Numerical and experimental investigation of a

AIAA 2015-2366 AIAA Aviation 22-26 June 2015, Dallas, TX 21st AIAA/CEAS Aeroacoustics Conference

Numerical and experimental investigation of a beveled trailing edge flow and noise field W.C.P. van der Velden∗ , S. Pröbsting∗ , A.T. de Jong† and A.H. van Zuijlen‡ Delft University of Technology, Kluyverweg 2, 2629 HT, Delft, the Netherlands

Downloaded by UNIVERSITY OF NOTRE DAME on October 24, 2016 | http://arc.aiaa.org | DOI: 10.2514/6.2015-2366

Y. Guan§ and S.C. Morris¶ University of Notre Dame, IN 46556, United States

Efficient tools and methodology for the prediction of trailing edge noise experience substantial interest within the wind turbine industry. In recent years, the Lattice Boltzmann method has received increased attention for providing an efficient alternative for the numerical solution of complex flow problems. Based on the fully explicit, transient, compressible Lattice Boltzmann solution in combination with the Ffowcs Williams and Hawking analogy, an estimation of the acoustic emission in the far field is obtained. To validate this methodology for the prediction of trailing edge noise, the flow around a plate with an asymmetric 25 degree beveled trailing edge in low Mach number flow is analyzed. Flow field dynamics are compared to experimental data obtained from Particle Image Velocimetry measurements and show similar trends and behavior for both the mean and fluctuating velocity. Results of the acoustic prediction are compared to acoustic measurements obtained through phased array beamforming in combination with a source power integration technique. Vortex shedding is captured, broadband noise is present on both the experimental and numerical obtained noise spectra and a typical cardioid-like directivity behavior is found.

Nomenclature β δ δij δ? δx,y κ ν ρ τ ω B b C c0 ci f# fi H

p

1 − M02 Boundary layer thickness Kronecker delta Displacement thickness Interrogation window size von Karman constant Viscosity Density Relaxation time parameter Angular frequency Law of the wall constant Domain width, span Bhatnagar-Gross-Krook collision term Speed of sound Discrete velocity vector Focal ratio Movement of the distribution of particles function Shape factor

∗ PhD

candidate, Faculty of Aerospace Engineering, Aerodynamics Department, [email protected] researcher, Faculty of Aerospace Engineering, Aerodynamics Department ‡ Assistant professor, Faculty of Aerospace Engineering, Aerodynamics Department § PhD candidate, Department of Aerospace and Mechanical Engineering ¶ Associate professor, Department of Aerospace and Mechanical Engineering, AIAA member † PhD

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h l Li Mi M0 p R r Re St T t ui xi yi

Thickness of beveled plate Length of beveled plate, chord Acoustic loading term Mach number vector Free stream Mach number Pressure Effective acoustic distance Geometric observer distance Reynolds number Strouhal number Temperature Time Velocity vector Observer position Source position

I.

