Axelle Tortosa
email: axelle.tortosa@inrae.fr

Brice Giffard

Luc L Barbaro

Jérémy Froidevaux

Sylvie Ladet

Jeanne Delhommel

Aude Vialatte

Nill Dietmar

© L Arthur Serotine

Arthur Kuhl's Pipistrelle

Gilles San

Martin Kuhl's Pipistrelle

Arthur 12% Lesser

Crop diversity in the landscape favors bats and biological control of some pests

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

Results

What are the implications for biological control of lepidoteran pests ?

•

  of these three crops at the landscape scale favorable to bats and does it favor biological control of pests ? 9 Higher species richness and activity VS SFE²-GFÖ-EEF 23 Novembre 2022 / Tortosa Axelle VS Is the mixture of these three crops at the landscape scale favorable to bats and does it favor biological control of pests ? ajoutée traitées par le CNES pour le pôle de données Theia www.theia.land.fr à partir de données Copernicus. Les traitements utilisent les algorithmes développés par les Centres d'Expertise Scientifique de Theia. of the three crops favorable to bats ? SFE²-GFÖ-EEF 23 Novembre 2022 / Tortosa Axelle 22 Results What are the implications for biological control of lepidoteran pests ? moths seems to attract bats, the associated predator-prey dynamics do not allow for biological control Bats were efficient biological pest control agent for the pine processionary moth (Charbonnier et al., 2014, 2021; Maine & Boyles, 2015)

Resources continuity within agricultural landscapes Lepidopteran pests are valuable food resources for bats (Mata et al, 2021;Tournayre et al., 2020) Bats may be considered as bioindicators through their activity (Alleva et al., 2006;Jones et al., 2009;Stahlschmidt & Brühl, 2012;Russo et al., 2021) Is the mixture of these three crops at the landscape scale favorable to bats and does it favor biological control of pests ?


Pietro Li
email: pietro.livolsi@parrot.com

Gianluigi Brogna
email: gianluigi.brogna@parrot.com

David Gomez-Ariza
email: david.gomez-ariza@sensefly.com

Romain Gojon
email: romain.gojon@isae-supaero.fr

Thierry Jardin
email: thierry.jardin@isae.fr

Hélène Parisot-Dupuis
email: helene.parisot-dupuis@isae-supaero.fr

Jean-Marc Moschetta
email: jean-marc.moschetta@isae-supaero.fr

Aerodynamic and Aeroacoustic Study of Low Reynolds Number Rotors: Influence of Pitch Angle, Airfoil Camber and Thickness

come

With the increase in the use of micro air vehicles (MAVs), the issue of noise pollution is becoming critical in both military and civilian fields. The main noise component in quadcopter MAVs is generated by the rotors and their interactions with the whole body. Here, a Non-Linear Vortex Lattice/Particle Method (NVLM), coupled with the Farassat formulation-1A of the Ffowcs-Williams and Hawkings (FW-H) acoustic analogy, is used to characterize the influence of the blade pitch and the airfoil shape on the tonal noise and the aerodynamic performance of low Reynolds number rotors typical of MAVs. The validation of both NVLM and FW-H codes is adressed with test cases from the literature and experiments on a 25cm diameter NACA0012 constant-pitch rotor in hovering conditions. The tonal noise of this MAV rotor is then studied in details. Finally, a parametric study of rotor pitch angle and of airfoil camber and thickness is performed, providing guidelines for the design of small rotors with low acoustic footprint and optimal aerodynamic performance.

I. Nomenclature c

= chord [m] c = speed of sound [m/s] (in Eq. 6-7)

C D = drag coefficient [-] C L = lift coefficient [-] C L /C D = lift-to-drag ratio [-] C Q = torque coefficient [-] C t = thrust coefficient [-] ∆ = Laplace operator dS = surface element [m 2 ] f = frequency [Hz] f + = dimensionless frequency, f + = f / f BPF [-] FM = figure-of-merit [-] Γ = vortex strength [m 3 /s] l r
= projection of the loading vector (l) onto the observer vector (r)

II. Introduction

D the last decade, the micro air vehicle (MAV) market has experienced a significant growth. Nowadays, MAVs are commonly used in both civil and military sectors. In the former they are a very good asset, for example in the cinema and photographic industry (see Fig. 1). In the military sector, their use is mostly for reconnaissance purposes. In any case, MAVs are increasingly used because of their low cost, low set-up time and the possibility to fly them in closed and difficult environments. For these reasons, they are intended to replace helicopters in many more applications. The quadcopter configuration of MAVs brings many advantages, but also has its own drawbacks. In fact, the rotor and body compactness leads to new aerodynamic and aeroacoustic challenges related to:

• the low Reynolds number flow that the rotors face;

• the highly coupled aerodynamic and aeroacoustic interactions between the rotors [START_REF] Lee | Rotor interactional effects on aerodynamic and noise characteristics of a small multirotor unmanned aerial vehicle[END_REF][START_REF] Alvarez | Rotor-on-Rotor Aeroacoustic Interactions of Multirotor in Hover[END_REF], and the rotors and the body [START_REF] Zawodny | Investigation of rotor-airframe interaction noise associated with small-scale rotary-wing unmanned aircraft systems[END_REF]; • the high sensitivity to wind gusts [START_REF] Ariza | Study of the sensitivity to the lateral wind of a Mini Unmanned Aerial Vehicle with VTOL flight capabilities[END_REF].

In the last few years, the research community has been focusing notably on the understanding of low Reynolds number effects on the aerodynamic performance of MAV rotors. The rotors compactness induces a chord-based Reynolds number ranging from Re c = 10 4 to 10 5 . In this flow regime, the blades suffer from increased viscous drag, hence reduced aerodynamic efficiency, and can experience separation due to increased viscous forces with respect to inertial forces in the boundary layer. In addition, MAVs rely on batteries with low energy storage. Thus, the aeroacoustic optimization towards acoustically stealthier MAV rotors, which operate at high rotational speeds, should be tightly coupled with aerodynamic optimization [START_REF] Serré | Analysis of the flow produced by a low-Reynolds rotor optimized for low noise applications[END_REF][START_REF] Serré | Towards silent micro-air vehicles: optimization of a low Reynolds number rotor in hover[END_REF][START_REF] Nana | Fast Multidisciplinary Optimization of a MAV propeller for noise reduction: from simulation to experimentation[END_REF][START_REF] Wisniewski | Designing small propellers for optimum efficiency and low noise footprint[END_REF][START_REF] Boyer | Multidisciplinary optimization of a MAV propeller for noise reduction[END_REF][START_REF] Wilke | Findings in Aero-Acoustic Simulations for Optimizations[END_REF][START_REF] Li-Volsi | Design of aeroacoustically stealth MAV rotors[END_REF][START_REF] Li-Volsi | Aeroacoustic optimization of MAV rotors[END_REF].

The objective of this paper is to analyze the effects of the airfoil camber and thickness and of the blade pitch angle on the aerodynamic and aeroacoustic footprints of a 25cm diameter rotor using a non-linear vortex lattice/particle method (NVLM) coupled with Ffowcs-Williams and Hawkings (FW-H) acoustic analogy. Before comparing different rotors, the NVLM code and FW-H coupling are validated upon data from the literature and experimental data obtained in the ISAE-SUPAERO anechoic chamber on a NACA0012 constant pitch rotor. The tonal noise emitted by this rotor is studied.

Since the actual acoustic code only computes the tonal component of the acoustic spectrum, the validation is done by comparing the 1st and the 2nd blade passing frequency (BPF) sound pressure level (SPL) at different microphone locations (shown in Fig. 2). The tests were performed in the ISAE-SUPAERO anechoic room using a directivity antenna of 13 microphones (see Fig. 2). Details about the experimental test campaign and the corresponding open database are presented in [START_REF] Gojon | Experimental investigation of low Reynolds number rotor noise[END_REF][START_REF] Gojon | JASA_2021_GOJON_JARDIN_PARISOT-DUPUIS[END_REF].

III. Methodology

A. Aerodynamic simulations

Vortex Lattice Method

The vortex lattice method (VLM) relies on potential flow theory, with the following assumptions:

• the flow is:

-incompressible - → ∇ • V = 0 ; -inviscid - → ν = 0 ; -irrotational - → ∇ ∧ V = 0 ;
• the lifting surfaces are considered thin (with respect to the chord).

These assumptions lead to a conservative vector field, that is, the velocity vector is equal to the gradient of a potential flow:

V = ∇(φ) (1)
which satisfies Laplace's equation (∆(φ) = 0). Since Laplace's operator ∆ is linear, different potential flows can be linearly added to create more complicated flows. These elementary potential flows can be freestream velocities, point sources and sinks, doublets and vortex filaments/panels. In the case of the VLM, the final flow field is a combination of elementary vortex rings (or panels with constant strength Γ). Since the rotor blade is discretized into N i × N j lattices, N i and N j being the number of lattices along the chord and span, respectively, the circulations of the vortex rings to be calculated are noted Γ i, j [START_REF] Katz | Low-Speed Aerodynamics[END_REF]. The application of the Kutta condition, alongside with the non-penetration condition at the blade surface (Eq.2) allows for the calculation of the circulations Γ i, j .

V in f low • n = (V induced(blade) + V induced(wake) + V ∞ -Ω × r) • n = 0 ( 2
)
where n is the normal to the surface of the panel, Ω is the rotor rotational speed, r is the radial position of the panel and V ∞ is the flight speed.

