Placide Jaohindy
email: placide.jaohindy@univ-reunion.fr

Habiba Ennamiri

François Garde

Alain Bastide

Numerical investigation

Keywords: VAWT, Savonius rotor, 1-DOF, Rigid-rotor, CFD, Vortex analysis

come

NOMENCLATURE

INTRODUCTION

The Savonius rotor is a vertical axis wind turbine (VAWT) that was originally invented by the Finnish engineer Sigurd J.

Savonius in 1922 [START_REF] Savonius | The S-rotor and its applications[END_REF]. The concept of a Savonius rotor consists of the principle formulated by Anton Flettner and Savonius [START_REF] Savonius | The S-rotor and its applications[END_REF]. It is formed by two semi-circular blades where the main axis is perpendicular to the wind direction and it accepts wind with any incidence without loss of performance. The Savonius rotor's operation is based on the differential drag between its two blades. The drag is the resistance generated by the fluid on the Savonius rotor blades and vice versa. Chauvin and Benghrib [START_REF] Chauvin | Drag and lift coefficients evolution of a Savonius rotor[END_REF] studied the lift and drag coefficients of a Savonius rotor with a tip speed ratio ranging from 0.2 to 1. They found that the drag coefficient of this rotor is always negative and the lift coefficient is negative for a low value of .

The Savonius rotor is usually considered to be a drag machine, but Modi and Fernando [START_REF] Modi | The Savonius-rotor and its applications[END_REF] and Kamoji et al. [START_REF] Kamoji | Performance test on helical Savonius rotor[END_REF] defend that it is a combination of a lift and a drag machine. Hence, at low angles of attack, the lift force contributes to torque production [START_REF] Kamoji | Performance test on helical Savonius rotor[END_REF]. The Savonius rotor is a slow running VAWT that starts at low wind speeds of 2 to 3 m/s. A large number of investigations have been conducted to identify the aerodynamic parameters of this wind turbine. Le Gourières [START_REF] Gourières | Energie éolienne, théorie, conception et calcul pratique des installations[END_REF] studied the geometrical parameters that affect the low efficiency of the basic Savonius rotor of 20 %. One of these parameters is the rotor aspect ratio ↵ [START_REF] Savonius | The S-rotor and its applications[END_REF][START_REF] Gourières | Energie éolienne, théorie, conception et calcul pratique des installations[END_REF]. For the conventional Savonius rotor of Ushiyama and Nagai [START_REF] Ushiyama | Optimum design configurations and performances of Savonius rotors[END_REF], it was suggested that ↵ = 4. This rotor can reach a maximum power coefficient C p of 30 % and an average static torque coefficient Cm of 33 %, with a rotor overlap ratio between 0.2 to 0.3 [START_REF] Ushiyama | Optimum design configurations and performances of Savonius rotors[END_REF]. In a two-dimensional study of a conventional rotor, Menet et al. [START_REF] Menet | Increase in the Savonius rotors efficiency via a parametric investigation[END_REF][START_REF] Menet | Prévision des performances aérodynamiques d'un nouveau type d'éolienne à axe vertical dérivée du rotor Savonius[END_REF] found that the optimal value of is 0.242. Before the Ushiyama and Nagai [START_REF] Ushiyama | Optimum design configurations and performances of Savonius rotors[END_REF] investigations, Blackwell et al. [START_REF] Blackwell Ben | Wind Tunnel Performance Data for TWO-and Three-Bucket Savonius Rotors[END_REF] performed several tests on the geometrical parameters of the Savonius rotor. Blackwell et al. [START_REF] Blackwell Ben | Wind Tunnel Performance Data for TWO-and Three-Bucket Savonius Rotors[END_REF] studied the performance of fifteen configurations of a Savonius rotor in wind tunnel conditions. The study by Blackwell et al. [START_REF] Blackwell Ben | Wind Tunnel Performance Data for TWO-and Three-Bucket Savonius Rotors[END_REF] varied the dynamic torque C t and the power cofficient Cp as functions of the tip speed ratio , the rotor overlap ratio , and the Reynolds number Re. The C p obtained by the three-blade Savonius rotor was 0.18 at = 0.10. For the same , the maximum dynamic torque coefficient C t obtained with a two-blade Savonius rotor is 0.35. They pointed out that increasing the Re number intensifies the performance of the Savonius rotor. The three-bladed Savonius rotor is known for its good starting torque [START_REF] Gupta | Comparative study of a three-bucket Savonius rotor with a combined three-bucket Savonius three-bladed Darrieus rotor[END_REF]. Blackwell et al. [START_REF] Blackwell Ben | Wind Tunnel Performance Data for TWO-and Three-Bucket Savonius Rotors[END_REF] recommended the use of a Savonius rotor with two-blades and a between 0.1 to 0.5 [START_REF] Ushiyama | Optimum design configurations and performances of Savonius rotors[END_REF]. The Savonius rotor of Run 35 and 37 of Blackwell et al. [START_REF] Blackwell Ben | Wind Tunnel Performance Data for TWO-and Three-Bucket Savonius Rotors[END_REF] was used in the present study because the uncertain analysis of the C t and Cp curves was measured in their experiments. The rotors used both cases tests are identical. The research based on the two-stages Savonius rotor shows that the starting torque coefficient of the rotor is never negative and it presents a good angular stability of the dynamic torque M t [START_REF] Menet | A double-step Savonius rotor for local production of electricity: a design study[END_REF]. To clarify the effect of on the flow mechanism, Fujisawa studied the flow fields in and around Savonius rotors at various in a water channel. Fujisawa compared the obtained experimental results with a two-dimensional numerical study using the discrete vortex method [START_REF] Fujisawa | Velocity measurements and numerical calculations of flow fields in and around Savonius rotors[END_REF]. The numerical investigations made by Fujisawa assumed that the studied problem was a high Reynolds number flow. Fujisawa concluded that the two-dimensional simulation approach of the flow is not valid. The numerical study of a Savonius rotor had to be run in a three-dimensional approach to have a better representation of the three-dimensional vortical structures around the rotor [START_REF] Fujisawa | Velocity measurements and numerical calculations of flow fields in and around Savonius rotors[END_REF]. The importance of the three-dimensional properties of the fluid flow in a case of a Savonius rotor bring us to lead this study in three-dimensions.

Wind energy problems involving airflow in and around a VAWT have traditionally been studied through wind tunnel testing facilities [START_REF] Modi | The Savonius-rotor and its applications[END_REF][START_REF] Blackwell Ben | Wind Tunnel Performance Data for TWO-and Three-Bucket Savonius Rotors[END_REF]. Computational Fluid Dynamics (CFD) techniques were used instead of a wind tunnel in order to make tests for several reasons. Wind tunnel experiments are expensive and the techniques used to visualize the flow fields are costly.

It was suggested that the CFD methods predict the flow around a VAWT rather accurately for the objectives of this work [START_REF] Mohamed | Optimization of Savonius turbines using an obstacle shielding the returning blade[END_REF].

The aims of this article are:

• To validate the developed numerical models based on the steady-state CFD simulation analysis and a solid-fluid coupling;

• To develop an approach to the study of a Savonius VAWT, which is based on the variable rotational speed method.

Few authors have opted for the variable rotational speed method applied to Savonius wind turbine modeling [START_REF] Montelpare | Unsteady Aerodynamics of a Savonius wind rotor: a new computational approach for the simulation of energy performance[END_REF]. Indeed, the constant rotational speed method is more regularly used to study the Savonius wind turbine rotor [START_REF] Menet | Increase in the Savonius rotors efficiency via a parametric investigation[END_REF]15]. However, a variable rotational speed method allows the new creation of a real Savonius rotor rotation by integrating its governing equation. The internal region concept at the interface between the two computational domain regions used in the sliding mesh method will shortly be described. Steady-state and dynamic studies have been conducted with the standard k-" and SST k-! models. Experimental studies of the Savonius rotor are in keeping with the steady-state simulation results. The dynamic study allows the identification of the mechanical behavior of the Savonius rotor and the flow fields around it. The analysis of the three fields of velocity, pressure, and vorticity are discussed.

SIMULATION OF A VAWT

Numerical method

In this study, the three-dimensional RANS equations were discretized in the computational domain by the finite volume method. The time integration was calculated by the implicit Euler scheme. This integration gives a temporal variation of the dynamic parameters in the simulation using the 1-DOF method. Two turbulence models for RANS closure were implemented: the standard k-" and SST k-! models [START_REF] Launder | The numerical computation of turbulent flow[END_REF][START_REF] Menter | Two-equation eddy-viscosity turbulence models for engineering applications[END_REF]. The standard k-" model is also known as a two equation model and is derived from the linear eddy viscosity model. With the standard k-" model approach, two transport equations are used to solve the Reynolds stress tensor [START_REF] Launder | The numerical computation of turbulent flow[END_REF]. These two equations are respectively the transport equation for the kinetic energy k and the dissipation rate ". The standard k-" model is generally used to model a turbulent flow far from the wall [START_REF] Launder | The numerical computation of turbulent flow[END_REF]. The SST k-! is a hybrid model developed by Menter and based on the standard k-" and k-! models [START_REF] Menter | Two-equation eddy-viscosity turbulence models for engineering applications[END_REF].

The Menter's first interest was to improve the performances for adverse pressure gradient (APG) flows, which are found during a flow around a static bluff body, such as VAWT. The second aim was to solve the problem of spurious sensitivity to free-stream conditions [START_REF] Yorke | Assessment of common turbulence models for an idealized adverse pressure gradient flow[END_REF]. Christopher and Susan [START_REF] Christopher | Prediction of high lift: review of present CFD capability[END_REF] argue that the k-! model shows poor performance regarding the wall-bounded separated flows caused by the APG [START_REF] Wilcox | Re-assessment of the scale-determining equation for advanced turbulence models[END_REF]. The improvements of the SST k-! approach in predicting many aeronautical flows motivate us to use this model in our study. For the standard k-" and the SST k-! models that were used, a log law function for smooth wall was used. For the classical two models, the maximum of the dimensionless wall distance y + is 5 at each simulation.

Modeling environment

Grid independence test

A mesh independence test was conducted for the steady-state and dynamic computational domains to improve the accuracy of the CFD results. These two computational domain meshes are then progressively refined until the observed physical quantities were stabilized. They were used to determine the rotor parameters in a steady and unsteady flow.

The two computational domains were meshed using Star-CCM+ software. The first one is used for a static simulation and requires a reasonable number of cells, in order to give consistent static values of moments M . The second one is used for the moving mesh methods. It is formed by internal and external grids. These two grids are separated by a fixed interface. The moving mesh methods is based on exchanging information at interfaces. The smooth wall boundary condition is assumed.

The near wall layers were prismatic elements with triangular bases. The length of the first layer from the blade surface was approximately 0.005. Two layers of prismatic elements were used and less than 1.15 growth ratio. The maximum height of the cells around the walls of the Savonius rotor blades and endplates obtained for y + was 5. The value of y + depends on the spatial variations of the rotating flow near the wind turbine rotor wall and the turbulence models.

The mesh independence study is checked by comparing the static torque M loads on the rotor blades. Fine mesh results are obtained with Grids 1, 2, and 3 in Table I for the static computational domain cells. Fine mesh results are also obtained with Grids 1 and 2 for the dynamic computational domain cells. For a given simulation, the number of cells in the chosen computational domain is of great importance for accurate simulation results. This number of cells for grid independence was taken in Table I, where there is no significant variation in the simulated parameter M . The computational domain of the steady-state study is composed of 77,836 cells. The minimum polyhedral cell width of this domain is equal to 0.015 m. The computational domain of the dynamic study comprises 242,198 cells. The minimum polyhedral cell width of this domain is equal to 0.0074 m. The mesh quality of the dynamic computational domain used in this study is established by an aspect ratio of 20 and a skewness of about 0.85.