Introduction

The trailing edge noise of a wind turbine blade is currently one of the most dominant noise sources on a wind turbine and therefore understanding and modeling of the physics associated with the generation and propagation is of main importance for the design of more silent wind turbines.1 The airfoil self-noise originates from unsteady flow over an airfoil. Brooks et al.2 defined some fundamental airfoil self-noise mechanism associated with the trailing edge, such as laminar and turbulent boundary layer noise and bluntness noise. With turbulent flow, the acoustic effects depend largely on the length scale of the turbulent eddies.3 For sharp trailing edges and a turbulent boundary layer, when turbulent eddies are by far smaller than the airfoil chord, this turbulent length scale is the local boundary layer displacement thickness. Local disturbances of the surface pressure, introduced and convected with the turbulent eddies, are scattered at the trailing edge. In the case of the turbulent boundary layer, the surface pressure is only affected over a highly localized area by a given turbulent eddy and thus the overall aerodynamic force acting on the airfoil remain comparatively constant. Due to the small length scale and high convection velocity of the eddies, this situation is typically encountered at high frequency and directivity pattern shows a bias towards the leading edge (i.e. in upstream direction).2, 3 A different situation is encountered at beveled trailing edges. Here, flow separation is observed at the beveled surface upstream of the trailing edge.4 This flow separation introduces a shedding component to the wake flow with large, coherent velocity fluctuations. The associated length scale is often characterized in terms of the wake thickness and associated to the bluntness of the trailing edge, for instance thickness of the plate. If this length scale is large compared to the boundary layer thickness, the tonal noise component associated to such coherent vortex shedding processes becomes a prominent feature of the acoustic emission.3 Beveled trailing edge geometries have served for validation purposes in the past, such as the study of Wang and Moin.5 Important characteristics of the unsteady surface pressure field are its auto-spectral density and spanwise coherence. Several authors, such as Amiet6 and Howe7 have discussed diffraction theory regarding trailing edge noise. Here, the auto-spectral density and spanwise correlation length of hydrodynamic pressure fluctuations were used to estimate the acoustic far field spectrum. Amiet6 and Howe7 assumed that the incident pressure fluctuations on the surface below the turbulent boundary convect over the trailing edge, acting as an impedance discontinuity, where the fluctuations are scattered in the form of acoustic waves. This theory forms the basis of multiple experimental and numerical studies, such as the Large Eddy Simulation study of Christophe,8 the surface pressure measurements of Brooks and Hodgson,9 and the recent study of Pröbsting et al.10 with high-speed tomographic Particle Image Velocimetry based pressure reconstruction. The present study is set to validate the applicability of a fully explicit, transient, compressible Lattice Boltzmann Method (LBM) based Computational Fluid Dynamics (CFD) solver for trailing edge noise prediction. The LBM solves the discrete Boltzmann equations in combination with a collision model to simulate the flow of a Newtonian fluid. By simulating streaming and collision processes across a limited number of particles, the flow is simulated. For this purpose, an asymmetric 25 degree beveled trailing edge test case in a low Mach number flow is selected. Validation is performed based on PIV and acoustic phased array

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experiments. Acoustic results are numerically obtained by means of Ffowcs Williams Hawking aeroacoustic analogy in the far-field and direct probes in the near-field. Perot11 studied aeroacoustic noise from a wind turbine before using similar methodology presented in this paper.

II.


A.

Measurement and simulation

Test case

The physical model used in the experiments was a 360 mm chord (l), 20 mm thick (h) flat plate with an asymmetrically beveled 25 degrees trailing edge, elliptical leading edge, and a span (b) of 400 mm. The particular trailing edge model used here is characterized by a radius of curvature R/h = 0, which is equivalent to a kink upstream of the slanted trailing edge section. To ensure a turbulent boundary at the trailing edge in the experimental setup, 3D randomly distributed roughness elements (nominal grain size 0.8 mm) were applied as tripping device on both sides of the plate at the quarter chord location. Similar experiments on a rounded trailing edge have been presented by Pröbsting et al.12 In the 2.56/h span (b) numerical model a fully developed turbulent boundary layer is obtained by use of a zigzag strip at a location of 5% of the chord with a height of 0.075h and a streamwise length of 0.015l. Its wavelength equals 0.11b mm, where a spanwise extent of nine wavelengths has been modeled. B.

Experimental measurements

PIV measurements were performed to provide an experimental reference for the flow field around the edge. In addition, acoustic phased array measurements in an anechoic facility were acquired for comparison with the computationally predicted noise levels. To assess the flow conditions in the anechoic facility, additional hot-wire measurements were performed. 1.

Flow field measurements

Time-resolved PIV experiments were performed in an open-jet, low-speed wind tunnel with a rectangular outlet of 0.4 × 0.4 m2 at Delft University of Technology. Side plates were mounted to the exit nozzle to restrict the expansion of the open jet at the spanwise ends of the model. PIV experiments were performed with a free stream velocity of 20m/s, equivalent to a chord based Reynolds number of 480, 000. The flow was seeded with evaporated water-glycol based fog fluid. The mean particle droplet is approximately 1 µm. A Quantronix Darwin Duo Nd:YLF laser (2 × 25 mJ at 1 kHz) illuminated the particles over the field of view (FOV). Two Photron Fastcam SA1.1 (1 M px, 12 bit resolution, 20 µm pixel pitch) were used for image acquisition in the planar PIV set-up. Both cameras were equipped with Nikon Micro-Nikkor 105 mm prime lenses and positioned on opposite sides of the test section with a small offset in the streamwise (x−)coordinate direction (see Fig. 1). The resulting FOV encompasses the beveled (suction side) of the trailing edge and the near-wake.

