A Level Set Method for Analysis of Film Boiling on an Immersed Solid Surface

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Numerical Heat Transfer, Part B:Fundamentals: An International Journalof Computation and MethodologyPublication details, including instructions for authors andsubscription information:http://www.tandfonline.com/loi/unhb20

A Level Set Method for Analysis of FilmBoiling on an Immersed Solid SurfaceGihun Son a & Vijay K. Dhir ba Department of Mechanical Engineering, Sogang University, Seoul,South Koreab Mechanical and Aerospace Engineering Department, University ofCalifornia, Los Angeles, California, USAVersion of record first published: 19 Jun 2007.

To cite this article: Gihun Son & Vijay K. Dhir (2007): A Level Set Method for Analysis of Film Boilingon an Immersed Solid Surface, Numerical Heat Transfer, Part B: Fundamentals: An InternationalJournal of Computation and Methodology, 52:2, 153-177

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A LEVEL SET METHOD FOR ANALYSIS OF FILMBOILING ON AN IMMERSED SOLID SURFACE

Gihun SonDepartment of Mechanical Engineering, Sogang University, Seoul,South Korea

Vijay K. DhirMechanical and Aerospace Engineering Department, Universityof California, Los Angeles, California, USA

A numerical method is presented for simulating film boiling on an immersed (or irregularly

shaped) solid surface. The level set formulation for tracking the phase interfaces is modified

to include the effect of phase change at the liquid–vapor interface and to treat the no-slip

condition at the fluid–solid interface. The boundary or matching conditions at the phase

interfaces are accurately imposed by incorporating the ghost fluid approach based on a

sharp-interface representation. The numerical method is tested through computations of

bubble rise in a stationary liquid, single-phase fluid flow past a circular cylinder, and film

boiling on a horizontal cylinder.

1. INTRODUCTION

Several numerical methods have been developed for simulation of incompress-ible liquid–vapor flows with phase change. While computing the interfacial motionassociated with boiling processes, Welch [1] and Son and Dhir [2] used moving-gridmethods in which the interface coincides with the computational grid points. Theequations governing the conservation of mass, momentum, and energy for eachphase were coupled through the matching (or jump) conditions of normal andtangential velocities and stresses at the interface. In those methods, the governingequations and the matching conditions are explicit and straightforward to discretize.However, the moving-grid methods are very hard to use for interface configurationswith large distortion or change in topology such as bubble breakoff and merger. Thislimitation can be avoided by using numerical methods formulated on a fixed grid.One of the major issues in implementing fixed-grid methods is the treatment ofmatching conditions at the interface not coinciding with the grid points. Juric andTryggvason [3] simulated film boiling flows employing a single-field formulationwhich consists of one set of conservation equations of mass, momentum, and energy

Received 24 September 2006; accepted 3 February 2007.

This work received support from NASA Microgravity Fluid Physics Program and the Micro

Thermal System Research Center through the Korea Science and Engineering Foundation.

Address correspondence to Vijay K. Dhir, 48-121 Engineering IV, 420 Westwood Plaza, University

of California, Los Angeles, CA 90095, USA. E-mail: [email protected]

153

Numerical Heat Transfer, Part B, 52: 153–177, 2007

Copyright # Taylor & Francis Group, LLC

ISSN: 1040-7790 print=1521-0626 online

DOI: 10.1080/10407790701347720

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with variable fluid properties, which is valid for the entire flow field. In the formu-lation, the matching conditions at the interface are implicitly imposed by the inter-facial source terms added to the governing equations as delta functions. Toprevent numerical instability arising from the steep gradients of fluid properties,the delta functions are smoothed (or smeared out) over several grid spacings. Thisimplies that the sharp interface separating two fluids is treated as a transition region,referred to as diffuse-interface modeling. The single-field formulation based onsmoothed delta functions is combined with a front-tracking method, in which theinterface is represented explicitly by the linked line or surface segments. Veryrecently, Esmaeeli and Tryggvason [4] have extended the front-tracking method tofilm boiling on horizontal cylinders while combining with an immersed-boundarymethod [5] to account for the velocity boundary conditions on irregular solid sur-faces. Although Tryggvason and co-workers have demonstrated the departure pro-cess of a bubble during film boiling, generally the Lagrangian method is notstraightforward to implement for the interface with change in topology.

Such difficulties can be overcome by employing Eulerian interface-capturingmethods such as the volume-of-fluid (VOF) method [6–8] and the level set (LS)method [9–11]. Welch and Wilson [12] applied the VOF method to computationof film boiling by incorporating the effect of phase change. The method is naturallymass-conservative, as it tracks the volume fraction of a particular phase in each cellrather than the interface itself. To determine (or reconstruct) accurately the interfacefrom the nonsmooth volume-fraction function, however, the method requires a com-plicated geometric calculation procedure known as a piecewise linear interface calcu-lation (PLIC) algorithm [7, 8]. Son and Dhir [13] extended the LS method forsimulation of film boiling by including the effect of phase change. The LS methodwas earlier developed by Sussman et al. [9] for solving incompressible two-phase

NOMENCLATURE

c specific heat

D cylinder diameter

g gravity

h grid spacing

hlv latent heat

H discontinuous step function

k thermal conductivity

lo reference length

_mm mass flux across the interface

n unit normal vector, (nx; ny)

