ILASS Americas, 19th Annual Conference on Liquid Atomization and Spray Systems, Toronto, Canada, May 2006

Computational investigation on the effect of velocity modulation on low-velocity round liquid jets

V. Srinivasan* , A. J. Salazar and K. Saito

Department of Mechanical Engineering University of Kentucky

Lexington, KY 40506-0028 USA

Abstract The growth of perturbations on the surface of a liquid jet has been sought to be the major mechanism behind liquid jet disintegration and atomization. The contributing factors to the initiation of disturbances are predominantly pre-sent in the nozzle interior in the form of pressure oscillations, turbulence and cavitation. These perturbations are further enhanced by the interaction of liquid jets with ambient aerodynamic forces. In the present study, periodic disturbances in the form of well defined velocity modulation were imposed on a cylindrical liquid jet exiting a noz-zle. The effect of such disturbances on the jet’s liquid-gas interface results in the formation of a wide variety of structures such as discs, bells and droplets. In the present paper, we investigate numerically the impact of these forced disturbances on the liquid jet behavior using a finite-volume-based code incorporating a Compressive Inter-face Capturing Scheme for Arbitrary Meshes (CICSAM) scheme on structured meshes to track the deformation of the liquid interface. The present study concerns low velocity jets and hence no turbulence model has been incorpo-rated. For small modulation amplitudes in the appropriate frequency range, bulging of the liquid jets with finite pe-riodicity is observed. Increasing the disturbance amplitude results in disc formation. Further increasing the modula-tion amplitude results in the break-up of the discs into rings that subsequently disintegrate into droplets due to the effect of aerodynamic forces. Computations with different combinations of modulation frequency, amplitude and liquid jet velocity are studied to identify their influence on liquid structures.

*Corresponding author

Intr tion oduc The studies concerning the disintegration of

round liquid jets are of critical importance in various industrial processes such as spray coating, spray drying, combustion and other chemical processes [1]. When a liquid jet emerges from a nozzle into ambience, the deformation of its liquid-gas interface and subsequent breakup processes are governed by a wide range of pa-rameters such as liquid jet velocity, interior nozzle de-sign (contributing to varying perturbation level), liquid and ambient gas properties, etc. Stability studies of low velocity laminar liquid jets have a long history of inves-tigation dating back to Rayleigh [2]. Rayleigh published mathematical descriptions of the breakup of jets and gave conditions under which disintegration of jets oc-curred. He described the instability of cylindrical, axi-symmetric inviscid liquid jets due to the growth of small disturbances on the jet’s interface, which disinte-grated into drops. Neglecting the ambient fluid, the viscosity of the liquid jet, and gravity, he demonstrated that a circular cylindrical liquid jet is unstable with re-spect to axisymmetric disturbances of wavelengths lar-ger than the jet circumference. Chandrasekhar [3] took into account the liquid viscosity and the liquid density, which was earlier neglected by Rayleigh[2], and showed mathematically that the viscosity tends to re-duce the breakup rate and increase the drop size. He also showed that the physical mechanism of breakup of a viscous liquid jet in a vacuum is capillary pinching. Weber [4] considered the effects of the liquid viscosity as well as the density of the ambient fluid and formu-lated analytical means to evaluate growth rates of freely issuing liquid jets. Taylor [5] showed that the density of the ambient gas has a profound effect on the form of the jet breakup. These studies indicate that the growth of finite perturbations present in the surface of the liquid jet is responsible for the disintegration of the liquid jet downstream. Many of the distortions and peculiar shapes of the liquid elements of the jet can be explained by a time and space development of weak initial distor-tions of momentum in travelling waves during propaga-tion. In carrying out analytical investigations, a distur-bance of finite magnitude is initiated in the jet. The growth/suppression of this artifically induced perturba-tion is commonly studied as a function of the jet pa-rameters such as jet velocity and radius. In real scenar-ios, it is not easy to quantify the finite value of the per-turbation that is initiated near the nozzle. Most often, these disturbances are caused due to several internal nozzle phenomena such as swirl, separation, flow cavi-tation and turbulence [6]. It might occur that most of them are highly stochastic in nature and hence, the pre-diction of liquid jet disintegration characteristics based on these quantities are highly non-deterministic.

In a different strategy, finite induced perturbations are imposed on liquid jets in the form of velocity modu-lations to study the resulting disintegration behavior. These forced liquid jets would carry with them, a well defined modulation amplitude and frequency. This ap-proach was first reported by Crane et. al [7]. They ex-perimented with the effect of modulations on the breakup of cylindrical water jets in air. His first account reported the break-up of a liquid-jets due to an inertial type mechanism induced by high amplitude, high fre-quency mechanical vibration of an orifice. The results clearly indicated that under high frequency vibrations inertial effects dominate the surface tension induced displacements. Extending the work of Crane et. al [7], McCormack et. al [8] carried out further experimental and theoretical analysis on velocity modulated jets. In their experiments, velocity modulation was realized by pressure variation upstream of the injector. They con-cluded that the application of mechanical vibration in the appropriate frequency range and at small vibration acceleration values can induce minute pressure fluctua-tions and trigger an capillary instability. With higher acceleration values, appreciable “bunching” effect in the liquid jet was observed. Formation of discs due to strong liquid bunching was evident. However, no de-tails of the aerodynamic interaction and the resulting spreading mechanism were discussed. Mass concentra-tion appeared with definite periodicity equal in com-parison to the imposed wavelength of modulation.

A modulated liquid jet issuing from a nozzle goes from a state of high shear inside the nozzle to a state of free shear in the ambience. A schematic of a modulated liquid jet interacting with the various force components is shown in Fig. 1.

