Flow mechanism study and geometrical parameters analysis of �uidic oscillators based on pressure-sensitive paint measurements and modal analysis

The present study optimized some novel �uidic oscillator designs, measured their internal-external �uid dynamics, and used modal analysis to reveal their underlying oscillation mechanisms. It also investigated the effects of structural parameters on their �uid dynamics. The time-resolved internal-external pressure �elds of the oscillators were determined by using pressure-sensitive paint (PSP) measurement. Proper orthogonal decomposition (POD) and dynamic mode decomposition (DMD) were used for modal analysis and phase reconstruction. The time-averaged pressure-eld and phase-reconstruction results reveal that recirculation bubbles inside the mixing chamber and the feedback �ow have underpinned the mechanism of formation of internal and external continuous sweeping �ows. The modal analysis results reveal the spatial modal structures and their time evolution, which dominated the internal-external �ow pattern. The inlet-wedge width and feedback-channel inlet diameter were found to be the structural parameters affecting feedback �ow and recirculation-bubble size, and thereby in�uenced �ow characteristics such as jet oscillation frequency and divergence angle. Different Coand ă surfaces altered the recirculation bubbles and feedback �ow of the mainstream, thereby in�uencing the formation mechanism of the sweeping jet.


Introduction
Fluidic oscillators generate spatiotemporally oscillating jets with characteristics that depend on their internal uid instabilities.Since being invented in the 1950s, uidic oscillators have been applied to sprinklers, nozzles, and other devices.Especially in the past two decades, uidic oscillators have drawn particular interest from aerodynamics researchers for applications in, for example, active ow control, jet thrust vectoring, and mixing enhancement, due to their simple geometries, unsteady blowing characteristics, wide range of working conditions, and robust performance (Raghu, 2013;Gregory et al., 2013;Hussain and Khan, 2022).Moreover, many studies have applied uidic oscillators for noise abatement, combustion control, and bluff-body drag reduction.
The distinctive ow features and excellent performance of uidic oscillators have attracted researchers to examine these devices' characteristics, and various types of uid oscillators have been developed.Woszidlo et al. (2019) categorized uidic oscillators based on their internal oscillation mechanisms as feedback-free oscillators, one-feedback-channel oscillators, and two-feedback-channel oscillators.Twofeedback-channel oscillators are robust and scalable, and rely on feedback channels and the occurrence of the Coandă effect in their mixing chambers to drive the mainstream to oscillate.This generates a selfsustaining spatially continuous sweeping jet at the outlet nozzle of two-feedback-channel oscillators (Fig. 1), and accounts for their also being called sweeping jet actuators (SJAs).Numerous researchers have used experimental and numerical methods to characterize the internal and external ow characteristics of SJAs.For example, Woszidlo et al. (2015) experimentally examined the internal and external ow elds of a SJA, and analyzed its time-resolved two-dimensional ow eld results to determine the underlying jet oscillation mechanisms.Ostermann et al. (2018) employed stereoscopic particle image velocimetry to obtain three-dimensional (3D) time-resolved velocity information of a sweeping jet emitted from a SJA into a quiescent environment and evaluated its phase-averaged velocities, forces, and entrainment.Furthermore, Ostermann et al. (2019) utilized 3D time-resolved information to investigate the spatially oscillating jet emitted by a SJA into an attached cross-ow and assessed how the jet was in uenced by various parameters, such as the velocity ratio and Strouhal number.Similar experimental studies have included those by Bobusch et al. (20131a, b), Hossain et al. (2017), Wen et al. (2018Wen et al. ( , 2020)), and Gaertlein et al. (2014).Moreover, SJAs have been examined in many numerical simulation studies.
For example, Pandey et al. (2020) used unsteady Reynolds-averaged Navier-Stokes analysis to evaluate the ow eld of an angled SJA in a quiescent and cross ow environment under various blowing ratios, and analyzed the interaction of its sweeping jet with the free stream in terms of vortex dynamics.Li et al.
(2021) numerically simulated the heat dissipation performance of a sweeping jet and used various metrics to qualitatively evaluate its dissipative properties.
The increase in application scenarios for uidic oscillators has led to a surge in studies on uidic oscillators under severe and complex working conditions, such as complex geometry, high-speed, hightemperature, or high-pressure conditions, which require uidic oscillators to exhibit high performance.Especially in aero-engine applications, such as turbine cascade, gas compressor, tip clearance ow, and air-lm cooling applications.Limited by the complex geometries, plate/shell structure, narrowed layout, severe working conditions, and strength requirements of the facilities, the size of the active ow control actuator is required to reach a millimeter-or sub-millimeter-sized level.More and more studies focused on the ow characteristics of uidic oscillators under extreme conditions or at microscales.For example, Shabnam et al. (2022) experimentally investigated the ow characteristics of an oscillating jet emitted by a uidic oscillator at a high nozzle-pressure ratio (4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16).Park et al. (2020) used the design of experiment method analyzed the statistical information of a two-feedback channel oscillator con guration under supersonic conditions.However, although classical SJAs have excellent scalability, they do contain some curved structures, e.g., the mixing chamber diaphragm and splitter structures indicated by the red dashed ellipse in Fig. 1 Nevertheless, despite great efforts to optimize the structure of uidic oscillators, all of the con gurations have contained spikes and thin-walled structures.Moreover, to the best of the authors' knowledge, there is no two-feedback channel oscillator design that is suitable for microscale machining.Thus, there is an urgent need to develop uid oscillators with simpli ed geometric structures, as this will allow their use in more extreme application scenarios.It is also important to examine the internal-external ow dynamics and key structural parameters that affect the ow characteristics of ow oscillators with simpli ed structures.
Furthermore, it remains challenging to measure high-spatiotemporal-resolution ow structures of uid oscillators.For example, particle image velocimetry uses air as the working uid, and the uneven illumination caused by the diaphragm inside an actuator leads to inaccurate results.Analogously, the Schlieren method has a limited ability to detect low-velocity ows, and it is di cult to use other experimental methods, such as water-ow and surface oil-ow visualization, for quantitative analysis.Fortunately, unsteady pressure-sensitive paint (PSP) can be used for the high-spatiotemporal-resolution measurement and quantitative analysis of the internal and external ow elds of uid oscillators.PSP Given the above-mentioned research, the main objective of the current study was to simplify the internal curvature structure of the classical dual-feedback-channel oscillator and provide a con guration suitable for microscale machining.An easy-to-operate PSP measurement and modal analysis method were used to obtain the internal and external ow structures of the oscillator, and characterize the effects of key structural parameters on the oscillation mechanisms and jet performance.
The remainder of this paper is organized as follows.Section 2 describes the experiment setup, geometric modi cations, and the PSP measurement technique.Section 3 introduces the modal analysis methods, namely proper orthogonal decomposition (POD) and dynamic mode decomposition (DMD), and the data processing procedure.Section 4 discusses the measurement and modal analysis results, and details the internal-external ow characteristics of the oscillator and how different con gurations of the oscillator in uence its key structural parameters.Section 5 summarizes the main conclusions of this paper.

