Accurate level set method for simulations of liquid atomization

Chinese Journal of Chemical Engineering 23 (2015) 597–604

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Chinese Journal of Chemical Engineering journal homepage: www.elsevier.com/locate/CJChE

Fluid Dynamics and Transport Phenomena

Accurate level set method for simulations of liquid atomization☆ Changxiao Shao, Kun Luo ⁎, Jianshan Yang, Song Chen, Jianren Fan State Key Laboratory of Clean Energy Utilization, Zhejiang University, Hangzhou 310027, China

a r t i c l e

i n f o

Article history: Received 25 November 2013 Received in revised form 19 May 2014 Accepted 10 July 2014 Available online 6 January 2015 Keywords: Computational fluid dynamics Level set method Spray atomization Interface capture Breakup

a b s t r a c t Computational fluid dynamics is an efficient numerical approach for spray atomization study, but it is challenging to accurately capture the gas–liquid interface. In this work, an accurate conservative level set method is introduced to accurately track the gas–liquid interfaces in liquid atomization. To validate the capability of this method, binary drop collision and drop impacting on liquid film are investigated. The results are in good agreement with experiment observations. In addition, primary atomization (swirling sheet atomization) is studied using this method. To the swirling sheet atomization, it is found that Rayleigh–Taylor instability in the azimuthal direction causes the primary breakup of liquid sheet and complex vortex structures are clustered around the rim of the liquid sheet. The effects of central gas velocity and liquid–gas density ratio on atomization are also investigated. This work lays a solid foundation for further studying the mechanism of spray atomization. © 2014 The Chemical Industry and Engineering Society of China, and Chemical Industry Press. All rights reserved.

1. Introduction In most of the propulsion devices, fuel is injected into combustion chamber in the form of liquid. It subsequently undergoes the process of atomization, evaporation and combustion in order to generate power for the engine. Since the atomization process governs the liquid droplet diameter distribution, it strongly affects both the evaporation and combustion processes. The atomization can be divided into primary atomization and secondary atomization. Primary atomization is the process wherein liquid jet breaks into filaments and droplets due to interaction with ambient air. Secondary atomization is the process wherein filaments and droplets continue to break into even smaller droplets. For the secondary atomization, some models have already been proposed, such as TAB model, WAVE model and so on. However, the primary atomization is hard to investigate. The measurement of atomization is difficult especially for the dense spray zone, but the computational fluid dynamics (CFD) has proven to be a promising method for atomization study. Nevertheless, CFD-based numerical simulation has difficulties in dealing with atomization [1]. Firstly, the gas–liquid interface is hard to locate. There are two main categories of methods for locating the interface, that is, the interface tracking method and interface capturing method. Interface tracking method is based on the Lagrange framework, in which the interface is handled as many particles, such as the Front-tracking approach. The main limitation of this ☆ Supported by the National Natural Science Foundation of China (51176170, 51276163) and the Zhejiang Provincial Natural Science Foundation for Distinguished Young Scholars (LR12E06001). This work is also partially supported by the Fundamental Research Funds for the Central Universities. ⁎ Corresponding author. E-mail address: [email protected] (K. Luo).

method is the lack of automatic topology modification and the difficulty in parallelization. Interface capturing method is based on the Euler framework, such as volume of fluid (VOF) method and level set method (LSM). VOF method tracks the volume fraction of liquid in each cell and has good mass conservation. However, since the interface is not smooth, a specific advection scheme is required, which adds constraints on the accurate VOF method. Additionally, the interface normal and curvature are difficult to calculate. In the LSM, the interface is represented implicitly as the zero level set of a continuous function. It is convenient to calculate the interface normal and curvature and the interface can be captured with sufficient resolution. The level set method has been used to many applications, such as [13,30,31]. Nevertheless, this method has no conservative properties inherently, that is, the liquid volume can be lost during the evolution of interface. Secondly, small length and time scales demand much cost of computation. The large eddy in the gas or liquid phase is several orders of magnitude bigger than the small ones. At the same time, droplets whose diameters may be smaller than the Kolmogorov scales must also be distinguished. Thirdly, the discretization of the Navier–Stokes equations becomes difficult because of the discontinuity in material properties across the interface, especially for the large ratio in density and dynamic viscosity. The other challenges in the simulation of atomization include the singularity of surface tension that just exists on the interface, the complexity of interaction between atomization and turbulence, and the frequent topological changes. LSM has many advantages over VOF but the mass conservation is not satisfactory. This drawback will seriously affect the accuracy of interface capturing. Recently, some approaches have been proposed to improve the mass conservation of level set method. Enright et al. [2] proposed the particle level set method, which uses massless

http://dx.doi.org/10.1016/j.cjche.2014.07.004 1004-9541/© 2014 The Chemical Industry and Engineering Society of China, and Chemical Industry Press. All rights reserved.

