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- 1. Trailing-edge scattering of spanwise-coherent structures Alex Sano∗ , Petrˆonio A. S. Nogueira† , Andr´e V. G. Cavalieri‡ Instituto Tecnol´ogico de Aeron´autica, S˜ao Jos´e dos Campos, SP, 12228-900, Brazil and William R. Wolf§ Universidade Estadual de Campinas, Campinas, SP, 13083-970, Brazil We study mechanisms of noise generation by an airfoil at zero angle of attack using ﬂow-acoustic correlations. We use a large-eddy simulation (LES) of the compressible ﬂow around a NACA0012 airfoil for M∞ = 0.115 and Rec = 408000. With this simulation we analyse ﬂow ﬂuctuations around the airfoil, such as pressure and velocity, and relate them to the far-ﬁeld sound using standard correlation techniques. Similar to the results of Cava- lieri et al.,3 who have obtained more signiﬁcant ﬂow-acoustic correlation for turbulent jets when the axisymmetric mode was isolated, we have noticed much higher correlations in the present problem when the two-dimensional mode, i.e. spanwise-averaged ﬂuctuations, was isolated from the turbulent ﬂow and subsequently correlated to the acoustic pressure. This result is justiﬁed theoretically by an analysis of the tailored Green’s function for a half plane, where we ﬁnd that a necessary condition for trailing edge scattering is |kz| < k, where kz is the spanwise wavenumber of turbulent disturbances and k is the acoustic wavenum- ber. Two-dimensional perturbations, associated to kz = 0, are always radiating. Isolation of spanwise-coherent disturbances is thus a means of ﬁltering non-radiating structures. Another feature that tends to increase correlation coeﬃcients is the use of the diﬀerence between ﬂuctuations in the upper and lower surfaces of the airfoil, which again is in line with theory: when disturbances are in phase opposition between the two sides of the airfoil acoustic scattering is maximal. Nomeclature δ Boundary-layer thickness xi Cartesian coordinate of observer yi Cartesian coordinate of source L Characteristic length c Chord gij Contravariant metric tensors ui Contravariant velocity components gij Covariant metric tensors ρ Density δ∗ Displacement-thickness R Distance between the source and observer R Distance between the source image and the observer ω Frequency G Green’s function qj Heat ﬂux ∗Msc. Student, Divis˜ao de Engenharia Aeron´autica, Instituto Tecnol´ogico de Aeron´autica. †MSc. Student, Divis˜ao de Engenharia Aeron´autica, Instituto Tecnol´ogico de Aeron´autica. ‡Assistant Professor, Divis˜ao de Engenharia Aeron´autica, Instituto Tecnol´ogico de Aeron´autica, AIAA Member §Assistant Professor, Faculdade de Engenharia Mecˆanica, Universidade Estadual de Campinas, Campinas, SP, Brazil 1 of 16 American Institute of Aeronautics and Astronautics DownloadedbyINSTTECDEAERONAUTICA(ITA)onMarch30,2016|http://arc.aiaa.org|DOI:10.2514/6.2015-2384 21st AIAA/CEAS Aeroacoustics Conference 22-26 June 2015, Dallas, TX AIAA 2015-2384 Copyright © 2015 by Alex Sano. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission. AIAA Aviation
- 2. He Helmholtz number Tij Lighthill tensor M∞ Mach number ∆˜x Non-dimensional distance n Surface normal vector xi Observer position vφ Phase velocity Pr Prandtl number p Pressure ˜p Fourier Transform of pressure r Radius γ Ratio of speciﬁc heats Re Reynolds number SG Lighthill’s tensor in the frequency domain ao Sound speed yi Source position Pi Dipole strength S Surface T Temperature E Total energy t Time ˜t Non-dimensional time u u-component of velocity in a Cartesian system v v-component of velocity in a Cartesian system w w-component of velocity in a Cartesian system τij Viscous stress tensor V Volume λ Wavelength k Wavenumber I. Introduction Noise emitted by airfoils and wings occurs due to the interaction of turbulence with the neighbouring solid surfaces. This kind of problem is fundamental for aeroacoustics, and there are several applications where this phenomenon is relevant, among which the design of low-noise aerodynamic proﬁles, rotor blades and fans. Brooks, Pope and Marcolini2 classify ﬁve diﬀerent mechanisms of sound generation by an airfoil, namely: turbulent boundary layer convected past the trailing edge noise; separation stall noise; laminar boundary layer leading to vortex shedding noise; tip vortex formation noise; and trailing edge bluntness vortex shedding noise. Most of the cases above correspond to the interaction between trailing edge and ﬂow, which is a result of the end of the proﬁle inducing an abrupt change of boundary condition for pressure and/or velocity. This was recognised early by Ffowcs Williams and Hall, who showed theoretically that a trailing edge scatters the neighbouring turbulent ﬂuctuations, leading to signiﬁcant sound radiation. This phenomenon has since been labelled trailing-edge noise. Unlike sound emission by compact surfaces, which tends to be dominated by lift