Comparative Study of Noise Control in Micro Turbojet Engines with Chevron and Ejector Nozzles Through Statistical, Acoustic and Imaging Insight

Abstract

In connection with subsonic jet noise production, this study investigates acoustic noise reduction in micro turbojet engines by comparing ejector and chevron nozzle configurations to a baseline. Through detailed statistical analysis, including assessments of stationarity and ergodicity, the current work validates that the noise signals from turbojet engines could be treated as wide-sense ergodic. This further allows to use time averages in acoustic measurements. Acoustic analysis reveals that the chevron nozzle reduces overall SPL by 1.28%, outperforming the ejector’s 0.51% reduction. Despite the inherent challenges of Schlieren imaging, an in-house code enabled a more refined analysis. By examining the fine-scale turbulent structures, one concludes that chevrons promote higher mixing rates and smaller vortices, aligning with the statistical findings of noise reduction. Schlieren imaging provided visual insight into turbulence behavior across operational regimes, showing that chevrons generate smaller, controlled vortices near the nozzle, which improve mixing and reduce noise. At high speeds, chevrons maintain a confined, high-frequency turbulence that attenuated noise more effectively, while the ejector creates larger structures that contribute to low-frequency noise propagation. Comparison underscores the superior noise-reduction capabilities of chevrons with respect to the ejector, particularly at high-speed. The enhanced Schlieren analysis allowed for new frame-specific insights into turbulence patterns based on density gradients, providing a valuable tool for identifying turbulence features and understanding jet flow dynamics.

1. Introduction

Attenuating sound in various industries, especially in aerospace and automotive engineering, is a critical concern as regulations on noise emissions become increasingly stringent (as identified for example by Airbus [1] in the early 2000s for various operating conditions and recently updated by Merino-Martinez et al. [2], theoretically described by Cumpsty [3] and Smith [4] in the 1970s and 1980s and updated by various authors, such as Moreau and Roger [5] or Sadeghian and Fleeter [6], correlated with environmental impact by Torija [7]). One of the key approaches to reduce noise is the use of passive devices, which are designed to mitigate sound without requiring external power sources (aerodynamic detuning by the replacement of alternate stator vanes with short chord splitter vanes [8], blade-tip modifications [9], entire leading edge [10,11,12] or trailing edge [13,14,15] modification using various serrations or cutouts). These devices, similar to passive blades, can be derived in elements like ejectors [16,17,18,19] and chevrons [20,21,22,23,24], which play a crucial role in noise control, especially in jet engines, where subsonic speed exhaust flows generate significant sound levels. Previous studies have examined both configurations separately [25,26], and a more in-depth statistical and acoustic analysis of both configurations would provide valuable insights. The objective of this paper is to assess the impact of two passive devices used for sound reduction in jet engines: the baseline nozzle paired with an ejector and the same baseline nozzle equipped with the chevron nozzle featuring 16 chevrons. The noise signature typically consists of a broadband noise, along with prominent spikes or tones at multiples of the blade-passing frequency. Although these discrete frequency tones may not significantly affect the overall noise level, they are the primary source of the irritating screech noise commonly associated with turbojets. Both studied configurations were found to be effective in reducing the jet noise signature [27]. Having no moving parts, minimizing jet noise and increasing thrust in an aircraft, the two above-mentioned passive devices need little maintenance and have low operational costs, coupled with no restriction on the working fluids and a long service life [28,29]. The chevron nozzles increase the turbulent shear layer’s rate of mixing [30,31,32], which is the layer of air between the hot, moving exhaust gas stream from the engine’s inner flux and the cold secondary air flux around the engine core. Chevron nozzles contribute to a significant reduction in jet noise and pressure variations by enhancing streamwise vortices that increase mixing within the plume to speed up the decay of the jet potential core. Enhancing mixing zones typically intensifies motion at smaller scales, which amplifies high frequency noise (e.g., the effects of different nozzle heads on turbulent jet noise [33]; triangular chevrons’ influence on acoustic spectra for several gas turbine engine regimes [20]; high bypass ratio engine gas turbine noise [34], the chevron-induced axial vorticity, which enhances mixing in the jet shear layer [35]; axial vortices induced by chevron serrations [23]; flow visualization for a fairly simple toothed nozzle [24] or flow mixing in a supersonic regime [16]).

The interaction between a plasma jet and targets with different physical properties was investigated by means of Schlieren optics [36]. Schlieren academic applications on acoustics allow visualizing sound itself with the help of this visualization technique [37,38,39]. Methods like Proper Orthogonal Decomposition (POD) [40] which is effective for identifying dominant coherent structures and their energy distribution, Dynamic Mode Decomposition (DMD) [41], which excels in tracking dynamic flow features and their temporal evolution, or Particle Image Velocimetry (PIV) [42], which offers richer visual or quantitative insights into extracting velocity field data, demand complex setups and computational resources. Instead, in this paper, the traditional Schlieren imaging paired with a code-based image processing is adopted, which represents a practical trade-off in directly linking flow patterns to acoustic measurements. This combination offers several advantages compared to modal extraction methods by providing a straightforward, visually intuitive representation of flow structures by highlighting iso-density gradients and flow features. The iso-level segmentation and contour mapping are tailored specifically to enhance Schlieren data, making it more accessible for qualitative and semi-quantitative visual insights into the interplay between turbulence and density gradients, which align well with the acoustic results and noise-reduction mechanisms explored in this paper. Unlike POD or DMD, this combination does not decompose the flow into modes, which could limit its ability to isolate dominant coherent structures.

In this study, ejector and chevron nozzle configurations are installed at the exhaust nozzle of a micro turbojet engine. The analysis focuses on the maximum operating regime, defined by the rotational speed, as it is the most critical. The baseline configuration (without chevrons or other passive control technique applied), the ejector configuration and the configuration with 16 triangular-shaped chevrons are examined. In addition to optical diagnostics, sound levels in the vicinity of the engine are recorded and subsequently processed for analysis. This paper explores the mechanics and effectiveness of these passive noise-reduction devices, emphasizing the performance of ejectors and chevrons in mitigating sound. Through comparative analysis and sound pressure level (SPL) measurements, the efficiency of these devices in reducing noise emissions, particularly in high-performance applications, such as jet engines, is assessed.

2. Theory Review

In engineering applications, recorded signals often display complex, unpredictable variations due to the stochastic nature of turbulent, high-speed flows. These random signals cannot be described by deterministic functions, necessitating statistical methods for modeling and interpretation. Key techniques include time averaging, which averages a signal over time, and ensemble averaging, which uses multiple samples of the same signal type. Parameters like the probability density function and autocorrelation function characterize these signals, describing amplitude variations and statistical evolution over time.

