Formation and characteristics of sprays from annular viscous liquid jet breakup

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Formation and Characteristics of Sprays

from A nnular Viscous Liquid Jet Breakup

by

Jihua Shea

B.Sc.. Southw estern Petroleum In stitu te, 1984 M.Sc.. University of New Brunswick. 1992

A D issertation S u b m itted in P artial Fulfillment of the R equirem ents for the Degree of

D O C T O R OF P H IL O S O P H Y

D epartm ent of Mechanical Engineering.

We accept this D issertation as conforming to th e required standard

Dr. X. Li. ^jipervisor (D ept, of M echanical Engineering)

Dr. S. Dost. D epartm ental M em ber (D ept, of Mechanical Engineering)

Dr. N. Djila ep^iLi^ieiltal M em ber (D ept, of M echanical Engineering) _____________________________________________ Dr. A. Weaver, O utside M em ber (School of E arth and Ocean Sciences)

Dr. P. G. Hill. E xternal Exam iner (U niversity of British Columbia)

© JiHUA Sh e n. 1997

U niversity of Victoria

All rights reserved. This D issertation may not be reproduced in whole or in part, by photocopy or oth er m eans, w ithout the permission of the author.

Supervisor; Dr. Xianguo Li

A bstract

T he form ation process and characteristics of sprays from an n u lar liquid je t breakup in m oving gas stream s have been inv estig ated . In the first p art of th e thesis, a linear instab ility ajiaiysis is carried out for th e instability and breaJcup of annular liquid je ts. A dispersion relation has been derived an d solved num erically by using M ullers m ethod. Temporal instability analysis shows th a t two independent unstable modes, para-sinuous and para-varicose, exist for th e an n u lar jet instability. The para-sinuous m ode outgrows the para-varicose o n e a t relatively low gas-liquid density ratios and large W eber num bers as typically en co u n tered in the twin-fluid atom ization. The curvature of the annular je t prom otes th e je t instability and m ay not be neglected for th e breakup processes of an n u la r liquid je ts. Not only th e velocity difference across each interface but also th e ab so lu te velocity of each fluid is im p o rtan t for the je t instability. Co-flowing gas a t high velocities is found to significantly improve atom ization performance.

A mesh-searching m ethod has b een developed to determ ine absolute m ode of instability. The num erical results in d ic a te th a t both absolute and convective insta­ bility exist for para-sinuous and para-varicose modes under certain flow conditions. Para-sinuous unstable waves outgrow para-varicose ones, and hence dom inate th e jet instab ility according to both absolute an d convective instability analysis. T h e liquid viscosity has a simple stabilizing effect on th e je t instability while the gcLS inertial force shows fairly complex influence on th e absolute instability of the je t. T he con­ vective growth rates for various inner gaa velocities indicate th a t not only th e velocity difference between, but also the a b so lu te velocity of the liquid an d gas. determ ine the je t breakup process.

In th e second part of this thesis, ex p erim en tal investigations have been conducted for th e breakup process of annular w ater je ts exposed to an inner air stream by pho­ tographic technique, and the ch aracteristics of th e resultant sprays by Phase Doppler P article Analyzer. Two annular nozzles of th e sam e stru ctu re b u t different dim en­ sions are designed and constructed especially to provide sm ooth contraction for the liquid flow. The test apparatus is co n stru c te d to produce th e annular liquid sheets or sprays of good quality.

Flow visualization reveals th a t th e re exist three regimes, i.e., bubble form ation, an n u lar je t formaion and ato m izatio n regim e for the jet breakup process. W ithin th e bubble formation regime, th e je t b reak u p characteristics m easured from th e pho­ tographs taken under various liquid a n d gas velocities show th a t uniform bubbles axe observed for various air-to-w ater velocity ratios. T he je t breakup and wave lengths

I l l

decrease w ith the air-to-w ater velocity ratio. T h e m easurem ents are com pared with th e predictions by the linear instability analysis, an d fair agreement is obtained.

Spray characteristics m easured by a Phase D oppler Particle .A.nalyzer indicate th at using atom izing air enhances th e jet breaJcup process and improves th e atom ization perform ance by producing fine sizes of droplets an d increasing the uniform ity of drop sizes. T he drop axial velocity has a je t-ty p e d istrib u tio n in the radial direction, and decreases m onotonically along th e spray axis. Increase in the water an d air velocities results in higher drop axial velocity. The droplet size described by its Sauter mean diam eter (SMD) reaches a m inim um value at th e central region of th e spray and increases towards the spray edge. The SMD has a com plex variation along the spray axis.

Exam iners:

Dr. X. Li<Sup/ervisor (D ept, of M echanical Engineering)

Dr. S. Dost. D epartm ental M ember (D ept, of M echanical Engineering)

Dr. N. Djilali. I ^ ^ r im e n ta l'M e m b e r (D ept, of M echanical Engineering)

Dr. .A. Weaver. O utside M ember (School of E a rth and Ocean Sciences)

IV

T a b le o f C o n te n ts

1.1 Scope and O b je c tiv e s ... 4

P A R T I: L I N E A R IN S T A B IL IT Y A N A L Y S IS 5

2 I n tr o d u c tio n 6

2.1 BéLsic C o n c e p t s ... 6 2.2 Small D isturbance T h e o r y ... 7 2.3 Linear Instability .A n aly sis... 8

3 L ite r a tu r e R e v ie w 12

3.1 Tem poral I n s t a b i l i t y ... 14 3.2 .Absolute and Convective I n s ta b ility ... 15

T A B L E O F C O N T E N T S v

4 L in ear I n s ta b ility A n a ly sis 18

4.1 Basic A s s u m p t i o n s ... 18

4.2 F o r m u la tio n ... 19

4.3 Dimensionless Forms of Dispersion R e la tio n s ... 25

4.3.1 A nnular Liquid Jets ... 25

4.3.2 A sym ptotic Relations for Limiting C a s e s ... 28

4.4 C o m putational .A lg o rith m ... 34 5 R e s u lts an d D is c u ss io n 36 5.1 Tem poral I n s t a b i l i t y ... 36 5.1.1 Effects of Geometric P a ra m e te rs ... 37 5.1.2 Effects of Flow P a ra m e te rs ... 49 5.2 .Absolute I n s ta b ility ... 60 5.3 Convective In s ta b ility ... 63 5.4 S u m m a r y ... 65 PA R T II: E X P E R IM E N T A L IN V E S T I G A T I O N 68 6 I n tr o d u c tio n 69 6.1 Basic C o n c e p t s ... 70