ﬂuctuations as shown by Curle,6 trailing-edge noise approaches the non-compact limit, modelled by Ffowcs Williams and Hall as a half plane with eddies on the vicinity of its edge. Follow-up modelling work by Amiet1 considered that trailing-edge noise is due to induced dipoles in the vicinity of the edge, due to advected turbulence, considered in statistical terms to remain stationary as it is advected. The objective of the present work is to use ﬂow simulation data to study in more detail the relevant sound-production mechanisms in the ﬂow around an airfoil, in particular with a view to extracting the relevant turbulent scales to the trailing-edge noise problem. To do this we study the large-eddy simulation (LES) of Wolf, Azevedo and Lele,13 of a NACA0012 airfoil at zero angle of attack, Mach number M∞ = 0.115 and Reynolds number based on the airfoil chord equal to Rec = 408000. We use the full simulation data to calculate ﬂow-acoustic correlations in order to examine the relevant ﬂuid motions related to acoustic 2 of 16 American Institute of Aeronautics and Astronautics DownloadedbyINSTTECDEAERONAUTICA(ITA)onMarch30,2016|http://arc.aiaa.org|DOI:10.2514/6.2015-2384
- 3. radiation. Some previous work have also used numerical simulations to study trailing-edge noise. As shown by Oberai, Roknaldin and Rughes9 using an LES, the source of acoustic noise in major part is concentrated on trailing edge due the interaction of the turbulent ﬂow and the end of the proﬁle. Sandberg and Sadham10 have conducted a direct numerical simulation (DNS) to investigate the interaction of the turbulent ﬂow past a trailing edge and shown that two-dimensional theory for low frequencies predicts the far ﬁeld acoustic pressure due the signiﬁcant spanwise coherence of pressure disturbances, showing that in this case the noise generation is predominately two-dimensional. A theoretical analysis, using the tailored Green’s function for a trailing edge derived by Ffowcs Williams and Hall, guides our investigation. We show in section II that only spanwise-elongated structures radiate to the far-ﬁeld, and subsequently we examine in section III how two-dimensional motions around the airfoil are related to the far-ﬁeld sound. In addition, we study what the inﬂuence on ﬂow-acoustic correlations when the quantities are manipulated doing a delta between the upper and lower surface, in other words, the diﬀerence of the properties between upper and lower surface of the airfoil. II. Theory A. Basic equations We review here the fundamental theory of noise generation by an airfoil. Lighthill8 rearranged the equations of motion of a ﬂuid in a way to obtain an inhomogeneous wave equation, with a source term called Lighthill tensor, Tij: ∂2 ρ ∂t2 − a2 o 2 ρ = ∂2 Tij ∂xi∂xj (1) Tij = ρvivj + pij − a2 oρδij (2) The solution of equation 1 neglecting solid boundaries makes use of the free-ﬁeld Green’s function to the problem, deﬁned as ∂2 G ∂t2 − a2 o 2 G = δ(x − y)δ(t − τ) (3) where the Green’s function G should satisfy the Sommerfeld boundary condition and also a causality condi- tion. Manipulation of eqs. (1) and (3) leads to ρ − ρo = 1 4πa2 o ∂2 ∂xi∂xj V Tij(y, t − |x−y| ao ) |x − y| dV (4) The previous equation does not account for solid boundaries to the problem, which are relevant for trailing-edge noise. Extension to Lighthill’s analogy accounting for surface eﬀects are presented by Curle6 and Ffowcs Williams and Hall.12 Curle’s theory employs a more general resolution to equation 1, as follows ρ − ρo = 1 4πa2 o v ∂2 Tij ∂xi∂xj dt |x − y| + 1 4π S 1 r ∂ρ ∂n + 1 r2 r n ρ + 1 aor ∂r ∂n ∂ρ ∂t dS(y) (5) With further manipulation, Curle rewrote the solution as ρ − ρo = 1 4πa2 o ∂2 ∂xi∂xj V Tij(y, t − r ao ) r dy − 1 4πa2 o ∂ ∂xi S Pi(y, t − r ao ) r dS(y) (6) where Pi = −njpij (7) 3 of 16 American Institute of Aeronautics and Astronautics DownloadedbyINSTTECDEAERONAUTICA(ITA)onMarch30,2016|http://arc.aiaa.org|DOI:10.2514/6.2015-2384
- 4. The term Pi corresponds to a surface distribution of dipoles and nj is the direction outward over the airfoil surface. Without knowledge of the normal tension Pi on the surface (approximately equal to the pressure for high- Re ﬂows), eq. (6) is actually an integral equation, which should be solved numerically using a boundary element method. However, a simpliﬁcation is possible if instead of the free-ﬁeld Green’s function G in eq. (3) one manipulates the corresponding equations using a tailored Green’s function Gt that satisﬁes the same boundary conditions as the pressure. For the trailing-edge noise problem, the Green’s function in a domain bounded by a rigid, semi-inﬁnite ﬂat plate was derived by Ffowcs Williams and Hall12 in the frequency domain as Gt(x, y, ω) = e 1 4 iπ √ π e−ikR R uR −∞ e−iu2 du + e−ikR R uR −∞ e−iu2 du , (8) where uR and uR are given by uR = 2 krr0 D + R 1 2 cos θ − θ0 2 , (9) uR = 2 krr0 D + R 1 2 cos θ + θ0 2 , (10) R is