Figure 1. Experimental set-up 2-component planar PIV flow field measurements


Images were acquired at 125 Hz in double frame mode for flow statistics. The image sequence is correlated with an iterative multi-grid, multi-pass correlation technique with window weighting, window deformation, and a final interrogation window size of 16 × 16 px2 . Parameters related to the planar PIV measurements are listed in Tab. 1. Table 1. Parameters for planar PIV measurements

Parameter


2

Field of view [mm ] Magnification [−] Interrogation window size [mm] Focal ratio [−] Free stream displacement [px] Sampling frequency [kHz] Number of samples [−]

2.

Symbol

Value

F OV

45 × 21 0.42 0.76 5.6 15 0.125 800

δx , δy f# dx fs −

Acoustic measurements

Acoustic phased array were carried out in the 0.6 × 0.6 m2 open jet Anechoic Wind Tunnel facility (AWT) at the University of Notre Dame over an extended range of Reynolds numbers (10 − 35 m/s). All surfaces of the surrounding chamber are treated with sound absorbent glass fiber wedges with an absorption coefficient greater than 99% for frequencies exceeding 100 Hz. Details on the experimental facility can be found in Mueller et al.13 The model is place vertically on the centerline of the tunnel. To adapt the dimensions of the test section to those of the model and avoid 3D flow effects affecting the measurement, a 19 mm wooden splitter plate with an elliptical leading edge is placed at a height of 0.4 m from the bottom surface of the exit nozzle and reaches 1.22 m downstream. The phased array consisted of 40 condenser microphones arranged in an streamwise elongated logarithmic spiral configuration. It had an aperture of approximately 120 × 60 cm2 and was placed parallel to the centerline of the wind tunnel at a distance of 2.38 m with its center aligned with the trailing edge of the model. Data was acquired at a frequency of 40 kHz in an ensemble of 64 windows with 32, 768 samples (total acquisition time 52.43 s). For beamforming a Cross- Spectral Matrix method (Conventional Beamforming) was used under the assumption of a point source distribution, taking into account the modified propagation path due to shear layer refraction effects.14 To distinguish the trailing edge noise source from parasitic noise sources, the source power was summed over an integration area around the trailing edge. The result of this integration procedure represents the average source power over the aperture of the array. Results presented later in the discussion are further scaled to the acoustic pressure auto-spectral density at the center location of the array. Details on the microphone array, the beamforming, and integration methodology can be found in Shannon & Morris.15 In addition, hot-wire measurements were performed to characterize the boundary layer upstream of the trailing edge at x/h = −1 and x/h = −2.5 on the pressure and suction side, respectively. A AA Lab System AN-1003 hot-wire system with low-pass filter at 14 kHz was used and the hot-wire probe was mounted on a 3D translation computer controlled stage. At each data point, the signal was sampled at a frequency of 40 kHz for a period of 26 s. Data was acquired for a free stream velocity of 20 m/s. C. 1.

Numerical simulation Source field simulation

Results are obtained from numerical simulations of the Lattice Boltzmann Method (LBM).16 Instead of solving the traditional set of partial differential equations, known as the Navier-Stokes equations, the discrete Boltzmann equations are solved for simulating complex fluid flows. The LBM method starts from mesoscopic kinetic equations, i.e. the Boltzmann equation, to determine macroscopic fluid dynamics. The commercial software package Exa Powerflow 5.0b is used for this study, allowing a three dimensional flow to be simulated in 19 discrete directions (D3Q19) within one single voxel. 4 of 13 American Institute of Aeronautics and Astronautics

