domain: phys spi

text:
Cécile Ghouila-Houri

Romain Viard

Quentin Gallas

Eric Garnier

Abdelkrim Talbi

Philippe Pernod

High temperature gradient wall shear stress micro-sensors for flow separation control

This paper reports CMOS-compatible calorimetric micro-sensors enabling the bidirectional measurement of wall shear stress in aerodynamic flows. Measuring both the amplitude and the sign of the wall shear stress at small length-scale and high frequencies, the micro-sensors are particularly suitable for flow separation detection and flow control. Their design is hot-wire like with three parallel micro wires suspended over a micro-cavity and mechanically supported using periodic perpendicular micro-bridges. The micro-sensors were manufactured using microelectronics techniques. Implemented on a flexible packaging, the micro-sensors were calibrated in static and in dynamic, in flat plate configuration, inside a turbulent boundary layer wind tunnel. After calibration, twelve micro-sensors have been implemented in a flap model for active flow control experiments. The work included the design and manufacturing of appropriate miniaturized electronics. The micro-sensors successfully detected the natural flow separation. The efficiency of the active flow control for avoiding separation or reattaching a separated flow was also evaluated in this configuration using the micro-sensors.

I. Nomenclature

II. Introduction

NE of the main objectives in research on future vehicles concerns the reduction of fuel consumption and CO2 or toxic NOx gases emission. A solution consists in improving the vehicles aerodynamic performances by the mean of flow control. MacMynowski and Williams [START_REF] Macmynowski | Flow Control Terminology[END_REF] define flow control as a modification of the natural state of the flow to a wanted state. Lord et al [START_REF] Lord | Flow control opportunities in gas turbine engines[END_REF] consider that this is a small-scale perturbation for a large-scale effect. All in all, it consists in manipulating free stream or wall bounded flows, using active or passive devices, in order to produce favourable changes in the flow such as preventing separation, delaying laminar-turbulent transition, reducing noise, enhancing mixing... Flow control is thereby a technical solution for both safety and ecology issues. Boundary layer separation is a fluidic phenomenon concerned by flow control. It is a mostly unwanted phenomenon for many technical applications such as automotive industry, and even dangerous in aeronautics. Indeed, separation occurring over flaps is responsible for large performance losses especially during take-off as it simultaneously drastically increases drag and reduces lift [START_REF] Gad-El-Hak | Flow Control: The Future[END_REF][START_REF] Gad-El-Hak | Separation Control: Review[END_REF]. Passive and active devices have been manufactured to encounter this phenomenon but only active closed-loop control enables to adapt the control to the configuration (take-off, landing and cruise are three different configurations for an airplane) and external perturbations (cross-wind, vehicle overtaking…). Indeed, closed-loop control considers electrically commanded actuators for manipulating the flow, and a set of sensors indicating the state of the flow [START_REF] Gad-El-Hak | Flow Control: The Future[END_REF]. In flow separation, the wall shear stress plays a key role: the point of separation is characterized by a shear stress null at the wall and in the recirculation area it stress is negative (change of flow direction). Therefore, accurate timeresolved wall shear stress measurement is crucial for implementing a closed-loop control. Flows at moderate Reynolds numbers exhibits typical length scales of 100 µm at the wall or less and time-scales require several kilohertz of bandwidth [START_REF] Sheplak | Characterization of a siliconmicromachined thermal shear-stress sensor[END_REF]. Micro-electro-mechanical systems are thereby a technical solution to resolve turbulent flows. Micromachined wall shear stress sensors are usually divided into two groups depending on their measurement techniques: direct techniques using floating element devices [START_REF] Schmidt | Design and calibration of a microfabricated floating-element shear-stress sensor[END_REF][START_REF] Chandrasekharan | Characterization of a MEMS-based floating element shear stress sensor[END_REF][START_REF] Lv | Design of a micro floating element shear stress sensor[END_REF] and indirect techniques such as thermal anemometry [START_REF] Sheplak | Characterization of a siliconmicromachined thermal shear-stress sensor[END_REF][START_REF] Bailey | Turbulence measurements using a nanoscale thermal anemometry probe[END_REF][START_REF] Talbi | A micro-scale hot wire anemometer based on low stress (Ni/W) multi-layers deposited on nanocrystalline diamond for air flow sensing[END_REF][START_REF] Jiang | A flexible micromachine-based shear-stress sensor array and its application to separation-point detection[END_REF][START_REF] Xu | IC-integrated flexible shear-stress sensor skin[END_REF]. Löfdahl and Gad-El-Hak [START_REF] Löfdahl | MEMS-based pressure and shear stress sensors for turbulent flows[END_REF] and Sheplak et al [START_REF] Sheplak | MEMS shear stress sensors: promise and progress[END_REF] present reviews on MEMS based shear stress sensor and discuss the advantages and drawbacks of each techniques. In this paper is reported a micromachined wall shear stress micro-sensor based on both thermal anemometry and calorimetric principle and enabling the bidirectional measurement of wall shear stress in aerodynamic flows. In the first part of the paper, the design and fabrication of the micro-sensors are presented, along with calibrations in a turbulent boundary layer wind tunnel. Then, a second part presents the implementation of twelve micro-sensors in a flap model along with miniaturized electronics. A large wind tunnel allowed to perform flow control experiments on this model and the first results are presented and discussed in the final part of the paper.

III. The micro-sensor design, fabrication and calibration

A. Fabrication and characterization in cleanroom

The micro-sensors at stake in this paper are based on both thermal anemometry and calorimetric principle. They are indeed composed of an electrically heated micro-wire, called a heater, and two cold-wires placed on both sides. The central wire is devoted to the measure of wall shear stress using thermal anemometry meaning that the heat losses between the hot-wire and the fluid due to forced convection gives an image of the flow velocity near the wall. Designed as identical in dimensions and materials, the two lateral wires are symmetrically heated by the central hot-wire. The flow direction will break the thermal boundary layer symmetry and the upstream lateral wire will be more cooled than the downstream one (Fig. 1). This principle is called calorimetric. Coupling both principles, the micro-sensors are able to detect the wall shear stress amplitude and sign. This property gives a direct advantage for flow separation detection, as separation is characterized by both a diminution of the wall shear stress amplitude and an inversion of its sign.

O

Fig. 1 Scanning Electron Microscopy image of the sensitive part of the micro-sensor.

The micro-sensors sensitive part is then composed of three parallel wires whose dimensions are 1 mm long, three µm wide and less than 1 µm thick. The lateral wires are composed of a SiO2 layer for mechanical support and Ni/Pt multilayer forming the metallic measurement wire. The central hot-wire is engineered with multiple layers including SiO2, Ni/Pt multilayer, another layer of SiO2 and Au layer. The heating current cross the Au layer and the Ni/Pt multilayer is used for measurement. Contrary to conventional hot-wire sensors [START_REF] Bailey | Turbulence measurements using a nanoscale thermal anemometry probe[END_REF][START_REF] Talbi | A micro-scale hot wire anemometer based on low stress (Ni/W) multi-layers deposited on nanocrystalline diamond for air flow sensing[END_REF]; this microstructure uncouples heating and measure in electronics [START_REF] Ghouila-Houri | High temperature gradient calorimetric wall shear stress micro-sensor for flow separation detection[END_REF]. The wires are free from the substrate, suspended over a 20 µm deep cavity. Suspended periodic perpendicular microbridges are disposed along the wire for mechanical support (Fig. 1). The fabrication process includes seven CMOS-compatible main steps (CMOS for Complementary Metal Oxide Semiconductor) and needs four photolithography masks. The different steps leading to the 140 micro-sensors manufactured on a 3 in. wafer (Fig. 2 (a)) are presented in details in [START_REF] Ghouila-Houri | High temperature gradient micro-sensor for wall shear stress and flow direction measurements[END_REF]. Electrical and thermal characterizations have been performed on each micro-sensors leading to a measured TCR (Temperature Coefficient of Resistance) of 2380 ppm/°C and a heating elevation of 9°C/mW for the central wire and 5 °C/mW for the lateral wires [START_REF] Ghouila-Houri | High temperature gradient calorimetric wall shear stress micro-sensor for flow separation detection[END_REF]. Thermal imaging presented in Fig. 2 (b) demonstrates the efficiency of thermal insulation as the heating located in the wires and there are few heat losses into the micro-bridges.

B. Calibration in wind tunnel

After design, fabrication and characterizations, the micro-devices were calibrated in a turbulent boundary layer wind tunnel. The test section is 30 cm x 30 cm and the wind velocity goes up to about 40 m/s. The boundary layer was characterized using a hot-wire probe mounted on a motorized support (Fig. 3). These measurements enables to deduce the momentum thickness and the skin friction coefficient using Coles-Fernholz equation [START_REF] Nagib | Approach to an asymptotic state for zero pressure gradient turbulent boundary layers[END_REF] and finally the experimental wall shear stress [START_REF] Ghouila-Houri | High temperature gradient calorimetric wall shear stress micro-sensor for flow separation detection[END_REF]. For upstream U∞ velocities reaching 40 m/s, the corresponding shear stress reach up to 2.5 Pa.

The micro-sensor is designed for flow control meaning that its purpose is to resolve shear stress going from -0.2 (separated flow) to 40 Pa (near jets). In practice, the wind tunnel enables to calibrate the sensor up to 2.5 pa. The micro-sensors were mounted on the wind tunnel wall for calibration presented in Fig. 4. A flexible packaging enables to recover contacts using surface mounted component welding technique. Figure 4 (a) presents the central wire response to the wall shear stress amplitude variation. The micro-sensor response fits a polynomial like conventional thermal wall shear stress sensors [START_REF] Sheplak | Characterization of a siliconmicromachined thermal shear-stress sensor[END_REF]. Figure 4 (b) present the sensitivity of the lateral wires to the flow direction. For this experiment, the difference of resistance between the lateral wires is measured as an image of the difference of temperature (via the TCR). The micro-sensor was also characterized dynamically at high velocity in Fig. 5. This figure presents the power spectral density measured for 𝑈 ∞ = 38 𝑚/𝑠. The response is relatively flat up to 1 kHz and starts to decrease as the boundary layer losses energy. In this experiment however the impact of the cavity of flexible packaging appears. Indeed, as contacts are front-side, the sensor is a bit under the wall. The signal is then damped for low frequencies as the cavity slightly filters it.

