deformation and secondary atomization of droplets in technical
TRANSCRIPT
Institute for Applied Sustainable Science, Engineering & TechnologyRoland Schmehl
Flow Problem Analysis in Oil & Gas Industry ConferenceRotterdam, 11 January 2011
Deformation and Secondary Atomization of Droplets in Techn ical Two-Phase Flows
Slide 1
Roland Schmehl Outline
• Introduction
• Basics
• Empirical description
• Normal Mode Analysis
• Nonlinear deformation analysis
• Potential Theory Breakup model
• Motion of deformed droplets
• Validation of deformation models
• Modeling of droplet breakup
• Validation of breakup models
• Summary
Slide 2
Roland Schmehl
Introduction
Droplet breakup in premix zone
Sheet breakupDroplet deformation
Bag formationBag breakup
Subsequent breakup
Spherical droplet
Dispersion
1mm
Air assisted pressure swirl atomizer Liquid: Tetradecane D0 = 60µm We0 ≈ 15
Slide 3
Roland Schmehl
Basics
Deformation and breakup phenomena
Low relative velocities
Shape oscillations
Forced deformations
Moderate to high relative velocities (top to bottom)
Bag breakup
Bag-plume breakup
Plume-sheet breakup
Sheet-thinning breakup
Image source: Wiegand (1987) Image source & terminology: Guildenbecher (2009)
Slide 4
Roland Schmehl
Basics
Forces acting on and in the droplet
Aerodynamic forces Viscous forces
Inertial forcesSurface tension
x1
x2
x3
Od
vrel
Slide 5
Roland Schmehl
Basics
Model mechanisms
Moderate velocities High velocities Extremly high velocities
Transverse deformation by aero- dynamic pressure distribution
Superposed separation of liquid by aerodynamic shear forces
Superposed hydrodynamic instability of front surface
Slide 6
Roland Schmehl
Basics
Dimensional analysis
t∗v t∗
t∗σ
t∗µ
Non-dimensional numbers
We =ρ v2
rel D0
σWeber number
Redef =vrel D0
µd
√ρ ρd Reynolds number of deformational flow
On =µd√
ρd D0 σdOhnesorge number
=
√We
Redef
Characteristic times
t∗ =
√
ρd
ρ
D0
vrelPressure distribution ↔ Inertia forces
t∗σ =
√
D30 ρd
σSurface tension ↔ Inertia forces
t∗µ =µd
ρ v2rel
Pressure distribution ↔ Dissipation
t∗v =D0
vrelFlow around droplet
Slide 7
Roland Schmehl
Basics
Influence of aerodynamic loading
0 5 10 15 20 25 30 35 40 45 50 55 60 65 700
2
4
6
8
We
Tσ = t/t∗σ
Droplet in shock tube flowDroplet in free fallDroplet in premix moduleDroplet in rocket engine preflow
Slide 8
Roland Schmehl
Basics
Classification of numerical models
Empirical description
Correlations and similarity laws fordescription of droplet deformation.
+ Arbitrary deformations+ Simple implementation− Limited to specific loading scenarios− No dynamic response to flow variations> Limited use for modeling
Simple mechanistic models
Deformation kinematics reduced to singledegree of freedom: Droplet shapesapproximated by spheroids.
+ Specific small & large deformations+ Description of nonlinear effects+ Low computational effort− Fitting of empirical constants required> Suitable for modeling
Normal mode analysis
Modal discretization of aerodynamicpressure distribution, kinematics anddynamics of deformation.
+ Small but arbitrary deformations+ Low computational effort− Neglects nonlinear effects− Neglects effect of aerodynamic shear> Suitable for modeling
Direct numerical simulation
Spatial and temporal discretization of theNavier-Stokes equations in both phases.
