PS16
フェーズフィールド LBM とゴーストノード法のカップリングによる液滴衝突解析
Numerical study of binary droplets collision using
coupling the phase-field lattice Boltzmann method andthe ghost node method.
濵田 航汰, 田川 俊夫
Kota HAMADA, Toshio TAGAWA
東京都立大学 大学院院システムデザイン研究科 航空宇宙システム工学域，
Department of Aeronautics and Astronautics, Tokyo Metropolitan University

1. Introduction
1.1. Background
Computational fluid dynamics is a field that uses computers to solve and analyze approximately the
Equation of continuity and Navier-Stokes equations. This field has developed with the improvement ofcomputer performance and has advantages, such as easier parameter settings and the ability to simulate
environments where experiments are difficult to perform. In addition, various methods have been developed
to date, and these methods have the advantage capable to analyze and consider multiple approaches
depending on the system being simulated. On the other hand, it is rare for simulation results to completely
match experimental results, and it is important to know how to evaluate differences.
Currently, the analysis of fluid phenomena is used in many places, and it is an important issue that is
related to the fuel efficiency of automobiles and airplanes, as well as the power generation performance ofhydroelectric and wind power plants. Among them, gas-liquid two-phase flow is a fluid motion in which gas
and liquid coexists, and it has a great impact on the core parts of systems such as inside engines and nuclear
power plants. In particular, droplet collision problems affect the formation of raindrops and spray coatings,
and the head-on collision between droplets, which is the subject of this study, plays an important role in thespray phenomenon and drug protection. There are various collision forms depending on the Weber number(𝑊𝑒), the collision parameter (𝐼), and the species of liquid 1).
𝑊𝑒 =
𝐼=

𝑏
𝐷

𝜌ℎ 𝐷𝑈02
𝜎

(1)

(2)

Brazier-Smith 2) conducted experiments on droplet collisions. Subsequently, Ashgriz and Poo 1) conducted
droplet collision experiments with the same and different sizes and concluded that collision dynamics can beclassified by the Weber number, collision parameters, and droplet size ratio. Here, the Weber number
represents the ratio of inertial force to surface tension, and the collision parameters represent the eccentricityof the droplet in the collision direction. Various models have been proposed for numerical analysis, including
the study of Nikolopoulos and Bergeles 3) using the VOF method and the study of Pan and Suga 4) using thelevel set method.
1 of 12Figure 1. Model of droplet collision.

Figure 2. We-I diagram collision outcome regimes, 1. SPC (soft permanent-coalescence): coalescence afterminor deformation, 2. B: Bouncing of the droplets, 3. HPC (hard permanent-coalescence): coalescence after
substantial deformation, 4. CFRS (coalescence followed by reflexive separation): coalescence followed byseparation for near head-on collisions, 5. CFSS (coalescence followed by stretching separation): coalescence
followed by separation for off-center collisions.

As can be seen from the above, simulations have been performed using a variety of methods, but there
are still some issues such as instability of the droplet interface, mass conservation, and instability due to large
density ratios. In addition, many methods currently used are not suitable for parallel calculation, and there
are high hurdles to high-speed calculations. Furthermore, when trying to solve these problems, models often
become complicated, and calculation costs are high. Thus, methods to solve these problems are required.
1.2. Purpose
Based on the above background, the purpose of this study is to perform numerical analysis using the
proposed method on collision phenomena related to head-on collisions of droplets, and to verify its validity.
The proposed method is a combination of the phase-field lattice Boltzmann method, which is suitable forparallel computing, and the ghost node method, which has a low implementation cost.
2.

Numerical Method

2.1. Calculation model
The calculation model used in this study is shown in Fig. 3. The characteristic length is the droplet
diameter, the characteristic velocity is the initial velocity, and the effect of gravity is ignored. The calculation
area is 2 × 2 × 4, where the droplet diameter is 1. In addition, because this study calculates a head-on collision,
the impact parameter (𝐼) is set to 0.

2 of 12Figure 3. Calculation model.

2.2. Phase-field lattice Boltzmann method
2.2.1. Macroscopic equation
The interface capturing method employs the phase-field method. The phase-field method assumes acertain width at the interface and solves the conservative Allen-Chan equation 5) for the order parameter 𝜙.
Using this method, it is possible to calculate complex interface shapes and surface tensions. The orderparameter 𝜙 is expressed as follows:𝜙=12𝑠(1 + tanh )
2
𝑊

(3)

Where 𝑊 is the interface width, 𝑠 is the signed distance from the interface. 𝜙 = 0 is the gas phase side,
𝜙 = 1 is the liquid-phase side. The model is shown below.

