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标题中的"A modified phase-field three-dimensional model for droplet impact with solidification"指的是一个改进的三维相场模型，用于模拟液滴撞击过程中的固化现象。在这个模型中，研究者着重考虑了液滴在撞击固体表面时的动态行为，以及在这个过程中发生的固态转变。描述中提到的液滴撞击涉及到多相流问题，特别是当液滴与固体表面接触时，可能发生的固化过程。固化的发生是由于液滴在接触到温度低于其熔点的表面时，会释放潜热，导致液态向固态转变。这个过程对热传递和流体动力学有显著影响，因此在模型中需要特别处理。文章的关键词包括"Phase-field"（相场）、"Heat source"（热源）和"Multiphase flow"（多相流），这些关键词揭示了研究的核心内容。相场方法是一种用于模拟相变的数值工具，它可以捕捉界面而不必明确定义其位置。热源则指的是液滴固化过程中释放的潜热，这是影响固液相变的关键因素。多相流则强调了模型需同时处理液态和固态的流动特性。在模型构建上，研究者基于Cahn-Hilliard方程（用于描述相界面演化）和Navier-Stokes方程（用于描述流体动力学）以及能量平衡方程（用于描述热传递）。他们将潜热作为能量方程中的源项来追踪固化轮廓，这种处理方式使得模型能更好地适应固液相变的模拟。文章通过五种情况的测试验证了模型的准确性，并与实验结果进行了对比。此外，还探讨了热接触电阻的影响，这在实际的液滴固化解析中是非常关键的，因为它影响热量的传递效率，从而影响固化的速度和形态。这个研究为理解和模拟液滴撞击固体表面时的固化现象提供了一个先进的计算工具，对于材料科学、热力学以及多相流领域具有重要意义，特别是在涉及冷凝、焊接、喷涂等技术中液滴固化的复杂过程。

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International Journal of Multiphase Flow 116 (2019) 51–66

Contents lists available at ScienceDirect

International Journal of Multiphase Flow

journal homepage: www.elsevier.com/locate/ijmulow

A modiﬁed phase-ﬁeld three-dimensional model for droplet impact

with solidiﬁcation

Mingguang Shen

, Ben Q. Li

b , ∗

, Qingzhen Yang

c , d

, Yu Bai

, Yu Wang

, Shaochong Zhu

Bin Zhao

, Tianqing Li

, Yongbao Hu

State Key Laboratory for Manufacturing Systems Engineering, Xi’an Jiaotong University, Xi’an Shaanxi 710049, PR China

Department of Mechanical Engineering, University of Michigan, Dearborn, MI 48128, USA

The Key Laboratory of Biomedical Information Engineering of Ministry of Education, School of Life Science and Technology, Xi’an Jiaotong University, Xi’an

Shaanxi 710049, PR China

Bioinspired Engineering and Biomechanics Center (BEBC), Xi’an Jiaotong University, Xi’an Shaanxi 710049, PR China

State Key Laboratory for Mechanical Behavior of Materials, Xi’an Jiaotong University, Xi’an Shaanxi 710049, PR China

State Key Laboratory of Electrical Insulation and Power Equipment, Xi’an Jiaotong University, Xi’an Shaanxi 710049, PR China

a r t i c l e i n f o

Article history:

Received 19 November 2018

Revised 1 April 2019

Accepted 6 April 2019

Available online 8 April 2019

Keywords:

Phase-ﬁeld

Heat source

Multiphase ﬂow

Thermal contact resistance

a b s t r a c t

A modiﬁed phase-ﬁeld three-dimensional model has been developed to simulate the spreading of an

impacting droplet undergoing solidiﬁcation. The model is based on the numerical solution of the Cahn-

Hilliard equation coupled with the Navier-Stokes equations for ﬂuid ﬂow and the energy balance equation

for heat transfer. The solidiﬁcation proﬁle is tracked by treating the latent heat as a source term in the

energy equation, which is modiﬁed to work with the Cahn-Hilliard equation. To verify the model, ﬁve

cases were tested and matched well with experiments. Also, the effect of thermal contact resistance on

the maximum spread factor of a solidifying droplet is discussed. One case was taken from practical ther-

mal spraying conditions where a solidifying ceramic droplet spreads on a cold surface at a supersonic

impact velocity; computed results are consistent with available measurements.