C. Shao et al. / Chinese Journal of Chemical Engineering 23 (2015) 597–604

marker particles to update the interface location predicted by an Eulerian transport equation. It was demonstrated that the particle level set method has great improvement over the original LSM in mass conservation, but the procedure of error correction and particle reseeding is complicated and the computational cost is high due to the large number of massless particles in each cell. Sussman et al. [3] and Menard et al. [4] developed the coupled level set and VOF (CLSVOF) method to enhance the mass conservation by combining the advantages of two methods, i.e. good mass conservation and easy surface curvature description. However, this method solves both advection equations and uses small time steps, leading to high computation cost. Herrmann [5] developed a balanced force refined level set grid (RLSG) method, which reduces the numerical errors in level set function advancement and re-initialization through the refined grid. This method has good efficiency in computation and good mass conservation but it is difficult to implement. An accurate conservative level set method proposed by Desjardins et al. [6] has good mass conservation properties by using hyperbolic tangent function and is easy to implement. In the present study, the accurate conservative level set method is further validated and applied to study fundamental phenomena in spray atomization. The liquid jet and crossflow have been extensively studied, as mentioned in [19–26]. However, the swirling liquid atomization has not been reported in the literature, which is applied to the aero engine and rocket propeller extensively and demands thorough study. In this paper, the conservative level set method is applied to the simulation of swirling liquid sheet atomization. This preliminary work is to reveal the mechanism of swirling liquid sheet atomization. The effects of central gas velocity and density ratio to atomization are also discussed. 2. Numerical Methods 2.1. Incompressible Navier–Stokes equations To describe gas–liquid flow, the incompressible Navier–Stokes equations are introduced:  h i ∂u 1 1 t þg þ u  ∇u ¼ − ∇p þ ∇  μ ∇u þ ∇u ρ ρ ∂t

the normal and curvature of the interface because |∇ϕ(x, t)| = 1. The normal and curvature can be expressed as ∇ϕ j∇ϕj

ð5Þ

κ ¼ −∇  n

ð6Þ

n¼

However, the signed distance function does not guarantee the mass conservation of each phase, and the volume of liquid may change during the advancement of the signed distance function. This will obviously bring numerical errors. Hence, the hyperbolic tangent function proposed by Olsson and Kreiss [7,8] is introduced following the study of Desjardins et al. [6]: ψðx; t Þ ¼

ð3Þ

where σ is surface tension, κ is curvature, and n is unit normal vector.

0.8

0.6

0.4

0 -1

2.2. Accurate conservative level set method In the level set method, the interface is usually defined as the isosurface of a smooth function. This function is defined as signed distance function:

-0.5

0

x

0.5

1

Fig. 1. Hyperbolic tangent function (dotted line) and liquid volume fraction (solid line) with interface located at x = 0.

From Fig. 1 it can be found that the hyperbolic tangent function is almost the same as a Heavyside function. Furthermore, the integral of the hyperbolic tangent function is equal to the liquid volume when ε goes to zero, i.e., Z

Z ψðx; t ÞdV ¼

lim

ε→0

jϕðx; t Þj ¼ minjx−xΓ j:

ð7Þ

0.2

In each phase, material properties are constant. The material properties jump at the interface, which can be represented as [ρ]Γ = ρl − ρg, [μ]Γ = μl − μ g. The velocity at the interface is continuous, [u]Γ = 0. The pressure at the interface is discontinuous, t

   ϕðx; t Þ tanh þ1 2ε

ð1Þ

ð2Þ

½ pΓ ¼ σ κ þ 2½ μ Γ n  ∇n  n



1

where u is velocity vector, ρ is density, p is pressure, g is gravity acceleration, and μ is dynamic viscosity. The continuity equation is ∂p þ ∇  ðρuÞ ¼ 0: ∂t

1 2

where ε is the thickness of the profile. The interface is located in the isosurface of ψ = 0.5. It is shown in Fig. 1 along with the liquid volume fraction when ε = 0.01.

