ATOMIZAÇÃO_phd6

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Chapter 6

DYNAMIC BREAKUP OF LIQUID-LIQUID JETS

Many, it may even be said, most of the still unexplained phenomenaof Acoustics are connected with the instability of jets of fluid. Forthis instability there are two causes; the first is operative in thecase of jets of heavy liquids, e.g., water, projected into air (whoserelative density is negligible), and has been investigated by Plateauin his admirable researches on the figures of a liquid mass, withdrawnfrom the action of gravity. It consists in the operation of the capillaryforce, whose effect is to render the infinite cylinder an unstableform of equilibrium and to favour its disintegration into detachedmasses whose aggregate surface is less than that of the cylinder.The other cause of instability, which is operative even when the jetand its environment are of the same material, is of a more dynamicalcharacter.

Lord Rayleigh, On the instability of jets(1879)

6.1 Background

The liquid-liquid jets discussed in chapter 5 eventually break up due to

increasing amplitude of disturbance waves on their surface. In this chapter we in-

vestigate the length of the resulting jets, which is dependent on many factors. The

axisymmetric, dynamic breakup of a Newtonian liquid jet injected vertically into

another immiscible Newtonian liquid at various Reynolds numbers is investigated.

When a liquid is injected into another liquid at a velocity above a certain critical

value, a jet forms, rises to a certain length, and then breaks up into drops (Meister,

1966; Meister and Scheele, 1967; Scheele and Meister, 1968; Meister and Scheele,

1969a, 1969b; Richards, 1978; Richards and Scheele, 1985; Richards et al., 1993;

Richards et al., 1994a). These drops result in the creation of large new surface

area, which leads to enhanced heat and mass transfer (Skelland and Walker, 1989).

107


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This has immediate implications for design of, for example, sieve-tray liquid-liquid

extractors. The prediction of the jet dynamics is thus important as a means to

explore both quantitative aspects of performance, e.g., to calculate the size of the

drops formed, and qualitative features, such as a test of stability theories.

The technique applied to steady laminar jets as described in chapter 5

has been slightly modified to represent better the physical stability limits of the

system, and is used in the present chapter to predict jet length and subsequent

breakup into drops.

Jet breakup has been extensively studied, both experimentally and theo-

retically, for jets injected into air with the air modelled as a vacuum or an inviscid

fluid; a brief review up to 1990 is given by Mansour and Lundgren (1990). Theo-retical developments started with the elegant analysis of Rayleigh (1879) for the

case of an infinite free inviscid jet injected into air. In the liquid-liquid jet problem

additional, non-trivial, effects enter due to the presence of an outside continuous

phase. The Rayleigh stability analysis was extended by Tomotika (1935) to a

viscous cylinder surrounded by a viscous fluid. However, he solved the general

eigenvalue relation only for the limiting case of a very viscous liquid in another

very viscous liquid. The general eigenvalue equation was first solved numerically

by Meister and Scheele (1967) and covered all the previous special cases such as a

viscous liquid jet in a gas by Weber (1931) and an inviscid liquid jet in an inviscid

liquid by Christiansen and Hixson (1955, 1957).

In a series of publications, Meister and Scheele (Meister, 1966; Meister and

Scheele, 1967; Scheele and Meister, 1968; Meister and Scheele, 1969a, 1969b)

worked to develop an understanding of the jet and drop formation based on

experiments obtained with 15 liquid-liquid systems. They described the general

behavior of the jets as follows. For low flow rates drops form, grow, and break

off from the nozzle at regular intervals. Above a certain critical velocity a jet

is formed that rises to a certain length from the nozzle at which point it breaks

up into drops. At still higher nozzle velocities, the jet disrupts into small drops.


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Much of the earlier literature on liquid-liquid jet breakup has been reviewed by

Meister and Scheele (1969a) and more recently by Skelland and Walker (1989) (up

to 1989).

Meister and Scheele (1969a), using the Tomotika (1935) stability equation,

developed an expression for the jet length at breakup that was an improvement

over that used by Smith and Moss (1917). This equation involved a Galilean

translation of the infinite Tomotika stationary jet at an average velocity of the

interface, calculated by assuming a certain form for the velocity profile inside the

jet and by solving for the unknown coefficients invoking overall momentum and

mass balances. We believe that this theory, although developed in the late 60s,

still represents the best one available for engineering design estimates of the jetlength.

For liquid jets injected into air, Sterling and Sleicher (1975) included in

the analysis the aerodynamic interaction between the jet and the air and found

that the presence of the air led to an enhanced growth rate of the axisymmetric

disturbances. In particular, they examined the case of a viscous liquid injected

into a stationary inviscid medium, and they developed an equation for the growth

rate which was an improvement over the one used by Weber (1931). Several

attempts have been made to include in the stability analysis the relative motion

of the continuous phase in liquid-liquid systems, the most recent being those of

Kitamura et al. (1982), Bright (1985), and Russo and Steen (1989). Kitamura et

al. (1982) experimentally varied the motion of the continuous phase to be either

faster, the same as, or slower than the jet down to the case of a stagnant continuous

phase. They found that the jet shortened as the absolute value of the continuous

phase velocity relative to the jet increased from zero. Moreover, the condition

of a zero relative phase velocity was found to give the best agreement between

Tomotikas stationary jet solution (translated at the average velocity of the jet) and

the experimental measurements. Thus, they attributed the discrepancy between

experiments and Tomotikas analysis in the nonzero relative velocity cases to the


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relative motion of the continuous phase, rather than to other factors, such as, for

example, a velocity profile relaxation.

Bright (1985) attempted to perform a linear viscous stability analysis as-

suming a constant, but unequal, velocity in each liquid phase. One difficulty with

this study rests on the inconsistency of having a discontinuity of velocity at the

interface for fluids that are not assumed to be inviscid. Even assuming that the

equations are formulated correctly, we have found that the particular solution pro-

vided by Bright (a nonhomogeneous modified Bessel ODE) is incorrect; its correct

form is much more complex than the one provided by his equation (18). Details

of the derivation can be found in appendix H.