Nu Nusselt number

p pressure

q heat flux

Ref Reynolds number (¼ qf uolo=mf )

S sign function

t time

u velocity vector, ðu; vÞuo reference velocity

U interface velocity vector, (U ;V )

Vv vapor volume

Wef Weber number (¼ qf u2olf =r)

x; y; z Cartesian coordinates

a q�1v � q�1

l

h dimensionless temperature

j interface curvature

m dynamic viscosity

q density

r surface tension coefficient

s artificial time

/ distance function from the liquid–

vapor interface

w distance function from the fluid–solid

interface

Subscripts

f liquid or vapor

I interface

l; v liquid, vapor

sat saturation

w solid

154 G. SON AND V. K. DHIR

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flows with large density ratios. In this method, the interface is tracked by the LSfunction defined as a signed distance from the interface. Compared to the PLICVOF method, the LS method using a smooth distance function is much simpler toimplement. The formulation of Son and Dhir [13] also employs diffuse-interfacemodeling based on smoothed delta functions similar to that presented by Juricand Tryggvason [3]. This approach is easy to implement, but it has some difficultiesin accurately imposing the interface temperature condition in the diffuse-interfaceregion, where the fluid velocities are smeared out. Also, it is not so efficient to extendon a nonuniform grid, where the ratio of the thickness of interface region to gridspacing is not constant. Recently, considerable progress has been made in the LS for-mulation for two-phase flows by introducing the ghost fluid method (GFM) [14–18]as a numerical technique for sharply enforcing the boundary or matching conditionsat the interface without being smeared out over several grid spacings. Gibou et al.[19] have applied the sharp-interface approach to film boiling on a horizontalsurface, but their formulation for viscosity terms is based on a smeared-out stepfunction. A similar concept, in the terms of a subcell model, has been used byLuo et al. [20, 21] for multiphase flows with interface temperature specified.

In this article, the level set method for incompressible liquid–vapor flows withphase change proposed in our earlier work [13] is improved by incorporating theghost fluid approach based on a sharp-interface representation. The method isfurther extended for computing film boiling on an immersed solid surface. Theno-slip conditions for velocity as well as specified wall and fluid temperatures arealso treated with the ghost fluid approach, which is different from the diffuse-interface modeling for two-fluid flows with immersed solid boundaries [5, 22].

2. MATHEMATICAL FORMULATION

2.1. Governing Equations

In this work, the following assumptions are made: (1) The flows are laminar;(2) the fluid properties, including density, viscosity, specific heat, and thermal con-ductivity, are constant in each phase; (3) the interface is maintained at the saturationtemperature. The effect of radiative heat transfer is possibly important for film boil-ing with high wall superheat, but it is not included in the present study. The interfaceseparating the two-fluid phases is tracked by a LS function, /, which is defined as asigned distance from the interface. The negative sign is chosen for the vapor phaseand the positive sign for the liquid phase. Thus, the interface is described as / ¼ 0.

The equations governing the conservation of mass, momentum, and energy foreach phase are written as

r � uf ¼ 0 ð1Þ

qf

quf

qtþ uf � ruf

� �¼ �ðrpÞf þ qf gþr � mf ðruþruT Þf ð2Þ

qf cfqTf

qtþ uf � rTf

� �¼ r � kf ðrTÞf ð3Þ

LEVEL SET METHOD FOR ANALYSIS OF FILM BOILING 155

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where the subscript f denotes the liquid phase for / > 0 and the vapor phase for/ < 0. The conservation equations for each phase are coupled through the matching(or jump) conditions at the interface [2]:

ul � uv ¼ a _mmn ð4Þ

n � mlðruþruTÞl � mvðruþruTÞv� �

� n ¼ 0 ð5Þ

�pl þ pv þ n � mlðruþruTÞl � mvðruþruTÞv� �

� n ¼ rj� a _mm2 ð6Þ

where a ¼ q�1v � q�1

l . The interface normal n, the interface curvature j, and the massflux _mm are defined as

n ¼ ðnx; nyÞ ¼ r/=jr/j ð7Þ

j ¼ r � n ð8Þ

_mm ¼ qf ðU� uf Þ � n ð9Þ

where U is the interface velocity. The interface temperature is specified as a Dirichletboundary condition:

Tl ¼ Tv ¼ Tsat ð10ÞThe mass flux _mm is evaluated from the energy balance at the interface:

_mm ¼ 1

hlvn � ðklrTl � kvrTvÞ ð11Þ

2.2. Level Set Equations

In the LS formulation, the interface is described as / ¼ 0. The zero level set of/ is advanced by the interface velocity while solving the equation

q/qtþU � r/ ¼ 0 ð12Þ

where U can be written from Eq. (9) as

U ¼ uf þ_mmn

qf

ð13Þ

For efficient implementation of Eq. (13), the mass flux _mm, which is evaluated at theinterface, is extrapolated into the entire domain, as described in Section 3.9. The sol-ution of Eq. (12) does not satisfy the condition that the LS function should be main-tained as a distance function, jr/j ¼ 1, to calculate the interface normal andcurvature accurately. Therefore, at each time step the LS function is reinitializedto a distance function from the interface by obtaining a steady-state solution ofthe equation

q/qs¼ Sð/oÞð1� jr/jÞ ð14Þ


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where

Sð/oÞ ¼/offiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

/2o þ h2

q

Here h is a grid spacing, S is a smoothed sign function, and /o is a solution ofEq. (12). In principle, the reinitialization equation does not change the zero levelset since Sð0Þ ¼ 0, but in numerical computations the zero level set tends to move.This results in the nonconservation of mass, which is still an important issue affect-ing the LS method [10, 11, 22–26]. For two-phase flow with a moving interface, theinterface (/ ¼ 0) is generally not located at the grid point where / is defined. If thegrid points near the interface have small but nonzero values for S, their level sets willbe slightly affected by Eq. (14). The calculation generally does not guarantee the zerolevel set to move under the mass conservation condition. From this reasoning, wechoose a near-zero level set rather than / ¼ 0 as the immobile boundary conditionduring the reinitialization procedure. The sign function is redefined as