V. Flow control experiments

Active flow control experiments were performed using the Festo® MHE2 valves as continuous or pulsed jets actuators and the micro-sensors for measurements. The first results are displayed in Fig. 9, 10 and 11 and deal with the natural separation detection, the control of separation and the control of a separated flow. For these last two point, the control objective differs according to the situation: if the flow is attached the goal will be to avoid or at least delay the separation (control of separation) whereas if the flow is separated the objective will be to reattach it (control of a separated flow).

All the presented results comes from the second individual sensor placed at 𝑥/𝑐 = 0.3, with an upstream velocity 𝑈 ∞ = 24.5 𝑚/𝑠 In Chabert's work, the flow separation point was deduced by comparing the hot-films responses two by two. Detecting both the amplitude and the flow direction, the micro-sensors developed here do not need such comparison: they can detect the flow separation by their own, independently one from the other. As shown on Fig. 9 in black symbols, the separation at 𝑥/𝑐 = 0.3 occurs for an angle of attack of about 17°: at this angle, the equivalent shear stress is zero and becomes negative for higher angles. The amplitude is measured using the central wire and the sign is deduced from the temperature difference between the two lateral wires. For steady blowing, it is even more important. This is explained easily considering the sensor position, which is not far from the actuators and the fact that the jets are supplied with 20 g/s of airflow. In future work, we are currently working on improving the electronics to enlarge the bandwidth up to several dozens of kilohertz. We also develop a new packaging to improve the signal to noise ratio and avoid damping signal. All these improvements are part of the development of sensors maturity in order to characterize the micro-sensors in other flow configurations.

x

  = position of the micro-sensor c = cord of the flap model τ = wall shear stress U∞ = flow velocity in the center of the wind tunnel test section

Fig. 2 (

 2 Fig. 2 (a) 3 in. wafer with 140 micro-sensors (b) Thermal imaging of the sensors supplied with 10 mW heating power.

Fig. 3

 3 Fig. 3 Boundary layer characterization using a hot-wire probe on a motorized supportThe micro-sensors are designed keeping robustness as an important parameter for they can measure real in flight with flow control shear stress. On one hand, shear stress on airplanes can reach a few Pascal, but introducing active flow control, it can reach up to several dozens of Pascal, near the actuation system. The micro-sensors are designed to support and accurately measure such shear stress however the wind tunnel used for calibration only reach up to 2.5 Pa meaning that future work on high shear stress values is needed.The micro-sensors work in constant temperature (CT) mode meaning that the central wire temperature is maintained constant by adapting the heating current crossing the Au layer in closed-loop control. The wire temperature is measured using the Ni/Pt multilayer resistance. The micro-sensors were mounted on the wind tunnel wall for calibration presented in Fig.4. A flexible packaging enables to recover contacts using surface mounted component welding technique. Figure4(a) presents the central wire response to the wall shear stress amplitude variation. The micro-sensor response fits a polynomial like conventional thermal wall shear stress sensors[START_REF] Sheplak | Characterization of a siliconmicromachined thermal shear-stress sensor[END_REF]. Figure4(b) present the sensitivity of the lateral wires to the flow direction. For this experiment, the difference of resistance between the lateral wires is measured as an image of the difference of temperature (via the TCR).

Fig. 4 (

 4 Fig. 4 (a) Central wire calibration in constant temperature mode (b) Difference of lateral resistances versus flow direction for 𝝉 = 2.4 Pa.

Fig. 5

 5 Fig. 5 Power Spectral Density of the micro-sensors at 𝑼 ∞ = 𝟑𝟖 𝒎/𝒔IV. Implementation of an array of micro-sensors in a flap model with miniaturized electronicsThe two following parts are dedicated to flow control experiments on a flap model. Flaps are used on airplanes (Fig.6(a)) to delay separation during take-off and landing. The flow control experiments planed using the micro-sensors follow the work of Chabert et al[START_REF] Chabert | Experimental detection of a periodically forced turbulent boundary layer separation[END_REF]. This work studies the natural flow separation on an inclined flap using tomoscopy flow visualizations and hot-film sensors measurements. Open loop control using Festo® MHE2 valves is also demonstrated. In this work, the same configuration is used but the hot-film sensors along the cord are replaced by the calorimetric micro-sensors (Fig.6(b)). The same actuators are used.

Fig. 6 (

 6 Fig. 6 (a) Flap on A340 (b) Schematic of the experiment with the flap model instrumented with the microsensors disposed along the cord. The experiment take place in the large L1 wind tunnel in ONERA Lille (Fig. 7 (a)). The wind tunnel test section has a diameter of 2.40 m and the length of the working area is about 10 meters. The flap model is placed between two plates to avoid side effects. Therefore, the flow is almost two-dimensional at the center of the flap. A 867 mm long flat plate stabilize the boundary layer before the flap. The flap is unslotted, has 𝑐 = 220 𝑚𝑚 long chord and is based on NACA 4412 airfoil shape. As seen on Fig. 7 (b), twelve micro-sensors have been implemented in the flap model with embedded miniaturized electronics designed and manufactured by Fluiditech. Two of them are individual sensors with proper CT electronics and are located close to the leading edge of the flap. A flexible 3 mm thick strip of ten sensors with mutual electronics takes place in the thinner part of the model and reaches up to 30 mm far from the trailing edge. The twelve sensors are flush-mounted in the model and the flexible strip enables to follow the airfoil shape camber.

Fig. 7 (

 7 Fig. 7 (a) Flap model with implemented micro-sensors (b) Zoom on the twelve micro-sensors: two individual sensors at the leading edge and a strip of ten sensors with common electronics Figure 8 shows in (a) a zoom on the leading edge sensors and in (b) their individual miniaturized CT electronics. The micro-sensors electronics is only 3 mm high and 50 mm long. The micro-sensors are flush-mounted at the flap model surface using PMMA support manufactured for following the flap camber. Figure 8 (c) shows the 115 mm long flexible strip with a focus on two micro-sensors. There are 10 micro-sensors embedded into the flexible strip, 8 mm away one from the other. Being flexible and 3 mm thick, the strip matches the flap camber. The small volume is a key parameter for the success of the micro-sensors implementation in the model. Figure 8 (d) shows an instrumentation electronics box. Three are needed: one for each individual sensor and one for the strip. The three boxes present dimensions small enough to be embedded inside the flat plate upstream of the flap.

Fig. 8 (

 8 Fig. 8 (a) The two leading edge micro-sensor (b) Individual CT electronics for the leading edge sensors (c) Zoom on three sensors from the strip (d) One of the three instrumentation electronics box embedded inside the flat plate.

Fig. 9

 9 Fig. 9 Control of separation: normalized response of the micro-sensor at 𝒙/𝒄 = 𝟎. 𝟑 without control and with pulsed jet control.

Fig. 10 (

 10 Fig. 10 (a) Natural separation: PSD at different angles of attack without control; (b) Control of separation: PSD at 2° (attached) 30° (separated) and 30° with pulsed jets.

Figure 11 (

 11 Figure11(a) presents the control of a separated flow. At 24° of angle of attack, the flow is naturally fully separated, and the shear stress amplitude is less important. When turning on the active flow control with either steady blowing or pulsed jets, the shear stress increases. For pulsed jet, the mean amplitude nearly reach the one of the attached flow. For steady blowing, it is even more important. This is explained easily considering the sensor position, which is not far from the actuators and the fact that the jets are supplied with 20 g/s of airflow. Figure11 (b)shows the separation of the controlled flow when the actuation is turned off.

  Figure11(a) presents the control of a separated flow. At 24° of angle of attack, the flow is naturally fully separated, and the shear stress amplitude is less important. When turning on the active flow control with either steady blowing or pulsed jets, the shear stress increases. For pulsed jet, the mean amplitude nearly reach the one of the attached flow. For steady blowing, it is even more important. This is explained easily considering the sensor position, which is not far from the actuators and the fact that the jets are supplied with 20 g/s of airflow. Figure11 (b)shows the separation of the controlled flow when the actuation is turned off.

Fig. 11 (

 11 Fig. 11 (a) Control of a separated flow: response of the micro-sensor at 𝒙/𝒄 = 𝟎. 𝟑 at 0°, 24° without control, 24° with steady blowing and 24° with pulsed jet control (b) Natural separation after turning off the actuators

VII. Acknowledgments

The French National Research Agency (ANR) in the framework of the ANR ASTRID "CAMELOTT" Project funded this work. It was supported by the regional platform CONTRAERO in the framework of the CPER ELSAT 2020 Project. The ELSAT 2020 project is co-financed by the European Union with the European Regional Development Fund, by the French State and the Hauts de France Region under the State Region Contracts (CPER). The authors also thank RENATECH, the French national nanofabrication network, and the ONERA Specific Devices and Models unit, which provided the design and fabrication of the test model for the micro-sensors implementation.