liquid volume fraction

598

Ω

Hðψðx; t Þ−0:5ÞdV

ð8Þ

Ω

where H is Heaviside function and dV is volume element. This equation means that conservation of hyperbolic tangent function is equivalent to conservation of liquid volume. The equation of level set function advection is

ð4Þ

Here ϕ(x, t) N 0 for liquid phase, ϕ(x, t) b 0 for gas phase and ϕ(x, t) = 0 at the interface. This function has the advantage to calculate

∂ψ þ ∇  ðuψÞ ¼ 0: ∂t

ð9Þ

C. Shao et al. / Chinese Journal of Chemical Engineering 23 (2015) 597–604

However, the level set profile will deteriorate during the transport of the level set function. Therefore, a re-initialization process is needed. The re-initialization equation is [7,8]: ∂ψ þ ∇  ðψð1−ψÞnÞ ¼ ∇  ðεð∇ψ  nÞnÞ ∂τ

ð10Þ

where τ is a pseudo time. Solving Eqs. (9) and (10) allows for advection of the interface by the fluid velocity, preserves the hyperbolic tangent profile, and maintains discrete conservation of ψ over the computational domain. High order upstream central (HOUC) scheme [6] is adopted in this work. nth-order level set transport scheme can be expressed as ∇  ðρuZ Þ ¼

  3  X ζi 1 δ2nd J ui ψ J δ2nd ζ i hi 2

ð11Þ

where n = 5, ζi is the computational spatial coordinate, which is related to the physical space coordinate xi through hi = dxi/dζi and J ¼ 3

also proposed. Tanguy and Berlenont [11] used LSM to simulate the binary drop collision and got excellent agreement with experiment. A new drop collision model was proposed by Ko et al. [12]. This model could predict the number of satellite droplets, the velocity and sizes of droplets. This model was compared with experiment to test its accuracy. This section aims at simulating three-dimensional binary drop collision in order to capture the main phenomena in the collision and compare with experiment. The experiment [10] is used here. The properties of liquid and gas are listed in Table 1. In this case, two binary drops of diameter 800 μm do the head-on collision. The relative velocity is 1.65 m ⋅ s− 1 and the Weber number is We = ρlU2D/σ = 30. The domain is 2 mm × 4 mm × 2 mm and the grid is 64 × 128 × 64, which satisfied mass conservation in [11]. The present simulations on We = 30 and We = 40 are shown in Figs. 2 and 3, respectively.

Table 1 The material properties of liquid and gas

∏ hi  δ2nd . is an operator for 2nd order derivative with respect to ζi of

i¼1

2nd order accuracy. Z is a scalar. The two terms in the re-initialization Eq. (10) can be expressed as ðψð1−ψÞnÞ ¼

  3  X 1 δ2nd J xi 2ndζ i ni ψð1−ψÞ J δ2nd ζ i hi i¼1

ðεð∇ψ  nÞnÞ ¼

  3  X 1 δ2nd J x x x εni i n i  ∇ i ψ : J δ2nd ζ i hi i¼1

599

Density/kg ⋅ m−3 Dynamic viscosity/kg ⋅ m−1 ⋅ s−1 Surface tension/N ⋅ m−1

Liquid

Gas

1000 1.137 × 10−3 0.0728

1.226 1.78 × 10−5

ð12Þ

ð13Þ

Accurate solution of Eq. (9) is performed using second order central differencing scheme for spatial discretization and second order semiimplicit Crank–Nicolson time integration. The velocity and pressure fields are computed using a projection method [6]. The pressure jump at the interface is discretized using ghost fluid method (GFM). The procedure of this method is briefly given as follows: Firstly, using time integration, advance the level set scalar by solving Eq. (9). Then, compute the interface normal and curvature using Eqs. (5) and (6). Finally, perform the re-initialization step using Eq. (10). The present simulation is based on the in-house code and the every step is not stopped until the velocity is convergent. The grid independence test is conducted through increasing the grid size. 3. Numerical Validations This method has been validated by Desjardins et al. [6] using Zalesak's disk, circle in a two dimensional velocity field and sphere in a three dimensional velocity field, but the velocity fields are all known in these cases and no practical cases were tested. To further validate the capability in managing actual problem for this method, binary drop collision and drop impacting on liquid film are studied, that always occur in spray atomization. 3.1. Binary drop collision The regime of coalescence and separation of binary drop collision was studied by Qian and Law [9]. They found five collision modes at different gas environments using water and hydrocarbon droplets. The five modes are coalescence after minor deformation, bouncing, coalescence after substantial deformation, coalescence followed by separation for near head-on collision and coalescence followed by separation for off-center collision. Ashgriz and Poo [10] studied the binary drop collision of different diameters and found two different separation modes, that is, reflexive and stretching separations, corresponding to head-on and off-center collision. The binary collision theoretical model was

