Wind Simulation Refinement: some New Challenges for Particle Methods

We present two new challenges related to the stochastic downscaling method (SDM) that we applied to wind simulation refinement in [1]. After setting the framework, we introduce the boundary forcing issue, and propose a numerical scheme adapted to Particle in Cell methods. Then we turn to the uniform density constraint raised by (SDM) and propose some new methods that rely on optimization algorithms. 1 The Stochastic Downscaling Method We are interested in the behaviour of an incompressible fluid in a domain D of R; D is such that the mass density ρ is supposed constant. We decompose the unknown functions as the sum of a large-scale component and a turbulent one. Rather than solving the Reynolds Averaged Navier Stokes (RANS) equations on the mean velocity 〈U〉 and pressure 〈P〉, we consider some stochastic differential equations (SDEs) that describe the stochastic dynamics of a fluid particle with state variables (Xt, Ut)t>0: dXt = Utdt, (1a) dUt =− 1 ρ ∇x〈P〉(t,Xt)dt− ( 1 2 + 3 4 C0 ) 〈ω〉(t,Xt) (Ut − 〈U〉(t,Xt)) dt + √ C0ε(t,Xt)dWt (1b) − ∑ 0≤s≤t 2Us− l l {Xs∈∂D} + ∑ 0≤s≤t 2Vext(s,Xs) l l {Xs∈∂D}, where ε is the turbulent kinetic energy dissipation rate, 〈ω〉 the turbulent frequency, and (Wt)t≥0 is a three dimensional Brownian motion. The fondation of ∗Article submitted in the proceedings of ECMI 2008, Springer Mathematics in Industry series. 1 in ria -0 03 37 39 6, v er si on 1 7 N ov 2 00 8 Author manuscript, published in "The European Consortium For Mathematics In Industry (2008) ?"