Sð/oÞ ¼ 0 if j/oj � dE

¼ /offiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi/2

o þ h2

q otherwise

where dE is the distance between the interface and the nearest grid point. This simplemodification of the sign function improves the LS method significantly [22].

To preserve mass conservation from any numerical errors occurring in numeri-cal implementation of the LS advection and reinitialization procedures, the followingvolume-correction step is added to the level set formulation:

q/qs¼ ðVv � Vv0Þjr/j ð15Þ

where Vv is a vapor volume computed form / and Vv0 is the vapor volume that satis-fies mass conservation. For two-phase flows with multiple bubbles rather than a sin-gle bubble, the volume-correction procedure can be extended for each bubble, asdescribed in a previous study [24].

3. NUMERICAL APPROACH

While discretizing the governing equations spatially, we adopt a staggered gridsystem in which the fluid velocity components (u; v) and the interface velocity com-ponents (U ;V ) are defined at cell faces, whereas the other dependent variables (/, p,T , n, and j) are defined at cell centers, as shown in Figure 1. The density at the cellcenters is written as

qi;j ¼ qv þ ðql � qvÞHð/i; jÞ ð16Þ

where H is not the smoothed step function varying over several grid spacings but the


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discontinuous step function defined as

Hð/Þ ¼ 1 if / > 0

¼ 0 if / � 0ð17Þ

3.1. Level Set Equations

The LS equations (12) and (14) are discretized by using a second-order, essen-tially nonoscillatory (ENO) scheme described as

Uq/qx

� �i

¼ maxðUi; 0Þq/qx

� ��i

þminðUi; 0Þq/qx

� �þi

ð18Þ

q/qx

� �2

i

¼ max sq/qx

� ��i

; �sq/qx

� �þi

; 0

� �2

ð19Þ

where Ui ¼ ðUiþ1=2 þUi�1=2Þ=2; s ¼ signðSÞ, and

q/qx

� ��i

¼ /i � /i�1

hþ 0:5h minmod

/iþ1 þ /i�1 � 2/i

h2;

/i þ /i�2 � 2/i�1

h2

� �q/qx

� �þi

¼ /iþ1 � /i

h� 0:5h minmod

/iþ1 þ /i�1 � 2/i

h2;

/i þ /iþ2 � 2/iþ1

h2

� �

with

minmodða; bÞ ¼ signðaÞ minðjaj; jbjÞ if ab > 0

¼ 0 otherwise

Figure 1. The staggered grid in two dimensions.


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It is noted that Eq. (19) is discretized to reconstruct the LS function to a signed dis-tance function by upwinding away from the interface described as / ¼ 0, which isused as the boundary condition for Eq. (14).

3.2. Ghost Fluid

The procedure discretizing the governing equations near the interface, which isnot coincident with the grid points, is simplified by using the ghost fluid approach[14, 16]. Each grid point occupied by one real fluid is defined as a ghost grid pointfor the other fluid (ghost fluid) that does not exist at the grid point. For example, ifvapor exists at a grid point, liquid is the ghost fluid at the point. The velocity ofghost fluid, uG, is evaluated from the velocity of real fluid, u, and the velocity jumpcondition given by Eq. (4), which is expressed as

uG ¼ u� a _mmn if / > 0

¼ uþ a _mmn if / � 0ð20Þ

where the values of / and _mmn at the cell faces are linearly interpolated from those atthe cell centers, e.g., /i�1=2;j ¼ ð/i;j þ /i�1;jÞ=2. Therefore, the liquid velocity and thevapor velocity are defined at every grid point of the computational domain as

ul ¼ u if / > 0

¼ uþ a _mmn if / � 0

uv ¼ u� a _mmn if / > 0

¼ u if / � 0

ð21Þ

3.3. Mass Conservation Equation

The ghost fluid approach is employed while discretizing the mass conservationequation for each phase on a fixed grid. For example, Eq. (1) can be discretized for acomputational cell ði; jÞ in Figure 2a as

ðr � uf Þi; j ¼ul;iþ1=2; j � ul;i�1=2; j

Dxþ

vl;i; jþ1=2 � vl;i; j�1=2

Dy¼ 0 ð22Þ

Using Eq. (21), we have

uiþ1=2; j � ui�1=2; j

Dxþ

vi; jþ1=2 � vi; j�1=2

Dy¼ða _mmnxÞi�1=2; jðHi; j �Hi�1=2; jÞ

Dxð23Þ

where Hi; j ¼ Hð/i; jÞ and Hi�1=2; j ¼ Hð/i�1=2; jÞ. In Eq. (23), ðHi; j �Hi�1=2; jÞ is zerounless the interface exists between point xi; j and point xi�1=2; j. For the general inter-face configuration, the mass conservation equation can be rewritten in a more usefulform including only real velocities,

r � u ¼ a _mmn � rH ð24Þ


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where the volume source term is discretized as

a _mmnxqH

qx

� �i; j

¼ða _mmnxÞiþ1=2; jðHiþ1=2; j �Hi; jÞ þ ða _mmnxÞi�1=2; jðHi; j �Hi�1=2; jÞ

Dx

It is noted that Eq. (24) is equivalent to the single-field formulation of massconservation derived in the previous study [13], but its descretiztion is based on asharp-interface representation.

