Noise Reduction Using Synthetic Microjet Excitation in Supersonic Rectangular Jets

Abstract

This work explores a potential methodology for rectangular jet noise reduction that employs nozzle unsteady microjet excitation. Using high-fidelity computational studies and spectral analyses, major jet noise sources impacted by the applied actuation are identified. A heated supersonic rectangular jet is considered with a nozzle aspect ratio of 2:1 at a Mach number of 1.5. The current study essentially validates the hypothesis of a previous reduced-order analysis that predicted jet noise reduction through jet excitation at the harmonic or subharmonic of the dominant frequency associated with jets’ large-scale structures. Such noise reduction was attributed to the excitation-induced nonlinear energy exchange between the coherent modes. In the current study, the synthetic microjet actuation of the jet plume shear layer using $1\%$ of the jet mass flow rate is implemented at the excitation ports located at the nozzle lip and directed along the jet axis. A resulting jet noise reduction of up to 4 dB at the peak radiation angle is predicted. An analysis of the near-field Spectral Proper Orthogonal Decomposition (SPOD) results provides further insights into the impact of jet actuation on the modification of jet flow structures, thus addressing the effectiveness of the proposed noise control methodology.

1. Introduction

Aircraft noise is an important limiter for commercial and military operations of aircraft. The FAA has been imposing stricter rules on noise affecting the community over the last few decades. For the military, noise provides an early warning while jets are flying in enemy territory but also creates an unhealthy environment for the ground crew working near fighter aircraft, e.g., on the decks of an aircraft carrier. Among the different noise components, jet noise is the loudest during take-off [1]. Rectangular nozzle shapes are particularly relevant, primarily in military aircraft applications. The current analysis focuses on the rectangular nozzle geometry first studied in [2,3]. The primary objective is to investigate how excitations imposed on the jet flow will affect the turbulent jet structures and how their alteration correlates with changes in the far-field acoustic spectrum. Such insights will pave the way for developing efficient closed-loop noise control strategies in future research.

1.1. Background

Jet engine noise arises primarily from the turbulent mixing of high-speed exhaust gases with the surrounding atmosphere within the jet shear layer, encompassing a spectrum of turbulent structures, from small, high-frequency-noise-generating ones to large, low-frequency noise propagating structures downstream. Concerns about jet noise arose with the advent of jet engines in the 1940s, with significant advancements in the 1950s when Sir James Lighthill introduced the Acoustic Analogy by transforming the Navier–Stokes Equations into an acoustic wave equation [4,5]. Such an analysis revealed the scaling laws for acoustic power. Large coherent turbulent structures in jet shear layers, emerging from nozzle exit instabilities, dominate the noise generation mechanism. These structures evolve downstream. In axisymmetric laminar jets, a shift from ring vortices to 3D turbulent formations downstream is evident, with a peak noise frequency around $S{t}_D=0.3$ [6]. A noise source model based on fine and large-scale turbulent structures has received validation from multiple studies [7]. In supersonic jets, distinct shock cell structures interact with these turbulent structures, producing broadband shock-associated noise (BBSAN). Earlier research in the 1970s indicated shock-expansion wave reflections as acoustic sources [8], with subsequent studies highlighting a spectrum of disturbances linked to turbulent scales [9]. Predicting screech noise amplitude, influenced by various factors, remains elusive, despite new insights into the wave modes in supersonic jets [10,11].

Stringent community noise regulations have catalyzed innovative jet noise suppression methods without impeding aircraft performance [12]. These techniques can be broadly classified into passive or active control measures. Passive strategies, including bypass flow alterations, the use of chevron nozzles, and nozzle corrugations, introduce design changes to modify airflow and influence far-field acoustics. Active methods, such as fluid injection, plasma actuation, and microjet excitation, utilize feedback control systems for dynamic and optimized noise reduction [13,14,15,16,17]. An advantage of active techniques is their ability to turn them off when they are not needed. Analyzing sound sources requires capturing unsteady fluctuations in the flow field. Direct Numerical Simulation (DNS) is a high-fidelity approach to simulate coupled aerodynamic and acoustic effects in “virtual laboratories”. DNS does not require subgrid-scale models or filtering, as it resolves all scales of turbulence directly. However, the computational demands of DNS are exceptionally high, making it impractical for high-speed and complex flows. Large Eddy Simulation (LES) offers an alternative that captures large scales directly while modeling small ones, thus balancing computational cost and accuracy. This approach has become the method of choice for jet aeroacoustic calculations [18]. Shortly after the first LES-based simulation of supersonic jet noise was reported [19], this technique was utilized to study noise generation and propagation. Hybrid methods, such as Detached-Eddy Simulation (DES) and Delayed Detached-Eddy Simulation (DDES), refine LES by allowing more flexibility in simulating flow regions with different turbulence characteristics [20]. Finally, the Reynolds-Averaged Navier–Stokes (RANS) equations, which yield time-averaged solutions, provide a lower-fidelity alternative that, while limited in capturing far-field pressure signals, can be coupled with empirical models to predict noise.