1. Introduction

Plasma spraying process has been widely used in manufactur-

ing metal and nonmetal coatings for a wide range of engineering

applications ( Padture et al., 2002; Stengl et al., 2009; Stathopou-

los et al., 2016 ). In a spraying coating process, a molten droplet

is accelerated to impinge onto, and subsequently, spread along a

substrate at a temperature much lower than the melting point of

the droplet, thereby resulting in a thin coating solidiﬁed on the

substrate. The process involves rather complex nonlinearly coupled

transport phenomena including ﬂuid ﬂow, droplet spreading, heat

transfer and moving boundaries. For a better understanding of the

process fundamentals, and for an optimized operation of the pro-

cess, many computational models of various degrees of sophistica-

tion have been developed in the past years. Because of the highly

nonlinear nature of the problem, numerical approach is often em-

ployed to obtain the required solutions.

Many numerical methods proposed in the literature have been

applied to treat the nonlinear problem of the moving boundaries in

∗

Corresponding author.

E-mail address: benqli@umich.edu (B.Q. Li).

the process, which include the marker-and-cell method, volume of

ﬂuid method, Lattice Boltzmann method and phase-ﬁeld method.

In a nutshell, the ﬁrst two methods may be classiﬁed as sharp in-

terface methods, while the phase-ﬁeld method is referred to as a

diffuse interface method. The Lattice Boltzmann method is viewed

as a mesoscale one.

Harlow and Shannon (1967) , using the “marker-and-cell” (MAC)

ﬁnite-difference method, studied droplet impact for the ﬁrst time,

with viscosity and surface tension being neglected for simplicity.

Later on, this approach was enhanced by Tsurutani et al. (1990) ,

who added these two elements, and considered heat transfer from

a hot droplet to a cold surface. He found that surface tension forces

could make the leading edge of a droplet more rounded during

spreading. However, solidiﬁcation was not accounted for, nor were

three dimensional simulations performed, in both works. The MAC

method belongs to the Lagrangian front-tracking method, where

the interface is explicitly tracked by Lagrangian particles, or mark-

ers. Modern front-tracking methods ( Unverdi and Tryggvason 1992;

Tryggvason et al., 2001 ), using surface markers, may give a more

precise interface description. Nevertheless, these methods are rela-

tively complex to implement.

Similar to the Lagrangian front-tracking methods, the front-

capturing methods also use structured grids. It is, however,

https://doi.org/10.1016/j.ijmultiphaseﬂow.2019.04.004

52 M. Shen, B.Q. Li and Q. Yang et al. / International Journal of Multiphase Flow 116 (2019) 51–66

required that an interface representation and advection algorithm

be clearly deﬁned. The volume of ﬂuid method falls into this

category. Two “volume of ﬂuid” (VOF) based codes, FLOW-3D

and RIPPLE, were employed by Trapaga and Szekely (1991) and

Liu et al. (1993) to simulate a thermal spray process. The former

did not take solidiﬁcation into consideration, but both chose par-

ticles of micrometers in diameter and impinging velocities up to

several hundred meters per second. Based on RIPPLE and a vol-

ume tracking algorithm, Bussmann et al. (1999) delved into the im-

pact of a droplet onto an oblique substrate and a sharp edge, with

surface tension converted to a volumetric force and contact angle

introduced as a boundary condition. However, heat transfer and

phase change during impact were ignored. An improved model was

proposed by Pasandideh-Fard et al. (2002) to track droplet free sur-

face and solidiﬁcation front. Drawing on an enthalpy method, they

also studied the heat released during phase transformations and

obtained results in good agreement with experiments. With a VOF

model, Wilden et al. (2006) examined the ﬂattening of particles.

Microstructure simulation was conducted as well in his work using

the phase-ﬁeld method. With the penalty and Eulerian-Lagrangian

VOF methods, Vincent et al. (2015) treated droplet solidiﬁcation,

and looking into thermal contact resistance, asserted that solidiﬁ-

cation would be accelerated by lowering it. Ramanuj et al. (2017) ,

via a VOF method, investigated splat morphology and solidiﬁca-

tion characteristics of a molten hypoeutectic alloy droplet. All these

variants of VOF methods, except that used by Vincent et al. (2015) ,

need interface reconstruction, which means the interface is not

guaranteed from local ﬂuid volume data, and request special care

to treat the advection term. Nevertheless, the inherent mass (and

volume) conservation is a salient feature of VOF methods.