Figure 4. Transition of order parameter (ϕ).

We solve the Conservative Allen-Chan equation.
𝜕𝜙 𝜕𝜙𝑢𝛼𝜕𝜕𝜙
1 − 4(𝜙 − 𝜙𝑎𝑣𝑒 )2+=𝑀[− 𝑛𝛼 [
]]
𝜕𝑡
𝜕𝑥𝛼
𝜕𝑥𝛼
𝜕𝑥𝛼
𝑊

(4)

Where 𝑀 is the mobility, 𝜙𝑎𝑣𝑒 is the arithmetic mean of the gas and liquid phases, which is set to 0.5 inthis study, and 𝑛𝛼 is the unit normal vector of the interface, which is expressed by the following formula.
𝑛𝛼 =

𝜕𝜙
𝜕𝜙
⁄|
|
𝜕𝑥𝛼 𝜕𝑥𝛽(5)Equation of continuity and Navier-Stokes equation are as follows:
𝜕𝑢𝛼
=0
𝜕𝑥𝛼

(6)
3 of 12

𝜌(

𝜕𝑢𝛼
𝜕𝑢𝛼
𝜕𝑝
𝜕
𝜕𝑢𝛼 𝜕𝑢𝛽
+ 𝑢𝛽
)=−
+
[𝜇 (
+
)] + 𝐹𝛼𝑠 + 𝐹𝛼𝑏𝜕𝑡𝜕𝑥𝛽𝜕𝑥𝛼 𝜕𝑥𝛽𝜕𝑥𝛽 𝜕𝑥𝛼(7)

Where 𝜌 is the density, 𝜇 is the viscosity.
𝜌 = 𝜌𝑙 + 𝜙(𝜌ℎ − 𝜌𝑙 )(8)

𝜇 = 𝜇𝑙 + 𝜙(𝜇ℎ − 𝜇𝑙 )(9)

The subscript ℎ means high and 𝑙 means low. 𝐹𝛼𝑠 is a surface tension term and 𝐹𝛼𝑏 is a gravity term,
which are defined by the following formula.
𝜕𝜙
𝜕𝑥𝛼

(10)

𝐹𝛼𝑏 = −𝜌𝑔𝛼(11)𝐹𝛼𝑠 = 𝜂𝜂 is the chemical potential and is defined as follows:𝜂 = 4𝛽𝜙(𝜙 − 1)(𝜙 − 𝜙𝑎𝑣𝑒 ) − 𝜅

𝜕2𝜙
𝜕𝑥𝛼 𝜕𝑥𝛼

(12)

𝛽=

12𝜎
𝑊

(13)

𝜅=

3𝜎𝑊
2

(14)

To avoid numerical instability, the pressure is calculated from the pressure equation instead of the state
equation. The pressure equation is given by the following formula 6).
𝜕𝑝𝜕𝑢𝛼+ 𝜌𝑐𝑠2
=0
𝜕𝑡
𝜕𝑥𝛼

(15)

2.2.2. Discretization
The conservative Allen-Chan equation is discretized using a weighted MRT model. The distributionfunction ℎ𝑖 for the order parameter is calculated as follows. We employ the D3Q27 model.
𝑒𝑞ℎ𝑖 (𝑥𝛼 + 𝑐𝑖𝛼 𝛿𝑡 , 𝑡 + 𝛿𝑡 ) = ℎ𝑖 (𝑥𝛼 , 𝑡) − 𝑴−𝟏 𝑺𝒉 𝑴 (ℎ𝑖 (𝑥𝛼 , 𝑡) − ℎ𝑖 (𝑥𝛼 , 𝑡))𝑒𝑞
Where ℎ𝑖𝑒𝑞

(16)

is the equilibrium distribution function, and in the phase-field method it is given as follows:ℎ𝑖 = 𝜙𝑤𝑖 [1 +𝑐𝑖𝛼 𝑢𝛼 (𝑐𝑖𝛼 𝑢𝛼 )2 𝑢𝛼2
1 − 4(𝜙 − 𝜙𝑎𝑣𝑒 )2 𝑤𝑖 𝑐𝑖𝛼
+
−
]
+
𝑀
[
] 2 𝑛𝛼
𝑐𝑠2
2𝑐𝑠4
2𝑐𝑠2
𝑊
𝑐𝑠

(17)

Where 𝑐𝑖 is the particle velocity and 𝑤𝑖 is the weighting coefficient. The D3Q27 model used in thisanalysis is described as follows:
Table 1. Particle velocity.

