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Hydrodynamic Instability in the Generation of Slat Noise
Available online at www.sciencedirect.com
ScienceDirect Procedia IUTAM 14 (2015) 344 – 353
IUTAM ABCM Symposium on Laminar Turbulent Transition
Hydrodynamic instability in the generation of slat noise Daniel Souzaa,∗, Carlos Pagania , Daniel Rodr´ıguez, Marcello Medeirosa a University
of S˜ao Paulo, Rua Jo˜ao Dagnone, S˜ao Carlos 13563-120, Brazil
Abstract Slat noise constitutes today a major contribution to the total noise emitted by regional jets, and the mechanisms responsible for their generation are not completely understood. In this paper, we are concerned with the low-frequency peaks that are commonly found in the far- and near-field spectra. Lattice-Boltzmann simulations of a standard three-element high-lift wing configuration are carried out, and attention is focused on the turbulent flow in the slat region. Near- and far-field data from the simulations are compared with wind-tunnel experiments and results in the literature, showing good agreement. Proper Orthogonal Decomposition is used to identify large-scale structures related to the two dominant frequency peaks in the noise spectrum. Different correlations are used in the POD: the usual one based on kinetic energy, and others based on the pressure fluctuations, inside and outside the slat region. Considering the latter increased the relative importance of the leading POD mode, reaching almost 100% of the correlation for the given frequency, and its physical structure resembles the fluctuations typically associated with mixing-layer instability. Additionally, the analyses indicate that the structures with highest turbulent kinetic energy are not necessarily the ones most correlated with the noise radiated to the far field. c 2015 2014The TheAuthors. Authors. Published by Elsevier © Published by Elsevier B.V.B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Selection and peer-review under responsibility of ABCM (Brazilian Society of Mechanical Sciences and Engineering). Selection and peer-review under responsibility of ABCM (Brazilian Society of Mechanical Sciences and Engineering)
Keywords: Aeroacoustics; High-Lift; Beamforming; Coherent Structures; Proper Orthogonal Decomposition
1. Introduction Owing to the reduction in engine noise attained in the last decades, in combination with the low-thrust conditions, airframe noise has become one of the dominant noise sources during the landing phase 1 . The most significant airframe noise sources in commercial jets are the landing-gear, the flap-side edge and the slat. In intercontinental airplanes, the landing-gear dominate the airframe noise generation and levels with the engine noise at landing phase. However, for regional, short range jets the slat generates a similar level of noise 2 . Experiments addressing the slat noise identified three different components in the radiated sound spectrum: a highfrequency tone, a broadband noise at low and medium frequencies and a series of tonal peaks above the low-medium frequency broadband noise 3 . An acoustic feedback loop, responsible for the discrete tones observed in the spectrum of open cavity acoustic field, known as Rossiter modes, is suggested to be the mechanism behind the the generation of the low-frequency peaks 4,5,6 . However, the existence of the such low-frequency peaks is controversial 7,8 . ∗
Corresponding author. Tel.: +55-016-3373-8195 E-mail address: [email protected]
2210-9838 © 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Selection and peer-review under responsibility of ABCM (Brazilian Society of Mechanical Sciences and Engineering) doi:10.1016/j.piutam.2015.03.058
Daniel Souza et al. / Procedia IUTAM 14 (2015) 344 – 353
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Correlation techniques can be an useful approach to understand the physics involved in noise generation and propagation. Successful examples exist in the literature on the use of correlation techniques in unveiling the underlying physics; of especial interest here is Proper Orthogonal Decomposition (POD), the utility of which in the eduction of large-scale coherent structures in turbulent flows has been widely demonstrated 9 . In this paper we address the generation of discrete tonal peaks of low-medium frequencies of the spectrum of the noise radiated by the slat. Time-resolved Lattice-Boltzmann simulations were carried out to provide detailed data of the flow around the slat. Near-field quantities and far-field acoustic emissions computed by the simulations were compared to our own wind-tunnel experiments and also with results available in the literature, in order to validate the numerical model applied. Then, the time-resolved flow data in the region of the slat were processed using the POD technique to identify the large-scale turbulent structures associated with the noise generation at the two leading tonal peaks. 2. Experimental procedures 2.1. Physical Setup Aeroacoustic measurements were performed with a high-lift airfoil in closed-section wind-tunnel of the S˜ao Carlos Engineering School (EESC). The closed-circuit wind-tunnel has working section measuring 1.3m high, 1.7m wide and 3.0m long in the stream-wise direction. Its contraction ratio is 1/8 and the air flow is driven by an eight blade axial fan, which safely provides flow speed up to 34 m/s. The test model was the three-element MD30P30N airfoil. The model stowed chord was 0.5m, with flap and slat extents being 30% and 15% of that chord, respectively. The airfoil, manufactured with aluminum alloy, was vertically mounted without sweep spanning the working section to simulate a 2D model. Figure 1(a) outlines the wind-tunnel experimental setup used for aeroacoustic measurements. Acoustic pressure data from the test model were measured with a 62-element wall-mounted array of microphones. The array geometry stems from optimizing the Archimedean Equation. The array panel was drilled with a precision machine to reduce microphone position uncertainties and prevent signal phase mismatching for acoustic beam-forming, and was mounted to smoothly adjust to the tunnel vertical wall. Microphones were flush-mounted to the array panel and the front grids used to protect their membranes were removed for measurements. The array position relative to the model in the tunnel exposes microphones directly to the acoustic pressure originated from the wing pressure side. This experimental setup optimally places the array to capture noise from potential aerodynamic sources acting into the slat cove and in the inner part of the slat upper trailing-edge. G lass
ࢁ
Ar r ay
Mo del
(a) Wind-tunnel setup
(b) POD regions
Fig. 1. (a) MD30P30N high-lift model vertically installed at 4◦ incidence angle for acoustic measurements in wind-tunnel. High-lift chord, array diameter and typical distances are depicted. (b) And scheme of the regions considered in the three different inner products defined for the POD analysis.
