Charles Goepfert

Jean-Louis Marié
email: jean-louis.marie@ec-lyon.fr

Delphine Chareyron

Michel Lance

Characterization of a system generating a homogeneous isotropic turbulence field by free synthetic jets

teaching and research institutions in France or abroad, or from public or private research centers.

Introduction

The concept of homogeneous and isotropic turbulence has proved to be particularly fruitful in the development of statistical theories of turbulence. Many researchers still use it as a powerful model to understand the basic mechanisms underlying more complex turbulent flows, whose analysis is often complicated by the existence of numerous physical couplings. This is for instance the case for flows in the presence of combustion, or two-phase phenomena. The simple framework offered by homogeneous and isotropic turbulence makes it possible to tackle these situations both experimentally and numerically. While Direct Numerical Simulations (DNS) now provide a very accurate and detailed description of turbulent phenomena, experiments still remain unchallenged in the domain of high Reynolds number flows. This is one of the reasons for this study, whose aim is to develop and test an experimental set-up to produce a quasi-isotropic turbulent field in order to investigate the dispersion of evaporating liquid droplets. A number of methods have been proposed in the literature to produce homogeneous turbulent flows as close as possible to this ideal situation.

Roughly speaking, most of them fall into two categories: those producing spatially decaying turbulence conveyed at a mean velocity, and those generating stationary turbulence in a box or a tank with zero mean velocity.

All the experiments with grids in wind or water tunnels like the well-known works of [START_REF] Comte-Bellot | The use of a contraction to improve the isotropy of gridgenerated turbulence[END_REF] or [START_REF] Uberoi | Effect of grid geometry on turbulence decay[END_REF] belong to the first category. In addition to the longitudinal non-homogeneity associated with the decay of turbulent energy, the turbulence fluctuations and the range of turbulent Reynolds numbers remain generally small at usual lab-scale wind tunnel velocities. The use of active grids instead of passive grids significantly increases the level of turbulence produced, at the cost of a loss in isotropy. The use of an oscillating grid in a tank is the most widely used method of the second category (De [START_REF] Silva | Oscillating grids as a source of nearly isotropic turbulence[END_REF]. Turbulence fluctuations thus generated remain small (of the order of 0.1m/s), but a fairly good degree of isotropy is reached without the need for a long development length. Such systems are used to produce spatially decaying turbulence under sustained oscillations and sometimes temporally decaying turbulence by stopping the forcing.

It is worthwhile noting that high Reynolds numbers can be reached using counter-rotating disks in liquid to produce a Von Karman flow [START_REF] Douady | Direct observation of the intermittency of intense vorticity filaments in turbulence[END_REF], counter-rotating cylinders in a square tank [START_REF] Friedman | Mean rise rate of droplets in isotropic turbulence[END_REF]. The advantage of these devices lies in the higher turbulence levels obtained, but the domain of homogeneity remains somewhat limited. Very recently Variano and Cohen (2008) employed a planar array of randomly actuated synthetic jets disposed on the bottom of a tank. This creates turbulence with a large Reynolds number, an excellent two-dimensional homogeneity and isotropy over a large region, which is well adapted for the study of turbulence at boundaries.

The search for high-efficiency mixing devices adapted to the study of premixed combustion has given rise to several interesting alternative methods. Among them, the systems using fans in the corners of a cubic box, all facing the centre, like in [START_REF] Birouk | Turbulence effects on the vaporization of monocomponent single droplets[END_REF] are worth noting. Velocity fluctuations as high as 1m/s can be obtained, as well as turbulent Reynolds numbers higher than a few hundred. More recently, Hwang and Eaton (2004a) designed a very simple and elegant method to produce high gas velocity fluctuations at reasonably high Reynolds numbers, while insuring spatial homogeneity and making it possible to perform Lagrangian and Eulerian measurements. The principle of such a facility improved the initial idea of [START_REF] Birouk | Turbulence effects on the vaporization of monocomponent single droplets[END_REF] by substituting loudspeakers for the eight variable speed fans initially used by these authors. The objective was to study turbulence attenuation by solid particles in zero gravity conditions, which required a compact, robust apparatus, and where there was no risk of moving mechanical parts breaking down (Hwang and Eaton 2004b). Like in [START_REF] Birouk | Turbulence effects on the vaporization of monocomponent single droplets[END_REF], loudspeakers were fixed in the corners of a cubic chamber. The membranes of the loudspeakers pushed air out of a plenum through an orifice and generated eight synthetic jets producing very strong turbulence at the centre of the chamber. Isotropic stationary or decaying turbulence was thus obtained within a homogeneity domain of 40 × 40 × 40 mm. The zero gravity conditions required the use of a box with solid walls, which introduced some confinement outside the domain where quasi-isotropy was met. In particular, Hwang and Eaton (2004b) noted the existence of a secondary motion in the chamber which generally remained weak, but could under certain conditions be more pronounced [START_REF] Hwang | A new facility for study of attenuation of homogeneous isotropic gas turbulence by solid particles[END_REF], [START_REF] Hwang | Homogeneous and isotropic turbulence modulation by small heavy (St∼50) particles[END_REF]). This motion was responsible for a small vertical mean flow inside the domain, typically equal to 10% of the r.m.s velocities and less than 30% of the r.m.s velocities in the less favourable case. As long as turbulence modulation by solid particles is concerned, like in [START_REF] Hwang | Homogeneous and isotropic turbulence modulation by small heavy (St∼50) particles[END_REF], this effect is not disruptive for the study. This might not be the case for other situations such as the one mentioned before, where Lagrangian tracking of the evaporating droplets is required. Indeed, the main problem in this case arises from droplets that leave the quasi-isotropic domain, might be transported outside, and due any possible recirculation generated by the confinement, be re-injected into the isotropic turbulence during their evaporating process. Moreover, the presence of the walls complicates the introduction of droplets into the domain of interest of the turbulent field and makes the use of optical methods more difficult due to the possible formation of a liquid film.

In this paper we extend the original idea of Hwang and Eaton to a non-confined facility, which is an attempt to improve their initial system for use in normal gravity conditions and in the presence of phase-change.

In the first section of this paper we describe the apparatus and the measuring techniques. Then we present the characterization of the stationary turbulence field generated in the open facility. The last paragraph is devoted to the results obtained in naturally decaying conditions.

Experimental setup and measuring techniques

Experimental facility

The configuration adopted here consists of an open "turbulence chamber" without any solid walls, displayed in Fig1a. The loudspeakers are mounted on a cubic frame (0.9m 3 ) made by the assembly of modular aluminium rods, and pointed towards the centre. Each loudspeaker is glued with a heat conducting silicone mastic onto an aluminium rod, which acts as a cooling device and ensures good thermal stability in time. Two basic geometries can be implemented: one with 8 loudspeakers mounted on the corners of the frame, like in the Hwang and Eaton turbulence chamber, one with 6 loudspeakers mounted on the lateral sides so as to form the vertices of a regular octahedron. It is also possible to work with all 14 loudspeakers arranged thus. With such a system, loudspeakers of different characteristics and sizes can easily be tested and the distance in between varied. The wide optical access resulting from the absence of walls allows velocity and droplet size measurements by using various simultaneous non-invasive techniques such as LDA, PIV, PDA, IPI (Interferometric Particle Imaging) or Numerical Holography. In particular, the development of the jets at the outlet of the loudspeakers and in the direction joining two loudspeakers can be investigated, which is more difficult in a closed chamber. The facility is operated in a large room, at ambient pressure and temperature, which minimizes the effect of confinement and therefore the re-circulations induced by the entrainment of air associated with the jets. Pressure and ambient temperature don't need to be precisely controlled to obtain flow repeatability, as far as the balancing procedure of the loudspeakers described thereafter is used.

The set-up with 6 loudspeakers (Fig1a) has been adopted for the present study, because it is simpler to drive. To obtain as large a homogeneous and isotropic domain as possible, the size of the synthetic jets has been maximized by using loudspeakers with large diameter membranes, closed by flat caps with multi-ejection orifices, a large distance apart (680mm).

The loudspeakers selected for preliminary tests are woofers with coated paper cones manufactured by SEAS (CA21REX: 200 mm in diameter, 80Wrms, free air resonance frequency 31 Hz). They have been then replaced by similar woofers with Carbone membrane manufactured by Davis Acoustics (20 MC8A: 200mm in diameter, free air resonance frequency 37Hz), which prove mechanically more resistant and waterproof. This latter point is crucial when working with droplets and smoke particles for LDA measurements.

Membranes push air out of the cavities formed by the caps through a network of 55 round holes, 6mm in diameter, arranged in a triangular mesh 20mm in size, covering an area 160mm in diameter Fig. 2. The pulsed jets thus generated merge rapidly and create large powerful turbulent jets at the centre of the frame. This holes network on the caps is chosen to provide both quite high outlet velocities and a uniform momentum flux over the major part of the membranes, without introducing too high a pressure drop which could affect their efficiency over time. Each woofer is driven with sine waves with the same 42 Hz frequency and phase, generated by a digital/analogical card (National Instrument PCI-6713) in a Labview environment. Driving with a frequency close to the resonance frequency ensures high membrane displacement, for a lower power supply and thus limited heating. Hwang and Eaton used sine waves with random frequencies and phases to avoid the formation of standing waves or other periodic structures within their chamber. This method did not prove to be necessary here with an open facility. The six signals are amplified by three standard 2-channel audio amplifiers. The loudspeakers never produce jets with exactly the same velocity when they are supplied with the same r.m.s voltage. The balance between the 6 jets at the centre of the domain is therefore optimized by finely adjusting these voltages at the beginning of each experiment. This is checked using the LDA technique described hereafter. The balance procedure starts when loudspeakers have reached a thermal stability that is after about 60 min.

functioning. The balance is achieved for each pair of loudspeakers facing each other, separately, and then for all the pairs together, so as to obtain well symmetrical velocity PDFs, zero mean velocities and equal fluctuations in the X and Y directions. This step can be long and delicate the first time that the loudspeakers are balanced, but the functioning point thus found can be rather easily recovered afterwards. When the loudspeakers are correctly working, the functioning point is generally obtained for r.m.s voltages which are close to each other. Once this is done, the loudspeaker heating being limited, the jets prove to be stable over time typically 60 min. and no further adjustment is thus necessary while running the experiment over such a length of time.