Geometric structures
In the current investigation, we optimized the geometric design of complex curvature parameters, such as the splitter and mixing chamber diaphragm, in a classical two-feedback channel oscillator and devised a simpli ed con guration, as shown in Fig. 2(a).Scaled-up versions of the con guration were used in subsequent experiments to improve the spatial resolution.The smallest cross-section of the con guration was the inlet nozzle width , and as this was a representative size of the actuator, other parameters were normalized to it, and the reference hydraulic diameter was de ned as being equal to .Other parameters were as shown in the subgraph of Fig. 2(a), the inlet-wedge width , the feedback channel had an inlet width and a diameter , notice this design eliminated the curvature splitter, which has a great in uence in microscale machining.For the outlet section, the outlet throat width and the half-divergent angle .Gaertlein et al. (2014) showed that the diffuser part of an outlet nozzle does not affect the switching behavior inside an oscillator, thus all the actuators used in the current study kept the same outlet-section con guration.Moreover, for ease of comparison, all the models had an identical cavity depth and thus an aspect ratio .
The in uence of critical structural parameters on the internal and external ow characteristics of the oscillator was determined by evaluating three parameters , , and the mixing-chamber Coandă surface.First, two variations of shown in Fig. 2(b) were investigated; i.e., inlet-wedge widths of and , with the other geometries the same as in the baseline con guration.Second, we investigated the in uence of by considering the two variations shown in Fig. 2(c); i.e., and .In addition, we investigated the difference between a curved and a straight Coandă surface in the mixing chamber by considering the con guration shown in Fig. 2(d).This con guration contained the curved Coandă surface structure of a classical dual SJA while its other parameters were consistent with the baseline design.This comparison can intuitively re ect the importance of the feedback ow effect and the Coandă wall-attachment effect in the formation of an oscillating jet.

Test conditions
Figure 3 shows the experimental setup that was used for pressure and PSP measurements.Highly pressurized air at a temperature of 25 ℃ was supplied by an air compressor (FENGBAO 265/7, China) at inlet mass ow rates of 100, 130, 150, 170, 190, and 210 SLPM, respectively.A mass ow meter (SMC Corp., PFMB7201) with an uncertainty of 2% was used to monitor and control the inlet jet ux The abovedescribed inlet conditions resulted in a nominal outlet-jet velocity of 37.04-77.78m/s (based on the hypothesis of incompressible ow) and a reference Reynolds number of approximately 17,800-37,400.
The internal and external ows were rst characterized using 10 symmetrically distributed discrete pressure transducers (Hanghua PSU64, with a full range of ± 10.3 kPa and an accuracy of ± 0.5% at full range) with a sampling frequency of 3 kHz, the locations of the ori ces are as the red dot indicated in Fig. 2(a).The transducers were mounted on the oor plate of the model (as shown in Fig. 3(a)).At least 30,000 samples were acquired to calculate time-averaged pressure values for each test case, and these discrete pressure measurements were also served as in-situ calibrations for PSP measurements.(2022), PSP materials were sprayed inside the oscillator and at the bottom of the outlet plane to capture the pressure uctuations of internal and external ow, respectively.For external measurements, we focused on the region of , , as shown by the red dotted line in Fig. 3(b).The upper surface of the model was made of optically accessible acrylic plates to allow image capture, and the upper and lower plates were sealed using colloids.This con ned geometry was expected to have a negligible in uence on the ow structure inside the uidic oscillator but to have a noticeable in uence on the external ow eld, and is discussed in detail in Section 4.1.3.