the separation between source (r0, θ0, z0) and observer (r, θ, z) and R is the distance between the source’s image (r0, −θ0, z0) and the observer in cylindrical coordinates (z and z0 are equivalent to x3 and y3, respectively). The far-ﬁeld assumption for the observers was taken into account and the relationship between the two coordinate systems can be seen in Figure 1. Trailing Edge ∞ ∞ ∞ z0 θ0 r0 θ r Plate z x1 y1 x2 y2 x3 y3 Figure 1. Relationship between diﬀerent coordinate systems of the problem The distances R, R and D can be obtained in cylindrical coordinates as R = r2 + r2 0 − 2rr0 cos (θ − θ0) + (z − z0)2 1 2 , (11) R = r2 + r2 0 − 2rr0 cos (θ + θ0) + (z − z0)2 1 2 , (12) D = (r + r0)2 + (z − z0)2 1 2 . (13) 4 of 16 American Institute of Aeronautics and Astronautics DownloadedbyINSTTECDEAERONAUTICA(ITA)onMarch30,2016|http://arc.aiaa.org|DOI:10.2514/6.2015-2384
- 5. Using the hypothesis of source near the plate (kr 1, i.e. the distance between trailing edge and source is much smaller than the acoustic wavelength), the leading-order Taylor-series approximation of the Green’s function (labelled here as Gs) is shown by Ffowcs-Williams and Hall to be Gs(x, y, ω) = 2e 1 4 iπ √ π 2krr0 r2 + (z − z0)2 1 2 cos θ 2 cos θ0 2 e−ikR R (14) Using either form of the tailored Green’s function, Gt or Gs, the far-ﬁeld sound at frequency ω can be obtained as ˆp(x, ω) = 1 4π y1 y2 y3 SG(y, ω)G(x, y, ω)dy3dy2dy1 (15) with Lighthill’s tensor in the frequency domain, SG(y, ω) = ˆ∂2Tij ∂yi∂yj (y, ω). (16) The source can be deﬁned throughout the domain, being deﬁned as zero outside the turbulent region, so that the integration limits eq. (15) go from −∞ to ∞ B. Radiating and non-radiating spanwise wavenumbers in the trailing-edge noise problem Further insight on the dominant turbulent structures leading to trailing-edge noise can be obtained by evaluating the contributions of diﬀerent spanwise wavenumbers to the far-ﬁeld sound. The source term can be deﬁned as a function of its Fourier transform in the spatial spanwise direction y3: SG(y1, y2, y3, ω) = 1 2π ∞ kz=−∞ ˆSG(y1, y2, kz, ω)e−ikzy3 dkz (17) Replacing in equation (15) we obtain ˆp(x, ω) = 1 8π2 ∞ y1=−∞ ∞ y2=−∞ ∞ kz=−∞ ˆSG(y1, y2, kz, ω) ∞ y3=−∞ G(x, y, ω)e−ikzy3 dy3 dkzdy2dy1 (18) where the expression inside the brackets can be deﬁned as: ∞ y3=−∞ G(x, y, ω)e−ikzy3 dy3 = ˆG(x, y1, y2, −kz, ω) (19) Replacing this result in equation (18) we can obtain the ﬁnal expression for the pressure ﬁeld in terms of the spanwise Fourier transform of both Green’s function and source, as ˆp(x, ω) = 1 8π2 ∞ y1=−∞ ∞ y2=−∞ ∞ kz=−∞ ˆSG(y1, y2, kz, ω) ˆG(x, y1, y2, −kz, ω)dkzdy2dy1 (20) Equation (20) shows that each spanwise wavenumber kz in the source, ˆS(y1, y2, kz, ω), will have its contribution to the far-ﬁeld sound weighted by a corresponding wavenumber −kz in the Green’s function, ˆG(x, y1, y2, −kz, ω). We will here evaluate analytically the Fourier transform in eq. (19) so as to determine the said weighting factors, which will show us the dominant source wavenumbers for far-ﬁeld radiation. To apply the spatial Fourier transform along the z axis for the source, we need to evaluate the integral ˆGs(x, y1, y1, kz, ω) = ∞ −∞ Gs(z0)eikzz0 dz0 (21) considering that only z0 varies (the observers are ﬁxed, as well as the other two components of the source). Replacing equation 14 into 21, taking out of the integral the terms with no z0 dependency and reorganizing, we obtain: ˆGs(x, y1, y1, kz, ω) = 2e 1 4 iπ+ikzz √ π (2krr0) 1 2 cos θ 2 cos θ0 2 ∞ −∞ 1 R (r2 + (z − z0)2) 1 4 e−ikz(z−z0)−ikR dz0 (22) 5 of 16 American Institute of Aeronautics and Astronautics DownloadedbyINSTTECDEAERONAUTICA(ITA)onMarch30,2016|http://arc.aiaa.org|DOI:10.2514/6.2015-2384
- 6. With the integral in this form, we can apply the following transformation on z0: z − z0 = q tan α, (23) dz0 = − q cos2 α dα, (24) with q deﬁned as q = r2 + r2 0 − 2rr0 cos (θ − θ0) = (x1 − y1)2 + (x2 − y2)2. (25) Since −∞ < z0 < ∞, α is a real number between −π/2 and π/2. If we apply the transform 23 on equation 22 and make the necessary changes, we obtain: ˆGs(x, y1, y1, kz, ω) = C π/2 −π/2 F(α)eiqψ(α) dα (26) where C = 2e 1 4 iπ+ikzz √ π (2krr0) 1 2 cos θ 2 cos θ0 2 (27) F(α) = 1 r2 + q2 tan2 α 1 4 cos α (28) ψ(α) = − k − kz sin α cos α (29) It is easy to see that both F and ψ are continuous in the integration domain, as well as their ﬁrst and second derivatives. The far-ﬁeld hypothesis corresponds to q → ∞; this allows an asymptotic solution to the integral using the stationary phase method.5 The points αs leading to constant phase at eiqψ(α) , which have the leading-order contribution to the integral, must satisfy ψ (αs) = 0. The only root is given by αs = arcsin −kz k , (30) and since α is a real number satisfying −π/2 < α < π/2, in order that the stationary phase point lie in the integration path we must have |kz| < k (31) which means that the only wave numbers that radiate sound correspond to supersonic phase velocities along the z axis, i.e. in the spanwise direction. This result is well known for the free-ﬁeld Green’s function,4,11,7 and is here extended to the tailored Green’s function related to the edge scattering problem. Using the far-ﬁeld approximation (q → ∞) on integral 26, the stationary phase method gives an asymp- totic approximation: ˆGs(x, y1, y1, kz, ω) ≈ C 2π q|ψ (α0)| 1 2 F(α0)eiqψ(α0)− 1 4 iπ (32) where F(α0) = 1 r2 + q2 k2 z k2−k2 z 1 4 √ k2−k2 z k (33) ψ(α0) = − k2 − k2 z (34) ψ (α0) = − k2 k2 − k2 z (35) and C is deﬁned by equation 27. 