Once the influence matrix is created and both circulation and the inflow velocity V in f low computed, the effective angle-of-attack (α e f f , j ) of each spanwise section is calculated through the following formula:

α e f f , j = θ pitch, j + θ twist, j + α induced, j (3)
Non-linearity is then introduced, as detailed in subsection III.A.2.

Non-Linear VLM

As described in subsection (III.A.1), the vortex lattice method derives from the potential flow theory and is, by definition, a linear method. Its application to wings and rotors under low Reynolds number conditions, where viscous effects may induce non-linear phenomena and strongly impact aerodynamic and aeroacoustic performance, is hence not fully appropriate. For this reason, Jo et al. [START_REF] Jo | Prediction of rotor flow for unmanned aerial system using nonlinear vortex lattice method[END_REF] developed a VLM code that takes into account part of the low-Reynolds-number-induced non-linearities through look-up tables (Fig. 3). The look-up table procedure is divided into five steps:

1) the sectional bound circulation strength is calculated from the linear VLM, as well as the contribution of each chordwise lattice (chordwise circulation ratio); 2) induced velocity V induced is computed at a reference point at each section; 3) the lift coefficient corresponding to the V induced and α e f f for each section are calculated through the look-up table; 4) the circulation value for each surface lattice is updated according to the lift coefficient; 5) if the convergence criterion is not satisfied, another iteration is necessary.

When the non-linear loop ends, the chordwise circulation strengths are calculated by taking the initial chordwise circulation ratii as reference and scaling them with the new sectional bound circulation strength. Once the non-linear correction is achieved, the forces are calculated and wake vortices are shed (see subsection III.A.3), a new time step is started.

Vortex Particle Method

One of the main advantages of NVLM codes is the implementation of meshless calculations to simulate the wake. As explained in [START_REF] Jo | Prediction of rotor flow for unmanned aerial system using nonlinear vortex lattice method[END_REF], [START_REF] Jo | Prediction of Noise from Low Reynolds Number Rotors with Different Number of Blades using a Non-Linear Vortex Lattice Method[END_REF] and [START_REF] Lee | Rotor interactional effects on aerodynamic and noise characteristics of a small multirotor unmanned aerial vehicle[END_REF], the code models the wake through a vortex particle method (VPM). This method consists in shedding vortex particles (or blobs, each with a circulation strength and a core radius) from the blade trailing edge into the wake. For convergence and stability purposes, a free wake model is also implemented and the vortex particles are shed after the third row of the free-wake lattices (red wake lattices of Fig. 4). These particles are convected into the wake and they influence each other. That is, the VPM computational time depends on the number of particles N p in the order of O(N 2 p ). To increase the efficiency of the code, Jo et al. [START_REF] Jo | Prediction of Noise from Low Reynolds Number Rotors with Different Number of Blades using a Non-Linear Vortex Lattice Method[END_REF] implemented a Fast Multipole Method which reduces the computational time cost to O(N p log N p ).

The vorticity field induced by the wake (composed of N p particles) at a location x in the flowfield is calculated as follows:

ω(x, t) = ∇ × V = N p i=1 Γ i (t)ζ σ i (x -x i (t)) (4)
With:

• x the location in the flowfield and x i the location of the i th particle;

• t the physical time;

• ζ = ζ(x/σ)/σ 3 is the regularized vorticity distribution function within the core of the particle (σ is the core size).

B. Far-field acoustic calculations

The experimental frequency spectrum [START_REF] Gojon | Experimental investigation of low Reynolds number rotor noise[END_REF][START_REF] Gojon | JASA_2021_GOJON_JARDIN_PARISOT-DUPUIS[END_REF] of a rotor noise measured at a far-field observer position of Fig. 5 shows discrete peaks at different frequencies (referred to as tonal noise) and a broadband noise [20][21]. The non-highlighted tonal peaks at the rotational speed (RPM) frequency and its odd harmonics are probably due to the non perfectly balanced rotor blades and will not be examined in this paper. The broadband noise is composed of different unsteady phenomena involving turbulence: the blade wake interaction (BWI), the trailing edge noise, the boundary layer separation noise and the vortex shedding noise. In real conditions (external to the anechoic chamber controlled environment), the turbulence ingestion noise is a very important part of the broadband noise. In this paper, the attention is focused on the tonal noise. The rotors tonal noise is generated by different sources:

• the thickness noise produced by the displacement of the fluid due to the passage of the rotor blades; • the loading noise characterized by the (steady and unsteady) pressure distribution along the chord and the span of the blades; • the blade vortex interaction (BVI) produced by the impact of the blade tip vortex on the next blade; • the high-speed impulsive noise present when the tip of the blade is transonic. Broadband noise Tonal noise Fig. 5 Experimental SPL spectrum of a NACA0012 rotor at 6000RPM taken from a microphone located 1.62m away from the rotor center at 60°angle [START_REF] Gojon | Experimental investigation of low Reynolds number rotor noise[END_REF][START_REF] Gojon | JASA_2021_GOJON_JARDIN_PARISOT-DUPUIS[END_REF]. The tonal peaks are highlighted by the red circles, while the high-frequency broadband noise is emphasized by the green rectangle.

To capture the far-field aeroacoustic noise emitted by the rotors and their directivity, the Farassat formulation-1A [START_REF] Farassat | The Prediction of Helicopter Discrete Frequency Noise[END_REF] of the FW-H acoustic analogy has been implemented. This formulation consists in calculating the near-and far-field noise generated by three different sources: the monopole, the dipole and the quadrupole. Because of the nature of the NVLM code used for this study, the tonal noise calculated is composed of monopole sources, previously called thickness noise, and dipole sources, or loading noise (the quadrupole noise is not calculated).Since lattices do not radiate to the observer at the same time, the 'retarded time' (τ) must be calculated and the acoustic pressure evaluated at this retarded-instant. The acoustic pressures of each lattice are added to calculate the whole rotor acoustic pressure. The rotor total tonal noise is therefore the contribution of both loading and thickness noise:

p (x, t) = p T (x, t) + p L (x, t) (5)
• The thickness noise is affected by the blade geometry (through the n normal vector) and the rotational speed (through the velocity (v) and Mach (M) vectors);

4πp T (x, t) = ∫ f =0 ρ 0 ( v n + v n ) r (1 -M r ) 2 ret dS + ∫ f =0 ρ 0 v n (r M r + cM r -cM 2 ) r 2 (1 -M r ) 3 ret dS (6)
• The loading noise depends on both steady and unsteady pressure distribution on the blade through the terms l and its derivative l;

4πp L (x, t) = 1 c ∫ f =0 l r r (1 -M r ) 2 r et dS + ∫ f =0 l r -l M r 2 (1 -M r ) 2 ret dS + 1 c ∫ f =0 l r (r M r + c(M r -M 2 )) r 2 (1 -M r ) 3 ret dS (7) τ = t - r c = t - |x -y| c (8)
With:

l r = l i ri l r = l i ri l M = l i Mi
In the following subsection, the aerodynamic and tonal noise calculations are assessed upon analytical solutions and real rotor tests taken from the literature.

C. Validation

Aerodynamic validation

The aerodynamic computational performance of the NVLM code was first validated [START_REF] Jo | Prediction of Noise from Low Reynolds Number Rotors with Different Number of Blades using a Non-Linear Vortex Lattice Method[END_REF] upon experimental data obtained on a commercial rotor by Zawodny and Boyd [START_REF] Zawodny | Investigation of rotor-airframe interaction noise associated with small-scale rotary-wing unmanned aircraft systems[END_REF]. Here, further comparison with homemade rotors is provided. The two-bladed NACA0012 rotor illustrated in Fig. 6 is used and operated at a rotational speed of 6000RPM. The corresponding rotor and simulation parameters are summarized in table 1. ∼0.00014

Average interval [revs] 20 -30 Table 1 Simulation parameters used for the validation of the NVLM code.

The thrust and torque obtained by NVLM computations for this configuration are 2.88N and 4.72 • 10 -2 N.m, which differ by 1.9% and 1.5% from the values reported by Jo et al. [START_REF] Jo | Prediction of Noise from Low Reynolds Number Rotors with Different Number of Blades using a Non-Linear Vortex Lattice Method[END_REF] using a similar approach, respectively.

Results are further compared with experimental data obtained in ISAE-SUPAERO anechoic chamber, reported by Gojon et al. [START_REF] Gojon | Experimental investigation of low Reynolds number rotor noise[END_REF] and available on [START_REF] Gojon | JASA_2021_GOJON_JARDIN_PARISOT-DUPUIS[END_REF], in terms of thrust and torque coefficients c T = T/(ρ f 2 D 4 ) and c Q = Q/(ρ f 2 D 5 ), respectively. The results in Table 2 show differences of 7.1% and 8.1% in thrust and torque coefficients, respectively. These differences are reasonable given the fact that different experiments on different specimen of a given rotor geometry can lead to discrepancies on the order of 10%, as previously highlighted by Deters et al. [START_REF] Deters | Reynolds number effects on the performance of ailerons and spoilers[END_REF] on commercial rotors. Similar differences were here found using homemade rotors with different manufacturing methods and materials and different test benches where interactions between rotor wake and mountings can be different.

Experimental data (Gojon et al. [START_REF] Gojon | Experimental investigation of low Reynolds number rotor noise[END_REF]) NVLM simulation

Thrust coefficient c T [-] 0.0649 0.0603 (-7.1%) Torque coefficient c Q [-]
0.00429 0.00394 (-8.1%) Table 2 Thrust and torque coefficients comparisons.