Figure 1: Schematic representation of various interact-ing forces in oscillatory free surface flows.

The liquid bulk in the exterior domain of the nozzle is subjected to the aerodynamic forces which interacts with the surface forces leading to liquid-gas interfacial instability. As the liquid jet enters the gas domain, shear layers are generated due to the velocity difference be-tween the liquid and gas phases. The viscous interaction of the two fluids, in addition to the shear layer instabil-ity, contribute to the enhancement of unstable surface waves. In the case of modulated liquid jets, local mass accumulation occurs at finite periodicity due to the transient variation in liquid velocity and as a result of momentum conservation, which was analyzed by Meier et. al [9].

We give a short introduction to critical dimen-sionless numbers which have been identified to influ-ence the breakup of liquid jets. The basic parameters that influence the disintegration of a liquid jet include the mean jet velocity, the liquid and ambient properties. Using dimensional analysis, researchers [1] have identi-fied two non-dimensionlized numbers: Reynolds num-ber (Re) and Weber number (We). They are represented based on the liquid properties, characteristic velocity and length scale as,

Re l

l

UDUP

(1)

2

e lU DW

UV

(2)

In discussing the disintegration of round liquid jets

[1], U and D in (1) and (2) refers to the injection liquid velocity and diameter of the orifice through which in-jection takes place, respectively. While the Reynolds number (Re) contributes to explaining the ratio of iner-tial forces to viscous forces, the latter Weber number (We) defines the ratio of inertial forces to adhesive (sur-face tension) forces.

Meier et. al [9] in their study of unsteady liquid jets showed that the conservation of initial momentum is one of the governing effects in jet instability and de-composition. They formulated the mass distribution along the axis of an modulated liquid jet without the effect of liquid properties such as surface tension and liquid viscosity. For large amplitudes and low frequen-cies of modulation, the jets are decomposed in very peculiar shapes, which can be very simply explained by the collision of packets of concentrated mass on the axis of the jet motion. Meier et. al [9] worked on large diameter and high velocity jets, which explains his ne-glection of surface tension forces. However, it is well known that liquid properties such as surface tension and viscosity are crucial in deciding the magnifica-tion/suppression of the initial disturbances [4]. In dif-

ferent experiments with water jets, they demonstrated breakup of very low velocity liquid jets with the effects of modulations. Formation of disc like waterfilms were observed, when a liquid jet was imposed with large amplitude distortions and concluded the existence of bowl shaped films. The instabilities occuring in such a system were seen reproducible. More recently, funda-mental research on the disintegration of a sinusoidally forced liquid jet was performed by Chaves et. al [10, 11]. Their experimental setup assured less effects of turbulence, swirl or any related effects inside the nozzle so that an unperturbed jet is made available for investi-gation in the nozzle exterior. In their investigation, a wide gamut of jet morphology was obtained by varying the modulation characteristics, amplitude and fre-quency, along with the mean jet velocity and liquid-gas density ratios. Classification of the disintegration of harmonically excited round liquid jets revealed surface waves, upstream directed bells, discs, downstream di-rected bells, multiple chains of droplets and phase jump phenomena. When turbulent conditions existed inside the nozzle, stochastic atomization process was ob-served. Extending this work, Geschner et. al [12] cre-ated a non-dimensional map for the appearance of spray structures of a periodically excited round liquid jet. The influence of modulation parameters on the detection of variety of highly reproducible structures ranging from discs to droplet chain indicate that spray formation can be controlled as a deterministic process. The experi-mental investigations on the modulated jets provide us with a global understanding of the phenomena. In order to identify and comprehend the basic underlying mechanism, details on the non-linear interaction be-tween various forces involved in a free shear flows of round liquid jets needs further clarification. Probing finer details of the liquid bulk disintegration using ex-perimental techniques is sophisticated and involves trial-and-error based methodology. It should also be noted that the scale factor involved in atomizer testing is very complicated. This situation can be overcome by resorting to accurate numerical simulations which can give more insight to the underlying physics by focusing on the domain of interest.

Accurate treatment procedures are required to re-solve the inter-phase dynamics engineered by the inter-action of aerodynamic forces present in the ambience with the inherent forces present in the liquid, due to the surface tension, viscosity etc. Recently, numerical simulations of liquid jet disintegration have been per-formed using Marker and Cell (MAC) method [13], Volume of Fluid (VOF) method [14], cubic interpolated pseudo-particle method [15], level set methods [16], etc. The range of applications involving VOF is quite general and has been used for the simulation of liquid droplet impact, jet and droplet breakup to cite a few [17]. The authors simulating jet breakup and atomiza-

tion have been restricted to low Reynolds number simu-lations due to difficulties in obtaining numerical con-vergence. Also, numerical simulations with high den-sity ratios present challenges in resolving the phase dynamics and their influence on the surrounding field [18]. In the current study, we use a VOF based interface tracking method for computing liquid jet behavior with imposed oscillations. In addition to solving an oscillat-ing liquid flow field accompanied by high gradients, the density ratios used in the current simulations exceed 500. We restrict our numerical testing to finding the effect of modulation amplitude and frequency on the behavior of the liquid jets. Testing the effects of ambi-ent gas density (liquid-gas density ratio), and mean liq-uid velocity on disintegration is of concern in our future studies. Testing the effect of liquid properties such as surface tension and viscosity have to be clarified since the growth and decay of perturbations can be affected by changes in these variables. The details of the nu-merical methodology and governing equations used in this investigation are discussed in the following sec-tions. Governing Equations

The present model treats the liquid-gas two-phase fields as a single incompressible continuum with an effective variable density ȡ and an effective viscosity ȝ, which can be discontinuous across the liquid-gas inter-face. This is possible by treating the phase boundary as an embedded interface and adding the appropriate source terms to the conservation laws. These source terms are delta functions localized at the interface and are selected to satisfy the correct matching conditions at the phase boundary. The effect of gravity is neglected. The effect of surface tension forces acting on the inter-faces preserves the curvature.

