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Chapter 5

STEADY LAMINAR FLOW OF LIQUID-LIQUID JETS

AT HIGH REYNOLDS NUMBERS

Let us take in the first instance the problem of the efflux of a liquidfrom a small orifice in the walls of a vessel which is kept filled up toa constant level, so that motion may be regarded as steady . . . Ex-periment shews however that the converging motion above spoken of

ceases at a short distance beyond the orifice, and that (in the case of a circular orifice) the jet then becomes approximately cylindrical . . .The calculation of the form of the issuing jet presents difficultieswhich have only been overcome in a few ideal cases of motion in twodimensions.

H. Lamb, Hydrodynamics (1932)

Among the systems we are interested in are ones that represent key elements

of one or more types of actual contactors. In this chapter we examine a liquid-

liquid jet that is representative of the situation above an orifice on a sieve tray.The object is to model the velocity and pressure fields as well as the interface

shape and location in this system that has complex free surfaces. Two regions are

considered for the liquid-liquid jet: the steady region near the nozzle is discussed

in this chapter, and the entire region from the nozzle to the breakup of the jet

into drops in chapters 6 and 7.

5.1 Background

Laminar liquid jets injected from a circular nozzle into another liquid have

been studied for many years (Addison and Elliott, 1950; Scheele and Meister,

1968; Meister and Scheele, 1969b; Meister and Scheele, 1969a; Yu and Scheele,

1975; Richards, 1978; Gospodinov et al., 1979; Richards and Scheele, 1985; Anwar

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72

et al., 1982). They are important as a means of heat and mass transfer due to

the creation of large new surface area. For example, since jetting occurs in sieve

plate columns, apparatuses have been specifically designed for mass transfer and

surface tension experiments with a single jet (Skelland and Huang, 1977; Skelland

and Huang, 1979; Skelland and Walker, 1989). Qualitative features of the jet

behavior are well known: for low flow rates, drops form, grow, and break off from

the nozzle at regular intervals. Above a certain critical velocity a stable laminar jet

is formed at the nozzle. The jet rises to a certain length, and then breaks up into

drops. At still higher nozzle velocities, the jet becomes turbulent and eventually

disrupts into small drops.

Brief overviews of the jet literature before 1987 can be found in Vrentas andVrentas (1982) and Gonzalez-Mendizabal et al. (1987). Previous experimental

and theoretical studies can be classified into those examining the steady-state

jet, such as the evaluation of the jet radius and the velocity profiles in the jet

(Addison and Elliott, 1950; Yu and Scheele, 1975; Richards, 1978; Gospodinov et

al., 1979; Richards and Scheele, 1985; Anwar et al., 1982), and those investigating

jet dynamics, such as drop volume before jetting and jetting velocity (Scheele and

Meister, 1968), drop sizes produced by the jet (Meister and Scheele, 1969b), and

jet length and disruption velocity (Meister and Scheele, 1969a). In the present

work we focus on the investigation of the steady-state laminar axisymmetric jet

flow. The major theoretical difficulty in calculating jet flows, in which numerical

methods are resorted to, is the unknown location of the free surface, which

must be found together with the pressure and velocity fields. The jet radius

and the velocity profiles of a cylindrical laminar liquid jet injected into air were

calculated numerically by Duda and Vrentas (1967). They used the stream-

function instead of the radial coordinate as an independent variable in their

equations, thus making the jet interface a constant coordinate surface. They

then simplified the steady equations of motion for the jet dispersed phase and the

jet surface using boundary-layer theory, and solved the resulting problem using



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a parabolic marching technique. Yu and Scheele (1975) extended this approach

to include the continuous phase equations to describe the liquid-liquid jet, but

assumed approximate velocity profile forms in their momentum integral treatment

of the two phases. Gospodinov et al. (1979) relaxed this assumption and solved

the boundary-layer equations for the velocity profile established within each phase,

also using a parabolic marching technique.

Experimentally, Richards and Scheele (1978, 1985) developed a flash pho-

tolysis dye technique to measure the velocity profiles in liquid-liquid jets and com-

pared their experimental profiles with the models of Yu and Scheele (1975) and

Gospodinov et al. (1979). Although they found reasonable agreement with the jet

radius and the velocity profiles with both models at the highest Reynolds numberexamined, the agreement deteriorated at lower Reynolds numbers as the interface

contraction increased. These issues are further explored in this chapter.

These previous numerical approaches relying on boundary-layer theory

have, by necessity, only limited regions of validity. They assume that the jet

is at steady-state, and the boundary-layer assumption is then used to simplify

the equations of motion. This assumption necessarily limits their validity to high

Reynolds numbers, where this approximation is considered valid. Vrentas and

Vrentas (1982) have shown by comparing boundary-layer results of Duda and

Vrentas (1967) and full equation simulations by Omodei (1980) for a free jet that

a Reynolds number greater than 1000 must be reached before the boundary-layer

limit is valid. This conclusion was confirmed by a more recent experimental and

numerical study by Gonzalez-Mendizabal et al. (1987). So far, similar comparisons

for liquid-liquid jets are not available since solutions to the full equations of motion

have not yet been performed for a liquid-liquid jet. It is the objective of the work

described in this chapter to fill this gap.

