domain: spi phys

text:
T Désert

T Jardin

H Bézard

J M Moschetta

Numerical predictions of low Reynolds number compressible aerodynamics

Keywords: Aerodynamics, low Reynolds number, compressible, low pressure, airfoils, rotors, Mars

Interest in low Reynolds number compressible flows is emerging due to prospective applications like flight on Mars and in the stratosphere. However, very little knowledge is available, both regarding the flow physics underlying this unique regime and the accuracy of numerical methods for its prediction. In this paper, low and high fidelity numerical approaches are compared with experimental measurements on both airfoils and rotors in the low Reynolds number compressible flow regime. It is shown that low fidelity approaches are suited to aerodynamic optimization despite high viscous and compressible effects. In addition, high fidelity approaches help reveal unique flow features of this regime.

Introduction

The exploration of the planet Mars with Rovers started more than a decade ago, allowing a more detailed description of the planet surface than what was possible so far with Mars orbiters. However, a major drawback of these Rovers is their relatively low speed (partly due to harsh terrain) which, until now, only allowed them to explore a few tens of kilometers -to be compared with the 21,000 km planet circumference. One way to facilitate surface exploration would be to use autonomous flying vehicles which could act as scouts for the Rovers. Unfortunately, the atmosphere of Mars is far from flight-friendly. On the one hand, its density is low which [START_REF] Munday | Nonlinear lift on a triangular airfoil in low-reynolds-number compressible flow[END_REF] results in low Reynolds number flows that promote both flow separation and high viscous drag and (2) requires to be compensated for by higher speed of the lifting surface to provide a sufficient lift force. On the other hand, its composition and temperature are such that the speed of sound is relatively low, which is conducive to supersonic flow regimes.

As a consequence, a typical flying vehicle designed for Mars exploration would operate in the low Reynolds number, compressible flow regime, for which very little knowledge is available.

Table 1 shows properties of the atmosphere of Mars at the ground level and compares it with properties of the atmosphere on Earth at sea level and 30 km above sea level (stratosphere). Note that the purpose here is to show orders of magnitude rather than precise time-averaged values of highly fluctuating properties. It is shown that the properties at 30 km above sea level are quite similar to those on Mars, at ground level, suggesting that vehicles flying in the stratosphere would operate under similar conditions to those described above, i.e. in the low Reynolds number, compressible flow regime, albeit with higher gravity force. The developement of stratospheric vehicles has also recently gained interest for applications like Earth observation, telecommunications and navigation.

Overall, low Reynolds number, compressible flow regimes are very poorly documented because of the relatively new applications to which they are re-lated. Apart from Mars and stratospheric flight, the development of high speed trains in low pressure tube (e.g. Hyperloop) could greatly benefit from deeper knowledge of these regimes, as well as liquid atomization where micro-sized droplets are formed and travel at high speed. Yet, a few authors brought to the fore unique features in low Reynolds number, compressible flows. For example, [START_REF] Munday | Nonlinear lift on a triangular airfoil in low-reynolds-number compressible flow[END_REF] numerically and experimentally investigated the flow past a triangular airfoil at Reynolds numbers 3000 and 10000 and Mach numbers 0.15 and 0.5. They showed that compressibility tends to elongate the wake, causing the transition from linear to non-linear lift and the subsequent vortex-induced lift to be delayed to higher angles of attack. [START_REF] Canuto | Two-dimensional compressible viscous flow around a circular cylinder[END_REF] demonstrated similar wake elongation for the flow past a two-dimensional circular cylinder, below and above the critical Reynolds number of the first, Hopf bifurcation. Above the critical Reynolds number, this results in a larger vortex-shedding wavelength, i.e. a lower shedding frequency. Wake elongation was found to be associated with higher drag and, in the stable state, with delayed separation.

In addition to little knowledge on the flow physics, it is not clear how conventional, numerical tools used to predict aerodynamic performance are suited to low Reynolds number, compressible flows. Yet, the design of small size unmanned vehicles flying in low pressure environment relies on the accuracy of such numerical tools.

In this paper, this issue is addressed by comparing predictions from low and high fidelity numerical tools with experimental results obtained under extreme conditions with typical Reynolds and Mach numbers in the range [100 -10000] and [0.1 -0.9] respectively. These numerical tools are applied to the flow past airfoils and rotors operating in such regimes. It is shown that while high fidelity approaches provide reasonable estimates of aerodynamic forces for all operating conditions, low fidelity approaches are limited to low angle of attack / pitch angle cases, which still make them suitable to aerodynamic optimization. In addition, unique features of low Reynolds number compressible flows are revealed, including, for example, wake elongation and subsequent damping of lift fluctuations, and displacement of shock foot far from the airfoil surface. Finally, experimental data for rotors (which are virtually unavailable in the literature) are reported and deeper insight into the resulting, enhanced aerodynamic forces (due to leading edge vortex stability) is provided by means of numerical approaches.

Numerical approaches

In this section, the theory and numerical procedures underlying three different models with increasing complexity are briefly described: (i) two-dimensional potential approaches, (ii) three-dimensional vortex lattice methods (VLM) and

(iii) numerical simulations of the two and three-dimensional unsteady Navier-Stokes equations (NS).

Two-dimensional potential flow methods

Aerodynamic performance of two-dimensional airfoils is first evaluated with a potential flow panel method combined with an integral boundary layer formulation. The latter treats both laminar and turbulent layers and empirically determines the transition point using an e N method. Preliminary linear stability analysis on triangular and cambered airfoils up to a Reynolds number of 6000 were performed and showed that transition was triggered for N < 1.

Therefore, in what follows, the empirical value for N is fixed sufficiently large to avoid transition. Moreover, compressibility is intrinsically accounted for in the compressible boundary layer formulation and applied as a Karman-Tsien correction in the potential method. Xfoil code [START_REF] Drela | Xfoil: An analysis and design system for low reynolds number airfoils[END_REF] is used to efficiently solve the problem via a global Newton method. Further details on the numerical procedure can be found in [START_REF] Drela | Xfoil: An analysis and design system for low reynolds number airfoils[END_REF]. The number of panels used to discretize an airfoil is such that the solution is converged with respect to spatial discretization and is on the order of 160 for all cases shown in this paper.

Three-dimensional vortex lattice methods

A vortex lattice approach is used to predict the aerodynamic performance of rotors with relatively low computational cost. Numerical simulations are performed using ONERA's in-house code PUMA, which combines lifting line and free wake models [START_REF] Boisard | Aerodynamic investigation of a helicopter rotor hovering in the vicinity of a building[END_REF]. The lifting line approach relies on two-dimensional airfoil polars (obtained from resolution of the Navier-Stokes equations, see next section) and applies 3D correction to account for blade sweep and unsteady phenomena (e.g. dynamic stall). The free wake approach relies on the theory developped by [START_REF] Mudry | La théorie des nappes tourbillonnaires et ses applications à l'aérodynamique instationnaire[END_REF] which describes the unsteady evolution of a wake modelled by a potential discontinuity surface. The blade and wake are discretized using 50 non-uniformely distributed radial stations (square root distribution for increased resolution near the tip). The computation is advanced in time using forward Euler scheme with a time step that corresponds to a 10 • rotation of the blades.

Four rotations are needed for initial transients to sufficiently decay.

Numerical simulations of the Navier-Stokes equations

The two and three-dimensional unsteady Navier-Stokes equations are numerically solved using a finite volume method. Compressible and incompressible formulations are considered, typically for Mach numbers above and below 0.2 respectively. Resolution is achieved using ONERA's in-house code elsA [START_REF] Cambier | elsA: An efficient object-oriented solution to CFD complexity[END_REF] and StarCCM+ commercial code [START_REF]Star-CCM+ user guide[END_REF]. Spatial and temporal discretization schemes are of second order in both codes, with explicit and implicit temporal marching for elsA and StarCCM+ respectively.

Airfoil simulations

For both two-dimensional and three-dimensional airfoil cases at Reynolds numbers on the order of 10 3 and subsonic Mach numbers, it was shown that a typical spatial resolution of ∆s/c = 0.01 allows for the Richardson extrapolated lift and drag [START_REF] Roache | Verification and Validation in Computational Science and Engineering[END_REF] to be approximated within 1.5%. An example of grid convergence is shown in figure 1a where the lift coefficient is displayed as a function of the typical grid spacing in the wake of the airfoil, ∆x/c. 2D and 3D results are obtained for a 10 • angle of attack triangular airfoil [START_REF] Munday | Nonlinear lift on a triangular airfoil in low-reynolds-number compressible flow[END_REF] at Reynolds and Mach numbers 3000 and 0.15, respectively. Figure 1b shows Q-criterion isosurfaces coloured by spanwise vorticity, obtained on the 3D configuration for typical grid spacings of c/50 (2 million cells) and c/160 (70 million cells). It is shown that although the largest grid spacing c/50 captures three-dimensional spanwise instabilities in the airfoil wake (leading to hairpin vortices), it is not able to capture them near the leading egde where the flow exhibits a nearly two-dimensional pattern. As such, it is interesting to note that for this spatial resolution, lift obtained from 3D simulation converges towards that obtained from 2D simulation. Note that the 3D case shown in figure 1 corresponds to a 3.3 aspect ratio wing with symmetrical boundary conditions at the tips. Other 3D configurations with non-slip walls at the tips and with a 0.3 aspect ratio wing will also be addressed in section 5.1. Subscripts 0.3 and 3.3 are used to denote aspect ratio 0.3 and 3.3 respectively. and unstructured (hexahedral trimmed cells) grids are used with elsA and Star-CCM+ respectively, resulting in a typical number of cells on the order of 10 5 for 2D cases and 2.10 6 and 2.10 7 for 3D cases with 0.3 and 3.3 aspect ratio wing, respectively.

Simulations are run for 100 convective times to ensure that initial transients have sufficiently decayed. Afterwards, time averaged quantities are obtained by averaging instantaneous values over 20 convective times.