6 of 16 American Institute of Aeronautics and Astronautics DownloadedbyINSTTECDEAERONAUTICA(ITA)onMarch30,2016|http://arc.aiaa.org|DOI:10.2514/6.2015-2384
- 7. From the results showed above we can deduce some features of the sound generation in this problem. Firstly, from equation 31 we ﬁnd that the only radiating turbulent disturbances are the ones with |kz| < k (or the perturbations with supersonic phase velocity, since we can deﬁne kz = ω vφ , where vφ is the phase velocity). In second place, the wavenumber kz for the source term is associated with the incidence angle of the perturbations on the trailing edge of the plate: for waves normal to the trailing edge we have kz = 0 (there is no component of the perturbation in z direction), with increasing wavenumber for increasing incidence angles. If this incidence angle is related to a wavenumber outside the supersonic region, only evanescent waves will be generated, with zero contribution to the far-ﬁeld sound. It is important to notice that two- dimensional disturbances, associated to kz = 0, are always radiating, whereas perturbations associated with other wavenumber may not generate sound at a given frequency. In the following section we will use this theoretical prediction to perform ﬂow-acoustic correlations in a diﬀerent manner. Besides taking the usual two-point correlation, we will isolate the two-dimensional component of the ﬂow ﬂuctuations, corresponding to kz = 0. This is always radiating, and by the evaluation of the correlation between kz = 0 components in the turbulence and in the acoustic ﬁeld we expect to highlight the dominant mechanism of trailing edge noise. III. Flow-acoustic correlations A. Database In this section we present the results of ﬂow-acoustic correlations in order to discern the regions in the ﬂow most relevant for sound radiation. The large-eddy simulation (LES) of Wolf et al. of a NACA0012 airfoil at zero angle of attack, M∞ = 0.115 and Rec = 4.08 · 105 was used for that matter. The simulation attempts to represent an inﬁnite-span airfoil by considering periodic boundary conditions at the spanwise extremities. This spanwise periodicity leads to discrete values of the spanwise wavenumber kz, given as kzc = 2πn Lz c, (36) where n is an integer number. The simulation domain has Lz = 0.1c, which is large compared to the boundary layer thickness, and was seen by Wolf et al. to be suﬃcient to obtain a decaying two-point coherence along the span. With this computational domain, besides the two-dimensional mode kz = 0 the ﬁrst non-zero wavenumber corresponds to kzc = 20π, which is large in terms of the airfoil chord, but small in terms of the boundary layer thickness. Table 1 shows the wavenumber in terms of boundary layer thickness. x/c 0.8540 0.8980 0.9170 0.9520 0.9700 0.9830 δ∗ 0.0039 0.0045 0.0048 0.0055 0.0059 0.0065 kzδ∗ 0.2471 0.2806 0.2995 0.3477 0.3728 0.4063 Table 1. Calculation of the wavenumber in terms of the boundary layer thickness. To determine ﬂow-acoustic correlations we have taken ﬂow ﬂuctuations throughout a body-ﬁtted O- grid block (Figure 11, referenced as grid 1). Fluctuations at all these positions were with a representative position in the acoustic ﬁeld, aligned with the trailing edge, at x/c = 1 and y/c = 2. Details on the numerical simulation can be found in the Appendix. The use of numerical data has the advantage of allowing the study of correlations using diﬀerent quantities, taken at various positions in the ﬂow. Moreover, it is possible to extract the expected dominant low spanwise wavenumbers satisfying kz < k, which would be an ambitious task to perform experimentally. However, the numerical simulation has a shorter duration (˜t = 15.98, ˜t = tao/L) compared to usual time series taken from experiments, which impact the calculation of correlations. The clearest eﬀect of this is an overall noise level in the correlations, whose order is 60%. We will nonetheless see that for some positions the correlation peaks greatly exceed the background noise, and are taken as signiﬁcant despite the relative short time series. B. Two-dimensionality of the acoustic ﬁeld A ﬁrst observation on the radiated sound can be made with regard to the power spectral density (PSD) at the point taken in the acoustic ﬁeld. The PSD was calculated at this position in two ways: the ﬁrst is the 7 of 16 American Institute of Aeronautics and Astronautics DownloadedbyINSTTECDEAERONAUTICA(ITA)onMarch30,2016|http://arc.aiaa.org|DOI:10.2514/6.2015-2384