PSP measurements
The PSP material was excited by an 18 W ultraviolet light-emitting diode at a wavelength of 395 nm (OP-C6U1S-HCI, YueKe Optic, China).The light intensity was su ciently uniform during the experiment, and any adverse effects of uneven light intensity were eliminated during the subsequent data-processing procedure.Images were captured at a sampling rate of 3 kHz using a high-speed complementary metaloxide-semiconductor camera (DIMAX HS4, PCO AG) equipped with a 650 ± 25 nm bandpass optical lter.At least 6,000 images were collected for each case, but given that this generated a large amount of data, only at least 100 cycles of data were used for post-processing, and other data were used for validation.The results show that this was su cient for the resolution of time-averaged and instantaneous ow data.The distributed pressure-transducer data were recorded in-situ and used to calibrate the PSP data.However, the high-speed camera was set to run freely, without synchronization with the pressure transducers.The temperature errors in the experiments were negligible because the ambient temperature and the inlet air temperature remained unchanged, and thus the in uence of temperature was considered to be negligibly small and was ignored in the post-processing procedure (Crafton et al., 2017;Peng et al., 2018).

Data Processing Methodology
Modal analysis extracts important energetic or dynamic characteristics of a ow eld and provides physical insights that enable the characterization of ow evolution and key ow phenomena.Commonly used data-based modal analysis methods include POD, balanced POD, and DMD, and such methods have often been used to extract the main features of a ow eld (e.g., Taira et al., 2017Taira et al., , 2019) ) or enhance the signal-to-noise ratio (SNR) of PSP data (e.g., Wen et al., 2018;Gößling et al., 2020).This section brie y introduces the basic concepts of POD and DMD methods, and the procedure that was used in the current study for the modal analysis of PSP data.
The input of the POD algorithm used in the current study is the unsteady pressure eld obtained from PSP measurement, which can be decomposed into a temporal mean component and an unsteady component .POD represents the unsteady components as a linear combination of a set of orthogonal modes and their corresponding time coe cients, as follows: 1 where is the mode arranged in the order of relative energy levels, and is the corresponding time coe cient of the jth mode.As the spatial scale is much larger than the temporal scale, we used the "snapshot" method devised by Sirovich (1987) to determine the POD modes.
The internal ow and the sweeping jets omitted from the oscillator were self-exciting and selfmaintaining, which meant that a phase-averaging analysis was necessary to obtain a high-quality timeresolved ow eld.The phase angle for each adjacent PSP image was calculated using the rst and second POD time coe cients, and , according to Unlike POD, DMD involves decomposition of a ow eld into several modes with single characteristic frequencies and growth/decay rates, which enables the importance of ow structures of various frequencies to be analyzed.Several characteristic modes may be exhibited by the internal-external ow of an oscillator and their periodic motion may have a determinative impact on ow patterns.Thus, DMD is used to analyze the speci c ow structure that affects the ow.
First, the pressure eld data obtained at a constant sampling frequency are arranged into the following two snapshot series matrices: The DMD algorithm provides a linear approximation of the two datasets, that is In practice, direct calculation of demands extensive resources, and thus it is approximated as the rank-reduced representation through the singular value decomposition of The DMD eigenvalues and modes are de ned as the eigenvalues and eigenvectors of the linear operator .Then, the ith low-rank approximations can be used to express the evolution of the dynamic system, as follows:

Data processing procedure
Figure 4 depicts the procedure used for processing the PSP measurement data.First, images sampled under wind-off (reference) conditions were averaged to obtain the reference value ( ). Next, instantaneous/time-averaged wind-on images ( ) captured under various conditions were used to calculate the instantaneous/time-averaged intensity ratio.Subsequently, camera noise-oor data ( ) sampled under light-off conditions were subtracted from both and to reduce the in uence of ambient light.The pressure ratio ( ) was then used in the form of the Stern-Volmer equation (Eq.( 6)), as follows: 6 where and are temperature-dependent calibration coe cients and were assumed to be constant, as an in-situ calibration method was used.The discrete pressure data obtained from the pressure transducers were used to calibrate the pressure ratio, i.e., the pressure data were directly compared with average pressure ratio data over a 10 × 10 pixel region surrounding each transducer.
As the modal analysis approach is computationally expensive, the calibrated pressure data were rst smoothed using a 2 × 2 pixel window and then half-subsampled to reduce the data size, and nally reformed into the snapshot data form required by the POD or DMD algorithm.The subsampled data effectively captured the details of the ow structure.As mentioned, the oscillation process was self- excited and self-sustaining, which meant that there were some random uctuations in the measurement process.These adversely affected analysis of the periodic characteristics of the oscillating jet.In addition, the SNR of the raw PSP data sampled under low pressure conditions was not su cient to clearly present the ow structures.As such, the POD-based modal reconstruction method was adopted to eliminate the random noise and construct the phase-averaged pressure eld.Unless otherwise speci ed, reconstructed data were the basis of the following results and discussions.

