A critical comparison of smooth and sharp interface methodsfor phase transitionCitation for published version (APA):Rajkotwala, A. H., Panda, A., Peters, E. A. J. F., Baltussen, M. W., Geld, C. W. M. V. D., Kuerten, J. G. M., &Kuipers, J. A. M. (2019). A critical comparison of smooth and sharp interface methods for phase transition.International Journal of Multiphase Flow, 120, [103093]. https://doi.org/10.1016/j.ijmultiphaseflow.2019.103093

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International Journal of Multiphase Flow 120 (2019) 103093

Contents lists available at ScienceDirect

International Journal of Multiphase Flow

journal homepage: www.elsevier.com/locate/ijmulflow

A critical comparison of smooth and sharp interface methods for

phase transition

A.H. Rajkotwala

a , A. Panda

a , E.A.J.F. Peters a , M.W. Baltussen

a , ∗, C.W.M. van der Geld

b , J.G.M. Kuerten

c , J.A.M. Kuipers a

a Multiphase Reactors Group, Department of Chemical Engineering and Chemistry, Eindhoven University of Technology, P.O. Box 513, Eindhoven 5600 MB,

the Netherlands b Interfaces with Mass Transfer Group, Department of Chemical Engineering and Chemistry, Eindhoven University of Technology, P.O. Box 513, Eindhoven

5600 MB, the Netherlands c Power and Flow Group, Department of Mechanical Engineering, Eindhoven University of Technology, P.O. Box 513, Eindhoven 5600 MB, the Netherlands

a r t i c l e i n f o

Article history:

Received 12 February 2019

Revised 25 June 2019

Accepted 11 August 2019

Available online 14 August 2019

Keywords:

Direct numerical simulation

Front tracking

Local front reconstruction method

Phase transition

Sharp interface approach

a b s t r a c t

In this study, the Local Front Reconstruction Method (LFRM) is extended to allow for the direct numerical

simulation of flows with phase transition. The LFRM is a hybrid front tracking method without connectiv-

ity, which can easily handle complex topological changes. The expansion due to phase change is incorpo-

rated as a non-zero divergence condition at the interface. The energy equation is treated with two differ-

ent approaches: smooth interface approach and sharp interface approach. The smooth interface approach

uses a one fluid formulation to solve the energy equation with an interfacial source term accounting for

phase change. This interfacial source term enforces the saturation temperature at the interface. However,

in the sharp interface approach, the thermal properties are not volume-averaged near the interface and

the saturation temperature is imposed as a boundary condition at the interface. A detailed mathematical

formulation and numerical implementation pertaining to both approaches is presented. Both implemen-

tations are verified using 1D and 3D test cases and produce a good match with analytical solutions. A

comparison of results highlights certain advantages of the sharp interface approach over the smooth in-

terface approach such as better accuracy and convergence rate, reduced fluctuations in the velocity field

and a physically bounded temperature field near the interface. Finally, both approaches are validated with

a 3D simulation of the rise and growth of a vapor bubble in a superheated liquid under gravity, where a

good agreement with experimental data is observed for the bubble growth rate.

© 2019 The Authors. Published by Elsevier Ltd.

This is an open access article under the CC BY-NC-ND license.

( http://creativecommons.org/licenses/by-nc-nd/4.0/ )

1

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. Introduction

Phase change phenomena, especially related to gas-liquid flows,

re seen in many industrial systems, such as refrigeration sys-

ems, cooling systems, boilers, heat exchangers and nuclear reac-

ors. Phase change is a relatively efficient mode of heat transfer

ecause of the observed high heat transfer rates. Another advan-

age is that a large amount of energy can be stored in the form

f latent heat making phase change important for thermal energy

torage applications. A detailed understanding and accurate predic-

ions of the phase change processes in such systems are important

or their safe and efficient operation.

∗ Corresponding author.

E-mail address: M.W.Baltussen@tue.nl (M.W. Baltussen).

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ttps://doi.org/10.1016/j.ijmultiphaseflow.2019.103093

301-9322/© 2019 The Authors. Published by Elsevier Ltd. This is an open access article u

In addition to experiments, numerical simulations have become

n important tool to study phase change phenomena.The advan-

age of Direct Numerical Simulations (DNS) is that, in contrast to

he experiments, the complete details on temperature and velocity

elds are available. This has facilitated the study of the transient

nd dynamic aspects of phase change.

The earliest attempt to simulate boiling flows can be traced

ack to Welch (1995) . In this study, a 2D moving mesh finite vol-

me method was developed. However, the method was limited to

mall topology changes of the interface. Since then, many different

nterface tracking techniques, such as Volume of Fluid (VOF), Level

et (LS), and Front Tracking (FT) methods, have evolved which can

andle large deformation of the liquid–vapor interface.

Welch and Wilson (20 0 0) studied a horizontal film boil-

ng problem using the VOF method. Schlottke and Weigand

2008) used the VOF method to study 3D droplet evaporation.

nder the CC BY-NC-ND license. ( http://creativecommons.org/licenses/by-nc-nd/4.0/ )

2 A.H. Rajkotwala, A . Panda and E.A .J.F. Peters et al. / International Journal of Multiphase Flow 120 (2019) 103093

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Tsui et al. (2014) simulated boiling bubbles emerging from a pla-

nar film and a circular film using VOF. More recently, Sato and

Ni ̌ceno (2013) used mass conservative VOF to simulate 3D nucle-

ation and growth of a single bubble. In the VOF method, the in-

terface is generally artificially smeared for the calculation of in-

terfacial properties resulting in a higher numerical error for phase

change calculations ( Sato and Ni ̌ceno, 2013 ).

The LS method has been extensively used by Dhir’s group ( Dhir,

1998; 2001; Dhir and Purohit, 1978; Li and Dhir, 2007 ) to study

different phase change phenomena such as nucleate boiling, film

boiling from a horizontal cylinder and sub-cooled pool boiling.

Apart from Dhir’s group, Gibou et al. (2007) used the LS method

to study film boiling and Tanguy et al. (2007) to simulate evapora-

tion of a moving and deforming droplet. However, the LS method

is a non-conservative approach and does not guarantee mass con-

servation for phase change calculations ( Sato and Ni ̌ceno, 2013 ).

The smearing of the interface, as encountered in the VOF

method, can be avoided by using the Front Tracking method,

where the interface is directly tracked with a Lagrangian mesh.

The Front Tracking method was first applied to study film boil-

ing by Juric and Tryggvason (1998) . They used an iterative pro-

cedure to set the correct temperature boundary condition at the

interface. Subsequently, Esmaeeli and Tryggvason (2004) improved

the method by eliminating the iterative algorithm and applied it

to simulate 3D film boiling from multiple horizontal cylinders. Re-

cently, Irfan and Muradoglu (2017) have developed an FT method

to simulate the species gradient driven phase-change process.

