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Axis-switching and breakup of rectangular liquid jets
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Axis-switching and breakup of rectangular liquid jets Mohammad Reza Morad , Mahdi Nasiri , Ghobad Amini PII: DOI: Reference:
S0301-9322(19)30538-5 https://doi.org/10.1016/j.ijmultiphaseflow.2020.103242 IJMF 103242
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International Journal of Multiphase Flow
Received date: Revised date: Accepted date:
20 July 2019 5 February 2020 8 February 2020
Please cite this article as: Mohammad Reza Morad , Mahdi Nasiri , Ghobad Amini , Axis-switching and breakup of rectangular liquid jets, International Journal of Multiphase Flow (2020), doi: https://doi.org/10.1016/j.ijmultiphaseflow.2020.103242
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Highlights
The evolution and disintegration of liquids jets ejecting from different rectangular orifices under various operating conditions were studied systematically.
Using a direct numerical solution and an adaptive meshing technique, the Navier-Stokes equations with a moving boundary have been solved for the two-phase flow problem.
The axis-switching, instability, and breakup of unperturbed and perturbed liquid jets in Rayleigh regime were investigated and the results were compared with experimental measurements and the linear stability analysis predictions.
Accounting for disturbances introduced to the jet for a wide range of wavenumbers and amplitudes, the effects of orifice geometry and inlet velocity are investigated.
Axis-switching and breakup of rectangular liquid jets Mohammad Reza Morada, Mahdi Nasirib, Ghobad Aminic, *
a.
Department of Aerospace Engineering, Sharif University of Technology, Tehran, Iran
b.
Department of Mechanical and Industrial Engineering, Concordia University, Montreal, Canada
c.
College of Engineering and Technology, American University of the Middle East, Kuwait
Abstract The behavior of low-speed liquid jets emerging from rectangular orifices into a quiescent air is studied numerically. After ejection, the rectangular cross-section transforms into an elliptical form along the jet and while axis-switching includes elliptical cross-sections only, the rectangular shape never establishes again. The optimum wavenumber, corresponding to the most dominant wave, is found to be greater in orifices with higher aspect ratios and, as a result, breakup length of the jet will be shorter. The breakup length decreases exponentially with the initial amplitude of disturbances. Moreover, it is observed that the form of final breakup leads to elimination of the satellite droplets at the optimum wavenumber with small and uniform main droplets. It is also shown that high-aspect ratio jets have a more extended range of effective disturbances. The results provide an in-depth insight into the effect of instabilities on axisswitching, breakup, and droplet formation.
Keywords: Rectangular Orifice; Liquid Jet; Axis-Switching; Instability; Droplet Formation
* Corresponding Author: [email protected] 2
1. Introduction
Characterizing of the liquid jet breakup ejecting from a plain orifice is very important in engineering due to its applications in systems such as combustion, thermal spray, agriculture, and inkjet printing. Disintegration of a liquid column into droplets increases the surface area of the liquid and subsequently increases the mass and heat transfer between the liquid and its surroundings. Based on the classical hydrodynamic theories, the breakup of the liquid jet has an instability origin (Rayleigh 1879). Accordingly, the jet is influenced only in a particular range of wavenumbers and there is an optimum wavenumber associate to the wave with the maximum growth rate which is responsible for the jet breakup. Although most of the relevant studies have been conducted for circular orifices, non-circular orifices also attracted the attention of several pioneer researchers; e.g., Rayleigh (1879), Bohr (1909), and Taylor (1960). However, the studies on non-circular jets are mostly limited to the elliptical orifices, mainly due to the usage of their axis-switching properties to measure dynamic surface tension of liquids (Bechtel et al. 1995, Saleh et al. 2019). Axis-switching is a phenomenon in elliptic jets which is manifestation of an alteration between the major and minor axes of the liquid jet along the injection direction. More recently, elliptic liquid jets were studied by Bechtel (1989), Kasyap et al. (2009), Amini et al. (2012, 2014), and Morad and Khosrobeygi (2019). Their results show that, compared to a circular orifice, flow physics associated with noncircular jets can change the breakup characteristics of a liquid column. For example, Amini and Dolatabadi (2011, 2012) showed that the elliptical jets have a higher perturbation growth rates and impose a higher range of effective perturbation wavelength than those of circular orifices. Among non-circular orifices, aside from elliptical ones, rectangular orifices were also found to be interesting due to their potential capabilities for passive flow control of mass entrainment and mixing (Miller et al. 1995, Chen and Yu 2014, Ghasemi 2019). While elliptical jets are intermediate configurations between commonly studied round and planar jets, the rectangular orifices present a special case; they incorporate the features of aspect ratio, similar to an elliptic orifice, and the corner effects, similar to triangular and square orifices (Sharma and Fang 2014). More recently, Wang and Fang (2015), Tadjfar and Jaberi (2019) and Jaberi and Tadjfar (2019) studied rectangular orifice shapes and investigated their characteristics under different ejecting 3
pressures. Their experimental data show that a rectangular jet has the shortest breakup length compared to other non-circular orifices, and axis-switching phenomenon is observed during its evolution as well. The majority of the studies on discharging jets are experimental and the theoretical analyses, mostly based on linear approaches, are much less. The linear theory of capillary breakup has been developed for small perturbations (Rayleigh 1879) and allows for a rather accurate prediction of the breakup time and length of capillary jets. However, as a result of the assumption that the perturbation magnitude remains small relative to the diameter of the jet, linear analysis fails near the point of jet breakup. Therefore, linear methods do not result in a prediction of the droplet sizes nor can they accurately forecast the production of satellite droplets (i.e., small droplets which are formed from liquid threads between the main drops). On the other hand, the theories of the nonlinear instability and breakup of liquid jets lead to rather cumbersome expressions. Therefore, the full nonlinear equations of motion for the jet need to be solved numerically. The numerical solution of the Navier-Stokes equation with a free surface is able to provide an indepth insight to the jet evolution and its breakup by modelling the formation of main and satellite drops. To the best of our knowledge, there is no numerical solution of the 3D problem for rectangular liquid jets in literature. The present study addresses the numerical analysis of instability of rectangular liquid jets, accounting for natural oscillations and disturbances introduced to the jet for a wide range of wavenumbers and amplitudes. The effects of orifice geometry and external disturbances on axis-switching, breakup and droplet formation are investigated. The wavelength of the axis-switching as well as breakup length obtained from the numerical results are validated with the experimental data available in the literature. In the rest of this article, the mathematical modeling and the solution method are presented first. Then, the results are presented to characterize axis-switching, breakup and drop formation for nonperturbed and perturbed jets.
2. Governing equations and numerical method
4
In order to simulate the ejection of the liquid jet from the nozzle into the quiescent air, a volume of fluid (VOF) method is employed. In this method, each cell at any instant has a value of γ equal to the volume fraction of water. The utilized VOF approach allows for simulating air-liquid flows including surface tension force by incorporating an interfacial compression flux term to mitigate the effects of numerical smearing of the interface (Rusche 2003). The main advantage of this method over other VOF methods commprises of intrinsic volume conservation and possibility of topological changes, e.g., breakup or merging of several entities. Governing equations are continuity, (
)
⃗
)1(
and conservation of momentum as, ( ⃗)
( ⃗⃗)
in which , v, , and
[ ( ⃗
⃗ )]
⃗
⃗⃗
)2(
represent the density, velocity, viscosity, and gravitational constant,
respectively. Density and viscosity could be written in terms of their value in the liquid and the (
gas as ⃗
)
(
and
) . The surface tension force is represented as
which is a function of the surface tension ( ), the curvature
(
⁄|
|), and
the volume fraction gradient. The volume fraction itself is obtained from the advection equation,
( )
( ⃗)
[⃗
(
)]
)3(
Here, the compression velocity (⃗ ) is obtained at the vicinity of liquid-gas interface through Rusche’s model (2003) as, ⃗
where
,
,
and
[
| | | |
| | ] | |
)4(
are the face volume flux, cell face area vector, compression coefficient
and face unit normal flux, respectively. Since the current study focuses on Rayleigh regime associated with low speed jets, simulations are performed with a laminar method.
