Guillaume Polidori

Fabien Legrand

Fabien Bogard

Farouk Madaci

Fabien Beaumont

Numerical Investigation of the impact of Kenenisa Bekele's cooperative drafting strategy on its running power during the 2019 Berlin marathon

In this paper we examined the aerodynamic drag and the resulting decrease of power output associated with the drafting positions used by Kenenisa Bekele at the 2019 Berlin marathon.

Examination of videos of the race revealed three different positions adopted by Bekele directly behind a frontline of three runners. These drafting positions were numerically studied using CFD simulations performed with the RANS equations and the SST k-ω model.

Depending on his position behind the pacers, our analyses indicated that Bekele's drafting strategy resulted in reductions of 38.5% to 57.3% of the drag resistance and aerodynamic power. In terms of metabolic power, Bekele's drafting positions yielded an estimated 1.91% to 2.84% reduction compared to the calculated metabolic power of 1427.89 W (or equivalent metabolic power relative to body mass of 25.50W/kg) that Bekele would have developed if he had run alone at the same speed of 5.75 m.s -1 .

INTRODUCTION

Widely used in sports such as cycling, open-water swimming, speed skating, cross-country skiing and triathlon [START_REF] Belloli | Drafting effect in cycling: Investigation by wind tunnel tests[END_REF][START_REF] Bilodeau | Effect of drafting on heart rate in cross-country skiing[END_REF][START_REF] Bilodeau | Effect of drafting on work intensity in classical cross-country skiing[END_REF]Blocken et al., 2013a;[START_REF] Brisswalter | Consequences of drafting on human locomotion: benefits on sports performance[END_REF][START_REF] Chatard | Drafting distance in swimming[END_REF][START_REF] Hagberg | The effects of drafting[END_REF][START_REF] Hausswirth | Effect of two drafting modalities in cycling on running performance[END_REF][START_REF] Millet | Effects of drafting on energy expenditure in in-line skating[END_REF][START_REF] Rundell | Effects of drafting during short-track speed skating[END_REF][START_REF] Silva | Analysis of drafting effects in swimming using computational fluid dynamics[END_REF][START_REF] Zouhal | Drafting's Improvement of 3000-m running performance in elite athletes: is it a placebo effect[END_REF], drafting in running is a strategy used in most long distance world record attempts such as marathons. In such competitive events, the term "drafting" describes the tactic of performing the activity in sheltered position, behind one or several pace-setters. It serves a dual purpose of reducing the mental effort required to (1) set and (2) monitor a challenging pace [START_REF] Renfree | The influence of collective behaviour on pacing in endurance competitions[END_REF].

Conserving energy is another advantage that has been associated with drafting because it can decrease air resistance, provided that the runner maintains his/her position in the close wake of those in front of him [START_REF] Kyle | Reduction of Wind Resistance and Power Output of Racing Cyclists and Runners Travelling in Groups[END_REF][START_REF] Zouhal | Drafting's Improvement of 3000-m running performance in elite athletes: is it a placebo effect[END_REF][START_REF] Barry | Aerodynamic drag interactions between cyclists in a team pursuit[END_REF][START_REF] Beaumont | How does aerodynamics influence physiological responses in middle-distance running drafting?[END_REF]. But, as illustrated by the two sub-2 hours marathon attempts by Eliud Kipchoge (Nike Breaking 2 event, Monza, May 2017 and Ineos 1:59 challenge, Vienna, October 2019), drafting advantage may vary depending on where the runner stands within the pack of leading runners. In his first attempt (Monza, 2017), Kipchoge was surrounded at all times by teams of seven pacemakers to minimize air resistance, with six forming an arrowhead in front of him and one more behind him. In the second event (Vienna, 2019), the aerodynamic pattern of the group was different with five pacers forming an inverted arrowhead in front of him, and two additional runners behind him.

The comparison of various drafting configurations in running for the reduction of energetic costs has been only poorly examined in the literature. As one of the pioneer investigators, [START_REF] Pugh | Oxygen intake in track and treadmil running with observation on the effect of air resistance[END_REF][START_REF] Pugh | The influence of wind resistance in running and walking and the mechanical efficiency of work against horizontal or vertical forces[END_REF] runde showed that at a speed of 4.46 m.s -1 and against a 6 m.s -1 wind, 80% of the oxygen cost of overcoming air resistance was eliminated by running 1 m behind another runner, resulting in a 6.5% reduction in VO2 [START_REF] Pugh | The influence of wind resistance in running and walking and the mechanical efficiency of work against horizontal or vertical forces[END_REF]. Later on, Kyle [START_REF] Kyle | Reduction of Wind Resistance and Power Output of Racing Cyclists and Runners Travelling in Groups[END_REF] found that energy consumption decreased about 2%-4% in middle and long distances for runners behind a group, leading to an increase of their running speed of about 0.1 m.s -1 . [START_REF] Zouhal | Drafting's Improvement of 3000-m running performance in elite athletes: is it a placebo effect[END_REF] have assessed the influence of drafting on a 3,000 m running performance within a sample of ten elite athletes. They observed that runners who drafted just behind two pacers completed the 3,000 m 1.59% faster than leading runners, corresponding to an average gain of 8.85 s over this middle distance. More recently, [START_REF] Beaumont | How does aerodynamics influence physiological responses in middle-distance running drafting?[END_REF] investigated the relationships between aerodynamic parameters (computer simulations) and physiological responses (experiments) to a submaximal exercise (1,000 m at 5.83 m.s -1 on an indoor running track) in an international level middle-distance runner who ran in two conditions (solo or in drafting position behind two pace-setters). The results indicated that compared to running solo, running in the aerodynamic shadow of two other runners resulted in reduced frontal drag area (-33%), which translated into decreased oxygen consumption (-6%), heart rate (-1%) and energy cost (-33%). There are, to our knowledge, no other reported studies on the influence of drafting in an elite long-distance runner from an aerodynamical perspective. Consequently, using a Computational Fluid Dynamics (CFD) code, the present study offers an analysis of the aerodynamic drag forces on the front face of Kenenisa Bekele at the 2019 Berlin marathon, where he clocked a time of 2:01:41. Kenenisa Bekele established his performance using a negative-split strategy, meaning that the second half of the race was executed at a faster pace than the first one (1:01:05 in the first half for an average speed of 5.75 m.s -1 and 1:00:36 in the second half for an average speed of 5.80 m.s -1 ) (Figure 1). This means that taking advantage of aerodynamics can lower the mechanical cost of the first part of the race, and be beneficial later on. At the Berlin 2019 marathon, Bekele benefited from the presence of pacers up to the 25th kilometer because only a very limited number of runners in the world can maintain this pace beyond this distance. As can be seen in the full video of the race (https://www.youtube.com/watch?v=vL_5YYpLPig) Bekele was assisted by 3 pacers positioned side-by-side along a frontal axis to form a kind of aerodynamic shield. It can be noted that Bekele did not maintain a fixed position behind the pacers' line. Actually, three privileged positions have been identified after thoroughly watching the video: (1) behind one of the lateral pacers, (2) between the two lead runners, or

(3) behind the central pacer.

The objective of the present study was to determine the aerodynamic drag forces exerting on Kenenisa Bekele in each of these three drafting configurations and to quantify the resulting improvements in terms of running economy compared to the situation where Bekele ran solo

NUMERICAL SETUP

A computational fluid dynamics study requires several distinct steps including the validation of the studied geometry as well as the computational domain that represents the area to be studied around the geometry. The simplified geometry of the runner's body in the running position has been designed using CAD software (Solidworks®), 3D printed and 3D scanned.

The CAD geometry was subsequently adapted to Bekele's body size. In order to focus only on the influence of a drafting position change on the aerodynamic parameters, the same CAD geometry was used for modeling Bekele and its pacers. Once the geometry defined, the virtual runners were disposed within a computational domain (Figure 2) the dimensions of which (i.e., 3 m upstream and on each side of the runner, 2.5 m above the runner's head and 10 m downstream the runner) were chosen according to the best practice guidelines [START_REF] Franke | Best practice guideline for the CFD simulation of flows in the urban environment[END_REF]. More detailed Information about geometry and computational domain are given in appendix1.

A careful examination of the video of the race during the first half-marathon showed that Bekele remained at an average distance of 1.3 m behind the front line of the three pacers. This distance can be judged to be "acceptable", though not optimal to fully benefit from the drafting effect [START_REF] Pugh | Air resistance in sport[END_REF]. However, this strategy allowed Bekele to be located in the aerodynamic shadow (i.e., area of adverse pressure gradients) of the leading pacers while decreasing the risk of falls due to tripping. The numerical simulations were therefore performed with this distance of 1.3 m between the pacer's back and Bekele's chest whereas the lateral distance from shoulder to shoulder was set to 0.3 m. Figure 2 [START_REF] Menter | Zonal Two Equation k-ω Turbulence Models for Aerodynamic Flows[END_REF]. Detailed CFD methodology is available in appendix 1.

CFD VALIDATION

Since no comparison of the numerical results could be made with the results obtained in a wind tunnel to validate the numerical procedure, the validation was carried out with the limited experimental data from the literature for a single runner, particularly with regard to the drag coefficient. [START_REF] Hill | The air-resistance to a runner[END_REF] demonstrated that the drag coefficient of a runner was 0.9, while Davies found that this value ranged from 0.82 to 0.91. [START_REF] Walpert | Aerodynamics of the human body in sports[END_REF] found that the drag coefficient of a runner was 0.79. More recently, [START_REF] Gourley | Faster: demystifying the science of triathlon speed[END_REF] reported that the drag coefficient of a runner ranged from 0.8 to 0.9 depending on speed and runner's frontal area.

The CFD value of 0.812 in the present numerical study is found to be consistent with the range of experimental drag coefficients reported in the literature.

