripped by a strip of carborundum particles placed 2.5 diameters
211 upstream of the nozzle exit so as to generate an initially turbulent jet. The same flow, i.e.,
212 nozzle geometry and boundary-layer trip, was studied in several previous works (see e.g.
213 Cavalieri *et al.* (2013); Jaunet *et al.* (2017); Maia *et al.* (2022); Audiffred *et al.* (2024a)).

214 A steel flat plate is installed parallel to the jet at the radial distance *H* = 1*D* from the nozzle
215 axis with the trailing edge positioned at an axial distance *x/D* = 2 from the nozzle exhaust.
216 The piezoelectric actuators are P876.A11 DuraAct Patch transducers and are driven by a
217 E-413.D2 DuraAct amplifier. They are installed at the TE of the plate and their streamwise
218 length, spanwise width and thickness are equal to 0.8 *D*, 0.7 *D* and 0.008 *D*, respectively.
219 For the results reported here, the five central actuation patches alone were used to control, as
220 this is where the jet-edge dipole is centred. We use a near-field microphone array as sensors.
221 The 6-microphone azimuthal array is placed at a radial distance of *r* = 0.9 *D* from the
222 nozzle axis to measure the pressure fluctuations in the near field and to extract the azimuthal
223 Fourier mode *m* = 0, which is the most efficient mode in the generation of installed jet noise
224 (Cavalieri *et al.* 2014). The near-field array can be moved along the streamwise direction.
225 Consistent with the results presented by Mancinelli *et al.* (2024), where the sensor-observer
226 transfer function was characterised in the same experimental setup with the aim of providing
227 guidance for sensor positioning for control, we consider two different streamwise positions of
228 the array, *x/D* = 0 and *x/D* = 1.7. Two microphones are placed in the acoustic field at the TE
229 axial position at a radial distance of 13 *D* from the trailing edge. The microphone observers
230 are installed at polar angles *ψ* = 90° and *ψ* = 270°, where the polar angle *ψ* is positive in the
231 downstream direction, on the shielded and scattered sides of the acoustic field, respectively,
232 where the dipole TE noise source has peak radiation. We mostly focus on control results
233 obtained using the microphone on the shielded side as observer position. The shielded side
234 has two advantages: the direct jet noise, which we do not attempt to control, contributes less;
235 the scattered sound does not radiate through the jet, minimising complications associated
236 with flow-sound interaction (refraction, scattering by turbulence). Control results obtained
237 on the unshielded side are addressed in §B.

238 Measurements are carried out in isothermal conditions for jet Mach numbers *M* = 0.2,
239 0.25 and 0.3, which correspond to nozzle-diameter-based Reynolds numbers, *Re*, of 2.2·10⁵,
240 2.8 · 10⁵ and 3.4 · 10⁵, respectively. Low-velocity conditions are dictated by limitations of
241 the actuation system. Pressure fluctuations are measured by PCB 46BP microphones, whose
242 frequency response is flat in the range 4 Hz-70 kHz. Sensor and observer readings and
243 actuation are performed using a DSPACE MicroLabBox DS1202 Controller Board. The
244 controller ran at a sampling rate, *f*<sub>s</sub>, of 10 kHz for an acquisition time *T*<sub>a</sub> = 60 s on the basis
245 of a Simulink plant model that was converted to the dSPACE Control Desk environment.
246 A sketch and a picture of the experimental setup and the microphone disposition with the
247 identification of the reference system adopted is shown in figure 3.

248 4. Results

249 4.1. *Actuation characterisation*

250 We characterise the actuator response to identify the operational frequency range and explore
251 actuation behaviour at different excitation amplitudes, *A*, so as to identify the linear actuation
252 regime, following Maia *et al.* (2021) and Maia *et al.* (2022). All the results shown below are
253 obtained with the jet off. We excite the actuators with a band-limited stochastic forcing, that is,
254 a band-pass filtered white noise, to explore different bandwidths and measure their response
255 via the microphones in the acoustic field. We set the lowest excitation frequency to *f* = 450 Hz
256 and consider different maximum excitation frequencies: *f* = 700 Hz, 1.1 kHz, 1.5 kHz and
<figure><img src="image_4.png" /><figcaption>Figure 3: Sketch and picture of the experimental setup and microphone disposition and identification of the reference system adopted.</figcaption></figure>