6.2 P hase Doppler Technique for Spray C h a r a c t e r i z a t i o n ... 72

7 L ite r a tu r e R e v ie w 75 7.1 Previous Experim ents on Cylindrical Jets and P lane S h e e t s ... 76

7.2 Previous E xperim ents on .Annular J e t s ... 79

8 E x p e r im e n ta l F a c ilitie s and T ech n iq u es 82 8.1 Design and S tru ctu re of N o z z l e ... 83

8.1.1 Nozzle Design ... 83

8.1.2 Nozzle S t r u c t u r e ... 83

8.2 S etup of Test A p p a r a t u s ... 85

8.3 In stru m e n ta tio n for M easurements ... 85

8.3.1 Photographic T e c h n iq u e ... 87

8.3.2 Particle Dynamics .Analyzer S y s te m ... 87

8.4 Test Procedure and Conditions ... 92

8.4.1 Flow V isualization P h o to g r a p h y ... 93

T A B L E OF C O N T E N T S vi

9 R e s u lts and D isc u ssio n 99

9.1 M echanism of J e t B r e a k u p ... 99

9.1.1 Flow R e g im e s ... 100

9.1.2 Je t Breakup Characteristics from P h o to g ra p h y ... 106

9.1.3 Predictions by Tem poral Instability .Analysis ( T .I ..A .) ... 108

9.1.4 Com parison of Mezisurements w ith T.I.A . R e s u l t s ... 112

9.2 Spray C h a ra c te ris tic s ... 115

9.2.1 S ym m etry of the Spray S t r u c t u r e ... 117

9.2.2 Spatial D istribution of the Drop Velocity and S i z e ... 119

9.2.3 Effects of the Liquid and Gas V elo cities... 125

9.2.4 U ncertainty of V le a s u re m e n ts ... 133

9.3 Sum m ary ... 137

10 C o n c lu sio n s 139 10.1 Linear Instability .A n a ly sis... 139

10.2 E xperim ental M e a su re m e n ts ... 141

11 R e c o m m e n d a tio n s for F u tu re W orks 144 R e fe r e n c e s 145 A C o m p a r iso n o f T h r e e I n s ta b ility M o d e s 153 B S u m m a r y o f P r e v io u s W orks 155 B .l Instability of .Annular Liquid J e t s ... 155

B.2 Experim ents for Sprays Formed by .Annular .Nozzles ... 156

C G e o m e tr ic a l C h a r a c te r istic s o f V arious J e t s 157 D M e sh -S e a r c h in g M e th o d 159 D .l I n tr o d u c tio n ... 159

D.2 Procedures to Find Pinch P o in ts ... 160

E S p e c ific a tio n s o f T e st F a c ilitie s 163 E .l Test .A p p a r a tu s ... 163

E.2 Photographic U n i t ... 163

E.3 P article Dynam ics .Analyzer System ( P D A ) ... 163

F O rig in a l E x p e r im e n ta l D a ta 167 F .l Je t B reakup C h a r a c te r is tic s ... 167

v i l

L ist o f T a b les

8.1 Main Dimensions of th e two A nnular Nozzles (U nit: m m ) ... 84

8.2 Working Fluids and C onditions for Photographic M easurements . . . 93

8.3 Working Conditions for P e n ta x -.\ 133 C a m e r a ... 93

8.4 W ater and .Air Flowrates Used for Photographic M easurements . . . . 94

8.5 Key P aram eters Set for PDA S y s t e m ... 95

8.6 Working Fluids and C onditions for PD.A. M ezisurem ents... 96

8.7 Test Conditions for PDA M easurem ents in two Orthogonal R adial Di­ rections ... 96

8.8 Test Conditions for PDA M easurem ents at Different Locations . . . . 97

8.9 Test Conditions for PD.A M easurem ents at Various Water and .A.ir Flowrates ... 98

9.1 Jet Breakup C haracteristics E stim ated from Photographic M easurem ents 108 9.2 Standard Deviation of M ulti-photo M e a su re m e n ts... 109

9.3 Dimensional P aram eters Corresponding to the Test Conditions . . . . 109

9.4 Dimensionless P aram eters for T.I..A. C a lc u la tio n s ... 110

9.5 Jet Breakup C haracteristics E stim ated from T .I.A ... I l l 9.6 Comparison of Photo M easurem ents with Temporal Instability .Analysis 111 9.7 Correlation Coefficient and U ncertainty for Drop .A.xial Mean Velocity 131 .A..1 Comparison of Three In stab ility M o d e s ... 154

B .l Previous Works on Tem poral Instability A n a l y s e s ... 155

B.2 Previous E xperim ental W orks on Jet Breakup ajid Spray Form ation . 156 C .l Geom etrical C haracteristics of Various J e t s ... 158

E .l Specifications for Test A p p a r a t u s ... 163

E.2 Specifications for Photographic Unit ... 165

L I S T OF T A B L E S viii

F .l M easurem ents from Magnified Images for Q( = 0.20 G P M (unit; mm) 168 F.2 M easurem ents from Magnified Images for Q( = 0.25 G P M (unit: mm) 170 F.3 M easurements from Magnified Images for Qi = 0.3 G P M (unit: mm) 171 F.4 M easurem ents from Magnified Images for Q/ = 0.4 G P M (unit: mm) 172 F.5 Interface Deformation E stim ated from Photographic M easurem ents 173 F.6 P D .\ M easurements in two O rthogonal Radial D ir e c tio n s ... 174 F.7 PD.A M easurements at Different Locations in the S p r a y ... 176 F.8 PD.A M easurements for Various W ater and .Air F lo w r a te s ... 181

IX

L ist o f F ig u res

4.1 Schem atic of an annular liquid jet exposed to inner and outer geis stream s 19 5.1 The cu rv atu re effects on the disturbance grow th ra te and am plitude

ratio for .A.. B para-varicose and C. D para-sinuous modes. We = 100. Æ = 1000./3 = 0.001. r/i = (rfc" — rg")/2 (hence rj, = -i-2) and Ta as show n... 40 5.2 T he effects of sheet curvature and Reynolds n u m b er on th e disturbance

grow th rate and am plitude ratio for A. para-sinuous and B. para- varicose modes. We = 1.025. Æ = 1000, p = 0.1. = (rf,* — ra*)/2 (hence rj, = - I - 2) and... as shown... 42

5.3 T he effects of outer radius rj on the disturbance grow th rate and am ­ plitude ratio for .A. B para-varicose and C. D para-sinuous modes.

We = 1000. Æ = 1000./J = 0.001. r,. = r^' (hence = 1) and as show n... 45 5.4 The effects of inner radius on the disturbance grow th rate and am ­

p litude ratio for A. B para-varicose and C. D para-sinuous modes.

We = 1000./fe = 1000./? = 0.001. r/i = r^," (hence = 1) and as show n... 48 5.5 Wave grow th rate for different velocities of inner gas stream with sta­

tionary o u te r gas medium, = 40.12. Æ = 4112. We = 19.25./? = 0.00129 and Ub = 0. .A: para-sinuous and B: para-varicose mode. . . . 51 5.6 Wave grow th rate of the para-sinuous mode for different velocities of

gas stream on either inner or outer side of th e je t. = 40.12, Æ = 4112. We = 19.25 and p = 0.00129. A: low and B: high gas velocity. . 53 5.7 Wave grow th rate of the para-sinuous mode for different velocities of

gas stream s on one side and both sides of the je t. = 40.12. Æ = 4112. We = 19.25 and p = 0.00129. A: high and B: low gas velocity. . 54 5.8 Effects of velocity difference between inner and o u te r gcis stream s on the

para-sinuous wave growth rate for r„ = 40.12, f t = 4112, We = 19.25 and p = 0.00129... 55