3.4. Viscous Diffusion Terms

When discretizing the moementum equation spatially, the convection termsare treated by a second-order ENO scheme, which can be implemented simply usingthe ghost fluid approach. The viscous diffusion terms are treated by a second-ordercentral difference scheme. For two-phase flows, however, they have to be discretizedto satisfy the matching conditions at the interface. Kang et al. [15] proposed anumerical formulation to treat the viscous diffusion terms accurately, but theirformulation is very complicated and is not easily extended to flows with phasechange. In this study, an effective viscosity formulation, which is also found as partof the formulation proposed by Kang et al. [15], is employed to discretize the viscousdiffusion terms. This formulation is derived by considering two extreme cases ofinterface configurations. For the vertical interface depicted in Figure 2a, assumingthat jqv=qxj >> jqu=qyj, the matching condition of the tangential stress at pointxi�1=2;j�1=2 is approximated as

mqv

qx

� �i�1=2; j�1=2

¼ ml

vl;i � vl;I

xi � xI

� �j�1=2

¼ mv

vv;I � vv;i�1

xI � xi�1

� �j�1=2

ð25Þ

Figure 2. Schematic of finite-difference mesh for the ghost fluid approach.


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where the subscript I denotes the interface. This results in

mqv

qx

� �i�1=2; j�1=2

¼ m̂mi�1=2; j�1=2

vi � vi�1 � ða _mmnyÞIDx

� �j�1=2

ð26Þ

where

m̂m�1i�1=2; j�1=2 ¼

ðxi � xIÞm�1l þ ðxI � xi�1Þm�1

v

xi � xi�1

The effective viscosity m̂m is evaluated from the LS function as

m̂m�1i�1=2; j�1=2 ¼ ml if /min > 0

¼ mv if /max � 0

¼ /maxm�1l � /minm

�1v

/max � /min

otherwise

where /max ¼ maxð/i; j�1=2;/i�1; j�1=2Þ and /min ¼ minð/i; j�1=2;/i�1; j�1=2Þ. Then,Eq. (26) can be expressed as

mqv

qx

� �f

¼ m̂mqv

qx� a _mmny

qH

qx

� �ð27Þ

Similarly, for the horizontal interface as shown in Figure 2b, we have

mqu

qy

� �f

¼ m̂mqu

qy� a _mmnx

qH

qy

� �ð28Þ

Extending Eqs. (27) and (28) for general interface configurations, the viscous stressterm for each phase is approximately formulated as

mf ðruþruTÞf ¼ m̂m½ru� ða _mmnrHÞT þ ðruÞT � a _mmnrH� ð29Þ

where m̂m is evaluated from /0s at the adjacent cell faces. It is noted from Eq. (29) thatthe viscous stress rather than its tangential component is approximated ascontinuous near the interface. Using Eq. (29), the momentum equation (2) can bewritten as

qf

quf

qtþ uf � ruf

� �¼� ðrpÞf þ qf gþr � m̂mðruþruTÞ

� r � m̂m½ða _mmnrHÞT þ a _mmnrH� ð30Þ

3.5. Temporal Discretization

While discretizing the governing equations temporally, the convection andsource terms are treated by a first-order explicit scheme and the diffusion terms


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by a fully implicit scheme as follows:

/nþ1 � /n

Dt¼ �Un � r/n ð31Þ

Tnþ1f � Tn

f

Dt¼ �un

f � rTnf þ

1

qf cfðr � krTÞnþ1

f ð32Þ

unþ1f � u�

Dt¼ � 1

qf

ðrpÞf þ1

qf

ðr � m̂mrunþ1 þ snuÞ ð33Þ

r � unþ1 ¼ ða _mmn � rHÞnþ1 ð34Þ

where

u� ¼ unf þ Dtð�un

f � runf þ gÞ

su ¼ r � m̂mruT �r � m̂m½ða _mmnrHÞT þ a _mmnrH�

It is noted that the source term Su is nonzero only near the interface. The semi-implicit treatment of the viscous terms is more efficient than a fully implicit treat-ment, which requires very time-consuming iterative procedures for a set of coupledmomentum equations and does not satisfy the condition that r � m̂mruT should bezero away from the interface. The semi-implicit scheme is unconditionally stable,as discussed by Li et al. [27].

3.6. Projection Method

The momentum equation (33) and the mass equation (34) are solved byemploying the projection method [16]. We decompose Eq. (33) into two steps,

u�� u�

Dt¼ 1

qf

ðr � m̂mru�� þ SnuÞ ð35Þ

unþ1f � u��

Dt¼ �1

qf

ðrpÞf ð36Þ

First, Eq. (35) is solved without the pressure term. Then, the resulting velocity, u��,which does not satisfy the mass conservation equation, is corrected as given byEq. (36). When substituting Eq. (36) into Eq. (35) and then comparing withEq. (33), the projection error can be estimated to be of the order of Dtr2p.


