Large-eddy simulations of noise reduction via fundamental-harmonic interactions in a supersonic rectangular jet

Abstract

This work explores the use of bi-modal excitation to reduce the noise in a supersonic rectangular jet. The excitation is guided by previously published reduced-order modelling, suggesting that harmonic interactions can reduce the dominant noise sources in the jet. An unexcited jet is considered first as the baseline case, which showed Strouhal number 0.15 as the most dominant near field coherent structure, which also appeared as a prominent peak in the far field noise at low emission angles and as a secondary peak at the peak emissivity angle, thus it was taken as the fundamental frequency to target with excitation. The jet is then excited with the single-mode harmonic, St 0.30. Other cases are considered with bi-modal excitation at St’s 0.15 and 0.30 with various phase lags. Both single-mode and bi-modal excitation show a considerable reduction of the near field coherent structures at St 0.15 and amplification at 0.30. Reduction of the fundamental in the bi-modal cases shows a dependence on the initial phase lag whereby reduction is maximized when harmonic amplification is maximized. All excited cases are also shown to reduce the eddy convection velocity. Reduction in far field sound pressure levels at St 0.15 is obtained for all excitation cases up to 13 dB. For all metrics, it is shown that bi-modal excitation is more effective than single-mode excitation given an optimized phase lag. All results ultimately support the use of fundamental-harmonic interactions as a noise reduction mechanism.

Keywords

Jet noise large eddy simulation rectangular jet jet excitation reduced order model aeroacoustics flow control

Introduction

Jet noise is a major concern in both civilian and military aviation. It is a nuisance to those living near airstrips and can cause adverse health effects on personnel working near aircraft. The focus of this work is active noise reduction in supersonic rectangular jets, which is of particular interest for military applications. This is partially because rectangular jets are easier to manufacture and integrate into an airframe. A specific use case for this work is to reduce noise of jets taking off from aircraft carriers. Carrier decks regularly see personnel working near aircraft during takeoff where noise levels can reach above 150 dB, which is beyond the level that personal protective equipment can protect against. The U.S. Department of Veteran Affairs spends approximately $1 billion per year on hearing loss cases with around 28% coming from the Department of the Navy.¹

Jet noise is a complex problem. In general, the three primary sources of noise are broadband shock noise, turbulent mixing noise, and screech. Additionally, at higher convective Mach numbers, the movement of large-scale structures can produce Mach wave radiation directed downstream. Screech is the result of a complex feedback process linking the acoustic wave produced by shock-vortex interaction to a receptivity process at the nozzle lip.² The dominant noise is generated by the large-scale coherent structures,³ which can be characterized as wave packets of various frequencies and azimuthal mode that can interact with each other as well as the mean flow and fine-scale turbulence.⁴ In supersonic jets, noise is distinguished between mixing noise and shock associated noise. Mixing noise is generated from convecting quadrupole sources as originally proposed by Lighthill.⁵ Shock associated noise is generated by the interaction between large-scale structures in the shear layer and the shock cells in the jet core.^6,7 For the same Mach 1.5 heated rectangular jet considered in this work, Gojon et al.⁸ show where these different types of noise originate in the near field and propagate to the far field. They show the mixing noise generated further downstream and appearing at low directivity angles. The shock associated noise appears further upstream and is dominant at higher polar angles and contains higher frequency components.

Rectangular nozzles offer an attractive feature for the design of aircraft, especially for airframe integration. Interest in studying rectangular jets has grown in recent years, highlighting their distinct flow characteristics. Experimental and numerical studies⁹ show that rectangular jets exhibit more complex structures due to the interaction of expansion waves from adjacent straight edges at the corners. The screeching frequency staging behavior is less prominent in rectangular jets compared to circular ones.¹⁰ Edgington-Mitchell et al.,¹¹ attributed the absence of staging behavior in large aspect ratio rectangular jets to the rapid decay of shock structures. A notable characteristic of rectangular supersonic jets is the oscillating mode. In the study by Powell¹² the antisymmetric mode corresponding to screech was visualized, and the symmetric mode related to the harmonic frequency of screech was also reported. Viswanath et al.¹³ investigated the flow structure and noise of ideally expanded supersonic rectangular and circular jets and observed that while plume spread near the nozzle differs slightly between rectangular and circular jets, both eventually converge to a similar rate. Significant differences exist in shock cell structures and jet core lengths. Double-diamond shock cells appear in the minor axis plane of the rectangular nozzle and the circular jet, but not in the major axis plane. Additionally, the effect of heated jets is investigated, and it is reported that when the operating temperature is increased, overall noise levels rise due to increased mixing noise, while shock-associated noise contributes less since shock cell structures near the nozzle exit are less temperature-dependent. At upstream and sideline locations, the sound pressure levels (SPL) of circular and rectangular jets are similar, particularly in broadband peaks. Downstream, the SPL profiles overlap until the circular jet SPL deviates further downstream. Both experimental and numerical results in this study suggest absence of screech tones for heated rectangular jets.

Noise mitigation techniques generally fall under two categories: passive and active control. Passive flow control methods involve introducing geometric changes that remain unchanged over time. An example of this type of flow control in jets are chevrons penetrating the jet plume^14,15 and using an airframe for shielding.^16–18 Of more relevance to this work is active noise control, which can be turned on and off to avoid any performance penalties when noise reduction is not needed. This has been ongoing in round jet experiments^19,20 and later by computation.²¹ In recent years, the use of some kind of fluid injection as an active noise cancellation device has attracted researchers, as a noise reduction approach to avoid any nozzle redesigns. Fluidic injectors disrupt shock cells, breaking them into smaller and weaker structures. They also influence the boundary layer near the nozzle, affecting instability wave growth and reducing large-scale mixing noise. Morris et al.²² designed fluidic insert achieving a 4 dB reduction in turbulent mixing noise and a 2 dB reduction in shock-associated noise. McLaughlin et al.²³ conducted experiments on both small-scale and larger-scale engines and observed the effectiveness of fluidic injectors in noise mitigation. Several numerical investigations^24–27 are conducted to elucidate the mechanisms of noise reduction using fluidic injections. More recent experiments at The Ohio State University have used plasma actuators to excite high speed jets with thermal perturbations and have found success in reducing the peak noise.^28–30 This same group has also found success with this excitation method for a single³¹ and twin rectangular jets.^32,33 Plasma actuators were implemented in computations with actuation strips by Gaitonde and Samimy.³⁴ Prasad and Unnikrishnan³⁵ excited a rectangular jet exiting a constant area duct using a similar computational methodology and found that the noise could be reduced depending on the forcing frequency. This work was followed up with a more realistic nozzle geometry and the excitation was able to reduce the noise by up to 5 dB when neglecting the actuation tone, but only 2 dB when the actuation tone was included.³⁶ Other experiments by Sinha et al.³⁷ found steady blowing to be effective for noise reduction, but unsteady blowing resulted in strong actuation tones. Employment of loudspeakers is considered as an alternative approach to fluidic injected by researchers in the past. Early experiments of Arbey and Ffowcs Williams¹⁹ on a circular jet subjected to simultaneous excitation by two distinct acoustic tones. In their work, acoustic excitation signals were supplied via loudspeakers. By adjusting the phase between two harmonically related frequencies, it is possible to control the harmonic generation process, sometimes nearly eliminating it. This effect is observed for both harmonic and subharmonic generation, although controlling subharmonic generation proves to be more challenging. Lepicovsky et al.³⁸ in detailed experimental data on the effects of upstream acoustic excitation on the mixing of heated jets with the surrounding air. It was reported in their findings that the sensitivity of heated jets to acoustic excitation varies significantly with jet operating conditions and that higher jet temperatures require higher excitation levels. Additionally, the preferred Strouhal number for excitation remains consistent across different operating conditions.