<table><caption>Table 1: Corresponding Strouhal number ranges to stochastic forcing excitation frequency bandwidths tested to characterise the actuator behaviour for the jet Mach numbers considered.</caption><thead><tr><th><i>f</i> [Hz]</th><th><i>M</i> = 0.2</th><th><i>M</i> = 0.25</th><th><i>M</i> = 0.3</th></tr></thead><tbody><tr><td>[450, 700]</td><td><i>St</i> ≈ [0.33, 0.51]</td><td><i>St</i> ≈ [0.26, 0.41]</td><td><i>St</i> ≈ [0.22, 0.34]</td></tr><tr><td>[450, 1100]</td><td><i>St</i> ≈ [0.33, 0.8]</td><td><i>St</i> ≈ [0.26, 0.64]</td><td><i>St</i> ≈ [0.22, 0.53]</td></tr><tr><td>[450, 1500]</td><td><i>St</i> ≈ [0.33, 1.09]</td><td><i>St</i> ≈ [0.26, 0.87]</td><td><i>St</i> ≈ [0.22, 0.73]</td></tr><tr><td>[450, 2000]</td><td><i>St</i> ≈ [0.33, 1.46]</td><td><i>St</i> ≈ [0.26, 1.16]</td><td><i>St</i> ≈ [0.22, 0.97]</td></tr></tbody></table>

2 kHz. The choice of the minimum excitation frequency was dictated by preliminary tests carried out using harmonic forcing signals that revealed that the actuators had a very weak response for $f < 400$ Hz. For the sake of brevity, these results are not reported herein. The corresponding Strouhal number ranges to the bandwidths tested for the *M* considered are reported in table 1, where the Strouhal number, *St*, is calculated based on nozzle diameter and jet velocity. Figure 4 shows the *PSD* on the shielded side for a given excitation amplitude. Consistent with the enlargement of the excited frequency band, the actuator response exhibits a spectrum with a non-zero amplitude on a larger frequency band. The actuators do not respond homogeneously throughout the excited frequency band.

We now explore the actuator behaviour for different excitation amplitudes, that is, $A = [0.5, 4]$ V. Figure 5 shows the standard deviation of the actuator response on the microphone on the shielded side as a function of the excitation amplitude. The standard deviation is calculated as the square root of the integral of the *PSD* response over the excited frequency band. As expected, the standard deviation amplitude increases as a consequence of the increase of the actuation bandwidth. We note that for the frequency range $f = [450, 2000]$ Hz the actuators tend to saturate at high amplitudes and start to behave non linearly. In light of this result, on the basis of the *St*-number range where JSI noise is dominant, that is, $St ≈ [0.1, 0.8]$ (Mancinelli et al. 2022, 2024), and following the results of Maia et al. (2021) and Maia et al. (2022) where control effectiveness was shown to degrade for increasing actuation bandwidths, we select the bandwidth $f = [450, 1100]$ Hz as the operational
<figure><img src="image_5.png" /><figcaption>Figure 4: Actuator response to band-limited white noise excitation signal measured by the microphone in the acoustic field on the shielded side for different excitation frequency bands. (a) excitation signal, (b) actuator response. Blue line refers to bandwidth $f = [450, 700]$ Hz, red line to $f = [450, 1100]$ Hz, black line to $f = [450, 1500]$ Hz, green line to $f = [450, 2000]$ Hz.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Figure 5: Standard deviation of the actuator response to band-limited white noise excitation signal measured by the pressure microphone in the acoustic field on the shielded side as a function of the excitation amplitude. (a) excitation frequency band $f = [450, 700]$ Hz, (b) $f = [450, 1100]$ Hz, (c) $f = [450, 1500]$ Hz, (d) $f = [450, 2000]$ Hz.</figcaption></figure>

<footnote n="277">277 frequency range of the actuators. All the results presented hereinafter are obtained using this operational frequency range.</footnote>

<footnote n="278">278 We finally assess that the actuation forcing has a dipole behaviour in the operational frequency range, which is what is required, following the study of Karban *et al.* (2023),</footnote>
<figure><img src="image_7.png" /><figcaption>Figure 6: Phase between the two microphones in the acoustic field on the shielded and unshielded sides for actuation excitation signal equal to a band-limited stochastic forcing in the frequency band $f = [450, 1100]$ Hz at different excitation amplitudes.</figcaption></figure>

to control JSI noise. Figure 6 shows the absolute value of the phase of the cross-spectrum between the two microphones in the acoustic field on the shielded and unshielded sides, $\angle S_{\zeta_1\zeta_2}$, as a function of frequency at different excitation amplitudes. Except for the smallest excitation amplitudes, $A = 0.5$ V and $1$ V, where the actuator response was weak, the phase remains roughly constant around $\pi$ over all the frequency band, thus implying a dipole source as required. A slight departure from phase opposition is observed close to the maximum excitation frequency for all the amplitudes tested. However, this drawback does not affect control effectiveness, as it is shown in what follows.