Hybrid methods have been developed combining/modifying

these standard methods: Coupled Level Set and Volume of Fluid

(CLSVOF) method, Level Contour Reconstruction Method (LCRM)

and Local Front Reconstruction Method (LFRM). Film boiling sim-

ulations were carried out by Shin and Juric (2002) using LCRM.

Tomar et al. (2005) developed a 2D CLSVOF method coupled with

phase change and applied it to simulate film boiling. Shin and

Abdel-Khalik (2007) studied the stability of an evaporating thin

film of liquid on a heated cylindrical rod with parallel and cross va-

por flow using improved LCRM. Recently, Shin and Choi (2016) fur-

ther improved the energy formulation in LCRM and simulated a

rising bubble with phase change. In this study, we use a hybrid

front tracking method, the modified Local Front Reconstruction

Method ( Rajkotwala et al., 2018; Mirsandi et al., 2018 ), which of-

fers better local and global mass conservation compared to other

Front Tracking methods. The LFRM method is capable of deal-

ing with breaking and merging of the interface and thus can be

used to simulate complex topological changes like droplet collision

( Rajkotwala et al., 2018 ). We extend the method to enable phase

change simulations, which has not been done before.

The most important aspect of the phase change simulation

methodology is the treatment of the variation of properties and of

the jump conditions across the interface. These are broadly clas-

sified into two approaches: in the first approach, the governing

equations are written in each phase separately and additional jump

conditions are imposed at the interface to satisfy the conservation

of mass, momentum and energy. This is known as the Jump Con-

dition Formulation or Sharp Interface Approach. This approach has

been coupled with VOF ( Sato and Ni ̌ceno, 2013 ), LS ( Gibou et al.,

2007; Tanguy et al., 2007; Lee and Son, 2017 ) and LCRM ( Shin and

Abdel-Khalik, 2007; Shin and Choi, 2016 ).

In the second approach,the jump conditions are expressed in

the governing equations by introducing appropriate source terms

using a (modified) Dirac delta function which is equal to zero ev-

erywhere in the domain, except at the interface. The interface is

normally smeared out across 2 or 3 computational cells by defin-

ing a smoothed Heaviside or Dirac function to represent the spa-

tial variation of physical properties across the interface, i.e . using a

one fluid approach. This approach is commonly used because of its

implicity in implementation. Numerous studies can be found us-

ng this approach with VOF ( Welch and Wilson, 20 0 0; Ding et al.,

017; Tsui et al., 2014 ), LS ( Li and Dhir, 2007 ), FT ( Juric and Tryg-

vason, 1998; Esmaeeli and Tryggvason, 2004 ) and LCRM ( Shin and

uric, 2002 ). However, the usage of this approach might result

n a reduced accuracy when the interface is too close to a wall

Shin and Abdel-Khalik, 2007 ). Moreover, it is difficult to enforce

he saturation temperature at the interface using a source term.

his affects the accuracy of the results as the interfacial mass flux

alculation is error prone if the temperature field is not accurate in

he vicinity of the interface ( Shin and Abdel-Khalik, 2007 ). Because

systematic comparison of the sharp and one fluid approaches is

learly missing in the literature, an attempt is made by this study

o bridge this gap.

The outline of the paper is as follows: In Section 2 , the re-

ormulation of the Navier–Stokes equation to accommodate jump

onditions for phase change is described. The different forms of

he energy conservation equation and closure equations are dis-

ussed based on the use of the sharp or smooth interface approach.

n Section 3 , the numerical methodology is explained with a spe-

ial focus on the implementation details of the sharp interface ap-

roach. In Section 4 , the methods are verified with the standard 1D

ests, namely the Stefan problem and the sucking interface prob-

em and with 3D bubble growth test. The errors are compared for

oth approaches. The section ends with a discussion of 3D bubble

ise simulations. Finally, conclusions are presented in Section 5 .

. Mathematical formulation

In this study, we solve the flow equations using a one fluid ap-

roach whereas the energy equation is treated by both a smooth

nterface approach and a sharp interface approach. Although both

pproaches correspond to the same continuum limit, they use dif-

erent numerical methods when dealing with the discontinuity at

he interface while solving the energy equation. In the next sec-

ions, these differences will be explained in detail.

.1. Mass conservation equation

We assume the liquid and the gas phases to be incompressible

nd single component. The mass density ρ in the one fluid formu-

ation is defined as

= ρg I + ρl (1 − I) , (1)

here ρg is the density of gas phase, ρ l is the density of liquid

hase and I is a Heaviside function which is one in gas phase and

ero in the liquid phase.

Due to the phase transition, there is a difference in the gas and

iquid velocities at the interface. The difference in the velocities is

elated to the evaporative mass flux where the relevant jump con-

ition can be obtained by integration of the continuity equation

cross the interface, which is as follows:

l ( u l − u f ) · n = ρg ( u g − u f ) · n =

˙ m , (2)

here u f is the interface velocity, and ˙ m is the evaporation rate at

he interface. Note that in absence of phase transition, the velocity

s continuous at the interface meaning u f = u g = u l .

It is safe to assume a divergence-free velocity field everywhere

xcept at the interface for low Mach number flows with low tem-

erature variations in the gas phase as well as in the liquid phase.

or flows with phase transition, the velocity field is not divergence-

ree at the interface because of fluid expansion/compression due to

hase change. Thus, an interfacial source term which accounts for

he expansion/compression has to be incorporated in the continu-

ty equation. This term can be derived by taking the volume inte-

ral of the divergence of the velocity over the interface thickness

A.H. Rajkotwala, A . Panda and E.A .J.F. Peters et al. / International Journal of Multiphase Flow 120 (2019) 103093 3

Fig. 1. Schematic of control volume around the interface.

(∫

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Fig. 1 ) as follows:

V f

∇ · u dV f =

∫ A f

( u g − u l ) · n f dA f . (3)

sing velocity jump condition 2 , we obtain:

V f

∇ · u d V f =

(1

ρg − 1

ρl

)∫ A f

˙ m d A f . (4)

The RHS of the above equation is non-zero at the interface only

nd can be written in terms of volume integral using the dirac-

elta function as follows:

V f

∇ · u d V f =

∫ V f

(1

ρg − 1

ρl

)∫ A f

δ( x − x f ) ˙ m d A f d V f , (5)

hich gives,

· u =

(1

ρg − 1

ρl

)∫ A f

δ( x − x f ) ˙ m dA f . (6)

.2. Momentum conservation equation

The momentum equation is formulated as follows:

∂ u

∂t + ρ

{∇ · ( u u ) − u

(∇ · u

)}= −∇p + ρg + ∇ · τ + F σ , (7)

here τ = μ(∇ u + ∇ u

T )

is the viscous stress tensor and F σ =∫ A f

δ( x − x f ) κ f n f dA f is the volumetric surface tension force.