The method discussed in chapter 3, after validation on several test prob-

lems, is used to calculate the axisymmetric steady flow of a liquid-liquid jet. The

results are compared with experimental data of Richards and Scheele (1985) and



74

the approximate numerical results of Yu and Scheele (1975) and Gospodinov et

al. (1979). Also, interface shapes are compared with a new macroscopic momen-

tum balance and previous macroscopic energy balances (Addison and Elliott, 1950;

Anwar et al., 1982). The steady-state jet is a special case of the more general prob-

lem of the analysis of the dynamic jet and the same numerical technique presented

is used in chapters 6 and 7 to examine this more general problem, which involves

not only the calculation of the steady jet radius and velocity, but also the transient

regions including drop volume before jetting, jetting velocity, drop sizes produced

by the jet, jet length, and disruption velocity.

5.2 Problem Definition and Formulation

The flow configuration is shown in Figure 5.1. It corresponds to the

experimental setup of Richards and Scheele (1985), details of which can be found

in that reference. Briefly, a jet of xylene is injected vertically from a circular nozzle

upwards into a tank of mutually saturated stationary immiscible water. Figure

5.1 shows the stationary tank of water (fluid 1) with density ρ1 and viscosity µ1

with the jet of xylene (fluid 2) flowing upward with density ρ2 and viscosity µ2,

from a nozzle of inner radius R and outer radius Ro with average velocity v. The

interfacial surface tension is σ, 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 L3 and the nozzle is of total length L4.

5.2.1 Continuum Formulation of the Equations using the CSF Method

It is assumed that the flow in each phase is axisymmetric, viscous, and

incompressible. The continuity equation is given in cylindrical, axisymmetric

coordinates (r, z) by equation (3.1) where (u, v) are the radial, axial components

of the velocity field respectively.2 The dynamic momentum equations are given

2 Note that (v, u) is used by Yu and Scheele (1975) for the velocity field.



75

Nozzle

Outflow Boundary

r

z

n

Axis ofSymmetry

ContinuousFluid 1F = 0

µ1, ρ1

DispersedFluid 2

F = 1µ2, ρ2

JetInterface

σ

Tank Wall

L3

InflowBoundary

g

R

v

L1

L2

L4

Ro

r=a(z)

Figure 5.1: Liquid-liquid jet flow configuration (not to scale).



76

by equations (3.2) and (3.3) with p the pressure, gr[= 0], gz [= −g] the radial and

axial components of the gravitational acceleration, and τ rr, τ zr, τ rz , τ zz are the

components of the Newtonian stress tensor given by equation (3.4). The curvature

of the liquid-liquid interface κ is given by equations (3.5) – (3.7) and (3.62) (Kothe

et al., 1991; Brackbill et al., 1992).

The basis of the VOF method is the fractional volume of fluid scheme for

tracking free boundaries. In this technique, a scalar function, F (r,z,t), is defined

by equation (2.44). The evolution equation (3.8) for the fluid function marker

field, F , shows that the interface moves with the fluid. The density and viscosity

fields are obtained from equations (3.9) and (3.10) from which it is seen that F

has the physical significance of being the relative volume fraction of fluid 1.Note that it is possible to relate the steady-state VOF function F (r, z) to an

equation of the interface used by other workers (Omodei, 1980; Reddy and Tanner,

1978; Georgiou et al., 1988; Malamataris and Papanastasiou, 1991; Adachi et al.,

1990) of the form r = a(z), where a(z) is the radial distance to the interface at

axial position z (see Figure 5.1):

F (r, z) = 1− H

(r−

a(z)) (5.1)

and H(x) is the Heaviside step function defined by:

H(x) ≡

1, x ≥ 00, x < 0

(5.2)

Equation (5.1) may be used in equations (3.5) to (3.7) to calculate curvature for

this case (Omodei, 1980; Reddy and Tanner, 1978):

κ =1

a

1 +

da

dz

2− d

dz

da

dz 1 +

da

dz

2

(5.3)



77

Massless marker particles can be placed in the flow to follow fluid elements

if necessary for comparison with experimental results. Local velocities (u p, v p)

of the marker particles are used to update the positions by use of the kinematic

relations (3.11). For 2-D axisymmetric flows the streamfunction ψ can be defined,

as usual, in order for the velocity to be divergence free (i.e., by satisfying the

incompressibility condition (3.1)) in equation (3.12). Then the streamfunction ψ

can be calculated from Poisson equation (3.13), given the velocity field.

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

in axisymmetric coordinates (r, z), with the boundary conditions as follows. For

the solid walls no-slip conditions are used:

u = v = 0 (5.4)

For the axis of symmetry at r = 0:

u = 0,∂v

∂r= 0 (5.5)

For inflow into the nozzle, fully developed flow is assumed:

u = 0, v = 2v

1 − r∗2

(5.6)

where v is the average velocity in the nozzle and r∗ ≡ r/R, the dimensionless axial

distance. The experimental ratio L4/(2R) was assumed to be sufficiently large

to guarantee this condition (Richards and Scheele, 1985). However, it is seen in

section 5.3 that this may not have always been the case.

The choice of the outflow boundary condition can pose a problem, partic-

ularly for low flow rate calculations, because it can influence the entire flow field

adversely. 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

it is assumed that there is no change in the axial direction (Nichols et al., 1980;



78

Hirt and Nichols, 1981):∂u

∂z=

∂v

∂z= 0 (5.7)

It is expected that the higher the speed of flow, the less influence this bound-

ary condition will have on the upstream flow, as pointed out by Patankar (1980).

Alternatively, a “free boundary condition” could be used (Malamataris and Pa-

panastasiou, 1991; Papanastasiou et al., 1992).