- 8. usual single-point PSD, and the second accounted solely for the kz = 0 component of the acoustic ﬁeld; this last approach amounts to taking a spanwise average prior to computing PSD. The results are shown in ﬁgure 2. 10 0 10 1 10 2 10 3 10 −20 10 −19 10 −18 10 −17 10 −16 10 −15 10 −14 10 −13 10 −12 10 −11 He PSD[dB] With spanwise averaged Without spanwise averaged He n=0 He n=1 He n=2 He n=3 Figure 2. Power spectral densities of pressure at x/c = 1 and y/c = 2 To help the analysis, the vertical lines in ﬁgure 2 show cut-on values of He for the successive spanwise wavenumbers, according to the analysis of section II: Hen = knc = 2πn Lz c (37) Helmholtz numbers satisfying He > Hen (i.e. to the right of the corresponding vertical line) are such that the nth spanwise wavenumber in the source is propagative. On the other hand, if He < Hen (left of the corresponding line) the nth spanwise wavenumber leads solely to evanescent waves. The two-dimensional mode kz = 0, or n = 0, is propagative for all He. Note that there is virtually no diﬀerence between spectra with and without spanwise averaging up to He corresponding to the cut-on condition for n = 1, which is consistent with the theoretical analysis. Above that value some diﬀerences appear, but the energy-containing part of the acoustic spectrum is clearly two- dimensional. It is interesting to notice that above the cut-on He for n = 2 a new peak also emerges in the PSD. These results are in line with the theory presented in section II, and suggest that the predominantly two-dimensional acoustic ﬁeld is generated by corresponding two-dimensional ﬂow ﬂuctuations. In a similar manner to the approach of Cavalieri et al.,3 who have obtained more signiﬁcant ﬂow-acoustic correlation for jets when the axisymmetric mode was isolated, we compare in what follows the usual two-point statistics with correlations where the two-dimensional component was extracted by taking spanwise averages in the turbulent and acoustic ﬁelds. C. Correlations between pressure in the near and acoustic ﬁelds We start the evaluation of correlation coeﬃcients by looking solely at pressure ﬂuctuations, with correlations between near-ﬁeld and acoustic pressure. The peak correlation, with acoustic pressure taken in the reference point, is shown in Figure 3, where both quantities were spanwise averaged. To aid the analysis the boundary- layer thickness and displacement thickness were plotted in the trailing edge region. Determination of the boundary layer thickness close to the trailing edge becomes diﬃcult due to the signiﬁcant gradients of the 8 of 16 American Institute of Aeronautics and Astronautics DownloadedbyINSTTECDEAERONAUTICA(ITA)onMarch30,2016|http://arc.aiaa.org|DOI:10.2514/6.2015-2384
- 9. outer velocity, and hence we show only values of δ up to x/c ≈ 0.94. There is a signiﬁcant correlation peak in outer portions of the ﬁgure, but this occurs outside the boundary layer, and is hence attributed to correlations between acoustic waves. Figure 3. Maximum correlation between near-ﬁeld and acoustic pressure. Both quantities are spanwise averaged. A zoom of ﬁgure 3 in the trailing edge region is shown in ﬁgure 4 (a). We observe high correlation peaks inside the boundary layer, extending all the way from the wall to the boundary layer thickness δ. The correlation coeﬃcients are roughly 40%. On the other hand, correlations without taking the spanwise average of the pressure, shown by ﬁgure (b), do not display visible peaks above the noise level. This result agrees with the theoretical considerations of section II, showing that higher correlations are obtained when kz = 0 is isolated in the turbulence ﬁeld. For the energy-containing part of the acoustic spectrum only the two-dimensional mode is radiating. Furthermore we observe that the source of sound is concentrated in the neighbourhood of the trailing edge (centred around x/c = 0.96), which is also predicted by the theory12 . Correlation coeﬃcients were seen to be oscillatory, and besides presenting a positive peak they also present negative peaks. In order to quantify these negative peaks, Figure 5 shows the minimum of ﬂow-acoustic correlation, again with a prior spanwise average of ﬂuctuations. To facilitate the comparison with ﬁgures of maximal correlation, the minimal were plotted using a reversed color scale. The results in Figure 5 also show signiﬁcant peaks (this time negative) of the correlation inside the boundary layer, in the vicinity of the trailing edge. This negative peak occurs around x/c = 0.93, which is slightly upstream of the region of positive peak. Furthermore, we also observe a region of signiﬁcant correlation further downstream, in the near-ﬁeld