Suction was applied at the main-element lee-side and at the slat leading edge in both airfoil ends to reduce the turbulent boundary layer effects on the uniform flow over the airfoil surface. The airfoil was instrumented to hold 110 chordwise static pressure tappings at the mid-span and two lines comprising 54 spanwise static pressure tappings in
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the main-element suction side. One line with 27 pressure tappings is laid near the main-element trailing edge and the other near the leading edge. The spanwise pressure tappings were used to monitor the effect of the applied suction over the time-averaged static pressure. 2.2. Phased-array Techniques Formulation Phased-array techniques were applied focusing on identifying and quantifying slat noise sources. Integrated slat noise spectra were obtained by using DAMAS (Deconvolution Approach for the Mapping of Acoustic Sources). DAMAS is a deconvolution technique used in the post-processing of conventional beam-forming results to reduce the array geometry influence on a map of sources, which leads to a substantial improvement in sources resolution and side-lobes suppression 10 . For an array composed of M microphones, the frequency-domain beam-forming formulation adopted in this work rests on the following equation M M ∗ | n=1,nm m=1,mn gm [Cm,n ]gn | b r s,o , ωl = M , (1) M n=1,nm m=1,mn | gm || gn | in which b r s,o , ωl is a squared pressure estimate of a source at focal point s, whose position relative to a reference point o is given by r s,o , Cm,n denotes the cross-spectrum component, at frequency ωl , for microphones m and n, gm and gn are transfer functions modeling the sound propagation from a monopole source at focal point s to microphones m and n, respectively. According to Eq. 1, the beam-former output stems from weighting each array microphone cross-spectrum component by a model-based transfer function at the corresponding frequency, adding all weighted terms and scaling the result. Each transfer function contains the essential phase-delay information associated with the path a sound wave follows from the assumed source position to a microphone position. The cross-spectrum Cm,n has information from all sources measured by the array. However, the signal phase arising of weighting Cm,n by gm and gn , according to the framework of Eq. 1, adds constructively for all pair of microphones when the focal point position matches a physical source one. This is the beam-forming strategy to associate each scanning point with a pressure estimate, which is high whether a source exists. In this work it is used a convective free-field green function for a monopole source as a model for the transfer function. Due to the applied normalization, the output in Eq. 1 nominally scales with a signal auto-power of a reference microphone as measured on the array plane. Entries for Eq. 1 are the array microphone cross-spectral matrix and a vector of transfer functions modeling sound propagation from a target point toward all array microphones. A beam-forming map stems from successively applying Eq. 1 throughout a grid of points featuring a discretized spatial domain. DAMAS approaches each conventional beam-forming output as the spatial convolution between the measured pressure field and the array point spread function in the frequency domain, so that a beam-forming map representing a distribution of K sources on a discretized spatial domain is modeled by the equations b ro,s , ωl = A s,1 X1 + A s,2 X2 + . . . + A s,k Xk + . . . + A s,K XK ,
(2)
with s=1,K, being Xk the source squared pressure at grid point of index k, as it would be measured without the array geometry bias and A s,k is the amplitude, at point k, of the array point spread function of a unitary point source at grid point s. DAMAS formulation assumes the measured pressure field comes from a distribution of incoherent point sources. The system of equations is solved iteratively by a Gauss-Seidel algorithm. At iteration (i), an approximation for Xk(i) holds as Xk(i)
⎤ ⎡ k−1 K ⎥⎥⎥ ⎢⎢⎢ (i) (i−1) = bk − ⎢⎢⎣⎢ Akk Xk + Akk Xk ⎥⎥⎥⎦ , Xk(i) ≥ 0, k =1
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k =k+1
A positivity constrain, Xk(i) ≥ 0, ensures only positive pressure values are taking into account and contribute to the convergence of the iterative procedure. After convergence, DAMAS yields a map of sources with improved resolution
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and side lobes suppression, from which a region of interest can be selected to integrate discrete squared pressure values associated with a spatially distributed source. This was done to integrate the slat sources along its span. 3. Lattice-Boltzmann model Numerical simulations were computed using the commercial code PowerFLOW 5.0a, based on the Lattice-Boltzmann method. This method solves the discrete Boltzmann equation with the collision term approximated by the BhatnagarGross-Krook model 11 , which is valid for low Mach numbers 12 . The velocity space is discretized in cubic lattices aligned with a Cartesian coordinate system, and a Chapman-Enskog expansion of the Lattice-Boltzmann equation leads to the Navier-Stokes equations with the perfect-gas relation, p = ρRT . Because of the restriction of the Cartesian discretization, the law-of-wall is used to construct boundary conditions at the volumes closest to no-slip walls. The effect of the smallest turbulent scales are accounted for by a modified version of the k- turbulence model in the Renormalization Group (RNG) form 12 . Two flow configurations were considered in the simulations, all based on the same MD30P30N airfoil geometry at 4◦ angle of attack. In the first one the configurations were chosen to match the wind-tunnel experiments described in section 2.1. Therefore, the stowed chord of the airfoil was 0.5m and the vertical dimension of the simulation domain was 1.7m, with free-slip wall condition representing the tunnel walls. At the inflow, uniform velocity equal to 34m/s was imposed. At the same boundary, the turbulence intensity was set to 0.0009 and turbulence length scale, to 1mm, whereas at the outflow the static pressure was prescribed as 1atm. This simulation was carried out for a Mach number of 0.1 and Reynolds number of one million, based on the free stream flow speed and the airfoil stowed chord, which are representative of the experimental conditions. The second simulation was set to match wind-tunnel experiments by Jenkins et al. 13 . In this case the stowed chord of the airfoil model was 0.457m and the vertical dimension of the domain was 0.711m. The inflow velocity was 56m/s, resulting in a Reynolds number of 1.7 million, based on the stowed chord, and Mach number of 0.17. This conditions represents closer the landing conditions. Moreover, besides the experiments by Jenkins et al. 13 , several Unsteady Reynolds Averaged Navier-Stokes (URANS) simulations were carried out accordingly and comparison with these results provides reliability to the model employed here. For the same configuration of this simulation, Sim˜oes et al. 14 showed that the slat boundary layer outside the cove has no significant influence on the slat noise. Therefore, here, free-slip wall boundary condition was considered in the slat surface outside the cove. Periodicity in the spanwise direction was imposed in the simulations, which, for a wavelength long enough, should represent an infinite wing. Statistical analyses by Choudhari and Khorrami 15 showed that a domain spanwise dimension of ∼40% of the slat chord is enough to capture the decorrelation of the vortical structures and pressure fluctuations in the slat cove. Previous tests with a number of spanwise lengths also showed that the spanwise extent used here, approximately 70% of the slat chord, was long enough for the results to be independent on this parameter. To reduce the initial transient of the simulations, the outer part of the domain was modeled as a high-viscosity fluid. 4. Proper Orthogonal Decomposition The Proper Orthogonal Decomposition (POD) is a correlation technique that delivers a basis of orthogonal functions {φk }, referred to as POD functions or modes, that optimally decompose a set of initial data {qk } based on a given correlation or metric. The maximization leads to the eigenvalue problem Rφ = λφ.