After the jet noise sources are predicted using LES-based techniques, several approaches can be used to extend the nonlinear CFD results to the acoustic far field [21]. One idea is the evaluation of Lighthill’s stress tensors using a volume integral, but this is memory-intensive. The Kirchhoff method [22] was the first method to allow the use of surface integrals and is much more efficient, and it has been applied to the evaluation of jet noise [23]. The classical formulation for the Kirchhoff method requires pressure and its time and normal derivatives. Other Kirchhoff-like methods have been proposed to avoid evaluation of the pressure normal derivatives, such as SIF (Surface-Integral Formulation) in the special case of cylindrical surfaces [24].

Lighthill’s work was also expanded to consider moving solid boundaries, resulting in the Ffowcs Williams–Hawkings (FWH) formulation covering areas such as rotor noise and impinging jet noise [25]. The FWH method requires pressure, velocity, density, and its time derivatives to be collected on a control surface that encloses most of the noise sources. At first, the formulation was proposed for an impermeable control surface coinciding with a radiating solid surface. The method was later extended to include a permeable control surface formulation which improved its versatility and is now the prevailing method to extend near-field CFD results to the acoustic far-field. The placement of the control surface for unsteady data collection and grid refinement play a vital role in the accuracy of acoustic prediction in these methods, and generally requires a parametric convergence study.

1.2. Purpose

Comparative analyses of numerical studies on noise control in rectangular jets reveal distinct approaches and outcomes achieved using various excitation methods. Previous investigations employed techniques such as passive control with flat and wavy plates [26], bi-modal excitation [27], plasma actuation [28], and periodic excitation [29]. Passive control focused on modifying the interaction between the jet and adjacent surfaces to reduce noise, achieving directionally dependent reductions due to shielding effects. However, this approach was constrained by reflections and induced fluctuations, resulting in limited broadband noise mitigation. Bi-modal excitation employed a reduced-order model to identify a frequency that redistributes energy from the dominant frequency to the rest of the spectrum. This method achieved significant noise reductions, with up to a 13 dB reduction at specific far-field angles. While effective in targeting particular modes, it showed limited capability for broadband noise control. Plasma actuation, tested at high Strouhal numbers around 1 and varying duty cycles, achieved its best performance with a 50% duty cycle, leading to approximately 1.5 dB noise reduction. This technique redistributed turbulent kinetic energy, suppressed large coherent structures, and enhanced streamwise vortical elements for localized noise mitigation, although actuator tones were prominent along the sideline direction. Similarly, the use of chevrons [30] and acoustic shielding [26] has elucidated unique features in rectangular jet flow compared to circular jets, attributed to the specific characteristics of the rectangular nozzle-induced free shear flows.

The present study introduces an active noise control method using synthetic microjet actuation to excite the jet shear layer, tuned to a specific frequency mode identified for its maximal influence on large-scale structures and energy transfer. The time-harmonic microjet velocity injection alters the jet flow structures and far-field acoustic signature. Previous numerical studies proposed various excitation methods and configurations for unsteady jet excitation. For instance, plasma actuation [31] effectively excited the jet at a very high frequency with temperature variation and demonstrated successful jet noise reduction at a Strouhal number of $S{t}_D=1$ [28]. A key aspect of this approach is that the excitation affects the jet’s shear layer, requiring the excitation port to be located inside the nozzle, with the excitation wavelength comparable to the size of the boundary layer. Another study [27] employed time-harmonic pressure excitation at the nozzle lip with an amplitude of around 0.4% of the jet exit mean pressure. Similarly to the current study, this noise control approach followed a theoretical analysis [32], showing that nonlinear energy transfer occurs from the most energetic (primary) mode to other modes when the jet is excited with the harmonic of the primary mode. In this case, the excitation frequency at around $S{t}_D=0.3$ was approximately double that of the far-field acoustic peak, targeting the jet’s large-scale structure development.

The present study revisits previous numerical simulations reported in [33] that utilized microjet-based jet shear layer excitation with the harmonic and subharmonic of the near-field pressure peak frequency. In contrast, a finer grid is employed in this study to enhance the accuracy of the results, enabling a more detailed exploration of the jet flow dynamics. The microjet-based excitation is implemented at the nozzle lip where the jet shear layer is most energetic. One of the objectives of this study is to compare the noise control results with those in [27], based on jet pressure excitation, while utilizing the same excitation frequency selection strategy proposed in the theoretical analysis of [32].