The kinetic equations are solved on a cartesian mesh, known as lattices, by explicit time-stepping and collision modeling. The equation has the general form: fi (x + ci ∆t, t + ∆t) − fi (x, t) = Ci (x, t),

(1)

where fi denotes the movement of the distribution of particles in the i th direction, according to a finite set of discrete velocity vectors {ci : i = 0, ..., N }. ci ∆t and ∆t are space and time increments respectively. The collision term on the right hand side of the LBM equation adopts the simplest and also the most popular form known as the Bhatnagar-Gross-Krook (BGK) form:17 ∆t (2) [fi (x, t) − fieq (x, t)] . τ In this collision term, τ denotes the relaxation time parameter, and fieq is the local equilibrium distribution function, which depends on local fluctuating properties. The basic fluid dynamic quantities, which do appear directly in the Navier-Stokes equations, such as fluid density ρ and velocity u are obtained using summations over the velocity vectors: X X ρ(x, t) = fi (x, t), ρu(x, t) = ci fi (x, t). (3)


Ci (x, t) = −

i

i

When considering the very low frequency and long wavelength limit, for a suitable choice of the set of discrete velocity vectors, the compressible Navier-Stokes equations can be recovered using the ChapmanEnskog expansion.16 The resulting equation of state obeys the ideal gas law: p = ρRT . The kinematic viscosity of the fluid is related to the relaxation time parameter, τ , according to:18 ν ∆t + . (4) RT 2 The viscosity model can be implemented through the relaxation time τ to locally adjust the numerical viscosity of the scheme.19 For the current study the turbulence model that is incorporated into the Exa Powerflow 5.0b package is used. The model consists of a two-equation k − Renormalization Group (RNG) modified to incorporate a swirl based correction that reduces the modeled turbulence in presence of large vortical structures. This approach is commonly known as a Very Large Eddy Simulation (VLES). Combining all previous equations and models will form the LBM scheme applied in this study. Fully resolving the near wall region is computationally too expensive for high-Reynolds number turbulent flow with the lattice concept of the LBM scheme. Therefore, a turbulent wall model is used to provide approximate boundary conditions. In the current study, the following wall-shear stress model based on the extension of the generalized law of the wall model is used:18, 20 + + y y 1 u+ = g = ln + B, (5) A κ A τ=

with

dp . (6) A=1+g dx This relation is iteratively solved to provide an estimated wall-shear stress for the wall boundary conditions in the LBM scheme. A slip algorithm,18 a generalization of bounce-back and specular reflection process, is then used for the wall condition. The LBM scheme inherently captures acoustic waves, since they recover the compressible, transient Navier-Stokes equations, including ideal gas equation of state. The scheme is solved on a grid composed of cubic volumetric elements, the lattices. A variable resolution is allowed, where the grid size changes by a factor of two for adjacent resolution regions. Due to the explicit time-stepping characteristics of the LBM scheme, the time-step size is increased with cell size in factors of two as well. Larger cells will therefore not be evaluated each timestep of the smallest cell. This gives rise to the notion of timestep equivalent number of cells, which are the number of cells scaled to operation at the shortest timestep in addition to the total number of cells. The total simulation domain size equals 15 chords squared, wherein the outer 2.5 chords are modeled as an anechoic outer layer to damp out any acoustic reflections. Multiple refinement regions are applied such that, near the boundary, the first cell is placed in the viscous sub layer, see Fig. 2. In total, around 160 million voxels were used to completely discretize the problem, with in total 100 million fine equivalent voxels. Data was acquired for a free stream velocity of 25 m/s, corresponding to a chord based Reynolds number of 600, 000. Sampling is started after a steady transient solution for 35 flow passes.



Figure 2. Close-up of the grid for the numerical simulation

2.