of the wake. We have seen that taking spanwise averages of ﬂow ﬂuctuations greatly enhances the correlation coeﬃ- cients. If we now consider that trailing edge scattering occurs due to advected disturbances in both upper and lower airfoil surfaces, it makes sense to extract the diﬀerence between upper- and lower-surface ﬂuctu- ations prior to the calculation of correlations. The underlying idea is that identical disturbances at both sides of the trailing edge would lead to destructive interference and hence low acoustic scattering. This idea is pursued in Figure 6, where we show the correlations of the pressure diﬀerence between upper and lower surfaces of the airfoil to the acoustic pressure; spanwise-averaged quantities were taken. Similar to the results previously presented, we observe high correlations inside the boundary-layer and close to the trailing edge. This time, correlation coeﬃcients are even higher, getting close to approximately 80%. We have seen that correlation coeﬃcients are greatly increased if spanwise averages and diﬀerences between upper and lower surfaces are taken in the near-ﬁeld, prior to the determination of correlations. This suggests that the relevant sound-producing ﬂuctuations are two-dimensional and in phase opposition in the trailing edge region. This will be further pursued in the next section, where velocity-pressure correlations are shown. D. Velocity-pressure correlations The results of the preceding section were of pressure-pressure correlations. Since velocity disturbances are more often studied in analyses of the turbulent ﬁeld, and are closely related to the source terms in acoustic analogies, we present here similar correlations, this time taken between components of the velocity vector with acoustic pressure. 9 of 16 American Institute of Aeronautics and Astronautics DownloadedbyINSTTECDEAERONAUTICA(ITA)onMarch30,2016|http://arc.aiaa.org|DOI:10.2514/6.2015-2384
- 10. (a) With spanwise average (b) Without spanwise average Figure 4. Maximum correlation between grid 1 pressure and far-ﬁeld pressure. Figure 5. Minimum correlation between grid 1 pressure and far-ﬁeld pressure, both quantities spanwise averaged. We plot (−1) times the minimal correlation coeﬃcient to allow direct comparison with the color scales of Figure 4 and other similar plots. Figure 7 shows the minimum velocity-pressure correlation, where the u component was taken. Fluctua- tions were spanwise-averaged as in the analysis of the previous section. No signiﬁcant positive correlation peaks were observed, hence we focus on the minimum correlation (which displays the negative peaks); again, 10 of 16 American Institute of Aeronautics and Astronautics DownloadedbyINSTTECDEAERONAUTICA(ITA)onMarch30,2016|http://arc.aiaa.org|DOI:10.2514/6.2015-2384
- 11. Figure 6. Maximum correlation between grid 1 pressure diﬀerence between the upper and lower surface of the airfoil and far-ﬁeld pressure; spanwise-averaged quantities. an inversion of the color scale was done to allow direct comparison with the other plots. We observe a thin region in the neighborhood of the trailing edge with high correlation with the radiated sound. This region is inside the boundary layer, around the displacement thickness. Figure 7. Minimum correlation between u-component and acoustic ﬁeld pressure. Fluctuations were spanwise- averaged. We plot (−1) times the minimal correlation coeﬃcient to allow direct comparison with the color scales of Figure 4 and other similar plots. When the spanwise average is not taken, the correlation coeﬃcients become negligible, as shown in Figure 8. This again shows that the two-dimensional part of the turbulent ﬁeld is most relevant for sound radiation. If the diﬀerence of the u-component between upper and lower surfaces is taken before correlation, we obtain the results of Figure 9. As was observed with the pressure, the correlation coeﬃcients become more signiﬁcant in the vicinity of the trailing edge. The two-dimensional part of the acoustic ﬁeld is the most correlated with the acoustic pressure, in spite of its relative low kinetic energy. This is shown in Figure 10, which compares the root-mean-square (RMS) of the u-component near the airfoil. We observe that downstream of the laminar-turbulent transition, in a region with approximately 0.2 ≤ x/c ≤ 1.2, we have RMS(u2D) << RMS(u3D). The two-dimensional part of the velocity ﬁeld is thus a small fraction of the overall turbulence; however, these low-amplitude two-dimensional structures are closely related with trailing-edge noise. Correlations using v- and w-components are not presented here due to lack of expressive ﬂow-acoustic correlations. 11 of 16 American Institute of Aeronautics and Astronautics DownloadedbyINSTTECDEAERONAUTICA(ITA)onMarch30,2016|http://arc.aiaa.org|DOI:10.2514/6.2015-2384