Russo and Steen (1989), in a study on liquid bridge stability, performednumerically a linear stability analysis for an infinite cylinder of viscous fluid with

a constant shear applied to the outer free surface. From their predicted growth

rates, they found qualitative agreement with the data of Meister and Scheele

(1969a), but they noted that they could not expect quantitative agreement since

they neglected not only the relaxation of the velocity profile along the jet, but also

the possibility of spatially growing instabilities (Russo and Steen, 1989). Linear

stability analyses can be further characterized as temporal or spatial. Temporal

stability analyses (e.g., Tomotika, 1935; Christiansen and Hixson, 1957; Mansour

and Lundgren, 1990) examine how spatially periodic disturbances grow in time. In

contrast, spatial stability studies assume that disturbances starting at the nozzle

tip are temporally periodic waves that travel spatially with the jet (Keller et al.,

1973; Busker et al., 1989; Russo and Steen, 1989). The most severe assumption

is that of a uniform (at least in the axial direction) base flow. Furthermore,

note that the usefulness of eigenvalue analysis, through linear stability analysis,

for a non-self-adjoint system of equations (e.g., the Navier-Stokes equations) has

been recently questioned as far as its relevance to the time evolution of finite

amplitude disturbances is concerned, by the discovery of solutions to the subcritical

linearized problem of Poiseuille and Couette flow that grow greatly even though


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all eigenmodes decay monotonically (Trefethen et al., 1993).

Thus, all the previous liquid-liquid stability theories, as based on rather

drastic simplifying assumptions, have limited validity. As was observed by Meister

and Scheele (1967) regarding previous stability analyses, the agreement between

experimental results and theoretical instability analyses has been good when

experiments are performed which closely approximate the assumptions made in

the theoretical development. The difficulties are associated with the various

effects (such as viscous, buoyancy, surface tension, inertial forces, jet contraction,

velocity profile relaxation, and relative motion of the continuous phase) that are

not all fully accounted for in any of the previous theories. This lack of an adequate

theoretical description which is driven by the inherent complexity of the problemled us to the present direct numerical simulation of the jet breakup process.

The only assumptions involved with the present approach are those of laminar

flow of the two Newtonian fluids with constant bulk densities and viscosities,

constant interfacial surface tension, and axisymmetric disturbances. Of these the

latter assumption has been shown experimentally to fail in the region above a

certain critical velocity where the jet lengthens appreciably to a maximum length

(Meister and Scheele, 1969a). However, in the region below this critical velocity,

experimental observation (Meister and Scheele, 1969a) shows that axisymmetric

disturbances dominate, and in consequence, this is the region of validity of the

present study (see Figure 7.1 which is Figure 1 of Meister and Scheele, 1969a).

The motivation for selecting the VOF method in this numerical study has

been elaborated previously in chapters 1 and 3 (Hirt, 1968; Nichols et al., 1980;

Hirt and Nichols, 1981; Kothe et al., 1991; Brackbill et al., 1992; Richards et al.,

1993). Primarily, the present numerical algorithm can capture the irregular shapes

of free surfaces, such as those that are realized during necking and detachment of

drops from the jet; this has not been possible with other methods devised for single

phase jets without considerable modification of the algorithms. For example,

Mansour and Lundgren (1990) followed the breakup of an initially stationary


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cylinder of inviscid liquid using a nonlinear boundary element method. They were

able to follow the solution until necking just before breakup; then, in a second stage

of calculation, they cut the forming satellite drop at the throat and continued the

development of the satellite shape. Boundary element techniques could be used,

but they would involve a restart after every necking-drop detachment and a rather

elaborate remeshing to capture the inertial effects. In contrast, the VOF method

allows very naturally arbitrary jet and drop shapes to be calculated.

The objectives of the present study are to use the VOF based numerical

method that we have developed for calculating liquid-liquid jet dynamics, from

the startup of their time evolution to their apparent (or pseudo) steady-state

involving the region from the nozzle exit to their breakup through necking anddetachment of drops. Our aim is to predict the jet length and the shape, as well as

the subsequent shape and size of drops that are formed. We compare the results

with the available experimental data of Meister and Scheele (1969a) and Bright

(1985) and the available linear stability theory of Meister and Scheele (1969a). We

also show that quantitative agreement of the present model is obtained with the

dynamic linear initial-value problem of Christiansen and Hixson (1955, 1957).

In section 6.2 we present the definition and formulation of the governing

equations for the solution to the problem of a Newtonian liquid jet injected verti-

cally into another Newtonian quiescent liquid. The analytical solution to the linear

inviscid liquid-liquid cylinder breakup problem is illustrated, and the stability anal-

yses of Tomotika (1935) and Meister and Scheele (1969a) for viscous liquid-liquid

jets are discussed. In section 6.3 we then discuss the numerical implementation

of the governing equations to the liquid-liquid jet problem. In section 6.4 the

numerical simulation is compared with the inviscid liquid-liquid cylinder breakup

solution. The present numerical simulation of free surface dynamics and jet lengths

is compared to the existing experimental data (Meister and Scheele, 1969a; Bright,

1985) and the Meister and Scheele (1969a) stability analysis. Finally, in section

6.5 we reach conclusions based on the results of the present work.


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6.2 Problem Definition and Formulation

The flow configuration investigated here is shown in Figure 6.1. It corre-

sponds to the experimental setup of Meister and Scheele (1966, 1969a) and Bright

(1985), details of which can be found in those references. Briefly, the problem

investigated involves the jet of an alkane (either n-heptane or n-decane) injected

vertically from a circular nozzle upwards into a tank of stationary mutually sat-

urated immiscible water. Figure 6.1 shows the stationary tank of fluid 1 (water)

with density 1 and viscosity 1 with the jet of fluid 2 (alkane) flowing upward

with density2 and viscosity 2, from a nozzle of inner radiusR and outer radius

Ro with average velocity v. The constant interfacial surface tension is , and the

gravitational acceleration, g, is directed downward. The distance from the nozzle

tip to the top of the tank is L1 and to the bottom is L2. The distance from the

centerline of the nozzle to the outer tank wall is L3and the nozzle is of total length

L4 with the velocity profile fully developed at length L5 from the nozzle tip. Dis-

turbances originating in the nozzle propagate with the jet flow and eventually lead

to breakup of the jet, with the length of the jet given by L, and the wavelength

of the growing disturbance given by .

6.2.1 Governing Equations

Equations (3.1) to (3.13) are solved with appropriate boundary conditions

as discussed in section 5.2.1. For the nozzle and the bottom of the tank no-slip

conditions are used:

u= v = 0 (6.1)

For the axis of symmetry at r = 0:

u= 0, v

r = 0 (6.2)


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Nozzle

rz

n

Axis of

Symmetry

ContinuousFluid 1F = 0

1, 1

DispersedFluid 2F = 1

2, 2

JetInterface

r=a(z)

Tank Wall

L3

g

RInflow

Boundaryv

L1

L2

L4

Ro

Outflow

Boundary

L

L5

Figure 6.1: Liquid-liquid jet flow configuration (not to scale).


