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Stochastic Lagrangian method for downscaling problemsin computational fluid dynamics

Published online by Cambridge University Press:  26 August 2010

Frédéric Bernardin ,
Mireille Bossy ,
Claire Chauvin ,
Jean-François Jabir  and
Antoine Rousseau
Frédéric Bernardin
Affiliation:
CETE de Lyon, LRPC, Clermont-Ferrand, France. Frederic.Bernardin@developpement-durable.gouv.fr
Mireille Bossy
Affiliation:
INRIA, TOSCA, Sophia Antipolis, France. Mireille.Bossy@sophia.inria.fr
Claire Chauvin
Affiliation:
INRIA, MOISE, Grenoble, France. Claire.Chauvin@inria.fr
Jean-François Jabir
Affiliation:
CMM Universidad de Chile, Blanco Encalada 2120, Santiago, Chile. jjabir@dim.uchile.cl
Antoine Rousseau
Affiliation:
INRIA & Laboratoire Jean Kuntzmann, 51 rue des Maths, BP 53, 38041 Grenoble Cedex 9, France. Antoine.Rousseau@inria.fr
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Abstract

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 usedfor the small scale computation (downscaling) of the consideredfluid. It isinspired 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.

Keywords

Langevin modelsPDF methodsdownscaling methodsfluid dynamicsparticle methods
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 44 , Issue 5: Special Issue on Probabilistic methods and their applications , September 2010 , pp. 885 - 920
Copyright
© EDP Sciences, SMAI, 2010

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References

Bernardin, F., Bossy, M., Chauvin, C., Drobinski, P., Rousseau, A. and Salameh, T., Stochastic downscaling methods: application to wind refinement. Stoch. Environ. Res. Risk. Assess. 23 (2009) 851–859. CrossRef
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/.
M. Bossy, J. Fontbona and J.-F. Jabir, Incompressible Lagrangian stochastic model in the torus. In preparation.
Carrillo, J.A., Global weak solutions for the initial-boundary-value problems to the Vlasov-Poisson-Fokker-Planck system. Math. Meth. Appl. Sci. 21 (1998) 907–938. 3.0.CO;2-W>CrossRef
C. Cercignani, The Boltzmann equation and its applications, Applied Mathematical Sciences 67. Springer-Verlag, New York (1988).
Chauvin, C., Hirstoaga, S., Kabelikova, P., Bernardin, F. and Rousseau, A., Solving the uniform density constraint in a downscaling stochastic model. ESAIM: Proc. 24 (2008) 97–110. CrossRef
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).
Degond, P., 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. CrossRef
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).
Di Francesco, M. and Pascucci, A., On a class of degenerate parabolic equations of Kolmogorov type. AMRX Appl. Math. Res. Express 3 (2005) 77–116. CrossRef
Di Francesco, M. and Polidoro, S., Schauder estimates, Harnack inequality and Gaussian lower bound for Kolmogorov-type operators in non-divergence form. Adv. Diff. Equ. 11 (2006) 1261–1320.
Drobinski, P., Redelsperger, J.L. and Pietras, C., Evaluation of a planetary boundary layer subgrid-scale model that accounts for near-surface turbulence anisotropy. Geophys. Res. Lett. 33 (2006) L23806. CrossRef
C.W. Gardiner, Handbook of stochastic methods, Springer Series in Synergetics 13. Second edition, Springer-Verlag (1985).
Guermond, J.-L. and Quartapelle, L., Calculation of incompressible viscous flows by an unconditionally stable projection FEM. J. Comput. Phys. 132 (1997) 12–33. CrossRef
F.H. Harlow and P.I. Nakayama, Transport of turbulence energy decay rate. Technical report (1968) 451.
J.-F. Jabir, Lagrangian Stochastic Models of conditional McKean-Vlasov type and their confinements. Ph.D. Thesis, University of Nice-Sophia-Antipolis, France (2008).
I. Karatzas and S.E. Shreve, Brownian Motion and Stochastic Calculus. Springer-Verlag, New York (1988).
Lachal, A., Les temps de passage successifs de l'intégrale du mouvement brownien. Ann. I.H.P. Probab. Stat. 33 (1997) 1–36.
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.
McKean, H.P., Jr, A winding problem for a resonator driven by a white noise. J. Math. Kyoto Univ. 2 (1963) 227–235. CrossRef
Minier, J.-P. and Peirano, E., The pdf approach to turbulent polydispersed two-phase flows. Phys. Rep. 352 (2001) 1–214. CrossRef
B. Mohammadi and O. Pironneau, Analysis of the k-epsilon turbulence model. Masson, Paris (1994).
Mora, C.M., Weak exponential schemes for stochastic differential equations with additive noise. IMA J. Numer. Anal. 25 (2005) 486–506. CrossRef
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).
Pope, S.B., P.D.F. methods for turbulent reactive flows. Prog. Energy Comb. Sci. 11 (1985) 119–192. CrossRef
Pope, S.B., On the relationship between stochastic Lagrangian models of turbulence and second-moment closures. Phys. Fluids 6 (1993) 973–985. CrossRef
Pope, S.B., Lagrangian pdf methods for turbulent flows. Annu. Rev. Fluid Mech. 26 (1994) 23–63. CrossRef
S.B. Pope, Turbulent flows. Cambridge Univ. Press, Cambridge (2003).
P.-A. Raviart, An analysis of particle methods, in Numerical methods in fluid dynamics, Lecture Notes in Mathematics 1127, Springer, Berlin (1985) 243–324.
Redelsperger, J.L., Mahé, F. and Carlotti, P., A simple and general subgrid model suitable both for surface layer and free-stream turbulence. Bound. Layer Meteor. 101 (2001) 375–408. CrossRef
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.
P. Sagaut, Large eddy simulation for incompressible flows – An introduction. Scientific Computation, Springer-Verlag, Berlin (2001).
D.W. Stroock and S.R. Varadhan, Multidimensional diffusion processes. Springer-Verlag, Berlin (1979).
R.B. Stull, An Introduction to Boundary Layer Meteorology. Atmospheric and Oceanographic Sciences Library, Kluwer Academic Publishers (1988).
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.