VIII. References


Antoine Hajczak
email: antoine.hajczak@onera.fr

Laurent Sanders
email: laurent.sanders@onera.fr

Vuillot François
email: francois.vuillot@onera.fr

Philippe Druault
email: philippe.druault@upmc.fr

Investigation of the Ffowcs-Williams and Hawkings Analogy on an Isolated Landing Gear Wheel

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Investigation of the Ffowcs-Williams and Hawkings Analogy on an Isolated Landing Gear Wheel

Antoine Hajczak * , Laurent Sanders † , François Vuillot ‡ ONERA -The French Aerospace Lab, F-92322 Châtillon, France Philippe Druault § Sorbonne Université UPMC Paris 6, CNRS UMR 7190, Institut Jean le Rond d'Alembert, F-75005 Paris, France This paper addresses the use of the so-called solid and permeable formulations of the Ffowcs-Williams & Hawkings (FW-H) equation in the scope of bluff body noise predictions. The aerodynamic turbulent flow is computed around and downstream of an isolated landing gear wheel from the LAGOON configuration by means of a Zonal Detached Eddy Simulation initiated in a previous study at a Mach number of 0.23. The aerodynamic field obtained with this simulation is then used to compute the FW-H integral respectively on the wheel solid walls and on a permeable surface that encloses the wheel and the most turbulent portion of its wake. The solid FW-H results are shown to be little influenced by the accurate resolution of the wake, with differences in the integrated levels of less than 2 dB only in the upstream direction. The comparison between the solid and permeable formulation results allows to check that at this Mach number, the quadrupolar contribution of the sources present in the wake of this landing gear wheel can be discarded for Strouhal numbers up to 19.3 (5 kHz). As previously observed in several studies, removing the endcap closure of the permeable surface reduces the spurious noise induced by the eddies passing through it. The influence of the FW-H surface discretisation is assessed by considering two levels of resolution. In this study, refining the outflow surface has shown to greatly reduce the spurious noise while the integration surface was kept closed.

I. Nomenclature

II. Introduction

T he aeronautical industry is currently facing the challenge of managing its environmental impact in order to accommodate the continued growth of both civilian air traffic and suburban areas. More and more people, as well as ecosystems, are being subject to noise pollution, and its mitigation has consequently become a major issue. Within the last decades, jet noise has been extensively addressed by the aeroacoustic community, leading to the design of much quieter turbofan engines. This reduction in engine noise has given way to other noise sources that were of secondary concern before, particularly at landing conditions, when aircraft noise is the most importunate. Indeed, the unsteady turbulent flow around high-lift devices and especially landing gears has proven to generate noise levels that could reach the ones related to the engines when operating at reduced thrust. The predominance of landing gear among airframe noise can be explained by its extremely complicated geometry that comprises a large number of blunt bodies of various size and shape. This particularity has two major consequences. First, complex aeroacoustic mechanisms take place due to massive flow detachments and multiple interactions of turbulent wakes with solid surfaces. Second, the computation of the turbulent flow in the vicinity of the gear can be challenging because of the grid work required, especially if a structured approach is adopted. At first, noise predictions of landing gears relied on the use of semi-empirical models (eg Fink [START_REF] Fink | Airframe noise prediction method[END_REF] and Guo [START_REF] Guo | A statistical model for landing gear noise prediction[END_REF]) but over the last decades, the progress achieved in computational fluid dynamics as well as the increase in computational power has allowed the simulation of the turbulent flow field around a landing gear of moderate complexity with very good accurracy [START_REF] De La Puente | Aeroacoustic simulations of landing gears with unstructured grids and a ZDES turbulence model[END_REF][START_REF] Casalino | Lattice-Boltzmann aeroacoustic analysis of the LAGOON landing-gear configuration[END_REF][START_REF] Murayama | Computational study of low-noise fairings around tire-axle region of a two-wheel main landing gear[END_REF][START_REF] Spalart | Initial noise predictions for rudimentary landing gear[END_REF]. In aeroacoustic studies, the quantity of interest is the far-field acoustic pressure radiated by the interaction of the landing gear with the flow. However, its direct computation still remains unaffordable and noise predictions of a particular landing gear geometry rely most of the time on hybrid methods. These methods consist in two steps that are (i) the computation of the unsteady turbulent flow in the vicinity of the gear thanks to a Navier-Stokes or Lattice-Boltzmann Method solver and (ii) the extrapolation of the turbulent flow to far-field acoustic perturbations by means of an acoustic analogy. Since the seminal work of Lighthill, several acoustic analogies have been derived. The most popular in the case of landing gear noise is the Ffowcs-Williams & Hawkings [START_REF] Williams | Sound generation by turbulence and surfaces in arbitrary motion[END_REF] (FW-H) analogy that consists in an inhomogeneous wave equation with source terms accounting for the presence of solid surfaces. The resolution of this equation leads to the integration of the source terms on an arbitrary closed control surface and in its outer volume. The interpretation of the FW-H equation can further lead to two formulations known respectively as the permeable and solid formulations, depending on how the integration surface is chosen. The most general is the permeable one and is very similar to the Kirchhoff method. This formulation takes into account the quadrupolar volume sources enclosed by the integration surface as well as the interaction with the solid surface of the body. Despite its general nature, this formulation has the main drawback of being computationally expensive (memory and CPU) and leads to problems related to the crossing of the surface by the wake of the body. This issue is well known and has been observed in many studies. Among others, Mendez et al. [START_REF] Mendez | On the use of the Ffowcs Williams-Hawkings equation to predict far-field jet noise from large-eddy simulations[END_REF] have established a "best practices" list in order to cope with the problems that can arise with the use of the permeable surface. The second formulation is the solid one and is the most popular in landing gear noise predictions as it avoids the problems related to the wake and is additionally less computationally expensive. These advantages come with the major hypothesis that the quadrupolar sound in the wake of the body is negligible. A specificity of landing gear noise is that, by definition, it is only relevant in landing phase where the Mach number upper bound for a commercial airplane is roughly 0.25. This argument is often invoked to justify the use of the solid formulation as it is commonly stated that, at low Mach number, the dipolar sound is much more efficient than the quadrupolar sound as their contributions scale respectively with M 6 and M 8 . This result, however, has been questioned, by Spalart [START_REF] Spalart | On the precise implications of acoustic analogies for aerodynamic noise at low Mach numbers[END_REF] who found that the scaling law for quadrupoles in the presence of dipoles was M 7 , rather than M 8 .

Furthermore, the use of a formulation instead of the other remains controversial in the scope of bluff body noise predictions as some studies have led to interesting results with the permeable one. For example, Spalart et al. [START_REF] Spalart | Initial noise predictions for rudimentary landing gear[END_REF] and Souliez et al. [START_REF] Souliez | Landing Gear Aerodynamic Noise Prediction Using Unstructured Grids[END_REF] have found a possible importance of near-field quadrupoles with the permeable surface for Mach numbers below 0.23. Additionally, Greschner at al. [START_REF] Greschner | Prediction of sound generated by a rod-airfoil configuration using EASM DES and the generalised Lighthill/FW-H analogy[END_REF] have even pointed out that the quadrupolar sound emitted by a rod airfoil configuration at M=0.2 was no longer negligible and that the permeable formulation better matched the experimental results of Jacob et al. [START_REF] Jacob | A rod-airfoil experiment as a benchmark for broadband noise modeling[END_REF] at high frequencies for an integration surface surrounding the whole rod-airfoil system and a portion of its wake.

Among the components of a landing gear, wheels are a major contribution to the acoustic far-field radiation [START_REF] Liu | Landing-gear noise prediction using high-order finite difference schemes[END_REF] and have gained considerable attention over the past few years [START_REF] Casalino | Facing rim cavities fluctuation modes[END_REF][START_REF] Wang | The noise generated by a landing gear wheel with hub and rim cavities[END_REF][START_REF] Wang | Interaction Noise from Tandem Landing Gear Wheels with Hub and Rim Cavities[END_REF][START_REF] De La Puente | Investigation on landing gear shallow round cavity flow field and noise signature[END_REF][START_REF] Ma | Aerodynamic and Acoustic Measurements of a Single Landing Gear Wheel[END_REF]. Moreover, the wheels of a landing gear contribute significantly to the turbulent wake of the whole gear, hence potentially inducing noise sources of quadrupolar nature. This is even more plausible in the case of a four-wheel landing gear, as the wheel-to-wheel interaction generates an intense turbulent wake.

The aim of this paper is to assess the use of the FW-H equation in the scope of landing gear noise predictions at low Mach number. In particular, this paper aims at clarifying (i) the influence of the wake resolution on the solid FW-H results, (ii) the role of the quadrupole noise in the case of an isolated wheel and (iii) the effects related to the problem of the outflow surface when the permeable surface approach is chosen.

It is organised in two sections. The first one is dedicated to the description of the test case and aerodynamic results. The second one presents the results of the FW-H computations and their interpretation.

III. Description of the test case and flow features

This paper further investigates the noise emitted by an isolated landing gear wheel from the LAGOON geometry initiated in a previous paper [START_REF] De La Puente | Investigation on landing gear shallow round cavity flow field and noise signature[END_REF]. In this section, the wheel geometry and main dimensions are first briefly recalled. Then, the numerical setup for the computation of the turbulent flow as well as the FW-H surface integral are described. Finally, an analysis of the wake downstream of the wheel is carried out in order to quantify the passage of the wake across the FW-H surface, which is known to be a critical issue in noise predictions.