+ Arbitrary deformation+ Includes all forces− Extremely high computational effort− Small scale processes problematic> Can not be used for modeling
Slide 9
Roland Schmehl
Empirical description
Load-based classification: On-We diagram
10-4 10-3 10-2 10-1 100 101 102 10310-1
100
101
102
103
We 0
On
Krzeczkowski (1980)Hsiang & Faeth (1995)
deformation < 5%
deformation 5-10%
deformation 10-20%
deformation> 20%shape oscillations
(oscillatory breakup)
bag breakupbag-plume breakup
transitional breakup
shear breakup Redef ,0 = 110100
ρd/ρ = 580−12000
Re0 = 240−16000
Slide 10
Roland Schmehl
Empirical description
Load-based classification: We-WeRe −0.5 diagram
Atmospheric data: Hinze (1955), Krzeczkowski (1980), Hsiang & Faeth (1995), Vieille (1995), Dai & Faeth (2001), Schmelz (2002).
Low pressure data: Zerf (1998). High pressure data: Vieille (1998).
10 20 30 40 50 60 70 80 90 100110
1
2
3
4
We0
We
Re−
0.5
bag breakupbag-plume breakuptransitional breakupshear breakup
p ≈ 0.1MPap≪ 0.1MPap > 0.1MPa
Ong =10−
2
10−3
10−4
Slide 11
Roland Schmehl
Empirical description
Temporal stages
Data source: Krzeczkowski (1980), Dai & Faeth (2001)
10 20 30 40 50 60 70 80 90 1000
1
2
3
4
5
6
7
8
We0
T=
t/t∗
bag
peripheral bags
breakup
breakup
breakup
ring
breakupplume
separation at equator
plume/core droplet
complex
larger fragments
plumebag
plume
transverse distortion + flattening
bumpligament formation at equator
Tb
Ti
Thmin
bag bag-plume plume-shear shear breakup
Slide 12
Roland Schmehl
Empirical description
Global secondary droplet properties
Root-normal distribution: f (x) =1
2σ√
2πxexp
−12
( √x−µσ
)2
withD0.5
D32= 1.2, µ = 1.0, σ = 0.22.
0 1 2 30
0.2
0.4
0.6
0.8
1
x = D/D0.5
V/V
0
bag breakupbag breakupF(x)We0=15
0 1 2 30
0.2
0.4
0.6
0.8
1
x = D/D0.5
V/V
0
bag-plume breakupbag-plume breakupF(x)We0=25
Data sources: Hsiang & Faeth (1992) and Chou & Faeth (1998).
Slide 13
Roland Schmehl
Empirical description
Global secondary droplet properties
Sauter diameter:D32
D0= 6.2 On0.5We−0.25
= Re−0.5def , On< 0.1 ,We0 < 103.
0 1 2 30
0.2
0.4
0.6
0.8
1
x = D/D0.5
V/V
0
transitional breakuptransitional breakupF(x)Fred(x)We0=40We0=70
0 1 2 30
0.2
0.4
0.6
0.8
1
x = D/D0.5
V/V
0
shear breakupshear breakupF(x)Fred(x)We0=125We0=250We0=375
Data sources: Hsiang & Faeth (1992) and Chou & Faeth (1998).