- 12. Figure 8. Minimum correlation between u-component and acoustic ﬁeld pressure. Fluctuations were not spanwise-averaged. We plot (−1) times the minimal correlation coeﬃcient to allow direct comparison with the color scales of Figure 4 and other similar plots. Figure 9. Minimum correlation between grid 1 u-component, with diﬀerence between the upper and lower surfaces of the airfoil, and acoustic ﬁeld pressure. See comments in the caption of Figure 7. IV. Conclusion We have used ﬂow-acoustic correlations to identify regions and mechanisms most eﬀective to generate sound in a turbulent ﬂow around a NACA 0012 airfoil with zero angle of attack and M = 0.115.. Theoretical results for the trailing-edge noise problem, modelled using the tailored Green’s function of Ffowcs Williams and Hall12 show that only wavenumbers satisfying kz < k, where kz is the spanwise wavenumber of turbulent disturbances and k is the acoustic wavenumber, radiate sound. In particular, two-dimensional disturbances (kz = 0) are always radiating. With this in mind we show, using ﬂow-acoustic correlations obtained using data from a large eddy sim- ulation, that when spanwise averages of ﬂow ﬂuctuations are taken, ﬂow-acoustic correlations signiﬁcantly increase. We have obtained signiﬁcant correlations in the trailing edge region for spanwise-averaged quanti- ties: when pressure-pressure correlations were taken, a region extending from the wall to the boundary layer thickness was found to present high correlation peaks; and for correlations between streamwise velocity and acoustic pressure, an elongated region around the displacement thickness was found to have signiﬁcant cor- relation peaks as well, indicating that in this region there are coherent structures inside the boundary-layer associated with far-ﬁeld sound. If the two-dimensional mode is not isolated, correlation coeﬃcients are lower, and even negligible in some cases. Flow-acoustic correlations also increase when a diﬀerence of ﬂuctuations between the upper and lower surfaces is taken, which agrees with theory: convection of disturbances in phase 12 of 16 American Institute of Aeronautics and Astronautics DownloadedbyINSTTECDEAERONAUTICA(ITA)onMarch30,2016|http://arc.aiaa.org|DOI:10.2514/6.2015-2384
- 13. (a) RMS of grid 1 u-component with spanwise-average (b) RMS of grid 1 u-component with no spanwise-average Figure 10. Comparison between the RMS of grid 1 u-component with spanwise-average and grid 1 u-component with no spanwise-average opposition across the trailing edge leads to maximal scattered sound. Some features of the analysis are particular to the kind of numerical simulation used here, which is an LES adopting periodic boundary conditions along the span. Similar simulations use the same approach (e.g. Sandberg and Sandham10 ), and, although validation of such simulations involve a demonstration of coherence decay along the spanwise direction of the computational domain, an artefact persists: only discrete values of spanwise wavenumbers are possible, and in some cases (such as in the simulation used in this work) most of the sound radiation is two-dimensional, as kz = 0 is the only radiating wavenumber for most of the acoustic spectrum. In experiments, we expect a much broader range of spanwise wavenumbers, and non-zero kz would probably radiate sound. However, the |kz| < k condition implies that for most Helmholtz numbers only spanwise-elongated structures, with characteristic dimension greater than the acoustic wavelength, radiate sound to the far ﬁeld. Further numerical and experimental work aiming at the characterization of such structures is promising. V. Acknowledgments The authors acknowledge the ﬁnancial support received from Conselho Nacional de Desenvolvimento Cient´ıﬁco e Tecnol´ogico, CNPq, under grant No. 402233/2013-1 and 382381/2014-9. A. V. G. Cavalieri was supported by Conselho Nacional de Desenvolvimento Cient´ıﬁco e Tecnol´ogico, CNPq, through a research scholarship. The authors also acknowledge the ﬁnancial support received from Funda¸c˜ao de Amparo `a Pesquisa do Estado de S˜ao Paulo, FAPESP, under Grant No. 2013/03413-4 and from Conselho Nacional de Desenvolvimento Cient´ıﬁco e Tecnol´ogico, CNPq, under Grant No. 470695/2013-7. We thank Peter Jordan for helpful discussions regarding the present work. References 1R. K. Amiet. Noise due to turbulent ﬂow past a trailing edge. Journal of Sound and Vibration, 47(3):387–393, 1976. 2T. F. Brooks, D. S. Pope, and M. A. Marcolini. Airfoil self-noise and prediction, volume 1218. National Aeronautics and Space Administration, Oﬃce of Management, Scientiﬁc and Technical Information Division, 1989. 3A. V. G. Cavalieri, D. Rodr´ıguez, P. Jordan, T. Colonius, and Y. Gervais. Wavepackets in the velocity ﬁeld of turbulent jets. Journal of Fluid Mechanics, 730:559–592, 2013. 13 of 16 American Institute of Aeronautics and Astronautics DownloadedbyINSTTECDEAERONAUTICA(ITA)onMarch30,2016|http://arc.aiaa.org|DOI:10.2514/6.2015-2384