A. Geometry and computational setup

Main dimensions of the LAGOON wheel

The LAGOON geometry (see figure 1a) is a 1:2.5 scaled canonical two-wheel nose landing gear which has been designed in the scope of the LAGOON experimental campaign led by Manoha et al. [START_REF] Manoha | LAGOON: an experimental database for the validation of CFD/CAA methods for landing gear noise prediction[END_REF][START_REF] Manoha | LAGOON: further analysis of aerodynamic experiments and early aeroacoustics results[END_REF] in 2007. This campaign then brought together several teams to compare their numerical computations to the experimental results in the framework of the Benchmark on Airframe Noise Computation (BANC). These computations included structured and unstructured methods, Chimera techniques, and Lattice-Boltzmann Methods (for a detailed review of these contributions, please refer to the summary presented by Manoha et al. [START_REF] Manoha | Summary of the LAGOON solutions from the Benchmark problems for Airframe Noise Computations-III Workshop[END_REF] in 2015). A single wheel from this geometry, without the axle, will be considered in this study (see figure 1b). Its external diameter is D w = 0.3m and the whole wheel can basically be considered as an installed shallow round cavity of internal radius r = 0.081m and depth h = 0.037m. Cavity noise is a phenomenon that has been extensively covered by the aeroacoustic community. Hence, cavities are classified according to their depth to diameter ratio κ. In the case of the LAGOON cavity, this parameter has a value κ = 0.23, which ranks it in the shallow cavities category. A common feature of cavity flows is that they often induce a strong tonal response that is attributed by several authors to the coupling between a depth mode and a Rossiter-like feedback loop [START_REF] Marsden | Investigation of flow features and acoustic radiation of a round cavity[END_REF][START_REF] Block | Noise response of cavities of varying dimensions at subsonic speeds[END_REF]. Casalino et al. [START_REF] Casalino | Facing rim cavities fluctuation modes[END_REF] have studied the two facing wheels of the LAGOON geometry and numerically recovered the tonal peaks at 1 and 1.5kHz already observed in the LAGOON experiments [START_REF] Manoha | LAGOON: an experimental database for the validation of CFD/CAA methods for landing gear noise prediction[END_REF]. Previous work on the isolated wheel showed that the 1.5kHz tone was still present, but much weaker than in the case of the two facing wheels [START_REF] De La Puente | Investigation on landing gear shallow round cavity flow field and noise signature[END_REF]. Fig. 1 LAGOON geometry and cut of the isolated wheel in a median plane

Navier-Stokes setup Computational domain and numerical resolution

The computational domain is a rectangular box of dimensions (L x , L y , L z ) = (40D w , 24D w , 18D w ). In order to compute the turbulent flow around the wheel, ONERA's in-house code CEDRE [START_REF] Refloch | CEDRE software[END_REF] has been used to perform a Z-DES mode II [START_REF] Deck | Recent improvements in the Zonal Detached Eddy Simulation (ZDES) formulation[END_REF] based on the one-equation turbulence model Spalart-Allmaras. A relaxation parameter has been applied on the boundaries in order to prevent any spurious reflexions on the boundaries of the computational domain. A 2nd order implicit Runge-Kutta temporal scheme has been used, coupled with a 2nd order HLLC flux scheme.

The upstream conditions are that of the LAGOON BANC-II problem statement in the CEPRA19 (ONERA's anechoic facility) configuration, namely P ∞ = 96772.3Pa, T ∞ = 288.39K, ρ ∞ = 1.18kg/m 3 and M ∞ = 0.23.

The calculation has been initialised with a RANS during 214ms before the Z-DES was launched. Then, after a Z-DES transient period, the flow has been solved for a total timelength of 328 ms with a time step ∆t C F D = 5.10 -6 s, producing a total number of 65536 samples. Otherwise specified, all the signals considered in this study will have a duration of 328 ms. As for the computation of the power spectral densities, the Welch estimator will be used by windowing the blocks with a Hann function and averaging them to obtain a frequency resolution ∆ f = 30H z.

Mesh description

In this study, two hybrid meshes M1 and M2 are considered. A cut of these two meshes in the median plane of the wheel is presented in figures 2a-2b. The M1 mesh is the one used in [START_REF] De La Puente | Investigation on landing gear shallow round cavity flow field and noise signature[END_REF] and directly stems from the mesh that has been corresponding to this refinement is such that D w /∆ > 120. This mesh has been designed with the idea that the flow at the direct vicinity of the wheel was mainly driven by the shear layer. It must be stressed that with the M1 mesh, the wake of the wheel is not resolved at all, as well as the flow on the external part of the cavity. The numerical resolution of the boundary layers is expected to give good results for the solid FW-H formulations. On the other hand, this mesh is expected to be poorly adapted to the permeable FW-H assessment as the resolution of the wake is very uncertain.

In this regard, a second mesh has been designed with emphasis on the accurate resolution of the wake of the wheel, as well as the correct propagation of the acoustic waves to the permeable surface. The second mesh M2 hence possesses exactly the same features than M1, except that, this time, the cell size downstream of the wheel allows an accurate resolution of the wake dynamics from the wheel to the endcap closure of the FW-H surface (a description of this surface will be given in the following paragraph).

The evolution of the cell diameter in the streamwise direction is depicted in figure 3. As we can see, the cell size has been roughly divided by 10 in the wake of the wheel from M1 to M2, hence giving a local cell size that ensures the propagation of acoustic waves up to a frequency of about 5 kHz (St D w = 19.3) with negligible dissipation and dispersion errors. Great care has been taken when choosing the cell stretching ratio in the streamwise direction in order not to generate spurious noise sources that would eventually be captured by the FW-H surface. This has led to a M2 mesh of 60 million elements (12.5 million nodes).

Ffowcs-Williams & Hawkings setup

ONERA's in-house code KIM [START_REF] Prieur | Aeroacoustic integral methods, formulation and efficient numerical implementation[END_REF] is used in this paper to compute the acoustic far-field. In this paragraph, we detail the setup of the FW-H calculations. In particular, the observation points, where the FW-H integral is computed are described. The design of the permeable surface is another crucial point that results from a trade-off between its discretisation (and eventually the memory cost associated) and its location relatively to the wake of the wheel as it is common that hydrodynamic perturbations crossing the surface induce spurious noise in the far field.

Observation surface

In all calculations, the observer points constitute a sphere of diameter 10D w . The center of this sphere coincides with the center of the wheel round cavity, in the plane grazing it. The whole sphere is composed of two half spheres, both having an azimuthal angle step ∆θ = 10°. The polar angle step is ∆φ = 10°on the half sphere over the cavity, while the other half (underneath the cavity) has a polar angle step equal to ∆φ = 11.5°. The flow is directed towards x > 0, hence coming from θ = 0 • while the direction φ = 0 • corresponds to the pure flyover direction with respect to the cavity. The observation surface thus defined is presented in figure 4a. In this paper, for the sake of clarity, only the results in the θ = 0 • plane will be presented, as it is the plane that presents the most significant results. In a whole landing gear configuration, this would correspond to the pure sideline plane.

Integration surfaces

A closed surface containing the wheel and a part of its wake has been defined. Its design has been guided by the concern of minimizing the effects related to the wake that crosses it, as will be seen in the following paragraph. Consequently, this surface has a shape that follows approximately the expansion of the wake downstream of the wheel and extends to a distance of 3.81D w from the wheel center. In the direction transverse to the flow, the dimension of the surface is 1.67D w . A cut of the permeable FW-H surface is presented in figure 4b, where the black dots correspond to numerical probes that will be used in the next paragraph for the assessment of the wake velocity and near-field pressure spectra. In order to investigate the influence of the surface discretisation, two cases are defined: a fine surface, with rectangular surface elements of about dS F = 2.5.10 -5 m 2 in order to be as close as possible to the CFD mesh cell size, and a coarse one, obtained by coarsening the surface elements downstream of the wheel such as dS C = 16dS F in the wake region, the wheel part being kept at the resolution dS F . A thorough analysis of the mean and unsteady flow results inside the cavity has already been carried out in [START_REF] De La Puente | Investigation on landing gear shallow round cavity flow field and noise signature[END_REF] and will not be recalled here for the sake of brevity. This analysis allowed to conclude that the LAGOON shallow circular cavity behavior compares well with the literature of grazing cavities as for the development of the shear layer, as well as the wall pressure inside the cavity. The computation on the M2 mesh allows to complete this analysis with some results concerning the development of the wake downstream of the wheel. Please note that, in this subsection, the results presented have all been obtained on the M2 mesh.

In order to address the wake dynamics, seven numerical probes have been placed downstream of the wheel, all equally spaced of 0.5D w from the location x = 0.53D w to x = 3.53D w . An eighth probe has also been placed on the endcap closure of the FW-H surface at x = 3.81D w . The power spectral densities of the transversal velocity v as well as the vertical velocity w are plotted in figure 5 as a function of the Strouhal number. As we can see, there is no evidence of a vortex-shedding like phenomenon in the wake of the wheel. The f -5/3 canonical Kolmogorov power law in the inertial subrange has also been plotted for reference.

The mean streamwise velocity profile U U ∞ has been plotted in spanwise and transverse cuts in figure 6 and allows to appreciate the decay of the wake inside the permeable surface (recall that the latter extends to x * = 3.81). In particular, the mean velocity at x * = 3.81 is about 10% of the upstream velocity. A slight asymmetry that tends to vanish with the distance is visible in the transverse direction in figure 6b (note that the z > 0 direction is oriented towards the inside of the cavity). The maximum velocity is obtained over the wheel (negative z values), where the recirculation inside the cavity takes place.

In order to assess the pressure field in the vicinity of the wake, a probe has been placed on the upper surface of the FW-H surface at the location x/D w = 2.71 and at a vertical distance of 1.2D w from the cavity floor. The PSD of the pressure is plotted in figure 7 and shows a wide emergence at 1850 Hz in the M2 computation that could hardly be detected on the M1 computation. This figure allows to appreciate how the M2 mesh has improved the propagation of the sound waves around the wake of the wheel compared to the M1 case. In particular, the CFD mesh cut-off seems here to be slightly greater than the 5 kHz announced in the first section.

The crossing of the surface by hydrodynamic fluctuations is known to generate spurious noise in the far-field when the permeable FW-H equation is considered. The RMS levels of the pressure on the endcap closure of the FW-H surface are represented in figure 8 in order to quantify this phenomenon. As we can see, only a small fraction of this surface is affected by significant RMS levels that may pollute the solution in the far-field. Among the two principal spots of high RMS levels, we can see that the most intense corresponds to positive values of z, ie under the wheel.