Acoustic validation

Before comparing the tonal noise code based on Farassat formulation-1A to real rotor cases, two test-cases allowed to separately validate both loading and thickness noise components.

As explained in section III.B, the loading noise depends on the loading distribution on the surface of the blade, its rotational speed and the microphone location. Two different test-cases (and different parameters) from Casalino [START_REF] Casalino | An advanced time approach for acoustic analogy predictions[END_REF] were used to validate the loading noise component of the tonal noise. These test-cases are based on radial and axial dipoles rotating and translating with respect to a fixed observer. As shown in both figures 7 and 8, for rotating and non-translating dipoles, the present FW-H simulation results are superposed to the analytical solution (see [25, pp. 269-275]) used by Casalino to validate his advanced time approach model . When the dipoles translate at high velocity (∼ 70m/s and ∼ 140m/s), small differences between the coded FW-H and the analytical solutions arise in the transition part. The capabilities of the present Farassat formulation-1A is validated for this study, that takes into account only free rotors in hovering flight conditions. The thickness noise validation test is based on the Brentner test-case. This one has been used by Hirsch et al. [START_REF] Hirsch | Fan Noise Simulation in the Time Domain : Validation Test Cases[END_REF], Ghorbaniasl and Hirsch [START_REF] Ghorbaniasl | Validation and application of a far-field time domain formulation for fan noise prediction[END_REF] and it is based on the thickness noise prediction of a conventional helicopter rotor obtained by Brentner [START_REF] Brentner | Prediction of Helicopter Rotor Discrete Frequency Noise[END_REF]. The helicopter rotor parameters are summed-up in table 3 and the comparisons in both temporal and frequency domains are shown in Fig. 9.

The tonal noise predicted by the NVLM code and the Farassat formulation-1A has been validated by comparing it with experiments performed previously at ISAE-SUPAERO [START_REF] Gojon | Experimental investigation of low Reynolds number rotor noise[END_REF][START_REF] Parisot-Dupuis | Experiments on UAV rotor noise at low Reynolds and low Mach numbers[END_REF] in the frequency domain. For comparison purposes, both experimental and numerical data are processed as follows. The experimental acoustic data are acquired at a sampling frequency of 51.2 kHz, during 16s. The acoustic narrow band spectra are obtained from time signals by dividing the main 16s signal into 0.10s duration signals with 50% overlap. The Hanning window is then applied, the fast Fourier transform (FFT) is computed and the amplitude correction factor is applied.

Finally, the SPL is obtained through the following formula:

SPL( f ) = 10 log 10 p 2 r ms ( f ) P 2 0 ( 9
)
with P 0 = 2 × 10 -5 Pa and p r ms is the root mean squared (RMS) acoustic pressure. The simulated frequency noise spectrum for a microphone in the rotor plane (1.62m away from the rotor center) is shown in Fig. 10.

Directivity plots of the 1st and 2nd BPFs comparing simulation and experimental results are shown in Fig. 11. This plot shows good predictions of the 1st BPF directivity (differences lower than 3dB) for microphones located in the rotor wake half-plane. Above the rotor, however, differences between 3dB and 8dB are observed. The 2nd BPF, instead, shows good agreement in a larger region, with differences lower than 5dB between -30°and +50°. On the other hand, discrepancies exist out of this range, particularly in the experimental directivity pattern below -40°(which may arise from experimental uncertainties). Comparisons between experimental results [START_REF] Gojon | Experimental investigation of low Reynolds number rotor noise[END_REF] and NVLM predictions.

Influence of NVLM parameters

In this subsection, we report on the influence of NVLM parameters on computational cost and aerodynamic and aeroacoustic performance. The aim here is to define relevant computational parameters to obtain accurate results with reduced computational costs. In that sense, we focus on three main parameters: the spatial resolution, number of rotations simulated and particle deletion.

Table 4 shows results obtained with and without vortex particle deletion and for 15 and 30 simulated rotations. Particle deletion is applied at a radial distance of 0.3m away from the rotor center (2.4 times the rotor radius), which is assumed from previous experiments on ground effect to have little impact on rotor performance [START_REF] Jardin | Aerodynamic performance of a hovering microrotor in confined environment[END_REF]. Time-averaging of thrust and torque signals is achieved during one-third of the total simulated time.

Configurations

Sim Differences in aerodynamic performance obtained from different parameter settings are found to be negligible while the computational time can be reduced up to 87% by applying particle deletion and reduced number of rotations. A directivity study, not shown here for the sake of conciseness, also demonstrated a negligible impact on acoustic signals.

Similarly, the influence of blade spatial discretization is shown in Table 5. Here again, the reduction in the number of both chordwise and spanwise lattices down to 5 × 10 has a negligible impact on the prediction of aerodynamic performance but significantly reduces computational costs.

Finally, Table 6 shows the SPL at the first and second BPF obtained for the reference simulation used in the previous comparison with experiments and the simulation with reduced spatial discretization, reduced number of rotations and particle deletion. Again, it is found that these changes in NVLM parameters have a negligible impact on the acoustics, at least regarding principal tonal components. For this reason, in order to make expensive parametric simulations "10-15 AV revs -w Del" have been chosen. This allowed to run more calculations without decreasing their precision.

IV. Results

As initially introduced, the main objective of this paper is to evaluate and analyze the influence of the blade pitch angle θ pitch and the blade sectional (airfoil) shape on rotor tonal noise, in addition to their aerodynamic performance. First, the NACA0012 constant pitch rotor tonal noise is studied in IV.A. Then a detailed analysis of the NACA0012 10°p itched rotor is conducted, and, is followed by the parametric studies of the airfoil thickness and camber. In order to test different rotor geometries, the simulation parameters used are those reported in subsection III.C.3 that minimize computational cost. As the 4-digits NACA series [START_REF] Jacobs | The Characteristics of 78 Related Airfoil Sections Sections From Tests In The Variable-Density Wind Tunnel[END_REF] airfoils allow the modification of both the airfoil thickness and camber independently, they have been chosen to analyze the influence of the airfoil shape.

Discretization Configurations Sim Time [hCPU]

Thrust 59.12 (+0.14dB) 41.24 (-0.18dB) Table 6 Tonal noise comparison for different simulation configurations for a microphone angle of 0°and distance of 1.62m away from the rotor center.

A. Acoustic decomposition of the rotor noise

The simulated 1st BPF SPL directivity plot of Fig. 11 shows a maximum located at a microphone angle of 10 • . To better understand this directivity shape, the total acoustic pressure is decomposed into three components: the thickness noise, the steady loading noise and the unsteady loading one. To calculate these two last components, the lattice loading has been split into two components: an averaged (called "steady") and a fluctuating ("unsteady") one, and the noise emitted by both is propagated:

l i (t) = |l i | average + l i (t) fluctuation (10)
With:

|l i | = 1 N timesteps t e n d t=t s t ar t l i (t) (11)
t st art = 20th rotation

t end = 30th rotation
Once the different acoustic pressures are extracted, the SPL is computed by applying the same post-processing used for the total signal. The directivity of these three new signals is plotted alongside the total signal in the left graph of Fig. 12. Here it is observed that the thickness noise has a maximum in the rotor plane and is the predominant component for a microphone angle range going from -80°up to 10°, where the steady loading noise matches the thickness noise level and becomes even higher at higher microphone angles. The unsteady loading noise has a minimum level in the rotor plane and maxima at -90°and +90°, showing a typical dipole directivity shape. However, the total noise is lower than the thickness noise between -80°and -20°and always higher than both unsteady and steady loading noise signals. To understand this phenomenon, on the right graph of Fig. 12, the phase has been plotted. Here, different phase directivities are observed. While the thickness noise phase is almost constant for all the microphone angles going from -80°to +80°, the steady loading noise shows a gradual decrease in the phase going from -90°to +80°. This gradual reduction of the steady loading noise phase, coupled with the constant and opposite thickness noise phase, explains why from -80°to -20°the total noise is lower than the thickness noise. In this range, in fact, a destructive interference between the steady loading and thickness noise components occurs. In the region [-10°; +80°], due to the change in phase of the steady loading noise, the interference evolves into a constructive one and the total signal becomes higher than both components.

