ILASS – Europe 2016, 27th Annual Conference on Liquid Atomization and Spray Systems, 4-7 September 2016, Brighton, UK

Mechanisms of Atomization of a Liquid-Sheet: A RegimeClassification

Camille Bilger∗, Stewart CantDepartment of Engineering, University of Cambridge, Cambridge, United Kingdom

*Corresponding author: cb757@cam.ac.uk

AbstractThe Robust Conservative Level Set (RCLS) method offers a novel state-of-the-art fully parallelized numerical capa-bility for efficient modelling of multiphase flows. It uses high-order WENO schemes on mixed-element unstructuredmeshes to solve the transport equation for the level-set variable. The present study applies the RCLS method to pri-mary breakup of liquid sprays and especially to understand the phenomenology observed during prefilming air-blastatomization of a planar liquid-sheet. In commercial aircraft engine applications, this is essential for effective controlof fuel atomization and hence for cleaner, more efficient combustion processes. For a fixed prefilmer geometry, wevaried parametrically the liquid and gas velocities to make an informed qualitative assessment of their impact on thetopological evolution of the liquid interface. The computations reproduced the transient nature of the fluid dynam-ics of the phase interface. The study has revealed that for higher liquid injection velocities the droplets generatedare smaller. An increase in gas velocity is found to be detrimental to efficient atomization. The interface exhibitsa behavior that we characterized as ligament-merging dynamics. Results also show accumulation of liquid at theprefilmer lip for lower liquid injection speeds. Preliminary regime maps that predict the prefilming liquid-sheet atom-ization behavior are constructed based on our numerical simulations. Three distinct types of “regime” are reported:accumulation, ligament-merging and 3D wave modes.

IntroductionImproving hydrocarbon fuel combustion is critical for the next generation of commercial aircraft in order to reducepollutant emissions and improve aircraft engine efficiency. To allow for optimized fuel injection into combustionchambers, effective control of fuel atomization is required. The mechanisms leading to the breakup in gas-assistedatomization and the nature of the spray produced are still poorly understood. This is true especially for primarybreakup, which occurs in the vicinity of the injection point. To achieve better understanding of primary atomizationunder subsonic flow conditions, developments in computational fluid dynamics techniques for multiphase flows areessential as experimental investigations involving a trial and error approach are expensive in both time and cost forindustry. With recent remarkable progress made in three-dimensional CFD, high-fidelity numerical simulations ofmultiphase flows with interface handling offer a promising alternative to experiments to gain insight into the injectionand atomization processes. However, due to the large density ratio and the intense multi-scale character of arealistic atomization process, numerical simulations are extremely expensive and challenging. Rather than computethe breakup phenomena in detail, models are often used instead in industry to reproduce the main features of therequired cloud of droplets. The boundary condition for the liquid fuel is imposed through the numerical injectionof droplet parcels with given droplet characteristics (size and velocity distribution). The strong intrinsic couplingbetween the gas and liquid phases and the resulting flow unsteadiness during the first instants of liquid injection arethus not taken into account.In this work, a novel state-of-the-art numerical capability for efficient multiphase flow modelling [1, 2] – the RobustConservative Level Set (RCLS) method – uses high-order WENO schemes on a three-dimensional mesh to solvefor the transport equation for the level-set variable. The present study aims to present numerical simulations usingthe RCLS method of prefilming liquid-sheet primary breakup at ambient conditions and attempts to discuss thedependence of the mechanism of primary breakup on the velocities of the air and liquid streams.Several excellent graphical classifications of regimes for two-phase flow dynamics have been reported in the liter-ature based on experimental observations: these cover for example secondary atomization of liquid drops [3] andjet breakup [4] based on the Weber number as well as non-prefilming liquid-sheet atomization [5] based on the mo-mentum flux ratio. A regime classification for prefilming atomization of planar liquid-sheets has yet to be formulated.A first step towards this was recently published by Déjean et al. [6]. Other recent experimental works [7, 8] help inthis respect. Inspired by the matrix of liquid film breakup phenomena proposed by Inamura et al. [9], the presentstudy sets out to classify the phenomenology observed during primary breakup of a planar liquid-sheet with a set ofhigh-fidelity numerical simulations. Using the results, we propose a mapping for prefilming liquid-sheet atomizationin velocity space.