IV. Far-Field Acoustics

In this section, the FW-H equation is first recalled and a brief description of its terms is made. Then, the effect of the time signal length is assessed in order to prove the convergence of the spectra that are shown in all the rest of the paper. Finally, the results of the FW-H calculation are presented for different configurations and discussed.

A. The Ffowcs-Williams & Hawkings equation

The process by which the energy carried by a turbulent flow is converted to acoustic perturbations is described by the Lighthill equation. This equation is obtained by rearranging the Navier-Stokes equations into an inhomogeneous wave equation that can then be solved with the Green's functions formalism. This methodology forms the basis of an acoustic analogy. By using an acoustic analogy, one thus avoids the high computational cost necessary to accurately propagate sound waves to the far-field without too much dissipation and dispersion. Ffowcs-Williams & Hawkings [START_REF] Williams | Sound generation by turbulence and surfaces in arbitrary motion[END_REF] have then extended this methodology to account for the presence of moving surfaces in the flow. Let (Σ : f = 0) be a mathematical surface enclosing the body of interest, such as f < 0 inside the surface, f > 0 outside and ∇ f = n. The surface is moving with respect to the reference frame at a velocity v Σ and the flow velocity is u. After the developments 8 Root mean square iso-contours of the pressure on the endcap closure of the FW-H surface of Di Francescantonio [START_REF] Di Francescantonio | A new boundary integral formulation for the prediction of sound radiation[END_REF] that allow the surface to be permeable, the FW-H equation writes:

2 c 2 ( ρ -ρ ∞ )H ( f ) = ∂ ∂t ρ ∞ U n δ( f ) - ∂ ∂x i L i δ( f ) + ∂ 2 ∂x i ∂x j T i j H ( f )
where

U n = u n + ( ρ/ρ ∞ ) -1 (u n -v n Σ ) L i = P i j n j + ρu i (u n -v n Σ ) T i j = P i j + ρu i u j -c 2 ( ρ -ρ ∞ )δ i j P i j = p δ i j -τ i j
The solution can be analytically expressed by convolution with the free-space 3D Green's function and thus writes:

4πc 2 ( ρ -ρ ∞ )H ( f ) = ∂ 2 ∂x i ∂x j V ( f >0) T i j r et r |1 -M r | (a) dV - ∂ ∂x i Σ [L i ] r et r |1 -M r | (b) dS + ∂ ∂t Σ ρ ∞ [U n ] r et r |1 -M r | (c) dS
where [] r et means that the time-dependant variable inside the brackets is evaluated at the retarded time τ r et = tr c 0 , r being the source-observer distance.

The calculation of the far-field acoustic pressure requires the evaluation of the three integrals described previously, that are respectively known as:

(a) : the volumic quadrupolar source term that accounts for the noise generated by the turbulence itself, ie the time variation of the vorticity in the regions of strong shear, (b) : the surface loading source term that describes the noise generated by the interaction between turbulent fluctuations and solid boundaries of the moving body, (c) : the surface thickness term related to the kinematics of the body. This equation has been derived from the Navier-Stokes equations without any loss of generality, under the assumption that the acoustic perturbations and the flow are not coupled. The major drawback is then the computational cost associated with the evaluation of the Lighthill stress tensor T i j in the volume exterior to the integration surface. While it is theoretically possible to compute the (a) term outside of the integration surface defined in the first section, the computational cost required is completely out of reach for industrial applications. In practice, two approaches are then possible.

• The first, and the most commonly used, is known as the solid formulation of the FW-H equation. The basic idea is to take the surface Σ coincident with the solid boundaries of the body considered so that only the pressure has to be saved on the rigid walls. Plus, two assumptions are made: the walls are impermeable and non-vibrant, and the Lighthill stress tensor is negligible everywhere. If these two assumptions are made, then the FW-H equation reduces to the Curle equation [START_REF] Curle | The influence of solid boundaries upon aerodynamic sound[END_REF] and the computational cost is significantly reduced. This formulation is then very convenient for industrial applications but one has to be careful that the assumptions made are valid when employing it. • The second approach is to design the Σ control surface so as it contains the body and all the most turbulent regions. The idea is that the Lighthill stress tensor vanishes outside the integration surface, so that the V ( f >0) term can be discarded. This actually results in a Kirchoff-like integration, provided that the flow leaving the surface on its endcap closure is not too strong, so the surface can be considered in the linear region. It has been suggested that the quadrupolar sources crossing the surface might be chopped into two dipoles that are much more efficient acoustically [START_REF] Greschner | Prediction of sound generated by a rod-airfoil configuration using EASM DES and the generalised Lighthill/FW-H analogy[END_REF], and the one inside the surface would eventually cause spurious noise generation. Several authors have derived corrective terms based on the calculations of the Lighthill source term flux across the surface in order to mitigate this phenomenon [START_REF] Rahier | Additional terms for the use of Ffowcs Williams and Hawkings surface integrals in turbulent flows[END_REF][START_REF] Ikeda | Quadrupole Corrections for the Permeable-Surface Ffowcs Williams-Hawkings Equation[END_REF]. Another idea brought by Shur et al. [START_REF] Shur | Noise prediction for increasingly complex jets. Part I: Methods and tests[END_REF] consists in averaging several outflow surfaces in order to remove the hydrodynamic component from the integration. These methods, however, have not been assessed in the present study but this constitutes an interesting perspective for landing gear noise as they have mainly been used for jet and trailing edge noise. In this paper, the solid and permeable formulations are alternatively considered in order to compare the results they provide at a typical landing Mach number of 0.23 and to assess the contribution of the quadrupolar sound.

B. Convergence of the spectra

First, three timelengths have been considered to compute the solid FW-H integral with the wall pressure perturbations obtained with the M1 mesh configuration in order to check the convergence of the spectra that will be presented in the rest of the paper. These three timelengths are hereafter referred to as T 1 , T 2 , and T 3 and equal respectively to 33.25, 85.20 and 118.45 T D w where T D w = D w U ∞ . The spectra at the location (θ = 0 • , φ = 30 • ) (that is, 30 • upstream from the flyover direction) are plotted in figure 9 where the dashed vertical line corresponds to a frequency f = 1571H z. As we can see, increasing the integration time has allowed a more tonal behavior to be detected at 1.5 kHz than what was stated in the previous paper [START_REF] De La Puente | Investigation on landing gear shallow round cavity flow field and noise signature[END_REF]. It should be highlighted that, even if the extension of the integration time length has led to a peakier spectrum, the latter still remains far from the classical tones usually observed in academic cavity studies, as the peak is still 400 Hz wide. This is consistant with the study made by Marsden et al. [START_REF] Marsden | Investigation of flow features and acoustic radiation of a round cavity[END_REF] who state that no strong acoustic tone can be generated in cavities with depth to diameter ratio such as κ ≤ 1. The T 3 spectrum is almost identical to the T 2 one for frequencies higher than 200 Hz, hence showing that a T 2 = 328ms integration time is enough in the present case. We suppose that this conclusion holds for the permeable formulation and for all the observation points and, from now on, a timelength of 328 ms will be used in the following subsections. In what follows, unless specified, all the spectra and integrated levels will be considered in the [200; 5000]Hz interval. The lower bound of this interval is thus limited to 200 Hz by the available 328 ms of time signals, and the upper bound by the CFD mesh cut-off frequency indicated in the first section.

Solid formulation: M1 vs M2

The aim of this paragraph is to study how an accurate resolution of the wake downstream of a bluff body influences the results obtained with the solid FW-H formulation. In this regard, the results obtained with the solid formulation in the case of the M1 and M2 computations will be compared. The surface mesh over the rigid walls has a similar resolution in both cases. The only difference between M1 and M2 is the refinement downstream of the wheel that allows an accurate resolution of the wake. The results at four observation points in the median plane are plotted in figures 10-11. The overall agreement is satisfying but, still, several observations can be made. First, the 1.5 kHz peak is equally well recovered in the direction φ = 30 • (see fig. 10b). As already pointed out in [START_REF] De La Puente | Investigation on landing gear shallow round cavity flow field and noise signature[END_REF], the most plausible explication for the occurrence of this tone is a coupling between a depth mode of the circular cavity and a Rossiter mode. This phenomenon is expected to be mainly driven by the shear layer, which is already accurately described by the M1 mesh. Another interesting point is that the noise predictions are in very good agreement in all directions except the ones parallel to the flow (figs 10a-11b) where a difference of several dB can be observed in the high-frequency range (typically, for f ≥ 2k H z).

Finally, we can observe that the 1850 Hz pseudo-tone already observed over the wake in the previous section in figure 7 is better described by the M2 computation (see figure 11) for downstream observers.

These observations allow us to conclude that the overall effect of resolving the wake on the acoustic predictions is small with the solid formulation of the FW-H equation. The differences here are mainly in the high-frequency range, and in the upstream direction. This is further confirmed by the overall sound pressure levels (OASPL) calculated between 200 Hz and 5 kHz and presented in figure 12. It should as well be stressed that the differences in the integrated levels are of about 2 dB at most and occur for observation angles that are of limited interest for industrial applications. The results obtained with the solid formulation on the M1 and M2 meshes seem to indicate that accurately resolving the wake gives very similar results in both cases, except in the upstream direction, where a slight overestimation of the levels is found with the coarse mesh. A possible explanation for this difference could be the generation of a spurious noise source due to a too sudden coarsening of the CFD mesh downstream of the wheel. This source could then induce spurious wall pressure perturbations that would eventually appear in the FW-H integral. This seems to indicate that, even if the quadrupolar sources are neglected, one still has to be careful about the resolution of the wake, and in particular, about the stretching ratio imposed during the mesh generation process.

At this stage, the quadrupolar contribution of the sound is still unknown as the permeable formulation has not been used. The aim of the next paragraph is to compare the results obtained with the solid and the permeable formulations.