L I S T OF F I G U R E S x

5.9 Density effects on th e para-sinuous wave growth rate for a = 40.12. f k = 1000 and We = 1000. A: w ith velocity discontinuity. L'a = 0 and Ub = 1

and B: w ithout velocity discontinuity. Ua = Ub = 1... 56

5.10 Effects of liquid viscosity on the para-sinuous wave growth rate without velocity discontinuity, = 40.12. We = 500. p = 0.001 and L'a = Ub = 1. 57 5.11 Surface tension effects on the para-sinuous wave grow th rate for = 40.12. Æ = 1000 and p = 0.001. .\: without velocity discontinuity. L'a = f 6 = 1 and B: w ith velocity discontinuity. Ua = 0 and Ub = 1- 58 5.12 D om inant wave num ber for the para-sinuous mode w ith = 40.12 and Ub = 0. (a) Be = 4112. We = 19.25 and (b) p = 0.001. Ua = 2. . . 59

5.13 Critical W eber num ber w ith various flow param eters for para-varicose mode. U( = \ and = 0. (a) = 1. f a = 0 and p = 0.001: (b) Ta = 40 (rg, = Ta + 2) and Re = 100... 61

5.14 Critical W eber num ber w ith various flow param eters for para-sinuous mode. Va = 40 (r*, = -f 2). U( = 1 and L\ = 0. (a) p = 0.001: (b) Re = 1000... 63

5.15 Convective wave grow th rate at different velocities of the inner gas stream , = 40 (rj, = 4- 2). Ui = 1. Ub = 0. Æ = 1000. We = 100 and p = 0.001. (a) para-sinuous mode: (b) para-varicose m ode... 64

8.1 Sectional view of an n u lar n o z z l e ... 84

8.2 Experim ental setup for the form ation of liquid jets or s p r a y s ... 86

8.3 Schem atic of the PD.A s e t u p ... 90

8.4 Sampling positions for PD.A m easurem ents ... 91

9.1 Different phases of breakup for annular water jets-bubble formation. Nozzle dimensions: l.D .= 9.525 mm. O.D. = 10.000 mm: W ater velocity: 2.165 m /s: .Air velocity: 3.854 m /s ... 102

9.2 Different phases of breakup for annular w ater jets-an n u lar jet forma­ tion. Nozzle dimensions: I.D .=9.525 m m . O .D .= 10.000 mm: W ater velocity: 2.165 m /s: .Air velocity: 20.000 m /s ... 104

9.3 Different phases of breakup for annular w ater jets-atom ization. Nozzle dimensions: l.D .=9.525 m m . O .D .=10.000 mm: W ater velocity: 2.165 m /s: Air velocity: 45.000 m /s ... 105

9.4 Schem atics of three flow regimes for annular w ater j e t s ... 106

9.5 Schem atic of bubbles form ed after th e jet b r e a k u p ... 107

9.6 Bubble diam eter versus air-to-w ater velocity ratio. Nozzle mean diam ­ eter: 9.7625 m m . Dash line: experim ent; Solid line: linear theory. . . 113 9.7 Jet breakup length versus air-to-w ater velocity ratio. Nozzle mean

L I S T OF FIG UR ES xi

9.8 U nstable wave length versus air-to-water velocity ratio. Nozzle m ean

diam eter: 9.7625 m m . Dcish line: experim ent: Solid line: linear theory. 115 9.9 Examples of drop velocity and size h is to g r a m s ... 116 9.10 Drop m ean axial velocities along two orthogonal radial directions. Wa­

ter velocity: U(’ = 10.15 m / s . air velocity Ua’ = 220.61 m / s . axiaJ location c = 151.57 m m ... 118 9.11 Drop Sauter Mean D iam eter along two orthogonal radial directions.

W ater velocity U(’ = 10.15 m / s . air velocity L'a~ = 220.61 m / s . axial location z = 151.57 m m ... 119 9.12 Spatial distribution of drop mean axial velocity. W ater velocity L'F =

10.15 m / s . air velocity Ua' = 220.61 m / s ... 121 9.13 Self-similarity of drop mean axial velocity. W ater velocity Ue' =

10.15 m / s . air velocity Ua' = 220.61 m / s . Solid line: correlation for cylindrical jet flow: Symbols: PD.A. m easurem ents... 121 9.14 Self-similarity for the turbulent component of drop axial velocity. Wa­

ter velocity UF = 10.15 m / s . air velocity L'a' = 220.61 m / s ... 122 9.15 Spatial distribution of turbulent intensity. W ater velocity Ug' = 10.15 m / s .

air velocity L'a' = 220.61 m / s ... 123 9.16 Spatial distribution of drop Sauter Mean D iam eter. W ater velocity

U(' = 10.15 m / s . air velocity L'a' = 220.61 m / s ... 124 9.17 Spatial distribution of num ber density. W ater velocity = 10.15 m /s .

air velocity L'a' = 220.61 m / s ... 125 9.18 Effects of w ater and air velocities on drop mean axial velocity along

the spray axis... 127 9.19 Correlation for drop m ean axial velocity. Ug' = 5 .4 1 m /s: Solid line:

correlation Eq. (9.5): Symbol: m easurem ent... 129 9.20 Correlation for drop mean axial velocity. Ug' = 10.15m /s: Solid line:

correlation Eq. (9.5): Symbol: m easurem ent... 130 9.21 Correlation for drop mean ajcial velocity. Ug' = 15.17m /s: Solid line:

correlation Eq. (9.5): Symbol: m easurem ent... 130 9.22 Effects of w ater and air velocities on drop Sauter M ean D iam eter along

the spray axis... 132 9.23 Effects of w ater and air velocities on num ber density along the spray

axis. W ater velocity Ug' = 5.41 m /s ... 134 9.24 Effects of w ater ajid air velocities on num ber density along the spray

axis. W ater velocity Ug' = 10.15 m /s ... 134 9.25 Effects of w ater and air velocities on num ber density along the spray

axis. W ater velocity UF = 15.17 m / s ... 135 D .l Solution Domain w ith com putational Meshes in Ar-plane... 162

L I S T OF F I G U R E S xii

X l l l

N o m e n c la t u r e

d d iam eter of individual droplet {jJ-m)

Db bubble d iam eter (m m )

i im aginary num ber, i =

Iq, Ii modified Bessel function of the first kind of o rd e r 0 and 1

k dimensionless wave num ber, k = ar^

Kq. K\ modified Bessel function of the second kind of order 0 and 1

Lb je t breakup length (m)

Lu.- wave length (m ) m r m agnification ra te

p pressure change induced by p erturbation i N/ r n^ )

p initial am p litu d e of pressure p ( N/ m~)

P. P pressure of p ertu rb e d and base flow {N/rrP)

Ta dimensionless inner radius, Ta* inner radius of annular sheet (m) rj, dimensionless o u ter radius, rj =

rb‘ ou ter radius of annular sheet (m)

r/i reference length scale (m ). r/, = r^ '. r&' or (rj" — r a ') /2

rm m ean sheet radius (m ), = (To* + r(,‘ )/2

N O M E N C L A T U R E xiv

t tim e (s)

u change of velocity vector induced by pertu rb atio n ( m/ s ) Û initial am plitude of velocity u ( m/ s )

U . U velocity vector of perturbed and base flow ( m/ s )

U' axial velocity of base flow ( m/ s )

U dimensionless axial velocity of base flow. U = U’ /L'h'

W e W eber num ber. We = p e i \ ' ~ ^ h /

z C oordinate along flow axial direction (m)

Greek Symbol

Q wave num ber ( l / m )

e initial am plitude of disturbance (m)

T) displacem ent of interface from its initial equilibrium position (m)

6 phase difference

A wavelength (m ). A = 2~/a

p dynam ic viscosity ( k g / m ■ s) u kinem atic viscosity (m~/ s)

Pn density ( k g/ m^ ) . subscript n = g.C

p density ratio, p = p g/ p(

a surface tension ( N / m )

sjZ dimensionless wave frequency, u,- = Q r h . / i \ '

N O M E N C L A T U R E xv

Subscript

a for inner gas m edium or at inner interface

b for outer gais m edium or at outer interface

i for liquid je t

g for both inner and outer gas medium

h for reference length and velocity scale

I for im aginary part of a complex r for real part of a complex

XVI

Acknowledgements

I would like to express m y deepest g ratitude to my supervisor. Dr. X. Li. for his continuous support, guidance and encouragem ent throughout all of this work. 1 would also like to thank Dr. S. Dost. Dr. X. Djilali. Dr. .A. Weaver and Dr. P. G. Hill for serving on my thesis exam ination com m ittee.

