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 :

Lii

L RutUtUD 12 222

L

L

TR

exp

0

dRT LL

221230

222 aaDL

L

L

T

uCD

2

02

2

22 2 uDL

2;; uTL L

tU

tU

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

μm250pd

740Re

μm14

06,1fp

Von Karman swirling flow of water

Normalized PDFs of the velocity increment

ms2,0

LD2

tLT

uC

2

0

2

9,2exp0 C

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

eddies1Y

2Y

vsY

5/13

Our objective

Compute this

experiment

740Re

xU LES

'x

x

LES

Problem: the small-scale strong inhomogeneity is filtered

xdxxGxUxU LESg ,

At sub-grid spatial scales, we need a model

UUU LESg

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

tU

tU dq

10

tUfofevolutiontUtU ;

tU

tU

At large times, this scenario reduces exactly to the Fokker-Planck-type equation

tUfU

UU

UU

UTt

tUf;ln

2

1ln

1; 2

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) ;

tdWT

dtT

Ud2

lnlnln

2

Langevin stochastic equation :

In the logarithmic space ---- usual form of Fokker-Planck equation

Uq ln

tqQqqTt

tqQ;ln

2

1ln

1; 22

2

refLl*2 lnlnln l

Finally : UUU LESlocal

Lagrangian tracking of solid particle : Stokes

plocalp VU

dt

dV

Lagrangian dispersion of light solid particle in a high Re number turbulence; LES with stochastic process at sub-grid scales

8/13

tVp Distribution of Lagrangian velocity :

velocity distribution follows Gaussian

Vp ( cm/s )

log1

0(

PD

F)

-300 -200 -100 0 100 200 300

-6

-5

-4

-3

-2LES + 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

22 tVtVD ppL Second order structure

function :

/k

DL 2(

)/

2 k

100 101 10210-3

10-2

10-1

100

101

102

a0(/k)2

C0/k

22/k

2

with sub-grid model

no sub-grid model

μm250pd

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

/k

DL 2(

)/(

)

100 101 1020

0.5

1

1.5

2

2.5

3

3.5

10/13

Expérience de Mordant & Pinton

with sub-grid model no sub-grid model

Experiment of Mordant & Pinton, 2001

22 tVtVD ppL Second order structure

function :

LES computation

9,2exp0 C

μm250pd

μm250pd

82,2mod0 C

Lagrangian dispersion of light solid particle in a high Re number turbulence; LES with stochastic process at sub-grid scales

a (m/s2)

log 10

(PD

F)

-5000 -4000 -3000 -2000 -1000 0 1000 2000 3000 4000 5000

-8

-7

-6

-5

-4

-3

11/13

from Kolmogov’s scaling

intermittency: weak accelerations alternated with “violent” accelerations

2m/s300k

kka

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

v / <(v)2>1/2

log 10

(PD

F)

-20 -10 0 10 20

-7

-6

-5

-4

-3

-2

-1

0

LES + sub-grid model Experiment of Mordant & Pinton, 2001

Statistics of the velocity increment, , at different time lags,

12/13

tVtV pp

at smaller times stretched tails with growing central peak

at integral times Gaussian distribution

no sub-grid model

v / <(v)2>1/2

log 10

(PD

F)

-20 -10 0 10 20

-7

-6

-5

-4

-3

-2

-1

0

Lagrangian dispersion of light solid particle in a high Re number turbulence; LES with stochastic process at sub-grid scales

13/1

3

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

stochLES UU

v / <(v)2>1/2

log 10

(PDF)

-20 -10 0 10 20

-7

-6

-5

-4

-3

-2

-1

0

Stokes

plocalp VU

dt

dV

a (m/s2)

log 10

(PD

F)

-5000 -4000 -3000 -2000 -1000 0 1000 2000 3000 4000 5000

-8

-7

-6

-5

-4

-3

Lagrangian dispersion of light solid particle in a high Re number turbulence; LES with stochastic process at sub-grid scales