Rotor simulations

For three-dimensional rotor cases, similar spatial resolution to that used for airfoil cases is used (∆s/c = 0.01) resulting in a typical number of cells on the order of 7.10 7 . Again, structured (hexahedral cells) and unstructured (polyhedral cells) grids are used with elsA and StarCCM+ respectively. On the other hand the time step is larger (∆tU/c = 0.08), which is made possible by the quasi-steady nature of the flow on the rotor considered in this paper (as will be shown in section 5.3). Also note that U is the velocity at the blade tip such that the blade span operates at velocities below U , hence at lower non-dimensional time step.

Simulations are run for 5 rotor rotations to ensure that initial transients have sufficiently decayed. Time averaged quantities are then obtained by averaging instantaneous values over one rotor rotation.

Experiments

Low pressure wind tunnel

Experimental data by [START_REF] Munday | Nonlinear lift on a triangular airfoil in low-reynolds-number compressible flow[END_REF] are used to assess the validity of numerical ap- Experiments are conducted at a pressure of 2000 Pa (minimum achievable pressure is 10 Pa). Both Air and CO 2 (96%) were tested and yielded similar results, in agreement with previous observations by [START_REF] Anyoji | Development of a Mars wind tunnel and its applications to low reynolds number and high-subsonic airfoil testing[END_REF]. Therefore, Air was used for all cases shown in this paper.

Data reduction

Results are analyzed in terms of global and local aerodynamic performance as well as flow quantities.

Global aerodynamic performance of airfoils is assessed using lift C L = 2L/ρSU 2 and drag C D = 2D/ρSU 2 coefficients, where ρ, U , L and D are the fluid density,

Results

Evaluation of numerical methods on triangular airfoil

The lift and drag coefficients and the lift-to-drag ratio obtained on a triangular airfoil using both potential flow (Xfoil) and Navier-Stokes (elsA and StarCCM+) solvers are first compared with experimental and numerical results

from [START_REF] Munday | Nonlinear lift on a triangular airfoil in low-reynolds-number compressible flow[END_REF]. The potential flow solution is two-dimensional while solutions to the Navier-Stokes equations are presented for 2D and 3D cases. 3D cases with aspect ratios 0.3 and 3.3 are considered. These aspect ratios are chosen to be consistent with numerical simulations (aspect ratio 0.3) and experiments (aspect ratio 3.3) from [START_REF] Munday | Nonlinear lift on a triangular airfoil in low-reynolds-number compressible flow[END_REF]. An additional case taking into account wind tunnel walls is also simulated. Figure 3 illustrates the airfoil profile and the aspect ratio 3.3, 3D case with wind tunnel test section. Note that this particular airfoil is here considered precisely because both numerical and experimental data in compressible, low Reynolds number flow conditions are readily available in the literature for comparison [START_REF] Munday | Nonlinear lift on a triangular airfoil in low-reynolds-number compressible flow[END_REF]. [START_REF] Munday | Nonlinear lift on a triangular airfoil in low-reynolds-number compressible flow[END_REF] selected this triangular airfoil as a potential candidate for the design of propellers for Martian aircrafts due to its simple geometry, with sharp edges and flat surfaces. These characteristics (i.e. sharp leading edges and flat surfaces) were previously found to promote aerodynamic performance in the low Reynolds number compressible flow regime [START_REF] Okamoto | An experimental study in aerodynamic characteristics of steady and unsteady airfoils at low reynolds number[END_REF]. is sufficient to capture three-dimensional instabilities and their influence on aerodynamic loads at high angles of attack. This is consistent with results from [START_REF] Munday | Nonlinear lift on a triangular airfoil in low-reynolds-number compressible flow[END_REF]. Conversely, taking into account wall boundary conditions has a significant impact on both lift and drag predictions and results in a much more accurate estimation for all three angles of attack tested.

Overall, these results suggest that 3D effects arising from the wind tunnel

Reynolds and Mach number effects on cambered airfoil

In light of the above conclusions, two-dimensional Navier-Stokes simulations of the flow past a cambered airfoil are performed to highlight Reynolds and Mach number effects on aerodynamic performance in the low Reynolds number compressible flow regime. The airfoil is a circular cambered plate and has 6.35% camber and 1% thickness with sharp leading edge and blunt trailing edge and was previously found to exhibit relatively good performance at low Reynolds number [START_REF] Bohorquez | Rotor hover performance and system design of an efficient coaxial rotary wing micro air vehicle[END_REF]. Computations are performed for a range of angles of attack α ∈ [0 • -15 • ] (with a step of 1 • ), for Reynolds numbers 100, 1000, 3000 and 10000 and Mach numbers 0.1, 0.5, 0.7, 0.8 and 0.9.

Mach number effects

Figure 5 shows the lift and drag coefficients and the lift-to-drag ratio obtained at Re=3000 for Mach numbers 0.1, 0.5, 0.7 and 0.8. First, it is observed that lift increases with Mach number for a broad range of angles of attack tested, i.e. α ∈ [2 • -10 • ]. Yet, the trend is opposite below 2 • and above 13 • . This point will be discussed in the next paragraph. Second, it is shown that drag increases with Mach number in a more significant amount than lift, leading to a decrease in lift-to-drag ratio. Finally, it can be observed that the angle of attack corresponding to maximum lift-to-drag ratio decreases with Mach number.

To provide insight into the mechanisms responsible for these trends, figure 6 displays spanwise vorticity contours obtained for Mach numbers 0.1, 0. x/c = 0.6. This bump is followed by a drop at the rear of the airfoil where vortices are shed and advected into the wake. This specific bump-drop pattern resembles that induced by a laminar separation bubble but is here the timeaveraged footprint of the unsteady formation and shedding of clockwise rotating vortices. Figure 7 also shows the C p distribution at M = 0.9. It is interesting to note that the distribution is similar to that obtained at M = 0.8 although the flow is transsonic (see figure 8). That is, it is shown that the presence of

Reynolds number effects

Figure 9 shows the lift and drag coefficients and the lift-to-drag ratio obtained at Mach=0.5 for Reynolds numbers 100, 1000, 3000 and 10000. The trends are in line with existing literature (under incompressible conditions, e.g. [START_REF] Kunz | Analysis and design of airfoils for use at ultra-low reynolds numbers[END_REF]) demonstrating a decrease and an increase in lift and drag coefficients with

Reynolds number, respectively. The increase in drag coefficient is particularly severe as the Reynolds number is decreased from 1000 to 100, where viscous drag becomes dominant. As a consequence, the lift-to-drag coefficient significantly increases with Reynolds number, being an order of magnitude greater at Re=10000 than at Re=100. In addition, it can be seen that the maximum lift-to-drag ratio is obtained at larger angles of attack as the Reynolds number is decreased. The dependance of aerodynamic performance on Reynolds number can here again be correlated with the vorticity flow fields depicted in figure 10. As α is increased, the flow at Re=100 remains roughly unchanged. Conversely, it becomes unsteady at Re=1000, with the shear layers interacting near the trailing edge.

At Re=10000 and α = 5 • , the upper shear layer rolls up into a clockwise rotating vortex near mid-chord. The latter is advected towards the trailing edge where it eventually interacts with the lower shear layer. At α = 10

• , the roll up of the upper shear layer is more severe, which leads to a strong interaction with the upper surface and the subsequent deflected wake pattern (previously described). While the interaction between the clockwise rotating vortex and the upper surface of the airfoil at Re=10000 becomes stronger as α is further increased, i.e. as the separation point moves upstream, the upper shear layer at Re=1000 is immune to roll up before interacting with the lower shear layer at the trailing edge. Therefore, while the wake at Re=1000 exhibits a relatively simple pattern characterized by alternate vortices up to α = 15 • , the flow at Re=10000 transits to a chaotic state (see also fluctuating forces and moments in the appendix).

The roll up of the upper shear layer at Re=10000 explains the sudden increase in lift observed at α = 8 • on figure 9. Conversely, the robust shear layers at Re=100 and 1000 explain the smooth increase of the lift coefficient with α.

That is, at the lowest Reynolds numbers, there is no drastic changes in the flow as α is increased, leading to a relatively simple relation between lift and the airfoil projected area c × sin α.

Overall, the present results show that, in the low Reynolds number compressible flow regime, both Reynolds and Mach numbers have significant impact on the flow pattern and the resulting aerodynamic performance. In particular, changes in aerodynamic performance due to variations in Reynolds and Mach numbers are on the same order of magnitude. Therefore, for a given characteristic dimension (e.g. wing chord) and environning pressure, there exists an optimal operating speed (i.e. an optimal Re-Mach couple) that leads to optimal aerodynamic performance. This is fundamentally different to conventional aerodynamics where large Reynolds and Mach number effects are usually uncoupled.

Evaluation of numerical methods on two-bladed rotor

In this section, the comparison between numerical approaches and experimental data is extended to the flow past a two-bladed rotor operating under hovering conditions. Based on the previous results that 2D numerical approaches are suited to the prediction of aerodynamic loads in the low Reynolds number compressible flow regime, around maximum efficiency, 3D vortex lattice methods are here applied using polars from 2D Navier-Stokes computations and compared to 3D Navier-Stokes computations and experiments (figure 11). The blades consist of a 6.35% camber, 1% thick airfoil (see section 5.2) with constant chord and constant twist angle β along the span. It is similar to that tested in [START_REF] Shrestha | Hover performance of a small-scale helicopter rotor for flying on Mars[END_REF]. The rotation speed and ambient pressure are such that the Reynolds and Mach numbers at the blade tip are on the order of 6000 and 0.35 respectively.