Except being tracked explicitly, the moving boundaries can also

be traced implicitly. Lattice Boltzmann method is such a strat-

egy, but it is on a mesoscale. A Lattice Boltzmann equation (LBE)

method for incompressible binary ﬂuids was invoked by Lee and

Liu (2010) to model contact line dynamics, solidiﬁcation not be-

ing considered. With a LBE method, Sun et al. (2015) , resting on

the pseudo-potential model and enthalpy formation, explored wa-

ter droplet solidiﬁcation. Though this method can capture inter-

faces implicitly and is simple to implement, yet it is impossible

to prescribe macroscopic ﬂuid properties as input parameters. In-

stead, they are input as functions of microscopic parameters, which

are to be adjusted in a model. Besides, hydrodynamic conditions

are diﬃcult to satisfy on a grid node exactly.

Retaining the appealing merit of implicitly capturing interfaces,

the phase-ﬁeld method has emerged as a powerful tool in simulat-

ing multiphase ﬂows, with the ease of numerical implementation

and ready extension to three dimensions. It can also be applied

to treat moving boundaries at both micro and macro scales. With

a phase-ﬁeld method, Lim and Lam (2014) studied the impinge-

ment and spreading of micro-sized droplets on surfaces of het-

erogeneous wettability at nonzero impacting velocities. For a bet-

ter understanding of the underlying physics behind droplet impact,

Zhang et al. (2016) , using a phase-ﬁeld method, examined different

impact phenomena, such as bouncing and splashing, under isother-

mal conditions. Luo et al. (2017) delved into droplet spreading on

topologically rough substrate surfaces, and moreover, over various

textures of substrate surface in three dimensions with a phase-

ﬁeld method. They focused on pure ﬂuid ﬂow, without consider-

ing heat transfer and solidiﬁcation. However, it is of crucial impor-

tance to incorporate these two thermal elements into a phase-ﬁeld

model to bring out its best in predicting droplet behavior during

plasma spraying, especially at supersonic velocities. Nevertheless,

to the authors’ best knowledge, few articles appear to have been

reported on the phase-ﬁeld modeling of coupled thermal and ﬂuid

dynamics of an impacting droplet. The paper is therefore devoted

to developing a phase-ﬁeld model for such purpose. At length, the

paper is organized as follows. First, the model was put forward and

veriﬁed against three low speed impacts, and then was applied to

a supersonic impact under practical conditions. The results show

good agreement with experiments.

2. Mathematical statement

During droplet impact in thermal spray, three physical pro-

cesses are coupled, i.e. , ﬂuid ﬂow, heat transfer, and solidiﬁcation.

Thus in this section the governing equations for these three ﬁelds

are given. Beginning from the ﬂow ﬁeld, these equations are to be

expounded, in turn and in a detailed manner.

2.1. Fluid ﬂow

As for two-phase ﬂows, the convective Cahn-Hilliard equation

( Jacqmin, 20 0 0 ) is employed to describe the evolution of the or-

der parameter c , which is a measure of phase, or a phase indicator.

Generally, two different integers will be assigned to c , each denot-

ing a distinct phase. A transition between the two phases is con-

sidered to occur over a thin region where c changes drastically but

continuously. With such a description, all physical quantities at a

ﬁxed point in space can be seen as functions of c . Once it is deter-

mined, the physical quantities can be calculated through a simple

liner relation as introduced later. The Cahn-Hilliard equation takes

the following form,

+ u · ∇ c − M∇

φ = 0 ∈  × T (1)

where  represents the geometric computational domain, T the

computing time duration, u the velocity ﬁeld, and M the strength

of diffusion. In this way, Eq. (1) describes the evolution of the or-

der parameter c via diffusion and convection, the ﬁrst mechanism

of which is due to the gradient of chemical potential φ = δ f /δc. It

is derived from the free energy density f per unit volume, a func-

tion of c ,

(

)

ξγα

∇c

+ ξ

−1

γα·

(

1 − c

)

∈  × T (2)

where ξ stands for a measure of interfacial thickness, γ surface

tension, and α is a constant equal to 6

√

2 . It is noted that in this

paper c = 1 is to denote a gas and c = 0 a liquid, with the value in

between indicating the interface. The ﬁrst term on the right hand

side of Eq. (2) accounts for the energy density due to phase gra-

dients and the second due to bulk phases. Additionally, the second

term has two stable minima, corresponding to two distinct phases.