Closed surface -Influence of the FW-H surface resolution

In this paragraph, the FW-H equation is used in its permeable formulation with the downstream closure kept closed. The aim of this paragraph is twofold. Firstly, the comparison between the permeable and solid results should allow to conclude on the contribution of the wake quadrupoles to the total radiated noise. Secondly, the cell size of the integration surface is a parameter that is rarely mentioned in the litterature. Consequently, the surface introduced in the first section is divided into two cases. The first surface has a spatial resolution such that each point is located approximately 5mm away from its neighbours, resulting in surface elements of about dS = 2.5.10 -5 m 2 . The second surface has been obtained by coarsening the first one downstream of the wheel to obtain a distance of about 20mm between each point of the grid, thus resulting in surface elements of about dS = 4.10 -4 m 2 . The results obtained with these surfaces will be referred to as M2C-Cl (Mesh2-Coarse-Closed) for the coarse surface, and M2F-Cl (Mesh2-Fine-Closed) for the finer one. The same observation points as in the previous paragraph are considered and the spectra in the far-field are presented in figures 13 -14, where the M2S results are also plotted for the sake of comparison.

The results obtained in the M2C-Cl configuration exhibit unphysically higher levels compared to the solid M2S configuration. The difference between these two configurations is shown to increase when the observation point 0 Fig. 13 Spectrum of the far-field pressure signal computed at two positions in the upstream direction with a closed integration surface considered gets closer to the downstream direction. This phenomenon has already been described by numerous authors in the aeroacoustic community and is most likely due to the crossing of the endcap surface by the wake of the wheel. Curiously, apart from small downstream observation angles, the M2F-Cl results agree very well with the the M2S results although the surface remains closed at its downstream section, the only different parameter between M2C-Cl and M2F-Cl being the FW-H surface elements size on the integration surface.

The OASPL has been computed in the median (zOx) plane and is plotted in figure 15. This figure allows to appreciate how increasing the resolution of the FW-H surface yields results in much better agreement with the M2S computation. Still, discrepancies appear and grow as the observer direction gets close to downstream angles. The OASPL in the M2F-Cl configuration are slightly higher than those obtained with the solid formulation. As we can notice in figures 13 and 14, the agreement between the M2S and M2F-Cl cases is good (less than 2 dB) for observation angles φ < 200 • and φ > 310 • . This asymmetry echoes the one already observed in figure 8, although the link is not straightforward. Finally, the comparison between the M2S and the M2F-Cl results seems to indicate that the quadrupolar sources in the wake of this isolated landing gear wheel have a negligible amplitude under 5kHz, the differences with the M2S results being attributed to the crossing of the integration surface by the wake, rather than a quadrupolar contribution. As highlighted in figure 15a, the surface discretisation has a major effect on the far-field results as it increases the integrated

Influence of opening the endcap closure of the surface

In the previous paragraph, the discretisation of the FW-H surface has been varied from a coarse resolution to a finer one (dS C = 16dS F ), which has allowed the far-field results to fall in much better agreement with the solid results for the fine resolution, except in the downstream direction. The coarse FW-H surface, on the other hand, has led to unphysically high noise levels in all directions. The aim of this paragraph is to further investigate this phenomenon by opening the downstream endcap of the FW-H surface in the M2C and M2F configurations. The cases thus defined will be referred to as M2C-Op and M2F-Op respectively.

The results are plotted in figures 16 -17 and show that, with an open downstream section, the coarse and fine FW-H surfaces exhibit almost exactly the same results (with differences of less than 1dB). As for the M2F-Cl configuration, the spectra fall in very good agreement with the M2S configuration, except at low frequency ( f ≤ 400H z) and in the flow direction where the discrepancies are very high (figure 17b). Here, the fact that the M2F-Op and M2C-Op configurations yield almost exactly the same results allows to conclude that the effect of the surface discretisation on noise predictions is actually very localised on the downstream section of the integration surface.

To summarize, the OASPL have been plotted for all the cases in figure 18. Coarsening the FW-H integration surface has, in the present case, very little influence on the integrated levels when the downstream end is open. This seems to indicate that the key parameter is, not only the crossing of the surface by the wake, but more precisely the discretisation of the surface closure and suggests the existence of a link between the size of the structures and the surface elements involved in the calculation of the FW-H integral.

A snapshot of the fluctuating density is plotted on the coarse surface (figure 19a) and on the fine surface (figure 19b) to illustrate this phenomenon. It is clear that the turbulent structures are way better described on the fine surface than on the coarse one where artifacts in the integration process are likely to appear in the far-field due to the undersampling of the downstream surface. Further work will focus on determining the precise implications of the surface discretisation on the numerical calculation of the FW-H surface integrals.

V. Conclusions

The far-field acoustic radiation generated by an isolated landing gear wheel from the LAGOON configuration has been addressed with a hybrid method. First, the turbulent flow field surrounding the wheel has been computed by means of a Zonal Detached Eddy Simulation that allowed to accurately solve the shear layer over the shallow round cavity of the wheel. A second mesh has been designed to additionally resolve the wake of the wheel and investigate the contribution of the quadrupolar volumic sources to the far-field. The wheel cavity has shown to behave like an academic cylindrical cavity as for the development of the shear layer and the pressure signature inside the cavity. The wake downstream of the wheel has also been addressed on an appropriate mesh M2 and it has been shown that no vortex-shedding mechanism occurs downstream of the wheel. As the crossing of the FW-H closure is known to cause some problems related to the wake of the body, the mean profile of the streamwise velocity has been scrutinised and it has been shown that the wake strength is considerably reduced on the endcap closure of the FW-H surface as the mean streamwise velocity deficit is under 10% of the free-stream velocity on the downstream end of the FW-H surface. The FW-H computations with the solid formulation have first been carried on with the M1 and M2 meshes. It has been shown that the overall agreement between the two computations is very satisfying, except in the upstream direction where the M1 computation tends to give slightly higher noise levels, most likely due to a too sudden coarsening of the mesh downstream of the wheel. This suggests that, even when using the FW-H in its solid formulation, one has to be cautious when meshing the wake in order to improve the accuracy of the noise predictions, specially for low observation angles relative to the flow. The permeable formulation has been assessed by keeping the endcap closure of the FW-H surface and varying its discretisation. This has led to the conclusion that the discretisation of the downstream closure of the surface seems to be a parameter that greatly influences the noise predictions. In this study, it has been found that refining the surface elements involved in the surface integral has allowed an almost perfect match with the solid formulation results over a large frequency interval, while the coarsely meshed surface closure generates spurious noise in every direction. Some discrepancies remain though, particularly at very low frequencies and in the directions close to the streamwise direction. The endcap closure of the surface has then been removed from the calculation. This has greatly improved the results obtained with the coarse integration surface, that were shown to match perfectly that of the fine integration surface with endcap open. However, as previously, opening the downstream surface induces discrepancies at low frequency and in the flow direction. The very good agreement between the solid and permeable results seems to indicate that the quadrupolar volumic source is negligible for the considered Mach number M = 0.23 and for frequencies under 5kHz (the CFD mesh cut-off frequency in the present case). The contribution of the quadrupolar noise relatively to the other terms decreases with the Mach number. We can then expect the solid formulation of the FW-H equation to give accurate results for Mach numbers between 0.18 and 0.23. This is a valuable result for landing gear noise predictions, as the computational cost (memory and CPU) related to the computation of the far-field pressure is significantly reduced by using this formulation.

PSD==

  Power Spectral Density φ = Azimuthal angle FW-H = Ffowcs-Williams & Hawkings θ = Polar angle RANS = Reynolds-Averaged Navier-Stokes f = Frequency Z-DES = Zonal Detached Eddy Simulation St D w = Strouhal number based on the wheel diameter = f D w U ∞ OASPL = Overall Sound Pressure Level Σ = FW-H integration surface D w = Wheel diameter n = Outward unit normal vector h = Cavity depth v Σ = FW-H surface velocity vector r = Wheel cavity radius H = Heavyside function κ = Cavity depth to diameter ratio = h Upstream flow velocity T D w = Wheel diameter based convective time = D w U ∞

  (a) The LAGOON geometry (b) Main dimensions of the LAGOON cavity (in millimeters)

Fig. 2

 2 Fig.2Cut of M1 and M2 in a median plane of the wheel used for the computation of the whole LAGOON gear in[START_REF] De La Puente Cerezo | Zonal Detached Eddy Simulation of a simplified nose landing-gear for flow and noise predictions using an unstructured Navier-Stokes solver[END_REF] which provided excellent results presented in the frame of the BANC[START_REF] De La Puente | Aeroacoustic simulations of landing gears with unstructured grids and a ZDES turbulence model[END_REF]. It consists in a hybrid 15 million element (3.9 million nodes) mesh, including 3.7 million prisms (21 layers, with a first cell height of D w /∆ = 30000) devoted to solving the attached boundary layers. A volume refinement is defined in order to correctly reproduce the shear layer that detaches on the upstream part of the tyre. The cell size ∆ corresponding to this refinement is such that D w /∆ > 120. This mesh has been designed with the idea that the flow at the direct vicinity of the wheel was mainly driven by the shear layer. It must be stressed that with the M1 mesh, the wake of the wheel is not resolved at all, as well as the flow on the external part of the cavity. The numerical resolution of the boundary layers is expected to give good results for the solid FW-H formulations. On the other hand, this mesh is expected to be poorly adapted to the permeable FW-H assessment as the resolution of the wake is very uncertain. In this regard, a second mesh has been designed with emphasis on the accurate resolution of the wake of the wheel, as well as the correct propagation of the acoustic waves to the permeable surface. The second mesh M2 hence possesses exactly the same features than M1, except that, this time, the cell size downstream of the wheel allows an accurate resolution of the wake dynamics from the wheel to the endcap closure of the FW-H surface (a description of this surface will be given in the following paragraph). The evolution of the cell diameter in the streamwise direction is depicted in figure3. As we can see, the cell size has been roughly divided by 10 in the wake of the wheel from M1 to M2, hence giving a local cell size that ensures