Data calibration and uncertainty
There were two sources of error in the PSP light intensity ratio data: the calibration process and the modal reconstruction process.In the calibration process, 10 distributed pressure-transducer results were used to calibrate the light intensity ratio obtained from PSP measurement.The mapping of light intensity ratio data to speci c pressure ranges was processed globally, as multiple pressure transducers were used, i.e., we pursued the best t between the light intensity ratio and pressure transducer data.For example, in the baseline case, the maximum error between the PSP sampling data and the pressure transducer data was , i.e., approximately 5.5% of the full pressure range ( ).In the modal reconstruction process, raw light intensity ratio data and the modal reconstruction data were sampled around to determine their evolution over time.Figure 5 illustrates that the modal reconstruction data were highly consistent with the raw data, with both sets of data re ecting the pressure uctuation process.The modal reconstruction results eliminated some peak values and random uctuation, which were likely caused by random oscillation of the self-oscillation process.The error in other cases was similar to that in the baseline case.

Results and Discussion
In this section, the pressure and PSP measurement results are presented and used to discuss the general ow characteristics and switching mechanisms of the uidic oscillator, a modal analysis is performed to examine the oscillator's internal and external ow dynamics, and the in uence of geometric parameters on the oscillator's ow characteristics are investigated.

Pressure measurement results
Figure 6 shows the time-evolution and corresponding frequency of the pressure measurement.For brevity, only data collected at the oscillator feedback-channel inlet ( , ) and the oscillator outlet ( ) are plotted.There was a substantial (i.e., near-180°) phase difference between the pressure values at and .The alternating pressure uctuations in the feedback channel indicate that the internal ow was transversely de ected, which might have caused the formation of the oscillating jets.There were two pressure peaks per cycle at and these corresponded to the moment at which the pressure in the two feedback channels reached equilibrium.This uctuation of indicates that oscillating jets formed at the outlet.The spectra of , , and exhibited similar frequency characteristics, and the dominance of the peak at 129 Hz in Fig. 6(b) indicates the robustness of the oscillating jet formation mechanism within the oscillator.The self-excited and self-sustaining nature of the oscillating jet accounts for the randomness in the pressure data in different periods, but all periods' data exhibited similar uctuation characteristics.As the pressure data and their spectral characteristics are similar at different ow rates, subsequent discussions are based on the inlet ow rate at 210 SLPM, unless otherwise speci ed.
Figure 7 shows the frequency characteristics of the sweeping jet at various inlet-mass ow rates, based on the data sampled at .As expected, the oscillation frequency increased linearly with the ow rate, even at very low ow rates.This has been observed in almost all studies on uid oscillators.Analogous results were obtained at all the pressure transducers arranged inside the oscillator, indicating that the generation of the oscillating jet was completely dependent on the oscillation mechanism of the internal feedback ow.

Time-averaged ow characteristics
The time-averaged pressure distribution of the baseline case at the maximum mass ow rate, which was obtained from PSP measurements, is shown in Fig. 8.The arrows in the ow channel indicate traces or scratches, which were caused by the ba e or other objects blocking the exciting light.These shadows had a negligible impact on the analysis of the overall ow dynamics and thus are ignored in our discussion.
It can be seen from the Fig. 8 that the internal structure of the oscillator caused the high-pressure inlet ow to rapidly dissipate after entering the mixing chamber of the oscillator.A low-pressure center was formed near the outlet of the mixing chamber, indicating the location of the vortex region.The recirculation bubble was strongly related to the jet de ection mechanism of the oscillator, which is detailed and discussed in combination with the phase-averaged ow eld in section 4.1.3.The differential pressure distribution in the feedback channel indicates the location of the ow that was recirculated to the inlet nozzle.The aforementioned ow characteristics inside the oscillator indicate the presence of a robust oscillation mechanism.Outside of the oscillator, a sweeping-jet in uence region formed at and .Far downstream of the outlet, the jet was rapidly dissipated by spanwise sweeping, and thus it was di cult to capture the ow structure approximately away from the outlet.We de ne the area where the pressure change reached 50% of the maximum pressure change as the jet coverage area, which is similar to the de nition of jet half-width that has been used by Tajik et al. (2021) and Woszidlo et al. (2015).The above-mentioned de nition showed that there was a ~ 75° fan-shaped jet divergence region, as indicated by the red dotted line in Fig. 8.This divergence angle is smaller than that of the classical con guration (which is approximately 110°), indicating that there was not a substantial Coandă effect at the outlet diffuser and that the jet did not attach to the outlet wall.Bobusch et al. (2013) showed that an adequate reduction of the divergent angle of a nozzle can cause attachment of the outlet jet, thereby increasing the angle of maximum de ection.The external ow eld of the traditional curved- edge double-feedback channel oscillator typically exhibits a "dual-peak" distribution, which is caused by a long dwelling time at the maximum jet de ection angle, as described by Ostermann et al. (2015b).In comparison, the external ow eld of our con guration was substantially different: inside the fan-shaped jet divergence region, the pressure pro le attenuated symmetrically along the central axis to both sides.
This fan-shaped distribution indicates that the external e ux of the oscillator had an equilibrium dwelling time at each de ection angle and thus, compared with the classical con guration, yielded a more homogeneous distribution and more even control performance in the external ow.These assessments were veri ed in the subsequent phase-averaging analysis.Furthermore, interestingly, the external ow eld of this con guration was very similar to that of the "angled oscillator" reported by Ostermann et al. (2015b).

Phase-averaged ow characteristics
The phase-averaging process eliminates random turbulence and potential small-scale ow characteristics to provide a basic visualization of major ow structures in external jets, such as their internal switching mechanisms and oscillatory modes.
The relative phase angle was obtained by the POD-based phase reconstruction method, and the timeresolved ow eld was then averaged with a phase-angle window of 6° to determine the pressure uctuation with 60 snapshots within a sweeping cycle.The moment when the oscillating jet de ection angle was 0° was arbitrarily regarded as the zero phase of an oscillation period.Figure 9 presents the phase-averaged pressure eld at 12 phase angles within a cycle.