The assistance from Mr. R. Katz in the installation of th e experim ental apparatus and from Mr. Jianm ing Cao in taking photograph for liquid je ts is highly appreciated.

The financial support from the University of Victoria and the N atural Sciences and Engineering Research Council of Canada is gratefully acknowledged.

X V II

To my dearest parents, supportive husband and lovely son who make my life full of happiness

C h a p te r 1

I n tr o d u c tio n

W hen a liquid is injected under pressure from a nozzle into a surrounding gas medium, a continuous liquid jet is form ed. Because of its inherent instability or its inability to sustain itself against even small perturbations, to which any physical system is subject, the liquid jet develops unstable waves, which am plify dow nstream , and even­ tually it disintegrates into a tra in of droplets. The process of the liquid jet breakup consists of two fundam ental steps. The first step is th a t the jet breaks up into liga­ m ents. The second is that th e ligam ents further disintegrate into fine droplets. This process of liquid jet breakup into ligam ents and then ligam ents into droplets of fine sizes is often referred to as liquid atom ization. The nozzle from which the liquid em anates is called atom izer, and the cluster of fine droplets so produced is usually term ed as a spray.

Sprays have wide applications in not only our daily life but also industries for decades [1. 2]. Hair sprays can be generated by sim ply forcing jelly through a fine orifice nozzle. .Agricultural spraying of herbicides, fungicides and insecticides is carried out on a huge scale in all countries using aircraft and tra cto rs. In m aterial processing industry, liquid m etal and ceram ic sprays are used to m anufacture tools, dyes, gear

C H A P T E R 1. I N T R O D U C T I O N 2

wheels and a wide vaxiety of objects with complex shapes. Instead of casting and machining m aterials, m aterial shapes are being formed by spraying layer upon layer of m aterials onto substrates. In food processing industry, spray drying is used very often to remove m oisture and produce dry packaged foods and powders. Sprays axe also widely used in pharm aceutical processes, and oral or nasal sprays are used by millions of people every day. In com bustion applications, liquid fuel sprays are necessary for the efficient an d effective com bustion of liquid fuels and th e control of pollutant emissions in power generation and propulsion systems.

In reality, alm ost every application has its own specific requirem ent of spray char­ acteristics. This makes th e design of atom izers and the organization of atom ization processes extrem ely im p o rtan t in order to produce sprays with the required specific spray angles, shapes and penetration as well as specified distributions of drop size, velocity, num ber density and liquid flux. T h ere are. baisically. three types of atom iza­ tion: pressure, rotary and twin-fluid atom ization. Pressure atom ization is achieved sim ply by forcing the pressurized liquid through an orifice, and a pressure atom izer is sim ple in construction and hence inexpensive as well. The disadvantage of pressure atom ization is its narrow operating range. This type of atom ization is extensively- used in our dafly life, such as hair and detergent sprays, and also in ind u stry such as in diesel engines, jet engines and ram jets. .A.S for a rotary atom ization, rotating and

speed controlling devices are required to generate the rotation motion for the liquid to be atomized. This type of atom ization is capable of handling slurries and often applied in spray drying and cooling systems.

The third category is so-called twin-fluid atom ization, that is th e focus of this thesis. In general, the relative motion between th e two fluids is utilized to disintegrate the liquid to be atom ized, and the other fluid can be a gas. its own vapor, or even another liquid which is im m iscible with the atom izing liquid. In practice, m ost

twin-C H A P T E R 1. I N T R O D U twin-C T I O N 3

fluid atom ization belongs to the air-assist or airblast atom ization. T h e m ain difference between th e two is th at the air-assist atom ization use relatively sm all quantities of air flowing at very high velocities (usually sonic), w hereas the airb last atom ization utilizes large am ounts of air flowing at much lower velocities. In com bustion system s, both air-assist and airblast atom ization are ideally su ited for dispersing liquid fuels, and the m ost common approach is that the liquid is first spread in to cylindrical or annular sheet called prefilming, and then exposed to high velocity air stream s on either one side or both sides of the sheet. .A.s a resu lt, an annular liquid sheet or jet is formed with two gas-liquid interfaces of finite radii of curvature. Such a liquid jet is un stab le against any disturbances, even infinitesim ally sm all ones, and the instability is m anifested through the onset and growth of unstable waves propagating dow nstream . When the am plitude of the unstable waves exceeds a certain critical value, the continuous liquid jet breaks up into discreet ligam ents and then individual droplets.

Twin-fluid atom ization is of significant fundam ental and practical im portance be­ cause of its extensive applications in pharm aceutical and chemical processing, spray drying operations, power generation and propulsion system s. T h e form ation and characteristics of sprays are strongly affected by the breakup process of th e annular liquid jets.

Therefore, the developm ent and growth of unstable waves on th e an n u lar liquid jet subject to inner a n d /o r o uter gas stream s are investigated in this thesis theoretically by using th e linear instability analysis, and experim entally by using th e photographic and video-graphic techniques. The resulting spray characteristics such as the distrib u ­ tion of droplet size, velocity, num ber density and liquid flux are m easured by Particle Dynamics A nalyzer (PD A ). The PD.A,. based on th e light scattering interferom etry and phcise Doppler principle, is a tim e-averaged, spatial-resolved an d non-intrusive

C H A P T E R 1. I N T R O D U C T I O N 4

instrum ent. Hence, it is a recognized state-of-art in stru m en t ideally suited for spray characterizat ion.

1.1

Scope and Objectives

The present work investigates interfacial instability of an annular viscous liquid jet subject to internal a n d /o r external gas stream s and th e characteristics of th e resulting sprays. In the first part of the thesis, a linear instab ility analysis of th e liquid jet is perform ed. T h e effects of various param eters such as finite curvature, liquid viscosity, surface tension, liquid and gas density on the je t breakup processes are examined qu an titativ ely by solving th e dispersion relations num erically.

In th e second part, an experim ental investigation is carried out for th e formation and the characteristics of sprays from th e annular liquid je t disintegration. Photo­ graphic technique is used to assess the je t breakup characteristics such as dom inant growth rate and wave length of the unstable surface waves and jet breakup length. T he m easured d a ta from th e photographs taken are com pared with the predictions by the linear instability theory. In order to stu d y atom ization mechanism of hollow-cone sprays form ed by twin-fluid atomizers, an Phase D oppler Particle .Analyzer is adopted to m easure drop sizes and velocities. Of special interest are spatial distributions of the drop axial velocity and size as well as the effects of liquid and gas flow rates on the drop velocity and size.