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For the inflow into the nozzle, at distance L5 from the nozzle tip, fully developed

Poiseuille flow is assumed:

u= 0, v= 2v

1

r

2 (6.3)

wherer r/R is the dimensionless radial distance, and v is the average velocity

in the nozzle. Dimensionless variables in this work are denoted with an asterisk

using R as the characteristic length and v as the characteristic velocity. The

experimental nozzle length ratio L4/R was assumed to be sufficiently large to

guarantee condition (6.3) (Meister and Scheele, 1969a). In the VOF approach,

fluid is allowed to flow through the mesh with a minimum of upstream influence.

For the outflow boundary at the top of the mesh, z = L1, it is assumed that there

is no change in the axial direction (Nichols et al., 1980; Hirt and Nichols, 1981;

Richards et al., 1993):u

z =

v

z = 0 (6.4)

To allow for the presence of a lateral wall far away from the nozzle

(L3/R 1) without increasing the computational grid size excessively, it is as-

sumed that there is no change along the radial direction at the outer boundary of

the computational grid at r = L3, L3 < L3:

u

r =

v

r = 0 (6.5)

The numerical implications of using the boundary condition (6.5) to represent the

outer lateral wall (as opposed to a no-slip condition used in the previous work,

Richards et al., 1993 ) are discussed in section 6.3. The jet interface is assumed to

be pinned at the nozzle lip, a fact that is observed experimentally, so no contact

angle needs to be specified in this problem.

6.2.2 Inviscid Free Liquid-Liquid Cylinder Breakup

To test the accuracy and performance of our method on the dynamics of

jet breakup, it was first applied to the problem of the capillary breakup of an


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initially stationary infinitely long inviscid liquid cylinder inside another inviscid

liquid within an outer free-slip wall. This problem was chosen because it has a

closed form analytical solution which was first derived for arbitrary values of the

initial conditions by Christiansen and Hixson (1955, 1957). The detailed derivation

is given in appendix F.1. We use the general solution of Christiansen and Hixson

(1957) to solve the initial value problem with initial conditionsa =R + a0 cos(kz)

and da/dt = 0 at t = 0, where a0 is the initial amplitude, s is the growth rate,

andk = 2/is the axial wavenumber of the disturbance. The particular solution

expressed in terms of the velocity potential i, where vi = i, i = 1, 2 in both

phases becomes:

a= R + a0 cosh(st)cos(kz) (6.6)

1 = a0 s

k

K0 (kr) + I0 (kr)K1 () I1 ()

sinh (st)cos(kz) (6.7)

2 =a0 s

k

I0 (kr)I1 ()

sinh (st)cos(kz) (6.8)

where:

s2 =

1 2

R3

2

I0()I1()

+ 1

K0() + I0()K1() I1()

(6.9) K1 ()/I1 (), kR, and kL3. Note that this equation reduces toRayleighs (1879) particular result for an inviscid liquid in a gas if the density of

the outer fluid is taken to be zero, 1= 0. Also note that as the outer wall radius

is increased, as 0.

6.2.3 Viscous Free Liquid-Liquid Cylinder Breakup

The linear stability analysis to calculate the growth rate, s, of a disturbance

on an initially stationary, infinitely long, viscous liquid cylinder in another viscous

liquid was first derived by Tomotika (1935). The detailed derivation is given in

appendix G. His solution is derived by the introduction of the streamfunction

u= (1/r) (/z),v= (1/r) (/r), and by the substitution of normal mode

solutions of the form a = R+ a (r)exp(ikz+ st) into the linearized equations,


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where a (r) is purely a function ofr. The determinant of the coefficients of the

equations for the growth rate is:

I1 () I1 () K1 () K1 ()

I0 () I0 () K0 () K0 ()

2k2

2

1

I1 ()

k2+m22

2

1

I1 () 2k

2K1 ()

k2 + m21

K1 ()

f1 f2 f3 f4

= 0

(6.10)

where:

f1 = 2k2

21

dI1 ()

d +

s21

I0 () + k

s1R2

2 1

I1 () (6.11)

f2 = 2km2

21dI

1()

d + k

s1R2

2 1

I1 () (6.12)

f3= 2k2 dK1 ()

d

s11

K0 () (6.13)

f4= 2km1dK1 ()

d (6.14)

m21= k2 +

s11

(6.15)

m22= k2 +

s2

2(6.16)

with Rm1, and Rm2. This quadratic equation (6.10) is solved for the

growth rate s and the dimensionless wavenumber is varied until the maximum

value of s is obtained. However, an iterative procedure is necessary since m1

and m2 depend on s (Meister, 1966). This maximum value ofs corresponds to

the most dangerous wavenumber . Note that the determinant equation (6.10)

reduces to the eigenvalue relation (6.9) of the previous section as 1 0 and

2 0 (Meister, 1966).

6.2.4 Meister & Scheeles Linear Stability Analysis for Jet Length

Meister and Scheeles (1969a) simplified model (MS) for the jet length, L

(see Figure 6.1), is now briefly reviewed, both as a convenience to the reader, and


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to describe the exact algorithm used in this chapter to estimate L based on this

theory. Further details can be found in Meister and Scheele (1969a). This theory

improves a previous estimate by Smith and Moss (1917) based on the amplification

of the critical disturbance obtained from the linear stability theory of the free jet

problem:

L=v

sln

R

a0

(6.17)

where v is the average nozzle velocity and s is the maximum growth rate obtained

from the Tomotika analysis, equation (6.10). By assuming that the disturbance

travels not with the nozzle velocity v as implied by (6.17), but at the speed of

the interfacial velocity vI and by adjusting the interfacial velocity to that of anoncontracting jet, Meister and Scheele (1969a) estimated the jet length to be:

L= v

2s

vIvA

z=5

+

vIvA

z=L

R

ln

R

a0

(6.18)

wherevIandvA v/a 2 are the axial interfacial and average velocity respectively,

a a/R is the dimensionless interfacial radius, z z/R is the dimensionless

axial distance. An iterative calculation is required to evaluate the ratio vI/vA

at z = L/R since this ratio is a function of L; only two iterations are usually

necessary. The evaluation of this ratio at z = 5 (which is approximatelyz /2)

andz =L/R represents an arithmetic average at these two characteristic points.

Here we have substituted Meister and Scheeles equation (21) for vI into their

equation (22) for L to show that the correction to the nozzle velocity does not

depend ona directly, as is implied by their jet contraction equation (15). Values

ofa

are necessary only if values ofvIare desired for comparison to experiment

using equation (6.26) to be presented below. IfvI = vA in equation (6.18), we

recover the equation of Smith and Moss (1917) (6.17).