## 4.2. System identification

The two control strategies involve two ingredients: estimation and actuation. In the estimation step, the noise radiated at the objective position, $\zeta(t)$, is provided via the transfer function $H_{\nu\zeta}$ from the near-field sensors $\nu(t)$. In the actuation step, the noise radiated in the acoustic field is zeroed, or at least reduced, based on the estimated $\zeta$, assuming that the effect of the actuators on the observer and the potential contamination of the sensors by the actuators might be described by the transfer functions $H_{u\zeta}$ and $H_{uv}$, respectively.

We here identify the transfer functions between sensors, observer and actuators used to calculate the kernels for both the IFFC and the W-H control, according to (2.7) and (2.15), respectively. The transfer functions are estimated in two successive steps. First, the sensor-observer TF is identified via simultaneous readings of sensors and observer with the actuators off and calculated using (2.9a). Then, the actuator-sensor and actuator-observer TFs are identified forcing the actuators with a band-limited stochastic forcing in the range $f = [450, 1100]$ Hz, performing simultaneous sensor-observer measurements. The TFs are calculated using (2.9c) and (2.9b). We note that the actuator-observer and actuator-sensor transfer function identification is not dependent on the type of forcing adopted. We chose a band-limited stochastic forcing for the present work for the sake of simplicity.

As mentioned above, the underlying assumption of the control strategy is linearity. We will use two-point coherence as a measure of the linearity of the system and as a means to characterise the transfer functions between sensors, observer and actuators. The accuracy of
the estimation and actuation processes, and, thus, the control performance, is underpinned by the magnitude of the sensor-observer and actuator-observer coherence. The coherence between two generic time signals a (t) and b (t) is defined as,

$$ \gamma_{ab}(f) = \sqrt{\frac{|\langle S_{ab}(f) \rangle|^2}{|\langle S_{aa}(f) \rangle| |\langle S_{bb}(f) \rangle|}} \quad (4.1) $$

### 4.2.1. Estimation

Figure 7 shows the PSDs of the axisymmetric pressure disturbance sensed by the near-field sensors, that successively constitutes the input for the controller, for both streamwise positions of the near-field array explored herein, $x/D = 0$ and $x/D = 1.7$. As an example, we show the results for jet Mach number $M = 0.2$. The $m = 0$ pressure mode is obtained as the average of the six near-field microphone readings along the azimuthal direction. The PSDs are presented hereinafter in dB/St, according to the expression reported in what follows (Pierce 2019; Mancinelli et al. 2019):

$$ \text{PSD}[\text{dB/St}] = 10 \log_{10} \left( \frac{\text{PSD}[\text{Pa}^2/\text{Hz}] U_j}{p_{\text{ref}}^2 D} \right), \quad (4.2) $$

where $U_j$ is the jet velocity and $p_{\text{ref}} = 20 \mu\text{Pa}$ is the reference pressure. The spectra change substantially with the streamwise position. As outlined by Mancinelli et al. (2024), the near-field sensors sense both the signature of the hydrodynamic wavepacket and the scattered acoustic pressure from the TE of the surface. Specifically, for array positions close to the nozzle exhaust the near-field sensors are dominated by the scattered acoustic pressure, whereas for positions downstream in the jet plume the hydrodynamic wavepacket signature becomes dominant. The physics just described explains the change of spectral shape observed in figure 7 and also affects control effectiveness, as will be shown in what follows. Figure 8 shows the coherence between the axisymmetric pressure mode educed by the near-field sensors and the noise at the observer position on the shielded side. We explore the effect on the coherence of both the streamwise position of the near-field sensors and the jet flow conditions. For this aim, we consider both the streamwise positions investigated herein, $x/D = 0$ and $x/D = 1.7$, and the Mach numbers $M = 0.2$ and $M = 0.3$. The coherence exhibits high values for both flow conditions for $x/D = 1.7$ in the range $St \approx [0.1, 0.8]$ where jet installation noise is dominant. Concerning the effect of the streamwise position of the near-field array, we selected as flow condition $M = 0.2$ and note that the coherence is slightly larger for $x/D = 0$ than that measured for $x/D = 1.7$ due to the stronger significance of the scattered acoustic field close to the nozzle exhaust (Mancinelli et al. 2024). The coherence values are sufficiently high to accurately estimate the noise at the observer position and attempt noise control for both positions of the near-field array.