RESULTS

We aimed at identifying all the aerodynamic drag forces involved in the various drafting configurations used by Kenenisa Bekele. We were especially interested in the changes in the distribution of pressure on Bekele's body as well as in his surroundings. In aerodynamics, drag is intrinsically related to the distribution of pressures around and in the wake of the runners [START_REF] Pugh | The influence of wind resistance in running and walking and the mechanical efficiency of work against horizontal or vertical forces[END_REF]. The pressure coefficient can be used to predict the fluid pressure at different locations. The pressure coefficient is defined as follows:

= 2

(2)

where is the static pressure, the reference static pressure (i.e. the atmospheric pressure)

and , the speed of the body through the fluid (m.s -1 ).

Figure 3 represents the pressure coefficient Cp distribution in a horizontal plane passing through the torso of the runners as well as on the whole-body surface of the drafting runner for each investigated position.

In order to better highlight changes in the pressure field, we have arbitrarily limited the color bar range even if the maximum and minimum absolute Cp values are much larger. Only the extreme Cp values relating to the drafting runner are given in Figure 3(a). Figure 3(a) clearly

shows the area of overpressure in front of the studied trailing runner and the area of underpressure behind him. Whatever the drafting position and compared to the isolated runner, there is a decrease in the extend of both overpressure and underpressure areas, yielding to a drag reduction of the trailing runner. A similar trend has been mentioned by Blocken et al. (2013b) for drafting in cycling. The best configuration is when the drafting runner is sheltered behind the central pacer, followed by the one behind a lateral runner. The worst of these 3 situations is the one where Bekele is between two leading pacers. This is confirmed by the mentioned values of Cp in Figure 3(a).

Consider, for example, the situation where Bekele is behind the middle pacer. In the study section passing through the runner's torso, it is locally observed a significant decrease in the maximum Cp value in front of the runner of about 95%, from Cp=1.0 when running solo to Cp=0.05 behind the middle pacer. A contrario, in this same study section, on the runner's back negative Cp values range from Cp=-0.56 to -0.24 for the same situations corresponding to a 57% difference. Obviously, what is preponderant is the gradient of this parameter between the front and the back of the runner to suggest a reduction in energy costs. [START_REF] Pugh | The influence of wind resistance in running and walking and the mechanical efficiency of work against horizontal or vertical forces[END_REF] had also described similar results by showing that the pressure differential was significantly reduced by running shielded in the wake of another runner, all runners starting from second position benefiting from a reduction in the absolute value of the overpressure in front of them as well as in the absolute value of the low-pressure behind them. Whatever the drafting configuration, the frontal surface pressure is always higher than the freestream static pressure as shown in Figure 3b. However, the comparison of all three configurations reveals that the observation of a transverse plane passing through the runner's torso (Figure 3a) can be largely transposed to the entire body. It is also important to note that drafting not only reduces the aerodynamic forces exerted on the draft runner (i.e. Bekele) but also that of the pacer [START_REF] Beaumont | How does aerodynamics influence physiological responses in middle-distance running drafting?[END_REF]. Thus, a pacer will also benefit from the positive effect of drafting because it will decrease his/her energy expenditure and will therefore push him/her forward in the race. This positive effect was also evidenced in other sports such as cycling [START_REF] Fitton | The impact of relative athlete characteristics on the drag reductions caused by drafting when cycling in a velodrome[END_REF]. If we compare the three drafting configurations, a clear-cut distinction can be made between the configuration in which the trailing runner is located between two leading pacers (Figure 4b) and those where he is sheltered directly behind one of the leading runners (Figures 4c andd). In the first case, Bekele does not benefit from the aerodynamic shelter formed by the wake of the runners in front of him. He must fight against rectilinear air displacement between the bodies of leading pacers as if he were alone, as shown in Figure 4a. The other two drafting configurations (Figures 4c andd) are the most beneficial from an aerodynamic point of view.

Air-flow behavior around

Indeed, in these drafting configurations, Bekele stands within the wake of the leading one, benefiting from the area of adverse pressure, which translates into higher running economy.

Several factors have to be considered when estimating the air resistance forces, or aerodynamic drag, on a runner [START_REF] Davies | Effects of wind assistance and resistance on the forward motion of a runner[END_REF]. The force experienced by a body moving through air is, roughly speaking, proportional to the kinetic energy in the air intercepted by the body. Aerodynamic drag FD (N) varies as a function of the dimensionless drag coefficient (CD), the frontal surface area A (m²), the air density (kg.m -3 ) and the approach flow velocity u (m.s -1 ).

=

(3)

The aerodynamic drag force FD is inferred from CFD calculations and the frontal area A (m²) of runner's body determined by equation ( 1).

The power required to overcome the aerodynamic drag is given by the following formula:

= × = (4)
Table 1 summarizes the various calculated aerodynamic parameters of the study that Bekele had to provide to fight air displacement for the different adopted drafting positions, as well as the aerodynamic power from equation ( 4 Results in table 1 perfectly reflect the fact that maintaining an optimal aerodynamic shelter position could have a positive impact on energy savings. Computer simulations showed that running solo resulted in an aerodynamic drag force of 7.77N acting on the runner, whereas this value was reduced to 3.32N in the best drafting position, which represents a gain of 57.3%. The aerodynamic power being proportional to the drag force, the same percentage gain of 57.3% is attributed to the aerodynamic power, varying from 44.69 W running solo to 19.15 W when optimally protected.

DISCUSSION

From an aerodynamic perspective, running directly behind the central leading pacer offers to the drafting runner the maximum reduction of air resistance. This can be explained in reference to phenomena associated with boundary layer flows. That is, the turbulent wake generated by the leading pacer imposes free-stream turbulence on the boundary layer of the trailing runner. This produces early transition from laminar to turbulent flow in its own boundary layer, the consequence of which is a downstream shift in the position of boundary layer separation. The more the drafting runner is sheltered, the more the separation lines at his body surface move downstream, inducing a reduction in wake spreading. The notion of runner-dependent aerodynamic power may seem complex and obscure to coaches or athletes, who are more familiar with the concept of mechanical and metabolic power outputs.

CFD simulations do not allow for a direct estimation of mechanical power. Derived from gold-standard measurements of power output in running by [START_REF] Cavagna | Mechanical work and efficiency in level walking and running[END_REF], the absolute mechanical power output taking into consideration the work to move the center of mass and to swing the limbs, plus the aerodynamic work can be expressed as = 9.42 + 4.73 + 0.266 .&& + ' (. )(

Where Pmecha is expressed in cal/(kg.min), u in km/h, m in kg and Paero is given in Table 1.

Total mechanical power output can be seen as the sum of running-induced mechanical power and aerodynamic power. Bekele having a mass of 56kg (https://www.sportsreference.com/olympics/athletes/be/kenenisa-bekele-1.html), the total mechanical power (including aerodynamic power) at a running speed of 20. From this it also appears that the contribution percentage of aerodynamic power to Pmecha is more than two times less in the most economical drafting configuration (2.19%) compared to running solo (4.97%). Interestingly, it can be noted that in the best drafting position a maximum gain of 57.3% on the aerodynamic power leads to a maximum gain of 2.84% on the total mechanical power.

These values are consistent with findings by [START_REF] Kyle | Reduction of Wind Resistance and Power Output of Racing Cyclists and Runners Travelling in Groups[END_REF], which highlighted that while air resistance is responsible for most of the power output in cycling (80%-90%), its impact is considerably lower in running.

The metabolic power is defined as the ratio between mechanical power and efficiency of locomotion. In [START_REF] Cavagna | Mechanical work and efficiency in level walking and running[END_REF], the efficiency of locomotion for a running speed of 20.7 km/h can be read to be 0.63 (see Figure 3 p. 475). Accordingly, the metabolic power can be predicted to range from 1427.89 W (i.e., 25.50W/kg) when running alone to 1387.35W (i.e 24.77, W/kg) when running behind the middle pacer (best drafting position). This metabolic power reduction of 40.54W (or 0.72 W/kg) also corresponds to a 2.84% gain. This also clearly shows that, in the absence of wind, deviating from the best drafting configuration leads to a reduction in energy gains, from 2.84% to 1.91% when the runner is located between two pacers. For sake of clarity, complete results on mechanical power and metabolic power are presented in Table 2. To complete these results, is also presented in Table 2 an Chris Boardman (holder of the cycling world hour record), produced an average mechanical power output of 442W (6.31W/Kg) for 60 min when he set the benchmark of 56.375 km [START_REF] Jeukendrup | The bioenergetics of world class cycling[END_REF]. Considering an efficiency of 0.25 [START_REF] Di Prampero | Cycling on Earth, in space, on the Moon[END_REF], Boardman performed his record with a high intensity metabolic power of 25.25 W/kg during one hour, slightly lower than Bekele's (25.50 W/kg) if he had to run alone during the first part of his marathon race, showing energy costs of the same order of magnitude.

The present study which explored the impact of aerodynamics on metabolic power paves the way for future investigations. For example, aerodynamic power is known to strongly depend on aerodynamic frontal area, which is anthropometry-dependent. A logical implication would be that smaller runners, because of a lower aerodynamic frontal area, would be more efficient in long-distance races. Compiling internet resources with print articles it appeared that the mean height for K. Bekele and the last five world record holders (E. Kipchoge, D. Kimetto, W. Kipsang, P. Makau, H. Gebreselassie) is 168 cm (standard deviation = 3.2 cm). Such relatively low values can be taken to support our assumption, but future studies are needed to further support this conjecture.

