Experiments were conducted to assess the aeroacoustic characteristics of a 30P30N three-element high-lift airfoil fitted with two different types of slat cove fillers in the Aeroacoustic Facility at the University of Bristol. The results are presented for the angle of attack α = 18° at a free-stream velocity of U = 30 m/s, which corresponds to a chord-based Reynolds number of Rec = 7 × 105. Simultaneous measurements of the unsteady surface pressure were carried out at several locations in the vicinity of the slat cove and at the far-field location to gain a deeper understanding of the slat noise generation mechanism. The results were analyzed using a higher-order statistical approach to determine the nature of the broadband hump seen at low frequencies for the 30P30N high-lift airfoil observed in recent studies and also to further understand the tone generation mechanism within the slat cavity. Intermittent events induced by the resonant mechanism between the slat and its trailing edge were investigated in detail. A series of correlation and coherence analysis of the unsteady surface pressure measurements was carried out to identify and isolate the low-frequency hump. A wavelet analysis was performed to investigate the nature of the slat-wing resonant intermittent events in both time and frequency domains. Novel approaches, dynamic mode decomposition, and multi-resolution dynamic mode decomposition were applied to the wavelet coefficient moduli to determine the nature of the noise.

The rapid growth of the aircraft industry constantly demands for more efficient and quieter aircraft than the ones currently in service. The impact of noise generated by aircraft during take-off and landing has come under scrutiny over the past decades, especially with the increasing popularity of air travel and expansion of airports. Although aircraft engine noise has been substantially reduced by the use of high bypass engines, the airframe noise remains a major contributor to aircraft noise. In order to reduce these prominent noise sources, several passive and active flow control methods1,2 have been investigated in the past, including morphing structures,3–9 porous materials,10–12 surface treatments,13 and serrations.14–17 High-lift devices, namely, slats and flaps, are one of the major contributors to airframe noise. Their noise generation mechanism is both broadband and narrowband in nature. Studies on conventional slat and wing configurations have shown that they mainly comprise broadband and tonal noise components. The emergence of discrete tones at a mid-frequency range has been reported in multiple studies.18–33 The number of these tonal peaks decreases with the angle of attack, and their amplitude decreases with increasing slat gap and overlap.27 However, their noise generation mechanism is yet to be fully understood.

Several successful studies34–44 were carried out over the past two decades to reduce the characteristic noise arising from the slat. This was achieved by enclosing the recirculation region within the slat cavity. The characteristic tones from the slat arise as a result of the feedback loop between the vortex shedding emanating from the slat cusp and its impingement on the slat lower surface. In order to eliminate this tonal and broadband noise from the shear layer instability and impingement, the slat cavity was enclosed with a solid filler having an aerodynamic profile. Horne et al.,34 from NASA, in order to eliminate the unsteady recirculation region within the slat cove, tested a solid Slat Cove Filler (SCF) and revealed that it was effective in reducing broadband slat noise up to 4 dB–5 dB. Streett et al.35 further demonstrated that the noise reduction was sensitive to the angle of attack and SCF modification. Furthermore, the SCF modification showed a reduction of 3 dB–5 dB over a wide spectrum. Imamura et al.38 and Ura et al.,39 from JAXA, showed that noise reduction of up to 5 dB could be achieved with the use of slat cove fillers, both experimentally and computationally.

Previous studies18–33 have indicated that the strong tonal peaks in the mid-frequency range (Sts = 1–5) are generated by the flow-acoustic cavity feedback mechanism of slat cove shear layer instabilities. Additionally, a possible quadratic interaction between the tonal peaks was observed.25–29 

In general, broadband humps observed at a low-frequency range (Sts < 1) in previous studies were assumed to be part of the cavity feedback mechanism. However, in a recent study, Kamliya et al.4 hypothesized that the first Rossiter mode is most likely to be concealed within the broadband hump 0.5 < Sts < 1 and that the source of that broadband hump is not solely due to the Rossiter modes. The Rossiter modes arise as a result of flow-induced oscillations from the vortex shedding at the slat cusp, and the resulting acoustic feedback mechanism portrays distinct narrowband peaks. A further higher-order spectral analysis of near-field pressure fluctuations demonstrated strong self-interaction of the broadband hump (0.5 < Sts < 1), with no cross-interaction, confirming the varying nature of the broadband hump. However, the other narrowband peaks showed high levels of quadratic coupling and cross-interaction reaffirming mode interactions. Moreover, the results from the persistence spectrum of the surface pressure fluctuations indicated that the broadband hump (0.5 < Sts < 1) was not persistent over the time period like the other narrowband peaks; instead the energy of the broadband spectra was evenly distributed in time over the entire frequency range.

Although several studies34–43 have reported the aerodynamic and aeroacoustic behavior of slat cove fillers, further fundamental physics of the noise generation mechanism of the slat and the noise suppression mechanism of the slat cove fillers are yet to be fully understood. It is well established from previous studies18–33 that the source of the tonal peak is due to the Rossiter modes that arise from the interaction between the slat cusp and its trailing edge; however, the nature and the source of the broadband hump at low-frequency is yet to be identified and documented.4 The present study aims to use a highly instrumented high-lift airfoil to establish the nature and identify the source of the broadband hump in question. Complementing the previous works on slat cove fillers by Kamliya Jawahar et al.,4 this study presents improved unsteady pressure measurements at several locations within the slat cove region for three different configurations. In addition to the standard Fourier transform based analysis, this paper uses new approaches including the wavelet analysis coupled with stochastic analysis to improve the understanding of this low-frequency broadband hump and the slat noise generation mechanism. Data were analyzed providing an exhaustive intermittency characterization including spectral quantities, cross correlations, and continuous wavelet transform (CWT). Furthermore, the Dynamic Mode Decomposition (DMD) technique has also been used as a data-driven analysis tool applied to the wavelet coefficient moduli from various sensors within the slat cove region to extract meaningful spatiotemporal patterns that dominate the dynamic activity in the broadband hump and the slat tone. Multi-resolution dynamic mode decomposition was used to identify the low-frequency pulsation of the broadband hump.

A three-element 30P30N high-lift airfoil with a retracted chord length of c = 0.35 m and a span length of l = 0.53 m was manufactured in aluminum using a computer-aided numerically controlled machine. The geometrical parameters of the high-lift airfoil are detailed in Table I and depicted in the schematic in Fig. 1. Furthermore, brackets between the main-element and slat were not used in the spanwise direction to retain two-dimensionality within the slat cove region. Instead, steel clamps were used on the side of the airfoil after the side plates to attach the three-elements. A zig-zag turbulator tape with a thickness of ht = 0.5 mm and a width of wt = 6 mm with a turbulator angle of 70° was used at x/c = −0.055 on the pressure side of the slat surface upstream of the slat cusp to trip the flow and to induce a turbulent shear layer.21 The aforementioned 30P30N high-lift airfoil model was thoroughly studied at different test conditions, and its aerodynamic and aeroacoustic performance was validated3,4 against the existing benchmark studies.23 

TABLE I.

Geometrical parameters in percentage of stowed airfoil chord, c = 0.35 m.