### 4.2.2. Actuation

Figure 9 shows the actuation effect on the near-field $m = 0$ pressure mode for jet Mach number $M = 0.2$ for both $x/D = 0$ and $1.7$. We remind the reader that for these tests the jet was on and, following the characterisation of the actuators presented in §4.1, we excited the actuators with a white noise band-pass filtered in the frequency range $f = [450, 1100]$ Hz, thus implying a $St$ band of interest $[0.33, 0.8]$ (see table 1). The actuation signature in the near-field sensor spectrum is clearly above the jet signature for $x/D = 0$, where the sensors mostly sense the acoustic pressure scattered by the TE of the plate. On the contrary, the sensors are less affected by the actuators when the array is moved further downstream in
<figure><img src="image_8.png" /><figcaption>Figure 7: Spectra of near-field $m = 0$ mode for $M = 0.2$ in the case of unforced jet: blue line refers to $x/D = 0$ and red line to $x/D = 1.7$.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Figure 8: Sensor-observer coherence. (a) Effect of jet flow conditions for the streamwise position of the near-field sensors $x/D = 1.7$: blue line refers to $M = 0.2$, red line to $M = 0.3$. (b) Effect of the streamwise position of the near-field sensors for $M = 0.2$: black line refers to $x/D = 0$, blue line to $x/D = 1.7$.</figcaption></figure>

351 the jet plume and the hydrodynamic pressure field generated by the jet becomes dominant.

352 The actuation effect on the observer on the shielded side for $M = 0.2$ and 0.3 is shown in figure 10. We note that the actuation signature in the spectrum becomes less evident for increasing $M$ and the noise emitted by the jet dominates over that generated by the actuators.

353 For the control to be efficient, the actuation signature needs to emerge above the jet signature, 354 since this ensures that the actuators have sufficient energy to generate pressure fluctuations 355 comparable to those created by the jet, thus being strong enough to cancel them. We thus 356 expect a reduction in control effectiveness at high $M$. 357
<figure><img src="image_10.png" /><figcaption>Figure 9: Actuation effect on near-field sensors for $M = 0.2$. Blue line refers to the spectrum with jet on and actuators off, red line to the spectrum with jet on and actuators forced with a band-limited stochastic forcing in the range $f = [450, 1100]$ Hz. (a) Streamwise position $x/D = 0$, (b) $x/D = 1.7$.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>Figure 10: Actuation effect on observer on the shielded side of the acoustic field. Blue line refers to the spectrum with jet on and actuators off, red line to the spectrum with jet on and actuators forced with a band-limited stochastic forcing in the range $f = [450, 1100]$ Hz. (a) Jet Mach number $M = 0.2$, (b) $M = 0.3$.</figcaption></figure>

The actuator-sensor and actuator-observer coherence is shown in figure 11. The coherence between the actuator and the observer is decreased with increasing Mach number. Where the actuator-to-sensor coherence is concerned, for the downstream sensor position, $x/D = 1.7$, the coherence drops to zero for both Mach numbers. This is because the hydrodynamic signature of coherent structures here largely dominates the signature of the scattered sound field, leading to a low signal-to-noise ratio between actuator and sensor, where 'noise' can
<figure><img src="image_12.png" /><figcaption>Figure 11: Actuator-sensor (blue lines) and actuator-observer (red lines) coherence functions for jet flow conditions $M = 0.2$ and 0.3 and streamwise positions of near-field sensor array $x/D = 0$ and $x/D = 1.7$. (a) $x/D = 0$ and $M = 0.2$, (b) $x/D = 1.7$ and $M = 0.2$, (c) $x/D = 1.7$ and $M = 0.3$.</figcaption></figure>

be here interpreted as the hydrodynamic component of the measurement. On the contrary, the large value of $\gamma_{uv}$ for $x/D = 0$ implies a significant contamination of the sensor readings by the actuators that might lead to Larsen effect (Sanfilippo & Valle 2013) for high actuation amplitudes; this aspect will be taken into account by the actuator-sensor transfer function $H_{uv}$ in the control kernel calculation according to (2.7) and (2.15).

### 4.3. Control

#### 4.3.1. Kernel calculation

In order to prevent undesired noise captured by the sensors/observer from affecting the kernel, a zero-phase band-pass filter for the frequency band for which the actuators were forced was
applied to the raw data before the calculation of the transfer functions. Following Audiffred et al. (2024b), a constant term $\epsilon$ was added to the PSDs of u and v to regularise the problem and avoid divisions by zero (or values close to zero) in the TF calculation in (2.9) and, thus, in the kernel calculation in (2.7) and (2.15). Choosing $\epsilon$ a couple of orders of magnitude lower than the typical value of the PSD in the frequency band of interest ensures a proper regularization of the calculation and that the transfer functions are well approximated in the frequency band. Values of $\epsilon$ of $10^{-6}$ and $10^{-5}$ were used for $S_{uu}$ and $S_{vv}$, respectively.