Figure 11 shows a relatively good agreement between NS and VLM approaches. These numerical methods, however, overstimate both experimental thrust and torque coefficients. Discrepancies in thrust and torque predictions somehow compensate each other, leading to a fair approximation of the thrustto-torque ratio. Despite these quantitative discrepancies, the trends in thrust and torque versus pitch angle are qualitatively similar for numerical and experimental approaches. Overall, C T increases to a maximum value at α ≈ 30 • and C Q continuously increases with α. As a consequence, C T /C Q exhibits an optimal value, which is found to be around 10 • pitch angle. This intrinsically three-dimensional mechanism cannot be predicted by VLM. shown that the flow separates at the leading edge and rolls up into a conical leading edge vortex (LEV). On the contrary to the LEV that develops on a two-dimensional airfoil and that eventually sheds into the wake, the LEV here remains stably attached to the blade, inducing a sustained low pressure suction region on its upper surface, which in turn generates thrust. Stability of the LEV is a common feature in the aerodynamics of low aspect ratio flapping and revolving wings (which operate in a comparable range of Reynolds numbers) and is hypothesized to result from rotational accelerations, making it a fundamentally three-dimensional phenomenon [START_REF] Lentink | Rotational accelerations stabilize leading edge vortices on revolving fly wings[END_REF][START_REF] Jardin | Coriolis effect and the attachment of the leading edge vortex[END_REF]. Because of this stability, sectional aerodynamic forces on the inboard part of the blade are steady (in contrast to those on two-dimensional airfoils, see appendix). In the outboard region of the blade, i.e. near the tip, it can be observed that the LEV bursts into smaller scale structures as it merges with the tip vortex. This phenomenon is also in line with previous numerical and experimental observations at low Reynolds numbers (e.g. [START_REF] Jones | Characterizing a burst leading-edge vortex on a rotating flat plate wing[END_REF][START_REF] Medina | Leading-edge vortex burst on a low-aspect-ratio rotating flat plate[END_REF][START_REF] Jardin | On the lift-optimal aspect ratio of a revolving wing at low reynolds number[END_REF]).

Figure 13 compares the sectional thrust distribution along the rotor blade obtained from NS computations with that obtained from VLM. While reasonable agreement between both approaches is observed up to 25 • pitch angle, non-negligible discrepancies are observed at 30 • pitch angle where the effect of the LEV on thrust is maximum. Discrepancies observed in the range 0.2 < r/R < 0.8 are partly compensated for by discrepancies at the tip where the dynamics of the vortical structures (merging between LEV and tip vortex) is also very complex and cannot be captured through VLM. As such, despite the relatively good agreement observed on figure 11, VLM should not be viewed as an accurate method for high pitch angles where separation occurs at the leading edge. Note however that special treatment of the two-dimensional polars used in VLM can be added to account for rotational effects on LEV stability [START_REF] Leng | Aerodynamic modelling of propeller forces and moments at high angle of incidence[END_REF].

Yet, because maximum efficiency is obtained at low pitch angles, present results suggest that VLM is suited to the aerodynamic optimization of rotors in the low Reynolds number compressible flow regime. For high angles of attack / pitch angles, these low fidelity approaches fail to predict massive flow separation and subsequent 3D effects, sometimes despite reasonable agreement with experimental data (which was shown to result from compensating errors). These results indicated that, in the low Reynolds number compressible regime, airfoil and rotor optimization can be achieved with reduced order models.

In addition, the influence of Reynolds and Mach number effects on the flow past airfoils was analyzed and significant impact on flow separation and subsequent wake patterns was demonstrated. In particular, results indicated that compressibility tends to elongate the airfoil wake and prevent early roll-up of the upper shear layer, in line with previous observations by [START_REF] Munday | Nonlinear lift on a triangular airfoil in low-reynolds-number compressible flow[END_REF][START_REF] Canuto | Two-dimensional compressible viscous flow around a circular cylinder[END_REF]. This resulted in a re-orientation of the incompressible deflected wake and to a smoother liftto-angle-of-attack curve and a damping of lift fluctuations. Moreover, unique features associated with the displacement of the shock foot away from the airfoil surface were revealed for transsonic operating conditions. On the other hand, Reynolds number effects were found to be qualitatively similar to those reported in extensive works at low Mach numbers. Overall, it was demonstrated that in the low Reynolds number compressible flow regime, the impact of both Reynolds and Mach numbers on aerodynamic performance are on the same order of magnitude which suggests that for a given characteristic dimension and environning pressure, there exists an optimal operating speed that leads to optimal aerodynamic performance. The precise reason for this non-monotonic trend is unclear at this point but it should be noticed that for such high angles of attack, strong non-linearities may trigger additional instabilities that may affect the whole shedding process. In particular, although not shown here for the sake of conciseness, the α = 15

Figure 1 :

 1 Figure 1: Lift coefficient as a function of grid spacing obtained for 2D and 3D flows past a triangular airfoil (a). Q-criterion isosurfaces coloured by spanwise vorticity contours obtained for grid spacings ∆x/c = 0.02 (top) and 0.00625 (bottom) on the 3D, aspect ratio 3.3 configuration (b).

Furthermore, decreasing the

  time step beyond ∆tU/c = 0.02 does not yield significant changes in the results. Therefore, in what follows, NS results will be shown for ∆s/c ≤ 0.01 and ∆tU/c ≤ 0.02. Structured (hexahedral cells)

3 . 2 .

 32 proaches for airfoils operating in the low Reynolds number, compressible flow regime. Data are obtained using the Mars Wind Tunnel (MWT) at Tohoku University. The MWT consists of an indraft wind tunnel housed inside a vacuum chamber that allows low-density compressible experiments. It has a 100 by 150 mm test section with typical turbulence intensity below 1%. The airfoil is triangular with 30 mm chord length and 100 mm span (hence aspect ratio 3.33) and a maximum thickness of 1.5 mm. A scheme of the airfoil is provided in section 5.1, figure3a. Typical operating conditions lead to a Reynolds number on the order of 10 3 -10 4 with Mach numbers on the order of 0.1 -1. More details on the experimental setup can be found in[START_REF] Munday | Nonlinear lift on a triangular airfoil in low-reynolds-number compressible flow[END_REF]. Low pressure chamber Experimental measurements of the aerodynamic performance of rotors operating in low pressure environment are conducted in a low pressure chamber of 18 m 3 volume at ONERA Fauga-Mauzac, figure2a. The test bench consists of a 0.457 m diameter rotor manufactured out of carbon and rotated by means of a Faulhauber 4490 H 024B brushless motor, figure2b. The propulsion set is fixed to a mast on top of which a 10 Newton strain gauge is mounted for thrust measurements. A similar gauge is used at the rear of the motor for torque measurements. Thrust and torque measurements are acquired at a frequency of 10 kHz during 10 seconds, which ensures statistical convergence. In addition, each measurement is performed four times and reported values are obtained by averaging these four measurements. Data accuracy is derived from the maximum deviation of each measurement to the average value of the four measurements and from discrepancies between measurements and calibration weights (under standard, atmospheric pressure conditions). It was estimated to be below 5% of reported values.

Figure 2 :

 2 Figure 2: Low pressure experimental facility (a), sketch of the test chamber (b) and of the rotor test bench embedded in the chamber (c).

Figure 3 :

 3 Figure 3: Triangular airfoil profile (a) and computational domain of the 'wind tunnel' configuration (b).

Figure

  Figure 4a compares two-dimensional numerical results with experimental results for Reynolds number 3000 and Mach number 0.5. Reasonable agreement

Figure 4c focuses on

  Figure4cfocuses on simulations with spanwise extent 3.3 chords (referred to as 3D 3.3 NS). Here, both symmetrical and wall boundary conditions at the wing tips are considered. Comparing with previous results, it is shown that for symmetrical boundary conditions, the spanwise extent of the computational domain has no significant influence on aerodynamic performance, at least for spanwise extents above 0.3 chord. In other words, a spanwise extent of 0.3 chord

Figure 4 :

 4 Figure 4: Comparison of lift and drag coefficients and lift-to-drag ratios obtained from numerical approaches and experiments on a triangular airfoil at Reynolds and Mach numbers 3000 and 0.5 respectively. 2D numerical approaches (a) and 3D numerical approaches with spanwise extent 0.3 (b) and 3.3 (c).

  test section tend to limit flow separation. Therefore, 2D Navier-Stokes computations and 3D Navier-Stokes computations with symmetrical boundary conditions predict early separation, which leads to lift coefficients rapidly diverging from experimental values. On the contrary, because 2D potential flow solvers fail to predict massive flow separation (at least without special empirical treatment), the potential flow solution converges towards a partially attached flow even at high angles of attack. That is, the solution is somehow similar to that arising from 3D effects induced by wall boundary conditions. Therefore, despite the seemingly accurate prediction of aerodynamic loads over the full range of angles of attack tested, potential flow theory should not here be viewed as an accurate approach for high angles of attack aerodynamics. Furthermore, because 3D Navier-Stokes computations with wall boundary conditions predict experimental data with reasonable accuracy, it can be inferred that 3D Navier-Stokes computations with symmetrical boundary conditions are suited to the prediction of aerodynamic loads on a nominally two-dimensional configuration at large angles of attack. Building on that, the similarity between 2D and 3D numerical results at low angles of attack suggest that all 2D methods are suited to the prediction of aerodynamic loads when the flow is mostly attached, i.e. when three-dimensional effects are limited and where the efficiency of the airfoil is maximum. A direct outcome is that potential flow theory, as well as 2D Navier-Stokes computations, can be used with reasonable accuracy for airfoil shape optimization in the context of low Reynolds number compressible flows.

Figure 5 :

 5 Figure 5: Lift and drag coefficients and lift-to-drag ratios obtained from 2D Navier-Stokes computations for a cambered airfoil at Reynolds number 3000 and Mach numbers 0.1, 0.5, 0.7 and 0.8.