Computational domain

The computational domain used in this study (see Figure 1) is a parallelepiped of 20 m ⇥ 10 m ⇥ 3 m with two regions.

A cylinder of 1 m radius and 3 m height is placed at the center of this domain. The Savonius rotor is interdependent with the cylindrical region and is placed at a height of 1.5 m on the rotational axis Oz. The parallelepiped and the cylindrical regions of the computational domain are geometrically independent from one another. The parallelepiped and the cylinder are centered at (0, 0, 1) m. The Savonius rotor is centered at (0, 0, 1.5) m. The cylindrical region rotates through the interface of 4,176 cells. The rotation of the internal region with respect to the parallelepiped region uses a sliding mesh method [START_REF]CD-adapco[END_REF]. Blackwell et al. [START_REF] Blackwell Ben | Wind Tunnel Performance Data for TWO-and Three-Bucket Savonius Rotors[END_REF] have highlighted in their report that the Savonius rotor was made of aluminum alloy type 6060-T6. However, there are many different types of aluminum in which densities differ. The density ⇢ of material of rotor construction is equal to 2700 kg/m 3 . knowing ⇢, we chose to set the thickness ep of the rotor by an inverse method. The initial condition of this method was to fix one point on the experimental curve. The stopping criterion, which gives the thickness of the rotor e p, is substantially equal to 0.0021 m, is the convergence of the experimental and the numerical points to the same value. This choice is arbitrary, but this does not affect the continuity of the calculations. Indeed, the thickness and density are involved only to calculate the moment of inertia J and in particular, they are involved to calculate the mass of the rotor. It is necessary to know the moment of inertia J to carry out the integration of the equation of motion of the rotor. In this study, the thickness e p of the rotor is considered as artificial.

The steady-state studies of the Savonius rotor are independent of the rotor thickness e p. A ep of 0.01 m was considered.

Boundary and initial conditions

The Reynolds number based on the rotor diameter D is given by

Re = ⇢UD µ ( 1
)
where Re is the Reynolds number, ⇢ is the air density, U is the free stream velocity, D is the rotor diameter and µ is the dynamic viscosity of the air.

In this study, the Reynolds number per metre is given by Re/m = 4. along Oz axis J Oz = 2 kg.m 2 were analytically estimated.

Computational methodology

In order to achieve the objectives of the study, the results of the three-dimensional steady-state and dynamic simulations were compared to experiments. The standard k-" and SST k-! models were used to predict the static torque coefficient C m, the dynamic torque coefficient, Ct and the power coefficient Cp.

Resolution methods

In this study, the steady-state simulation is solved by a direct method. The flow chart in Figure 3 is made of three main processes : the Fluid Resolution process, the Motion Resolution Process and the Mesh rotation process. In this section, the aerodynamic parameters and the formulation of the equation of motion for the VAWT rotor are analyzed. The present study was performed by using a CFD solver with steady-state and dynamic simulation capabilities [START_REF]CD-adapco[END_REF]. The motion resolution of a Savonius rotor was obtained by the coupling of the fluid and rigid-rotor equations. This method applied to a two-dimensional numerical investigation of a Savonius rotor was initiated by D'Alessandro et al. [START_REF] Montelpare | Unsteady Aerodynamics of a Savonius wind rotor: a new computational approach for the simulation of energy performance[END_REF][START_REF] Mctavish | Steady and rotating computational fluid dynamics simulations of a novel vertical axis wind turbine for small-scale power generation[END_REF]. The method was used in the present study to lead a full three-dimensional numerical investigation of a Savonius rotor. The implicit Euler scheme was used for the dynamic simulation as it is numerically stable in comparison to other schemes. Figure 3 gives the flow chart of the dynamic studies of the Savonius rotor.

In Figure 3, the rotor angle ✓ and the rotational velocity ! parameters have been initialized by ✓ 0 and !0. At a time t, these two parameters are obtained by the integration of Eq. ( 7) and are carried out in the Motion Resolution process of Mt are low and vice-versa. ! and Mt of this study are in close link with the weight of aluminum. This stopping criterion is called a periodic steady state (PSS), a state in which all the respective wind turbine rotor dynamic parameters remain stable over a given period. The time step t of the study is invariable for a particular dynamic simulation, but changes according to the intensity of M f were applied to the wind turbine rotor at each simulation.

In this section, we have to determine the static torque coefficient C m, the dynamic torque coefficient Ct and the power coefficient C p. They are the main VAWT parameters and their expressions are given by the following relations:

Cm = M 1 4 ⇢DAU 2
(2)

Ct = Mt 1 4 ⇢DAU 2 (3)
Cp = P 1 2 ⇢AU 3 (4)
where A is the rotor projected area, D is the rotor diameter, U is a free stream velocity. M is the static torque, Mt is the dynamic torque and P is the aerodynamic power.

Equation of motion of the Savonius rotor

In this section, we give a numerical treatment of the rotor rotation in a fluid. By neglecting some motion of the rigidrotor, we reduced the complexity of the system to a 1-DOF problem. The rotor inertia tensor J is part of the equation of motion of the rigid-rotor and its expression is given below:

J = 2 6 6 6 6 6 6 4
J Ox Jxy 0 J xy JOy 0 0 0 J Oz 3 7 7 7 7 7 7 5 (5)
where JOx, JOy and JOz are the main moments of inertia of the rotor. Jxy is the product of inertia of the rotor.

The equation of motion of the rigid-rotor is given by the following relation: where JOz is the component of the rotor inertia tensor J along Oz axis. MOz is the moment of forces acting on the Savonius rotor. G Oz represents the forces acting on the rotor. Assuming that the Savonius rotor makes a perfect pivot with the rotational axis Oz of the rotating mesh, this implies that G Oz is equal to zero. The reduced form of the rotation equation of the Savonius rotor is then:

JOz ✓ = MOz + GOz (6)
JOz ✓ = MOz (7)
MOz = (Mp + M⌧ + M f )fr (8)
where Mp corresponds to the moments of pressure, M⌧ is the moment of the viscous stresses, M f is the moment of external forces. f r is the ramping factor and it was applied to MOz and GOz on the body of the rotor. It was used to minimize the numerical shock caused by the coupling between structural and fluid dynamics at the start of the simulation [START_REF]CD-adapco[END_REF].

f r = 8 > > > > > > < > > > > > > : 0 :t = 0 t tr : 0  t  tr 1 :t tr (9)
where t is the time. tr is the duration of the ramp. This kind of numerical strategy reduces the instabilities at the beginning of the simulations.

The specificity of the dynamic study compared to the steady-state study takes into account the rotor inertia tensor J and the rotor angular acceleration ✓ in problem formulation. Eq. ( 7) of the rigid-rotor is then solved at each time step t [START_REF]CD-adapco[END_REF].

The variation of the intensity of M p and M⌧ at each rotor angle ✓ influences the oscillation of ! curve in Figure 5. ! and ✓ result from the integration of the rotor angular acceleration ✓. ! and M t evolve according to the computational time T and the rotor angle ✓.

Time step

The time step t varies according to the minimal cell size of the computational domain, the rotor angular velocity ! and the CFL number. At each dynamic simulation, the stabilized ! was obtained in a PSS of the rigid-rotor by averaging the oscillations of !(t). The same method was used to determine the average value M t of the dynamic torque Mt(t). Here, the rotor revolution number is calculated from 45 s. If the PSS is reached, the average value of the two dynamic quantities ! and M t are stable and calculated over twelve period. The more the averaging period approaches the total simulation time T, the higher the accuracy of the average values. Figure 5 shows that the rotor performed 88 revolutions before reaching the PSS calculated from a simulation time of 45 deg. 88 revolutions corresponds to the numbers of peaks of ✓ until a simulation time of 45 s. This number of revolutions corresponds to an asymptotic convergence of average output variables. t plays an important role in a numerical simulation and it permits the determination of dynamic parameters of the rotor along the computational time T . A wrongly estimated t can lead the rigid-rotor to a false PSS.

The CFL is related to t, R, and ! parameters [START_REF] Lui | Influence of time step length and sub-iteration number on the convergence behavior and numerical accuracy for transient CFD[END_REF][START_REF] Ahuja | Optimization of fuel-air mixing for a scramjet combustor geometry using CFD and geneticalgorithm[END_REF]. The maximum CF L value should be low for the consistency of the simulation results [START_REF]CD-adapco[END_REF]. 5 is the maximum value of CFL number that was obtained for the dynamic simulation and implicit Euler integration. CFL and y + are the two parameters that control the accuracy of simulation results.

Vortex identification

In dynamic simulations, the analysis of dynamic phenomenon that occurs during the Savonius rotor rotation requires a robust visualization method. Among the existing methods, Jeong and Hussain [START_REF] Jeong | On the Identification of a Vortex[END_REF] have presented the 2 approach which describes the formation and destruction of the vortices through the Savonius rotor blades and endplates. The 2 method is based on two criteria : a vortex core must have a net vorticity and the geometry of the identified vortex core must be a Galilean invariant [START_REF] Escudié | Characterization of trailing vortices generated by a Rushton turbine[END_REF]. Taking the gradient of Navier-Stokes equations and decomposing them into a symmetric and an antisymmetric part, they derive the vorticity transport equation and the strain rate transport equation. The latter is represented as follows:

DS ij Dt ⌫S ij,kk + ⌦ ik ⌦ kj + S ik,kj = 1 ⇢ pij ( 10
)
where pij is the hessian pressure, S the deformation rate tensor

Sij = 1 2 ✓ @ui @xj + @uj @xi ◆ (11)
and ⌦, the rotation rate tensor If the two eigenvalues of pij are positive, this indicates the presence of a minimum local pressure and corresponds to a vortical region [START_REF] Escudié | Characterization of trailing vortices generated by a Rushton turbine[END_REF]. Discarding the contribution of the unsteady irrotational straining and viscous effects in Eq. [START_REF] Gupta | Comparative study of a three-bucket Savonius rotor with a combined three-bucket Savonius three-bladed Darrieus rotor[END_REF] allowed us to obtain a better contribution for the existence of a vortex [START_REF] Kolář | Vortex identification: New requirements and limitations[END_REF]. Jeong and Hussain [START_REF] Jeong | On the Identification of a Vortex[END_REF] have pointed out that the presence of a minimum pressure is the basis of the vortex identification technique, but that it is not a sufficient detection criterion for a vortex core in general. The two terms removed from Eq. ( 10) are considered to be the main cause of this inaccuracy [START_REF] Jeong | On the Identification of a Vortex[END_REF][START_REF] Kolář | Vortex identification: New requirements and limitations[END_REF]. Jeong and Hussain define a vortex as a connected region where S 2 +⌦ 2 are two negative eigenvalues (S and ⌦ are respectively the symmetric and antisymmetric parts of the velocity gradient tensor ru). The tensor S 2 +⌦ 2 is symmetric and only admits real eigenvalues. If the eigenvalues are in the order of 1 > 2 > 3, the vortex identification criterion is equivalent to 2 < 0.

⌦ij = 1 2 ✓ @ui @xj @uj @xi ◆ (12

RESULTS AND DISCUSSION

In order to validate the VAWT model based on the fully coupled fluid rigid-rotor interaction, our numerical results were compared to the experimental investigations of Blackwell et al. [START_REF] Blackwell Ben | Wind Tunnel Performance Data for TWO-and Three-Bucket Savonius Rotors[END_REF]. The difficulty with the dynamic simulation comes from the important computational time T and the choice of the resolution methods that must be adopted. The integration of the rotation equation of the Savonius rotor gives information about its mechanical behavior. The analysis of the airflow and the vortex generation during the dynamic simulation allowed better understanding concerning the mechanism of the Savonius rotor rotation.