5
To facilitate the validation of the numerical results, the orifices in the present study are chosen to have the same configurations as those used in the previous studies (Wang and Fang 2015, Farvardin and Dolatabadi 2013, Kasyap et al. 2009). The aspect ratio
, defined as the ratio of
the length (major axis) to the width (minor axis) of orifices is 1, 1.5, 3, and 4.5 for the orifices REC0, REC1, REC2, REC3, respectively (Table 1). In addition, an elliptical orifice E3 with aspect ratio of 4.5 and a circular orifice C0 will be examined as well. Having the same crosssectional area, the equivalent diameter for all orifices, is
mm.
Table 1. Dimensions of the orifices used in the numerical simulation Orifice name
Aspect ratio
Major axis (mm)
Minor axis (mm)
REC3
4.5
0.5588
0.1270
REC2
3.0
0.4580
0.1527
REC1
1.5
0.3260
0.2170
REC0
1.0
0.2667
0.2667
E3
4.5
0.6303
0.1400
C0
1.0
0.3000
0.3000
The numerical domain of the problem is a cube with dimensions of 4×4×50 mm (Fig. 1). The domain inlet, which is the orifice exit, is accommodated on the top wall. A no-slip boundary condition is applied over the solid walls. Boundary conditions on the sides and the outlet are atmospheric pressure. In addition, at the inlet of the numerical domain, a top-hat velocity profile, associated to a short nozzle, is introduced. The coupled set of Navier-Stokes equations is solved implicitly by a PISO (pressure implicit with splitting of operators) method at each time step. The initial time step is 10-6 sec with the possibility of adjustment by Courant number with the maximum value of 0.5. Density of air and water is assumed to be 1.2 kg/m3 and 103 kg/m3, respectively. The air-water surface tension is 0.07 N/m. In addition, viscosity of air and water is assumed to be 1.8 x 10 -5 N.s/m2 and 10-3 N.s/m2 , respectively. The inlet velocity used in this study ranges from 1.25 to 5.6 m/s associated to Weber numbers ( and Reynolds numbers (
) from 4 to 80
) from 250 to 1000.
6
On the x-y plane, a mapped mesh is used to form the shape of the orifice cross-section. The numerical domain is discretized initially with 0.07 mm grids in a Cartesian coordinate system. However, a dynamic mesh refinement is used in order to precisely capture the interface of the liquid and small droplets. The dynamic mesh refinement process is controlled by the value of γ. Consequently, the cells which have 0.001 < γ < 1 are refined one level in the x, y, and z directions. For the next iteration, the cells which contain liquid are refined again. Figure 2 demonstrates an example of dynamic mesh refinement method in the solution. For mesh dependency checking, the primary simulations were performed with two, three and four levels of refinement. Although in 2-level and 3-level refined cases, axis-switching phenomenon was captured, their wavelength, surface ruffles, and breakup point were different. Figure 3 shows breakup length of the liquid jet ejecting from orifice REC3 for three different mesh refinement levels at different Weber numbers. It can be seen that the accuracy of predicting breakup length in four-level mesh refinement is higher than the others. However, three-level mesh refinement has also an acceptable accuracy compared to two- and four- level mesh refinement. Due to a high computational time of four-level mesh refinement, three-level mesh refinement has been chosen for simulation. It is noted that in order to calculate the breakup length and droplet diameter, image processing is performed via Paraview and Matlab to extract the free-surface profile from numerical data. To minimize the effects of randomness, the results obtained from several simulations for each case are averaged within two standard deviations.
3. Results and discussion
Using the open-source code OpenFOAM6, the governing equations (1) - (4) are solved with a finite volume method for free and perturbed jets. Accordingly, two types of boundary conditions are imposed at the inlet of the numerical domain; for free jets the inlet velocity is constant while for the forced jets there will be a perturbed term added to the constant velocity. The results of the free jets are shown first and, after validating against experiments, the perturbed jets results are presented and discussed.
3.1 Free jets
7
Figure 4 displays a snapshot of numerical simulation of evolution of rectangular jet REC3 showing how the major axis of the liquid cross-section substitutes frequently with the minor axis, a property called axis-switching. We define the axis-switching wavelength,
as the distance
between any two consecutive axis-switching. Similarly, axis-switching number is defined as the number of repeating this phenomenon. Accordingly, Fig. 5 shows the magnitude of pressure, surface tension, and vorticity along the z axis at different axial locations within the occurrence of axis-switching. Once the liquid-gas interface is highly curved, the surface tension force becomes greater at the interface and, as a result, local pressure would be higher in those areas either. In order to eliminate the pressure difference, the liquid jet cross-section tends to change from rectangular to elliptical shape with a lower curvature at the interface. The elliptical shape causes a high pressure gradient at the two extremes of the major axis, where the radius of curvature is smaller. Such pressure gradient coupled with the effect of surface tension results in transformation of the cross-section towards a circular shape with the minimum surface energy in order to reach a balance of forces. Simultaneously, there is a velocity gradient that, using inertial forces in the bulk liquid, changes the cross-section to an ellipse with its major axis normal to the original one. This axis-switching occurs for a number of times in which the jet switches its minor and major axis within a wavelength until either the oscillations are damped by viscosity or the jet breaks up due to capillary instability. The vorticity around liquid-gas interface is another parameter that should be taken into account. The cross-section transformation in rectangular jets changes the vorticity contours in the proximity of the interface and alters the outside flow pattern and pressure as the jet progresses. Figure 6 presents the total vorticity magnitude and zcomponent vorticity for rectangular and circular liquid jets at the same axial position. Both total and z-component vorticities are more intensified around rectangular jets compared to those of circular jets which are more uniform. The vorticity intensity unbalances the pressure difference and assists the cross-section to deform easier. Figure 7 shows variations of dimensionless axis-switching wavelengths with Weber number for all jets. It can be seen that the axis-switching wavelength increases with Weber number almost linearly with the same slope for REC1, REC2 and REC3 jets. The results are validated against the experimental measurements of Wang and Fang (2015) for REC3 orifice. Based on experimental measurements for high aspect-ratio rectangular jets, Wang and Fang (2015) suggested the above slope to be 3.71. However, based on a linear stability analysis, Amini and 8
Dolatabadi (2012) calculated the wavelength of axis-switching for a non-circular liquid jet as,
√ (
)
,
(5)
where n is an integer representing the jet shape and is assigned as 2, 3, and 4, for elliptical, triangular and square jets, respectively. For a rectangular orifice, it can be assumed n=2, similar to an elliptic orifice. Therefore, for elliptical or rectangular orifices, the dimensionless wavelength is proportional to square root of We number by a factor of 2.56 (Amini and Dolatabadi 2012). However, this proportionality is based on the linear theory assuming departure from circularity of the cross-section is small, i.e., valid mostly for low aspect ratio jets. We note that nonlinear effects are strong at locations of the surface that are close to the jet axis (Gertsenshtein and Shkadov 1973) which is the case for jets with high aspect ratios. In addition, corner effects as an important feature of the rectangular orifice, might also be responsible for the slope deviation. We note that axis-switching is under the influence of different parameters such as viscosity, surface tension and inertia. The wavelength of the axis-switching increases with the jet velocity because the axial velocity directly increases the axis substitutions time intervals. The jet with the highest aspect ratio, REC3, has the longest axis-switching wavelength in comparison with REC2 and REC1 jets for all Weber numbers. The reason could be attributed to the time required for cross-section transformation that affects the axis-switching wavelength directly. For high aspect ratio jets, the lateral-inertial is stronger than surface tension forces and can resist longer. Therefore, with the same axial velocity, the completion of axis-switching cycle takes a longer time which leads to a longer wavelength. However, viscosity would constrain the liquid jet crosssection adaptation so that for REC3 orifice, it takes more time to substitute the axes. Figure 8 demonstrates how axis-switching at We
decays with distance down the orifice
along jet progression. The major axis of REC3 orifice is longer than that of other orifices, while its minor axis is the smallest. Down the jet, before disintegration, the major axis decreases while the minor axis increases until the liquid jet alters into a circular shape with the minimum surface tension energy. This phenomenon is repeated four to five times for each jet at all Weber numbers. It is noticed that at low Weber numbers, the liquid column cross-section reshapes into a 9