- 14. 4D. G. Crighton. Basic principles of aerodynamic noise generation. Progress in Aerospace Sciences, 16(1):31–96, 1975. 5D. G. Crighton, A. P. Dowling, J. E. Ffowcs-Williams, M. Heckl, and F. G. Leppington. Modern methods in analytical acoustics: lecture notes. Springer, 1992. 6N. Curle. The inﬂuence of solid boundaries upon aerodynamic sound. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 231(1187):505–514, 1955. 7P. Jordan and T. Colonius. Wave packets and turbulent jet noise. Annual Review of Fluid Mechanics, 45(1), 2013. 8M. J. Lighthill. On sound generated aerodynamically. i. general theory. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 211(1107):564–587, 1952. 9Assad A. Oberai, Farzam Roknaldin, and Thomas J. R. Hughes. Computation of trailing-edge noise due to turbulent ﬂow over an airfoil. AIAA journal, 40(11):2206–2216, 2002. 10Richard D. Sandberg and Neil D. Sandham. Direct numerical simulation of turbulent ﬂow past a trailing edge and the associated noise generation. Journal of Fluid Mechanics, 596:353–385, 2008. 11Christopher K. W. Tam. Supersonic jet noise. Annual Review of Fluid Mechanics, 27(1):17–43, 1995. 12J. E. Williams and L. H. Hall. Aerodynamic sound generation by turbulent ﬂow in the vicinity of a scattering half plane. Journal of Fluid Mechanics, 40(04):657–670, 1970. 13W. R. Wolf, J. L. F. Azevedo, and S. K. Lele. Convective eﬀects and the role of quadrupole sources for aerofoil aeroacoustics. Journal of Fluid Mechanics, 708:502–538, 2012. A. Numerical Simulation The Large-Eddy Simulations (LES) in this work are the result of Wolf et al.13 for which time-resolved data is available for all points in the grid. In what follows we summarize some of the simulation characteristics. To perform the numerical simulation two diﬀerent grids were used. The ﬁrst allows accurate resolution of the airfoil boundary layer. This grid is body-ﬁtted O-grid block composed by 1536 x 125 x 128 nodes (Figure 11 (b)). The second grid is more appropriate for acoustic waves emanating from the airfoil. This grid is a Cartesian background grid block and it is composed by 896 x 511 x 64 (Figure 11 (a)). Wolf et al. performed the solution of general curvilinear form of the compressible Navier-Stokes equations, given as ∂ρ ∂t + ∂(ρui ) ∂t = 0, (38) ∂(ρui ) ∂t + (ρui uj + gij p−τij) ∂xj = 0, (39) ∂E ∂t + ∂[(E + p)ui − τijgikuk + qj] ∂xj = 0, (40) To close the set of equations we deﬁne the total energy equation, E, express the heat ﬂux using Fourier’s law and the viscous stress tensor using the expression for a Newtonian ﬂuid. We obtain thus E = p γ − 1 + 1 2 ρgikui uk , (41) qj = − µ RePr gij ∂T ∂xi (42) τij = µ Re (gjk ∂ui ∂xk + gik ∂uj ∂xk − 2 3 gij ∂uk ∂xk ) (43) Finally, the gas is taken as calorically perfect, hence p = γ − 1 γ ρT (44) 14 of 16 American Institute of Aeronautics and Astronautics DownloadedbyINSTTECDEAERONAUTICA(ITA)onMarch30,2016|http://arc.aiaa.org|DOI:10.2514/6.2015-2384
- 15. (a) (b) (c) Figure 11. Mesh details; (a)Full view of Cartesian grid with composed 896 x 511 x 64 nodes; (b)Body-ﬁtted O-grid block with composed 1536 x 125 x 128 nodes. (c)Detailed rounded trailing edge. In Table 2 there is a summary of detailed properties used on current simulation. To force transition to turbulence in the boundary layer, a boundary layer trip was simulated using artiﬁcial blowing and suction. Figure 12 illustrates the region of boundary layer tripping. Rec M∞ AoA[deg.] BL tripping ∆t NS NpS 408000 0.115 0 Top and bottom sides 0.0004 3 512 Table 2. Summary of ﬂow conﬁguration analyzed. Here, Rec is the Reynolds number based on the aerodynamic proﬁle chord, AoA is the angle of attack, δt is the dimensionless time step for time marching of the LES equations, NS is the number of datasets for acoustic processing, and NpS is the number of samples in each dataset. Figure 12. Illustrative description of aerodynamic proﬁle used on LES simulation and the region of boundary layer tripping. Figure 13 (a) shows a general view of the turbulent ﬁeld around the airfoil by plotting the iso-surfaces of λ2. Figure 13 (b) illustrates the boundary layer transition region and the noise generation due the airfoil. 15 of 16 American Institute of Aeronautics and Astronautics DownloadedbyINSTTECDEAERONAUTICA(ITA)onMarch30,2016|http://arc.aiaa.org|DOI:10.2514/6.2015-2384
- 16. (a) (b) Figure 13. LES of ﬂow past a NACA0012 airfoil at AoA = 0 deg, M∞ = 0.115 and Rec = 408000; (a) iso-surfaces of λ2 colored by vorticity magnitude; (b) iso-surfaces of vorticity magnitude colored by streamwise momentum with contours of dilatation in the background (a) (b) Figure 14. (a) Points used to correlate with center at x/c = 0.5; (b) Points used to correlate with center at x/c = 1.0 16 of 16 American Institute of Aeronautics and Astronautics DownloadedbyINSTTECDEAERONAUTICA(ITA)onMarch30,2016|http://arc.aiaa.org|DOI:10.2514/6.2015-2384