T=0.006 0.0045

0.002

0.0012

0.0006

0.0

Fig. 2. Drop collision at We = 30 (above: experimental results [10]; below: the present simulations).

As can be seen from Figs. 2 and 3, the level set results agree well with the experimental observations. When the Weber number is small, the two drops can develop into a flying saucer shape. Then it comes into being the shape of torus when the inertia force decreases. Later, bounce appears and two drops go away from each other. Two separate drops form when inertia force overcomes the surface force due to surface tension. For the case of bigger Weber number, ligament forms between the two drops, and the ligament separates from the two drops and forms satellite drop between two primary drops. The simulations catch well the phenomena that appear in the experiments. As mentioned in [11], quantitative comparisons can be biased by a lack of experimental operating conditions or detailed measurement data. Here we just compare our results with experiment [9] and simulation [14] for the transition between coalescence regime and separation regime for drop head-on collision. The result is shown in Fig. 4.

600

C. Shao et al. / Chinese Journal of Chemical Engineering 23 (2015) 597–604

1 We=40 We=30

V/Vo

0.98

0.96

0.94

0.92

T=0.006 0.0047

0.0026

0.0012

0.0006

0.0 0.9

Fig. 3. Drop collision at We = 40 (above: experimental results [10]; below: the present simulations).

0

0.001

0.002

0.003

T

0.004

0.005

0.006

Fig. 5. Time evolution of volume fraction for binary drop collisions in Figs. 2 and 3.

60 55 50 45 40

We

35 30 25 20 Sim. Rieber & Frohn Exp. Qian & Law Simulation

15 10 5 0

0

0.01

0.02

0.03

Oh

0.04

0.05

0.06

Fig. 4. Comparison between numerical and experimental results for the transition between coalescence and separation in Oh–We space for drop head-on collision.

pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffi The Ohnesorge number is defined as Oh ¼ μ l = ρl σD ¼ We=Re. The agreement between our numerical results and experimental results is satisfactory and confirms that the present level set method is an accurate tool to deal with the drop collision. Fig. 5 displays the time evolution of the drop volume fraction for binary drop collision. The mass loss is about 3% for We = 30 case and about 4% for We = 40 case, indicating that the mass conservation of the method is reasonable.

3.2. Drop impacting on liquid film There are many experiments and simulations about the drop impacting on liquid film. Cossali et al. [15] studied the structure of crown formed when single drop impacts on liquid film. The height and diameter of crown were analyzed for different Weber numbers and non-dimensional heights of film. Lee et al. [16] used LSM to simulate

the experiment done by Cossali et al. [15]. It was found that the data coincided with experiment at early time but didn't long after, which indicated that drop impacting on liquid film took on symmetry at early time while it presented three-dimensional nature after certain time. Liang et al. [17] studied the impact of viscosity and surface tension on the crown using CLSVOF method and revealed the mechanism of neck jet of drop impacting on liquid film. The breakup and coalescence of drop impacting on liquid film were discussed using PIV technology by Kassim and Longmire [18]. In order to test the present method's capability of capturing complex phenomena, the case of drop impacting on liquid film on a solid surface is studied. The material properties are as the same as in Table 1. The computational domain is 30 mm × 15 mm and the grid is 512 × 256. The drop of diameter 3.8 mm is initially at the surface of the film of height 3 mm. The drop velocity is 2.4 m·s− 1. The numerical results are shown in Fig. 6. It is obvious to see that the crown or the splashing lamella forms in Fig. 6. The rim forms at the top of crown at t = 0.001 s because of surface tension. The rim gradually escapes from the crown at t = 0.002 s and then develops into secondary droplets. The height of crown gradually increases and diameter magnifies. The rim, crown, and secondary droplet are all captured in our simulation. The evolution of dimensionless crown radius r/D with time tV/D is quantitatively compared with the numerical results in [19]. As shown in Fig. 7, the trend of the radial growth of the splashing lamella is consistent with each other although the parameters are different between the two simulations.

