View
213
Download
0
Category
Tags:
Preview:
Citation preview
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
Recommended