Florian Bernard

Angelo Iollo

Gabriella Puppo

Accurate Asymptotic Preserving Boundary Conditions for Kinetic Equations on Cartesian Grids

Keywords: Boltzmann equation, BGK model, Asymptotic preserving schemes, Cartesian grid Equation de Boltzmann, modèle BGK, grille cartésienne Inria

A simple second-order scheme on Cartesian grids for kinetic equations is presented, with emphasis on the accurate enforcement of wall boundary conditions on immersed bodies. This approach preserves at the discrete level the asymptotic limit towards Euler equations up to the wall, thus ensuring a smooth transition towards the hydrodynamic regime. We investigate exact, numerical and experimental test cases for the BGK model in order to assess the accuracy of the method.

Introduction

A typical feature of complex gas flows is the presence of both continuum and rarefied regimes in the same field. The Boltzmann equation is the main tool for modelling rarefied gas regimes [START_REF] Cercignani | The Boltzmann Equation and Its Applications[END_REF]:

∂f ∂t (x, ξ, t) + ξ • ∇ x f (x, ξ, t) = Q(f, f ) (1)
where f is the distribution function with an initial data given by f 0 (x, ξ, t = 0), x is the space variable of dimension d, ξ is the microscopic velocity variable of dimension N and Q(f, f ) is a bilinear operator that models the collisions between particles. Rarefied flow regimes are characterized by a high Knudsen number (> 10 -2 ) defined as follows:

Kn =
λ L where λ is the mean free path of the particles and L is the characteristic length of the problem. When the Knudsen number goes to zero, the continuum regime is approached. In this regime the flow can be modelled thanks to Navier-Stokes equations for small Knudsen numbers, or Euler equations when Kn → 0.

The main fields of application of Boltzmann equation are those where the mean free path becomes large compared to the scales of the phenomenon under consideration. This is typically the case for space re-entry problems where the density dramatically decreases in the upper layers of the planetary atmosphere. More recently, applications are also found in nano systems, even in standard gas conditions, [START_REF] Karniadakis | Microflows and nanoflows : fundamentals and simulation[END_REF].

Directly solving the Boltzmann equation is computationally prohibitive because of highdimensionality and of the complexity of the collision operator Q(f, f ), see for instance [START_REF] Degond | Modeling and Computational Methods for Kinetic Equations[END_REF]. However, several numerical models exist to get around this difficulty.

One of the most popular model is the Direct Simulation Monte Carlo (DSMC, [START_REF] Bird | Molecular Gas Dynamics and the Direct Simulation of Gas Flows[END_REF]). This is a statistical approximation of the Boltzmann equation that is numerically viable for high Knudsen numbers, but becomes very costly for low Knudsen numbers. For these regimes it also has the disadvantage of generating noisy results.

The BGK equation [START_REF] Bhatnagar | A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems[END_REF] is a computationally efficient approximation of the Boltzmann equation for kinetic regimes at relatively low Knudsen numbers. It is based on a simple model for the collision term, that is expressed as a relaxation of the distribution function towards an equilibrium Maxwellian distribution. In this work, we will consider the BGK equation as kinetic model, but the structure of the boundary condition we propose can be extended to more general kinetic models.

The BGK model is by construction consistent with the hydrodynamic limit at the continuous level, i.e., for very small Knudsen numbers the Euler limit is recovered. The numerical schemes used to solve the BGK model that respect this property are called asymptotic preserving (AP), after the pioneering work [START_REF] Jin | Efficient asymptotic-preserving (AP) schemes for some multiscale kinetic equations[END_REF]. Many schemes available today can be proven to be AP, see for instance the deterministic methods of [START_REF] Pareschi | Efficient asymptotic preserving deterministic methods for the Boltzmann equation[END_REF]; the Exponential Runge Kutta schemes of [START_REF] Dimarco | Exponential Runge-Kutta methods for stiff kinetic equations[END_REF]; the Micro-Macro decomposition of [START_REF] Bennoune | Uniformly stable numerical schemes for the Boltzmann equation preserving the compressible Navier-Stokes asymptotics[END_REF], the fully implicit scheme in [START_REF] Pieraccini | Microscopically Implicit-Macroscopically Explicit schemes for the BGK equation[END_REF], or the Unified Gas-kinetic scheme of [START_REF] Xu | A unified gas-kinetic scheme for continuum and rarefied flows[END_REF]. See also the review in [START_REF] Jin | Asymptotic preserving (AP) schemes for multiscale kinetic and hyperbolic equations: a review[END_REF]. However the same attention has not been devoted to the AP enforcement of boundary conditions. Also the task of preserving accuracy up to the wall is not trivial [START_REF] Filbet | An inverse Lax-Wendroff method for boundary conditions applied to Boltzmann type models[END_REF], and moreover the behaviour of kinetic boundary conditions, as the Knudsen number approaches zero, to our knowledge has not yet been dealt with. In this work we show that a naive implementation of standard kinetic boundary conditions creates a spurious energy flux at the boundary, which is inconsistent with the Euler set up. We propose a simple modification that ensures that the scheme remains AP up to the boundary. In this sense, two contributions are present in this paper. We start from the consideration that standard specular reflection at solid walls is a consistent asymptotic preserving boundary condition at the continuous level. However, we show that at the discrete level specular reflection introduces spurious boundary layers also in the limit of very small Knudsen numbers. Therefore, we first consider the problem of devising a numerically consistent boundary condition of the kinetic scheme in the hydrodynamic regime. Next, we investigate a second-order accurate method to enforce this boundary condition on solid wall arbitrarily crossing a Cartesian mesh.

The use of Cartesian grids is motivated by the fact that they allow a drastic reduction of the computational setup for complex unsteady geometries. For body-fitted grids, in the case of moving geometries, at each time step a new mesh has to be generated. Accurate moving mesh techniques have been recently developed for diluted gas flows [START_REF] Jin | A unified moving grid gas-kinetic method in Eulerian space for viscous flow computation[END_REF], [START_REF] Chen | A unified gas kinetic scheme with moving mesh and velocity space adaptation[END_REF]. Nonetheless, body-fitted schemes for moving and possibly deforming multiple bodies evolving in a rarefied flow do not seem a computationally viable approach in general. The schemes resulting from Cartesian methods are easily parallelizable and they can efficiently be mapped to high-performance computer architectures. They avoid dealing with grid generation and grid adaptation, a prohibitive task when the boundaries are moving. However, in order to preserve adequate spatial accuracy at the boundaries arbitrarely crossing, the effort is transferred to the enforcement of the boundary condition near the body.

Peskin [START_REF] Peskin | Flow patterns around heart valves: A numerical method[END_REF] first introduced the immersed boundary method (IBM) to study biological flow problems governed by the incompressible Navier-Stokes equations. The idea was to add singular forcing terms in the momentum equation to take into account the effect exerted by the solid on the fluid. Another class of immersed boundary methods is the so-called Cartesian cut-cell methods [START_REF] Yang | A cartesian cut cell method for compressible flows[END_REF], [START_REF] Ingram | Developments in Cartesian cut cell methods[END_REF]. In this approach, the boundary cells are cut to fit the body. This may lead to singular cells that are not square any more but polygons. If the size of the polygons becomes too small or if a polygon is too deformed, a heavy restriction is imposed on the time step and accuracy. To avoid this, problematic cells are merged to neighbours and new cells are created, but this task is still a delicate issue in general geometries. Another possibility is to introduce ghost cells in the solid where fictitious data are computed in order to enforce boundary conditions at cells interfaces [START_REF] Fedkiw | Coupling an Eulerian fluid calculation to a Lagrangian solid calculation with the ghost fluid method[END_REF], [START_REF] Fedkiw | A non-oscillatory Eulerian approach to interfaces in multimaterial flows (the ghost fluid method)[END_REF]. Then the fluxes are modified to take into account the presence of the body. This idea has already been developed to second-order accuracy for Euler equations [START_REF] Gorsse | A simple second order cartesian scheme for compressible Euler flows[END_REF] and elliptic equations [START_REF] Cisternino | A Parallel Second Order Cartesian Method for Elliptic Interface Problems[END_REF], [START_REF] Gibou | A second-order-accurate symmetric discretization of the Poisson equation on irregular domains[END_REF]. In the present work it is extended to the BGK equation with a special care on preserving the asymptotic limit towards Euler equations.

Governing equations

In the following we will present in some detail the model equations and the discrete integration scheme for sake of completeness and to fix notation.

The dimensionless BGK model

The BGK model gives a simple representation of the collision term Q(f, f ):

∂f ∂t (x, ξ, t) + ξ • ∇ x f (x, ξ, t) = 1 τ (M f (x, ξ, t) -f (x, ξ, t)) ( 2
)
where τ is the relaxation time and M f is the Maxwellian distribution function, that is obtained as follows:

M f (x, ξ, t) = ρ(x, t) (2πRT (x, t)) N/2 exp - | ξ -U (x, t) | 2 2RT (x, t) (3)

Inria

where R is the universal gas constant, T (x, t), U (x, t) and ρ(x, t) are the macroscopic values of temperature, velocity and density, respectively. Macroscopic quantities are calculated from the moments of f defined by:

  ρ(x, t) ρ(x, t)U (x, t) E(x, t)   = R N f (x, ξ, t)m(ξ)dξ with m(ξ) =    1 ξ 1 2 | ξ | 2    ( 4
)
Here E is the total energy obtained as follows:

E(x, t) = N 2 ρRT + 1 2 ρ | U (x, t) | 2 (5)
We consider a monoatomic gas for which the ratio of specific heats γ can be calculated as:

γ = 1 + 2 N
In the following, N = 3 and therefore γ=5/3.