Past theoretical work has given insight to the mechanisms by which waves of different modes in a jet interact with one another, the mean flow, and background turbulence. A Reduced-Order Model (ROM) developed by Mankbadi³⁹ used an integral technique. The Navier-Stokes Equations were reduced to a set of Ordinary Differential Equations (ODEs) that describe the evolution of the mean flow and two harmonically related modes. The ODEs only require a mean flow, which can be obtained by analytic expressions or Reynolds Averaged Navier Stokes, and disturbance shape functions, which can be obtained using affordable computational tools such as Linear Stability Theory (LST) or the Linearized Euler Equations (LEE). It was shown that adding subharmonics could amplify or reduce the fundamental depending on the initial phase angle. This theory was later used by Raman and Rice²⁰ to guide jet excitation in experiments. They found that two-frequency excitation can be more effective than single-frequency and that the initial phase lag has a significant impact on whether subharmonics are amplified or suppressed. A similar ROM is shown by Lee and Liu,⁴⁰ which considers five interacting modes. This theory has also been used to describe the development of a single mode in a round jet using LST⁴¹ and LEE.⁴² This ROM was rederived in cartesian coordinates and applied to a 2D compressible shear layer operating at the same conditions as the jet considered in this work,⁴³ which used LST. Malczewski and Mankbadi⁴⁴ extended this for the 3D rectangular jet. Both recent works found a most amplified frequency that was consistent with the peak noise in the jet column mode in Samimy et al.³³ They both also concluded that adding the harmonic frequency can be an effective mechanism for reducing the most amplified frequency, which could theoretically reduce the peak noise. This finding is consistent with the general trend that higher excitation frequencies are more effective in noise reduction.^31,35

The objective of this work is to perform a high-fidelity computational study to apply the theory from Malczewski and Mankbadi⁴⁴ to see if harmonic interactions are effective in reducing the peak noise in the jet. Evidence for this theory is significant because the ROM offers an affordable method for performing quick parametric studies to predict effective excitation parameters. The ROM relies on low-fidelity linear methods (i.e., LST and LEE) that are then brought to the nonlinear regime using the resultant ODEs. The ROM is not meant to be a replacement to high-fidelity studies, but rather to be a precursor method to guide their setup. The ROM has been shown to reproduce certain trends from experiments and high-fidelity computations, but there are no studies for a 1:1 comparison. This work is also significant in that it is one of only a handful of computational studies of excited high speed rectangular jets. Computational setup and methodology of this work covers the computational methods used for both Large-Eddy Simulations (LES) and Ffowcs Williams-Hawkings solver. Baseline case and determination of the fundamental frequency presents the results for the unexcited baseline case and concludes the fundamental frequency that needs to be reduced. Excited LES results presents the results for the excited cases, first for the case of single-mode excitation, and then for bi-modal excitation. Conclusions are presented in Conclusions.

Computational setup and methodology

To apply the theory developed by Malczewski and Mankbadi,⁴⁴ Large-Eddy Simulations (LES) are used. This work considers the same jet used in the aforementioned work, which is also depicted in Figure 1. The same jet is used in other studies at the University of Cincinnati^8,13,16,45 and at The Ohio State University,^31,33 establishing a strong basis for comparison. The nozzle geometry is a converging-diverging nozzle with an aspect ratio of 2:1 ( $12.95 m m \times 25.91 m m$ ) and a design Mach number of 1.5. The operating conditions considered in this work are that the jet is perfectly expanded with a nozzle pressure ratio of 3.67 and is also heated with a total temperature of 3. To non-dimensionalize quantities: lengths are non-dimensionalized by the height of the jet, $H = 12.95 m m$ , velocities by the jet exit velocity, $U_{j e t} = 750 m / s$ , density by the ambient density $ρ_{\infty} = 1.225 k g / m^{3}$ , and temperature by ambient temperature $T_{\infty} = 300 K$ . Frequencies are non-dimensionalized into Strouhal numbers using the height of the jet, $H = 12.95 m m$ , and jet exit velocity (i.e., $S t_{H} = f H / U_{j e t}$ ).

Figure 1.

Schematic showing computational grid (left) and nozzle schematic (right); the computational grid shows the minor plane (red) and the major plane (blue). A. LES Methodology.

LES Methodology

The LES code used in this work is Scalable Agile Fluid Framework (SAFF), which is a derivative of the code originally developed by Visbal and Gaitonde.⁴⁶ This code solves the compressible Navier-Stokes equations in strong conservative form with a curvilinear coordinate system.

\frac{\partial}{\partial τ} (\frac{q}{J}) + \frac{\partial \hat{F}}{\partial ξ} + \frac{\partial \hat{G}}{\partial η} + \frac{\partial \hat{H}}{\partial ζ} = \frac{1}{R e} [\frac{\partial {\hat{F}}_{ν}}{\partial ξ} + \frac{\partial {\hat{G}}_{ν}}{\partial η} + \frac{\partial {\hat{H}}_{ν}}{\partial ζ}]

(1)

Above,

J

is the Jacobian of the curvilinear transform given by

J = \partial (ξ, η, ζ, τ) / \partial (x, y, z, t)

. The solution vector is denoted by

q = {[ρ, ρ u, ρ v, ρ w, ρ E]}^{T}

. Fluxes are denoted by

\hat{F}

\hat{G}

, and

\hat{H}

for the

ξ

η

, and

ζ

directions, respectively. The flux vectors are further described in references.^47,48 The subscript,

(v)

, denotes viscous flux vectors. Energy is defined as:

E = \frac{T}{γ (γ - 1) M_{\infty}^{2}} + \frac{1}{2} (u^{2} + v^{2} + w^{2})

(2)

The SAFF code has been successfully used in several prior efforts with jet flows.^47,49–51 As described by Chakrabarti et al.,⁴⁷ the code uses third-order upwind-biased reconstruction in conjunction with a Roe flux-difference split scheme⁵² and a harmonic limiter.⁵³ Viscous terms are discretized using second-order finite differencing. No explicit turbulence model is used, thus classifying this as implicit-LES (ILES). Temporal integration is performed using a second-order approximately factored Beam-Warming scheme with two sub-iterations.⁵⁴

In the code, velocities are normalized by the ambient speed of sound, $c_{\infty}$ , density by ambient density, $ρ_{\infty}$ , and pressure by $ρ c_{\infty}^{2}$ . The ideal gas law is considered, $p = ρ T / γ$ , and a constant Prandtl number of 0.7 is used. Sutherland’s law is used to relate viscosity as a function of temperature and the Stoke’s hypotheses is assumed for the bulk viscosity coefficient, $λ = - \frac{2}{3} μ$ . For all simulations, the Reynolds number is set to 570,000 based on the jet width and ambient speed of sound.

The computational domain here uses the same structured grid as in Chakrabarti et al.,⁴⁷ who previously confirmed dominant flow features to be grid independent. The grid is a single block composed of $915 \times 529 \times 629$ nodes in the x, y, and z directions respectively, giving the grid approximately 300 million nodes in total. The domain extends $64 H$ in the x-direction, and $20 H$ in the y and z directions. Two planes of interest are defined: the minor plane, which is along the short end of the jet (height) and the major plane along the long end (width). Slices of the minor and major planes are shown in Figure 1.

Excitation is imposed using actuation strips along the upper and lower nozzle lips near the exit, which are shown in Figure 1. The actuation strips span the entire width of the nozzle. This is different from other works with excitation, which have smaller actuators placed along the width, giving a z-periodicity in the major plane.^35,49 This work is putting into practice the ROM developed by Malczewski and Mankbadi,⁴⁴ which did not include z-periodicity in the disturbance shape functions, thus the actuators span the entire nozzle width for consistency.

The actuators function by imposing a small periodic pressure fluctuation. Two types of excitations are considered: single-mode excitation, and bi-modal excitation, which take the form of equations (3) and (4) respectively and are depicted graphically in Figure 2. For all cases, the upper and lower actuators work in-phase with each other. For the single-mode case, the jet is excited with twice the frequency of the most amplified mode in the nearfield. As is later shown in Baseline case and determination of the fundamental frequency, $S t_{H} = 0.15$ is the most amplified coherent structure in the near field, which will be referred to as the fundamental, $f$ . From the ROM,⁴⁴ interactions with $2 f$ should reduce the fundamental, so the single-mode case is excited with $S t_{H} = 0.30$ . The amplitude of the single-mode forcing is chosen such that it on the order of $S t_{H} = 0.15$ near the nozzle exit, which works out to 0.4% of the mean pressure in that region. For the bi-modal cases, excitation is prescribed with two frequencies, $S t_{H} = 0.15$ and $0.30$ . This is done to control the phase lag, $β_{0}$ , between the two frequencies, which is shown to have an effect by Malczewski and Mankbadi.⁴⁴ The forcing amplitudes here have been increased to 1% of the time-averaged pressure for better control over the initial phase lag. In either excitation case, the amplitude is very small, on the order of acoustic fluctuations and the actuators used are representative of a speaker.

p_{a c t} = \bar{p} (1 + A \sin (ω_{2 f} t))

(3)

p_{a c t} = \bar{p} (1 + A \sin (ω_{f} t) + A \sin (ω_{2 f} t + β_{0}))

(4)

Figure 2.

Actuation for all excited cases.