Understanding concepts such as random variables, stationary and ergodic processes and power spectral density (PSD) is critical for effective signal analysis. While a detailed explanation of these concepts is beyond the scope of this paper, readers are encouraged to consult references [43,44,45] for further information. Stationary processes, with time-invariant statistical properties, simplify tasks like filtering and spectral analysis. However, real-world signals are often non-stationary, requiring careful assessment of stationarity assumptions to ensure valid results.

This paper explores statistical testing of experimental data to identify patterns and trends, analyzing acoustic signals from an ejector nozzle and 16 chevron nozzles. The results demonstrate that trimmed signals are adequate for calculating acoustic metrics, even though real signals often exhibit non-stationary behavior. The study emphasizes verifying stationarity assumptions for meaningful analysis.

Stationarity. Ergodicity. Statistical analysis is essential when working with real data, as they represent random signals and processes. A key focus is on ergodic signals, where statistical properties can be derived from a single realization over time rather than averaging across multiple realizations. For a process to be ergodic, it must be strictly stationary, though not all stationary processes are ergodic. Ergodicity allows determining moments like mean and autocorrelation through time or ensemble averages, making it a reasonable assumption for many physical processes unless specific exceptions arise.

In this study, the non-stationary portions of signals, such as engine start and stop phases, are excluded to ensure reliable results, like calculating SPL for the engine’s highest regime. While strict ergodicity cannot be proven, a wide-sense assumption is often reasonable.

For a process $p\left(t\right)$, wide-sense stationarity (WSS) requires a constant mean, $\mu \left(t\right)$, a time-independent variance, ${\sigma }^2\left(t\right)$ and an autocorrelation function $R_{PP}\left(\tau \right)$ dependent only on time lag $\tau$. A WSS process is ergodic if time and ensemble averages are equivalent, simplifying the analysis to specific metrics like mean and autocorrelation.

This paper first verifies WSS and ergodicity before extracting meaningful insights from experimental data, demonstrating their significance in signal analysis.

Acoustics metrics. In this acoustics data analysis, metrics like time-variant sound pressure level $L_p\left(t\right)$, X-weighted equivalent continuous sound pressure level $L_{\mathrm{Xeq},T}$, (e.g., often X-weighted A, C, Z), peak sound pressure level $L_{peak}$ and maximum sound pressure level $L_{\max }$ are calculated to understand sound behavior over time, assess risks and take precautions to safeguard human health and safety. When the integration period is the entire period of discussion(e.g., the trimmed period for which the process is ergodic), $T=T_0$, then one refers to overall X-weighted sound pressure level, denoted by $\mathrm{SPL}(X)$ (i.e., X = A, C, Z) or OXSPL.

The WSS random process does not have finite energy and, hence, does not possess a Fourier Transform (FT) [39]. Such signals have finite average power and therefore Power Spectrum Density (PSD) analysis is used to investigate if signals that show a correlation in the time domain are also correlated in the frequency domain. According to the Wiener–Khintchine–Einstein theorem [40], the PSD[W/Hz] for WSS random process $p\left(t\right)$, denoted by $S_{PP}\left(f\right)$, is the FT of the signal autocorrelation function:

$$S_{PP}\left(f\right)={\displaystyle \underset{-\infty }{\overset{\infty }{\int }}{R}_{PP}(\tau )e^{-i2\pi f\tau }d\tau }$$

(1)

can be determined from the following relation:

$$S_{PP}\left(f\right)=\underset{T\to \infty }{\lim }{\displaystyle \frac{1}{T}}E \left[{\left|{\displaystyle \underset{-T/2}{\overset{T/2}{\int }}p(t)e^{-i2\pi ft}dt}\right|}^2\right]=\underset{T\to \infty }{\lim }{\displaystyle \frac{1}{T}}E \left[{\left|{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}{p}_T(t)e^{-i2\pi ft}dt}\right|}^2\right]=\underset{T\to \infty }{\lim }{\displaystyle \frac{1}{T}}E \left[{\left|P_T\left(f\right)\right|}^2\right]$$

(2)

where $P_T\left(f\right)$ is the Fourier transform of the truncated version of the process for which $p_T\left(t\right)=p\left(t\right),t\in \left[0,T\right]$, and $p_T\left(t\right)=0,t\notin \left[0,T\right]$.

3. Experimental Test Bench

In the current study, a JetCAT P80 micro turbojet engine [26], which is part of the Aerospace Engineering Faculty’s equipment, was employed. The engine (see Figure 1) is operated by a command-and-control panel provided by GUNT Germany that is responsible for engine startup and throttling. It is also instrumented with several transducers for thrust, combustion chamber pressure and temperature, outlet compressor temperature, fuel mass flow, air mass flow and shaft rotational speed. These parameters are displayed instantaneously at run time on dedicated displays installed on the command box, but they are also recorded with 1 Hz acquisition frequency via dedicated PC software. The data post-processing allows for precise characterization of the working regimes, either steady or transient.

Acoustic measurements are performed using a ½ 40 AQ microphone (from G.R.A.S. in Holte, Denmark) placed on a tripod 0.4 m from the engine, as shown in Figure 2. This high-precision condenser microphone is designed according to IEC 61094-4 [46] standards and is optimized for accurate sound measurements in random, diffuse and reverberant environments. It offers exceptional durability and reliability, capable of measuring sound pressure levels from 3.15 Hz to 16 kHz, with a maximum of 148 dB. Each 40 AQ microphone is individually calibrated at the factory and comes with a calibration chart detailing its unique open-circuit sensitivity and pressure frequency response. The 40 AQ is specifically engineered to provide precise sound-level measurements in diffuse sound fields, which can occur when multiple sound sources or reflective surfaces are present.

The optical diagnosis of the exhaust jet plume is performed by a Z-type Schlieren setup. The system comprises two f/6 parabolic mirrors with 60″ focal length, an adjustable slit diaphragm (knife) for light cutoff, a point light source and a Phantom VEO710 high-speed camera. For sufficient time resolution, the camera is operated at 48 k fps at 320 × 320 resolution with 1 µs exposure time. To limit the optical distortions, the off-axis beam angles are set below 20° [26,35]. The experimental test bench with the Schlieren system aligned at the exhaust of the micro turbojet engine is presented in Figure 3.