The relaxation time for the BGK model can be written as:

τ -1 = cρT 1-ν with c = RT ν 0 µ 0
where ν is the exponent of the viscosity law of the gas, µ 0 is the reference viscosity of the gas at the reference temperature T 0 . Let the following dimensionless parameters:

t = t t 0 x = x L ξ = ξ (RT 0 ) 1/2 Û = U (RT 0 ) 1/2 ρ = ρ ρ 0 T = T T 0 f = f ρ 0 /(RT 0 ) N/2 Mf = ρ (2π T ) N/2 exp - | ξ -Û | 2 2 T
then, the dimensionless form of the BGK equation is (hats are dropped for simplicity):

Sh ∂ t f + ξ.∇ x f = 1 Kn ∞ ρT 1-ν (M f -f ) (6)
with

Sh = L (RT 0 ) 1/2 t 0 Kn ∞ = λ L with λ = µ 0 √ RT 0 ρ 0 ( 7
)
where Sh is the so-called Strouhal number usually set to 1 by choosing an adapted time scale and Kn ∞ is the Knudsen number in the reference conditions. In the following it will be useful to define a local Knudsen number that corresponds to the relaxation time in dimensionless form:

1 τ = 1 Kn local = 1 Kn ∞ ρT 1-ν

BGK reduced model

Let us consider the dimensionless BGK model with Sh = 1. Thanks to the Chu reduction [START_REF] Chu | Kinetic-theoretic description of the formation of a shock wave[END_REF], in 1D and 2D it is possible to reduce the number of independent variables in velocity space. For the 1D case, let:

φ(x, ξ u , t) = R 2 f (x, ξ, t)dξ v dξ w ψ(x, ξ u , t) = R 2 1 2 (ξ 2 v + ξ 2 w )f (x, ξ, t)dξ v dξ w
and multiply the full model by m

1 (ξ) = (1, ξ u , 1 2 ξ 2 u ) T .
After integration with respect to ξ v and ξ w , we have:

     ∂ t φ(x, ξ u , t) + ξ u ∂ x φ(x, ξ u , t) = 1 τ (M φ (x, ξ u , t) -φ(x, ξ u , t)) ∂ t ψ(x, ξ u , t) + ξ u ∂ x ψ(x, ξ u , t) = 1 τ (M ψ (x, ξ u , t) -ψ(x, ξ u , t)) (8)
where

M φ = R 2 M f dξ v dξ w and M ψ = R 2 1 2 (ξ 2 v + ξ 2 w )M f dξ v dξ w .
In the continuous case, these expressions are:

       M φ (ξ) = ρ(x, t) 2πT (x, t) exp - (ξ u -U (x, t)) 2 2T (x, t) M ψ (ξ) = (N -1)T (x, t) 2 ρ(x, t) 2πT (x, t) exp - (ξ u -U (x, t)) 2 2T (x, t) = (N -1)T (x, t) 2 M φ
where U (x, t) and T (x, t) are the macroscopic velocity and macroscopic temperature, respectively.

The same procedure can be applied in 2D, integrating over the third component of the velocity. We obtain:

φ(x, y, ξ u , ξ v , t) = R 2 f (x, y, ξ, t)dξ w ψ(x, y, ξ u , ξ v , t) = R 2 1 2
ξ 2 w f (x, y, ξ, t)dξ w and then:

     ∂ t φ(X, ξ 2 , t) + ξ 2 • ∇φ(X, ξ 2 , t) = 1 τ (M φ (X, ξ 2 , t) -φ(X, ξ 2 , t)) ∂ t ψ(X, ξ 2 , t) + ξ 2 • ∇ψ(X, ξ 2 , t) = 1 τ (M ψ (X, ξ 2 , t) -ψ(X, ξ 2 , t)) (9)
where

ξ 2 = (ξ u , ξ v ) T , M φ = R 2 M f dξ w and M ψ = R 2 1 2 ξ 2 w M f dξ w .
In the continuous case, these expressions are:

       M φ = ρ(X, t) 2πT (X, t) exp - |ξ 2 -U (X, t)| 2 2T (X, t) M ψ = (N -2)T (X, t) 2 ρ(X, t) 2πT (X, t) exp - |ξ 2 -U (X, t)| 2 2T (X, t) = (N -2)T (X, t) 2 M φ Inria 2.

The discrete model in velocity space

By construction, the distribution function f and the Maxwellian distribution function M f satisfy:

R 3 M f m(ξ)dξ =   ρ(x, t) ρ(x, t)U (x, t) E(x, t)   = R 3 f m(ξ)dξ
This is an essential property to ensure conservation of mass, momentum and energy.

In the discrete case, a grid must be introduced in velocity space and integrals are evaluated by quadrature. Let ., . denote the quadrature rule. Based on the work of Cabannes et al. [START_REF] Cabannes | The discrete Boltzmann equation[END_REF] on entropic Maxwellian states, Mieussens proved in [START_REF] Mieussens | Discrete-Velocity Models and Numerical Schemes for the Boltzmann-BGK Equation in Plane and Axisymmetric Geometries[END_REF] that a discrete Maxwellian can be expressed as M f = exp(α • m(ξ)), such that:

M f (ξ), m(ξ) =   ρ ρU E  
at the discrete level.

The discrete Maxwellian distribution function M f can be computed as the solution of a non-linear system solved with a Newton-Raphson algorithm.

We use a uniform velocity grid symmetric with respect to 0 and such that f is negligible outside the grid. Hence, the trapezoidal quadrature rule is used because it has spectral accuracy for smooth and periodic functions on a uniform grid. In 1D:

G v = (ξ i ) i=-n..n with ξ i = i∆ξ
For multidimensional cases, the same discretization is independently performed in all directions.

The numerical method

The numerical method will be described for one equation of the 2D reduced model. The two equations of the reduced model have the same form. Hence we employ the same scheme for both equations (on two different distribution functions).

The space discretization

The physical space Ω is discretized on a Cartesian grid with n × m cells:

Ω = i=1..n j=1..m Ω i,j = i=1..n j=1..m [x i-1/2 , x i+1/2 ] × [y j-1/2 , y j+1/2 ]
such that (x i , y j ) are the coordinates of the center of cell (i, j) and (x i+1/2 , y j ) are the coordinates of the center of the interface between cells (i, j) and (i+1, j). On a space cell

Ω i,j = x i - ∆x 2 , x i + ∆x 2
× y j -∆y 2 , y j + ∆y 2 , eq.( 9) is integrated with a finite volume method:

∂f i,j ∂t + ξ • ∂Ωi,j f n ∂Ωi,j dσ = 1 τ i,j (M fi,j -f i,j ) (10)
where f i,j = 1 |Ω i,j | Ωi,j f dxdy and M fi,j = 1 |Ω i,j | Ωi,j M f dxdy.

RR n°8471

Since a uniform Cartesian grid is considered, the equation can be simply rewritten in terms of fluxes at each numerical interface (between two cells):

∂f i,j ∂t + 1 ∆x (F i+ 1 2 ,j -F i-1 2 ,j + F i,j+ 1 2 -F i,j-1 2 ) = 1 τ (M fi,j -f i,j ) (11)
with

F i+ 1 2 ,j
the numerical flux between cell Ω i,j and cell Ω i+1,j (with a similar notation for the other fluxes) which is expressed as :

F i+ 1 2 ,j = max(0, ξ u )f l + min(0, ξ u )f r (12)
with f r and f l the values of f on the two sides of the interface and ξ u the first component of the microscopic velocity. The numerical expression of the distribution functions f l , f r depends on the reconstruction used at the numerical interface. For a first order reconstruction, f l = f i,j and f r = f i+1,j . For second order accuracy, a MUSCL reconstruction with slope limiters (MinMod for example) is employed:

     f l = f i,j + 1 2 MinMod(f i+1,j -f i,j , f i,j -f i-1,j ) f r = f i+1,j - 1 2 MinMod(f i+1,j -f i,j , f i+2,j -f i+1,j ) (13)

The time discretization

The time discretization can be performed for all terms explicitly. But in this case, the time step will be determined by the space discretization (∆x), the maximum velocity of the velocity grid and the relaxation time τ . For small Knudsen numbers, the relaxation part becomes very stiff (τ very small) and imposes a very strong restriction on the time step. Asher et al. [START_REF] Ascher | Implicit-explicit Runge-Kutta methods for timedependent partial differential equations[END_REF] first presented IMEX schemes to cure this issue. Here, the IMEX scheme [START_REF] Kennedy | Additive Runge-Kutta schemes for convection-diffusionreaction equations[END_REF], [START_REF] Pareschi | Implicit-Explicit Runge-Kutta Schemes and Applications to Hyperbolic Systems with Relaxation[END_REF] is chosen. The relaxation term is treated implicitly while the convective part is non stiff but highly non linear which means that an explicit scheme is more efficient.

The time integration for a ν-stages IMEX Runge-Kutta scheme reads as follows:

f n+1 i,j = f n i,j -∆t ν k=1 ωk ξ∇ x f (k) i,j + ∆t τ ν k=1 ω k (M (k) fi,j -f (k) i,j ) f (k) i,j = f n i,j -∆t k-1 l=1 Ãk,l ξ∇ x f (l) i,j + ∆t τ k l=1 A k,l (M (l) fi,j -f (l) i,j ) f (1) i,j = f n i,j + ∆t τ A 1,1 (M (1) fi,j -f (1) i,j ) (14)
where A and à are ν × ν matrices, with Ãi,s = 0 if s ≥ i and A i,s = 0 if s > i. These coefficients are derived from a double Butcher's tableaux:

à ωT A ω T

Inria

All the quantities until stage k -1 are known so the equation for stage k becomes:

f (k) i,j = τ A k,k ∆t + τ f n i,j -∆t k-1 l=1 Ãk,l ξ∇ x f (l) i,j + ∆t τ k-1 l=1 A k,l (M (l) fi,j -f (l) i,j )+ A k,k ∆t τ M (k) fi,j where f (k)
i,j can be then computed explicitly since all the right hand side is known.In fact the Maxwellian at stage k, M (k) fi,j can be computed using the macroscopic variables at the previous stages, see [START_REF] Pieraccini | Implicit-Explicit schemes for BGK kinetic equations[END_REF].

We are interested in first and second order schemes, hence we use the IMEX schemes given by the following tableaux:

First-order scheme:

0 1 1 1
Second-order scheme:

0 0 0 0 0 0 0 1 0 0 1 2 1 2 1 2 0 0 -1 2 1 2 0 0 1 2 1 2 0 1 2 1 2

The level set function

When an immersed solid is considered on a Cartesian grid, one needs to apply the wall boundary condition on a surface that is arbitrarily crossing the grid. The discretization scheme we employ, directly uses geometric information delivered by the distance function defined on the Cartesian grid to reconstruct the solution near the solid boundaries.