Steady-state study of the Savonius rotor

In this section of the present article, results of the steady-state three-dimensional CFD simulation were validated with the steady-state experimental data [START_REF] Blackwell Ben | Wind Tunnel Performance Data for TWO-and Three-Bucket Savonius Rotors[END_REF]. So, the shape, the maximums, and the minimums of the experimental C m curve are accurately predicted by the turbulence models that are used [START_REF] Blackwell Ben | Wind Tunnel Performance Data for TWO-and Three-Bucket Savonius Rotors[END_REF]. In Figure 4, we can observe an improvement of both three-dimensional steady-state cases compared to a two-dimensional steady-state case. It also appears that, when comparing the SST k-! and standard k-" models, a better approximation of the experimental C m curve is obtained with the SST k-! model.

Dynamic study of the Savonius rotor

The dynamic study of the Savonius rotor gives the dynamic torque coefficient Ct curves and power coefficient Cp curve in Figures 6, 7 and8. These curves are obtained by the superposition of several PSS of the rotor. To define these curves, simulations were carried out in a range of moment of external forces M f in Tables II and III below. Generally, for each dynamic simulation, the rotor angular velocity ! of the rigid-rotor starts at 0 rad/s before reaching its PSS. Each rigid-rotor PSS corresponds to a specific !. The time step t was fixed by the dynamic torque coefficient of experimental data. The number of iterations at each t was set to 20 and the sub-iteration number for the 1-DOF solver must be higher than 10 to attest the convergence of the residuals. The convergence criteria imposed for all residual parameters of continuity, momentum, turbulent kinetic energy (Tke) and turbulent dissipation rate (Tdr) were fixed at 10 6 . Higher values of this criterion produce invalid results. The t is a parameter which intervenes in the modeling of the rigid-rotor motion. The choice of t is based on the compromise between accuracy and the computational time T . A higher t may compromise the validity of results and a low t increases computing resources. The ! of the rigid-rotor during the transitory state and in a PSS was imposed by the Eq. ( 7). The ! and the dynamic torque M t curves of the rigid-rotor are shown in Figure 5. These curves correspond to the moment of external forces M f = -2.6 Nm. At T = 0 s, the rigid-rotor is positioned at the rotor angle ✓ = 0 deg and, after this time, it starts moving. As T increases, the ! and M t parameters tend to a stable average value in their PSS. The increase of the rotor angular velocity ! in Figure 5 corresponds to the decrease of its dynamic torque Mt. These two parameters oscillate around an average value of 14.67 rad/s and 2.6 Nm in their PSS. Some of the results obtained by D'Alessandro et al. [START_REF] Montelpare | Unsteady Aerodynamics of a Savonius wind rotor: a new computational approach for the simulation of energy performance[END_REF] on the dynamic torque M t have also been underlined in this study (see Figure 6). The present results show that the dynamic torque curves can have negative values depending on the rotor angle ✓. This is explained by the fact that, in Figure 5, the results have a negative part from a time of 30 s until the end of the computational time T . The negative values of the dynamic torque coefficient C t curves of a Savonius rotor in polar coordinates is subjected to a moment of external forces M f near 0 Nm. on the tip speed ratio . Between M f of -4 Nm to -3.9 Nm, the Ct curve of the standard k-" model is in harmony with the experimental C t curve with a small perturbation on the point corresponding to M f = -3.9 Nm [START_REF] Blackwell Ben | Wind Tunnel Performance Data for TWO-and Three-Bucket Savonius Rotors[END_REF]. From = 1.2 to the end, the C t curve of the standard k-" model underestimates the experimental Ct curve [START_REF] Blackwell Ben | Wind Tunnel Performance Data for TWO-and Three-Bucket Savonius Rotors[END_REF]. In this range of , the dynamic torque coefficient C t curve exceeds the margin of error. If we analyze the data in Tables II andIII The power coefficient Cp is an important parameter in a wind turbine study and it contributes to the wind turbine rotor design. A Savonius rotor is often designed for a tip speed ratio near 1 [START_REF] Gourières | Energie éolienne, théorie, conception et calcul pratique des installations[END_REF]. The optimum obtained by the present study is around 0.86. The C p curves of the standard k-" model, the SST k-! model, and the experimental data are close and have a common maximum of 0.21 [START_REF] Blackwell Ben | Wind Tunnel Performance Data for TWO-and Three-Bucket Savonius Rotors[END_REF]. From = 0 to 1, the C p curves of the standard k-" and the SST k-! models slightly overestimate the experimental C p curve, but they remain in the uncertain intervals except for = 0.21 [START_REF] Blackwell Ben | Wind Tunnel Performance Data for TWO-and Three-Bucket Savonius Rotors[END_REF]. From = 1 to 1.47, the C p curve obtained by the standard k-" model curve underestimates the experimental Cp curve, but it remains in the uncertain intervals [START_REF] Blackwell Ben | Wind Tunnel Performance Data for TWO-and Three-Bucket Savonius Rotors[END_REF]. The same behavior is observed for the SST k-! model. At maximum of 1.54, the power coefficient C p and the dynamic torque coefficient Ct curves obtained with this model tolerably overestimate the margins of error, but they remain interesting compared to those obtained with the standard k-" model. Figures 7 and8 show that the C t and Cp curves obtained are in agreement with the experimental results.

Analysis of the flow field

In the operation, the Savonius rotor generates a complex flow. Chauvin et al. [START_REF] Chauvin | Évolution du coefficient de puissance d'un rotor Savonius[END_REF] argued that, in their experiments in a water channel with this rotor, the retreating blade generates a turbulent wake and the advancing blade is the source of an intense trailing edge vortex rotating in the same direction as the rotor [START_REF] Béguier | instabilité et décollement[END_REF]. They also specify that the motive vortices occur twice per revolution [START_REF] Chauvin | Évolution du coefficient de puissance d'un rotor Savonius[END_REF]. Analyzing the physics of flow in the Savonius rotor revolution, it was deduced that the the rotor. For this observation, the fluid flow study in this paper is restricted to a half-turn of the rotor. Figure 9(a) shows the velocity field around a Savonius rotor taken at a rotor angle ✓ = 90 deg with a stagnation point on the extrados of the retreating blade. The 2 < 0 criterion of Jeong and Hussain [START_REF] Jeong | On the Identification of a Vortex[END_REF] is the vortex analysis and the visualization method used here and, at this position, the emptying of the advancing blade vortices begins. This phenomenon can be seen in From the analysis of the vortex generated by the Savonius rotor rotation, we suggest that the swirling zone in and around the rotor partly comes in part from the low pressure formed by the vacuum created by the shear stress during its rotation. The validation results show that both models that were used have difficulties in predicting the high rotor angular velocity.

The present work is the preliminary study of a single VAWT simulated with a variable rotational speed !; in the future we will focus on the CFD coupling of the VAWT rotor model with buildings. This study allowed us to improve the efficiency of the VAWT, which is strongly influenced by the building shape.

Figure 1 . 3 .

 13 Figure 1. Geometrical characteristics of the computational domain. The radius of the internal region of the computational domain is 2.22R. The wind flow direction at the computational domain is along the Ox axis.

Figure 2 .

 2 Figure 2. Sketch and geometric parameters of the Savonius rotor.

32⇥10 5 ,

 5 the air density by ⇢ = 1.2157 kg/m 3 and the dynamic viscosity of the air by µ = 1.777⇥105 Ns/m 2 . The variation of these two parameters influences the aerodynamic and mechanical parameters of the wind turbine rotor. For the velocity inlet boundary conditions, the free stream velocity U = 7 m/s. The pressure outlet boundary conditions specify the static pressure at flow outlets. The inlet and outlet boundaries were placed far enough apart to ensure their minimal influence on the flow in the immediate vicinity of the rotor. A slip wall was used for the top, lower, left and right parts of the computational domain. The Savonius rotor is 7 of aluminum with a ⇢ of 2700 kg/m 3 . The rotor mass Mr = 17.74 kg and the component of the rotor inertia tensor J

Figure 3 .

 3 Figure 3. Flow chart of the dynamic simulation.

Figure 3 .

 3 Figure 3. Both parameters U and M f are used to initialize the Fluid resolution process of Figure 3. From the initial state, the rotor is considered to be stationary and ! 0 = 0. These initial conditions allow each t to trigger the processes: Fluid Resolution and Motion Resolution of Figure 3. The moments of pressure M p and the moments of the viscous stresses M⌧ determined by the Fluid Resolution and the Motion Resolution Process used first the integration of Eq. (7) and the solution of this equation makes it possible to re-calculate the values of !(t) and M t(t) of the Savonius rotor. t depends on CF L, and U . t of Tables II and III have been tested according to the initial values of Figure 3 and calculated according to the dynamic mode. The parameters !(t) and M t(t) highly depend on inertia J of the rotor. If J weight is heavy, !max and

  Numerical investigation of airflow through a Savonius rotor P. Jaohindy et al.

  The static torque coefficient C m curve was obtained at different rotor angle ✓. ✓ evolving from 0 to 360 deg with a rotor angular step ✓ of 15 deg. The optimum position for the Savonius rotor was identified at ✓ = 30 deg[START_REF] Gourières | Energie éolienne, théorie, conception et calcul pratique des installations[END_REF][START_REF] Menet | Prévision des performances aérodynamiques d'un nouveau type d'éolienne à axe vertical dérivée du rotor Savonius[END_REF]. The numerical C m curves of the present study have a minimum at ✓ = 124.6 deg. Menet and Leiper[START_REF] Menet | Prévision des performances aérodynamiques d'un nouveau type d'éolienne à axe vertical dérivée du rotor Savonius[END_REF] conducted a two-dimensional steady-state simulation with a standard k-" model and compared their results with experimental data (see the Figure4below)[START_REF] Blackwell Ben | Wind Tunnel Performance Data for TWO-and Three-Bucket Savonius Rotors[END_REF]. The experimental C m curve does not last over a whole period, but it is sufficient to make a comparison of results[START_REF] Blackwell Ben | Wind Tunnel Performance Data for TWO-and Three-Bucket Savonius Rotors[END_REF]. Among these four curves, the minimum of the numerical C m curves of the present study are close to the experimental C m curve[START_REF] Blackwell Ben | Wind Tunnel Performance Data for TWO-and Three-Bucket Savonius Rotors[END_REF]. The minimums of the experimental Cm curve and the SST k-! model C m curve are (respectively) 0.05 for ✓ = 124.6 deg and 0.043 for ✓ = 135 deg [9]. The minimum of the 12 Cm curve with the standard k-" model is 0.038 for ✓ = 120 deg. The maximums of the two-dimensional C m curve of Menet and Leiper [8] and the three-dimensional Cm curve for the present study with the standard k-" model are 0.79 and 0.7979 respectively for 210 deg. The experimental C m curve and the SST k-! model curve have the same maximum of 0.82 at ✓ = 204.33 deg and 210 deg respectively [9]. For ✓ from 170 deg to 253 deg, the C m curves of the standard k-" and SST k-! models have an acceptable evolution. For ✓ in range of 0 deg to 253 deg, we deduce that the experimental C m curve was accurately predicted by the SST k-! model in comparison with the standard k-" model [9]. The average C m values obtained with the standard k-" and SST k-! models are respectively 39.5 % and 41.3 %.

Figure 4 .

 4 Figure 4. Comparison of the experimental and the numerical static torque coefficient.

Figures 7 and 8

 8 Figures 7 and 8 depend on of the Savonius rotor and its expression is defined as follows:

Figure 5 .