The jet interface is assumed to be “pinned” at the nozzle lip, a fact that

is observed experimentally, so no contact angle need be specified in this problem.

However, for problems involving the effects of wall adhesion characterized by a

given equilibrium or dynamic contact angle, θ, such as the evaluation of the

interface separating two stationary immiscible fluids, this condition can be imposed

by requiring that the unit normal to the interface at the wall contact line, n, satisfy

(Kothe et al., 1991):

n = nw cos θ + tw sin θ (5.8)

where nw and tw are the unit normal and tangent to the wall at the contact line

respectively.

The original SOLA-VOF 2-D program (Nichols et al., 1980) is well suited

for high Reynolds number flows, including those involving free surfaces. Among

the latter, however, it is better suited for gas-liquid than for liquid-liquid systems,

and relaxing this limitation has been an important part of our efforts. Thus, we

have implemented extensive modifications in the original program. Both planar

and axisymmetric 2-D flows can be simulated. The momentum equations are

finite-differenced on a locally variable, staggered mesh using the control volume

approach, as illustrated in Figure 3.1. As Figure 3.1 shows, the radial velocity

ui+ 1

2,j and axial velocity vi,j+ 1

2are centered at the right face and top face of each

cell respectively, whereas the pressure, pi,j , and marker function, F i,j , are located

at the center. Further details on the finite-difference expressions used can be found

chapter 3.



79

5.2.2 Macroscopic Balance Approximations

Simple relations for the radial interface position as a function of axial posi-

tion can be obtained by integrating the continuum equations of the previous section

over the volume of the liquid-liquid jet, resulting in the jet macroscopic momen-

tum and energy balances (Bird, 1957). All dimensionless variables in this chapter

use R as the characteristic length and v as the characteristic velocity. Equations

(3.1) to (3.11) can be simplified if it is assumed that the jet is at steady-state,

boundary-layers exist at the interface in the continuous and dispersed phases, the

interface corresponds approximately to a cylinder with dimensionless local radius

a∗(z∗) (see Figure 5.1) and dimensionless axial position z∗ ≡ z/R, a parabolic

velocity profile exists at the nozzle exit, a fully relaxed, flat, velocity profile existsdownstream, and viscous terms are neglected. Using these assumptions, and inte-

grating equation (3.3) for the dispersed phase over the volume of the jet, we have

the macroscopic momentum balance (derived in appendix D):

0 =N jζ

2a∗4 +

2

W e

a∗3 − a∗2

+

4

3a∗2 − 1 (5.9)

where ζ ≡ z∗/Re2 is the axial position downstream scaled with the dispersed

phase Reynolds number, Re2 ≡

2R ρ2

v/µ2

, F r≡

v2/2Rg is the Froude number,

W e ≡ 2Rv2ρ2/σ is the Weber number, and N j ≡ ±Re2F r

ρ1ρ2

− 1

is the buoyancy

number appropriate for a liquid-liquid jet pointed upwards (+) or downwards (−)

respectively, which reduces to N j = ∓Re2F r

for a free jet (ρ1 = 0). This simple

form of the momentum balance for a liquid-liquid jet has not to our knowledge

appeared before in the literature. Equation (5.9) can be obtained as a limiting

case of equation (14) in Anwar et al. (1982) by neglecting the viscous terms and

integrating over ζ from the nozzle tip to a flat profile downstream.

Equation (5.9) reduces to the Slattery and Schowalter (Slattery and Scho-

walter, 1964; Duda and Vrentas, 1967) and Gavis (1964) result for N j = 0:

0 =2

W e

a∗3 − a∗2

+

4

3a∗2 − 1 (5.10)



80

Finally, equation (5.10) in turn reduces to the Harmon (1955) result that a∗ =√

3/2 as W e → ∞. Equation (5.9) has an advantage over other macroscopic

balance equations in that its limiting form, equation (5.10), was found to be in

excellent agreement with the calculations performed by Duda and Vrentas (1967).

A similar approach has been used to develop the macroscopic mechanical

energy balance for free jets by Addison and Elliott (1950) by assuming an initially

flat velocity profile at the nozzle, and by neglecting viscous dissipation terms:

0 = N jζa∗4 +4

W e

a∗4 − a∗3

+ a∗4 − 1 (5.11)

Scriven and Pigford (1959) obtained the limiting form of this equation for W e →∞. Anwar et al. (1982) assumed a parabolic profile at the nozzle and obtained:

0 = N jζa∗4 +4

W e

a∗4 − a∗3

+ 2a∗4 − 1 (5.12)

This equation, with N j = 0, was also derived earlier by Gavis (1964), but he has

an error in the exponents of a∗ in the surface tension term. Note that in equation

(5.12) as W e → ∞, a∗ = 1/ 4√

2, a slightly different asymptotic result to that given

by equation (5.9). A more recent attempt to predict interface position at low Re2

is discussed by Adachi et al. (1990).

Interface positions predicted by equations (5.9) (MOM), (5.11) (AE), and

(5.12) (ABDW) are compared in section 5.3 with experimental data and numeri-

cal results from the present simulations (R), as well as from previous approximate

results of Duda and Vrentas (1967) (DV), Yu and Scheele (1975) (YS) and Gos-

podinov et al. (1979) (GRP).