To this end, the domain is decomposed in a fluid part and a solid part. In the solid the values of the physical variables are imposed in each cell since there is no calculation to perform. Such cells are called penalized cells. To decide whether or not a cell is penalized on a Cartesian mesh and to improve accuracy at the boundaries, we use the signed distance between a grid point and the immersed body. This distance is given by a specific level set function. Introduced by Osher and Sethian [START_REF] Osher | Fronts propagating with curvature-dependent speed : Algorithms based on hamilton jacobi formulations[END_REF], the level set function implicitly defines the solid interface Σ in the computational domain by its zero isoline. It is defined by:

φ(x) = dist Σ (x) outside the solid -dist Σ (x) inside the solid ( 15
)
where dist Σ (x) is the minimum distance between the point considered (with coordinates x) and the solid interface Σ.

Thanks to this function it is possible to compute the unit normal of the distance isoline through x, pointing towards the fluid as

n(x) = ∇φ(x) |∇φ(x)| (16)
For φ = 0, we have the unit normal to the interface, n w (x).

In case of moving geometries, the level set function is convected with the imposed boundary velocity u φ :

∂ t φ + u φ • ∇φ = 0 (17)
For the numerical test cases, when needed, this equation is solved with a WENO5 [START_REF] Jiang | Efficient Implementation of Weighted ENO Schemes[END_REF] discretization scheme in space and a standard Runge-Kutta 4 scheme for the integration in time. Integrating [START_REF] Gorsse | A simple second order cartesian scheme for compressible Euler flows[END_REF] in time does not preserve the distance property of φ. Therefore, a reinitialisation step is performed after each time integration step starting from the boundary (φ = 0). In our case, this is done via a Fast Marching algorithm [START_REF] Sethian | Fast Marching Methods[END_REF].

Wall boundary conditions

Two kinds of boundary conditions for kinetic models are usually found in the literature: the diffuse boundary condition and the specular reflection.

The diffuse boundary condition model assumes that the solid is in equilibrium with the fluid in contact with the wall. The distribution function for the fluid is therefore described by a wall Maxwellian distribution function, M w , computed with given temperature and velocity of the wall.

The specular reflection models a wall at which particles are merely reflected. There is no mass and energy fluxes through the wall (impermeability condition). The distribution function f s , corresponding to this boundary condition is a reflection of the distribution function coming from the fluid.

Both models can be taken into account using an accommodation coefficient α ∈ [0, 1] to create the boundary (Maxwell) model:

f b = αM w + (1 -α)f s ( 18
)
where f b is the distribution function representing the actual wall model. Next, we consider the discretization of these boundary conditions.

The diffuse boundary condition

We use this wall model in the formulation of the Euler-AP boundary condition, see section 5.2.2. The diffuse condition is imposed through a Maxwellian distribution function. Boundary conditions prescribe temperature, velocity and zero mass flux. The wall Maxwellian is first computed with a density of 1, temperature and velocity of the wall. Then by invoking mass conservation through the wall, one can recover the density ρ w . Indeed, mass conservation near the wall can be written

F in + F out = 0
with F the mass flux with subscript "in" to denote the flux going towards the wall and "out" the flux going towards the fluid at the physical interface. Then F out corresponds to the quantity of mass going towards the fluid with the wall characteristics (density, temperature, velocity) and with a Maxwellian distribution M w , called the wall Maxwellian. We have:

ξ•nw<0 ξ • n w f w dξ + ξ•nw>0 ξ • n w M w dξ = 0
with f w the distribution function near the wall (for example the one in the closest fluid cell), M w the wall Maxwellian and n w the normal to the wall pointing towards the fluid. This equation Inria can also be expressed in terms of the wall macroscopic quantities (velocity U w , temperature T w and density near the wall ρ w ):

ξ•nw<0 ξ • n w f w dξ + ρ w ξ•nw>0 ξ • n w 1 (2πT w ) 3/2 exp - | ξ -U w | 2 2T w dξ = 0 ξ•nw<0 ξ • n w f dξ + ρ w ξ•nw>0 ξ • n w M w dξ = 0
with M w the wall Maxwellian corresponding to a density of 1. Then, M w can be computed with a Newton-Raphson algorithm (discrete Maxwellian).

Then the density ρ w can be calculated as :

ρ w = - ξ•nw<0 ξ • n w f dξ ξ•nw>0 ξ • n w M w dξ (19)
and

M w = ρ w M w

Impermeability boundary condition

Here the impermeability condition is considered in the sense of Euler equations. Through the boundary there is no mass flux and no energy flux. We first describe how this boundary condition is typically imposed by specular reflection of the distribution function. Then a novel Euler-AP condition method will be introduced for the inviscid limit.

Specular reflection

Each particle hitting the wall is immediately reflected by the wall with the same tangential velocity and the opposite normal velocity :

ξ ref l = ξ -2((ξ -U w ) • n w )n w
with ξ ref l the particle velocity after reflection, ξ the particle velocity before reflection, U w the wall velocity and n w the normal to the wall. This holds true for each particle such that ξ •n w > 0.

For ξ • n w < 0, the distribution function on the boundary is already known and equal to the one in the fluid cell. The distribution function for the boundary condition has to be computed only for ξ • n w > 0. The entire distribution function f s enforcing the boundary condition is then:

f s = f for ξ • n w < 0 f (ξ ref l ) for ξ • n w > 0 (20)
In the reduced model, the same procedure is applied to φ and ψ. This guarantees zero mass and energy fluxes (here U w is set to zero for simplicity):

F mass = ξ•nw<0 ξ • n w f (ξ)dξ + ξ•nw>0 ξ • n w f (ξ ref l )dξ = ξ•nw<0 ξ • n w f (ξ)dξ + ξ•nw<0 -ξ • n w f (ξ)dξ = 0 RR n°8471 F energy = ξ•nw<0 |ξ| 2 ξ • n w f (ξ)dξ + ξ•nw>0 |ξ| 2 ξ • n w f (ξ ref l )dξ = ξ•nw<0 |ξ| 2 ξ • n w f (ξ)dξ + ξ•nw<0 -|ξ| 2 ξ • n w f (ξ)dξ = 0
However, because of the discretization of the velocity space, one needs to compute f (ξ ref l ) where in general ξ ref l does not correspond to a collocation point. In other words, even though ξ is a grid point, ξ ref l is not a velocity grid point except if the wall is parallel to the grid. Therefore ξ ref l must be interpolated. The numerical experiments show that the interpolation must be higher-order accurate to guarantee zero mass and energy fluxes at the wall to an acceptable degree of approximation.

In the 1D case, this problem does not arise. It is enough to have a velocity grid symmetric with respect to ξ = 0. Specular reflection reduces then to choose f s (ξ) = f (-ξ) in order to build the reflected distribution function.

In 2D, due to interpolation errors, mass and energy fluxes are not identically zero at the walls. This error leads to a boundary layer in the limit of inviscid gas simulations that should not exist. As shown in the numerical illustrations, to remove this spurious effect it is possible to use either a finer velocity grid or a higher-order interpolation, at the price of significantly larger computational costs.

A new discrete Euler-AP impermeability condition

Let us assume that the distribution function is a Maxwellian (Kn number close to 0). Then, imposing the impermeability condition at the wall corresponds to impose a Maxwellian distribution function at the wall exactly as described in section 5.1. However, in this case the velocity must have the same tangential component of the fluid next to the boundary with zero wall-normal component and the temperature must be the same of the fluid. Therefore, in contrast with what is done in section 5.1, tangential velocity and temperature are extrapolated from the fluid to the wall.

Eventually, the diluted fluid next to the wall can be considered Maxwellian only in the limit of the continuum regime. To build a fully asymptotic preserving boundary condition valid in more rarefied regimes, this model is included in equation [START_REF] Ingram | Developments in Cartesian cut cell methods[END_REF] with a new coefficient β:

f b = αM w + (1 -α)(βf s + (1 -β)f M ) (21)
with β ∈ [0, 1] and such that it is close to zero in the inviscid limit for Kn → 0 and f M is the distribution function built along the lines described above. If Kn is not close to zero, the classic specular reflection (β → 1 and f b → αM w + (1 -α)f s ) correctly takes over.

To set the value of β we emphasize that f M corresponds to the specular reflection only when the distribution function in the fluid is close to a Maxwellian. If it is not the case, the specular reflection is computed with f s (in particular in the rarefied regime). Thus, in our model, β is set as follows:

β = min 1, ||f -M f || L 2 max(f )tol (22)
with tol a tolerance on the distance in L 2 norm between the closest interior domain distribution function f and its corresponding Maxwellian. Thus, if

||f -M f || L 2 max(f )
<< tol, the specular reflection fully corresponds to the Euler-AP boundary condition. In the following, tol = 10 -3 .

Inria

Wall boundary condition on Cartesian grid

In this section we describe the scheme to impose the Euler-AP boundary condition of section 5.2.2 to immersed boundaries on Cartesian grids. The diffuse and the specular reflection schemes are similar. Also, it should be noted that this technique can be easily applied to the case of body fitted grids.

In the case of a solid immersed in the flow, a fictitious state has to be created in the solid to compute the transport step numerically between a fluid cell and a cell containing the solid. The idea is first to compute the equivalent distribution function at the solid interface satisfying the imposed boundary condition and then create a fictitious state in the neighbour solid cell called ghost cell that respects the boundary value at a given approximation order. To do so, a few parameters on the boundary are needed. These parameters are presented in figure 1.

Figure 1: Immersed interface on a Cartesian mesh.

In each grid point, the shortest distance to the boundary φ is known through the levelset function (eq.15). One can also compute the normal n to the boundary in each grid point thanks to [START_REF] Gibou | A second-order-accurate symmetric discretization of the Poisson equation on irregular domains[END_REF]. In the following, we distinguish the interface between two cells and the interface between the fluid and the solid. The first interface will be called numerical interface (where the numerical fluxes are computed) while the latter will be called the physical interface (where the boundary condition is imposed).

The problem reduces to detemine a Maxwellian distribution function on the boundary and extrapolate or interpolate it to the numerical interface. The method will be described for the new impermeability condition but the second part of the method (transport at the numerical interface) can be easily transposed for any condition where the distribution function is known at the boundary (specular reflection, diffuse condition).

The point A will be used to compute the interpolation at the interface i + 1/2, j1. The fluxes at interface i1, j -1/2 will be computed by calculating the wall Maxwellian on point B. Since the same scheme is used on points A and B, the method will be explained on point A.