The above equation is cast into non-conservative form to avoid

umerical instabilities while solving the equations especially at

igh density ratio ( Sato and Ni ̌ceno, 2013 ). However, the convec-

ion term is formulated in a semi-conservative form to allow dis-

retization by a Total Variation Diminishing (TVD) scheme.

.3. Energy conservation equation

A simplified energy conservation equation, based on an en-

halpy formulation, is used to predict temperature variations in the

wo phases. In this study, the viscous heating and pressure effects

re neglected.

In the smooth interface approach, the interfacial energy flux ˙ q is

ncorporated in the energy equation by means of a source term:

ρC p ∂T

∂t + ρC p

{∇ · ( u T ) − T (∇ · u

)}= ∇ · ( k ∇T ) −

∫ A f

δ( x − x f ) ̇ q dA f . (8)

The energy flux at the interface is related to the interfacial mass

ux ˙ m by the following equation:

˙ h =

˙ q (9)

here h = h lg + (C pl − C pg )(T f − T sat ) with h lg is the latent heat

easured at the equilibrium saturation temperature T sat ( P ), corre-

ponding to the reference ambient system pressure and T f is the

nterface temperature.

The final form of the energy equation is obtained by substitut-

ng above equation in ( 8 ):

ρC p ∂T

∂t + ρC p

{∇ · ( u T ) − T (∇ · u

)}= ∇ · ( k ∇T ) −

∫ A f

δ( x − x f ) ˙ m h dA f . (10)

In the sharp interface approach, the jump conditions are im-

osed on the interface as a boundary condition, and hence, the

nterfacial source term is absent from the corresponding energy

quation:

C p ∂T

∂t + ρC p

{∇ · ( u T ) − T (∇ · u

)}= ∇ · ( k ∇T ) . (11)

.4. Additional equations

In the above presented equations, additional closure equations

re required to calculate the interface temperature T f and interfa-

ial mass transfer ˙ m .

A standard way to derive these closures is assume that the

hase transition occurs at a uniform temperature and the tem-

erature depends only on the system pressure. Thus, the interface

emperature is set to the equilibrium saturation temperature cor-

esponding to the system pressure.

f = T sat (P ∞

) (12)

The corresponding mass flux is calculated from the energy

ump condition as follows:

˙ =

k g ∂T ∂n

| g − k l ∂T ∂n

| l h lg

(13)

This form of closure is used in many studies involving flow boil-

ng ( Mukherjee et al., 2011; Li and Dhir, 2007 ), film boiling ( Shin

nd Juric, 2002; Welch and Wilson, 20 0 0; Esmaeeli and Tryggva-

on, 2004 ), pool boiling ( Dhir, 2001 ) and evaporation ( Irfan and

uradoglu, 2017 ).

A counter argument to the above assumption can be made that

ecause of the pressure jump across the interface, the equilibrium

aturation temperatures corresponding to those pressures cannot

e equal i.e. T sat ( p g ) � = T sat ( p l ). To test the validity of Eq. (12) , Juric

nd Tryggvason developed a general expression of the deviation of

he interface temperature from the saturation temperature using

he entropy jump condition ( Juric and Tryggvason, 1998 ). The im-

act of each term in this general expression was quantified using

imulation of film boiling of hydrogen and it was concluded that

f = T sat is a valid assumption except when the interface comes

lose to the heated wall. Near the heated wall, the simple expres-

ion derived from kinetic theory of gases by Tanasawa (1991) was

ecommended.

4 A.H. Rajkotwala, A . Panda and E.A .J.F. Peters et al. / International Journal of Multiphase Flow 120 (2019) 103093

Fig. 2. Schematic of the normal probe method for interfacial mass flux calculation

Shin and Juric (2002) .

w

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T

A different approach to calculate the interfacial mass flux is

using semi-empirical relations. For example, Schrage (1953) de-

veloped a model using kinetic theory of gases. In this model,

the interfacial mass flux is calculated by relating the flux of

molecules crossing the interface during phase change to the tem-

perature and pressure of the phases.The empirical parameters in

such models are generally calculated by comparing the mass flux

to experimental data. This limits the general applicability of the

model. A summary of such models is given in the recent paper by

Kharangate and Mudawar (2017) . To avoid introduction of empiri-

cal parameters in the method, the mass flux calculation from the

energy jump condition, Eq. (13 ), is selected in this study.

3. Numerical methods

3.1. Fluid flow solver

The interface is represented by an unstructured mesh of tri-

angular markers, which are formed by the connection of the La-

grangian points situated at the corners of these markers. The

markers enable advection of the interface and surface tension cal-

culation. F σ is calculated on the Eulerian grid using the hybrid

method of Shin et al. (2005) , which combines the advantage of

accurate curvature calculation (similar to the pull force model

( Dijkhuizen et al., 2010 ) in Front Tracking methods) and the proper

balance of pressure jump and surface tension force at the dis-

crete level (similar to the continuum surface model ( Brackbill et al.,

1992 ) in front capturing methods).

Subsequently, the governing equations are solved by a frac-

tional step method to obtain the flow field at the next time step

( Das et al., 2016 ). The method consists of two steps: In the first

step, a projection of the velocity u

n ∗ is calculated using information

from the previous time step

u

n ∗ = u

n +

�t

ρn

[−ρ∇ · ( u u ) + ρu

(∇ · u

)−∇ p + ρg − ∇ · τ + F σ

]n .

(14)

This equation is spatially discretized on a staggered grid using

a finite volume approach. The convective term is discretised with

a second order flux-delimited Barton scheme whereas the diffu-

sion term with a second order central scheme ( Rajkotwala et al.,

2018 ).The convection term is treated explicitly and diffusion term

is treated semi-implicitly. The term ρu

(∇ · u

)is evaluated by av-

eraging the calculated divergence at the faces of velocity cells.

After the projected velocities are calculated, the velocity field is

corrected in the pressure correction step to satisfy the continuity

equation including interfacial mass flux.

∇ · �t

ρ∇ (δp) = ∇ · u

∗ − ∇ · u

n +1 . (15)

The pressure equation is solved using a robust and efficient paral-

lel Block-Incomplete Cholesky Conjugate Gradient (B-ICCG) solver

( Das et al., 2016 ).The term ∇ · u

n +1 is non-zero at the interface

in case of phase change and is calculated using Eq. (6) . The in-

terfacial mass flux is calculated on the Lagrangian interface using

Eq. (13) and distributed on the Eulerian grid using a discrete distri-

bution function. In this study, we use a trilinear distribution func-

tion.