The ratio vI/vA is obtained from an assumed velocity profile in the jet


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evaluated at the interface:

vIvA

= 1 + eA(z)z

1 eB(z)z

(6.19)The authors assumed velocity distributions both in the jet and in the continuous

phase involving four unknown functions of the axial distance, z. They used con-

tinuity of mass in the jet, continuity of axial velocity at the interface, continuity

of tangential shear stress at the interface, and an approximate axial jump momen-

tum balance across the interface to evaluate these unknown functions. They then

arrived at the following equations:

1 + e Az

11

2e Bz

= 1 (6.20)

G

1 e Bz

2 2B+ 2zdBdz

A+z dAdz

1 e Bz

(2 e Bz) e Bz

= 1 (6.21)

where G Rv (1 2) 1/22. Since the method of solution of these two equa-

tions was not given the following approach was taken here. Equation (6.20) isdifferentiated with respect to z, and the resulting equation and equation (6.21)

are solved directly for the derivatives of the unknown functions A(z), B(z) as

follows:

dA

dz=

4AGe4Bz

AGeAz + 11AG e

3Bz + (3AG2) eAz + 11AG 2 e

2Bz

(3AG1) eAz

+ 5AG 1

eBz

+ AGeAz

+ AG

Gz

4e4Bz

eAz

+ 11

e3Bz

+

3eAz

+ 11

e2Bz

3eAz

+ 5

eBz

+ eAz

+ 1

(6.22)


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dB

dz=

4BGe4Bz

BGeAz + 11BG + 4 e3Bz + 3BGeAz + 11BG + 4 e2Bz

3BGeAz

+ 5BG + 1

eBz

+ BGeAz

+ BG

Gz

4e4Bz

eAz

+ 11

e3Bz

+

3eAz

+ 11

e2Bz

3eAz

+ 5

eBz

+ eAz

+ 1

(6.23)

These ODEs are integrated numerically forz >0 subject to initial conditions as

z 0:

A= 3G

1

3

z2

3 (6.24)

B=

3

2G

13

z2

3 (6.25)

The interfacial velocityvI/v can then be calculated from:

vIv

= 1

a2

vIvA

(6.26)

where the dimensionless jet radiusa a/Ris obtained by using the macroscopic

mechanical energy balance for free jets developed by Addison and Elliott (1950)

(assuming an initially flat velocity profile at the nozzle, and by neglecting viscous

dissipation terms), which was found useful as a rough estimate close to the nozzle

in chapter 5.

Nja4 +

4

W e

a4 a3

+ a4 1 = 0 (6.27)

HereNj Re2F r

12 1, the buoyancy number appropriate for a liquid-liquid

jet pointed upwards (+) or downwards () respectively, Re2 2R2v/2, is

the dispersed phase Reynolds number, F r v2/2Rg, is the Froude number,

W e 2Rv22/, is the Weber number, and z/Re2 is the axial position

downstream scaled with the dispersed phase Reynolds number. Note that equation


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(6.27) differs from the one used by Meister and Scheele (1969a) who erroneously

used an 8 instead of a 4 multiplying the second term.

6.3 Numerical ImplementationThe method discussed in chapter 3, after validation on several test prob-

lems, is used to calculate the axisymmetric dynamic flow and breakup of a liquid-

liquid jet. For the breakup studies, a temporal sinusoidal perturbation of ampli-

tude a0 and angular frequency was introduced in the F function at the nozzle

lip. Other disturbances such as perturbations on the inlet velocity profile (6.3) are

also possible but have not been used here.

Two important factors in the numerical calculation of jet breakup that must

be mentioned here are the method used to calculate the surface tension volumetric

forces, and the upwinding method in the finite difference approximation of the

inertial terms contribution to the momentum equations (3.2) and (3.3). The

original SOLA-VOF of Nichols et al. (1980) algorithm for liquid-liquid surface

tension force calculation was found to be completely unstable for liquid-liquid jet

calculations. This motivated the incorporation of the CSF algorithm (Kothe et

al., 1991) into the code, which resulted in a stable calculation for the steady-

state region of the nozzle (Richards et al., 1993). A critical feature for numerical

stability is some smoothing of the interface, especially important given the discrete

character of the interface shape approximation. Spatial smoothing of the F

function for calculation of the curvature, equation (3.5), is possible in the CSF

algorithm (see section 3.5), but this can lead to excessive stabilization of the jet,

preventing its breakup. A minimal amount of smoothing (one pass through the

spatial filter in the CSF algorithm) was used in the present work. Similarly, if the

inertial terms are approximated by second-order accurate central differencing, the

algorithm is numerically unstable (Hirt, 1968) If they are approximated by first-

order accurate upwinding or donor-cell differencing, the algorithm is stable

but it might be overstable due to the presence of excessive artificial viscosity


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(see section 3.2).

6.4 Results and Discussion

Several free surface test problems that have known analytical or numericalsolutions were used to test the accuracy and validate the algorithm; the results for

the inviscid free liquid-liquid cylinder breakup of Christiansen and Hixson (1955,

1957) discussed in section 6.2.2 are discussed in more detail in section 6.4.1 below.

As previously discussed in section 6.1, Meister and Scheele (1969a) mea-

sured the jet length, L, vs. the dispersed phase Reynolds number, Re2, for five

nozzle diameters. Their results, obtained by averaging the lengths of jets mea-

sured from four photographs per point, are summarized in Figure 7.1 which is

Figure 1 of Meister and Scheele (1969a). We have used their data as well as the

photograph in Figure 1 of Bright (1985) for comparison purposes. The parameters

corresponding to the six base cases that we have used in this work are summarized

in Table 6.1, encompassing six nozzles and two liquid-liquid systems. The systems

have been allowed to become mutually saturated (to avoid mass transfer between

the phases) before the experiments, and the physical parameters for the saturated

systems are given in Table 6.2. In Table 6.2 note the substantial differences be-

tween the experimentally measured and the literature values for the interfacial

surface tension. The most probable cause for this difference is the presence of

(unknown) contaminants. Since the emerging jet forms a new, clean surface that

under the experimental circumstances is not anticipated to have had enough time

to become saturated with the contaminant, the literature values for pure compo-

nents were used for the interfacial surface tension in this work. However, possible

additional implications are discussed later in this section.

Five independent dimensionless groups result from the dimensionless form

of equations (3.1)-(6.1). Those are listed in Table 6.1 as Re2 2R2v/2,

the dispersed phase Reynolds number, Re1 2R1v/1, the continuous phase

Reynolds number, F r v2/2Rg, the Froude number, W e 2Rv22/, the


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Table 6.2: Physical properties.