First order Euler-AP scheme

To compute the fictitious state at first order, the Maxwellian distribution function at the wall is built as presented previously in 5.1 with the tangential velocity U A • τ i,j1 and temperature T A taken from the fluid cell and a zero relative normal velocity

(U A -U w ) • n i,j1 :      T A = T i,j1 (U A -U w ) • τ i,j1 = (U i,j1 -U w ) • τ i,j1 (U A -U w ) • n i,j1 = 0
The density ρ A is calculated thanks to the distribution function in cell (i, j1) invoking mass conservation through the wall as in [START_REF] Jiang | Efficient Implementation of Weighted ENO Schemes[END_REF]. The Maxwellian built with ρ A , U A , T A is then simply imposed as the state in the first solid cell (i + 1, j1).

Second order Euler-AP scheme

The main idea is now to impose the boundary condition on the physical interface and reconstruct the conditions at the numerical interface with second-order accuracy.

The impermeability boundary condition is applied at the physical interface by imposing a Maxwellian distribution function. This distribution function depends on the temperature, the velocity and the distribution function in the fluid. All these information are extrapolated from the fluid. To find the position of the wall, the levelset function is used and the distance d A between the cell center and the wall is known as:

d A = ∆x|φ i,j1 | |φ i,j1 | + |φ i+1,j1 |
where φ i,j1 (respectively φ i+1,j1 ) is the distance between the point (i, j1) (respectively (i + 1, j1)) and the boundary and ∆x is the space grid step. The normal can also be computed by

n A = n i,j1 + d A ∆x (n i+1,j1 -n i,j1 )
where n i,j1 (respectively n i+1,j1 ) is the normal to the boundary in point (i, j1) (respectively

(i + 1, j1)).
The temperature and the velocity now can be extrapolated with a second order polynomial using T i,j1 , T i-1,j1 , T i-2,j1 and U i,j1 , U i-1,j1 , U i-2,j1 . The wall temperature T ext is the result of the extrapolation while the wall velocity U ext is only equal to the tangential part of the extrapolated velocity such that the impermeability condition ((U A -U w ).n A = 0) is enforced:

T A = T ext U A = U ext -((U ext -U w ) • n A ) n A
To extrapolate the distribution function at the boundary in order to compute the density, an upwind reconstruction is used. For each microscopic velocity ξ such that ξ

• n A < 0 f (ξ) is set as: f A (ξ) = f n,m (ξ) such that x A -x n,m ||x A -x n,m || • ξ ||ξ|| = max (k,l)∈Υi,j1 ( x A -x k,l ||x A -x k,l || • ξ ||ξ|| )
Inria with x A the position of the boundary (x A = x i,j1 + d A ) and Υ i,j1 containing all the fluid neighbours of cell (i, j1). A graphic illustration is given in figure 2. In this case, the velocity grid has 8 grid points. The distribution function is required in A for ξ 1 , ξ 6 , ξ 7 , ξ 8 . Here Υ i,j is represented in blue. One can create f A for ξ • n A < 0 as :

         f A (ξ 1 ) = f i,j (ξ 1 ) f A (ξ 6 ) = f i+1,j+1 (ξ 6 ) f A (ξ 7 ) = f i+1,j+1 (ξ 7 ) f A (ξ 8 ) = f i,j+1 (ξ 8 )
Figure 2: Graphic illustration of the reconstruction for a 8 velocities grid.

It is worth to remark that this reconstruction avoids interpolation but is not formally secondorder accurate. A MUSCL-type reconstruction could be applied instead but an upwind reconstruction has the advantage of taking into account the main direction of the flux. Indeed, a reconstruction with slopes chosen as in [START_REF] Fedkiw | A non-oscillatory Eulerian approach to interfaces in multimaterial flows (the ghost fluid method)[END_REF] does not prevent the possibility of selecting a downwind reconstruction which might lead to high extrapolation errors.

The part of f for ξ • n A > 0 corresponds to microscopic velocities coming from the wall and is not required for the computation of the wall Maxwellian f MA .

Finally, the distribution function used as boundary condition can be computed as follows:

f b = f A for ξ • n A < 0 f MA for ξ • n A > 0 (23)
Once the distribution for the boundary condition is known at the wall, it has to be interpolated or extrapolated to the numerical interface with a linear reconstruction to preserve second order.

One can use the same slope as in the fluid cell and the exact distance between the numerical (between cells) and physical interface (d A on figure 1):

f i+1/2,j1,r = f b + d A ∆x MinMod(f i,j1 -f i-1,j1 , f i-1,j1 -f i-2,j1 ) (24)
The fluxes at the numerical interface can be now computed as usual.

Numerical results

The accuracy of the Euler-AP boundary condition is initially tested in 1D for cases where the reference solution is exact, section 6.1. In 2D, we consider well known test cases both on a body fitted mesh (section 6.2) and on a Cartesian mesh (section 6.3), in the continuum regime. Secondorder accuracy of the method is illustrated on the Ringleb flow (section 6.4). Furtheremore, we compare the Euler-AP wall model to numerical results in the literature in the rarefied regime (section 6.5). Finally, we simulate a nozzle plume with coexisting continuum and rarefied regime to qualitatively validate the Euler-AP boundary condition against experimental data.

Reflection of a rarefaction wave and a shock wave

Here, we will test the precision of the new Euler-AP method for typical 1D problems for which the exact solution is known. In particular, we consider the reflection of a rarefaction wave and a shock wave on a solid wall for a monoatomic gas at equilibrium. The boundary condition applied at the wall is the impermeability condition with the Euler-AP method. One can note that the classical way to implement it (specular reflection) perfectly works here since there is no need to interpolate in the velocity space. For the rarefaction wave, the exact solution is computed using Riemann invariants with a velocity v = -0.35 assigned to the left of the rarefaction wave (see Figure 3, left). To the right, where the gas is steady, the pressure is 1 and the speed of sound is 1. For the shock, the Rankine-Hugoniot conditions and an exact Riemann problem at the wall are used (see Figure 3, right). The shock wave propagates at Mach 1.2. Kn ∞ is set to 10 -10 . The domain goes from 0 to 3 and the wave is generated at t = 0 in 0. The wall position is set to x = 3. In order to compute the error of our scheme with respect to the analytical solution, the grid points (center of each cell) are shifted with respect to the wall position as follows:

x i,j = x i,j + ψ∆x with ∆x the space step and ψ a coefficient varying from 0 to 1. When ψ = 0.5, the position of the wall exactly corresponds to a numerical interface. In this case, the error with respect to the analytical solution is optimal.

Numerical simulations are performed with 100 space grid points and 50 grid points in velocity, with an initial data (t s = 0) computed analytically (for Euler equations) at t=1.8 for the rarefaction wave and t=1 for the shock wave. The simulation is stopped at t s = 3 and t s = 2 respectively, when the wave is completely reflected by the wall. This is equivalent to the analytical solution at t = 4.8 for the rarefaction wave and t = 3 for the shock wave.

Errors in L 1 , L 2 and L ∞ -norm for the velocity as a function of ψ (the position of the wall), have been computed and shown on Figure 4.

We observe that the errors in L 1 and L 2 norms increase as ψ goes to 0 because the macroscopic quantities are extrapolated over larger distances. However, they remain close to the optimal error. For the three norms, the error for ψ = 0 and ψ = 1 is the same because it represents the same configuration.

The oblique shock

Here, we will compare the specular reflection described in sect.5.2.1 with the new Euler-AP method on a 2D exact test case, using a body-fitted grid. A wedge reflect an incoming horizontal flux at M = 2.324. The grid is body fitted to deal with the single issue of the boundary condition without the influence of the Cartesian grid scheme. The angle of the wall is 10 degrees. The initial condition is u = 3, v = 0, p = 1, T = 1 in dimensionless form and Kn ∞ = 10 -8 . The same condition is imposed at inlet and the impermeability condition is applied on the body. A shock is created on the body and moves in the fluid. The analytical solution for the angle of the shock (δ), Mach number, density, pressure and temperature after the shock is known:

               δ = 35.2326 M = 1.818 ρ = 1.4987 P = 1.9969 T = 1.3324
Two specular reflections schemes have been tested with different interpolation methods and compared to the new Euler-AP method. The first one uses a bilinear interpolation. The second one uses a bicubic interpolation. Results are shown for a grid 100×100 in space and 21×21 in velocity space (and 81×81 for the bilinear interpolation).

The layer developing close to the wall with the bilinear interpolation (see figure 5) is due to a spurious energy flux across the wall (100 times larger than for the present method, see figure 8). This energy flux is caused by interpolation errors. In fact, when the velocity grid is refined, the spurious layer tends to decrease as we can see using 81×81 points on the velocity grid (see figure 6).

The layer is much smaller in the case of a bicubic interpolation (see figure 6.2) because the spurious energy flux across the wall is much lower and gives results in better accordance with the theory. Again, it shows that this spurious layer is only a numerical artefact.

The Euler-AP method does not show this kind of layer on the boundary (see figure 8). The solution is uniform after the shock. On a horizontal line (y = 0.3), one can compare the different results with respect to the analytical solution. Figure 9 shows the comparison of the solutions for the pressure and the temperature given by each method with 21×21 points in velocity. The exact solution (for Euler equations) is represented with a solid line.

For all methods, the position of the shock is correct but the values for the pressure and the temperature after the shock are different. The Euler-AP method and the specular reflection with a bicubic interpolation are clearly more accurate. Zooming on these pictures (see figure 10) we can see that the results obtained by the Euler-AP method are even more accurate than the ones obtained with the bicubic interpolation.

Note that these spurious overestimated post-shock values arise from the error in the boundary condition and propagate in the bulk of the fluid affecting the whole solution.

The blunt body problem

The Euler-AP boundary condition has been validated on the oblique shock test case for a body fitted grid. In this section accuracy of the specular reflection with bicubic interpolation and Euler-AP method are compared on a Cartesian grid. A cylinder is immersed in a flow at M = 3.09

(u = 4, v = 0, p = 1, T = 1 in dimensionless form). The domain is [-0.3,0]×[0,0.45] and the cylinder radius is 0.1 centred in [0,0].
The impermeability condition is first computed with a purely specular reflection. The velocity Inria grid goes from -10 to 10 in each direction and Kn ∞ = 10 -8 . The boundary conditions at the border of the domain are free flow on north and east, inlet with the initial condition on west and specular reflection on south. Since the velocity grid is Cartesian, imposing the specular reflection on south can be done exactly. The impermeability condition is applied on the body. The present method and the specular reflection with a bicubic interpolation are tested on a spatial grid 80×120 and a velocity grid 31×31. Figure 6.3 shows the solution with the Euler-AP method on the left and the normalized difference in the density between the two methods on the right.