The calculation of the interfacial mass flux term on the

Lagrangian mesh is done using the normal probe method of

Udaykumar et al. (1996) as shown in Fig. 2 . The right hand side

of Eq. (13) is discretized using a first order approximation as fol-

lows:

˙ m =

1

h f g

1

�n

[ k g (T g − T sat ) − k l (T sat − T l ) ] , (16)

u

here T l and T g are found by interpolating the temperature using

wo normal probes which originate at the phase boundary and ex-

end a distance �n into the liquid and the gas. A tri-linear function

s used for the interpolation of temperature on probing points from

xed grid points and the normal distance is usually taken equal

o the grid step size ( Shin and Juric, 2002; Esmaeeli and Tryggva-

on, 2004 ). The choice of tri-linear interpolation is motivated by

he observation that there is a negligible difference in the results

hen a second order discretization is used ( Esmaeeli and Tryggva-

on, 2004 ).

Once the flow field is solved, the interface is advected, i.e . the

agrangian points are moved. The equation of motion for marker

oints is

d x f

dt = u n n f (17)

here u n is the normal velocity. The equation of this normal ve-

ocity, derived from Eq. 2 , reads

n = u · n − ˙ m

2

(1

ρl

+

1

ρg

). (18)

n the above equation, the normal velocity at the interface has two

omponents; one due to fluid advection (the first term) and an-

ther one due to phase change (the second term). The first term

s interpolated from the Eulerian grid (Cubic spline interpolation is

sed) and the second term is directly calculated on the Lagrangian

esh using normal probe method of Udaykumar et al. (1996) as

xplained before. The equation is integrated in time using the

ourth order Runge–Kutta method.

Due to the advection of Lagrangian points, the mesh quality de-

reases, a reconstruction procedure is required to overcome this.

n this study, we use the modified LFRM ( Rajkotwala et al., 2018;

irsandi et al., 2018 ). Finally, the physical properties are updated

sing local phase fractions in each cell. From the calculated phase

ractions, the average mass density and viscosity are calculated in

ach grid cell by algebraic and harmonic averaging, respectively.

he local phase fraction can be calculated by geometric analysis

sing the position of the interface ( Dijkhuizen et al., 2010 ) which

A.H. Rajkotwala, A . Panda and E.A .J.F. Peters et al. / International Journal of Multiphase Flow 120 (2019) 103093 5

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s exact and computationally more efficient than solving the Pois-

on equation.

.2. Energy solver

At each time step, the energy equation is solved using the up-

ated velocity and physical properties. However, the form of en-

rgy equation varies with the numerical approach as explained

n Section 2.3 . In both approaches, the convective term and the

iffusion term are discretised with a second order flux-delimited

arton scheme ( Rajkotwala et al., 2018 ) and a second order cen-

ral scheme, respectively. The time step restriction is relaxed by

reating the diffusion term implicitly. The discretization of term

C p T (∇ · u

)for cell ( i, j, k ) uses the temperature in the cell cen-

er times the divergence that is also naturally defined in the cell

enter.

The main differences in the two approaches lie in the treatment

f the interfacial source term and the calculation of thermal prop-

rties. The specific details of the two approaches are given in fol-

owing sections.

.2.1. Smooth interface approach

The treatment of the interfacial source term in the energy con-

ervation Eq. (8) is similar to that in mass conservation Eq. (6) , i.e .

he interfacial mass flux will be calculated on the Lagrangian in-

erface using Eq. (13) and distributed on the Eulerian grid using a

iscrete distribution function. In this study, we use a trilinear dis-

ribution function. From the updated phase fractions, the average

olumetric heat capacity and conductivity are calculated in each

rid cell by algebraic and harmonic averaging, respectively.

.2.2. Sharp interface approach

In the sharp interface approach, the saturation temperature is

mposed as a boundary condition at the interface. This is handled

t the discrete level by use of irregular stencils in the discretization

f diffusion terms. The thermal properties are treated sharply, i.e .

he cell centers lying in the liquid and gas phase are assigned cor-

esponding phase properties. Thus, the first step in this approach is

hase-wise flagging of all cells and calculation of the stencil length.

As shown in Fig. 4 , the cells are marked with four different

ags:“gas” (cell center in gas phase), “liquid” (cell center in liquid

hase), “I-gas” (gas cell adjacent to interface) and “I-liquid” (liquid

ell adjacent to interface). As the interface is represented using a

riangulated mesh, the exact location of the interface is known and

an be used for the flagging of the cells. This is done by identifying

he triangles which intersect the grid lines ( i.e . the lines passing

hrough neighboring cell centers) as shown in Fig. 3 . A fast trian-

Fig. 3. Schematic of line triangle intersection for flagging.

p

u

3

r

le line intersection algorithm ( Pant et al., 2001 ) is used, which

ses the side product of the line with the edges of the triangle.

The dot product of the line with the normal of the triangle in-

icates whether a cell center is to be flagged as I-gas or I-liquid

ell. Once the interface cells are flagged, the cells which are com-

letely filled with gas (gas fraction I = 1 ) are flagged as gas cells

nd the rest of the cells are flagged as liquid cells. Next, the nor-

alized distance of the interface (normalized by the grid size) with

espect to the closest cell center in the gas phase is calculated

e.g. ξXn and ξ Yn in Fig. 4 ). This is done by calculating the inter-

ection point of the triangle (plane containing triangle) and the

ine passing through the neighboring cell centers. These distances

re stored in a linked-list data structure where path to the data is

iven by flag of “I-liquid” cells.

Subsequently, the thermal properties are assigned to the cells

ased on cell flags, i.e . cells with “gas” or “I-gas” flags are given gas

roperties and cells with “liquid” or “I-liquid” flags are given liq-

id properties. As mentioned previously, the convection term is ex-

licitly calculated using the Barton scheme. The temperature gra-

ient in the convection term can be computed in the sharp for-

ulation ( Sato and Ni ̌ceno, 2013 ) or conventional form (similar to

mooth approach). Similar to Shin and Choi (2016) , we found that

he sharp energy formulation for the convection term gives similar

ccuracy of the solution as the conventional form. Therefore, the

onventional form is used for the discretization of the convection

erm.

The diffusion terms are discretized by a second order central

ifference scheme. The complete discretized energy equation is

epresented as a set of algebraic equations as follows:

C T C +

∑

nb

A nb T nb = B C (19)

here T is temperature, the subscript ‘C’ indicates cell center and

nb’ indicates the six neighbouring cells. The coefficients A C and

nb depend on thermal properties, grid size and time step size and

C contains all explicit terms. For all the interface cells, the satu-

ation temperature is imposed as a boundary condition by chang-

ng the coefficients and B C term. The implementation is based on

he immersed boundary method of Das et al. (2016, 2018) and

een et al. (2012) .