PIV measurements and data processing

Two-dimensional two-component instantaneous turbulent velocity fields are obtained by PIV using a Dantec system including a transmitting arm with light sheet optic and a Flow Map

LDA measurements

The turbulence characterization performed with PIV was confirmed and completed by 1-point 2-component and 2-point LDA spatial explorations. The emitting optic is a Dantec system including a beam expander (expansion ratio: 1.94), mounted on a three-dimensional displacement system. The light source is a 5W argon-ion laser (Coherent, Innova70) emitting beams of diameter 2.2mm, at two wave lengths (514.5 nm and 488.0 nm). The beam separation at the output of the optics and the focal length are respectively 58.3mm and 310mm for the 1-point measurements, and 41.3 mm and 600 mm for the 2-point measurements. The 2-point measurement method is described in detail in [START_REF] Belmabrouk | Taylor length scale measurement by laser Doppler velocimetry[END_REF] and consists in generating two probe volumes, the first one being kept fixed, while the second one is placed at various separation distances thanks to the rotation of a mirror. The integral length scale is deduced from the transverse velocity correlation function thus measured. The flow is seeded with the same tracers and following the same procedure as for PIV. Doppler signals are processed in coincidence by a Dantec system (57N20 and 57N35 BSA). The mean flow velocities and turbulence characteristics are determined from ensemble-averages over 1,000000 samples, processed with the transit time weighting method proposed by George (1978). This correction proves to be necessary in order to remove the velocity bias likely to occur in the domain of interest, where the mean velocities are small and fluctuations large. This method is efficient when the sampling time is much smaller than the eddy turnover time of the turbulence (Eq.8), which is the case here.

Results

Development of synthetic jets

The use of perforated caps to increase the spreading of synthetic jets was inspired from the technique of multiple jets found in practical applications such as, multi-jet burners, liquid propellant rocket engines or agricultural sprays. This technique is not common for synthetic jets and to our knowledge, no extensive study has been reported on the subject. The results available in the literature mainly concern the formation and development of single synthetic jets (Glezer andAmitay 2002, Bera et al. 2001), their use as actuators for flow control, and more rarely the interaction of 2 adjacent jets [START_REF] Smith | The formation and evolution of synthetic jets[END_REF]). In the present experiment, the geometrical configuration of the triangular network of pulsed jets described above is very close to that of the co-flowing jets investigated by [START_REF] Villermaux | Periodically arranged co-flowing jets[END_REF]. A sample of the mean velocity field obtained in our working conditions (driving frequency 42Hz, applied rms voltage 10 volts) in the P1 plane located behind the network is presented in Fig. 3. Symbol ) ( denotes the local ensemble average over all the images. The peak velocity of the jets originating from each hole is equal to 5m/s, corresponding to a

Reynolds number based on d of 2000. These elementary jets expand with distance from the cap to merge with their neighbours and form a single axisymmetric turbulent flow. The merging operates at a distance L/M between 3 and 4, comparable to that reported by [START_REF] Villermaux | Periodically arranged co-flowing jets[END_REF] at this Reynolds number. The centreline velocity m U of the resulting single jet at location L/M = 5 from the holes is 3m/s and the width of the jet where

5 0. U U m =
is about 110mm. This spreading is greater than that which could be obtained with a synthetic jet from a single orifice.

Phase-averaged velocity fields provide the average time evolution of the ring vortices issuing from each hole for various phases of the injection cycle. Figure 4, which corresponds to the beginning of blowing, shows that the released vortices present good axisymmetry and approximately equal strength all over the surface of the perforated cap. However, the direction of emission of the vortices progressively inclines towards the centre of the jet away from the centreline of the network. The velocity fields for the other phases of the injection cycle show that this effect is caused by the surrounding air which is drawn inwards, parallel to the cap, during the second half of the blowing, and towards the holes during suction. This air movement limits the expansion of the jets located on the periphery of the network and is responsible for the flow convergence appearing in Fig. 3. Phase average fields confirm that the mixing of vortices takes place at a distance L/M between 3 and 4, which was checked on the temporal energy spectra by the absence of peaks at the release frequency of vortices and its higher harmonics. Before mixing, the vortices are conveyed in the longitudinal direction with an average speed of 5.4m/s, which corresponds to a Strouhal number based on the diameter of the hole, of the order of 0.05.

Properties of the stationary turbulence

The results presented correspond to a turbulence with Re λ ~ 240, comparable to that documented in detail by Hwang and Eaton (2004). The turbulence levels and the Reynolds number can be adjusted by changing the driving voltage of the subwoofers. Each new setting of the driving voltage requires the six woofers to be re-balanced and the new turbulence conditions to be characterized in detail. In these experiments, the woofers are balanced under the control of LDA measurements, performed at the intersecting point of the six jets ( X = Y = Z = 0 ). Increasing the driving voltage increases the Reynolds number and the intensity of fluctuations. [START_REF] Hwang | A new facility for study of attenuation of homogeneous isotropic gas turbulence by solid particles[END_REF] were able to reach Re λ as high as 450, but noted the poor repeatability of the flow. This fact is confirmed with the present device, which makes it possible to produce highly turbulent Reynolds number fluctuating fields, but the stationarity of the flow proves to be difficult to maintain. In our case, the reason is that working with high voltages causes an important overheating of the loudspeakers and as a result, the balance of the loudspeakers, which is crucial to get a well homogeneous isotropic domain, is not very stable in time. This difficulty could be overcome by installing a more efficient cooling system of the loudspeakers, like it was done in [START_REF] Hwang | Homogeneous and isotropic turbulence modulation by small heavy (St∼50) particles[END_REF].

The mean velocity fields obtained by PIV with the large field of view (plane P2) are displayed in Fig. 5. The small triangular area visible in the right-hand upper corner of the field of view corresponds to the boundary of the laser sheet. The size of the spatial domain where the mean velocity is close to zero within ± 0.2m/s limits is approximately 50 mm in the vertical Y direction and 80mm in the horizontal X direction. The fact that it is smaller along the vertical axis is the consequence of the flow generated by the pair of woofers aligned with this direction (see Fig. 1b). The size along the Y direction is fixed by the depth of mutual penetration of the facing jets and can hardly be extended, even when working with a broad jet cross-section like in this experiment. This phenomenon constitutes one of the main limitations for this type of turbulence production. As a rule, the homogeneity domain is smaller in the three directions joining the subwoofers and subsequently, the domain of interest is not spherical.

A more precise insight into the domain of interest obtained with the small field of view (plane P3) is shown in Fig. 6. It can be checked on the mean flow velocities and the rms fluctuations that the flow globally exhibits good symmetry in the main X and Y directions. Such a result requires a careful balance of the six jets, obtained by adjusting the driving voltages of each subwoofer. The region where the mean velocity is homogeneous and close to zero within ± 0.1 m/s limits, is a roughly rectangular box of 60mm in the X direction by 40 mm in the Y direction. As confirmed on the V component, its size in the Y direction is limited by the 2 incoming flows generated by the vertically placed woofers. Inside this "turbulence box", the rms fluctuations u', v' are fairly homogeneous in space, with values 0.84 and 0.9 m/s respectively.

Maps of the main flow characteristics inside the turbulence box are reported in Fig. 7. The local mean velocities vary between -0.05 and 0.1 m/s for U , and between -0.1 and 0.05 m/s for V . The rms fluctuations divided by their spatial average over the domain, designated by < > , lie in the 0.94-1.04 range for u' and 0.94-1.06 for v', indicating a fairly good degree of homogeneity in space. The isotropy ratio u'/v' has a value comprised between 0.85 and 1.05, which means that the isotropy requirement is also well satisfied. The map of the kinetic energy calculated by assuming isotropy in the Z direction, and normalized by its spatial average in the whole domain, exhibits the same good homogeneity in space. This assumption was validated a posteriori by LDA measurements in the third direction. The spatially average skewness > < u S , > < v S , and flatness

> < u F , > < v F
were also calculated from the fluctuations maps. They were found to be close to the values typical of a Gaussian distribution, that is, 0 and 3 respectively.

The velocity PDFs for U (horizontal) and V (vertical) velocity components were obtained at all locations by PIV and LDA. A sample of these PDFs, normalized by the local rms velocity, is provided in Fig. 8a and compared to those measured by Hwang and Eaton in 2 directions perpendicular to the faces of their box (i-e not-aligned with the speakers). The PDFs presented by these authors (closed symbols) closely follow the same Gaussian profiles (solid line) in these two directions, with skewness ≈ 0 and flatness ≈ 3. Our results exhibit a significant difference in that the distribution in the direction of the speaker (V component) deviates from

Gaussian profiles. This tendency is confirmed by the PDFs obtained at the centre of the domain by PIV and LDA (Fig. 8b). However, the deviation remains weak. We believe that it is linked to the flow behaviour at the stagnation point and can hardly be avoided with this way of generating a THI.

Two component LDA measurements of mean velocity, rms fluctuations, isotropy ratio u'/v', correlation cofficient uv/u'v', skewness and flatness in the vertical direction, which is the most restrictive regarding homogeneity, agree with the results of PIV. They confirm that the domain where the turbulence can be considered as homogeneous isotropic with a high degree of confidence extends in this direction between -20 mm and +20 mm. However, for some parameters, such as the mean flow velocity or the correlation coefficient, ±30 mm limits a domain, which is still acceptable.

The flow characteristics are compared in Table 1 with those of Hwang and Eaton (2004).

Although the two configurations are slightly different (6 free jets for the present study, 8 confined jets for these authors), the characteristics of the two turbulences are very close to each other. The horizontal and the vertical components of the mean velocity are slightly more uniform in the present case, the cause probably being the channel balancing procedure.

Indeed, the driving signals all have the same amplification in Hwang and Eaton, while here they are set to optimize the balance between the jets. Also, the absence of a higher vertical mean flow in our case might be an indication that there is no significant re circulation motion outside the homogeneous domain. The spatially averaged level of turbulence, the skewness

> < u S , > < v S , and flatness > < u F , > < v F
present similar values, which means that these small differences in the mean flow have no effect on the turbulence parameters. The degree of spatial homogeneity is comparable, but within a domain which turns out to be larger in the present work (60 mm × 40 mm).

Fig. 9 provides an example of the one-dimensional turbulent kinetic Eulerian frequency spectrum measured by LDA at the centre of the turbulence domain. For flow without mean velocity, this spectrum cannot be related to the spatial one, because the classical Taylor's frozen turbulence hypothesis linking frequency and wave number domain does not hold.