One can observe that the difference is mostly located on the shock and on the body. The relative error is about 10% of the solution.

The same test case is solved with Euler equations. BGK simulations are compared to this Euler solution with emphasis on the boundary. The structure of the error is very close in the two cases. The largest difference is located on the shock because the solution of the BGK equation is more diffusive than the one calculated with Euler equations. Thus, the shock is less sharp. On the cylinder, the error is much smaller. For the bicubic interpolation, the biggest difference is about 10% while for the present method it is less than 5% of the Euler solution. As in the oblique shock case, the Euler-AP method is more accurate on the boundary (see top of the cylinder in fig. 12). It correctly preserves the asymptotic properties towards Euler equations even on Cartesian grid with immersed boundaries. Moreover the computational time is dramatically different (9 hours for the Euler-AP method and 19 hours for the bicubic interpolation with 128 processors). A quantitative comparison of the computational time for the two methods is performed on 64 processors and is shown in figure 13. The Euler-AP method is clearly faster than the specular reflection with a bicubic interpolation. Also, the computational time increases faster for the bicubic interpolation as the number of velocity grid points increases (linear regression of the two curves: 1.73 against 1.55 for the present method).

Bernard & Iollo & Puppo

Ringleb flow

Now that we have validated the boundary condition on body fitted grids and Cartesian grids, we show that our Cartesian method with this boundary condition is second-order accurate. Ringleb flow is a 2D steady state test case where the analytical solution can be calculated for Euler equations and will be used as a reference solution. This is a potential flow and the exact solution is obtained with the hodograph method [START_REF] Shapiro | The Dynamics and Thermodynamics of Compressible Fluid Flow[END_REF].

Setting (θ,V) such that u = V cosθ and v = V sinθ, the stream function is given by Ψ = sinθ V .

The streamline equations are:

x = 1 2ρ ( 1 V 2 -2Ψ 2 ) + L 2 and y = sinθcosθ ρV 2
with (for γ = 5/3, monoatomic gas):

L = ln 3V 6 + 2 √ 9 -3V 2 - √ 9 -3V 2 (V 2 -4) V 2 -3 , c 2 = 1 - γ -1 2 V 2 , ρ = c 2 γ -1

Inria

The computational domain is [-0.5,-0.1]×[-0.6,0] and the flow is solved between 2 streamlines : Ψ 1 =0.8 and Ψ 2 =0.9. Since u • n = 0 on a streamline (with n the normal to the streamline), any streamline can be considered as a solid boundary where the Euler impermeability condition is enforced. The boundary conditions in inlet (y=-0.6) and outlet (y=0) are supersonic and exactly imposed.

The finest grid used is 128×192 in space and 241×241 in velocity. To compute the space order of convergence, the solution is calculated on five different grids in space (8×12, 16×24, 32×48, 64×96) but the velocity grid is kept 241×241. Thus, the result is not perturbed by convergence in velocity space. Figure 14 shows the solution for the pressure on the left and the error map for the density on the right. The solid black lines are the two streamlines between which the BGK equation is solved. Outside, the exact Euler solution is shown. The flow is not perturbed by the boundary condition along the streamlines and the error map on the right shows that the error with respect to the analytical solution is less than 1%.

RR n°8471

Figure 15 shows the error with respect to the analytical solution for L 1 and L ∞ norms. Second order is obtained in L 1 norm. However, on the finest grid, it is possible to notice a slight deterioration in accuracy with respect to previous grids which actually show a convergence order bigger than 2. This phenomenon is probably due to the fact that at this degree of refinement the space error is of the same order of magnitude of the error in velocity space, where the reconstruction of the distribution function at the boundary is indeed carried out with first order accuracy. In L ∞ norm the order is smaller (1.5 for the pressure) but it occurs on a tiny subdomain and it does not spoil the L 1 norm order of convergence.

Flow past a vertical plate

Several test cases have been so far presented to show the accuracy of the Euler-AP boundary condition in the continuum regime, for body-fitted and Cartesian meshes. We now validate the Euler-AP wall condition in the rarefied regime.

The flow past a vertical plate was first introduced by Bird [START_REF] Bird | Molecular Gas Dynamics and the Direct Simulation of Gas Flows[END_REF] to study the formation of vortices at different Knudsen numbers. This case highlights the rarefied effects for a simple geometry, with a specular boundary condition applied on the vertical plate. Our results are compared to the numerical ones obtained by Bird [START_REF] Bird | Molecular Gas Dynamics and the Direct Simulation of Gas Flows[END_REF] with DSMC and by Chen et al. [START_REF] Chen | A unified gas kinetic scheme with moving mesh and velocity space adaptation[END_REF] with AUGKS.

A vertical plate is immersed in a flow at Mach 0.53 of Argon. The undisturbed initial flow is imposed at inlet (x = -4/3), outlet (x = 2), at the upper boundary (y = 2), U ∞ = 0.69 and a temperature of 1 (in dimensionless values). At y = 0, a symmetric boundary condition is enforced corresponding to the specular reflection. The plate is placed at x = 0 and goes from y = 0 to y = 1 (height h=1). It has zero thickness and is situated exactly at the interface between two cells. The Euler-AP condition is imposed on both sides of the plate with our second order scheme.

The steady state of this problem shows a vortex behind the plate. As mentioned by Bird, the solution strongly depends on the boundaries of the domain. However, for the same computational domain and boundary conditions, comparisons can be made with the results obtained by Bird on the velocity profile along a vertical axis passing through the center of the vortex for different Knudsen numbers. We note that for this test case, the results are presented with the value of the Knudsen number in Bird's sense (Kn B ). To fit with our model, we actually take Kn ∞ =

Kn B 5 √ 2π 16
.

-0. Figures 16 and17 show that the velocity profiles obtained with the Euler-AP boundary condition and a second-order scheme are in good agreement with Bird's results for different Knudsen numbers. The results show that the vortex becomes stronger as the Knudsen number decreases. Small variations of the position of the vortex center definition induce large variations of the velocity at y = 0. In this case, the value of the parameter β is 0 at the initial state but goes quickly towards 1 for all boundary points since the Knudsen number is large.

A nozzle plume

We consider a final qualitative validation against experimental results. A flow expands at the outlet of a nozzle in a low pressure atmosphere. The spreading jet has been studied for example by Latvala et al. in [START_REF] Latvala | Experimental Determination of Jet Spreading from Supersonic Nozzles at High Altitudes[END_REF] and [START_REF] Latvala | Studies of the Spreading of Rocket Exhaust Jets at High Altitudes[END_REF]. Experiments were performed for different ambient pressures to determine the angle of the jet at the outlet of the nozzle.

The area ratio between the throat and the outlet of the nozzle is 4.8. Here, we impose the total pressure (P tot = 1) and the total temperature (T tot = 0.6) at the inlet of the nozzle. Considering that Mach 1 is reached at the throat, one can determine the Mach number, the pressure and the temperature at the outlet under the assumption of a one dimensional isentropic flow. In our case, this gives M=3.7763, T=0.1738 and p=0.0126. This pressure is called the adaptation pressure P c . If the ambient pressure P atm is lower than P c the flow inside the nozzle is independent of P atm .

We obtain the jet angle by tracking the contact discontinuity between the gas coming from the nozzle and the gas initially outside the nozzle with a level set function keeping fixed the point at the extremity of the nozzle (x = 2). At each time step, the level set function is transported according to the velocity of the fluid with the procedure described in section 4 for moving boundaries. All the viscous effects (mixing layer) are supposed to be concentrated across the contact discontinuity represented by the isoline zero of the level set function. The velocity of the contact discontinuity is computed thanks to a Riemann problem where only the ambient pressure P atm is imposed.

At initial state, the nozzle is filled with a gas at rest with p = 1 and T = 0.6. Outside of the nozzle, the gas is also at rest, with p = P atm and T = 0.6. The initial configuration with the zero isoline of the level set function is shown on figure 18. The dark area above the nozzle is penalized, no computations are performed unless the level set evolves and includes this region. The computational domain is [-1,5]×[0,4] and is discretized with 120×80 cells. The velocity space [-10,10]×[-10,10] is discretized with 101 points in each directions. Figure 23 shows the angle δ of the jet at the outlet for different pressure ratio P c /P atm and for BGK and compressible Euler models. When this ratio is equal to 1, the angle of the flux at the outlet should be the same as the angle of the nozzle outlet. The figure shows that in this configuration, δ is actually larger. This is due to the calculation of P c through 1D relations. Thus, the pressure at the boundary of the nozzle at the outlet is not equal to P c but it is larger. If this pressure is imposed outside the nozzle, we obtain an angle δ equal to the one of the nozzle exit (first point on figure 23). These results can be qualitatively compared with the experimental results obtained by Latvala et al. in [START_REF] Latvala | Experimental Determination of Jet Spreading from Supersonic Nozzles at High Altitudes[END_REF] where it is shown, for γ = 7/5, that the evolution of the jet angle is linear with the logarithm of the pressure ratio. The same behaviour is observed in figure 23 for the BGK model. The compressible Euler model progressively detaches from a straight line as the rarefaction increases. The quantitative results cannot be directly compared to experiments since in our case, since γ = 5/3.

For small pressure ratios (< 10) the BGK and compressible Euler models give the same angle. When the ratio increases, the difference becomes larger and the kinetic model stays very close to a straight line. For this kind of pressure ratio, the local relaxation time increases outside the nozzle and becomes too large to consider the fluid at equilibrium. Thus, the continuum model tends to give a different solution. This emphasizes the necessity of using a kinetic model with an AP boundary condition since this problem cannot be solved with a continuum model for high pressure ratio. Also, a solution computed with a specular reflection wall condition in the nozzle would significantly pollute the simulation because of a net reduction of the efficient section of the nozzle.