The governing mass and momentum conservation equations are the Navier-Stokes equations, which in the basic form can be represented as:

0i

i

u

x

w

w (3)

i i i is

i j j

u u u uPF

t t x x x

U UQ§ ·w w ww w

� � � �¨ ¸¨ ¸w w w w w© ¹ (4)

where ui (i=1,2,3) represent the velocity components, P the pressure field and Fs is the force term arising due to surface tension. It can be expressed as,

(5) � �( )

' ' ' ,s

S t

F n x xV N G �³ dS

which is computed based on the curvature of the inter-

face denoted by ț’ and the unit normal vector n’ on the interface S(t)at x’. The surface tension term in the NS equations creates the most obvious difficulties, since it is a singular term. In several implementations of the method, these difficulties are manifest in both numeri-cal instabilities and/or noise, and in poor accuracy of capillary effects.

The location of the transient interface S(t) is deter-mined by using a Volume of Fluid (VOF) surface cap-turing methodology employing the volume fraction indicator function equation. The volume fraction Ȗ used in the VOF formulation is defined as:

0 1

0

1 for a point inside the liquid

for a point in the transition region

for a point completely outside the liquid

JJ � �

­°®°̄

(6)

The liquid-gas interface resides on the transition

region defined in the above definition. The indicator function, a Lagrangian invariant, obeys the transport equation of the form,

� � 0i

i

ut x

J Jw w�

w w (7)

From the definitions of the phase indicator function

Ȗ as described above, the effective local density and viscosity of the fluid can be estimated as: � �1l gU JU J U � � (8) � �1l gP JP J P � � (9) where the subscripts l and g represent the liquid and gas phase, respectively. Since the interface is within the transitional zone, the exact shape and location are not explicitly known, and the surface integral in (5) cannot be evaluated directly. This problem is solved using the Continuum Surface Force (CSF) approach [19], which represents the surface tension forces as continuous volumetric force acting in the transition region. The interfacial surface phenomena are replaced by smoothly varying volumetric forces derived by integration of the surface tension forces over the interface. By means of the CSF approach, the surface tension force is trans-formed into a volumetric force yielding, � �' ' ' ,

S

n x x dSVN G VN J� | �³ (10)

where the curvature N is given by:

. JNJ

§ ·� � ¨̈

�© ¹¸̧ (11)

The curvature is the divergence of the normal vec-

tor to an interface’s surface element. The normal vec-tors are determined by the non-dimensional gradient of the volume fraction Ȗ. Numerical Methodology

The numerical simulations were performed within the framework of OpenFoam [20] C++ libraries avail-able for continuum mechanics. A finite volume method for arbitrary cell-shapes in combination with a segre-gated approach is used to discretize the equations. Fi-nite volume based computations are performed with non-overlapping cells in the domain. These equations are iterated over time using a multi-step method. To reconstruct the fluxes from variables of adjacent cells interpolation of convective fluxes and difference ap-proximations for inner derivates are required. The con-vective fluxes are determined by Gamma scheme [21] which guarantees a bounded solution minimising the numerical diffusion of sharp changes in the gradient of the variable. The gamma scheme is a high resolution second order convection-diffusion differencing scheme based on Normalized Variable Diagram (NVD) [22] specially developed for unstructured meshes. The dis-cretization of diffusive fluxes present in (4) is carried out to minimize the non-orthogonality errors. For this reason, the viscous fluxes are decomposed in orthogo-nal and non-orthogonal parts based on the cell configu-ration. Central difference approximations are applied to the orthogonal parts while the face interpolation of the gradients of the dependent variables are used for the non-orthogonal parts. Details of this decomposition procedure have been elaborated by Jasak [21].

The pressure velocity coupling is handled with a Pressure Implicit Splitting of Operators (PISO) proce-dure [23] based on a modified Rhie-Chow interpolation for cell centered data storage. Pressure velocity cou-pling using PISO algorithm provides more accurate adjustment of the face mass flux correction according to the normal pressure correction gradient in transient simulations. The equations are solved sequentially with iteration over the coupling terms with time marching using a second order explicit scheme. The time step is limited by the CFL number, which in the present case is restricted to 0.25 to ensure accuracy and to resolve the dynamics created by highly fluctuating flow fields.

Symmetric matrices arising from the discretization of the governing equation using Finite Volume method were solved using Incomplete Cholesky Conjugate Gradient (ICCG) based iterative solvers while the asymmetric matrices were solved using Incomplete-Cholesky preconditioned biconjugate gradient methods.

The code was assembled using C++ class libraries available for computational continuum mechanics in openFoam. The discretization of (7) and hence the in-terface tracking is the key feature of this paper. Hence, we briefly present the underlying schemes used in this study.

In general, the volume fraction indicator equation should be able to preserve sharp interface with negligi-ble diffusion. However, some available classical VOF schemes have problems in maintaining the bounded nature of the volume fraction scalar. Due to numerical inaccuracy and grid refinement, discontinuity prevails over the reconstructed interface. VOF is an interface compression method which has been enhanced by Normalised Variable Diagram (NVD) based bounded compression schemes.