Slide 14
Roland Schmehl
Empirical description
Scondary droplet properties - differentiated by origin
10 20 30 40 50 600
0.2
0.4
0.6
0.8
1
ringplumebagcore
We0
V/V
0
10 20 30 40 50 600
0.2
0.4
0.6
0.8
1
ringplumecoreglobal
We0
D32/D
0
Data source: Dai & Faeth (2001)
Slide 15
Roland Schmehl
Normal Mode Analysis
Langrangian description of flow kinematics
r0d
r
r0
O0
vrel
U
ud
ud
v
u
g
P
t
Od
Slide 16
Roland Schmehl
Normal Mode Analysis
Linear normal mode decomposition
x2
x3
δr
θa,vrel
n = 0 n = 1 n = 2 n = 3 n = 4 n = 5
Decomposition of an arbitrary axisymmetric droplet shape into orthogonal deformation modes
Slide 17
Roland Schmehl
Normal Mode Analysis
Derivation of dynamic deformation equations
Linearized Navier-Stokes equations
1
r2
∂
∂r(r2vr)+
1r sinθ
∂
∂θ(vθ sinθ) = 0
ρd∂vr
∂t= −
∂p∂r+µd
[
∇2vr−2
r2vr−
2
r2sinθ
∂
∂θ(vθ sinθ)
]
+ρda cosθ
ρd∂vθ∂t= −
∂pr∂θ+µd
[
∇2vθ+2
r2
∂vr
r∂θ−
1
r2sinθvθ
]
−ρda sinθ
Series expansion ( v = ∇ψ)
ps− p∞ =ρv2
rel
2
∞∑
n=0
Cn Pn(cosθ)
p =ρdv2
rel
2
∞∑
n=0
βn
( rR
)nPn(cosθ)
δr = R∞∑
n=0
αn
( rR
)n−1Pn(cosθ)
Deformation equations ( n ≥ 2) formulated on nondimensional time scale Tσ = t/t∗σ
d2αn
dT2σ
+ 8n(n−1)Ondαn
dTσ+ 8n(n−1)(n+2)αn = −2nCnWe (Hinze 1948)
d2αn
dT2σ
+ 8(2n+1)(n−1)Ondαn
dTσ+ 8n(n−1)(n+2)αn = −2nCnWe (Isshiki 1959)
{ Viscous term different in both theories !
Slide 18
Roland Schmehl
Normal Mode Analysis
Pressure distribution on spherical surface
0 30 60 90 120 150 180-1.5
-1
-0.5
0
0.5
1
1.5
θ [◦]
p s−
p ∞ρ
v2 rel/
2
Re=50, Tomboulides und Orszag (2001)Re=100, Tomboulides und Orszag (2001)Re=500, Bagchi et al. (2001)Re=104, Constantinescu und Squires (2000)Re=1.62·105, Achenbach (1972)potential flow
Slide 19
Roland Schmehl
Normal Mode Analysis
Modal representation of pressure boundary condition
10-1 100 101 102 103 104 105-0.2
0
0.2
0.4
0.6
0.8
1
1.2
Re
Cn
C2
C3
C4
C5
flow separationwake asymmetricwake unsteadytransition lam.-turb.
rigid sphere,in uniform flow
Slide 20
Roland Schmehl
Normal Mode Analysis
Stationary deformation
Water droplet in vertical air flow (Pruppacher et al. 1970)
D0 [mm]: 8.00 7.35 5.80 5.30 3.45 2.70vrel [m/s]: 9.20 9.20 9.17 9.13 8.46 7.70We: 11.1 10.2 8.0 7.3 4.1 2.6Re: 4723 4340 3413 3105 1873 1334
Slide 21
Roland Schmehl
Normal Mode Analysis
Deformation from shock loading
Water droplet in horizontal shock tube flow: D0 = 1mm, On= 3.38·10−3
We: 1 5 12Redef : 296 662 1025Re: 491 1099 1701vrel: 7.7 m/s 17.2 m/s 26.68 m/s
Slide 22
Roland Schmehl
Nonlinear deformation analysis
Motivation
Linear analysis: First order theory { Forces and displacements at undeformed droplet
Nonlinear analysis: Second order theory { Forces and displacements dependend on deformation
Nonlinear phenomena:
• Mode coupling, excitation of higher modes
• Oscillation dynamics depends on amplitude (frequency and period)
• Nonlinear resonance effects
• Hydrodynamic instabilities
Slide 23
Roland Schmehl
Nonlinear deformation analysis
Kinematics and basic equations
x1
x2
x3
y
vrelx1
x2
x3
ys
ζ
10−1
ζ = 0 :∂x3
∂s= 1, vs = vs,max
ζ = ±1 :∂x3
∂s= 0, vs = 0
Dynamic equilibrium of mechanical energy contributions
ρd
2ddt
∫
Vv2dV + σ
dSdt= −
∮
Spsv · ndS −
∫
VΦdV =⇒ F(y, y, y) = 0
Slide 24
Roland Schmehl
Potential Theory Breakup model
Energy contributions
Potential energy of surface: polynomial approximation
σdSdt= σ
dSdy
dydt
,1
S 0
dSdy=
9.98y3−30.34y2+33.94y−13.58 , 0.5< y < 1 ,
0.67y3− 4.01y2+ 9.21y− 5.67 , 1≤ y < 2.3 .