At phase angle
, the main ow fed through the inlet wedge into the mixing chamber and formed a low-pressure center in the mixing chamber.This center was biased toward the upper feedback channel, indicating that the inner jet was clearly de ected.At this time, the pressure in the upper feedback channel was higher than that in the lower feedback channel.As the ow developed, the low-pressure center gradually moved toward the inlet of the upper feedback channel.The distinct separation-bubble footprint indicates that most of the main ow ejected from the outlet nozzle and a portion of the ow that impinged on the lower converging wall of the outlet nozzle owed into the upper feedback channel.Thus, the pressure in the lower feedback channel decreased signi cantly, which caused a reverse ow in the upper feedback channel ( ).In addition, when the main ow entered the mixing chamber, a small amount of uid owed into the feedback channel due to the presence of the split wedge inlet, thereby forming a forward ow in a direction opposite to that of the feedback ow.When the feedback ow was stronger than the forward ow, the feedback ow returned to the power nozzle through the upper feedback channel, driving the main ow to further de ect and reach a maximum de ection at .This caused a new low-pressure center to begin forming in the mixing chamber near the lower diaphragm, which caused the pressure difference in the upper and lower feedback channels to begin to decrease.Consequently, the main jet began to enter the lower feedback channel, and as the separation bubble grew, the amount of the jet entering the lower feedback channel increased to form the feedback ow in the lower feedback channel and became dominant at .This process drove another ∘ internal ow diversion in the direction opposite to the former process, at -.At the aforementioned process repeated, thereby driving another circulation of the ow.Repetition of the above-described jet switching process drove the internal jet to turn continuously and form an external sweeping jet.At all phase angles, the sweeping jet ow formed a continuous lowpressure area in a region approximately downstream of the outlet.At , the jet became incoherent and moved downstream in a plume-like pattern.This occurred because the ejection of the continuously turning jet caused the static uid in the external domain to be continuously entrained into the jet (due to the restriction of the wall and the strong mixing effect), resulting in dissipation and breakdown of the persistent ow.This entrainment and dissipation reduced the velocity of the jet, leading to a phase difference between the jet near the exit region and the jet in the far-eld region.In the timeaveraged ow eld, this ow phenomenon was manifested in the two high-pressure areas on the edge of the fan-shaped area at downstream of the outlet (Fig. 8).Similar entrainment characteristics of a spatially oscillating jet ejecting into a non-con ned quiescent environment have been discussed in Li et al. (2021) and Ostermann et al. (2018).In the current study, the extremely con ned geometry increased the friction between the jet and wall surface and thus increased the energy consumption, such that all dissipations were expected to be larger than those occurring in open space.More details on the effects of con ned structures on the ow and heat transfer characteristics of sweeping jets are available in Mohammadshahi et al. (2020Mohammadshahi et al. ( , 2021)).
A few more aspects are as follows.The geometry of the proposed design prevented the ejected oscillating jet from attaching to the divergent nozzle wall, even at the maximum jet-de ection angles ( = 90° and 270°).Wen et al. (2018) and Koklu et al. (2016) have noted that the Coandă effect on a diverging exit wall is the primary cause of increased jet residence time at the jet's maximum de ection angle.
Therefore, the time-averaged external ow eld formed a fan-shaped structure rather than a "dual-peak" structure.

POD analysis
POD and DMD methods were used to further characterize the internal and external ow dynamics.
Figure 10 shows the energy proportion of the rst 20 order POD modes.It is evident that the rst 5 order modes contained most of the kinetic energy and the rst 20 order modes accounted for 35% of the kinetic energy distribution.Considering the measurement noise in the PSP experiment, such energy distribution results are convincing and indicate the presence of a large coherent structure or a dominant ow pattern, i.e., oscillatory behavior.
Figure 11 shows the contours of the rst 12 order POD modes.The rst-and second-order modes were basically symmetrically distributed inside the oscillator and captured the jet oscillating motion.The change in the spatial form of the rst 2 order modes in the feedback loop indicates that the strong feedback-ow caused energy transfer that ultimately led to oscillation of the main jet.The rst-and second-order modes in external space also captured the main ow characteristics of the external sweeping jet but were asymmetrical, due to the jet mixing caused by the con ned walls.The rst few modes re ect the average turbulence characteristics, and as the mode order increased, the characteristic scales decreased, re ecting the transfer of turbulent energy from the large turbulent scale to the small scale.At the same time, the in uence of measurement noise gradually increased as the number of modes increased, with this being most apparent when the mode number was greater than 7.Although there was much less energy contained in these higher-order modes (i.e., modes 7-12) than in the rst 2 order modes, the scale of these higher-order modes was comparable to the size of the separation bubble inside the mixing chamber.Thus, they also likely re ected turbulent energy transfer inside the oscillator.The turbulent structure at this scale was captured by using at least the rst 30 order modes to reconstruct the phase-averaged pressure eld.
The time coe cients of the rst 2 order modes ( and ), and are shown in Fig. 12(a).In an ideal harmonic oscillation between two modes, the time coe cients are sinusoidal with a 90° phase shift (Ostremann, 2015a).In the current study, the phase difference of the time coe cients was close to 90°( the time-coe cient pairs of , and , showed the same characteristics) but was not fully consistent with ideal harmonic characteristics.The phase portrait of the rst 2 order time coe cients shown in Fig. 12(c) also showed some distortion.This phenomenon has been observed in other investigations of sweeping jets (e.g., Ostermann et al., 2015a;Wen et al., 2019) and is due to four factors: (1) the self-excited and self-sustaining characteristics of the sweeping jet, which cause the uctuation of periodic motion; (2) the capture of multiple frequency components in a single POD mode; (3) the change in the ow pattern caused by the entrainment of the sweeping jet and the dissipation of its kinetic energy; and (4) the in uence of measurement noise.
These deviations reduced the accuracy of the phase-averaged results.However, as Fig. 12(b) shows, the frequency spectra of the rst 6 order time coe cients exhibited prominent peaks with harmonics at 129 Hz (the same frequency as the pressure measuring results), and the peak energy decreased as the mode order decreased, indicating that all the rst 6 order modes captured key ow structures.Moreover, Fig. 12(d) indicates that the rst 3 order time coe cients formed a limit cycle in high dimensions.Thus, although it is not visible in the graphs, it is believed that in high-order POD space, the completeness of the ow structure captured by POD modes increased as the number of POD modes increased.