 5 Figure 5. Time-dependent dynamic torque, rotor angular velocity and rotor angular positions.

Figure 5 also shows that the SST 14 F

 14 model has a rapid convergence compared to the standard k-" model. The ! and the dynamic torque Mt curves of the SST k-! model are consistent with those obtained by the standard k-" model. These results are consistent with the data given in Blackwell et al.[START_REF] Blackwell Ben | Wind Tunnel Performance Data for TWO-and Three-Bucket Savonius Rotors[END_REF] and this confirms the robustness of the SST k-! model.

Figures 6 (

 6 Figures 6(a) and 6(b) show that the evolutions of the dynamic torque coefficient C t curves in polar coordinates. The moments of external forces M f of these curves are in the range of -0.1 Nm to -4.2 Nm for the standard k-" and SST k-! models. The rigid-rotor behaviors reported by these two curves show similarities, except for the case where it was loaded by M f = -0.1 Nm. The rigid-rotor starts off with the standard k-" model and M f = -4.2 Nm. The rigid-rotor also starts off with the SST k-! model and the two M f values of -4 Nm and -4.2 Nm. The first full rotation of the rigid-rotor in Figure 6(a) with M f = -4 Nm and a time step t = 0.001 s was obtained with the standard k-" model. The first full rotation of the rigid-rotor where it is subject to M f = -3.95 Nm and a t = 0.001 s in Figure 6(b) was obtained with the SST k-! model. These observations correspond to the high maximum values of C t = 0.64 obtained with the standard k-" model and C t = 0.61 obtained with the SST k-! model. Using the standard k-" model and M f = -4.2 Nm, the rigid-rotor oscillates between the rotor angles ✓ of 73.48 deg and 77.75 deg and tends to infinity in ✓ = 75.44 deg. With the SST k-! model and M f = -4 Nm, the rigid-rotor oscillates between ✓ = 10.10 deg and 90 deg and tends to infinity in ✓ = 41.27 deg. The C t curve obtained with the SST k-! model and M f = -4.2 Nm shows that the rigid-rotor oscillates between ✓ = 29.05 deg and 90 deg and tends to infinity in ✓ = 47.55 deg. With M f of -0.1 Nm, -2 Nm and -2.6 Nm, the Ct curves obtained by the standard k-" and SST k-! models present positive and negative values.

Figure 6 .Figure 7

 67 Figure 6. Dynamic torque coefficients of the Savonius rotor in polar coordinates.

  , it is found that the SST k-! model predicts the rotor behavior more accurately than the standard k-" model. With the same conditions as the SST k-! model (moment of external forces M f = -0.1 Nm and the time step t = 0.0004 s), the Ct curve of the standard k-" model is unable to reach an angular rotor velocity ! higher than 22.77 rad/s. TablesII and IIIshow that a high ! of 23.89 rad/s was obtained with the SST k-! model.

Figure 7 .

 7 Figure 7. Tip speed ratio depending on the dynamic torque coefficient.

Figure 8 .

 8 Figure 8. Tip speed ratio depending on the power coefficient.

  et al.[START_REF] Chauvin | Évolution du coefficient de puissance d'un rotor Savonius[END_REF] are consistent with the numerical simulation of the vortex formation in and around

Figure 9 (

 9 Figure 9(c) in which the fluids expelled from the intrados of the advancing blade create a low pressure area at the trailing edge of the blade as shown in Figures 9(b) and 9(e). This phenomenon has already been studied by Aouachria[START_REF] Aouachria | Eolienne Savonius comportement mécanique et aérodynamique[END_REF] and he states that it is a result of the pressure drop between the intrados and extrados at the blade tip. The importance of this fact can be shown in Figure9(f) by expanding the diameter of the vorticity field (low pressure zone) at the trailing edge of the advancing blade. After ✓ = 180 deg, the functionalities of the Savonius rotor blades were reversed. The advancing blade becomes the retreating blade and the retreating blade becomes the advancing blade. This inversion of the Savonius rotor blades functionality is confirmed by the increase of v 0 from ✓ = 106 deg to 206 deg and the decrease of v 0 from ✓ = 206 deg to 265 deg in TableIV. We can also see in this table that the v 0 = 0.33 of ✓ = 106 deg is close to the v 0 = 0.36 of ✓ = 265 deg. According to Figure9, the motive vortex period specified by Chauvin et al.[START_REF] Chauvin | Évolution du coefficient de puissance d'un rotor Savonius[END_REF] is in this range of ✓ = 0 deg to 180 deg and 180 deg to 360 deg.

Figure 9 .

 9 Figure 9. Visualization of the velocity, pressure and vorticity fields on a symmetry plane of the Savonius rotor. The rotor angle were taken at ✓ of 90 deg and 123 deg. The method of Jeong and Hussain ( 2 < 0) was used for the visualization of vorticity fields.

Figure 10 accurately

 10 Figure 10 accurately gives the emptying phenomenon in Figures 9(c) and 9(f).Figure 10 also shows vorticity fields through

Figure 10

 10 Figure 10 accurately gives the emptying phenomenon in Figures 9(c) and 9(f).Figure 10 also shows vorticity fields through the Savonius rotor filtered with 2 = -8000, -600, and 0. The restriction of 2 enabled us to obtain a better visualization of the swirling zone generated by the rotor in three dimensions. In this figure, the 2 = -8000 and 0 show the extreme distribution of the vorticity fields. The 2 = -600 shows a better visualization of the swirling flow induced by the rotor

Figure 10 .

 10 Figure 10. Visualization of vorticity fields in and around the Savonius rotor in three-dimensions using Jeong and Hussain's technique ( 2 = -8000, -600, and 0).

  dimensional CFD model of a Savonius rotor has been developed in this study. The model has been validated in steady-state and dynamic conditions in comparison with experimental results. The numerical models employed are based on the Reynolds averaged Navier Stokes equations with standard k-" and SST k-! models. The results of the steadystate and dynamic simulations of the Savonius rotor show that the best approximation of the static torque coefficient, the dynamic torque coefficient and the power coefficient are obtained with the SST k-! model rather than with the standard k-" model. The dynamic study of the Savonius rotor allowed the modeling of the dynamic torque coefficient with the application of the moment of external forces M f on the rigid-rotor. This study reveals that, at start-up, the dynamic torque coefficient curves of the Savonius rotor C t oscillate around fixed values in polar coordinates. In a PSS, the Ct curves of the Savonius rotor that are subjected to the moment of external forces M f lower than -2.6 Nm have strictly positive values. The C t curves subjected to M f higher than -3 Nm have positive and negative parts. The dynamic study permitted the study of the fluid flow in and around the Savonius rotor by application of the Jeong and Hussain technique. Links between velocity, pressure, and vorticity fields have been identified. The analysis of the airflow through the Savonius rotor shows that half-period of the formation and dissipation of the vortex generated by the Savonius rotor blades rotation is in the range of the rotor angle ✓ = 0 deg to 180 deg. Due to the axial symmetry of the Savonius rotor, coherent structures have been observed at the upper and lower parts of the rotor.

  Figure1 83x66mm (300 x 300 DPI)

Table I .

 I Grid independence test for the static and dynamic computational domain

	Grids	cells (static)	M (Nm)	cells (dynamic)	M (Nm)
	Grid1	94,139	5.712	266,201	4.71
	Grid2	77,836	5.711	242,198	4.71
	Grid3	45,404	5.710	93,442	4.52
	Grid4	44,921	5.683	66,147	4.40
	Grid5	38,784	5.002	51,965	4.20

Table II .

 II Summary of data obtained from the dynamic study of the Savonius rotor with the standard k-" model.

	M f (Nm)		Mt (Nm)	Ct	! (rad/s)	P (W)	Cp	t (s)
	-4.2	0	4.20	0.323	0	0	0	0.02
	-4.0	0.22	4.56	0.376	3.40	15.48	0.082	0.001
	-3.9	0.42	4.35	0.359	6.55	28.48	0.151	0.001
	-3.6	0.58	4.37	0.360	9.01	39.39	0.209	0.001
	-3.0	0.81	3.21	0.265	12.57	40.34	0.214	0.001
	-2.6	0.95	2.60	0.214	14.67	36.16	0.202	0.0005
	-2.0	1.09	2.01	0.166	16.90	33.99	0.181	0.0005
	-0.1	1.47	0.59	0.049	22.77	13.44	0.072	0.0004

Table III .

 III Summary of data obtained from the dynamic study of the Savonius rotor with the SST k-! model.

	M f (Nm)		Mt (Nm)	Ct	! (rad/s)	P (W)	Cp	t (s)
	-4.2	0	4.12	0.340	0	0	0	0.02
	-4.0	0	4.12	0.340	0	0	0	0.001
	-3.95	0.19	4.44	0.359	2.83	12.35	0.065	0.001
	-3.9	0.57	4.57	0.370	8.84	40.41	0.207	0.001
	-3.6	0.69	3.70	0.305	10.71	39.61	0.210	0.001
	-3.0	0.85	2.65	0.256	13.19	39.56	0.210	0.001
	-2.6	0.95	2.65	0.219	14.75	39.10	0.208	0.0005
	-2.0	1.11	2.00	0.165	17.22	34.44	0.180	0.0005
	-0.1	1.54	0.28	0.023	23.89	6.69	0.036	0.0004

Table IV .

 IV Rotor angle depending on the dimensionless axial velocity.

	✓ (deg) 106 deg 139 deg 173 deg 206 deg 235 deg 265 deg
	v 0	0.33	0.43	0.66	0.76	0.56	0.36

Contract/grant sponsor

This work was partially supported by European Union (EU) and the Regional Council of Reunion Island. Received . . .


Nils Caillerie
email: nc691@georgetown.edu

Large deviations of a forced velocity-jump process with a Hamilton-Jacobi approach

Keywords: Kinetic equations, Hamilton-Jacobi equations, large deviations, perturbed test function method, Piecewise Deterministic Markov Process. AMS Class. No: 35Q92, 45K05, 35D40, 35F21

We study the dispersion of a particle whose motion dynamics can be described by a forced velocity jump process. To investigate large deviations results, we study the Chapman-Kolmogorov equation of this process in the hyperbolic scaling (t, x, v) → (t/ε, x/ε, v) and then, perform a Hopf-Cole transform which gives us a kinetic equation on a potential. We prove the convergence of this potential to the solution of a Hamilton-Jacobi equation. The hamiltonian can have a C 1 singularity, as was previously observed in this kind of studies. This is a preliminary work before studying spreading results for more realistic processes.

Introduction

In this paper, we study the dispersion in R d of a particle whose motion dynamics is described by the following piecewise deterministic Markov process (PDMP). During the so-called "run phase" (i.e the deterministic part), the particle is moving in R d and is submitted to a force whose intensity and direction are given by the vector Γ, which only depends on the insantaneous velocity of the particle. Therefore, its position X s and velocity V s at time s are given by the following system of ODEs   

• X s = V s , • V s = Γ(V s ).
We shall call the measure space (V, ν) the set of admissible velocities. After a random exponential time with mean 1, a "tumble" occurs: the particle chooses a new velocity at random on the space V , independently from its last velocity. The law of the velocity redistribution process is given by the probability density function M . The particle enters a new running phase which will again last for a random exponential time of parameter 1, and so on.

The Kolmogorov equation of this process is the following conservative kinetic equation:

∂ t f + v • ∇ x f + div v (Γf ) = M (v)ρ -f, (t, x, v) ∈ R + × R d × V, (1.1) {{eq1}}
where ρ is the macroscopic density of f : ρ(t, x) := ˆV f (t, x, v)dν(v).