𝒊

𝑐𝑖

𝑤𝑖

0
1~6
7~18

(0,0,0)(±1,0,0), (0, ±1,0), (0,0, ±1)(±1, ±1,0), (0, ±1, ±1), (±1,0, ±1)

8/27
2/27
1/54

19~26

(±1, ±1, ±1)

1/216

𝑺𝒉 is a diagonal matrix, which is expressed as follows:
𝑺𝒉 = diag(0, 𝑠ℎ , 𝑠ℎ , 𝑠ℎ , 1.5,1.5, … ,1.5)(18)
4 of 12𝑠ℎ =1
𝜏ℎ + 0.5(19)

𝑀 = 𝜏ℎ 𝑐𝑠2 𝛿𝑡
𝑐𝑠 =

(20)

1

(21)

√3

𝛿𝑡 = 𝛿𝑥 = 1(22)

𝑴 is the transformation moment, which will be shown at the end of this section.
Equation of continuity and Navier-Stokes equation are discretized using the velocity-based lattice
Boltzmann method. The weighted MRT model is used to calculate the collision term.
𝑒𝑞𝑓𝑖 (𝑥𝛼 + 𝑐𝑖𝛼 𝛿𝑡 , 𝑡 + 𝛿𝑡 ) = 𝑓𝑖 (𝑥𝛼 , 𝑡) − 𝑴−𝟏 𝑺𝒇 𝑴 (𝑓𝑖 (𝑥𝛼 , 𝑡) − ̅̅̅̅̅
𝑓𝑖 (𝑥𝛼 , 𝑡)) + 𝐹𝑖

(23)

1
𝑒𝑞
𝑒𝑞
̅̅̅̅̅
𝑓𝑖 = 𝑓𝑖 − 𝐹𝑖2(24)

Where 𝑓𝑖follows 6):
𝑓𝑖

𝑒𝑞

𝑒𝑞

is the equilibrium distribution function, 𝐹𝑖 is the external force term, which is given as
𝑐𝑖𝛼 𝑢𝛼 (𝑐𝑖𝛼 𝑢𝛼 )2 𝑢𝛼2
+
− 2]
𝑐𝑠2
2𝑐𝑠4
2𝑐𝑠

(25)

𝑐𝑖𝛼 𝐹𝛼𝜌𝑐𝑠2(26)= 𝑤𝑖 [𝐹𝑖 = 𝛿𝑡 𝑤𝑖

𝐹𝛼 is a macroscopic external force term and is as follows:𝑝𝜇𝐹𝛼 = 𝐹𝛼𝑠 + 𝐹𝛼𝑏 + 𝐹𝛼 + 𝐹𝛼

(27)

𝑝

𝜇

𝐹𝛼 is the pressure term and 𝐹𝛼 is the modified viscosity term, as follows:𝑝𝐹𝛼 = −

𝜕𝑝
𝜕𝑥𝛼

(28)
𝜇

𝜇

𝐹𝛼 = −(𝜏𝑓 + 0.5)𝜌𝑐𝑠2 𝛿𝑡𝑒𝑞[∑ 𝑐𝑖𝛼 𝑐𝑖𝛽 (𝑓𝑖 − 𝑓𝑖 )]
𝑖

𝜕𝜌
𝜕𝑥𝛽

(29)

𝑺𝒇 is a diagonal matrix, expressed as follows:
𝑺𝒇 = diag(1,1,1,1, 𝑠𝑓 , 𝑠𝑓 , 𝑠𝑓 , 𝑠𝑓 , 𝑠𝑓 , 1,1, … ,1)(30)

1𝜏𝑓 + 0.5(31)

𝜇 = 𝜏𝑓 𝜌𝑐𝑠2 𝛿𝑡

(32)