(

  second half of the race). *****************************Figure 1 here***********************************

  shows the four different situations under investigation. For the reference case simulating non-drafting conditions, Bekele is running alone (a). The other cases are studied in a drafting situation, where Bekele is located in the wake of the lateral left pacer (b), between two pacers (c) and behind the central one (d). Note that with no crosswind conditions, running behind the left or right lateral pacer is similar. *****************************Figure 2 here*********************************** A constant and uniform horizontal speed (5.75 m.s -1 or 20.70 km.h -1 ) was used at the entrance of the fluid domain, corresponding to Kenenisa Bekele's average running speed during the first part of the race (distance 25 km; time 1:12:30). Simulations were performed using ANSYS Fluent © 2019 R3 CFD code. The SST k-ω model was used to solve the 3D Reynolds Averaged Navier-Stokes (RANS) equations

*

  ****************************Figure 3 here***********************************

  Kenenisa Bekele is displayed in Figure 4 (streamline patterns colored by velocities). For each drafting configuration, streamlines are presented in the runner's sagittal plane. The wake downstream of the runner representing Bekele is clearly visible. *****************************Figure 4 here***********************************

  ) and deduced corresponding power gain. *****************************Table 1 here***********************************

  estimation of the metabolic power developed by C. Boardman in cycling, during his world hour record in 1996. *****************************Table2 here***********************************

  7 km/h in non-drafting conditions can be estimated at Pmecha = 230.36 cal/(kg.min) or, expressed in W/kg, at Pmecha =

	16.06W/kg.	Bekele	having	a	mass	of	56kg	(https://www.sports-
	reference.com/olympics/athletes/be/kenenisa-bekele-1.html),	the	absolute	estimated
	mechanical power output results in Pmecha = 899.57 W when he is not in drafting position.
	Looking at our estimations for aerodynamic power presented in Table 1, Pmecha can be split as
	follows: 854.88W for running-induced mechanical power, and 44.69 W for aerodynamic
	power when running solo.
	Since the different drafting strategies were run at the same speed of 20.7 km/h, it is assumed
	that the running-induced mechanical power was similar regardless of the drafting position and
	that only the aerodynamic power differed. Referring to Table 1 again, it can be deduced that
	Bekele's total mechanical power in drafting positions ranged from 874.03 W in the best
	drafting configuration (854.88 + 19.15) to 882.34 W between two pacers (854.88 + 27.46).

LIMITATIONS

The results reported here should be interpreted in the context of several limitations. First of all, the runner geometry used in this numerical study has been simplified and does not consider anthropometric differences between runners and pacers that could affect aerodynamic drag values. In addition, the CFD calculations were performed in a static position that did not consider the body's movement while running. Finally, this study was conducted in a windless environment.

CONCLUSION

This study represents an original effort to deepen our understanding of the aerodynamic drafting strategies used in marathons. Based on the negative-split strategy adopted by Kenenisa Bekele during the 2019 Berlin marathon, we numerically highlighted that the appropriate choice of sheltered position behind a front-line of three pacers allowed him to gain up to 57.3% in drag and deduced aerodynamic power. In terms of metabolic power, Bekele's drafting positions yielded an estimated 1.91 % to 2.84 % reduction compared to the calculated metabolic power of 1427.89 W (or equivalent metabolic power relative to body mass of 25.50 W/kg) that Bekele would have developed if he had run alone at the same speed of 20.7 km/h (i.e., 5.75 m.s -1 ).

DISCLOSURE

The authors report no conflicts of interest in this work.

Appendix 1: CFD methods

GEOMETRY

It is worth noting that for simplification purposes, the runner is considered as a static body in the computation process. In such a case, the running movements of the arms and legs are not modelled. However, cycling studies have shown that leg movement has a negligible influence on the aerodynamic forces acting on the cyclist/bicycle system [START_REF] Crouch | Dynamic leg-motion and its effect on the aerodynamic performance of cyclists[END_REF]. Based on this finding, we assume that this is also true for running. An accurate estimation of the frontal area of the runner is required to calculate the correct values of force-coefficients. It is possible to estimate Bekele's frontal area from its anthropometric data (https://www.sportsreference.com/olympics/athletes/be/kenenisa-bekele-1.html) using the following relationship [START_REF] Vaughan | Simulation of a sprinter. Part I. Development of a model[END_REF]):

A is the projected frontal area (m²) which is estimated at 0.475 m², H is the runner's size (1.65 m) and M the runner's weight (56 kg). The value of the projected frontal area deduced from equation ( 1) is in accordance with the results obtained with CAD software (Solidworks ® ).

COMPUTATIONAL DOMAIN AND GRID

The boundary layer mesh requires a very fine mesh near the wall to fully resolve the viscous/laminar sublayer. In order to correctly reproduce the boundary layer separation, an inflation layer composed of 40 layers of incremental thickness (growth ratio: 1.2) was generated around the body wall. To achieve a Y* value of less than 1, necessary for accurate boundary layer resolution, the size of the cells adjacent to the wall around the runner's body has been set to a value of 10 μm. The three-dimensional grid was generated using the ANSYS Workbench Meshing ® software and is composed of prism layers for inflation in the near-wall region and tetra elements in the free-stream. The average size of the mesh elements in the computational domain around and in the wake of the runners is 0.05 m. The mesh for the runner alone is made up of about 33 million cells, whereas it has nearly 67 million cells when there are 4 runners.

BOUNDARY CONDITIONS AND CFD SIMULATIONS

A turbulence intensity of 0.5% was chosen according to wind-tunnel experiments [START_REF] Inoue | Wind-tunnel experiment on aerodynamic characteristics of a runner using a moving-belt system[END_REF]. At the outlet, a pressure outlet condition with ambient static pressure was imposed, and the mass conservation law (all the gradients equal zero) was applied. A symmetry condition was imposed on the upper and side surfaces of the fluid domain. An air density of 1.219 kg/m 3 was used as a reference value for CFD calculation. The latter corresponds to the average temperature and humidity (14.8°C and 77.6% RH) recorded in Berlin during K.

Bekele's running time (https://www.infoclimat.fr/observationsmeteo/archives/29/septembre/2019/berlin-tegel/10382.html).

Focusing primarily on the aerodynamic forces acting on the runner's body, it was necessary to find a compromise between the computation cost and the accuracy of the findings. So, we performed a first calculation with the two methods RANS (Reynolds Averaged Navier-Stokes) and URANS (unsteady RANS). The monitoring of the drag coefficient value throughout the calculation showed in Figure 3 that there were no significant differences in the value of the forces acting on the runner. Moreover, compared to the RANS method, the URANS method requires a significant computation time (65 h vs. 14 h for a calculation performed on 48 processors). We therefore used the RANS (Reynolds Averaged Navier- The coupled algorithm was used for pressure-velocity coupling with second-order discretization schemes and gradients which were computed using the Green-Gauss cell-based method. The SST k-ω turbulence model is a two-equation eddy-viscosity model that is used for many aerodynamic applications. This turbulence model is fully adapted for adverse pressure gradients and separating flow, hence perfectly suited for aerodynamic studies in the field of sports [START_REF] Blocken | Aerodynamic drag in cycling pelotons: new insights by CFD simulation and wind tunnel testing[END_REF][START_REF] Fintelman | CFD simulations of the flow around a cyclist subjected to crosswinds[END_REF]. To improve convergence criterion, the pseudo-transient under-relaxation approach was used with 300 time-steps of 0.01s. The drag coefficients were monitored all along the calculation process for each runner, and stable convergence of results was achieved when the values no longer varied over time.

When the calculation was stopped, the residuals showed values close to 1 × 10


Yinshi Zhou

Vincent Valeau

Jacques Marchal

François Ollivier

Régis Marchiano

Three-dimensional identification of flow-induced noise sources with a tunnel-shaped array of MEMS microphones

Keywords:

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The aeroacoustic noise resulting from the impingement of a flow on a rigid body is encountered widely in engineering domains, such as aeronautics, ground transport and wind turbines whereas it is not a positive effect and should be reduced. For example, noise nuisance is of main concern for the public facing nowadays the increasing aeronautical activities, and thus more and more stringent civil aircraft noise regulations have been proposed by the International Civil Aviation Organization (ICAO) in Annex 16 to the Convention on International Civil Aviation [1]. To reduce noise emission, the noise source identification and the aeroacoustic optimization should be carried out during the conception phase of aircrafts, ground vehicles, wind turbines, etc. This is usually carried out in anechoic wind-tunnels by using scale models [START_REF] Oerlemans | Quantification of airframe noise using microphone arrays in open and closed wind tunnels[END_REF], albeit the vital achievements of computational aeroacoustics [START_REF] Wang | Computational prediction of flowgenerated sound[END_REF]. However, the identification of noise sources in terms of position, strength, directivity and generation mechanism still remains a challenging task because this kind of complex measurements rely highly on wind-tunnel facilities, microphone arrays (a set of arranged microphones, typically several tenths, with known spatial positions), flow measurement techniques and data processing strategies.

The standard experimental method widely used by research studies and industries for imaging noise sources is the beamforming technique with phased microphone arrays, which enables the directional scanning on an area consisting of a set of discretized grid points through an acoustic signal processing approach called delay & sum [START_REF] Johnson | Array signal processing: concepts and techniques[END_REF][START_REF] Allen | Aeroacoustic measurements[END_REF]. The beamforming results are plotted on this scan area, which generates what is usually called a sound map. Some of its applications, most of which were conducted with planar microphone array, planar scanning area and monopole noise source propagation model assumption, can be found in experimental investigations of the aeroacoustic sources of flight and ground vehicles [START_REF] Oerlemans | Quantification of airframe noise using microphone arrays in open and closed wind tunnels[END_REF][START_REF] Siller | Fly-over source localisation on a boeing 747-400[END_REF][START_REF] Fleury | Extension of deconvolution algorithms for the mapping of moving acoustic sources[END_REF][START_REF] Zhang | Source contribution analysis for exterior noise of a high-speed train: Experiments and simulations[END_REF]. Additionally, the beamforming technique can be associated with a deconvolution method such as DAMAS [START_REF] Brooks | A deconvolution approach for the mapping of acoustic sources (damas) determined from phased microphone arrays[END_REF] and CLEAN-SC [START_REF] Sijtsma | Clean based on spatial source coherence[END_REF] in order to improve the spatial resolution of the sound maps.