We assume that Γd -→ S is the null measure, where the vector -→ S (v) is the normal vector to ∂V at point v ∈ ∂V . This condition guarantees that the total mass of the system is conserved thanks to Ostrogradsky's Theorem. We also assume that V is a compact manifold of R d with a (possibly empty) boundary. In the case where the boundary of V is not empty, for every function g from V to R, we define (when possible) Γ • ∇ v g on ∂V as follows:

(Γ • ∇ v g)(w) = d ds g(γ s ) s=0 ,
for γ in V such that γ(0) = w and

• γ s = Γ(γ s ) for all s in [-δ, δ]. For a function G from V to R d , we define div v (ΓG) on ∂V (when possible) as

div v (ΓG)(w) = i (Γ∇ v G i )(w).
The function M ∈ C 0 (V ) is assumed to satisfy min v∈V M (v) > 0.

(1.2) {{infM}}

The so-called force term Γ is a lipschitz-continuous function of v. We assume that there exists α > 0 such that 1 + divΓ (v) ≥ α > 0, ∀v ∈ V.

(1.3) {{divgamma}}

We introduce the flow of -Γ:

• φ v s = -Γ (φ v s ) , φ v 0 = v, (1.4

) {{flowgamma}

We assume that it satisfies a Poincaré-Bendixson condition in the sense that, for all v ∈ V , the limit set of orbit of v is either a zero of -Γ or a periodic orbit of -Γ. In other words, ∀v ∈ V, ∃w 0 ∈ V and T ≥ 0 such that φ w 0 T = w 0 and t>0 {φ v s , s ≥ t} = {φ w 0 s , 0 ≤ s ≤ T } . (1.5) {{Poincare}} Finally, we assume the following mixing property:

∀v ∈ V, ∃w(v) ∈ V such that ∀F ∈ C 0 (V, R), lim t→+∞ 1 t ˆt 0 F (φ v s )ds = F (w). (1.6) {{mixing}}
Note that, thanks to the Poincaré -Bendixson condition (1.5), we already get the existence of a w(v) in the convex hull of V such that 1

t ´[0,t] F (φ v s )ds → F (w). Here, we assume furthermore that this "representative" of v can be chosen in V , even when V is not convex.

In order to study large deviations results for this process, we use the method of geometric optics [START_REF] Evans | A PDE Approach to Geometric Optics for Certain Semilinear Parabolic Equations[END_REF][START_REF] Mark | Geometric Optics Approach to Reaction-Diffusion Equations[END_REF]. We study the rescaled function f ε (t, x, v) := f t ε , x ε , v , which satisfies

∂ t f ε + v • ∇f ε + 1 ε div v (Γf ε ) = 1 ε (M (v)ρ ε -f ε ) , (t, x, v) ∈ R + × R d × V.
The function f ε quickly relaxes towards M , the solution of

div v Γ(v) M (v) = M (v) ´V M (v )dν(v ) -M (v), ´V M (v )dν(v ) = 1. (1.7) {{tm}}
We introduce the following WKB ansatz

ϕ ε (t, x, v) := -εlog f ε (t, x, v) M , or equivalently, f ε (t, x, v) = M e -ϕ ε (t,x,v) ε .
Then, ϕ ε satisfies

∂ t ϕ ε + v • ∇ϕ ε + Γ ε • ∇ v ϕ ε = M M ˆV M (v ) 1 -e ϕ ε -ϕ ε ε dν(v ), (1.8) {{main}}
where (here and until the end) ϕ ε stands for ϕ ε (t, x, v ).

The main result of this paper is that (ϕ ε ) ε converges to the viscosity solution of some Hamilton-Jacobi equation.

Motivations and earlier related works

The motivation of this work comes from the study of concentration waves in bacterial colonies of Escherichia coli. Kinetic models have been proposed to study the Run & Tumble motion of the bacterium at the mesoscopic scale in [START_REF] Alt | Biased random walk models for chemotaxis and related diffusion approximations[END_REF][START_REF] Stroock | Some stochastic processes which arise from a model of the motion of a bacterium[END_REF]. More recently, it has been established that these kinetic models are more accurate than their diffusion approximations to describe the speed of a colony of bacteria in a channel of nutrient [START_REF] Saragosti | Directional persistence of chemotactic bacteria in a traveling concentration wave[END_REF]. This has raised some interest on the study of front propagation in kinetic models driven by chemotactic effect [START_REF] Calvez | Chemotactic waves of bacteria at the mesoscale[END_REF] but also by growth effect [START_REF] Bouin | A Hamilton-Jacobi approach for front propagation in kinetic equations[END_REF][START_REF] Bouin | Spreading in kinetic reaction-transport equations in higher velocity dimensions[END_REF][START_REF] Bouin | Propagation in a Kinetic Reaction-Transport Equation: Travelling Waves And Accelerating Fronts[END_REF]. Our goal is to explore those studies further, by considering kinetic equations with a force term, in view of studying propagation of biological species with an effect of the environment (one could think of fluid resistance of water for bacteria, for example). A physically relevant force term may not satisfy all the assumptions of the present paper, mostly because of (1.3), but our result and methods can be adapted for different force terms. Therefore, our study should be considered as a preliminary work before studying more realistic models.

When Γ ≡ 0, a convergence result for (ϕ ε ) ε already exists. The question has originally been solved in [START_REF] Bouin | A kinetic eikonal equation[END_REF] by Bouin and Calvez who proved convergence of (ϕ ε ) ε to the solution of a Hamilton-Jacobi equation with an implicitly defined hamiltonian. Their result, however, only holds in dimension 1, since the implicit formulation of the hamiltonian may not have a solution. It was then generalized to higher dimensions by the author in [START_REF] Caillerie | Large deviations of a velocity jump process with a Hamilton-Jacobi approach[END_REF]. The proof relied on the establishment of uniform (with respect to ε) a priori bounds on the potential ϕ ε , which may not hold in our situation. If one requires that divΓ = 0, the proof of [START_REF] Caillerie | Large deviations of a velocity jump process with a Hamilton-Jacobi approach[END_REF] can be adapted to our situation since one can establish those a priori bounds (see [START_REF] Caillerie | Stochastic and deterministic kinetic equations in the context of mathematics applied to biology[END_REF], Chapter 3).

When the velocity set is unbounded and Γ ≡ 0, one observes an acceleration of the front of propagation, which highlights the difference between the kinetic model and its diffusion approximation. Due to this acceleration, the hyperbolic scaling is no longer the right one to follow the front. In the special case where M is gaussian and for the scaling (t, x, v) → ( t ε , x ε 3/2 , v ε 1/2 ) the Hamilton-Jacobi limit was performed by Bouin, Calvez, Grenier and Nadin in [START_REF] Bouin | Large deviations for velocity-jump processes and non-local Hamilton-Jacobi equations[END_REF].

As was previously mentioned, spreading can also been driven by growth effect. Propagation in a similar model, without the force term but with a reaction-term of KPP-type was investigated by Bouin, Calvez and Nadin in [START_REF] Bouin | Propagation in a Kinetic Reaction-Transport Equation: Travelling Waves And Accelerating Fronts[END_REF]. They established the existence of travelling wave solutions in the one-dimensional case. Interestingly enough, the speed of propagation they established differed from the KPP speed obtained in the diffusion approximation. Their result was generalized to the higher velocity dimension case by Bouin and the author in [START_REF] Bouin | Spreading in kinetic reaction-transport equations in higher velocity dimensions[END_REF]. In the present paper, we will use the method of geometric optics [START_REF] Evans | A PDE Approach to Geometric Optics for Certain Semilinear Parabolic Equations[END_REF][START_REF] Mark | Geometric Optics Approach to Reaction-Diffusion Equations[END_REF], the half-relaxed limits method of Barles and Perthame [START_REF] Barles | Exit Time Problems in Optimal Control and Vanishing Viscosity Method[END_REF] and the perturbed test function method of Evans [START_REF] Evans | The perturbed test function method for viscosity solutions of nonlinear PDE[END_REF] in a similar fashion as in [START_REF] Bouin | Spreading in kinetic reaction-transport equations in higher velocity dimensions[END_REF][START_REF] Bouin | A Hamilton-Jacobi approach for a model of population structured by space and trait[END_REF]. The Hamilton-Jacobi framework can also be used in other various situations involving population dynamics (not necessarily structured by velocity) [START_REF] Bouin | Invasion fronts with variable motility: Phenotype selection, spatial sorting and wave acceleration[END_REF][START_REF] Bouin | A Hamilton-Jacobi approach for a model of population structured by space and trait[END_REF][START_REF] Gandon | A Hamilton-Jacobi method to describe the evolutionary equilibria in heterogeneous environments and with non-vanishing effects of mutations[END_REF][START_REF] Mirrahimi | Asymptotic analysis of a selection model with space[END_REF][START_REF] Mirrahimi | Time fluctuations in a population model of adaptive dynamics[END_REF].

Main result

To identify a candidate for the limit, let us assume formally that this limit ϕ 0 := lim ε→0 ϕ ε is independent of the velocity variable and that the convergence speed is of order 1 in ε, which would mean that there exists a function η such that

ϕ ε (t, x, v) = ϕ 0 (t, x) + εη(t, x, v) + O(ε 2 ).
(1.9) {{hilbert}} Plugging (1.9) into Equation (1.8), we get formally at the order ε 0 v) . Then, Q satisfies

∂ t ϕ 0 + v • ∇ x ϕ 0 + Γ • ∇ v η = M M ˆV M 1 -e η-η dν(v ). ( 1
:= ∇ x ϕ 0 (t, x), H := -∂ t ϕ 0 (t, x) and Q(v) := e -η(
HQ(v) = v • p -M (v) M (v) Q(v) -Γ(v) • ∇ v Q(v) + M (v) M (v) ´V M Q dν(v ), v ∈ V, Q > 0.
(1.11) {{probspec}}

For fixed p, this is a spectral problem where Q and H are viewed as an eigenvector and the associated eigenvalue. We will discuss the resolution of this spectral problem in Section 2. This resolution motivates the introduction of the following hamiltonian. {defiprincip Definition 1.1. For all p ∈ R d , we set

H(p) := inf H ∈ R, ˆV M (v )Q p,H (v )dν(v ) ≤ 1 , (1.12) {{hamiltonia where Q p,H (v) := ˆ+∞ 0 M (φ v t ) M (φ v t ) exp - ˆt 0 M (φ v s ) M (φ v s ) + H -φ v s • p ds dt, (1.13) {{defQ}}
and φ is the flow of -Γ:

• φ v s = -Γ (φ v s ) , φ v 0 = v. (1.14) {{flow}}
As in [START_REF] Coville | Singular measure as principal eigenfunction of some nonlocal operators[END_REF], this spectral problem may not have a solution in C 1 (V ) and one may need to solve it in the set of positive measures (one can refer to [START_REF] Caillerie | Large deviations of a velocity jump process with a Hamilton-Jacobi approach[END_REF] where a similar situation occurs). Therefore, let us define the so-called singular set of M and Γ, that is the set where the spectral problem has no solution in C 1 (V ): Definition 1.2. We call "Singular set of M and Γ" the set

Sing (M, Γ) := p ∈ R d , H ∈ R, 1 < ˆV M Q p,H dν(v ) < +∞ = ∅ . ( 1
ϕ ε (0, x, v) := ϕ 0 (x) ≥ 0.
Then, the function ϕ ε converges uniformly locally toward some function ϕ 0 which is independent from v. Moreover, ϕ 0 is the viscosity solution of the Hamiton-Jacobi equation

∂ t ϕ 0 + H ∇ x ϕ 0 = 0, (t, x) ∈ R * + × R d , ϕ 0 (0, •) = ϕ 0 . (1.16) {{HJ}}
where H is defined as in Definition 1.1.

Remark 1.4. The sufficient conditions on H that guarantee the uniqueness of the solution of the Hamilton-Jacobi Equation (1.16) will be proven in Proposition 2.10.

The rest of the paper is organized as follows: in Section 2, we describe how we obtain the hamiltonian and prove some results that we will use later on. In particular, we solve the spectral problem (1.11). Section 3 is devoted to the proof of Theorem 1.3.