Slat chord cs 0.15c 
Main-element chord cme 0.83c 
Flap chord cf 0.3c 
Slat deflection angle δs 30° 
Flap deflection angle δf 30° 
Slat gap gs 2.95% 
Flap gap gf 1.27% 
Slat overhang os −2.5% 
Flap overhang of 0.25% 
Slat chord cs 0.15c 
Main-element chord cme 0.83c 
Flap chord cf 0.3c 
Slat deflection angle δs 30° 
Flap deflection angle δf 30° 
Slat gap gs 2.95% 
Flap gap gf 1.27% 
Slat overhang os −2.5% 
Flap overhang of 0.25% 
FIG. 1.

Geometric parameters of the 30P30N three-element high-lift airfoil.

FIG. 1.

Geometric parameters of the 30P30N three-element high-lift airfoil.

Close modal

1. Slat cove-filler design

In the present study, half slat cove filler (H-SCF) and slat cove filler (SCF) configurations were tested for the purpose of slat noise reduction. The H-SCF configuration was observed to exhibit favorable aerodynamic and noise reduction properties in computational studies by Tao and Sun43 and experimental studies by Kamliya Jawahar et al.4 The SCF configuration tested in this study was similar to that introduced by Imamura et al.38,39 The SCF profile was designed using the slat cusp shear layer trajectory possessing high turbulent kinetic energy at the angle of attack α = 8° extracted from Reynolds-averaged Navier–Stokes (RANS) steady-state simulation of the baseline configuration. Both the slat cove fillers were manufactured using 3D printing technology in four different sections such that they could be slid along the span of the slat cove. It should be noted that although the SCF is designed for a free-stream angle of attack of α = 8°, the aerodynamic characteristics of the high-lift device remain unaffected over a wider range of angles of attack, as shown by Kamliya Jawahar et al.4 

Aeroacoustic measurements for the 30P30N high-lift airfoil were carried out at the University of Bristol Aeroacoustic Facility.45 The closed-circuit open-jet anechoic wind tunnel has a nozzle exit of 0.5 × 0.775 m2 with a contraction ratio of 8.4:1. The wind tunnel is capable of velocities up to 40 m/s with turbulence levels as low as 0.25%.45 The flow turbulence intensity (TI) was calculated using urms/Umean, where urms is the root-mean-square of the velocity fluctuations and Umean is the mean velocity. The effect of the large-scale facility unsteadiness from the turbulence intensity measurements was eliminated by applying a high-pass filter to the velocity fluctuations at f = U/2LTS, where LTS is the largest test section dimension. A schematic of the airfoil installed in the wind tunnel is shown in Fig. 2. The airfoil was placed at a distance of 1c from the nozzle exit (see Fig. 2). The tests were carried out at the geometric angle of attack α = 18° for a free-stream velocity of U = 30 m/s, which corresponds to a chord-based Reynolds number of Rec = 7.0 × 105. It is important to note that the high-lift airfoils cause high-flow deflection and a circulation region, resulting in a difference of about 8.5° between the corrected and geometric angles of attack for high-lift airfoils in an open-jet configuration, as previously shown in other studies.4,46

FIG. 2.

Schematic of the experimental setup.

FIG. 2.

Schematic of the experimental setup.

Close modal

1. Acoustic measurements and instrumentation

Several FG-3329-P07 pressure transducers from Knowles Electronics were installed on the main-element of the wing to measure the unsteady surface pressure around the slat region on the main-element (FG1-FG5, see Table II). This pressure transducer was selected for the measurement of the near-field pressure as it was proven successful in the previous experiments carried out by Garcia-Sagrado and Hynes47 on a NACA 0012 airfoil. The transducer placement on the main-element was based on the previous studies by Murayama et al.23 The transducer has a diameter of 2.5 mm and a height of 2.5 mm with a sensing area of 0.8 mm. Additionally, remote sensing microphone probes connected to the pressure tap brass tube using a plastic tubing with an inner and outer diameter of 0.8 mm and 4 mm were employed to measure the unsteady surface pressure fluctuations on the slat and at various streamwise locations on the main-element (see Fig. 4 and Table II).48 The remote sensors were made using Panasonic WM-61A microphones attached to 1.6 mm diameter brass tubes with 0.4 mm pinholes (see Fig. 3). In order to eliminate the spurious tones related to the standing waves in the flexible tubing, an acoustic termination tube of 2 m (determined from a parametric study) was used for each port (see Fig. 3). The far-field noise was measured using a Panasonic WM-61A microphone placed at an angle of 90° at a distance of 1 m away from the slat trailing edge. The near- and far-field microphones were calibrated using a GRAS 40PL microphone that is factory calibrated with a known manufacturer sensitivity.49 

TABLE II.

Streamwise and spanwise unsteady pressure measurement location on the 30P30N three-element high-lift airfoil.

No.x (mm)y (mm)z (mm)
S1 −6.918 −11.622 265 
M1 17.347 −10.019 265 
M2 15.126 −5.839 265 
M3 17.622 265 
M4 23.520 5.485 265 
FG1 22.414 −11.356 277 
FG2 22.414 −11.356 280.6 
FG3 22.414 −11.356 288.4 
FG4 22.414 −11.356 301.4 
FG5 22.414 −11.356 319.6 
No.x (mm)y (mm)z (mm)
S1 −6.918 −11.622 265 
M1 17.347 −10.019 265 
M2 15.126 −5.839 265 
M3 17.622 265 
M4 23.520 5.485 265 
FG1 22.414 −11.356 277 
FG2 22.414 −11.356 280.6 
FG3 22.414 −11.356 288.4 
FG4 22.414 −11.356 301.4 
FG5 22.414 −11.356 319.6 
FIG. 3.

Schematic of the remote sensor configuration.

FIG. 3.

Schematic of the remote sensor configuration.

Close modal

A beamforming study using an array of 63 free-field microphones was carried out to evaluate the airfoil and side-plate junction noise. The junction noise showed an insignificant contribution to the slat frequencies [>3600 Hz (Sts ≈ 6.2)] that are of interest in the study. In contrast, the beamforming revealed that the primary noise source was clustered at the slat cove and well distributed over the span of the wing. Furthermore, the results obtained from an individual far-field microphone followed a trend similar to that of the beamforming while accurately capturing the characteristic tonal peaks and broadband humps from the slat. Previous studies4,29 have also established that individual far-field microphones were sufficient for the measurement of the dominant characteristic features of the slat noise.

National Instruments PXIe-4499 modules inserted in a National Instruments PXIe-1062Q chassis were used for data acquisition. The data were simultaneously collected for 120 s at a sampling frequency of fs = 215 Hz for all the near- and far-field microphones. The power spectrum results were obtained using the power spectral density (PSD) of the pressure signals with the Hanning window and were further averaged 220 times to yield a frequency resolution of Δf = 2 Hz. The statistics of the pressure fluctuation were evaluated using the following equation to obtain the Sound Pressure Level (SPL):

SPL=20log10prmspref,
(1)

where prms is the root-mean-square of the acoustic pressure and pref = 20 μPa is the reference pressure. The SPL and other results are presented in terms of Strouhal number (Sts), which is defined as

Sts=fcsU,
(2)

where cs is the slat chord and U is the free-stream velocity. The results of the unsteady surface pressure auto-spectra are presented in Fig. 6 for all the remote sensors (see Table II) and the tested slat configurations.