Kinetic energy: viscous potential flow
ρd
2ddt
∫
Vv2dV =
85πρdR5
13
(
1+2
y6
)
d2y
dt2− 2
y7
(
dydt
)2
dydt
Viscous dissipation: viscous potential flow
∫
VΦdV = 12µd
∫
V
(
∂v2
∂x2
)2
dV = 16πR3µd
(
1y
dydt
)2
Slide 25
Roland Schmehl
Potential Theory Breakup model
Work performed by aerodynamic forces
Velocity potential on surface: stationary external flow
ψs =2
2−γ0vrel x3 , γ0 =
21− e2
e2
(
artanhee−1
)
, y < 1 ,
2
e2
(
1−√
1− e2 arcsinee
)
, y > 1 .
Pressure distribution on surface
ps− p∞ρ/2v2
rel
= 1−∆pmax
(
∂x3
∂s
)2
,
(
∂x3
∂s
)2
=1− ζ2
1− (1− y6)ζ2,
s : Surface coordinate
ζ : Non-dimensional axial coordinate
Total work performed by aerodynamic pressure forces
∮
Spsv · ndS = −
ρv2rel
22πR3
∆pmaxλ0
ydydt
, λ0 =
∫
+1
−1
1−4ζ2+3ζ4
1− (1− y6)ζ2dζ =
21− e2
e4
[
(3− e2)artanhe
e−3
]
, y < 1
21− e2
e4
[
3−2e2
√1− e2
arcsinee−3
]
, y > 1
Slide 26
Roland Schmehl
Potential Theory Breakup model
Nonlinearity of aerodynamic load term
0.5 1 1.5 20
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.5
1
1.5
2
y
λ0,λ
0/y,
C2λ
0/y
C2
C2 = 2/3∆pmax, potential theory
λ0, exactλ0/y, exact
C2 = 2/3∆pmax, CFD-simulationC2λ0/y, with C2 from CFD-simulationC2λ0/y, polynomial approximation
Slide 27
Roland Schmehl
Potential Theory Breakup model
Comparison of linear and nonlinear models
Taylor Analogy Breakup (TAB) model
d2y
dT2σ
+ 40Ondy
dTσ+ 64(y−1) = 2C2 We
Potential Theory Breakup (PTB) model
13
(
1+2
y6
)
d2y
dT2σ
−2
y7
(
dydTσ
)2
+ 40On1
y2
dydTσ
+ 201
S 0
dSdy=
154
C2λ0
yWe
Slide 28
Roland Schmehl
Motion of deformed droplets
Empirical description
Equation of motion
mddud
dt=π D2
8ρ cD vrel vrel + md g
Exposed cross section of droplet
π D2= π D2
0 y2
Aerodynamic drag coefficient
cD = f cD,sphere + (1− f )cD,disc
cD,sphere = 0.36+5.48Re−0.573+
24Re
, Re. 104
cD,disk = 1.1+64πRe
f = 1−E2
h
2y E =h2y
101 102 103 104
0.5
1
1.5
2
2.53
Re
c D
E-based interpolationSpheroid E = 0.5, exp.