5.3 Results and Discussion

If equations (3.1)-(3.11) are made dimensionless only five independent di-

mensionless groups result (see appendix E). Those used by Yu and Scheele (1975)

are the dispersed phase Reynolds number Re2, Froude number F r, Weber num-

ber W e, buoyancy number N j, and the continuous phase Reynolds number,



81

Re1 ≡ 2Rρ1v/µ1. The Re2, Re1 numbers represent the ratio of inertial to viscous

forces, F r represents the ratio of inertial to gravitational forces, W e represents

the ratio of inertial to surface forces, and N j represents the ratio of buoyancy to

viscous forces. In the single phase jet, only three groups, Re2, W e, and N j are

needed. Other possibilities for dimensionless groups, such as the viscosity ratio

µ2/µ1, or the capillary number, Ca2 ≡ We/Re2 representing the ratio of viscous

to surface forces may also be used.

A theory of how these groups affect the radius and velocities in the liquid-

liquid jet was developed by Yu and Scheele (1975). Briefly, they concluded that

“an increase in jet momentum resulting from a force acting in the same direction

as the jet motion will increase jet contraction, while any force that opposes jetmotion will reduce jet contraction.” Their results show that jet contraction should

increase with increasing values of Re2, N j, W e, µ2/µ1, and decreasing values of

Re1.

Table 5.1 illustrates the experimental cases studied by Richards and Scheele

(Richards, 1978; Richards and Scheele, 1985) and Duda and Vrentas (1967). For

comparison purposes, the previous approximate numerical solutions were either

obtained by running the actual programs (Yu and Scheele, 1975; Gospodinov et

al., 1979) or have been published (Duda and Vrentas, 1967) and are indicated in

Table 5.1. All jets in the present study contract; however, Goren and Wronski

(1966) showed experimentally that horizontal (N j = 0) free jets expand below

Re2 of about 14 to 20. Although not shown here, the present simulation (R) does

reproduce this result. Non-Newtonian jets can also expand and are discussed by

Metzner et al. (1961).

In the numerical simulation of the flow in this work, the flow rate at the

nozzle was instantaneously increased from zero to its final value from the initial

condition with xylene filling the nozzle and water filling the tank, in much the same

way as the physical system was started (Richards, 1978; Richards and Scheele,

1985) and the simulation followed in time until no further appreciable changes



83

were noted in the solution. Thus, an advantage of the dynamic simulation is that

generally no special initial guess (therefore, no parameter continuation procedure

such as from an asymptotic solution) is necessary to arrive at the final steady-state

solution.

A typical (coarse) mesh is shown in Figure 5.2, with the increased mesh

refinement confined, as illustrated, in the dispersed phase. A finer local mesh is

used inside the jet region, whereas a coarser mesh is employed in the much more

slowly moving outer region. The actual mesh size used for all runs involved 34 cells

in the radial direction (with 25 cells being confined to the interval 0 ≤ r∗ ≤ 1) and

68 cells in the axial direction. This cell distribution has been optimized through

mesh sensitivity studies. In addition, the effects of the effluent boundary distance,L1, the bottom distance, L2, and the outer wall distance, L3, were investigated

by sensitivity studies. The actual tank dimensions were L1 = 23 cm, L2 = 9 cm,

and L3 = 10.2 cm. Yu and Scheele (1975) found experimentally that placing an

annular disk flush with the nozzle tip (L2 = 0) had little effect on the measured

jet radii. In system 11, L3 = 5.25 cm due to placing a glass cylinder around the

jet, which had no observed effect on the experimental jet radii or velocity profiles

(Richards and Scheele, 1985) as compared to systems 5 to 12. These dimensions

result in dimensionless lengths of L1/R > 40, L2/R > 16, and L3/R > 9 for the

three nozzles.

It was found that, for these conditions, the use of L1/R = 10, L2/R = 1,

and L3/R = 3 were large enough to establish insensitivity of the results to the

actual values of L1/R, L2/R, and L3/R. For example, comparing L3/R = 3 to

6, no significant difference was found in the dispersed phase solution, which is

the sole region of the experimental measurements. Likewise, because of the high

speed of the jet, comparing L1/R = 5 to 10, no significant difference was found

in the dispersed phase solution so that L1/R = 10 was found to be sufficient not

to influence the upstream flow field, even though the jet velocity profile is not

completely relaxed. We can conjecture that the reason that boundary condition



84

0

-1

10

0 1 3

Wall

SymmetryAxis

Nozzle

r*

z*

Figure 5.2: Coarse mesh layout.



85

(5.7) works well at finite L1/R distances is that upwinding is used in the inertial

terms so that very little information is used at the outflow in the calculation

(Patankar, 1980). Note that not all of the flow domain is shown in Figures 5.5 to

5.10, since the region of experimental results is generally L1/R < 6. The present

simulations were calculated to a length of L1/R = 12.5 in Figure 5.3. It may be

that by using the “free boundary condition” of Malamataris et al. (Malamataris

and Papanastasiou, 1991; Papanastasiou et al., 1992) instead of equation (5.7) as

the outflow boundary condition, this distance L1/R = 10 for Figures 5.5 to 5.10

could be further decreased. This shortening of the computational domain could

become important if operating at Reynolds numbers Re2 lower than 1000.

Table 5.2: Key to data figures 5.3 – 5.16.