3.1. Ejector Nozzle Configuration

The ejector is a device that creates a secondary airflow due to the phenomenon of ejection. Ejection refers to the process by which a passive fluid is entrained by an active fluid. This is possible because the primary fluid creates a low-pressure area, turning that zone into a suction zone, drawing in surrounding air with higher pressure and thus generating the secondary jet. In jet engines, the ejector is typically mounted as an extension of the reaction nozzle, where it helps entrain air from the atmosphere to form a secondary flow, which mixes with the primary exhaust flow, leading to a single, quieter exhaust stream. The ejector depicted in Figure 4a was accommodated at the exhaust of the JetCAT P80 micro turbojet engine.

3.2. Chevron Nozzle Configuration

Chevrons are serrated patterns applied to the trailing edges of engine nozzles. Their primary function is to smooth out the mixing of high-speed exhaust gases with the ambient air, thus reducing turbulence and, consequently, noise. By controlling the interaction between the two airflow streams, chevrons help in reducing both high-frequency and low-frequency noise, making them a common feature in modern engine designs.

The main chevron parameters which geometrically describe them are chevrons number$N$, chevron length $L$ and the immersion angle $I$ [26,43]. According to the study presented in [44], it is recommended that the chevron’s length be within 5 and 10 percent of the equivalent diameter [26]. The main dimensions of the chevron nozzle, as derived from the baseline one, are reported in Figure 5a,b. The micro turbojet engine with 16-chevron nozzle installed is presented in Figure 5c.

4. Results and Discussions

In the current experimental campaign, the turbojet engine was operated at a high RPM regime (approximately 115,000 RPM) for all the tests under analysis. At this operating regime, the exhaust mass flow rate is approximately 0.187 kg/s, with an outlet Mach number around 0.53 (based on ideal cycle computations assuming optimum expansion) corresponding to a 291 m/s exhaust velocity. The experiments were conducted using the Jet CAT P80 [45] micro turbojet engine described in the previous section with all its auxiliary optical and acoustic instrumentation. This engine, in the same instrumented arrangement, has been utilized in previous experimental campaigns and research efforts [47,48]. Three exhaust configurations were employed, namely the baseline one with flat nozzle, the chevron nozzle and the ejector augmented system.

The issue of noise reduction for a micro turbojet engine is extremely difficult due to its small dimensions, which could not allow to reduce noise by producing jets with greater mass flow but lower velocity using larger nozzles. To reduce noise, we will consider two different configurations: the case of an ejector engine [25] and the case of a 16-chevron engine [26]. The cited publications conducted just a basic spectral analysis in which the random signal was assumed to be ergodic without any supporting proof. Therefore, we believe that to find new characteristics and patterns in the signal noise, a much deeper signal analysis is needed to finish the study.

4.1. Ejector Nozzle

The first dataset analyzed involves microphone recordings near a micro turbojet in two configurations: baseline engine and engine with an ejector. Data were collected at 115,486 rpm (baseline) and 115,179 rpm (ejector) at a sampling frequency of $f_s=\mathrm{48,000} \mathrm{samples}/\mathrm{s}$. Both signals, shown in Figure 6, represent 300 s of recording (the engine is started, reaches the maximum regime and stops).

To calculate acoustic metrics, the most stationary portions of these non-stationary signals were extracted during maximum regime of the engine: [130–165] seconds for the baseline and [100–150] seconds for the ejector. The segments’ durations differ, but each was selected to satisfy stationarity and ergodicity tests. Once verified, the extracted data were used for further processing, treating the signals as wide-sense ergodic stationary processes.

The procedure to check the stationarity and ergodicity of a general sampled random signal, ${\left(p\left(t_i\right)\right)}_{i=1,N}$ on a given period of time, $T_0={\displaystyle \frac{N}{f_s}}$ is as follows:

• Compute the mean, the variance, the standard deviation and the autocorrelation for the entire signal of length $N$;
• Divide the signal into $M$ overlapping segments (n.b. the so-called shorter realizations) with 50% overlapping. Each segment should be of a reasonable length to capture meaningful data behavior. One chooses the segment length in the range of $N_s=\left[{10}^4,{10}^5\right]$ samples, with step ${10}^4$;
• Examine if the random signal is stationary if the conditions presented in the theory section are satisfied. The expected value, the variance and the autocorrelation functions of the random signal in a discrete formulation are considered uniformly distributed. Numerically, it means that across all $M$ segments of size $N_s$, at the same time that $t_k, k=1,N$, one calculates the mean and the variance with the same probability (n.b. independent of time):

  $$\begin{array}{c}\mu \left(t_k\right)=E\left[p(t_k)\right]={\displaystyle \frac{1}{M}}{\displaystyle \sum _{n=1}^M{P}_n\left(t_k\right)},\hfill \\ {\sigma }^2\left(t_k\right)=E\left[{\left(p(t_k)-\mu \left(t_k\right)\right)}^2\right]={\displaystyle \frac{1}{M}}{\displaystyle \sum _{n=1}^M{\left[P_n\left(t_k\right)-\mu \left(t_k\right)\right]}^2}\hfill \end{array}$$

  (3)

  where $p_n\left(t\right)$ represent the signal’s value at the n-th realization of the random process $p\left(t\right)$. In the acoustic field, the signal represents the fluctuation in the pressure with regard to standard atmospheric pressure. Thus, one expects that the constants for the expectation and variance of the random process to be zero or very close to zero. As it is impossible to get the value zero in a real case, one should verify if the mean and the standard deviation are below a certain tolerance $\epsilon$ for all the random variables at each time:

  $$t_k\left|\mu \left(t_k\right)\right|\le \epsilon , \sigma \left(t_k\right)\le \epsilon k=1,N$$

  (4)

The examination continues with the discrete autocorrelation function for different time lags ($N_{lag}\le N_s$) and for each segment (realization) $n$ of size $N_s$ given by

$$R_n\left({\tau }_{k+1}\right)={\displaystyle \frac{1}{N_s-k}}{\displaystyle \sum _{j=1}^{N_s-k}{P}_n\left(t_j\right)}{P}_n\left(t_j+{\tau }_{k+1}\right),k=0,N_{lag}$$

(5)

where the ${\tau }_k=k/f_s$ is the time lag. The mean of the autocorrelation function, at each delay ${\tau }_k$, across all $M$ realizations, is given by

$$R\left({\tau }_{k+1}\right)={\displaystyle \frac{1}{M}}{\displaystyle \sum _{n=1}^M{R}_n\left({\tau }_{k+1}\right)},k=0,N_{lag}-1$$

(6)

and the condition to be a stationary process is expressed as follows:

$$\left|R\left({\tau }_{k+1}\right)\right|=ct\left(k\right),k=0,N_{lag}-1$$

(7)

This condition should be understood in the following way: the mean of the autocorrelation function of a signal is constant for the same delayed version of itself. However, the constant could depend on different lags.