A brief summary of the Dynamic Mode Decomposition (DMD) technique50–53 employed in the current study is presented here. The exact DMD52,53 method based on the following equations was applied on the wavelet coefficient moduli to understand the dynamic characteristic of the local oscillating features of the surface pressure measurement. As described by Kutz et al.,50 the data used for DMD involve two parameters: the number of spatial measurements per time snapshot (n) and the number of snapshots taken in time (m). In the present study, at first, the wavelet coefficient moduli (|Wx|) for a wide range of Strouhal numbers (Sts = 0.2–8) were calculated using the continuous wavelet transform detailed in Sec. IV C 2. The calculated wavelet coefficient moduli vary in intensity over time depending on the fluctuations of the pressure signal. The snapshot matrix xk of dimension nx × ny is then generated, with nx = 4 remote sensor locations (M1–M4, see Fig. 4) and ny = 171 as the Strouhal numbers of interest, distributed between Sts = 0.5–7. The snapshot matrix (xk) is reshaped into an n-dimensional column vector (nxny × 1) with 684 rows. The data were collected at regularly spaced time intervals (Δt) given by

tk+1=tk+Δt,
(3)

where the data snapshots are collected at tk from k = 1, 2, …, m for a total of m measurement times. The data snapshots are arranged into two n × (m − 1) matrices,

X=[x1,x2,,xm1],
(4)
X=[x2,x3,,xm].
(5)

As can be seen from Eqs. (4) and (5), the columns of X′ are shifted one Δt from those in X. X and X′ can be written as

X=AX,
(6)

where A is a best fit linear operator that can be described as a high-dimensional linear regression of the nonlinear dynamics that relate X to X′.52 The DMD of the data matrix pair (X and X′) is given by the eigendecomposition of A. The present snapshot X could be related to the future snapshot X′ using the operator A. This approach aims to solve the nonlinear dynamical system using the approximated optimal linear dynamics. An approximation of A can be achieved using Single Value Decomposition (SVD) of the data matrix X,

XUΣV*,
(7)

where * denotes the conjugate transpose, UCn×r, ΣCr×r, VCm×r. Here, rm denotes either the exact or approximate rank of the reduced SVD approximation to X. The left singular vectors U are proper orthogonal density (POD) modes. The columns of U are orthonormal, so U * U = I; similarly, V * V = I.50,51

FIG. 4.

Location of the unsteady pressure measurement probes: (a) baseline, (b) half slat cove filler (H-SCF), and (c) slat cove filler (SCF).

FIG. 4.

Location of the unsteady pressure measurement probes: (a) baseline, (b) half slat cove filler (H-SCF), and (c) slat cove filler (SCF).

Close modal

According to Eq. (6), the full matrix A can be obtained by computing the pseudo-inverse of X,

A=XUΣ1V*.
(8)

To obtain the leading r eigenvalues and eigenvectors of A, A could be projected onto the POD modes U. By replacing Eqs. (7) and (8) in (6), we obtain

Ã=U*AU=U*XVΣ1.
(9)

Thus, the eigendecomposition of the matrix à can be computed by

ÃW=Wλ,
(10)

where W is the matrix with the eigenvectors and λ contains the corresponding eigenvalues λk, respectively. Each non-zero eigenvalue λk is a DMD eigenvalue. Then, the exact DMD modes Φ are given by

Φ=XVΣ1W.
(11)

The approximate solution at all future times [xDMD(t) is given by

xDMD(t)k=1mbkϕkexp(ωkt)=Φexp(Ωt)b,
(12)

where b is the vector of the coefficient, bk is the initial amplitude of each mode, Φ is the matrix whose columns are the eigenvectors ϕk, Ω = diag(ω) is a diagonal matrix whose entries are the eigenvalues ωk, and ωk = ln(λk)/Δt.

In the present study, a new approach called multi-resolution Dynamic Mode Decomposition (mrDMD)50 is explored to capture the frequency modulation by applying it on the wavelet coefficient moduli on a single remote sensor. This technique is based on a recursive approach to remove the low-frequency or slowly moving features from a snapshot series. The calculated wavelet coefficient moduli from the remote sensor signals were considered as a series of snapshots m. These m snapshots contain a series with low- and high-frequency signatures.

Kutz et al.50 have mathematically shown that the mrDMD separates the DMD approximate solution [Eq. (12)] at the first level as follows:

xmrDMD(t)=k=1mbkϕk(1)exp(ωkt)=k=1m1bkϕk(1)exp(ωkt)slow modes+k=m1+1mbkϕk(1)exp(ωkt)fast modes,
(13)

where ϕk(1) are the DMD modes evaluated from the full m snapshots. We identify the slow modes with the first summation on the right side, whereas the second one defines the fast modes. Once again, the process is iteratively repeated, removing the slow modes at each step. The algorithm stops when the desired decomposition level is achieved, obtaining at the end a series of matrices with modes at different frequencies. The final DMD approximation reads as

xmrDMD(t)=k=1m1bk(1)ϕk(1)exp(ωk(1)t)+k=1m2bk(2)ϕk(2)exp(ωk(2)t)+k=1m3bk(3)ϕk(3)exp(ωk(3)t)+.
(14)

To summarize, ϕk(i) and ωk(i) are the DMD modes and DMD eigenvalues obtained at the ith level of decomposition, bk(i) are the initial projections of the data onto the time interval of interest, and mk is the number of slow modes retained at each level. Similarly, long, medium, and short term trends in data are identified through successive levels containing faster modes. For detailed information on this method, the readers are referred to Ref. 50.

Aerodynamic characteristics of the 30P30N three-element high-lift airfoil tested at the University of Bristol with suggested slat modifications (H-SCF and SCF) have been well documented in a previous study by Kamliya Jawahar et al.4 It has been established that the aerodynamic characteristics of the three-element high-lift airfoil were not affected by the slat modifications. The observations from the current experimental campaign will be presented in the following order. First, the far-field acoustic characteristics of the tested slat configurations are presented. Subsequently, single- and two-point spectral analyses of the near-field unsteady pressure measurements are analyzed in detail at various streamwise locations. A thorough wavelet analysis for various remote sensors is carried out, and the time-frequency analysis is performed using statistical mechanics. Finally, the dynamic characteristics of the time-frequency behavior of the pressure fluctuations are presented and analyzed.

The far-field noise levels for the 30P30N three-element high-lift airfoil with various slat cove filler configurations were measured using a single Panasonic WM-61A microphone away from the slat trailing edge at an angle of 90° and at a distance of 1 m. The far-field pressure fluctuations were analyzed in the frequency domain using the SPL through Eq. (1). The SPL spectra for all the tested configurations are reported in Fig. 5. Evidently, the persistence of the tonal peaks close to Sts = 1.6 and Sts = 3.2 demonstrated the characteristic behavior of the slat noise. The SPL spectra for the surface mounted pressure sensors at FG1 are presented in Fig. 5(b) to show the general trend of the near-field pressure characteristics over a wide range of frequencies of up to f = 10 kHz (Sts = 17.5). It is apparent that the high-frequency dissipation within the tiny tubes of the remote sensors yields accuracy only over a narrow frequency range. In the current experimental setup, the remote sensors are accurate and valid only up to f = 6 kHz (Sts = 10). Accordingly, it is evident from Fig. 5(b) that a frequency of about f = 6 kHz (Sts = 10) was sufficient to capture the dominant characteristics of slat noise.