The steady loading noise SPL and phase directivities are different from the usual dipole and monopole general shapes. Since the understanding of the steady loading directivity is not straightforward, its simplification can lead to a better comprehension of the phenomena involved. The acoustic pressure of the steady loading component takes into account the force vectors applied at each lattice composing the blades. The NACA0012 rotor reference simulation, has an amount of 440 blade lattices (11 chordwise lattices × 20 spanwise lattices × 2 blades). Since the rotor external radius is very small compared to the microphones' distance (0.125m vs 1.62m), the blades can be considered as two compact sources rotating around the rotor axis. Both rotor thrust and torque have been applied at 0.1m, i.e. 80% of the rotor external radius. This value has been arbitrarily chosen, but it is coherent with the literature, where it is indicated that the blade thrust/torque overall distribution that can degenerate into a point source at around 75%-80% of the blade span (see Leishman [START_REF] Leishman | Principles of helicopter aerodynamics[END_REF]). Fig. 13 shows a comparison between the steady loading noise coming from the NVLM blade force distribution and the rotating dipoles (L i V E decomposition). Both SPL and phase directivities match very well, which validates the • An axial force vector pointing upward in order to simulate a pure thrust force. The total thrust absolute value corresponds to the full rotor simulated thrust; • A tangential force vector, in the direction of the dipole velocity vector (but of opposite sign) to simulate a pure torque and whose force value is equal to the full-rotor torque divided by the distance of the dipole from the rotation axis (0.1m). The new signals, as well as the previous rotating dipoles with the rotor force vector, are plotted in Fig. 14. Both pure-thrust and torque signals have a symmetrical directivity shape centered at 0°, while the total steady loading noise directivity is not symmetric. This is, again, due to constructive/destructive interactions between the pure-thrust and torque signals. From -90°to 0°, both pure-force signals have opposite phases, hence the total signal is lower then the two others because of a destructive interference. At 0°, the pure-thrust signal drastically changes phase and its interaction with the pure-torque noise signal becomes constructive. This explains the maximum located at 20°. In conclusion, for this NACA0012 constant pitch rotor configuration, the 1st BPF total tonal noise is the contribution of different components: the thickness noise and the loading noise. The first one is predominant upstream the rotor and is lower than the steady loading noise downstream. The total noise, due to destructive interferences upstream the rotor, is lower than the thickness noise. However, this rotor is not aerodynamically, nor aeroacoustically optimized and the NACA0012 profile is very thick compared to the ones used in the commercial rotors (E63/Clark Y in APC rotors as an example). For this reason, in the following subsections, the influence of pitch angles, thickness and camber of the airfoil are studied.

B. Influence of the Pitch angle

We first look at the influence of the pitch angle (θ pitch ). It is straightforward that an increase in θ pitch induces an increase in effective angle of attack α e f f (e.g. Equation 3), hence an increase in rotor thrust, providing that local blade sections do not stall.

Since rotor thrust is dependent on the square of the rotational speed (i.e. T = c T ρΩ 2 D 4 ), a quadratic fit is used to interpolate data at any values of RPMs and study all the rotor configurations at iso-thrust. Note that this is strictly valid for constant c T values, or linear dependency of c T with respect to RPM, which depends on Reynolds number effects. Hence care should be taken in the low Reynolds number regime typical of MAV rotors, i.e. where Reynolds number effects may be strong. To reach the target thrust, a calculation at 6000RPM is run, the "a" coefficient of the quadratic law (T = a • Ω 2 ) is calculated, the rotation speed Ω target thrust is deduced and another aerodynamic calculation is run.

The tonal noise is also related to the rotational speed. In this case, it is straightforward that an increase in pitch angle leads to an increase in sectional lift coefficient (again providing that stall does not occur), hence to a higher lattice load l and an increase in the loading noise component (Eq.7). In addition, for a fixed pitch angle, an increase of the rotational speed implies an increase of the blade lattice velocities v and, consequently, of the thickness noise component (Eq.6).

The design of a quadcopter MAV rotor in hovering conditions generally relies on a thrust constraint, with each rotor thrust equal in magnitude to one fourth of the MAV weight. In what follows, the influence of blade geometrical parameters is analyzed for a given target thrust of 2N, corresponding to a hovering MAV total mass <800 grams.

Given a target thrust, a first parameter to look at and optimize is the power loading PL = T/P (or its dimensionless counterpart, the Figure-of-Merit) which gives an indication on how much power is needed to sustain a given weight. That is, the higher the PL, the lower the power needed to sustain the same amount of weight.

In light of the above considerations, the following Fig. 15 shows the rotational speed Ω, the 1st BPF SPL of the total, thickness and loading noise components for a microphone located in the rotor plane, and the Power Loading as a function of the pitch angle θ pitch at 2N isothrust. Results from this graph can be summarized as follows:

• The rotational speed required for a 2N thrust decreases when the pitch angle increases. The ratio between the rotational speed of the 6°pitch angle and that at 19°pitch angle is almost two; • The total noise SPL of the 1st BPF decreases with a similar trend to that of the rotational speed. The ∆SPL of the total noise between the 6°pitch angle and the 18°pitch angle is equal to 15dB; • The efficiency of the rotor increases between 6°and 14°. Beyond 15°, the power loading monotonically decreases. The lower rotational speed at a higher pitch angle is due to an increased α e f f , hence a higher local lift coefficient of the blade sections. As Fig. 15 shows, the thickness noise is highly dependent on the rotational speed of the rotor. This component is higher than the loading noise for pitch angles going from 6°to 10°and the total noise is driven by the thickness noise. After 10°angle, the loading noise becomes preponderant. However, the ∆SPL between the 6°and 19°p itch value is less than 5dB. This suggests that in the loading noise, and for a microphone angle located in the rotor plane, the total thrust of the rotor plays a more important role than the rotational speed. On the other hand, the rotational speed is the key parameter in the generation of thickness noise.

In conclusion, for a constant pitch rotor with a NACA0012 constant section, the pitch angle to choose for aerodynamic purposes is 15°, because the power loading is the highest, hence its endurance is the highest. However, the total noise emitted by this rotor is higher than the least noisy one, the 19°angle, by about 3dB.

C. Influence of the Airfoil Thickness

The interest in the airfoil thickness for low Reynolds numbers has recently grown, due to the introduction of MAVs. Previous works [START_REF] De Paula | The thickness effects on symmetrical airfoil flow characteristics at low Reynolds number[END_REF][START_REF] Jacobs | Airfoil section characteristics as affected by variations of the Reynolds number[END_REF] have shown that the thickness plays a prominent role in the aerodynamics of symmetric NACA airfoils at Reynolds numbers in the order of 10 4 -10 5 , with significant influence on the occurrence of laminar separation bubbles (LSBs) and advanced stall for airfoils with large thicknesses. Because of mass and inertia constraints and low Reynolds number encountered, MAV rotors airfoil thicknesses are generally low.

To understand the implications of the airfoil thickness on MAV rotors from both aerodynamic and aeroacoustic points of view, the performance of different airfoil thicknesses are presented, ranging from the NACA 0007 to the NACA 0015 airfoils (the NACA 0012 being the reference one). On NACA airfoils, the maximum thickness is located at ∼ 30% of the chord. As done in subsection IV.B, the rotational speed, the power loading and the 1st BPF SPL (of the total noise and loading/thickness noise components) at 0°microphone angle are plotted as function of the maximum thickness in Fig. 16, for a given target thrust of 2N and a constant pitch angle of 10°for all rotors.

The following observations are made: • The rotational speed steadily decreases (∼ -400RPM) due to thicker airfoils generating more thrust when rotating at a certain rotational speed; • The 1st BPF total SPL increases when the airfoil thickness increases (∼ +1.8dB). The thickness noise follows the same trend of the Total SPL, with a higher range (+3dB), while the loading noise does not significantly change for the 0°microphone angle; • The PL reaches a maximum value for 8% of the airfoil thickness and then it monotonically decreases. To understand why the aerodynamic performance of the NACA 0008 airfoil are better compared to the others, the lift coefficient, the drag coefficient, as well as the lift-to-drag ratio of the different sections and airfoil thicknesses are displayed in Fig. 17. This graph is made by using the look-up tables described in III.A.2. After having run the aerodynamic calculation, both induced sectional Reynolds number and angles-of-attack are averaged and used as inputs of the look-up table, where the sectional C L and C D values are calculated.

As mentioned in IV.A, rotor thrust and torque are mainly produced at 75-80% of the blade span. This is verified in Fig. 17, where the highest C L values are produced at 78% and 87% of the blade span (independently of the airfoil thickness). On the other hand, the drag coefficients of the outboard sections are lower than the ones near the root. These considerations explain the higher lift-to-drag ratios for the external sections of the blade. An increase in the airfoil thickness near the tip sections (where most of the thrust/torque are produced), leads to higher C L values. This is the main reason why the rotational speed needed to reach 2N of thrust decreases. C D values at 87% of the span section reaches a minimum for an airfoil thickness of 8% of the chord. Above this maximum thickness value, the C D value increases. This trend is mirrored in the power loading graph of Fig. 16, where the maximum efficiency is reached by the rotor carrying the NACA 0008 airfoil.

From the aeroacoustic point of view, it is remarkable that the increase in the airfoil thickness induces an increase in the thickness noise by 3dB even if the rotational speed decreases. Higher airfoil thickness, in fact, means higher amount of fluid displaced by the rotor blade and, consequently, higher thickness noise component. The loading noise at 0°microphone, however, is not impacted by the airfoil thickness increase. The change in both torque and rotational speed is not high enough to affect the loading noise. It is interesting to note that loading and thickness noise graphs intersect each other for a maximum thickness value of 10%. Below this value, the loading noise is preponderant while, above, the thickness noise becomes higher. This explains why increasing the maximum thickness value from 7% to 15% increases the thickness noise by 3dB, and the total noise by only 1.8dB. This airfoil thickness parametric study on the 10°pitched MAV rotors at 2N iso-thrust shows that the airfoil thickness is important for the aerodynamic performance of the rotor, but also for the aeroacoustic tonal noise footprint. A 8% thick airfoil, in fact, enhances the aerodynamic performance, but also reduces the thickness noise component of the tonal noise compared to the NACA0012 reference rotor. This thinner airfoil reduces the rotor weight by 33% compared to the NACA 0012. Even if the weight of the rotors is negligible compared to the full drone weight, it is important to limit it in Thickness, [% of the Chord] the design phase. Lower rotor weight due to thinner airfoil (at the same chord distribution) means also lower inertia. This impacts also the mass of the motor since less torque will be required to spin the rotor. The gain of few grams can also allow the addition of other important components useful to better accomplish the missions (cameras, sensors, slightly heavier batteries to increase the flight endurance, etc) or to comply with regulations.