If the signal is stationary, one could check if the random signal is ergodic. Essentially, all the time averages and statistical averages have to be equal at the limit with probability. However, in reality, only the mean, variance and autocorrelation functions are of relevance to us. So, we have to implement the following inequalities which must be satisfied simultaneously:

$$\left|\mu \left(t_k\right)-\overline{p\left(t\right)}\right|\le \epsilon ,\left|{\sigma }^2\left(t_k\right)-\overline{{\left[p\left(t\right)-\overline{p\left(t\right)}\right]}^2}\right|\le \epsilon , k=1,N$$

(8)

$$\left|R\left({\tau }_{j+1}\right)-\overline{p\left(t\right)p\left(t+{\tau }_{j+1}\right)}\right|\le \epsilon , j=0,N_{lag}-1$$

(9)

where the time averages over the entire signal are approximated with the finite sums:

$$\overline{p\left(t\right)}={\displaystyle \frac{1}{N}}{\displaystyle \sum _{j=1}^{N}p\left(t_j\right)},\overline{{\left[p\left(t\right)-\overline{p\left(t\right)}\right]}^2}={\displaystyle \frac{1}{N}}{\displaystyle \sum _{j=1}^N{\left[p\left(t_j\right)-\overline{p\left(t\right)}\right]}^2}$$

(10)

$$\overline{p\left(t\right)p\left(t+{\tau }_k\right)}={\displaystyle \frac{1}{N}}{\displaystyle \sum _{j=1}^{N}p\left(t_j\right)p\left(t_j+{\tau }_k\right)}, k=0,N_{lag}-1$$

(11)

The root mean square error (RMSE) between two arrays $x\left(k\right),y\left(k\right),k=1,N$

$$RMSE\left(x,y\right)=\sqrt{{\displaystyle \frac{1}{N}}{\displaystyle \sum _{k=1}^N{\left[x\left(t_k\right)-y\left(t_k\right)\right]}^2}}$$

(12)

is a discrete version of RMS, which is frequently employed in analytics to compare two arrays, such as time averages and statistical averages. This is because the squaring of the values tends to exaggerate greater disparities, making it particularly beneficial when we want to draw attention to significant discrepancies. Therefore, from now on, we shall employ RMSE rather than conventional norms when comparing distinct arrays.

4.1.1. Stationarity and Ergodicity

Table 1. Mean, variance and standard deviation for both configurations.

	Segment Length [Samples]	Overlapped Segments	Mean for All Segments	Standard Deviation of the Mean	Mean of the Variance of All Segments	Standard Deviation of the Variance
B	10,000	335	−0.0000022	0.00476377	0.00789353	0.0005773
	20,000	167	−0.0000011	0.00677244	0.00787076	0.0008311
	40,000	83	−0.0000017	0.00973911	0.00782580	0.0011910
	80,000	41	−0.0000015	0.01347090	0.00774198	0.0016813
	100,000	32	−0.0000022	0.01558355	0.00768273	0.0019063
E	10,000	479	0.0000068	0.00371549	0.00652053	0.0004123
	20,000	239	0.0000066	0.00521679	0.00650899	0.0005810
	40,000	119	0.0000072	0.00745629	0.00648578	0.0008262
	80,000	59	0.0000061	0.01070496	0.00643085	0.0011721
	100,000	47	0.0000055	0.01172369	0.00640985	0.0013073

evaluates whether the process can be regarded as wide-sense stationary for both the baseline (B) and ejector nozzle (E) configurations by presenting the mean, variance and standard deviation of the segments.

The absolute mean for all segments is less than 10⁻⁵, which is close to zero, indicating the data are centered around zero. This could suggest that the signal oscillates around zero, with no significant drift (n.b. which is true as the data are normalized to the maximum value). The variance of all segments is relatively stable and is less than 0.008, indicating that the spread of the data remains consistent across different number of segments.

The standard deviation of the mean increases slightly, suggesting moderate variability in the signal over larger segments. A low standard deviation of the variance indicates consistent spread within segments, while an increase reflects differing spread across segments.

The variance and autocorrelation values fluctuate within 0.002, indicating minimal variability. Signals with near-zero mean and variance are centered and show little fluctuation, suggesting relative constancy.

Overall, if the mean and variance of a signal are both close to zero, it indicates that the signal is centered around zero and exhibits very little variability or fluctuation.

This could imply that the signal is relatively flat or constant, lacking significant changes or dynamics. Now, regarding the autocorrelation values, they are the represented along with the overall mean and overall variance for each segmentation (n.b. from Table 1) in Figure 7. A total of $N_{lag}=100$ lags is considered, represented by ${\left({\tau }_k\right)}_{k=0,N_{lag}}$, and we randomly select 10 lags to represent the mean of the autocorrelation for each segmentation based on segment length. The graphical representation shows that the mean remains nearly constant across all segmentations, with small fluctuations suggesting minimal repeating patterns or periodic behavior. Thus, if the mean, variance and autocorrelation remain stable over time, the signals can be reasonably considered stationary, or at least stationary in the wide sense.

The analysis confirms that the signals demonstrate characteristics of wide-sense ergodicity (WSE), which implies they are also wide-sense stationary (WSS). This property allows statistical parameters, such as mean, variance and autocorrelation, to be derived from a single sample function, making it feasible to analyze real-world signals where ensemble averaging is impractical.

From

Table 2. Pressure time averages over the entire trimmed signal.

Configuration	Mean	Variance
B	−0.00000068	0.00791683
E	0.00000668	0.00653186

, the comparison between time averages and ensemble averages shows that the RMSE values for mean and variance are sufficiently small. This indicates strong agreement between these averages, validating the ergodic nature of the signals in terms of their central tendency and spread.

Table 3. Verifying the ergodicity using root mean square error (RMSE) for statistical quantities.

Configuration	RMSE Mean	RMSE, Variance	RMSE, Autocorrelation	Is the Signal Ergodic?
B	0.00000131	0.00014134	0.00016418	YES
E	0.00000065	0.00007448	0.00057095	YES

further examines autocorrelation, revealing slightly higher RMSE values compared to the mean and variance. While this suggests minor variations in the evolution of signal patterns, the values remain small enough to support the conclusion that the signals are ergodic, or at least ergodic in a wide sense.

Overall, the results justify treating the signals as wide-sense ergodic, allowing time averages to be reliably used for further acoustic analysis.