FIG. 5.

Sound pressure level for all the tested configuration (a) far-field measurements FF1 at a distance of 1 m away from the slat trailing edge and (b) near-field measurement FG1 on the main-element.

FIG. 5.

Sound pressure level for all the tested configuration (a) far-field measurements FF1 at a distance of 1 m away from the slat trailing edge and (b) near-field measurement FG1 on the main-element.

Close modal

The tonal characteristics of the wall pressure spectra with multiple distinct narrowband peaks indicate the presence of cavity oscillations for the baseline configuration; see Fig. 5. Additionally, these strong tonal peaks are also seen in the far-field noise measurements. The observed tonal peaks are due to the flow-acoustic coupling between the slat cusp shear layer and its impingement on the slat trailing edge, leading to distinct tonal peaks. These tones can be accurately predicted using a simplified Rossiter mode equation [Eq. (15)], as previously proven by Terracol et al. and other studies.20,24,54 In line with Terracol's investigation,24 the discrete tonal frequencies from the flow interaction with the slat were predicted using

fn=nULa1M+ηl/κv.
(15)

A simplified schematic and the parameters used in Eq. (15) by Terracol et al.24 are shown and listed in Table III along with the parameters used in the current study. The local mean flow velocity data required for this prediction model were acquired from the particle image velocimetry (PIV) measurements by Kamliya Jawahar et al. in previous studies.3,5,8,49 It should be noted that according to Terracol et al.,24 the vortices originating from the slat cusp are convected along the main shear layer surrounding the main vortex core inside the slat cove at a mean convection velocity (Uv). Although the flow is highly unsteady in the slat shear layer, following Terracol et al.,24 in the present study, the convecting velocity of the vortices in the slat shear layer was assumed to be constant and the average velocity was used in the Rossiter formula.

TABLE III.

Simplified schematic of the tonal frequency prediction model by Terracol et al.24 and the parameters used for the tonal peak frequency prediction in the present study for angle of attack α = 18°.

SchematicVariablesParametersValuesUnits
 Lv Shear layer path length 0.037 
La Acoustic path length 0.035 
Uv Mean convection velocity 18 m/s 
U Free-stream velocity 30 m/s 
M Free-stream Mach number 0.09 … 
κv = Uv/U 0.60 … 
ηl = Lv/La 1.06 … 
SchematicVariablesParametersValuesUnits
 Lv Shear layer path length 0.037 
La Acoustic path length 0.035 
Uv Mean convection velocity 18 m/s 
U Free-stream velocity 30 m/s 
M Free-stream Mach number 0.09 … 
κv = Uv/U 0.60 … 
ηl = Lv/La 1.06 … 

The analytical prediction for the baseline configuration at an angle of attack α = 18° is presented in Table IV using Eq. (15) by Terracol et al.24 It was observed that Eq. (15) can predict the modes accurately, thereby confirming that the modes are generated from the acoustic feedback mechanism. In Fig. 5 and Table IV, the narrowband peaks from St2 to St6 are distinct with high energy. However, the St1 peak observed in the results is more of a broadband hump that is well distributed between Sts = 0.5 and 1. This combined nature of the St1 peak was identified and documented by Kamliya Jawahar et al.4 However, the nature and source of this broadband hump were not identified. In Secs. IV B–IV D, the nature and source of this broadband hump will be further analyzed and its difference from the Rossiter mode will be documented.

TABLE IV.

The tonal peaks observed for the baseline case in the near-field and far-field measurements at angles of attack of 18°.

No.StnStsDescriptionExperiment f (Hz)Analytical f (Hz)
St1 0.857 Mode 1 489.7 461.7 
St2 1.611 Mode 2 920.6 923.3 
St3 2.303 Mode 3 1316 1385 
St4 3.212 = 2St2 1835.4 1846.6 
St5 4.056 = St3 + St2 2317.7 2308.3 
St6 4.841 = 3St2 2766.2 2769.9 
No.StnStsDescriptionExperiment f (Hz)Analytical f (Hz)
St1 0.857 Mode 1 489.7 461.7 
St2 1.611 Mode 2 920.6 923.3 
St3 2.303 Mode 3 1316 1385 
St4 3.212 = 2St2 1835.4 1846.6 
St5 4.056 = St3 + St2 2317.7 2308.3 
St6 4.841 = 3St2 2766.2 2769.9 

1. Power spectral density

As a part of this experimental campaign to isolate the broadband hump at St1, several remote sensors (M1–M4 and S1) were connected to the surface pressure ports on the main-element and slat in the slat cove region. The locations of these remote sensors are detailed in Fig. 4 and Table II. The SPL spectra for the remote sensors on the main-element M1–M4 and on the slat S1 for the tested baseline, H-SCF, and SCF configurations are presented in Fig. 6. At first glance, the baseline configuration clearly shows several tonal peaks related to the Rossiter modes arising from the feedback between the slat cusp and the slat trailing edge. The sound pressure level increases from remote sensor location M1 to M3 and decreases past the slat gap at M4. Overall, the highest sound pressure levels were observed at M3. At location S1 on the pressure side of the slat, lower sound pressure levels were observed compared to the other remote sensor locations. When considering the broadband hump between Sts = 0.5 and 1, the highest sound pressure levels were observed at both the M2 and M3 locations with similar magnitude, and the lowest sound pressure levels were observed at the slat sensor location S1.

FIG. 6.

Near-field energy spectra for the surface pressure transducers shown in Fig. 4: (a) baseline, (b) half slat cove filler (H-SCF), and (c) slat cove filler (SCF).

FIG. 6.

Near-field energy spectra for the surface pressure transducers shown in Fig. 4: (a) baseline, (b) half slat cove filler (H-SCF), and (c) slat cove filler (SCF).

Close modal

The use of slat cove fillers eliminated the tonal peaks observed in the baseline configuration. Overall, the remote sensors M1 and M4 have similar spectral levels at Sts < 1.5 with M4 having higher spectral levels at Sts > 1.5 for both the presented slat cove filler configurations. The remote sensors M2 and M3 show similar levels at Sts < 1, but at Sts > 1, M3 shows higher spectral levels for both the cove filler configurations. Ultimately, the unsteady surface pressure results have shown high levels of energy at M3 compared to the other measurement locations for all the presented configurations. The higher energy levels observed at M3 cannot confirm the location at which the broadband hump is being generated within the slat cove region; therefore, a further analysis is required to isolate the source of the broadband hump (St1).

1. Cross-correlation analysis

Two-point statistics of the pressure signals were computed using the auto and cross correlation between two remote sensors. The auto- and cross correlations were computed using

Rpipj(τ)=pi(t+τ)pj(t)¯pirmspjrmsforpi=M1andpj=M1, M2, M3, M4,
(16)

where pi and pirms are the surface pressure and surface pressure root-mean-square of the reference remote sensor signals (M1), pj and pjrms are the surface pressure and surface pressure root-mean-square of the downstream remote sensor signals at various locations along the chord, τ is the time-delay, and the time average is represented by the overbar.