2. Methodology

The current research investigates a jet noise control approach that focuses on exciting the frequency modes of coherent structures through unsteady fluidic injection directed downstream of the nozzle lip. Such excitation of the jet flow affects the instability modes near the nozzle lip. As those propagate downstream, they evolve into large turbulent structures that appear as the major sources of radiated noise.

This section presents the numerical setup for 3D high-fidelity simulations of a rectangular jet flow subject to unsteady microjet excitation localized at the longer nozzle lip, as it was found to be the region of greater turbulent kinetic energy amplitude. A time-periodic velocity injection in the downstream direction is considered. This study focuses on a heated, supersonic, fully expanded Mach 1.5 case and compares the baseline scenario without excitation to an excited case at a forcing frequency of $S{t}_F=0.30$ but with a constant excitation velocity amplitude. The geometry and the flow condition match with a previous experiment [2,3]. A hybrid LES-URANS approach is employed to model the flow, utilizing URANS near wall boundaries and LES in the regions of free shear flow. The analysis involves a comparison of the far-field noise and near-field parameters between the excited and non-excited cases, thus examining the effectiveness of the active noise control methodology, which was previously also explored for a rectangular jet in [33] using a coarse-grid analysis, and in [34] for a circular jet. The proposed computational setup builds on prior work involving numerical simulations of rectangular jet acoustics shielded by an airframe, which was successfully validated against experimental measurements [26].

2.1. Problem Statement

The simulations utilized a computational domain encompassing the nozzle and the surrounding fluid region. The geometry consisted of a cross-section with exit dimensions of 12.95 mm× 25.91 mm (aspect ratio of 2), an exit Mach number of $1.5$, a nozzle pressure ratio (NPR) of $3.67$, and a nozzle temperature ratio (NTR) of 3, as reported in the experimental paper [2,3]. The ratio of the throat-exit area was $1.18$ in the plane parallel to the minor axis, while the nozzle walls in the major axis plane were straight from the inlet to the nozzle exit. The Reynolds number for the experiment was $Re\approx 277,500$ for air. The jet flow was expected to be fully expanded when operating under these conditions.

This rectangular jet nozzle has been extensively studied experimentally [2,3], making it a suitable reference for comparison in this study. Also, a previous study utilizing the same computational setup performed a similar simulation [34].

In the following discussion, the simulation results will be presented and discussed for a baseline case, as was carried out in our previous work [33]. Figure 1 shows the rectangular jet nozzle, showing its key characteristics and highlighting its dimensions. Furthermore, this computational investigation considers the nozzle’s shorter lip length (D) to be the characteristic length for determining the Strouhal Number ($St$). This facilitates further comparison with geometries at different aspect ratios, with D kept constant. Note that if the nozzle’s effective (hydraulic) diameter ($D_{\mathrm{eff}}=0.020665$) is considered for the definition of $St$, then the values of $St$ should be multiplied by $1.5957$.

2.2. Computational Setup

The computational setup is similar to that employed in our previous numerical simulations [33]. The previous computations are now repeated on a finer grid to achieve more accurate results. A density-based compressible OpenFOAM v1912 solver [35] using the total-variation-diminishing (TVD) scheme is employed to simulate the flow field of a supersonic, fully expanded heated jet. The current computational study adopts the $k-\omega$ Shear Stress Transport (SST) turbulence model to conduct DES. This reduces the computational costs compared to the full LES, which would require extensive near-wall treatment. In the current simulations, a statistically steady-state solution is first achieved with the $k-\omega$ SST RANS model. Then, DES simulations are marched in time using the second-order backward time scheme and the second-order central upwind scheme in space, with the RANS results applied as an initial solution. The initial RANS solution is obtained using the OpenFOAM pseudo-transient first-order accurate time marching scheme “localEuler” based on the Euler implicit time marching scheme, with the cell-based time step set by the Local Time Stepping (LTS) solver.