Fig. 3

 3 Fig.3Evolution of the ratio between the wheel diameter D w and the local cell size ∆ behind the wheel, at y = 0.

  (a) Observation surface (b) Permeable surface and numerical probes

Fig. 4

 4 Fig. 4 Description of the FW-H setup: observation surface (left) and permeable surface (right) used for the computation

Fig. 5 Fig. 6

 56 Fig. 5 Power spectral density of the transverse velocity components at eight x/D w locations in the wake. Dashed black line: Kolmogorov's power law f -5/3

Fig. 7

 7 Fig. 7 Power spectral density of the CFD pressure time signal for an observer located at x * = 2.71 and at a vertical distance 1.2D w from the cavity floor. The dashed line indicates f = 1850H z

3 Fig. 9

 39 Fig. 9 Spectrum of the far-field pressure signal computed with a solid FW-H at the position (θ = 0°, φ = 30°) for three different recorded time signal durations: 128ms, 328ms and 456ms

  Fig. 10 Spectrum of the far-field pressure signal computed at two positions in the upstream direction with the solid formulation of the FW-H equation

  Microphone (θ = 0 • ; φ = 80 • ) Microphone (θ = 0 • ; φ = 30 • )

Fig. 15

 15 Fig. 14 Spectrum of the far-field pressure signal computed at two positions in the downstream direction with a closed integration surface

  Fig. 16 Spectrum of the far-field pressure signal computed at two positions in the upstream direction with an open integration surface

Fig. 18

 18 Fig. 18 Effect of opening the downstream end of the surface on the OASPL (dB) for the ( ) M2S FW-H computation, ( ) M2F-Op and ( ) M2F-Cl (on the left), ( ) M2C-Cl and ( ) M2C-Op (on the right) in the (zOx) plane. The dashed lines represent the angle covered by the outflow surface

Fig. 19

 19 Fig. 19 Snapshot of the fluctuating density ρ on the downstream closure of the FW-H integration surface


Olivier Verneau

Frédéric Thomas

Anne De Meeüs

François Catzeflis

François Renaud

Frdddric Anne Thomas

Francois De Meeiis

Catzeflis

Francois Renaud

-primer polymerase chain reaction random amplified polymorphic DNA fingerprinting

Keywords:

published or not. The documents may come

Evidence of two genetic entities inBothriocephalus funiculus (Cestoda) detected by arbitrary-primer polymerase chain reaction random amplified polymorphic DNA fingerprinting

Introduction

Until recently, information provided by the development of molecular methodologies in evolutionary parasitology O. Verneau 9 F. Thomas 9 E Renaud (~) Laboratoire de Parasitologie Comparde (URA 698, CNRS); Universit~ Montpellier II, Pl. E. Bataillon, 34095 Montpellier CEDEX 5, France A. de Meeiis Centre de Recherches de Biochimie Macromoldculaire (UPR 9008, CNRS) BP 5051, Route de Mende, 34033 Montpellier CEDEX, France E Catzeflis Institut des Sciences de l'Evolution (URA 327, CNRS), Universit~ Montpellier II, P1. E. Bataillon, 34095 Montpellier CEDEX 5, France and analyzed through population-genetics concepts has revealed an increase in parasite diversity [START_REF] Curran | Use of restriction fragment length differences to identify nematode species[END_REF][START_REF] Piessens | Highly repeated DNA sequences as species specific probes from Brugia[END_REF][START_REF] Rishi | Genomic cloning of human Echinococcus granulosus DNA: isolation of recombinant plasmids and their use as genetic markers in strain characterization[END_REF][START_REF] Unnasch | DNA probes to identify Onchocerca volvulus[END_REF]. Indeed, narrow links (i.e., specificity) between hosts and parasites appear more and more frequently, suggesting parallel evolution in these biological pairs. Such a case has been demonstrated in bothriocephalids of teleostean fishes [START_REF] Renaud | Le complexe Bothriocephalus scorpii (Mueller, 1776): diff6renciation par 61ectrophorbse enzymatique des espaces parasites du turbot (Psetta maxima) et de la barbue (Scophthalmus rhombus)[END_REF][START_REF] Renaud | Geographical divergence in Bothriocephalus (Cestoda) of fishes demonstrated by enzyme electophoresis[END_REF][START_REF] Renaud | Polymorphisme enzymatique de pop-uIations du groupe Bothriocephalus scorpii (Mueller, 1776). Etude des parasites de divers Tdl6ost6ens des c6tes du Fin-ist~re[END_REF][START_REF] Ortega | Differenciacion genetica entre dos poblaciones del complejo Bothriocephalus scorpii (Pseudophyllidea: Bothriocephalidae)[END_REF]. Using multilocus enzyme electrophoresis (MEE), these different authors showed that far from representing a nonspecific species displaying a large geographical distribution, the original species Bothriocephalus scorpii (Mueller 1776) constituted a complex of biological species highly specific for their host species. Moreover, it has been shown that these species present different biological cycles with strong consequences on the epidemiological transmission and population dynamics of these parasites [START_REF] Robert | Importance of the paratenic host in the biology of Bothriocephalus gregarius (Cestoda, Pseudophyllidea), a parasite of the turbot (Psetta maxima)[END_REF]). However, on the basis of MEE analysis it was impossible to discriminate between individuals of B. funiculus parasitizing two related species of teleost fishes belonging to the genus Onos, (i.e., O. mustela and O. tricirratus: Gadidae, Gadiforms). As MEE analyses were conducted on 12 enzymatic loci, the resolving power of the methodology in investigating the genetic divergence between parasite populations was put into question. Indeed, despite constant methodological progress, the use of genomic markers to study the ecology of natural populations was limited by technical considerations [START_REF] Hadrys | Applications of random amplified polymorphic DNA (RAPD) in molecular ecology[END_REF].

Recently, DNA polymorphisms have been detected by random amplified polymorphic DNA (RAPD; [START_REF] Welsh | Fingerprinting genomes using PCR with arbitrary primers[END_REF]Williams etal. 1990). RAPDs are generated by the amplification of genomic DNA with a single primer of arbitrary nucleotide sequence. The polymorphisms generated function as genetic markers that can be used in taxonomy and population biology studies [START_REF] Demeke | Potential taxonomic use of random amplified polymorphic DNA[END_REF][START_REF] Hadrys | Applications of random amplified polymorphic DNA (RAPD) in molecular ecology[END_REF]. Indeed, RAP-Ds have been used for the biosystematic analyis of bacteria (Fekete etal. 1992;Brikun etal. 1994), protozoa (Steindel etal. 1993), yeast (Lehmann etal. 1992), plants [START_REF] Vierling | Use of RAPD markers to determine the genetic diversity of diploid wheat genotypes[END_REF][START_REF] Demeke | Potential taxonomic use of random amplified polymorphic DNA[END_REF][START_REF] He | Detection of DNA sequence polymorphisms among wheat varieties[END_REF]), nematodes (Caswell-Chen et al. 1992), insects [START_REF] Chapco | A feasibility study of the use of random amplified polymorphic DNA in the population genetics and systematics of grasshoppers[END_REF][START_REF] Hunt | Patterns of inheritance with RAPD molecular markers reveal novel types of polymorphism in the honey bee[END_REF], and mammals (Welsh etal. 1991), Concerning parasitic helminths, RAPDs have been used in sexing Schistosoma larvae [START_REF] Gasser | Sexing larval stages of Schistosoma mansoni by polymerase chain reaction[END_REF] and to identify Schistosoma strains and species [START_REF] Barral | Genetic variability and evolution of the Sehistosoma genome analysed by using random amplified polymorphic DNA markers[END_REF][START_REF] Neto | The random amplification of polymorphic DNA allows the identification of strains and species of schistosome[END_REF].

In view of these new results, the aim of the present study was to analyze and discuss polymorphisms evidenced by RAPD multilocus probes within and between two samples of the species B. funiculus [START_REF] Renaud | Polymorphisme enzymatique de pop-uIations du groupe Bothriocephalus scorpii (Mueller, 1776). Etude des parasites de divers Tdl6ost6ens des c6tes du Fin-ist~re[END_REF] parasitizing two sympatric teleostean species.

Materials and methods

Hosts and parasites

Hosts [i.e., Onos mustela (L., 1758) and O. tricirratus (Brfinnich, 1768)] were captured by hand under rocks at low tide in front of the biological station of Roscoff (Brittany, France). Fishes were killed using the anesthetic 3-aminobenzoic acid ethyl ester (Sigma A5040) and dissected. Parasites (i.e., Bothriocephalus funiculus)

were removed from the host gut, washed in a saline solution (0.9% NaCI), and individually stored in liquid nitrogen before DNA analysis. A total of 72 adult worms were obtained, 36 from O. mustela and 36 from O. tricirratus.

Bothriocephalus genomic DNA preparation

Genomic DNA extraction of Cestoda was individually obtained by phenol extraction and ethanol precipitation according to the protocol defined by [START_REF] Verneau | DNA reassociation kinetics and genome complexity of a fish (Psetta maxima: Teleostei) and its gut parasite (Bothriocephalus gregarius: Cestoda)[END_REF].

Primers

A total of 68 arbitrary 10-bp primer used for DNA amplifications were synthesized according to the theoretical parameters (i.e., greater than 60% G-C content, no dyad symmetry, and non-selfcomplementary) and purchased from Operon Technologies Inc. (Alameda, Calif., USA). They were from Operon kits B (15 primers, B2-B16), E (20 primers, El-E20), F (20 primers, F1-F20), and L (13 primers, L3-L8 and L13-L20).