Acoustic analogy

Due to the fact that LBM is inherently compressible and provides a time dependent solution, the sound pressure field can directly be extracted from the computational domain, provided that there is sufficient resolution to capture the acoustic waves. In addition, for most trailing edge noise problems, an acoustic analogy is used to obtain the far-field noise. Therefore, the directly obtained sound field from the simulation is compared with an acoustic analogy based on the simulation fluid dynamics. To recover the acoustic far-field, the Ffowcs Williams and Hawkings21 (FW-H) equation is employed. The time-domain FW-H formulation developed by Farassat known as formulation 1A,22 and extended based on the convective form of the FW-H equation is used to predict the far-field sound radiation of the beveled trailing edge in a uniformly moving media.23 The solver is able to handle a motion of the noise generating solid geometry, either in a fluid at rest or in a uniform flow. The formulations are implemented in the time domain using a source-time dominant algorithm also referred to as an advanced time approach.23 The input to the FW-H solver is the time-dependent flow field on a surface mesh provided by the transient LBM simulations. This surface mesh is defined either as a solid surface corresponding to the physical body or as a permeable surface surrounding the solid body. For practical reasons, the present study uses the solid formulation, with pressure information recorded on the airfoil. Hence, dipole sources are the only source term for the current analogy, which simplifies the integral relation to the following form:23 # " Z î L˙ i R 1 4πp‘Q (x, t) = dS ˆ i )2 c0 g=0 R(1 − Mi R ret " # Z ˆ i − Li Mi Li R + dS (7) 2 ˆ i )2 g=0 R (1 − Mi R ret # " Z ˆ i (Mi R î − M 2) Li R + dS. ˆ i )3 R2 (1 − Mi R g=0 ret The subscript ret denotes the evaluation of the integrand at the time of emission. The distance between the observer (x) and the source position (y), R is defined as: −M0 (x1 − y1 ) + R? , β2

(8)

p (x1 − y1 )2 + β 2 [(x2 − y2 )2 + (x3 − y3 )2 ],

(9)

R= with R? = and

β=

q

1 − M02 .


(10)

Technically, R represents the effective acoustic distance rather than the geometric distance between the source and the observer in terms of time delay between emission and reception. The unit radiation vector then reads: ? ˆ i = −M0 R + (x1 − y1 ) , x2 − y2 , x3 − y3 . R (11) β2R R R Finally, the acoustic dipole source Li is defined as: Li = (p − p0 )δij nj

(12)


with p − p0 the fluctuating pressure on the solid surface, dij being the Kronecker delta and nj the surface normal. Sufficient accuracy is obtained when considering at least 12 cells per wavelength.24 Probes are equally distributed in a circle around the trailing edge. The direct probes are located at a two chord distance of trailing edge, whereas the FW-H analogy results are obtained ten chords away from the trailing edge.

III. A.

Results and discussion

Flow and acoustic source field

In the simulation, the boundary layer was tripped using a zig-zag tripping device. The flow field around and downstream of this device is visualized in Fig. 3 using iso-contours of λ2 , which is defined as the second invariant of the velocity tensor. The zig-zag strip generates a clear shear layer which separates from the tip, which contains wave like appearances of spanwise vortices which spanwise wavelength according to the size of the zig-zag strip. The spanwise vortices break up into arches and subsequently growth legs to form hairpin-like structures and full packages of hairpins. Close to the trip, the vortices appear to be correlated and of equal size. But further downstream when the turbulent boundary layer develops, vortices break up and new smaller-scaled vortical structures appear to finally form an uncorrelated, fully developed turbulent boundary.25 In general, for the sharp beveled edge, the flow experiences first a favorable pressure gradient followed by an adverse pressure gradient as soon the knuckle starts.

Figure 3. Isosurfaces of λ2 contours depicting the tripping mechanisms in the numerical simulation

The location of separation and recirculation can be examined in Fig. 4, where a normalized contour plot is shown of the mean streamwise velocity for both the numerical (Re = 600, 000) and experimental solution (Re = 480, 000). Colormaps as well as contour lines are similar for both plots. Clearly, the separation is fixed at the kink on the suction (top) side and at the trailing edge on the pressure (bottom) side. These separation points are independent of Reynolds number, making it possible to compare the experimental en numerical results. A large recirculation region is observed over the knuckle around the trailing edge. The contours of both the numerical and PIV results show close resemblance and display the separation and recirculation at similar locations and amplitudes. 7 of 13 American Institute of Aeronautics and Astronautics

