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).