D. Influence of the Airfoil Camber

In this last section, the influence of the airfoil maximum camber onto the aerodynamic and aeroacoustic performance of MAV rotors is investigated. For the sake of conciseness, both thickness and position of the airfoil maximum camber are fixed to 12% and 50% of the chord, respectively. The first digit, corresponding to the maximum camber, is the investigated variable. As for the pitch and thickness parametric study, the rotors are studied at 2N thrust, the pitch angle is fixed to 10°and the microphone is in the rotor plane (at 0°angle). The results in Fig. 18 show that, when the airfoil camber increases:

• The rotational speed steadily decreases (∼ -800RPM), meaning that more cambered airfoils generate more thrust at the same rotational speed; • 1st BPF total SPL and both loading and thickness noise components decrease monotonically; • The power loading exhibits a peak for 1% maximum camber and then it decreases.

As the results of Fig. 18 show, from the aerodynamic point of view, adding 1% camber to the 12% thick NACA airfoil improves the aerodynamic efficiency and, as a consequence, increases the flight time. Winslow et al. [START_REF] Winslow | Basic understanding of airfoil characteristics at low Reynolds numbers (104-105)[END_REF] studied the influence of the maximum camber on NACA X403 airfoils and pointed out that, at low Reynolds numbers (2 × 10 4 and 1 × 10 5 ), the more cambered airfoils got higher lift peaks, higher drag and, overall, higher efficiency. They concluded that, at these Reynolds numbers, 4% cambered airfoils are the ones to choose when efficiency is the main objective. Here, higher cambered airfoils led to rotor rotating at lower speeds, which means that the more the airfoil is cambered, the higher is the lift created by the airfoil and the larger the rotor thrust at the same rotational speed. However, the most efficient airfoil in this study is the one with 1% camber. XFOIL simulations on rotors section at 87% of the blade span showed that, by increasing the maximum camber from 0% to 1%, the LSB and the turbulent transition are pushed towards the trailing edge. It increases the low pressure region on the suction side that leads to overall higher lift coefficient. This, coupled to the unchanged drag coefficient, conducts to higher lift-to-drag ratio and more efficient rotors. The LSB are bigger for maximum cambers above 2%, and the trailing edge separation occurs at high cambers. This is the main reason why both lift and drag coefficients are increased, but the overall efficiency is not. While the NACA 1512 is the most efficient airfoil from the aerodynamic point of view, it is not from the aeroacoustic one. Since higher cambered airfoils produce more lift, the rotational speed needed to reach 2N thrust is lower and, as a direct consequence, 1st BPF SPL of total, loading and thickness noise components are reduced. In this case, even if the airfoil thickness distribution is unchanged, the thickness noise decreases. In light of the results presented in this subsection, MAV propellers working at low Reynolds number conditions are more efficient with cambered airfoils. For thick airfoils like the NACA x512 series, the ideal camber is 1%. Above that value, the drag coefficient increases at a higher rate than the lift coefficient, leading to less efficient rotors. Attention must be paid when choosing the camber of the airfoil. In fact, this design input is linked to the Reynolds number at which the rotor will operate, but also on the airfoil thickness.

V. Conclusion

The aerodynamic performance and the aeroacoustic footprint of quadcopter rotors are highly dependent on the rotor geometry and their operating Reynolds number. The low Reynolds number typical of micro air vehicles (MAVs) leads to new design challenges that the research community has tried to overcome these last decades by developing new low-order fidelity simulation codes and optimization methodologies.

Here, a non-linear vortex lattice/particle method (NVLM/VPM) code coupled with the Farassat formulation-1A of the Ffowcs-Williams & Hawkings (FW-H) acoustic analogy is validated upon results from the literature and experimental results on a NACA0012 constant pitch rotor. The NVLM code is then used to analyze the directivity of the 1st blade passing frequency (BPF) sound pressure level (SPL). The influence of the blade pitch angle and the airfoil shape on the aerodynamic performance and the tonal noise emitted by the rotor is also studied.

Preliminary results presented in this paper showed the tonal noise directivity shape of the 1st and 2nd BPF. The 1st one was studied more in details and destructive/constructive interferences phenomena between the thickness and the steady loading noise are demonstrated. The steady loading noise is also shown to be more important than the unsteady loading noise and its directivity is equivalent to the directivity of a rotating dipole with same rotational speed, total thrust and torque as the full rotor. In order to reproduce the same directivity, the dipoles should be positioned at a radius equal to 80% of the blade radius. Another constructive/destructive interferences are shown to be the causes of the non-symmetrical shape of the steady loading noise.

The influence of the pitch of the blades is then studied. It affects the sectional lift coefficient, hence the rotational speed needed to withstand the 2N thrust decreases as well as the tonal noise. The power loading reaches a maximum value at 14°pitch angle and decreases beyond this value.

Then, different airfoil thicknesses going from 7% to 12% are studied, with a constant pitch angle of 10°. For an isothrust of 2N, the best configuration is the one with 8% maximum thickness. Increasing this value affects the airfoil performance and, as a consequence, the rotor one. The overall lift coefficient increases with the thickness, resulting in lower rotational speed to reach 2N. However, the torque increases as well and, above 8% thickness, it increases more than the thrust, leading to lower power loading. As expected, the thickness noise increases with the airfoil thickness while the loading noise does not and becomes even lower than the thickness noise at 11% thickness. This means that, decreasing the thickness has more benefits than drawbacks. But thinner airfoils lead also to lighter and less stiff blades. Last, a camber can be added to the airfoil in order to increase the efficiency of the airfoil and, overall, of the whole rotor. A more pronounced camber pushes the laminar separation bubble towards the trailing edge. This increases the low pressure region on the suction side and leads to higher lift coefficients and more efficient airfoils up to a certain limit. In fact, while for thin airfoils (3% thick) the camber that gives the most efficient rotor is 4%, in the present investigated case of 12% thick airfoils the most efficient airfoil has a camber of 1%. Highly cambered airfoils produce more thrust, thus the rotational speed to reach 2N thrust is decreased. This reduction in the speed leads to lower SPL. Cambered airfoils should be chosen to obtain aerodynamic and aeroacoustic improvements.

In the rotor design phase, different parameters need to be taken into account and can help in reaching the objective performance. The pitch angle and the airfoils are two important parameters. Increasing the first one can easily lead to more efficient and less noisy rotors at the same time. The second one is more complex: the aerodynamic performance of the airfoil can be improved by playing on both thickness and camber distribution. However these parameters are non-linearly dependent on the Reynolds number. This means that there is not an airfoil that optimize the aerodynamic performance over a wide range of Reynolds number. In the 25cm diameter constant pitch rotor studied in this paper, thinner cambered airfoils are the best one for both aerodynamic and aeroacoustic performance. The study presented here takes into account only tonal noise in the aeroacoustic analysis. Even if the tonal part of the acoustic spectrum has the highest peaks, the broadband noise contains much more frequencies that disturb the human ear. In the future, the full acoustic spectrum will be approximated by adding semi-empirical methods to the present NVLM/VPM code.

  Pa] l r = projection of the derivative of the loading vector (l) onto the observer vector (r) [Pa/s] l M = projection of the loading vector (l) onto the Mach vector (M) [Pa] M = Mach number [-] (M = the Mach vector) n = surface normal vector ∇ = gradient function ν = kinematic viscosity [m 2 /s] ω = vorticity [1/s] Ω = rotational speed [RPM] P = aerodynamic power [W] p = total acoustic pressure [Pa] p L = loading noise acoustic pressure [Pa] p T = thickness noise acoustic pressure [Pa] PL = power loading [N/W] Q = torque [Nm] r = observer vector [m] r = radial position of the panel [m] (in Eq. 2) R = rotor radius [m] R cut-off = cut-off radius / rotor radius [-] Re c = chord-based Reynolds number, Re c = (ΩRc)/ν [-] ρ = density [kg/m 3 ] T = thrust force [N] θ = directivity angle [°] θ pitch = pitch angle [°] θ twist = twist angle [°] v = lattice velocity [m/s] V ∞ = freestream velocity [m/s] x = lattice position [m] y = observer (microphone) position [m]

Fig. 1

 1 Fig. 1 On the left: The PARROT ANAFI Ai (800 grams) mostly used in the civilian sector. On the right: The PARROT ANAFI USA (500 grams), used in the military field, but also in the industry. [Courtesy of PARROT Drones]

Fig. 2

 2 Fig. 2 ISAE-SUPAERO anechoic chamber and microphone locations ranging from θ = -60°to θ = 60°at a constant distance of 1.62m away from the rotor center. Positive microphone angles indicate the induced flow direction.

Fig. 3

 3 Fig. 3 Look-up table of the NACA0012 profile computed via XFOIL[17] (with N crit = 8 [18]). c L coefficients (on the left) and c D coefficients (on the right), as functions of the Angle-of-Attack for three different Reynolds numbers.

Fig. 4

 4 Fig. 4 Simulation snapshot showing the blade lattices (in black), the wake lattices (in red) and the vortex particles (in green).