4. Applications to Primary Atomization 4.1. Swirling liquid sheet atomization Primary atomization offers a rich physical phenomenology which is still poorly understood. With the increase of computer capability, CFD numerical simulation of primary atomization has been developed rapidly in recent years, such as jet atomization [20–24] and crossflow atomization [25–27]. However, there is little study on swirling liquid atomization. In this section, the swirling liquid sheet atomization in laminar regime is investigated using the accurate level set method. The sketch map of the swirling liquid sheet atomization is shown in Fig. 8. The gas is injected at the center with the nozzle of diameter D in . The swirling liquid flows coaxially with the annular gap

C. Shao et al. / Chinese Journal of Chemical Engineering 23 (2015) 597–604

601

Fig. 6. The interface evolution of drop impacting on liquid film (black line represents the interface).

membrane with thickness 1/2(D out–D in ). The liquid film ejecting from the annulus is laminar flow. The time is non-dimensioned by Dout/Vl, that is, T = tVl/Dout, where V1 is the axial velocity of liquid. At an early time of T = 1.50, the tip mushroom shape is observed. A recirculation region appears behind the jet front and instantaneous breakup occurs from the mushroom edge. With the evolution of atomization, liquid sheet is observed. As no disturbances are added upstream, the liquid sheet keeps its smooth surface to a certain distance. It is observed that transverse azimuthal perturbation occurs in the tip of liquid sheet at T = 3.00. Villermaux et al. [28] have proposed that transient acceleration in the direction normal to the liquid at the rims triggers a Rayleigh–Taylor instability, which produces the azimuthal perturbation. The transverse azimuthal perturbation grows in amplitude producing ligaments at the wave crests at T = 3.75. Later, these ligaments are stretched and droplets are produced at the tips of ligaments.

2

r/D

1.5

1

the present simulation Rieber & Frohn

0.5

4.3. The relationship between interface and vortex structure

0

0

0.5

1

1.5

T

2

2.5

3

Fig. 7. Comparison of simulation results about radial growth of the splashing lamella Fig. 6.

thickness (Dout–Din)/2. The detailed parameters in this case are listed in Table 2. 4.2. Interface evolution of the swirling sheet atomization Fig. 9 shows the interface evolution of the swirling liquid sheet atomization. The initial liquid is set to the shape of semi-sphere

Fig. 8. Sketch of the swirling sheet atomization.

The vortex structures characterized by the Q criterion are shown in Fig. 10. The second invariant of the velocity gradient tensor defined as Table 2 Summary of parameters for swirling liquid sheet atomization Outer diameter Dout/mm Inner diameter Din/mm Liquid swirl Gas swirl Density of liquid/kg ⋅ m−3 Density of gas/kg ⋅ m−3 Dynamic viscosity of liquid/kg ⋅ m−1 ⋅ s−1 Dynamic viscosity of gas/kg ⋅ m−1 ⋅ s−1 Axial velocity of liquid/m ⋅ s−1 Swirl velocity of liquid/m ⋅ s−1 Velocity of gas/m ⋅ s−1 Surface tension/N ⋅ m−1 Rel (based on Dout) We (based on lip thickness 0.1 mm) Dynamic viscosity ratio Kinematic viscosity ratio Density ratio Ohnesorge number Domain Number of CPU Number of cells Δx/μm Time step size/s

0.4 0.2 Yes No 800 40 1.6 × 10−3 1.6 × 10−4 15 15 30 0.036 3000 500 10 0.25 20 0.015 10Dout × 5Dout × 5Dout 12 × 6 × 6 = 432 512 × 256 × 256 = 33.55 million 7.8 1 × 10−8

602

C. Shao et al. / Chinese Journal of Chemical Engineering 23 (2015) 597–604

T=0.00

T =2.25

T =3.75

T=1.50

shape and ligament-shape structures are seen from these complex structures. In the front of liquid sheet, ligaments and droplets formed undergo involute movement, which are controlled by inertial force, centrifugal force, aerodynamic force and surface tension. The local flow patterns, associated with velocity gradient, have an effect on the deformation and movements of ligaments and droplets. The interface and vortex structures characterized by the Q criterion are simultaneously shown in Fig. 10. It is clear to see that torus-shape vortex structures warp the droplet, and the filamentary vortex tubes are along with the rim of liquid sheet and a large number of vortex tubes are aligned with axis. During the movement of liquid sheet, anti-parallel vortex pairs in Fig. 11 occur in the sides of the top of liquid sheet. In the outside of liquid sheet, the vortex is counterclockwise, while the vortex is clockwise inside of liquid sheet. It is obvious that vortex inside is larger than that outside, which indicates that the two eddies are not equal in the capability of entrainment. Therefore, the liquid sheet tends to bend to inside zone because of the strength of vortex inside. Moreover, the intense shear force around the liquid sheet should lead to the breakup of bulk liquid.