P A R T I: L I N E A R I N S T A B I L I T Y

A N A L Y S IS

C h a p te r 2

I n tr o d u c tio n

In this chapter, classic linear stab ility theory or small disturbance theory in hydrody­ namics is introduced starting from some basic concepts for the theory, m athem atical procedure involved in stability or instability analysis, and finally the dispersion rela­ tion which governs the characteristics of unstable wave evolution.

2.1

Basic Concepts

The hydrodynam ic equations, in spite of their complexity, allow some simple p attern s of flow as stationary solutions. However, these patterns of flow can be realized only for certain ranges of the param eters characterizing them . Outside these ranges, they cannot be observed in real hydrodynam ic system s because of their inherent instability, in another word, their inability to sustain themselves against small pertu rb atio n s to which any physical system is su bject. Then the question is whether th e perm issible p attern of flow is stable or not when it is disturbed, even slightly. There are three possible responses; the disturbance may gradually dam p such th a t the flow system

C H A P T E R 2. I N T R O D U C T I O N 7

returns to its original sta te : the disturbance may persist with sim ilar m agnitude or oscillate w ith time: the disturbance may grow in am plitude such th a t th e flow system progressively departs away from its initial sta te and never reverts to it. These th ree responses are named (asym ptotically) sta b le , n e u tra lly s ta b le , and u n sta b le , respectively. .A. system m u st be considered as unstable even if there is only o n e particular disturbance w ith respect to which it is unstable. On the other h and, a system can not be considered as stable unless it is stable with respect to e v e r y possible disturbance to which it can be subject. .Among these three classes of states, the neutral state which separates the stable from unstable states is also called m a r g in a l s ta t e and is always one of th e prim e objects of hydrodynam ic stability studies.

2.2

Small Disturbance Theory

Small D isturbance Theory is a theory used to study the instability of a hydrody­ nam ic flow system by applying small disturbances to the base flow and then observ­ ing whether the disturbed flow is stable, neutrally stable or unstable. For a given hydrodynam ic system , the base flow is first obtained from the governing equations of hydrodynam ics. Then the base flow is assum ed to be disturbed by small (infinitesi­ m al) disturbances, and the equations governing the disturbed flow are obtained from th e governing equations of hydrodynam ics. By linear stability theory, all term s which involve higher orders than th e first order of the disturbances are neglected, an d only the linear term s of the distu rb an ce are retained. For non-linear stability theory, the finite am plitudes of the disturbances are allowed. T he evolution of disturbances are then followed.

The m athem atical procedure involved in linear instability analysis is now alm ost stajidardized as listed below [3]:

C H A P T E R 2. I N T R O D U C T I O N 8

(a) Select a base flow

(b ) Add an infinitesim al disturbance to the base flow ( c ) Find th e disturbance equations

(d ) Linearize the obtained disturbance equations

(e) Solve the linearized equations by assuming a wave form solution (e.g. a traveling wave form)

(f) Solve for th e eigenvalue problem, i.e.. find th e dispersion relation

( g ) In terp ret the stability conditions and draw a chart showing th e neutral curves and

grow th rates, find the m axim um wave growth rate or dom inant wave num ber from the dispersion relation.

.As for the linear stability analysis for a steady simple base flow, the disturbance (or solution) is typically assumed in the form of norm al mode. i.e.. exp[z(oz — fit)]. This is because any disturbance can be resolved into independent com ponents or modes in the form of exp[f(oz — Of)], and for a linear system each mode can be treated separately. T he solutions obtained in this m anner are often called n o rm a l m o d e solutions.

2.3

Linear Instability Analysis

In linear instability ajialyses. the dispersion relations for liquid or gas jets are usually solved to determ ine the values of Q. and the spatial variations of corresponding wave com ponents q as eigenvalues or eigenfunctions. Here. and a could all be com plex

C H A P T E R 2. I N T R O D U C T I O .\ 9

or im aginary parts may vanish. Depending on the signs of Q. and q satisfying the dispersion relations, three modes of instability may be possible for liquid or gas jets, which are t e m p o r a l, c o n v e c tiv e and a b s o lu te instability.

Corresponding to the wave form. e.xp[f(Qc — Qf)j. a te m p o r a l in s ta b i l i t y mode would exists if a is real or q , = 0 and Q is complex w ith Q, > 0 representing the

tem poral growth rate of a certain disturbance (of wave num ber q^). In this case, the disturbance will grow exponentially with tim e until it is so large th a t nonlinearity becomes significant. The disturbance is. therefore, said to be u n s ta b le . If fl, = 0. it means th a t the disturbance stays as it is. and it is said to be n e u t r a l ly s t a ­ ble. However if fl, < 0. the disturbance is dam ped exponentially until its complete disappearance, which is said to be (asym ptotically) s ta b le .

Usually, there will be a critical value of th e wave num ber Or. which separates the unstable from the stable region of disturbance wavenumbers. That is D, = 0 at such a critical value of the wave num ber Qr often nam ed s ta b i l i t y lim it. W ithin the unstable region, there usually exists a particu lar mode of disturbance whose growth rate D, reaches a maximum. This mode is term ed the m ode of m axim um instability or fastest growing mode or d o m in a n t m o d e . It is this dom inant unstable wave mode th a t could actually be observed in reality if all modes of disturbances have com parable initial magnitude.

In practice, a disturbance will be not only one norm al mode, but usually some superposition of many normal modes determ ined by th e nature of th e initial distur­ bances. For an unstable system, a localized initial disturbance not only will grow, but also may propagate and spread, w ith each unstable component growing at its own ra te and moving at its own phase velocity. Generally, the space-tim e evolution of a localized initial disturbance in an unstable system could be classified into two physically distinct categories:

C H A P T E R 2. I N T R O D U C T I O N 10

(a ) th e disturbance can grow as its center propagates away from its origin, such th a t eventually at a fixed point in space the disturbance decays with tim e. T his is called c o n v e c tiv e in s ta b ility :

(b ) th e growing disturbance can engulf more and m ore of space as tim e goes on. such th a t eventually at every spatial location the disturbance grows. This is referred to as a b s o lu te in s ta b ility .

To determ ine w hether absolute an d /o r convective instabilities exist for a p articu lar system with disturbances in the form of .4q exp (i(QC — Of)], th e analysis usuaJly sta rts from th e dispersion relation. .According to Briggs [4] and Bers [5]. it is essential to search for pinch points (oo.flo) on the complex o-plane with qq., < 0 and f2o,, > 0

for absolute instabilities. Notice that one pinch point with fio.i > 0 for an absolute instability represents one normal mode, the norm al mode w ith the largest positive Qq., is th e majcimum instability mode which dom inates the tim e-asym ptotic response of the system .

T he pinch points (qq. flo) may have the other two possibilities. One is th at Qo.i = 0 stands for a critical s ta te which separates absolute from convective instability. T he other is flo.i < 0. m eaning that disturbances at every spatial location will die away w ith tim e, such th a t the original unperturbed s ta te retains. This indicates th a t there are no absolute instabilities. However, spatially growing waves or convective instability may exist. Since convectively unstable waves possess the character of a w a v e p a c k e t rather th a n a single wave, it is th e wavepacket th a t propagates and spreads out while it grows. The m ethod to find convective instabilities is to solve the dispersion relation for a(H r) with fl, = 0. If a , solved is of negative values, the disturbances grow and propagate through the space in the positive z direction, as it is usually set in the direction of jet flow originated from a nozzle, and vice versa.