The auto-correlation results [Rpipj(τ)] as a function of the normalized time-delay τ* = τU/cs for the three tested configurations at various remote sensor locations on the main-element are presented in Fig. 7. At first glance, it is evident that the flow exhibits different characteristics at different remote sensor locations for the three presented configurations. For the baseline, M1 and M4 exhibit a fast decaying periodic behavior, whereas M2 and M3 exhibit a slow decaying periodic behavior with a Gaussian shape having a low decay rate. Therefore, this behavior is suggestive of a strong vortex shedding. Furthermore, this can be confirmed by calculating the frequency (τvs*) from the two peaks of Rpipj(τ) as demonstrated by Kamliya Jawahar et al.4 The calculated time-delay τvs*=0.6626 corresponds to Stvs = 1.51 ≈ St2, which is the vortex shedding frequency. This frequency further corresponds to the tone seen in the SPL plots in Figs. 5 and 6. In the case of the H-SCF configuration, at M2 and M3, the periodic behavior decays rapidly, indicating the very low energy of the vortex shedding. Moreover, the periodic behavior is completely absent at all the remote sensor locations, suggesting the absence of a vortex shedding for the SCF configuration.

FIG. 7.

Auto-correlation of the surface pressure fluctuations for the remote sensors on the main-element: (a) baseline, (b) H-SCF, and (c) SCF.

FIG. 7.

Auto-correlation of the surface pressure fluctuations for the remote sensors on the main-element: (a) baseline, (b) H-SCF, and (c) SCF.

Close modal

The results of the cross correlation of the surface pressure fluctuations with reference to the microphone on the main-element M1 are presented in Fig. 8. As expected, the streamwise cross-correlation results for the remote sensors on the main-element (M1–M4) confirm the difference in the time-lag between the various remote sensors. For the baseline configuration, the time-lag increases with the distance between the remote sensors. The H-SCF configuration shows a considerable increase in the time-lag between the remote sensors on the main-element and also has increased the time-lag than that of the baseline. In the case of the SCF configuration, the time-lag further increases compared to both the baseline and H-SCF configurations. Additionally, the cross-correlation magnitude between M1 and M4 is reduced substantially for the H-SCF configuration and is completely absent for the SCF configuration. This is likely due to the interference of the cove fillers on the coherent structures passing between M1 and M4. The use of cove fillers appears to interfere with the turbulent structures, making the cross-correlation shape larger (τ* = τU/cs) seemingly due to the breaking of coherent structures.

FIG. 8.

Cross correlation of the surface pressure fluctuations for the remote sensor with M1 as the reference: (a) baseline, (b) H-SCF, and (c) SCF.

FIG. 8.

Cross correlation of the surface pressure fluctuations for the remote sensor with M1 as the reference: (a) baseline, (b) H-SCF, and (c) SCF.

Close modal

2. Coherence analysis

The coherence function calculates the phase correlation between two different remote sensors averaged over time. The coherence between the surface remote sensors can be obtained using the following equation:

γpipj2(f)=Φpipj(f)2Φpipi(f)Φpjpj(f) for pi=M1 and pj=M1, M2, M3, M4,
(17)

where the symbol |◻| denotes the absolute value, pi is the reference remote sensor signal (M1), pj is the remote sensor signal at various locations within the slat cove region, and Φpipj is the cross-spectral density between two pressure signals, pi and pj. The results for the near-field coherence of the pressure signals collected within the slat cavity region with M1 as the reference transducer are presented in Fig. 9.

FIG. 9.

Coherence of the surface pressure fluctuations between the remote sensor M1 and the following remote sensors on the main-element: (a) baseline, (b) H-SCF, and (c) SCF.

FIG. 9.

Coherence of the surface pressure fluctuations between the remote sensor M1 and the following remote sensors on the main-element: (a) baseline, (b) H-SCF, and (c) SCF.

Close modal

The results of the streamwise coherence of the remote sensors present a high level of coherence for the even narrowband peaks (St2,4,6) and low coherence for the odd narrowband peaks (St3,5) between the main-element remote sensors (M1–M4). In line with the discussion from Secs. IV B and IV C 1, from the following results shown in Fig. 9, it was observed that the broadband hump was clearly of a different nature. Accordingly, in the case of M1–M2/M3, the results exhibit a high coherence level between Sts = 0.6 and 1, whereas in the case of M1–M4, a narrower peak with high levels of coherence is observed exclusively between Sts = 0.8 and 1. This shows that even though the peak St1 might be influenced by the Rossiter modes, the broadband hump is not generated by them and is further confirmed by the cove filler coherence results. For the H-SCF and SCF configurations, high coherence at Sts < 1 for M1–M2 and M1–M3 is observed. Interestingly, in the case of M1–M4, the narrow peak observed for the baseline configuration is absent for both the cove filler configurations with the SCF configuration having very low coherence. However, for both the H-SCF and SCF configurations, a peak in the coherence level can be found at Sts = 1.6.

In order to understand the far-field propagation effects of the identified wall pressure signatures, the magnitude-square coherence between the wall pressure and far-field signals was calculated and presented in Fig. 10. The results of the near- to far-field coherence presented in Fig. 10 show a very high level of coherence for a series of narrowband peaks for the baseline configuration. Furthermore, high coherence levels were observed for the even narrowband peaks (St2,4,6) relative to the odd narrowband peaks (St1,3,5) at all the remote sensor locations. Additionally, high levels of coherence of up to 0.6 were observed for Sts = 0.7–0.9, thus clarifying the presence of the Rossiter mode at St1. Moreover, this gives an insight into the broadband hump as the far-field coherence is narrower when compared to the streamwise coherence results seen earlier in Fig. 9. In the case of the slat cove filler configurations, low coherence levels are detected for the entire bandwidth with values less than 0.2. For the H-SCF configuration, marginally increased coherence levels with values of up to 0.4 are found at about Sts = 1.6. The SCF configuration has the lowest coherence levels compared to all the configurations, except for a small broadband hump at about Sts = 1.6. It is noteworthy that for the SCF configuration, the broadband hump at St1 has no coherence with the far-field noise, which suggests that the broadband hump is due to the non-propagating hydrodynamic energy field within the slat cove region.

FIG. 10.

Coherence of the surface pressure fluctuations between the far-field microphone FF1 located at 1 m away from the slat trailing edge and the various remote sensors on the high-lift airfoil: (a) baseline, (b) H-SCF, and (c) SCF.

FIG. 10.

Coherence of the surface pressure fluctuations between the far-field microphone FF1 located at 1 m away from the slat trailing edge and the various remote sensors on the high-lift airfoil: (a) baseline, (b) H-SCF, and (c) SCF.