In Fig. 4 , I-liquid cell C has two neighbour I-gas cells X n and Y n .

hile calculating the temperature at I-liquid cell C, the saturation

emperature is imposed at the interface by changing the value of

oefficients of the neighbouring I-gas cells X n and Y n . A quadratic

t (between S 1 − C − X p and S 2 − C − Y p ) is used to modify the co-

fficients. However, when the interface is very close to C, a linear

t (between S 1 − X p and S 2 − Y p ) is used to modify the coefficients.

Once all the coefficients are modified, the temperature is com-

uted by solving the resulting modified system of linear equations

sing a parallel B-IICG solver ( Das et al., 2016 ).

.3. Summary of the solution algorithm and the time step size

estriction

The solution procedure can be summarized as follows:

1. Calculate the mass flux ˙ m using temperature field at time level

n, Eq. (16) .

2. Update the hydrodynamic properties (based on interface posi-

tion of time level n)

3. Calculate the surface tension force F σ4. Update the velocities and pressure to time level n+1 using frac-

tional step method, Eqs. (14) and ( 15 ).

5. Advect the interface to the next time level n+1, Eq. (17) .

6. Update the phase fraction using updated interface.

7. Update the thermal properties (based on interface position of

time level n+1)

6 A.H. Rajkotwala, A . Panda and E.A .J.F. Peters et al. / International Journal of Multiphase Flow 120 (2019) 103093

Fig. 4. Schematic of boundary condition treatment at the interface. The notation X p , X n , Y p , Z p and Z n is used for six neighbouring cells where X, Y, Z stand for co-ordinate

direction and subscripts ‘p’ and ‘n’ indicates positive and negative direction from cell center, respectively.

Fig. 5. Schematic of the Stefan problem ( Irfan and Muradoglu, 2017 ).

T

d

q

x

w

β

w

8. Update the temperature field to time level n+1 either using

smooth interface approach, Eq. (10) or sharp interface approach,

Eq. (11) .

9. Advance to time level n+1.

The time step size is restricted by CFL condition and the stabil-

ity criteria due to surface tension as follows:

�t = min

(

c CF L �x

| u | max ,

(ρg �x 3

2 πσ

)1 / 2 )

(20)

where c CFL is the dimensionless safety factor.

4. Results and discussion

4.1. Stefan problem

4.1.1. Problem definition

The Stefan problem is a well-known test case to verify the

phase change problems, which is schematically shown in Fig. 5 . As

seen in the figure, the vapor and liquid phases are separated by

a vertical interface. The temperature of the left wall (next to the

vapor phase) is kept above the saturation temperature. The liquid

starts vaporizing and the interface moves towards the right. A free

flow boundary condition is applied at the right boundary for the

liquid to flow out. However, the liquid temperature stays fixed at

the saturation value. Because the vapor remains stationary, diffu-

sion is responsible for heat transfer from the wall to the interface.

The analytical solution is given in literature ( Welch and Wilson,

20 0 0; Irfan and Muradoglu, 2017 ).

The energy equation needs to be solved only in the vapor

phase, and is given by

∂T

∂t = αg

∂ 2 T

∂x 2 (21)

where x varies from 0 to x �( t ), with x �( t ) the position of the inter-

face at time t .

The above equation is solved with the following boundary con-

ditions:

T (x = 0 , t) = T w

and T (x = x �(t ) , t ) = T sat (22)

he heat flux at the interface is computed using energy jump con-

ition:

˙ � = −k g

∂T

∂x

∣∣∣∣�

g

(23)

The analytical solution for the interface location is given by

�(t) = 2 β√

αg t (24)

here β is the solution of the transcendental equation

exp β2 erf β =

C p,g (T w

− T sat )

h lg

√

π(25)

here h lg is latent heat of vaporization.

A.H. Rajkotwala, A . Panda and E.A .J.F. Peters et al. / International Journal of Multiphase Flow 120 (2019) 103093 7

Table 1

Fluid properties.

Property Gas Liquid

ρ (kg/m

3 ) 0.25 2.5

μ (Pas) 7 . 0 × 10 −3 9 . 8 × 10 −2

k (W/mK) 0.0035 0.0015

C p (J/kgK) 10 10

h lg (J/kg) 100 -

T sat (K) 10 -

b

T

4

(

w

s

5

i

m

t

i

i

4

p

d

u

w

H

t

(

t

a

r

fl

4

a

a

a

r

l

s

4

a

t

t

d

t

t

L

d

ε

Fig. 6. Comparison of the analytical solution and numerical results obtained by

smooth interface approach for the Stefan problem.

w

t

t

t

f

p

t

p

b

fi

t

E

4

4

v

s

The analytical solution of temperature in the gas phase is given

y

g (x, t) = T w

+

(T sat − T w

erf β

)erf

(

x

2

√

αg t

)

(26)

.1.2. Problem setup

The test parameters are chosen from Irfan and Muradoglu

2017) . The properties of the fluids are summarized in Table 1 . The

all temperature T w

is set to 12 K. As the problem is one dimen-

ional, the computational domain is replicated as 1D by choosing

cells in Y and Z directions respectively. The length ‘L’ along X

s taken as 1 m. Three grid sizes are selected: coarse ( 4 × 10 −2 m),

edium ( 2 × 10 −2 m) and fine ( 1 × 10 −2 m). The analytical solu-

ion is evaluated using MatLab. The problem is initialized with an

nterface at x �, 0 = 0 . 1 m. The temperature and velocity fields are

nitialized using the analytical solution corresponding to x �,0 .

.1.3. Smooth interface approach

The Stefan problem is solved using the smooth interface ap-

roach for different grid sizes. The temperature variation along the

omain length and the evolution of the interface location and liq-

id velocity with time are given in Fig. 6 . The solution agrees well

ith the analytical solution and converges upon grid refinement.

owever, it can be seen in Fig. 6 (a) that the temperature at the in-

erface is not completely maintained at the saturation temperature

especially on the coarse grid). This is a well-known problem of

his technique ( Shin and Abdel-Khalik, 2007; Shin and Choi, 2016 )

s the interface temperature is implicitly enforced to be the satu-

ation temperature by a source term, which also leads to velocity

uctuations in the domain ( Fig. 6 (c)).

.1.4. Sharp interface approach

Similarly, the Stefan problem is solved using the sharp interface

pproach for 300 s for different grid sizes. The temperature vari-

tion along the domain length is given in Fig. 7 (a). The solution

grees well with the analytical solution and converges upon grid

efinement. The variation of the interface location ( Fig. 7 (b)) and

iquid velocity ( Fig. 7 (b)) with time is compared to the analytical

olution and a very good agreement is observed.