0.2

0.4

0.6

0.8

∞

0

u/u [−]

−0.2

20

10

0

−10 −60

30 Wall normal direction [mm]

Wall normal direction [mm]

30

−50

−40

−30 −20 −10 0 Streamwise direction [mm]

10

20

20

10

0

−10 −60

30

−50

−40


10

20

30

0.005

0.01

0.015

0.02

v‘v‘/u∞ [−]

The visualization of the shear layer can be shown by means of contour lines for the y-velocity fluctuations, depicted in Fig. 5. A clear thin shear layer appears at the knuckle for both the simulation and experimental results. At the trailing edge, the wall-normal velocity fluctuations depict a large area where vortex shedding is present, directly located behind the recirculation zone. The size of the vortex shedding area in Fig. 5 differ a bit between the simulation and experiment, possibly because of the difference in chord based Reynolds numbers considered between both. A grid resolution study is required for the numerical simulation to confirm this discrepancy. 0.025

0.03

20

10

0

−10 −60




Figure 4. Contour plot of mean streamwise velocity for numerical simulation (left) and PIV experiment (right)

−50

−40


10

20

30

20

10

0

−10 −60

−50

−40


10

20

30

Figure 5. Contour plot of fluctuating y-velocity for numerical simulation (left) and PIV experiment (right)

Far upstream of the edge, the boundary layer on both sides shows characteristics of a generic, flat plate, turbulent boundary. The magnitude of the normalized mean and root mean square (rms) velocity as a function of vertical distance from the wall obtained by simulation and experiment are shown in Fig. 6 and Fig. 7, for pressure side and suction side, respectively. The comparison shows good agreement between simulation and experiment although Reynolds number and tripping device were not identical. The experimental results were obtained from chord based Reynolds number of 480, 000 HWA results whereas the numerical results were obtained from the simulation introduced before (Re = 600, 000). When observing the mean boundary layer profile on both the pressure and suction side, the boundary layers in both the experiment and simulation attain a turbulent state near the trailing edge with slightly different shape factors (H = 1.5 and H = 1.3 for experiment and simulation, respectively). The rms velocity shows the expected behavior of a turbulent boundary layer, with larger fluctuations in the inner layer. Further away from the boundary, the simulation starts to deviate from the experiment. A difference in displacement thickness, h/δ ? = 15 and 13 for the pressure side experiment and simulation respectively and h/δ ? = 19 and 15 for the suction side experiment and simulation respectively could characterize the differences in Fig. 6 and Fig. 7. The shedding of larger vortical structures, appearing from the trailing edge is visualized in Fig. 8 using iso-contours of λ2 , which is defined as the second invariant of the velocity tensor. The incoming boundary layer merges with the wake as soon as the shear layer is formed. The shedding is visualized by the sinusoidal movement of the iso-surfaces behind the trailing edge. The streamwise distance between the upper and lower wave is measured to be 3.8 cm in the far wake, where convective speed is assumed to be similar to the free u∞ 25 = 0.038×2 = 330 Hz, stream flow conditions. This will result in a shedding frequency of approximately 2∆ x 8 of 13 American Institute of Aeronautics and Astronautics

0.8 Simulation HWA

Simulation HWA

0.7

1

0.6

y/δ [−]

0.5

0.6

0.4 0.3

0.4

0.2 0.2

0 0

0.1

0.2

0.4

0.6 umag/u∞ [−]

0.8

0 0

1

0.02

0.04

0.06 0.08 u /u [−] rms

0.1

0.12

0.14

∞

Figure 6. Streamwise mean and rms velocity at the pressure side measurement location x/h = −1

0.8

Simulation HWA

Simulation HWA

0.7

1

0.6

0.8 y/δ [−]

0.5

y/δ [−]


y/δ [−]

0.8

0.6

0.4 0.3

0.4

0.2

0.2

0 0

0.1

0.2

0.4

0.6 umag/u∞ [−]

0.8

0 0

1

0.02

0.04

0.06 0.08 urms/u∞ [−]

0.1

0.12

Figure 7. Streamwise mean and rms velocity at the suction side measurement location x/h = −2.5


0.14


h = 0.26, which is close to corresponding to a Strouhal number based on the plate thickness (h) of St = uf∞ 3 26 the experimental determined values of Blake, Bearman and Greenway and Wood27 which vary between 0.24 and 0.28 for high and low Reynolds numbers for beveled trailing edges respectively.