To conclude, the trimmed selected signals from the original ones are ergodic, and all the results concerning acoustic quantities could use just the trimmed data signal. Assuming ergodicity, one can further assert that the signal is not merely a hypothesis but rather a predictable outcome based on the theory. Nevertheless, the fact that ergodicity analysis might be more complex and involve additional aspects is not ignored. However, the presented procedure can be used as a basic approach to examine ergodicity based on average behavior.

4.1.2. Time and Frequency Domain Analysis

The comparison between (B) and (E) cases in

Table 4. Acoustic time metrics for baseline and ejector engine.

Type	SPL(Z) [dB]	OASPL [dB(A)]	SPL(C) [dB(C)]	LAeq [dB(A)]	LCeq [dB(C)]	LZmax [dB]	LAmax [dB(A)]	LCmax [dB(C)]
B	118.49	117.33	118.08	117.34	118.08	119.59	118.15	119.27
E	117.90	116.47	117.60	116.47	117.60	119.91	117.66	119.73
%	0.51%	0.74%	0.41%	0.74%	0.42%	0.27%	0.41%	0.39%

Green—enhanced performance for ejector configuration; red—detrimental performance for ejector configuration.

shows that the (E) reduces OXSPL and L_Xeq for all weightings X = A, C, Z. Thus, the SPL(Z) (zero-weighted for all frequencies) shows a reduction of 0.51%, while in the human frequency sensitive hearing range the OASPL decreases by 0.74%. Similarly, the SPL(C) shows a smaller reduction of 0.41%, reflecting a decrease in low-frequency range noise.

However, the peak SPL values (L_Zmax and L_Cmax) show slight increases of 0.27% and 0.39%, respectively, in the (E) case. This suggests that while the average noise levels decrease, occasional high-intensity noise spikes may still occur. The L_Amax exhibits a minor reduction of 0.41% in peak noise perceived by humans. Overall, the ejector configuration effectively reduces general noise levels but may not completely eliminate peak sound events. Figure 8a,b and Figure 9a,b show the time-variant sound pressure level $L_p\left(t\right)$ and the three time-average sound pressure level, $L_{\mathrm{Aeq},T}$ $L_{\mathrm{Ceq},T}$ and $L_{\mathrm{Zeq},T}$ considering a period of time $T=0.125s$ for capturing short-term variations in sound levels. As a quantitative observation, the time evolution of $L_p\left(t\right)$ provides no meaningful information, as it is nearly identical for both configurations. However, when looking at the time-equivalent X-weighted sound pressure levels, one can observe a significant improvement, as the values for the ejector are consistently just under 118 dB. Figure 8c and Figure 9c depict the power spectrum using the periodogram to identify what frequencies remain along the test period. The PSD provides valuable information about the distribution of signal power across different frequencies, especially for understanding the frequency content of a signal. In Figure 8d,e and Figure 9d,e, by means of the PSD, the spectrum SPL is computed in four different ways. For numerical implementation, the data are divided into $N_{seg}$ overlapped segments (n.b. 50% overlapping between the segments), and the power spectral density is calculated in four different ways: (a) using the PSD provided by the spectrogram routine with a Hamming window of $N_f$ number of samples, (b) using the short-time Fourier transform from the spectrogram routine with a Hamming window of the same sample size, $N_f,$ (c) using the Welch periodogram and (d) using an in-house routine. The discrete expression of PSD is given by

$$PSD(f_k)={\displaystyle \frac{1}{T}}{\left|\Delta T{p}_T(n)e^{-j2\pi f_{k}n\Delta T}\right|}^2={\displaystyle \frac{1}{N\Delta T}}{\left|\Delta T{P}_T\left(f\right)\right|}^2$$

(13)

where the time step is $\Delta T=1/f_s$, and $f_s$ is the frequency sampling. Equation (15) is adapted for each segment of length $T_w$ having $N_w$ samples. Thus, the total power spectral density averaged over $N_{seg}$ segments is given by

$$PSD(f)={\displaystyle \frac{1}{N_{seg}}}{\displaystyle \sum _{r=1}^{N_{seg}}\frac{{\left|\Delta T{X}_r(f)\right|}^2}{T_w}}={\displaystyle \frac{1}{N_{seg}}}{\displaystyle \sum _{r=1}^{N_{seg}}\frac{{\left|X_r(f)\right|}^2}{f_s{N}_w{U}_w}}$$

(14)

where $U_w$ is the power of the window function needed for truncation. Note that for a rectangular window $U_w=1$. Finally, the spectrum sound pressure level $SPL\left(f\right)$ can be determined by

$$SPL(f)=10\lg \left[{\displaystyle \frac{1}{p_{ref}^2{N}_{seg}}}{\displaystyle \sum _{r=1}^{N_{seg}}\frac{{\left|X_r(f)\right|}^2}{f_s{N}_w{U}_w}}\right]$$

(15)

In Figure 8d and Figure 9d, the four algorithms gave very similar results, starting from 50 Hz. Comparing Figure 8d,e and Figure 9d,e, it is noticeable that a frequency appears around 11528 Hz. It is also visible in both the periodogram, Figure 8c and Figure 9c, and the frequency spectrum in Figure 8f and Figure 9f. The explanation for this is that the signal selection was made at the maximum regime of the turbo microjet, operating at a rotational speed of 115,179 RPM (n = 1919 rev/s) for the baseline and 115,486 RPM (n = 1924 rev/s) for ejector. The micromotor is equipped with a centrifugal compressor with B = 6 blades, and the theoretical frequency is $f_{teoretical}=B\cdot n=\mathrm{11,549} \mathrm{Hz}$. The theoretical frequency calculated based on the RPM and the compressor’s blade count aligns closely with the measured frequency, confirming that this feature is a result of the compressor’s operation. To conclude, the detected frequency is a characteristic signature of the centrifugal compressor at the specified operating conditions. Moreover, the ejector increases slightly the magnitude of the spectrum SPL from 83 dB to 86 dB, which is not a bad thing because frequency is higher than 8 kHz. On the other hand, in the case of peaks in the lower frequency range, a decrease in SPL is observed. For example, at a frequency of 270 Hz, there is a peak of 102.4 dB in the baseline configuration, while in the configuration with the ejector, at approximately the same frequency, it is 101.8 dB. Similarly, at 1289 Hz, which is the fundamental harmonics of the compressor, the baseline shows an SPL = 98.3 dB (n.b. the magnitude of the spectrum is 0.547 Pa), whereas the ejector configuration registers SPL = 94.5 dB (n.b. the magnitude of this peak is decreased up to 0.226 Pa).