Close modal

1. Continuous wavelet transform

In the present study, the continuous wavelet transform (CWT) is employed to capture the local oscillating features of the remote sensor signals in the time–frequency plane, thereby enabling us to observe the temporal characteristics of the pressure fluctuations along with their associated frequency. The CWT consists of a signal projection over a basis of support functions provided by the translation and dilatation of a mother wavelet function, and it is defined55,56 as

W(s,t)=Cψ12p(τ)ψ*tτs,
(18)

where s is the scale dilatation parameter corresponding to the width of the wavelet, τ is the translation parameter corresponding to the position of the wavelet, Cψ12 is a constant that takes the mean value of ψ(t) into account, and ψ*tτs is the complex conjugate of the dilated and translated mother wavelet ψ(t). The Morlet kernel was chosen as the mother wavelet, as reported analytically in previous studies.4,55 The contours of the wavelet coefficient moduli for the near-field remote sensor on the main-element (M3) are presented in Fig. 11 for all the tested configurations. Although the analysis was carried out for all the remote sensors, the results are presented only for M3 due to increased fluctuations at this location as will be shown later. In principle, the wavelet coefficients remain small if the fluctuations at a given time are smooth, and their value increases locally for large fluctuations, i.e., irregular structures. As a consequence, wavelet coefficients do not correspond to the entire signal's strength but to the fluctuations at a given scale and a given point. Following that, the baseline configuration shows a high wavelet coefficient at the vortex shedding frequency of Sts = 1.61 for all the remote sensor locations. For the frequencies of interest, St1 and St2, the highest wavelet coefficient was found at the location M3 among all the other remote sensor locations. For the H-SCF and SCF configurations, it is observed that the peak with increased fluctuations at St2 is absent. Overall, the results show high wavelet coefficient levels at M3 compared to all the other remote sensor locations for all the tested configurations.

FIG. 11.

Wavelet scalogram of a segment of a pressure signal obtained at remote sensor location M3: (a) baseline, (b) H-SCF, and (c) SCF.

FIG. 11.

Wavelet scalogram of a segment of a pressure signal obtained at remote sensor location M3: (a) baseline, (b) H-SCF, and (c) SCF.

Close modal

The importance of phase-space characteristics to analyze turbulence was well demonstrated by Farge.56 Accordingly, the phase-space characteristics of the pressure fluctuations along with the real part of the wavelet coefficients for the remote sensor M3 on the main-element are presented in Fig. 12. Throughout the discussion, it is preferred for the readers to refer to the real part of the wavelet coefficients [see Figs. 12(a)–12(c)] along with the phase-space characteristics [see Figs. 12(d)–12(f)] for better interpretation of the results. To better visualize the phase-space characteristics, the results are presented for a relatively short time period of 0.33 s. At first glance, all three configurations evidently show a different phase-space characteristic pattern with a lack of background noise. These patterns can further be separated into characteristic classes (repetitive and cone-like patterns) and can then be analyzed by class. The baseline configuration at St2 shows multiple repetitive patches in both the real part of the wavelet coefficients and the phase-space. This phase-space behavior is expected of a turbulent field with a superposition of wave-packets, such as eddies.56 As defined by Farge,56 the horizontal length of these patches corresponds to the spatial support of the eddies and the vertical length corresponds to the frequencies characterizing each excited wave-packet. The phase-space behavior at St2 with repetitive patches also corresponds to the vortex shedding frequency. The phase-space behavior at the broadband hump Sts = 0.5–1 displays energy density distributed among cone-like patterns, suggestive of the presence of coherent structures with each cone pointing to an excited structure.56 The spatial support (width) of these cones at the broadband hump Sts = 0.5–1 is significantly larger than the repetitive patches seen at St2, yet again clarifying the different nature of the broadband hump at Sts = 0.5–1. In the case of the H-SCF and SCF configurations, the energy density is distributed in cone-like patterns of different sizes at varying bandwidths that point to the presence of various coherent structures. The amplitude of the real part of the wavelet coefficients shows that these coherent structures possess lower energy for the slat cove filler configurations compared to the baseline configuration.

FIG. 12.

The real part of the wavelet coefficient [(a) baseline, (b) H-SCF, and (c) SCF] and the phase-space characteristics [(d) baseline, (e) H-SCF, and (f) SCF] of the surface pressure fluctuations at M3 from the continuous wavelet transform.

FIG. 12.

The real part of the wavelet coefficient [(a) baseline, (b) H-SCF, and (c) SCF] and the phase-space characteristics [(d) baseline, (e) H-SCF, and (f) SCF] of the surface pressure fluctuations at M3 from the continuous wavelet transform.

Close modal

The power spectral density of the wavelet coefficient moduli (|WSt|) at frequencies of interest would further aid in the understanding of the tone generation mechanism. The results of the amplitude of the wavelet coefficient moduli [E(|WSt|)] are presented in terms of the slat chord-based Strouhal number (Sts) in Fig. 13 at frequencies listed in Table IV. For the baseline configuration, the results show that modes St3, 4–6 are modulated by ΔSts = 1.61. A broadband hump with a peak at Sts = 0.7 is observed for the mode St3 at the remote sensor location M3. Overall, these results identify the Rossiter mode concealed within the broadband hump. On the contrary, the H-SCF and SCF configurations do not show any amplitude modulation at any of the remote sensor locations for all the frequencies considered here.

FIG. 13.

Power spectral density of the time signal and wavelet coefficient at selected resonance frequencies at location M3: (a) baseline, (b) H-SCF, and (c) SCF.

FIG. 13.

Power spectral density of the time signal and wavelet coefficient at selected resonance frequencies at location M3: (a) baseline, (b) H-SCF, and (c) SCF.

Close modal

2. Stochastic analysis

To better understand the fluctuating characteristics of the surface pressure at various remote sensors, the statistical distribution of the fluctuating properties using the wavelet coefficient moduli (|Wx|) at every frequency is presented.17,57 A simple schematic of the statistical analysis employed in this study is shown in Fig. 14. It is well known that the Fourier spectra of a given signal can be recovered by integrating the wavelet coefficient moduli in time, providing spectra less affected by statistical uncertainties.56 Following a similar strategy, in the present study, a detailed statistical analysis of the wavelet coefficient moduli was carried out over a time period of 120 s. The statistical representation of the wavelet coefficient moduli is presented in terms of the arithmetic mean [μ(|Wx|)], standard deviation [σ(|Wx|)], and coefficient of variation [CV(|Wx|)] to characterize the fluctuations in the flow. They are employed in the current study, with N as the total number of time steps,

μ(|Wx|)=Σi=1N|Wx|N,
(19)
σ(|Wx|)=Σi=1N(|Wx|iμ(|Wx|))2N,
(20)
CV(|Wx|)=σ(|Wx|)μ(|Wx|)×100.
(21)
FIG. 14.

Schematic of the statistical analysis of the wavelet coefficient moduli.

FIG. 14.

Schematic of the statistical analysis of the wavelet coefficient moduli.