In the present study, interface capturing has been performed by using CICSAM (Compressive Interface Capturing Scheme for Arbitrary Meshes)[24], a fully conservative technique based on finite volume method. The CICSAM scheme based on Normalised Variable Diagram (NVD), switches between different high reso-lution differencing schemes depending on the orienta-tion of the interface to the flow direction, to yield a bounded scalar field and preserve both the smoothness and sharpness of the interface. The derivation of the scheme is based on the recognition that no diffusion of the interface can occur. No explicit reconstruction of the interface is needed. This scheme is particularly ap-plicable to sharp fluid interfaces. Further, the scheme does not need any operator splitting and is implicit with second-order temporal accuracy. The capability of the scheme in handling interface rupture and coalescence has been tested by Ubbink et. al [24].

The CICSAM scheme of Ubbink was formulated based on the idea of the donor-acceptor concept, i.e. as a scheme that varies as a function of the interface-cell face angle. The discretization depends on the interface velocity direction and the angle it makes with the inte-gration cell face. However, rather than choosing as base schemes the downwind and upwind schemes between the ULTIMATE QUICKEST scheme and Hyper-C scheme [25] are chosen, with Hyper-C scheme used when the cell face is perpendicular to the interface’s normal vector and the ULTIMATE-QUICKEST scheme used when the cell face normal vector is aligned with the normal vector of the interface. Mathe-matical description of the scheme can be found in [25]. In the case of highly transient flow fields interacting with the free surface as in the present scenario, use of a blended scheme can lead to stable and accurate solu-tions while preserving smoothness of the solutions [26]. Boundary Conditions

The computational domain with relevant boundary conditions used in the numerical simulation is shown in

Fig. 2. Since we are interested in observing the liquid jet behavior in the nozzle exterior, no internal portion of the nozzle domain is included. However, it is to be re-minded that the nozzle interior regions play an impor-tant role in determining the disturbance levels of exiting jets [1]. The simulation domain extends 2.4 mm × 1.6 mm in the X and Y directions respectively. The diame-ter of the liquid inlet orifice is 0.2 mm. The simulations are performed in an axisymmetric fashion with one cell thickness extending in the azimuthal direction. Hence, in the analysis of the results, we observe discs and sheets in a three-dimensional sense. The extra cell in the azimuthal direction helps obtain a quantifiable third dimension in providing details for the finite volume formulation. Periodic boundary conditions are assumed on either side of the faces. Pressure outlets representing the free stream conditions are imposed at the top and downstream boundary faces. The reference pressure in the system is set to 1 bar, standard pressure value at sea level. No-slip conditions are imposed on the wall boundaries. Velocity inlet conditions indicating the issuing liquid jet bounded by wall conditions are as-sumed.

Figure 2: Schematic of the computational domain used in the simulation with relevant boundary conditions.

In our simulations, the relevant perturbation parameters viz., modulation amplitude and frequency, are applied at the velocity inlet in addition to the specification of mean jet velocity. In specifying the liquid jet velocity, a uniform velocity profile distribution is assumed, al-though in reality a finite boundary layer exists and re-sults in a parabolic velocity profile distribution in case of laminar regime. Several studies [27] indicate that the velocity profile distribution affects the radial spread of the liquid jet downstream and hence its disintegration characteristics. This fact is supplemented by the fact that the dynamics of induced vorticity layers changes appreciably when the liquid emerges into a state of free shear. Details on the effect of velocity profile relaxation

in addition to velocity modulation will be taken up in future studies. The injection velocity of the liquid jet at the inlet is held constant at U = 20 m/s throughout the simulation unless stated otherwise. The liquid jet prop-erties used in the current study match liquid-ethanol properties, while the ambience is assumed to be filled with air. Standard density and viscosity properties of ethanol and air at STP have been assumed in the current simulation. The surface tension of ethanol in air is as-sumed to be 0.0225 N/m. Notice that the Reynolds number (Re) for the issuing jet corresponds to a value ~ 2600 which is just above the Laminar regime ( < 2300 ) [1]. In the present simulations, no turbulence model has been included. The importance of resolving the turbu-lent fluctuations is, however, highlighted in the later part of the discussions. Before proceeding to the discus-sion of the results obtained from the numerical simula-tion, we classify the parameter space using non-dimensional parameters as explained below.

Besides the liquid, gas properties and jet parame-ters such as velocity and diameter, our current analysis includes additional variables in the form of modulation frequency and amplitude. Applying Buckingham ʌ-theorem, the following set of dimensionless parameters are derived. The definition of Re* and We* are not the usual definitions as given by (1) and (2), since f, D and ȡl have been used as the fundamental flow quantities by which all the other standard ones have been non-dimensionalized [10,11].

DfSr

U

S ,

3 2

* l D fWe

UV

, 2

* D fRe

Q (12)

u

UH ' , * g

l

UU

U

Adding to the available set of non-dimensional numbers represented in (11), a modified weber number Wew is obtained by dividing the Weber number (We) by the Strouhal number (Sr).

3

w

UWe

f

UV

(13)

This definition given by (13), has been used [10] to

provide a clearer map of the regions of appearance of structures in liquid jets than the standard definition given in (11). In the present study, two discrete fre-quency values (f = 50 KHz, 100 KHz) in the given range of modulation amplitudes (İ = 0.1, 0.2, 0.3) have been chosen to study their effect on the disintegration of round liquid jets. In the high frequency range, a value of İ = 0.4 has been tested. The following section

details the results of numerical simulations and argu-ments concerning the formation of structures and the associated interface dynamics are discussed. Results and Discussions

In the experiments performed by Chaves et. al [10,11], they identified different possible jet structures such as waves, discs, upstream directed bells, down-stream directed bells, touching bells, droplet chains, stochastic disintegration, and a phase jump based on the Sr, Wew and İ parameters. Geschner [12] observed that the basic parameter is the modulation amplitude which is representative of the structures that are formed. The Strouhal number by its definition directs to Rayleigh’s theory of magnification of perturbation amplitudes. Identify Strouhal number as,