However, [START_REF] Tennekes | Eulerian and Lagrangian time microscales in isotropic turbulence[END_REF] proposed a model of the Eulerian frequency spectrum in homogeneous isotropic turbulence without mean flow, by using a variant of Taylor's hypothesis. Tennekes' basic argument is that, at high Reynolds numbers, the small dissipative eddies are randomly advected, or swept, past the observer by the large energy -containing eddies. If the Reynolds number is large enough for there to be a great separation in scales between the frequencies of the energy containing eddies and the frequencies of the small eddies thus advected, then an inertial subrange on the frequency spectrum with a -5/3 power law can be expected to exist [START_REF] Tennekes | A first course in turbulence[END_REF]. Although this in theory requires Re λ to be very high, the existence of such an inertial subrange has been experimentally confirmed in different works, both by DNS, as in Yeung and Sawford (2001) for Re λ =140 and experimentally, as still recently in [START_REF] Variano | A random-jet-stirred turbulence tank[END_REF] for Re λ ~ 314. Fig. 9 shows that it is also the case here, and that the -5/3 power law extends over about one decade. This result seems consistent with the value of Re λ that has been estimated in our experiment (Table 2).

On the other hand, no peak at the frequency or the harmonics of the subwoofer is detected.

This proves that the initial periodic structure of the synthetic jets has totally disappeared.

The longitudinal and transverse one-dimensional spatial energy spectra, respectively E uu (k x ), E vv (k y ) and E uu (k y ), E vv (k x ), have been computed from our PIV data. They have been obtained by performing a Fourier transform of each image line and ensemble averaging over all the image pairs, as described in [START_REF] Liu | Evolution and modelling of subgrid scales during rapid straining of turbulence[END_REF], Hwang and Eaton 2004[START_REF] Jong | Dissipation rate estimation from PIV in zero-mean isotropic turbulence[END_REF].

According to [START_REF] Kolmogorov | The local structure of turbulence in an incompressible viscous fluid for very large Reynolds numbers[END_REF], these spectra in the inertial range can be modelled as

( ) ( ) 3 5 3 2 55 18 - = α α αα ε k C ) (k E [1] ) (k E ) (k E α ββ α αα 4 3 = [2]
where C is a universal constant ≈ 1.5 [START_REF] Pope | Turbulent Flows[END_REF]. The spectra, as calculated from our data, are plotted in Fig. 10, together with the models. Since PIV data are discretized over a grid with a finite size, larger than the smallest scales of the turbulence eddies, and cover a limited spatial domain (the domain of homogeneity), the largest wave numbers corresponding to the dissipative scales, as the smallest wave numbers corresponding to the integral scales, are not accurately resolved in the spectra (Hwang andEaton 2004, Variano andCohen 2008). Despite this important bias, we see in Fig. 10 that the collapse between E uu (k x ), and E vv (k y ) is rather satisfactory, confirming the isotropic character of the flow. The relation between the longitudinal and tranverse spectra is rather well verified. Also, the spectra exhibit a -5/3 power-law decay at the smaller wave numbers, suggesting that an inertial subrange is partially observed. The best match between Kolmogorov's model and the data in that range is obtained for a dissipation ε ≈ 3.5 -4m 2 s -2 .

Hwang and Eaton ( 2004) calculated in addition the two dimensional spatial energy spectrum from their PIV data and found that it well agreed with Kolmogorov's spectra at the low wave numbers.

Two-point spatial longitudinal (X direction) and lateral (Y direction) velocity correlations coefficients were measured both by PIV and LDA. They are respectively defined as:

2 2 r,y) u(x u(x,y) / r,y) u(x,y)u(x R (r) F x uu uu + + = = [3] 2 2 r) v(x,y v(x,y) / r) y v(x,y)v(x, R (r) F y vv vv + + = = [4] 2 2 r) u(x,y u(x,y) / r) y u(x,y)u(x, R (r) G y uu uu + + = = [5] 2 2 r,y) v(x v(x,y) / r,y) v(x,y)v(x R (r) G x vv vv + + = = [6]
These correlation coefficients are plotted in Fig. 11. In the case of PIV, each coefficient is an average over all pair of points of the homogeneity domain, separated by r. In the case of LDA, each coefficient is an average of the values measured by positioning the second probe volume at ± r from the centre of the domain. The data obtained by the two methods are in a fairly good agreement, even for separations r where the second LDA probe volume is outside the homogeneity domain (r > 20 mm along the X-axis, r > 30mm along the Y-axis). Both show that isotropy is best verified for G than for F, where a slight difference is noted in the direction aligned with the speakers. Whatever the method, the spatial resolution is not sufficient to calculate the Taylor microscale λ from the curvature at the origin. In particular, the 2-point LDA method suffers from the finite size of the two probing volumes (typically 300 × 50 µm), which makes the measurement inaccurate when the Taylor length scale is smaller than, say, 1 mm. On the other hand, since the two-point correlation curves Fuu, Fvv don't reach zero, the integral length scale f Λ is calculated by fitting a function on these curves using a non-linear least square algorithm. The best fit is obtained with a sum of Gaussians. Integrating the fitting functions yields an integral length scale of the order of 40mm for Fuu, 36mm for Fvv, to be compared with the 56mm value for Hwang and Eaton. A fit with an exponential is also possible, but overestimates the data for large separations r. The relationship between the longitudinal and lateral velocity correlations that are expected from isotropic turbulence theories is written:

r F(r) r F(r) G(r) ∂ ∂ + = 2 [7]
Fig. 11 shows that this relation is well confirmed experimentally, in agreement with the findings.

The main properties of the turbulence are summarized in Table 2 and compared to those of the turbulence box of Hwang and Eaton. Accurate measurement of the viscous dissipation is very difficult, even in isotropic flows, because eddies having a size in the range of the Kolmogorov's scale are spatially filtered by PIV or LDA. So, dissipation and all the properties of the turbulence were first estimated from the turbulent kinetic energy q 2 and the integral length scale Λ f inferred from the data, by using the scaling laws valid in homogeneous isotropic turbulence [START_REF] Batchelor | The theory of homogeneous turbulence[END_REF]):

( )

        Λ = f q C 2 / 3 2 3 ε ε , ( ) ε τ ε 2 2 q ≅ , 2 / 1       ≅ ε ν τ k , 4 / 1 3         ≅ ε ν η [8]
τ ε denotes the eddy turnover time, τ k the Kolmogorov time, η the Kolmogorov length and C ε the normalized dissipation rate, which is of order unity. The Taylor microscale λ and the associated Reynolds number are given by

2 / 1 2 5         ≅ ε ν λ q , ( )         ≅ ν λ λ 2 / 1 2 3 Re q [9]
The universality of the value of C ε at high Reynolds numbers has been extensively discussed in numerous studies [START_REF] Sreenivasan | An update on the energy dissipation rate in isotropic turbulence[END_REF][START_REF] Burattini | On the normalized turbulent energy dissipation rate[END_REF][START_REF] Bos | Spectral imbalance and the normalized dissipation rate of turbulence[END_REF]. For homogeneous isotropic turbulence kept stationary by injecting energy at the large scales, they

show that C ε is close to 0.5. Since our flow resembles this situation this value seems to be realistic here. Taking C ε to be 0.5 yields ε = 6.7 m 2 /s 3 . With that value, we see in Table 2 that the two turbulences have very similar properties and characteristic scales. It is worth noting that Hwang and Eaton determined their characteristics from the turbulent kinetic energy, with the third fluctuation component measured separately by LDA and an estimation of the dissipation calculated from PIV measurements, together with the LES analogy of [START_REF] Sheng | A large eddy PIV method for turbulence dissipation rate estimation[END_REF] to resolve the small scales of turbulence. In order to check the assumption on the dissipative rate C ε , ε has also been estimated from the second-order longitudinal structure function:

( )

) r) v(x,y v(x,y) (r) D vv 2 2 + - =
[10] computed from our PIV data and spatially averaged over all the domain of homogeneity.

Referring to Variano and Cohen (2008) and De Jong et al. (2009), structures functions can be accurately resolved by PIV and do not suffer from the above mentioned spatial filtering which affects the energy spectra. As a result, they appear to be a more robust way for estimating ε than a simple spectrum fitting. Kolmogorov's second similarity hypothesis [START_REF] Kolmogorov | The local structure of turbulence in an incompressible viscous fluid for very large Reynolds numbers[END_REF] implies that, for high Reynolds-number turbulence, in the inertial subrange,

( )3 2 2 r C (r) D vv ε = [11]
where the universal constant C 2 lies in the range 2.0-2.2 [START_REF] Pope | Turbulent Flows[END_REF]. We see that the secondorder longitudinal structure function, plotted in Fig. 12 under the usual compensated form, exhibits a small plateau in agreement with the theory, from which we deduce ε = 5.8 m 2 /s 3 when taking the value C 2 = 2 . This corresponds to C ε of the order of 0.44, which agrees rather well with the above assumption, given the uncertainty on the constant C 2 . The characteristics of the turbulence computed with this second estimation of ε have been listed in Table2.

Decay of a turbulent spot

Stopping the woofers causes the natural decay of the turbulence. The statistics of this decay were measured using PIV. The procedure was as follows. The woofers were run for approximately 60s, the time needed to reach stationary turbulence, and then turned off, until the turbulence totally vanished and ambient air was at rest. This cycle was repeated one hundred times by driving the signal of the woofers automatically. For each cycle, pairs of images with a time separation of 300µs were taken at time intervals of 0.25s, starting from 3s before the woofers were turned off, and ending 12s after. The kinetic energy was then calculated at each of these 60 time steps, from an ensemble average of the 100 velocity fields obtained from the image pairs, assuming isotropy on the third velocity component W.

When the forcing by the loudspeakers is stopped, the turbulent energy contained in the measurement volume decays under the combined action of viscous dissipation and diffusion.