(4)
R is a correlation matrix with elements Ri j = q j , qi , where the brackets define a correlation or inner product. The POD functions are ranked according to the POD eigenvalues λk , that serve as a measure of the relative importance of the POD mode k in the flow. Here we perform POD analysis on sets of data extracted from the Lattice-Boltzmann simulations described in section 3. As discussed, this simulations considered homogeneity of the solution in the spanwise direction allowing a Fourier decomposition in this direction. We also assume that the non-steady turbulent flow that generates the slat noise is ergodic, and the time-series were divided in blocks and each block was Fourier-decomposed. Consequently,
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the realizations {qk } employed in the POD analysis corresponded to the spanwise Fourier modes and the temporal Fourier modes for each block k of data. In the present work we perform POD analyses based on two different inner products or correlations. The first one considers a correlation of velocity components, leading to the usual turbulent Kinetic Energy (TKE) norm: q j , qi tke = (ui )∗ · u j dxdy, (5) x
y
where the superscript ∗ denotes complex conjugate and u = [u v w]T is the velocity vector. For this inner product, the integration was carried out over the rectangular region of figure 1(b) containing the slat and leading-edge of the main element. The TKE norm is adequate to educe structures which are dominant in the flow dynamics, but is not necessarily the best choice in order to study the structures more relevant to noise generation. We thus considered two other inner products. One is based on the fluctuations of static pressure inside the same rectangular area as the TKE inner product and is defined as q j , qi p = (pi )∗ · p j dxdy, (6) x
y
and the other one considers the pressure correlation integrated along a line located outside the slat cove (horizontal line in figure 1(b)), where the dominant fluctuations are expected to be related to the acoustic waves propagated towards the ground. This is formally represented by q j , qi p f f = (pi )∗ A(x, y)p j dxdy, (7) x
y
where A(x, y) is zero everywhere except in the line below the airfoil. 5. Results 5.1. Simulation with Reynolds number 1 million To assess the sensitiveness of the numerical solution to the spatial resolution, two simulations were carried out with different grid resolutions for the conditions of the EESC wind-tunnel (Reynolds number 1 million). In the coarse mesh, the smallest cubic volumes had edges of δxmin =0.20mm, while in the fine mesh these edges had δxmin =0.14mm. The other regions of the domain were refined proportionally. Figure 2 shows the time-averaged pressure coefficient, C p , on the airfoil surface and the Power Spectral Density (PSD) of the far-field pressure fluctuations as function of the Strouhal number based on the free-stream flow velocity and the slat chord. The experimental PSD of the far-field fluctuations were computed as the integration of the deconvoluted beam-forming maps over a region of interest centered at the slat with 0.8m at the spanwise direction and 0.18m in the streamwise direction. This spectrum represents the signal of an ideal microphone at the center of the array. The numerical data for the far-field was calculated at this same location using the Ffowcs Williams-Hawkings analogy 16 The static pressure distribution in figure 2(a) shows that, with the grid refinement, the separation of the boundary layer on the flap suction side moves downstream and the aerodynamic load on this element increases. The predicted load on the other two elements also increases slightly. Good agreement in the C p is found between the numerical and wind-tunnel measurements. The far-field spectra in figure 2(b) show the multiple tonal peaks of low-frequency typical of the slat noise. The sensitiveness of the solution to spatial resolution is restricted to the two tonal peaks of lowest frequencies and the comparison with wind-tunnel measurements is good, building confidence in the consistence of the numerical solution. 5.2. Simulation with Reynolds number 1.7 million The pressure coefficient C p distributions calculated by our LBM simulation and measured by Jenkins et al. 13 are compared in figure 3. The data from the numerical results were taken from the central plane of the spanwise computational domain, and similarly, the experimental data were measured in the center of the airfoil model. Overall good
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Fig. 2. Comparison between numerical and experimental results for the flow conditions of the EESC wind-tunnel, Reynolds number 1 million and Mach number 0.1. (a) Chordwise distribution of pressure coefficient and (b) Power Spectral Density (PSD) of the acoustic fluctuation.
agreement is observed. The greatest difference is seen in the suction side of the flap: since the simulation captures a premature separation of the boundary-layer close to the flap trailing-edge in comparison with the experiment, it predicts smaller load for this element. Similarly, the circulation of the main element calculated by the simulation is lower than the observed in the wind-tunnel measurements. This discrepancy is probably related to a poor representation of boundary-layer by the code PowerFLOW. However, the mean static pressure distribution on the slat and the suction peak of the main element are very well captured by the simulation if one takes the experiment as reference. -6
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-5 -4 Cp
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Fig. 3. Comparison of chordwise distribution from our LBM simulation with wind-tunnel measurements by Jenkins et al. 13 .
The pressure-fluctuations spectra at two locations on the slat surface (Fig. 4(a)) was calculated based on our LBM simulation and compared with results of the URANS computations by Lockard and Choudhari 17 . As seen in figure 4, the agreement is good for the dominant frequency range of the fluctuations. The two simulations recover very similar results for the frequency peaks and their amplitudes, especially for the probe p2. Since the probe p2 is outside the cove region, the fluctuations there are expected to be dominated by acoustic waves emanating from the cove through the gap between the slat and the main element. In fact, the spectrum at this point is dominated by low-frequency tonal peaks similar to those seen in figure 2(b). Far-field pressure spectra was calculated from the LBM simulation using the Ffowcs Williams-Hawkings (FW-H) analogy Figure 5 shows the PSD at a point below the airfoil (in the direction of noise emited to the comunities in airport region) at a distance from the cove equal to ten times the airfoil stowed chord. The spectrum is similar to the one shown in figure 2(b). 5.3. POD analysis The POD methodology described in section 4 allows us to study the flow structures related to fluctuations at a specific frequency. We thus analyse in this section fluctuations at the frequencies of two of the peaks seen in figure
Daniel Souza et al. / Procedia IUTAM 14 (2015) 344 – 353
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LBM simulation Lockard & Choudhari (2010) 10 St
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5, one at St∼1.75 and one at St∼2.42 using the data of the simulation with Mach number 0.17. The data used in the POD analyses were comprised of 0.112s with a time resolution of 4.361×10−5 s. The time-averaged was subtracted from the data, and the Fourier modes in spanwise direction were calculated based on 33 points equaly spaced in this direction. The data time-series were then divided in 19 blocks with 50% overlap and the time-Fourier modes of each block for the two frequencies of interest were calculated, with a frequency resolution of 89.6Hz. Figure 6 shows the spectra of POD eigenvalues for St∼1.75, for the three inner-products described in section 4, as function of the spanwise wavenumber βz . The eigenvalues are normalized by the sum of all eigenvalues for the entire range of βz considered ( n βz λnβz ). The spectrum of the TKE inner product, q j , qi tke , is dominated by the first POD mode for βz =123.7rad/m. This indicates that the structures at St∼1.75 with highest turbulent kinetic energy inside the slat cove have a finite typical wavelength in the spanwise direction, equal to 74% of the slat chord. Since 123.7rad/m is the lowest non-zero wavenumber for our spanwise length, this typical wavelenght is unaccurate and data with better wavenumber resolution is needed to determine this typical wavelength precisely. Considering the correlations involving pressure fluctuations on both near- and far-field, we see that these fluctuations are typically homogeneous in the spanwise direction, since the spectrum for inner products q j , qi p and q j , qi p f f are dominated by the first POD mode for βz =0. For the near-field pressure inner product, the dominant POD mode corresponds to almost 60% of the correlation at St∼1.75, while the dominant POD mode for q j , qi p f f amounts to almost 100% of the correlation at this frequency. The streamwise velocity and pressure components of the leading POD modes at St∼1.75 for the three inner products are shown in figure 7. The mode representing the most energetic structures (Fig. 7a) corresponds to fluctuations inside the slat cove with a characteristic length of the order of the cove dimension. On the other hand, the dominant modes related to correlations of pressure fluctuations are mainly represented by spanwise vortices typical of mixinglayer instability. We note that the shape of the dominant modes for q j , qi p and q j , qi p f f are the same, while the relative POD eigenvalues were very different. This indicates that, although a small number of modes have significant coherence with pressure fluctuations inside the slat cove, only the first one is significantly correlated to the pressure fluctuations in the far-field. Moreover, this mode is not directly related to the structures with higher turbulent kinetic energy.
