Gada and Sharma Level Set

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Numerical Heat Transfer, Part B:

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On Derivation and Physical

Interpretation of Level Set MethodBased

Equations for Two-Phase FlowSimulationsVinesh H. Gada

a& Atul Sharma

a

aDepartment of Mechanical Engineering, Indian Institute of

Technology Bombay, Mumbai, India

Published online: 03 Nov 2009.

To cite this article: Vinesh H. Gada & Atul Sharma (2009): On Derivation and Physical Interpretation

of Level Set MethodBased Equations for Two-Phase Flow Simulations, Numerical Heat Transfer, Part B:Fundamentals: An International Journal of Computation and Methodology, 56:4, 307-322

To link to this article: http://dx.doi.org/10.1080/10407790903388258

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ON DERIVATION AND PHYSICAL INTERPRETATIONOF LEVEL SET METHODBASED EQUATIONS FORTWO-PHASE FLOW SIMULATIONS

Vinesh H. Gada and Atul SharmaDepartment of Mechanical Engineering, Indian Institute of Technology Bombay,

Mumbai, India

The level set (LS) method is one of the most popular and recent methods for two-phase flow

simulation. However, its origin and governing equations rely heavily on mathematical

sources. The present work attempts mass and volume conservation law-based derivation

of the LS advection equation and the continuity equation, respectively, for two-phase flow

with phase change. Physical interpretation of the Heaviside function and the Dirac delta

function, used in the LS method, is done here in a novel way and the resulting expressions

are used in the derivations. The diffused interface, used in the LS method, is shown to

introduce negligible error on a sufficiently fine grid.

1. INTRODUCTION

Two-phase flows are widely studied because of their engineering relevance,such as in mold design for casting in manufacturing, effective cooling in electronics,various thermal-hydraulic situations in nuclear industries, and as a means to trans-port large amounts of heat in the form of latent heat in power-generation industries.Researchers have obtained various correlations from experimental investigations,which are, however, limited to certain operating conditions and geometry. An alter-native approach, namely, computational multifluid dynamics (CMFD), has openedup new ways of detailed investigation to study the interface transport mechanisms.Some of the foremost interface tracking=capturing techniques to simulate separatedtwo-phase flow are those based on front tracking [1, 2], the volume-of-fluid (VOF)method, [35] and the level set (LS) [626] method.

In the LS and VOF methods, the region occupied by a particular fluid isfollowed with time rather than the exact interface, avoiding the logical statementsand other difficulties encountered with the moving-mesh method and explicit inter-face tracking. Sussman et al. [6] proposed a level set method based on a single fieldformulation for simulation of incompressible airwater two-phase flow, wherein a

Received 17 July 2009; accepted 14 August 2009.

The present work is part of a research project funded by the Board of Research in Nuclear Sciences

(India) under project no. 2007=36=14 BRNS=718.

Address correspondence to Atul Sharma, Department of Mechanical Engineering, Indian Institute

of Technology Bombay, Mumbai 400076, India. E-mail: [email protected]

Numerical Heat Transfer, Part B, 56: 307322, 2009

Copyright # Taylor & Francis Group, LLC

ISSN: 1040-7790 print=1521-0626 onlineDOI: 10.1080/10407790903388258

307


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LS function is used to represent the region occupied by different fluids. For the

LS method, Chang et al. [7] derived the volumetric source term of the surfacetension force at the interface added to the momentum equation. Compared tothe VOF method, the strengths of the LS method are easier numericalimplementation and accurate calculation of interfacial information, i.e.,location, normal, and curvature. The occurrence of mass error is its majorshortcoming, which has been reduced by certain improvements in the LSmethod [814]. Son [15] presented an LS-based immersed boundary methodfor solving complex domain problems using Cartesian grids. Son and Hur [16]presented a contact angle modeling technique combined with an immersedboundarybased LS formulation to simulate the motion of droplets over

inclined wall. Furthermore, Suh and Son [17] simulated a thermal inkjet pro-cess where three fluid interaction was modeled using the LS method. Moreover,the LS method was also used to solve problems involving phase change atinterfaces, such as solidification [1820] and boiling [2124].

Thus, from the literature survey, it is found that the LS method is one of themost popular recent (compared to the VOF method) methods for simulation oftwo-phase flow in addition to various other applications in different fields of scienceand engineering [25, 26]. However, conservation lawbased derivation of the equa-tions, used in the LS method, is not found in the published literature, which is themajor objective of the present work. Furthermore, the objective is to give morephysical interpretation to the functions and the reinitialization equation used in

the LS method.

NOMENCLATURE

Cp specific heat

E error

~FFST surface tension force vector~GG body force vector

h12 latent heat

H(/) Heaviside function

_mm; ~mm mass flux and mass flux vectoracross interface

~nn; nn normal vector and unit normalvector of interface

p pressure

~qq heat flux vector

R radius of circular interface

S(/) sign function

t time~uu velocity vector

a thermal diffusivity ( k=qCp)d(/) Dirac delta function

DS, DV surface area and volume

Dx, Dy size of representative cell in the

horizontal and vertical directions

e interface half-thickness

j interface curvature

m dynamic viscosity

q density

r coefficient of surface tensionn specific volume

/ level set function

r gradient operator

SubscriptsA area

c center

f cell face center

i interface

m mean=volume averaged

o irregular

p phase (liquid or vapor)PC phase change

s pseudo

t total

V volume

x,y horizontal and vertical directions

e smoothened

1,2 liquid (fluid 1), vapor (fluid 2)

308 V. H. GADA AND A. SHARMA


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2. FUNCTIONS USED IN THE LEVEL SET METHOD

The level set method is a Eulerian computational technique for capturingmoving boundaries or interfaces. In this method, there are three functions: A level

set function to define the interface, a Heaviside function to calculate fluid properties,and a Dirac delta function to model the effect of two-phase flow as interfacial masstransfer in the continuity and surface tension in the momentum equations.

