Stochastic modeling of primary atomization : application to Diesel spray.
J. Jouanguy & A. Chtab & M. Gorokhovski
CNRS – UNIVERSITE et INSA de Rouen
Introduction.
Atomization: main phenomena:
Turbulence in liquid and gaz phase, cavitation, cycle by cycle variations
Deterministic description of such atomization is very diffcult task.
Stochastic approach
=> Application to primary atomization.
Sacadura ,CORIA
Floating cutter particles.
r
axial direction
radial direction
r
radial direction
axial direction
axial direction
r
radial direction
Liq.
Liq.
Liq.
Time t1 Time t2
r
axial direction
radial direction
r
radial direction
axial direction
axial direction
r
radial direction
Liq.
Liq.
Liq.
The main asumption: scaling symetry for thickness.
x
r(x,t)liq.
y
r(x,t) r(x,t)r(x,t + dt )
txrdttxr ,, 10
q
Fragmentation intensity spectrum
1. Scaling symetry for thickness of liquid core.
2. Life time.
3. Ensemble of n particles.
Theory of Fragmentation under Scaling symetry (Gorokhovski & Saveliev 2003, Phys. Fluids).
Evolution equation
1
0
,, ,
f r t r df t q f r t
t
qKnowledge of
Fokker Planck type equation
2ln,1
ln , ,2
f r tr f r t r r f r t
t r r r
Knowledge of and
ln 2ln
Langevin type equation
2ln ln / 2r r r t
Equation for the distribution function
Equation for one realisation
Log brownian stochastic process
*
lnlnrrCONST c
rc = critical length scale
r* = typical length scale
y
xliqr
*
2
lnln
ln
rrc
Identifiction of main parameters.
Formation of droplets through a minimal size rcr
Transverse instability
=
Rayleigh Taylor instability
Shearing instability
Liquid
rc = critical radius rcr
r* = Rayleigh Taylor scale=>
RTλ
Time scale RT
crlggcr
ruuWe
2
(Reitz 1987)
06.067.1
2
7.05.0
87.01
4.0145.9102.9 r
We
TZRT
02
001 ruuWe lgl
02
002 ruuWe lgg
llg ruu 0001Re
15.0
1 ReWeZ
5.02ZWeT
6.0
5.12
5.030
4.11138.034.0
TZWerl
RT
Realization of stochastic process; « Stochastic floating cutter particles ».
Motion of particules
Grid
Axial direction
Radial direction
Stochastic particle position
Path of stochastic particle
Liquid zone
Gaz zoneInjector radius
Staitistics of liquid core boundary=>
In the radial direction: 2ln ln / 2ip
ip
V r r t
d rV
d t
In the axial direction:
ipip
ipg
ipgl
gip
Udtxd
ruu
uudtUd
21
Experimental setup.
C. Arcoumanis, M. Gavaises, B. French SAE Technical Paper Series, 970799 (1997).
injU
cmR 2
ux,
vr,
Gaz initial conditions:
Atmospheric conditions
Liquid initial conditions:
KT 300
barp 1
3/8.0 cmgp
cmRinj 009.0
mstinj 85.0
mgminj 2.3
KTinj 300
smtUU injinj /2600
Probability to get liquid core; formation of discret blobs using presumed distribution:
x (cm)
r(c
m)
0 0.2 0.4 0.6 0.8 10
0.02
0.04
0.06prob0.990.90.80.70.60.50.40.30.20.10.01
t = 0.02 ms
x (cm)
r(c
m)
0 0.2 0.4 0.6 0.8 10
0.02
0.04
0.06prob0.990.90.80.70.60.50.40.30.20.10.01
t = 0.2 ms
x (cm)
r(c
m)
0 0.2 0.4 0.6 0.8 10
0.02
0.04
0.06prob0.990.90.80.70.60.50.40.30.20.10.01
t = 0.4 ms
Statistics of liquid core boundary
Motion of droplets
Initial conditions
tUu injp
Injection velocity
p
gliquid
pp K
radv
=> Standard KIVA procedure with velocity conditionned on the presence of liquid
typr Radius of the injector
typtyp rr
rrf exp1
Radius:
Injection of droplets
Mass flow rate conservation
x (cm)
r(c
m)
0 0.2 0.4 0.6 0.8 10
0.02
0.04
0.06prob0.990.90.80.70.60.50.40.30.20.10.01
t = 0.2 ms
x (cm)
r(c
m)
0 0.2 0.4 0.6 0.8 10
0.02
0.04
0.06prob0.990.90.80.70.60.50.40.30.20.10.01
t = 0.4 ms
x (cm)
r(c
m)
0 0.2 0.4 0.6 0.8 10
0.02
0.04
0.06prob0.990.90.80.70.60.50.40.30.20.10.01
t = 0.02 ms
Example of distribution of formed blobs.