Fig. 6

 6 Fig. 6 The rotor used in the simulation for the correlation with the experimental data.

Fig. 7

 7 Fig. 7 Comparison between Casalino test cases B and the FW-H acoustic simulations.

Fig. 8

 8 Fig. 8 Comparison between Casalino test cases C and the FW-H acoustic simulations.

Fig. 9 Fig. 10 Fig. 11

 91011 Fig. 9 Comparison between Brentner test-case and present FW-H Farassat formulation-1A of the thickness noise.

Fig. 12 Fig. 13

 1213 Fig. 12 Directivity of the amplitude of the 1 st BPF (graph on the left) and its phase (graph on the right). Influence of the thickness noise and both unsteady and steady loading noise.

Fig. 14

 14 Fig. 14 Directivity of the amplitude of the 1 st BPF (Graphs on the left) and its phase (Graphs on the right) corresponding to the two dipoles rotating at Ω = 6000RPM. The blue curve simulates two dipoles with the same thrust and torque as the full rotor. The orange and green curves show a pure thrust and a pure torque, respectively.

Fig. 15

 15 Fig. 15 Rotational speed (blue line), SPL of the 1 st BPF (red line) for a microphone located in the rotor plane and power loading (green line) plotted as function of the blade pitch angle at 2N isothrust.

Fig. 16

 16 Fig. 16 Rotational speed (blue line), SPL of the 1 st BPF (red line) for a microphone located in the rotor plane and power loading (green line) plotted as function of the airfoil thickness for 2N isothrust and θ pitch = 10°.

Fig. 17

 17 Fig. 17 Sectional lift coefficient, drag coefficient and lift-to-drag ratio as function of the maximum airfoil thickness for different blade spanwise sections.

Fig. 18

 18 Fig. 18 Rotational speed (blue line), SPL of the 1 st BPF (red line) for a microphone located in the rotor plane and power loading (green line) plotted as function of the airfoil maximum camber for 2N isothrust and θ pitch = 10°.

Table 3 Simulation parameters used for the thickness noise validation.

 3

	Brentner rotor parameters
	Rotor	1/4th scale UH-1 rotor
	# of blades	2
	Airfoil Section	NACA0012
	Chord Length [m]	0.1334
	Radius [m]	1.829
	Pitch [°]	15.0
	Twist [°/m]	-10.9
	Root AoA (θ root ) [°]	14.07
	Tip AoA (θ tip ) [°]	4.1
	Root cut-off (R cut-off )	8.5% of the Radius
	Simulation parameters
	Revolutions simulated [-]	10
	Rotational speed [RPM]	1296
	Step angle [°]	0.9
	Average interval [revs]	0 -10
	Microphone location
	Microphone distance [m]	3.0
	Directivity angle [°]	0.0

Table 4 Aerodynamic trust and torque and computational cost comparison between different configurations: AV revs = Average intervals in revolutions; wo Del = Without vortex particle deletion; w Del = With vortex particle deletion.

 4

	Time [hCPU]	Thrust [N]	Torque [Nm]

Table 5 Aerodynamic main results and computational cost comparison between different configurations. Configurations 1st BPF SPL [dB] 2nd BPF SPL [dB]

 5

	[N]	Torque [Nm]

Acknowledgments

This work has been financed by Parrot Drones SAS and the Association Nationale Recherche Technologie (ANRT), realized at ISAE-SUPAERO and performed using HPC resources from PANDO (ISAE-SUPAERO), GENCI [CCRT-CINES-IDRIS] (Grant 2021-[A0102A07178] and 2022-[A0122A07178].


Romain Gojon
email: romain.gojon@isae-supaero.fr

Nicolas Doué
email: nicolas.doue@isae-supaero.fr.

Hélène Parisot-Dupuis
email: helene.parisot-dupuis@isae-supaero.fr

Bertrand Mellot
email: bertrand.mellot@isae-supaero.fr.

Thierry Jardin
email: thierry.jardin@isae-supaero.fr.

Aeroacoustic radiation of a low Reynolds number two-bladed rotor in interaction with a cylindrical beam

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

Aeroacoustic radiation of a low Reynolds number two-bladed rotor in interaction with a cylindrical beam

Romain Gojon * , Nicolas Doué † , Hélène Parisot-Dupuis ‡ , Bertrand Mellot § , and Thierry Jardin ¶ ISAE-SUPAERO, Université de Toulouse, Toulouse, France

In this paper, a characterization of a low Reynolds number two-bladed rotor in interaction with a beam is performed using both an experimental campaign performed in an anechoic room and implicit large eddy simulations (iLES) associated with the Ffowcs-Williams and Hawkins (F-WH) analogy. The rotor studied has a NACA0012 blade section profile extruded in the radial direction with a constant chord of 25 mm, a constant pitch of 10 • and a diameter of 200 mm. This rotor is tested in interaction with 3 cylindrical beams of different diameters positioned at a distance of 20 mm below the rotor disk plane. The blade passing frequency (BPF) and the high frequency broadband noise appear to not be affected by the presence of the beam. On the contrary, the magnitude of the 2×BPF to 15×BPF harmonics increases up to 30 dB compared to the case without beam. For the dominant harmonics of the BPF, a dipolelike pattern aligned with the beam can be observed. Numerical simulations were successfully validated against the experimental data for the dominant BPF harmonics and were then used to study the contribution of the rotor and the beam to the noise radiated. It is found that the beam is responsible for the increase in BPF harmonics and their characteristic directivity pattern.

Nomenclature

I. Introduction

Quadcopter unmanned air vehicles (UAVs) are increasingly used in both civilian and military fields for their versatility and their low operating cost. They perform videos for the cinema industry, inspection of cracks in dams, or reconnaissance mission, just to name a few. This evolution may increase city noise pollution in the near future and lead FAA and EASA to put in place more stringent certification standards to protect civilian population from UAV noise. It will then become important to understand UAV noise generation mechanisms in order to develop UAV designs suitable for these new standards. In quadcopter UAVs, main noise components are due to the rotors and rotor-rotor and rotor-frame interactions. The evaluation and prediction of rotor noise began few decades ago (e.g. George [START_REF] George | Helicopter noise: State-of-the-art[END_REF], Brentner and Farassat [START_REF] Brentner | Helicopter noise prediction: the current status and future direction[END_REF]) but most studies conducted on this topic were mainly concerned with helicopters. Compared to helicopters, quadcopter UAVs operate at lower Reynolds and Mach numbers and may feature complex interactions between multiple rotors. Hence, extensive knowledge on helicopter noise may not be directly transposable to UAVs.

Recently, low Reynolds and low Mach numbers hovering isolated rotors have been studied experimentally and numerically. Zawodny et al. [START_REF] Zawodny | Acoustic characterization and prediction of representative, small-scale rotary-wing unmanned aircraft system components[END_REF] reported an experimental analysis of small scale off-the-shelf rotors. Jo et al. [START_REF] Jo | Prediction of Noise from Low Reynolds Number Rotors with Different Number of Blades using a Non-Linear Vortex Lattice Method[END_REF] used a non-linear vortex lattice method (NL-VLM) coupled with a Ffowcs-Williams and Hawkins (F-WH) analysis to study the effect of the number of blades on the aeroacoustics of hovering rotors. Serré et al. [START_REF] Serré | A design methodology for quiet and long endurance MAV rotors[END_REF][START_REF] Serré | Towards silent micro-air vehicles: optimization of a low Reynolds number rotor in hover[END_REF] performed an aeroacoustic optimization of a hovering rotor through low order models (blade element, FW-H and broadband models). They showed that the overall rotor noise could be significantly reduced as compared with conventional off-the-shelf rotors. At TU-Delft, Grande et al. [START_REF] Grande | Aeroacoustic Investigation of a Propeller Operating at Low Reynolds Numbers[END_REF] studied experimentally the performances and the noise emitted by low Reynolds number rotors for different advance ratios thanks to an aeroacoustic wind tunnel. Romani et al. [START_REF] Romani | Performance and noise prediction of low-Reynolds number propellers using the Lattice-Boltzmann method[END_REF] performed Lattice-Boltzmann simulations on the same setup to better understand noise source mechanisms. Finally, Gojon et al. [START_REF] Gojon | Experimental investigation of low Reynolds number rotor noise[END_REF] provided the community with an open experimental database on 3-D printed academic low Reynolds number rotors. In addition to isolated rotors, studies have been performed on rotor-frame and rotor-rotor interactions. Zawodny and Boyd Jr [START_REF] Zawodny | Investigation of rotor-airframe interaction noise associated with small-scale rotary-wing unmanned aircraft systems[END_REF] investigated rotor-frame interaction noise in simplified configurations representative of hovering rotors, through both experiments and numerical simulations. The authors showed that rotor-frame interactions produce additional noise that is highly dependent on frame geometry, position of the frame with respect to the rotor disk plane (below or above) and distance to the rotor disk plane. Similarly, for wind turbine applications, Cotte et al. [START_REF] Cotte | Experimental Characterization and Analytical Modelling of Rotor Tonal Noise[END_REF] studied the tonal noise due to the interaction of a simplified rotor and a circular tower and Wu et al. [START_REF] Wu | Propeller-strut interaction tone noise[END_REF] analyzed the tonal noise produced by an UAV rotor operating with a circular beam placed in the wake. Overall, in all the preceding studies, the presence of the beam increase the acoustic emissions at the harmonics of the blade passing frequency. More complex configurations with multiple rotors and airframes were investigated by Intaratep et al. [START_REF] Intaratep | Experimental study of quadcopter acoustics and performance at static thrust conditions[END_REF], Yoon et al. [START_REF] Yoon | Computational aerodynamic modeling of small quadcopter vehicles[END_REF], Tinney and Sirohi [START_REF] Tinney | Multirotor drone noise at static thrust[END_REF], Lee and Lee [START_REF] Lee | Rotor interactional effects on aerodynamic and noise characteristics of a small multirotor unmanned aerial vehicle[END_REF], among others.