System Re2 Interface Dye Axial Radial

Position Trace Velocity Velocity

a∗(z∗) z∗(r∗, t) v∗(r∗) u∗(r∗)

Duda & Vrentas (1967) 1038 5.3 5.4

8 2079 5.5 5.8 5.11 5.14

2 1957 5.6 5.9 5.12 5.15

13 1160 5.7 5.10 5.13 5.16

A key to the Figures 5.3 – 5.16 showing the obtained simulation results,

is given in Table 5.2. The comparison of the results of the present simulations

(R), of case 4 from Duda and Vrentas (1967) (DV), and the interface positions

predicted by equations (5.9) (MOM), (5.11) (AE) (Addison and Elliott (1950),

and (5.12) (ABDW) (Anwar et al., 1982) with the experimental interfacial data

of Duda and Vrentas (open squares) is shown in Figures 5.3 and 5.4. The

comparison of the results of the present simulations (R) to experiments of Richards

and Scheele (Richards, 1978; Richards and Scheele, 1985) (RS data points),



86

and simulations using the programs of Yu and Scheele (1975) (YS model) and

Gospodinov et al. (1979) (GRP) and the interface positions predicted by equations

(5.9) (MOM), (5.11) (AE), and (5.12) (ABDW) are shown in Figures 5.5 to 5.16.

The experiments did not study the five dimensionless groups systematically, but

represent a sampling of typical jetting conditions for three nozzle diameters and

several flow rates. System 8 was replicated by systems 5 to 12 and system 13 was

replicated by system 14, while system 2 has no replicate. As Table 5.2 shows,

numerical results of Yu and Scheele (1975) are available for systems 1 to 14, and

numerical results of Gospodinov et al. (1979) are available only for systems 2 and 8.

However, system 13 has the lowest Reynolds number and greatest contraction. For

these reasons we focus our analysis on systems 2, 8, and 13. These three systemsalso represent typical experimental results of all 14 systems. The main comparison

with experimental data involves the interface position and the evaluation of the

location of marker particles released above the nozzle that are then tracked in time

(equations (3.11)). These marker particle traces compare directly to the laser dye

trace experiments from which the velocities have been evaluated (Richards and

Scheele, 1985). The jet radius and dye trace positions inside the jet, obtained

from high speed motion pictures, were measured. The dye trace positions wereoptically corrected for the difference in index of refraction between xylene and

water. Axial and radial velocity profiles were then calculated by a mass balance

technique that is described in subsection 5.3.4.

5.3.1 Duda and Vrentas Case 4

Case 4 from Duda and Vrentas (1967) was selected as a strict validation test

of the numerical method developed in the present work. This case corresponds to

a water jet injected downward into air, with the conditions used shown in Table

5.1. This case corresponds to a gas-phase outer liquid and is characterized by a

moderate Re2, low Re1, moderate W e, high µ2/µ1, and low N j. However, it also

has high surface tension, 73 dyne/cm, with large viscosity and density differences.



87

The case is simulated as a two fluid flow with the continuous phase having the

physical properties of air, L1/R = 12, L2/R = 0, and L3/R = 3. Thus, comparison

of the results with those presented by Duda and Vrentas could be used to roughly

estimate the errors introduced by the boundary-layer assumptions and neglecting

the air phase.

Figure 5.3 compares the interface position a∗(z∗) for the DV model and

the present simulation (R), and those predicted by equations (5.9) (MOM), (5.11)

(AE), and (5.12) (ABDW) as well as the interface position corresponding to the

available experimental data. Excellent agreement is shown for the DV and R

models and the experimental data. The DV model results exhibit a slightly greater

contraction than the R model, which is consistent with the Re2 = 1000 numericalsimulations of Omodei (1980) as reported in Vrentas and Vrentas (1982). The

slight deviations between the current method predictions and those of DV can

be attributed to the fact that ρ1 = 0 and µ1 = 0 for the present simulation,

and that the DV model is based on boundary-layer approximations obtained by

eliminating certain small (as Re → ∞) terms in the momentum equations. The

MOM, AE, and ABDW equations are qualitatively accurate, which is somewhat

unexpected considering their simplicity. The MOM and ABDW models are close

to the DV model at large axial position, where the assumptions made in deriving

them are more appropriate, while the AE model is more accurate near the nozzle

since it is guaranteed to match exactly at the nozzle lip. Gonzalez-Mendizabal

et al. (1987) state that “it is remarkable that [their equation (9), which is the

AE model] is generally more accurate than the more elaborate Duda and Vrentas

model” based on their experimental data. This is clearly not true in the present

case by examining Figure 5.3, and it is unreasonable to expect these macroscopic

balances to be quantitatively accurate considering the severity of the assumptions

listed in section 5.2.

Figure 5.4 shows the axial velocity v∗(r∗) as a function of radial coordinate

for axial positions z∗ = 5.19 and 10.38, which corresponds to axial positions scaled



88

z*

r*

DV

ABDW

AE

MOM

R

DV Case 4

0.5 0.6 0.7 0.8 0.9 1.0

0

10

20

30

40

50

Figure 5.3: Jet interface experimental data (squares) and numerical results (DV)of Duda and Vrentas case 4 compared to the present model (R),equations (5.9) (MOM), (5.11) (AE), and (5.12) (ABDW).

by the Reynolds number ζ ≡ z∗

/Re2 = 0.005 and 0.01, respectively. Here again

good agreement is obtained between the DV and R models, with the DV model

slightly faster due to the increased jet contraction predicted by the DV model, as

mentioned above.