.Df USr Sr D

U f

S S§ ·

� ¨ ¸¨ ¸© ¹

(14)

If Ȝ is the representative wavelength, /U fO ,

and a finite modulation amplitude be applied , Sr indi-cates if the wavelength of the imposed perturbation exceeds the circumference of the liquid jet. If the ampli-tude is too small, no structures can be generated. Rayleigh’s theory [2] predicts that in the range Sr > 1, where the wavelength is shorter than the circumference of the jet, perturbations are damped. Linear theories predict bunching of the liquid under certain modulation amplitudes [8]. However, the linear theories cannot be held valid for İ > 0.003. From the experiments of Chaves [10,11] and Geschner [12], it was observed that prominent structures were produced with İ = 0.07 and above. A non-dimensional map created from the ex-periments of Geschner [12] constructed using dimen-sionless numbers, İ, Sr, Wew, is shown in Figs. 3 and 4. These graphs are used to compare our numerical results with actual experimental results in a qualitative sense.

Figure 3: Variety of structures observed from the ex-

periments of Geschner et. al [12] presented in the İ-Wew plane for comparing with our numerical simulations:ż

waves, Ÿupstream directed bells, Ƒ discs, ¸ down-stream directed bells, Ɣ droplet chains.

Figure 4: Variety of structures observed from the ex-periments of Geschner et. al [12] presented in the İ-Sr plane for comparing with our numerical simulations:ż

waves, Ÿupstream directed bells, Ƒ discs, ¸ down-stream directed bells, Ɣ droplet chains.

The first numerical test case presented here illus-trates the transient evolution of liquid jet ethanol in-jected with a mean jet velocity of 20 m/s into air me-dium with an imposed fluctuation frequency of 50000 Hz and fluctuation amplitude, İ = 0.1. These values depict a system with a strouhal number, Sr = 1.57 with a Reynolds number Re* = 1315 and a modified Weber number Wew= 5612. The applied velocity perturbation creates fast and slow sections along the axis of each liquid shell [28]. This results in accumulation of mass at finite spacing as defined by the wavelength of the su-perimposed perturbation. The same effect is shown in the computed solution, which indicates generation of surface waves and the bulging of the liquid jet with finite periodicity. Experimental observations [8] shown in Fig. 5 confirm our test case. The transient variation in axial velocity components as the jet proceeds down-stream induce radial velocity components which intrude into the gaseous medium. At this juncture, the interac-tion of aerodynamic forces with the capillary forces is crucial in deciding the fate of the protrusions.

Figure 5: Plot of Volume fraction of liquid ethanol

injected into air: Sr = 1.57, Wew=5612, İ = 0.1.

For low modulation amplitudes, the pressure distri-bution on the amplified wave due to surface tension effects, ı/D, overcomes the gas pressure ȡgUo

2 compo-nents and hence, the disintegration of the filament from the liquid base is averted. From Fig. 5, it is clear that the evolved discs are bent backwards due to the drag forces of the medium. The growth of the discs from the surface of the liquid jet are strong functions of the Rey-nolds number and the fluctuation amplitude.

The present study is based on a single fluid model with interface construction performed only in regions where a liquid-gas interface exist. Plotting the overall flow field existing in the domain would help us resolve the magnitude of flow field variation induced by the high density liquid flow in a low density gas medium. In essence, we present a contour plot of x-velocity com-ponent in the fluid domain.

Figure 6: Contour plot of x-component of velocity in

the domain, Sr = 1.57, Wew= 5612, İ = 0.1.

From Fig. 6, notice strong recirculation regions near the cusps formed by the modulated liquid jet. The axial velocity components are good representation of the existing local flow fields and the behavior of the liquid-gas interface. In the event that a strong recircula-tion zone is identified from the plot, we can conclude that this region can lead to increased pressure distur-bances on the liquid bulk and based on its magnitude a breakup event can be closely monitored. Also, clearly note the localized region of highly transient flow field close to the liquid-gas interface. This is attributed to the fact that the given set of modulation parameters do not contribute to intensified radial spread and hence lesser influence on the gas entrainment dynamics.

The next logical step implemented in the numerical simulation is to increase the modulation amplitude while retaining the frequency of modulation. The struc-

tures formed due to increased modulation amplitude, İ = 0.2 are shown in Fig. 7.

Figure 7: Observed liquid jet disintegration with an

increased modulation amplitude İ = 0.2 with Sr = 1. 57, Wew = 5612.

As the modulation amplitude is increased to a cur-rent value of İ = 0.2, the jet elongation along the radial direction is seen pronounced. The propagation of the liquid jet downstream involves radial spreading. As the liquid travels downstream with a modulation wave im-posed on it, the central core responds quickly to these pressure waves while the outer region, formed by the liquid-gas interface, lags behind due to viscous effects. This induces an shear instability close to the interface and leads to shedding of small eddies in the vicinity of the interface. From Fig. 7, observe that the intact nature of the discs close to the injector exit due to the dominat-ing surface tension is overcome by the aerodynamic forces present in the medium downstream.

Figure 8: Capturing pinch off mechanism at the edge of

the disc occuring resulting in an overall upstream di-rected bell structure.

Also, as the amplified surface wave projects into the gas medium, an instability arising from the base connecting the disc to the liquid core is triggered. The breakup occuring near the rim of the projecting disc

from the liquid bulk can be related to a Rayleigh-Taylor type instability. As the liquid segment expands radially, sheet thinning occurs due to the effect of capillary forces.