Identification of the hamiltonian {identificat

Positivity and boundedness of M

As a first step, we will prove that 0 < M < +∞. Proof. Let v min bet the point where M reaches its minimum. Suppose v min is in the interior of V . Then,

∇ v M (v min ) = 0, thus M (v min )divΓ(v min ) = M (v min ) -M (v min ), which implies M (v min ) = M (v min ) 1 + divΓ(v min ) ≥ min M 1 + divΓ(v min ) > 0. If v min ∈ ∂V , we can conclude likewise. Indeed, if Γ(v min ) = 0, then the result is trivial. If v min ∈ ∂V and Γ(v min ) = 0, since Γ(v) • d - → S (v) = 0 for all v ∈ ∂V , there exists v 0 ∈ V , v 1 ∈ V and δ > 0 such that      φ v min s ∈ V, ∀s ∈ [-δ, δ], φ v min -δ = v 0 , φ v min δ = v 1 .
The extremal property of (v min ) now implies that

Γ(v min ) • ∇ v M (v min ) = - d ds M (φ v min s ) s=0 = 0. {Lemmma} Lemma 2.2.
Let us assume that (1.3) holds. Then,

sup v∈V M (v) < +∞.
Proof. We use the same ideas as in Lemma 2.1 to prove that

M (v max ) = M (v max ) 1 + divΓ(v max ) ≤ M ∞ α ,
thanks to (1.3).

The spectral problem {subpart1}

Here, we discuss the resolution of the spectral problem, that is: for all p ∈ R d , find H and a function Q > 0 such that

HQ(v) = v • p - M (v) M (v) Q(v) -Γ(v) • ∇ v Q(v) + M (v) M (v) ˆV M Q dν(v )
holds, for all v ∈ V .

To find such a solution, we use the method of characteristics: let us define φ as the flow of -Γ (see (1.14)). Then, we have

d ds Q(φ v s )exp - ˆs 0 M (φ v σ ) M (φ v σ ) + H -φ v σ • p dσ = -exp - ˆs 0 M (φ v σ ) M (φ v σ ) + H -φ v σ • p dσ M (φ v s ) M (φ v s ) + H -φ v s • p Q(φ v s ) + Γ(φ v s ) • ∇ v Q(φ v s ) = -exp - ˆs 0 M (φ v σ ) M (φ v σ ) + H -φ v σ • p dσ M (φ v s ) M (φ v s ) ˆV M Q dν(v ).
Suppose that

lim s→+∞ exp - ˆs 0 M (φ v σ ) M (φ v σ ) + H -φ v σ • p dσ = 0.
Then, integrating between 0 and +∞ gives

Q(v) = ˆ+∞ 0 M (φ v t ) M (φ v t ) exp - ˆt 0 M (φ v s ) M (φ v s ) + H -φ v s • p ds dt ˆV M Q dν(v ).
(2.17) {{vecteurpro

Integrating Equation (2.17) against M finally gives

1 = ˆV M (v) ˆ+∞ 0 M (φ v t ) M (φ v t ) exp - ˆt 0 M (φ v s ) M (φ v s ) + H -φ v s • p ds dtdν(v).
In other terms, solving the spectral problem is equivalent to finding H ∈ R such that ´V M Q p,H dν(v ) = 1 holds, where

Q p,H (v) = ˆ+∞ 0 M (φ v t ) M (φ v t ) exp - ˆt 0 M (φ v s ) M (φ v s ) + H -φ v s • p ds dt.
It is straight-forward to check that H → ´V M Q p,H dν(v ) is monotically decreasing and continuous. As a result, if there exists H such that ´V M Q p,H = 1, then such H is unique. Let us recall the definition of our hamiltonian:

H(p) := inf H ∈ R, ˆM (v )Q p,H (v )dν(v ) ≤ 1 . {resolution} Proposition 2.3. Resolution of the spectral problem in C 1 (V ) (i) If p ∈ Sing(M, Γ) c , then ´V M Q p,H(p) dν(v ) = 1, i.e. the couple (Q p,H(p) , H(p)) is a solution to the spectral problem (1.11). (ii) If p ∈ Sing(M, Γ), then sup v∈V Q p,H(p) = +∞, i.e.
there is no solution of the spectral problem

(1.11) in C 1 (V ).
Proof. (i) Let p ∈ Sing(M, Γ) c . By definition, there exists

H 0 ∈ R such that ´V M Q p,H 0 dν > 1.
By continuity and monotonicity of H → ´V M Q p,H dν , this means that, for all H 0 < H < H(p),

+∞ > ˆV M Q p,H 0 dν > ˆV M Q p,H dν > 1,
the last inequality being true by definition since H < H(p) (recall (1.12)). Finally,

1 ≥ ˆV M Q p,H(p) dν = lim H H(p) ˆV M Q p,H dν ≥ 1,
which proves (i).

(ii) Suppose that p ∈ Sing(M, Γ) and that Q p,H(p) is bounded. We let δ > 0. Then, sup v∈V Q p,H(p)-δ = +∞. Indeed, in the opposite case Q p,H(p)-δ is bounded and hence, integrable on V which is not possible since p ∈ Sing(M, Γ). Where defined, the function

Z δ := Q p,H(p)-δ -Q p,H(p) satisfies M (v) M (v) + H(p) -δ -v • p Z δ + Γ • ∇ v Z δ = δQ p,H(p) ≤ δ Q p,H(p) ∞ .
By the method of characteristics, this implies that

Z δ (v) ≤ δ Q p,H(p) ∞ ˆ+∞ 0 exp - ˆt 0 M (φ v s ) M (φ v s ) + H(p) -δ -φ v s • p ds dt = δ Q p,H(p) ∞ M (v) M (v) Q p,H(p) (v) ≤ δ Q p,H(p) ∞ max M min M Q p,H(p)-δ (v),
for all v where Q p,H(p)-δ (v) < +∞. Hence,

Q p,H(p) (v) ≥ 1 -δ Q p,H(p) ∞ max M min M Q p,H(p)-δ (v).
Since Q H(p) is bounded and sup v∈V Q p,H(p)-δ = +∞, this is absurd.

We now discuss the resolution of the spectral problem in the set of positive measures.

{utile} Lemma 2.4. Let p ∈ Sing(M, Γ) c and v ∈ V \ D(Q p,H(p) ), i.e. Q p,H(p) (v) = +∞. Thanks to the Poincaré-Bendixson condition (1.5), either φ v t converges to some v 0 ∈ V or the limit set of (φ v t ) t is the periodic orbit of some v 0 ∈ V . Either way, Q p,H(p) (v 0 ) = +∞.
Proof. The result holds since

lim t→+∞ 1 t ˆt 0 M (φ v s ) M (φ v s ) + H(p) -φ v s • p ds = lim t→+∞ 1 t ˆt 0 M (φ v 0 s ) M (φ v 0 s ) + H(p) -φ v 0 s • p ds, which implies that exp - ˆt 0 M (φ v s ) M (φ v s ) + H(p) -φ v s • p ds ∼ t→+∞ exp - ˆt 0 M (φ v 0 s ) M (φ v 0 s ) + H(p) -φ v 0 s • p ds . {utile2}
Lemma 2.5. Let now v 0 be defined as in Lemma 2.4 and let w(v 0 ) ∈ V be the vector defined after the mixing property (1.6). Then, M (w) M (w) + H(p) -w • p = 0.

Proof. Let us assume that M (w) Let p ∈ Sing(M, Γ). Let v 0 be defined after Lemma 2.4.

M (w) + H(p) -v • p = δ > 0. Then, exp - ˆt 0 M (φ v 0 s ) M (φ v 0 s ) + H(p) -φ v 0 s • p ds ∼ t→+∞ e -δt , hence Q p,H(p) (v 0 ) = ´[0,+∞) M (φ v 0 t ) M (φ v 0 t ) exp -´[0,t] M (φ v 0 s ) M (φ v 0 s ) + H(p) -φ v 0 s • p ds dt < +∞,
(i) If v 0 is such that Γ(v 0 ) = 0 and M (v 0 ) M (v 0 ) + H(p) -v 0 • p = 0, then the measure µ := δ v 0 , where δ v 0 is the dirac mass at v 0 satisfies M M + H(p) -v • p Q + Γ • ∇ v Q = 0 (2.18) {{distributi
in the sense of distributions.

(ii) If v 0 ∈ V belongs to a periodic orbit of period T and

φ v 0 t ∈ V \ D(Q p,H(p)
) for all 0 ≤ t ≤ T , then the uniform probability measure µ on the set {φ v 0 t , 0 ≤ t ≤ T } satisfies (2.18) in the sense of distributions.

(iii) Either way, the positive measure μ := Q p,H(p) dν + 1 -´V Q p,H(p) dν µ associated with the eigenvalue H(p) is a solution to the spectral problem (1.11).

Proof. (i) This is trivial since

M (v 0 ) M (v 0 ) + H(p) -φ v 0 t • p ψ(v 0 ) + Γ(v 0 ) • ∇ v ψ(v 0 ) = 0 × ψ(v 0 ) + 0 • ∇ v ψ(v 0 ) = 0, for all ψ ∈ C ∞ (V ). (ii) Let ψ ∈ C ∞ (V ). Then, ˆ{φ v 0 t , 0≤t≤T } M M -H(p) -v • p ψ dµ + ˆ{φ v 0 t , 0≤t≤T } Γ • ∇ v ψ dµ = ˆT 0 M (φ v 0 t ) M (φ v 0 t ) + H(p) -φ v 0 t • p ψ(φ v 0 t )dt + ˆT 0 Γ(φ v 0 t ) • ∇ v ψ(φ v 0 t )dt. Now, ˆT 0 Γ(φ v 0 t ) • ∇ v ψ(φ v 0 t )dt = - ˆT 0 • φ v 0 t • ∇ v ψ(φ v 0 t )dt = [ψ(φ v 0 t )] t=T t=0 = 0,
since φ v 0 is periodic with period T . Moreover, since

1 nT ˆnT 0 M (φ v 0 t ) M (φ v 0 t ) + H(p) -φ v 0 t dt = n nT ˆT 0 M (φ v 0 t ) M (φ v 0 t ) + H(p) -φ v 0 t dt,
for all n ∈ N, and

1 nT ˆnT 0 M (φ v 0 t ) M (φ v 0 t ) + H(p) -φ v 0 t dt -→ n→+∞ M (w) M (w) + H(p) -w • p, we finally get ˆT 0 M (φ v 0 t ) M (φ v 0 t ) + H(p) -φ v 0 t • p ψ(φ v 0 t )dt = T M (w) M (w) + H(p) -w • p ψ(w) = 0,
which ends the proof.

(iii) This is straight-forward since µ solves Equation (2.18) and since

M M + H(p) -v • p Q p,H(p) + Γ • ∇ v Q p,H(p) = M M , for all v ∈ D(Q p,H(p) ).
Remark 2.7. We do not necessarily have uniqueness of the spectral problem (1.11) in the set of positive measures as there might exist several points in V \ D(Q p,H(p)) , which do not belong to the same orbit. However, we will only use the solutions of Proposition 2.3 and Proposition 2.6 in the rest of the paper. We will use the perturbed test function method of Evans [START_REF] Evans | The perturbed test function method for viscosity solutions of nonlinear PDE[END_REF] to build a sub-and a supersolution of the Hamilton-Jacobi equation (1.16). When p ∈ Sing(M, Γ) c , we will use the C 1 solution of Proposition 2.3 to build the perturbed test function in question. It is worth mentioning that when p ∈ Sing(M, Γ), we will use the solution in the set of positive measures given by Proposition 2.6. However, we will only use the regular part Q p,H(p) in the super-solution procedure, whereas we will only use the singular part µ in the sub-solution procedure.