E
0
0.25
0.5
0.751
Slide 29
Roland Schmehl
Motion of deformed droplets
Droplets falling into horizontal free jet flow
Wiegand (1987), experiment Normal Mode Analysis
Slide 30
Roland Schmehl
Motion of deformed droplets
Computed motion and deformation
-0.016
-0.014
-0.012
-0.01
-0.008
-0.006
-0.004
-0.002
0
y[m
]
Wiegand (1987)Versuch 17-3W
Slide 31
Roland Schmehl
Validation of deformation models
Effect of viscous damping on free shape oscillations
0 2 4 6 8 10 120.6
0.8
1
1.2
1.4
1.6
1.8
2√
2Tσ
E
Becker et al. (1994)
NMNLTAB3
a
bc
a: 0.00707b: 0.0707c: 0.707
On
Aspect ratio: E =1+α2
1− 12α2
, Nomal Mode Analysis
E =1
y3, Spheroid-based models
Slide 32
Roland Schmehl
Validation of deformation models
Effect of increasing amplitude on free shape oscillations
0 0.1 0.2 0.3 0.40.9
0.92
0.94
0.96
0.98
1
α2,0
ω/ω
0
NLTAB3 model:
On=0.0089On=0.0139On=0.0406On=0.0631
0 0.1 0.2 0.3 0.40.9
1
1.1
1.2
α2,0
∆Tσ/∆
Tσ,0
On=0.0089On=0.0139On=0.0406On=0.0631
prolatespheroid
oblatespheroid
Frequency shift Asymmetry of oscillation period
Slide 33
Roland Schmehl
Validation of deformation models
Stationary deformation of droplets in free fall
0 1 2 3 4 5 6 7 80.5
0.6
0.7
0.8
0.9
1
D0 [mm]
E∞
experimentcorrelationNMTABC2 = 4/3
C2
4/31.0
0.7
0 1 2 3 4 5 6 7 85
6
7
8
9
10
D0 [mm]
u d,∞
experimentcD,spherecD(Re,We)NM modelNLTAB3 mod.,C2 = 4/3
vrel prescribed coupled with droplet deformation
Slide 34
Roland Schmehl
Validation of deformation models
Maximum transverse distortion of droplets in shock tube flow
1 3 5 7 9 11 13 15 17 19211
1.2
1.4
1.6
1.8
2
2.2
20.5 4We0
y max
Hsiang & Faeth (1992) Exp.Hsiang & Faeth (1995) Exp.Temkin & Kim (1980) Exp.Dai & Faeth (2001) Exp.Haywood et al. (1994) CFDLeppinen et al. (1996) CFDHase (2002) CFD
TAB, C2 = 2/3TAB, C2 = 1.15TAB, C2 = 4/3NM, D0 = 1 mmNLTAB3 C2 = 1.15PTB C2 = 1.15PTB C2λ0/y, Polynom
yc
yM
Slide 35
Roland Schmehl
Validation of deformation models
Deformation of droplet in shock tube flow
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.11
1.05
1.1
1.15
t [s]
S/S
0
VOF method, Hase (2004)NM modelTAB modeloutput times
We
4.92
9.83
VOF method
NM model
Slide 36
Roland Schmehl
Validation of deformation models
Droplet falling through horizontal jet flow at We 0 = 0.5
0 0.01 0.02 0.03 0.04 0.05-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
xd
y d
Test 7-6WcD,spherecD(Re,We)NLTAB3, y0=1.0NLTAB3, y0=1.3NLTAB3, y0=1.5
We0 = 0.5
0 0.01 0.020.8
1
1.2
1.4
0 0.01 0.020.6
0.8
1
1.2
0.2
0.3
0.4
0.5
t
t
yc D
y0=1.3y0=1.0
y0=1.3:cDWe
We
Trajectories in laminar core flow Trajectory data NLTAB3 model
Slide 37
Roland Schmehl
Validation of deformation models
Droplet falling through horizontal jet flow at We 0 = 3.3
0 0.01 0.02 0.03 0.04 0.05-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
xd
y d
Test 10-1WcD,spherecD(Re,We)NLTAB3, y0=1.0NLTAB3, y0=1.3NM, y0=1.0NM, y0=1.3
We0 = 3.3
0 0.005 0.01 0.0150.8
1
1.2
1.4
0 0.005 0.01 0.015
0.5
0.6
0.7
1
2
3
4
t
t
yc D
y0=1.3y0=1.0
y0=1.3:cDWe
We
Trajectories in laminar core flow Trajectory data NLTAB3 model