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The spectra of POD modes for fluctuations at St∼2.42 are shown in figure 8, showing the dominance of the spanwise-homogeneous structures. The leading POD mode for the q j , qi tke inner product captures ≈ 12% of the fluctuating kinetic energy at this frequency; the one corresponding to the near-field pressure inner-product amounts to 55% of the correlation, and the leading POD mode for the far-field pressure correlation corresponds to more than 70%. Figures 9 shows the streamwise velocity component and pressure leading POD modes for the three inner products, which are shown to be dominated by the mixing-layer vortices. However, the vortices with St∼2.42 have smaller wavelength compared to the structures corresponding to St∼1.75 (figures 7b and 7c). In the POD spectrum for farfield pressure inner product, we observe that the relative value of the second POD mode with βz =0 is greater than 0.2. The shape of the real part of this POD mode is ploted in figure 9c and suggests that this mode also corresponds to mixing-layer vortices. Together, the two leading POD modes represent almost 100% of the correlation of the far-field pressure fluctuations.
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−0.008
0.4
0.2 0 −0.2 −0.4 −0.6 −0.8
(a) βz =0 rad/m, q j , qi tke
(b) βz =0 rad/m, q j , qi p f f
(c) βz =0 rad/m, q j , qi p f f
Fig. 9. Real part of streamwise velocity component (upper row) and pressure (lower row) of the leading POD modes obtained for St∼2.42. (a) first POD mode for q j , qi tke ; (b) first POD mode for q j , qi p f f ; (c) Second POD mode for q j , qi p f f .
6. Conclusions Lattice-Boltzmann simulations and wind-tunnel experiments were carried out to study the aeroacoustic noise generated by the slat of MD30P30N high-lift geometry at 4◦ . Good comparison between the numerical results and the tunnel measurements were achieved both for the averaged static pressure distribution on the airfoil surface and for the far-field acoustic spectra in the frequency range of maximum amplitude. Particularly, the match of the frequencies and the levels of the peaks indicates that the simulations captures adequately the physics responsible for their generation and are therefore appropriate for our goal of analyzing the flow structures related to theses peaks. To further evaluate the quality of the numerical results, another simulation was carried out at a flow condition extensively studied by other research groups, with Reynolds number of 1.7 million and Mach number of 0.17, finding a good agreement with experimental (C p distribution) and numerical results (spectrum of pressure fluctuations on the slat surface) available on the literature. Based on the results of the simulation with Reynolds number equal to 1.7 million, analyses with the Proper Orthogonal Decomposition were made, focused on the turbulent structures inside the slat cove. The simulation data was decomposed into spanwise and temporal Fourier modes prior to their use in the POD. Attention was focused on two frequencies, corresponding to the dominant low-frequency peaks identified in the spectra: St∼1.75 and St∼2.42. Be-
Daniel Souza et al. / Procedia IUTAM 14 (2015) 344 – 353
sides the usual norm that identifies the structures according to the turbulent kinetic energy, norms based on the static pressure fluctuations in the near- and far-field were also considered. For the two frequencies considered, the use of the norm based on the far-field pressure increased remarkably the correlation associated to the dominant POD mode. For St∼1.75, the dominant eigenvalue, normalized by the sum of all eigenvalues at all spanwise wavenumbers considered, was almost 1 (100% of the correlation), while for the turbulent kinetic energy norm, the respective normalized eigenvalue was smaller than 0.13. For St∼2.42, the leading POD eigenvalue was greater than 0.7 for the far-field pressure norm and approximately 0.12 for the turbulent kinetic energy norm. This indicates that the turbulent kinetic energy at these frequencies is distributed over a number of large-scale structures with different shapes, but structures of one specific shape are far more correlated to the pressure dynamics in the far-field than all other coherent structures. For the two frequencies analyzed, these structures were similar to the spanwise vortices generated by the mixing-layer instability. This agrees with the hypothesis that the peaks are generated by the excitation of the mixing-layer in the context of an acoustic feedback loop, similar to the Rossiter modes in open cavities. Acknowledgements C. C. P. received support from CNPq/Brasil - Conselho Nacional de Desenvolvimento Cient´ıfico e Tecnol´ogico. D. S. S. received funding from CAPES - Coordenac¸a˜ o de Aperfeic¸oamento de Pessoal de N´ıvel Superior. M. A. F. M. received support from CNPq/Brasil - Conselho Nacional de Desenvolvimento Cient´ıfico e Tecnol´ogico. The authors acknowledge EMBRAER for providing licence of the commercial code PowerFLOW. The authors are grateful to Dr. Meelan Choudhari for allowing the use of data published by his group to compare with our results. References 1. Crighton, D.. Airframe noise. In: Aeroacoustics of flight vehicles:theory and practice; vol. 1. Nasa Publication; 1991, p. 391–447. 2. Dobrzynski, W.. Almost 40 years of airframe noise research: What did we achieved? Journal of Aircraft 2010;47(2):353–367. 3. Imamura, T., Ura, H., Yokokawa, Y., Yamamoto, K.. A far-field noise and near-field unsteadiness of a simplified high-lift-configuration model (slat). In: Proceedings of the 47th AIAA Aerospace Sciences Meeting. Orlando, USA; 2009, . 4. Roger, M., P´erenn`es, S.. Low-frequency noise sources in two-dimensional high-lift devices. In: Proceedings of the 6th AIAA/CEAS Aeroacoustics Conference. Lahaina, USA; 2000, . 5. Kolb, A., Faulhaber, P., Drobietz, R., Grunewald, M.. Aeroacoustic wind tunnel measurements on a 2d high-lift configuration. In: Proceedings of the 13th AIAA/CEAS Aeroacoustics Conference. Rome, Italy; 2007, . 6. Terracol, M., Manoha, E.., Lemoine, B.. Investigation of the unsteady flow and noise sources generation in a slat cove: hybrid zonal rans/les simulation and dedicated experiment. In: Proceedings of the 20th AIAA Computational Fluid Dynamics Conference. Honolulu, Havaii; 2011, . 7. Dobrzynski, W., Pott-Pollenske, M.. Slat noise source studies for farfield noise prediction. In: Proceedings of the 7th AIAA/CEAS Aeroacoustics Conference. Maastricht, The Netherlands; 2001, . 8. Pott-Pollenske, M., Delfs, J., Reichenberger, J.. A testbed for large scale and high reynolds number airframe noise research. In: Proceedings of the 19th AIAA/CEAS Aeroacoustics Conference. Berlin, Germany; 2013, . 9. Holmes, P., Lumley, J., Berkooz, G.. Turbulence, Coherent Structures, Dynamical Systems and Symmetry. Cambridge University Press; 1996. 10. Dougherty, R.. Extensions of DAMAS and benefits and limitation of deconvolution in beamforming. In: Proceedings of the 11th AIAA/CEAS Aeroacoustics Conference. Monterey, USA; 2005, . 11. Bhatnagar, P., Gross, E., Krook, M.. A model for collision processes in gases. i. small amplitude processes in charged and neutral one-component systems. Physical Review 1954;94(3):511–525. 12. Fares, E.. Unsteady flow simulation of the Ahmed reference body using a lattice Boltzmann approach. Computers and Fluids 2006; 35(8-9):940–950. 13. Jenkins, L., Khorrami, M., Choudhari, M.. Characterization of unsteady flow structures near leading-edge slat: Part 1. piv measurements. In: Proceedings of the 10th AIAA/CEAS Aeroacoustics Conference. Manchester, United Kingdom; 2004, . 14. Sim˜oes, L., Souza, D., Medeiros, M.. On the small effect of boundary layer thicknesses on slat noise. In: Proceedings of the 17th AIAA/CEAS Aeroacoustics Conference. Portland, USA; 2011, . 15. Choudhari, M., Khorrami, M.. Effect of three-dimensional shear-layer on slat cove unsteadiness. AIAA Journal 2007;45(9):2174–2186. 16. Br`es, G., P´erot, F., Freed, D.. A Ffowcs Williams-Hawkings solver for Lattice-Boltzmann based computational aeroacoustics. In: Proceedings of the 16th AIAA/CEAS Aeroacoustics Conference. Stockholm,Sweden; 2010, . 17. Lockard, D., Choudhari, M.. The effect of cross flow on slat noise. In: Proceedings of the 16th AIAA/CEAS Aeroacoustics Conference. Stockholm,Sweden; 2010, .