In the absence of a mean velocity field, the balance of kinetic energy density ( )

2 2 2 2 2 1 2 w' v' u' q + + =
, discarding viscous diffusion, is expressed as:

        + ∂ ∂ - - = ∂ ∂ ρ p'u' q u' x ε t q i i i 2 2 [12]
where the RHS term is the sum of the dissipation rate ε and the diffusion by the triple correlation and pressure-velocity correlation, which is significant only at the boundaries of the domain where gradients become important. The latter term acts on the self-diffusion of the turbulent cloud and causes its spatial expansion with time, resulting in a decrease of the local energy density. Therefore, decay for 2 q with time t is expected to be faster than for classical isotropic turbulence such as in [START_REF] Comte-Bellot | The use of a contraction to improve the isotropy of gridgenerated turbulence[END_REF], [START_REF] Uberoi | Effect of grid geometry on turbulence decay[END_REF]. In the forced turbulence experiments presented above, this non-homogeneous edge effect is also present, but is balanced by the production due to the jets. This faster decay has been observed

by Hwang and Eaton (2004a). It confirms that the secondary motion in the chamber is weak and so, quickly dies out after turning out the loudspeakers. However, the decay is not so fast for the case Re λ ~ 415 documented in [START_REF] Hwang | A new facility for study of attenuation of homogeneous isotropic gas turbulence by solid particles[END_REF], as the secondary mean flow is more pronounced. Due to the absence of confinement, the situation here turns out to represent a slightly different situation, i.e. the evolution of an initially isotropic turbulent "spot" in a quiescent fluid. The time evolution of the total turbulent kinetic energy at the centre of the turbulent box is given in Figs. 13 and 14. In Fig. 14, the ratio of the energy density 2 q at time t to the initial turbulent energy density 2 0 q at time t 0 , when the loudspeakers are stopped, is plotted in log-log coordinates versus the time t -t 0 normalized by the eddy turnover time τ ε , while Fig. 13 shows a linear plot of the normalized turbulent energy with time from the beginning of the cycle. The slight discontinuities appearing in data in Fig. 13 for times greater than 7s arise from the fact that the image pair separation time is kept fixed, equal to 300µs. Indeed, the latter, which is well adapted for the beginning of the turbulence decay, becomes too small to achieve particle displacements of a sufficient number of pixels at the end of the decay. As expected, the decay follows a power law n t q -∝ 2 , where the exponent n is about 1.6 ± 0.1 here. This value is slightly smaller than the one (n=1.86) found by Hwang and Eaton (2004a), but since the fitting of the exponent proves to be sensitive to the value of t 0 , the results are quite consistent in the domain of accuracy. We can conclude that the two situations are very similar and that the decay in the chamber is not affected by the confinement effects in that case. Despite the uncertainty on the exponent, it is clear that the decay is faster in the present configuration than in isotropic grid-turbulence, which is commonly observed for diffusive turbulence.

Conclusion

The original idea proposed by Hwang and Eaton (2004) to generate quasi homogeneous isotropic turbulence in a cubic box has been extended to a non-confined facility, in order to investigate the dispersion of evaporating liquid droplets. Turbulence is produced there, by the mixing of six free synthetic jets working in ambient atmosphere, and set out so they form a regular octahedron. However, a configuration with 8 jets as in these authors or 14 (6 + 8) can also be implemented. With this new arrangement, evaporating droplets leaving the turbulence domain will hardly return to it, as it might be the case with the possible secondary motions existing in the box. This is the condition required to study the evaporation process in controlled turbulence.

The turbulent field has been qualified by combining non invasive PIV and LDA techniques for the case Re λ = 240, which is comparable to the situation documented in detail by these authors. The statistical properties of the confined and non-confined turbulent flows in the stationary state have been found to be close to each other, with a slightly larger region of homogeneity, on average, for the present facility. This region proves to be smaller in the direction joining the subwoofers (40mm) than in the direction of the bisectors between the subwoofers (60mm). This shows that the size of the homogeneity domain is determined by the depth of mutual penetration of the facing jets. It constitutes the main limitation of this technique for generating turbulence. Increasing the size seems difficult and our attempt to use large synthetic jets to reach this objective is a posteriori not a very efficient solution. The mean flow velocity, the rms turbulence fluctuations and the integral length scale corresponding to the Reynolds number investigated, are of the order of 0.04 m/s, 0.9 m/s and 40 mm respectively. Measurements in the naturally decaying turbulence created by stopping the subwoofers have also been performed. The decay follows a power law of the form n t q -∝ 2 , with an exponent n = 1.6 which is slightly smaller than the value n = 1.86 found by Hwang and Eaton. This decay is faster than in isotropic grid turbulence, which is the mark of a spatially diffusing turbulence.

  1500 processor. Two coupled YAG laser sources (2 × 120 mJ Quantel lasers) provide pairs of laser pulses at a frequency of about 10 Hz. The laser light sheet is 0.3mm in width. The jets are seeded with micrometer-sized droplets generated by a smoke generator. Seeding is carried out in the surrounding fluid while the woofers are running. Measurements start once the tracers have uniformly diffused all over the working chamber. The light scattered by the tracers illuminated by laser pulses is recorded with a Dantec FlowSense camera (1600 x 1186 pixels, 10 bits). Image pairs are processed according to the adaptive correlation method. The interrogation window is fixed at 16 × 16 pixels with 50% overlap, resulting in instantaneous velocity fields with 159 × 127 vectors. The main planes of view that have been investigated are presented in Fig.1b, with the system of coordinates. The origin ( X = Y = Z = 0 ) is taken at the centre of the octahedron formed by the loudspeakers. The plane labelled P1 (130mm x103 mm) has been chosen to visualize the flow behind the speakers and to characterize the synthetic jet produced. Planes P2 (162.1 × 218.7mm, spatial resolution 2mm × 2mm) and P3 (62.3 × 46.2 mm, spatial resolution 0.6 mm × 0.6 mm) are used to determine the limits of the domain of interest and characterize the turbulence field inside. All these planes are located at Z = 0. The mean flow maps and turbulence characteristics presented in the next section is computed by ensemble averaging over 5,000 instantaneous velocity vector fields. To study the average time evolution of the ring vortices behind the speakers (Plane P1) phase-average velocity fields are obtained by averaging 1000 instantaneous velocity fields, for a given phase in the injection cycle. The whole cycle ("blowing" and "suction") is covered by 16 equally spaced phases. The turbulence domain is also characterized in planes Z = 20 mm and Z = 40 mm and in a plane with an angle of 35°, formed by the X-axis and the line joining the centres of opposite faces of the octahedron.

Acknowledgements

The authors acknowledge the CNRS (Centre National de la Recherche Scientifique) and the French Ministry of Research and Technology (for their financial support. They express their gratitude to M. Massot for his scientific collaboration. They also acknowledge N. Grosjean and E. Jondeau for their efficient technical assistance in the handling of measuring techniques.


trailing-edge noise model dedicated to low-speed fans operating in free field.

The model is intrinsically related to the aerodynamics of the blades and should lead to a useful fast-running tool to be included in a blade design process in an industrial context. The investigations are made on a twobladed low-speed axial fan without shroud, installed inside an anechoic room. The blades are instrumented with two sets of embedded small-size microphones (2.5-mm diameter) and the wall-pressure signals are acquired via a slip ring mounted on the fan axis. The chord-based Reynolds number is about 200000 and the tip Mach number about 0.07. The data base is completed by far-field measurements made with a single microphone on a moving support. The analytical model is based on a previously published extension of Amiet's trailing-edge noise theory. A blade is split into several strips in the spanwise direction and the model is applied to each strip. For this the input data are interpolated from the measurements performed with the aforementioned sets of microphones. The trailing-edge noise model is more reliable for observer positions within ±30 • from the fan rotation plane.

Larger differences with the measurements are found when the observer gets closer to the fan axis.

I. Introduction

Considering a simple fan made of a subsonic open rotor, the aerodynamic noise produced by the rotating blades can be divided into tonal noise at the multiples of the blade passing frequency, due to a possible stationary inflow distortion, and broadband noise associated with random fluctuations in the flow. When the ingested flow is highly disturbed, the broadband noise is mainly generated by the scattering of incident turbulence. This mechanism referred to as turbulence-interaction noise has been for instance investigated experimentally for a helicopter rotor and compared with analytical models by Paterson & Amiet. [START_REF] Paterson | Noise of a Model Helicopter Rotor Due to Ingestion of Turbulence[END_REF] In the absence of upstream disturbances, a rotating blade also radiates self noise due to three mechanisms: the generation of vortices at the blade tip, the vortex-shedding possibly occuring due to blunt trailing-edge thickness and the scattering of the boundary-layer turbulence as sound at the trailing edge. The present paper deals with the third mechanism called trailing-edge noise, which corresponds to the minimum noise level radiated by rotating blades without any installation effect or tip clearance, in low-turbulence inflow conditions.

Several analytical trailing-edge noise models applying to airfoils were developed during the seventies, reviewed by Howe. [START_REF] Howe | A review of the theory of trailing-edge noise[END_REF] Later on, Brooks, Pope & Marcolini 3 proposed the use of a large database for empirical airfoil self-noise prediction. The trailing-edge noise prediction was found accurate for the high Reynolds number flows at low to moderate angles of attack. A new interest on trailing-edge noise emerged with the recent progress in Computational Fluid Dynamics (CFD) combined with the acoustic analogy. The detailed description of the flow around the airfoil trailing edge of an airfoil has provided the strength of the acoustic sources. [START_REF] Wang | Computation of Trailing-Edge Flow and Noise using Large-Eddy Simulation[END_REF][START_REF] Manoha | Trailing-Edge Noise Prediction using Large-Eddy Simulation and Acoustic Analogy[END_REF] Given the source distribution, various numerical or analytical approaches to evaluate the trailing-edge noise have been developed. Casper & Farassat 6 proposed a timedomain formulation considering a flat plate in a non-uniform flow, based on the fluctuating surface pressure distribution. Zhou & Joseph 7 considered airfoils of arbitrary geometries in a uniform mean flow, resorting to a formulation in the frequency domain, and extended the numerical procedure to a rotating blade. [START_REF] Zhou | Frequency-Domain Method for Rotor Self-Noise Prediction[END_REF] Kim & George 9 have also proposed a trailing-edge noise model for open rotors, based on a moving point dipole formula with spanwise loading corrections and assessing the strength of the dipole given by the theory of Amiet. They concluded that trailing-edge noise was important in low inflow turbulence conditions.

The present investigation is similar to the study by Schlinker & Amiet, 10 later followed by Moreau et al. [START_REF] Moreau | Competing Broadband Noise Mechanisms in Low-Speed Axial Fans[END_REF] In both references, only the far-field sound was measured and the wallpressure sources were modeled. The difference is that in the present case, a two-bladed fan is instrumented with wall-pressure sensors and a microphone is placed in the far field to measure the radiated noise. The experimental set-up,which can be used to validate trailing-edge noise models, is described in section II. Dedicated sensors have been manufactured and a special care has been taken for their calibration. The wall-pressure statistics, corresponding to the equivalent acoustic sources in the sense of the acoustic analogy applied in various analytical models, is described in section III. The far-field results are presented in section IV. Finally the analytical model is presented in section V, and the trailing edge noise predictions are compared with the acoustic measurements.