Conclusion

In recent years, the notion of Asymptotic Preserving schemes [START_REF] Filbet | A class of asymptotic-preserving schemes for kinetic equations and related problems with stiff sources[END_REF] has been introduced answering the need for numerical methods that automatically converge to discretizations of appropriate reduced models, as the Knudsen number changes within the flow. This work shows that such methods are not AP in presence of interior boundaries, unless a proper treatment of the boundary condition is introduced at the price of coslty higher order interpolations in the velocity space.

Here we propose an efficient boundary condition which ensures that an AP scheme remains AP up to the boundary. We illustrate this result by comparing several numerical schemes to model the impermeability condition for the BGK model with emphasis on asymptotic preserving properties in the Euler limit. We have also shown how to recover second order accuracy on Cartesian meshes using this new wall condition and simulated non-trivial rarefied regime test cases. In future work, we will concentrate on the asymptotic limit of wall models towards Navier-Stokes equations using an ES-BGK model.

Inria

Figure 3 :

 3 Figure 3: Characteristics for the reflection of a rarefaction wave (left) and a shock wave (right).

Figure 4 :

 4 Figure 4: Velocity error for the reflection of a rarefaction wave (left) and a shock wave (right).

Figure 5 :

 5 Figure 5: Density solution for the oblique shock with bilinear interpolation on 21×21 grid points.

Figure 6 :

 6 Figure 6: Density solution for the oblique shock with bilinear interpolation on 81×81 grid points.

Figure 7 :

 7 Figure 7: Density solution for the oblique shock with a bicubic interpolation with 21×21 velocity grid points.

Figure 8 :

 8 Figure 8: Density for the oblique shock with the present method (21×21).

Figure 9 :

 9 Figure 9: Comparison of the specular and Euler-AP conditions for the pressure (right) and the temperature (left).

  0

Figure 10 :

 10 Figure 10: Zoom on post-shock values for the pressure (left) and the temperature (right).

Inria

Figure 11 :

 11 Figure 11: Left: Solution for the density with the Euler-AP method. Right: map of the difference between the two methods

Figure 12 :

 12 Figure 12: Error with respect to Euler solution for the Euler-AP method (left) and the bicubic interpolation (right).

Figure 13 :

 13 Figure 13: Computational time for the two methods with respect to the number of velocity grid points.

Figure 14 :

 14 Figure 14: Left: Solution on a grid 128×192 for the pressure. Right: Error map for the density on a grid 128×192.

Figure 15 :

 15 Figure 15: Error in L 1 (left) and L ∞ norm (right).

Figure 17 :

 17 Figure 17: Left: Kn B = 0.147. Right: Kn B = 0.293.

Figure 18 :

 18 Figure 18: Computational domain at t=0.

Figure 21 :Figure 22 :Figure 23 :

 212223 Figure 21: Mach number and velocity vectors at t=11 for P c /P atm = 2000.

1 2 . 3 7 3numerical method 7 3 . 1 3 . 2 8 4level set function 9 5 14 6results 16 6 . 1

 12373132891461 The dimensionless BGK model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 BGK reduced model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 The discrete model in velocity space . . . . . . . . . . . . . . . . . . . . . . . . . The The space discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 The time discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Wall boundary conditions 10 5.1 The diffuse boundary condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 5.2 Impermeability boundary condition . . . . . . . . . . . . . . . . . . . . . . . . . . 11 5.2.1 Specular reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 5.2.2 A new discrete Euler-AP impermeability condition . . . . . . . . . . . . . 12 5.3 Wall boundary condition on Cartesian grid . . . . . . . . . . . . . . . . . . . . . 13 5.3.1 First order Euler-AP scheme . . . . . . . . . . . . . . . . . . . . . . . . . 14 5.3.2 Second order Euler-AP scheme . . . . . . . . . . . . . . . . . . . . . . . . Numerical Reflection of a rarefaction wave and a shock wave . . . . . . . . . . . . . . . . . . 16 6.2 The oblique shock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176.3 The blunt body problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 6.4 Ringleb flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 6.5 Flow past a vertical plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 6.6 A nozzle plume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 7 Conclusion 29 RR n°8471 RESEARCH CENTRE BORDEAUX -SUD-OUEST 200 avenue de la Vieille Tour 33405 Talence Cedex Publisher Inria Domaine de Voluceau -Rocquencourt BP 105 -78153 Le Chesnay Cedex inria.fr

  Figure 16: Left: Kn B = 0.044. Right: Kn B = 0.088.

		1										1
		0.8										0.8
	y/h	0.6									y/h	0.6
		0.4										0.4
		0.2					Euler-AP				0.2				Euler-AP
							DSMC (Bird)								DSMC (Bird)
							AUGKS (Chen et al.)							AUGKS (Chen et al.)
		-0.4 0	-0.2		0	0.2 U x /U ∞	0.4	0.6	0.8			-0.4 0	-0.2	0	0.2 U x /U ∞	0.4	0.6	0.8
				1										1
				0.8										0.8
			y/h	0.6									y/h	0.6
				0.4										0.4
				0.2					Euler-AP				0.2		Euler-AP
									DSMC (Bird)						DSMC (Bird)
									AUGKS (Chen et al.)					AUGKS (Chen et al.)
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RR n°8471


FOOD REFORM MOVEMENTS

Nicolas Larchet Social historians have broadly defined two cycles of American history characterized by an efflorescence of social movements aiming to reform both the individual and the society at large: the Jacksonian Era, from the 1830s to the 1850s, and the Progressive Era, from the 1890s to the 1920s. The reform impulse thrived wherever there was a perceived vice, abuse or corruption of industrial civilization that needed to be changed, corrected or improved. Studies of "antebellum" and "progressive" reforms have thus been made across a very broad spectrum of interests, from temperance and anti-prostitution crusades to housing and sanitation laws. However, it is only fairly recently, starting in the 1970s and 1980s, that historians have rediscovered the figure of the "health reformer" [START_REF] Whorton | Crusaders for Fitness: The History of American Health Reformers[END_REF], frequently specializing in "food reform" -or depending on the context "diet", "dietetic" or "dietary reform" -that is the zealous drive to change the way Americans eat or grow foods, based on religious or secular claims to truth (Kirkland, 1973;[START_REF] Levenstein | The New England Kitchen and the Origins of Modern American Eating Habits[END_REF]Aronson, 1982;[START_REF] Schwartz | Never Satisfied: A Cultural History of Diets, Fantasies, and Fat[END_REF][START_REF] Shapiro | Perfection Salad: Women and Cooking at the Turn of the Century[END_REF].

Not incidentally, as historians question the past from the vantage point of their present, it can be argued that this surge of interest in historical attempts to change American food habits is related to the emergence of a contemporary food reform movement, taking shape in the 1980's and well established by the end of the 2000's decade, centered around the promotion of fresh, locally-grown and organic foods.

"Simpler, plainer and more natural": Dietary reformers and the agrarian myth in the Jacksonian Era.

Aroused by the Second Great Awakening, this religious revival movement that spread through the young republic in the first half of the 19 th century, confronted with the making of an urban and industrial society, religious leaders were quick to embrace physiological reform, preaching a moral and physical salvation instead of a theological one and taking a prominent role in what has been called the Popular Health Movement of the 1830s-1850s. Health reformers usually embraced a large variety of causes: hydrotherapy, dress reform, sex hygiene, temperance, and among them all, dietary reform.

The most eminent dietary reformer of the era was the Presbyterian minister Sylvester Graham, who began his career on the temperance circuit in Pennsylvania in 1830. In his view there was no reason to limit temperance to drink, as gluttony also had his toll on human misery: the worthy individual had to live a temperate life in his drinking, eating and sexual activities. He advocated a "simpler, plainer and more natural" diet, advising against the consumption of meats, white flour, condiments and alcoholic stimulants and calling for greater consumption of fresh fruit and vegetables at a time when city councils prohibited their sale -rotten fruits were widely held to be one of the causes of the 1832 cholera epidemic. He soon found allies in the medical and teaching professions, such as Dr. William Alcott, a prolific author and educator with whom he founded the First American Physiological Society in 1837. Graham's ideas were circulated in books, pamphlets and periodicals such as The Library of Health (1835-1843) and the Graham Journal of Health andLongevity (1837-1839). Whereas the Graham diet, with its endorsement of whole wheat and high-fiber food, has made its way to the regimens of our day, one must not forget that slimming did not matter for Grahamites: the problem with the American diet was gluttony and indigestion, not fatness; therefore the solution was to be found in moral perfectionism rather than physical culture.

Even the foremost religious leaders of the time preached some measures of dietary reform, such as Joseph Smith who in 1835 in his Doctrine and Convenants prohibited the use of tobacco, alcoholic and hot drinks (coffee, tea) and advised for the consumption of whole grains, fruit and vegetables -if meat was allowed, it should be eaten "sparingly". Another reformer who engaged in a mission to change the American diet after a divine revelation was Ellen G. White, one of the founders of Seventh-Day Adventism and a fervent advocate of vegetarianism who established the Western Health Reform Institute in Battle Creek, Michigan in 1866, which was to be made famous by John Harvey Kellogg at the end of the century. One of the most striking features of the Popular Health Movement was the role of women in the promotion and adoption of hygienic reforms: for instance, a third of the American Physiological Society's members were women and at its second annual meeting the society acknowledged women's role in health promotion, deemed "only second to the Deity in the influence that [they] exert on the physical, the intellectual and the moral interests of the human race". Indeed, health reformers joined together with early women's rights movements, as they did with abolitionists. If Mary Gove Nichols lectured on vegetarianism and sanitary education to women across New England, she also spoke and wrote against marriage, which she saw as the "annihilation of women", and was acquainted with radical groups espousing the views of Fourier. Oberlin College, a hotbed for abolitionism and women's rights, adopted the Graham diet in its dining room in 1835 -though they had to abandon it in 1841 amidst rumors of mass starvation.

As far as one was concerned with food reform, there was also more than just diet to be considered: finding the right way to eat supposed a right way to grow food. Agricultural reformers also flourished in the Jacksonian Era, advocating the proper way to fertilize the soil or to breed livestock, drawing on the agrarian myth of a republic of small, self-sufficient yeoman farmers. Little has been written on the subject of agricultural reformers in their relation to the dietary reformers of the day -one can assume that they didn't appeal to the urban middle-class intellectuals who embraced health reform. Nevertheless some contemporary accounts seem to indicate a proximity of interests between both groups, such as the famous lecture on "New England Reformers" that Ralph Waldo Emerson -an occasional vegetarian himself -read in 1844 before the American Anti-Slavery Society:

What a fertility of projects for the salvation of the world! One apostle thought all men should go to farming; and another, that no man should buy or sell: that the use of money was the cardinal evil; another, that the mischief was in our diet, that we eat and drink damnation. These made unleavened bread, and were foes to the death to fermentation. It was in vain urged by the housewife, that God made yeast, as well as dough, and loves fermentation just as dearly as he loves vegetation; that fermentation develops the saccharine element in the grain, and makes it more palatable and more digestible. No; they wish the pure wheat, and will die but it shall not ferment.