The ILES solver was run with a time step of $5 \times 10^{- 4}$ based on the ambient speed of sound and jet major axis length (jet width). For all statistics presented, snapshots are collected every 50 time-steps for a total of 4200 samples. This gives a sampling frequency of $536 k H z$ and a duration of $453 H / U_{j e t}$ . For excited cases, a delay of $76 H / U_{j e t}$ is used before collecting samples. A cutoff frequency was computed assuming 12 points per wavelength required to resolve any wave. A previous study by Salehian and Mankbadi¹⁷ used 15 points per wavelength with a second order scheme. Tam and Webb⁵⁵ show that high order compact schemes can resolve waves with less than eight points per wavelength. The present study lies somewhere in between being a 3rd order scheme, thus using 12 points per wavelength. Using the grid spacing where the peak noise sources originate, this gives a maximum resolved St of 1.92.

Ffowcs Williams-Hawkings methodology

To compute the far field noise, a Ffowcs Williams-Hawkings (FWH) solver was developed and employed. Formulations I and II of the FWH equations given in Brentner and Farassat⁵⁶ are first considered. Lyrintzis⁵⁷ shows the conversion of these formulations to the frequency domain with a Fourier transform. Mendez et al.⁵⁸ manipulated this into a new frequency domain formulation, which is ultimately used in this work. The far field pressure signal can be decomposed as follows:

p^{'} (\vec{x}, t) = p_{T}^{'} (\vec{x}, t) + p_{L}^{'} (\vec{x}, t) + p_{Q}^{'} (\vec{x}, t)

(5)

Above, $\vec{x}$ is the observer location. The first two terms are the loading and thickness noise, which are monopole and dipole sources, respectively. The third term is the quadrupole noise, which is neglected in this work. It has been shown by Spalart and Shur⁵⁹ that the quadrupole term is more compact in pressure FWH formulations, and less error is introduced when neglecting it. They exhibit that the density formulation is less accurate when entropy fluctuations cross the FWH surface, which is particularly the case for heated jets like the one used in this work. Hence, the fluctuating density is replaced by $ρ^{*} = ρ_{\infty} + p^{'} / c_{\infty}^{2}$ in this work. Mendez et al.⁵⁸ also neglect the quadrupole term and show good agreement for jet far field acoustics. The condition here is that the sampling surface lies outside sources of nonlinearity (i.e., vorticity). Figures of the FWH surfaces and time-averaged vorticity from the baseline case are shown in Figure 3.

Figure 3.

Ffowcs Williams-Hawkings surfaces plotted with time-averaged vorticity in (a) minor plane, (b) major plane, (c) isometric view showing end caps.

The time history of the primitive variables is collected on each of the surfaces. For a given observer, source terms are computed as functions of time and space using the formulation from Mendez et al.⁵⁸

F_{1} = \frac{p^{'} {\hat{n}}_{j} \cdot {\hat{r}}_{j} + ρ u_{j} u_{n} \cdot {\hat{r}}_{j}}{c_{\infty} r} + \frac{ρ u_{n}}{r}; F_{2} = \frac{p^{'} {\hat{n}}_{j} \cdot {\hat{r}}_{j} + ρ u_{j} u_{n} \cdot {\hat{r}}_{j}}{r^{2}}

(6)

In equation (6),

p^{'}

is the instantaneous pressure fluctuation,

{\hat{n}}_{j}

is the unit normal on the FWH surface at index

j

{\hat{r}}_{j}

is the unit vector pointing from the

j^{t h}

index on the FWH surface to the observer of interest,

u_{j}

is the three component velocity vector at location

j

u_{n}

is the normal component of the velocity vector (i.e.

u_{j} \cdot {\hat{n}}_{j}

c_{\infty}

is the ambient speed of sound, and

r

is the magnitude of

{\hat{r}}_{j}

. These source terms are then transformed to the frequency domain using Fast Fourier Transform (FFT) before integration. This makes the FWH formulation a frequency domain formulation, which avoids accounting for retarded time as with the time domain formulations.^56,57 The total of 4200 collected samples are separated into five segments with a 75% overlap. A Hanning window is applied to each segment, and they are averaged using Welch’s method.⁶⁰ Similar signal processing techniques are used in Prasad and Unnikrishnan.³⁵ The far field pressure is then computed as follows:

4 π \hat{p} (\vec{x}, ω) = \iint_{S} Ψ (y) (i ω {\hat{F}}_{1} (y, ω) + {\hat{F}}_{2} (y, ω)) e^{- \frac{i ω r}{c_{\infty}}} d S

(7)

In the above notation,

y

corresponds to locations on the FWH surface and

\vec{x}

corresponds to observer locations. The weighting term,

Ψ (y)

, is used to treat the end caps shown in Figure 3(c), which have bulk flow passing through them. This is described by Mendez et al.⁵⁸ and was originally used by Shur et al.⁶¹ The weighting function is simply one if the surface is not an end cap, and

1 / N_{c a p s}

if the surface is an end cap. In this work, four end caps are used beginning at a streamwise location of

28.7 H

and spaced every

3.2 H

. The number of end caps and cap spacing was chosen to minimize the error in the St range of 0.15-0.30, which is also described by Mendez et al.⁵⁸ Sound pressure level (SPL) spectra and overall sound pressure levels (OASPL) are then computed using the equations below from Mendez et al.⁵⁸

S P L (\vec{x}, S t) = 10 \log_{10} [\frac{2 \hat{p} (\vec{x}, ω) {\hat{p}}^{*} (\vec{x}, ω)}{S t_{\min} p_{r e f}^{2}}]

(8)

O A S P L (\vec{x}) = 10 \log_{10} [\sum_{S t = S t_{\min}}^{S t_{\max}} \frac{2 \hat{p} (\vec{x}, ω) {\hat{p}}^{*} (\vec{x}, ω)}{p_{r e f}^{2}}]

(9)

The FWH solver is validated using the unexcited baseline case and comparing directly with Mora et al.¹⁶ and Viswanath et al.,¹³ which consider the same jet in both experiment and computation, respectively. They also use the same observer locations, which are also adopted in this work. For all cases, observers are placed in an arc at a radial location of $63 H$ , measured from the jet exit plane at the centerline. Arcs are taken in both the minor ( $ψ = 0 °$ ) and major ( $ψ = 90 °$ ) planes. Observer definitions can be visualized in Figure 4. Solver validation is shown later in Baseline case and determination of the fundamental frequency.

Figure 4.

FWH observer coordinate system.

Baseline case and determination of the fundamental frequency

The results begin with the unexcited, baseline case, which provides a basis for comparison of the excited case and to identify a dominant coherent structure to target for the excitation cases. The near field pressure spectra at the lower nozzle lip jet exit on the minor plane is shown in Figure 5. The spectrum shows a distinct peak at $S t_{H} = 0.15$ , which is consistent with the most amplified St from the ROM.⁴⁴ This peak is not considered a screech tone. Screech is more likely to occur in the presence of strong shocks, which is not the case here as the jet is near-perfectly expanded. The experimental work by Mora et al.¹⁶ only observed a screech tone for the perfectly expanded unheated jet whereas the jet used here is heated. They showed that increasing the temperature ratio of the jet reduced screech and no screech tone appeared in the far field at a temperature ratio of 3. It is also later shown that this St does not appear as a tone in the far field, but rather as a peak in the broadband noise. Figure 6 shows contours of the pressure magnitude for a range of St’s between 0.075 and 0.30. These were computed using a sample of 4200 snapshots and Welch’s method with four segments with 25% overlap. St 0.15 is shown to significantly amplify downstream and spread into the far field. Peak amplification is shown in the region associated with the broadband mixing noise.

Figure 5.

Near nozzle spectra of baseline case.

Figure 6.

Pressure magnitudes at various St’s for baseline case.

Far field overall sound pressure levels for the baseline case given in Figure 7 show excellent agreement with the previously published results.^13,16 Peak values in the directivity differ by less than 1 dB in both planes of interest. The spectra shown in Figure 8 exhibits decent agreement with experiment.¹⁶ The spectra in the major plane compared with Viswanath et al.¹³ are shown in Figure 9 and have better agreement than with Mora et al.¹⁶ Here, the St’s above 0.07 have good agreement around the peak directivity angle. There are notable differences in the low frequency range and at high directivity angles. This is expected as the end cap spacing was chosen to reduce error between St 0.15-0.30. This work is more focused around the peak directivity range between, so this is allowable. The peak noise in the present study occurs at a polar angle of approximately $50 °$ in both minor and major planes.

Figure 7.

Directivity comparison for baseline case in the minor (left) and major (right) planes.

Figure 8.

Spectra comparison in minor plane for baseline case.

Figure 9.