𝑠𝑓 =

The first- and second-order differentials that appear in the formula are calculated as follows. 𝜃 isarbitrary physical quantity.
𝜕𝜃1=∑ 𝑐𝑖𝛼 𝑤𝑖 𝜃(𝑥𝑖 + 𝑐𝑖𝛼 𝛿𝑡 , 𝑡)𝜕𝑥𝛼 𝑐𝑠2 𝛿𝑥

(33)

𝜕2𝜃
2
= 2 2 ∑ 𝑤𝑖 [𝜃(𝑥𝑖 + 𝑐𝑖𝛼 𝛿𝑡 , 𝑡) − 𝜃(𝑥𝑖 , 𝑡)]2𝜕𝑥𝛼 𝑐𝑠 𝛿𝑥

(34)

𝑖

𝑖

5 of 12

To perform a fully explicit calculation, the pressure equation is solved using the following finite difference
method.
𝑒𝑞𝑝 ∗ (𝑥𝛼 , 𝑡 + 𝛿𝑡 ) = 𝑝(𝑥𝛼 , 𝑡) + 𝛿𝑡 𝜌𝑐𝑠2 ∑(𝑓𝑖 − 𝑓𝑖 )

(35)

𝑖

In addition, to improve the stability near the interface, the pressure is calculated using a second-order
spatial average.
𝑝(𝑥𝛼 , 𝑡 + 𝛿𝑡 ) = ∑ 𝑤𝑖 𝑝∗ (𝑥𝛼 + 𝑐𝑖𝛼 𝛿𝑡 , 𝑡 + 𝛿𝑡 )

(36)

𝑖

The transformation moment is defined by the following formula 7).
𝑴 = [𝒎𝟎 , 𝒎𝟏 , … , 𝒎𝒊 , … , 𝒎𝟐𝟓 , 𝒎𝟐𝟔 ]𝑇(37)

𝑚0𝑖 = 1(38)

𝑚1𝑖 = 𝑐𝑖𝑥(39)

𝑚2𝑖 = 𝑐𝑖𝑦(40)

𝑚3𝑖 = 𝑐𝑖𝑧(41)

𝑚4𝑖 = 𝑐𝑖𝑥 𝑐𝑖𝑦(42)

𝑚5𝑖 = 𝑐𝑖𝑦 𝑐𝑖𝑧(43)

𝑚6𝑖 = 𝑐𝑖𝑧 𝑐𝑖𝑥

(44)

2
2
𝑚7𝑖 = 3𝑐𝑖𝑥
− 𝑐𝑖𝛼

(45)

2
2
𝑚8𝑖 = 𝑐𝑖𝑦
− 𝑐𝑖𝑧

(46)

2
𝑚9𝑖 = 𝑐𝑖𝛼
−1

(47)

2
𝑚10𝑖 = 3𝑐𝑖𝛼
𝑐𝑖𝑥 − 5𝑐𝑖𝑥

(48)

2
𝑚11𝑖 = 3𝑐𝑖𝛼𝑐𝑖𝑦 − 5𝑐𝑖𝑦(49)

2
𝑚12𝑖 = 3𝑐𝑖𝛼
𝑐𝑖𝑧 − 5𝑐𝑖𝑧

(50)

𝑚13𝑖 = (𝑐𝑖𝑦 𝑐𝑖𝑦 − 𝑐𝑖𝑧 𝑐𝑖𝑧 )𝑐𝑖𝑥

(51)

𝑚14𝑖 = (𝑐𝑖𝑧 𝑐𝑖𝑧 − 𝑐𝑖𝑥 𝑐𝑖𝑥 )𝑐𝑖𝑦(52)

𝑚15𝑖 = (𝑐𝑖𝑥 𝑐𝑖𝑥 − 𝑐𝑖𝑦 𝑐𝑖𝑦 )𝑐𝑖𝑧

(53)

𝑚16𝑖 = 𝑐𝑖𝑥 𝑐𝑖𝑦 𝑐𝑖𝑧

(54)

4
2
𝑚17𝑖 = 1.5𝑐𝑖𝛼− 3.5(𝑐𝑖𝛼− 1) − 2.5

(55)

2
2
2
2
2
(3𝑐𝑖𝑥
) − 4(3𝑐𝑖𝑥𝑚18𝑖 = 3𝑐𝑖𝛼
− 𝑐𝑖𝛼
− 𝑐𝑖𝛼
)

(56)