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Substituting Eq. (36) into the velocity jump condition given by Eq. (4) andimposing u�l � u�v ¼ a _mmn, we have

1

qv

ðrpÞv ¼1

ql

ðrpÞl ð37Þ

This can be discretized at point xi�1=2; j in Figure 2a as

1

qf

qp

qx

� �i�1=2; j

¼ 1

qv

pv;I ; j � pi�1; j

xI � xi�1¼ 1

ql

pi; j � pl;I ;j

xi � xIð38Þ

which results in

1

qf

qp

qx

� �i�1=2; j

¼ 1

q̂qi�1=2; j

pi; j � pi�1; j þ ðrj� a _mm2Þxi � xi�1

ð39Þ

where

q̂qi�1=2; j ¼ðxi � xI Þqi; j þ ðxI � xi�1Þqi�1; j

xi � xi�1

which is evaluated from the LS function as

q̂qi�1=2; j ¼ qi; j if Hð/i; jÞ ¼ Hð/i�1; jÞ

¼/i; jqi; j � /i�1; jqi�1; j

/i; j � /i�1; j

otherwise

Then, Eq. (39) can be generalized as

1

qf

ðrpÞf ¼1

q̂q½rpþ ðrj� a _mm2ÞrH� ð40Þ

where the q̂q are evaluated from /0s at the adjacent cell points. Using Eq. (40), thediscretized momentum equations (35) and (36) can be rewritten as

u�� u�

Dt¼ 1

q̂qðr � m̂mru�� þ sn

uÞ ð41Þ

unþ1 � u��

Dt¼ � 1

q̂q½rpþ ðrk � a _mm2ÞrHnþ1� ð42Þ

Substituting Eq. (42) into the mass conservation equation (34), the governingequation for pressure is obtained as

r� 1q̂qrp ¼ r � u

�� ða _mmn � rHÞnþ1

Dt�r � 1

q̂qðrj� a _mm2ÞrHnþ1 ð43Þ

3.7. Energy Equation

The energy equation (32) is discretized spatially enforcing a Dirichlet boundarycondition Tf ¼ Tsat at the interface (/ ¼ 0). The discretization procedure isdeveloped by considering the one-dimensional case, as shown in Figure 3a. The


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diffusion term is discretized from [17] as

qqx

kqT

qx

� �fi

¼ 1

Dxkf

Tf ;iþm � Tfi

xiþm � xi� kf

Tfi � Tf ;i�m

xi � xi�m

� �ð44Þ

which results in

qqx

kqT

qx

� �fi

¼ 1

Dxk̂kf ;iþ1=2

Tf ;iþm � Tfi

xiþ1 � xi� k̂kf ;i�1=2

Tfi � Tf ;i�m

xi � xi�1

� �ð45Þ

where

Tf ;i�m ¼ Tf ;i�1 if /i/i�1 > 0

¼ Tsat otherwise

k̂kf ;i�1=2 ¼ kf if /i/i�1 > 0

¼ kf =max E;xi � xi�m

xi � xi�1

� �otherwise

Here, ðxi � xi�mÞ=ðxi � xi�1Þ is approximated as /i=ð/i � /i�1Þ, and E is a smallvalue, say, 10�2, which is introduced to prevent the numerical singularity at/i ¼ 0. When discretizing the convection term, a second-order ENO scheme is used

Figure 3. Schematic for implementation of boundary conditions: (a) a Dirichlet boundary condition;

(b), (c) immersed solid boundary conditions for /i=ð/i � /i�1Þ < wi=ðwi � wi�1Þ and /i=ð/i � /i�1Þ >wi=ðwi � wi�1Þ.


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except at the grid point which is the first downstream node from the interface. Forthe point satisfying the condition, (ufi > 0 and /i/i�1 � 0) or (ufi < 0 and/i/iþ1 � 0), we use

qTf

qx

� �i

¼ Tf ;iþm � Tf ;i�m

xiþm � xi�mð46Þ

3.8. Modification for an Immersed Solid Surface

In order to treat an immersed solid surface, we introduce another LS function,w, which is defined as a signed distance from the fluid–solid interface. The negativesign is chosen for the solid region and the positive sign for the fluid region. In thisstudy, we assume that the solid is stationary and is maintained at a constant tem-perature, Tw. The procedure discretizing the energy equation is easily extended fromEqs. (45) and (46), which are still valid if adding the following conditions forwiwi�1 � 0 (refer to Figures 3b and 3c):