Continuous, 1 water

Dispersed, 2 n-heptane n-decane

, pure,a dyne/cm 51.1c 52c

, exp.,b dyne/cm 36.2d 22.5e

2, g/cm3 0.683d 0.732e

1, g/cm3 0.996d 0.999e

2, g/(cm s) 0.00393d 0.0099e

1, g/(cm s) 0.00958d 0.011e

a Pure value used in this work.b Measured value used in MS model.

c Johnson (1966)

d Scheele and Meister (1968)

e Bright (1985)

of buoyancy to viscous forces. Alternatively, one or more of these dimensionless

groups can be replaced by other dimensionless parameter combinations, such as

the viscosity ratio 2/1, or the capillary number, Ca2 We/Re2, representing

the ratio of viscous to surface forces.

In the numerical simulations, the flow rate at the nozzle was instantaneously

increased from zero to its final value with the alkane filling the nozzle and water

filling the tank, in much the same way as the physical system was started. Then

the simulation followed the time evolution of the flow until the jet length vs.

time profile reached pseudo-steady behavior. A typical (coarse) mesh is shown

in Figure 6.2, with the increased mesh refinement confined, as illustrated, to the

dispersed phase. A locally refined, but uniform, mesh is used inside the jet region,

whereas a variable, coarser, mesh is employed in the much slower moving outer


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region. The mesh size that was used for the liquid-liquid jet runs involved 55

cells stacked in the radial direction (with 40 cells being confined to the interval

0 r 2) and 255 cells in the axial direction, with the exception of nozzle 1 and

nozzle 4, which involved 65 cells in the radial direction. This cell distribution was

optimized through mesh sensitivity studies. In addition, the effects of the effluent

boundary distanceL1, the bottom distance,L2, the fully developed velocity profile

distance, L5, and the outer wall distance L3 were investigated separately by

sensitivity studies. The actual tank dimensionsL1/Rand L3/Rare given in Table

6.1. The bottom distance was not given in the reference, but was estimated as at

least L2/R >10. In any case, Yu and Scheele (1975) have found experimentally

that placing an annular disk flush with the nozzle tip (L2 = 0) had little effect onthe measured jet radii, so this dimension appears to be of little importance.

For the conditions of Table 6.1, it was found that L1/R = 50, L2/R = 1,

L3/R = 12, and L5/R = 1 were large enough to establish insensitivity of the

results to the actual values ofL1/R, L2/R, and L3/R. For example, comparing

results obtained with L3/R = 6 to 12, no significant difference was found in the

breakup length. Likewise, because of the high speed of the jet, comparing results

corresponding toL1/R= 25 to 50 for nozzle 5, no significant difference was found

in the dispersed phase solution as long as the jet breakup occurs far from the

outflow, as was mostly the case.

For the high velocity jet cases it was found necessary to modify the outflow

condition (6.4) by forcing only flow out of the mesh at the outflow boundary

z= L5, i.e., by requiringv 0, a method suggested by Hirt (1993). This was found

to alleviate a problem with the jet initially hitting the outflow reflective boundary

and causing a significant backflow and consequent jet disruption. Further, it was

found that it was preferable to use the boundary conditions (6.5) at the far outer

radial region rather than the no-slip condition (e.g., (6.1)) used in chapter 5. More

specifically, it was found that since the axial domain is longer in this study than the

previous one, use of a no-slip condition led to the development of a recirculation


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0

-1

50

0 1 12

Symmetry Axis

Nozzler*

z*

JetInterface

Figure 6.2: Coarse mesh layout.


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in the outer continuous phase. Since this outer boundary at L3 is actually closer

to the nozzle than the actual tank wall at L3, the recirculation does not exist

in the physical system. Thus, a more reasonable physical choice, which does not

produce this recirculation (i.e., (6.5)), was to use the fact that the radial gradients

of velocity diminish rapidly with distance from the nozzle.

The breakup of the jets occurs experimentally as a result of naturally

occurring perturbations. Given the uncertainty of its exact form in the reported

experiments, we added a numerical perturbation to the free surface at the nozzle

lip as discussed in section 6.3. The angular frequencies of the perturbations applied

are given in Table 6.1. As was first pointed out by Rayleigh (1879), the most

dangerous frequency, corresponding to the fastest growing disturbance on thejet, dominates all others, and it becomes the one observed in experiments. We

have estimated this frequency in the following way. We calculated from the linear

theory equation (6.10) the most dangerous wavenumber = kR. Then assuming

that the average interfacial velocityvIwas about half the nozzle velocity (Bright,

1985), the angular frequency was estimated as = k vI. This estimate is close

enough to the most dangerous frequency since a sensitivity study on the nozzle 3

case revealed that the jet length decreases by only 19% as is varied from 28 to

114 rad/s, with 57 rad/s being the base angular frequency.

The amplitude of the perturbation, a0/R = 0.01, was fixed for all the

present simulation cases. Its variation has a varying influence on the results,

depending on the Reynolds number. In fact, for the high Reynolds number,

Re2, cases such as nozzle 5, no perturbation was necessary, there being enough

numerical noise to break up the jet. However, the lowRe2 jets, such as nozzle 1,

required a finite perturbation to break within L/R 50; the larger the amplitude

of the perturbation, the shorter the jet length, in agreement with the logarithmic

dependence of the perturbation given in equation (6.18). In calculating the

predicted jet lengths from the linear theory of Meister and Scheele (1969a), we

have used their recommended value, ln(R/a0) = 6, corresponding to a value


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of a0/R = 0.0025. The use of a0/R = 0.01 would have resulted in a wider

disagreement of the MS model and the data. The variation of this parameter

in the present simulation is discussed in section 6.4.4.

The value of interfacial surface tension used in the present simulations was

the pure value taken from the literature and listed in Table 6.2. Meister and Scheele

(1968) used the Harkins and Brown (1919) drop volume technique to measure

the interfacial surface tension, which is an equilibrium or static measurement.

However, Bohr (1909) recognized that a dynamic measurement of surface tension

was needed to describe new, freshly formed surfaces, which are presumably free

from surface active contaminants, and developed a relationship in terms of the

undulations on a jet exiting from a circular nozzle into a gas. Later workers(Addison and Elliott, 1950; Garner and Mina, 1959) used a contracting jet

technique to measure the surface tension as a function of surface age for liquid-

liquid jets. Garner and Mina (1959), in a study of aqueous solutions of n-hexyl

alcohol jetting into mutually saturated medicinal paraffin, showed in their Figure 9

that when the surface age approached 0.4 s for low concentrations of alcohol the

dynamic interfacial tension decreased from near the water/paraffin value to a lower

value as the surface active agent (alcohol here) adsorbed on the surface. Thus,

instead of the lower interfacial tension values measured by Meister and Scheele

and Bright, and listed in Table 6.2, which are probably due to slight impurities,

we have used the pure values, since the surface ages of the breaking jets in this

study are all less than about 0.1 s. However, to calculate the predicted jet lengths

of the linear theory of Meister and Scheele (1969a), we have used their measured

surface tension values. Again, use of the pure surface tension values would have

resulted in a higher discrepancy between their predictions and the data. Further

discussion of dynamic surface tension can be found in the work of Edwards et al.