circular one and the oscillations that appear over its surface, lead to the jet breakup. In contrast, at high Weber numbers, the liquid jet is short of time to reach the circular shape and breakup occurs after fourth or fifth substitution process. For the case of REC0 orifice with the aspect ratio of unity, axis-switching could hardly be observed but it exists with short swings. This result is in agreement with the measurements of Rajesh et al. (2016). We also note that in an elliptical jet the axes switch and, as the jet proceeds, the cross-section changes from elliptic to circular (Kasyap et al. 2009, Amini and Dolatabadi 2012). However, in a rectangular jet, the liquid cross-section reforms from rectangular to elliptic shape and then acts the same as an elliptical jet without establishing a rectangular cross-section again. Results show that the cross-section of REC0 jet decays to a circular shape faster. By the end of axis-switching, when the initial disturbance grows enough, diameter of the jet decreases and breakup occurs. Figure 9 shows a comparison between the present simulations for REC3 orifice and the captured images by Wang and Fang (2015) for the same operating conditions. It could be seen that the breakup length increases with Weber number as a result of higher liquid inertia, which overcomes capillary instability and delays the disturbances growth rates. Figure 10 shows numerical simulation of liquid jet development into quiescent air for all orifices at
and
. The orifice with the highest aspect ratio, REC3, has the
shortest breakup length, whereas REC1 orifice, with the lowest aspect ratio, holds the longest breakup length. However, the breakup length of REC1 is still shorter than those of square jet REC0 and the circular jet C0. It is concluded that rectangular jets with high aspect ratios have higher perturbation growth rates, causing the jet to disintegrate at a shorter length compared to circular or square jets. Figure 11 shows breakup length for all jets at Weber numbers from 4 to 80 in conjunction with the experimental results of Wang and Fang (2015). The dependency of breakup length on aspect ratio and Weber number can be clearly seen in this figure. The liquid jets with higher aspect ratios show a shorter breakup length. As the aspect ratio decreases, the breakup length increases until it reaches its peak assigned to the circular jet. We observe a decrease of breakup length at higher We numbers, associated to the first wind-induced regime. Basically, at higher jet velocities, the aerodynamic forces become effective and shear stress of the surrounding gas assists capillary forces on more instability and, therefore, a faster disintegration of the liquid 10
column occurs. 3.2. Perturbed jets
It is well known that imposing perturbations on circular liquid jets could manipulate the breakup length and size and uniformity of the generated droplets and even merging of satellite droplets to main drops. In literature, numerous experiments have been conducted on circular jets to take advantage of instabilities for altering the liquid jet breakup characteristics (Strayer and Rankin 2001, Rohani et al. 2010, 2012). Based on linear stability analysis (Rayleigh 1879, Amini 2011), the disturbances imposed on the surface of a liquid jet with radius Req are described by a surface wave (Fig. 12) as,
(
in which and
)
(6)
represents the initial amplitude,
is the wavenumber,
is the wavelength,
is the growth rate of disturbances. The breakup length of the liquid jet, zb is defined as the
distance between the orifice exit to the breakup point, where amplitude of the oscillations grows and exceeds the unperturbed radius of the jet, i.e.,
. Accordingly, the breakup length and
growth rate are related as,
√ ⁄
(
) (7)
where β
dimensionless growth rate defined as, √
.
(8)
The linear stability analysis is based on imposing a surface disturbance on liquid jet at the nozzle exit ejecting with a constant velocity U0. However, in many cases, it is more practical to apply velocity disturbance while maintaining a constant jet radius. In the current work, in order to study the behavior of the jet with respect to perturbations, a variable velocity is imposed at the 11
(
inlet as and
). Here,
is the main jet velocity (equal to 4.0 m/s for all cases)
is the amplitude of initial velocity perturbation. By analysis of the mechanical energy that
enters into the jet, Moallemi and Mehravaran (2016) showed that velocity modulation radius modulations
and
are related to each other via, (9)
In our calculation, we employ equation (9) assuming
. Figure 13 shows the
evolution of the liquid jet emerging from REC3 orifice perturbed at different wavenumbers. Starting from zero, by increasing the wavenumber the jet breakup length reduces to a minimum value. From this point on, breakup length increases with wavenumber until the liquid column achieves its initial breakup length. The optimum disturbance wavenumber for which the liquid jet disintegrates with the shortest length is equal to
. Accordingly, we can expect that
at the optimum wavenumber the number of axis-switching be minimum. Figure 14 shows deviation of breakup length with respect to its average value for C0, E3, REC2, and REC3 jets at different disturbance wavenumbers. With distance down the jet, small oscillations appear on the liquid column that grow until disintegration point. The breakup length variation has a minimum value at the optimum wavenumber for each jet. As shown in Fig. 13, for wavenumbers in the range of 1.0 to 1.8, breakup length of the liquid jet issuing from REC3 orifice is the shortest. Based on equation (7), this is a result of higher growth rate of perturbation, β, which decreases the breakup length,
.
Using equation (7) and by measuring breakup length, the disturbance growth rates have been calculated and the results are presented in Fig. 15. For all orifices, the growth rate increases with wavenumber and then, after reaching a peak, decreases to the initial value. The optimum wavenumber and the range of wavenumbers that influences the liquid jets are increased at higher aspect ratios. For REC3 jet, and similarly E3, the optimum wavenumber is obtained to be around 1.5, whereas it is 1.2, 1.0, 0.9, and 0.8 for REC2, REC1, REC0 and C0 jets, respectively. The REC3 orifice also has the longest effective wavenumber range from 0 to 2.4, while this range is between 0 to 2.2, 1.6, 1.4 and 1.2 for REC2, REC1, REC0 and C0 orifices, respectively. The 12
maximum disturbance growth rate, occurred at optimum wavenumber, shifts to higher values for higher aspect ratio jets. Compared to the linear theory results (Amini et al. 2014), numerically obtained data in the present study predicts the previous experimental data better due to covering nonlinear effects particularly raised when the aspect ratio increases. Figure 16 depicts the variation of breakup length (
⁄
by amplitude of disturbances
⁄ ). Results show these two parameters have a linear relationship at small values of δ0,
corresponding to small perturbation amplitudes, while at higher δ0 values a deviation will occur. This result implies that the disintegration length decreases for large initial amplitudes of perturbation and the jet behavior will be nonlinear. Figure 17 shows the moment of breaking of non-perturbed jets C0 and REC3, and perturbed jet REC3. It can be seen that for non-perturbed jets, before liquid breakup, a thin ligament is generated between the main jet and the tip droplet which is still attached. This ligament stretches until breaking from one of its tips, while the other side is still attached to the jet column. At this position, the separated side moves toward the attached side to reunion. However, the second breakup also occurs at the remaining attached side and ligament separates completely making a satellite droplet. From Fig. 16 it could be seen that the length of ligaments in non-perturbed circular jets is longer than that of non-perturbed rectangular jets. This is due to the higher growth rate of disturbances in rectangular jets which reduces the time of shrinking of the liquid surface to about one-fourth so that the ligament has not enough time to be formed compared to that of a circular jet. As a result, it could be expected that formation of satellite droplets in rectangular jets is limited and, if appear, they are smaller in size. Additionally, for the circular jet perturbed at the optimum wavenumber, it can be seen that the breakup length has the shortest value, and the size of the produced droplets are very close to each other. Satellite droplets are clearly visible in Fig. 17 for the case of circular jet breakup in almost all wavenumbers except at the optimum wavenumber. Figure 18 shows the size of droplets formed after jet breakup for perturbed circular jet C0 as well as two rectangular jets REC2, REC3. It is seen that near the optimum wavenumber, droplets have smaller diameters compared to those of non-perturbed or perturbed jets at non-optimum wavenumbers. The smallest droplets exist in perturbed jets with the optimum disturbance 13
wavenumbers. Furthermore, jets emerging from higher aspect-ratio orifices produce smaller droplets which could be due to the short length of the dominant wave at higher growth rates. Mathematically, the droplet volume is equal to the integral of the jet cross-section over its wavelength, and a high wavenumber, corresponding to a short wave, produces a small droplet. The standard deviation of droplet diameter with respect to its average at different wavenumbers can be a measure of uniformity of generated droplets. This parameter has been calculated and results are presented in Fig. 19. Results confirm that the deviation of droplets generated from liquid jets perturbed with optimum wavenumber is lower than the value of deviation when wavenumbers are close to beginning and end of the range of effective wavenumbers. This is also a result of higher growth rate at the optimum wavenumber that shortens the breakup length leading to formation of less satellite droplets. In addition, at maximum growth rate, the variation of the jet boundary is faster and, therefore, long ligaments do not have enough time to be formed, be separated from the main body of the jet, and generate satellite droplets. Therefore, similar to circular jets (Rohani et al. 2012), the breakup length and the size of the main and satellite droplets are minimum at optimum wavenumber in rectangular jets.