To analyze the simulation data, 3072 samples are collected with a sampling rate of 200 kHz ($dt=5\times {10}^{-6}$ s). The minimum resolved frequency from Equation (1) is calculated based on the maximum simulation time of approximately 593 “flow times”, with the “flow time” $\tau$ defined as the physical time t non-dimensionalized by the jet exit velocity ($U_j$) and the characteristic length scale D, so that $\tau =t{\displaystyle \frac{U_j}{D}}$. This translates to a simulation physical time of $0.01024$ s. Considering the utilization of the second-order backward time scheme, a resolution of approximately $N=12$ points per cycle should suffice, ensuring that the minimum resolvable frequency corresponds to $St=0.0135$. If the signal is divided into windowed blocks, the minimum resolvable frequency increases as the number of windows increases because the time duration of each window shortens. It is desired to utilize three windows of 1024 samples; thus, the minimum resolvable frequency corresponds to $St=0.0404$.

$$S{t}_{min}={\displaystyle \frac{ND}{\tau U_j}}$$

(1)

The inlet conditions are prescribed for the desired $NPR$ and $NTR$. The advective far-field condition is imposed at the rest of the domain boundaries. This boundary condition, also called “waveTransmissive”, is implemented using the Navier–Stokes Characteristic Boundary Condition (NSCBC) [36]. The non-reflective boundary condition is based on the characteristic wave relationship derived from the Euler equations and reformulated in the orthogonal coordinate system such that one of the coordinates is normal to the boundary [36]. Finally, the nozzle inner walls are prescribed as an adiabatic no-slip condition so that RANS simulations near the wall can resolve the boundary layer with a specified $y^{+}$ value.

It is important to note that the mean-flow velocity at the exit is $U_j=750$ m/s and the characteristic length $D=0.01295$ m. The Reynolds number is defined by Equation (2) and matches the one from the experiments [2,3].

$$Re={\displaystyle \frac{\rho U_{j}D}{\mu }}=1.73\times {10}^5$$

(2)

The current computational study utilizes a coarse and a fine grid of 43 and 112 million cells, respectively. These are composed predominantly of hexahedral cells. The main reason for this is that it offers superior accuracy when solving acoustic waves that travel away from the sources. By using hexahedral cells, one can ensure that the code does not lose accuracy and provides precise results even when dealing with complex acoustic wave phenomena. Additionally, it offers better mesh quality and a higher aspect ratio, which results in a better representation of the geometry and fluid flow physics. Therefore, it is essential for obtaining accurate and reliable simulation results.

The core of the computational domain extends up to $64D$ downstream, $3D$ upstream, and $20D$ radially. Surrounding this core, a buffer zone is implemented to dissipate fluctuations before they reach the domain boundaries, extending up to $125D$ downstream, $16D$ upstream, and $40D$ radially. The grid spacing close to the nozzle walls is selected to achieve a $y^{+}$ value of 50 on both coarse and fine grids. The recommended values from the literature range between 30 and 100. This is because the wall models for $\omega$ in the $k-\omega$ SST model do not perform well when they are either too close to or too distant from the wall [37]. The value for $y^{+}$ is calculated based on flow results from the DES simulation itself.

Figure 2 offers a general view of the fine mesh structure, while Figure 3 zooms in on the nozzle exit area. The cell size varies the grid spacing from $D/65$ at the nozzle wall to $D/25$ at $32D$ away along the jet axis to resolve the jet potential core and turbulent mixing, thus utilizing a growth ratio of 1.01. It then gradually increases to $D/6.475$ to capture the jet decay and growth of large-scale structures until it reaches a distance of $64D$ downstream. The computational domain’s outer region extends from the nozzle exit to $12D$ in the radial direction and to $64D$ downstream, and maintains a maximum grid spacing of $\Delta =D/6.475$. This grid has a buffer zone from $64D$ to $125D$ downstream of the nozzle. This is necessary to dissipate the fluctuations before they reach the outflow boundaries, and thus ensures that fewer numerical reflections bounce back into the domain to adversely affect the predicted results.

The FWH method is used to capture the far-field acoustics [21]. The external surface of the outer region serves as the FWH surface to capture the flow information for acoustic calculations (see Figure 4). The maximum grid spacing ensures a grid cut-off frequency with the consideration of $N=20$ grid points per wave chosen based on the second-order central upwind scheme adopted in this simulation. The cut-off frequency is estimated using the line-of-sight (LOS) method [38]. Instead of temporal sampling, the maximum resolved frequency is limited by the local grid spacing on the acoustic data surfaces for the FWH method. The LOS method assumes that the noise source is at the end of the potential core, and the grid cut-off is determined by the largest grid spacing where the LOS intersects the FWH surface. The grid spacings in every direction are considered. The cut-off frequency is calculated using Equation (3), where $\Delta$ is the maximum value between ${\Delta }_1$, ${\Delta }_2$ and ${\Delta }_3$ defined in Equation (4):

$$S{t}_{max}={\displaystyle \frac{1}{N\Delta M_d}}\sqrt{{\displaystyle \frac{T_{\infty }}{T_j}}}$$

(3)

$${\Delta }_1={\displaystyle \frac{\Delta x}{D}}cos\varphi {\Delta }_2={\displaystyle \frac{\Delta y}{D}}cos\varphi {\Delta }_3={\displaystyle \frac{\Delta z}{D}}sin\varphi ,$$