T=3.00

T=4.50

Fig. 9. Interface evolution of swirling liquid sheet atomization.

Q = (AijAij − SijSij)/2 is used to visualize the vortex structures, where Aij and Sij denote antisymmetric and symmetric parts of the velocity gradient tensor, respectively. It is observed that the complex vortex structures are mainly clustered around the front of liquid sheet. The torus-

Fig. 11. Velocity field in the sides of top of liquid sheet (black line for interface location; white arrows for x–y plane velocity vector; continuous colors for distance to interface, G N 0 for liquid and G b 0 for gas, abs(G) N threshold, G = constant).

Fig. 10. Interaction between interface and vortex structures characterized by Q criterion (color for Q value).

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603

4.4. The effect of central gas velocity on atomization Similar to Kulkarni et al. [29], the different central gas velocity is performed to demonstrate the effect of central gas velocity on atomization. As shown in Fig. 12, for large central gas velocity, the central air pulls the liquid sheet more intensively, so that the cone angle is smaller for larger central gas velocity, which accords with the experimental result by Kulkarni et al. [29]. In addition, it is almost the same for the breakup length, which is defined as the distance between the orifice exit and the location for liquid sheet breakup. It is apparent that the thickness of liquid film decreases with increasing distances from the orifice exit and beyond a certain distance, the liquid sheets begin to disintegrate.

Fig. 13. The interface location for different liquid–gas density ratios.

Fig. 12. The interface location for different central gas velocity.

4.5. The effect of liquid–gas density ratio on atomization For different liquid fuel applied to the combustion, it is apparent that properties of liquid fuels can have effect on atomization. In order to demonstrate the effect of liquid–gas density ratio on the atomization, the density of gas is changed. As shown in Fig. 13, the liquid sheet is more wrinkled for little liquid–gas density ratio. This may be attributed to the stronger aerodynamic force to the liquid sheet. In addition, the breakup length is shorter for ρl/ρg = 8.

5. Conclusions In this study, the accurate level set method is applied to the computation of gas–liquid interface capture, which is validated using two cases of simulation. Simulations of binary drop collisions are well in agreement with experimental results and the critical Weber number accords to that conducted by Qian & Law [9]. The simulation of drop impacting on liquid film captures the small structures of liquid such as rim, crown, and secondary droplet. The swirling liquid sheet atomization is studied to reveal some phenomena about primary atomization. From the interface

evolution, Rayleigh–Taylor instability in the azimuthal direction causes the primary breakup of liquid sheet. It is found that complex vortex structures are clustered around the rim of the liquid sheet. The anti-parallel vortex pairs are also observed near the rim of liquid sheet and the liquid sheet tends to bend to inner side since strong recirculation zone. In addition, the presence of the central air pulls the liquid sheet toward the spray axis. The liquid sheet is more wrinkled for higher gas density. The development shown in this work is not sufficient to reveal the mechanism of primary atomization. In addition, the droplet diameter distribution is not included in present simulation. The quantitative analysis needs to be conducted further to consider the parameters that affect the primary atomization. In summary, the accurate level set method presented here shall be a promising tool to study the primary atomization. Nomenclature g gravity acceleration, kg ⋅ m ⋅ s−1 n interface normal pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffi Oh Ohnesorge number (Oh ¼ μ l = ρl σ D ¼ We=Re) P pressure, Pa u velocity, m ⋅ s−1 V volume of fluid We Weber number (We = ρlU2D/σ) X location of cell ε thickness of the profile κ curvature, m−1 μ dynamic viscosity, kg ⋅ m−1 ⋅ s−1 ρ density, kg ⋅ m−3 σ surface tension, N ⋅ m−1 ϕ(G) signed distance function ψ hyperbolic tangent function

Subscripts G Gas L Liquid Γ interface

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