4. Conclusion
Noncircular jets are found to be interesting due to their potential capabilities for passive flow control. A rectangular orifice is able to impose inherently a passive alteration on instabilities. To take advantage of a combination of active and passive instabilities, the present work addresses the behavior of rectangular liquid jets under the effect of disturbances introduced at the nozzle exit. The evolution and disintegration of liquids jets ejecting from different rectangular orifices under various operating conditions were studied systematically. Using a direct numerical solution and an adaptive meshing technique, the Navier-Stokes equations with a moving boundary have been solved for the two-phase flow problem. To the best of our knowledge, this is the first numerical solution for a two-phase jet with a rectangular free surface. The axisswitching, instability, and breakup of unperturbed and perturbed liquid jets in Rayleigh regime were investigated and the results were compared with experimental measurements and the linear stability analysis predictions. The flow-field around the liquid-gas interface including, pressure, 14
surface tension and vorticity contours are captured. Accounting for disturbances introduced to the jet for a wide range of wavenumbers and amplitudes, the effects of orifice geometry and inlet velocity are investigated. Studying the liquid jet evolution reveals that, similar to elliptical liquid jets, the axis-switching phenomenon is visible for rectangular jets. The wavelength of axis-switching increases linearly with the jet velocity and the slope shows a deviation from the theoretical results obtained for an elliptical jet. Additionally, by increasing the aspect ratio, the wavelength of axis-switching is increased which results in a shorter breakup length. For a rectangular jet, with aspect ratio of unity, the axis-switching is hardly visible. The breakup length and growth rates were computed under the effect of a velocity modulation at the nozzle exit, and the results were examined for a wide range of effective wavenumbers. For perturbed jets, the growth rate is found to be strongly dependent on the wavenumber of disturbances. The results indicate that the maximum growth rate which causes the shortest breakup occurs at wavenumbers around 0.9, 1.1, and 1.2 for the jets with aspect ratios of 1.5, 3, and 4, respectively. The range of influence of disturbances increases as the aspect ratio of the orifice increases. At a fixed wavenumber, a higher aspect ratio increases the inherent instability effect, causing the breakup length to be decreased. It is also observed that the satellite droplets are almost eliminated at the optimum wavenumber. Furthermore, liquid jets perturbed at the optimum wavenumber produce smaller and more uniform droplets compared to non-perturbed and perturbed jets at non-optimal wavenumbers. Author statement
Attached please find our revised manuscript IJMF-2016-727 entitled “Axis-switching and breakup of rectangular liquid jets” to be considered for publication in International Journal of Multiphase Flow.
Conflict of Interest The authors certify that they have NO affiliations with or involvement in any organization or entity with any financial interest or non-financial interest in the subject matter or materials discussed in this manuscript.
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Morad M. R., Khosrobeygi, H., 2019. Penetration of elliptical liquid jets in low-speed crossflow. Journal of Fluids Engineering 141, 1.
[17]
Rajesh, K., Sakthikumar, R., Sivakumar, D., 2016. Interfacial oscillation of liquid jets discharging from non-circular orifices. International Journal of Multiphase Flow 87, 1-8.
[18]
Rayleigh, L., 1879. On the capillary phenomena of jets. Proc. R. Soc. 71-97.
[19]
Rohani, M., Jabbari, F., Dunn-Rankin, D., 2010. Breakup control of a liquid jet by disturbance manipulation. Physics of Fluids 22, 107103.
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Rohani, M., Dabiri, S., Jabbari, F., Dunn-Rankin, S., 2012. Modeling the Breakup of Liquid Jets Subjected to Pure and Composite Disturbances. Atomization and Sprays 22.
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Rusche, H., 2003. Computational fluid dynamics of dispersed two-phase flows at high phase fractions. Imperial College London.
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Saleh, A., Amini, G., and Dolatabadi, A., 2019. Measurement of dynamic surface tension of suspension sprays based on oscillating jet technique. Atomization and Sprays 29 (1), 39-57.
[23]
Sharma, P.,
Fang, T., 2014. Breakup of liquid jets from non-circular orifices.
Experiments in Fluids 55, 1666. [24]
Strayer, B., Dunn-Rankin, D., 2001. Toward a control model for manipulating the breakup of a liquid jet. Atomization and Sprays 11.
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Tadjfar, M., Jaberi, A., 2019. Effects of Aspect Ratio on the Flow Development of Rectangular Liquid Jets Issued into Stagnant Air. International Journal of Multiphase Flow 115, 144-157.
[26]
Taylor, G. I., 1960. Formation of thin flat sheets of water. Proc. R. Soc. 259, 1-17.
[27]
Wang, F., Fang, T., 2015. Liquid jet breakup for non-circular orifices under low pressures. International Journal of Multiphase Flow 72, 248-262.
17
Fig. 1. Numerical domain (left) and the dimensions of orifices (right). All orifices have the same cross-sectional area.
18
Fig. 2. A three-level mesh for simulation of breakup of the REC3 jet at We=33.64.
19
Fig. 3. Mesh-independent checking of numerical solution for breakup length of REC3 jet in different refinement levels at different Weber numbers.
20
Fig. 4. Definition of axis-switching wavelength and axis-switching number (left), and the rectangular jet ejecting along z-axis from REC3 orifice (right)
at
.
21
Fig. 5. Pressure, surface tension and vorticity contours at different axial locations for REC3 jet.
22
Fig. 6. Distribution of total vorticity and z-component vorticity for rectangular jet REC3 (left) and circular jet C0 (right) at axial position z=0.7 mm.
23
Fig. 7. Variation of axis-switching wavelength by Weber number.
24
Fig. 8. Deformation of cross-section along the jet at We=33.64.
25
Fig. 9. Evolution and breakup of REC3 jet at different We numbers. Numerical results have been compared to experiments of Wang and Fang (2015) (Courtesy of Elsevier).