Stop, dear nature, these incessant advances of thine; let us scotch these ever-rolling wheels! Others attacked the system of agriculture, the use of animal manures in farming; and the tyranny of man over brute nature; these abuses polluted his food. The ox must be taken from the plough, and the horse from the cart, the hundred acres of the farm must be spaded, and the man must walk wherever boats and locomotives will not carry him. Even the insect world was to be defended,that had been too long neglected, and a society for the protection of ground-worms, slugs, and mosquitos was to be incorporated without delay.

The quest for "Health and Efficiency": Nutrition science and social reform in the Progressive Era.

Reformers at the turn of the 20 th century were less likely to be found among preachers, when the early successes of the social gospel paralleled the institutionalization of social science as a legitimate cure for society's ills. In the same way that antebellum food reform couldn't be separated from moral and spiritual renewal, food reform in the Progressive Era was inextricably linked with social reform: a social problem oriented the first scientific investigations on human nutrition, defined by a loose alliance of biochemists, economists, statisticians, social workers, and philanthropists who were to craft a solution to the "social question" (that is, labor unrest) by attempts at rationalizing the working-class eating habits. what foods they should select to the best advantage of "health and purse". In the Bureau report published in 1886, confronted with the classic instance of those who bought luxury foods for status (i.e., best cuts of beef), Atwater blamed the "extravagance of the poor", resting on a "curious foundation made up of pride, ignorance and indifference", and called for "a reform in the dietary habits of (…) the classes who work for small wages", consisting "in many instances in the use of less food as a whole, and in many more cases in the use of relatively less meat and larger proportions of vegetable foods". Suggesting minimum protein and calorie requirements, Atwater followed Atkinson in the claim that nutrition research could solve the labor problem: one could improve workers' productivity and prevent labor agitation without raising wages, if only the working-class learned to eat scientifically and rationalized its domestic economy… Atwater exposed his views in various articles for The Century, a widely circulated periodical among the opinion-forming middle class. Lobbying for federal support, in 1894 he obtained funding from Congress for nutrition studies to be conducted at Agricultural Experiment Stations, with a suggestion to determine "rations less wasteful and more economical than those in common use".

From 1894 to 1911 more than 400 dietary studies would be conducted by the Office of Experiment Stations at Atwater's instigation, engaging some of the foremost social reformers of the day such as pioneer of social work Jane Addams, who would collaborate on a dietary study of immigrant families living in the "congested districts" of Chicago, and black educator Booker T.

Washington, authorizing a dietary study of black sharecroppers in the Tuskegee vicinity.

Whereas food reformers needed nutrition science to develop objective norms and standards of food needs, research had to be coupled with an education project to effectively change the people's eating habits according to those standards. To put this strategy to work, Atkinson joined in 1889 with Mary H. Abel and Ellen S. Richards, two leading figures in the burgeoning specialty of domestic science, to open a public kitchen in Boston. Established in an immigrant neighborhood, selling at a low price broths slow-cooked in the Aladdin Oven (a fuelefficient, asbestos-insulated stove of Atkinson's invention), the New England Kitchen struggled to attract manual laborers and immigrants. Irish and Italians were especially disregarding the standard New England dishes served there -an Irish mother pressed to take an Indian pudding home famously replied "My boys says, 'Oh ! You can't make a Yankee of me that way !' ".

Meanwhile other public kitchens opened in New York, Philadelphia and most notably in Chicago at Jane Addams' Hull House. The culmination of the movement came in 1893 at Chicago World's Fair, where Massachusetts erected a small cottage with a model kitchen serving more than 10, 000 meals over a 2-month period to educate the public about "scientific cooking".

While workers and immigrants resisted reformers' endeavors to rationalize their diet, union leaders such as Eugene V. Debs opposed nutrition investigations on the ground that they could be used to maintain the U.S. laborer "at a cost as low as Chinamen are subjected to" (some workers ate more than usual when participating in the Office of Experiment Stations surveys to raise the nutritionists' estimates of reasonable food expenditures). On the contrary, the movement had a large audience at the other end of the social spectrum: followers of food faddists Dr. John Harvey Kellog, a Seventh-Day Adventist vegetarian who ran the Battle Creek Sanitarium, and Horace Fletcher, a San Francisco art dealer and dietician who prescribed that food needed to be chewed thirty two-times before being swallowed, included 27 th U.S. president eaten in a car, the "pastoral", which he divides in two meals, one coming from "big organic" corporations, the other from small organic farms, and the "personal", consisting in a "perfect meal" entirely made of food he hunted, gathered and grew himself. Both Schlosser and Pollan also narrated the 2008 documentary Food, Inc., exposing America's corporate controlled food industry. While Schlosser called for "reforming the food industry" in the 2009 companion book to Food, Inc., Pollan called for "a reform of the entire food system" in a New York Times open letter to the future U.S. president in 2008. The most distinctive feature of the contemporary food reform movement is thus its holistic approach: nowadays reformers attempt not merely to change the American diet but rather to transform its globalized, industrial food system, intervening at every stage in the process of production, transportation, distribution, and consumption of foods. However, if Waters, Schlosser and Pollan may be the public faces of this social movement, they did not play a central role in its emergence: the backbone of today's food movement is provided by a network of non-profit organizations engaging in various local initiatives such as farmer's markets, food co-ops, Community-Supported Agriculture farms, community gardens, urban agriculture and food education programs in schools and community centers, which all intend to promote the use of fresh, local or organic foods in underserved communities. A growing number of Food Policy Councils have been established nationwide to develop local food policies at the city, regional or state-level, bringing together various actors having a stake in foodrelated issues: anti-hunger advocates, nutritionists, farmers, retailers, community gardeners, urban planners, bankers, philanthropists… Most of these projects originated in the 1980s through government and private funding, primarily as a result of an alliance between anti-hunger advocates, sustainable agriculture groups and environmental approaches to public health.

As with earlier food reform efforts, a social problem oriented the early development of the movement: the discovery of underserved communities in American inner cities where lowincome, minority groups had a limited access to supermarkets and paid a higher cost for food.

Whereas sociologists and Civil Rights activists had long denounced a phenomenon of retail redlining, showing that the inner city poor paid more for lower quality goods, it was only at the end of the seventies that local governments started to tackle the issue. In 1977, two simultaneous actions were undertaken to decrease the cost of food and improve its access for low-income consumers. In Hartford, Connecticut, a non-profit named the Hartford Food System was established by the city to create "an alternative urban food system, based on greater resident selfsufficiency", formed by an alliance of environmentalists rooted in the back-to-the-land movement with black and Hispanic neighborhood organizations. The city contracted with community organizer Catherine Lerza, one of the authors of Food for People, Not for Profit, to prepare "a strategy to reduce the cost of food", devising four "self-help approaches": community gardening, solar greenhouses, food distribution systems and a food processing center. On that same year in Knoxville, graduate students and faculty of the department of urban and regional planning at the University of Tennessee completed a study of the city's food system, focusing on inequities in the food supply. Using data from this study the local Community Action Committee, an anti-poverty organization, received a two-year grant from the federal government to develop community gardens and food assistance programs. These initiatives led to the creation of the country's first Food Policy Council, established by a Knoxville city council resolution in 1982 to ensure "that all citizens have access to an adequate and nutritious food supply" (government officials were also sensitive to the need to monitor the city's capacity to supply and dispose of food for the coming World's Fair).

Meanwhile, one of the first systematic efforts to study the U.S. food system was undertaken by Rodale Press, the publishing company founded by Jerome I. Rodale, an author, editor and health reformer who introduced organic farming to the United States in the 1940s via its magazine Organic Gardening and Farming. In 1980, Rodale Press launched the Cornucopia Project, a two-year intensive research program to "create an accessible body of information about the U.S. food system", distributing a free newsletter to 25,000 people, publishing op-ed ads in various newspapers and issuing reports based on audits of the food system in several states.

Headed by Medard Gabel, an international consultant and disciple of futurist Buckminster Fuller, the project ended in 1982 with the publishing of a white paper on food reform, Empty Breadbasket? The Coming Challenge to America's Food System and What We Can Do About It. Most of today's food movement intellectual toolkit is to be found in the Cornucopia Project agenda, which made recommendations such as "consume more fresh, locally-grown fruits and vegetables", "support local farmers", "develop a prudent diet", "grow more food" (directed at consumers), "encourage sustainable farming methods", "minimize energy and material use" (directed at the food industry), "establish a department of food", "facilitate access to food", (directed at cities), etc.

Detouring to Canada, this problematization of the food system was soon to be translated by philanthropic and international organizations in the new field of "health promotion" (otherwise known as the "new public health"), which objectives were stated at the 1977 World Health Assembly as "the attainment by all the citizens of the world by the year 2000 of a level of health that will permit them to live socially and economically productive lives". In 1984 in Toronto, the international conference "Beyond Health Care" paved the way for the establishment of a Food Policy Council as a sub-committee of the city's Board of Health, following recommendations made in the workshop "Healthy Toronto 2000", an event recognized as a starting point of the World Health Organization's Healthy Cities project, which acknowledged the role of the urban environment in shaping a healthy lifestyle. Among the prime supporters of the Healthy Cities project was the W.K Kellogg Foundation, created in 1930 by breakfast-cereal manufacturer Will Keith Kellogg -John Harvey's brother -to "promote the health, happiness and well-being of children". In 1987, the foundation established the Agriculture, Food and Human Values Society, associated with the academic journal Agriculture and Human Values, to foster agricultural studies in liberal arts colleges, and started funding research such as The Local Food System Project in 1994-1997, providing technical assistance to six sites developing food policy structures based on the experiences of Hartford and Knoxville, among others. From this time on, the foundation would support grassroots initiatives to change the food system: in 2000, the Food & Society Program (renamed Food & Community Program in 2009) was launched to fund "projects supporting the creation of community-based food systems that support local, healthy, sustainably grown food", organizing yearly conferences, delivering grants and providing fellowships to leading food advocates. From the mid-1990s to 2009, the foundation's total funding to community food projects amounted to more than 230 million dollars, counting as the single largest contributor to the movement.