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Mathematical formulationSolution of the incompressible Navier-Stokes equationsThe RCLS method is implemented within the open source CFD toolkit OpenFOAM® [10]. The governing equationsfor multiphase flows with interface modelling are expressed over the whole multiphase domain. The ContinuumSurface Force (CSF) description of the surface tension due to Brackbill et al. [11] is used. The conservationformulation of the incompressible Navier-Stokes equation in each phase is given by:

∇ · u = 0 (1)

∂

∂t(ρu) + (u · ∇)(ρu) = −∇p+ ρg +∇ · (2µD) (2)

where u is the fluid velocity, ρ its density, D is the rate-of-strain tensor and g is the gravitational acceleration.We consider a liquid-gas two-phase flow with constant densities ρg and ρl and viscosities µg and µl. The sharpinterface Γ represents a discontinuous steep change of the properties (density, viscosity, etc) of the two fluids. TheYoung-Laplace condition determines the dynamics of the interface at each timestep:

[−pI + D]Γ · n̂Γ = σκn̂Γ (3)

for a constant surface tension σ along the interface. The normal to the interface is n̂Γ, κ its curvature and σ thesurface tension coefficient.

Computational framework for the RCLS methodThe RCLS implicit interface capturing method of Pringuey and Cant [1, 2] is based on the conservative level-setmethod (CLS) of Olsson et al. [12, 13] that transports the liquid-vapor interface accurately while conserving mass.Instead of transporting a signed distance function, the method transports a hyperbolic tangent function ψ(x, t) thatcan be initialized with the signed distance function φ as follows:

ψ =1

2

(tanh(

φ

2ε) + 1

)(4)

The conservative level set variable ψ goes smoothly from 0 (in the gas phase) to 1 (in the liquid phase) with theinterface location defined by the level set ψΓ = 0.5. All the volume encompassed by that surface can be consideredas filled with liquid. Olsson et al. [12] introduced ε in (4) as a control parameter relating to the thickness of theinterface, kept constant once defined, where ε = ∆x/2 and ∆x is the mesh spacing. The function ψ is advected ina conservative manner, in a zero-divergence velocity field, with an arbitrarily high-order WENO scheme developedfor three-dimensional mixed-element unstructured meshes in order to handle complex real engine geometries [1]:

∂ψ

∂t+∇ · (uψ) = 0 (5)

In addition, a re-initialization algorithm is applied:

∂ψ

∂τ+∇ · (ψ(1− ψ)n̂Γ)− ε∇ · (∇ψ) = 0 (6)

where τ is the artificial time along which the equation is solved until the initial level set profile is recovered [2].A third-order Runge-Kutta (RK) scheme for temporal discretization is used for both the advection and re-initializationsteps.The addition of the Multidimensional Universal Limiter with Explicit Solution (MULES) of Weller et al. [10] guaranteesthat the liquid volume fraction remains physical everywhere in the computational domain (0 < ψ < 1). The pressure-velocity coupling is handled within the framework of OpenFOAM using the standard Pressure Implicit with Splitting ofOperators (PISO) method of Issa [14]. In addition, a n-halo parallelization method was implemented in OpenFOAMto perform the computations at the required order of accuracy.The resulting method conserves mass to machine accuracy. Its ability to capture the physics of atomization of aliquid-sheet sandwiched between two walls and two fast co-flowing air streams was demonstrated in Pringuey andCant [2].