Examples

Such a hamiltonian was already studied in a less general setting. Here are two examples taken from [START_REF] Bouin | A kinetic eikonal equation[END_REF][START_REF] Caillerie | Large deviations of a velocity jump process with a Hamilton-Jacobi approach[END_REF] and [START_REF] Caillerie | Stochastic and deterministic kinetic equations in the context of mathematics applied to biology[END_REF].

Example 2.8. The special case Γ ≡ 0. Suppose that V is a compact set such that 0 belongs to the interior of the convex hull of V . In this case, it is straight-forward to check that M is a solution to (1.7) hence M = M . Moreover,

Sing(M, Γ) = p ∈ R d , ´V M (v ) µ(p)-v•p ≤ 1
, where µ(p) := max v∈V {v • p}. The hamiltonian is then defined by: ˆV

M (v) 1 + H(p) -v • p dν(v) = 1, if p / ∈ Sing(M, Γ), (2.19) {{bouincalve H(p) = µ(p) -1, if p ∈ Sing(M, Γ). ( 2

.20) {{caillerie}

When V = [-1, 1] and M ≡ 1 2 , then Sing(M, Γ) = ∅ and H(p) = p-tanhp tanhp for all p ∈ R d .

One can refer to [START_REF] Bouin | A kinetic eikonal equation[END_REF][START_REF] Caillerie | Large deviations of a velocity jump process with a Hamilton-Jacobi approach[END_REF] for more details. Let us emphasize that the hamiltonian (2.19)-(2.20) is consistent with ours. Indeed, when Γ ≡ 0, then M ≡ M and φ v s = v, for all s, hence ˆV M (v)

ˆ+∞ 0 M (φ v t ) M (φ v t ) exp - ˆt 0 M (φ v s ) M (φ v s ) + H(p) -φ v s • p ds dtdν(v) = ˆV M (v) ˆ+∞ 0 exp - ˆt 0 M (v) M (v) + H(p) -v • p ds dtdν(v) = ˆV M (v) 1 + H(p) -v • p dν(v).
It is also straightforward to check that the hamiltonian from [START_REF] Bouin | A kinetic eikonal equation[END_REF][START_REF] Caillerie | Large deviations of a velocity jump process with a Hamilton-Jacobi approach[END_REF] and ours coincide on Sing(M, Γ).

{chap3}

Example 2.9. Let d = 3, V be the unit sphere that we parametrize with the usual spherical coordinates: One can find a proof of this result in [START_REF] Caillerie | Stochastic and deterministic kinetic equations in the context of mathematics applied to biology[END_REF], Chapter 3. Let us emphasize that the addition of the force term is a singular perturbation in our Hamilton-Jacobi framework since the hamiltonian of Example 2.9 is different, at least on Sing(M, Γ), from the one obtained when Γ ≡ 0.

V = {(θ, ϕ) ∈ [0, 2π] × [0, π]}, v(θ,

Properties of the hamiltonian {lipschitz}

Proposition 2.10. The hamiltonian has the following properties:

(i) 0 ∈ Sing(M, Γ) c and H(0) = 0. (ii) H is continuous on R d and C 1 on R d \ Sing(M, Γ). (iii) H is lipschitz-continuous. Proof. (i) This result is trivial once one notices that ˆV M (v) ˆ+∞ 0 M (φ v t ) M (φ v t ) exp - ˆt 0 M (φ v s ) M (φ v s ) ds dtdν(v) = ˆV M (v) M (v) M (v) dν(v) = 1.
(ii) On Sing(M, Γ) c , the function H is implicitly defined by the relation

ˆV M (v)Q p,H(p) (v)dν(v) = 1. (2.21) {{implicit1} On • Sing(M, Γ), H(p) is implicitly defined by the relation ˆV M (v)Q p,H(p) (v)dν(v) -max H∈B(p) ˆV M (v)Q p,H (v)dν(v) = 0, (2.22) {{implicit2} where B(p) = H ∈ R, ´V M (v)Q p,H (v)dν(v) < +∞ .
Hence, by the implicit function Theorem,

H is C 1 on • Sing(M, Γ) ∪ Sing(p) c = R d \ ∂Sing(M, Γ).
Moreover, since for all p ∈ ∂Sing(M, Γ), max

H∈B(p) ˆV M (v)Q p,H (v)dν(v) = 1,
we also conclude that H is continuous on R d . (iii) Differentiating (2.21) and (2.22) with respect to p and recalling (1.13), we get for all

p ∈ R d \ ∂Sing(M, Γ), ˆV M (v) ˆ+∞ 0 M (φ v t ) M (φ v t ) ˆt 0 (∇H(p) -φ v s ) ds exp ˆt 0 M (φ v s ) M (φ v s ) + H(p) -φ v s • p ds dtdν(v) = 0.
Hence, ∇H ∞ ≤ sup v∈V |v| on R d \∂Sing(M, Γ), from which we deduce that H is lipschitz-continuous.

3 Convergence to the Hamilton-Jacobi limit 3) hold and that M satisfies (1.7). Let ϕ ε satisfy Equation (1.8). Let us assume that the initial condition is well-prepared:

ϕ ε (0, x, v) = ϕ 0 (x) ≥ 0.
Then, ϕ ε is uniformly bounded with respect to x, v, and ε. More precisely, for all 0 ≤ t ≤ T ,

0 ≤ ϕ ε (t, •, •) ≤ ϕ 0 ∞ + max M min M T (3.23)
Proof. Let us recall that, under assumptions (1.2) and (1.3), the results of Lemma 2.1 and Lemma 2.2 hold. Let (X ε , V ε ) be the characteristics associated with (1.8):

             • X x,v s,t = V x,v s,t , X x,v t,t = x, • V x,v s,t = Γ(V x,v s,t ) ε , V x,v t,t = v.
Here, we dropped the ε for readability reasons. Using the method of characteristics, we get the following relation

ϕ ε (t, x, v) = ϕ 0 (X x,v 0,t ) + ˆt 0 M V x,v s,t M V x,v s,t ˆV M (v ) 1 -exp ϕ ε s, X x,v s,t , V x,v s,t -ϕ ε s, X x,v s,t , v ε dν(v )ds ≤ ϕ 0 (X x,v 0,t ) + ˆt 0 M V x,v s,t M V x,v s,t ˆV M (v )dν(v )ds ≤ ϕ 0 ∞ + ˆT 0 max M min M ˆV M (v )dν(v )ds = ϕ 0 ∞ + max M min M T,
so we have an upper bound on ϕ ε . We get the lower bound by noticing that 0 trivially satisfies (1.8).

Proof of Theorem 1.3

In this Section, we prove Theorem 1.3 using the half-relaxed limits method of Barles and Perthame [START_REF] Barles | Exit Time Problems in Optimal Control and Vanishing Viscosity Method[END_REF] in the same spirit as in [START_REF] Bouin | Spreading in kinetic reaction-transport equations in higher velocity dimensions[END_REF]. Additionally, we use the method of the perturbed test function of Evans [START_REF] Evans | The perturbed test function method for viscosity solutions of nonlinear PDE[END_REF] using the same ideas as in [START_REF] Bouin | Spreading in kinetic reaction-transport equations in higher velocity dimensions[END_REF][START_REF] Bouin | A kinetic eikonal equation[END_REF][START_REF] Caillerie | Large deviations of a velocity jump process with a Hamilton-Jacobi approach[END_REF]. Thanks to Proposition 3.1, the sequence (ϕ ε ) ε is uniformly bounded in L ∞ with respect to ε. We can thus define its lower and upper semi continuous envelopes:

ϕ * (t, x, v) = lim sup ε→0 (s,y,w)→(t,x,v) ϕ ε (s, y, w), ϕ * (t, x, v) = lim inf ε→0 (s,y,w)→(t,x,v) ϕ ε (s, y, w).
(3.24) {{envelopes}

We will prove that ϕ * and ϕ * are respectively a sub-and a super-solution of the Hamilton-Jacobi equation. In order to do that, we need to prove that neither functions depend on the velocity variable. For this, we will use a similar proof to [START_REF] Bouin | Spreading in kinetic reaction-transport equations in higher velocity dimensions[END_REF]. We write it here for the sake of self-containedness.

{indep} Lemma 3.2. Both ϕ * and ϕ * are constant with respect to the velocity variable on

R * + × R d . Proof. Let (t 0 , x 0 , v 0 ) ∈ R * + × R d × V and ψ ∈ C 1 R * + × R d × V be a test function such that ϕ * -ψ has a strict local maximum at (t 0 , x 0 , v 0 ). Then, there exists a sequence (t ε , x ε , v ε ) such that ϕ ε -ψ attains its maximum at (t ε , x ε , v ε ) and such that (t ε , x ε , v ε ) → (t 0 , x 0 , v 0 ). Thus, lim ε→0 ϕ ε (t ε , x ε , v ε ) = ϕ * (t, x, v). Moreover, at point (t ε , x ε , v ε ), we have: ∂ t ψ + v ε • ∇ x ψ + Γ(v ε ) ε • ∇ v ψ = M (v ε ) M (v ε ) ˆV M (v ) 1 -e ϕ ε -ϕ ε ε dν(v ).
From this, and using the fact that

0 < min M ≤ M ≤ max M < +∞, 0 < min M ≤ M ≤ max M < +∞, we deduce that ε ´V M (v )e ϕ ε (t ε ,x ε ,v ε )-ϕ ε (t ε ,x ε ,v ) ε dν(v ) is uniformly bounded for all V ⊂ V . By the Jensen inequality, εexp 1 ε |V | M ˆV M (v ) ϕ ε (t ε , x ε , v ε ) -ϕ ε (t ε , x ε , v ) dν(v ) ≤ ε |V | M ˆV M (v )e ϕ ε (t ε ,x ε ,v ε )-ϕ ε (t ε ,x ε ,v ) ε dν(v ),
where

|V | M := ˆV M (v )dν(v ). We deduce that lim sup ε→0 ˆV M (v ) ϕ ε (t ε , x ε , v ε ) -ϕ ε (t ε , x ε , v ) dν(v ) ≤ 0 We write ˆV M (v ) ϕ ε (v ε ) -ϕ ε (v ) dν(v ) = ˆV M (v ) (ϕ ε (v ε ) -ψ(v ε )) -ϕ ε (v ) -ψ(v ) + ψ(v ε ) -ψ(v ) dν(v ) = ˆV M (v ) (ϕ ε (v ε ) -ψ(v ε )) -ϕ ε (v ) -ψ(v ) dν(v ) + ˆV M (v ) ψ(v ε ) -ψ(v ) dν(v )
We can thus use the Fatou Lemma, together with

-lim sup ε→0 ϕ ε (t ε , x ε , v ) ≥ -ϕ * (t 0 , x 0 , v ) to get ˆV M (v )dν(v ) ϕ * (v 0 ) - ˆV M (v )ϕ * (v )dν(v ) = ˆV M (v ) ϕ * (v 0 ) -ϕ * (v ) dν(v ) ≤ ˆV M (v ) lim inf ε→0 ϕ ε (v ε ) -ϕ ε (v ) dν(v ) ≤ lim inf ε→0 ˆV M (v ) ϕ ε (v ε ) -ϕ ε (v ) dν(v ) ≤ lim sup ε→0 ˆV M (v ) ϕ ε (v ε ) -ϕ ε (v ) dν(v ) ≤ 0,
We shall deduce, since the latter is true for any |V | that

ϕ * (t 0 , x 0 , v 0 ) ≤ inf V ϕ * (t 0 , x 0 , •)
and thus ϕ * is constant in velocity.

To prove that ϕ * is constant with respect to the velocity variable, we use the same technique with a test function ψ such that ϕ ε -ψ has a local strict minimum at (t 0 , x 0 , v 0 ).