(1991).


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6.4.1 Inviscid Free Liquid-Liquid Cylinder Breakup

The problem of the capillary breakup of an initially stationary infinitely

long free inviscid liquid cylinder in another inviscid liquid with an outer free-

slip wall (Christiansen and Hixson, 1957) was chosen as a validation test of

the numerical method developed in the present work. This problem involves

a transient flow, a non-constant surface curvature, and admits an analytical

solution at least for small times obtained from a linear analysis as given in

section 6.2.2. Figure 6.3 shows the complete breakup time sequence as obtained

with the VOF numerical calculation for one axial wavelength using an initial

cosine amplitude perturbation with a0/R = 0.001 on the interface and the most

dangerous wavenumber kR = 0.6773 as calculated from equation (6.9). A halfwavelength was used for the actual computational domain with free-slip conditions

at z z/R = 0 and z = 4.6384, and the distance from the centerline to the

free-slip wall is L3/R= 3; similar results were obtained for two wavelengths.

Figure 6.4 shows the early time sequence for a half wavelength calculated

using the linear theory of Christiansen and Hixson (1957) (CH) (equations (6.6)

and (6.9)) compared to the present nonlinear simulation. The densities used were

those of n-heptane in water given in Table 6.2,1= 2= 0, and = 51.1 dyne/cm.

The uniform mesh used for the simulation comprised 75 cells in the radial and

120 cells in the axial directions. This cell distribution was optimized through

mesh sensitivity studies. The breakup time of 0.023 s predicted by the simulation

compares well with that predicted by the linear theory of 0.024 s (when a = R

in equation (6.6)). This result is consistent with the calculations of Mansour and

Lundgren (1990), who also found good agreement with breakup times between

their nonlinear model and Rayleighs (1879) linear theory, but, in addition, found

significant interface deviation from the linear theory near breakup time, leading

to the formation of the satellite drop. The excellent agreement of the simulation

results with the analytical theory validates the numerical code under conditions

that share many factors with the actual physical problems studied in this work.


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Figure 6.3: Inviscid liquid-liquid cylinder breakup. Heptane in water, t =0.002 s.


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0 1 2 3

r*

0

1

2

3

4

5

z*

CH

This work

Interface at t = 0 s

t = 0.014 s

t = 0.018 s

t = 0.022 s

Free Slip Wall

Axis ofSymmetry

Figure 6.4: Inviscid liquid-liquid cylinder. Heptane in water with interface po-sitions at four times before breakup. Linear theory of Christiansenand Hixson (1957) (CH) compared to the present nonlinear model.

6.4.2 Long-Time Free Surface Dynamics

The main body of the results presented here are comparisons between

the experiments of Meister and Scheele (Meister, 1966; Meister and Scheele,

1969a) and Bright (1985) and our numerical simulations. Figures 6.5, 6.6, and

6.7 represent free surface comparisons of nozzle 5, 3, and 6 respectively (see

Table 6.1). The experimental interface data, traced from photographs of the

interface, are located in the lower right corner frame of each figure (labeled with

the experimenters name). Each frame of the figures is a contour of the interface


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(F = 1/2) at equal time intervals, starting from startup at t = 0.

Figure 6.5, nozzle 5, represents a short jet at high Re2 with only one

undulation before breakup into drops. Our numerical simulation also shows one

undulation before breakup, and about the same jet length before breakup as the

data. Notice the cap-like drop formation and the general trend to oscillating

ellipsoid-like drops in both the numerical and experimental data. There is some

evidence of non-axisymmetric drops in the experimental frame on Figure 6.5; this

is obviously not captured by the simulations, but appears to be a minor feature in

the overall picture. Qualitatively the drop sizes predicted by our simulation can

be seen to be about the same as the data, but the quantitative comparison of drop

sizes is the subject of chapter 7, and is not discussed here. Our simulation also

indicates a significant jet interface contraction before breakup, also in agreement

with the data.

Figure 6.6, nozzle 3, represents a jet of intermediate length, obtained at

intermediate Re2 with again only one undulation before breakup into drops. The

numerical simulation also shows one undulation before breakup, and about the

same jet length before breakup as the experimental data. The oscillating ellipsoid-

like drops can also be seen in some of the simulation frames. The drop sizes

predicted by the simulation can again be seen to be about the same as the data,

with more shape variety in the simulation results. The simulation also indicates

some jet interface contraction before breakup, in agreement with the data.

Figure 6.7, nozzle 6, represents the dynamic behavior of a long jet, obtained

at the lowest Re2 with, in general, three undulations observed before breakup.

The numerical simulation shows about two undulations before breakup, and again

about the same jet length before breakup as the data. Notice the pear-like drops

formed, which can also be seen in the numerical simulations. The present simu-

lation indicates very little jet interface contraction before breakup, in agreement

with the data.


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Figure 6.5: Free surface comparison. Experimental data of Meister (1966) com-pared to the present simulation. Heptane into water, nozzle 5, R =0.344 cm, v = 15.48 cm/s.


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Figure 6.6: Free surface comparison. Experimental data of Meister (1966) com-pared to the present simulation. Heptane into water, nozzle 3, R =0.127 cm, v = 21.6 cm/s.


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Figure 6.7: Free surface comparison. Experimental data of Bright (1985) com-pared to the present simulation. Decane into water, nozzle 6, R =0.08 cm, v = 44 cm/s.


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6.4.3 Short-Time Free Surface Shape Sequence

Meister and Scheele (1969a) presented a short-time sequence of frames for

the liquid jet obtained with nozzle 3 to illustrate that alternating larger and smaller

drops could be produced from a liquid-liquid jet. Their tracings of the free surface

taken from a high-speed movie (their Figure 6) are shown in Figure 6.8. For

comparison, a jet case with the same conditions, corresponding to nozzle 3 and

Figure 6.6, was simulated numerically and is illustrated in Figure 6.9.

In Figure 6.8 the data indicate a pattern corresponding to a large drop

breaking off, a small drop breaking off about five frames later, followed by another

large drop breaking off about 18 frames later. The numerical simulation indicates

about the same pattern and interval between frames being produced. It turnsout that in the simulation this pattern did not always recur, with several large

drops being produced before the large-small-large pattern appeared again. It is

interesting also that most of the simulation frames do not contain satellite drops

which would be formed from the neck of the fluid drawn out during drop break off.