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ScienceDirect Procedia IUTAM 14 (2015) 344 – 353
IUTAM ABCM Symposium on Laminar Turbulent Transition
Hydrodynamic instability in the generation of slat noise Daniel Souzaa,∗, Carlos Pagania , Daniel Rodr´ıguez, Marcello Medeirosa a University
of S˜ao Paulo, Rua Jo˜ao Dagnone, S˜ao Carlos 13563-120, Brazil
Abstract Slat noise constitutes today a major contribution to the total noise emitted by regional jets, and the mechanisms responsible for their generation are not completely understood. In this paper, we are concerned with the low-frequency peaks that are commonly found in the far- and near-field spectra. Lattice-Boltzmann simulations of a standard three-element high-lift wing configuration are carried out, and attention is focused on the turbulent flow in the slat region. Near- and far-field data from the simulations are compared with wind-tunnel experiments and results in the literature, showing good agreement. Proper Orthogonal Decomposition is used to identify large-scale structures related to the two dominant frequency peaks in the noise spectrum. Different correlations are used in the POD: the usual one based on kinetic energy, and others based on the pressure fluctuations, inside and outside the slat region. Considering the latter increased the relative importance of the leading POD mode, reaching almost 100% of the correlation for the given frequency, and its physical structure resembles the fluctuations typically associated with mixing-layer instability. Additionally, the analyses indicate that the structures with highest turbulent kinetic energy are not necessarily the ones most correlated with the noise radiated to the far field. c 2015 2014The TheAuthors. Authors. Published by Elsevier © Published by Elsevier B.V.B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Selection and peer-review under responsibility of ABCM (Brazilian Society of Mechanical Sciences and Engineering). Selection and peer-review under responsibility of ABCM (Brazilian Society of Mechanical Sciences and Engineering)
Keywords: Aeroacoustics; High-Lift; Beamforming; Coherent Structures; Proper Orthogonal Decomposition
1. Introduction Owing to the reduction in engine noise attained in the last decades, in combination with the low-thrust conditions, airframe noise has become one of the dominant noise sources during the landing phase 1 . The most significant airframe noise sources in commercial jets are the landing-gear, the flap-side edge and the slat. In intercontinental airplanes, the landing-gear dominate the airframe noise generation and levels with the engine noise at landing phase. However, for regional, short range jets the slat generates a similar level of noise 2 . Experiments addressing the slat noise identified three different components in the radiated sound spectrum: a highfrequency tone, a broadband noise at low and medium frequencies and a series of tonal peaks above the low-medium frequency broadband noise 3 . An acoustic feedback loop, responsible for the discrete tones observed in the spectrum of open cavity acoustic field, known as Rossiter modes, is suggested to be the mechanism behind the the generation of the low-frequency peaks 4,5,6 . However, the existence of the such low-frequency peaks is controversial 7,8 . ∗
Corresponding author. Tel.: +55-016-3373-8195 E-mail address: [email protected]
2210-9838 © 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Selection and peer-review under responsibility of ABCM (Brazilian Society of Mechanical Sciences and Engineering) doi:10.1016/j.piutam.2015.03.058
Daniel Souza et al. / Procedia IUTAM 14 (2015) 344 – 353
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Correlation techniques can be an useful approach to understand the physics involved in noise generation and propagation. Successful examples exist in the literature on the use of correlation techniques in unveiling the underlying physics; of especial interest here is Proper Orthogonal Decomposition (POD), the utility of which in the eduction of large-scale coherent structures in turbulent flows has been widely demonstrated 9 . In this paper we address the generation of discrete tonal peaks of low-medium frequencies of the spectrum of the noise radiated by the slat. Time-resolved Lattice-Boltzmann simulations were carried out to provide detailed data of the flow around the slat. Near-field quantities and far-field acoustic emissions computed by the simulations were compared to our own wind-tunnel experiments and also with results available in the literature, in order to validate the numerical model applied. Then, the time-resolved flow data in the region of the slat were processed using the POD technique to identify the large-scale turbulent structures associated with the noise generation at the two leading tonal peaks. 2. Experimental procedures 2.1. Physical Setup Aeroacoustic measurements were performed with a high-lift airfoil in closed-section wind-tunnel of the S˜ao Carlos Engineering School (EESC). The closed-circuit wind-tunnel has working section measuring 1.3m high, 1.7m wide and 3.0m long in the stream-wise direction. Its contraction ratio is 1/8 and the air flow is driven by an eight blade axial fan, which safely provides flow speed up to 34 m/s. The test model was the three-element MD30P30N airfoil. The model stowed chord was 0.5m, with flap and slat extents being 30% and 15% of that chord, respectively. The airfoil, manufactured with aluminum alloy, was vertically mounted without sweep spanning the working section to simulate a 2D model. Figure 1(a) outlines the wind-tunnel experimental setup used for aeroacoustic measurements. Acoustic pressure data from the test model were measured with a 62-element wall-mounted array of microphones. The array geometry stems from optimizing the Archimedean Equation. The array panel was drilled with a precision machine to reduce microphone position uncertainties and prevent signal phase mismatching for acoustic beam-forming, and was mounted to smoothly adjust to the tunnel vertical wall. Microphones were flush-mounted to the array panel and the front grids used to protect their membranes were removed for measurements. The array position relative to the model in the tunnel exposes microphones directly to the acoustic pressure originated from the wing pressure side. This experimental setup optimally places the array to capture noise from potential aerodynamic sources acting into the slat cove and in the inner part of the slat upper trailing-edge. G lass
ࢁ
Ar r ay
Mo del
(a) Wind-tunnel setup
(b) POD regions
Fig. 1. (a) MD30P30N high-lift model vertically installed at 4◦ incidence angle for acoustic measurements in wind-tunnel. High-lift chord, array diameter and typical distances are depicted. (b) And scheme of the regions considered in the three different inner products defined for the POD analysis.