DMD analysis
DMD analyzes a ow eld from the perspective of frequency and provides a unique view of the eld's ow pattern and its growth rate.The standard (exact) DMD algorithm does not specify the mode arrangement; thus, in practice, the DMD modes are placed in order based on their respective energies.Accordingly,  The real and imaginary parts of each DMD mode eigenvalue (the Ritz value) are shown in Fig. 13(a).The fact that most of the eigenvalues are located on the unit circle indicates that the main ow structure in the ow eld was largely stable.However, the 1 to 12 order DMD modes (indicated by the red dots) are far from the unit circle, re ecting the periodic instability of the ow eld, which was caused by measurement noise and the uctuation of periodic motion.
Figure 13(b) shows the amplitude of the rst 12 order DMD modes.The rst-order mode (the static mode) was consistent with the average pressure eld and had a frequency of 0 Hz, and thus its contour is not shown.In addition, the rst-order mode contained the highest energy, as its mode amplitude was 1 × orders greater than those of the other modes (Fig. 13(c)).This result was not unexpected, as the experiment was conducted under a relatively low-pressure condition and the pressure sensitivity of the PSP was approximately 0.7%.Thus, the pressure uctuation signal was much smaller than the average pressure.The other modes were complex conjugate pairs with the same stability characteristics and frequencies but different signs.In the following discussion, each mode pair is identi ed only by the mode with the positive-valued frequency.As shown in Fig. 13(c), the 2 to 12 order modes also had a higher amplitude than the other modes and contained abundant energies; thus, they had a substantial in uence on the ow structures.
Figure 13(b) shows that the time coe cient and kept sinusoidal varied throughout the measurement period.In addition, exhibited a peak at (Fig. 13(d)), which was the secondary harmonic of the self-excited oscillation frequency.However, Fig. 14 shows that the corresponding modal structure was submerged in the noise.Correspondingly, mode 6 (7) had a clear mode contour but a rapidly decaying time coe cient.Therefore, mode 6 (7) and mode 8 (9) had a limited in uence on the overall dynamic development of the oscillating jet.Moreover, Fig. 14 indicates that mode 4 (5), mode 10 (11), and mode 12 (13) were noise modes, and that their corresponding time coe cients decayed rapidly (Fig. 13(b)) and apparently had no peak frequency (Fig. 13(d)).These modes re ect the in uence of measurement noise on the dynamics and contributed to some of the errors in the DMDreconstructed results.Various DMD variants focus on eliminating the impact of measurement noise, but a detailed examination of these was beyond the scope of this study.
Overall, the DMD results indicate that the uid dynamics of the oscillator were dominated by a single ow structure (mode 2(3)), with the contour of mode 2 (3) being very similar to the rst POD mode (Fig. 11).This shows that the oscillation formed by the feedback ow dominated the ow dynamics in terms of both frequency characteristics and energy.