It can be observed from Fig. 8, that the width of the pinched off liquid bulk is larger than the base ligament thickness. We studied the pinch off mechanism with detail and observed that liquid accumulation at the tip of these discs occurred until the aerodynamic forces overcame the stiff resistance of the capillary forces re-sulting in breakup [29].

Due to the effect of surface tension, the liquid sur-face tends to pull back the extending ligament leading to accumulation of fluid near the tip of the liquid fila-ment. Thinning of the cross wise extending ligament occurs due to drag forces which pull the ligament away from the jet surface to which it is attached, while the surface tension force acts to stabilize this surface insta-bility [30]. The thinning process continues as long as the ligament is connected to the liquid surface. At the same instant, the surface tension near the edge of the ligament acts to stabilize this thinning process by creat-ing a drop. When the aerodynamic forces overcome their surface tension counterpart, liquid sheet disinte-gration occurs. The stretching of the ejected ligament and surface deformation rate is a strong function of the local fluctuating velocity components. Also observe that the length of the disc element from the base liquid is longer compared to the previous case with a modula-tion amplitude of İ = 0.1. The increase in elongation of the liquid filament can be attributed to the increased rate of change of radial velocity components towards the disc periphery [31].

Disc disintegration at the rim obtained from our numerical simulations is shown in Fig. 9. The tip of these liquid sheets extends into the gas domain inducing vorticity components as clearly seen in Fig. 9(a). The continuing interaction of the eddies with the liquid tip results in instability similar to the Rayleigh-Taylor type. The core of the eddies forms a depression and pulls the liquid element away from the attached sheet base. This stretching mechanism is shown in Fig. 9(b). The insta-bility propagates along the sheet and at this juncture, the aerodynamic forces overcome the surface energy of the liquid disc. The surface tension at this juncture helps in the detachment phenomenon Fig. 9(c). Our current simulation can predict the formation of rings in the three dimensional sense, although in reality, drop formation occurs.

(a) (b)

(c) (d)

Figure 9: Breakup of liquid jet near the disc rim: Sr = 1.57, Wew= 5612, İ = 0.2.

We can, hence, predict the thickness of the cylin-drical rings which can be closely approximated to the diameter of the droplets. Even as disintegration occurs, the waves on the sheet continue to grow and shed off ligaments [32] (portion cut off from the rim of the liq-uid discs), while obtaining a continuous supply of liq-uid from the liquid core.

Plotting the contours of axial velocity components reveals increased radial spread and entrainment of gases into the liquid domain. Further, increased recirculation regimes can be identified from Fig. 10.

Figure 10: Contour plot of x-velocity component indi-cating the presence of gas vortices between the bulging

liquid discs: Sr = 1.57, Wew = 5612, İ = 0.2.

Although with the current set of parameters disin-tegration of the liquid sheets takes place, the flow field is not seen very much distorted.

A further increase in the modulation amplitude to İ = 0.3, leads to increased intensity of jet disintegration. Contour plot (11) reveals decreasing liquid core diame-ter as we progress downstream.

Figure 11: Disc disintegration into ligaments and drop-

lets: Sr = 1.57, Wew = 5612

As seen from Fig. 11, the instability of the liquid discs is enhanced by the increased modulation effect. The disc thickness is reduced due to increased radial fluctuations. Another striking feature that demands at-tention is the alignment of the disc with respect to the flow direction. The liquid bulk forming thick sheets, extending into the gas interface, are not bent back-wards. Rather, the alignment is almost orthogonal to the liquid flow direction. This feature is not seen in the immediate vicinity near the injector but when we pro-ceed a few diameters downstream. Referring to the chart in Fig. 3 representing different structures in the İ-Wew plane, we identify our current configuration with İ = 0.3 and Wew ~ 5612 stays close to the upper region where the upstream directed bells exist. Similar identi-fication can be made by viewing at İ-Sr plane from Fig. 3 corresponding to a value of İ = 0.3 and Sr ~1.57. It should be noted that an increase in the Strouhal number or a decrease in the modified Weber number would result in downstream directed bells. This investigation will be taken up in later studies.

Increased modulation amplitude results in an in-creased radial spread of the liquid jet. The disintegra-tion mechanism induces more complicated flow struc-tures. Fig. 12 portrays distorted flow field structure near the rim of the liquid sheet, disintegrating under the in-fluence of aerodynamic forces.

Figure 12: Contour plot of axial components of veloc-

ity: Sr = 1.57, Wew= 5612, İ = 0.3.

Increased vorticity components prevail near the tip of the sheets which can be identified from the recircu-lating regimes in Fig. 12. Notice that as we proceed downstream, the plot indicates presence of vorticity components (recirculation zones) away from the central liquid core. These represent the shed-off vortices from the tip of the liquid sheet rim. They travel downstream convected by the induced flow and interact in a non-linear fashion with evolving recirculation regions. The identification of these vortex pairs is crucial in order to establish the path of low diameter droplets which get caught in these low pressure vortical structures [33]. These shed-off gas vortices, on interacting with trans-lating bigger droplets, induce stretching and destabiliza-tion due to shear, leading to secondary breakup phe-nomena.

In the following computation, we retain the low modulation amplitude İ = 0.1 while doubling the initial frequency value (50 KHz). This leads to doubling the Strouhal number, Sr = 3.14 and a modified Weber number, Wew=2806. Since we have reduced the modu-lation amplitude to initial value, a decrease in the value of disc elongation cross-stream wise is expected. The plots indicate intact disc formation which can be seen as an extended surface wave with surface tension forces dominating over the gas pressure forces. The spacing of the bulging in the jet exactly matches the wavelength of the superimposed modulation wave.