26
Fig. 10. Disintegration of the jets and droplet formation at
.
27
Fig. 11. Variations of breakup length for all jets at different Weber numbers.
28
FIG. 12. Modulation of jet velocity at orifice exit.
29
Fig. 13. Effect of disturbances with different wavenumbers imposed on REC3 jet at We
.
30
Fig. 14. Standard deviation of breakup length versus disturbance wavenumber at We
31
Fig. 15. Growth rate of disturbances versus wavenumber at We = 33.64.
32
Fig. 16. Dimensionless breakup length versus amplitude of the initial disturbance at optimum wavenumber for REC3 jet at
.
33
FIG. 17. The moment of breakup and formation of satellite droplets for non-perturbed jet C0 (left), non-perturbed jet REC3 (middle) and jet REC3 perturbed at kReq=1.3 (right).
34
FIG. 18. Variation of diameter of the generated droplets with wavenumber of the imposed disturbances.
35
FIG. 19. Standard deviation of diameter of the generated droplets versus wavenumber of the imposed disturbances.
36
Axis-switching and breakup of rectangular liquid jets Mohammad Reza Morad , Mahdi Nasiri , Ghobad Amini PII: DOI: Reference:
S0301-9322(19)30538-5 https://doi.org/10.1016/j.ijmultiphaseflow.2020.103242 IJMF 103242
To appear in:
International Journal of Multiphase Flow
Received date: Revised date: Accepted date:
20 July 2019 5 February 2020 8 February 2020
Please cite this article as: Mohammad Reza Morad , Mahdi Nasiri , Ghobad Amini , Axis-switching and breakup of rectangular liquid jets, International Journal of Multiphase Flow (2020), doi: https://doi.org/10.1016/j.ijmultiphaseflow.2020.103242
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Highlights
The evolution and disintegration of liquids jets ejecting from different rectangular orifices under various operating conditions were studied systematically.
Using a direct numerical solution and an adaptive meshing technique, the Navier-Stokes equations with a moving boundary have been solved for the two-phase flow problem.
The axis-switching, instability, and breakup of unperturbed and perturbed liquid jets in Rayleigh regime were investigated and the results were compared with experimental measurements and the linear stability analysis predictions.
Accounting for disturbances introduced to the jet for a wide range of wavenumbers and amplitudes, the effects of orifice geometry and inlet velocity are investigated.
Axis-switching and breakup of rectangular liquid jets Mohammad Reza Morada, Mahdi Nasirib, Ghobad Aminic, *
a.
Department of Aerospace Engineering, Sharif University of Technology, Tehran, Iran
b.
Department of Mechanical and Industrial Engineering, Concordia University, Montreal, Canada
c.
College of Engineering and Technology, American University of the Middle East, Kuwait
Abstract The behavior of low-speed liquid jets emerging from rectangular orifices into a quiescent air is studied numerically. After ejection, the rectangular cross-section transforms into an elliptical form along the jet and while axis-switching includes elliptical cross-sections only, the rectangular shape never establishes again. The optimum wavenumber, corresponding to the most dominant wave, is found to be greater in orifices with higher aspect ratios and, as a result, breakup length of the jet will be shorter. The breakup length decreases exponentially with the initial amplitude of disturbances. Moreover, it is observed that the form of final breakup leads to elimination of the satellite droplets at the optimum wavenumber with small and uniform main droplets. It is also shown that high-aspect ratio jets have a more extended range of effective disturbances. The results provide an in-depth insight into the effect of instabilities on axisswitching, breakup, and droplet formation.
Keywords: Rectangular Orifice; Liquid Jet; Axis-Switching; Instability; Droplet Formation
* Corresponding Author: [email protected] 2
1. Introduction
Characterizing of the liquid jet breakup ejecting from a plain orifice is very important in engineering due to its applications in systems such as combustion, thermal spray, agriculture, and inkjet printing. Disintegration of a liquid column into droplets increases the surface area of the liquid and subsequently increases the mass and heat transfer between the liquid and its surroundings. Based on the classical hydrodynamic theories, the breakup of the liquid jet has an instability origin (Rayleigh 1879). Accordingly, the jet is influenced only in a particular range of wavenumbers and there is an optimum wavenumber associate to the wave with the maximum growth rate which is responsible for the jet breakup. Although most of the relevant studies have been conducted for circular orifices, non-circular orifices also attracted the attention of several pioneer researchers; e.g., Rayleigh (1879), Bohr (1909), and Taylor (1960). However, the studies on non-circular jets are mostly limited to the elliptical orifices, mainly due to the usage of their axis-switching properties to measure dynamic surface tension of liquids (Bechtel et al. 1995, Saleh et al. 2019). Axis-switching is a phenomenon in elliptic jets which is manifestation of an alteration between the major and minor axes of the liquid jet along the injection direction. More recently, elliptic liquid jets were studied by Bechtel (1989), Kasyap et al. (2009), Amini et al. (2012, 2014), and Morad and Khosrobeygi (2019). Their results show that, compared to a circular orifice, flow physics associated with noncircular jets can change the breakup characteristics of a liquid column. For example, Amini and Dolatabadi (2011, 2012) showed that the elliptical jets have a higher perturbation growth rates and impose a higher range of effective perturbation wavelength than those of circular orifices. Among non-circular orifices, aside from elliptical ones, rectangular orifices were also found to be interesting due to their potential capabilities for passive flow control of mass entrainment and mixing (Miller et al. 1995, Chen and Yu 2014, Ghasemi 2019). While elliptical jets are intermediate configurations between commonly studied round and planar jets, the rectangular orifices present a special case; they incorporate the features of aspect ratio, similar to an elliptic orifice, and the corner effects, similar to triangular and square orifices (Sharma and Fang 2014). More recently, Wang and Fang (2015), Tadjfar and Jaberi (2019) and Jaberi and Tadjfar (2019) studied rectangular orifice shapes and investigated their characteristics under different ejecting 3
pressures. Their experimental data show that a rectangular jet has the shortest breakup length compared to other non-circular orifices, and axis-switching phenomenon is observed during its evolution as well. The majority of the studies on discharging jets are experimental and the theoretical analyses, mostly based on linear approaches, are much less. The linear theory of capillary breakup has been developed for small perturbations (Rayleigh 1879) and allows for a rather accurate prediction of the breakup time and length of capillary jets. However, as a result of the assumption that the perturbation magnitude remains small relative to the diameter of the jet, linear analysis fails near the point of jet breakup. Therefore, linear methods do not result in a prediction of the droplet sizes nor can they accurately forecast the production of satellite droplets (i.e., small droplets which are formed from liquid threads between the main drops). On the other hand, the theories of the nonlinear instability and breakup of liquid jets lead to rather cumbersome expressions. Therefore, the full nonlinear equations of motion for the jet need to be solved numerically. The numerical solution of the Navier-Stokes equation with a free surface is able to provide an indepth insight to the jet evolution and its breakup by modelling the formation of main and satellite drops. To the best of our knowledge, there is no numerical solution of the 3D problem for rectangular liquid jets in literature. The present study addresses the numerical analysis of instability of rectangular liquid jets, accounting for natural oscillations and disturbances introduced to the jet for a wide range of wavenumbers and amplitudes. The effects of orifice geometry and external disturbances on axis-switching, breakup and droplet formation are investigated. The wavelength of the axis-switching as well as breakup length obtained from the numerical results are validated with the experimental data available in the literature. In the rest of this article, the mathematical modeling and the solution method are presented first. Then, the results are presented to characterize axis-switching, breakup and drop formation for nonperturbed and perturbed jets.