Tf ;i�m ¼ Tsat if /i/i�1 � 0 and/i

/i � /i�1

<wi

wi � wi�1

¼ Tw otherwise

xi � xi�m

xi � xi�1¼ /i

/i � /i�1

if /i/i�1 � 0 and/i

/i � /i�1

<wi

wi � wi�1

¼ wi

wi � wi�1

otherwise

Also, the momentum equations (41) and (42) can be extended for two-fluidflows with an immersed solid boundary condition as

u�� ¼ HðwÞ½u� þ Dt

q̂qðr � ^̂mm̂mmru�� þ sn

uÞ� ð47Þ

unþ1 ¼ u�� HðwÞDt

q̂q½rpþ ðrj� a _mm2ÞrHð/nþ1Þ� ð48Þ

where

^̂mm̂mmi�1=2 ¼ m̂mi�1=2 if wiwi�1 > 0

¼ m̂mi�1=2=max E;wi

wi � wi�1

� �otherwise

Here, for convenience, the subscript i refers to the grid point where u or v is defined.Substituting Eq. (48) into the mass conservation equation (34), the governing

equation for pressure is obtained as

r �HðwÞ þ Eq̂q

rp ¼ r � u�� ½a _mmn � rHð/Þ�nþ1

Dt

�r �HðwÞq̂qðrj� a _mm2ÞrHð/nþ1Þ ð49Þ


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Here, E is introduced to prevent the numerical singularity at w ¼ 0 occurring in thematrix calculation procedure for pressure equation.

3.9. Mass Flux at the Interface

As the major variable to account for the effect of phase change, the mass flux _mmacross the interface is included in the LS advection equation (12), the mass conser-vation equation (24), and the momentum equation (30). For its efficient implemen-tation, _mm, defined at the interface (/ ¼ 0), is required to be extended to the entiredomain (or a narrow band near the interface). For this, n � rTf (or qTf =qn), whichis included in Eq. (11), is evaluated at the interface and then extrapolated to theentire domain, as described in [17]. First, qTf =qn is calculated at the grid points nearthe interface, which satisfies the condition, /i; j 6¼ 0 and (/i; j/i�1; j � 0 or/i; j/i; j�1 � 0), using the following equations (refer to Figure 4):

qT

qx

� �fi

¼ qT

qx

� ��fi

if /i/i�1 � 0 or j/i�1j < j/iþ1j

¼ qT

qx

� �þfi

if /i/iþ1 � 0 or j/i�1j > j/iþ1j

¼ 1

2

qT

qx

� ��fi

þ qT

qx

� �þfi

" #if /i/i�1 � 0 and /i/iþ1 � 0

where

qT

qx

� ��fi

¼ Tfi � Tf ;i�m

xi � xi�mif

xi � xi�m

xi � xi�i� 0:01

¼ Tf ;im � Tf ;i�m

xim � xi�motherwise

Figure 4. Schematic for discretization of qT=qx near the interface: (a) /i/i�1 < 0; (b) /i/iþ1 < 0; (c)

j/i�1j < j/iþ1j; (d) j/i�1j > j/iþ1j.


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Then, the computed qTf =qn is extrapolated into the entire domain by obtaining asteady-state solution of the equation

qqs

qTf

qn

� �þ signð/Þn � r qTf

qn

� �¼ 0 ð50Þ

This equation can be solved efficiently using the fast marching method [28] based ona first-order upwind scheme.

4. NUMERICAL TESTS

The LS formulation based on a sharp-interface representation is tested throughcomputations of two-dimensional, axisymmetric, and three-dimensional problems.In carrying out the calculations, the following dimensionless parameters are defined:

Ref ¼qf uolo

mf

Wef ¼qf u2

olo

rPrf ¼

mf cf

kfJaf ¼

cf DT

hlvh ¼ T � Tsat

DT

where lo is a reference length and uo is a reference velocity, which are determinedfrom the characteristics of each test problem.

4.1. Bubbles Rising in a Liquid

The computations are performed for bubbles rising in a liquid to validate theLS formulation for appropriately imposing the matching conditions of pressure andviscous stress at a sharp interface. The effect of phase change is not included in thistest. The radius of an initial bubble is chosen for lo and

ffiffiffiffiffiffiglop

for uo. The parametersused in the calculations are qv=ql ¼ 10�3, mv=ml ¼ 10�2, Rel , and Wel as listed inTable 1. The computational domain is taken to be a cylindrical region of0 � r � 12 and �12 � y � 12. Initially, a spherical bubble is placed in a stationaryliquid. The calculation is carried out until the bubble rise velocity attains an asymp-totic value. The results for grid resolutions of h ¼ 0:1 and h ¼ 0:05 are plotted inFigure 5. The bubble shapes at the steady state for both meshes are observed to haveno significant differences. The computed terminal velocities are listed in Table 1. Incomparison to the results obtained by Ryskin and Leal [29] using a body-fittedmethod, the present results differ by less than 3.5% when using h ¼ 0:1. The differ-ence is reduced to less than 1% as the grid spacing is halved.

Table 1. Comparison of terminal velocities of rising bubbles

Case Rel Wel

Ryskin and

Leal [29]

Present work

h ¼ 0:1

Present work

h ¼ 0:05

(a) 1.34 14.3 0.374 0.377 0.372

(b) 5.87 4.13 0.852 0.878 0.849

(c) 27.6 0.61 1.814 1.752 1.813


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Figure 5. Bubble shapes at the steady state for three cases: (a) Rel ¼ 1:34, Wel ¼ 14:3; (b) Rel ¼ 5:87,

Wel ¼ 4:13; (c) Rel ¼ 27:6, Wel ¼ 0:61.

Figure 6. Streamlines of steady flow past a circular cylinder: (a) Re ¼ 20; (b) Re ¼ 40.