The final analysis of this topic focuses on the observation that, at first glance, Figure 8d and Figure 9d reveal very similar results across all four algorithms. However,

Table 5. SPL comparison obtained in time and frequency domain for baseline and ejector engine.

Configuration	SPL(Z) [dB]	SPL[dB] Spectrogram Power	SPL[dB] STFT from Spectrogram	SPL[dB] Welch Periodogram	SPL[dB] In-House
B	118.49	118.49	118.75	118.49	118.49
E	117.90	117.90	118.15	117.90	117.90

shows very small differences in the SPL values obtained using the four methods for both configurations, (B) and (E).

Although the values are extremely close, the spectrogram and Short-Time Fourier Transform (STFT) methods tend to give slightly higher values, especially for configuration “E”, while the in-house code, the power spectrogram and the SPL(Z) calculated using the pressure fluctuations, provide almost identical results.

4.2. Chevron Nozzle

4.2.1. Stationarity and Ergodicity

For the sake of brevity, the signal selection procedure used to check that the data are stationary and ergodic is skipped over, as the outcomes are quite similar for all setups.

4.2.2. Time and Frequency Domain Analysis

Similar to the last topic, where the baseline and the ejector configurations were examined, this section analyzes experimental data, in both the time and frequency domains, from the other two configurations—the baseline nozzle and the nozzle with 16 chevrons—at the same maximum operating regime of the microjet engine. Finally, the configurations are compared to identify the most effective noise reduction. Comparing the 16-chevron nozzle to the baseline,

Table 6. Acoustic time metrics for baseline and 16-chevron nozzle.

Type	SPL(Z) [dB]	OASPL [dB(A)]	SPL(C) [dB(C)]	LAeq [dB(A)]	LCeq [dB(C)]	LZmax [dB]	LAmax [dB(A)]	LCmax [dB(C)]
B	118.42	117.80	116.68	117.80	116.68	118.81	118.27	117.13
C	116.89	116.37	115.34	116.37	115.33	117.75	117.40	116.31
%	1.28%	1.21%	1.15%	1.21%	1.15%	0.89%	0.73%	0.70%

Green—enhanced performance for chevron configuration.

demonstrates a considerable reduction in noise. OASPL (A-weighted) declines by 1.21%, and SPL(Z) drops by 1.28%, indicating less noise in the region that is most audible to humans. SPL(C), or low-frequency noise, drops by 1.15 percent. The chevrons effectively reduce sound levels in both the low-frequency range (C-weighted) and the human hearing range (A-weighted), by 1.21% and 1.15%, respectively, shown by both L_Aeq and L_Ceq, which integrate sound levels over time with the fast time weighting (using a 125 ms time constant). L_Zmax, L_Amax and L_Cmax all decrease by 0.89%, 0.73%, and 0.70%, respectively, at peak noise levels, indicating that the chevrons aid in mitigating peak noise events.

Figure 10a,b and Figure 11a,b present the time-variant sound pressure level $L_p\left(t\right)$ and the three time-averaged sound pressure levels $L_{\mathrm{Aeq},T}$ $L_{\mathrm{Ceq},T}$ and $L_{\mathrm{Zeq},T}$ over a defined period to capture short-term variations in sound levels. The nozzle with 16 chevrons shows a clear reduction in SPL across all X-weighted levels (LXeq) compared to the baseline. The reductions range from approximately 1 dB to 1.8 dB, particularly in the A-weighted and C-weighted frequencies. The most noticeable frequency appears around 10,175 Hz, which is detected in both the periodogram from Figure 10c and Figure 11c and in the frequency spectrum (Figure 10d–f and Figure 11d–f). This can be explained similarly to the case of the ejector nozzle, where the signal was selected at the maximum operating regime of the turbo microjet, running at a rotational speed of 101,520 RPM (or 1692 revolutions per second), as the microjet engine is equipped with a centrifugal compressor with six blades, giving a theoretical frequency of 10,152 Hz. The fundamental harmonics of the centrifugal compressor are located at 1700 Hz (n.b. with the magnitude of the spectrum of 1.2 Pa) for the baseline configuration and 1685.21 Hz (n.b. with the magnitude of the spectrum of 0.99 Pa) for the chevron nozzle, further confirming the relationship between these frequencies and the operating characteristics of the compressor.

Similar to the ejector, the baseline and the chevron nozzle configurations are compared by looking at the Sound Pressure Level (SPL) readings from the Baseline (B) and Chevron (C) setups using various techniques.

Table 7. SPL comparison in time and frequency domains for baseline and chevron configurations.

Type	SPL(Z) [dB]	SPL [dB] Spectrogram Power	SPL [dB] STFT from Spectrogram	SPL[dB] Welch Periodogram	SPL [dB] In-House
B	118.42	118.42	118.67	118.42	118.42
C	116.89	116.88	117.14	116.88	116.88

reveals small but measurable differences in the overall SPL values produced by these techniques.

The SPL calculated from the pressure signal serves as the reference for comparing FFT-based techniques. The in-house method, Welch periodogram and spectrogram power methods produce nearly identical results, confirming their reliability for consistent SPL calculations. The STFT method, while close, shows slightly higher SPLs, especially for configuration (C), likely due to its sensitivity to transient features and short-duration variations.

4.3. Schlieren Imaging Analysis

In the following section, the processing of images obtained using the Schlieren technique is addressed. A significant challenge with this visualization technique is primarily because Schlieren flow visualization typically provides only an external or integrated view of the flow with density gradients, being challenging to extract relevant information of velocity or turbulence data directly. Moreover, the resulting images are sensitive to setup alignment and become complex to interpret without advanced image processing.

Despite the limitations of the Schlieren technique, the authors previously disseminated some interesting findings in [26], along with preliminary conclusions on the impact of chevrons on the flow pattern relative to the baseline configuration, solely using image processing with the Phantom VEO710 camera software (PCC 3.9.40). To explore whether the Schlieren approach can provide additional insights into flow dynamics, a code for image processing was developed. This method offers a new perspective by creating color contours of equal-density gradients across grayscale subdomains (range: [0, 255]), enabling a more detailed analysis of density variations. While Schlieren imaging primarily highlights density gradients, this enhanced processing allows us to infer flow characteristics indirectly.

The development of this method included several steps, specifically designed for this study, to enhance the extraction of flow characteristics:

• Images were normalized to enhance contrast across the density gradient spectrum, allowing for a more precise delineation of flow structures;
• Grayscale intensity values were segmented into multiple iso-levels, each corresponding to specific density gradients, enabling detailed visualization of subtle flow features;
• An apropiate color scheme was applied to the iso-levels, providing visual clarity and distinguishing fine-scale turbulence from larger coherent structures;
• Contours were superimposed on the grayscale images to highlight regions of interest, such as shear layers and potential flow areas, automatically generating consistent visualizations for all configurations.