(4)

where $M_d$ is the jet exit Mach number, $T_j$ is the jet exit static temperature, and $T_{\infty }$ is the freestream temperature. The plane wave propagation angle ($\varphi$) is between the jet axis and the observer. The worst-case scenario occurs when the observer is positioned at either $\varphi =0^{\circ }$ or $\varphi ={90}^{\circ }$, where the grid cut-off frequency is $S{t}_{\max }=0.30$. In contrast, the cut-off frequency reaches its peak value of $S{t}_{\max }=0.42$ when the observer is at $\varphi ={45}^{\circ }$.

The end cap of the FWH surface might create spurious signals as strong eddies pass through it [39]. For reducing spurious low-frequency noise generated by vorticity traveling near the caps of the FWH surface, it is recommended to carefully configure both the extent and number of end caps used for the permeable surface. A recent study by Ribeiro et al. [40] has shown that the distance between the first and last end cap, or the “end cap region”, should be sufficiently large to allow for the substantial decorrelation of coherent flow structures as they pass through the caps. Generally, maintaining an end cap region size that is about 5% to 10% of the overall permeable surface length in the primary flow direction proves effective [40]. Within this region, using between 5 to 8 end caps for averaging is optimal, helping to smooth out the noise signal without excessive computational cost. The study also showed that the spacing between these end caps has a minimal impact on the effectiveness of noise reduction, as doubling the cap spacing does not significantly change the results. However, it is essential to recognize that strong streamwise vortices may still affect noise levels even with a large end-cap region.

In this investigation, five end caps are placed at equal distances from each other, from $60D$ to $64D$. Their noise contribution is averaged at each observer to eliminate the low-frequency spurious noise compared to the approach with no end caps. The averaging method reduces the artificial non-zero contribution [39]. The placement of the FW-H surface must be studied further because this surface should be properly placed to account for all (or most) noise sources, avoiding regions with coarse mesh where the solution is not properly resolved.

The simulations are run using 896 processors, each operating at 2.3 GHz. Considering a sampling frequency of 200 kHz, 2048 samples would require 120,000 and 180,000 processor-hours for the cases with 43 and 112 million cells, respectively. The small difference between both cases compared to the number of grid cells is due to the CFL number being limited by the ratio between cell length and flow velocity. The finer grid has more points because its growth ratio from the nozzle exit is smaller, which improves the LES simulation accuracy. Still, the cells become larger as they approach the jet centerline, which improves the simulation speed.

Post-processing involves sampling surfaces to collect flow data at every cell in the mid-span cuts in both the X-Y and X-Z planes and on the FWH surface. Once gathered, these data are processed separately using an in-house code, where they are subjected to either a Spectral Proper Orthogonal Decomposition (SPOD) or permeable-surface FW-H integration using the Farassat 1A formulation [41]. The obtained time histories of the unsteady pressure fluctuations at the far-field observer locations are transformed into frequency spectra by applying the Fast Fourier Transform (FFT) method [42]. Upon gathering the FWH surface data, the results at the far-field observer points can be analyzed as required. The nomenclature for the coordinates of these observer locations can be discerned by referring the diagram illustrated in Figure 5.

For both data processing tools, a single-period sample is divided into three segments, each containing 1024 samples with a Hanning window with 50% overlap. FFT is then applied to each segment, and the resulting spectra are averaged. This procedure may help avoid significant ‘spectral leakage’ and reduce the amplitude of the discontinuities at the boundaries of each finite segment. Although using more window blocks reduces the frequency resolution, this method smooths out the signal and filters it without causing phase distortion, thereby providing a reasonable spectrum estimate.

In this research, SPOD enables some valuable insights into the fundamental physical processes of noise generation and improves our understanding of energy transfer between structures of different frequencies. Before carrying out a 2-dimensional SPOD, the DES data must be interpolated onto a structured Cartesian grid of equally stretched cells. This approach does not raise concerns regarding the accuracy of the analysis, as the main emphasis lies in decomposing the coherent structures, which are considerably larger than the maximum cell length of the interpolation grid.