.1.5. Error comparison

As the grid is refined, the results of both methods approach the

nalytical solution. To compare the accuracy of both approaches,

he temperature distributions for grid size L/ �x = 100 obtained at

ime t = 219 . 02 s are shown in Fig. 8 . As discussed before, an un-

ershoot below the saturation temperature is observed in case of

he smooth approach, which is not physical in nature, whereas for

he sharp approach the interface is at saturation temperature. The

2 norm of the error in the temperature at time t = 219 . 02 s at

ifferent grid sizes is calculated using the following expression

T =

√ √ √ √

∑ N x i =1

(T num,i −T ana,i

T ana,i

)2

N x (27)

w

here T num

is the temperature obtained from simulation, T ana is

he temperature obtained from the analytical solution and N x is

he number of grid points along domain length L .

The convergence of the L2 norm of the error in the tempera-

ure with decreasing grid size is shown in Fig. 9 . The error is lower

or the sharp interface approach than for the smooth interface ap-

roach. Also, the average order of convergence for the sharp in-

erface approach (1.24) is higher than for the smooth interface ap-

roach (1.05). The reason for the reduced order of convergence can

e the boundary condition treatment and mass flux calculation ( i.e .

rst order temperature gradients calculation and linear interpola-

ion of temperature field onto Lagrangian mesh and mass flux onto

ulerian grid).

.2. Sucking interface problem

.2.1. Problem definition

The sucking interface problem is also a benchmark case used to

erify phase change models. The schematic of the test case can be

een in Fig. 10 . In this test case, a vapor layer is attached to the left

all of the domain while the rest is filled with liquid as the same

8 A.H. Rajkotwala, A . Panda and E.A .J.F. Peters et al. / International Journal of Multiphase Flow 120 (2019) 103093

Fig. 7. Comparison of the analytical solution and numerical results obtained by

sharp interface approach for the Stefan problem.

Fig. 8. Comparison of the temperature variation at grid size L/ �x = 100 obtained

by both approaches for the Stefan problem.

Fig. 9. Errors in the temperature variation at different grid sizes obtained by both

approaches for the Stefan problem.

Fig. 10. Schematic of the sucking interface problem ( Irfan and Muradoglu, 2017 ).

x

w

β

w

species as vapor. The vapor phase is at saturation temperature T sat ,

and stays at rest throughout the simulation. The liquid tempera-

ture is higher than the saturation value, therefore phase change oc-

curs at the interface. This causes the interface to move towards the

right and a flow develops in the liquid phase. The heat of vapor-

ization, absorbed at the interface due to phase change, comes from

the liquid, which results in the formation of the thermal boundary

layer at the interface. The analytical solution is given in literature

( Welch and Wilson, 20 0 0; Irfan and Muradoglu, 2017 ).

The energy equation needs to be solved only in the liquid

phase, given by

∂T

∂t + u

∂T

∂x = αl

∂ 2 T

∂x 2 (28)

with x �( t ) the position of the interface at time t .

The above equation is solved with the following boundary con-

dition:

T (x = x �(t ) , t ) = T sat (29)

The analytical solution for the interface location is given by

�(t) = 2 β√

αg t (30)

here β is the solution of the transcendental equation

−(T ∞

− T sat ) C p,g k l √

αg exp

(−β2 ρ

2 g αg

ρ2 l αl

)h lg k g

√

παl erfc

(β

ρg √

αg

ρl √

αl

) = 0 , (31)

here h lg is latent heat of vaporization.

A.H. Rajkotwala, A . Panda and E.A .J.F. Peters et al. / International Journal of Multiphase Flow 120 (2019) 103093 9

Fig. 11. Comparison of the analytical solution and numerical results obtained by

smooth interface approach for the sucking interface problem.

i

T

4

T

p

m

r

g

a

M

m

a

Fig. 12. Comparison of the analytical solution and numerical results obtained by

sharp interface approach for the sucking interface problem.

4

a

t

a

l

w

t

4

g

t

b

4

d

s

o

p

The analytical solution for the temperature in the liquid phase

s given by

l (x, t) = T ∞

−

⎡

⎣

T ∞

− T sat

erfc

(β

ρg √

αg

ρl √

αl

)⎤

⎦ erf

(x

2

√

αl t +

β(ρg − ρl )

ρl

√

αg

αl

)

(32)

.2.2. Problem setup

The physical properties used in this test case are given in

able 1 . The wall temperature T w

is set equal to saturation tem-

erature T sat . Similar to the Stefan problem, the computational do-

ain is replicated as 1D by choosing 5 cells in Y and Z directions

espectively. The length ‘L’ in the X-direction is taken as 1 m. Three

rid sizes are selected: coarse ( 2 × 10 −2 m), medium ( 1 × 10 −2 m)

nd fine ( 5 × 10 −3 m). The analytical solution is evaluated using

atLab. The problem is initialized with an interface at x �, 0 = 0 . 1

. The temperature and velocity fields are initialized using the an-

lytical solution corresponding to x �,0 .

.2.3. Smooth interface approach

For the smooth interface approach, a good agreement with the

nalytical solution is observed for the temperature variation and

he evolution of interface location as shown in Fig. 11 . The overall

greement in the liquid velocity variation with the analytical so-

ution is good. Significant fluctuations of the velocity are observed

hich are due to implicit enforcement of the saturation tempera-

ure at the interface using source term as explained before.

.2.4. Sharp interface approach

The results obtained using the sharp interface approach are

iven in Fig. 12 . The results converge to the analytical results as

he grid is refined. The minor oscillation in the velocity is caused

y the mass flux source term in the pressure correction equation.

.2.5. Error comparison

To compare the accuracy of both approaches, the temperature

istributions for grid size L/ �x = 200 at time t = 219 . 02 s are

hown in Fig. 13 . Similar to the Stefan problem, the convergence

f both methods is compared by calculating the error in the tem-

erature at time t = 219 . 02 s at different grid sizes as shown in

10 A.H. Rajkotwala, A . Panda and E.A .J.F. Peters et al. / International Journal of Multiphase Flow 120 (2019) 103093

Fig. 13. Comparison of the temperature variation at grid size L/ �x = 200 obtained

by both approaches for the sucking interface problem.

Fig. 14. Errors in the temperature variation at different grid sizes obtained by both

approaches for the sucking interface problem.

Fig. 15. Schematic of the 3D bubble growth problem.

Fig. 16. Comparison of evolution of bubble radius with analytical solution.

w

g

Fig. 14 . The results are qualitatively similar to the Stefan problem,

i.e . the error is lower for the sharp interface approach than for the

smooth interface approach and the average order of convergence

for the sharp interface approach (1.74) is higher than the smooth

interface approach (1.07).

4.3. 3D bubble growth under zero gravity condition

4.3.1. Problem definition

In this verification test case, a simulation of a 3D spherical bub-

ble in a super-saturated liquid under zero gravity is carried out.