Over the last ten years, this approach has been extended to the noise source identification in 3D areas for more efficient sound imaging. Some studies intended to perform 3D sound source imaging by using a monopolar noise source assumption and planar arrays, although the spatial resolution in the direction perpendicular to a planar array is known to be bad. [START_REF] Brooks | Three-dimensional applications of damas methodology for aeroacoustic noise source definition[END_REF] firstly demonstrated the feasibility of using planar microphone arrays with 3D beamforming and the DAMAS technique, with an application to the case of a flap in flow. It was found that a microphone array with a larger size provided a beam pattern more "focused" and less elongated. However, the computational cost of the DAMAS method in 3D applications is very high. Using the CLEAN-SC method, Sarradj [START_REF] Sarradj | Three-dimensional acoustic source mapping with different beamforming steering vector formulations, Advances in Acoustics and Vibration[END_REF] conducted 3D beamforming tests with a planar microphone array of 64 microphones for simulated noise sources with different depth. Although four different steering vectors were examined, none of them could provided both correct positions and strength of the noise sources. Geyer et al. [START_REF] Geyer | Application of a beamforming technique to the measurement of airfoil leading edge noise[END_REF] identified the noise sources on an airfoil located in the turbulent flow of an open jet wind-tunnel, using a planar array of 56 microphones. The sound maps calculated on a 3D scanning grid including the potential noise source area of nozzle exit and airfoil were obtained using the CLEAN-SC method. However, no correction of the noise source position due to the shear layer of the wind-tunnel flow was made, and the leading edge noise sources, located near the midspan of the airfoil, spread in the direction perpendicular to the airfoil.

Brooks and Humphreys

Other studies focusing on 3D beamforming by developing non-planar arrays were also reported. Using four sub-arrays of 192 microphones mounted on the closed test section wall of a wind-tunnel, Padois et al. [START_REF] Padois | 3d source localization in a closed windtunnel using microphone arrays[END_REF] identified the noise source generated by a cylinder in a flow without considering the dipolar nature of the aeolian tone [START_REF] Mimani | An experimental appli-54 cation of aeroacoustic time-reversal to the aeolian tone[END_REF]. It was found that the CLEAN-SC provided a better spatial resolution and less side lobes. Similarly, Dobler et al. [START_REF] Döbler | On 3d-beamforming in the wind tunnel[END_REF] used a larger 3D microphone array of 576 microphones consisting of three sub-arrays for the noise source identification of a real car in a wind-tunnel. Although the sub-arrays were used to measure the acoustic data simultaneously, they were not considered as one unique array during the beamforming processing. The 3D sound maps were obtained by merging the beamforming results of each sub-array before using the CLEAN-SC method in a 3D scanning grid. Padois and Berry [START_REF] Padois | Two and three-dimensional sound source localization with beamforming and several deconvolution techniques[END_REF] examined the performance of four deconvolution methods including the DAMAS and CLEAN-SC methods for synthetic monopole noise sources in terms of accuracy of position and sound intensity, and their computational cost. This was accomplished by using a single or two perpendicular planar arrays for the beamforming processing in two-dimensional (2D) and 3D scanning zones. They found that the advantage of using the 3D array with the CLEAN-SC method are twofold, i.e., (i) a significant performance improvement of beamforming processing in 3D scanning area; (ii) a good trade-off between less computation time and good accuracy of noise source identification.

However, the beamforming technique with monopole radiation assumption can not identify correctly sources of dipolar nature if their orientation is not nearly perpendicular to the planar array [START_REF] Liu | Beamforming correction for dipole measurement using two-dimensional microphone arrays[END_REF]. It then becomes necessary to implement another noise source model, especially for non-planar arrays, when identifying the flow-induced noise of complex geometries such as cylinders [START_REF] Mimani | An experimental appli-54 cation of aeroacoustic time-reversal to the aeolian tone[END_REF] or airfoils [START_REF] Moreau | Flow-induced noise of a wall-mounted finite airfoil at low-to-moderate reynolds number[END_REF]. Hence, an approach where the dipolar noise source model was adapted for 3D beamforming algorithms, i.e., conventional beamforming and multiplicative cross-spectral beamforming and the CLEAN-SC method was proposed by Porteous et al. [START_REF] Porteous | Threedimensional beamforming of dipolar aeroacoustic sources[END_REF], for which the beamforming performances were compared for the aeroacoustic noise sources identification of a cylinder and an airfoil using a non-planar microphone array of 62 microphones. Their simulations suggested that the conventional beamforming associated to the CLEAN-SC method is the most efficient method in spite of its computational cost. However, the two perpendicular sub-arrays only enclosed partially the noise source area. Due to this array geometry and the relatively weak number of microphones, the sound maps for experimental sources remained difficult to interpret, especially concerning a finite airfoil in a wind-tunnel flow.

Despite the advantage of the traditional microphone array systems, the high cost and the complexity of such devices limit evidently the implementation of large arrays of microphone. In recent years, some efforts based on using the cheap MEMS microphones were made. It involves the work of Arnold et al. [START_REF] Arnold | A directional acoustic array using silicon micromachined piezoresistive microphones[END_REF], which demonstrated the cost reduction and the data processing efficiency of a modular directional acoustic array system based on 16 hybrid-packaged MEMS silicon piezoresistive microphones installed on a printed board. The system, comparable to conventional ones, provided a higher signal to noise ratio and a larger measuring frequency range than that of a single microphone. However, extensive calibration of the system in terms of directivity and accuracy was not conducted. Furthermore, Humphreys et al. [START_REF] Humphreys | Application of mems microphone array technology to airframe noise measurements[END_REF] used a planar array of 128 MEMS microphones with comparable performance to the current generation of array for the aeroacoustic radiation study of a scaled landing gear. With the design of a unique array mounting architecture to the closed test section of a wind-tunnel, the measurement system presented lower cost and complexity than the traditional directional array system.

To our knowledge, the use of the 3D beamforming technique associated to a microphone array with a very large number of microphones enclosing the wind-tunnel test section was not reported before. Therefore, the main objective of this paper is to demonstrate the effectiveness of such a massive 3D array of MEMS microphones for the identification of aeroacoustic sources in wind-tunnels. This is achieved by the identification of the noise sources of a wall-mounted airfoil yielding complex aeroacoustic noise radiation at different test conditions. The 3D beamforming results are compared with the published ones of experimental and numerical investigations. The organization of the papers is as follows. The experimental apparatus including the anechoic wind-tunnel, the test model and the microphone array are presented in Sec. II. The 3D beamforming method, the deconvolution technique CLEAN-SC and its application to simulated sources, and the modeling of the flow effects are highlighted in Sec. III. Finally, the results in terms of noise sources identification for one, and then two wall-mounted airfoils using the massive 3D array are presented in Sec. IV, followed by some concluding remarks.

Experimental set-up

Anechoic wind-tunnel and sound source configurations

The experiments presented in this paper were carried out in the anechoic wind-tunnel of the PPRIME Institute (Fig. 1). The flow is accelerated through a nozzle with a contraction ratio of 10, enters an anechoic room with a section outlet of (0.7 × 0.7) m 2 , and is then sucked through a collector of dimensions (width × height) (1.00 × 0.87) m 2 . The streamwise length of the test section is 1.44 m, and the anechoic chamber has a volume of 90 m 3 and a cutoff frequency of 200 Hz. The flow is bounded on its lower part by a plank of wood, and the dimension of the boundary layer is of the order of 10 mm at the nozzle exit. The maximum flow speed of the wind-tunnel is 60 m/s (216 km/h), and the turbulence intensity of the flow is about 1.5% at the nozzle exit. The coordinate system used for this study is represented in Fig. 1. The origin of the streamwise coordinate x is located at the nozzle exit, while the origin of the vertical coordinate z is located at the lower boundary of the flow. be rotated around an axis located at 66.6 mm (a third of the chord length) from the leading edge; the angle of attack is then controlled by using a motorized rotating table, allowing a precision of 0.05 • on the angle of attack.

Special care has been taken to ensure a perfectly smooth surface of the airfoil. For all tests, the airfoil was tripped using a rough band (sandpaper) with a width of 15 mm placed at a distance of 15 mm from the leading edge along the span on one side of the airfoil surface, which allowed a forced transition of boundary layer to occur on this side and a natural transition of boundary layer to occur on the other side of the airfoil surface. The sign of the angle of attack of the finite wall-mounted airfoil is defined as positive when the tripping device is on the pressure side of the airfoil. In this paper, results are given for positive angles of attack at 0 • , +4 • , +10 • and +20 • , and a flow speed of 40 m/s, corresponding to a chord-based Reynolds number of 5.33 × 10 5 . At these testing conditions, tonal noise generation is eliminated and broadband noise dominates the noise radiation, which is similar to formerly reported experimental results [START_REF] Pröbsting | Regimes of tonal noise on an airfoil at moderate reynolds number[END_REF][START_REF] Paterson | Vortex noise of isolated airfoils[END_REF][START_REF] Migliore | Wind tunnel aeroacoustic tests of six airfoils for use on small wind turbines[END_REF].

The MegaMicros system

The acoustic field is recorded by a 3D array of microphones. The development of a 3D antenna requires the use of a high number of microphones for the acoustic field to be correctly sampled and to lower spatial aliasing artifacts.

Microphones dedicated to measurement are generally very reliable but they require a non versatile installation: power supply, analog/digital conversion module, many cables and in addition they are expensive. On the contrary, digital MEMS microphones embed their conditioning circuitry and analog to digital conversion stage. They require very low power supply and deliver noise free numerical signals . Consequently they provide great compactness and allow to build very versatile arrays. In addition, these microphones being used in mobile telephony are massively produced which leads to significantly low costs. As a counterpart, these microphones might be believed to provide poorer measurements. Nevertheless, Vanwynsberghe et al. [START_REF] Vanwynsberghe | Design and implementation of a multi-octave-band audio camera for realtime diagnosis[END_REF] have shown that their characteristics are largely sufficient for acoustic imaging applications by using an antenna of 128 ADMP441 microphones developed by Analog Device. Indeed, these microphones are quasi-omnidirectional (1.7 dB maximum variation between -90 • and + 90 • ). The standard deviation of their sensitivity is low (0.8 dB). The variations of the phase are acceptable because they remain lower than π/20 rad between 70 and 3000 Hz, and lower than 0.24 rad between 3000 and 10000 Hz. Finally the frequency response is not flat but since it does not vary much from one element to the other, it is known and can be corrected a posteriori.