2.3. Excitation Method

A configuration for active jet noise control is proposed that involves the placement of multiple microjet actuators distributed around the nozzle lip along the major axis. In the numerical setup, this configuration represents a hollow rectangle segment at the nozzle exit, encompassing the nozzle lip surface. The primary concept involves exciting the flow instabilities at a specific frequency by introducing unheated air along the lip line in the downstream direction. Therefore, the approach would be to directly inject the unsteady flow in the shear layer, which is situated downstream from the nozzle lip (see Figure 6). The thickness of the lip measures 0.5 mm, and the injection zone corresponds to the major axis of the nozzle lip, excluding the edges. Hence, the area ratio between the microjet actuator and the nozzle exit obtained using the original geometry is $A_t/A_e=0.07722$. The injection velocity is adjusted to achieve the desired amplitude of the sinusoidal microjet velocity profile. The peak injection mass flow rate is determined with an injection-to-jet ratio of $1\%$ ($m_i/m_j=0.01$); this was previously selected for a study on active noise control in axisymmetric jets [43], and provided successful outcomes. Consequently, it serves as a starting point for this current work.

When analyzing the pressure spectra at the near nozzle lip in the minor and major planes, the spectra in the minor plane of the jet show considerably more fluctuations than those in the major plane, which show very few fluctuations (see Figure 7). This is in accordance with what has been previously observed [27]. Differences in the pressure fluctuations between planes are not entirely unexpected because the expansion in the nozzle occurs in the minor plane, whereas the nozzle contour in the major plane is flat. For this reason, excitation occurs only along the major axis. Additionally, the peak at around $S{t}_D=0.15$ already provides evidence that the near-field and SPOD analysis might give the Strouhal number range to focus on when attempting to manipulate coherent structures to reduce noise.

Various excitation frequencies corresponding to different Strouhal numbers can be examined. However, in the current work, the excitation frequencies are selected based on the methodology reported in reference [27], where the authors utilize a reduced-order model that indicates the potential frequencies that would suppress the growth of specific turbulence scales responsible for the most prominent part in the jet noise spectrum. Previously, a Linearized Euler Equation solver (LEE) was used to evaluate this strategy [32], followed by high-fidelity simulations [27], concluding that the harmonic of the near-field peak frequency should be considered. The examination of the baseline case determined that the peak frequency in the near field was at $S{t}_D=0.15$. Thus, it was concluded that the noise sources could be minimized by introducing an excitation at its harmonic, $S{t}_F=0.30$. In the current study, a careful analysis of the baseline case guides the selection of frequencies that may influence the turbulence structures and the acoustic characteristics of the jet.

3. Results

The preliminary coarse-grid simulations reported in [44] provided the first insights into jet flow-field characteristics. The current study concentrates on revisiting the instantaneous contour plots and SPOD results obtained using fine-grid simulations. Following this, an analysis of the near-field and far-field noise spectra, along with acoustic directivity plots, is conducted to determine the effectiveness of the unsteady microjet excitation noise control method.

3.1. Time-Averaged Flow

Figure 8 presents a comparison of the non-dimensional time-averaged velocity at the center line and upper lip line for the baseline case against previous LES simulations under the same operating conditions as is Viswanath et al. [45] using the jet noise-reduction (JENRE) code developed at the U.S. Naval Research Laboratory. The LES grid utilized 65 million points. The results show agreement at the center line up to $15D$ from the nozzle exit (see Figure 8a). However, the fine grid improves the prediction when considering amplitude at the lip line (see Figure 8b,c).

As shown in Figure 8a, the current simulation predicts a potential core length of $9.5D$ for the rectangular jet. Very weak diamond shock cells are visible through the potential core in the center line plot, and Figure 8b reveals that remnants of the throat shock are visible in the minor axis plane lip-line plot. However, the major plane, with its straight walls, displays only the lip shock. The near-field shock cell locations, size, and strength are similar for the rectangular planes, and Figure 9 serves as a complementary comparison to the previous analysis as it provides contour plots of the time-averaged streamwise velocity at different stations. This figure displays how the jet plume develops further downstream, changing its transverse section shape from rectangular to near-circular. The excitation increases the jet spread along the minor and major axes, resulting in cross-shaped sections.

Figure 10 compares the jet decay, a ratio between the jet exit and centerline velocities, and the jet half-width, a ratio between the jet exit length and the length where the jet plume reaches half of the centerline velocity at the specific station. It is evident that the excitation shortens the jet potential core length and increases the momentum thickness further downstream. Also, the jet spread is higher for the excited case. The jet decay lines are also a good indicator of whether the simulation reaches convergence to initiate acoustic data collection. These lines must be as straight as possible from the end of the jet potential core to the end of the grid or until the buffer zone is reached. A commonly used term to describe this condition is the flow-through time. It represents the average time taken for the flow to travel from the nozzle exit to the end of the computational domain, and can be determined using the grid spacing along the jet axis and the mean flow velocity at the centerline. The flow-through time is calculated as $T_{FT}=0.0048$ s based on the given jet decay. Therefore, the DES simulation of the jet should run for approximately 10 flow-through times before analyzing the acoustic data or time-averaged quantities. In this simulation, 8 flow-through times are completed before the acoustic data collection is conducted.