5 and 0. 8 at

 8 Reynolds number 3000. For α = 2 • , the flow separates near the trailing edge and the upper and lower surface, opposite sign vorticity layers interact at the trailing edge and roll up into opposite sign vortical structures, leading to a von Kármán vortex street. Alternatively, one can understand the emission of trailing edge vortices as the unsteady response of bound circulation to the emission of vortices induced by separation on the upper surface. It can be seen that the width of the vortex street slightly increases with the Mach number, indicating that the separation point moves upstream. The corresponding Strouhal numberdecreases from 2.17 to 2.07 and 1.79 at Mach 0.1, 0.5 and 0.8 respectively (see appendix for further analysis on unsteady response of forces and moments to vortex shedding). Similar observations can be made at α = 5 • and 8 • . At this point, the effect of increasing the Mach number at a given α can thus be viewed as similar to increasing α at a given Mach number. This tends to increase lift (in the range of α considered), which supports previous observations on lift trend. For Mach=0.1 and α = 10 • the upper vorticity layer interacts with the upper surface of the airfoil and rolls up into a clockwise rotating vortex before reaching the trailing edge. This 'early' interaction leads to an upward deflected wake. This phenomenon is more clearly visible at α = 12 • . For Mach=0.5, such a wake pattern is not observed at α = 10 • but appears at higher α, see for example α = 12 • and 15 • . Conversely, the flow at Mach=0.8 still exhibits a non-deflected wake up to α = 15 • . For the latter, it can be seen that the upper vorticity layer does not roll up into a vortex prior to interacting with the trailing edge, although the separation point appears very close to the leading edge. A direct outcome is that increasing the Mach number reduces the fluctuating loads on the airfoil. For instance, the relative standard deviation of the α = 15 • lift coefficient is equal to 0.104, 0.098, 0.039 and 0.019 for Mach numbers 0.1, 0.5, 0.7 and 0.8 respectively (see appendix). The absence of 'early' vorticity layer

Figure 6 :

 6 Figure 6: Instantaneous spanwise vorticity flow fields obtained from 2D Navier-Stokes computations for a cambered airfoil at Reynolds number 3000 and Mach numbers 0.1, 0.5 and 0.8. Snapshots are shown for six angles of attack α = 2 • , 5 • , 8 • , 10 • , 12 • and 15 • .

Figure 7 :

 7 Figure 7: Distribution of time-averaged pressure coefficients Cp obtained from 2D Navier-Stokes computations for Mach numbers 0.1, 0.8 and 0.9.

  shocks do not significantly affect the pressure distribution in this low Reynolds number compressible flow regime. Here, enhanced viscous effects (i.e. smooth velocity gradients associated with thick boundary/shear layers) tend to move the shock foot away from the airfoil, which is a unique feature of low Reynolds number compressible flow regimes. In other words, extensively investigated phenomenon such as shock wave boundary layer interaction may not apply here, resulting in completely different responses of aerodynamic performance to transsonic/supersonic regimes.

Figure 8 :

 8 Figure 8: Instantaneous spanwise vorticity flow fields obtained from 2D Navier-Stokes computations for a cambered airfoil at Reynolds number 3000 and Mach number 0.9. Snapshots are shown for three angles of attack α = 2 • , 8 • and 12 • . Transparency is applied to vorticity contours, on which contours of pressure gradient magnitude are superimposed to highlight shock waves.

Figure 9 :

 9 Figure 9: Lift and drag coefficients and lift-to-drag ratios obtained from 2D Navier-Stokes computations for a cambered airfoil at Mach number 0.5 and Reynolds numbers 100, 1000, 3000 and 10000.

Figure 10

 10 Figure 10 displays spanwise vorticity contours obtained for Reynolds numbers 100, 1000 and 10000 at Mach number 0.5. For α = 2 • , the flow at Re=100 is characterized by two thick, opposite sign vorticity layers on the upper and lower surfaces of the airfoil, exhibiting a steady pattern. The flow at Re=1000 is rather similar, yet with thinner and stronger shear layers. As Re is increased to 10000, the upper and lower shear layers interact near the trailing edge, leading to an unsteady wake characterized by opposite sign vortical structures. These flow patterns are qualitatively similar at α = 5 • .

Figure 10 :

 10 Figure 10: Instantaneous spanwise vorticity flow fields obtained from 2D Navier-Stokes computations for a cambered airfoil at Mach number 0.5 and Reynolds numbers 100, 1000 and 10000. Snapshots are shown for six angles of attack α = 2 • , 5 • , 8 • , 10 • , 12 • and 15 • .

Figure 11 :

 11 Figure 11: Comparison of thrust and torque coefficients and thrust-to-torque coefficients ratio obtained from numerical approaches and experiments on a two-bladed rotor.

Figure 12

 12 Figure 12 displays iso-surfaces of Q-criterion obtained from NS simulations for pitch angles 15 • , 19 • , 25 • and 30 • . At higher pitch angles (β > 20 • ), it is

Figure 12 :

 12 Figure 12: Surface pressure contours and Q-criterion isosurfaces obtained from 3D Navier-Stokes computations for a two-bladed rotor with pitch angle 15 • , 19 • , 25 • and 30 • .

Figure 13 :

 13 Figure 13: Sectional thrust coefficient as a function of the non-dimensional rotor radius obtained from 3D NS computations and VLM for pitch angles 15 • , 19 • , 25 • and 30 • .

Finally, experimental data

  for rotors operating in the low Reynolds number compressible regime were provided and, along with numerical results, similar flow pattern to that observed on bio-inspired flapping and revolving wings (i.e. stability of the leading edge vortex and enhanced thrust production) were analyzed.These results provided insight into the physics of low Reynolds number compressible flows and into the accuracy of numerical approaches for their prediction, thereby helping future design of Mars and stratospheric flying vehicles.Appendix -Unsteady forces and moments on cambered airfoilUnsteady responses of lift, drag and pitching moment obtained on a cambered airfoil at Reynolds number 3000 and Mach numbers 0.1, 0.5 and 0.8 are further analyzed for angles of attack 2 • , 8 • and 15 • . Figure14plots the instantaneous fluctuating parts of lift, drag and pitching moment coefficients (C L , C D and C m respectively) as a function of the convective time. Note that the pitching moment coefficient is defined as C m = 2m/ρScU 2 , where m is the pitching moment about a spanwise axis located a quarter chord downstream of the leading edge.At α = 2 • , it is seen that amplitudes of force and moment fluctuations decrease as the Mach number is increased from 0.1 to 0.5 but then increase slightly as the Mach number is further increased to 0.8. As observed in section 5.2, this non-monotonic trend may result from two competing mechanisms. On the one hand, increasing the Mach number at a given Reynolds number tends to move the separation point upstream, hence a larger portion of the airfoil is under the influence of the unsteady, separated leading edge shear layer. On the other hand, it tends to stabilize the separated leading edge shear layer which seems immune to 'early' roll-up. At higher angles of attack α = 8 • and 15 • , the first mechanism is relatively weaker since separation already occurs close or at the leading edge even at low Mach numbers. Hence force and moment fluctuations monotonically decrease with Mach number as a result of the stabilization of the leading edge shear layer.In addition, it is shown that the Strouhal number generally decreases with increasing Mach number. At α = 2 • , St = 2.17, 2.07 and 1.79 for Mach number 0.1, 0.5 and 0.8 respectively. At α = 8 • , St = 1.78, 1.65 and 1.33 for Mach number 0.1, 0.5 and 0.8 respectively. At α = 15 • , the Strouhal number first decreases from 0.75 to 0.59 as the Mach number is increased from 0.1 to 0.5, and then increases to 0.78 as the Mach number is further increased to 0.8.

  figure 10, force and moment signals are steady at Re = 100 and may exhibit chaotic fluctuations as the Reynolds number is increased to 10000.

Figure 14 :

 14 Figure 14: Fluctuating parts of lift, drag and pitching moment coefficients obtained from 2D Navier-Stokes computations for a cambered airfoil at Reynolds number 3000 and Mach numbers 0.1, 0.5 and 0.8.

Table 1 :

 1 Properties of Earth and Mars atmospheres

			Mars	Earth
		Sea level	Ground level	Altitude 30 km
	Gas	N 2 (78%) O 2 (21%)	CO 2 (96%)	N 2 (78%) O 2 (21%)
	Gravity (m.s -2 )	9.81	3.72	9.71
	Density (kg.m -2 )	1.225	0.014	0.018
	Pressure (Pa)	10 5	600	1154
	Temperature (K)	288	210	227
	Speed of sound (m.s -1 )	340	238	302

  • , 25 • and 30 • .

		very prospective nature of these applications, very little knowledge is available
		on the physics of low Reynolds number, compressible flows. Furthermore, it is
		not clear whether conventional numerical methods used to predict aerodynamic
		forces on moving bodies are suited to such flow regime.
		In this paper, the accuracy of low and high fidelity numerical approaches in
		the prediction of aerodynamic forces at low Reynolds numbers and moderate
		subsonic Mach numbers was assessed. Assessment was achieved by comparing
		numerical results with experimental data on airfoils from [1] and with present
		experiments on rotors. It was shown that high fidelity approaches, i.e. numerical
		resolution of the 3D Navier-Stokes (NS) equations, provide results in generally
		good agreement with experiments for the whole range of operating conditions
		tested. On the other hand, it was shown that numerical resolution of the 2D
		6. Conclusion
		Flight on Mars and in the stratosphere, high speed trains in low pressure
	470	tubes and liquid atomization share unique atmodynamic features in that they
		involve bodies moving in a compressible viscous flow, i.e. they operate in a
		low-to-moderate Reynolds number, compressible flow regime. Because of the

Navier-Stokes equations, potential flow approaches and Vortex Lattice Methods (VLM) provide reasonable estimation of aerodynamic forces when the angle of attack / pitch angle is low and the flow is attached (or weakly seperated).


Houda Oufaida
email: houfaida@esi.dz

Philippe Blache
email: blache@lpl-aix.fr

Omar Nouali
email: onouali@cerist.dz

A coherence model for sentence ordering

Keywords: coherence features, coherence model, sentence ordering, automatic summarization, genetic algorithm

Text generation applications such as machine translation and automatic summarization require an additional post-processing step to enhance readability and coherence of output texts. In this work, we identify a set of coherence features from different levels of discourse analysis. Features have either positive or negative input to the output coherence. We propose a new model that combines these features to produce more coherent summaries for our target application: extractive summarization. The model use a genetic algorithm to search for a better ordering of the extracted sentences to form output summaries. Experimentations on two datasets using an automatic coherence assessment measure show promising results.