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4.2. Flow past an Immersed Circular Cylinder

The formulation for an immersed solid boundary condition, which is based onanother LS function for the fluid–solid phases, is tested through the computations offluid flow past an immersed cylinder. The cylinder diameter is used for lo and thefree-stream velocity for uo. The computational domain is chosen as a region of�63 � x � 63 and 0 � y � 63. A circular cylinder is located at ðx; yÞ ¼ ð0; 0Þ insidethe domain. Uniform meshes with h ¼ 0:05 are used near the cylinder, jxj � 1:5 andy � 1:5, whereas nonuniform meshes with the ratio of two adjacent intervals of 1.04are used for the other regions to save computing time. Figure 6 shows the streamlinesobtained at the steady state. It is seen that the wake formed behind the cylinder isexpanded as Re increases. The computed wake lengths and drag coefficients, CD,are listed in Table 2. The present results are in good agreement with the resultsobtained in [30, 31] using modified polar-coordinates.

4.3. Simple Phase-Change Problems with Immersed Solid Boundaries

To validate the present LS formulation for two-phase flows including the effectof phase change and immersed solid boundaries, the computations are performedfor a simple phase-change problem in a two-dimensional channel, 0 � x; y � 1:2.The effect of gravity is neglected for this case. The computational domain containsa fluid region of 0:2 < y < 1 and a solid region of y � 0:2 and y � 1. The solid tem-perature is taken to be Tsat þ DT for y � 0:2 and Tsat � DT for y � 1, which meansthat both evaporation and condensation occur at the same time. The slip condition isimposed at the side boundaries, x ¼ 0 and x ¼ 1:2, except one cell face at x ¼ 1:2and 1� h � y � 1, which is treated as an open boundary to allow for volume expan-sion or contraction due to phase change. The parameters used in the calculations areql=qv ¼ 100, ml=mv ¼ 10, cl=cv ¼ 1, kl=kv ¼ 3, Rel ¼ 1:9, Wel ¼ 1, Prl ¼ 100, andJal ¼ 0:05. The liquid–vapor interface is initially specified as / ¼ y� 0:7. Thetemperature profiles in the vapor and liquid layers are chosen to be linear, and fluidvelocity is set to be zero. Figure 7 shows the temporal variation of the interface atx ¼ 0 and x ¼ 1:2, which is caused by the imbalance between the heat fluxes throughthe liquid and vapor layers. As time elapses, the oscillation decays due to fluid vis-cosity and thereafter the interface becomes stationary. The temperature profileobtained at the stationary state matches perfectly with the analytical solution,h ¼ ð0:4� yÞ=0:2 for y � 0:4 and h ¼ ð0:4� yÞ=0:6 for y � 0:4, as shown inFigure 8. The relative errors in local values of temperature for h ¼ 0:08, h ¼ 0:04,and h ¼ 0:02 are all less than 2:7� 10�8. When using the original LS method,

Table 2. Comparison of wake lengths and drag coefficients

Re ¼ 20 Re ¼ 40

Wake length CD Wake length CD

Dennis and Chang [30] 0.94 2.05 2.35 1.52

Fornberg [31] 0.91 2.00 2.24 1.50

Present work 0.95 2.02 2.33 1.53


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the error is increased up to 1:9� 10�4 for h ¼ 0:04. The increased error is caused bythe diffuse modeling of boundary conditions at the interfaces, as shown in Figure 8.Also, the new LS formulation based on the ghost fluid method reduces the comput-ing time by 10% compared with the original LS method, in which the temperaturefields have to be solved separately in the liquid and vapor layers.

The LS formulation is also tested for a three-dimensional channel,0 � x; y; z � 1:2, where a fluid occupies the region 0:2 < rð¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 þ y2

pÞ < 1.

Initially, the liquid–vapor interface is given as / ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 þ y2

p� 0:7. The other

computational conditions are similar to those in the previous two-dimensional case.Figure 9 shows that the steady-state result compares well with the analyticalsolution, h ¼ lnðr=rI Þ= lnð0:2=rIÞ for r � rI and h ¼ lnðr=rI Þ= lnðrI Þ for r � rI , where

Figure 8. Comparison of exact and numerical temperature profiles in a two-dimensional channel with

h ¼ 0:04.

Figure 7. Temporal variation of interface location in a two-dimensional channel with h ¼ 0:04.


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rI ¼ 0:20:75. The relative errors for h ¼ 0:08, h ¼ 0:04, and h ¼ 0:02 are 1:19� 10�3,2:88� 10�4, and 8:94� 10�5, respectively.

4.4. Film Boiling on a Horizontal Cylinder

The present numerical method is applied to simulate on a horizontal cylindersaturated film boiling of water at 1 atm pressure. We define lo ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir=gðql � qvÞ

pand

uo ¼ffiffiffiffiffiffiglop

. Based on the properties of saturated water at 1 atm, the following para-meters are evaluated:

D

lo¼ 1

qv

ql

¼ 6:24� 10�4 mv

ml

¼ 4:26� 10�2 Rev ¼ 19:4

Wel ¼ 1 Prv ¼ 0:99 Jav ¼ 0:18 ðfor DT ¼ 200 KÞ

where D is the cylinder diameter. For computation of saturate film boiling, in whichthe liquid temperature is maintained as the saturation temperature, kl (or Prl) is notrequired.