Figure 11 displays the contours of the time-averaged turbulence kinetic energy (TKE) normalized by the jet exit velocity for every case. This quantity is calculated from the first Reynolds stress tensor component $u_{\mathrm{xx}}^{\prime 2}$. The square root of this component is known as RMS velocity fluctuation, and isnormalized by the jet exit velocity $U_j$. The excitation modifies the development of the jet plume by increasing the turbulent mixing. The TKE peak occurs at the lip line, close to the nozzle exit.

3.2. Instantaneous Flow

Although the time-averaged flow data might show some trends and indications of turbulence amplification or reduction, it is important to take a closer look at the instantaneous flow field to understand how the noise sources are interacting and generating the acoustic waves that travel to the far-field. Snapshots of the velocity gradient overlayed by the contours of vorticity, together with divergence of the normalized velocity field, are provided in Figure 12.

Overall, the acoustic waves appear to propagate in the preferred directivity angle. Some waves are traveling upwards and upstream, and these might be related to shock-associated noise. From the comparison, it is apparent that the radiation pattern is affected by the unsteady excitation. Somewhat higher upstream noise radiation appears as a result of the applied active noise control.

Based on the observation that the waves that propagate in the preferred direction for the $S{t}_F=0.30$ case are weaker than those in the baseline case, we may infer that the excitation does reduce the far-field noise. We can also infer that the peak acoustic frequency is being directly affected by the excitation as the wavelengths seem wider and better organized than the ones in the baseline case, which means that a distinct, lower-frequency tone is dominant. The enhanced jet spread seems to affect the preferred radiation angle by slightly pushing it away from the jet axis.

3.3. SPOD Modes

The 2D SPOD analysis focusing on pressure fluctuations is performed over the minor- and major-plane mid-span cut. Figure 13 and Figure 14 show the mode energy spectra for each mode, with the first mode being the most energetic, encompassing around 80% to 90% of the cumulative energy of all modes [46]. Interestingly, the rectangular minor plane exhibits a prominent peak close to $S{t}_D=0.17$. This agrees with LES simulations of [27] that found a peak frequency at $S{t}_D=0.15$. It is important to recognize that [27] employed a higher-order code and a much finer mesh when compared to the current work. Since the baseline case near-field pressure peaks at around $S{t}_D=0.15$, we choose to excite the jet at the harmonic of the peak frequency, $S{t}_F=0.3$.

The mode energy spectral analysis helps identify key frequencies for further investigation, including those associated with spectrum peaks, excitation, and harmonics. Figure 15 illustrates a comparison of the first SPOD mode of fluctuating pressure at $S{t}_D=0.1$, $0.15$, $0.2$, and $0.3$. Each mode is dimensionalized by scaling the first mode with its corresponding expansion coefficient (A) at the respective frequency, i.e., $A_1\left(f\right)$.

This expansion coefficient is often utilized to reconstruct the SPOD data into a time series of the pressure field, which means that the product of SPOD modes and these coefficients for each frequency provide comparisons between modes and frequencies resulting from different simulations [47]. The contour plot of the first pressure SPOD mode reconstructed at different frequencies, covering from the nozzle exit to $30D$ downstream and $10D$ radially, suggests that the $S{t}_F=0.30$ case suppresses the compact wavepacket from the Kelvin–Helmholtz (KH) shear-layer instability at $S{t}_D=0.17$ (see Figure 15). This is evidenced by the $S{t}_D=0.17$ frequency mode in the baseline case, where alternating structures display an organized pattern, and the contour strength is higher in the jet potential core than further downstream. In contrast, the same frequency mode in the excited case appears chaotic, with the downstream contour strength matching that in the jet potential core. Additionally, for the excited case, the $S{t}_D=0.30$ mode is symmetric, as expected due to the excitation method, and displays a much stronger contour. The excitation breaks down large-scale structures at $S{t}_D=0.17$ and establishes a strong axisymmetric noise source at the excitation frequency, which predominantly propagates in the preferred direction.

It seems that the excitation reduced the energy from other frequencies as well, which might impact the broadband noise. However, we might have peaks in the excitation frequency and its harmonics. In summary, the $S{t}_F=0.30$ case reduces the noise in the far field through the manipulation of coherent structures in the near field.

3.4. Far-Field Acoustics

The far-field acoustics were calculated using the FWH methodology described in Section 2.2. The SPL spectrum and OASPL directivity results were compared against experimental data collected at 40 equivalent diameters ($D_{\mathrm{eff}}=0.020665$) from the nozzle exit. These comparisons aimed to validate the numerical simulations and assess the impact of excitation on the far-field noise characteristics.