Suction was applied at the main-element lee-side and at the slat leading edge in both airfoil ends to reduce the turbulent boundary layer effects on the uniform flow over the airfoil surface. The airfoil was instrumented to hold 110 chordwise static pressure tappings at the mid-span and two lines comprising 54 spanwise static pressure tappings in
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the main-element suction side. One line with 27 pressure tappings is laid near the main-element trailing edge and the other near the leading edge. The spanwise pressure tappings were used to monitor the effect of the applied suction over the time-averaged static pressure. 2.2. Phased-array Techniques Formulation Phased-array techniques were applied focusing on identifying and quantifying slat noise sources. Integrated slat noise spectra were obtained by using DAMAS (Deconvolution Approach for the Mapping of Acoustic Sources). DAMAS is a deconvolution technique used in the post-processing of conventional beam-forming results to reduce the array geometry influence on a map of sources, which leads to a substantial improvement in sources resolution and side-lobes suppression 10 . For an array composed of M microphones, the frequency-domain beam-forming formulation adopted in this work rests on the following equation M M ∗ | n=1,nm m=1,mn gm [Cm,n ]gn | b r s,o , ωl = M , (1) M n=1,nm m=1,mn | gm || gn | in which b r s,o , ωl is a squared pressure estimate of a source at focal point s, whose position relative to a reference point o is given by r s,o , Cm,n denotes the cross-spectrum component, at frequency ωl , for microphones m and n, gm and gn are transfer functions modeling the sound propagation from a monopole source at focal point s to microphones m and n, respectively. According to Eq. 1, the beam-former output stems from weighting each array microphone cross-spectrum component by a model-based transfer function at the corresponding frequency, adding all weighted terms and scaling the result. Each transfer function contains the essential phase-delay information associated with the path a sound wave follows from the assumed source position to a microphone position. The cross-spectrum Cm,n has information from all sources measured by the array. However, the signal phase arising of weighting Cm,n by gm and gn , according to the framework of Eq. 1, adds constructively for all pair of microphones when the focal point position matches a physical source one. This is the beam-forming strategy to associate each scanning point with a pressure estimate, which is high whether a source exists. In this work it is used a convective free-field green function for a monopole source as a model for the transfer function. Due to the applied normalization, the output in Eq. 1 nominally scales with a signal auto-power of a reference microphone as measured on the array plane. Entries for Eq. 1 are the array microphone cross-spectral matrix and a vector of transfer functions modeling sound propagation from a target point toward all array microphones. A beam-forming map stems from successively applying Eq. 1 throughout a grid of points featuring a discretized spatial domain. DAMAS approaches each conventional beam-forming output as the spatial convolution between the measured pressure field and the array point spread function in the frequency domain, so that a beam-forming map representing a distribution of K sources on a discretized spatial domain is modeled by the equations b ro,s , ωl = A s,1 X1 + A s,2 X2 + . . . + A s,k Xk + . . . + A s,K XK ,
(2)
with s=1,K, being Xk the source squared pressure at grid point of index k, as it would be measured without the array geometry bias and A s,k is the amplitude, at point k, of the array point spread function of a unitary point source at grid point s. DAMAS formulation assumes the measured pressure field comes from a distribution of incoherent point sources. The system of equations is solved iteratively by a Gauss-Seidel algorithm. At iteration (i), an approximation for Xk(i) holds as Xk(i)
⎤ ⎡ k−1 K ⎥⎥⎥ ⎢⎢⎢ (i) (i−1) = bk − ⎢⎢⎣⎢ Akk Xk + Akk Xk ⎥⎥⎥⎦ , Xk(i) ≥ 0, k =1
(3)
k =k+1
A positivity constrain, Xk(i) ≥ 0, ensures only positive pressure values are taking into account and contribute to the convergence of the iterative procedure. After convergence, DAMAS yields a map of sources with improved resolution
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and side lobes suppression, from which a region of interest can be selected to integrate discrete squared pressure values associated with a spatially distributed source. This was done to integrate the slat sources along its span. 3. Lattice-Boltzmann model Numerical simulations were computed using the commercial code PowerFLOW 5.0a, based on the Lattice-Boltzmann method. This method solves the discrete Boltzmann equation with the collision term approximated by the BhatnagarGross-Krook model 11 , which is valid for low Mach numbers 12 . The velocity space is discretized in cubic lattices aligned with a Cartesian coordinate system, and a Chapman-Enskog expansion of the Lattice-Boltzmann equation leads to the Navier-Stokes equations with the perfect-gas relation, p = ρRT . Because of the restriction of the Cartesian discretization, the law-of-wall is used to construct boundary conditions at the volumes closest to no-slip walls. The effect of the smallest turbulent scales are accounted for by a modified version of the k- turbulence model in the Renormalization Group (RNG) form 12 . Two flow configurations were considered in the simulations, all based on the same MD30P30N airfoil geometry at 4◦ angle of attack. In the first one the configurations were chosen to match the wind-tunnel experiments described in section 2.1. Therefore, the stowed chord of the airfoil was 0.5m and the vertical dimension of the simulation domain was 1.7m, with free-slip wall condition representing the tunnel walls. At the inflow, uniform velocity equal to 34m/s was imposed. At the same boundary, the turbulence intensity was set to 0.0009 and turbulence length scale, to 1mm, whereas at the outflow the static pressure was prescribed as 1atm. This simulation was carried out for a Mach number of 0.1 and Reynolds number of one million, based on the free stream flow speed and the airfoil stowed chord, which are representative of the experimental conditions. The second simulation was set to match wind-tunnel experiments by Jenkins et al. 13 . In this case the stowed chord of the airfoil model was 0.457m and the vertical dimension of the domain was 0.711m. The inflow velocity was 56m/s, resulting in a Reynolds number of 1.7 million, based on the stowed chord, and Mach number of 0.17. This conditions represents closer the landing conditions. Moreover, besides the experiments by Jenkins et al. 13 , several Unsteady Reynolds Averaged Navier-Stokes (URANS) simulations were carried out accordingly and comparison with these results provides reliability to the model employed here. For the same configuration of this simulation, Sim˜oes et al. 14 showed that the slat boundary layer outside the cove has no significant influence on the slat noise. Therefore, here, free-slip wall boundary condition was considered in the slat surface outside the cove. Periodicity in the spanwise direction was imposed in the simulations, which, for a wavelength long enough, should represent an infinite wing. Statistical analyses by Choudhari and Khorrami 15 showed that a domain spanwise dimension of ∼40% of the slat chord is enough to capture the decorrelation of the vortical structures and pressure fluctuations in the slat cove. Previous tests with a number of spanwise lengths also showed that the spanwise extent used here, approximately 70% of the slat chord, was long enough for the results to be independent on this parameter. To reduce the initial transient of the simulations, the outer part of the domain was modeled as a high-viscosity fluid. 4. Proper Orthogonal Decomposition The Proper Orthogonal Decomposition (POD) is a correlation technique that delivers a basis of orthogonal functions {φk }, referred to as POD functions or modes, that optimally decompose a set of initial data {qk } based on a given correlation or metric. The maximization leads to the eigenvalue problem Rφ = λφ.