Major plane spectra comparison with Viswanath et al. (2017) for baseline case.

A contour of SPL versus directivity angle and St is shown in Figure 10. For both planes, there is a distinct peak for $S t_{H} = 0.25$ at the peak directivity angle of $θ = 50 °$ , with a secondary peak around St 0.15. For lower polar angles between $32 ° - 46 °$ , the peak St is in the vicinity of 0.15, which is consistent with the dominant near field coherent structure. This peak St jumps to 0.25 for polar angles between $47 ° - 52 °$ , which is where the overall sound pressure level peaks in Figure 7. Another observable trend is that as the polar angle increases, the peak St also increases.

Figure 10.

Contour of SPL versus directivity angle and Strouhal number; shown are the minor plane (left) and major plane (right).

There are several other works that have covered the same jet used here and there are some variations in the peak directivity and dominant frequencies. Most of these studies report St based on the equivalent nozzle diameter, $S t_{D} = f D_{e q} / U_{j}$ , whereas this work uses the height, $S t_{H} = f H / U_{j}$ . For consistency, results from others have been converted to use the $S t_{H}$ convention. Experiments by Mora et al.¹⁶ found a peak directivity between $28 ° - 44 °$ in the minor plane with peak St’s ranging from $S t_{H} = 0.06 - 0.13$ . Their peak directivity is at a slightly lower polar angle than the present study, but the peak frequency at those lower angles is comparable as is shown in Figure 8. This study compares very well with the results of Viswanath et al.,¹³ but with a slightly higher peak directivity angle. Their peak frequency was similar to Mora et al.¹⁶ at $S t_{H} = 0.13$ . This study finds the same peak directivity of Crawley et al.³¹ at a polar angle of $θ = 50 °$ . The baseline case by Mankbadi and Salehian¹⁸ found a lower peak frequency at $S t_{H} = 0.06$ . Gojon et al.⁸ considered the same jet at a slightly overexpanded operating condition and found $S t_{H} = 0.10$ as the dominant noise. They also showed that the peak shock associated noise corresponded with $S t_{H} = 0.27$ . Prasad and Unnikrishnan³⁵ considered the same jet, but issued from a constant area duct, removing the shocks, and found the peak noise between $S t_{H} = 0.076 - 0.152$ . This work ultimately finds a similar peak directivity to the previous works and peak St’s in the same range. An earlier study using near field FFT and SPOD analysis also highlighted the significance of $S t_{H} = 0.15$ and corelated it to the primary radiation angle in the far field.⁶²

Figure 6 shows St 0.15 to be the most amplified coherent structure in the near field, which shows up as a secondary spectrum peak at the peak emission angle and as the peak frequency for emission angles slightly less than the peak one. The theory proposed by Malczewski and Mankbadi⁴⁴ also demonstrated the role of natural amplification to initiate fundamental-harmonic interactions, and it is evident that St 0.15 is more amplified. Even though 0.25 is the peak St at the peak emissivity angle, St 0.15 still plays a major role and is more suitable to target with excitation. Thus, St 0.15 is taken as the fundamental that should be reduced by interactions with the harmonic, St 0.30, which will be added via excitation. Based on the results shown in Fig. and 10, reduction of St 0.15, should generate noise reduction for polar angles between $30 °$ and $52 °$ .

Excited LES results

Single-mode excitation case

The first case considered is that of a single forcing mode at $S t_{H} = 0.30$ , which takes the form given in equation (3). In this case, the phase between St’s 0.15 and 0.30 is not controlled. Malczewski and Mankbadi⁴⁴ showed that on average, the addition of the harmonic should reduce the fundamental since it is not very amplified naturally. It is recalled that the amplitude of the forced pressure disturbance is small, on the order of 0.4% of the time-averaged pressure at the nozzle exit.

Figure 11 shows a comparison of pressure contours for St’s 0.15 and 0.30 between the excited and baseline case. Qualitatively, there is considerable reduction of St 0.15. The forced St, 0.30, does not considerably amplify downstream for the excited case, so actuation tones should be absent in the far field spectra. The symmetric structure of the excitation is also clearly visible up to $X / H = 4$ . To better quantify changes in the near field coherent structure, the contours in Figure 11 were integrated along the y-direction using equation (10).

I (x, S t_{H}) = \int_{- \infty}^{\infty} | \hat{p} (x, y, S t_{H}) | d y

(10)

Figure 11.

Near field minor plane contours of $| \hat{p} |$ for St’s 0.15 and 0.30; compared are the baseline case and single-mode excited case.

This integration is shown in Figure 12. The reduction of St 0.15 is again very apparent at streamwise locations of $X / H > 10$ . St 0.30 sees some amplification over a broad range. The peak is only slightly larger than in the baseline case and the peak occurs sooner, slightly before $X / H = 10$ .

Figure 12.

Integrated pressure comparing baseline and single-mode excited cases; St’s 0.15 (left) and 0.30 (right) are displayed.

Positions of the near field large-scale structures were manually tracked, and convection velocities were backed out. The eddy displacement, $D$ , and convection velocities are shown in Figure 13. Here, it is shown that the eddy convection velocity is reduced for the case of single-mode excitation. It was originally proposed by Lighthill⁶³ that higher eddy convection speed should increase sound intensity. Tam⁶⁴ later stated that higher eddy convection velocity is more favorable for eddy Mach waves. Kandula and Vu⁶⁵ show that higher convective Mach numbers increase acoustic intensity for supersonic jets. The reduction of eddy convective velocity here should indicate there is some reduction in the far field noise.

Figure 13.

Eddy displacement (left) and eddy convection velocity (right) comparing single-mode excitation and baseline case.

Far field noise directivity is shown in Figure 14. In the minor plane, there is a slight increase in the peak noise, about 0.3 dB. However, in the major plane, there is a peak noise reduction of 1 dB. In both planes, the peak emissivity angle remains $50 °$ . The differences in OASPL across polar angles are plotted in Figure 14. In the minor plane, there is up to a 1 dB decrease in the noise for low polar angles between $21 °$ and $39 °$ . The major plane exhibits noise reduction across almost the entire range of polar angles. Notably it sees a reduction of approximately 1 dB for all angles between $30 °$ and $62 °$ . In Baseline case and determination of the fundamental frequency, it was shown that St 0.15 had a strong presence for polar angles between $30 °$ and $52 °$ either as the primary or secondary peak. The excited case shows a reduction in the strength of this St and is showing reduction at the polar angles where St 0.15 was dominant Figure 15.

Figure 14:

Minor (left) and major (right) plane OASPL comparison with single-mode excitation.

Figure 15.

Minor (left) and major (right) plane $Δ O A S P L$ for single-mode excitation; negative values indicate noise reduction.

Spectra at the peak emissivity angle are shown in Figure 16. In the minor plane, there is an increase in the noise between St’s 0.15 and 0.20, which is where the rise in OASPL comes from. The major plane sees a drastic decrease in noise around the peak of St 0.25, resulting in the 1 dB of OASPL. It is noted that at the peak emissivity angle, St 0.15 was not the dominant peak. An additional spectra comparison is shown for $θ = 35 °$ in Figure 17 where St 0.15 was the dominant frequency in the unexcited case and there was almost a 1 dB overall noise reduction. In the minor plane, there is a reduction of St 0.15, which led to a reduction of the OASPL. The excitation is intended to target St 0.15 and the SPL at this St is shown in Figure 18. Here, it is clearly demonstrated that the minor plane peak is reduced. Peak-to-peak reduction is observed up to 9 dB at this St There is also a broad reduction between directivity angles of $20 °$ - $55 °$ . In the major plane, there is considerably less reduction at this St, only achieving a peak-to-peak reduction of up to 5 dB. The region of reduction is also much more limited than in the minor plane. This is not surprising because excitation is only imposed in the minor plane. This aligns with what is shown in the near field where a St 0.15 is reduced and further supports the theory presented by Malczewski and Mankbadi⁴⁴ whereby exciting with harmonics can reduce the fundamental.

Figure 16.

Comparison of spectra at peak emissivity angle for single-mode excitation.

Figure 17.

Comparison of spectra at $θ = 35 °$ for single-mode excitation.

Figure 18.

Comparison of SPL at St 0.15 for single-mode case.

Spectral proper orthogonal decomposition (SPOD) is used to separate the spectra into energy containing modes.⁶⁶ The objective of SPOD analysis is to identify wave packets based on their associated frequencies. These modes capture the variations in wave packet shapes for each St These variations are crucial to the jet’s acoustic properties and identifying the effectiveness of excitation frequency, and later phase, on reducing jet noise. The same set of 4200 snapshots are used in the minor plane. These snapshots are split into 31 blocks, each containing 256 snapshots with a 50% overlap. This struck a balance between having a well-resolved spectra and having distinct spatial modes.

