Stochastic lagrangian method for downscaling problems in computational fluid dynamics
ESAIM: Modélisation mathématique et analyse numérique, Tome 44 (2010) no. 5, pp. 885-920.

This work aims at introducing modelling, theoretical and numerical studies related to a new downscaling technique applied to computational fluid dynamics. Our method consists in building a local model, forced by large scale information computed thanks to a classical numerical weather predictor. The local model, compatible with the Navier-Stokes equations, is used for the small scale computation (downscaling) of the considered fluid. It is inspired by Pope's works on turbulence, and consists in a so-called Langevin system of stochastic differential equations. We introduce this model and exhibit its links with classical RANS models. Well-posedness, as well as mean-field interacting particle approximations and boundary condition issues are addressed. We present the numerical discretization of the stochastic downscaling method and investigate the accuracy of the proposed algorithm on simplified situations.

DOI : 10.1051/m2an/2010046
Classification : 65C20, 65C35, 68U20, 35Q30
Mots clés : Langevin models, PDF methods, downscaling methods, fluid dynamics, particle methods
@article{M2AN_2010__44_5_885_0,
     author = {Bernardin, Fr\'ed\'eric and Bossy, Mireille and Chauvin, Claire and Jabir, Jean-Fran\c{c}ois and Rousseau, Antoine},
     title = {Stochastic lagrangian method for downscaling problems in computational fluid dynamics},
     journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
     pages = {885--920},
     publisher = {EDP-Sciences},
     volume = {44},
     number = {5},
     year = {2010},
     doi = {10.1051/m2an/2010046},
     mrnumber = {2731397},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an/2010046/}
}
TY  - JOUR
AU  - Bernardin, Frédéric
AU  - Bossy, Mireille
AU  - Chauvin, Claire
AU  - Jabir, Jean-François
AU  - Rousseau, Antoine
TI  - Stochastic lagrangian method for downscaling problems in computational fluid dynamics
JO  - ESAIM: Modélisation mathématique et analyse numérique
PY  - 2010
SP  - 885
EP  - 920
VL  - 44
IS  - 5
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/m2an/2010046/
DO  - 10.1051/m2an/2010046
LA  - en
ID  - M2AN_2010__44_5_885_0
ER  - 
%0 Journal Article
%A Bernardin, Frédéric
%A Bossy, Mireille
%A Chauvin, Claire
%A Jabir, Jean-François
%A Rousseau, Antoine
%T Stochastic lagrangian method for downscaling problems in computational fluid dynamics
%J ESAIM: Modélisation mathématique et analyse numérique
%D 2010
%P 885-920
%V 44
%N 5
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/m2an/2010046/
%R 10.1051/m2an/2010046
%G en
%F M2AN_2010__44_5_885_0
Bernardin, Frédéric; Bossy, Mireille; Chauvin, Claire; Jabir, Jean-François; Rousseau, Antoine. Stochastic lagrangian method for downscaling problems in computational fluid dynamics. ESAIM: Modélisation mathématique et analyse numérique, Tome 44 (2010) no. 5, pp. 885-920. doi : 10.1051/m2an/2010046. http://www.numdam.org/articles/10.1051/m2an/2010046/

[1] F. Bernardin, M. Bossy, C. Chauvin, P. Drobinski, A. Rousseau and T. Salameh, Stochastic downscaling methods: application to wind refinement. Stoch. Environ. Res. Risk. Assess. 23 (2009) 851-859.

[2] M. Bossy, J.-F. Jabir and D. Talay, On conditional McKean Lagrangian stochastic models. Research report RR-6761, INRIA, France (2008) http://hal.inria.fr/inria-00345524/en/.

[3] M. Bossy, J. Fontbona and J.-F. Jabir, Incompressible Lagrangian stochastic model in the torus. In preparation.

[4] J.A. Carrillo, Global weak solutions for the initial-boundary-value problems to the Vlasov-Poisson-Fokker-Planck system. Math. Meth. Appl. Sci. 21 (1998) 907-938. | Zbl

[5] C. Cercignani, The Boltzmann equation and its applications, Applied Mathematical Sciences 67. Springer-Verlag, New York (1988). | Zbl

[6] C. Chauvin, S. Hirstoaga, P. Kabelikova, F. Bernardin and A. Rousseau, Solving the uniform density constraint in a downscaling stochastic model. ESAIM: Proc. 24 (2008) 97-110. | Zbl

[7] C. Chauvin, F. Bernardin, M. Bossy and A. Rousseau, Wind simulation refinement: some new challenges for particle methods, in Springer Mathematics in Industry series, ECMI (to appear).