Introduction

Coherence and cohesion are key elements for text comprehension [START_REF] Slakta | L'ordre du texte (The Order of the Text)[END_REF]. Coherence involves logical flow of ideas around an overall intent. It reports a conceptual organization of discourse and can be observed at the semantic level. Coherence is essential to text comprehension. Indeed, with a lack of coherence, the text loses quickly its informational value. Dealing with text coherence remains a difficult issue for several NLP applications such as machine translation, text generation and automatic summarization. Most of automatic summarization systems rely on extractive methods which extract complete sentences from source texts to form summaries. This ensures that the summary is grammaticality correct but in no case its coherence. Considering coherence of extractive summaries involves dealing with sentence informativness input against summary's flow. Several elements contribute to text coherence such as discourse relations [START_REF] Barzilay | The Rhetorical Parsing of Natural Language Texts[END_REF], sentences connection by mean of common entities patterns [START_REF] Barzilay | Modeling Local Coherence: An Entity-Based Approach[END_REF] and thematic pregression [START_REF] Barzilay | Inferring strategies for sentence ordering in multidocument news summarization[END_REF]. In the automatic summarization task, it is fundamental to generate intelligible summaries. Extractive techniques succeed in selecting most relevant information but mostly fail to guarantee their coherence. Only few of these techniques considered coherence as an additional feature in the summary extraction process. It is a difficult task which tackles with multi level discourse analysis: syntactic level which connectors are used to improve text cohesion, semantic level in which textual segments are regrouped around common concepts and finally, global level in which sentences are presented in a logical flow of ideas. In this paper we deal with coherence as an optimization problem. We identify a set of coherence features that have positive or negative impact on summaries coherence. The intuition is that positive input features such as original thematic ordering in the source text/texts and shared entities of adjacent sentences contribute to local and global coherence. These features should be maximized and negative input features such as redundancy should be minimized. The rest of the paper is organized as follows: we first introduce a review of the very few works in the field. Second, we describe how our coherence model combines between coherence features to better ordering sentences within system summaries. Details and discussion of our experiments are presented, the coherence model is introduced as a post processing step. Finally, we conclude our work with some interesting perspectives.

Related work

Early approaches of automatic summarization use sentence compression techniques to improve summaries' coherence. The main idea is to reproduce human summarization process, namely: i-identify relevant sentences ii-compress and reformulate relevant iii-reorder sentences iv-add discourse elements to make a cohesive summary. Probably the most referenced work is Rhetorical Structure Theory (RST) discourse analysis [START_REF] Barzilay | The Rhetorical Parsing of Natural Language Texts[END_REF]. A set of discourse relation markers from an annotated corpus are used to define two elements for each relation: nucleus and satellite. The analysis generate a tree in which the nucleus parts of the top levels are the most relevant ones. [START_REF] Knight | Summarization Beyond Sentence Extraction: A Probabilistic Approach to Sentence Compression[END_REF] train an algorithm on collections of (texts, summaries) to discover compression rules using a noisy-channel framework. The assumption is that the compressed form is the source of a signal which was affected by some noise, optional text. The model learns how to restore the compressed form and assesses the probability that it is grammaticality correct . More recently, [START_REF] Molina | Discursive sentence compression[END_REF] define the concept of textual energy of elementary discourse units. It reflects the degree of each segments informativeness: the more the segment shares words with other segment the more it is informative. Less informative segments are eliminated and the remaining segments grammaticality is estimated by mean of a language model. [START_REF] Barzilay | Inferring strategies for sentence ordering in multidocument news summarization[END_REF] study the thematic progression in the source texts and identify which thematic ordering is better for the output summaries. The authors define three strategies for sentence ordering: (1) majority ordering which is a generalization of ordering by sentence position and reflects, for each couple of themes, how many source texts sentences from the first theme precede the sentences from the second one (2) chronological ordering in which themes are ordered by their publication date and (3) Augmented ordering which add a cohesion element that regroups themes whose sentences appear in the same blocks of texts. Sentences in the output summary are assigned to themes and follow the thematic ordering. Augmented ordering seems to be the best alternative for news articles. [START_REF] Barzilay | Modeling Local Coherence: An Entity-Based Approach[END_REF] define local coherence as a set of sentence transitions required for textual coherence. An entity-based representation of the source text is used to model coherent transitions. The intuition is that consecutive segments (sentences) about same entities are more coherent. The model estimates transition patterns probabilities from a collection of coherent texts. More recently, [START_REF] Christensen | Towards Coherent Multi-Document Summarization[END_REF] introduce a joint model that combines between coherence and sentence salience in the sentence extraction process. A discourse graph is first generated in which vertices correspond to sentences and positive edges weights to coherent transitions between each couple of sentences i-e the second sentence could be placed after the first sentence in a coherent text. It is based on syntactic information such as deverbal noun reference, event/entity continuation and RST discourse markers. The success of deep learning architectures in various NLP tasks including coherence models was recently investigated. [START_REF] Li | A Model of Coherence Based on Distributed Sentence Representation[END_REF] train a three level neural network to model sentences composition to form coherent paragraphs. Here, positive examples are coherent sentence windows and negative examples are sentences windows in which a sentence was randomly replaced. Sentence vectors are induced from the sequence of its word embeddings using recurrent neural networks. The neural network is trained using pairs of original articles and randomly permuted sentences, window size is three consecutive sentences. [START_REF] Nguyen | A Model of Coherence Based on Distributed Sentence Representation[END_REF] propose to generalize the entity based coherence model initially proposed by [START_REF] Barzilay | Inferring strategies for sentence ordering in multidocument news summarization[END_REF] using a neuronal architecture. The model maps grammatical roles within entity grid to a continuous representation (a real valued vector learned by back propagation). Entity transition representations of a given sentence sequence are used by convolution, pooling and linear projection layers to finally compute a coherence score. The model is trained on a set of ordered coherent/less coherent document pairs and compared to several coherence models for three tasks: sentence ordering and summary coherence rating. In the previous work, various features are used to improve output coherence. RST discourse analysis is certainly of value to define a global coherence model. However, it requires deep text analysis which is not available for most languages. In this work, we have selected a set of coherence features. Each feature is supposed to help the model to give higher or lower coherence score according to a particular sentence ordering. The model combines between features and selects an ordering that maximises the coherence score. We assume that these features, once applied together, complement each other and lead to better coherence. We use genetic algorithm to select a coherent ordering. The advantage is that the model can be easily alimented by additional and language specific features. Features can be added to the fitness function by specifying its contribution to the output ordering. The next section describes, in detail, the proposed coherence model.

Coherence model

In our coherence model, we propose to combine state-of-the-art features using a genetic algorithm. These features are domain independent and could be automatically extracted for a large number of languages.

Coherence features

Positive input features positive input features are features who should be maximized in the output summary. They are assumed to help the model to produce more coherent summaries.

Sentence position: sentence position feature is based on the assumption that sentence ordering in source text is coherent and a coherent summary should follow the initial ordering. In multi-document summarization, this ordering is generalized using publication date in a way that the first sentence in the first document is given the label "1" and the last sentence in the most recent document is given the label "n", "n" being the number of sentences in all source documents. Shared entities: it is an important feature based on the assumption that sentences discussing same entities should appear in the same textual segment. [START_REF] Charolles | Introduction aux problmes de la cohrence des textes: Approche thorique et tude des pratiques pdagogiques[END_REF] defines textual continuity as "a linear progression of elements with strict recurrence" which puts forward that coherent development of text should not introduce a sudden break. Shared entities feature was introduced by [START_REF] Barzilay | Modeling Local Coherence: An Entity-Based Approach[END_REF], it requires part of speech tagging. In practice, noun phrases tag set depends on target language and the Part of Speech tagger used (NN, NNP, NNS, NNPS, etc. for English Peen Treebank tag set). We use the number of shared noun phrases between each couple of adjacent sentences in the candidate summary as a positive input feature (1) [START_REF] Barzilay | The Rhetorical Parsing of Natural Language Texts[END_REF].

Common Entities(S 1 , S 2 ) = 2 × |Entities(S 1 ) ∩ Entities(S 2 )| |S 1 | + |S 2 | (1) Score Entities(R) = i=1..|R|-1 Common Entities(S i , S i+1 ) (2)
Thematic ordering: thematic progression is a key factor in information ordering and text comprehension. Presenting information in a logical progression is important especially in summaries which are size limited. Following [START_REF] Barzilay | Inferring strategies for sentence ordering in multidocument news summarization[END_REF], we want to make summaries thematic progression similar to source texts. We define a precedence matrix (PM) of topics. Each entry P M [c i , c j ] corresponds to the percentage of sentences from topic i which appears before sentences from the second topic j in source texts.

        Topics T 0 T 1 T 2 T 3 T 4 T 5 T 0 0.
       
Different possible strategies for thematic ordering could be considered. A first strategy is to order topics according to their precedence value. We define precedence value of a target topic as the sum of remaining topics precedence value to the target topic (sum per column) (3). Topic with minimum precedence will be the first topic to be mentioned in the summary thematic ordering.

P recedence Score(C j ) = i=1..|C| P recedence(C i , C j ) (3)
Another strategy is to build thematic ordering gradually. The algorithm starts with couple of topics with a strong precedence score (T 2 and T 5 in the example). Then the algorithm search for another couple of topics that maximizes precedence scores for the just selected topics at the beginning/end of the previous ordering. Algorithm 1 repeats these steps until finding a complete ordering which includes all topics. We compare system summary ordering against source Return: Ordering R Algorithm 1: Pseudo algorithm for thematic ordering extraction texts thematic ordering using using the distance between the two ordering vectors (4). System summary is likely to be not complete, we complete the shortest vector by the value of the last item (last topic number)

T hematic Ordering Score = 1 Distance(Sum Ord, Source Ord) (4)

Negative input features

Redundacy: in addition to the size constraint, redundancy is not recommended. Bringing new information in each sentence is essential to the semantic coherence of any text. In the context of automatic summarization, it is critical to present new relevant information in each single sentence. We use a sentence similarity measure proposed in [START_REF] Oufaida | Using Distributed Word Representations and mRMR Discriminant Analysis for Multilingual Text Summarization[END_REF] to compute sentence relatedness between each couple of sentences.