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Numerical setup and conditionsIn order to reduce the complexity of the configuration in this parametric study, the realistic annular geometry ofmost air-blast commercial atomizers is simplified into a three-dimensional planar geometry with cyclic boundaryconditions on either side (Figure 1), where the liquid is injected as a thin planar sheet. This design is becomingincreasingly popular as a benchmark flow to study primary atomization among experimentalists, particularly dueto the relative simplicity of experimental visualization. In addition, the basic atomization mechanisms are believedto be the same in both axisymmetric and planar configurations [15]. Our computational domain is as follows. Thex-direction corresponds to the streamwise direction, the y-direction is upward and the z-direction is spanwise. Ourcomputational domain is a cuboid of 13× 18× 4 mm3, meshed with 1.56× 106 hexahedra, where the prefilmer is aflat plate of 4mm width, 1mm height and 8mm length.A continuous airflow with no swirl is supplied at the inlet plane. The air emerging is split into two streams thatpass the prefilmer on each side. The liquid and the gas are both entering the simulation domain on the left handside with different heights Hl=0.1mm and Hg,top=Hg,bottom=5.5mm; and at different mean velocities ul and ug. Aparabolic profile is adopted for the inlet air streams in order to account for upstream flow development in the airsupply channels (Figure 2). The liquid is continuously fed onto the prefilmer through an injector slit of 4mm×0.1mmwith a uniform velocity profile. The liquid forms a thin film traveling downstream. The flow conditions chosen forthis study are listed in Table 1. All numerical simulations use atmospheric conditions for pressure and temperature.Above the prefilmer plate, the liquid (ψ = 1) is injected with a mean speed varied parametrically between 0.5 and10m/s. A small amplitude turbulent inlet fluctuation is applied at the liquid inlet plane. Above and below the liquidinjection channel, the gas (ψ = 0) enters the domain with a speed varied parametrically between 5 and 50m/s.Gravity is ignored in the context of primary atomization as it has no effect over the lengthscales and timescalesinvolved.

Figure 1. View of the computational domain with the liquid interface shown in grey, the gas inlet planes in blue and the prefilmer(rectangular parallelepiped) in black.

Phase ρ [kg/m3] µ [Pa s] u [m/s] σ [N/m]Gas 1.2 1.18× 10−5 5-50 0.0283

Liquid 998 1.0× 10−3 0.5-10

Table 1. Simulation parameters

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0 5 10 15 20 25 30 35 40−1

0

1

2

3

4

5

5.5x 10

−3

U [m/s]

y [

mm

]

Top air channel

Fuel channel

Figure 2. Example of one of the inflow axial velocity profiles for the top channel, above the prefilmer plate. The liquid velocity(uniform profile with ul = 2m/s) is shown in red and the gas velocity (parabolic profile with ug = 20m/s) in blue. The grey box

corresponds to the location of the prefilmer.

Results and DiscussionInterfacial behaviorUpon injection, the liquid-sheet is accelerated downstream by the air flow and a shear-driven liquid wall film ad-vances. In every flow configuration studied in this work, a two-phase unidirectional channel Poiseuille flow developsrapidly downstream in accordance with two-phase flow theory over a solid plate. The accurate application of bound-ary layers is important to predict the onset of the breakup and the growth of perturbations at the interface, as theinteraction of both velocity fields, and as a result, the shear stress, are believed to play a significant role in theatomization process. The works of Lozano and Barreras [15] and Marmottant and Villermaux [16] concluded thatthe viscous boundary layer height that develops upstream has a major influence on the strength of the observedinstabilities.For a mean liquid injection speed greater than 2m/s, the simulations picked up surface wrinkling due to the growthof linear instabilities at the interface location: these include longitudinal instabilities as well as transverse instabili-ties and the development of both spanwise and streamwise ligaments disintegrating into smaller liquid structures.Instantaneous snapshots in Figure 3 and Figure 4 illustrate this behavior.The development of longitudinal surface instabilities is mainly dependent on the shearing intensity exerted by theair on the liquid-sheet. The longitudinal instabilities are driven by the shear between the two fluid phases, a resultof the aerodynamic forces from the high-speed co-flowing gas, believed to arise from a Kelvin–Helmholtz instabilityof the mixing layer, as predicted by stability analysis and weakly non-linear theories. This characterizes the initialgrowth of the fundamental undulations of the liquid-sheet.In addition to longitudinal undulations, transverse instabilities appear. In the liquid-sheet reference frame, a liquidparticle propagates in a “wavy corridor” [17] due to the previously mentioned longitudinal instabilities. This liquidparticle is periodically accelerated upwards and downwards. It experiences transient accelerations perpendicular tothe plane of the sheet whose intensity and direction depends on the curvature of the sheet. These accelerations trig-ger a secondary instability responsible for the amplification of spanwise thickness modulations of the sheet. Indeed,Villermaux and Clanet [18] proposed a scenario in which the key effect responsible for the initiation of digitationsperpendicular to the interface is due to the liquid-sheet being periodically accelerated by the incoming longitudinalwaves via a Rayleigh–Taylor-like instability [19] [20]. Such a combination of surface waves in both orthogonal direc-tions leads to a periodic variation of the sheet thickness, which is responsible for the sheet indentations, digitationsat the interface, the formation of ligaments, and drops (Figure 3 and 4).Similarly, Bhayaraju and Hassa [8] attempted to explain this spanwise thickness modulation of the sheet by notingthat as the spacing between the longitudinal waves increases – the waves ahead travel at a higher velocity than thewaves that are following – stresses are induced on the liquid surface. The surface tension forces try to balance theinduced stresses by accumulating liquid mass and this causes the formation of ligament-like structures, wherebythe sheet thickness is greater than its mean height, and cell-like structures, enclosed in-between two ligaments,whereby the sheet thickness is reduced (Figure 4a). This phenomenon had also been observed by the authors in aprevious work [21].Spanwise and streamwise ligaments can disintegrate via either aerodynamic tearing or as a result of a capillaryinstability of a liquid cylinder that resembles the Plateau–Rayleigh instability (Figure 3b and Figure 4c), as indicatedby theoretical predictions [19] and observed in numerical [22] and experimental works [5]. This event is called