C H A P T E R 2. I N T R O D U C T I O N 11

The differences am ong tem p o ral, absolute and convective instability modes are sum m arized in .\p p en d i.\ .A.

12

C h a p te r 3

L ite r a tu r e R e v ie w

The instability and breakup process of liquid jets have been the research subject in connection with liquid atom ization for m ore th an a century. Since Rayleigh [6. 7. 8] Ccirried out the first stability analysis for cylindrical liquid columns in vacuum, many studies have contributed to th e instability analyses of jets of various cross sectional shapes under different flow conditions (see .\p p en d ix C). Most of the earlier studies focused on tem poral instability analyses. For exam ple, for a cylindrical liquid jet. the first stab ility analysis by Rayleigh was tem poral one [6. 7. 8]. Then W eber [9], Sterling and S lei cher [10]. Lin and Kang [11] and m any others [12] also conducted tem poral analyses by taking into account th e effects of different param eters such as gas density, liquid velocity and viscosity. T h e spatial instability was first introduced to je t instability studies by Keller et al. [13] for a cylindrical liquid jet. although the idea of spatially amplifying disturbances has been known in m any other problems in hydrodynam ics [14. 15. 16. 17] and plasm a instab ility [4. 5].

The instability and breakup of annular liquid sheets, which axe often referred to as annular liquid jets as well, axe of significant scientific and practical im portance, and extensive studies have been conducted in the past in relation to the form ation of

C H A P T E R 3. L I T E R A T U R E R E V I E W 13

spherical shells [18. 19. 20. 21]. water bells [22. 23. 24] and acoustical barriers [25]. and only a lim ited num ber of studies are related to liquid atom ization and spray application [26. 27]. For a given liquid and annular nozzle, the annular je t converges, at relatively low velocities, to become an ordinary liquid jet some distance dow nstream of the nozzle prim arily due to the capillary effects. T he geom etric configuration and instability mechanism of annular jets [24. 28] and the related water bells [23] as well as com pound je ts [29. 30] have been studied in great details, including th e effects of gravity [23]. surface tension [24]. buoyancy [31] and pressure (or velocity) differences [22. 24] between the inner and outer gas regions.

At relatively high liquid velocities, unstable waves develop at the two interfaces of

an annular liquid jet. The growth of these waves eventually leads to the breakup of the jet into ligam ents and finally individual droplets. This process of the je t disintegration is often employed for the form ation of liquid sprays [26. 27]. One typical ty p e of sprays, hollow-cone sprays formed from conical liquid sheet disintegration, has been extensively used in practical applications, ranging from pharm aceutical and chemical processes, spray drying operations to power generation and propulsion system s. Such a conical liquid sheet involves three im portant characteristics. F irst, th e liquid sheet is very thin. Second, it has two interfaces with finite radius or curvature. .A.nd last its thickness changes with th e dow nstream distance. To fully understand the m echanism of hollow-cone spray form ation, the instability of such a conical liquid sheet has to be investigated. However, because of the com plexity of th e problem , such a liquid sheet has been conventionally m odeled by a constant (but very sm all) thickness with infinite radius, which is a plane liquid sheet case [32. 33. 34. 35, 36, 37. 38]. T he instability and breakup characteristics of thin and radically moving liquid sheets whose thickness reduces eis the distance from the nozzle increases have also been studied [39. 40]. In order to assess the effects of finite curvature on the sheet instability processes.

C H A P T E R 3. L I T E R A T U R E R E V I E W 14

annular liquid sheets with well-defined curvature need to be studied. This is one of the m otivations for the present work. not her m otivation of this work is th a t an annular liquid sheet can be regarded as a generalization including three well-known lim iting Ccises. a cylindrical liquid jet. th in planar liquid sheet and cylindrical géis jet (A ppendix C). The investigations on th e annular sheet instability is. therefore,

necessary and essential for the understanding of liquid jet disintegration processes.

3.1

Temporal Instability

.\ few studies have been carried out to investigate th e breakup process of annular liquid jets based on t e m p o r a l i n s ta b ility analyses. .As sum m arized in Table B .l. C rapper et al. [41] analyzed theoretically th e instability of an inviscid annular liquid sheet moving in an inviscid stationary gas m edium . T h e tem poral wave growth rates were obtained for two unstable wave modes, para-varicose (sym m etric), and para- sinuous (anti-sym m etric) with the approxim ation of very thin liquid sheets. Meyer and Weihs [42] investigated the capillary instability of a static viscous liquid sheet in a moving gas stream with a particular type of disturbances by assum ing the am plitude ratio of initial disturbances at the outer interface to th a t at the inner interface equal to the ratio of th e inner to o u ter radii of th e je t. The instability of a statio n ary viscous annular liquid sheet with unequal gas velocities for th e inner and outer gas stream s was form ulated by Lee and Chen [43]. Two dispersion relations corresponding to each interface were derived. However, only cases for inviscid liquids were theoretically exam ined in th eir study. Obviously, in th e previous works, either statio n ary liquid or gets was considered. However in twin-fluid atom ization, the velocity of air stream s on one side [43. 20] or both sides [44. 45] of liquid sheets is very im p o rtan t for the breakup process of the liquid sheets. Using high-velocity gas can prom ote th e breaJcup

C H A P T E R 3. L I T E R A T U R E R E V I E W 15

processes of th e liquid sheets and improve atom ization perform ance. On the other hand, since th e G alilean transform ation of coordinate often changes the characteristics of th e tem p o ral-sp atial evolution of unstable waves [5. 46]. th e absolute velocities of both liquid and gas should be taken into account.

Therefore as one part of this work, a tem poral in stab ility analysis is carried out for an an n u lar viscous liquid jet exposed to both inner and outer geis stream s of unequal velocities [47. 48]. The general forms of the dispersion relation and the equations for th e am plitude ratio of initial disturbances at the two interfaces are derived by considering absolute velocity of each flow. T h e effects of geometrical and flow p aram eters are examined based on num erical results ob tain ed from th e dispersion relations.