Sim(S 1 .S 2 ) = i M atch(w i ) + j M atch(w j ) |S 1 | + |S 2 | (5)
We define a redundancy score for each system summary as the sum of all relatedness scores of included sentences ( 6). This feature is competing with the continuity defined by the shared entities feature. Indeed, if two sentences mention the same entities, they are similar to a certain degree.

Redundancy Score = i,j=1..|R||i =j Relatedness(S i , S j ) (6)

Coherence model

Our problem is to order most relevant sentence in most possible coherent way. We have defined a set of positive/negative input features that improve/degrade summary coherence. Obviously, evaluating a coherence score for each possible ordering is not feasible. Indeed, a summary of 250 words in English contains approximately 13 to 17 phrases (A sentence contains, in average, 15 to 20 words).

In the fitness function, each coherence feature is an objective to be attended (maximize or minimize) in the output summary ordering. Figure 1 presents an overview of the coherence model steps.

Model parameters

Fitness Function: each coherence feature is integrated to the fitness function according to its sense of contribution. For example, (Shared entities, +), (Thematic ordeing, +), (Sentence similarity, -1) is a fitness function. We define several possible combinations and evaluate coherence for each target fitness function.

Ordering codification: each candidate summary ordering is represented by a vector of sentences IDs. Vector size is equal to the number of sentences included in the system summary with respect to the summary's size. Coherence assessment: Each feature value is calculated for each ordering (chromosome) in the population. An ordering is better than another if it has higher feature values.

Selection: it consists of selecting best coherent ordering from the population to form the next generation. Each ordering which fits the best fitness function (coherence features) is more likely to be selected in the next generation. We use the tournament selection method since it tends to converge quickly towards satisfactory output [START_REF] Razali | Genetic algorithm performance with different selection strategies in solving TSP[END_REF]. Each selected ordering will be a parent of the next generation orderings. Tournament selection is repeated n times until having the complete set of parents.

Crossover: the parents are used to form new orderings using the crossover operator. Two parents are randomly selected and a two-point crossover operator is applied to merge parts of parents and form new orderings. We believe that two points crossover is sufficient for summaries (less then 20 sentences for a summary of 250 words).

Crossover operation may generate invalid orderings in the case of duplicate sentences or surpassed size of desired summary. In this case, invalid children are ignored and the crossover operation is repeated until the desired number of orderings is reached.

Mutation: it consists of randomly switching couple of sentences in the target ordering to create a new one. Besides the crossover operator, mutation assists in genetic diversity. It does not generate invalid summaries since it keeps the same sentences.

Final output: the purpose of the development stage is to make sentence orderings more coherent across generations until reaching the maximum number of generations to be explored. Here, the ordering which fits, the most, fitness function is selected from the last generation as the final output.

Experimentation

The main goal of the experimentation is to assess the input of each coherence feature to enhance output coherence. We have implemented our solution under DEAP Package [START_REF] Fortin | DEAP : Evolutionary algorithms made easy[END_REF] which implements a set of evolutionary algorithms for optimisation problems: genetic algorithms, particle swarm optimization and differential evolution. We have opted for a dynamic fitness function that allows users to define couples of (feature, input sense) to be considered.

Coherence assessment

It is a difficult task to assess text coherence from different levels; local and global coherence and in all its aspects: rhetorical organization, cohesion and readability. Using a coherence metric is a first quick option to assess coherence features input.

We use Dicomer metric [START_REF] Lin | Combining coherence models and machine translation evaluation metrics for summarization evaluation[END_REF] which is based on a model that captures statistical distribution of intra and inter-discourse relations. The model uses a matrix of discourse role transitions of terms from adjacent sentences. The nature of transition patterns and their probability are used to train an SVM classifier. The classifier learns how to rank original texts and texts in which sentence ordering is shuffled. Three collections of texts and summaries from TAC conferences are used to train the classifier.

Datasets

Since our target task is text summarization, we use two summarization datasets. The MultiLing 2015 dataset [START_REF] Giannakopoulos | MultiLing 2015: Multilingual Summarization of Single and Multi-Documents, On-line Fora, and Call-center Conversations[END_REF] is a collection of 15 document sets of news articles from the WikiNews website. Each document set contains 10 news texts about the same event such as 2005 London bombings or the 2004 tsunami. The task is to provide a single fluent summary of 250 words maximum. The second dataset is DUC 2002 single document summarization dataset 4 . In our experiment, we use random 100 news articles and produce system summaries that not exceed 100 words. For each document , a human made summary is provided as a reference.

Summarization system

We use a multilingual summarizer [START_REF] Oufaida | Using Distributed Word Representations and mRMR Discriminant Analysis for Multilingual Text Summarization[END_REF] to generate extractive summaries. The summarizer first performs sentence clustering to identify main topics within source texts. Second, terms are ranked according to their relevance to each topic using minimum Redundancy and Maximum Relevance feature selection algorithm [START_REF] Peng | Feature selection based on mutual information criteria of max-dependency, max-relevance, and min-redundancy[END_REF]. Finally, a score is assigned to each sentence according to the terms mRMR scores. The system summary keeps top relevant sentences up to the summary maximum size. Top relevant sentences could be extracted from different source documents and paragraphs which necessarily affects summaries coherence. Finding a better ordering of output sentences will improve summary's coherence

Genetic algorithm parameters

In addition to fitness function, there is a set of parameters that should be fixed such as crossover and mutation probability, population size and number of generations. For our experimentations, we have fixed population size at 300 individuals, the number of generations at 300, mutation probability at 0.001 and crossover probability at 0.01. We deliberately decrease the crossover probability since crossover operator generated invalid individuals (summaries that contain duplicate sentences or exceed desired size).

Evaluation protocol

As described in 1, we define eight configurations for output summary generation: Baseline, thematic ordering and genetic ordering.

Baseline the first configuration represents our baseline: ordering sentences following the original source text ordering. We assume that baseline ordering introduces gaps between sentences since sentences' sequence is broken.

Topline we consider as a topline, Dicomer scores of reference summaries. Since reference summaries are human made, we assume that it is an upper bound for Dicomer coherence scores.

Rule this configuration combines between our baseline (original ordering) and thematic ordering (see pseudo algorithm 1). Sentences follow first thematic ordering and within each topic, sentences are ordered following their positions.

Coherence model ordering we define several configurations according to the number of positif/negatif input features and the number of sentences to be considered as an input. Here, shared entities feature is combined with thematic ordering, sentence position in the fitness function. Sentence relevance and redundancy penalty features are considered when the model take as an input sentences that exceed the size limit (125% and 150% in our configurations). Then, the model selects a subset of sentences that optimize fitness function score with respect to summary size.

Results and discussion

Figures 2 and 3 report Dicomer coherence scores for each configuration. Topline (Human reference summaries) coherence scores reaches an upper bound of 1.9 for MultiLing 2015 dataset and 1.87 for DUC 2002 dataset.

Baseline system summaries following original orderings (SUMBA) coherence scores is 1.41 for Multiling dataset and 1.29 for DUC 2002 Dataset. Thematic ordering combined with shared entity (SUMG2,SUMG4) present best coherence score for system summaries for both DUC 2002 dataset with a value of 1.34 and Multiling dataset with a value of 1.59. It is the maximum coherence value of system summaries. However, coherence model scores are average and range from 1.27 when five features are considered (SUMG5, SUMG6) to a value of 1.38 when shared entities are considered along with thematic ordering and sentence position feature for the Multiling dataset (SUMG1). Baseline coherence scores are particularly high compared to other configuration results. When we examine output summaries of the TopN configuration, we find that TopN sentences are similar (contain most relevant terms) leading to some degree of topical coherence.

Conclusion

Dealing with text coherence is a challenging task in the NLP field. Taking into account coherence is critical to design efficient tools for text generation which is essential to a range of NLP tasks such as automatic summarization, dialog systems and machine translation. Modeling coherence involves syntactic and semantic levels of discourse analysis: entity-transition patterns, thematic ordering and rhetorical discourse relations. The difficulty with is in defining coherence features and operating all its aspects in a single model. In this work, we have defined a first model of coherence which combines features that, we assume, have positive/negative input and enhance/affect text coherence.

We have designed a genetic algorithm model that take into account a set of coherence features: shared entities, thematic ordering, sentence position, relevance and redundancy. The last three features are useful for target task: extractive summarization. We have experimented different combinations of features thanks to the flexibility of the model and its ability to easily include/exclude features. Due to the nature of source texts (news texts which contains significant amount of date phrases), the results are strongly affected by the dissolution of temporal sequences. Temporal relations are also an important aspect of global coherence and should be considered for future experimentations [START_REF] Muller | Annotating and Measuring Temporal Relations in Texts[END_REF]. Another possible interesting direction is to make the model task independent. Some features that we have defined, such as sentence position and relevance, are task-related and could not be considered for other NLP tasks.