89

v*

r*

z* = 0

z* = 5.19

z* = 10.38

DV

R

DV Case 4

0.0 0.2 0.4 0.6 0.8 1.0

0

1

2

3

Figure 5.4: Axial velocity profiles for VOF (solid line) compared to Duda andVrentas case 4 (dashed line) for axial positions z∗ = 0, 5.19, and10.38. Interface positions are indicated by circles.

5.3.2 Interface Positions

The remainder of the results presented here are based on the experiments of

Richards and Scheele (1985) and the corresponding simulations. Figures 5.5, 5.6,

and 5.7 compare the experimental interface positions a∗(z∗) computed with the

R, YS, GRP programs and the MOM, AE, and ABDW equations for systems 8,



90

2, and 13 respectively. Note that the experimental uncertainty in radial position

at a given axial position was about ±1% for experiments using nozzles 1 and

3 (Richards and Scheele, 1985). Thus, for systems 8 and 2, the R, YS, and

GRP models are all within experimental data uncertainty and not significantly

different from each other, although for system 2 the R and the GRP model are

closer to, and lie on the same side of, the data. For system 13, whereas the

R results are within experimental error and cannot be distinguished from the

experimentally determined interfacial position, there is a greater disagreement

with the YS model at large axial position. As with Figure 5.3, Figures 5.5, 5.6,

and 5.7 show qualitative agreement of the macroscopic balance equations MOM,

AE, and ABDW with the data, but no pattern of which one of the three, if any,is best seems to emerge. All that can be said in reviewing these data is that

the MOM, AE, and ABDW equations give a qualitative estimate of where the

interface is with very little computational effort, but they should not be used to

replace the more rigorous estimates of the present simulation, DV, YS, and GRP

as is perhaps suggested by Gonzalez-Mendizabal et al. (1987).

Anwar et al. (1982) simplified Yu and Scheele’s analysis, reducing the

problem to solving two nonlinear, coupled ODE’s, but we have found by direct

numerical comparison that this approach gives similar results to the YS model. We

do not report the Anwar et al. numerical simulation results since their approach

does not appear to represent an improvement in accuracy over that of the YS

model.

Although it is not apparent from Figures 5.6 or 5.7, the interfacial position

is not as smooth for the present numerical simulation results as for the other two

approximate models for systems 2 and 13. This roughness has been smoothed out

by a smoothing spline before plotting and appears as slight wiggles on the order

of the mesh spacing. It is more pronounced with higher jet contraction with the

same mesh spacing, and can be reduced by further mesh refinement. Figure 5.5 has

not been smoothed with a spline and illustrates the magnitude of the roughness.



91

z*

r*

R

GRP

YS

AE

ABDW

MOM

System 8

0.5 0.6 0.7 0.8 0.9 1.0

0

1

2

3

4

5

Figure 5.5: Jet interface experimental data of Richards and Scheele (squares)compared to the present model (R), Yu and Scheele (YS), Gos-podinov et al. (GRP), equations (5.9) (MOM), (5.11) (AE), and(5.12) (ABDW) for system 8.

The accuracy obtained is consistent with the fact that the present simulations are

accurate to between first and second order in mesh spacing h and to first order

in time increments ∆t. To minimize this roughness, and in practice to minimize

the number of iterations taken for each cycle, some smoothing of the F marker



92

z*

r*

ABDW

AE

MOM

YS

GRP

R

System 2

0.5 0.6 0.7 0.8 0.9 1.0

0

1

2

3

4

5


function is done internally in the program only to calculate the curvature κ from

equation (3.5), as discussed in the CSF references (Kothe et al., 1991; Brackbill

et al., 1992). However, it has been found that too much smoothing can lead to

nonphysical curvatures.



93

z*

r*

ABDW

AE

MOM

YS

R

System 13

0.5 0.6 0.7 0.8 0.9 1.0

0

1

2

3

4

5


5.3.3 Dye Trace Positions

Another approach to comparing experimental data with the numerical

procedure involves a direct comparison of the dye trace positions. Dye traces

were used in the Richards and Scheele (1985) experiments as a means to identify



94

the location of particles on the streamlines at given increments of time. These

data were later analyzed to obtain estimates of the velocities as discussed in the

next section. Since this last step involves certain simplifying assumptions, we

consider as a more faithful test of the comparison between the numerical and the

experimental results the direct comparison of the shape of a dye trace.

Figures 5.8, 5.9, and 5.10 show comparisons of dye trace and interface

position data with the present simulated dye traces using massless marker particles

(equations (3.11)) for systems 8, 2, and 13 respectively. As can be seen from

Table 5.1, system 8 has the highest Re2, Re1, W e, and lowest N j, indicating that

overall, inertial, surface tension and buoyancy forces are much larger than viscous

forces. System 2 has about the same Re2 as system 8, but a lower W e and higherN j, with a resulting larger contraction in the interface. Apparently the threefold

increase in N j (which increases contraction) affects jet contraction more than the

approximately twofold decrease in W e (which decreases contraction) for these two

systems. System 13 is notable for having the lowest Re2, Re1, W e, and highest N j,

with the result that it has the largest contraction in the interface. Since the flow is

mainly in the axial direction, parabola-like shapes are observed in the dye traces

for each time step. Experimental dye trace data do not exist near the interface

due to interference by optical refraction.