In this paper, we report on an experimental and numerical investigation of the noise generated by an isolated small scale two-bladed rotor interacting with different cylindrical beams and operating under hovering conditions. The experimental and numerical setups are first described in section II. The aerodynamics and acoustics results are then presented in section III where the effects of the blade on the rotor performance are studied.

II. Experimental and numerical setups A. Experimental setup

The rotor is positioned at the center of the ISAE-SUPAERO anechoic room at the top of a rotor test stand, as it can be seen in Figure 1. This facility is acoustically treated in the frequency range 80 Hz -16000 Hz and has wedge-tip to wedge-tip dimensions of 5.02 m long, 5.24 m wide and 5.34 m high. A beam can be positioned below the rotor at several distances from the rotor disk plane. The rotor is driven by means of a Faulhaber 3274G024BP4 3692 electric brushless motor, which presents low noise emissions as compared to conventional off-the-shelf motors [START_REF] Gojon | Experimental investigation of low Reynolds number rotor noise[END_REF]. Real time revolution per minute (RPM) are directly given by the associated motion controller MC5010 which provides a precise speed control. A six-axis ATI Nano17 load cell is placed beneath the motor for measuring aerodynamic loads. A directivity antenna with 13 1/4" GRAS 40PH microphones is used to measure the farfield noise radiated 162 cm away from the rotor center, for latitude angles θ every 10 • from -60 • to 60 • , where 0 • corresponds to the rotor disk plane. Moreover, in order to be able to study the directivity along the rotor disk plane, the test stand is positioned on a turntable so that one can rotate in azimuth with a ∆φ = 10 • the whole rotor-beam setup. With the couple directivity antenna/turntable, the almost complete spherical directivity can be investigated, only the poles (|θ| > 60 • ) being missing. A near-field microphone is located 25 cm below the rotor disk plane and 25 cm away from the rotor center, in the radial direction. Acoustic data are acquired at a sampling frequency of 51.2 kHz, during 16 s. This experimental setup has been validated in a previous paper [START_REF] Gojon | Experimental investigation of low Reynolds number rotor noise[END_REF] without the turntable.

In this experimental campaign, the rotor has been 3D-printed using stereolithography and features a NACA0012 blade cross-section profile extruded in the radial direction with a constant chord of 25 mm, a constant pitch of 10 • and a diameter of 200 mm. The rotor is rotating in the clockwise direction as seen from above. This rotor is tested in interaction with 3 cylindrical beams of different diameters positioned at a distance L = 20 mm below the rotor disk plane, with the rotor blowing in the downstream direction towards the beam. This distance is defined as the distance between the rotor disk plane, considered as the plane passing through the blades at mid-chord, and the center of the round beam. Three diameters have been tested : 10 mm, 15 mm and 20 mm. The beams will therefore be labelled D10, D15 and D20 respectively. The rotational speed of the rotor varies from 4000 to 8000 RPM.

B. Numerical methods and meshes

The simulations are performed with the CFD commercial software Fluent v2020 R1. The rotor geometry includes the two blades, the hub and the beam, as shown in Figure 2. The computational domain is split into two volumes: an inner rotating volume including the rotor and the rotating part of the test rig, and an outer static volume for the static part of the test rig, the beam and the farfield boundary conditions. The separation is represented by a black line in Figure 2 (right). Both volumes are meshed with polyhedral cells. Prism layers are generated at walls for boundary layers discretization. The thickness of the near wall prism layer is set to target a value of 1 for Y + at blade walls. The inner and outer volumes communicate through interfaces. All walls are modelled as nonslip walls. Upstream, downstream and lateral boundaries are modelled as stagnation inlet, pressure outlet and slip walls, respectively in the outer volume. The total number of cells for each geometrical configuration is around 7 millions. Since the flow is incompressible, a segregated solver with constant density is selected. Note that the local Reynolds number based on the local blade velocity and chord length increases from about 25.000 at the blade root to 130.000 at the blade tip. The laminar boundary layers will be predominant, and fully turbulent URANS simulations are not appropriate with such Reynolds numbers. We therefore used implicit large eddy simulation (implicit LES). Navier-Stokes (N-S) equations are solved directly without turbulence or subgrid model and only the largest turbulent structures are resolved. The spatial and temporal discretization schemes are of second order, and the time-step for the implicit unsteady solver is defined to obtain a 1 degree rotation every time-step. The mean and maximal values of the convective Courant number in the rotating part are around 1 and below 10, respectively.

Three simulations are performed at a rotation speed of 8000 RPM : one without beam, one with the D10 beam located at a distance L = 20 mm below the rotor disk plane and one with the D20 beam at the same location. An isocontour of Q criterion is proposed in Figure 3 for the case with the D20 beam and L = 20 mm. The tip vortex and its interaction with the beam are clearly visible.

III. Results

A. Aerodynamic results

The results are first analyzed in terms of time-averaged thrust and torque as functions of rotating speed. Figure 4 (left) shows that the thrust of the isolated rotor increases quadratically with RPM, as expected. Slight variations with respect to the quadratic fit shown as dashed-dotted line may arise, for example, from Reynolds number effects or blade deformation with RPM which may slightly affect the trust coefficient. Similar observations can be made with the torque shown in figure 4 (right).

Effects of the beam

Results obtained for the rotor in interactions with the cylindrical beam are also depicted in figure 4. It can be observed that the presence of the beam has a negligible impact on both thrust and torque, for all RPMs and beam diameters.

The maximum thrust obtained at 8000 RPM is about 2.1 N. According to Froude theory, the average, axial induced velocity over the rotor disk is v i = (T/(2ρA)) ≈ 5.3 m/s, where A is the rotor disk area and ρ is the fluid density. The resulting flow interacts with the cylindrical beam placed beneath the rotor, with a characteristic Reynolds number Re = ρv i D/µ (hence neglecting tangential induced velocity) in the range [3.10 4 -7.10 4 ], where D is the cylinder diameter and µ is the dynamic viscosity of the fluid. In this flow regime, the flow past the cylinder is characterized by an alternating vortex shedding in the wake, at Strouhal number St around 0.2. The corresponding frequency f = Stv i /D lies between 5 to 10 Hz (for different values of D), which is far below the rotor rotation frequency and is out of the scope of the present study.

Validation of the numerical simulations

Similarly to experimental results, numerical simulations do not predict significant changes in aerodynamic performance with beam diameter. Specifically, thrust values of T = 2.08, 2.08 and 2.07 N are obtained for the cases without beam and with 10 and 20 mm diameter beams at 8000 RPM, respectively. For that reason, only the case with D = 20mm is shown in figure 4.

B. Acoustic results

For both numerical simulations and experimental measurements, narrowband acoustic spectra are obtained from time signals by computing the fast Fourier transform (FFT) with a Hanning window applied on 100 segments, using a 50% overlap and an amplitude correction factor. The frequency resolution is ∆ f = 3.125 Hz.

For the numerical simulation, the Ffowcs-Williams and Hawkins (F-WH) analogy has been used from the fluctuating pressure on different surfaces in order to calculate the farfield acoustics. First, the total fluctuating pressure (i.e. that of the rotor, beam, and test stand) is considered in order to compare the numerical results with the experimental results. Second, only certain surfaces will be used in the F-WH analogy to better analyze the noise source mechanisms.

Effects of the beam

The first experimental results presented concern a rotational speed of 8000 RPM, with and without beams D10, D15, and D20 positioned at L = 20 mm. Farfield spectra at a distance of 1.62m, latitude angles of θ = 0 • and θ = 40 • with respect to the rotor disk plane, and at an azimuth angle of φ = 90 • (where φ = 0 • corresponds to the beam positioned oppositely to the directivity antenna) are compared to the case without beam in Figure 5. Please note that BGN stands for background noise ans corresponds to the noise with a zero rotational speed.

For the four cases and the two angles, the tone with the highest level corresponds to the blade passing frequency (BPF), which is equal to 267 Hz for this configuration. The rotational frequency (133 Hz) is also visible. Harmonics of those two tones can be seen up to 10.000 Hz. Finally a high-frequency broadband noise can be observed in the frequency range 1000 -16.000 Hz. These results are similar to those obtained in a recent study conducted on isolated rotors [START_REF] Gojon | Experimental investigation of low Reynolds number rotor noise[END_REF]. It can be seen that the interaction of the rotor with the beam does not seem to affect the amplitudes of the BPF, its first harmonic and the high-frequency broadband noise. However, the interaction of the rotor with the beam significantly increases the BPF harmonics 2×BPF to 15×BPF, up to 30 dB for the fifth harmonic of the BPF (5×BPF). This amplification of the harmonics of the BPF increases with the diameter of the beam. The same behavior has been observed recently [START_REF] Zawodny | Investigation of Rotor-airframe Interaction Noise Associated with Small-Scale Rotary-Wing Unmanned Aircraft Systems[END_REF] for a similar configuration.

In order to study the effects of the rotational speed on the noise produced by the interaction of the rotor with the beam, farfield spectra at latitude angles of θ = 0 • and θ = 40 • with respect to the rotor disk plane and at an azimuth angle of φ = 90 • are compared to the case without beam in Figure 6 for the D20 beam positioned at L = 20 mm and for rotational speeds of 4000 : 1000 : 8000 RPM.