We first explore the effect of the penalisation factor $R$ used for the kernel calculation in (2.7) and (2.13a) on the control effectiveness to identify the value $R$ that maximizes noise reduction. To do so, we iteratively calculate the control kernel for both the IFFC and W-H control for different values of $R$ and implement an off-line control. Following the expression in (2.11), the controlled observer signal in the time domain can be estimated as the linear superposition of the open-loop behaviour and the actuation effect:

$$ \zeta_c(t) = \zeta_{unc}(t) + h_{u\zeta}(t) * u(t) = (h_{v\zeta}(t) + h_{u\zeta}(t) * k(t)) * v(t), \quad (4.3) $$

where $h_{v\zeta}$ is the sensor-observer transfer function in the time domain. Figure 12 shows the band-limited Over-All Sound Pressure Level difference, $\Delta BLOASPL$, between the controlled and uncontrolled observer signals as a function of the penalisation factor $R$ used to compute the control kernel. As an example, we show the results for $M = 0.2$ and the near-field sensors at $x/D = 1.7$. The $BLOASPL$ is calculated using the formula

$$ BLOASPL = 20 \log_{10} \frac{\sigma}{p_{ref}}, \quad (4.4) $$

where $\sigma$ is the standard deviation of the controlled/uncontrolled signal. The standard deviation is calculated from the variance $\sigma^2$, which is estimated as

$$ \sigma^2 = \int_{f_1}^{f_2} \text{PSD}(f) \, df, \quad (4.5) $$

where $f_1$ and $f_2$ correspond to the minimum and maximum forcing frequencies of the actuators, that is, 450 Hz and 1.1 kHz, respectively. We compare the noise reduction achieved in the off-line tests using the control kernel obtained for both the W-H and the IFFC approaches as well as using the non-causal kernel, as formalised in (2.2). The non-causal kernel provides the largest noise reductions, but, as discussed above, this implies the actuation dependence on sensor readings in the future, which is not feasible in real applications. As expected, the achievable noise reductions tend to zero for $R \to \infty$ for all the kernels considered and this is related to the fact that the actuation effect on the observer becomes negligible for high control penalisation values. We note that there exists a range of penalisation factor values for which the W-H and the IFFC approaches exhibit high effectiveness, the W-H control being more effective. The range of optimal penalisation values is slightly different between the IFFC and W-H control. Specifically, the largest noise reductions are achieved for $R = [10^{-5}, 10^{-4}]$ and $R = [10^{-4}, 10^{-3}]$ for the IFFC and the W-H control, respectively. Hence, the $R$ values are chosen within these ranges for the two controllers. We note that in the W-H framework the noise reduction trend for increasing values of the penalisation factor should be monotonically decreasing (Martini et al. 2022). The performance loss for low $R$ values is likely related to a low signal-to-noise ratio, the undesired measurement noise captured by the sensors and the observer being larger than $R$.

Figure 13 shows the IFFC and W-H control kernels for $M = 0.2$ and $R$ values of $10^{-4}$ and $10^{-3}$, respectively, for both the streamwise positions of near-field sensors, $x/D = 0$ and
<figure><img src="image_13.png" /><figcaption>Figure 12: Noise reduction achieved in the off-line control for $M = 0.2$ and near-field sensors at $x/D = 1.7$. Blue ◆ refer to results obtained using the truncated, causal IFFC kernel, red ◦ to W-H control kernel, black △ to non-causal kernel.</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>Figure 13: Control kernels for IFFC (blue lines) and W-H (red lines) approaches for $M = 0.2$: (a) near-field sensors at $x/D = 0$, (b) $x/D = 1.7$.</figcaption></figure>