The processing workflow allowed the extraction and comparison of multiple consecutive frames, identifying features such as the formation, interaction and dissipation of vortices, even when subtle differences were not visible in raw images. The novelty lies in the tailored combination of normalization, segmentation and visualization techniques, which provided enhanced insights into turbulence features, such as vortex formation, interaction and dissipation. These tools proved particularly valuable for comparing configurations like chevrons and ejectors, which exhibit subtle but critical differences in flow behavior. The aim of the current work is to improve the recognition of flow features, such as counter-rotating streamwise vortices and coherent structures, while identifying factors contributing to noise reduction observed in the acoustic analysis. This study compares the flow characteristics of two different nozzle configurations, including baseline and chevron nozzles across three operating regimes. Additionally, it examines chevron nozzles with and without ejectors, focusing on the maximum regime due to limited data availability for other cases. This refined approach strengthens the ability to validate and expand upon conclusions from previous research.

The experiments were conducted using a micro turbojet in its baseline configuration, equipped with the standard nozzle, to capture Schlieren visualizations of the jet at operational conditions of 35k, 55k and 115k RPM. Subsequently, a nozzle with 16 chevrons replaced the baseline nozzle, and measurements were repeated under identical conditions. Finally, an ejector was mounted while retaining the baseline nozzle, and measurements were performed for this configuration exclusively at 115k RPM.

The frames were captured at intervals of five consecutive frames, corresponding to 20.83 μs apart, amounting to approximately 0.1 ms in total. Based on reference [26], presenting and analyzing consecutive frames is challenging due to the minimal visible differences between them. However, with the current image-processing code employed in this study, subtle density variations can now be identified and analyzed. Figure 12 shows the common characteristics that are present in the baseline configuration which remain valid for every regime. At the nozzle exit, located on the middle-left side in each processed frame, the flow first emerges and encounters the surrounding environment.

The yellow triangle frame signifies the inner region where the potential core is located. In between the triangle and the wall of the central cone, one can see initial small-scale turbulent structures that begin to form as the high-speed jet exits the nozzle due to the initial shear layer. For regimes with higher velocities, the rate at which surrounding air mixes into the jet is lower. This is because the jet has more momentum to counteract the entrainment of surrounding air. As a result, the potential core is preserved over a longer distance, as the surrounding air does not mix in as quickly to decelerate or disrupt the high-speed jet flow.

As the flow progresses downstream, these small-scale turbulent structures (eddies) grow in size, expanding and interacting more extensively with the surrounding air. This interaction leads to larger coherent flow structures that eventually break down into fully developed, chaotic turbulence. The lower-right region of the frame depicts this transition, where turbulent energy is dispersed across various scales, contributing to the overall chaotic motion.

This visualization also captures the detachment of coherent eddies from the central cone as they reach its end. These detached eddies coalesce into larger vortical structures that move downstream and eventually dissipate. Such large-scale structures are prominent sources of low-frequency noise due to their organized, energy-containing motion.

In the baseline nozzle configuration without modifications, the shear layer has more freedom to develop these large-scale vortices shortly after separating from the wall. Without additional elements like serrations or chevrons to disrupt this natural growth, the vortices roll up and combine into coherent structures that persist downstream. These large structures are less attenuated and propagate low-frequency noise over longer distances.

It is worth noting that the acoustic signals generated by the flow are stationary and ergodic, which implies that the turbulent flow captured through imaging exhibits statistically consistent properties over time. This allows for the random selection of video frames for analysis, as the correlation of a stochastic function remains invariant under uniform time translations. This feature simplifies the imaging process while ensuring robust analysis. Accordingly, the instantaneous Schlieren visualization of the exhaust jet for both configurations, baseline and chevrons, for the three regimes are presented in Figure 13 and Figure 14.

Figure 13 displays two sets of five consecutive frames, one for the baseline configuration and the other one for the chevron configuration for the first regime (note that both the original and processed pictures are displayed for consistency). The baseline nozzle frames, Figure 13a′–e′, show a more organized flow structure near the nozzle exit, with large-scale coherent vortical structures developing along the shear layers. These structures persist as they move downstream, gradually transitioning into turbulence. This promotes the formation of prominent low-frequency turbulent regions, a characteristic of flows without additional mixing features. Consequently, the baseline nozzle produces low-frequency noise, which travels farther due to minimal attenuation.

In contrast, the chevron nozzle disrupts the flow immediately at the nozzle exit, breaking up large-scale coherent structures earlier. This accelerates the transition to smaller-scale turbulence, resulting in a more fragmented and chaotic flow pattern. Smaller eddies form sooner and dominate the flow downstream, leading to a finely distributed turbulence field. By enhancing the mixing, the chevron nozzle shifts the energy spectrum toward higher frequencies, reducing low-frequency noise that propagates over long distances. This high-frequency turbulence is less susceptible to travel forward, as it attenuates more quickly in the atmosphere, making the chevron configuration quieter in the far field. Although this shift may slightly increase high-frequency noise near the source, the overall effect is beneficial in reducing the low-frequency noise that travels farther. Thus, while the baseline nozzle promotes larger, low-frequency structures, the design of the chevron nozzle effectively enhances mixing and reduces noise impact by favoring smaller-scale, high-frequency turbulence.

At a higher rotation speed of 55k RPM, Figure 14 points out differences between the baseline and chevron nozzle configurations. In the baseline nozzle, larger, more prominent vortex structures develop downstream, as highlighted by the red frames in Figure 14b′,e′. These structures are larger and more dispersed compared to the 35k RPM case, exhibiting a typical free-jet expansion. In contrast, the chevron nozzle disrupts these large structures immediately at the nozzle exit, fragmenting the flow into smaller vortices and promoting finer-scale turbulence, as seen in the red frames from Figure 14a–e. This enhances mixing and shifts the energy spectrum toward higher frequencies, reducing low-frequency noise.

At the highest speed of 115k RPM (Figure 15), the baseline nozzle exhibits even more densely packed large-scale vortices near the cone end, maintaining coherence further downstream. This results in stable, low-frequency turbulence that contributes to significant noise propagation. In the chevron nozzle, however, turbulence is finer and more confined near the nozzle wall. Localized high-shear regions and reverse flow induced by the chevrons generate smaller vortices that remain close to the nozzle, leading to smoother mixing and smaller-scale turbulence. This structure increases high-frequency noise near the source but effectively reduces low-frequency noise that travels farther.