6 of 122
2
2
2
2
𝑚19𝑖 = 3𝑐𝑖𝛼
(𝑐𝑖𝑦
− 𝑐𝑖𝑧
) − 4(𝑐𝑖𝑦
− 𝑐𝑖𝑧
)

(57)

2
𝑚20𝑖 = 3𝑐𝑖𝑥 𝑐𝑖𝑦 𝑐𝑖𝛼− 7𝑐𝑖𝑥 𝑐𝑖𝑦(58)

2
𝑚21𝑖 = 3𝑐𝑖𝑦 𝑐𝑖𝑧 𝑐𝑖𝛼− 7𝑐𝑖𝑦 𝑐𝑖𝑧(59)2
𝑚22𝑖 = 3𝑐𝑖𝑧 𝑐𝑖𝑥 𝑐𝑖𝛼− 7𝑐𝑖𝑧 𝑐𝑖𝑥

(60)

4
2
𝑚23𝑖 = 4.5𝑐𝑖𝑥 𝑐𝑖𝛼− 5.5(3𝑐𝑖𝛼𝑐𝑖𝑥 − 5𝑐𝑖𝑥 ) − 14.5𝑐𝑖𝑥

(61)

4
2
𝑚24𝑖 = 4.5𝑐𝑖𝑦 𝑐𝑖𝛼− 5.5(3𝑐𝑖𝛼𝑐𝑖𝑦 − 5𝑐𝑖𝑦 ) − 14.5𝑐𝑖𝑦

(62)

4
2
𝑚25𝑖 = 4.5𝑐𝑖𝑧 𝑐𝑖𝛼− 5.5(3𝑐𝑖𝛼𝑐𝑖𝑧 − 5𝑐𝑖𝑧 ) − 14.5𝑐𝑖𝑧

(63)

6
4
2
2
𝑚26𝑖 = 4.5𝑐𝑖𝛼− 12(1.5𝑐𝑖𝛼− 3.5(𝑐𝑖𝛼− 1) − 2.5) − 25.5(𝑐𝑖𝛼− 1) − 14.5

(64)

2.3. Ghost node method8)
In this study, the ghost node method is used to perform a numerical calculation of droplets bouncing. A
ghost node layer is used on the collision surface between droplets, and Dirichlet boundary conditions areapplied to it. In previous research 8), the method was used in combination with the level set method, but themethod is unsuitable during parallel calculation. Therefore, by combining it with the phase-field lattice
Boltzmann method, high-speed calculations are achieved in the present study.

Figure 5. Illustration of ghost node method.

2.4. Energy calculation, radial expansion and algorithm
The energy conservation equation is shown below.
𝐸𝑡𝑜𝑡𝑎𝑙 = 𝐾𝐸𝑖𝑛𝑖𝑡𝑖𝑎𝑙 + 𝑆𝑇𝐸𝑖𝑛𝑖𝑡𝑖𝑎𝑙 = 𝐾𝐸 + 𝑆𝑇𝐸 + 𝑇𝐷𝐸(65)

Where 𝐾𝐸 is kinetic energy, 𝑆𝑇𝐸 is surface tension energy, and 𝑇𝐷𝐸 is dissipated energy.
In this study, the kinetic and surface tension energy are calculated. Note the kinetic energy is calculated
for the entire region, while the formula for surface tension energy is calculated only for the interface region.
1 2
𝜌𝑉 𝑑𝑣𝑝𝑝2

𝐾𝐸 = ∫(66)𝜅 𝜕𝜙 2
𝑆𝑇𝐸 = ∫ (𝛽𝜙 2 (1 − 𝜙)2 + | | ) 𝑑𝑣𝑝
2 𝜕𝑥

(67)

Radial Expansion represents the expansion and contraction in the radial direction of the droplet.
7 of 12

𝐷
𝐷0

𝑅𝐸 =

(68)

The algorithm is as follows:
Step 1. Initial conditions
Step 2. Translation process calculation
Step 3. Boundary conditions
Step 4. Calculation of macroscopic physical quantities
Step 5. Application of the ghost node method
Step 6. Pressure calculation
Step 7. Collision process calculation
Step 8. Outputs
Step 9. Return to Step 2
3.

Figure 6. Illustration of Radial Expansion. (above:
initial condition, below: Calculation in progress.)