The array of microphones used in this study is similar to the system described in [START_REF] Vanwynsberghe | Design and implementation of a multi-octave-band audio camera for realtime diagnosis[END_REF], but uses 256 MEMS microphones. It is not a commercial device, and was developed at the d'Alembert Institute. The microphones are grouped by beams of 8 microphones connected to a concentrator which makes it possible to synchronize all the signals and to transfer them to a PC via a USB serial bus (see description in [START_REF] Vanwynsberghe | Design and implementation of a multi-octave-band audio camera for realtime diagnosis[END_REF]). The beams are connected to the concentrator by a RJ45 cable. Thus each beam can be deported over large areas. For this configuration, the RJ45 cables had a length of 10 m.

The beams of microphones are arranged on rigid rectilinear bars on which they are spaced regularly (every 15 cm). This mechanical constraint involves that the microphones are arranged into parallel lines of 8 regularly spaced microphones, creating a regular array. The 3D antenna is made of three flat panels: two vertical panels (left and right-side array on Fig. 2 The measurement by hand of the 3D positions of 256 microphones is impractical for the array geometry used in this study. For this reason, the microphones coordinates are assessed by two methods. First we used the blind alignment method developed by Ono et al. [START_REF] Ono | Blind alignment of asynchronously recorded signals for distributed microphone array[END_REF]. Later on the microphone relative positions were verified using the robust method developed by Vanwynsberghe et al. [START_REF] Vanwynsberghe | Geometric calibration of very large microphone arrays in mismatched free field[END_REF]. More exactly, these methods provide the relative positions between the 256 microphones. To obtain the absolute positions of the microphones in the coordinate system associated with the wind-tunnel (Fig. 1), the calibration process should be completed by using a reference noise source with known position. By using the 3D beamforming results associated to a monopolar radiation assumption (the algorithm will be presented in Section 3.1), and applying it to the case of the artificial monopolar noise source at position (0.68 m, 0 m, 0.35 m) with no flow, the estimated position was compared to the actual position of the source. The absolute coordinates of the 256 microphones were then obtained by subtracting to their coordinates the position errors in x, y and z-directions. Some differences are observed in terms of spectra between the reference and the MEMS microphone, especially at high frequencies. It indicates a trend of the MEMS microphones to overestimate the noise level, from around 1 dB at 300 Hz, increasing up to around 5 dB at 3 kHz. The overestimation drops down around 3.5 kHz, and increases again up to 8-10 dB for frequencies higher that 5 kHz. Such differences were already reported in [START_REF] Vanwynsberghe | Design and implementation of a multi-octave-band audio camera for realtime diagnosis[END_REF], based on a sample of 128 microphones. With a high number of microphones, it is not possible to perform an individual amplitude calibration for each microphone. However, it is possible to perform some frequency equalization with the mean frequency response of the MEMS microphones. In this study, such a frequency equalization was carried out, using the frequency variation of the average sensitivity obtained from the test of a set of microphones (model ADMP441) by Vanwynsberghe et al. [START_REF] Vanwynsberghe | Design and implementation of a multi-octave-band audio camera for realtime diagnosis[END_REF]. This will allow the levels in dB ref. 2.10 -5 Pa, that are given in different octave bands, to be compared to each other.

Array processing technique

Three-dimensional beamforming

Processing techniques. The beamforming method used in this study is based on the delay & sum process. Consider an array of M microphones (described by their position x m , with 1 ≤ m ≤ M ) located in the pressure far-field generated by noise sources. A scanning domain, that is a plane (generally parallel to a planar array) for 2D beamforming or a volume for 3D beamforming, is chosen to enclose the potential noise sources. During the beamforming process, it is discretized and represented by a scanning grid (described by the position x f ). For each point x f , the "beamformed signal" in the frequency domain is given by the following equation:

Z(x f , ω) = 1/M M m=1 P (x m , ω)e * (x m , x f , ω), (1)
where ω = 2πf is the angular frequency, P (x m , ω) the Fourier transform of the far-field pressure signals recorded by the m th microphone, the symbol * the complex conjugate operator, and e(x m , x f , ω) the steering function.

This steering function describes the propagation between x m and x f , and is dependent on the assumption that is made on the source radiation: most generally, a monopolar radiation is considered.

Using the vector notation P = (P (x 1 , ω)...P (x M , ω)) T and e = (e(x 1 , x f , ω)...e(x M , x f , ω)) T

with the symbol T indicating the transpose operator, the frequency-domain beamforming function at each scanning point x f is given by:

A(x f , ω) = Z(x f , ω)Z(x f , ω) H = e H Ce e 4 , (2)
where the Cross Spectral Matrix (CSM) of size (M × M ) is defined as C = P P H , H being the Hermitian transpose operator. The CSM is a Hermitian

matrix (C ij (ω) = C * ji (ω)).
The steering vector e enables the microphone array to "steer" to different scanning points x f . Most generally, a monopolar assumption is used, and the elements e of the corresponding steering vector can be written:

e(x m , x f , ω) = exp(-jωr m /c 0 ), (3)
where r m = |x m -x f | and c 0 is the speed of sound. In this case, the term e 4 is equal to M 2 . In the case of a quiescent medium, the time delay τ (x m , x f ) can be simply given by r m /c 0 . When the influence of flow on propagation is significant such as in a wind-tunnel, methods based on ray tracing [START_REF] Amiet | Refraction of sound by a shear layer[END_REF][START_REF] Koop | Investigation of the systematic phase mismatch in microphone-array analysis[END_REF] could be used to calculate the steering vector. The way that the flow effect on propagation is taken into account in this study is presented in Section 3.3.

For 3D beamforming, the monopolar noise source assumption is usually not appropriate, because aeroacoustic noise sources induced by the interaction of a solid body tend to have a radiation that can be approximated by a dipolar radiation [START_REF] Howe | The influence of solid boundaries upon aerodynamic sound[END_REF]. The steering function for a point dipole source in the beamforming technique can be written by introducing a correction term to the monopolar steering vector of Eq. ( 3) according to [START_REF] Porteous | Threedimensional beamforming of dipolar aeroacoustic sources[END_REF]:

e d (x m , x f , ω) = e(x m , x f , ω) ξ • x m -x f r m , (4)
where ξ is the orientation of the dipole source. The advantage of using such a steering vector is that the frequency-domain beamforming function exhibits a maximum at the source location for a dipolar source, so that the sound maps can be readily interpreted. However, Eq. ( 4) shows that an assumption on the dipole orientation ξ is required. In the present case, the wall-mounted airfoil has a symmetric section of NACA 0012 profile, for which the orientation of the dipole is known to be perpendicular to the airfoil surface.

The matrix A calculated for a given frequency is a (N × N ) matrix in 2D domain, or a (N × N × N ) matrix for the 3D domain. N is the grid point number per dimension. The elements of A are real and positive by definition. The so-called sound map is produced by plotting the matrix A for a given frequency with respect to the coordinates of the grid points.

To obtain sound maps with a satisfactory dynamical range, the so-called trimmed version of the CSM is used in this study (the diagonal elements of the CSM being set to zero), in order to remove the contribution of the noise induced by the interaction of the wind with the microphones [START_REF] Allen | Aeroacoustic measurements[END_REF]. Indeed in this study the array is located out of the flow, but close enough to the test section, so that hydrodynamic pressure perturbations could be expected due to flow re-circulation at low velocity.

Deconvolution. The sound map obtained by beamforming is the result of the convolution of the sound source distribution with the array response to a point source (monopole or dipole) with the corresponding radiation model. The sound map calculated using beamforming consists of a main lobe centered on the source position and of side lobes. As a result, the sound source imaging of the beamforming technique can only provide approximately the source location and its distribution for the reason that the size of the lobes are frequency-dependent and the side lobes can be identified as sound sources. In order to improve the spatial resolution and to make the sound maps more readily interpretable, the so-called deconvolution methods, which deconvolve the array response from the beamforming maps, can be associated to the beamforming algorithm. The advantage are twofold: (i) the side lobes are removed from sound maps; (ii) a good resolution of source distribution can be achieved in all frequency bands.

The DAMAS method (Brooks et al. [START_REF] Brooks | A deconvolution approach for the mapping of acoustic sources (damas) determined from phased microphone arrays[END_REF]) is based on Point Spread Functions (discussed in the next section) for which the noise source at each scanning grid is supposed to be a point monopole source with uniform radiation pattern. This technique is computationally expensive because in 3D applications, the number of equations to solve is of the order of N 3 (N being the number of scanning points per dimension). In contrast, the CLEAN-SC method (Sijtsma [START_REF] Sijtsma | Clean based on spatial source coherence[END_REF]) is less demanding as it searches the peaks of the sound map at each iteration, and an adaptation of this technique to 3D applications, using monopolar and dipolar radiation models, is used in this paper.

The method is similar to the one developed by Porteous et al. [START_REF] Porteous | Threedimensional beamforming of dipolar aeroacoustic sources[END_REF].

The CLEAN-SC technique works as follows: by finding the peak level in the sound map of beamforming results (or "dirty map" for the CLEAN-SC method), this method allows to determine the main lobe, its values and its coherent source spots including its side lobes. A portion of their amplitude, determined by a safety factor in this approach, are then removed from the "dirty map". This process is done iteratively with a stop criterion. All the main lobes identified in the process are replaced by so-called clean beams with a predefined beamwidth. A sound map called "clean map" with better spatial resolution is finally obtained. In this paper, the 3D sound maps obtained by using the CLEAN-SC technique contain only clean beams.