3.2

A bsolute and Convective Instability

T here are very lim ited previous works on absolute and convective instability analy­ ses for liquid je ts or sheets. Especially for annular liquid sheets or jets, there is no any work published so far. The first work by Keller et al. [13] waa for a cylindrical liquid je t. For th e first time. Keller et al. pointed out th a t th e tem poral instability theory im plies th a t the disturbance wave grows in am p litu d e everyw here along the je t. even in th e im m ediate neighborhood of a nozzle, which is contrary to the ex­

perim ental observations [33. 49. 35]. The instability and breakup of a liquid jet or sheet discharged from a nozzle was due to spatiaily ra th e r th a n tem porally growing disturbances. T hey also found th a t the tem poral and sp a tia l instability a t sufficiently large W eber num bers are related by C a ste r’s relation [50]. Leib and Goldstein studied th e absolute in stab ility of a cylindrical inviscid [51] and viscous liquid jet [52]. and showed th a t th e je t is absolutely unstable for W eber num bers below a certain critical

C H A P T E R 3. L I T E R A T U R E R E V I E W 16

value and is convectively unstable when the W eber number is above th e critical value. The effect of the am bient gas density on th e absolute instability of cylindrical liquid jets W cis studied by Lin and Lian [53]. Spatial mode of instability was also identified

for plane liquid sheets, as Lin et al. [54] did in their studies for a viscous liquid sheet in a stationary gas m edium . They reported th a t the sinuous mode of disturbances is neutrally stable below a critical Weber num ber of one . and in the sense of Briggs [4] and Bers [5] th ey term ed it as pseudo-absolute instability. For sinuous mode at W eber numbers higher th at the critical value of one and for varicose mode at any W eber num ber, convective instability exists in the system for a non-zero gas density. Li [37] further analyzed the problem and pointed out that liquid viscosity plays a duaJ role of stabilizing and destabilizing for sinuous mode at low W eber num bers, while for varicose mode and sinuous mode at higher W eber numbers, it is cdways stabilizing. The sam e problem has also been addressed by Ibrahim [55] and the subject has been reviewed by Li [37].

As a part of this work, absolute and convective instability are reported for an annular viscous liquid je t with its inner c in d o u te r sides exposed to inviscid gas streams

of unequal velocities [56]. .An efficient mesh-searching m ethod over the complex plane of wave num ber is used to determ ine the absolute modes of instability [57]. The effects of geom etrical and various flow param eters are exam ined. It is found that both absolute and convective instability exist for para-sinuous and -varicose modes under certain flow conditions. For para-sinuous mode, the annular liquid je t with an inner gas moving a t relatively small velocity can have convective or absolute instability depending on specific flow conditions. However, the jet héis only absolute instability if the inner gas is e ith e r stationary or moves a t sufficiently large velocity. Para-sinuous unstable waves outgrow para-varicose ones, and hence dom inate the je t instability according to both absolute and convective instability analysis. The liquid viscosity

C H A P T E R 3. L I T E R A T U R E R E V I E W 17

has a sim ple stabilizing effect on the je t instability while the gas inertial force shows fairly complex influence on the absolute instability of th e jet. T he convective growth rates for various in n er gas velocities indicate that not only th e velocity difference between, but also th e absolute velocity of the liquid and gas. determ ines th e jet breakup process.

18

C h a p te r 4

L inear I n s ta b ility A n a ly s is

An annular liquid jet is form ed by discharging liquid from an annular nozzle into surrounding gas stream s of unequal velocities as shown in F igure 4.1. The formed annular liquid je t is subjected to the influence of surface tension and the gaseous pressure difference between its two sides. The linear instability for the liquid jet is then studied by imposing sm all two-dimensional disturbances at the two interfaces based on Small D isturbance Theory as introduced in C hapter 2.

4.1

Basic Assum ptions

To simplify th e problem, th e following assum ptions are m ad e w ithout a loss of the main characteristics of the problem ;

(a ) Both liquid je t and ga.seous m edia are assum ed to be incom pressible since both velocities are presum ed to be small com pared to the velocity of the sound; (b ) Surrounding gas is inviscid. but liquid is viscous with co n stan t viscosity ///;

C H A P T E R 4. L I N E A R I N S T A B I L I T Y A N A L Y S I S 19

( c ) Fluid properties such as density pe and pg and surface tension <r are assum ed to be constant:

(d ) G ravity effect is neglected:

( e ) T he base liquid and gas flows are sem i-infinitely long and ajcisymmetric with different uniform velocities Uf'. L'a'• and Ub' for the liquid, inner gas and outer gas. respectively, in th e axial direction:

(f) Two-dimensional dispersive wave is supposed to propagate in th e axial direction.

Outer gas

► Inner gas

► Liquid jet

Figure 4.1: Schematic of an annular liquid jet exposed to inner and outer gcis stream s

4.2

Formulation

Figure 4.1 shows an sem i-infinitely long an n u lar liquid je t w ith inner radius r^" and outer radius r&". To derive th e dispersion relation and th e equation for the am plitude ratio of initial disturbances a t two gas-liquid interfaces, two-dim ensional infinitesim al disturbances are applied a t the two interfaces. Since th e gravity is neglected, the

C H A P T E R 4. L I N E A R I N S T A B I L I T Y A N A L Y S I S 20

pressure fields for the base flows are constant within the liquid and gets, respectively, and have a ju m p across the two gas-liquid interfaces, due to the effect of surface tension a. W hen disturbances develop in th e liquid je t. resulting in th e interface deform ation and deviation away from its equilibrium configuration, th e flow field is d isturbed with the perturbed flow velocity u and pressure p superim posed on the bcise flow velocity Ü and pressure P. Then in a cylindrical coordinate system (:. r. 9). the p erturbed flow fields become:

U„ = Ü „ -f u„. u„ = ( u„. r„. 0). - I - (4.1)

where the subscript n = (. a and 6 correspond to the liquid je t. the inner and outer gas stream s, respectively. T he base flow quantities are given by:

Ü , =

(tv.0.0).

Pf = Pa — crjrY = Ph + cr/rb’

The equations governing the motion of th e perturbed flow are the continuity equa­ tion for incompressible flow and balance of m om entum , which become, upon lineariza­ tion.

V • u„ = 0 (4.2)

+ = —V 4- V^u„ (4.3)

T he boundary conditions th a t the solutions of the above governing equations have to satisfy are the kinematic and dynamic conditions at the inner and o u te r interfaces, which are represented by r = r Y + r)a{:.t) and r = rb' + r)b{^~ f). respectively. In linear instability theory, these conditions need not be applied a t the distu rb ed liquid-gas interfaces. R ather, they can be linearized in th e same m anner as done above for the governing equations. Then th e linearized boundary conditions can be applied at the

C H A P T E R 4. L I N E A R I N S T A B I L I T Y A N A L Y S I S 21

unperturbed interfaces. Because the interfaces are m aterial surfaces, the kinem atic boundary conditions are

at r = r ~ and

at r = r/,'

The dynam ic condition implies th at th e shear stress must vanish at the interfaces because of th e inviscid assum ption for th e gas phase, and normal stresses across the interfaces m ust be continuous with allowance for the effect of surface tension cr. M athem atically, these dynam ic conditions can be expressed as follows:

IJ.( h ——^ = 0 (at r -=■ ra and r = rf, ) (4.6)

Pa - p e + = -cr (at r = r / ) (4.7)

P f , - p e A 2 p t ^ = + ( a t r = r6*) (4.S) Further, in th e am bient gas phase, th e effects of disturbances should physically remain bounded, w hether it is at the centerline or far away from the liquid jet. T hat is.

Ua and Pa bounded as r —*• 0 (4.9)

C H A P T E R 4. L I N E A R I N S T A B I L I T Y A N A L Y S I S 22

T he solutions to the governing equations are sought in term s of the norm al m ode in th e following form:

(Un.Pn-r]a.T]b) = [Û„ ( T ). ( T ). . Cfc] exp {z(oZ - Qt)} (4.11) where n = I . a and 6. and tf, are the am plitudes of initial disturbances at the inner and o u ter interfaces, and are regarded to be much sm aller than the inner and o u te r radius as well zis th e thickness of the annular liquid jet. The real part Qr of a is axial w avenum ber of th e disturbance and is related to the disturbance wavelength A by th e relation Q r = 2~/A . The im aginary part q , stands for the rate of grow th or decay

of th e disturbance through space. The imaginary part fl, of represents the rate of growth or decay of the disturbance with tim e, the real part fir is equal to 2~ tim es the d isturbance frequency, and —f lr /o represents the wave propagation velocity of the disturbance.

