Deformation and Secondary Atomization of Droplets in Technical ...

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 aerodynamic 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


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


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


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


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


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


Slide 14

Roland Schmehl


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


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


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


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


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


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


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


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


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


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


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


Droplets falling into horizontal free jet flow

Wiegand (1987), experiment Normal Mode Analysis

Slide 30

Roland Schmehl


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


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


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


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


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


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



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


Slide 38

Roland Schmehl



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


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


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


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


Slide 43

Roland Schmehl


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


Simulation using load-based breakup criterion

0

0

0

0

1010

1010

2020

2020

Slide 45

Roland Schmehl


Simulation using deformation-based breakup criterion

0

0

0

0

1010

1010

2020

2020

Slide 46

Roland Schmehl


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

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