The equilibrium phase proﬁle is determined through minimizing

the total free energy F =



f d V .

Now we turn to ﬂuid motion. With an assumption about the in-

compressibility of Newtonian ﬂuids, the equations governing mass

and momentum conservation are obtained as follows:

∇ · u = 0 (3)

∂u

∂t

+ u · ∇u = −

∇p

(

)

∇ · σ

(

)

+ g +

φ∇c

(

)

+ S (4)

in which φ∇c , signifying surface tension, has been converted,

equivalently, into a body force. S is a source term. σ = μ(c)[ ∇u +

(∇u )

] is the Newtonian stress tensor. Also, g is the local gravita-

tional acceleration. Note that density and viscosity have been re-

cast as functions of the order parameter c in such a form that

(

)

(

− ρ

)

c + ρ

(

)

(

− μ

)

c + μ

and ρ

indicate the densities of a gas and a liquid, respec-

tively, in this paper. If physical properties change much after solidi-

ﬁcation, ρ

, for example, should be further interpreted as functions

M. Shen, B.Q. Li and Q. Yang et al. / International Journal of Multiphase Flow 116 (2019) 51–66 53

of the liquid fraction, F

, as ρ

= ( ρ

2 l

− ρ

2 s

) F

+ ρ

2 s

, with subscripts

s and l standing for solid and liquid, respectively.

Flow near a solidifying front, or in an artiﬁcial mushy region

where solid and liquid coexist, is modeled as a porous media ﬂow.

To be speciﬁc, a source term S is introduced into the momentum

Eq. (4) , taking the form of

S = A u = −d

(

1 − F

)

+ b

where d is a rather large number to sweep out all other momen-

tum sources when the liquid fraction F

, expressing the volume

percentage of the ﬂuid component, is coming close to zero. Specif-

ically, when a liquid is near the completion of solidiﬁcation, its ve-

locity is made to vanish. If F

is unity, namely, when no solidiﬁca-

tion occurs, this term drops out, exerting no effect on ﬂuid motion

as expected. While b is a quite small number to avoid division by

zero. In this paper, d and b are chosen to be on the order of 10

and 10

−3

, respectively.

As such, though the Cahn-Hilliard phase ﬁeld method has its

appeal in selecting arbitrary interface thickness and removing con-

tact line singularity, yet some crucial problems of actual numerical

implementation remain to be solved. For instance, adequate grid

points are needed in the interfacial region. In other words, this

diffuse band must be ﬁnely resolved; otherwise unreliable results

will arise. Finer meshes are required when dealing with drastic

interface changes. This may present no major diﬃculties in two-

dimensions, but that is not the case in three-dimensions, even

with the aid of parallel computing or of some advanced numeri-

cal technique, for example, adaptive mesh reﬁnement ( Zhou et al.,

2010 ). Nevertheless, despite the high demand on computational re-

sources, the merit of simplifying a two-phase problem into a sin-

gle phase model by evolving a phase parameter has made it very

useful in simulating droplet coalesce and breakup, and more im-

portantly, capturing interfaces implicitly.

2.2. Heat transfer with phase change

In dealing with heat transfer with phase change, a source term

accommodating latent heat was added to the energy balance equa-

tion. Assuming negligible viscous dissipation and laminar ﬂow, the

modiﬁed heat equation turns up as

(

)

(

)



∂

∂t

+ u · ∇T



= ∇ ·

[

(

)

∇T

]

− ρ

(

)

∂F

∂t

(5)

With the last term taking care of the latent heat released,

this equation can be seen as an extension of the formulation by

Zheng et al. (2015) , who coupled the heat equation with the Cahn-

Hilliard equation without considering phase change. It is readily

seen that the thermal conductivity k ( c ) is handled in the same

way as density ρ( c ) and heat capacity c

( c ). L embodies latent

heat of freezing. The liquid fraction F

, deﬁned over a temperature

range and showing the volume percentage of the ﬂuid component,

changes continuously with temperature as shown below



1 T > T

+ ε

T −

(

−ε

)

2 ε

− ε ≤ T ≤ T

+ ε

0 T < T

− ε

where T

is the melting point of a solid, or more exactly, tin, in

this paper, and ɛ is a quite small temperature interval. Note that

the heat equation is applied to the entire computational domain,

but that the heat source term is updated only where c ≤0.5, with

c = 0 . 5 being a threshold demarcating tin and the surrounding

gas. For c > 0.5, F

is ﬁxed at unity.