II. Experimental Set-Up

The present study is aimed at validating isolated-airfoil models for the prediction of low-speed fan trailing-edge noise. The statistics of the aerodynamic pressure induced in the trailing-edge region by the turbulent boundary-layer is most often required for such models. On airfoils, it can be deduced either from dedicated experiments, using for instance remote microphone probes, [START_REF] Moreau | Effect of Airfoil Aerodynamic Loading on Trailing-Edge Noise Sources[END_REF][START_REF] Garcia Sagrado | Experimental Investigation Into Trailing Edge Noise Sources[END_REF][START_REF] Rozenberg | Effect of Blade Design at Equal Loading on Broadband Noise[END_REF] or from detailed numerical simulations such as Large-Eddy Simulations (LES). [START_REF] Wang | Computation of Trailing-Edge Flow and Noise using Large-Eddy Simulation[END_REF][START_REF] Manoha | Trailing-Edge Noise Prediction using Large-Eddy Simulation and Acoustic Analogy[END_REF][START_REF] Wang | LES prediction of pressure flcuctuations on a low speed airfoil[END_REF][START_REF] Moreau | Influence of Turbulence Modeling on Airfoil Unsteady Simulations of Broadband Noise Sources[END_REF][START_REF] Moreau | Validation of Unstructured-Mesh LES of the Trailing-Edge Flow and Noise of a Controlled-Diffusion Airfoil[END_REF] Dealing with a rotating fan blade, LES is still too computationally intensive to model the acoustic sources accurately. One has then to resort to specific detailed measurements to characterize the noise sources. The experimental set-up designed to that goal and and the instrumented fan are described in sections A and B below. Finally, the calibration method is presented in section C.

A. Description of the Test Fan

The selected fan is a typical industrial ventilation fan (see Figs. 1 & 3). Its moderately twisted blades are made of plates with a circular-arc mean camber line. The number of blades fixed on the hub can be varied between 1 and 12. To avoid additional blade to blade interactions and thus get closer to the behavior of an isolated-blade, only two blades are mounted as described in the next section on a 260 mm-diameter hub. The blade span of 270 mm leads to a tip radius of 400 mm. For a rotational speed of 600 rpm, the blade tip Mach number is M t = 0.07. The maximum thickness of the blade is 4 mm and the thickness close to the trailing edge is 3 mm (at 95% of the chord length). The chord length slowly varies between 12 cm and 13.5 cm from hub to tip. The stagger angle of the blades is 48 • at the hub and 60 To focus on the self-noise mechanism only, great care has been taken to reduce other possibly contributiong mechanisms. Therefore, the fan is placed in a quiet environment in the middle of the ECL-LMFA (Laboratoire de Mécanique des Fluides et Acoustique de l'Ecole Centrale de Lyon) anechoic chamber far from any additional solid surface close to the blades, as shown in Fig. 1. The motor and the support struts are assumed to have no significant effect. Nevertheless, hot-wire measurements have been carried out in a plane upstream of the fan to check that the incoming flow is not turbulent enough to induce turbulence-interaction noise. Some tuft flow visualizations (see Fig. 20 in Rozenberg et al. [START_REF] Rozenberg | Rotating Blade Self Noise: Experimental Validation of Analytical Models[END_REF] ) have also shown that at the present stagger angle, the tip vortex detaches from the blade tip without interacting with the trailing edge. Furthermore the limited number of blades ensures that the tip vortex from one blade is convected by the mean flow without interaction with the following blade. Finally, trailing-edge noise and vortex-shedding noise remain as the only possible contributors. However, the stagger angle of the blade leads to a relatively high local angle of attack and consequently vortex-shedding noise (also called blunt trailing-edge noise) is unlikely to occur, as observed by Hutcheson & Brooks. [START_REF] Hutcheson | Effects of Angle of Attack and Velocity on Trailing Edge Noise[END_REF] It is then considered that the measured broadband noise in the present experiment is essentially trailing-edge noise.

The anechoic room is designed to ensure nearly-zero reflection above the low-frequency limit of 100 Hz. A half-an-inch Brüel and Kjaer 4181 free-field microphone is mounted on a rotating arm, at a distance R 0 = 1.7 m away from the center of the fan (cf. Fig. 1). This distance leads to a value of kR 0 = 3 for the lowest frequency of interest (100 Hz), compatible with the condition of acoustic far-field at higher frequencies. This condition is implicitly assumed in the models. The microphone sensitivity has been previously calibrated using a Brüel and Kjaer pistonphone at 1 kHz. The set-up allows to investigate the noise directivity within the range -130 • < Θ < 130 • , where Θ is the angle between the axis of rotation and the microphone. The far-field spectra could be reproduced with an accuracy below 1 dB. Reliable and repeatable measurements are achieved for 30 • < |Θ| < 130 • . For a microphone angle less than 30 • , the microphone is in the fan exhaust jet and the acoustic measurements are spoiled by pseudo-sound.

B. Fan Blade Instrumentation

Recent improvements in electronics make the mounting of small-size microphones on moving fan blades possible. Here the microphones selected for the unsteady wall-pressure measurements are FG-3329-P07 manufactured by Knowles Acoustics. They have a diameter of 2.5 mm and a length of 2.5 mm, small enough to be inserted inside the blade parallel to the surface but not enough to be normally flush mounted. The connections also prevent such a mounting perpendicular to the surface.

The microphones and their connection cables are maintained in aluminum tubes along grooves just beneath the surface and communicate with the surrounding air through a 0.6 mm-diameter lateral pinhole which is the actual measuring point. This mounting features a small cavity between the active section of the microphone and the hole, acting as a Helmholtz resonator at high frequencies, as shown on the cross-section view of Fig. 2. The pinholes are small enough to resolve most scales of the turbulent boundary layer as shown in section C. The complete system, including the microphone and the perforated cavity will be called a sensor later on in the paper. A 14-channel slip ring on the fan axis allows transmitting the signals from the rotating sensors to the processing unit. Its effect on the signal is found to be insignificant in the frequency range of interest. All the measurements are collected with a 16-channel HP3565 Paragon acquisition system. The processing is made between 0 and 12800 Hz with a constant bandwidth of 4 Hz and provides the power spectral density (PSD) of the wall-pressure fluctuations in dB/Hz with a reference pressure of 2.10 -5 Pa. The number of averages (N i = 400) has been chosen to provide an experimental uncertainty of 0.2 dB on the wall-pressure spectra. Reliable and repeatable measurements are achieved for all sensors over a range 100 Hz-10 kHz, except for sensors 6, E and F, which were damaged during their insertion inside the blade. So the wall pressure spectra given by these three sensors are not used in the present study.

The sensors are arranged in two networks of 6 for the sake of determining the statistics of the aerodynamic pressure, as shown in Fig. 3. The mid-span network is placed at a distance r 1 = 270 mm from the fan axis on one blade and the tip network at r 2 = 357 mm on the other one.

The sets of sensors (1 to 4) and (A to D) distributed along the span provide the spanwise coherence length, whereas streamwise aligned sensors with appropriate post-processing provide information on the convection speed of the turbulent eddies. Special care was taken to select the distance of the U-shaped networks from the trailing edge. On the one hand, the measurement of the incident field must not be affected by the scattering process occurring at the trailing edge. On the other hand, it must be representative of the turbulence properties past the trailing edge. According to Ffowcs-Williams & Hall 20 and Brooks & Hodgson, 21 the distance must be more than the hydrodynamic wavelength d ≈ λ h = U/f . It can be larger if the wall-pressure fluctuations are statistically homogeneous in the aft part of the blades. Here, the closest sensors are 10 mm away from the trailing edge, which corresponds to a typical hydrodynamic wavelength for a frequency of 570 Hz at a radius R = 270 mm, for a rotating speed of 200 rpm. At the blade tip, it corresponds to 740 Hz. So, it can be considered that the measured incident surface pressure is not contaminated by the scattered pressure field above these frequencies. First, the sensors are calibrated using a pistonphone generating broadband noise. A reference microphone (Brüel & Kjaer 1/4") provides the transfer function for each sensor. To check the ability of the sensor to resolve the turbulent boundary layer scales, a comparison is provided for a zero-pressure-gradient wall-pressure spectrum. The results obtained with the sensor are compared with the spectrum given by a Brüel & Kjaer 1/4" microphone. The attenuation at high frequencies due to the spatial averaging of small hydrodynamic wavelengths is taken into account for the 1/4" microphone, using the correction proposed by Corcos [START_REF] Corcos | Resolution of Pressure in Turbulence[END_REF] with an equivalent radius 23 r eq = 0.62r and a convection velocity U c = 0.7U , U being the flow velocity in the middle of the wind tunnel section and r the actual radius of the microphone. The wall-pressure spectra measured with the sensor and the reference microphone perfectly match between 300 Hz and 3000 Hz. Between 3000 Hz and 6000 Hz a difference of approximately 3 dB is observed. It corresponds to the Helmholtz resonance frequency range of the cavity. At higher frequencies, the agreement is acceptable (discrepancies less than 1 dB). Anyway, the sensor technology leads to specific calibration issues. They have been discussed by Rozenberg et al. [START_REF] Rozenberg | Rotating Blade Self Noise: Experimental Validation of Analytical Models[END_REF] who demonstrated by additional tests that sensors mounted on the blades have a non linear behavior when the fan rotates at 600 rpm. The time signals, not shown here, exhibited a clear saturation. Facing this difficulty, an alternative calibration procedure has been defined as follows. The idea is to decrease the rotational speed down to a value for which the linear behavior of the sensor is obviously recovered becaue the mesaured fluctuations are weaker, and the transfer function is valid. The measured wall-pressure spectrum is then made non-dimensional using mixed variables as proposed by Keith et al., [START_REF] Keith | A Comparison of Turbulent Boundary Layer Wall-Pressure Spectra[END_REF] namely the tangential velocity U , the displacement thickness δ * and the wall shear stress τ w . Finally, it is rescaled to a higher speed for which the measurements are not reliable. The displacement thickness and wall shear stress are deduced from the 1/7 th -power velocity distribution law, which is the most common power law for the turbulent boundary layer without any pressure gradient. The non-dimensional spectrum reads:

Φ pp = Φ pp U δ * τ 2 w . (1)
Plotting the direct measurements of Φ pp as a function of the Strouhal number St based on the displacement thickness and the tangential velocity U = ΩR of the fan blades for different rotational speeds should make the curves collapse, assuming the self-similarity of the flow. This is carried out for 6 rotational speeds between 200 rpm and 594 rpm. The inspection of the time signals confirmed that the saturation occurs clearly for the three highest rotational speeds. As a consequence, the normalized spectra do not collapse for these configurations, as observed in Fig. 5 for sensor 1. For the lower velocities, the discrepancies are reduced and the resonance due to the cavity is attenuated and even avoided at 200 rpm. This is why the data collected at a rotational speed of 200 rpm are best suited for the calibration procedure. Then they are normalized and extrapolated to 600 rpm using formula (1). The results are presented in the following section. It must be noted that the procedure is based on the assumption of self-similar flows at different speeds. This has been confirmed by inspection of the measured sound spectra, which follow a clear similarity scaling law.