Figure 13: Computations with Sr = 3.14, Wew=2806 results in thickening disc formation as the liquid jet

propagates downstream.

The contours of axial velocity in the domain reveal a very narrow perturbed flow field caused by the ex-panding liquid bulk. This configuration representing the presence of surface waves has been experimentally verified from Figs. 2 and 3.

Figure 14: Axial velocity components for low modula-

tion, high frequency laminar jets.

Fig. 14 highlights presence of high velocity regions spaced very closely in the vicinity of the liquid-gas in-terface. For identifying recirculation regions and their interaction with liquid elements a closer look is essen-tial.

The high frequency modulation results in bulging of liquid streams in close vicinity due to reduced im-posed wavelengths. The surface wave amplification due to velocity modulation creates radial velocity compo-nents with less relaxation time for their spread. This shortened time period greatly helps the surface tension effects in distorting any radial propagation of instabil-ity.

Figure 15: Velocity vectors near the growing liquid jet. Notice the small scale vortices near the rim of the disc due to shear generated by the liquid element intrusion

into the gaseous medium.

The presence of a thicker rim as the jet proceeds downstream is due to the accumulation of liquid by the action of capillary forces. In the current simulation do-main, no breakup of the liquid jet along the core or in the periphery was observed for the chosen set of pa-rameters.

With the strong bulging effect prevailing in the simulation, we introduce an increase in modulation amplitude İ = 0.2 with the same frequency of perturba-tion. The Strouhal number and modified Weber number are retained as the same value in the preceding simula-tion.

Figure 16: Developing disc configuration of the modu-

lated liquid jet. Eventually, the discs disintegrate in a rim-mode type under the influence of radial velocity

components. Simulation performed with Sr = 3.14, Wew = 2806, İ = 0.2.

Observe that the thickness of the rim at the edge of the liquid disc in the present case is higher compared to our simulation with Sr=1.57 and Wew=5612 with the same modulation amplitude İ = 0.2. From Figs. 3 and 4, we can conclude that the parameters, Sr=3.14 and

Wew=2806, İ=0.2, represent zones where disc struc-tures are frequently observed.

A detailed look in the vicinity of rim-mode breakup of the liquid sheet can be viewed in Fig. 17. The necking phenomenon just before the event of pinch off can clearly be visualized in Fig. 17.

Figure 17: Disintegration at the tip of the disc for Sr =

3.14, Wew = 2806.

Disintegration of the liquid disc results in forma-

tion of filaments which interact with the drag compo-nents of the medium can be visualized in Fig. 18.

Figure 18: Disintegrating liquid sheets and their inter-action with local flow field is shown. The trajectory of the disintegrated liquid bulk is modified based on the

decelerating drag and may result in secondary breakup.

The flow field associated with the disintegration of modulated liquid jets, Fig. 16, is depicted in Fig. 19. As previously performed, the plots indicate contours of axial velocity components. The pinched off liquid fila-ments from the extruded liquid sheets undergo stretch-

ing and deformation induces a variety of localized flow features. Increased radial spread is observed. However, as the liquid progresses downstream the disintegration mechanism induces more secondary flow structures and cross-wise fluctuations.

Figure 19: Axial velocity components depicting more

complicated flow structure as the liquid jet expands downstream. The radial fluctuations perturb the global

flow structure and hence the interface dynamics be-tween the gas medium and the teared off ligaments.

In the final case presented in this study, the modu-

lation amplitude is increased to İ = 0.4 with no varia-tion in frequency of modulation as in our previous computation.

Figure 20: Increased modulation amplitude results in

change in orientation of the bunching liquid sheets: Sr = 3.14, Wew = 2806, İ = 0.4.

The parameters opted for testing the current con-

figuration penetrate into the downstream directed bell regime similar to Fig.8. However in the current scenario a closer look at the İ-Sr reveals slow transition into the downstream directed bell regime in addition to the İ-Wew plane. Encountering this regime in the previous simulations, only the İ-Wew plane showed any correla-tion to structures where the bells were directed down-

stream. This indicates that the modulation amplitude plays a dominant role in transition between structures in comparison with the effect of modulation frequency. The plot of axial-velocity components clearly indicates complicated flow structures as the liquid jet interacts with the surrounding gas domain downstream. As al-ready discussed, there is a change in the orientation of the radially expanding liquid sheets and increased thin-ning of the liquid core along the axis. Note that, the currently simulated parameters with increased modula-tion amplitude moves towards regions where down-stream directed bells are formed (Figs. 2 and 3). This is similar to the previous case where structures aligning towards downstream were observed. However, in the current scenartio, the inclination of the liquid discs downstream is more predominant.

Figure 21: Contour plot of axial velocity components

for high amplitude, high frequency regime correspond-ing to Fig. 20.

As the liquid jet traverses downstream, a redistri-bution of the internal energy of the liquid jet takes place depending on the extent to which the velocity profile of the liquid jet varies from the uniform velocity at the central core. This redistribution process may create rigorous radial components by itself in addition to the superimposed modulation. In this concern, the viscosity of the liquid jet and the gas plays a vital role in deter-mining the relaxation parameters.

A plot of velocity components along the central liquid jet axis is shown in Fig. 22. The increase in ve-locity along the centerline and the decrease in velocity around the edges is largely the result of the non-uniform aerodynamic pressure forces on the jet surface. Classical aerodynamic theories of breakup are based on the concept that flow velocities in the radial direction are increased for the disc emerging from the liquid sur-face due to the acceleration of the gas over the tip of the liquid sheet. This results in a radial pressure drop across

the sheet which causes an increase in the radial veloci-ties near the liquid-gas interface. Over the depression present in the interface, a localized increase in pressure, combined with increased effects of surface tension, results in the necking experienced by the jet. The laws of conservation of momentum result in an increased axial components due to thinning of liquid core be-tween the extruded sheets resulting in low radial veloci-ties. The non-linear interaction and uneven distribution of the pressure forces on the interface results in core breakup due to the action of surface tension which tends to minimize the surface energy of the liquid bulk.