We shall now prove the following fact {viscosity} Proposition 3.3. Let ϕ ε be a solution of (1.8) and let ϕ * and ϕ * be defined by (3.24). Proof. (i) Let ψ be a test function such that ϕ * -ψ has a local minimum at point (t 0 , x 0 ) ∈ R * + ×R d . We set p 0 := ∇ x ψ(t 0 , x 0 ). For all H ≥ H(p 0 ), let us define ψ ε H := ψ+εη H , where η H := -log(Q p 0 ,H ) and

Q p 0 ,H (v) := ˆ+∞ 0 M (φ v t ) M (φ v t ) exp - ˆt 0 M (φ v s ) M (φ v s ) + H -φ v s • p 0 ds dt, ∀v ∈ V. (3.25) {{QQ}}
For all H > H(p 0 ), by construction of η H , we have

ˆV M e -η H dν(v ) = ˆV M Q p 0 ,H dν(v ) < ˆV M Q p 0 ,H(p 0 ) dν(v ) = 1, if p 0 / ∈ Sing(M, Γ), or ˆV M e -η H dν(v ) = ˆV M Q p 0 ,H dν(v ) < ˆV M Q p 0 ,H(p 0 ) dν(v ) ≤ 1, if p 0 ∈ Sing(M, Γ).
Moreover, Q p 0 ,H ∈ C 1 (V ) and

Q p 0 ,H M (v) M (v) + H -v • p 0 + Γ • ∇ v Q p 0 ,H = M (v) M (v) , ∀v ∈ V. ( 3
, v ε ) ∈ R * + × R d × V , satisfying t ε → t 0 and x ε → x 0 . The extremal property of (t ε , x ε , v ε ) implies that ∂ t ϕ ε (t ε , x ε , v ε ) = ∂ t ψ ε H (t ε , x ε , v ε ), ∇ x ϕ ε (t ε , x ε , v ε ) = ∇ x ψ ε H (t ε , x ε , v ε ).
Moreover, we have

Γ(v ε ) • ∇ v ϕ ε (t ε , x ε , v ε ) = Γ(v ε ) • ∇ v ψ ε H (t ε , x ε , v ε ). Indeed, if v ε ∈ • V or Γ(v ε ) = 0, then the result is trivial. If v ε ∈ ∂V and Γ(v ε ) = 0, since Γ(v) • d - → S (v) = 0 for all v ∈ ∂V , there exists v 0 ∈ V , v 1 ∈ V and δ > 0 such that      φ v ε s ∈ V, ∀s ∈ [-δ, δ], φ v ε -δ = v 0 , φ v ε δ = v 1 .
The extremal property of (t ε , x ε , v ε ) now implies that

Γ(v ε ) • ∇ v (ϕ ε -ψ ε H )(t ε , x ε , v ε ) = - d ds (ϕ ε -ψ ε H ) (t ε , x ε , φ v ε s ) s=0 = 0.
Finally, since V is a compact set, we know that there exists v * ∈ V and a subsequence of (v ε ) ε , which we will not relabel, such that v ε → v * .

At point (t ε , x ε , v ε ), we have:

∂ t ψ + v ε • ∇ x ψ + Γ(v ε ) • ∇ v η H = ∂ t ψ ε H + v ε • ∇ x ψ ε H + Γ(v ε ) ε • ∇ v ψ ε H ≥ ∂ t ϕ ε + v ε • ∇ x ϕ ε + Γ(v ε ) ε • ∇ v ϕ ε (3.27) {inneqa} = M (v ε ) M (v ε ) 1 -ˆV M e ϕ ε -ϕ ε ε dν(v ) .
By the minimal property of (t ε , x ε , v ε ), we can estimate the right-hand side of the last equation, such that

∂ t ψ + v ε • ∇ x ψ + Γ(v ε ) • ∇ v η H ≥ M (v ε ) M (v ε ) 1 -ˆV M (v )e η H (v ε )-η H (v ) dν(v ) (3.28) {inneqb} ≥ M (v ε ) M (v ε ) 1 -e η H (v ε ) (3.29) {inneqc} = M (v ε ) M (v ε ) 1 - 1 Q p 0 ,H (v ε ) , so we have at point (t ε , x ε , v ε ), Q p 0 ,H (v ε ) M (v ε ) M (v ε ) -∂ t ψ -v ε • ∇ x ψ + Γ(v ε ) • ∇ v Q p 0 ,H (v ε ) ≤ M (v ε ) M (v ε ) .
Taking the limit ε → 0, we get at point (t 0 , x 0 , v * ),

Q p 0 ,H (v * ) M (v * ) M (v * ) -∂ t ψ -v * • p 0 + Γ(v * ) • ∇ v Q p 0 ,H (v * ) ≤ M (v * ) M (v * ) . ( 3
∂ t ψ(t 0 , x 0 ) + H ≥ 0.
Since this is true for any H > H(p 0 ), we finally have

∂ t ψ(t 0 , x 0 ) + H(p 0 ) ≥ 0,
which proves (i).

(ii) Let ψ be a test function such that ϕ * -ψ has a global strict maximum at a point (t 0 , x 0 ) ∈ R * + × R d . We still denote p 0 = ∇ x ψ(t 0 , x 0 ). # First case: p 0 / ∈ Sing (M, Γ)

Then, from the very definition of Sing(M, Γ) (check Definition 1.1), there exists

H 0 < H(p 0 ) such that, for all H 0 < H < H(p 0 ), +∞ > ˆV M Q p 0 ,H dν(v ) > ˆV M Q p 0 ,H(p 0 ) dν(v ) = 1, (3
.31) {{thanksto}} using the same notation as earlier. We can then conclude using the same arguments as in the proof of (i). We emphasize that the Estimates (3.27) and (3.28) are reverted in this "maximum" case and that (3.29) is reverted thanks to (3.31).

# Second case: p 0 ∈ Sing (M, Γ)

Thanks to Lemma 2.4, there exists v 0 ∈ V such that Q p,H(p) (v 0 ) = +∞ and that either v 0 is a fixed point of the flow of -Γ, i.e. Γ(v 0 ) = 0, or v 0 belongs to a periodic orbit of the flow.

Suppose that v 0 is a fixed point, then, after Lemma 2.4, we have

M (v 0 ) M (v 0 ) + H(p) -v 0 • p = 0. (3.32) {{fatal}}
Moreover, the function (t, x) → ϕ ε (t, x, v 0 ) -ψ(t, x) has a local maximum at a point (t ε , x ε ) and, by definition of ϕ * , we have t ε → t 0 and x ε → x 0 . By the maximal property of (t ε , x ε ), we have at point (t ε , x ε , v 0 ),

∂ t ψ(t ε , x ε ) + v 0 • ∇ x ψ(t ε , x ε ) + 0 = ∂ t ψ(t ε , x ε ) + v 0 • ∇ x ψ(t ε , x ε ) + Γ(v 0 ) ε • ∇ v ϕ ε (t ε , x ε , v 0 ) = M (v 0 ) M (v 0 ) ˆV M 1 -e ϕ ε -ϕ ε ε dν ≤ M (v 0 ) M (v 0 ) .
Taking the limit ε → 0 and recalling, (3.32), we get ∂ t ψ(t 0 , x 0 ) + H ∇ x ψ(t 0 , x 0 ) ≤ 0, which proves that ϕ * is a viscosity subsolution of (1.16). Suppose now that v 0 belongs to a periodic orbit. At point (t, x, φ v 0 s ), we have

∂ t ϕ ε (φ v 0 s ) + φ v 0 s • ∇ x ϕ ε (φ v 0 s ) + Γ(φ v 0 s ) ε • ∇ v ϕ ε (φ v 0 s ) ≤ M (φ v 0 s ) M (φ v 0 s )
Applying lim t→+∞ ˆt 0 (•)ds to the latter expression gives

∂ t ϕ ε (t, x, w) + w • ∇ x ϕ ε (t, x, w) ≤ M (w) M (w) ,
where w(v 0 ) is the representative of the orbit defined by the mixing property (1.6). Indeed,

1 t ˆt 0 Γ(φ v 0 s ) ε • ∇ v ϕ ε (φ v 0 s )ds = - 1 t ˆt 0 • φ v 0 s ε • ∇ v ϕ ε (φ v 0 s )ds = - ϕ ε (φ v 0 t ) -ϕ ε (φ v 0 0 ) tε -→ t→+∞ 0.
After Lemma 2.5, we know that M (w) M (w) + H(p) -w • p = 0 so we can conclude as in the previous case by considering the function (t, x) → ϕ ε (t, x, w) -ψ(t, x).

We can now conclude the proof of Theorem 1.3.

Proof of Theorem 1.3. We refer to Section 4.4.5 in [START_REF] Barles | Solutions de viscosité des équations de Hamilton-Jacobi. Number 17 in Mathématiques & applications[END_REF] and Theorem B.1 in [START_REF] Evans | A PDE Approach to Geometric Optics for Certain Semilinear Parabolic Equations[END_REF] for arguments giving strong uniqueness (which means that there exists a comparison principle for sub-and supersolution) of Equation (1.16) in the viscosity sense. We emphasize that the lipschitz-continuity proven in Proposition 2.10 is sufficient for these results. Thanks to Proposition 3.3, as ϕ * and ϕ * are respectively a sub-and a super-solution of the Hamilton-Jacobi Equation (1.16), the comparison principle yields ϕ * ≤ ϕ * . However, from their definitions, it is clear that ϕ * ≥ ϕ * . Hence, the function ϕ 0 := ϕ * = ϕ * is the viscosity solution of Equation (1.16) and (ϕ ε ) ε converges uniformly locally as ε → 0 to ϕ 0 , which concludes the proof.

{borneinf} Lemma 2 . 1 .

 21 Let us assume that (1.2) and (1.3) hold and let M satisfy (1.7). Then, min v∈V M (v) > 0.

  ϕ) := (sin(ϕ)cos(θ), sin(ϕ)sin(θ), cosϕ) and let M depend only on ϕ and Γ(θ, ϕ) = (sinϕ, 0).If p / ∈ Sing(M, Γ), then H(p) is implicitly defined by ˆV M (ϕ) ˆ+∞ 0 exp -ˆt 0 (1 + H(p) -v(θ -s, ϕ) • p) ds dt sin(ϕ)dθdϕ 4π = 1,and if p ∈ Sing(M, Γ), then H(p) = |p • e 3 | -1, where e 3 = (0, 0, 1).

Proposition 3 . 1 .

 31 Let us assume that (1.2) and (1.

  {prop:semili (i) The function ϕ * is a viscosity super-solution to the Hamilton-Jacobi equation (1.16) on R * + × R n . (ii) The function ϕ * is a viscosity sub-solution to the Hamilton-Jacobi equation (1.16) on R * + ×R n .

  .30) {{finQQ}} Combining (3.26) and (3.30) at v = v * , we get

  .10) {{prespec}} When t and x are fixed, (1.10) is a differential equation in the variable v. Let us set p

  which is absurd after Lemma 2.4. Resolution of the problem in the set of positive measures.

{inthesenseo Proposition 2.6.

  .26) {{QQQ}} By uniform convergence of ψ ε H toward ψ and by the definition of ϕ * , the function ϕ ε -ψ ε H has a local minimum located at a point (t ε , x ε

Acknowledgement

The author would like to thank Julien Vovelle and Vincent Calvez for presenting this problem to him and for the many discussions that followed. The author would also like to thank Emeric Bouin and Serge Parmentier, who were very helpful. The author has also received help from Nicolas Champagnat and Jérôme Coville who, when reviewing the author's PhD thesis, gave important advice on the redaction of the proof. This work was performed during the author's PhD in Université Claude Bernard Lyon 1.