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(a) (b) (c)

Figure 3. Interface location colored by liquid velocity for ug = 20m/s, ul = 5m/s. Formation and elongation of longitudinalligaments that rupture into drops.

(a) (b) (c)

Figure 4. Interface location colored by liquid velocity for ug = 40m/s, ul = 5m/s. Membrane puncturing and formation oftransverse ligament that ruptures into drops under capillary breakup.

secondary atomization. Fragmentation of the ligaments starts along the streamwise direction when ligaments havebeen generated and elongated in the streamwise direction – under aerodynamic tearing due to the strong stream-wise air velocity field (Figure 3) – whereas ligaments seem to be mostly unstable due to capillary effects and breakapart in the transverse direction when first generated in the transverse direction (Figure 4).A higher liquid velocity induces a higher liquid Weber number. Those forces act on the ligaments and cause themto be destabilized over shorter time periods and thus lead to earlier fragmentation. This is conducive to better atom-ization in terms of timescales. The ligaments are also smaller in size, which leads to smaller drops being generated(Figure 3). However, since the liquid phase is so fast, the liquid core rim has had time to reach the end of theprefilmer before the first atomization events take place; the liquid-sheet “shoots-off” the prefilmer trailing edge andatomization happens further downstream, which is problematic with regards to jet-engine combustion.As the mean air velocity is increased from 20m/s (Figure 3) to 40m/s (Figure 4) – and ul is held at 5m/s – ahigher amplitude sinuous mode develops on the liquid film. This longitudinal wave mode makes the bulk liquid flapdownstream of the lip and helps promote atomization.When the injection speed of the liquid is decreased down to 0.5m/s, due to the lower liquid flow rate, the bulk liquidaccumulates close to the atomizing edge, as previously witnessed also in the experiments of Gepperth et al. [7]and Dejean et al. [6]. Figure 5 shows a series of snapshots for the liquid-sheet evolution with a bulk air velocity ofug = 20m/s and ul= 0.5m/s. Prior to breakup, at 13.6ms (Figure 5a), an initial cell-like structure – or membrane – inthe sheet is visible. This membrane is a result of the thickness modulations previously discussed on the liquid film,whose amplitude grows until it becomes of the order of the liquid film thickness and the film perforates, as witnessedin the experimental works of Bremond et al. [17]. This membrane grows in time before bursting, building the onsetof the primary breakup at 15.4ms (Figure 5b). This is another indication of the surface tension effect, as well as theshearing action of the gas on the liquid at the lip. The bursting of this membrane is accompanied by the creationof a streamwise ligament that remains attached to the rim of the liquid bulk, which facilitates the pinch-off eventsthrough aerodynamic tearing. At 27ms (Figure 5c), a large ligament stretches away from the liquid core rim in thestreamwise direction and pinches off into a droplet that is accelerated into the air. We also note that the pinch-off ofmembranes at the troughs creates regions of very low velocity and non-wetting of the plate (Figure 5d, which showsthe same time instant as Figure 5b).When both the liquid and gas are injected at very low speeds (ug=5m/s and ul=2m/s), it seems that not enoughshearing intensity is exerted onto the liquid film to provoke the formation of strong surface waves, let alone ligamentsand droplets (Figure 6). The authors postulate that the timescale required for the initiation of shearing instabilitiesis longer and is happening further downstream, outside the modeled computational domain.For a fixed liquid injection velocity of 2m/s, we varied the mean air velocity, spanning a range from 20m/s to 50m/s. In