Spectra showing the first and second mode energies are shown in Figure 19. These spectra are normalized by the flow energy integrated over the spectra, allowing the spectra to be thought of as a percentage of that energy represented by that mode.⁶⁷ A notable feature of the spectra for the single-mode case is that there is an increased gap between the first and second mode around St 0.30, indicating the introduction of a new wave packet at that frequency from the excitation. It is also observed that there is a narrowing of the gap between the first and second mode in the lower frequency range for St<0.15, also indicating an effect on those frequencies.

Figure 19.

SPOD spectrum comparing baseline and single-mode case showing first and second mode energies; mode number increases moving downward.

The real component of the first and second modes are shown in Figures 20 and 21, respectively. The figures show that the excitation increases the coherence of the wave packet structures at St 0.15. Then in the second mode the structures at St 0.15 are significantly broken up. In the baseline case, they formed into coherent structures at $X / H = 7$ , but in the excited case, this is delayed to $X / H = 12$ . Looking at St 0.30, the first mode shows stronger structures than in the baseline case, which is expected based on the gap between mode energies highlighted in Figure 19. In the first mode, St 0.30 begins to amplify and radiate into the far field around $X / H = 6$ , whereas this did not occur until $X / H = 8$ in the baseline case.

Figure 20.

First mode SPOD contours comparing baseline and single-mode excited case; shown are St’s 0.15 and 0.30.

Figure 21.

Second mode SPOD contours comparing baseline and single-mode excited case; shown are St’s 0.15 and 0.30.

Turbulent kinetic energy (TKE) is computed along the shear layer and displayed in Figure 22. At X/H = 0 (middle), the increase in initial TKE is apparent and the difference from the baseline is due to the excitation at St 0.30. This mode proceeds to amplify until reaching a peak around X/H = 2.5, where the TKE peak of the excited case is slightly higher than that of the baseline case. This location corresponds to the streamwise location where the integrated pressure greatly increases at St 0.30 in the excited case as was shown in Figure 12. As was earlier suggested, St 0.30 is amplified by absorbing energy from St 0.15, but the overall energy between all modes is conserved. Since the added energy at X/H = 0 is small, the peak energy around X/H = 2.5 is comparable between the two cases.

Figure 22.

Shear layer turbulent kinetic energy between baseline and single-mode excitation cases; shown are magnitudes for the full shear layer (left), initial shear layer (middle), and at the peak (right).

By investigating the TKE evolution in the shear layer, along with SPOD breakdown of fluctuations, the mechanism involved in the excitation can be explained as energy transfer between the fundamental (St = 0.15) and harmonic (St = 0.30) modes. The input energy introduced near the nozzle energizes the fluctuations up to X/H = 5, which show themselves as coherent structures shown in Figure 20. However, such structures have the potential to absorb energy in the region 5 < X/H<8. The energy is absorbed from the dominant frequency as shown in Figure 21. Which leads to reduction of fluctuations for St 0.15 frequency located around X/H = 10, which has been previously show as the dominant source of noise specifically in the maximum radiation angle shown in Figure 8. Hence, from the energy conservation prospect, the introduced energy due to excitation by energy transfer mechanism between the fundamental (St = 0.15) and harmonic (St = 0.3) modes. Energy conservation is satisfied when accounting for energy input via the excitation introducing additional energy into the system. Which leads to redistribution dynamics, where interactions through nonlinear processes redistribute the input energy between modes. Any leftover energy at higher frequency dissipates into the environment in physical case, however, would be out of the cut-off frequency of interest in this work. Although this mechanism provides small differences in peak TKE when evaluated along with the SPOD modes but provides the basis for introduction of bi-modal excitations by providing more effective noise reduction.

Bi-modal excitation cases

The cases of bi-modal excitation are now considered. Here, the jet is excited with two frequencies, St’s 0.15 and 0.30, in accordance with equation (4). This is done to exert control over the phase lag between the two frequencies. Four cases are run with different phase lags, $0$ , $π / 2$ , $π$ , and $3 π / 2$ . Again, there is no phase difference between the upper and lower actuation strips. The forcing amplitude here has been increased to a 1% fluctuation of the mean pressure, which is imposed for both frequencies.

Figure 23 shows the near field pressure contours for the bi-modal cases. Qualitatively, each of the bi-modal cases appear to have reduced St 0.15. However, the extent of reduction is significantly different depending on the initial phase lag. St 0.30 shows a clear amplification for all excited cases, with a sharp initial peak occurring at $X / H = 1.5$ . This is quantified in Figure 24 showing the integrated pressures using equation (10). Interestingly, all cases show a reduction of St 0.15 for $X / H > 10$ . There are varying levels of reduction with phase lags of $β_{0} = π / 2$ and $β_{0} = 3 π / 2$ showing the most reduction. Incidentally, these two cases also show the most amplification of St 0.30. This further supports the theory that the noise reduction mechanism is a transfer of energy from the dominant noise source to its harmonic.

Figure 23.

Near field minor plane contours of $| \hat{p} |$ for St’s 0.15 and 0.30; compared are the baseline case and all bi-modal excited cases with different phase lags.

Figure 24.

Integrated pressure comparing baseline and bi-modal excited cases; St’s 0.15 (left) and 0.30 (right) are displayed.

Eddy displacements and convection velocities are once again computed and shown in Figure 25. It is shown that the bi-modal excitation reduced the peak convection velocity for all cases. The greatest overall reduction comes from the $β_{0} = 3 π / 2$ case, which also had the greatest near field reduction of St 0.15. The case of $β_{0} = π$ exhibits a similar average convection velocity to the baseline case, but the peak instantaneous velocity is reduced. Again, the reduction of the eddy convection velocity should indicate acoustic reductions.

Figure 25.

Eddy displacement (left) and eddy convection velocity (right) comparing bi-modal excitation and baseline case.

Figures showing $O A S P L$ and $Δ O A S P L$ for the bi-modal cases are shown in Figures 26 and 27, respectively. Like the case of single-mode excitation, peak noise reduction occurred primarily in the major plane of the jet for all bi-modal excitation cases. In the minor plane, phase lags of $β_{0} = π$ and $β_{0} = 3 π / 2$ increased the peak noise with $β_{0} = 3 π / 2$ increasing the peak noise by around 1 dB. However, these two phase lags see some noise reduction at low polar angles between $25 °$ and $35 °$ . It was shown in Figures 23 and 24 that $β_{0} = 3 π / 2$ reduced St 0.15 the most in the near field. This same phase lag generated the most reduction for directivity angles with St 0.15 as the dominant frequency. This supports the claim that reducing the near field structures can reduce the far field noise. The other two phase lags, $β_{0} = 0$ and $β_{0} = π / 2$ , had little to no impact on the peak noise in the minor plane. Overall, the case of $β_{0} = 0$ had the best noise reduction performance with minimal amplification in the minor plane and a reduction up to 1.8 dB in the major plane for a wide range of emissivity angles. Ultimately for all cases, noise reduction is greatest in the major plane between polar angles of $θ = 20 ° - 60 °$ .

Figure 26.

Minor (left) and major (right) plane OASPL comparison for bi-modal excitation cases.

Figure 27.

Minor (left) and major (right) plane $Δ O A S P L$ for bi-modal excitation cases; negative values indicate noise reduction.

Figures 28 and 29 show the peak emission far field spectra in the minor and major planes, respectively. In the minor plane, each case still sees an amplification around St 0.15, which is likely a result of the actuation. However, there are stark contrasts between the cases around St 0.30. Particularly for the case of $β_{0} = 0$ , there is a large reduction between St’s 0.20 and 0.35, however, the amplification around St 0.15 prevents the overall noise from being reduced. It is recalled from the near field analysis that this phase lag had some of the smallest amplification of St 0.30. The major plane shows a similar trend where all cases reduced the noise around St 0.25, but to varying degrees with $β_{0} = 0$ performing the best.

Figure 28.

Far field spectra at peak emissivity angle in minor plane comparing bi-modal excited cases to baseline case.

Figure 29.

Far field spectra at peak emissivity angle in major plane comparing bi-modal excited cases to baseline case.

It was noted earlier that St 0.15 is dominant at lower emission angles, where Figures 26 and 27 show reduction for all bi-modal cases in both planes. Looking at the spectra for $θ = 35 °$ in Figures 30 and 31, there is a clear reduction of St 0.15 for all cases, which results in an $O A S P L$ reduction for that emission angle. Again, there are varying levels of effectiveness with $β_{0} = 0$ giving the most reduction of St 0.15. The phase lag, $β_{0} = 3 π / 2$ , still exhibits an overall reduction, but there remains a prominent peak at St 0.15.