As can be observed from the shapes of the traces in Figure 5.8, a trend

appears of increasing error with time, about −2%. This error can be explained

by an error in the experimentally measured flow rate (by rotameter), which had

a reproducibility of about ±1.8% (Richards, 1978). Also, as a check on the

experimental data reduction done by Richards and Scheele (1985), a flow rate

percentage error defined as Qerr ≡ 100(Qcalc − Qexp)/Qerr was calculated from the

velocity profile data vs. the experimentally measured flow rate. As can be seen in

Table 5.1, this error ranged from about 0.01% to 8.9% in absolute value for the

14 systems investigated experimentally (Richards and Scheele, 1985), with system

8 having a low calculated error of +0.1%. So, as can be seen from Figure 5.8,



95

z*

r*

Dye Trace

Interface

System 8

0.0 0.2 0.4 0.6 0.8 1.0

0

1

2

3

4

5

Figure 5.8: Dye trace (crosses) and interface (squares) experimental data of Richards and Scheele compared to the present model (R) for system8 with ∆t = 0.003735 s.

for system 8, the actual flow rate may be higher than the rotameter reading, and

from Figure 5.10 for system 13 lower than the rotameter reading. The experimental

traces appear to have the same shape as observed in the present simulation for

systems 8 and 13.

However, in system 2, the experimental traces also appear to be flatter



96

z*

r*

Dye TraceInterface

System 2

0.0 0.2 0.4 0.6 0.8 1.0

0

2

4

6


than the present simulation prediction would suggest, Figure 5.9. This flatness

appears to be associated only with nozzle 2 for system 2, and to some extent

for system 1. A possible explanation is that this nozzle was not long enough to

develop the velocity profile fully. Langhaar (1942) has solved the entrance flow in



97

z*

r*

Dye Trace

Interface

System 13

0.0 0.2 0.4 0.6 0.8 1.0

0

1

2

3

4

5


a tube approximately assuming a flat profile at the inlet and found that:

u∗ = 0, v∗ = I 0 (γ ) − I 0 (γr∗

)I 2 (γ )

(5.13)

where I 0, I 2 are modified Bessel functions of the first kind of orders 0 and 2

respectively, and γ is an decreasing function of 2L4/(R Re2), a table of which



98

is given by Langhaar (1942). The parabolic profile (equations (5.6)) is recovered

when 2L4/(R Re2) = ∞ (γ = 0), and a flat profile when 2L4/(R Re2) = 0 (γ = ∞).

In order for the dimensionless centerline velocity v∗(0) to be at least 1.993 (γ =

0.4), Langhaar found that 2L4/(R Re2) had to be at least 0.304. All experimental

systems are above 2L4/(R Re2) = 0.4, except systems 1 and 2 (see Table 5.1). This

again suggests that the flatness may be due to an incompletely developed profile.

To test this hypothesis, the profile indicated by equation (5.13) was used as an

initial condition for the present simulation instead of the parabolic equation (5.6).

It was found that it is possible to reproduce the flatness in the traces in Figure

5.9 by using γ = 2.25, such that v∗(0) was 1.83, and simultaneously increasing the

flow rate by 3%.

5.3.4 Axial Velocities

Figures 5.11, 5.12, and 5.13 show the axial velocity v∗(r) as a function

of radial position at axial position z∗ = 2, 4, and 2, for systems 8, 2, and

13 respectively. Velocities were calculated using the dye trace measurements,

corrected for optical refraction at the interface using a mass balance technique

(Richards and Scheele, 1985). Briefly, the technique assumes a parabolic velocity

distribution at the nozzle exit, then constructs streamline position on each dye

trace to satisfy the mass balance between that streamline and the axis. Streamline

polynomials are then fit from these positions, and differentiated to give axial and

radial velocities. This procedure makes the velocities more prone to error than the

dye trace measurements. The maximum standard deviation in the axial velocities

expressed as a percentage was ±7.9% and ±5.8% for experiments using nozzle 1

and nozzle 3 respectively (Richards and Scheele, 1985). However, unlike the dye

trace data, the velocity profile data can be compared to all three models. As

can be seen in Figure 5.11, all three models appear to be within experimental

error and almost indistinguishable in system 8. Differences are observed to occur

among the predictions of the three models for systems 2 and 13, however. Here



99

the present simulation and the GRP model are much closer to the data (which are

much flatter) and to each other than the YS model. This can be explained by the

failure of the parabolic approximation of the velocity profile used in the YS model

momentum integral method to represent the actual profile adequately. Also, the

present simulation and the GRP model differ from each other near the interface,

with the GRP model approaching zero faster with radial position.

5.3.5 Radial Velocities

Figures 5.14, 5.15, and 5.16 show the radial velocity profile u∗(r∗) at axial

position z∗ = 2, 4, and 2, for systems 8, 2, and 13 respectively. Since radial

velocities are about two orders of magnitude lower than axial velocities, the error

in their measurement is considerable. The maximum standard deviation error in

the radial velocities was about ±100% for nozzles 1 and 3 (Richards and Scheele,

1985). Therefore they are to be used only as a measure of the order of magnitude

of the radial velocities. All three models show similar trends and magnitudes,

except near the interface where the GRP model shows a more pronounced kink in

the profile than the present simulation or the YS model, which could be due to

mesh refinement.