The source based method for phase change problems has

gained popularity over the past years. It can be easily accommo-

dated into existing computational codes, easy to implement and

accurate especially in dealing with phase change problems over

a ﬁnite temperature range. For phase change of pure substances,

special numerical techniques must be taken to handle the source

term, which contains a step change of latent heat at the melting

point. More often, it can be modeled through introducing an artiﬁ-

cial mushy region, over which the latent heat changes continuously

with respect to temperature. The source based method per se be-

longs to the weak solutions, where both phases need not be con-

sidered separately and the solidiﬁcation front is captured implicitly

after the solution is obtained in a ﬁxed grid. This ﬁxed grid ap-

proach ﬁts well with the phase ﬁeld model. However, poor perfor-

mance may be led if it is used to deal with substances with large

ratios of latent heat to sensible heat, such as water. The larger the

ratio, the more severe the poor performance of the weak solutions

will be. Fortunately, common metals do not fall into this category,

and given the large undercoolings in this paper, it is justiﬁed to

employ this method.

Above is for heat transfer in the two-phase ﬂow. Eq. (5) is, how-

ever, also applicable to a solid substrate by dropping out the con-

vective and source terms. Besides, the thermal contact resistance,

mostly caused by surface roughness, or equivalently, by nonideal

contact between a ﬂuid and a substrate must be taken into con-

sideration. It is not realistic to assume perfect contact between an

impacting droplet and a substrate during spraying. A temperature

discontinuity always exists because of an insulating layer created

by the imperfect contact that is related to surface roughness, con-

tact pressure, and materials involved. In this paper, it is accounted

for in the junction between a substrate and a ﬂuid, that is, layers

J = 0 and J = 1 as in Fig. 1 , two rows of nodes in the computa-

tional plane. Between any vertically adjacent control volumes in

these two layers, a heat ﬂux exists due to the temperature discon-

tinuity mentioned before, and can be related to the contact resis-

tance ( Bot and Arquis, 2009 ) as



tot

= R



t,c

(

)

y

2 k

y

2 k

y

T



(6)

where T is the temperature drop and q



the heat ﬂux. Also, y

denotes the height of a control volume and R



t,c

thermal contact resistance, with a moderate value between a liquid

and a substrate, usually determined for a certain pair of liquid and

substrate, and a much bigger one between a gas and a substrate.

The terms, k

( c ) and k

, are the thermal conductivities of a ﬂuid

mixture and a substrate, respectively. Special attention is given to

, the effective conductivity, which arises again in Section 3 . It

is actually this effective conductivity that embodies the effect of

thermal contact resistance.

By considering the participation of a gas in heat transfer

and the thermal contact resistance between it and a substrate,

it is no longer necessary to apply adiabatic conditions along

the deforming gas-liquid interface and the non-wetting part of

the substrate, as done by Pasandideh-Fard et al. (2002) and

Ganesan et al. (2015) . Excluding heat transfer with the surround-

ing gas may be unrealistic when temperature gradient in between

is sharp, or when gas entrapment occurs. Additionally, made of

stainless steel, the substrate is of comparable thickness with the

height of the ﬂow ﬁeld, justifying the use of no heat ﬂux on the

bottom.

2.3. Boundary conditions

Last but not least, boundary conditions closing a mathemati-

cal system must be prescribed. The zero gradient condition on all

boundaries is applied to the chemical potential φ. It also ﬁts for

the order parameter c , but only when the static contact angle is

◦

. If it is deviated from 90

◦

, Eq. (2.14) in ( Jacqmin, 20 0 0 ) is to

be imposed on the bottom where a droplet interacts with a sub-

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