z
x
0 1 2 3 4 50
0.1
0.2
0.3
0.4
0.5Smr(mm)
0.10.090.080.070.060.050.040.030.020.010
t = 0.2 ms
z
x
0 1 2 3 4 50
0.1
0.2
0.3
0.4
0.5Smr(mm)
0.10.090.080.070.060.050.040.030.020.010
t = 0.4 ms
z
x
0 1 2 3 4 5 6 7 8 9 100
0.2
0.4
0.6
0.8
1Smr(mm)
0.10.090.080.070.060.050.040.030.020.010
t = 0.6 ms
z
x
0 2 4 6 8 100
0.2
0.4
0.6
0.8
1Smr(mm)
0.10.090.080.070.060.050.040.030.020.010
t = 0.8 ms
z
x
0 5 10 150
0.2
0.4
0.6
0.8
1
1.2
1.4 Smr(mm)0.10.090.080.070.060.050.040.030.020.010
t = 1.0 ms
z
x
0 5 10 150
0.2
0.4
0.6
0.8
1
1.2
1.4 Smr(mm)0.10.090.080.070.060.050.040.030.020.010
t = 1.2 ms
Computed mean sauter diameter.
max
min
max
min
2
3
32 D
D
D
D
dDDfD
dDDfDD
Sauter Mean Diameter
time (ms)
velo
city
(m/s
)
0 0.5 1 1.50
50
100
150
200
250
300
x = 30 mm
time (ms)
velo
city
(m/s
)
0 0.5 1 1.50
50
100
150
200
250
300
x = 50 mm
time (ms)
velo
city
(m/s
)
0 0.5 1 1.50
50
100
150
200
250
300
x = 60 mm
time (ms)
velo
city
(m/s
)
0 0.5 1 1.50
50
100
150
200
250
300
x = 40 mm
time (ms)
velo
city
(m/s
)
0 0.5 1 1.50
50
100
150
200
250
300
x = 20 mm
time (ms)
velo
city
(m/s
)
0 0.5 1 1.50
50
100
150
200
250
300
x = 10 mm
Centerline droplet mean axial velocity.
Symbols = experiment
Line = simulation
time (ms)
SM
D(
m)
0 0.5 1 1.50
70
140
210
x = 20 mm
time (ms)
SM
D(
m)
0 0.5 1 1.50
70
140
210
x = 40 mm
time (ms)
SM
D(
m)
0 0.5 1 1.50
70
140
210
x = 50 mm
time (ms)
SM
D(
m)
0 0.5 1 1.50
70
140
210
x = 30 mm
time (ms)
SM
D(
m)
0 0.5 1 1.50
70
140
210
x = 60 mm
time (ms)S
MD
(m
)0 0.5 1 1.50
70
140
210
x = 10 mm
Centerline Sauter Mean Diameter (SMD).
Symbols = experiment
Line = simulation
Application to Air-Blast atomization.
Experiment => mean liquid volume fraction (Stepowski & Werquin 2001)
Simulation => statistics of liquid core boundary
Ug = 60 m/s, Ul = 0.68 m/s
Ug = 60 m/s, Ul = 1.36 m/s
2
2
ll
gg
uuM
Key parameter:
Drop injection and lagrangian tracking.
3
52/51
2typ crg
r We
Typical size resulting from primary atomization
Motion of the drops injected.
Modification of the gas velocity field:
lllglg PuPuu 1
pp u
dtdx
plgp
p uuStf
dtdu
Lagrangian tracking :
Pl probability of presence of liquidf Drag coefficient , Stp particle Stokes number
Distribution of formed blobs.
Pl10.9285710.8571430.7857140.7142860.6428570.5714290.50.4285710.3571430.2857140.2142860.1428570.07142860
0 0.5 1 1.5 2 2.5 3x/Dg
-0.75
-0.5
-0.25
0
0.25
0.5
0.75
y/Dg
Ul = 0.13 m/s
Ul = 2.8 m/s
Experiments (Lasheras & al 1998)
Ug = 140 m/s
Examples of instantaneous distribution of formed droplets with instantaneous
conditionned velocity of gaz and liquid core
Pl10.9285710.8571430.7857140.7142860.6428570.5714290.50.4285710.3571430.2857140.2142860.1428570.07142860
0 1 2 3x/Dg
-0.75
-0.5
-0.25
0
0.25
0.5
0.75
y/Dg
=> Secondary processes:
0
0
0.05
0.05
0.1
0.1
x
-0.02 -0.02
-0.01 -0.01
0 0
0.01 0.01
0.02 0.02
y
Computation in the far field.
Shearing
Turbulence
Collisions
Example of instantaneous distribution of formed droplets
ug = 140 m/s , ul = 0.55 m/s
221 )( rrSeff
tVrel
2
1Fragmentation
or
coalescence
Comparison in the far field.
10 20 30 40x/Dg
30
40
50
60
70
80
90
D32
(m
)
ul = 0.13 m/s modelul = 0.55 m/s modelul = 0.31 m/s modelul = 0.13 m/s Lasheras 1998ul = 0.55 m/s Lasheras 1998ul = 0.31 m/s Lasheras 1998
Ug = 140 m/s
Conclusion.
Simple enginering model for primary atomization is proposed.
This allows to form the blobs in the near-injector region.
Future work.
Comparison with experiments in Brighton (trying different main mechanisms for fragmentation).