2. Governing equations and numerical method
4
In order to simulate the ejection of the liquid jet from the nozzle into the quiescent air, a volume of fluid (VOF) method is employed. In this method, each cell at any instant has a value of γ equal to the volume fraction of water. The utilized VOF approach allows for simulating air-liquid flows including surface tension force by incorporating an interfacial compression flux term to mitigate the effects of numerical smearing of the interface (Rusche 2003). The main advantage of this method over other VOF methods commprises of intrinsic volume conservation and possibility of topological changes, e.g., breakup or merging of several entities. Governing equations are continuity, (
)
⃗
)1(
and conservation of momentum as, ( ⃗)
( ⃗⃗)
in which , v, , and
[ ( ⃗
⃗ )]
⃗
⃗⃗
)2(
represent the density, velocity, viscosity, and gravitational constant,
respectively. Density and viscosity could be written in terms of their value in the liquid and the (
gas as ⃗
)
(
and
) . The surface tension force is represented as
which is a function of the surface tension ( ), the curvature
(
⁄|
|), and
the volume fraction gradient. The volume fraction itself is obtained from the advection equation,
( )
( ⃗)
[⃗
(
)]
)3(
Here, the compression velocity (⃗ ) is obtained at the vicinity of liquid-gas interface through Rusche’s model (2003) as, ⃗
where
,
,
and
[
| | | |
| | ] | |
)4(
are the face volume flux, cell face area vector, compression coefficient
and face unit normal flux, respectively. Since the current study focuses on Rayleigh regime associated with low speed jets, simulations are performed with a laminar method.
5
To facilitate the validation of the numerical results, the orifices in the present study are chosen to have the same configurations as those used in the previous studies (Wang and Fang 2015, Farvardin and Dolatabadi 2013, Kasyap et al. 2009). The aspect ratio
, defined as the ratio of
the length (major axis) to the width (minor axis) of orifices is 1, 1.5, 3, and 4.5 for the orifices REC0, REC1, REC2, REC3, respectively (Table 1). In addition, an elliptical orifice E3 with aspect ratio of 4.5 and a circular orifice C0 will be examined as well. Having the same crosssectional area, the equivalent diameter for all orifices, is
mm.
Table 1. Dimensions of the orifices used in the numerical simulation Orifice name
Aspect ratio
Major axis (mm)
Minor axis (mm)
REC3
4.5
0.5588
0.1270
REC2
3.0
0.4580
0.1527
REC1
1.5
0.3260
0.2170
REC0
1.0
0.2667
0.2667
E3
4.5
0.6303
0.1400
C0
1.0
0.3000
0.3000
The numerical domain of the problem is a cube with dimensions of 4×4×50 mm (Fig. 1). The domain inlet, which is the orifice exit, is accommodated on the top wall. A no-slip boundary condition is applied over the solid walls. Boundary conditions on the sides and the outlet are atmospheric pressure. In addition, at the inlet of the numerical domain, a top-hat velocity profile, associated to a short nozzle, is introduced. The coupled set of Navier-Stokes equations is solved implicitly by a PISO (pressure implicit with splitting of operators) method at each time step. The initial time step is 10-6 sec with the possibility of adjustment by Courant number with the maximum value of 0.5. Density of air and water is assumed to be 1.2 kg/m3 and 103 kg/m3, respectively. The air-water surface tension is 0.07 N/m. In addition, viscosity of air and water is assumed to be 1.8 x 10 -5 N.s/m2 and 10-3 N.s/m2 , respectively. The inlet velocity used in this study ranges from 1.25 to 5.6 m/s associated to Weber numbers ( and Reynolds numbers (
) from 4 to 80
) from 250 to 1000.
6
On the x-y plane, a mapped mesh is used to form the shape of the orifice cross-section. The numerical domain is discretized initially with 0.07 mm grids in a Cartesian coordinate system. However, a dynamic mesh refinement is used in order to precisely capture the interface of the liquid and small droplets. The dynamic mesh refinement process is controlled by the value of γ. Consequently, the cells which have 0.001 < γ < 1 are refined one level in the x, y, and z directions. For the next iteration, the cells which contain liquid are refined again. Figure 2 demonstrates an example of dynamic mesh refinement method in the solution. For mesh dependency checking, the primary simulations were performed with two, three and four levels of refinement. Although in 2-level and 3-level refined cases, axis-switching phenomenon was captured, their wavelength, surface ruffles, and breakup point were different. Figure 3 shows breakup length of the liquid jet ejecting from orifice REC3 for three different mesh refinement levels at different Weber numbers. It can be seen that the accuracy of predicting breakup length in four-level mesh refinement is higher than the others. However, three-level mesh refinement has also an acceptable accuracy compared to two- and four- level mesh refinement. Due to a high computational time of four-level mesh refinement, three-level mesh refinement has been chosen for simulation. It is noted that in order to calculate the breakup length and droplet diameter, image processing is performed via Paraview and Matlab to extract the free-surface profile from numerical data. To minimize the effects of randomness, the results obtained from several simulations for each case are averaged within two standard deviations.
3. Results and discussion
Using the open-source code OpenFOAM6, the governing equations (1) - (4) are solved with a finite volume method for free and perturbed jets. Accordingly, two types of boundary conditions are imposed at the inlet of the numerical domain; for free jets the inlet velocity is constant while for the forced jets there will be a perturbed term added to the constant velocity. The results of the free jets are shown first and, after validating against experiments, the perturbed jets results are presented and discussed.
3.1 Free jets
7
Figure 4 displays a snapshot of numerical simulation of evolution of rectangular jet REC3 showing how the major axis of the liquid cross-section substitutes frequently with the minor axis, a property called axis-switching. We define the axis-switching wavelength,
as the distance
between any two consecutive axis-switching. Similarly, axis-switching number is defined as the number of repeating this phenomenon. Accordingly, Fig. 5 shows the magnitude of pressure, surface tension, and vorticity along the z axis at different axial locations within the occurrence of axis-switching. Once the liquid-gas interface is highly curved, the surface tension force becomes greater at the interface and, as a result, local pressure would be higher in those areas either. In order to eliminate the pressure difference, the liquid jet cross-section tends to change from rectangular to elliptical shape with a lower curvature at the interface. The elliptical shape causes a high pressure gradient at the two extremes of the major axis, where the radius of curvature is smaller. Such pressure gradient coupled with the effect of surface tension results in transformation of the cross-section towards a circular shape with the minimum surface energy in order to reach a balance of forces. Simultaneously, there is a velocity gradient that, using inertial forces in the bulk liquid, changes the cross-section to an ellipse with its major axis normal to the original one. This axis-switching occurs for a number of times in which the jet switches its minor and major axis within a wavelength until either the oscillations are damped by viscosity or the jet breaks up due to capillary instability. The vorticity around liquid-gas interface is another parameter that should be taken into account. The cross-section transformation in rectangular jets changes the vorticity contours in the proximity of the interface and alters the outside flow pattern and pressure as the jet progresses. Figure 6 presents the total vorticity magnitude and zcomponent vorticity for rectangular and circular liquid jets at the same axial position. Both total and z-component vorticities are more intensified around rectangular jets compared to those of circular jets which are more uniform. The vorticity intensity unbalances the pressure difference and assists the cross-section to deform easier. Figure 7 shows variations of dimensionless axis-switching wavelengths with Weber number for all jets. It can be seen that the axis-switching wavelength increases with Weber number almost linearly with the same slope for REC1, REC2 and REC3 jets. The results are validated against the experimental measurements of Wang and Fang (2015) for REC3 orifice. Based on experimental measurements for high aspect-ratio rectangular jets, Wang and Fang (2015) suggested the above slope to be 3.71. However, based on a linear stability analysis, Amini and 8