(4)
R is a correlation matrix with elements Ri j = q j , qi , where the brackets define a correlation or inner product. The POD functions are ranked according to the POD eigenvalues λk , that serve as a measure of the relative importance of the POD mode k in the flow. Here we perform POD analysis on sets of data extracted from the Lattice-Boltzmann simulations described in section 3. As discussed, this simulations considered homogeneity of the solution in the spanwise direction allowing a Fourier decomposition in this direction. We also assume that the non-steady turbulent flow that generates the slat noise is ergodic, and the time-series were divided in blocks and each block was Fourier-decomposed. Consequently,
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the realizations {qk } employed in the POD analysis corresponded to the spanwise Fourier modes and the temporal Fourier modes for each block k of data. In the present work we perform POD analyses based on two different inner products or correlations. The first one considers a correlation of velocity components, leading to the usual turbulent Kinetic Energy (TKE) norm: q j , qi tke = (ui )∗ · u j dxdy, (5) x
y
where the superscript ∗ denotes complex conjugate and u = [u v w]T is the velocity vector. For this inner product, the integration was carried out over the rectangular region of figure 1(b) containing the slat and leading-edge of the main element. The TKE norm is adequate to educe structures which are dominant in the flow dynamics, but is not necessarily the best choice in order to study the structures more relevant to noise generation. We thus considered two other inner products. One is based on the fluctuations of static pressure inside the same rectangular area as the TKE inner product and is defined as q j , qi p = (pi )∗ · p j dxdy, (6) x
y
and the other one considers the pressure correlation integrated along a line located outside the slat cove (horizontal line in figure 1(b)), where the dominant fluctuations are expected to be related to the acoustic waves propagated towards the ground. This is formally represented by q j , qi p f f = (pi )∗ A(x, y)p j dxdy, (7) x
y
where A(x, y) is zero everywhere except in the line below the airfoil. 5. Results 5.1. Simulation with Reynolds number 1 million To assess the sensitiveness of the numerical solution to the spatial resolution, two simulations were carried out with different grid resolutions for the conditions of the EESC wind-tunnel (Reynolds number 1 million). In the coarse mesh, the smallest cubic volumes had edges of δxmin =0.20mm, while in the fine mesh these edges had δxmin =0.14mm. The other regions of the domain were refined proportionally. Figure 2 shows the time-averaged pressure coefficient, C p , on the airfoil surface and the Power Spectral Density (PSD) of the far-field pressure fluctuations as function of the Strouhal number based on the free-stream flow velocity and the slat chord. The experimental PSD of the far-field fluctuations were computed as the integration of the deconvoluted beam-forming maps over a region of interest centered at the slat with 0.8m at the spanwise direction and 0.18m in the streamwise direction. This spectrum represents the signal of an ideal microphone at the center of the array. The numerical data for the far-field was calculated at this same location using the Ffowcs Williams-Hawkings analogy 16 The static pressure distribution in figure 2(a) shows that, with the grid refinement, the separation of the boundary layer on the flap suction side moves downstream and the aerodynamic load on this element increases. The predicted load on the other two elements also increases slightly. Good agreement in the C p is found between the numerical and wind-tunnel measurements. The far-field spectra in figure 2(b) show the multiple tonal peaks of low-frequency typical of the slat noise. The sensitiveness of the solution to spatial resolution is restricted to the two tonal peaks of lowest frequencies and the comparison with wind-tunnel measurements is good, building confidence in the consistence of the numerical solution. 5.2. Simulation with Reynolds number 1.7 million The pressure coefficient C p distributions calculated by our LBM simulation and measured by Jenkins et al. 13 are compared in figure 3. The data from the numerical results were taken from the central plane of the spanwise computational domain, and similarly, the experimental data were measured in the center of the airfoil model. Overall good
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Fig. 2. Comparison between numerical and experimental results for the flow conditions of the EESC wind-tunnel, Reynolds number 1 million and Mach number 0.1. (a) Chordwise distribution of pressure coefficient and (b) Power Spectral Density (PSD) of the acoustic fluctuation.
agreement is observed. The greatest difference is seen in the suction side of the flap: since the simulation captures a premature separation of the boundary-layer close to the flap trailing-edge in comparison with the experiment, it predicts smaller load for this element. Similarly, the circulation of the main element calculated by the simulation is lower than the observed in the wind-tunnel measurements. This discrepancy is probably related to a poor representation of boundary-layer by the code PowerFLOW. However, the mean static pressure distribution on the slat and the suction peak of the main element are very well captured by the simulation if one takes the experiment as reference. -6
LBM simulation Jenkins et al.
-5 -4 Cp
-3 -2 -1 0 1 -0.2
0
0.2
0.4
0.6
0.8
1
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Fig. 3. Comparison of chordwise distribution from our LBM simulation with wind-tunnel measurements by Jenkins et al. 13 .
The pressure-fluctuations spectra at two locations on the slat surface (Fig. 4(a)) was calculated based on our LBM simulation and compared with results of the URANS computations by Lockard and Choudhari 17 . As seen in figure 4, the agreement is good for the dominant frequency range of the fluctuations. The two simulations recover very similar results for the frequency peaks and their amplitudes, especially for the probe p2. Since the probe p2 is outside the cove region, the fluctuations there are expected to be dominated by acoustic waves emanating from the cove through the gap between the slat and the main element. In fact, the spectrum at this point is dominated by low-frequency tonal peaks similar to those seen in figure 2(b). Far-field pressure spectra was calculated from the LBM simulation using the Ffowcs Williams-Hawkings (FW-H) analogy Figure 5 shows the PSD at a point below the airfoil (in the direction of noise emited to the comunities in airport region) at a distance from the cove equal to ten times the airfoil stowed chord. The spectrum is similar to the one shown in figure 2(b). 5.3. POD analysis The POD methodology described in section 4 allows us to study the flow structures related to fluctuations at a specific frequency. We thus analyse in this section fluctuations at the frequencies of two of the peaks seen in figure
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LBM simulation Lockard & Choudhari (2010) 10 St
(b) Slat trailing-edge (p1)
(c) Slat suction side (p2)
Fig. 4. PSD of pressure fluctuations against Strouhal number for two locations (a) on the slat surface, at the (b) trailing-edge and at the (c) suction side.