As shown schematically in Fig. 15 , a gas bubble is placed in the

middle of the domain and is surrounded by super saturated liq-

uid. Similar to the sucking interface problem, the gas phase is at

saturation temperature T sat , and stays at rest throughout the sim-

ulation. The phase change at the interface causes growth of the

bubble, which will push the liquid out of the domain. The heat of

vaporization comes from the liquid, and thus, a thermal boundary

layer is formed in the liquid phase near the interface. The analyt-

ical solution is given in literature ( Scriven, 1959; Sato and Ni ̌ceno,

2013 ).

The analytical solution for the bubble radius R as a function of

time is given by

R = 2 β

√

k l ρl C pl

t , (33)

where k l , C pl and ρ l are the thermal conductivity, the specific

heat capacity and the density of liquid, respectively, and β is the

“growth constant“. β is obtained by solving the following implicit

equation

ρl C pl ( T ∞

− T sat )

ρg

(h lg +

(C pl − C pg

)( T ∞

− T sat ) )

t

= 2 β2

∫ 1

0

exp

(−β2

(( 1 − ζ )

−2 − 2

(1− ρg

ρl

)ζ − 1

))dζ , (34)

here C pg and ρg are the specific heat capacity and the density of

as, respectively, h lg is the latent heat and T ∞

is the super satura-

ion temperature of the liquid.

A.H. Rajkotwala, A . Panda and E.A .J.F. Peters et al. / International Journal of Multiphase Flow 120 (2019) 103093 11

Fig. 17. Final bubble shape at time t = 83 . 68 s for the coarsest grid.

g

T

4

c

a

a

T

p

c

c

Fig. 18. Temperature distribution in the center plane at time t = 53 . 68 s for the

coarsest and finest grid obtained by the sharp interface approach.

4

i

w

a

c

s

i

t

The analytical solution for the temperature field at time t is

iven by

=

⎧ ⎪ ⎨

⎪ ⎩

T ∞

− 2 β2

(ρg ( h lg + ( C pl −C pg ) ( T ∞ −T sat ) )

ρl C pl

)∫ 1

1 − R r

exp

(−β2

(( 1 − ζ )

−2 − 2

(1 − ρg

ρl

)ζ − 1

))dζ r > R

T sat r ≤ R

(35)

.3.2. Problem setup

The properties of the fluids are the same as in the previous test

ases. A bubble of radius R 0 = 0 . 15 m is initialized in the center of

cubic domain with size 1 m

3 . The temperature and velocity fields

re initialized using the analytical solution corresponding to R 0 .

he surrounding liquid temperature is set to T ∞

= 12 K. A constant

ressure ( P = 0 bar) and zero heat flux ( dT dn

= 0 K/m) boundary

onditions are applied to all walls. Three grid sizes are selected:

oarse ( 4 × 10 −2 m), medium ( 2 × 10 −2 m) and fine ( 1 × 10 −2 m).

.3.3. Results and comparison

The bubble growth is simulated for time 70 s starting from the

nitial bubble of radius R 0 . The evolution of the computed radius

ith time is shown in Fig. 16 . The results closely agree with the

nalytical solution and grid convergence can be observed with de-

reasing grid size. The final shape of the bubble at the end of the

imulation is shown in Fig. 17 . The figure shows that the spher-

cal shape of the bubble is retained by both approaches even at

he coarsest grid. The sphericity parameter � for the final bubble

12 A.H. Rajkotwala, A . Panda and E.A .J.F. Peters et al. / International Journal of Multiphase Flow 120 (2019) 103093

Fig. 19. Temperature distribution in the center plane at time t = 53 . 68 s for the

coarsest and finest grid obtained by the smooth interface approach.

Fig. 20. Comparison of the temperature variation along the radial direction at t =

53 . 68 s obtained by both approaches.

c

i

i

t

a

T

r

n

H

s

t

p

t

b

i

a

T

s

f

d

f

a

t

b

r

p

fi

a

s

p

c

t

b

r

t

T

b

4

4

p

a

o

l

t

shape can be calculated using the following expression

� =

π1 3 (6 V b )

2 3

A b

(36)

where A b and V b are bubble area and volume respectively. At the

coarse grid, the area and volume of the bubble time t = 83 . 68 s for

the sharp interface approach are 1.6127 m

2 and 0.1925 m

3 , respec-

tively and the corresponding sphericity � = 0.9999. Similarly, the

area and volume of the bubble time t = 83 . 68 s for the smooth in-

terface approach are 1.8465 m

2 and 0.2359 m

3 , respectively and the

orresponding sphericity � = 0.9999. In both cases, the sphericity

s very close to 1 confirming that the spherical shape of the bubble

s retained by both approaches even at the coarsest grid.

The temperature distribution in the center plane obtained by

he sharp interface approach is shown in Fig. 18 . For both coarse

nd fine grids, the temperature distribution is radially symmetric.

he temperature profile in the thermal boundary layer is not fully

esolved for the coarse grid, which is indicated by the disconti-

uities in the temperature distribution near the interface region.

owever, these discontinuities are not present in the fine grid re-

ult indicating that the thermal boundary layer is fully resolved.

Similar observations can be made for the temperature distribu-

ion of the bubble growth obtained by the smooth interface ap-

roach ( Fig. 19 ). However, an important difference in the tempera-

ure distribution obtained by the two approaches is that the bub-

le does not remain at the saturation temperature in the smooth

nterface approach. The bubble is at temperature T = 10 . 089 K

nd T = 10 . 012 K for the coarse and the fine grid, respectively.

his (small) non-physical deviation in temperature occurs in the

mooth interface approach due to the implicit forcing of the inter-

ace to the saturation temperature. As expected, this non-physical

eviation is not observed in the results obtained by sharp inter-

ace approach. For further comparison, the temperature variation

long radial direction t = 53 . 68 s obtained by both approaches for

he coarsest and finest grid is shown in Fig. 20 . It can be seen that

oth methods converge towards the analytical solution with grid

efinement. Also, the non-physical deviation from saturation tem-

erature in case of the smooth interface approach is clearly visible.

Finally, the velocity distribution in the center plane for the

nest grid obtained by both approaches is shown in Fig. 21 . Ide-

lly, the vapor phase in the bubble should be at rest. However, a

mall vortical velocity field is observed in the results indicating the

resence of parasitic currents. However, the magnitude of these

urrents is in the range of 10 −6 m/s and, hence, does not affect

he temperature field. In the velocity distribution corresponding to

oth approaches, a similar magnitude and pattern of parasitic cur-

ents is observed which is the expected behaviour since the surface

ension term is treated in a similar manner in both approaches.

his finding suggests that the parasitic currents are not influenced

y how the temperature field is calculated.