The SPL spectra for baseline and excited cases at multiple observer angles are shown in Figure 16. While experimental data were only available at $\theta =0^{\circ }$ and $\theta ={90}^{\circ }$ [2], analyzing the $\theta ={45}^{\circ }$ case provides additional insight into how excitation alters noise radiation along diagonal planes. This is particularly important as the flow-field results indicate that enhanced mixing primarily occurs along the minor and major planes, with reduced effects in diagonal directions (refer to Figure 9). These comparisons reveal that excitation introduces a distinct tonal component in the far-field noise spectrum while reducing broadband noise across the spectrum. These findings highlight how turbulence intensity and coherent structures are modified by excitation, as observed in the coarse-mesh simulations reported previously [48].

The SPL spectrum comparisons also serve to validate the numerical setup. The fine-grid results demonstrate improved agreement with the experimental measurements, addressing discrepancies observed in prior coarse-grid simulations. Proper grid stretching in the fine-grid setup likely contributed to resolving these differences.

The OASPL results, shown in Figure 17, compare the baseline and excited cases at three azimuthal angles. This analysis underscores the overall noise reduction achieved through excitation. For example, at $\theta =0^{\circ }$, the $S{t}_F=0.30$ case reduces the peak OASPL by up to 4 dB compared to the baseline, primarily due to broadband noise reduction. While the excitation introduces a tonal peak, the overall reduction in broadband noise dominates the far-field noise characteristics.

The comparisons in Figure 16 and Figure 17 are significant because they provide a comprehensive understanding of how excitation influences noise radiation patterns across a range of observer angles. The SPL spectra highlight the interaction between excitation frequencies and coherent structures, while the OASPL comparisons offer a clear picture of the overall noise reduction. Together, these analyses demonstrate the efficacy of the excitation strategy and validate the simulation methodology against experimental benchmarks. Any discrepancies, such as those observed at upstream angles, may stem from limitations in grid resolution or the placement of the FWH surface. However, the focus on peak noise radiation angles ensures that the findings are meaningful for practical applications.

Unlike previous methods relying on pressure fluctuations or plasma-based actuation, this approach employs velocity-based synthetic jets, offering a novel mechanism to influence turbulence dynamics and acoustic fields. While building on the reduced-order modeling methodology utilized in [27], this work adopted a greater energy amplitude for the excitation. The unsteady microjet excitation achieved up to a 4 dB reduction in OASPL, attributed to the suppression of Kelvin–Helmholtz instabilities and decreased energy of coherent structures across the spectrum. This technique demonstrated effectiveness in both broadband noise reduction and modifying acoustic directivity, presenting a balanced solution for near- and far-field noise control. While all of the mentioned methods contribute to noise reduction in rectangular jets, the current work distinguishes itself by achieving broadband noise mitigation through tailored unsteady excitation. In contrast to passive methods [26], which are limited to geometrical shielding, and frequency-specific active methods [27,28], the excitation in this study directly targets the jet plume shear layer, offering a broader spectrum of noise reduction and enhanced practical applicability.

4. Conclusions

Rectangular jet nozzles, noted for their potential advantages over traditional round nozzles, i.e., reduced drag, simpler design, and improved exhaust mixing, are becoming increasingly relevant for both civilian and military applications [33]. In the current work, a high-fidelity investigation of supersonic rectangular jet flow and noise was conducted. The impact of the synthetic microjet actuation exciting the jet shear layer with the frequency mode that appeared to have the maximum effect on large-scale structure evolution and nonlinear energy transfer (as suggested in the previous study [27]) was examined.

The flow and acoustic characteristics of the rectangular jet were analyzed, focusing on the effects of unsteady microjet excitation on both near-field and far-field noise. The time-averaged flow results reveal that the applied excitation altered the jet spread and the potential core while enhancing the jet turbulence kinetic energy, particularly in the lip-line regions. The instantaneous flow field confirmed that the jet excitation modifies the propagation and structure of the acoustic waves, reducing the generation of noise sources while slightly shifting the preferred acoustic directivity angles. The SPOD analysis further highlighted that the applied excitation reduced the Kelvin–Helmholtz instability at the peak frequency and the overall energy of the coherent structures throughout the spectrum.

In conclusion, the obtained results suggest that the investigated unsteady microjet excitation approach presents a promising solution for the rectangular jet noise control strategy. Indeed, the observed noise reduction of up to 4 dB in OASPL in the $S{t}_F=0.30$ case highlights such potential, and is primarily attributed to the impact of the jet excitation on the overall broadband noise reduction across the acoustic spectrum.