III. Measured Wall-Pressure Statistics

The embedded sensors are used to collect the statistical features of the wall-pressure fluctuations. Wall-pressure spectra are first studied here to check the turbulence homogeneity in both instrumented blade areas. Then the convection velocity of the turbulent eddies is deduced from phase measurements between streamwise-aligned sensors. Finally, a length scale is defined from the coherence between spanwise sensors.

A. Wall-Pressure Spectra

The wall-pressure spectra from the mid-span and the blade-tip sensor networks, after calibration according to the procedure described in section II.C, are shown in Figs. 6 and7. The spectra measured by the mid-span network and plotted in Fig. 6 collapse, confirming the expected turbulence homogeneity. They exhibit different slopes in three frequency ranges: a first one (f -1 ) attributed to large eddies, a second one (f -7/3 ) to middle-size eddies and a third one (f -5 ) to smaller eddies. Despite the lack of local aerodynamic data, the present slopes can be compared with those observed around airfoils by previous investigators. The (-1) slope is observed in the overlap region, corresponding to Bradshaw's inactive motion. [START_REF] Bradshaw | Inactive' motion and pressure fluctuations in turbulent boundary layers[END_REF] Its frequency range depends on the Reynolds number. The higher this number is, the wider the frequency domain of the overlap region. In this region, the wall-pressure spectra can be normalized by both the inner and the outer boundary-layer variables. These results found theoretically by Bradshaw [START_REF] Bradshaw | Inactive' motion and pressure fluctuations in turbulent boundary layers[END_REF] have also been observed experimentally by McGrath & Simpson. [START_REF] Mcgrath | Some Features of Surface Pressure Fluctuations in Turbulent Boundary Layers with Zero and Favorable Pressure Gradients[END_REF] The second slope (-7/3) has been deduced by Monin & Yaglom 28 from a theoretical study on the locally isotropic turbulence. Finally, the (-5) slope is considered by Blake as typical of the inner-layer turbulence, [START_REF] Blake | Mechanics of Flow-Induced Sound and Vibration[END_REF] i.e. very close to the wall. It has been experimentally observed by Gravante et al. [START_REF] Gravante | Characterization of the Pressure Fluctuations Under a Fully Developed Turbulent Boundary Layer[END_REF] In contrast, a significant local non-homogeneity of the pressure field near the blade tip is suggested by the wall-pressure spectra measured in that region and prevents them from convincingly collapsing, as shown in Fig. 7. This makes the definition of characteristics slopes from the spectra questionable. The blade-tip spectra are very different from the mid-span spectra both in terms of spectral shape and level. In this configuration, the inertial sublayer characterized by the (f -5 ) slope has almost disappeared. Goody & Simpson 32 in their study of a wing-body junction flow observed wall-pressure spectra with higher levels when the flow is fully three-dimensional, which seems to be the case in the blade tip. Therefore, the nonhomogeneity of the wall-pressure spectra suggests the presence of a three-dimensional flow, possibly due to the induction of the expected detached vortex formed near the blade tip. Nevertheless the scatter in the data does not exceed 4 to 6 dB depending on the frequency, and remains compatible with a use as input data for noise evaluation, provided that it is considered as responsible for a margin of uncertainty.

B. Convection Velocity

Since the lifespan of an eddy is proportional to its scale, [START_REF] Brooks | Trailing Edge Noise Prediction from Measured Surface Pressures[END_REF] only the larger eddies remain coherent between two widely separated points. Moreover, larger eddies are farther from the surface and are convected at a higher velocity. So the convection velocity is an increasing function of the distance ξ between sensors, as pointed out by Brooks & Hodgson. [START_REF] Brooks | Trailing Edge Noise Prediction from Measured Surface Pressures[END_REF] Because the number of embedded sensors is limited, only a single streamwise separation distance (ξ = 5 mm) is available to evaluate this quantity. Corcos 33 defined an average convection velocity U c based on the cross-spectral phase φ ij between the sensors i and j separated by a distance ξ as:

φ ij = ωξ U c (ω, ξ) , (2)
The cross-spectral phase is plotted in Fig. 8 for each configuration. On each set, two separate cross spectra are post-processed giving more confidence in the results summarized in Table 2.

The convection velocity deduced from the tip network is found higher than the tangential velocity (U = 22.4 m/s), highlighting the three-dimensional effect of the vortical structures near the blade tip.

C. Spanwise Correlation Length Scale

The spanwise correlation length is defined as:

l y (ω) = ∞ 0 γ 2 (ω, ξ = 0, η)dη, (3)
where γ 2 is the coherence function, an example of which is plotted in Fig. 9. Between 300 Hz and 3000 Hz, the coherence decreases with both frequency and sensor separation. This agrees with Corcos' model, postulating an exponential decrease with both parameters:

γ 2 (ω, ξ = 0, η) = e - 2ω b c U c η . (4)
Knowing U c and plotting the logarithm of the coherence as a function of the frequency, a linear regression of the experimental data provides the coefficient b c . The values determined this way are compared in Table 2 with the reference value for a turbulent boundary layer over a flat plate obtained by Corcos 33 (b c = 1.4) and the data collected on stationary airfoils at the LMFA-ECL facilities (see Table 3). As the blade profile is basically a plate with a circular mean line, the values obtained at the mid-span for b c and the ratio U c /U are similar to the results obtained on the flat plate and the CD airfoil, which is also characterized by a circular arc mean line. In Fig. 9, a poor agreement is observed between the measured coherence on the mid-span set and Corcos' model. The predictions are in a better agreement with the coherence determined from the blade-tip set. The maximum error made on Corcos' coefficient b c is 30%. Since the coherence length scale defined by the model can be written l y = b c U c /ω, the maximum error on l y is also 30%. This error will be taken into account in the section V.B to evaluate the numerical inaccuracy of the prediction induced by the data uncertainties. At low frequencies, the decrease observed in the different plots is attributed to the finite size of the turbulent eddies. Indeed, the spanwise correlation length is characteristic of the size of turbulent eddies, restricted by the boundary layer thickness.

At high frequencies, the measured coherence is very low and is no longer decreasing with frequency. But the measurement is not reliable below a threshold imposed by data processing constraints. The larger the number of averages is, the lower the threshold is. Due to this limitation, Corcos' assumption of continuously decreasing coherence with frequency is more relevant, once confirmed by the measurements at lower frequencies. So, an adhoc model for the spanwise coherence length is used here. At low frequencies (between 100 Hz and 800 Hz), the length scale is defined by the experimental results using Eq. (3). Between 1200 Hz and 10 kHz, the coherence function is below the confidence limit roughly defined by 1/ √ N i , where N i = 400 is the number of averages. Corcos' model is retained in this range. Finally, between 800 Hz and 1200 Hz, a linear interpolation is defined between the low-frequency model and the high-frequency model. The spanwise coherence length detemined this way is presented in Fig. 10. Flow visualizations using tuft have shown evidence of large structures at the blade tip. [START_REF] Rozenberg | Rotating Blade Self Noise: Experimental Validation of Analytical Models[END_REF] These structures are supposed to have an influence on the blade-tip set of sensors. As a result, the maximum of the coherence length is obtained at a lower frequency in this region.

IV. Far-Field Results

The experimental set-up described in section II and in Fig. 1 allows studying the acoustic far-field in reliable anechoic conditions. In the present section the scaling of the far-field pressure with the rotational speed is addressed first. Then an investigation of the directivity is described.

A. Scaling with Rotational Speed

According to the acoustic analogy, the overall sound intensity radiated by a dipole is proportional to the 6 th power of a characteristic velocity. Hence, a scaling law of the pressure PSD normalized by U 5 as a function of the Strouhal number St = f c/U based on the chord length should be found. This is based on the assumptions of self-similar flows and compact acoustic sources. Recent experimental results on airfoils [START_REF] Moreau | Effect of Airfoil Aerodynamic Loading on Trailing-Edge Noise Sources[END_REF][START_REF] Oerlemans | Aeroacoustic Wind Tunnel Tests of Wind Turbine Airfoils[END_REF] suggest a scaling with U n where the exponent n varies between 4 and 5. Brooks & Hodgson 21 and Schlinker & Amiet 10 obtained n = 5. Deviations from this exponent emphasize the chordwise non-compactness of the airfoil at the measured frequencies. The same trend and values for n have been observed by Stephens & Morris 37 on a ducted rotor for different operating points. For the present fan, the far-field acoustic spectra in the rotational plane were measured for 7 rotational speeds between 400 rpm and 1000 rpm. The normalization by U 5 provides a good collapse as can be seen in Fig. 11. This result provides a first a posteriori justification of the assumptions made for the sensor calibration in section II.C, namely the self-similarity of the flow.

It should be stressed that the trailing-edge noise model 38 assumes relatively high frequencies and holds in the limit of low frequencies down to kc ≈ 1/4, thus typically 100 Hz in the present application. Despite the non-compactness, the 5 th -power law remains acceptable also at higher frequencies. This can be explained by the progressive concentration of the induced lift fluctuations closer to the trailing edge as the frequency increases, as stated for a flat plate by Amiet's model. [START_REF] Amiet | Noise due to turbulent flow past a trailing edge[END_REF] It must also be noted that the low frequencies in the present experiment correspond to a range in which the leading-edge back-scattering correction shown below in section V.A is needed. [START_REF] Roger | Back-scattering correction and further extensions of Amiet's trailing edge noise model. Part 1: Theory[END_REF]

B. Directivity Measurements

The only difference between a first observer at a position (R 0 ; Θ) and a second one at (R 0 ; -Θ) results from the rotation of the blades. But considering an averaged sound pressure level for random sources with homogeneous properties along their path, the noise perceived by both observers must be the same. This symmetry is highlighted in Fig. 12 10 dB over almost the whole frequency range of interest, typicallty between 100 Hz and 10 kHz. The data can be used unambiguously as a basis for validation with no need for any subtraction procedure, and preserving uncertainties below 1 dB. Only some limited tonal noise components coming from the electric motor contaminate the fan noise measurements, but they are not prejudicial to the present study.

