Lagrangian dispersion of light solid particle in a high Re number turbulence; LES with stochastic process at sub-grid scales CNRS – UNIVERSITE et INSA de Rouen

Lagrangian dispersion of light solid particle in a high Re number turbulence; LES with stochastic process at sub-grid scales 2/13  Kolmogorov, 1941 : dissipative zone inertial range large scales Gaussian distribution of particle velocity Dispersion of fluid particle. Classical results  Taylor, 1921 :

3/13 Measurements of Lagrangian statistics of light particle in the high Re turbulence (from Mordant and Pinton, ENS of Lyon, 2001, 2004) Second order structure function PDF of particle acceleration Von Karman swirling flow of water Normalized PDFs of the velocity increment Lagrangian dispersion of light solid particle in a high Re number turbulence; LES with stochastic process at sub-grid scales

In Diesels: Re is very high flow is very intermittent at small scales  Strong local gradient of velocity: strong gradients of concentration, “spontaneous” ignition or extinction sites, burning spray extinction, “preferential” concentration, large spectrum of interacting spatial scales of liquid fragments  Need of droplet dynamics low in the highly intermittent flow ! 4/13 Importance of intermittency effects Lagrangian dispersion of light solid particle in a high Re number turbulence; LES with stochastic process at sub-grid scales high stretching between two eddies

5/13 Our objective Compute this experiment LES Problem: the small-scale strong inhomogeneity is filtered At sub-grid spatial scales, we need a model Lagrangian dispersion of light solid particle in a high Re number turbulence; LES with stochastic process at sub-grid scales

6/13 Turbulent cascade as fragmentation under scaling symmetry  Frequency of smaller eddies formation is high enough that the portion of energy transmitted to the smaller eddy does not depend on the energy of parent eddy (scaling symmetry assumption) * Gorokhovski (2003) CTR, Stanford, Annual Briefs ** Gorokhovski & Chtab (2005), Lecture Notes in Computational Science and Engineering series, Springer  At large times, this scenario reduces exactly to the Fokker-Planck-type equation Lagrangian dispersion of light solid particle in a high Re number turbulence; LES with stochastic process at sub-grid scales

7/13 Log-brownian stochastic process with constant force  diffusivity / drift = ln(spatial scale) ;  Langevin stochastic equation :  In the logarithmic space ---- usual form of Fokker-Planck equation  Finally :  Lagrangian tracking of solid particle : Lagrangian dispersion of light solid particle in a high Re number turbulence; LES with stochastic process at sub-grid scales

8/13 Distribution of Lagrangian velocity :  velocity distribution follows Gaussian LES + sub-grid model Lagrangian dispersion of light solid particle in a high Re number turbulence; LES with stochastic process at sub-grid scales

9/13 Second order structure function : with sub-grid model no sub-grid model  inertial range is very restricted Lagrangian dispersion of light solid particle in a high Re number turbulence; LES with stochastic process at sub-grid scales

10/13 Expérience de Mordant & Pinton with sub-grid modelno sub-grid model Experiment of Mordant & Pinton, 2001 Second order structure function : LES computation Lagrangian dispersion of light solid particle in a high Re number turbulence; LES with stochastic process at sub-grid scales

11/13  from Kolmogov’s scaling  intermittency: weak accelerations alternated with “violent” accelerations with sub-grid model no sub-grid model Distribution of Lagrangian acceleration LES computation Experiment of Mordant & Pinton, 2001 Lagrangian dispersion of light solid particle in a high Re number turbulence; LES with stochastic process at sub-grid scales

LES + sub-grid modelExperiment of Mordant & Pinton, 2001 Statistics of the velocity increment,, at different time lags, 12/13  at smaller times stretched tails with growing central peak  at integral times Gaussian distribution no sub-grid model Lagrangian dispersion of light solid particle in a high Re number turbulence; LES with stochastic process at sub-grid scales

13/13 Conclusion  Strong velocity gradients at small sub-grid scales Stochastic modeling on sub-grid non-resolved scales  LES computation + stochastic model + solid light particle tracking  Experimental observations of non-Gaussian behavior of solid light particle were reproduced in computation Lagrangian dispersion of light solid particle in a high Re number turbulence; LES with stochastic process at sub-grid scales