<footnote>x/D = 1.7. The non-causal part of the IFFC kernel is much more significant at x/D = 0 than at x/D = 1.7. This is consistent with the physics described above and with the results presented by Mancinelli et al. (2024), where the sensor-observer coherence was shown to be dominated by the acoustic pressure scattered by the trailing edge of the surface. The higher significance of the non-causal part at x/D = 0 is further supported by the larger peak of the W-H kernel for τ ≈ 0 for such streamwise position, which is how the W-H kernel compensates for the anti-causal part.</footnote>
<figure><img src="image_15.png" /><figcaption>Figure 14: Noise control at the objective position on the shielded side of the acoustic field for $M = 0.2$ and near-field sensors at $x/D = 0$ and $x/D = 1.7$: (a) $x/D = 0$, (b) $x/D = 1.7$. Reduction and amplification kernels are used for both the IFFC and W-H controls: bold black lines refer to the uncontrolled case, blue lines to IFFC, red lines to W-H. Solid lines refer to results obtained using the reduction kernel, dashed lines to the amplification kernel.</figcaption></figure>

4.3.2. *Control results*

Figure 14 shows the control results at the objective position on the shielded side for $M = 0.2$ and near-field sensors placed at $x/D = 0$ and $x/D = 1.7$ for both the IFFC and W-H kernels. In order to prove control authority, we compare the uncontrolled case with controlled cases using both the reduction and amplification kernels. The reduction kernels are obtained using (2.7) and (2.15) for the IFFC and W-H, respectively, whereas the amplification kernel is obtained by multiplying the reduction kernel by $-1$ to create a dipole that interferes constructively, instead of destructively, with the waves scattered at the TE. We note that noise control is achieved for both streamwise positions of the near-field sensors and for both the IFFC and W-H kernels over the whole frequency range we sought to control. Noise control is more effective for $x/D = 1.7$ and using the W-H kernel. The dependence of the control performance on the sensor position is essentially related to the nature of the pressure disturbances measured by the near-field sensors. Control is more effective for positions where the hydrodynamic component related to the convected wavepacket dominates the sensor reading, that is, for downstream positions in the jet plume of the near-field array. Nevertheless, consistent with the results obtained by Bychkov et al. (2019) and Kopiev et al. (2020), where jet installation noise was successfully controlled using wave cancellation in a feedback scheme, and as suggested by Sipp & Schmid (2016), actuators have a certain effectiveness and control is achieved at $x/D = 0$ even if most of the IFFC kernel information is non-causal due to a certain spatio-temporal coherence of the scattering wavepacket.

The higher effectiveness of the W-H control with respect to the IFFC is apparent for both the streamwise positions of near-field sensors considered. This result is consistent with what was found by Audiffred et al. (2023) for the control of Tollmien-Schlichting waves over a wing profile. The maximum noise reductions and amplifications and the $\Delta BLOASPL$ achieved with both the IFFC and W-H kernels for both the streamwise positions of sensors are visualised in figure 15.
<figure><img src="image_16.png" /><figcaption>Figure 15: Noise reductions and amplifications with respect to the uncontrolled case at the observer position obtained using both IFFC and WH approaches for *M* = 0.2 and streamwise positions of near-field sensors *x*/*D* = 0 and *x*/*D* = 1.7. (a) Band-limited OASPL difference, (b) maximum spectral amplitude difference at a given *St*.</figcaption></figure>

We now explore the effect of jet flow conditions on the control effectiveness. We carried out this analysis only for the streamwise position of the near-field sensors where we obtained the largest noise reductions/amplifications, that is, *x*/*D* = 1.7. Figure 16 shows the PSD at the objective position on the shielded side of the acoustic field for *M* = 0.2, 0.25 and 0.3. Consistent with what has been shown above, we compare the spectra of the uncontrolled case with the results obtained using both the IFFC and W-H and both the amplification and reduction kernels. As outlined above, we kept the actuation frequency range fixed in the band where the actuators were found to be more effective. As a consequence, the Strouhal number range in which control is achieved is different for the *M* explored. Noise control is achieved over all the actuation frequency range and for all jet flow conditions. We note that the control effectiveness is slightly higher at high frequencies, where the actuators showed the largest response (cf. figure 4), and reduces with increasing *M*. The degradation of the control efficiency with increasing *M* is quantified in figure 17, where the maximum noise reductions/amplifications and the ΔBLOASPL with respect to the uncontrolled case are reported. The higher performance of the W-H approach than that of the IFFC is evident for all the jet flow conditions considered herein.