Slide 38
Roland Schmehl
Validation of deformation models
Droplet falling through horizontal jet flow at We 0 = 11.8
0 0.01 0.02 0.03 0.04 0.05-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
xd
y d
Test 17-3WcD,spherecD(Re,We)NLTAB3, y0=1.0NLTAB3, y0=1.3NM, y0=1.0NM, y0=1.3NM, y0=1.5
We0 = 11.8
0 0.002 0.004 0.0060.8
1
1.2
1.4
1.6
1.8
2
0 0.002 0.004 0.0060.5
0.6
0.7
0.8
0.9
4
6
8
10
12
t
t
yc D
y0=1.3y0=1.0
y0=1.3:cDWe
We
Trajectories in laminar core flow Trajectory data NLTAB3 model
Slide 39
Roland Schmehl
Modeling of droplet breakup
Breakup criterion based on critical deformation
0 0.5 1 1.5 2 2.50.8
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
T = t/t∗0
y
Leppinen et al. (1996), We0 = 2, LLT-CKim (1977), We0 = 12.6Dai & Faeth (2001), We0 = 15Dai & Faeth (2001), We0 = 20y, NLTAB3y, empiricalymax, PTBymax, NLTAB3ymax, TAB
We0
yM
yc
disk- bag-
bulging expansiondydT = 0 dy
dT = 3.2
20
15
13
10
2
Slide 40
Roland Schmehl
Modeling of droplet breakup
Simulation of the On-We diagram
10-3 10-2 10-1 100 101 102
100
101
102
We 0
On
1.05
1.1
1.2
1.71.9ymax
multimode breakup We|y=1.8 = 30
shear breakup We|y=1.8 = 69
stability limitymax = 1.8
critical damping
PTB, cD = cD(Re,A)
Hsiang & Faeth (1995), Exp.Hsiang & Faeth (1992), Exp.
PTB, cD = cD(Re)
Slide 41
Roland Schmehl
Modeling of droplet breakup
Setup of modeling framework
10 20 30 40 50 60 70 80 90 100 0
1
2
3
4
5
6
7
8
We0
T
Dynamic deformation models
Dynamic boundary layer models
Empirical models
Slide 42
Roland Schmehl
Validation of droplet breakup
Size distribution computed from differentiated model
We0 = 15 (bag breakup) We0 = 125 (shear breakup)
0 1 2 30
0.2
0.4
0.6
0.8
1
x = D/D0.5
V/V
0
experimentexperiment
root-normalcomputed
0 1 2 30
0.2
0.4
0.6
0.8
1
x = D/D0.5
V/V
0
experimentroot-normalcomputed
Data sources: Hsiang & Faeth (1992) and Chou & Faeth (1998).
Slide 43
Roland Schmehl
Validation of droplet breakup
Single droplet falling in horizontal free jet
• Air & Ethanol
• We = 68.5, On = 0.0076, Re = 4362
• Software: OpenFOAM & Ladrop
• Dispersion model switched off
• Experimental data:Guildenbecher (2009)
Slide 44
Roland Schmehl
Validation of droplet breakup
Simulation using load-based breakup criterion
0
0
0
0
1010
1010
2020
2020
Slide 45
Roland Schmehl
Validation of droplet breakup
Simulation using deformation-based breakup criterion
0
0
0
0
1010
1010
2020
2020
Slide 46
Roland Schmehl
Validation of droplet breakup
Comparison of load- and deformation-based simulations
breakup for We> We0,c breakup for y > ymax
0
0
0
0
1010
1010
2020
2020
0
0
0
0
1010
1010
2020
2020
Slide 47
Roland Schmehl Summary
Analytical models for description of linear and nonlinear d eformation dynamics
Simple mechanistic approach for coupling of droplet deform ation and motion
Empirical stability criteria, classification and kinemati cs of breakup process
Systematic validation and assessment of models based on fun damental test cases
Future: Test modelling framework within practical Euler-L agrange simulations
Slide 48