Figure 8. λ2 iso-surfaces of the flow around the beveled plate, colored by means of spanwise vorticity

B.

Acoustic emission

Estimations of the acoustic emission are obtained directly from the numerical solution by means of pressure probes placed in the measurement volume. Moreover, the FW-H analogy is applied as described in section 2. The beamforming results reflect the average source power over the aperture of the array.28 In the numerical results, this is modeled by averaging the source power over the complete set of angles which are covered by the microphone array, i.e., between 75 and 105 degree above the trailing edge. A similar procedure for comparison between model equations and beamforming results was described by.10 The raw and corrected sound pressure level values are shown in Fig. 9 for the chord based Reynolds number of 600, 000 case. Due to the cyclic boundary conditions and limited span, the SPL measured directly in the simulation contains contributions from mirrored coherent image sources of the airfoil arriving through the cyclic domain boundaries to the microphone location. To correct for this, Oberai29 derived a three-dimensional correction for low Mach number flows: 2 fb , (13) ∆Lcyclic = 10 log10 u∞ r where b is the limited domain width, r the observer distance and f shows the frequency dependence in the equation. For both the experimental as well as the acoustic analogy the sound pressure levels are scaled by: r2 b1 ∆Lnorm = 20 log10 + 10 log10 , (14) r1 b2 The acoustic simulation results refer to an observation distance of r1 = 1 m and wetted span of b1 = 1 m. The results show a good agreement between the experimental acoustic data and the simulated data. The peak, at a thickness based Strouhal number of 0.26, corresponding to vortex shedding at the trailing edge, is present in all three models. At low frequency, the beamforming results underestimate the actual source power due to spectral leakage through the boundaries of the fixed integration domain. The numerical results on the other hand, show less of a decay below the shedding frequency but could not be fully converged yet. The direct method includes another peak at a lower frequency possibly due to standing waves between the c = 125 Hz. The second top and bottom frictionless walls, which results in multiple of frequencies of 7.5l standing wave, i.e. St = 0.2 Hz is visible on the acoustic spectra of the direct probe. In the mid-frequency regime, where a broadband noise spectra is present, a good match between simulation and experiments is observed. Furthermore, the analogy predicts a decay similar to the results from the microphone array, with a maximum deviation of 2 dB as can be seen from Fig. 9. On the other hand, the sound pressure levels of the direct microphone probes inside the computational volume starts to increase again above St = 0.8. This increase is possibly caused by the refinement regions which are getting coarser away from the beveled plate. A mesh resolution should confirm this hypothesis in a next study. Another possibility of over-prediction in the high frequency regime could be assigned by inflow turbulence, resulting in hydrodynamic pressure fluctuations at the locations of the direct probes. 10 of 13 American Institute of Aeronautics and Astronautics

−2

10

60

FWH Solid Direct Experiment

50

−4

Corrected ASD [dB]

10

2

Uncorrected ASD [Pa /Hz]

FWH Solid Direct Experiment

−6

10

−8

10

40

30

20

10

−10


10

0.4

0.8 fh/u∞ [−]

1.6

3.2

0

0.4

0.8 fh/u∞ [−]

1.6

3.2

Figure 9. Uncorrected (left) and corrected (right) auto-spectral density levels above the trailing edge

A 360 degrees acoustic field was measured around the trailing edge to obtain the acoustic emissions for the beveled plate. With this information, it is possible to discuss directivity effects for different ranges of frequencies. In Fig. 10, the (scaled) SPL for four different frequency regimes are depicted. It is known from experiments and analytical investigations that the acoustic radiation of trailing edge noise has the highest sound pressure level in an oblique upstream direction.2 Also the maximum radiation appears at higher upstream angles with increasing frequency. This behavior is also observed in Fig. 10, where the mean pressure levels between Strouhal numbers of 0.2 and 0.4 show compact dipole behavior whereas at higher frequency, non-compactness appear and a typical cardioid-like directivity pattern is present. 105°