## 5. Conclusions

We presented an experimental study of real-time, linear reactive control of jet installation noise. We considered a setup where a flat plate is installed parallel to the nozzle axis of a subsonic turbulent jet at a Reynolds number of [2.2, 3.4] · 10⁵. The configuration is intended to be a simplified representation of a jet-wing interaction. In this configuration, jet-surface interaction noise is underpinned by the scattering of turbulent wavepackets by the trailing edge of the surface. The control strategy is based on wave cancellation and involves the use of near-field sensors to drive edge-mounted actuators to control the noise radiation. The near-field sensors and the observer in the acoustic field consist of a 6-microphone azimuthal array to educe the *m* = 0 pressure disturbances and a microphone located in the shielded region of the acoustic field, respectively, and the actuators are piezoelectric patches. Following the results obtained by Karban et al. (2023) in a simplified model, the control mechanism amounts to the imposition of an edge dipole driven in phase opposition to the jet-edge-interaction dipole.
<figure><img src="image_17.png" /><figcaption>Figure 16: Mach effect on noise control effectiveness for near-field sensors at streamwise position $x/D = 1.7$: (a) $M = 0.2$, (b) $M = 0.25$, (c) $M = 0.3$. Reduction and amplification kernels are used for both the IFFC and W-H controls: bold black lines refer to the uncontrolled case, blue lines to IFFC, red lines to W-H. Solid lines refer to results obtained using the reduction kernel, dashed lines to the amplification kernel.</figcaption></figure>

<figure><img src="image_18.png" /><figcaption>Figure 17: Noise reductions and amplifications with respect to the uncontrolled case at the observer position obtained using both the IFFC and W-H approaches for $x/D = 1.7$ and $M = 0.2$, 0.25 and 0.3. (a) Band-limited OASPL difference, (b) maximum spectral amplitude difference at a given $St$.</figcaption></figure>

Two control strategies were tested: (i) inverse feed-forward control (Maia et al. 2021, 2022) and (ii) Wiener-Hopf control (Martini et al. 2022; Audiffred et al. 2024b). Both approaches are based on measured transfer functions. Based on the results presented by Mancinelli et al. (2024), where optimal sensor positioning for implementation of closed-loop control of JSI noise was assessed via the investigation of the effectiveness of noise estimation based on near-field sensor readings, we implemented control for two streamwise positions of the near-field sensors, $x/D = 0$ and $x/D = 1.7$. We showed that control efficiency depends on sensor positioning due to the stronger relative presence of the scattered sound field in upstream sensor measurements where the hydrodynamic signature of the convecting coherent structures is
weak. Noise control was successfully achieved using both IFFC and W-H approaches and we showed that the W-H approach generates the largest noise reductions/amplifications. OASPL reductions and amplifications over all the actuation frequency range of up to 3.5 dB and 4 dB, respectively, and a maximum noise reduction/amplification of up to 6 dB/St were achieved at $x/D = 1.7$ using the W-H approach.

Future developments of the present activity will involve an improvement of the actuation technology to test reactive control at higher jet velocities and the exploration of control robustness to jet-flow condition variations. We finally note that for the present academic study we were not interested in assessing the control cost, but rather in addressing the question whether broadband control was possible. The assessment of the control applicability considering the control effort as a constraint will be the objective of future studies.

**Acknowledgements.** The authors acknowledge Redouane Kari for his support in the design of the experimental setup.

**Funding.** M.M. and P.J. acknowledges the support of the EU project DJINN funded by the European Union's Horizon 2020 research and innovation programme under grant agreement No 861438. D. A. acknowledges the financial support from the São Paulo Research Foundation (FAPESP), grants 2022/03279-5 and 2019/26546-6.

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

**Author ORCIDs.** M. Mancinelli, https://orcid.org/0000-0002-7332-4335; D. Audifredd, https://orcid.org/0000-0003-3120-9782; E. Martini, https://orcid.org/0000-0002-3144-5702; P. Jordan, https://orcid.org/0000-0001-8576-5587; A.V.G. Cavalieri, https://orcid.org/0000-0003-4283-0232; A. Lebedev.

## Appendix A. Wiener-Hopf approach

The Wiener-Hopf formalism is adopted for problems where a half-domain condition needs to be satisfied (Noble & Weiss 1959), such as the following half-convolution problem,

$$ \mathbf{b}(t) = \int_{0}^{+\infty} \mathbf{w}_{+}(\tau) \mathbf{s}(t-\tau) \, d\tau, \quad t > 0, \qquad (A 1) $$

where $\mathbf{w}$ is an unknown function and $\mathbf{s}$ and $\mathbf{b}$ are known functions which in the jet installation noise control framework presented herein are related to the sensor and actuation signals, respectively. Extending this equation to $t < 0$ leads to the Wiener-Hopf equation (Noble & Weiss 1959) in the frequency domain:

$$ \mathbf{S}(f) \mathbf{W}_{+}(f) = \mathbf{W}_{-}(f) + \mathbf{B}_{+}(f), \qquad (A 2) $$