.4. Bubble rise with phase change

.4.1. Problem definition

A 3D simulation of the rise and growth of a vapor bubble in su-

erheated liquid under gravity is carried out. This test case serves

s a validation case for both approaches. The hydrodynamic part

f the current method has already been validated with the simu-

ation of the rise of a bubble under gravity in isothermal condi-

ions ( Rajkotwala et al., 2018 ). The rise of the bubble with phase

A.H. Rajkotwala, A . Panda and E.A .J.F. Peters et al. / International Journal of Multiphase Flow 120 (2019) 103093 13

Fig. 21. Velocity distribution in the center plane at time t = 53 . 68 s for the finest

grid obtained by the sharp and smooth interface approaches.

c

t

f

a

4

C

i

p

h

Table 2

Ethanol vapor and liquid properties.

Property Gas Liquid

ρ (kg/m

3 ) 1.435 757.0

μ (Pas) 1 . 04 × 10 −5 4 . 29 × 10 −4

k (W/mK) 0.020 0.154

C p (J/kgK) 1830 3000

h lg (J/kg) 9.63 × 10 5 -

σ (N/m) 0.018 -

T sat (K) 351.45 -

Fig. 22. Bubble shape, temperature and velocity distribution after t = 60 ms ob-

tained by sharp and smooth interface approaches.

i

t

o

hange is more complicated as the vapor bubble not only rises due

o buoyancy but at the same time expands due to boiling at its sur-

ace. The evolution of the growth of an ethanol bubble is calculated

nd compared to the experimental data of Florschuetz et al. (1969) .

.4.2. Problem setup

The simulation settings are taken from the study by Shin and

hoi (2016) . The thermodynamic properties of the working fluids,

.e . liquid ethanol and ethanol vapor, are selected at the system

ressure of 101.3 kPa and are given in Table 2 . The liquid super-

eat �T is set to 3.1 K. A spherical bubble of radius 210 μm is

nitialized in a 5 × 5 × 5 mm

3 cubic domain. The initial position of

he bubble is (2.5,2.5,3) mm; i.e . it is located along the central axis

f the domain but at the height of 3 mm from the bottom face.

14 A.H. Rajkotwala, A . Panda and E.A .J.F. Peters et al. / International Journal of Multiphase Flow 120 (2019) 103093

Fig. 23. Comparison of the evolution of the bubble radius with the experimental

data ( Shin and Choi, 2016 ).

p

u

t

b

t

t

o

c

p

s

m

t

p

b

a

t

p

n

t

a

t

s

3

p

g

i

A

g

n

(

T

D

t

S

f

1

R

B

D

D

D

D

D

D

D

D

The temperature is initialized with the analytical solution for the

zero gravity condition ( Eq. (35) ) corresponding to the initial bubble

size. A moving window concept ( Rajkotwala et al., 2018 ) is used in

this simulation such that the bubble retains its initial position with

respect to the domain. A free slip boundary condition is applied at

the side walls and an outflow condition is applied at the remain-

ing walls. The simulations are carried out for grid dimensions of

250 × 250 × 250 and 500 × 500 × 500 corresponding to a grid size

of 20 and 10 μm, respectively.

4.4.3. Results and comparison

In Fig. 22 , it can be seen that the bubble shape changes from

spherical to ellipsoidal and bubble shape is well preserved by both

approaches. As expected, a region of low superheat can be seen at

the trailing edge of the bubble and the thermal boundary layer on

the top and side of the bubble is thinner than at the bottom of

the bubble. Also, a winding vortex can be seen at the side of the

bubble.

The evolution of the non-dimensional bubble radius with time

obtained by both approaches is compared to experimental data in

Fig. 23 . At a fine grid, the results obtained by both approaches

agree well with the experimental data. At a given instant in time,

the bubble radius obtained by the smooth interface approach is

larger than in the sharp interface approach. This is because the

temperature of the bubble is not maintained at the saturation tem-

perature in case of the smooth interface approach. In the current

case, it is lower than the saturation temperature giving rise to ad-

ditional mass flux at the interface. Note that this is not apparent

in Fig. 22 (b) as the temperature range is modified to be same as

Fig. 22 (a).

5. Conclusion

In this paper, we have extended the Local Front Reconstruc-

tion Method to allow for phase transition using two different ap-

proaches to solve the energy equation. Both phases are assumed to

be incompressible; however, the expansion due to phase change is

incorporated by modifying the divergence-free velocity field con-

dition at the interface. This modification requires the addition of

a source term in the continuity equation which is related to mass

flux generated due to phase transition. The derivation of this mod-

ified continuity equation was presented in Section 2.1 .

The energy equation has been treated with two different ap-

proaches: the smooth interface approach and the sharp interface

approach. The smooth interface approach uses a one fluid formu-

lation to solve the energy equation with an interfacial source term

accounting for phase change. In the smooth interface approach this

interfacial source term is used to enforce the saturation tempera-

ture at the interface. In the sharp interface approach, the thermal

roperties are not volume-averaged near the interface, and the sat-

ration temperature is imposed as a boundary condition at the in-

erface. Thus, a different form of energy equation is required for

oth approaches as explained in Section 2.3 . Based on a litera-

ure study and relevance of application of the method, we chose

o keep the interface at saturation temperature and a model based

n the energy jump condition at interface was selected.

Finally, both approaches were verified with two 1D phase

hange problems: the Stefan problem and the sucking interface

roblem. The obtained results closely agree with the analytical

olution, and the error comparison showed the sharp interface

ethod to be more accurate with a better order of convergence

han the smooth interface approach. A 3D bubble growth in a su-

erheated liquid under zero gravity was also simulated. The bub-

le growth rate agreed well with the analytical solution for both

pproaches upon grid convergence. A symmetric temperature dis-

ribution was seen during the whole simulation time, and the im-

act of the parasitic currents on the simulation was found to be

egligible. However, a small unphysical rise in the temperature of

he bubble was observed in the results of the smooth interface

pproach. This is an artifact of the implicit forcing of the satura-

ion temperature at the interface and is a serious drawback of the

mooth interface approach. In addition to the verification tests, a

D validation simulation of the rise and growth of an ethanol va-

or bubble in superheated liquid ethanol was carried out, and a

ood agreement was obtained between the calculations and exper-

ment for the growth rate of the bubble.

cknowledgments

This work is part of the research programme Open Technolo-

ieprogramma with project number 13781, which is (partly) fi-

anced by the Netherlands Organisation for Scientific Research

NWO) Domain Applied and Engineering Sciences (TTW, previously

echnology Foundation STW). This work was carried out on the

utch national e-infrastructure with the support of SURF Coopera-

ive.

upplementary material

Supplementary material associated with this article can be

ound, in the online version, at doi: 10.1016/j.ijmultiphaseflow.2019.

03093 .

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