S u b stitu tin g Eq. (4.11) into the governing differential equations. Eqs. (4.2) and (4.3). yields the required general solutions with unknown integration constants which can be determ ined by using the boundary conditions. Eqs. (4.4). (4.5). (4.6) and two lim iting conditions Eqs. (4.9) and (4.10). The solutions are

=

Pa for the inner gas flow.

Ub Vb = Pb = lAocVa-(4.12] I \ ( ar^' ) eaPglo(ar) . _ ^>2 .(a.—nq A i ( a r6')

iebly{ar)^ b ’

C H A P T E R 4. L I N E A R I N S T A B I L I T Y A N A L Y S I S 23

for the o u ter gas flow, and

u/ = = [.4 i5 /o (5 r) — A2 S Ko ( S r ] + ,43Q/o(Qr) — .44QA'o(Qr)]e'*“*~^‘' r cfr

V ( ~ --- — z Q e ‘ ^“ ' ^ ^ * [ , 4 i / i ( 5 r ) + , 4 2 A i ( 5 r ) + . 4 3 / i ( a r ) + . 4 4 A i ( q t O | . 1 4 )

r oz

P ( — — p ( i o i L ( ' — n ) [ . 4 3 / o ( o r r ) — . 4 4 A o ( c k r ) ] e ‘*“ *

for the liquid flow, where

• 4 i =

■42 = •43 =

44 =

22Qf//[e6A'i(5rg*) - £gA i(5rj,')]

[Ii {Sra~)Ki {Srb’ ) - I i ( Srb' )Ki {Sra' ) ] 2iai/f[tqIi(Srb') -

I i { S r ^ ' ) Ki ( S r b " ) - Ii(Srb’ ) h \ ( S r a ' ) ll'dS'^ + Q^)[eqA'i(Qrj,~) — £6A'i(Qra‘ )]

a [ / i ( a r a ‘ )A'i(Qr(,-) - Ii(ctrb')K\(Qra')]

ii/dS^ 4- a^) [t bI i (arY) - Cg/i(Qrfe~)]

Q [/i(Q r„-)A 'i(Q n*) - /i(o ri* )A 'i(Q ra* )]‘

S u b stitu tin g the above solutions of each flow. Eqs. (4.12). (4.13) and (4.14). into the dynam ic boundary conditions at th e two interfaces. Eqs. (4.7) and (4.8). leads to two equations with two unknowns Ca and £>,• Since these two equations are ho­ mogeneous. the coefflcients of the equations must satisfy a condition for non-trivial solutions to exist. T h at is the d eterm in a n t of the coefficient m atrix must vanish, which gives the following dispersion relation between f2 and a.

((5^ -f q")"A4Aj - 4Q^5A3Ae - - ^ ( a L Y - Q)

I ra’ l'l

+ - P g l o ( a r Y )

Pi h ( a r a ' ) { a i V - ^ Ÿ 2ia

X < (5"^ 4- )^A4A2 — 4q^SA3A5 H {ciL’t — LI) rb vi + -vi “ Oc<7 ^ 1 pe r& pe K'i(arb') a

1

C H A P T E R 4. L I N E A R I N S T A B I L I T Y A N A L Y S I S 24

T h e relation betw een the wave am plitudes at the inner and outer surface, as the part of the solution, is expressed cis:

tf, aG^ (4.16) G3 = + a 2ia ( j4 = ( A ct — (o f f — Cl) — 4o ra'l'f + ^ ( - L - a " ) + ("(“ ’•'■'’ ( o r . - - n i^ 6 Or. (4.17)

Pii^r rg"' p(i/p IiiaVa

— — ta bGe Ct Gq = ( + Q*)^A4A2 H—~ — (o f / — n ) — 4q^5A3A5

w

- n)=

where, the dim ensional variables are q and Cl, wave num ber and frequency: i ' Y - I a'

and L'f)~• the velocities of liquid je t. inner gas and outer gas streams: p( and Pg. liquid and gas density: and r ^ . inner and outer radius of two interfaces: cr. surface tension of liquid an d gas: U(. kinem atic liquid viscosity: 5 = ; / and A’, modified first an d second kind Bessel functions: and

A i = I o( ar Y) Ki { ct r C) + Ko{ara‘ )Ii{ocri,'): A2 = Io{ari,' )Ki{ara' ) + KQ(arb')I\{Qra'): A3 = [h{Sra' )I<i{Srb’ ) - h i S r O K i i S r Y ) ] - ' : A4 = [/i(Q ra*)A 'i(or6") - /i(Q r6")A 'i(o ra ’ )]"^ As = Io{Srf,’ ) Ki ( S r a ' ) + A o(5r6”) /i ( 5 r a ‘ ): As = /o (5 r,-)A 'i(5 r6 * ) + A o (5 r,-)/i(5 r6 * ):

C H A P T E R 4. L I N E A R I N S T A B I L I T Y A N A L Y S I S 25

4.3

Dimensionless Forms o f Dispersion Relations

The dim ensional dispersion relation. Eq. (4.15). and in itial am p litu d e ratios at two gas-liquid interfaces. Eqs. (4.16) and (4.17). for annular liquid jets are necessary to be in non-dim ensional forms for further num erical analyses. T he dim ensionless forms are presented in the following section. .A.s a partial check for the equations derived above, the dimensionless forms of dispersion relations for three lim iting case (see •■\ppendix C) of plane liquid sheets, cylindrical gas jets an d cylindrical liquid jets are retrieved from th a t for annular liquid jets and also presented below.

4 .3 .1

A n n u la r L iquid J e ts

T he dim ensional variables in Eq.(4.15) are non-dim ensionalized as follows:

(a ) Length scale is denoted by r/,. which could be one of three choices- half sheet thickness (rj,* — rY)/'2. inner r Y and o u te r radius r^" of an n u lar sheet

(b ) Velocity scale is denoted by L’h'- which could be one of three choices- liquid jet

i / ‘ . inner L'a' and outer gas stream velocity t V

(c) T im e (t) is normalized by [rh/L'h') (d ) Wave num ber k = avh

(e) Reynolds num ber. Æ = Uh'Thli'i (f) W eber num ber. We = pei'k’'rh.l(T (g) s =

Srh

iRt{kUt

C H A P T E R 4. L I N E A R I N S T A B I L I T Y A N A L Y S I S 26

(i) The geom etric sizes, ajid are norm alized by r^. i.e. = r ^ '/ r ^ . r& =

rb'/rh

(j) Density ratio , p =

p^jpt-S ubstituting th e above non-dim ensional variables into Eq. (4.15) and tim es both sides of the equation by we have:

is~ fc^)

4k^

N-

N

The am p litu d e ratios of th e initial disturbances. Eqs. 4.16 and 4.17. at the inner and outer interfaces become:

or | ( s ^ - I - A '2) ^ A i A 4 — 4A ^ s A 3A e 4- B a | 4 P A3 — (s^ 4- k’^YN.^/k (4.19)

Tb

k^)

Bb'^

'■Sîfeî