This appears to be consistent with the experiments of Meister and Scheele (1969a)

who found that only in some instances were satellite drops formed. This is in

contrast to single phase jets where satellite drops are frequently formed (Donnellyand Glaberson, 1966).

6.4.4 Jet Length Comparisons and Discussion

The jet length,L/R, was calculated as a function of time for each simulation

case, and a typical result is shown in Figure 6.10. As can be seen in that figure,

the general pattern is a rise to a certain length, detachment of a drop, with a

buildup in length to form the next drop. This results in a sawtooth pattern

as seen in the figure, a result that was confirmed by Fourier analysis of the time

series data. Superimposed on that, a longer-term trend appears in that the mean

height between drops rises from zero at startup and levels off to a pseudo-steady-

state (limit cycle) value. However, it can be seen that at long times the length is


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Figure 6.8: Free surface sequence. Experimental data of Meister and Scheele(1969a) Heptane into water, nozzle 3,R = 0.127 cm, v= 21.6 cm/s,t = 0.00416 s.


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Figure 6.9: Free surface sequence the present simulation. Heptane into water,nozzle 3,R = 0.127 cm, v = 21.6 cm/s, t= 0.00416 s.


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not periodic, but rather chaotic. We have found that the variation in amplitude

and period generally increases with increasing Reynolds number, Re2, with the

fluctuations becoming more random. We thus conjecture that the aperiodicity

is due to the increasing importance of the nonlinear terms in the momentum

equations, causing a chaotic response.

0 1 2

t (s)

0

10

20

30

L/R

Figure 6.10: Jet length vs. time. Heptane into water, nozzle 3, R = 0.127 cm,v = 21.6 cm/s.

The average, minimum, and maximum lengths at the pseudo-steady-state

predicted by the numerical simulation are shown in Figures 6.11 to 6.13 for all six

cases. Also shown are the experimental length data of Meister and Scheele (1969a)


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and Bright (1985) and the numerical results of Meister and Scheele (1969a) (MS)

from equations (6.18) (6.25). Figure 6.11 shows the predicted lengths for nozzles

1, 3, and 5 as a function of Reynolds number, Re2. Figure 6.12 shows the predicted

lengths for nozzles 2 and 4 as a function of Reynolds number, Re2. Figure 6.12

also shows the predicted lengths of the current simulation for nozzle 2 at two

perturbation levels, a0/R = 0.01 and 0.001. Figure 6.13 shows predicted length

vs.experimental length for all six nozzles. As pointed out by Meister (1966), since

the average measured height of the jet is increased by /2 due to the forming and

detaching of the drops (see, e.g., Figure 6.10), a value of/2 was added to the MS

model length predictions to be comparable to the photographic data.

Several points regarding the origin and nature of the experimental datashould be noted assessing the agreement of theory with experiment. One is

that Meister and Scheele (1969a) took four photographs of the jet and averaged

them to obtain their experimental length (see Figure 1 of the reference). No

information is available on the accuracy (or long-term reproducibility) of this

approach, except the regularity and frequency of the data points themselves, but it

can be conjectured by considering the numerically simulated length in Figure 6.10

that only four points of this curve taken at random times would not be a very

representative sample. Only one photograph was available for the data point of

Bright (1985) which was used in Figure 6.13. One can then imagine minimum

and maximum bars on the experimental data about as large as those included in

Figures 6.11 to 6.13 for the numerical simulations.

Another aspect of the data inadequately accounted for in the simulations is

the fact the magnitude and frequency of the perturbations is experimentally un-

known. We have kept the amplitude constant for all cases for both the MS model

(a0/R= 0.0025) and the numerical simulations (a0/R= 0.01). A better fit of the

data could be obtained by varying this parameter for each nozzle but little is to be

gained from this approach in terms of improving the predictive capability of the

simulations. As previously mentioned, the frequency of the perturbation was also


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500 1000 1500 2000

Re2

0

10

20

30

40

50

L/R

MS model

This work

MS Data, R = 0.04065 cm

MS Data, R = 0.127 cm


Figure 6.11: Jet lengthvs.Reynolds number data of Meister and Scheele (1969a)(heptane into water) and numerical results of Meister and Scheele(1969a) (MS) compared to the present simulation for nozzles 1, 3,and 5.

varied in the numerical simulations to evaluate the sensitivity of the results, which

proved in this case to be small. The MS model used the maximum in the growth

rate according to equation (6.10) to estimate the most dangerous wavelength. To

eliminate this ambiguity a future experimental study could be undertaken with

forced amplitude and frequency perturbations for liquid-liquid jets analogous to

the experiments of Donnelly and Glaberson (1966) who investigated liquid jets in

air. Recently we became aware of an experimental study by Tadrist et al. (1991),


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500 1000 1500 2000

Re2

0

10

20

30

40

50

L/R

MS model

This work, ao/R=0.001This work, ao/R=0.01


MS Data, R = 0.166

Figure 6.12: Jet lengthvs.Reynolds number data of Meister and Scheele (1969a)(heptane into water) and numerical results of Meister and Scheele(1969a) (MS) compared to the present simulation for nozzles 2 and4.

in an attempt to characterize the instability as completely as possible, measured

fluctuations of the interface of a liquid-liquid jet and found them to be charac-

teristic of a weak-bad random signal with the power spectra densities of typical

signals presenting a band of a given width around a characteristic frequency. Near

the nozzle the jet amplitude fluctuations were slight, but fluctuations increased

rapidly with distance from the nozzle. This study also found that the measured

amplitude of the disturbance on the jet was (a0/R 0.01), a value that we found


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0 10 20 30 40 50

L/R Experimental

0

10

20

30

40

50

L/RP

redicted

MS model, MS Data

MS model, Bright Data

This work, MS Data

This work, Bright Data

Figure 6.13: Jet length predicted vs. experimental data of Meister and Scheele(1969a) (heptane into water) and Bright (1985) (decane into wa-ter) and numerical results of Meister and Scheele (1969a) (MS)compared to the present simulation for nozzles 1 6.

independently by our simulations that best fit the data.

As was mentioned previously in this section, a surface contaminant was

probably present in the experiments as evidenced by the discrepancy between the

pure and measured surface tension values listed in Table 6.2. Since the jet is

generating a clean surface at the nozzle tip, the contaminant concentration will

be low. As surface age increases, the contaminant will diffuse to the surface,

therefore resulting in an axial concentration gradient with a subsequent surface


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tension gradient resulting. This can lead to surface forces along the axial direction

resulting in a Marangoni flow (Edwards et al., 1991). However, in the present

cases, the magnitude of this gradient is probably small since the jet breaks up

rather quickly, and the measured value for the interfacial tension of n-heptane is

only 21% lower than the pure value.