Close modal

The statistical representation of the fluctuating properties calculated from the continuous wavelet analysis is presented in Fig. 15 in terms of frequency. The mean [μ(|Wx|)] and standard deviation [σ(|Wx|)] of the wavelet coefficient moduli for various remote sensors are presented in the first and second row of Fig. 15, respectively. At first glance, the remote sensor M3 shows a substantial increase for μ(|Wx|) and σ(|Wx|) compared to the other remote sensor locations. For the baseline case, increased values of μ(|Wx|) at M3 are observed with a peak at St2. Although all the remote sensors record this primary peak, M3 possesses the highest fluctuation compared to the other locations. In addition to this, the broadband hump at Sts = 0.5–1 also shows increased μ(|Wx|) levels at M3 compared to the readings at other locations. The results for the H-SCF configuration [Fig. 15(b)] shows reduced fluctuations compared to the baseline, whereas the SCF configuration shows further reductions compared to the H-SCF configuration. The σ(|Wx|) for the baseline case displays two peaks at St1 and St2 for the M2 and M3 remote sensor locations. Interestingly, this double peak [Fig. 15(d)] is not observed at any other remote sensor location, indicating that the pressure fluctuations for the broadband hump (≈St1) are spread far from the mean distribution at M2 and M3 with relatively higher values at M3. The σ(|Wx|) for M3 at St1 shows marginally larger values than at St2. Ultimately, the σ(|Wx|) results show that the generation of the broadband hump is at the proximity of M3 adjacent to the slat gap. Conversely, these two peaks at St1 and St2 in σ(|Wx|) are not detected at M4. Contrary to the baseline configuration, the σ(|Wx|) for the H-SCF and SCF configurations do not show any characteristic behavior, instead follow the trends similar to that of the μ(|Wx|).

FIG. 15.

Stochastic analysis for the various frequencies of interest at various remote sensor locations: (a) μ(|Wx|), baseline; (b) μ(|Wx|), H-SCF; (c) μ(|Wx|), SCF; (d) σ(|Wx|), baseline; (e) σ(|Wx|), H-SCF; (f) σ(|Wx|), SCF; (g) CV(|Wx|), baseline; (h) CV(|Wx|), H-SCF; and (i) CV(|Wx|), SCF.

FIG. 15.

Stochastic analysis for the various frequencies of interest at various remote sensor locations: (a) μ(|Wx|), baseline; (b) μ(|Wx|), H-SCF; (c) μ(|Wx|), SCF; (d) σ(|Wx|), baseline; (e) σ(|Wx|), H-SCF; (f) σ(|Wx|), SCF; (g) CV(|Wx|), baseline; (h) CV(|Wx|), H-SCF; and (i) CV(|Wx|), SCF.

Close modal

The coefficient of variation CV(|Wx|) is the ratio of the standard deviation to the mean, which essentially shows the fluctuation intensity of the pressure signals at a given frequency in terms of percentage.17,58 The higher the CV(|Wx|), the greater the level of dispersion around the mean. The CV(|Wx|) is presented to compare the pressure fluctuations at various pressure transducers on the same relative scale (ratio). The CV(|Wx|) in Fig. 15 is presented in terms of percentage and shows the relative dispersion of the pressure fluctuations. The frequencies with smaller CV(|Wx|) are less dispersed compared to the frequencies with a larger CV(|Wx|). For the baseline, it is evident from the CV(|Wx|) that St1 with larger values (up to 65%) is of a different nature compared to the St2 with lower values (up to 20%) for all the locations on the slat and main-element. Interestingly, at M4, the CV(|Wx|) is low at St1 compared to all the sensors. In the case of the H-SCF configuration, the changes in CV(|Wx|) remain low for all the frequencies and remote sensor locations except for S1 at frequencies Sts > 5. Finally, for the SCF configuration, the changes in CV(|Wx|) remain relatively low for the entire bandwidth at all the locations compared to the baseline and H-SCF configurations. These observations reaffirm that the application of the slat cove filler changes the flow structure within the slat region entirely, and more importantly, the St1 and St2 are of a different nature.

The phase-space analysis (see Fig. 12) and stochastic analysis (see Fig. 15) of the wavelet coefficient moduli evidently show that the St1 and St2 are of a different nature. Furthermore, from σ(|Wx|), it is clear that the broadband hump (Sts = 0.5–1) is generated at the vicinity of the slat gap [see Figs. 4 and 15(d)], and it should also be noted that M3 showed the highest energy in Sec. IV B 1 (see Fig. 6) and also in the continuous wavelet transform scalogram (see Fig. 11).

3. Mode Dynamics

Although the underlying physics of the slat tones and the broadband hump generation observed for the baseline configuration appear to be complex, it can be better identified by using lower-order models governed by a few modes to capture the dominant behavior of the system. These modes are commonly referred to as coherent spatiotemporal structures.59 In order to extract the most dominant characteristics of the system, to find the underlying patterns, and to predict the system dynamics, the data-driven technique Dynamic Mode Decomposition (DMD) was employed. It should be emphasized that in the present study, the DMD technique is not used in the conventional way commonly used in fluid mechanics, but it is used as a data-driven analysis tool to capture the time dependence of a point in geometrical space, as demonstrated by Kutz et al.50,51 The DMD technique is applied on large time evolution datasets of the wavelet coefficient moduli (|Wx|) calculated from different remote sensors within the slat cove region where the slat Strouhal number (Sts) and sensor locations are treated as spatial measurements. The DMD analysis aids us to better interpret the Sts evolution in time as the derived modes of the Sts would be associated with their damped (or driven) sinusoidal behavior in time and geometrical space. It is well known that the standard DMD algorithm is sensitive to its eigenvalue spectrum to noisy data.50,51 Therefore, the method of averaging the forward and backward DMD introduced by Dawson et al.60 to remove the systematic bias in the DMD eigenvalues caused by the measurement noise was also considered. However, the characteristics of the modes of interest showed a negligible difference compared to the standard DMD procedure as a result of the intensity of the dominant characteristics being higher than the signal noise levels. Ultimately, the standard DMD algorithm was applied to 8.19 × 105 snapshots of the wavelet coefficient moduli from various normalized remote sensor signals of the baseline configuration. For the sake of brevity, the results are presented only for the baseline configuration since the slat cove filler configurations lacked dominant dynamic behavior that the DMD procedure could capture.

The magnitude of the Ritz value (λr and λi) for the first 30 modes relative to the unit circle is presented in Fig. 16. The discrete-time eigenvalues typically lie on the unit circle, indicating the neutral stability of the corresponding DMD spatial mode Φ, as seen in the continuous-time plot. The color of the markers indicates the amplitude of the eigenvalue in the sequential data where darker colors indicate higher amplitude. Furthermore, the oscillatory frequency of the modes could be found using the phase of the λi. The eigenvalues λ are mapped logarithmically and normalized by Δt to obtain ω = log(λ)/(2πΔt), resulting with the imaginary component of ωi as the frequency of oscillation in units of cycles per second. Although the dynamics of transients are captured quite well, the eigenvalues with small real parts decay quickly, which is in line with previous studies.61 

FIG. 16.

DMD eigenvalues for the first 30 modes of the Sts evolution shown in Fig. 17. The discrete-time eigenvalues λ relative to the unit circle and the eigenvalues presented in continuous time as ω = log(λ)/(2πΔt) colored by their amplitude: (a) Ritz value and (b) DMD spectrum.

FIG. 16.

DMD eigenvalues for the first 30 modes of the Sts evolution shown in Fig. 17. The discrete-time eigenvalues λ relative to the unit circle and the eigenvalues presented in continuous time as ω = log(λ)/(2πΔt) colored by their amplitude: (a) Ritz value and (b) DMD spectrum.