First, two-dimensional calculations are performed in a computational domainof jxj � 3:3 and �3:6 � y � 11:5 including a circular cylinder located atðx; yÞ ¼ ð0; 0Þ. The fluid and solid regions are represented by the LS function,w ¼


p� 0:5. The slip condition is imposed at all of the computational

boundaries except the top of the domain, which is treated as an open boundary.Initially, the LS function for the liquid and vapor regions is specified as/ ¼


p� 0:6. The vapor temperature profile is chosen to be linear, and fluid

velocity is set to be zero. To save computation time, we use nonuniform meshes withthe ratio of two adjacent intervals of 1.05 except near the cylinder, jxj � 0:8 and

Figure 9. Comparison of exact and numerical temperature fields in a three-dimensional channel with

h ¼ 0:04.


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y � 0:8, where the grid spacing, ho, is uniform. Convergence for grid resolutions aretested with ho ¼ 0:02, ho ¼ 0:01, and ho ¼ 0:005. The results are plotted in Figures 10and 11. The interface shapes for two finest grids show insignificant differences except

Figure 11. Effect of mesh size on Nusselt number during the first computational cycle of film boiling.

Figure 10. Effect of mesh size on the liquid–vapor interface shape at t ¼ 5:6 during film boiling; (a) over

the computational domain; (b) in the expanded region near the cylinder.


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for the elongated neck of the detaching bubble. It is also noted from Figure 10b thatthe computed vapor film on the lower part of cylinder is comparable with the theor-etical prediction of Bromely [32] under the assumption that the slip condition existsbetween vapor and liquid. The vapor film thickness, d, can be expressed for D ¼ lo as

dlo¼

2qv Jav

R u0 sin1=3 u du

ðql � qvÞRe2v Prvð1þ 0:4 JavÞ sin4=3 u

" #1=4

where the angle, u, is measured from the bottom of the cylinder. Figure 11 shows thetemporal variation of Nusselt number during the computational period prior to the

Figure 12. Two-dimensional evolution of the liquid–vapor interface during the fourth computational cycle

of film boiling.


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first bubble pinch-off. The Nusselt number is defined as Nu ¼ D�qqw=kv DT , where �qqw

is the wall heat flux averaged over the surface area of the cylinder. As the grid spa-cing decreases, the relative difference of the Nusselt numbers between successivemesh sizes becomes small. For ho ¼ 0:01 and ho ¼ 0:005, the Nusselt number aver-aged over the computational period differs by less than 3%.

Since film boiling is a cyclic process, the computations need to be carried outover several cycles until the effect of ambiguity in specification of initial conditionsdisappears. Figure 12 shows the evolution of the liquid–vapor interface during thefourth computational cycle. After a bubble pinches off at t ¼ 12:3, the vapor filmon the cylinder grows with time. As the buoyancy force is dominant over the surfacetension force with the increase in vapor volume, another bubble starts to form andthen pinches off at t ¼ 16:5. The interface shapes at t ¼ 12:3 and t ¼ 16:5 have nosignificant difference near the cylinder, which implies the effect of initial conditionson heat transfer almost vanishes. The Nusselt number for several cycles is plotted inFigure 13. The temporal variation of Nusselt numbers is found to be nearly cyclicafter a few bubble release periods. The Nusselt number averaged over one cyle is28.1. This value is in good agreement with 27.0 and 28.0, which are obtained fromthe correlations given in [32] and [33], respectively.

The computation is performed for three-dimensional simulation of film boil-ing on a horizontal cylinder using the domain, jxj � 3:3, �3:6 � y � 11:5 andjzj �

ffiffiffi3p

p, where 2pffiffiffi3p

lo corresponds to the most unstable wavelength for film boil-ing on a flat plate. The grid spacing in the z direction is chosen as hz ¼ 0:085. Weuse the same parameters as in the two-dimensional case except for the cylinderdiameter, which is reduced to D ¼ 0:5lo. Initially the interface is disturbed sinusoid-ally as / ¼


p� ½0:4þ 0:015 cosðz=

ffiffiffi3pÞ�. Figure 14 shows three-dimensional

evolution of the liquid–vapor interface. During the early period, discrete vaporbubbles are released alternatively at the node, z ¼ 0, and antinodes, jzj ¼

ffiffiffi3p

p,of the upper portion of the cylinder. However, after t ¼ 4:7, bubble releaseoccurs at jzj ¼ 0:57

ffiffiffi3p

p rather than at jzj ¼ffiffiffi3p

p. This means that the most

Figure 13. Variation of Nusselt number during several computational cycles of film boiling.


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unstable wavelength for film boiling shrinks on a small-diameter cylinder, asobserved experimentally in [34].

5. CONCLUSIONS

A level set method was developed for computing film boiling on an immersedsolid surface. The formulation, based on a sharp-interface representation for incom-pressible two-fluid flows, was developed and extended to include the effect of phasechange at the liquid–vapor interface and to treat the no-slip condition at the fluid–solid interface. The method was tested through the computations of bubble rise andsingle-fluid flow past a circular cylinder. The numerical results show a good agree-ment with those obtained from body-fitted methods. Also, the computations werecarried out for liquid–vapor phase-change problems with immersed solid bound-aries. The numerical results are observed to compare well with the exact solutionsor the experimental data available in the literature.

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Figure 14. Three-dimensional evolution of the liquid–vapor interface during film boiling.


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Documents

A Level Set Method for Analysis of Film Boiling on an Immersed Solid Surface