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(a) (b)

(c) (d)

Figure 5. Interface location colored by liquid velocity for ug = 20m/s, ul = 0.5m/s. Development of a membrane puncturing theliquid-sheet (a) that breaks up (b) into a streamwise ligament that elongates and sheds a droplet into the air stream. (d)

Visualization of velocity regions above the mean liquid injection speed (red) and below (blue) at the same time instant as (b).Perforating of the liquid film through membrane puncturing is nicely visible.

Figure 6. Interface location colored by liquid velocity for ug = 5m/s, ul = 2m/s. The velocity pattern highlights minor longitudinaland transversal instabilities; the shearing intensity is not enough to destabilize the liquid film.

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every flow configuration, longitudinal ripples exhibit higher wavelengths (Figure 7) than transverse ripples (Figure 8).The former also seem not to have an impact on the thickness modulation of the liquid film and they can be seen onlyin the velocity field mapped onto the liquid/vapor interface contour. The full growth of the amplitude of longitudinalwaves is not visible for this prefilmer length. For a longer prefilmer, the authors had witnessed in previous work[21] thickness modulations of the liquid film in the streamwise direction, due to the growth of the Kelvin–Helmholtzinstability mode. Transverse waves on the other hand do induce thickness modulations from which liquid structureselongated in the streamwise direction emerge. These streamwise ligaments seem to merge and pair up, beforesplitting off the remaining liquid stream, as depicted in Figure 8. Surface tension holds those ligaments together. Asmore liquid is injected, it feeds the growth of those ligaments, which reach for the prefilmer lip, remain attached toit and break up. A clear wavy-surface mechanism with three-dimensional perpendicular wave modes is no longerpresent. As the liquid/gas velocity ratio increases – for a fixed liquid velocity and increasing gas velocity – theair co-flow seems to constrain the growth of sinusoidal longitudinal waves, a comment made also by Lozano andBarreras [15]. Here, surface tension is winning over the inertia of the liquid stream. This leads to an inhibitionof the development of instabilities in the liquid-sheet and a delay in the onset of atomization. This delay causesatomization to occur further downstream from the injection channel. Liquid is drawn out from the atomizing lip inthe form of long streamwise ligaments, which are able to grow bigger. The result from such an atomization regimeis that drop sizes tend to be bigger (Figure 7a). These results might be of high relevance for other applicationssuch as the surface treatment of substrates, in which wetting with washing or etching liquids is critical. A similarligament-merging behavior occurs for smaller mean gas velocities and a fixed liquid injection velocity of 1m/s. Thistime the liquid stream is not fully wetting the prefilmer as the ligaments develop and merge with one another quiteearly on.

(a) (b)

Figure 7. Interface location colored by liquid velocity above the mean liquid injection speed ul = 2m/s (red) and below (blue) withug = 40m/s (a) and ug = 30m/s (b). Longitudinal and transverse surface ripples are visible.

(a) (b) (c)

Figure 8. Interface location colored by liquid velocity for ug = 30m/s and ul = 2m/s outlaying the ligament-merging dynamicsencountered for an increasing gas injection velocity.