The CLEAN-SC method is not efficient for the identification of noise sources that are spatially close, overlapping and forming a single peak level in sound maps. This problem usually arises at low frequencies. In addition, a source area containing coherent noise sources limits the application of the CLEAN-SC method (Bahr and Cattafesta [START_REF] Bahr | Wavenumber-frequency deconvolution of aeroacoustic microphone phased array data of arbitrary coherence[END_REF]). However, this method has been demonstrated to provide a good compromise between computation cost and efficiency [START_REF] Porteous | Threedimensional beamforming of dipolar aeroacoustic sources[END_REF][START_REF] Padois | Two and three-dimensional sound source localization with beamforming and several deconvolution techniques[END_REF].

Numerical simulations

Some simulations were conducted by using synthetic white noise sources (monopole, dipole and distributed dipole noise sources). The actual 3D array geometry used in the wind-tunnel measurements is used in these simulations.

The aim is to carry out a first evaluation of the 3D array performance, and to introduce the corresponding notions of 3D sound maps for dipole noise sources. The resolution is generally defined as the beamwidth of the main lobe of the PSF at a value of 3 dB below the peak level for 2D beamforming, and gives an order of magnitude of the minimum distance between two sources for being discriminated. In 3D cases associated with the 3D array geometry in the study, the isosurface below 3 dB of the peak level is an ellipsoid. In order to estimate its size, an average of the three major principle axis of the ellipsoid was computed for each case from the results of Fig. 4. This value for the 3D sound maps of 1 kHz, 2 kHz and 4 kHz is respectively 176 mm, 88 mm and 43 mm (obtained by a 3D scanning grid with spatial resolution of 2 mm), from which an inverse proportionality between the size of the ellipsoid and the beamforming frequency is expected; the results show indeed that the beamwidth at -3 dB is very close to half-a-wavelength. This result was explained by Cassereau et al. [START_REF] Cassereau | Time-reversal of ultrasonic fields-part iii : Theory of the closed time-reversal cavitv[END_REF] in the ultrasound domain. The major principal axis of the ellipsoid is nearly in the x-direction to which no sub-array is perpendicular, which yields a lower spatial resolution in this direction;

Three-dimensional

practically, this direction is the streamwise direction, as no microphone can be located in the flow. Logically, the results show that the size of the main and side lobes decreases at increasing frequency, allowing a better spatial resolution in the medium frequency range ([2 kHz; 4 kHz]). Results for a point dipole source. Fig. 5 shows the 3D beamforming results for a simulated broadband dipolar source, located at (0.5 m, 0 m, 0.5 m), and oriented along the y-direction using different steering vectors. Fig. 5(a) shows the sound map obtained using a monopolar steering vector at an analysis frequency of 3 kHz. Two dominant lobes are observed near the real source position. Indeed, it yields the existence of a dipole source (Liu et al. [START_REF] Liu | Beamforming correction for dipole measurement using two-dimensional microphone arrays[END_REF]) whereas its exact location is ambiguous: the lobes are in fact two out-of phase monopoles creating the dipolar radiation. Fig. 5(b) shows the sound map obtained using a dipolar steering vector, correctly oriented by setting ξ = (0, 1, 0) (Eq. 4). In spite of two side lobes near the noise source position, the dipole is clearly identified by the main lobe with a maximum coinciding exactly with the simulated dipole location with comparison to Fig. 5(a). The result of searching for a dipole with incorrect orientation ξ =

(1, 0, 0) (perpendicular to the real orientation) is shown in Fig. 5(c). In this case four main lobes appear around the source location. Knowing that incorrect orientation can lead to an incorrect sound map, one disadvantage of beamforming using a dipolar steering vector is that the orientation of dipole should be assessed priorly.

Modeling of the flow effects on sound propagation

The beamforming technique is based on a propagation model from a sound source to a microphone. The most straightforward way of implementing the beamforming technique is to use a model based on the Green function in a quiescent medium. However, in the present application, the sound source is located in a flow while the array is outside the flow, so that the wavefronts undergo convection and refraction effects. If the flow effect is not properly taken into account by using appropriate "flow effect corrections", the beamforming output produces an apparent shift of the source, upstream the actual position of the source [START_REF] Bahr | Shear layer time-delay correction using a non-intrusive acoustic point source[END_REF][START_REF] Padois | Numerical validation of shear flow corrections for beamforming acoustic source localisation in open wind-tunnels[END_REF].

In this study, such effects have been assessed experimentally by using the artificial monopolar noise source presented in Section 2.1. The 3D sound maps were produced by using a monopolar steering vector; the clean beamwidth for the CLEAN-SC technique is set to 0.05 m, and the safety factor is set to 0.99. The position with maximum level of the 3D beamforming sound map was considered to be the estimated position of the monopolar source. A 3D scanning grid with a spatial resolution of 5 mm was set for the 3D beamforming concerning the two test cases in this section.

A sample example is given in Fig. 7, which shows the 3D beamforming results without flow effect correction (Green function with a quiescent medium assumption) concerning the monopolar noise source embedded in the floor of the wind-tunnel at flow speed 40 m/s and position (0.68 m, 0 m, 0 m). First, it is observed that the raw beamforming sound map (Fig. 7(a)

is difficult to interpret due to the presence of ambiguous side lobes, while the use of the CLEAN-SC deconvolution technique allows the unambiguous identification of a source located at z = 0 m (Fig. 7(d)). However, as expected, the estimated position is significantly shifted downstream the actual position (6.5 cm in this case). This apparent spatial shift in the streamwise direction x is noted down ∆x 3D in Fig. 7(d). Let us specify here that the number of iterations using the CLEAN-SC method is 16 in the present case, and that only the corresponding clean beam is represented in the sound map of Fig. 7(d). It means that 16 sources were identified before reaching the stop criterion. It can be observed that the peak level of the "dirty map" in Fig. 7(b), obtained after removing the main lobe identified at the first iteration and its coherent side lobes, is 20 dB lower than that of Fig. 7(a). The Amiet's model [START_REF] Amiet | Refraction of sound by a shear layer[END_REF] is a well-known model for taking into account the refraction and convection effects of flow on sound propagation. It is based on a ray-tracing approach and on the assumption that the shear layers are infinitely thin. In this paper, an implementation of the Amiet's method in three dimensions is used, following the equations derived by Bahr et al. [START_REF] Bahr | Shear layer time-delay correction using a non-intrusive acoustic point source[END_REF].

In the present work, the shear layers are modeled according to three planar and infinitely thin shear layers, as depicted in Fig. 8, and only the angle correction is addressed. To illustrate the interest of the Amiet's method, a noise source is supposed to be at the position (x 0 , y 0 , z 0 ). Three microphones were considered around the shear layers of the wind-tunnel. The ray paths from the noise source to the three microphones were computed at a wind speed of 40 m/s. Fig. 8 shows that the calculated ray paths (intersecting with the shear layers at point I) are not straight from the noise source to the microphones, which involves a modification of the propagation time from the source to the microphone, compared to a rectilinear path. An evaluation of the implemented "flow effect correction" technique is now considered, for an experimental case using the artificial noise source at height z = 0.35 m and for a flow speed of 40 m/s. Fig. 9 shows the 3D sound beamforming sound maps of this test case. Without flow effect correction, Fig. 9(a) shows that the main lobe of the sound map is not centered at the free end of the metal pipe as expected. Similar to Fig. 7(d), the sound map after deconvolution reveals clearly the shift downstream (6 cm in this case) of the noise source with respect to the expected position shown by the red cross (see Fig. 9(b)). Note that the apparent shift ∆x 3D of the noise source due to the flow effect is not the same at different heights in the test section.

According to Fig. 9(c) with flow effect correction, the contribution of the aeroacoustic sources of the cylinder seems to be weak at the beamforming frequency. In addition, the merging of a main lobe and of some side lobes at the top area of the pipe makes it difficult to identify the real noise source. By using the 3D CLEAN-SC code, the noise source is well located at the top of the pipe (Fig. 9(d)). The position error of noise source in 3D sound map is less than 8 mm. It is then concluded that the 3D microphone array provides good performances in terms of noise source position accuracy for isolated noise sources, and that the modeling of the flow effects on propagation is satisfactory, as the source apparent shift (previously observed in Fig. 7(d))

is not present anymore.

Results and discussion

Three-dimensional dipolar beamforming implementation

Acoustic data from the 256 channels were recorded at a sampling frequency of 50 kHz with a sampling time of 5 s for each test case. During the array signal processing, the signal of each channel was filtered by a 256 th order numerical high-pass filter with a cutoff frequency of 20 Hz. The CSM was obtained by using the Welch's method, for which each channel data set was divided into 100 blocks containing 2500 sampling points without overlap and a Hanning window. The resulting frequency resolution was 20 Hz, and the computed CSM for 3D beamforming was then a matrix of dimension (256×256×2500), 256 being the number of microphones and 2500 the number of discrete frequency values. A 3D scanning grid containing 226981 points (61 3 ) in a cube of dimension 0.6 m × 0.6 m × 0.6 m enclosing the wall-mounted airfoil, was used for all the sound maps in this section (Fig.

2(a)

). The grid was then composed of cubes with an edge length of 10 mm.

The safety factor of the CLEAN-SC method was set to 0.99 and the clean beamwidth was chosen as 0.05 m [START_REF] Sijtsma | Clean based on spatial source coherence[END_REF]. The dipolar nature of the trailing edge noise [START_REF] Moreau | Flow-induced noise of a wall-mounted finite airfoil at low-to-moderate reynolds number[END_REF] and leading edge noise sources [START_REF] Gershfeld | Leading edge noise from thick foils in turbulent flows[END_REF] was taken into consideration by making the assumption that the dipole orientation is perpendicular to the airfoil symmetry axis.

Beamforming results are given in third-octave bands in the following.