Analysis of structural parameters 4.3.1 Inlet wedge width ( )
The inlet wedge width substantially affects internal feedback ow and external ow-eld morphology (Bobusch et al., 2013b;Wen et al., 2020), primarily by in uencing the volume ow ratio between the mixing chamber and the feedback channel.We investigated two con gurations-B-1, and B-2-to We found that the inlet wedge width affected the ow dynamics by changing the proportion of feedback ow.A comparison of the internal ow topology of three con gurations with a small inlet wedge width (the B-1 con guration) revealed that a large portion of the mainstream was separated and guided into the feedback channel after entering the oscillator inlet throat.This resulted in a forward ow in the feedback channel and rapid energy dissipation in the mixing chamber.Therefore, the low-pressure vortex bubble in the mixing chamber was more implicit and closer to the outlet in the B-1 con guration than in the baseline con guration.The uid traveling in the reverse direction that entered the feedback channel needed additional kinetic energy to counteract the forward ow, resulting in the outlet jet having a small sweeping range.With an increase in inlet wedge width (i.e., in the B-2 con guration), there was an increase in the number of inlet streams entering the mixing chamber, leading an increase in the size of the low-pressure zone that was formed.This zone subsequently extended into the feedback channel inlet, indicating a dominant feedback ow.Thus, the transverse de ection of the internal jet increased in magnitude, leading to an increase in the sweeping motion of the external ow.The frequency characteristics of the three con gurations (Fig. 15(b)) clearly show that the sweeping jet frequency gradually increased as the inlet-wedge width increased, although the lateral de ection amplitude from B-1 to B-2 increased.This was due to the aforementioned counteraction in the feedback channel, which increased the energy dissipation and thus delayed de ection of the internal ow.
For the external ow eld, the jet diffusion angle increased from ~ 60° to ~ 90° as the inlet wedge width increased, and this was closely related to the ow transverse de ection inside the oscillator.Moreover, although the external jet of the con guration with a smaller inlet-wedge width was concentrated more along the centerline than that of the con guration with larger inlet-wedge width, the three con gurations had a similar penetrating force.This was because a decrease in caused an increase in energy dissipation inside the oscillator, thereby decreasing the kinetic energy contained in the emitted jet.were provided herein to investigate the in uence of feedback-channel inlet width ( ) on the ow dynamics of our oscillator.
Figure 16 shows the time-averaged pressure distributions of C-1 and C-2.A comparison of these distributions with that of the baseline revealed that all three con gurations formed stable feedback-ow topologies in internal ow, but exhibited differences in pressure distributions in the feedback channel.We quantitatively described these differences using the pressure difference between and ( ) and the standard deviation of ( ) (which characterizes the magnitude of pressure uctuations).The for the C-1, baseline, and C-2 con gurations were 820, 956, and 1115 Pa, respectively.The gradual increase in the magnitude of indicates that the feedback ow gradually increased as the feedback-channel inlet width increased.Similarly, the of the three con gurations were 121, 198, and 235 Pa, respectively.The increase in the pressure uctuation in the mixing chamber was caused by the enhancement of feedback ow.It also indicates that mainstream de ection intensi ed as the feedback-channel inlet width increased, as indicated by the transient process (although this intensi cation was di cult to observe in the time-averaged contours).The increase in the feedback ow and the angle of de ection increased the scale of the ow in the mixing chamber, resulting in an increase in energy consumption.This ow pattern generated a hysteresis in the angular de ections of the mainstream and the outlet jet, and decreased the ow velocity to a certain extent.Thus, the oscillation frequency in the C-2 con guration was less than that in the C-1 con guration, as shown in Fig. 16(b).
For the external pressure eld, the angular de ection hysteresis of the internal ow led to an increase in the sweeping angle and to dissipation of the external jet.Therefore, as shown in Fig. 16(a), the feedback channel diameter increased from to (in con gurations C-1 to C-2, respectively) and the fan-shaped time-average pressure contour gradually transitioned to an approximate dual-peak distribution.The external jet emitted from the C-2 con guration showed some stagnation characteristics at its maximum de ection angle, although these were much milder than those in the typical two-feedback con guration.The continuity of the external jet in the C-2 con guration was substantially disrupted and exhibited no pulsating characteristics during sweeping motion, and thus would require special attention in practical applications.

Mixing chamber Coandă surface
The design of the D-1 con guration increased the oscillator's mixing-chamber volume, which facilitated the formation of unsteady separation bubbles.Moreover, it caused the distance between the mainstream and the wall downstream of the inlet wedge to be much larger than in the baseline con guration, which decreased the number of conditions that were consistent with the Coandă effect.These factors led to the oscillation mechanism of this design depending more on the instability of the ow inside the chamber itself.
The time-averaged pressure distribution in Fig. 17(a) shows that there was an uneven pressure distribution inside the two-sided feedback channel and the amesial separation bubble close to the exit outlet (as indicated by the red dotted line).This indicates that once the mainstream entered the mixing chamber, the energetic ow dissipated rapidly and accumulated inside the mixing chamber.Only a small amount of feedback uid (driven by the pressure difference) entered the feedback channel, which had little momentum available to de ect the high-energy mainstream.Thus, the internal oscillation mechanism of this con guration was not strong enough to drive the internal and external jets to generate a substantially de ected large-scale sweeping motion.These ow characteristics were also re ected in the pressure uctuation characteristics.Figure 17(b) shows the frequency spectrum characteristics of , revealing that it had no signi cant peak.These random uctuations and asymmetric pressure distributions may be attributable to small errors in machining of components and the interaction between the jet and the environmental uid.
However, the Coandă effect of the main jet entering the chamber was weak, which meant that the movement of the vortex bubbles was concentrated in the center of the main chamber and could not be fully de ected to either side.It was therefore unsurprising that the external jet exhibited weak unsteady characteristics with a smaller de ection angle and slower dissipation than the baseline case, akin to a steady jet.Therefore, based on the investigation of this con guration, if a robust oscillating jet is desired with a Coandă surface design, the volume ow rate of the feedback ow must be increased.This could be achieved by increasing the feedback channel inlet width or setting a splitter structure at the feedback channel inlet, thereby returning to the classical two-feedback SJA design.