V. Trailing-Edge Noise Model for a Rotating Blade

A. Model Background

In 1976, Amiet published an isolated-airfoil trailing-edge noise model 39 based on a theoretical background named Schwarzchild's technique, previously proposed for predicting the broadband noise of an airfoil in an incident turbulent flow. [START_REF] Amiet | Acoustic radiation from an airfoil in a turbulent stream[END_REF] More recently Roger & Moreau [START_REF] Roger | Back-scattering correction and further extensions of Amiet's trailing edge noise model. Part 1: Theory[END_REF] extended the trailing-edge noise formulation to account for the effects of a finite chord length, by including a leading-edge back-scattering correction. Schwarzschild's technique provides the chordwise unsteady lift distribution which acts as the acoustic sources. The far-field noise is then evaluated by applying the classical formula for the radiation of a dipole in a uniform flow, via a statistical analysis. The final result, namely the far-field noise PSD produced by the scattering of the turbulent boundary layer past the trailing edge, is expressed below in the reference frame centered at the trailing edge of the airfoil at mid-span. x 1 is aligned with the inflow velocity, x 3 is perpendicular to both the trailing edge and x 1 , with x 3 = 0 + on the suction side. Finally, x 2 is aligned with the trailing edge to obtain a direct system. The turbulence is supposed frozen in a small region near the trailing edge, so that the streamwise wavenumber of its pressure signature at the wall is equal to K 1 = ω/U c . Assuming a large aspect ratio (L/c), the PSD of the sound pressure is derived as: 39 Essentially it is determined by the wall-pressure statistics upstream of the trailing edge, including the wall-pressure spectrum Φ pp and the corresponding spanwise correlation length l y , and by the aeroacoustic transfer function L derived analytically. L = L 1 + L 2 where L 1 is the main term as defined by Amiet [START_REF] Amiet | Noise due to turbulent flow past a trailing edge[END_REF] and L 2 is the aforementioned back-scattering correction. [START_REF] Roger | Back-scattering correction and further extensions of Amiet's trailing edge noise model. Part 1: Theory[END_REF] The expressions of L 1 and L 2 are reproduced in appendix A.

S pp ( x, ω) = ωcx 3 2πc 0 S 2 0 2 L 2 L ω U c , kx 2 S 0 2 Φ pp (ω)l y (ω) . (5)
Eq. ( 5) is slightly different from the one derived by Amiet. Indeed an asymptotic analysis at low Mach number and at high frequency [START_REF] Moreau | Back-scattering correction and further extensions of Amiet's trailing edge noise model. Part 2: Application[END_REF] suggests that an extra factor 4 is needed to make Amiet's asymptotic solution coincide with Howe's model. [START_REF] Howe | A review of the theory of trailing-edge noise[END_REF] The presence of this factor in eq.( 5) has also been validated by comparison with experimental results on airfoils. A possible explanation based on the Kutta condition has been recently proposed by Moreau Original Amiet's trailing-edge noise model for an isolated airfoil was extended by Schlinker & Amiet 10 to a low solidity helicopter rotor blade, based on a strip-theory approach. The same is made in the present study. Essentially the rotating blade is split into n segments. The airfoil theory is applied to each segment, assimilating the circular motion to its locally tangent translating motion. This is only acceptable for sound frequencies much higher than the rotational frequency. [START_REF] Paterson | Noise of a Model Helicopter Rotor Due to Ingestion of Turbulence[END_REF] More precisely, considering a blade segment according to the sketch of Fig. 13, the single-airfoil formulation (5) is first applied to calculate the radiated sound from one azimuthal location with no account of the relative motion with respect to the observer, but with the actual flow parameters relative to the segment according to the local velocity triangle. It must be noted that formulation (5) has been preferred to the more general one for arbitrary aspect ratio, involving a sine cardinal function. Indeed, it has been recently observed that using the general formulation, the noise predicted for a single airfoil can be slightly different from the cumulated noise as calculated by the sum of the contributions from a given number of sub-parts of the same airfoil. [START_REF] Christophe | Numerical Issues in the Application of an Amiet Model for Spanwise-Varying Incoming Turbulence[END_REF] The relative motion of the blade is taken into account afterwards by adding a Doppler-factor correction expressed by Eq. ( 7) below. The mean velocity is assumed parallel to the chord line according to the weakly-loaded airfoil assumption of the linearized unsteady-aerodynamics theory. The observer can be placed in the (XZ) plane (see Fig. 13) with no loss of generality. R 0 is defined in the fixed reference frame by (X obs = R 0 sin Θ, Y obs = 0, Z obs = R 0 cos Θ). The first step consists in calculating observer's coordinates in the reference frame of the appropriate blade segment. The change of reference frame takes into account the angle of attack, the sweep angle Ψ in the rotational plane and the stagger angle γ s = π/2 -β. The stagger angle is defined with respect to the rotational plane, so that the blade is exactly in that plane when the stagger angle is zero. In the present application, the angle of attack is assumed zero in accordance with the unloaded-airfoil assumption and the blade is purely radial.

& Roger. 34 R 0 U z X Y Z Θ Ψ x 1 x 2 x 3
Observer's position in the moving reference frame is defined by the vector sum x = R 0 -R A where R A = (0; R; 0) denotes the middle of the trailing-edge segment in the (x 1 , x 2 , x 3 ) coordinate system (see Fig. 13). Eq. ( 5) gives the sound radiated by the segment, provided that observer's coordinates are expressed in the rotating reference frame and the frequency is corrected by the Doppler factor due to the relative motion. The instantaneous emitted frequency ω e (Ψ) at the current position Ψ = Ωt is related to the received frequency ω by: 42

ω ω e (Ψ) = 1 + -→ M t . OS 1 - -→ M r . OS , (6)
where M r is the Mach number of the source relative to the fluid, M t is the Mach number of the source relative to the observer and OS is the unit vector from the retarded source position to the observer. The present study applies to low-Mach number fans operating in a medium at rest. In this particular case, U Z = 0 and thus M r = M t , leading to the simplified expression:

ω e (Ψ) ω = 1 + M t sin Ψ sin Θ, (7)
in which M t = ΩR/c 0 is the rotational Mach number. The sound heard by the observer at a frequency ω is emitted at a frequency ω e (Ψ) function of the angular position. The resulting spectrum must be calculated by averaging over all possible angular locations of the blade segment and by weighting with the Doppler ratio to be physically consistent with the necessary conservation of energy. This yields the following far-field noise PSD for a fan with B independent blades, assuming no blade-to-blade correlation for such a low-solidity fan:

1 S pp ( -→ X , ω) = B 2π 2π 0 ω e (Ψ) ω S Ψ pp ( -→ x , ω e )dΨ. (8)
S Ψ pp is the far-field pressure PSD due to a blade segment and is given by the single-airfoil theory (Eq. 5), where the observer coordinates are defined in the (x 1 , x 2 , x 3 ) coordinate system. The integration over Ψ is calculated by a recursive Newton-Cotes rule. In the following section, the model of reference 10 with the back-scattering correction of reference [START_REF] Roger | Back-scattering correction and further extensions of Amiet's trailing edge noise model. Part 1: Theory[END_REF] is assessed by comparing with the experimental results.

B. Validation

Using the wall-pressure statistics measured in the aft part of the blades as input data according to the methodology discussed in section III, the analytical model is applied to the low-speed fan test case. The implementation of the model requires providing the blade geometry, the mean flow speed and the wall-pressure statistics for each blade segment, assuming local homogeneity. In the present application the blade is at least divided in three segments: one close to the hub, one around the mid-span and one close to the blade tip. The wall-pressure statistics for the hub and mid-span segments are both provided by the measurements made at mid-span, extrapolated on the basis of self-similar flows, whereas for the blade-tip segment they are directly provided by the blade-tip data. Then the extrapolation procedure is extended to a splitting into a larger number of segments. Firstly, the influence of the number of segments is reported in Fig. 14. A blade is divided in 3, 6, 9 and 12 segments. The spectrum obtained with 12 segments is chosen as a reference and the differences with this spectrum are plotted only for Θ = 140 • , other angles leading to the same conclusions. At each blade segment, the spectra are transposed to the local rotational speed, displacement thickness and wall shear stress using eq. ( 1). The convection speed is evaluated by the ratio U c /U from Table 2 and the local tangential speed U = ΩR. The coherence length scale l y is deduced from a fitted with Corcos' model (l y (ω) = b c U c /ω). Since the blade geometry is almost constant at different radii, the far field spectra do not depend a lot on the number of segments. For n = 3, the predicted sound level is roughly 0.5 dB lower than for the other three cases. Therefore, the blade will be divided only into 6 segments in the next, because a finer discretization would be time-consuming without being more accurate.

The variation of the acoustic spectrum with the rotational speed is now evaluated, rescaling the normalized wall-pressure spectra at various rotational speeds (cf. Fig. 15). The coherence length scale is obtained by Corcos'model, so that it decreases with increased velocity. For frequencies below 500 Hz, this coherence length scale and consequently the trailing-edge noise prediction could be overestimated. The good agreement between the experimental results and the prediction could then be fortuitous and attributed to another unidentified mechanism. The good agreement at high frequencies show that the scaling laws used for the input data are valid over the whole speed range investigated here. The analytical model is more widely compared with the experimental results on the radiation maps of Fig. 16, which show an overall good agreement. The cuts along the vertical dash-dotted lines produce the spectra at four angular positions plotted in Fig. 17 and the cuts along the horizontal dotted lines lead to the directivity pattern at four frequencies shown in Fig. 18.