where the subscripts + and – indicate that the associated functions are analytical in the upper and lower complex half planes, respectively. This is ensured for $\mathbf{W}_-$ and $\mathbf{W}_+$ given that these functions are zero for positive and negative times, respectively. The solution of the problem described by (A 1) is found by using the multiplicative and additive factorisations (Daniele & Lombardi 2007) defined in the following,

$$ \mathbf{S}(f) = \mathbf{S}_{-}(f) \mathbf{S}_{+}(f), \qquad (A 3a) $$

$$ \mathbf{S}_{-}^{-1}(f) \mathbf{b}_{+} = \left( \mathbf{S}_{-}^{-1}(f) \mathbf{b}_{+}(f) \right)_{+} + \left( \mathbf{S}_{-}^{-1}(f) \mathbf{b}_{+}(f) \right)_{-}, \qquad (A 3b) $$

which lead to the solution (Audifredd *et al.* 2023),
<figure><img src="image_19.png" /><figcaption>Figure 18: Noise spectra on the unshielded side for M = 0.2 and near-field microphone array at x/D = 1.7. Reduction and amplification kernels are used for both IFFC and W-H approaches. Bold black line refers to the uncontrolled case, blue lines to IFFC and red lines to W-H: solid lines refer to results using the reduction kernel, dashed lines to the amplification kernel.</figcaption></figure>

$$ \mathbf{W}_{+}(f) = \mathbf{S}_{+}^{-1}(f) \left( \mathbf{S}_{-}^{-1}(f) \mathbf{B}(f) \right)_{+} \quad (A 4a) $$

$$ \mathbf{W}_{-}(f) = -\mathbf{S}_{-}(f) \left( \mathbf{S}_{-}^{-1}(f) \mathbf{B}(f) \right)_{-} \quad (A 4b) $$

## Appendix B. Control results on the unshielded side

We here explore jet installation noise control on the unshielded side. We carried out this analysis only for the streamwise position of the near-field sensors where we obtained the largest noise reductions/amplifications, that is, $x/D = 1.7$. Figure 18 shows uncontrolled and controlled noise spectra for $M = 0.2$ on the unshielded side using both IFFC and W-H approaches. Control results are consistent with those found on the shielded side, the W-H approach being more effective than IFFC. We note that control effectiveness is lower than that achieved on the shielded side. As mentioned in §3, this behaviour is likely due to the influence of direct jet noise, whose physical mechanism is different from JSI noise and cannot be controlled with the application developed herein, and flow-sound interactions, such as refraction and scattering by turbulence. The comparison of control authority between the shielded and unshielded sides is quantified in figure 19, where we show the maximum noise reductions and amplifications and the $\Delta BLOASPL$ with respect to the uncontrolled case achieved in both regions of the acoustic field. The shielded and unshielded sides are identified via the polar angle $\psi = 90^{\circ}$ and $\psi = 270^{\circ}$, respectively. Control effectiveness is reduced of 40% to 50% on the unshielded side with respect to the shielded side.

The effect of jet flow conditions on control effectiveness on the unshielded side is explored in figure 20, where we show the maximum noise reductions/amplifications and the $\Delta BLOASPL$ with respect to the uncontrolled case for all $M$ considered herein. The degradation of control efficiency with increasing $M$ is consistent with what observed on the shielded side (cfr. figure 17).
<figure><img src="image_20.png" /></figure>

Figure 19: Comparison between noise reductions and amplifications with respect to the uncontrolled case achieved on the shielded and unshielded sides ($\psi = 90^\circ$ and $270^\circ$) using both IFFC and W-H for $M = 0.2$ and near-field sensors at $x/D = 1.7$. (a) Band-limited OASPL difference, (b) maximum spectral amplitude difference at a given $St$.

<figure><img src="image_21.png" /></figure>

Figure 20: Noise reductions and amplifications with respect to the uncontrolled case on the unshielded side as a function of $M$ using both IFFC and W-H approaches for near-field sensors at $x/D = 1.7$. (a) Band-limited OASPL difference, (b) maximum spectral amplitude difference at a given $St$.

We note that the control results on the unshielded side shown above are obtained using the microphone on the shielded side as observer to calculate the control kernel. To explore whether the choice of the microphone used to calculate the kernel might affect the control effectiveness, we consider the off-line control and compare the noise reductions achieved on the unshielded side using both the microphones on the shielded and unshielded sides to estimate the kernel. The results are shown in figure 21 as a function of the penalisation factor $R$ and reveal that the choice of the observer for kernel estimation either on the shielded or unshielded side does not affect control authority.

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