Conclusion
In this study, we devised several novel uid-oscillator designs, experimentally examined their internalexternal uid dynamics, and revealed their oscillation mechanisms by using pressure and PSP measurement methods.Modal analysis methods (both POD and DMD) were used to obtain the timeresolved and phase-averaged internal-external pressure eld topologies.In addition, we employed structural parameter analysis, namely inlet wedge width , feedback-channel inlet width , and mixing-chamber Coandă surface analysis, to comprehensively investigate the impact of internal geometry on a uid oscillator's internal-external ow mechanisms.Our major ndings are summarized below.
1.The pressure and time/phase-averaged pressure eld measurements reveal that the pressure difference between the mixing chamber and the feedback channel drove the recirculation bubble inside the mixing chamber, thereby causing transverse motion in the mainstream and the feedback ow inside the feedback channel that ultimately formed an external continuously sweeping ow.
There was a linear relationship between the oscillation frequency and the inlet mass ow rate.2. The modal analysis characterized the spatial structures and their time evolution, which dominated the internal-external ow pattern.Where the POD results revealed the most energetic ow structure and corresponding time evolution, while the DMD results extracted the dominant mode containing a single frequency.Although performed from different perspectives, the POD and DMD results exhibited signi cant similarities because of the remarkable periodicity of the ow. 3. The structural parameter analysis showed that the inlet wedge width affected the working mechanism of the oscillator by in uencing the proportion of feedback ow.Increasing the from to led to an increase in the proportion of feedback ow and the de ection of the 5. Unlike the above-mentioned two parameters, which were found to affect the jet ow pattern, the mixing-chamber Coandă surface caused the mainstream to accumulate inside the mixing chamber and prevented the development of recirculation bubbles and Coandă effects, thereby breaking the feedback ow mechanism.
Overall, this study constituted a comprehensive investigation of simpli ed uid-oscillator designs.It may support increased use of oscillating jets in extreme application scenarios and will serve as a valuable reference for the selection of actuators.
. The sub-millimeter manufacturing of such complicated internal structures is highly challenging, which means that different internal structures must be explored to simplify oscillator design.Various studies have focused on the parameter analysis and structural design of oscillators.Melton et al. (2016) and Koklu et al. (2017) have parametrically evaluated the ow control effects of SJAs using an adverse pressuregradient ramp and a NACA 0015 model, respectively.Jurewicz et al. (2018) and Kara et al. (2017) have numerically evaluated the effects of feedback channels and Coandă surfaces on the performance of SJAs.Tajik et al. (2021) used distributed hot-wire measurements to characterize the ow behavior of an oscillator in various con gurations.Wen et al. (2020) experimentally examined the in uence of the internal geometry of a SJA on its working mechanism and devised several novel oscillator designs.Kwon et al. (2021) and Tomac et al. (2020) have designed several inspiring uidic oscillator pairs (including uidic oscillator pairs, synchronized stacked arrangement, and phase-synchronized uidic oscillator pairs) through various types of shared feedback channels to break through the defects of the classical con guration.
measurement methods have been comprehensively reviewed by Liu et al. (2021), Peng et al. (2019), and Gregory et al. (2014), and have been used for the measurement of uidic oscillator ow elds.Gregory et al. (2001, 2007) rst utilized the PSP method to measure the unsteady external ow eld of a micro uidic oscillator and characterize the structure of its sweeping jet ow.More recently, Zhou et al. (2022) applied PSP measurement to obtain the time-resolved ow dynamics inside a uidic oscillator with a jet speed of up to Mach 0.7.
Ostermann et al. (2015a) and Bobusch et al. (2013), as follows: the diagonal matrix composed of , and the vector contains the initial modal amplitudes.Kutz et al. (2016) and Tu et al. (2013) have comprehensively surveyed the above-mentioned DMD algorithm.
j t) b j = \varvecΦexp (\varvecΩt) \varvecb ω j = ln (λ j ) /Δt \varvecΩ = diag (ω) based on Tu et al. (2013) and Ali et al. (2016), the DMD modes were sorted by the -norms .The physical meaning of the energy contained in DMD modes differs from that contained in POD modes and corresponds to the contribution of speci c ow structures to the ow eld.
uence of various inlet-wedge widths on the devised con gurations.The time-average pressure contours of the B-1 and B-2 con gurations are shown in Fig.15.
Wen et  al. (2020)  discussed the in uence of inlet wedge width on the classical two-feedback channel con guration over a wider range than that considered in the current study.Overall, the impact of inlet wedge width on the uid dynamics of our con guration is consistent with the results obtained byWen  et al. (2020) for classical designs.4.3.2Feedback-channel inlet width ( ) Tajik et al. 2021 and Bobusch et al. 2013 have shown that the ow dynamics of an oscillator can also be affected by feedback ow strength.However, the underlying mechanisms by which this parameter affects ow dynamics remain unclear.Accordingly, two con gurations, C-1 and C-2 (as detailed in Section 2.1) mainstream inside the oscillator, such that the external jet-de ection angle increased from ~ 60° to ~ 90° and the sweeping frequency increased.4.The feedback-channel inlet widthaffected the magnitude of transverse de ection and the topology of the internal-external jet.An increase in from to caused the feedback ow strength and energy consumption to increase, resulting in hysteresis of the external jet de ection, a change in the ow topology, and a reduction in the oscillation frequency.

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Figure 4 Post
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Figure 5 Light
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Figure 8 Time
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Figure 9 Phase
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Figure 10 Energy
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Figure 12 See
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Figure 13 Characteristics
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Figure 15 Flow
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Figure 16 Flow
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Figure 17 Flow
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