90°

75°

120°

60°

135°

45°

150°

30°

165°

15°

1×10−9 0.8

0.6

0.4

±180°

0.2

0°

−165°

−15°

−150°

−30°

St = 0.2 − 0.4 St = 0.4 −45°− 0.8 St = 0.8 − 1.6 −60° St = 1.6 − 3.2

−135° −120° −105°

−90°

−75°

Figure 10. Acoustic sound pressure directivity plot for different frequency regimes obtained from the numerical simulation using the FWH analogy


IV.

Conclusion


Based on the fully explicit, transient, compressible Lattice Boltzmann solution in combination with the Ffowcs Williams and Hawking analogy, an estimation of the acoustic emissions in the far field is obtained. To validate this methodology for the prediction of trailing edge noise, the flow around a beveled plate with an asymmetric 25 degrees trailing edge in low Mach number flow is analyzed. Flow field dynamics are compared to experimental data obtained from HWA and PIV measurements and show similar trends and behavior for both the mean and fluctuating velocity. The shedding frequency is correctly captured and agrees well with various experimental studies.3, 26, 27 Results of the acoustic prediction are compared to acoustic measurements obtained through phased array beamforming. Vortex shedding is in perfect line with the experimental results, broadband noise is in good agreement with a maximum deviation of 2 dB and a typical cardioid-like directivity behavior is found for the acoustic pressure obtained from the FW-H analogy. The probes directly placed in the measurement volume need further investigation, since an overestimation is observed for Strouhal numbers over 0.8.

Acknowledgments This research is funded and supported by Siemens Wind Power A/S, Brande, Denmark. This research is supported by the European Community’s Seventh Framework Programme (FP7/20072013) under the AFDAR project (Advanced Flow Diagnostics for Aeronautical Research). Grant agreement No.265695. The authors would like to acknowledge A. Gupta for his contribution.

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21 Ffowcs-Williams, J. and Hawkings, D., “Sound generation by turbulence and surfaces in arbitrary motion,” Philosophical Transactions of the Royal Society od London, Vol. 264, 1969, pp. 321–342. 22 Farassat, F. and Succi, G., “The prediction of helicopter discrete frequency noise,” Vertica, Vol. 7, No. 4, 1983, pp. 309– 320. 23 Bres, G., Perot, F., and Freed, D., “A Ffowcs Williams-Hawkings solver for Lattice-Boltzmann based computational aeroacoustics,” 16th AIAA/CEAS Aeroacoustic Conference, Vol. 3711, 2010. 24 Habibi, K., Gong, H., Najafi-Yarzdi, A., and Mongeau, L., “Numerical Simulations of Sound radiated from Internal Mixing Nozzles with Forced Mixers using the Lattice Boltzmann Method,” 19th AIAA/CEAS Aeroacoustic Conference, , No. 2143, 2013. 25 Elsinga, G. and Westerweel, J., “Tomographic-PIV measurement of the flow around a zigzag boundary layer trip,” Experiments in Fluids, Vol. 52, 2012, pp. 865–876. 26 Bearman, P., “Investigation of flow behind a two-dimensional model with a blunt trailing edge and fitted with splitter plates,” Journal of Fluid Mechanics, Vol. 21, 1965, pp. 241–255. 27 Greenway, M. and Wood, C., “The effect of a beveled trailing edge on vortex shedding and vibration,” Journal of Fluid Mechanics, Vol. 61, 1973, pp. 323–335. 28 Oerlemans, S. and Sijtsma, P., “Determination of absolute levels from phased array measurements using spatial source coherence,” Tech. rep., National Aerospace Laboratory NLR, 2002. 29 Oberai, A., Roknaldin, F., and Hughes, T., “Trailing edge noise due to turbulent flows,” 02-002, Boston University, 2002.


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