Dolatabadi (2012) calculated the wavelength of axis-switching for a non-circular liquid jet as,
√ (
)
,
(5)
where n is an integer representing the jet shape and is assigned as 2, 3, and 4, for elliptical, triangular and square jets, respectively. For a rectangular orifice, it can be assumed n=2, similar to an elliptic orifice. Therefore, for elliptical or rectangular orifices, the dimensionless wavelength is proportional to square root of We number by a factor of 2.56 (Amini and Dolatabadi 2012). However, this proportionality is based on the linear theory assuming departure from circularity of the cross-section is small, i.e., valid mostly for low aspect ratio jets. We note that nonlinear effects are strong at locations of the surface that are close to the jet axis (Gertsenshtein and Shkadov 1973) which is the case for jets with high aspect ratios. In addition, corner effects as an important feature of the rectangular orifice, might also be responsible for the slope deviation. We note that axis-switching is under the influence of different parameters such as viscosity, surface tension and inertia. The wavelength of the axis-switching increases with the jet velocity because the axial velocity directly increases the axis substitutions time intervals. The jet with the highest aspect ratio, REC3, has the longest axis-switching wavelength in comparison with REC2 and REC1 jets for all Weber numbers. The reason could be attributed to the time required for cross-section transformation that affects the axis-switching wavelength directly. For high aspect ratio jets, the lateral-inertial is stronger than surface tension forces and can resist longer. Therefore, with the same axial velocity, the completion of axis-switching cycle takes a longer time which leads to a longer wavelength. However, viscosity would constrain the liquid jet crosssection adaptation so that for REC3 orifice, it takes more time to substitute the axes. Figure 8 demonstrates how axis-switching at We
decays with distance down the orifice
along jet progression. The major axis of REC3 orifice is longer than that of other orifices, while its minor axis is the smallest. Down the jet, before disintegration, the major axis decreases while the minor axis increases until the liquid jet alters into a circular shape with the minimum surface tension energy. This phenomenon is repeated four to five times for each jet at all Weber numbers. It is noticed that at low Weber numbers, the liquid column cross-section reshapes into a 9
circular one and the oscillations that appear over its surface, lead to the jet breakup. In contrast, at high Weber numbers, the liquid jet is short of time to reach the circular shape and breakup occurs after fourth or fifth substitution process. For the case of REC0 orifice with the aspect ratio of unity, axis-switching could hardly be observed but it exists with short swings. This result is in agreement with the measurements of Rajesh et al. (2016). We also note that in an elliptical jet the axes switch and, as the jet proceeds, the cross-section changes from elliptic to circular (Kasyap et al. 2009, Amini and Dolatabadi 2012). However, in a rectangular jet, the liquid cross-section reforms from rectangular to elliptic shape and then acts the same as an elliptical jet without establishing a rectangular cross-section again. Results show that the cross-section of REC0 jet decays to a circular shape faster. By the end of axis-switching, when the initial disturbance grows enough, diameter of the jet decreases and breakup occurs. Figure 9 shows a comparison between the present simulations for REC3 orifice and the captured images by Wang and Fang (2015) for the same operating conditions. It could be seen that the breakup length increases with Weber number as a result of higher liquid inertia, which overcomes capillary instability and delays the disturbances growth rates. Figure 10 shows numerical simulation of liquid jet development into quiescent air for all orifices at
and
. The orifice with the highest aspect ratio, REC3, has the
shortest breakup length, whereas REC1 orifice, with the lowest aspect ratio, holds the longest breakup length. However, the breakup length of REC1 is still shorter than those of square jet REC0 and the circular jet C0. It is concluded that rectangular jets with high aspect ratios have higher perturbation growth rates, causing the jet to disintegrate at a shorter length compared to circular or square jets. Figure 11 shows breakup length for all jets at Weber numbers from 4 to 80 in conjunction with the experimental results of Wang and Fang (2015). The dependency of breakup length on aspect ratio and Weber number can be clearly seen in this figure. The liquid jets with higher aspect ratios show a shorter breakup length. As the aspect ratio decreases, the breakup length increases until it reaches its peak assigned to the circular jet. We observe a decrease of breakup length at higher We numbers, associated to the first wind-induced regime. Basically, at higher jet velocities, the aerodynamic forces become effective and shear stress of the surrounding gas assists capillary forces on more instability and, therefore, a faster disintegration of the liquid 10
column occurs. 3.2. Perturbed jets
It is well known that imposing perturbations on circular liquid jets could manipulate the breakup length and size and uniformity of the generated droplets and even merging of satellite droplets to main drops. In literature, numerous experiments have been conducted on circular jets to take advantage of instabilities for altering the liquid jet breakup characteristics (Strayer and Rankin 2001, Rohani et al. 2010, 2012). Based on linear stability analysis (Rayleigh 1879, Amini 2011), the disturbances imposed on the surface of a liquid jet with radius Req are described by a surface wave (Fig. 12) as,
(
in which and
)
(6)
represents the initial amplitude,
is the wavenumber,
is the wavelength,
is the growth rate of disturbances. The breakup length of the liquid jet, zb is defined as the
distance between the orifice exit to the breakup point, where amplitude of the oscillations grows and exceeds the unperturbed radius of the jet, i.e.,
. Accordingly, the breakup length and
growth rate are related as,
√ ⁄
(
) (7)
where β
dimensionless growth rate defined as, √
.
(8)
The linear stability analysis is based on imposing a surface disturbance on liquid jet at the nozzle exit ejecting with a constant velocity U0. However, in many cases, it is more practical to apply velocity disturbance while maintaining a constant jet radius. In the current work, in order to study the behavior of the jet with respect to perturbations, a variable velocity is imposed at the 11
(
inlet as and
). Here,
is the main jet velocity (equal to 4.0 m/s for all cases)
is the amplitude of initial velocity perturbation. By analysis of the mechanical energy that
enters into the jet, Moallemi and Mehravaran (2016) showed that velocity modulation radius modulations
and
are related to each other via, (9)
In our calculation, we employ equation (9) assuming
. Figure 13 shows the
evolution of the liquid jet emerging from REC3 orifice perturbed at different wavenumbers. Starting from zero, by increasing the wavenumber the jet breakup length reduces to a minimum value. From this point on, breakup length increases with wavenumber until the liquid column achieves its initial breakup length. The optimum disturbance wavenumber for which the liquid jet disintegrates with the shortest length is equal to
. Accordingly, we can expect that
at the optimum wavenumber the number of axis-switching be minimum. Figure 14 shows deviation of breakup length with respect to its average value for C0, E3, REC2, and REC3 jets at different disturbance wavenumbers. With distance down the jet, small oscillations appear on the liquid column that grow until disintegration point. The breakup length variation has a minimum value at the optimum wavenumber for each jet. As shown in Fig. 13, for wavenumbers in the range of 1.0 to 1.8, breakup length of the liquid jet issuing from REC3 orifice is the shortest. Based on equation (7), this is a result of higher growth rate of perturbation, β, which decreases the breakup length,
.