Although the predictions from the MS model come close to the data at

L/R 15, it can be seen from Figure 6.13 that their slope is much less than

the 45 line (which corresponds to perfect agreement). On the other hand, while

the results from the simulations show the same general trend of the 45 line,

they also show a small bias to over-prediction of the length. This should be

attributed to the presence of numerical viscosity (Hirt, 1968) which tends tomake the numerical solution more stable than the physical one. For example, in

numerical calculations where HNwas set to unity (full donor-cell differencing),

the nozzle 5 case would not break in the region L/R 20. As was mentioned

previously,HNwas set to 1.2 times the minimum required for numerical stability

to minimize this viscosity effect. As a smaller value for HN would have made

the code numerically unstable, it appears that the only remedy is mesh refinement

that the present availability of computer resources has limited to about 65 255

cells. A stringent numerical stability criterion (Hirt and Nichols, 1981) is used to

adjust the time step automatically to a small value (typically 1 105 s), and

combined with the large number of mesh cells, this results in about 10 teraflop of

work for a typical calculation such as those reported in Figures 6.11 to 6.13.

An exception to the above observation for overprediction of jet-length

are the results obtained for nozzle 2, which are substantially under-predicted.

A possible explanation is that this nozzle may correspond to an exceptionally

low value of a0/R experimentally. To test this we simulated the nozzle 2 case

with a ten-fold lower value of a0/R = 0.001 and found the results to be more

consistent with the data as expected from linear stability analysis. As can be seen

in Figure 6.12, the simulation approaches the data as the perturbation is lowered.


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Keeping the preceding discussion in mind, we conclude that, in most cases, the

present simulation agrees satisfactorily with the data, within experimental error.

To be more quantitative, a more careful control of the noise present during the

experiments appears to be necessary.

The MS model has many assumptions, as mentioned in section 6.2.4. As

can be seen by examining the jet length equation (6.18), two of the critical steps

are the calculation of the most dangerous growth rate s and the velocity at

which the disturbance is propagated, the interfacial velocity vI. As discussed

in section 6.1, Kitamura et al. (1982) experimentally varied the relative motion of

the continuous phase. They found that the jet length shortened as the absolute

value of the continuous phase velocity relative to the jet increased from zero withcondition of zero relative phase velocity giving the best experimental agreement

with Tomotikas stationary jet solution translated at the average velocity of the jet.

They concluded that the discrepancy between experiment and Tomotikas analysis

in the nonzero relative velocity cases was due primarily to the relative motion of

the continuous phase, and not to other factors such as velocity profile relaxation.

In our case, the outer phase is relatively stagnant away from the interface, and

this could result in a shearing type instability (Russo and Steen, 1989) that would

result in an incorrect s, since the Tomotika equation (6.10) assumes that the two

phases have the same axial velocity. To correct this, an analysis similar to that

of Sterling and Sleicher (1975), Bright (1985), or Russo and Steen (1989) would

have to be performed, but since gravity (buoyancy) and the outer phase viscosity

would have to be included, the modifications would be nontrivial and, if feasible,

would result in a rather complex solution procedure. The value of the Tomotika

analysis is that it is currently the only approach that incorporates in the outer

phase the physical properties of density and viscosity.

Another source of error in the MS model is the calculation of the axial

interfacial velocity,vI. It is noteworthy that Kitamura et al. obtained good results

at zero relative phase velocity, using the average velocity of the jet in their jet


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length equation, the same equation used by Smith and Moss (1917) (6.17), with the

maximum growth rates calculated using Tomotikas equation (6.10). This can be

explained by the fact that if the jet is long (their shortest heptane into water jet was

L/R= 16), the velocity profile will have relaxed so that the interfacial velocity will

be comparable to the average velocity. However, if the velocity profile has not had

a chance to relax, this would not be true. Bright (1985) found experimentally that

the surface wave velocity was about half the nozzle velocity for decane into water

jets at v= 150 cm/s. Meister and Scheele (1969a) reported only two experimental

point checks of liquid-liquid interfacial velocities predicted by equations (6.19)

(6.27) as measured by observing disturbance node velocities on the surface of the

jet. To test these ideas we compared the MS model calculation of jet radii a/R(equation (6.27)) and interfacial velocitiesvI/v (equation (6.26)) with the current

simulation for nozzle 3; the results are shown in Table 6.3. The comparison is

quite good until just before jet breakup, which is consistent with the experimental

checks by Meister and Scheele (1969a). Therefore, we conclude that the error in

the calculation of the interfacial velocities is not a major contributor to the overall

discrepancy between the MS model and the data. Thus, given this fact, coupled

with the experimental observations by Kitamura et al., and by examining the MSjet length equation (6.18), the most likely reasons for the MS disagreement are

(1) the maximum growth rate s is assumed independent of the jet velocity v, and

(2) the entire analysis is a local one, not taking into account the velocity profile

relaxation in the axial direction away from the nozzle exit. Still, the MS analysis

can be used as a first estimate, its major drawback being its relative insensitivity

towards Reynolds number changes.

6.5 Conclusions

The robust and stable VOF-based numerical method developed in this

dissertation has been used here successfully to simulate high Reynolds number,

high buoyancy number, and low Weber number flows, up to and beyond the


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to include dynamic surface tension, which has an influence on the predicted jet

lengths. Predominant among the issues that the present study revealed is the sen-

sitivity to the external disturbances. An experimental study further characterizing

these ambiguities could be undertaken with pure materials and forced amplitude

and frequency perturbations for liquid-liquid jets, analogous to the experiments

of Donnelly and Glaberson (1966), who investigated liquid jets in air. However, a

broader implication for real liquidliquid contactors is that totally predictive sim-

ulations will remain difficult, even with the great improvements in computational

power that continue to become available.

The physical problem studied here is a challenging one because of the

existence of moving complex free surfaces and the significant roles of surfacetension, inertia, and gravity, as well as experimental disturbances of unknown

frequency and amplitude. It particularly places stringent requirements of stability

and accuracy on our numerical technique in fact, any numerical technique

aiming to handle these types of high Reynolds number free surface problems.

As in all numerical computations, our current calculations are able to provide

an approximate answer only. Much more extensive mesh refinement and/or

use of adaptive (local) mesh refinement is needed for more accurate answers.

This additional computational effort is not currently justified, however, given the

existing uncertainty in the magnitude of the experimental disturbances driving

the jet breakup.

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