Figure 30.

Far field spectra at $θ = 35 °$ in minor plane comparing bi-modal excited cases to baseline case.

Figure 31.

Far field spectra at $θ = 35 °$ in major plane comparing bi-modal excited cases to baseline case.

Figure 32 shows the SPL at St 0.15 in the minor and major planes. Again, there is a noticeable peak-to-peak noise reduction in the minor plane up to 8.5 dB for $β_{0} = π$ and $3 π / 2$ . The case for $β_{0} = 3 π / 2$ sees the greatest broad reduction between directivity angles of $20 °$ and $50 °$ with up to a 13 dB reduction at $35 °$ . The other two phase lags of $β_{0} = 0$ and $π / 2$ still reduce the peak, but only by 3 dB. This is very similar to what is predicted by the ROM presented by Malczewski and Mankbadi.⁴⁴ They showed that for the case of harmonic interactions, two of the phase lags would reduce the fundamental and the other two would not cause significant amplification. Here, amplification is not observed, but the reduction is relatively minor. Like with the single-mode case, St 0.15 was not significantly altered in the major plane.

Figure 32.

Comparison of SPL at St 0.15 for bi-modal excited cases.

The results for the bi-modal cases are a mix of the expected and unexpected. It has long been observed that the phase lag affects the interaction between noise sources.^19,20 It was predicted by the ROM of Malczewski and Mankbadi⁴⁴ that some phase lags would decrease the noise, while others would amplify it. This is best shown when analyzing the SPL at St 0.15 in the minor plane where the excitation is imposed. The trends between the ROM and LES agree and they only differ in the specific phase lags that reduced the noise. Minor plane peak-to-peak noise was not considerably reduced by any of the bi-modal cases; however, some reduction is observed at the lower directivity angles where St 0.15 is dominant. The greatest reduction in this region came from $β_{0} = 3 π / 2$ . Near field analysis shows that this case had the greatest reduction of St 0.15 and the greatest reduction of the eddy convection velocity, which are both indicators of noise reduction. This case went on to reduce St 0.15 by the greatest amount in the far field.

SPOD analysis is once again performed. Mode energy spectra are shown in Figure 33.

Figure 33.

SPOD spectrum comparing baseline and bi-modal cases showing first and second mode; mode number increases moving downward.

Like with the single-mode case, there are considerable differences in the gaps between the first and second modes. For all bi-modal cases, there is a reduction in the mode gap around St 0.15 indicating a redistribution of mode energy to the second mode. A general comment on the shape of the spectra here is that for all excited cases, there is a flattening of the peak in mode one around St 0.15. Qualitatively, the greatest flattening occurs for the $β_{0} = 3 π / 2$ case. This case also has the tightest gap at St 0.15, meanwhile there is not an additional peak at St 0.30. The tightening of the gap at St 0.15 indicates that it redistributed the most energy from the first to second mode, which is an order of magnitude smaller. This aids in explaining why this specific phase lag saw the greatest reduction of St 0.15 in both the near field and far field.

Contours of the first and second mode are shown in Figures 34 and 35, respectively. With the forcing amplitude being increased for the bi-modal cases, the effects are more pronounced than for the single-mode cases. The actuation at St 0.30 shows in the first mode for all bi-modal cases with the distinct symmetric structures. The streamwise amplification of 0.30 changes with the initial phase lag. The phase lag of $β_{0} = 0$ has St 0.30 peaking at $X / H = 10$ and $β_{0} = π / 2$ causes the peak to be at $X / H = 7$ . The phase lag of $β_{0} = π$ , delays the peak to $X / H = 9$ , and $β_{0} = 3 π / 2$ does not show a distinct peak, but rather a broad amplification along the streamwise direction. Previous analysis by Salehian et al.⁶² show the peak noise sources originate from $X / H = 7$ for the same jet at St 0.15. Focusing on St 0.15 in Figure 34, it is observed that the coherent structures are weaker for the bi-modal cases of $β_{0} = 0$ and $β_{0} = 3 π / 2$ , which also saw the most noise reduction at the lower emission angles where St 0.15 is dominant.

Figure 34.

First mode SPOD contours comparing baseline and bi-modal excited cases; shown are St’s 0.15 and 0.30.

Figure 35.

Second mode SPOD contours comparing baseline and bi-modal excited cases; shown are St’s 0.15 and 0.30.

Shear layer TKE for the bi-modal cases are finally shown in Figure 36. Again, the initial difference in energy at X/H = 0 is very clear. This difference is also greater than that of the single-mode case, which is expected since the forcing amplitude is higher here. For all excited cases, the shear layer TKE reaches a similar peak to that of the baseline case, which is not surprising. Interestingly, it is observed that two of the bi-modal cases achieve peaks higher than baseline, and two are lower. This has a correlation to the integrated pressure displayed in Figure 24, specifically at the fundamental frequency, St 0.15. The two phase lags of $β_{0} = 0$ and $π$ produced the highest peaks in integrated pressure at St 0.15 out of the bi-modal cases, which are also the cases that increased peak shear layer TKE the most. It stands to reason that differences in the peak TKE are a direct result of the amplification or reduction of St 0.15, which is the dominant near field frequency.

Figure 36.

Shear layer turbulent kinetic energy between baseline and bi-modal cases; shown are magnitudes for the full shear layer (left), initial shear layer (middle), and at the peak (right).

The bi-modal cases suggest that energy is conserved in the proposed theory, albeit tracking how that energy is distributed is difficult. Two frequencies are considered, St’s 0.15 and 0.30. The energy added by the excitation is small, and only small differences are observed in the shear layer TKE. But these small differences are significant and correspond to previous observations made with near field pressure and the proposed theory of Malczewski and Mankbadi.⁴⁴ In the cases where the integrated pressure at St 0.15 was most reduced (i.e., $β_{0} = π / 2$ and $3 π / 2$ ), St 0.30 was the most amplified. These cases also correspond to peaks in the shear layer TKE that are smaller than that of the baseline case. The underlying reason behind this observation follows the proposed theory. Higher frequencies, thus finer structures have more efficient dissipation,⁶⁸ so as energy is transferred to St 0.30, dissipation also increases, thus resulting in a lower peak in TKE compared to the baseline case.

As described previously in Single-mode excitation case, the excitation mechanism involves energy transfer between the fundamental (St = 0.15) and harmonic (St = 0.30) modes, as seen in the evolution of TKE and SPOD analysis of fluctuations. Input energy introduced near the nozzle initially energizes fluctuations forming coherent structures. These structures absorb energy in the 5 < X/H<8 region, reducing fluctuations at St = 0.15 downstream around X/H = 10, previously identified as the dominant noise source at the maximum radiation angle (Figure 8). This redistribution aligns with the observations from Figures 34 and 35. In the bi-modal cases, streamwise amplification at St = 0.30 changes with the phase lag ( $β_{0}$ ), indicating how nonlinear energy transfer redistributes the input energy among modes. This process satisfies energy conservation by balancing input energy, redistribution dynamics, and downstream suppression effects.

Conclusions

This work began by considering the large eddy simulation of an unexcited rectangular jet as a basis for analysis of bi-modal excitation to reduce the dominant frequency noise. Near field analysis revealed St 0.15 as the most amplified. This St appeared in the far field noise as the dominant peak for low emission angles between $32 °$ and $46 °$ . This mode also appeared as a secondary peak in the far field at the peak emission angle of $50 °$ . Thus, St 0.15 was taken as the fundamental that needed reduced. A ROM from Malczewski and Mankbadi⁴⁴ predicted that fundamental-harmonic interactions can reduce the dominant coherent structure, which meant the addition of St 0.30. This was introduced by exciting the jet with a small pressure fluctuation. The noise reduction mechanism being that St 0.15 can transfer energy to its harmonic, St 0.30, and be reduced. A case of single-mode excitation was first considered where only St 0.30 was excited. A second set of cases was considered with bi-modal excitation, where both St’s 0.15 and 0.30 were forced with different phase lags with respect to one another.