Regime mappingA key benefit of the findings presented above is that we can now construct a preliminary regime diagram that quan-tifies under what velocity conditions a liquid film will atomize following a certain interfacial behavior. A representativediagram is given in Figure 9, which shows the results of the set of high-fidelity numerical simulations we conductedat ambient pressure and temperature. The authors considered it convenient to present these simulation results ina dimensional diagram to facilitate comparisons with experimental runs, where normally variables such as velocity

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rather than non-dimensional groups are controlled [7, 9]. Because our geometry is fixed as well as the thermody-namic state of the fluids, we have explicitly chosen not to use non-dimensional numbers for the characterisation ofthe boundaries of regimes. Presently, it is not yet clear what those non-dimensional numbers should be. The use ofnon-dimensional numbers will be the subject of future work.Interestingly, there seems to be a stronger change in the atomization mechanism going along the vertical axis ofincreasing liquid injection velocity. Only under higher liquid injection speeds does there appear to be an atomizationscenario with an overall sheet-to-small drop transition that takes place. Contrary to what the authors had anticipated,our velocity matrix suggests that the small-droplets spray phenomenon does not occur simply due to an increased airvelocity. Instead, the injected liquid suffers a diminished sinusoidal interfacial structure in the streamwise direction;only transverse wave modes are visible, in which surface tension forces dominate over aerodynamical forces leadingto ligament merging behavior. The droplets generated in this instance are quite big.Thus, the best atomization characteristics seem to occur in the “3D wave modes” zone, as shown in Figure 10, witha dominant longitudinal oscillation of the liquid-sheet and rapid development of perpendicular transverse waves. Inthis atomization scenario, breakup does not occur in the vicinity of the prefilmer lip, which is essential for aircraftengine combustion applications, but further downstream. However, the liquid drops generated are rather small.Finally, for very low liquid flow rates, the accumulated liquid at the atomizing edge experiences deformation by whichan intermittent liquid release occurs. The liquid structures and droplets generated are rather big. This is becausethe liquid entrained into the vortex at the prefilmer lip is rather slow and bulky. The air-stream is not fast enough toempty the liquid reservoir in a way to promote fine atomization. We call this regime “accumulation”.

10

𝑈"𝑙𝑖𝑞𝑢𝑖𝑑[m/s]

5

2

1

0.5

2010 255 30 50𝑈"𝑔𝑎𝑠 [m/s]

40

Figure 9. Matrix representation of the observed liquid film breakup phenomena over our velocity parametric space.

ConclusionsThe calculations captured the interaction of the liquid-sheet with the surrounding ambient air, whereby rapidly grow-ing waves are superimposed on the sheet. The development of surface instabilities is mainly dependent on theintensity of the shearing exerted by the gas on the liquid-sheet. We observed qualitative similarities between ourhigh-fidelity numerical simulations and the atomization mechanisms described in the literature at atmospheric con-ditions, which have been investigated experimentally by, for example, Gepperth et al. [7] and Bhayaraju and Hassa

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0 5 10 15 20 25 30 35 40 45 50Ugas [m/s]

-2

0

2

4

6

8

10

Uliqu

id[m

/s]

3D wave modes

Accumulation

Ligament mergingat lip

Ligament mergingon plate

Figure 10. Regime map obtained from our numerical simulations for prefilming liquid-sheet atomization.

[8].A regime diagram was introduced that quantifies the relationship between the liquid and gas velocity conditionsunder which the interfacial behavior of the liquid film varies. When aerodynamic forces on the liquid core aresignificant, they expand the liquid film against the contracting effect of surface tension and three-dimensional wavemodes co-exist, leading to the breakup of the liquid. We witnessed this physical behavior for sufficiently high liquidinjection speeds (ul > 2m/s). We called this atomization scenario “3D wave modes”.However, for an increasing mean air velocity and fixed liquid velocity of ul = 2m/s, transverse instabilities seem todominate over longitudinal ones. Generated streamwise ligaments tend to pair up, held strongly by an overpoweringsurface tension, resulting in bigger droplets being shed into the main air stream. The latter symbolizes the growthof “ligament-merging” which also appears in our regime diagram.Finally, for very low liquid-flow rates, accumulated liquid at the atomizing edge undergoes deformation by which thedroplets are generated.

AcknowledgementsThis work was funded by Rolls-Royce Group plc and EPSRC.

Nomenclatureu velocity [m/s]H channel height [mm]u mean velocity [m/s]µ dynamic viscosity [kgm−1s−1]ν kinematic viscosity [m2s−1]ψ conservative level set function (ψΓ = 0.5)

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ρ density [kgm−3]σ surface tension [N/m]g gas phasel liquid phaseΓ interface location

References

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