1 :

 1 Input: P recedence[, ] : precedence matrix 2: Initialise: Ordering = {} 3: Ordering= (CMax i , CMax j ) = M ax{P recedence(Ci, Cj), ∀i, j < |C|} 4: do 5: M axi=M ax{P recedence( * , Cj), ∀j < |C|} 6: M axj=M ax{P recedence(Ci, * ), ∀i < |C|} 7: Ordering = Ordering ∪ {((CMax i , Cj))} 8: Ordering = Ordering ∪ {((Ci, C M ax j ) )} 9: while |Ordering| < |C| 10:

Fig. 1 :

 1 Fig. 1: Coherence model

Fig. 2 :

 2 Fig. 2: MultiLing 2015 Dicomer coherence scores

Fig. 3 :

 3 Fig. 3: DUC 2002 Dicomer coherence scores

Table 1 :

 1 Configurations for output summaries orderings

	Baseline SUMBA [TopN, Position]
	Topline SUMMA Model summary A MultiLing 2015
	Topline SUMMB Model summary B MultiLing 2015
	Topline SUMMC Model summary C MultiLing 2015
	Topline SUMMD Model summary C DUC 2002
	Rule	SUMTP [Thematic, Position]
	Genetic SUMG1 [+Entity,+Thematic,+Position]
	Genetic SUMG2 [+Thematic]
	Genetic SUMG3 [+Entity]
	Genetic SUMG4 [+Entity,+Thematic]
	Genetic SUMG5 [125%, +Entity,+Thematic,+Position,+Relevance,-Redundancy]
	Genetic SUMG6 [150%, +Entity,+Thematic,+Position,+Relevance,-Redundancy]

*https://duc.nist.gov/duc2002/


Roques Noémie

Experice

Paris

Fictional plays broadcast on YouTube: meet the Playmobil family

Keywords: Playmobil, fictional play, YouTube, play culture

The 2.0 web, also called participative web, is rife with pro-amateur productions, including some created by children who have access to connected devices like laptops, tablets, and smartphones. This paper focuses on children's web videos which depict them involved in fictional plays with toys, primarily Playmobil. These kinds of videos share strongly normative displays like home-made introductions, descriptive video titles, temporal formats, and channel fiction internal coherency.

Children's play web videos constitute a permanent archive of game practices, which differentiates them from offline game practices which are mainly transmitted via oral culture and consequently have limited traceability. The publication of these videos, initiated by the children themselves, allows their game practices to exist beyond their bedrooms and playrooms. This case study aims to demonstrate that web videos are indeed media of play culture [Brougère, 2005] but also witness culture of convergence with the incorporation of external elements from other media and multimodal communication to sustain the channel. This contribution is extracted from ongoing research regarding learning game play through web videos in which YouTube video content is analysed, including the comments section.

Videos were selected of users fitting specific profiles (French-speaking child under 16 years old without adult participation), toys (Playmobil), and a predetermined number of channel subscribers. A video corpus was created, composed of nine channels. Given the importance of the context, play videos were not segregated from other videos on the channel, like FAQ videos, DIY videos, etc.

On MussoMusso's YouTube channel, a video entitled "L'anniversaire de Léa" (Léa's birthday) sets up four Playmobil characters with a Playmobil house in the background. Musso, the young girl who creates the video, orally titles her video and continues: "First I'm going to introduce you to the characters; the father is called Bastien, the mother, Sonia, then little Lea, and little Caramel, let's go!" She punctuates each presentation with gentle movements to place the character at the centre of each frame. The image is generally unstable. She then cuts the shot and allows her fingers to appear in front of the camera lens. The video immediately continues with the father and mother talking in the kitchen. The characters are static, Musso produces the dialogue herself.

"Honey, it's our daughter Lea's birthday soon." The father replied, "But I know she is called Lea (exasperated breath)." "Waf, waf! (dog tongue sounds)". The mother screams "Shut up, Toffee!".

Another scene, viewed slightly from above, follows the exchange. The mother is in the living room, near the dog's basket, and she calls, "Toffee, heel, right now!" The following shot shows the dog in the basket, emitting plaintive squeaks.

Extract from L'anniversaire de Léa, Musso Musso1 . MussoMusso's channel is one of many dedicated to Playmobil on the YouTube platform. Mostly run by children, theses channels stage Playmobil stories and adventures, through short videos, which are primarily recorded at home. The children create their own histories and scenarios using personal toys, cutting and splicing their rushes together to create coherent videos. Arguably two 'cultures', which social sciences researchers have associated with juvenile practices, coexist in the production of fictional plays with Playmobil filmed and broadcast: "screen culture" and "bedroom culture".

As Dominique Pasquier and Josiane Jouët (1999) stated after a national study, young people develop what they call "screen culture". The characterization of younger people's culture continued with "bedroom culture", a phrase coined by Sonia Livingston (2002). "Bedroom culture" describes the growing importance of digital uses, by younger people, within a domestic environment. Sylvie Octobre's work (2003Octobre's work ( , 2004) ) explores family dynamics around digital objects, focussing on intra-family negotiations necessary for young people to assert their digital autonomy. Indeed, access to equipment (camera, telephone, computer, etc.) dictates children's habits. The term script denotes not a rigid dictation of performed action but, rather, a necessary openness to resistance, interpretation, and improvisation. When I describe elements of material culture as "scripting" human actions, I am not suggesting that people lack agency. Rather, I am proposing that agency, intention, and racial subjectivation co-emerge through everyday physical encounters with the material world. I use the term script as a theatrical practitioner might: to denote an evocative primary substance from which actors, directors, and designers build complex, variable performances that occupy real time and space. [...] That which I call a "scriptive thing," like a play script, broadly structures a performance while simultaneously allowing for resistance and unleashing original, live variations that may not be individually predictable. (Bernstein, 2009, 68) Through the analysis of a particular children's book and arcade photographs, Bernstein shows that toys, or a background set, contain a script, i.e. a defined usage and set of representations -this notion enlightens the phenomenon of fictional play videos. This paper develops, firstly, the dramatization of everyday life in Playmobil web-videos and, secondly, focuses on the question of material and media culture.

Data collection methods

The results presented are extracted from a Master's Degree research program which explored the dramatization of plays on the Internet. Due to the exploratory nature of this research, the statements presented are more a series of hypotheses than actual conclusions. The data analysed was based on 1059 web-videos, extracted from nine YouTube channels selected dependent on; the producer's age, the absolute nonparticipation of an adult, and the subscriber's number -to avoid the selection of only popular channels. The participation of an adult was sometimes difficult to ascertain; indeed, the last line of the table is in red due to the late realisation that an adult was involved in the creative process.

Reaching children through the Internet to carry out research is problematic, however, interviews were conducted with two producers of Playmobil web videos, girls, aged 12 and 15 years old. While these interviews may be insufficient to produce strong research results, nevertheless shed some light on certain production dynamics. The video corpus examined revealed that 43.6% of the material selected was about everyday life -for eight of the nine channels, it is the most significant content. The locations are primarily the familial home or classroom, occasionally a school or family trip. Given the producer's socio-demographic profile, in the corpus, it is unsurprising that everyday life was the primary frame of reference. Indeed young girls, who are encouraged to play with everyday objects and situations and naturally engage in interpersonal relationships, are the main creators of these web videos, as opposed to boys who are arguably motivated by war games (Brougère, 2003a). The term "frame" is used purposefully, since regular routines structure many of the web video stories: sleep or meal times are present in 71% of the 'Daily Life' category.

Dramatization of everyday life

Despite its apparent triviality, daily life content is overtly dramatized. Conflict, fights, punishment between family's members, friends, classmates or money and social distress are included in these web videos.

Conflicts between children and parents display obvious patterns: the child did something wrong, had a bad attitude, or the parent did not appreciate the language used etc,, The young video producer may dramatise events from their own experiences, those of their peers, or create stories initiated by audience requests Fights between brothers are often regarding perceived authority or the sharing of space and resources, whereas, conflict between friends is primarily about trust and envy.

A child may include elements, into their plays, from observation or direct personal experience. During one particular interview, Marine was with her parents, while they appeared talkative and often interjected, allowed their daughter to build on her own memories:

Interviewer: Do you talk about things that happen to you?

Marine: Sometimes, yes.

Marine's father: Like fights.

Marine: Yeah, fights, but not that much.

Marine's father: I remember some conflicts you told us about.

Marine's mother: Bullying too… Marine: But that wasn't happening to me.

Marine's mother: Yes it was not but you've heard about it.

Marine's father: At school.

Marine: Yeah, I heard about it.

Interview Marine, 12 years old girl.

Irrespective of the dramatised conflict, it is invariably resolved by the end of, or in the next, episode -if in a serialised format. The resolutions tend to follow a prescribed pattern; awareness of the conflict's causes, a specific solution, and moral lessons learnt by the protagonists.

Extraordinary events, such as travel, provide realistic material to be incorporated into the fiction. Though no reproductions of an actual trip were found in this study's dataset, elements from various outings were incorporated in the videos. For example, one episode of Laurie's Playmobil family story, introduced details about the use of coins during a camping holiday.

Interviewer: By the way you made a video about camping; it was with your Pe-tzShop, does that means you have been camping? Laurie: Yeah.

Interviewer: I was wondering since you were very specific with the coin story.

Laurie: (laughing) Yeah absolutely, how could I have known?

Interview Laurie, 11 years old girl

The action/adventure category, of the video corpus, included stories with the following structure: initial scenario, a disruptive element, adventures, resolution and an epilogue. Two series of MussoMusso videos follow this format, taking place in a fairy universe with princes and princesses. On the whole, daily life videos follow the traditional action/adventure story structure, presumably due to the influence of children's cartoons which adopt this narrative scheme and consequently simple to understand and implement, and affords story serialisation.

Serial format

The serialising of Playmobil stories may create viewer anticipation as they wish The work of Anne Besson (2004) describes how 'seriality', or "serial narration" -a series of independent scenarios -may involve cyclical narration i.e. episode series based on the same scenario. In the corpus, serial narration is the most frequently used format, with episodes broadcasted more or less regularly.

The nine channels selected for this study have similar content -beyond the theme -and often framed by a generic which present the same Playmobil protagonist and characters in a video series, producing a fictional internal coherency.

Story titles permit easy theme and series identification, while numeric identifiers -for example, 'Episode 1 Season 1' -allows the producer to alternate stories.

Interviewer: Are you aware of viewers' demands? Marine: Oh yes.

Marine's Father: I can confirm. If we do not listen to them, it's the flameware 3 .

Interviewer: Like they're unhappy? Marine's father: I think so, obviously they follow this format. I don't watch Playmobil videos myself, but other producers adopt this format -they don't show their hands in the video.