Figure 22: Plot of axial and radial velocity components

along the central axis of the jet. Notice that for high thinning rates of the liquid jet at the center, the axial velocity increases rapidly as the jet interacts with un-even aerodynamic forces. Sr = 3.14, Wew = 2806, İ =

0.4.

The radial spread characteristics of the modulated jets under a variety of conditions are shown in Figs. 23 and 24. The spread angle or the half-cone angle in this study is measured as the angle between the liquid jet axis and the tip of the stretched liquid sheet or the disin-tegrated liquid element. The trajectory of the disinte-grated sheets are not linear due to the competing aero-dynamic and surface forces. Larger liquid disintegrated elements carry more inertia and follow their initial tra-jectory while smaller ligaments follow the induced gas flow field. Entrainment of droplets into gas vortices more frequently observed with smaller droplets. Details of secondary breakup are not discussed in this paper.

Figure 23: Plot of spread angle as a function of modu-lation amplitude for St = 1.57, Wew = 5612.

Figure 24: Plot of developing spread angle as a func-

tion of modulation amplitude for St = 3.14, Wew = 2806.

Note that, the decrease in the slope of measured spread angle as the modulation amplitude is increased. As simulated before, a variety of structure transitions are observed as the modulation amplitudes are increased under constant Strouhal and modified Weber numbers. The nature of the curves presented in Figs. 23 and 24 indicates the influence of frequency in determining the spread of the jets. High frequency and high amplitude jets clearly result in higher disintegrated jet spreading. Maintaining a specified modulation amplitude and re-ducing the frequency creates a smaller reduction in spread angle and more importantly affects the spacing of the protruding sheets. However, the modulation am-

plitude has a more profound effect on the jet disintegra-tion as it affects the overall disintegration characteris-tics. Higher frequency perturbations lead to high non-linearity in the disintegration mechanism and poten-tially increases mixing between the gas and liquid phases. Conclusions

The current study has focused on investigating the effect of modulation amplitude and frequency on the jet morphology. The mean jet velocity has been kept con-stant during our simulations. The results have identified fundamental interface dynamics that take place when modulations on an otherwise unperturbed jet is en-forced. The results show non-linear interaction of ed-dies with the growing liquid filaments. Variation in modulation amplitude under constant Strouhal numbers lead to intensified disintegration process, while the ef-fect of Strouhal number is to define the periodicity and mutual interaction of the bunching liquid streams. The spread angle of the liquid jet increases as the modula-tion amplitude and frequency are increased. For higher modulation amplitudes under low frequency, jet breakup along the core is numerically concluded. Since the interface interaction with the surrounding gas me-dium invokes a wide range of length and time scales, inclusion of a good turbulence model is required to es-timate these interactions. The disintegration characteris-tics would would be greatly modified by the turbulence of the liquid and the gas medium. The compression scheme used in this study yields some wrinkling of the surface due to presence of very steep gradients of flow field resulting in sharp gradients of volume fraction induced at a very low time scale. More sophisticated studies need to be carried out to quantify the error that might creep into the simulation due to these dynamics. A complete three dimensional simulation is required to simulate discrete droplet formation and instabilities present in the growing sheets (discs). Since liquid prop-erties modify the disintegration rates, future studies would concentrate on simulating the effect of modula-tions on a variety of liquids with varying density ratios between the gas and liquid medium. In addition, the simulation domain would be extended farther down-stream in order to identify secondary breakup mecha-nisms and ligament trajectories. u velocity P pressure F force t time n normal vector Ȗ volume fraction ț curvature Sr Strouhal number Ȝ wavelength

Re Reynolds number 15. Yabe, T., Ishikawa, T., Wang, P.,-Y., Aoki, T., Kadota, Y., and Ikeda, F., Comput. Phys. Commun. 66: 233-242, 1991.

We Weber number Wew modified Weber number Re* Reynolds number based on Ȝ 16. Sussman, M., Fatemi, E., Smereka, P. and Osher, S.,

Comp. Fluids 27: 663-680, 1998. We* Weber number based on Ȝ Re Reynolds number 17. Zaleski, S., Li, J. and Succi, S., Physical Review

Letters 75: 244-247, 1995. ȡ effective density ȡ* density ratio 18. Scardovelli, R., and Zaleski, S., Annual Review of

Fluid Mechanics 31: 567-603, 1999. İ modulation ratio 19. Brackbill, J., Kothe, D.B., and Ruppel, H.M., J.

Computational Physics 100:335-354, 1992. U average velocity f frequency

20. Weller, W.O.H., Tabor G., Jasak, H., and Fureby, C., Computers in Physics 12(6):620-631, 1998.

ı surface tension of the liquid ǻu modulation amplitude

21. Jasak, H., PhD Thesis, Imperial college, University of London, 1996.

D diameter of nozzle ȝ effective viscosity

22. Leonard, B.P, Int. J. Numer. Methods Fluids 8: 1291- 1318, 1988.

Subscripts

23. Issa, R.I., J. Computational Phys. 62: 40-65, 1985. i,j index 24. Ubbink, O., and Issa., R., J. Comput. Phys. 153: 26-

50, 1999. s surface g gas

25. Leonard, B.P, Computer Methods in Applied Me-chanics and Engineering 88: 17-74, 1991.

l liquid

26. Ubbink, O., PhD Thesis, University of London, 1997.

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