The velocity profile differences noted in the previous sections between the

YS model and the data can best be understood in the following way. Both the

GRP model and YS model assumed boundary-layer theory, which will be correct

at high Reynolds numbers. However, the YS model assumed forms for the velocity

profiles in the two phases, and solutions were obtained using a momentum integral

approach. The jet phase profiles were assumed to be parabolic, which can be

seen clearly in Figure 5.13. The GRP model relaxed this assumption, but kept

the boundary-layer assumption. The lack of fit is most probably not due to the

boundary-layer assumptions, which are valid at least at Reynolds number greater

than 1000 (Vrentas and Vrentas, 1982), but due to the assumption of the form

of the velocity profiles, since the GRP model does about as well as the present



100

v*

r*

YS

R

GRP

System 8

z* = 2

0.0 0.2 0.4 0.6 0.8 1.0

0

1

2

3

Figure 5.11: Axial velocity profile data of Richards and Scheele (triangles) com-pared to the present model (R), Yu and Scheele (YS), and Gos-podinov et al. (GRP) at axial position z∗ = 2 for system 8. Inter-face positions are indicated by circles.

simulation in predicting the data, and the present simulation assumes neither

boundary-layer theory nor forms for the profiles. This study also illustrates that

interfacial shapes alone are not sufficient to distinguish between models, since in

most cases the fits between YS and the data were quite good with the exception,



101

v*

r*

YS

R GRP

System 2

z* = 4

0.0 0.2 0.4 0.6 0.8 1.0

0

1

2

3

Figure 5.12: Axial velocity profile data of Richards and Scheele (triangles) com-pared to the present model (R), Yu and Scheele (YS), and Gos-podinov et al. (GRP) at axial position z∗ = 4 for system 2. Inter-face positions are indicated by circles.

perhaps, of Figure 5.7.



102

v*

r*

YS

R

System 13

z* = 2

0.0 0.2 0.4 0.6 0.8 1.0

0

1

2

3

4

Figure 5.13: Axial velocity profile data of Richards and Scheele (triangles) com-pared to the present model (R), and Yu and Scheele (YS) at axialposition z∗ = 2 for system 13. Interface positions are indicated bycircles.

5.4 Conclusions

A robust and stable direct numerical simulation has been developed by

combining the SOLA-VOF two-fluid algorithm (Hirt and Nichols, 1981) with

the Continuum Surface Force (CSF) algorithm (Kothe et al., 1991). It has



103

u*

r*

R

GRP

YS

System 8

z* = 2

0.0 0.2 0.4 0.6 0.8 1.0

-0.05

-0.04

-0.03

-0.02

-0.01

0.0

Figure 5.14: Radial velocity profile data of Richards and Scheele (triangles)compared to the present model (R), Yu and Scheele (YS), andGospodinov et al. (GRP) at axial position z∗ = 2 for system 8.Interface positions are indicated by circles.

successfully simulated high Reynolds number, high buoyancy number, and low

Weber number flows, up to the limit of physical instability. The use of this direct

numerical simulation allowed us to ascertain the validity of the other approximate

steady-state numerical schemes. We have also shown that the present simulation



104

u*

r*

R

GRP

YS

System 2

z* = 4

0.0 0.2 0.4 0.6 0.8 1.0

-0.05

-0.04

-0.03

-0.02

-0.01

0.0

Figure 5.15: Radial velocity profile data of Richards and Scheele (triangles)compared to the present model (R), Yu and Scheele (YS), andGospodinov et al. (GRP) at axial position z∗ = 4 for system 2.Interface positions are indicated by circles.

approaches the steady-state in a time-dependent fashion, thus allowing it to be

used in chapter 6 for dynamic studies. We have also introduced a new macroscopic

momentum balance predicting interface position that shows qualitative agreement

with the experiments.



105

u*

r*

R

YS

System 13

z* = 2

0.0 0.2 0.4 0.6 0.8 1.0

-0.20

-0.15

-0.10

-0.05

0.0

Figure 5.16: Radial velocity profile data of Richards and Scheele (triangles)compared to the present model (R), and Yu and Scheele (YS)at axial position z∗ = 2 for system 13. Interface positions areindicated by circles.

The following four conclusions can be drawn about the data and model

comparisons:

(1) The present simulation, the GRP and YS models, and the RS data

are almost indistinguishable from each other and agree well with the experimental



106

data at Reynolds numbers greater than about 2000 and with low contraction. This

is consistent with the assumptions (that is the boundary layer limit) made in the

models.

(2) The present simulation and the GRP models are within experimental

error and are within good agreement with each other for all the cases studied. All

cases studied, which have Reynolds numbers greater than about 1000, are expected

to be within the range of the assumptions of boundary layer theory. However,

even the GRP model is not expected to remain accurate as the Reynolds number

is decreased. The present program has been independently tested to Reynolds

numbers as low as 1, and no limiting high Reynolds number assumptions are

made.(3) The YS model differs significantly from the data and the present sim-

ulation results at high interface contraction, with the difference becoming more

pronounced the greater the contraction. This can be explained by the use of

parabolic forms in the YS model for the velocity profiles as used in the momen-

tum integral method.

(4) Interface positions predicted by macroscopic momentum and energy

balances give quantitative agreement with the data, but cannot be expected to

provide more accurate results than the more comprehensive numerical simulations.

In conclusion, we have applied our extended VOF and CSF methods to the

liquid-liquid jet problem. This problem has indeed been challenging for several

reasons. It involves a moving complex free surface with high surface tension, at

high Reynolds number, in the presence of gravity, a difficult problem even for

one-phase calculations. In addition, the experiments bear out the fact that the

operating region for the jets in this study places them at the limit of physical

stability (Richards and Scheele, 1985).

atomizaÇÃo_phd5

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