This was achieved by calculating the beamforming sound maps for each frequency bin contained in a given third-octave band, and then summing the Hz], with respective nominal midband frequencies of 0.8, 2.5, 4 and 5 kHz.

Sound source localization results for a wall-mounted airfoil

As mentioned in Section 2.1, this paper focuses on the sources of broadband noise produced by a wall-mounted airfoil in a flow. Several noise sources, including trailing edge noise and wing tip noise (Brooks et al. [START_REF] Brooks | Airfoil self-noise and prediction[END_REF]), leading edge noise (Geyer et al. [START_REF] Geyer | Application of a beamforming technique to the measurement of airfoil leading edge noise[END_REF]) and junction noise (Moreau et al. [START_REF] Moreau | Wall-mounted finite airfoil-noise production and prediction[END_REF]), are expected to be observed in the 3D sound maps. In all the sound maps presented in this section, the 3D isosurface at 9 dB below the maximum is Geyer et al. [START_REF] Geyer | Measurement of the noise generated by wall-mounted airfoils of different thickness[END_REF]), with an elongated source evenly distributed along the trailing edge. In addition, at higher frequency (4 kHz), the apparent diame-ter of the line source along the trailing edge slightly increases close to the tip (Fig. 10(d)), which means that the intensity of the source is higher at the tip. Moreau et al. [START_REF] Moreau | Wall-mounted finite airfoil-noise production and prediction[END_REF], similarly, reported on a similar tripped NACA 0012 airfoil at angle of attack 4 • (with a similar length/chord ratio L/C = 3, but at a higher flow speed of 60 m/s) the apparition of tip noise around 4 kHz, but also that trailing edge noise almost disappears at this frequency. The case of the airfoil at angle of attack α = 0 • is now considered, in order to investigate more specifically the lowest and higher frequency bands (midband frequency 0.8 and 5 kHz). Fig. 13(a) presents the beamforming result at low frequency, 0.8 kHz. This range of frequency generates some main and side lobes that have very significant extents, so that the airfoil is totally enclosed in the main lobe, making impossible the identification of the source of noise. Fig. 13(b) shows the sound map after applying the deconvolution by using the CLEAN-SC technique, from which a dominant source concentrated in the middle of the junction and a source concentrated at the trailing edge were identified. Similarly, the dominance of junction noise for a wall-mounted airfoil at low frequencies was reported before (Moreau and Doolan [START_REF] Moreau | Tonal noise production from a wall-mounted finite airfoil[END_REF]).

Considering the higher frequency band (5 kHz), Fig. 13(c) displays periodic side lobes centered around the leading edge, similarly to Figs. 10 and 11. A difference is however noticeable, with the apparition of lobes at x = 0.2 m; these lobes are due to spatial aliasing occurring at high frequencies. Interestingly, the deconvolution process removes the aliasing effect and reveals the existence of the source of trailing edge noise in this frequency band.

Sound source localization results for two wall-mounted airfoils

To demonstrate the potentiality of 3D array of microphones, a last configuration with two airfoils is investigated. Two wall-mounted airfoils of chord length 0.2 m but with different span lengths (458 mm and 400 mm) and angles of attack of 0 • are located in an incoming flow with velocity U ∞ = 40 m/s. The leading edge of the shorter airfoil is located 10 cm closer to the wind-tunnel nozzle than the leading edge of the longer airfoil (see Fig. 14).

Each airfoil has the same characteristics (apart from the span length) as the airfoil described in Sec. 2.1, with a similar boundary layer tripping device.

Hence, the generation of broadband noise is expected in the case. This geometry is a real 3D configuration and cannot be treated by a conventional 2D approach. To investigate the sound produced by this configuration, we used a second array of microphones based on the same technology but with more channels [START_REF] Zhou | Design and use of a three-dimensional array of mems microphones for aeroacoustic measurements in wind-tunnels[END_REF]. However, to be consistent with the first experiments, we used only 256 channels among the 1024 available. Fig. 14 shows the array geometry with 16 bars of 16 microphones distributed uniformly (every 72 mm) on each side of the antenna. The methods to avoid it could be the optimization of the array geometry to minimize the side lobes. Note that the distribution of the trailing edge noise sources of the airfoils is not as uniform as that shown in Fig. 10.

This may be due mainly to three reasons: (i) the existence of the objects, and in particular their possible mutual diffraction, was not considered in the array signal processing technique; (ii) the two airfoils are shorter than that in Fig. 10; (iii) strong side lobes may cause CLEAN-SC to attribute noise to the wrong airfoil and subtract coherent noise (i.e., actual sources) from the other one. Nevertheless, the large array system can be associated to more advanced signal processing methods to account for the solid boundaries of objects for sound imaging. The results in this study show that the use of a massive 3D array of MEMS microphones, associated to a 3D array processing adapted to dipolar radiation could locate the noise sources of the wall-mounted airfoil in a 3D domain. This 3D beamforming method can reveal accurately the distribution of the noise sources, and is promising for the investigation of the flow-induced noise of objects in flows with a strong 3D configuration. When the existence of the solid boundaries of objects presents some significant influence on the sound propagation in the flow, the calculation of the corresponding Green function could be associated to the beamforming process in future works [START_REF] Bousabaa | Sparse green's functions estimation using orthogonal matching pursuit: Application to aeroacoustic beamforming[END_REF].

Conclusion

This prove that a tunnel-type array including several hundreds of microphones associated to appropriate 3D array processing techniques performs very well, and that cheap MEMS microphones are good candidates for measuring the sound radiation efficiently. In the future, the 3D signal processing should be further developed in order to take into account sources with non-constant orientation of the dipole, installation effects such as reflection or diffraction, and non-stationary acoustic events such as in Ref. [START_REF] Fischer | Beamforming of aeroacoustic sources in the time domain: an investigation of the intermittency of the noise radiated by a forward-facing step[END_REF]. Additionally, the large array system can be associated to more advanced array signal processing methods like the time-reversal technique for experimental aeroacoustic investigations.

Figure 1 :

 1 Figure 1: Diagram of the test section of the anechoic wind-tunnel, with (from left to right): the nozzle, the wall-mounted airfoil, the collector. The dark band on the airfoil stands for the boundary layer tripping device.

  (a)) and one quasi-horizontal panel (top-side array on Fig. 2(a)). As shown in the picture of Fig. 2(b), the 3D array totally encloses the 3/4 open test section of the wind-tunnel. These panels are placed in such a way that they form a rectangular base cone of size (width × height) about 1.2 m × 1.0 m near the air inlet and 2.0 m × 1.2 m near the outlet. The 32 beams are positioned horizontally in the direction of the flow, and each of them is fixed on one of the three panels (beams 0 to 10 on left-side array, beams 11 to 20 on the top-side and beams 21 to 31 on the right-side array). The gap between two neighbouring beams is 10 cm (except for beams 11-12 and 19-20 for which the gap is 15 cm). In this way, the acoustic field measured by the three panels is mostly meshed with rectangles of dimension 10 cm (streamwise direction) × 15 cm (spanwise direction).

Figure 2 :

 2 Figure 2: (a) Schematic depicting the 3D microphone array and the wall-mounted airfoil. The black points indicate the positions of the microphones of the 3D microphone array. The cube with dashed edges enclosed by the 3D microphone array shows the 3D scanning grid for the beamforming calculation in a volume of dimension 0.6 m × 0.6 m × 0.6 m. (b) Picture of the experimental setup, taken from the collector of the wind-tunnel. The NACA airfoil and the collector appear at the center of the picture, and the three sub-arrays are on the top, left and right sides of the picture.

Fig. 3

 3 Fig. 3 shows the far-field spectra of the wall-mounted airfoil measured by a MEMS microphone and a measurement microphone (type GRAS 40F mounted with a Preamplifier 26AK) with a flat frequency response up to frequency 10 kHz, at the same location. The spectrum of the GRAS microphone is taken as a reference result. The considered case is the noise radiated by the airfoil at angle of attack 10 • and flow speed 40 m/s. As expected, the noise radiated by the airfoil is of broadband nature, with no tonal components. The energy content decreases with frequency, with a sudden drop around frequency 3.5 kHz that could be attributed to a wind-tunnel effect. A similar spectral behaviour was observed by Moreau et al. for a wall-mounted NACA 0012 airfoil at 35 m/s [19].

Figure 3 :

 3 Figure 3: Comparison of the far-field acoustic power spectral density for the wall-mounted finite airfoil at velocity U = 40 m/s for angle of attack α = 10 • using a MEMS microphone (dotted line) and a GRAS microphone (plain line).

  Point Spread Function. The sound map showing the beamforming results for a point monopole source is generally called the point spread function (PSF). Fig. 4 yields the simulated 3D PSF at frequencies 1 kHz, 2 kHz and 4 kHz of a broadband point monopole noise source located at position (0.5 m, 0 m, 0.5 m), using the experimental 3D array geometry of Fig. 2(b). Similarly to 2D beamforming results, a main lobe shown by the isosurfaces of ellipsoid shape centered on the noise source position is observed, together with side lobes evidenced by isosurfaces with lower level.At higher frequencies, the side lobes appear periodically in the y-direction, pointing out the need for an efficient deconvolution technique. The presence of such lobes is due to the the structure of the array, in particular the two parallel vertical arrays facing each other.

Figure 4 :

 4 Figure 4: Sound maps of a synthetic monopole noise source at the position (0.5 m, 0 m, 0.5 m) showing 3D point spread function at frequency (a) 1 kHz, (b) 2 kHz, (c) 4 kHz. The 3D isosurfaces at 3 dB, 6 dB and 12 dB below the peak level are plotted in each sound map from the inside to the outside. The symbols '×' denote the exact location of simulated monopole noise source.

Figure 5 :

 5 Figure5: Sound maps at beamforming frequency f = 3 kHz of a synthetic dipole noise source at the position (0.5 m, 0 m, 0.5 m) , (a) using a monopole steering vector, (b) using a dipole steering vector with ξ = (0, 1, 0), (c) using a dipole steering vector with wrong orientation of the dipole ξ = (1, 0, 0). The symbols '×' denote the exact location of simulated dipole noise source. The 3D isosurface in each sound map is plotted at 3 dB below the maximum.