Overall, the chevrons enhance mixing and break down large-scale structures, reducing the impact of low-frequency noise compared to the baseline nozzle.

This limitation near the wall is key to maintaining smaller vortex scales for a longer distance. When the boundary layer encounters an adverse pressure gradient, it is forced to slow down. This can cause the boundary layer to thicken, making it more susceptible to separation and instability. As the boundary layer slows and becomes unstable, vortices within it may detach from the surface because they no longer have enough momentum to remain attached and migrate downstream. As the flow is dominated by chaotic and disordered motion, one common finding that applies to all regimes is that most of the vortices or eddies inside the jet are short-lived; they form, interact with other eddies, as in Figure 15a′,b′, and quickly dissipate or merge with other structures, as in Figure 15c′,e′.

Eventually, Figure 16a–j shows consecutive frames which belong to the ejector nozzle operated at 115k RPM. Twice as many consecutive frames presented in both original and processed images were analyzed in comparison to the chevron configuration. This approach highlights the advantage of advanced image processing in capturing and analyzing the instantaneus flow behavior more comprehensively, especially in cases where turbulence is less structured.

The additional frames clearly illustrate the broader, less confined nature of turbulence in the ejector configuration. The processed images reveal larger-scale vortices distributed over a wider area, indicating that the mixing process is less localized and less efficient compared to the chevron nozzle. These frames also emphasize the absence of fine-scale turbulence near the nozzle wall, a feature that is consistently observed in the chevron configuration, where turbulence remains confined and structured.

When compared to the chevron nozzle, the ejector configuration generates larger, less organized vortices that dominate the flow field, leading to a noisier low-frequency profile. In contrast, the chevrons promote earlier mixing and finer turbulence scales by disrupting large coherent structures immediately at the nozzle exit. This localized mixing shifts the energy spectrum toward higher frequencies, effectively reducing low-frequency noise that propagates farther.

This comparison aligns with the findings from the acoustic analysis, which demonstrated the chevrons’ superior efficiency in noise reduction. The high-frequency turbulence generated by the chevrons attenuates more rapidly, leading to reduced far-field noise compared to the ejector’s broader low-frequency mixing. This consistency between the visualizations and acoustic results reinforces the conclusion that the chevron nozzle provides better overall performance in controlling noise and turbulence.

5. Conclusions

To our knowledge, this study is the first to comprehensively compare ejector and chevron configurations for noise reduction in micro turbojet engines using an integrated analysis of statistical, acoustic and Schlieren imaging techniques. Due to the compact design of these engines, traditional noise-reduction methods, such as increasing mass flow and reducing jet velocity, are impractical. Instead, modifications to jet flow dynamics provide alternative solutions. This research highlights the distinct mechanisms and relative effectiveness of ejector and chevron configurations in mitigating noise.

The first critical step was ensuring the suitability of the recorded signals for meaningful analysis. Statistical validation demonstrated that the signals representing flow properties were stationary and ergodic, allowing time averages to reliably represent ensemble averages. This property ensured the robustness of the selected trimmed segments for noise analysis. Evaluations of mean, variance and autocorrelation confirmed consistency across segments, supporting the use of time-independent visualizations for analyzing jet dynamics.

The chevron configuration proved to be more effective than the ejector in reducing noise across the frequency spectrum. Specifically, for the chevron configuration, the chevron nozzle reduced overall SPL(Z) by 1.28% and OASPL by 1.21%. Low-frequency noise (SPL(C)) decreased by 1.15%, with specific reductions in harmonics, such as 270 Hz and 1289 Hz, where SPL dropped by 4.1 dB and 3.8 dB, respectively. These reductions were attributed to the chevron’s ability to disrupt large-scale vortices, promoting fine-scale turbulence and shifting energy to higher frequencies that dissipate more quickly. Conversely, the ejector configuration achieved modest noise reductions, with SPL(Z) decreasing by 0.51% and OASPL by 0.74%. In spite of the fact that it generated secondary airflow through ambient air entrainment, the resulting larger turbulence scales, clearly identified inside the frames, contributed to low-frequency noise. This noise propagates farther, and it is more difficult to be attenuated. This comparison highlights the effectiveness of the chevron configuration in mitigating jet noise by modifying flow dynamics and enhancing the dissipation of acoustic energy.

Schlieren imaging supported these findings by providing visual evidence of jet flow behavior. Thus, for the baseline configuration, larger, uncontrolled vortices were identified as they detached quickly from the nozzle, contributing to chaotic, low-frequency turbulence.

In contrast, the chevron configuration revealed smaller, tightly packed vortices that remained near the nozzle wall, enhancing mixing and noise control by promoting high-frequency turbulence. For the ejector nozzle, larger coherent structures were identified, with less effective mixing and greater contributions to low-frequency noise, particularly during high-speed operations.

The analysis of the post-processed Schlieren images confirmed that chevrons produce finer turbulence scales, reducing noise more effectively than the ejector configuration, especially at higher regimes. These findings were consistent across the statistical and acoustic analyses, validating the chevron’s ability to shift energy toward higher frequencies and attenuate low-frequency noise.

Additionally, the study underscored the value of integrating advanced image processing with statistical and acoustic methods. The newly created code-based image-processing techniques provided enhanced insights into turbulence features, such as vortex formation, interaction and dissipation, enabling a more detailed understanding of jet flow behavior. This methodological integration demonstrates the potential of Schlieren imaging to complement acoustic measurements in characterizing flow dynamics.

While the current study focused on Schlieren imaging for qualitative and semi-quantitative insights, the authors recognize the value of combining this approach with quantitative techniques like Proper Orthogonal Decomposition (POD) or Dynamic Mode Decomposition (DMD) in future research. These methods would allow for a more detailed decomposition of flow structures and their dynamic behavior, complementing the Schlieren findings. Additionally, Particle Image Velocimetry (PIV) could provide velocity field data to enhance the understanding of the relationship between flow features and noise generation mechanisms.

Future work will explore the integration of Schlieren techniques with these quantitative methods to enhance the understanding of turbulence structures and their contribution to noise mitigation. This combination of advanced techniques could further refine the insights obtained from the current methodology and extend its applicability to broader contexts.

In conclusion, the combination of statistical, acoustic and imaging analyses offers a holistic approach to understand and mitigate noise in micro turbojet engines. The chevron configuration emerges as a superior noise-reduction tool, particularly in high-speed operations, due to its ability to generate structured, high-frequency turbulence. Future work will focus on further enhancing Schlieren post-processing to extract additional turbulence features and investigate their correlation with acoustic performance in high-speed jet applications.