40
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10 St
Fig. 5. PSD of acoustic fluctuations for a point below the slat cove at a distance of ten times the airfoil stowed chord.
5, one at St∼1.75 and one at St∼2.42 using the data of the simulation with Mach number 0.17. The data used in the POD analyses were comprised of 0.112s with a time resolution of 4.361×10−5 s. The time-averaged was subtracted from the data, and the Fourier modes in spanwise direction were calculated based on 33 points equaly spaced in this direction. The data time-series were then divided in 19 blocks with 50% overlap and the time-Fourier modes of each block for the two frequencies of interest were calculated, with a frequency resolution of 89.6Hz. Figure 6 shows the spectra of POD eigenvalues for St∼1.75, for the three inner-products described in section 4, as function of the spanwise wavenumber βz . The eigenvalues are normalized by the sum of all eigenvalues for the entire range of βz considered ( n βz λnβz ). The spectrum of the TKE inner product, q j , qi tke , is dominated by the first POD mode for βz =123.7rad/m. This indicates that the structures at St∼1.75 with highest turbulent kinetic energy inside the slat cove have a finite typical wavelength in the spanwise direction, equal to 74% of the slat chord. Since 123.7rad/m is the lowest non-zero wavenumber for our spanwise length, this typical wavelenght is unaccurate and data with better wavenumber resolution is needed to determine this typical wavelength precisely. Considering the correlations involving pressure fluctuations on both near- and far-field, we see that these fluctuations are typically homogeneous in the spanwise direction, since the spectrum for inner products q j , qi p and q j , qi p f f are dominated by the first POD mode for βz =0. For the near-field pressure inner product, the dominant POD mode corresponds to almost 60% of the correlation at St∼1.75, while the dominant POD mode for q j , qi p f f amounts to almost 100% of the correlation at this frequency. The streamwise velocity and pressure components of the leading POD modes at St∼1.75 for the three inner products are shown in figure 7. The mode representing the most energetic structures (Fig. 7a) corresponds to fluctuations inside the slat cove with a characteristic length of the order of the cove dimension. On the other hand, the dominant modes related to correlations of pressure fluctuations are mainly represented by spanwise vortices typical of mixinglayer instability. We note that the shape of the dominant modes for q j , qi p and q j , qi p f f are the same, while the relative POD eigenvalues were very different. This indicates that, although a small number of modes have significant coherence with pressure fluctuations inside the slat cove, only the first one is significantly correlated to the pressure fluctuations in the far-field. Moreover, this mode is not directly related to the structures with higher turbulent kinetic energy.
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Fig. 7. Real part of streamwise velocity component (upper row) and pressure (lower row) of the first POD modes obtained for St∼1.75 corresponding to the three different inner products.
The spectra of POD modes for fluctuations at St∼2.42 are shown in figure 8, showing the dominance of the spanwise-homogeneous structures. The leading POD mode for the q j , qi tke inner product captures ≈ 12% of the fluctuating kinetic energy at this frequency; the one corresponding to the near-field pressure inner-product amounts to 55% of the correlation, and the leading POD mode for the far-field pressure correlation corresponds to more than 70%. Figures 9 shows the streamwise velocity component and pressure leading POD modes for the three inner products, which are shown to be dominated by the mixing-layer vortices. However, the vortices with St∼2.42 have smaller wavelength compared to the structures corresponding to St∼1.75 (figures 7b and 7c). In the POD spectrum for farfield pressure inner product, we observe that the relative value of the second POD mode with βz =0 is greater than 0.2. The shape of the real part of this POD mode is ploted in figure 9c and suggests that this mode also corresponds to mixing-layer vortices. Together, the two leading POD modes represent almost 100% of the correlation of the far-field pressure fluctuations.
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Fig. 9. Real part of streamwise velocity component (upper row) and pressure (lower row) of the leading POD modes obtained for St∼2.42. (a) first POD mode for q j , qi tke ; (b) first POD mode for q j , qi p f f ; (c) Second POD mode for q j , qi p f f .
6. Conclusions Lattice-Boltzmann simulations and wind-tunnel experiments were carried out to study the aeroacoustic noise generated by the slat of MD30P30N high-lift geometry at 4◦ . Good comparison between the numerical results and the tunnel measurements were achieved both for the averaged static pressure distribution on the airfoil surface and for the far-field acoustic spectra in the frequency range of maximum amplitude. Particularly, the match of the frequencies and the levels of the peaks indicates that the simulations captures adequately the physics responsible for their generation and are therefore appropriate for our goal of analyzing the flow structures related to theses peaks. To further evaluate the quality of the numerical results, another simulation was carried out at a flow condition extensively studied by other research groups, with Reynolds number of 1.7 million and Mach number of 0.17, finding a good agreement with experimental (C p distribution) and numerical results (spectrum of pressure fluctuations on the slat surface) available on the literature. Based on the results of the simulation with Reynolds number equal to 1.7 million, analyses with the Proper Orthogonal Decomposition were made, focused on the turbulent structures inside the slat cove. The simulation data was decomposed into spanwise and temporal Fourier modes prior to their use in the POD. Attention was focused on two frequencies, corresponding to the dominant low-frequency peaks identified in the spectra: St∼1.75 and St∼2.42. Be-
Daniel Souza et al. / Procedia IUTAM 14 (2015) 344 – 353
sides the usual norm that identifies the structures according to the turbulent kinetic energy, norms based on the static pressure fluctuations in the near- and far-field were also considered. For the two frequencies considered, the use of the norm based on the far-field pressure increased remarkably the correlation associated to the dominant POD mode. For St∼1.75, the dominant eigenvalue, normalized by the sum of all eigenvalues at all spanwise wavenumbers considered, was almost 1 (100% of the correlation), while for the turbulent kinetic energy norm, the respective normalized eigenvalue was smaller than 0.13. For St∼2.42, the leading POD eigenvalue was greater than 0.7 for the far-field pressure norm and approximately 0.12 for the turbulent kinetic energy norm. This indicates that the turbulent kinetic energy at these frequencies is distributed over a number of large-scale structures with different shapes, but structures of one specific shape are far more correlated to the pressure dynamics in the far-field than all other coherent structures. For the two frequencies analyzed, these structures were similar to the spanwise vortices generated by the mixing-layer instability. This agrees with the hypothesis that the peaks are generated by the excitation of the mixing-layer in the context of an acoustic feedback loop, similar to the Rossiter modes in open cavities. Acknowledgements C. C. P. received support from CNPq/Brasil - Conselho Nacional de Desenvolvimento Cient´ıfico e Tecnol´ogico. D. S. S. received funding from CAPES - Coordenac¸a˜ o de Aperfeic¸oamento de Pessoal de N´ıvel Superior. M. A. F. M. received support from CNPq/Brasil - Conselho Nacional de Desenvolvimento Cient´ıfico e Tecnol´ogico. 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