In Fig. 17 the uncertainties have also been plotted. The errors are due to the experimental uncertainties and to the procedure to extrapolate the data on the 6 segments. Concerning the wall-pressure spectra, the experimental uncertainty is 0.2 dB as explained in section II.B. It is recalled that the procedure to rescale the wall-pressure spectra is based on a mean normalized wall-pressure from the mid-span set or the blade-tip set depending on the radial position. Based on the Fig. 5, it can be deduced that the error due to the rescaling is bigger in the low frequency range and can be evaluated to ±1 dB maximum. The coherence length scale l y is also a source of error that has been evaluated to 30%. Finally, the accuracy of the model is evaluated to ±3 dB and the dashed lines in Fig. 17

VI. Conclusion

Analytical models aimed at predicting the trailing-edge noise radiated by stationary isolated airfoils have already been validated in the literature. [START_REF] Brooks | Trailing Edge Noise Prediction from Measured Surface Pressures[END_REF][START_REF] Roger | Broadband Self-Noise from Loaded Fan Blades[END_REF] In a previous attempt to extend the validation of one of them to a rotating blade, Schlinker & Amiet 10 concluded that accurate predictions should refer to measured surface pressure data obtained near the trailing edge as input. In the continuation of the same methodology, the present dedicated experimental work resorts to a low-speed fan with instrumented blades to collect the wall-pressure statistics and the far-field acoustic pressure. A technology of embedded sensors has been developed and a particular calibration method has been found necessary to overcome the non-linearity of the sensors. The presently used analytical model is an extension of Schlinker & Amiet's formulae including the back-scattering correction proposed by Roger & Moreau. [START_REF] Roger | Back-scattering correction and further extensions of Amiet's trailing edge noise model. Part 1: Theory[END_REF] The data collected on the blades provide the necessary inputs. An overall agreement is obtained between the far-field measurements and the model predictions, with some overestimate as the observer gets closer to the fan axis. The effect of the number of segments has also been addressed, and in this particular case where the spanwise variation of the blade geometry is negligible (small twist and nearly constant chord), it does not affect the prediction of the far-field acoustic pressure. Finally, the present experiment provides a full database including wall-pressure statistics and far-field pressure to be compared to and/or used to validate later numerical simulation of the noise sources and of the acoustic field.

Appendix A

The extended solution presented by Roger & Moreau [START_REF] Roger | Back-scattering correction and further extensions of Amiet's trailing edge noise model. Part 1: Theory[END_REF] has been implemented for this study. A main trailing edge contribution L 1 and a first-order correction from the leading edge back-scattering L 2 make up the total acoustic transfer function L.

It leads to the following results :

L 1 = - e 2i C i C    (1 + i )e -2i C B B -C E * [2(B -C)] -(1 + i )E * [2B] + 1    (9)

  k = acoustic wave number K 1 = streamwise hydrodynamic wavenumber L = blade strip span l y = spanwise correlation length L = aeroacoustic transfer function M t = tip Mach number N = fan rotational speed r 1 = distance between the fan axis and the mid-span network r 2 = distance between the fan axis and the tip network R = radial distance R 0 = distance between observer position and fan center R A = position vector of the middle of the trailing-edge segment in the moving reference frame R e = Reynolds number S 0 = x 2 1 + β 2 (x 2 2 + x 2 3 ) = corrected distance for convection effects S pp = acoustic pressure power spectral density S Ψ pp = acoustic pressure power spectral density due to one blade segment U = tangential velocity U c = convection velocity (x 1 , x 2 , x 3 ) = moving reference frame x = observer position in the moving reference frame (X, Y, Z) = fixed reference frame X = observer position in the fixed reference frame α g = airfoil angle of attack β = √ 1 -M 2 = compressibility factor δ * = displacement thickness γ 2 = coherence function γ s = stagger angle η = spanwise distance φ ij = cross spectral phase between signals i and j Φ pp = wall-pressure power spectral density Φpp = normalized wall-pressure power spectral density Θ = azimuthal observer angle ρ = fluid density τ w = wall shear stress ω = radian frequency ω e = emission radian frequency Ω = fan angular velocity ξ = streamwise distance

Figure 1 :

 1 Figure 1: Experimental set-up for far-field studies: the slip ring is removed and the sensors embedded in the blades are not connected.

Figure 2 :

 2 Figure 2: Description of the sensor geometry. Cross-section view.

Figure 3 :

 3 Figure 3: Fan geometry and sensor set-up in each blade

10 - 5

 105 Figure 4: Wall-pressure spectrum beneath a zero-pressure-gradient turbulent boundary layer measured by a B&K 1/4" microphone (-) and a Knowles sensor (•). Mean velocity: U = 20 m/s.

Figure 5 :

 5 Figure5: Wall-pressure spectra from the sensor 1 on the rotating blade normalized using eq. (1).

  and McGrath & Simpson 27 and in a direct numerical simulation of a pipe-flow with adverse pressure gradient by Na & Moin.[START_REF] Na | The structure of wall-pressure fluctuations in turbulent boundary layers with adverse pressure gradient and separation[END_REF] Thus, the present mid-span wall-pressure spectra are quite similar to several canonical flows.

Figure 6 :

 6 Figure 6: Wall-pressure PSD transposed to N = 600 rpm from measurements at N = 200 rpm. Mid-span network.

Figure 7 :

 7 Figure 7: Wall-pressure PSD transposed to N = 600 rpm from measurements at N = 200 rpm. Blade-tip network.

Frequency

Figure

  Figure 8: φ ij : cross spectral phase between sensor i and sensor j. Black thick lines correspond to Eq. (2).

Frequency

Figure 9 :Figure 10 :

 910 Figure 9: Coherence function γ 2 . Experimental results (symbols) and Corcos' model (lines).

Figure 11 :

 11 Figure 11: Effect of the rotational speed on the sound radiated in the rotation plane.

  Background noise +40 • Background noise -40 • (a) Θ = ±40 • Background noise +60 • Background noise -60 • (b) Θ = ±60 •

Figure 12 :

 12 Figure 12: Far field acoustic pressure PSD (dB/Hz ref. 2.10 -5 Pa). Results at R = 1.7 m for microphone angle Θ = 40 • (a) and Θ = 60 • (b). Assessment of the symmetry pattern of the fan noise: positive angle (plain) and negative angle (dash). Fan noise (black) and background noise (gray).

  Background noise +120 • Background noise -120 • (d) Θ = ±120 •

Figure 12 :

 12 Figure 12: (Cont'd) Far field acoustic pressure PSD (dB/Hz ref. 2.10 -5 Pa). Results at R = 1.7 m for microphone angle Θ = 90 • (c) and Θ = 120 • (d). Assessment of the symmetry pattern of the fan noise: positive angle (plain) and negative angle (dash). Fan noise (black) and background noise (gray).

Figure 13 :

 13 Figure 13: Coordinate system used in the rotating blade model.

2 SFigure 14 :Figure 15 :

 21415 Figure 14: Influence of the blade segment number (n) on the prediction of acoustic pressure PSD at Θ = 140 • .

  correspond to this accuracy. The prediction overestimates the noise radiated by the fan in the mid-frequency range as the observer position gets closer to the fan axis (Θ = 40 • and Θ = 120 • ).

Figure 16 :

 16 Figure 16: Angle-frequency directivity maps. Left: experiments. Right: analytical prediction. The vertical dash-dotted lines represent the spectra at 4 angular positions plotted in Fig. 17and the horizontal dotted lines represent the directivity at 4 frequencies shown in Fig. 18.

  Figure 16: Angle-frequency directivity maps. Left: experiments. Right: analytical prediction. The vertical dash-dotted lines represent the spectra at 4 angular positions plotted in Fig. 17and the horizontal dotted lines represent the directivity at 4 frequencies shown in Fig. 18.

Figure 17 :

 17 Figure 17: Far field acoustic pressure PSD (dB/Hz ref. 2.10 -5 Pa). Results at R = 1.7 m for four microphone angles Θ = [40 • ; 60 • ; 90 • ; 120 • ]. -Analytical prediction (with numerical accuracy --). 2 Experimental results.

Figure 18 :

 18 Figure 18: Fan directivity plot at 600 rpm for 4 frequencies: f = [300; 1000; 3000; 6000] Hz. Experimental results (symbols) and analytical results (thick lines).

L 2 ≈- 1 / 2 G√

 212 with B = K1 + μ(M + 1) , C = K1μ (x 1 /S 0 -M ) and μ = K1 M/β 2 (10) H e 4i μ (1 -(1 + i )E * [4μ]) ce 2i D + i D + K + μ(Mi )e -4i μ(1 -Θ ′2 ) 2 √ π(α -1) K1 √ B , D = μ (1 -x 1 /S 0 ) , ǫ = 1 + 1 4μ = (1 + ǫ)e i (2μ+D) sin(D -2μ) D -2μ + (1 -ǫ)e i (-2μ+D) sin(D + 2μ) D + 2μ ... + (1 + ǫ)(1i ) 2(D -2μ) e 4i μE * [4μ] -(1 -ǫ)(1 + i ) 2(D + 2μ) e -4i μE[4 μ]... + e 2i D 2 2μ D E * [2D] (1 -ǫ)(1 + i ) D + 2μ -(1 + ǫ)(1i ) 2πt dt, [x + iy] c = x + iǫy

Table 1 :

 1 • at the tip. More geometrical data are given in Table1. Geometrical data of the blade.

	Radius r (m)	0.155 0.195 0.235 0.275 0.315 0.355 0.395
	Chord length (m) 0.12 0.12 0.125 0.13 0.13 0.135 0.135
	Stagger angle ( • )	47.8 49.7 51.9 53.8 55.6 57.7	59

Table 2 :

 2 Convection velocity and Corcos' model constant in the present experiment.

		U c (m/s) U c /U	b c
	mid-span set	12.7	0.75 1.56
	blade tip set	24	1.07	0.9
		U 0	U c	U c /U 0 b c
	Flat plate 34 (α g = 5 • ) LMFA small wind-tunnel	40 m/s 32 m/s		0.8	2
	CD airfoil 12 (α g = 15 • ) LMFA large wind-tunnel	16 m/s 12 m/s		0.75 1.2
	CD airfoil 12 (α g = 8 • ) LMFA large wind-tunnel	16 m/s 11.2 m/s	0.7	1.5
	CD airfoil 35 (α g = 13 • ) LMFA small wind-tunnel	16 m/s 9.6 m/s		0.6	1.5
	V2 airfoil 14 (α g = 20 • ) LMFA small wind-tunnel	16 m/s 10.4 m/s 0.65 1.4

8: φ ij : cross spectral phase between sensor i and sensor j. Black thick lines correspond to Eq. (

2

).

Table 3 :

 3 Convection velocity and Corcos' model constant from typical airfoil experiments.[START_REF] Moreau | Effect of Airfoil Aerodynamic Loading on Trailing-Edge Noise Sources[END_REF][START_REF] Rozenberg | Effect of Blade Design at Equal Loading on Broadband Noise[END_REF][START_REF] Moreau | Back-scattering correction and further extensions of Amiet's trailing edge noise model. Part 2: Application[END_REF][START_REF] Roger | Broadband Self-Noise from Loaded Fan Blades[END_REF]

Acknowledgements

The authors would like to thank Alain Guédel for his help during the experimental campaign. The present work was achieved as a collaboration between Valeo, CETIAT, SNECMA and Ecole Centrale de Lyon. The study was supported by the CIRT (Consortium Industrie-Recherche en Turbomachine), together with the French Ministry of Research.