Both sets of excitation cases provide strong support for the theory of Malczewski and Mankbadi.⁴⁴ Near field analysis showed that the fundamental, St 0.15, could be reduced with the addition of the harmonic, 0.30. The underlying mechanism is that energy is transferred between the modes, which was shown by amplification of 0.30 as 0.15 was reduced. The initial phase lag between the two modes was shown to have an effect in the bi-modal cases, and greater reduction of St 0.15 could be achieved depending on the phase lag. In the bi-modal cases, it is shown that the more amplified St 0.30 gets, the greater 0.15 gets reduced, further supporting the theory energy exchange between the two modes. This also corresponded to peak shear layer TKE whereby the more that St 0.30 was amplified, the lower the peak TKE. Eddy convection speed was also computed, and it is shown that all excitation cases result in reduction, which is known to reduce acoustic intensity. In the minor plane, the peak noise was not significantly reduced and was even amplified with bi-modal excitation for $β_{0} = 3 π / 2$ . At lower emission angles in the minor plane where St 0.15 was dominant, there was reduction for all cases. Investigation of those spectra showed a reduction at St 0.15. When looking at only the SPL at St 0.15, it is shown that the excitation causes substantial reduction up to 13 dB, again for $β_{0} = 3 π / 2$ . There is also a strong correlation between near and far field reductions. For all metrics evaluated in the near field, the bi-modal case of $β_{0} = 3 π / 2$ achieved the greatest reduction. This persisted in the far field with the greatest reduction in the low-angle far field noise and in St 0.15 for a broad range of directivity angles. This case of bi-modal excitation was found to be more effective than the single-mode excitation. This work ultimately concludes that the noise-causing coherent structures can be targeted and reduced by exciting with harmonics. The reduction of the targeted structure can reduce the far field noise. Forcing interaction between the fundamental and harmonic with bi-modal excitation can be more effective than single-mode, given a favorable phase lag between the two.

Footnotes

Acknowledgements

Financial support is provided by the Office of Naval Research under Grant No. N00014-21-1-2102 monitored by Dr Steven Martens. Computations were carried out using DoD HPC. The authors would like to thank Dr Chitrarth Prasad for assistance in using the SAFF LES code.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Office of Naval Research (N00014-21-1-2102).

ORCID iDs

Benjamin J Malczewski

Saman Salehian

Vladimir V Golubev

Reda R Mankbadi

References

Doychak

. Department of Navy jet noise reduction project overview. In: Partners in environmental technology technical symposium & workshop. Defense Technical Information Center, 2010.

Edgington-Mitchell

Honnery

Soria

. Staging behaviour in screeching elliptical jets. Int J Aeroacoustics 2015; 14(7): 1005–1024. DOI: 10.1260/1475-472X.14.7.1005.

Mankbadi

Liu

JTC

. Sound generated aerodynamically revisited: large-scale structures in a turbulent jet as a source of sound. Phil Trans Roy Soc Lond Math Phys Sci 1984; 311(1516): 183–217. DOI: 10.1098/rsta.1984.0024.

Mankbadi

. Multifrequency excited jets. Phys Fluid Fluid Dynam 1991; 3: 595–605. DOI: 10.1063/1.858121.

Lighthill

. On sound generated aerodynamically I. General theory. Proc Roy Soc Lond Math Phys Sci 1952; 211(1107): 564-587. DOI: 10.1098/rspa.1952.0060.

Tam

CKW

Tanna

. Shock associated noise of supersonic jets from convergent-divergent nozzles. J Sound Vib 1982; 81(3): 337–358. DOI: 10.1016/0022-460X(82)90244-9.

Tam

CKW

. Stochastic model theory of broadband shock associated noise from supersonic jets. J Sound Vib 1987; 116(2): 265–302. DOI: 10.1016/S0022-460X(87)81303-2.

Gojon

Gutmark

Mihaescu

. Antisymmetric oscillation modes in rectangular screeching jets. AIAA J 2019; 57(8): 3422–3441. DOI: 10.2514/1.J057514.

Menon

Skews

. Shock wave configurations and flow structures in non-axisymmetric underexpanded sonic jets. Shock Waves 2010; 20: 175–190. DOI: 10.1007/s00193-010-0257-z.

10.

Zaman

KBMQ

. Spreading characteristics and thrust of jets from asymmetric nozzles. In: 34th Aerospace Sciences Meeting and Exhibit, Reno, Nevada, January 15-18, 1996, p. 200.

11.

Edgington-Mitchell

Liu

, et al. A unifying theory of jet screech. J Fluid Mech 2022; 945: A8. DOI: 10.1017/jfm.2022.549.

12.

Powell

. On the noise emanating from a two-dimensional jet above the critical pressure. Aeronaut Q 1953; 4(2): 103–122.

13.

Viswanath

Johnson

Corrigan

, et al. Flow statistics and noise of ideally expanded supersonic rectangular and circular jets. AIAA J 2017; 55(10): 3425–3439. DOI: 10.2514/1.J055717.

14.

Rask

Kastner

Gutmark

. Understanding how chevrons modify noise in supersonic jet with flight effects. AIAA J 2011; 49(8): 1569–1576. DOI: 10.2514/1.J050628.

15.

Heeb

Gutmark

Kailasanath

. Investigation of chevron penetration’s effect on supersonic jet noise reduction. AIAA J 2016; 54(3): 1135–1139. DOI: 10.2514/1.J054361.

16.

Mora

Baier

Gutmark

, et al. Acoustics from a rectangular C-D nozzle exhausting over a flat surface. In: 54th AIAA aerospace sciences meeting, San Diego, California, USA, 4–8 January 2016. DOI: 10.2514/6.2016-1884.

17.

Salehian

Mankbadi

. Numerical simulation of acoustic shielding effect on supersonic jets. In: AIAA Propulsion and Energy 2019 Forum, Indianapolis, IN, 19–22 August 2019. DOI: 10.2514/6.2019-3822.

18.

Mankbadi

Salehian

. A proposed wavy shield for suppression of supersonic jet noise utilizing reflections. Int J Aeroacoustics 2021; 20(1–2): 4–34. DOI: 10.1177/1475472X20978385.

19.

Arbey

Williams

JEF

. Active cancellation of pure tones in an excited jet. J Fluid Mech 1984; 149: 445–454. DOI: 10.1017/S0022112084002743.

20.

Raman

Rice

. Axisymmetric jet forced by fundamental and subharmonic tones. AIAA J 1989; 29(7): 1114–1122. DOI: 10.2514/3.10711.

21.

Mankbadi

Shih

Hixon

, et al. Direct computation of sound radiation by jet flow using large-scale equations. In: 33rd aerospace sciences meeting, Reno, Nevada, January 9–12, 1995. DOI: 10.2514/6.1995-680.

22.

Morris

McLaughlin

Kuo

. Noise reduction in supersonic jets by nozzle fluidic inserts. J Sound Vib 2013; 332(17): 3992–4003. DOI: 10.1016/j.jsv.2012.11.023.

23.

McLaughlin

Morris

Martens

. Scaled demonstration of fluid insert noise reduction for tactical fighter aircraft engines. J Aircraft 2019; 56(5): 1935–1941. DOI: 10.2514/1.C034976.

24.

Coderoni

Lyrintzis

Blaisdell

. Noise reduction analysis of supersonic unheated jets with fluidic injection using large eddy simulations. Int J Aeroacoustics 2018; 17(4-5): 467–501. DOI: 10.1177/1475472X18778285.

25.

Cuppoletti

Gutmark

Hafsteinsson

, et al. Elimination of shock-associated noise in supersonic jets by destructive wave interference. AIAA J 2019; 57(2): 720–734. DOI: 10.2514/1.J057402.

26.

Prasad

Morris

. Effect of fluid injection on turbulence and noise reduction of a supersonic jet. Philos Trans A Math Phys Eng Sci 2019; 377(2159): 20190082. DOI: 10.1098/rsta.2019.0082.

27.

Patel

Miller

. Analysis of supersonic jet turbulence, fine-scale noise, and shock-associated noise from characteristic, bi-conic, faceted, and fluidic injection nozzles. J Acoust Soc Am 2021; 150(1): 490–505. DOI: 10.1121/10.0005626.

28.

Samimy

Adamovich

Webb

, et al. Development and characterization of plasma actuators for high-speed jet control. Exp Fluid 2004; 37: 577–588. DOI: 10.1007/s00348-004-0854-7.

29.

Samimy

Kearny-Fischer

Kim

, et al. High-speed and high-Reynolds-number jet control using localized arc filament plasma actuators. J Propul Power 2012; 28(2). DOI: 10.2514/1.B34272.

30.

Cluts

Kuo

Samimy

. Effects of excitation frequency and azimuthal mode on the flow, pressure, and acoustic fields of supersonic jets. In: 22nd AIAA/CEAS aeroacoustics conference, Lyon, France, 30 May–1 June, 2016. DOI: 10.2514/6.2016-2940.