Using equation (7) and by measuring breakup length, the disturbance growth rates have been calculated and the results are presented in Fig. 15. For all orifices, the growth rate increases with wavenumber and then, after reaching a peak, decreases to the initial value. The optimum wavenumber and the range of wavenumbers that influences the liquid jets are increased at higher aspect ratios. For REC3 jet, and similarly E3, the optimum wavenumber is obtained to be around 1.5, whereas it is 1.2, 1.0, 0.9, and 0.8 for REC2, REC1, REC0 and C0 jets, respectively. The REC3 orifice also has the longest effective wavenumber range from 0 to 2.4, while this range is between 0 to 2.2, 1.6, 1.4 and 1.2 for REC2, REC1, REC0 and C0 orifices, respectively. The 12
maximum disturbance growth rate, occurred at optimum wavenumber, shifts to higher values for higher aspect ratio jets. Compared to the linear theory results (Amini et al. 2014), numerically obtained data in the present study predicts the previous experimental data better due to covering nonlinear effects particularly raised when the aspect ratio increases. Figure 16 depicts the variation of breakup length (
⁄
by amplitude of disturbances
⁄ ). Results show these two parameters have a linear relationship at small values of δ0,
corresponding to small perturbation amplitudes, while at higher δ0 values a deviation will occur. This result implies that the disintegration length decreases for large initial amplitudes of perturbation and the jet behavior will be nonlinear. Figure 17 shows the moment of breaking of non-perturbed jets C0 and REC3, and perturbed jet REC3. It can be seen that for non-perturbed jets, before liquid breakup, a thin ligament is generated between the main jet and the tip droplet which is still attached. This ligament stretches until breaking from one of its tips, while the other side is still attached to the jet column. At this position, the separated side moves toward the attached side to reunion. However, the second breakup also occurs at the remaining attached side and ligament separates completely making a satellite droplet. From Fig. 16 it could be seen that the length of ligaments in non-perturbed circular jets is longer than that of non-perturbed rectangular jets. This is due to the higher growth rate of disturbances in rectangular jets which reduces the time of shrinking of the liquid surface to about one-fourth so that the ligament has not enough time to be formed compared to that of a circular jet. As a result, it could be expected that formation of satellite droplets in rectangular jets is limited and, if appear, they are smaller in size. Additionally, for the circular jet perturbed at the optimum wavenumber, it can be seen that the breakup length has the shortest value, and the size of the produced droplets are very close to each other. Satellite droplets are clearly visible in Fig. 17 for the case of circular jet breakup in almost all wavenumbers except at the optimum wavenumber. Figure 18 shows the size of droplets formed after jet breakup for perturbed circular jet C0 as well as two rectangular jets REC2, REC3. It is seen that near the optimum wavenumber, droplets have smaller diameters compared to those of non-perturbed or perturbed jets at non-optimum wavenumbers. The smallest droplets exist in perturbed jets with the optimum disturbance 13
wavenumbers. Furthermore, jets emerging from higher aspect-ratio orifices produce smaller droplets which could be due to the short length of the dominant wave at higher growth rates. Mathematically, the droplet volume is equal to the integral of the jet cross-section over its wavelength, and a high wavenumber, corresponding to a short wave, produces a small droplet. The standard deviation of droplet diameter with respect to its average at different wavenumbers can be a measure of uniformity of generated droplets. This parameter has been calculated and results are presented in Fig. 19. Results confirm that the deviation of droplets generated from liquid jets perturbed with optimum wavenumber is lower than the value of deviation when wavenumbers are close to beginning and end of the range of effective wavenumbers. This is also a result of higher growth rate at the optimum wavenumber that shortens the breakup length leading to formation of less satellite droplets. In addition, at maximum growth rate, the variation of the jet boundary is faster and, therefore, long ligaments do not have enough time to be formed, be separated from the main body of the jet, and generate satellite droplets. Therefore, similar to circular jets (Rohani et al. 2012), the breakup length and the size of the main and satellite droplets are minimum at optimum wavenumber in rectangular jets.
4. Conclusion
Noncircular jets are found to be interesting due to their potential capabilities for passive flow control. A rectangular orifice is able to impose inherently a passive alteration on instabilities. To take advantage of a combination of active and passive instabilities, the present work addresses the behavior of rectangular liquid jets under the effect of disturbances introduced at the nozzle exit. The evolution and disintegration of liquids jets ejecting from different rectangular orifices under various operating conditions were studied systematically. Using a direct numerical solution and an adaptive meshing technique, the Navier-Stokes equations with a moving boundary have been solved for the two-phase flow problem. To the best of our knowledge, this is the first numerical solution for a two-phase jet with a rectangular free surface. The axisswitching, instability, and breakup of unperturbed and perturbed liquid jets in Rayleigh regime were investigated and the results were compared with experimental measurements and the linear stability analysis predictions. The flow-field around the liquid-gas interface including, pressure, 14
surface tension and vorticity contours are captured. Accounting for disturbances introduced to the jet for a wide range of wavenumbers and amplitudes, the effects of orifice geometry and inlet velocity are investigated. Studying the liquid jet evolution reveals that, similar to elliptical liquid jets, the axis-switching phenomenon is visible for rectangular jets. The wavelength of axis-switching increases linearly with the jet velocity and the slope shows a deviation from the theoretical results obtained for an elliptical jet. Additionally, by increasing the aspect ratio, the wavelength of axis-switching is increased which results in a shorter breakup length. For a rectangular jet, with aspect ratio of unity, the axis-switching is hardly visible. The breakup length and growth rates were computed under the effect of a velocity modulation at the nozzle exit, and the results were examined for a wide range of effective wavenumbers. For perturbed jets, the growth rate is found to be strongly dependent on the wavenumber of disturbances. The results indicate that the maximum growth rate which causes the shortest breakup occurs at wavenumbers around 0.9, 1.1, and 1.2 for the jets with aspect ratios of 1.5, 3, and 4, respectively. The range of influence of disturbances increases as the aspect ratio of the orifice increases. At a fixed wavenumber, a higher aspect ratio increases the inherent instability effect, causing the breakup length to be decreased. It is also observed that the satellite droplets are almost eliminated at the optimum wavenumber. Furthermore, liquid jets perturbed at the optimum wavenumber produce smaller and more uniform droplets compared to non-perturbed and perturbed jets at non-optimal wavenumbers. Author statement
Attached please find our revised manuscript IJMF-2016-727 entitled “Axis-switching and breakup of rectangular liquid jets” to be considered for publication in International Journal of Multiphase Flow.
Conflict of Interest The authors certify that they have NO affiliations with or involvement in any organization or entity with any financial interest or non-financial interest in the subject matter or materials discussed in this manuscript.
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17
Fig. 1. Numerical domain (left) and the dimensions of orifices (right). All orifices have the same cross-sectional area.
18
Fig. 2. A three-level mesh for simulation of breakup of the REC3 jet at We=33.64.
19
Fig. 3. Mesh-independent checking of numerical solution for breakup length of REC3 jet in different refinement levels at different Weber numbers.
20
Fig. 4. Definition of axis-switching wavelength and axis-switching number (left), and the rectangular jet ejecting along z-axis from REC3 orifice (right)
at
.
21
Fig. 5. Pressure, surface tension and vorticity contours at different axial locations for REC3 jet.
22
Fig. 6. Distribution of total vorticity and z-component vorticity for rectangular jet REC3 (left) and circular jet C0 (right) at axial position z=0.7 mm.
23
Fig. 7. Variation of axis-switching wavelength by Weber number.
24
Fig. 8. Deformation of cross-section along the jet at We=33.64.
25
Fig. 9. Evolution and breakup of REC3 jet at different We numbers. Numerical results have been compared to experiments of Wang and Fang (2015) (Courtesy of Elsevier).
26
Fig. 10. Disintegration of the jets and droplet formation at
.
27
Fig. 11. Variations of breakup length for all jets at different Weber numbers.
28
FIG. 12. Modulation of jet velocity at orifice exit.
29
Fig. 13. Effect of disturbances with different wavenumbers imposed on REC3 jet at We
.
30
Fig. 14. Standard deviation of breakup length versus disturbance wavenumber at We
31
Fig. 15. Growth rate of disturbances versus wavenumber at We = 33.64.
32
Fig. 16. Dimensionless breakup length versus amplitude of the initial disturbance at optimum wavenumber for REC3 jet at
.
33
FIG. 17. The moment of breakup and formation of satellite droplets for non-perturbed jet C0 (left), non-perturbed jet REC3 (middle) and jet REC3 perturbed at kReq=1.3 (right).
34
FIG. 18. Variation of diameter of the generated droplets with wavenumber of the imposed disturbances.
35
FIG. 19. Standard deviation of diameter of the generated droplets versus wavenumber of the imposed disturbances.
36