PCR amplification

The final protocol-adopted reactions (25 gl) for polymerase chain reaction (PCR) contained 1.0 unit Taq DNA polymerase (ATGC), 10 mM TRIS-HCL (pH 8.3), 50 mM KCL, 1.5 mM MgC12, 80 gM of each deoxynucleoside triphosphate (dNTP; Boehringer), 0.5 gM primer (Operon), and 10 ng genomic DNA. This reaction was overlaid with mineral oil and amplification was performed on a thermocycler (Crocodile II, Appligene) for 1 cycle of 95 ~ C for 5 min (denaturation), 35 ~ C for 1 min annealing), and 72 ~ C for 2 min (extension), which was followed by 40 cycles of 95 ~ C for 1 rain, 35 ~ C for 1 min, and 72 ~ C for 2 min. Reaction products were resolved by electrophoresis at 75 V for 4 h in a 1.5% agarose gel containing TBE lx and were visualized with ethidium bromide fluorescence. The size of amplified fragments was estimated by comparison with various standard DNA-marker sets (bacteriophage 7, double-digested by EcoRI-HindIII). Controls were routinely used with no genomic DNA. No amplification occurred in these controls, thus eliminating possible artefactual products and contamination. We also included a control for host contamination, Fig. 1 Polymorphic patterns obtained from Bothriocephalus funiculus DNA at three RAPD primers (L8, F15, B8). Three different patterns can be observed for the L8 primer: the first is represented by amplification fragment A1 at 500 pb; the second, by fragment B1 at 460 pb; and the third, by fragments A1 and B1 at 500 and 460 pb, respectively, which represent a combination of the two others. In the same way, we can distinguish three different patterns for the F15 primer: the first is characterized by fragment A2 at 320 pb; the second, by fragment B2 at 380 pb; and the third, by fragments A2 and B2 at 320 and 380 pb, respectively. The polymorphism observed for the B8 primer reveals only two patterns, one of which is represented by fragment A3 at 280 pb, whereas the other, named B3, revealed no amplification product.

Arrows point to shared bands (A1, B1, A2, B2, A3, B3) referred to above which always shows amplification patterns quite different from those obtained with parasites. Moreover, to test the quality and the reproducibility of amplification reactions, we repeated each reaction at least once. Markers were scored as being present or absent, and only amplified markers present in a reproducible way were taken into account.

Results

For the 72 Bothriocephalus individuals analyzed, 65 monomorphic amplification patterns were obtained among the 68 primers tested. Amplified product sizes ranged from 0.2 to 1.6 kb. However, the 3 remaining primers L8, F15, and B8 revealed polymorphic patterns. The primer sequences were as follows: L8 (5'-AG-CAGGTGGA-3', F15 (5'-CCAGTACTCC-3'), and B8 (5'-GTCCACACGG-3'). According to the different patterns observed at each of these 3 primers (Fig. 1), the analyzed Bothriocephalus can be divided into 8 classes (Table 1). In all, 60 (83%) Bothriocephalus can be distributed into 2 genetic groups: 30 are included into type A (fragments A1, A2, and A3 for the three respective primers; Fig. 1) and 30, into type B (fragments B1, B2 and B3 for the three respective primers; Fig. 1). Most type A flatworms (28 among 30) were collected in Onos mustela, whereas the majority of type B (29 among 30) were collected in O. tricirratus. The last 12 (27%) Bothriocephalus represent combinations of types A and B (Table 1). These results were confirmed by a factorial correspondence analysis (FCA; [START_REF] Benzecri | DNA sequence divergence among derivatives of Escherichia coli K-12 detected by arbitrary primer PCR (random amplified polymorphic DNA) fingerprinting[END_REF][START_REF] Lebart | Genotypic identification and characterization of species and strains within the genus Candida by using random amplified polymorphic DNA[END_REF], whereby each individual is characterized for each existing fragment (i.e., A1, A2, A3, B1, B2, B3) present at the considered primer. Individuals are then analyzed as active variables using FCA. Thus, this method Fig. 2 Variability of different cestodes as studied using FCA for the first two axes. Axes I and H represent 92.62% and 4.83% of the overall respective variability. A, B, and C correspond to the genetic groups defined in Table 1 of analysis characterizes each sample simultaneously according to all the genetic variables (amplified fragments) and shows the contribution of each fragment to the overall variability of the samples.

Figure 2 shows the projection on the plane defined by the two major axes (axes I and II), which represent 97.45% of the overall variability of the samples. Axis I, which represents 92.62% of the overall variability, clearly opposes type A against type B. The other individuals (C1-C6, Table 1) that present combined patterns are distributed between the two groups (A and B).

Discussion

The large amount of monorphic amplification products observed for each primer reveals the very close genetic relationships that link these Bothriocephalus. However, at 6 independent sites of RAPD primers, this study clearly shows the existence of 2 related genetic entities (i.e., types A and B; 30 individuals for each type among the 72 analyzed) displaying a narrow host specificity.

A question remains concerning the 12 Bothriocephalus that present different combination patterns between 593 the A and B types. Because RAPD fingerprints act as dominant genetic markers [START_REF] Hadrys | Applications of random amplified polymorphic DNA (RAPD) in molecular ecology[END_REF], heterozygosity is normally not detectable and allelic frequencies might be difficult to estimate in diploid organisms such as Bothriocephalus. However, since Bothriocephalus species seem to present random mating [START_REF] Renaud | Le complexe Bothriocephalus scorpii (Mueller, 1776): diff6renciation par 61ectrophorbse enzymatique des espaces parasites du turbot (Psetta maxima) et de la barbue (Scophthalmus rhombus)[END_REF], it is clear that strong linkage disequilibria occur between the A and B entities. Indeed, we noticed that a large portion of the possible combinations were not encountered during the present study (i.e., only 8 genetic combinations were observed among the 32 expected; Table 1), reflecting a nonrandom mating process between the 2 genetic entities. Two hypotheses can be proposed to explain such a situation: (1) there is no gene flow between the two entities, and the patterns observed for the 12 Bothriocephalus could be the reflect of ancestral polymorphisms present in the 2 entities; and (2) a limited gene flow between the 2 entities does occur and, thus, the 3 Bothriocephalus presenting the mixed patterns [type C2, Table 1: fragments (A1, B1), (A2, B2), and A3, respectively] could be considered as F1 and the 9 others, as recombinant patterns. It is worth emphasizing that the two host species O. mustela and O. tricirratus live in sympatry and, eventually, in syntopy, as they are currently found together under the same rocks (shelter) at low tide. Thus, there is no geographical barrier to a potential gene flow between the two populations of Bothriocephalus parasites.

Although RAPD markers can formally be treated as Mendelian alleles (Arnold etal. 1991;Carlson etal. 1991), we cannot settle our question because the genetic determinism of RAPD markers is difficult to elucidate without experimental crosses between the two types A and B. However, our work as well as that of several other investigators [START_REF] Hu | Identification of broccoli and cauliflower cultivars with RAPD markers[END_REF][START_REF] Hadrys | Applications of random amplified polymorphic DNA (RAPD) in molecular ecology[END_REF] shows that if the RAPD amplification is repeated two or more times, the majority of markers are clearly reproducible and scorable. Thus, this study undoubtedly reveals the existence of two genetic entities preferentially parasitizing one or the other of the two teleostean species. Furthermore, the close genetic relatedness observed between these two entities is in accordance with a previous analysis conducted by one of us (E R.). Indeed, an electrophoretics study realized on 12 enzymatic loci was not capable of revealing any genetic difference within or between Bothriocephalus from the two teleosteans (i.e., O. mustela and O. tricirratus). Such a case has been encountered in plants [START_REF] Hu | Identification of broccoli and cauliflower cultivars with RAPD markers[END_REF][START_REF] Martin | Rapid identification of markers linked to a Pseudomonas resistance gene in tomato by using random primers and near-isogenic lines[END_REF] and in Schistosoma [START_REF] Barral | Genetic variability and evolution of the Sehistosoma genome analysed by using random amplified polymorphic DNA markers[END_REF][START_REF] Neto | The random amplification of polymorphic DNA allows the identification of strains and species of schistosome[END_REF], where allozyme studies have not proved to be sensitive enough. Thus, RAPD markers detected by PCR amplification of small inverted repeats scattered through the genome seem to be a new molecular tool in parasitology to investigate the biodiversity and biosystematics of closely related biological entities, whereas morphoanatomical and/or traditional genetics studies are not informative enough. Moreover, this methodology, which necessitates small quantities of DNA, can thus be very useful for the analysis of parasite larvae (e.g., cora-cidia, plerocercoids, and cercariae, among others) in an epidemiological context.

Table 1

 1 RAPD types observed at 3 primers for the 72 Bothriocephalus analyzed

	Number of	Primer L8 Primer F 15	Primer B 8
	Bothriocephalus a	types	types	types
	(A) 26 M and 4 T	A1	A2	A3
	(B) l M and 29 T	B1	B2	B3
	(C1) 1 M and 2 T	A1, B1	A2, B2	A3
	(C2) 5M	A1	A2	B3
	(C3) 1M	A1, B1	A2	A3
	(C4) 1M	A1	A2, B2	A3
	(C5) 1M	A1	B2	B3
	(C6) 1T	B1	A2, B2	B3
	a Letters M and T characterize host species of Bothriocephalus (M
	for Onos mustela, T for O. tricirratus). Boldface letters in paren-
	theses correspond to the different groups observed after the FCA
	analysis (Fig. 2)
			Axis 2
			4.83 %
	A	C3
				B
		C4	C1	Axis I __
				92.62%
			C6
		C2	CS

Acknowledgements We thank Dr. T. De Meefis for his helpful comments and discussions. The laboratory facilities were partly funded by Services Communs de Biosyst6matique, Direction de la Recherche et des Etudes Doctorales (Minist6re de l'Education National, France).