Marine: I'm putting my hands when I'm playing alone, I don't have to think about my hands… I tried to do the same in front of the camera but… (she remains quiet), Interviwer: The viewers don't like it? Marine's father: My god no, not at all. Marine: So now I make cuts 4 in the video, cutting is right.

Interview Marine, 12 years old girl

Marine's father is clearly aware there are specific audience-approved techniques to producing videos -to which he and his daughter conform to avoid 2 https://www.youtube.com/watch?v=Cq0-T0yi5pc&t=81s 3 Indigenous term ("engueulade" -pronounced with emphasis) which means a fight through negative comments.

4 'Cutting' is a technical term used in the video editing process.

'flamewares' (negative comments and conflict). It is noteworthy that although he is familiar with the norms which govern production: "…,obviously, they follow this format", he claims not to watch Playmobil videos. To accept his claim assumes viewers have made similar comments about particular details i.e. the visibility, or not, of hands in videos. Viewer's feedback tends to normalise content, however, the YouTube system permits observation of a child's video content and processes as these evolve over time.

Material and media culture

Toy's affordance

In addition to similar format or differing video norms, particular storylines reoccur: holidays spent at campsites, hotels, or the family home. While a child may have experienced these locations, their narrative may be enhanced or constrained by the Playmobil at their disposal since the toy is sold in themed packaged sets.

The limited variety of themed play sets along and the norms in using the YouTube platform tends to result in children producing similarly scripted material, this in addition to the toy's affordance. The affordance may be defined as including the toy's physical attributes and user's perceived cultural propositions i.e. all the actions which may be performed with the object. A Playmobil figurine is unable to spread its arms horizontally (and other motion and geometric limitations) as well as the affordance of the thematic set.

Modifications and mix

The required realism and limited diversity of the Playmobil system was fundamental for the young video producers since it is difficult to physically modify the various action figures or tools, the producers naturally resort to innovation.

Extract from Roxy's video5

Media consumption and integration in Playmobil web-videos

A specific though uncommon video model was found within the study's video corpus -3% of the video studied -mise en abyme. This formal technique involves duplication of an image or concept within itself; a play of signifiers within a text, of sub-texts, mirroring each other. Playmobil video stories in which the action figures themselves, are making a film or video.

An illustration of this video technique was created by Kingsolaya, a 12 year old girl video producer, in which her main fictional protagonist became pregnant. Immediately on return from the maternity ward, the protagonist shoots a "maternity haul" video -a popular style of YouTube video displaying all the recent acquisitions for the baby, for example, a cot, clothes, etc.. The action figure is shown talking to an audience about their shopping.

That's it, YouTube. I will show all the stuff I bought for my baby, I will open my channel. Let's go. That's so great. "Hello everyone, I just gave birth to a baby girl and I will show you all I bought. A pink high chair with a little bird. A little play carpet with pink fishes. A pink bed. And a changing table. Here you go. See you." I'm too lazy to cut all this, pfff.

Extract from the video "la naissance de Salomé"6

While the conception and delivery is depicted in the girl's fictional play, arguably the level of detail regarding maternity accessories and the manner of 5 https://www.youtube.com/watch?v=A5hDNi6SS6o&t=82s 6 https://www.youtube.com/watch?v=m-BMQNBYKvI talking to an audience interested in maternity items, may appear surprising.

Though perhaps understandable considering the number of famous women on YouTube -followed by a young audience -who fall pregnant and adapt their online content due to their life changes, including maternity haul videos. Perhaps young girls who followed these women on YouTube for their makeup tutorials, for example, are remixing [START_REF] Willett | Children, Media and Playground Cultures: Ethnographic Studies of School Playtimes[END_REF]) by interpreting through their own frames of reference and subjectivity, the content they view. Children using digital media are often active subjects, they are critical observers; they have critical distance, about the content.

Young video producers appear to not simply reproduce or imitate, with their toys particular media content. On occasions, the child may possess limited memories about a show they are attempting to interpretively recreate. For example, one study participant conducted a video based on the television show 'Supernanny'7 . She was asked:

Interviewer: The awful little girl in your first episode, her name's Peggy, is she from a 'Supernanny' episode? Marine: Mmm… Nop.

Iinterviewer: Oh, you invented her? Same thing for Jean-Pierre -the one who insults his mom and bites her once? Marine: Nooop, I invented, that's too gross for 'Supernanny'.

Marine 's father: I bet there are kids like this in the show.

Marine 's mother: We saw it, sweetie.

Marine: You're right, I remember one who was putting his middle finger up.

Interview with Marine, 12 years old girl.

Clearly elements which are introduced in Marine's 'Supernanny' video are a bit fuzzy and her recollections differ from that of her parents. The problematic child depicted in her video was not an exact representation of a child present on the show, but rather an amalgamation of various bad attitudes displayed during the television program and re-enacted chronologically in her narrative. Marine's 'Supernanny' is structurally similar to that of the television show: family presentation, explanation of the difficulties with the child, observation time, rules enact-ment, 'Supernanny' examples, 'Supernanny' leaving, parents facing their child, the return of 'Supernanny' and the review.

Replicating the television show structure creates a suitable framework on which to base the web video. The video represents original contents though loosely based on observed media material supports the relevant notion of "play script", as described by Robin Bernstein. The child's narrative is supported by media content, what is significant in the fiction, or, the spectacularization of the play (Brougère, 2003a), is the correct identification of the characters (like 'Supernanny'), their function (a strict guardian), and the relationships developed ('Supernanny' is the authority figure).

Conclusion

The themes and stories children developed, as captured in this study's video corpus, are arguably influenced by scripts pre-existing in the Playmobil package designs and the toy's affordances. These dramatized narratives are often in response to the demands of the video's audience, constrained by the toy's inherent limitations, and strategies employed to overcome the drawbacks. Playmobil web videos appear to draw on the child's personal experiences, observances -through various media channels and real-life -and invariably include a moral lesson to be learnt.

It has been noted that YouTube video technical processes may influence these video productions as well as reality television shows -this screen content may inform the reception by children. Children do not tend to passively assimilate the content, but rather re-interpreting and repeating, enhancing the material with their own fictional elements. William Corsaro developed the concept of "interpretive reproduction" (2011), i.e. the children's capacity to be an active member of society : I mean interpretive to suggest the innovative and creative aspects of children's participation in society, and, in fact, children as young as two create and participate in their peer cultures by appropriating information from the adult world to address their unique peer concerns. I mean by reproduction the idea that children do not simply internalize society and culture, but they actively contribute to cultural production and change. I think the two words of the term together also imply that children are-in their very participation in society-constrained by the existing social structure and by processes of social reproduction. That is, children and their childhoods are affected by the societies and cultures of which they are members [START_REF] Corsaro | Reproduction interprétative et culture enfantine. Universalité et diversité de l'expression[END_REF].

Thus the notion of "interpretive reproduction" finds a particular resonance with the research presented in this paper, given the child has an active participation, and an active function, in society.

Fictional play videos broadcasted on YouTube reach beyond questions of numeric traces and visibility of domestic practices, but may enlighten observers on child cultural construction. Likewise, fictional plays on YouTube are not analysable like a real witness of what occurs in the privacy of a bedroom. If, as Stephen Kline states, "Children have an internal fantasy and internal dialog very elaborated" (Kline, 1995, 142), Playmobil videos are not the means by which to observe this phenomenon, on account of the strong normative displays discussed in this paper.

However, Gilles Brougère asserted, "Playing with a toy, it's receive through an artefact complex cultural data, interpret them, and produce a new cultural action, which, even if it reacts to the object's prescription, is always a singular production in a specific context." (Brougère, 2003a, 323). This exploratory research presented highlights that Playmobil fictions constructed, filmed, broadcasted by children are composed of various cultural elements, from numeric entertainment culture and social, familial, and technical contexts.

Fictional

  plays are mostly private taking place between children in a playground or in the privacy of a bedroom. Fictional play web-videos constitute a permanent archive of game practices and share strongly normative displays like home-made introductions, descriptive video titles, temporal format, and channel fiction internal coherency. All these elements invite an understanding as to what kinds of play culture are displayed; convergence culture with the incorporation of external elements from other media and multimodal communication to sustain the channel. Robin Bernstein introduces the notion of "play script" (2009):

  to know what happens next to the protagonists: "I want to know what happens neeeeeeext." Review from the video -On Va Avoir Un Bébé 1/2 [2018] 2

Channel's name Sub- scribers Produc- er's name A ge Gen- der First con- tent published Number of videos Playmo- bil_Tuto&Film 42K Lauryn 15 Girl August 2012 275 Romain 20K Romain 12 Boy February 2014 379 MussoMusso 1K Rebecca (?) 9 Girl Décember 2013 48 PlayWithCamille 3.3K Camille 10 Girl January 2016 90 Kingsolaya 12K Sophie 12 Girl July 2015 30 Lola !!! 2.5K Lola 13 Girl July 2016 17 Lounaplaymobil 587 Louna 12 Girl August 2015 37 Roxymovie82 925 Roxy ( ?) 13 Girl May 2015 7 Les histoires d'Océane 9028 Océane 12 Girl April 2015 176

Table 1 :

 1 topics present in the videos studied

	Channel YouTube TV
			serie Events Dailylife Holidays Action Unclassifiable TOTAL
	Play-
	mo_Tut
	o&Films			34	77	26	16		153
	Romain		9	12	77	44	93		235
	Play-
	withCa
	mille			3	34	3	12		52
	King-solaya	1	1	1	11				14
	Lola !!
			1	1	5		2		9
	Loun-
	aplay-
	mobil				4	2	2	5	13
	Rox-
	iemov-
	ie82				3	1	3		7
	Musso-
	musso		4	3	7	6	3		23
	Les
	His-
	toires
	d'Océan
	e		1	6	23	4	11	1	46
	TOTAL 1	16	60	241	86	142	6	552
	Percent-
	age	0,1	2,9	10,8	43,6	15,5	25,7	1,08	99.68

https://www.youtube.com/watch?v=61ffiPybsag

Supernanny is a popular reality television show featuring families experiencing behavioural difficulties with their young children.