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.
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://archive.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://archive.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://archive.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, Special Issue on Probabilistic methods and their applications, Tome 44 (2010) no. 5, pp. 885-920. doi : 10.1051/m2an/2010046. http://archive.numdam.org/articles/10.1051/m2an/2010046/
[1] , , , , and , Stochastic downscaling methods: application to wind refinement. Stoch. Environ. Res. Risk. Assess. 23 (2009) 851-859.
[2] , and , On conditional McKean Lagrangian stochastic models. Research report RR-6761, INRIA, France (2008) http://hal.inria.fr/inria-00345524/en/.
[3] , and , Incompressible Lagrangian stochastic model in the torus. In preparation.
[4] , 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] , The Boltzmann equation and its applications, Applied Mathematical Sciences 67. Springer-Verlag, New York (1988). | Zbl
[6] , , , and , Solving the uniform density constraint in a downscaling stochastic model. ESAIM: Proc. 24 (2008) 97-110. | Zbl
[7] , , and , Wind simulation refinement: some new challenges for particle methods, in Springer Mathematics in Industry series, ECMI (to appear).
[8] , 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] and , Existence of solutions and diffusion approximation for a model Fokker-Planck equation. Internal report, École Polytechnique, Palaiseau, France (1985). | Zbl
[10] and , On a class of degenerate parabolic equations of Kolmogorov type. AMRX Appl. Math. Res. Express 3 (2005) 77-116. | Zbl
[11] and , 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] , and , Evaluation of a planetary boundary layer subgrid-scale model that accounts for near-surface turbulence anisotropy. Geophys. Res. Lett. 33 (2006) L23806.
[13] , Handbook of stochastic methods, Springer Series in Synergetics 13. Second edition, Springer-Verlag (1985). | Zbl
[14] and , Calculation of incompressible viscous flows by an unconditionally stable projection FEM. J. Comput. Phys. 132 (1997) 12-33. | Zbl
[15] and , Transport of turbulence energy decay rate. Technical report (1968) 451.
[16] , Lagrangian Stochastic Models of conditional McKean-Vlasov type and their confinements. Ph.D. Thesis, University of Nice-Sophia-Antipolis, France (2008).
[17] and , Brownian Motion and Stochastic Calculus. Springer-Verlag, New York (1988). | Zbl
[18] , Les temps de passage successifs de l'intégrale du mouvement brownien. Ann. I.H.P. Probab. Stat. 33 (1997) 1-36. | Numdam | Zbl
[19] , and , 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] , Jr, A winding problem for a resonator driven by a white noise. J. Math. Kyoto Univ. 2 (1963) 227-235. | Zbl
[21] and , The pdf approach to turbulent polydispersed two-phase flows. Phys. Rep. 352 (2001) 1-214. | Zbl
[22] and , Analysis of the k-epsilon turbulence model. Masson, Paris (1994).
[23] , Weak exponential schemes for stochastic differential equations with additive noise. IMA J. Numer. Anal. 25 (2005) 486-506. | Zbl
[24] , and Eds., Adaptive Mesh Refinement - Theory and Applications, Lecture Notes in Computational Science and Engineering 41. Springer, Chicago (2003). | Zbl
[25] , P.D.F. methods for turbulent reactive flows. Prog. Energy Comb. Sci. 11 (1985) 119-192.
[26] , On the relationship between stochastic Lagrangian models of turbulence and second-moment closures. Phys. Fluids 6 (1993) 973-985. | Zbl
[27] , Lagrangian pdf methods for turbulent flows. Annu. Rev. Fluid Mech. 26 (1994) 23-63. | Zbl
[28] , Turbulent flows. Cambridge Univ. Press, Cambridge (2003). | Zbl
[29] , An analysis of particle methods, in Numerical methods in fluid dynamics, Lecture Notes in Mathematics 1127, Springer, Berlin (1985) 243-324. | Zbl
[30] , and , A simple and general subgrid model suitable both for surface layer and free-stream turbulence. Bound. Layer Meteor. 101 (2001) 375-408.
[31] , , , and , Stochastic particle method applied to local wind simulation, in Proc. IEEE International Conference on Clean Electrical Power (2007) 526-528.
[32] , Large eddy simulation for incompressible flows - An introduction. Scientific Computation, Springer-Verlag, Berlin (2001). | Zbl
[33] and , Multidimensional diffusion processes. Springer-Verlag, Berlin (1979). | Zbl
[34] , An Introduction to Boundary Layer Meteorology. Atmospheric and Oceanographic Sciences Library, Kluwer Academic Publishers (1988). | Zbl
[35] , 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
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