Numerical calculation results

3.1. Calculation conditions
The calculation conditions are listed in Tables 2 and 3. The following values are used for the Reynolds
number and Weber number for comparison with the experimental results of previous research. The calculation
was performed up to 2.0ms in dimensional time.
Table 2. Calculation condition9)

Parameter

Case1

Case2

𝑅𝑒
𝑊𝑒

111
9.33

55.27
2.27

Table 3. Common parameters.

Parameter

Value

Grid
Mach number
mobility32
0.01√30.05

Interface width5

Impact parameter0.0

Density ratio

760

Viscosity ratio

118.75The Reynolds number and Weber number are defined as follows:
𝑅𝑒 =

𝑈0 𝜌𝐷
𝜇ℎ

(69)

𝜌ℎ 𝐷𝑈02
𝜎

(70)

𝑊𝑒 =

3.2. Results
First, the results of Case 1 are presented. Figure 7 shows the time series of droplet collisions. Comparing
the simulation results obtained in this analysis with the experimental results9), we can confirm that the shapesof Cases 1 is roughly consistent. It can also be seen that there are gaps between the droplets. This is because
the ghost node method requires ghost layer on the collision surface.
8 of 12Figure 7. Result of Case1 (Left: Numerical result, Right: Experimental result). (𝑡 =0.0, 0.20, 0.35, 0.49, 0.61, 0.70, 0.80, 0.99 ms.) 9)

The volume conservation was less than 0.02% in both cases, making use of the advantages of the lattice
Boltzmann method. You can also see that the graph of volume change is oscillating. This is because thepressure equation is solved explicit method. Figures 8 and 9 show the results for the maximum droplet radius.
First, the droplet is stretched in the radial direction due to collision. Then, it can be confirmed that it repeatedly
expands and contracts due to recoil.

Figure 8. Volume change rate graph.

Figure 9.

Radial expansion graph.

Figure 10 shows the changes in the kinetic energy and surface tension energy over time. As the droplet
collides, the kinetic energy decreases and, the surface tension energy increases. After that, the kinetic energy
increases again, and the surface tension energy decreases. We can also see that the graph of STE change isoscillating. This is because the pressure equation is solved explicit method. In addition, since the calculationof STE is greatly influenced by the interface shape, we believe that even slight interface vibrations are reflectedin the STE.

9 of 12Figure 10. Energy graph.

Next, the results of Case 2 are presented. As in Case 1, Comparing the simulation results obtained in thisanalysis with the experimental results9), we can confirm that the shapes of Cases 1 is roughly consistent.

Figure 11. Result of Case2 (Left: Numerical result, Right: Experimental result). (𝑡 =0.0, 0.30, 0.60, 0.70, 0.90, 1.05, 1.15, 1.25, 1.30, 1.50 ms.) 9)

The volume conservation was less than 0.02% in both cases, making use of the advantages of the lattice
Boltzmann method. Figures 12 and 13 show the results for the maximum droplet radius. First, the droplet is
stretched in the radial direction due to collision as in Case 1. Then, it can be confirmed that it repeatedly
expands and contracts due to recoil. On the other hand, the droplet bound time is slower than in Case 1 because
the Weber number is smaller.

10 of 12Figure 12. Volume change rate graph.

Figure 13. Radial expansion graph.

Figures 14 and 15 show the changes in the kinetic energy and surface tension energy over time. As thedroplet collides, the kinetic energy decreases, and the surface tension energy increases as in Case 1. On theother hand, since the Weber number is smaller than in Case 1, the ratio of kinetic energy to total energy issmaller. This trend is consistent with previous research9), and it demonstrates the validity of the proposed
method.

Figure 14. Kinetic energy graph.

4.

Figure 15. Surface tension energy graph.

Conclusion

The purpose of this study was to use the proposed method to perform numerical calculations of thecollision phenomenon related to droplet rebound and to verify its validity. The proposed method is a couplingof the phase-field lattice Boltzmann method, which is suitable for parallel calculations, and the ghost node
method, which has low implementation costs. Calculations were performed and compared with previous
studies9), proving the validity of the proposed method. It was also found that the advantages of the lattice
Boltzmann method, such as good volume conservation and suitability for parallel calculations using GPUs,
were able to be utilized.
In future, we plan to expand the proposed method to calculations when droplets are misaligned, i.e.,collision parameter 𝐼 is other than 0.
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