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Section: New Results

Keywords : interacting particle system, confinement of Langevin system, discrete mass-transport problem, pdf approach.

Interacting particle systems in Lagrangian modeling of turbulent flows

Participants : Mireille Bossy, Jean-François Jabir, Pierre-Louis Lions.

This section presents the results obtained by Tosca on the modeling of turbulent flows. Actually, it could be a part of the previous section on stochastic modeling and applications. We have chosen to present this work in a distinct coherent section.

In the statistical approach of Navier-Stokes equation in Im12 ${\#8477 }^3$ , the Eulerian properties of the fluid (such as velocity, pressure and other fundamental quantities) are supposed to depend on possible realizations $ \omega$$ \in$$ \upper_omega$ . In the incompressible case, in order to compute the averaged velocity, one needs to model the equation of the Reynolds stress. A direct modeling is for example the so-called k -epsilon turbulence model. An alternative approach [63] consists in describing the flow through a Lagrangian stochastic model whose averaged properties are linked to those of the Eulerian fields by conditional means w.r.t. particle position. In the computational fluid dynamics literature, those models are referred to as stochastic Lagrangian models. For example, the Simplified Langevin model [64] , which characterizes particle positions-velocity Im13 ${(X_t,\#119984 _t)}$ in the case of homogeneous turbulent flow, is defined as

Im14 $\mtable{...}$(3)

where C0 , Im15 ${\mfenced o=〈 c=〉 w{(t,x)}}$ and k( t, x) are positive real quantities supposed to be known, and W is a Brownian motion. The pressure gradient Im16 ${\mfenced o=〈 c=〉 \#119979 {(t,x)}}$ is treated as solution of a Poisson equation in order to satisfy the constant mass-density (i.e. particle positions are uniformly distributed) and divergence free conditions. Moreover in the presence of physical boundaries evaluated at Im17 ${\#8706 \#119967 }$ , the model is submitted to a limit condition of the form: Im18 ${\#120124 \mfenced o=( c=) \#119984 _t\#8739 X_t=x=0}$ , for Im19 ${x\#8712 \#8706 \#119967 }$ . The corresponding discrete algorithm is reported in Section  7.1 .

Analysis of the Langevin equations in turbulent modeling

Keywords : Stochastic Lagrangian Models, Vlasov-Fokker-Planck equations.

In his Ph.D. thesis supervised by M. Bossy and D. Talay, J.-F. Jabir studies theoretical problems involved by ( 3 ) in the case where the pressure gradient is removed while k and Im20 ${\#9001 \#969 \#9002 }$ are supposed to be constants.

M. Bossy and J.-F. Jabir study the well-posedness of a simplified version of equation ( 3 ) (in particular, without the pressure term) where Im21 ${\#119984 _t-\#120124 _\#8473 \mfenced o=( c=) \#119984 _t{~|~}X_t}$ is replaced by the kernel Im22 ${\#120124 _\#8473 \mfenced o=( c=) b{(v-\#119984 _t)}{~|~}X_t{|}_{v=\#119984 _t}}$ . The conditional expectation involved in ( 3 ) implies that this equation is a McKean-Vlasov equation with singular kernels. When b is a bounded continuous function, we proved the well-posedness of the Lagrangian system. In particular, the existence result has been obtained by studying related smoothened interacting particle systems and by proving a propagation of chaos result.

Motivated by the downscaling application (see Section  7.1 ), we construct and investigate the well-posedness of Langevin system confined within a regular domain Im23 $\#119967 $ of Im24 $\#8477 ^d$ with the imposed boundary condition

Im25 ${\#120124 \mfenced o=( c=) \mfenced o=( c=) \#119984 _t·n_\#119967 {(X_t)}{~|~}X_t=x=0~\mtext for~x\#8712 \#8706 \#119967 }$(4)

for Im26 $n_\#119967 $ denoting the outward normal unit vector related to Im23 $\#119967 $ . The basic idea of our construction is to add a “confinement” term, Im27 ${-2\mfenced o=( c=) \#119984 _t^-·n_\#119967 {(X_t)}n_\#119967 {(X_t)}}$ , to the velocity dynamic at each time the particle hits the boundary. Assuming that the law of the Lagrangian system Im28 ${(X,\#119984 )}$ admits a trace component along the boundary Im17 ${\#8706 \#119967 }$ with suitable integrability properties, the law of Im28 ${(X,\#119984 )}$ satisfies the specular boundary condition (see [50] )

Im29 ${\#961 {(t,x,u)}=\#961 {(t,x,u-2{(u·n_\#119967 {(x)})}n_\#119967 {(x)})}~for~{(t,x,u)}\#8712 {(0,T)}×\#8706 \#119967 ×\#8477 ^d}$(5)

and show that ( 5 ) implies ( 4 ).

When Im30 ${\#119967 =\#8477 ^d×\#8477 ^+}$ , (strong) existence and uniqueness for the “confined” Langevin system have been established in the simplified case of classical SDE with a constant diffusion thanks to the results of Lachal [57] . Moreover, under suitable conditions on $ \rho$0 and the drift coefficient we also show that the law of Im28 ${(X,\#119984 )}$ satisfies ( 4 ) and ( 5 ). These results have been also extended for the class of McKean-Vlasov equations with smooth and bounded kernels.

When Im23 $\#119967 $ is a bounded regular domain of Im24 $\#8477 ^d$ , in collaboration with P.-L. Lions, we study the Fokker-Planck equation associated to Im13 ${(X_t,\#119984 _t)}$

Im31 $\mfenced o={ \mtable{...}$

When b is bounded, we establish the well-posedness of this equation using Maxwellian super-solutions and sub-solutions which give upper and lower bounds for the solution.

The position correction step in the Lagrangian simulation of constant mass density fluid

We continue our collaboration with A. Rousseau (MOISE Inria Grenoble – Rhône-Alpes) and F. Bernardin (CETE Clermont-Ferrand) on this topic (see section 7.1 ). We combine a “fractional step” method and a “particles in cell” method to simulate the time evolution of the Lagrangian variables of the flow described by equation ( 3 ).

The most noticeable problem arises from the mass-density conservation between two time steps within a given cell. The aim is to move the particles in a box such that:

This problem is not classical at all and is referred in the literature as a problem of discrete optimal transportation, which is known to be nonlinear and numerically very hard to solve in dimension 3.

During summer 2007, A. Rousseau managed a research program at CEMRACS, and the group adapted an algorithm from D. Bertsekas [42] , in order to compute the solution of an optimal transport problem that concerns the particles involved in SDM : see [Oops!] . This research program was funded by CEMRACS and INRIA (Service de Formation par la Recherche). Next January, C. Chauvin, from the CEMRACS group, will start a 12-months post-doc position in the TOSCA and MOISE teams.


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