[8] P. Degond, Global existence of smooth solutions for the Vlasov-Fokker-Planck equation in 1 and 2 space dimensions. Ann. Sci. École Norm. Sup. 19 (1986) 519-542. | Numdam | Zbl

[9] P. Degond and S. Mas-Gallic, Existence of solutions and diffusion approximation for a model Fokker-Planck equation. Internal report, École Polytechnique, Palaiseau, France (1985). | Zbl

[10] M. Di Francesco and A. Pascucci, On a class of degenerate parabolic equations of Kolmogorov type. AMRX Appl. Math. Res. Express 3 (2005) 77-116. | Zbl

[11] M. Di Francesco and S. Polidoro, Schauder estimates, Harnack inequality and Gaussian lower bound for Kolmogorov-type operators in non-divergence form. Adv. Diff. Equ. 11 (2006) 1261-1320. | Zbl

[12] P. Drobinski, J.L. Redelsperger and C. Pietras, Evaluation of a planetary boundary layer subgrid-scale model that accounts for near-surface turbulence anisotropy. Geophys. Res. Lett. 33 (2006) L23806.

[13] C.W. Gardiner, Handbook of stochastic methods, Springer Series in Synergetics 13. Second edition, Springer-Verlag (1985). | Zbl

[14] J.-L. Guermond and L. Quartapelle, Calculation of incompressible viscous flows by an unconditionally stable projection FEM. J. Comput. Phys. 132 (1997) 12-33. | Zbl

[15] F.H. Harlow and P.I. Nakayama, Transport of turbulence energy decay rate. Technical report (1968) 451.

[16] J.-F. Jabir, Lagrangian Stochastic Models of conditional McKean-Vlasov type and their confinements. Ph.D. Thesis, University of Nice-Sophia-Antipolis, France (2008).

[17] I. Karatzas and S.E. Shreve, Brownian Motion and Stochastic Calculus. Springer-Verlag, New York (1988). | Zbl

[18] A. Lachal, Les temps de passage successifs de l'intégrale du mouvement brownien. Ann. I.H.P. Probab. Stat. 33 (1997) 1-36. | Numdam | Zbl

[19] E. Lanconelli, A. Pascucci and S. Polidoro, Linear and nonlinear ultraparabolic equations of Kolmogorov type arising in diffusion theory and in finance, in Nonlinear problems in mathematical physics and related topics, Int. Math. Ser., Kluwer/Plenum, New York (2002) 243-265. | Zbl

[20] H.P. Mckean, Jr, A winding problem for a resonator driven by a white noise. J. Math. Kyoto Univ. 2 (1963) 227-235. | Zbl

[21] J.-P. Minier and E. Peirano, The pdf approach to turbulent polydispersed two-phase flows. Phys. Rep. 352 (2001) 1-214. | Zbl

[22] B. Mohammadi and O. Pironneau, Analysis of the k-epsilon turbulence model. Masson, Paris (1994).

[23] C.M. Mora, Weak exponential schemes for stochastic differential equations with additive noise. IMA J. Numer. Anal. 25 (2005) 486-506. | Zbl

[24] T. Plewa, T. Linde and V.G. Weirs Eds., Adaptive Mesh Refinement - Theory and Applications, Lecture Notes in Computational Science and Engineering 41. Springer, Chicago (2003). | Zbl

[25] S.B. Pope, P.D.F. methods for turbulent reactive flows. Prog. Energy Comb. Sci. 11 (1985) 119-192.

[26] S.B. Pope, On the relationship between stochastic Lagrangian models of turbulence and second-moment closures. Phys. Fluids 6 (1993) 973-985. | Zbl

[27] S.B. Pope, Lagrangian pdf methods for turbulent flows. Annu. Rev. Fluid Mech. 26 (1994) 23-63. | Zbl

[28] S.B. Pope, Turbulent flows. Cambridge Univ. Press, Cambridge (2003). | Zbl

[29] P.-A. Raviart, An analysis of particle methods, in Numerical methods in fluid dynamics, Lecture Notes in Mathematics 1127, Springer, Berlin (1985) 243-324. | Zbl

[30] J.L. Redelsperger, F. Mahé and P. Carlotti, A simple and general subgrid model suitable both for surface layer and free-stream turbulence. Bound. Layer Meteor. 101 (2001) 375-408.

[31] A. Rousseau, F. Bernardin, M. Bossy, P. Drobinski and T. Salameh, Stochastic particle method applied to local wind simulation, in Proc. IEEE International Conference on Clean Electrical Power (2007) 526-528.

[32] P. Sagaut, Large eddy simulation for incompressible flows - An introduction. Scientific Computation, Springer-Verlag, Berlin (2001). | Zbl

[33] D.W. Stroock and S.R. Varadhan, Multidimensional diffusion processes. Springer-Verlag, Berlin (1979). | Zbl

[34] R.B. Stull, An Introduction to Boundary Layer Meteorology. Atmospheric and Oceanographic Sciences Library, Kluwer Academic Publishers (1988). | Zbl

[35] A.-S. Sznitman, Topics in propagation of chaos, in École d'Été de Probabilités de Saint-Flour XIX - 1989, Lecture Notes in Mathematics 1464, Springer, Berlin (1991) 165-251. | Zbl

Cité par Sources :