31.

Crawley

Kearney-Fischer

Samimy

. Control of a supersonic rectangular jet using plasma actuators. In: 18th AIAA/CEAS aeroacoustics conference, Colorado Springs, CO, 04 June 2012–06 June 2012. DOI: 10.2514/6.2012-2211.

32.

Esfahani

Webb

Samimy

. Coupling modes in supersonic twin rectangular jets. AIAA SciTech forum. Virtual Event, 2021. DOI: 10.2514/6.2021-1292.

33.

Samimy

Webb

Esfahani

, et al. Perturbation-based active flow control in overexpanded to underexpanded supersonic rectangular twin jets. J Fluid Mech 2023; 959(A 13): A13. DOI: 10.1017/jfm.2023.139.

34.

Gaitonde

Samimy

. Coherent structures in plasma-actuator controlled supersonic jets: axisymmetric and mixed azimuthal modes. Phys Fluids 2011; 23: 095104. DOI: 10.1063/1.3627215.

35.

Prasad

Unnikrishnan

. Effect of plasma actuator-based control on flow-field and acoustics of supersonic rectangular jets. J Fluid Mech 2023; 964(A 11). DOI: 10.1017/jfm.2023.354.

36.

Prasad

Unnikrishnan

. Flow-field and acoustics of over-expanded rectangular jets subjected to LAFPA based control. AIAA SciTech Forum 2024: 8–12. DOI: 10.2514/6.2024-2304.

37.

Sinha

Towne

Colonius

, et al. Active control of noise from hot supersonic jets. AIAA J 2018; 56(3): 933–948. DOI: 10.2514/1.J056159.

38.

Lepicovsky

Ahuja

Salikuddin

. An experimental study of tone-excited heated jets. J Propul Power 1986; 2(2): 149–154. DOI: 10.2514/3.22859.

39.

Mankbadi

. On the interaction between fundamental and subharmonic instability waves in a turbulent round jet. J Fluid Mech 1985; 160: 385–419. DOI: 10.1017/S0022112085003536.

40.

Lee

Liu

. Multiple coherent mode interaction in a developing round jet. In: AIAA 2nd shear flow conference, Tempe, AZ, USA, 13–16 March 1989. DOI: 10.2514/6.1989-967.

41.

Dahl

Mankbadi

. Analysis of three-dimensional, nonlinear development of wave-like structure in a compressible round jet. In: 8th AIAA/CEAS aeroacoustics conference & exhibit, Breckenridge, Colorado, 17–19 June 2002. DOI: 10.2514/6.2002-2451.

42.

Dahl

Hixon

Mankbadi

. Computation of large-scale structure jet noise sources with weak nonlinear effects using linear euler. In: 9th AIAA/CEAS aeroacoustics conference and exhibit, Hilton Head, South Carolina, 12–14 May 2003, pp. 12–14. DOI: 10.2514/6.2003-3254.

43.

Mankbadi

Malczewski

Golubev

. Coherent fundamental-harmonic interactions in a compressible shear layer via integral nonlinear instability approach. AIP Adv 2022; 12: 045127. DOI: 10.1063/5.0077291.

44.

Malczewski

Mankbadi

. Modeling symmetric coherent mode interactions in a supersonic rectangular jet. AIAA J 2023; 62(4): 1550–1562. DOI: 10.2514/1.J063421.

45.

Heeb

Mora

Gutmark

. Investigation of the noise from a rectangular supersonic jet. In: 19th AIAA/CEAS aeroacoustics conference, Berlin, Germany, May 27–29, 2013. DOI: 10.2514/6.2013-2239.

46.

Visbal

Gaitonde

. High-order-accurate methods for complex unsteady subsonic flows. AIAA J 1999; 37(10): 1231–1239. DOI: 10.2514/2.591.

47.

Chakrabarti

Gaitonde

Unnikrishnan

, et al. Turbulent statistics of a hot, overexpanded rectangular jet. J Propul Power 2022; 38(3). DOI: 10.2514/1.B38073.

48.

Pletcher

Tannehill

Anderson

. Computational fluid mechanics and heat transfer. CRC Press, 2012.

49.

Gaitonde

. Analysis of the near field in a plasma-actuator-controlled supersonic jet. J Propul Power 2012; 28(2): 281–292. DOI: 10.2514/1.B34289.

50.

Gaitonde

. Analysis of plasma-based flow control mechanisms through large-eddy simulations. Comput Fluid 2013; 85: 19–26. DOI: 10.1016/j.compfluid.2012.09.004.

51.

Unnikrishnan

Gaitonde

. Acoustic, hydrodynamic and thermal modes in a supersonic cold jet. J Fluid Mech 2016; 800: 387–432. DOI: 10.1017/jfm.2016.410.

52.

Roe

. Approximate riemann solvers, parameter vectors, and difference schemes. J Comput Phys 1981; 43(2): 357–372. DOI: 10.1016/0021-9991(81)90128-5.

53.

Van Leer

. Towards the ultimate conservative difference scheme. V. A second-order sequel to godunov’s method. J Comput Phys 1979; 32(1): 101–136. DOI: 10.1016/0021-9991(79)90145-1.

54.

Pulliam

Chaussee

. A diagonal form of an implicit approximate-factorization algorithm. J Comput Phys 1981; 39(2): 347–363. DOI: 10.1016/0021-9991(81)90156-X.

55.

Tam

CKW

Webb

. Dispersion-relation-preserving finite difference schemes for computational acoustics. J Comput Phys 1993; 107: 262–281. DOI: 10.1006/jcph.1993.1142.

56.

Brentner

Farassat

. Analytical comparison of the acoustic analogy and Kirchhoff formulation for moving surfaces and Kirchhoff formulation for moving surfaces. AIAA J 1998; 36(8): 1379–1386. DOI: 10.2514/2.558.

57.

Lyrintzis

. Surface integral methods in computational aeroacoustics - from the (CFD) near-field to the (Acoustic) far-field. Int J Aeroacoustics 2003; 2(2): 95–128. DOI: 10.1260/147547203322775498.

58.

Mendez

Shoeybi

Lele

, et al. On the use of the Ffowcs Williams-Hawkings equation to predict far-field jet noise from large-eddy simulations. Int J Aeroacoustics 2013; 12(1–2): 1–20. DOI: 10.1260/1475-472X.12.1-2.1.

59.

Spalart

Shur

. Variants of the Ffowcs Williams-Hawkings equation and their coupling with simulations of hot jets. Int J Aeroacoustics 2009; 8(5): 477–491. DOI: 10.1260/147547209788549280.

60.

Welch

. The use of Fast fourier transform for the estimation of power spectra: a method based on time averaging over short, modified periodograms. IEEE Trans Audio Electroacoust 1967; 15(2): 70–73. DOI: 10.1109/TAU.1967.1161901.

61.

Shur

Spalart

Strelets

. Noise prediction for increasingly complex jets. Part I: methods and tests. Int J Aeroacoustics 2005; 4(3–4): 213–245. DOI: 10.1260/1475472054771376.

62.

Salehian

Malczewski

Mankbadi

. Coherent structure in a rectangular supersonic jet and its role in peak noise radiation. AIAA Sci Tech Forum 2024. DOI: 10.2514/6.2024-2631.

63.

Lighthill

. On sound generated aerodynamically II. Turbulence as a source of sound. Proc R Soc Lond A Math Phys Sci 1954; 222(1148): 1–32. DOI: 10.1098/rspa.1954.0049.

64.

Tam

CKW

. On the noise of a nearly ideally expanded supersonic jet. J Fluid Mech 1972; 51(1): 69–95. DOI: 10.1017/S0022112072001089.

65.

Kandula

. On the scaling laws for jet noise in subsonic and supersonic flow. In: 9th AIAA/CEAS aeroacoustics conference and exhibit, Hilton Head, South Carolina, 12–14 May 2003. DOI: 10.2514/6.2003-3288.

66.

Schmidt

Colonius

. Guide to spectral proper orthogonal decomposition. AIAA J 2020; 58(3): 1023–1033. DOI: 10.2514/1.J058809.

67.

Towne

Schmidt

Colonius

. Spectral proper orthogonal decomposition and its relationship to dynamic mode decomposition and resolvent analysis. J Fluid Mech 2018; 847: 821–867. DOI: 10.1017/jfm.2018.283.

68.

Mankbadi

Liu

JTC

. A study of the interactions between large-scale coherent structures and fine-grained turbulence in a round jet. Phil Trans Roy Soc Lond Math Phys Sci 1981; 298(1443): 541–602. DOI: 10.1098/rsta.1981.0001.