It is found that the magnitudes of the BPF, its first harmonics and the high frequency broadband noise increase with the rotational speed, following results obtained on isolated rotors [START_REF] Gojon | Experimental investigation of low Reynolds number rotor noise[END_REF]. For the higher BPF harmonics, as rotational speed increases, their magnitudes increase too, and the envelope of BPF harmonics (dashed lines in Figure 6) is shifted towards higher frequencies, so that the highest BPF harmonic remains between 5×BPF and 6×BPF.

Validation of the numerical simulations

In order to validate our numerical setup for farfield acoustic radiation, the case without beam at a rotational speed of 8000 RPM is analyzed first. Farfield spectra at a distance of 1.62m and latitude angles of θ = 0 • and θ = 40 • with respect to the rotor disk plane are plotted in Figure 7. Note that for the numerical simulation, the spectra are obtained by averaging spectra in the azimuthal direction, which is a homogeneous direction for this configuration without beam. For the BPF and its first harmonic, differences of less than 1 dB are observed. For the second harmonic of the BPF, differences of up to 5 dB are observed. For higher harmonics and the broadband noise, the iLES presented here is not able to recover the experimental values, as expected with the rather coarse grid used here.

The case with the D10 beam at a rotational speed of 8000 RPM is then analyzed. Farfield spectra at a distance 1.62m, latitude angles of θ = 0 • and θ = 40 • with respect to the rotor disk plane and at an azimuth angle of φ = 90 • are plotted in Figure 8.

The top Figures correspond to the comparison of experimental results with F-WH calculations based on the fluctuating pressures obtained on all the surfaces (rotor, beam and tests stand). When all the surfaces radiate, the magnitude of the BPF is well predicted with differences less than 1 dB. BPF harmonics are quite well predicted with errors inferiors to 5dB between the numerical results and the experimental ones, except for the first and the second harmonic of the BPF at θ = 40 • . As already pointed out in the spectra comparisons for the case without beam, the broadband noise is not well reproduced by the simulation.

Once the numerical results are validated, it is interesting to isolate some radiating surfaces of the simulation to relate the amplitudes of the BPF and its harmonics to specific surfaces. The farfield acoustic radiation of the two surfaces assumed to be the main acoustic sources, namely the rotor and the beam, are presented on bottom figures. The strong levels of high BPF harmonics appears to be due to the beam radiation. Errors on the first and second harmonics magnitudes at θ = 40 • seem to be due to errors associated to rotor radiation in the coupled iLES / F-WH approach.

Finally, the case with the D20 beam at a rotational speed of 8000 RPM is studied in order to assess the robustness of these tendencies. Farfield spectra at a distance 1.62m, latitude angles of θ = 0 • and θ = 40 • with respect to the rotor disk plane and at an azimuth angle of φ = 90 • are plotted in Figure 9.

Same conclusions can be drawn from numerical and experimental comparison with this beam diameter. The increase of the beam diameter tends to increase the difference between the contribution of the rotor and the beam to high BPF harmonics magnitudes, the rotor contributing mainly only to the BPF magnitude.

Overall, the simulations permit to recover rather nicely the envelope of the BPF harmonics due to the interaction of the rotor with the beam, permitting thus to look at the directivity patterns obtained from simulations to try to discriminate the role of the rotor and the beam in the noise radiating field.

Directivity patterns

In order to study the influence of the beam in a more quantitative way, several specific noise components are analyzed as functions of the directivity angles in Figure 10 for the D10 beam at 8000 RPM: the overall sound pressure level (OASPL), the amplitude of the BPF and its fifth harmonic, and the amplitude of the high frequency broadband noise. The overall sound pressure level is computed over a frequency range of 80 Hz < f < 16000 Hz and the broadband noise is computed over a frequency range of 1000 Hz < f < 16000 Hz (for the latter, tones were cleaned from the spectra before computing the integral value). For the OASPL, the BPF and the broadband noise, axisymmetric radiation patterns are observed. The OASPL is omnidirectionnal, whereas the BPF presents a maximum level at a latitude angle of about 10 • , and the broadband noise a minimal amplitude at a latitude angle of -10 • . These results are similar to the ones obtained in the case without beam [START_REF] Gojon | Experimental investigation of low Reynolds number rotor noise[END_REF]. Those noise sources seem thus not to be affected by the presence of the beam, as already qualitatively observed in Figure 5.

On the contrary, the directivity of the fifth harmonic of the BPF seems to be affected by the presence of the beam. A dipole like pattern is observed with minimum values at azimuth angles of 0 • and 180 • and latitude angle of 0 • . The same pattern is observed for all the BPF harmonics which are increasing in the presence of the beam.

The numerical results are then used to better understand the directivity patterns of the dominant fifth harmonic of the BPF in particular, by separating rotor and beam contributions in Figure 11.

This comparison highlights that the dominant fifth harmonic of the BPF is mainly due to radiation coming from the rotor for latitude angles superior to 30 • and from the beam near the rotor disk plane. The farfield acoustic radiation predicted from the beam surface seems to be very reliable, as it fits experimental results for noise components for which it is the main contributor. However, it is not the case for farfield acoustic radiation predicted from the rotor surface. This discrepancy will be investigated in the future. As already done for previous experiments without beam [START_REF] Gojon | Experimental investigation of low Reynolds number rotor noise[END_REF], all those results will be made available shortly as an open database, on https://dataverse.isae-supaero.fr/.

IV. Conclusion

The radiation characteristics of rotor-airframe interaction noise are studied experimentally and numerically for a low Reynolds two-bladed small-scale rotor in interaction with three cylindrical beams. The rotor-beam airflow interaction does not have any significant impact on aerodynamic loads. In particular, the thrust is found to grow quadratically with the rotational speed (RPM) for all cases. In terms of acoustics, the presence of the beam does not seem to affect the blade passing frequency (BPF) and the high frequency broadband noise sound pressure levels (SPL) radiation patterns. For a fixed beam diameter, the SPL of BPF and high frequency broadband noise simply increases with the rotational speed while being shifted towards higher frequencies. However, the beam significantly increases (up to 30 dB) the BPF harmonics 2×BPF to 15×BPF SPL. This increase grows with the rotational speed and the beam diameter with a highest SPL for BPF harmonics typically between 5×BPF and 6×BPF. The most intense BPF harmonics exhibit a horizontal dipole-like radiation pattern aligned with the beam axis. The experimental campaign allows also to assess the reliability of the proposed coupled iLES / F-WH approach, which is validated for the dominant BPF harmonics. This allows to use the ability of numerical simulations to predict the radiated acoustic field related to different parts of the setup in order to discriminate the noise contributions. It shows that the level increase and the characteristic directivity pattern of the dominant BPF harmonics are beam-related.

φ

  = Azimuth angle ( • ) θ = Latitude angle ( • ) BGN = Background noise (dB) BPF = Blade passing frequency (Hz) O ASPL = Overall sound pressure levels (dB) Q = Torque (N.mm) T = Thrust (N) RPM = Rotation per minute (RPM) SPL = Sound pressure levels (dB) U AV = Unmanned air vehicles

Fig. 1

 1 Fig. 1 Photograph of the experimental setup: two-bladed rotor with the D20 beam located L = 20 mm downstream of the rotor disk plane.

Fig. 2

 2 Fig. 2 Rotor and beam surface meshes (left) and cut of the mesh along the vertical direction (right) for the beam D20 positioned at L = 20 mm.

Fig. 3

 3 Fig. 3 Isocontour of Q criterion (Q = 10 6 1.s -2 ) for a rotational speed of 8000 RPM and the beam D20 positioned at L = 20 mm.

Fig. 4

 4 Fig. 4 Evolution of the Thrust (T) and the Torque (Q) as functions of the RPM. Comparisons with experiments.

Fig. 5 Fig. 6

 56 Fig. 5 Sound pressure level (SPL) at a latitude angle of θ = 0 • (top) and θ = 40 • (bottom) and at an azimuth angle of φ = 90 • for a rotational speed of 8000 RPM and the beams positioned at L = 20 mm.

Fig. 7

 7 Fig. 7 Sound pressure level (SPL) at latitude angles of θ = 40 • (left) and θ = 0 • (right) for a rotational speed of 8000 RPM, without beam.

Fig. 8 Fig. 9

 89 Fig. 8 Sound pressure level (SPL) at a latitude angles of θ = 40 • (left) and θ = 0 • (right) and at an azimuth angle of φ = 90 • for a rotational speed of 8000 RPM, D10 beam.

Fig. 10 3D

 10 Fig. 10 3D directivity of different noise components measured for the two-bladed rotor in interaction with the D10 beam at L = 20 mm and 8000 RPM: overall sound pressure level (OASPL), blade passing frequency (BPF) and its fifth harmonic, and the high frequency broadband noise.

Fig. 11 3D

 11 Fig. 11 3D directivity of the two-bladed rotor fifth blade passing frequency (BPF) harmonic in interaction with the D10 beam at L = 20 mm and 8000 RPM: comparison with numerical simulations.

Acknowledgments

The authors gratefully acknowledge financial support from the DGA. They are grateful to the entire ISAE-SUPAERO technical team for the success of the experimental campaign. The numerical part of this work was performed using resources from GENCI [CCRT-CINES-IDRIS] (Grant 2021-[A0102A07178] and 2022-[A0122A07178]). The authors thank Michel Roger and Stephane Moreau for their insights and advices.