Figure 6 :

 6 Figure 6: Sound maps for a distributed noise sources of 11 synthetic dipoles at a thirdoctave band of nominal midband frequency of 2.5 kHz. (a): 3D dipolar beamforming; (b): CLEAN-SC technique. The symbols '×' denote the exact location of simulated dipole sources. The shadows and the small black symbols '×' correspond respectively to the projection of the 3D sound source imaging and the real noise source position in different directions. The 3D isosurface in each sound map is plotted at 9 dB below the maximum.

Fig. 7 (

 7 Fig. 7(c) shows that the peak level of the "dirty map" after 2 iterations is 26.3 dB, indicating a much lower background noise level with respect to the synthetic noise source.

Figure 7 :

 7 Figure 7: Sound maps without flow effect correction at frequency 3 kHz of the synthetic noise source at (0.68 m, 0 m, 0 m) generated by a compression chamber at an incoming velocity U ∞ = 40 m/s in the wind-tunnel. (a): 3D monopole beamforming with a peak level at 51.3 dB; (b): 3D "dirty map" after the first iteration with a peak level at 31.3 dB; (c): 3D "dirty map" after the second iteration with a peak level at 26.3 dB; (d): CLEAN-SC technique with maximum value of 51.3 dB on the sound map. The red symbols '×' denote the exact location of synthetic monopole noise source. The black circles indicate the metal pipe position. The shadows and the black symbols '×' correspond respectively to the projection of the 3D sound source imaging and the real noise source position in different directions. The 3D isosurface in the sound maps is plotted at 9 dB below the maximum.

Figure 8 :

 8 Figure 8: Sketch of the sound rays in the wind-tunnel flow. The calculated sound rays from the noise source to the microphones are indicated by solid black lines, while the rectilinear paths are depicted by dashed lines.

Figure 9 :

 9 Figure 9: Sound maps at frequency 3 kHz of the synthetic noise source at (0.68 m, 0 m, 0.35 m) generated by a compression chamber at an incoming velocity U ∞ = 40 m/s in the windtunnel. (a): 3D dipolar beamforming without flow effect correction with maximum value of 52.2 dB on the sound map; (b): CLEAN-SC technique without flow effect correction with maximum value of 52.2 dB on the sound map; (c): 3D dipolar beamforming with flow effect correction with maximum value of 52.2 dB on the sound map; (d): CLEAN-SC technique with flow effect correction with maximum value of 52.2 dB on the sound map. The red symbols '×' denote the exact location of synthetic monopole noise source. The white cylinder represents the pipe connected to the compression chamber. The shadows and the black symbols '×' correspond respectively to the projection of the 3D sound source imaging and the real noise source position in different directions. The 3D isosurface in each sound map is plotted at 9 dB below the maximum value of the sound map.

  3D sound maps. Four third-octave bands are considered in this study, i.e., [707 Hz; 891 Hz], [2245 Hz; 2828 Hz], [3564 Hz; 4490 Hz] and [4490 Hz; 5657

  Figs.10(a) and 10(c) present the results at angle of attack α = 4 • using the 3D beamforming technique, for two third-octave bands with respective midband frequency 2.5 and 4 kHz. At frequency 2.5 kHz, one lobe at the corner of the junction and the leading edge can be observed in Fig.10(a), which may correspond to dipolar leading edge noise generated by the impingement of the wall turbulent boundary layer on the leading edge[START_REF] Gershfeld | Leading edge noise from thick foils in turbulent flows[END_REF][START_REF] Moreau | Wall-mounted finite airfoil-noise production and prediction[END_REF]. At both

Figure 10 :

 10 Figure 10: Sound maps for the wall-mounted finite airfoil at incoming velocity U ∞ = 40 m/s and angle of attack α = 4 • , at two third-octave bands of nominal midband frequency 2.5 kHz (top figures) and 4 kHz (bottom figures). (a) and (c): 3D dipolar beamforming with respectively a maximum value of 35.7 dB and 29.8 dB on the sound map; (b) and (d): CLEAN-SC technique with respectively a maximum value of 33.1 dB and 29.8 dB on the sound map. The 3D isosurface in each sound map is plotted at 9 dB below the maximum value of the sound map.

Fig. 11 presents

 11 Fig. 11 presents the sound maps of the wall-mounted airfoil at angle of attack α = 10 • and flow speed 40 m/s. As shown in Figs. 11(a) and 11(c), the

Figure 11 :

 11 Figure 11: Sound maps for the wall-mounted finite airfoil at incoming velocity U ∞ = 40 m/s and angle of attack α = 10 • , at two third-octave bands of nominal midband frequency 2.5 kHz (top figures) and 4 kHz (bottom figures). (a) and (c): 3D dipolar beamforming with respectively a maximum value of 35.2 dB and 30.2 dB on the sound map; (b) and (d): CLEAN-SC technique with respectively a maximum value of 34.4 dB and 30.2 dB on the sound map. The 3D isosurface in each sound map is plotted at 9 dB below the maximum value of the sound map.

Fig. 12 plotsFigure 12 :

 1212 Fig. 12 plots the sound maps of the wall-mounted finite airfoil at angle of attack α = 20 • . In this case, the beamforming result in Fig. 12(a) still shows

Figure 13 :

 13 Figure 13: Sound maps for the wall-mounted finite airfoil at incoming velocity U ∞ = 40 m/s and angle of attack α = 0 • , at two third-octave bands of nominal midband frequency 0.8 kHz (top figures) and 5 kHz (bottom figures). (a) and (c): 3D dipolar beamforming with respectively a maximum value of 41.9 dB and 35.0 dB on the sound map; (b) and (d): CLEAN-SC technique with respectively a maximum value of 40.8 dB and 35.3 dB on the sound map. The 3D isosurface in each sound map is plotted at 9 dB below the maximum value of the sound map.

Figure 14 :

 14 Figure 14: Schematic depicting the 3D microphone array and the two wall-mounted airfoils under study. The black points indicate the positions of the microphones of the 3D microphone array. The cube with dashed edges enclosed by the 3D microphone array shows the 3D scanning grid for the beamforming calculation in a volume of dimension 0.6 m × 0.6 m × 0.6 m.

Fig. 15

 15 Fig. 15 yields the sound maps of the two wall-mounted finite airfoils at angles of attack α = 0 • and flow speed 40 m/s, in two third-octave bands of midband frequencies 2.5 and 4 kHz. It is found that the overlap of the main lobes and side lobes from the two airfoils makes the beamforming results without deconvolution impossible to interpret (see Fig. 15(a) and 15(c)).

Fig. 15 (

 15 Fig. 15(b) and 15(d) show clearly the trailing edge noise sources of the two airfoils using the CLEAN-SC method. The dominant noise source of

Figure 15 :

 15 Figure 15: Sound maps for the two wall-mounted finite airfoils at incoming velocity U ∞ = 40 m/s and angles of attack α = 0 • , at two third-octave bands of nominal midband frequency 2.5 kHz (top figures) and 4 kHz (bottom figures). (a) and (c): 3D dipolar beamforming with respectively a maximum value of 41.9 dB and 36.2 dB on the sound map; (b) and (d): CLEAN-SC technique with respectively a maximum value of 40.5 dB and 35.5 dB on the sound map. The 3D isosurface in each sound map is plotted at 9 dB below the maximum value of the sound map.

  paper has presented the development of a tunnel-shaped array for imaging aeroacoustic sources in the open section of an anechoic wind-tunnel, together with the associated signal processing techniques, and an application to the case of a wall-mounted airfoil. The antenna is made of three perpendicular planar arrays enclosing the test section, the lower part of the flow being bounded by a rigid planar surface. The 256 digital MEMS microphones of the 3D array are mounted on 32 bars surrounding the test section of the wind-tunnel, uniformly distributed on each bar. The source under consideration is a wall-mounted NACA 0012 airfoil located in the wind-tunnel flow, for several angles of incidence ranging from 0 • to 20 • . Such a configuration results in a complex superimposition of several broadband sources of noise, the main source being the trailing edge noise, but also the tip noise, the leading edge noise and the junction noise. A test case concerning two wall-mounted airfoils with different span lengths in a flow using a similar large array system was also addressed.The data processing is based on a 3D implementation of the beamforming technique associated to a deconvolution method (CLEAN-SC) developed in 3D in order to remove the numerous side lobes that make the interpretation of the sound maps ambiguous. The wall-mounted airfoil being known to generate dipolar sources, the beamforming technique is implemented by using a dipolar radiation model perpendicular to the airfoil. The simulations indicate that the data processing technique allows to identify a linear distribution of non coherent dipoles, similar to the radiation that should be expected from the trailing edge. The flow effect on propagation is taken into account in the beamforming technique by using the Amiet's method in terms of angle correction.In spite of the relatively low metrological qualities of the MEMS microphones used in this experimental study (compared to measurement microphones), in particular in terms of variability of the amplitude and phase response, the performances of the 3D source localization method are excellent, due to the high number of microphones and to the tunnel-type geometry allowing to surround the sources in the flow. For the wall-mounted airfoil, the different sources of noise that are to be expected are accurately identified in the third-octave bands under investigation, and are in good agreement with experimental results published in the literature. To our knowledge, no 3D beamforming identification of aeroacoustic sources with such level of accuracy have been published formerly in the literature. The presented results

Acknowledgements

This work is supported by Agence Nationale de la Recherche (ANR), the CPER FEDER project of Région Nouvelle Aquitaine, the Civil Aviation University of China and the China Scholarship Council. The authors wish to thank Pascal Biais, Dominique Busquet, P. Challande, Janick Laumonier, H. Moingeon, Christian Ollivon, Laurent Philippon, Philippe Szeger and Jean-Christophe Vergez for their technical support to this work.