Welfare, charity or self-sufficiency? Hunger relief and the transformations of social provision.

Whereas Rodale's Cornucopia Project, the World Health Organization and the W.K. Kellogg Foundation all helped to unite concerns for sustainable agriculture with health promotion, the social problems of hunger and poverty were also to be reframed in the 1990s in accordance with the sustainability and healthy living agenda. We have already seen that the first municipal efforts at reforming the food system in Hartford and Knoxville were related to the framing of the problem of low-income access to food -one doesn't have to stretch the imagination too far to see that what was defined as a problem of financial access to a necessity was to be aptly reframed as a problem of physical access to healthy, locally-grown foods, turning a right to the satisfaction of a basic human need into a duty to manage risks for health and the environment. How did this framing come to be?

One has to look a few years back in time to grasp the conversion of anti-hunger advocates from disillusioned efforts at federal lobbying to grassroots initiatives, directed at emergency hunger relief on the one hand, and at community food projects on the other hand. Following the rediscovery of hunger in the Mississippi Delta in 1969, a generation of activists trained in the Civil Rights movement and the War on Poverty successfully lobbied the federal government to develop income redistribution policies through food assistance programs, increasing participation in the Food Stamp Program and pressing for new legislation, such as the Special Supplemental Nutrition Program for Women, Infants and Children (WIC) -federal expenditures on food assistance grew by 500 percent in the following decade. Skilled at working within the legislative process, this group known as the "anti-hunger lobby" was compelled to rethink its strategy entirely in the early 1980's. Following Ronald Regan administration's severe cuts to welfare programs and fiscal austerity measures, anti-hunger activists acknowledged that a continuous focus on advocacy for federal assistance programs held little chance of success. As new breadlines were forming in inner cities, a majority of activists were drawn to what sociologist Janet Poppendieck calls the "emergency food movement", providing charity to families moving off of the welfare rolls via a growing network of food banks, food pantries and soup kitchens.

However, what was intended to be an en emergency measure endured to our day. Anti-hunger activists who could not settle for managing poverty and still wanted to make a change were thus forced to reframe the hunger problem by forging an alliance with sustainable agriculture groups through the mediation of urban planners.

The 1992 Los Angeles riots brought a major impulse to this strategy: the destruction of many food businesses in South Central Los Angeles that followed the Rodney King verdict prompted the department of urban planning at UCLA to conduct a study of the neighborhood's food system, Seeds of Change: Strategies for Food Security for the Inner City, focusing on the issue of lowincome access to food and proposing "a framework for food security planning that is equitable, economically efficient and environmentally sound". The synthesis between urban anti-hunger interests and rural sustainability perspectives was to be laid around the notion of "food security", defined after the United Nations Food and Agricultural Organization as "the state in which all persons obtain a nutritionally adequate, culturally acceptable diet at all times through nonemergency sources". Andy Fisher, one of the study's authors, called for a comprehensive reform of hunger alleviation in a policy paper considered a cornerstone of the community food security approach, emphasizing a shift from compensation to prevention: "Food security differs from hunger in certain crucial ways. First, food security represents a community need rather than an individual's plight, as with hunger (…) Second, whereas hunger measures an existing condition of depravation, food security is decidedly prevention-oriented, evaluating the existence of resources -both community and personal -to provide an individual with adequate acceptable food (…) A food system offering security should have sustainability such that the ecological system is protected and improved over time, and equity, meaning as a minimum, dependable access for all social groups". In 1994, Andy Fisher joined with Mark Winne, then executive director of the Hartford Food System to build a national network of local food projects, the Community Food Security Coalition (Both Fisher and Winne were later to become W.K. Kellogg's Foundation Food & Society Fellows). The coalition successfully introduced legislation supporting community food projects as part of the 1996 Farm Bill, securing federal funding ever since. In a political climate averse to social welfare and labor rights, dissolving the hunger problem in the more consensual notion of food security seemed like a prerequisite for food advocates: the call for social justice had to be translated into an appeal to self-sufficiency and environmental responsibility to get external support.

Conclusion : The food movement between populism and elitism.

The contemporary food reform movement has emerged from an uneasy alliance between a plurality of professions, all engaged in a competition for material and symbolic rewards to be obtained from the government, non-profit or philanthropic organizations through the mobilization of practical and scientific knowledges. This plurality is evident in the many concurring terms commonly used to make sense of these efforts: "alternative food", "(community) food security", "food justice", "food sovereignty", "good food", "local food", "real food", "sustainable food", "true food" -or simply just the "food movement".

Nevertheless, unifying themes pervade much of this social movement, forming a set of shared beliefs, assumptions and values which can be traced back to earlier periods: the romanticized ideal of a pre-industrial past when man lived closer to nature, a sense of moral responsibility and civic virtues tied to agrarian living, anti-monopoly sentiments in the form of support to small farms, a yearning for self-sufficiency coupled with a denunciation of entrepreneurial and political elites -as it appears in the diatribe against agribusiness interests ("Big Agriculture") and federal farm policies… All these familiar themes would identify the food reform movement with a particular brand of American populism.

However, the movement often faces the charge of elitism, for the already high cost of fresh, local or organic foods continues to rise while the cost of sodas, snacks and other energydense processed foods has been decreasing in the long term. In the words of epidemiologist Adam Drewnowski, who has spent much of his career researching how American's food choices correlate to social class, "food has become the premier marker of social distinctions, that is to say-social class" ("Divided We Eat", Newsweek). Ironically, the fact that many farmers' markets, food co-ops or CSA farms that are established in inner cities end up catering to a majority white, upper-middle class clientele tend to lend weight to this charge of elitism, and to belie the assumption that there is an unmet demand for fresh produces in those communities. Critics have also pointed to the self-indulgence contained in the idea that consumer action would substitute for political action, as in the slogan "vote with your fork". At the risk of turning into a classprejudiced moral crusade, or of degenerating into a fad, the food movement would have to confront the eventuality that the problems with our current food system and diet may be mere symptoms of a wider social problem, namely the persistence of structural inequalities of wealth in a land of abundance. While in 1984 the Physician's Task Force on Hunger in America denounced a "hunger epidemic" in American inner cities, this phenomenon has been eclipsed by growing social fears about the obesity epidemic, which in recent years has been used as an ultimate argument for the need to reform. Then, despite much evidence suggesting that both ills are related, a conspiracy of silence still surrounds the problems of hunger and poverty. ASHMAN, Linda, Jaime de la Vega, Marc DOHAN, Andy FISHER, Rosa HIPPLER and Bill

  Wilbur O. Atwater, the son of a Methodist pastor and temperance advocate, earned a doctorate in chemistry from Yale in 1869, became professor of chemistry at Wesleyan University in 1873 and was named director of the Connecticut Agricultural Experiment Station in 1875. He struggled to get funding for investigations in the then-new field of human nutrition until he met with Edward Atkinson in 1885, the man who was to become his patron. An influential Boston industrialist, philanthropist and self-taught economist, Atkinson was interested in solving the wage problem by reducing the budget workers spent on food so that they would have more money for shelter and clothing without calling for higher wages, a move which liberal economic theory deemed impossible anyway. He introduced Atwater to Carroll D. Wright, U.S. Commissioner of Labor and chief of the Massachusetts Bureau of Labor Statistics, who mandated Atwater to analyze statistics on food consumption of factory workers and to determine

  William H. Taft, oil magnate John D. Rockefeller, liberal economist Irving Fisher, muckrakers Upton Sinclair and S. S. McClure. In 1909, Fletcher and Fisher organized the Health and Efficiency League of America, promoting educational campaigns through its organ Good Health magazine, treating food as fuel for the human body and engaging in a crusade against waste. As a matter of fact, food reform had close affinities with managerial values and the doctrine of efficiency: while Frederick W. Taylor sought to eliminate waste of time and energy in the work of laborers, home economists sought to eliminate waste of money and nutritious materials in the housewives' selection and preparation of foods, translating techniques of scientific management from the factory to the home -from the production of goods to the reproduction of the labor force.If somewhat unsuccessful in the short term, progressive food reformers were influential in the birth of two new professions, that of nutritionist (or dietician) and that of home economist.With the creation of the American Home Economics Association in 1908, Ellen S. Richards defined a legitimate area for women's career in science, who had been barred from entering other fields, and helped spread nutritional knowledge among generations of students."Fresh, local and organic": Sustainability and healthy living in today's holistic food reform movement.In 2009, The New York Times ran an article about the rise of the "sustainable-food movement". Entitled "Is a Food Revolution Now in Season?", it acknowledged the growing political weight of advocates of fresh, local and organic foods, emboldened by Michelle Obama's campaign against childhood obesity. Three public figures are most commonly associated with this social movement: celebrity chef Alice Waters and journalists Eric Schlosser and Michael Pollan.All three are members of Slow Food, an international organization founded in Italy in 1989 to counter the fast-paced, fast food lifestyle.Alice Waters is credited with renewing the taste for local, seasonal products, which have been on the menu of her Berkeley restaurant "Chez Panisse" since its 1971 opening. She also started the "Edible Schoolyard" program at a Berkeley middle school in 1995, where students learn to grow, harvest and prepare produces from an organic garden as part of the school curriculum. Edible Schoolyards have since been duplicated in New Orleans, Los Angeles, Greensboro and Brooklyn. Waters' lobbying efforts also led First Lady Michelle Obama to plant an organic garden on the white house lawn in 2009, the first to do so since Eleanor Roosevelt's victory garden during World War II. Published in 2001, Erich Schlosser's Fast Food Nation is a work of investigative journalism that has been compared to Upton Sinclair's The Jungle. Denouncing the social, economic, environmental and health impacts of the fast food industry, from the exploitation of migrant workers in the meat packing industry to corporate sponsorship of school cafeterias, it had been required reading for the incoming freshman class at several universities and was adapted into a film in 2006. Michael Pollan's The Omnivore Dilemma was named by the New York Times one of the ten best books of 2006. It traces the origins of four meals, following three food chains: the "industrial", taking the example of a McDonalds meal

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