Close modal

The result of the first six DMD modes of the Sts evolution over time at various remote sensor locations for the baseline configuration is presented in Fig. 17. Since the Sts evolution over time has complex translation dynamics, it would require several modes to get the precise dynamics with all the embedded low-rank modes. Nevertheless, only the first few modes are presented here to represent the most important dynamic characteristics of the Sts. The results of mode 1 show the sinusoidal behavior of St2 = 1.6 at location M2 oscillating between Sts ≈ 1.1 and 2.1, which is due to the vortex shedding phenomenon. At mode 2, phase oscillations corresponding to the vortex shedding were observed at location M1. As expected, none of the vortex shedding associated phase oscillation could be observed at locations M3 and M4 for any of the presented modes as the shear layer encompassing the vortex shedding impinges on the slat trailing edge and does not pass through the slat gap. The results of the remaining modes show that St1 (Sts = 0.5–1) moves periodically with a fixed phase relation between M2 and M3. This yet again supports the hypothesis that the St1 is different to St2 and that the broadband hump (St1) translates in geometrical space through the slat gap past location M3, whereas the dominance of St2 is not seen past M2 in the first few modes.

FIG. 17.

The contours of the first six DMD modes of the Sts evolution in time at various remote sensor locations for the baseline configuration: (a) mode 1, (b) mode 2, (c) mode 3, (d) mode 4, (e) mode 5, and (f) mode 6.

FIG. 17.

The contours of the first six DMD modes of the Sts evolution in time at various remote sensor locations for the baseline configuration: (a) mode 1, (b) mode 2, (c) mode 3, (d) mode 4, (e) mode 5, and (f) mode 6.

Close modal

It is well established that the standard DMD method is incapable of capturing transient phenomena such as the on–off nature of the modes,50,51 as seen from the DMD results presented above. The multi-resolution DMD (mrDMD)50 has been shown to effectively decompose the dynamics into different timescales, isolating the transient and intermittent patterns. Additionally, it has also been proven to be very successful in separating slow and fast modes even with the presence of background noise. Therefore, the mrDMD, capable of separating the multi-scale spatiotemporal features to uncover multi-scale structures, was applied to the judiciously truncated dataset consisting of the wavelet coefficient moduli of the various remote sensors. The mrDMD was first applied to the dataset truncated between Sts = 0.5 and 1 (window 1), which encompasses the first slat mode of interest St1 = 0.857. In a similar fashion, the mrDMD was then applied to the data truncated between Sts = 1 and 2 (window 2), which encompasses the slat second mode St2 = 1.6 for a time period of 3 s. Finally, the mrDMD results of both the judiciously truncated windows 1 and 2 for the baseline configuration at remote sensor location S1 are presented in Fig. 18 for a model number and decomposition level of 10 and 8, respectively [ϕ10(8,j)]. For window 1, the mrDMD results show that the St1 has a strong pulsation with a frequency of 10 Hz–21 Hz at two instances over a time period of 3 s. The St1 also portrays a slightly faster pulsation with a frequency of 21 Hz–41 Hz but with reduced modal intensity occurring in pairs at random instances over the evaluated time period. Subsequently, the mrDMD results for window 2 (St2) show high modal energy at the first level. This was expected as the tonal characteristics were present throughout the time signal. In some instances, a low-intensity pulsation of the St2 occurs with a frequency of 42 Hz–85 Hz. The mrDMD results have shown that there exists a strong pulsation of the broadband hump. Overall, the slat modes (St1 and St2) possess distinct intermittent characteristics, which were identified by the mrDMD method.

FIG. 18.

The multi-resolution view of power of the recurring events at (a) window 1 (Sts = 0.5–1) and (b) window 2 (Sts = 1–2) as it varies in time and in frequency for the baseline configuration at remote sensor location on the slat S1.

FIG. 18.

The multi-resolution view of power of the recurring events at (a) window 1 (Sts = 0.5–1) and (b) window 2 (Sts = 1–2) as it varies in time and in frequency for the baseline configuration at remote sensor location on the slat S1.

Close modal

An experimental study with a 30P30N three-element high-lift airfoil fitted with two different types of slat cove fillers was carried out in the Aeroacoustic Facility at the University of Bristol. Unsteady surface pressure measurements within the slat cove were carried out using several remote sensors. The results are presented for all the tested configurations at the angle of attack α = 18° for a free-stream velocity of U = 30 m/s, corresponding to a chord-based Reynolds number of Rec = 7 × 105. As expected, the slat cove filler configurations show reduced noise levels in both the near- and far-field measurements at low frequencies and elimination of the characteristic tonal noise generated by the slat. The near-field spectra confirmed that the tones were indeed a result of cavity resonance and Rossiter modes. Intermittent events induced by the resonant mechanism between the slat and its trailing edge demonstrated a substantial difference for all the configurations with notable behavior close to the slat gap. The auto-correlation of the near-field surface pressure fluctuations for both the slat cove filler configurations indicated suppression of the strong vortex shedding. Furthermore, in the slat cove region, the streamwise correlation between the remote sensors pointed out possible interference of the slat cove fillers with the turbulent structures, making the cross-correlation shape larger probably due to the break-up of coherent structures. Interestingly, the streamwise coherence indicated that the broadband hump is different from that of the high-order Rossiter modes. The near- to far-field pressure coherence results exhibited high coherence for the narrowband peaks for the baseline configuration, whereas for the slat cove filler configurations, an overall low coherence was observed. To identify the source of the broadband hump, a wavelet analysis using continuous wavelet transform was carried out. The results from the wavelet analysis showed that high levels of wavelet coefficient moduli for the frequencies of interest (Sts = 0.5–1) were observed at the remote sensor M3 located at the slat gap. The distinctive characteristics of the broadband hump at St1 and the slat tone at St2, as well as the different characteristics between the three configurations, were demonstrated by the phase-space analysis of the wavelet coefficients. The power spectral density of the wavelet coefficients at the frequencies of interest pointed out the modulation frequency to be the vortex shedding frequency (St2). Additionally, the statistical analysis of the wavelet coefficients indicated that the St1 was of different nature and the source of the broadband hump was located at M3 close to the slat gap. The results from the wavelet analysis showed the intermittent nature of the slat tones hidden behind the seemingly deterministic resonances. The wavelet coefficients from various remote sensors were subject to the DMD analysis, which highlighted the difference between the nature of St1 and St2. Furthermore, the slat based Strouhal numbers were segregated into two windows and were subject to the mrDMD analysis, which revealed the modulation frequency of St1 and St2. The application of these techniques provided the characterization of the dynamics of the slat tones. Overall, this study demonstrates that the St1 and St2 tones are relatively different in nature and that St1 is not solely generated due to Rossiter cavity modes, concurring with the previous hypothesis. The source of St1 was identified to be at the slat gap at location M3 possibly due to the high-velocity flow passing through it. The identified slat noise nature could lead to an improved noise reduction mechanism, and the new comprehensive database within the slat cove region provides a reference for future experimental and numerical studies.

The high-lift device used in this study was manufactured as part of a completed project sponsored by Embraer.

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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