2.1. Level Set Function

The level set interface representation is based on the concept of implicitsurfaces, wherein a level set function (/) is defined in a domain having a fixed valueat the interface. The LS function is defined as a signed normal distance function mea-sured from the interface and is equal to zero at the interface. For a circular interface ofradius 5 units, Figure 1a shows the level set function / 0 at the interface with nega-

tive values in fluid 2 and positive values in fluid 1. Figure 1b shows the level setcontours for the circular interface as concentric circles with increasing absolutevalues away from the interface. The LS field is smooth and the exact instantaneousinterface position can be captured by locating the zero level set, thus avoiding logicaldifficulties encountered during interface reconstruction.

2.2. Heaviside Function

In the present work, a single field formulation [1, 21] is used, where theNavier-Stokes equations consider individual material properties in different fluidsand mean properties at the interface. Fluid density and viscosity in the domainare thus calculated as

qm q1H/ q21 H/

mm m1H/ m21 H/1

Figure 1. Level set method; (a) interface representation; (b) level set contours for a circular interface of

radius 5 units.

DERIVATION AND PHYSICAL INTERPRETATION OF LS EQUATIONS 309


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where the subscripts m, 1, and 2 represent the mean, fluid 1, and fluid 2, respectively.H(/) is a unit step function or Heaviside function, defined [6, 27] as

H/

1 if / > 0

0:5 if / 00 if / < 0

8 0.5, and Hs< 0.5. The above equation requires Hfto be calculated geometrically. However, in the LS method, the Heaviside functionis defined at the cell center and its value at a face center H

fis interpolated from

neighboring cell center values.In summary, the Heaviside function is dimensionless, and its value at the

centroid of a CV is physically interpreted as fraction of the volume occupied by thefluid 1 in a CV [Eq. (3)]. Furthermore, its value at a face center of a CV is interpretedas fraction of the face area occupied by the fluid 1 [Eq. (4)]. These expressions will beused for the derivation of the equations in the level set method. Note thatthe second-order approximation for volume and surface averaging used above inthe physical interpretation of H at cell and face centers, respectively, is commonlyused in the finite-volume method.

In the LS method, the sharp Heaviside function calculated from Eq. (2) results

in numerical instability [6, 7], and geometric calculation of the volume of fluid 1 in a



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CV [Eq. (3)] needs to be avoided. Thus, a smoothed Heaviside function [6, 7] result-ing in continuous variation of the Heaviside function across the interface is used,defined as

He/ 0; if / < e/e

2e 1

2p sinp/e

if j/j e1; if / > e

8 0, p 2 if/< 0, and T TSAT if/ 0.

3.4. Reinitialization Equation

In this section, the physical interpretation of the reinitialization equation isdone. The level set field obtained after solving the advection equation [Eq. (14)],in general, will not remain a normal distance function field, as shown in Figure 5a.For accurate calculation of the Heaviside function [Eq. (5)] and the Dirac deltafunction [Eq. (8)], it is necessary to maintain a constant width of the diffused inter-

face along the interface (Figure 5b) at all times. This is ensured by reinitializing theadvected level set function field to a signed normal distance function field withoutaltering the location of the interface obtained after the advection step (/ 0 remainsunchanged in Figures 5a and 5b). In level set methods for two-phase flows,a PDE-based reinitialization procedure [622] is generally used, wherein an irregularlevel set function (/o) is reinitialized to a signed normal distance function byobtaining the steady-state solution of the following equation:

q/

qts Se /o r/j j 1 0 17

where ts is pseudo-time and Se /o /o=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi/2o e

2

qis the smoothed sign function

[6]. The steady-state solution of Eq. (17) ensures that j r/j 1, i.e., the LS

Figure 5. Diffused interface (a) after advection and (b) after reinitialization.



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function is set as the normal distance function. For easy implementation, Eq. (17)is written as

q/

qts Se /o ^

nn r/ Se /o

where j r/jj r/j r/ r/ and the unit normal vector nn r/= r/j j. The aboveequation is a hyperbolic equation with characteristics=advecting velocity as Se /o nn.It is seen that at the interface, the advecting velocity is zero [because S

e(/o) 0]

and away from the interface it is normal to the interface, pointing away fromthe interface into both phases (because of the sign function). Such an advectingvelocity field carries the interface information from the interface into both fluidsin the normal direction. This PDE-based reinitialization technique has the advan-tage that it does not require an exact interface position to recover the normal

distance function.

4. CONCLUSION

The LS method is one of the most common and recent methods for simulationof two-phase flow. The functions used in the LS method have been interpretedphysically: the level set function as a normal distance function, the Heaviside func-tion as the volume fraction or surface-area fraction occupied by fluid 1, and theDirac delta function as the ratio of the interface area in a CV to the volume ofthe CV. Furthermore, the expressions corresponding to these interpretations havebeen used for conservation law-based derivations of the equations used in the LS

method for simulation of two-phase flow with phase change. The continuity equa-tion was derived from volume conservation, and the level set advection equationwas derived from mass conservation laws. The term for the heat=mass transfer(due to phase change) and surface tension force (in the momentum equation) atthe interface was also derived. The energy equation for the LS method-based simula-tion of film boiling was discussed, and finally, the physical interpretation and work-ing of the reinitialization equation was presented. From a numerical test, it has beenshown that the error incurred upon smoothing the Heaviside and Dirac delta func-tion is negligible on a sufficiently fine grid. The interpretation of the Heaviside andDirac delta function and conservation lawbased derivations done here is not found

in the published literature.

REFERENCES

1. D. Juric and G. Tryggvason, Computations of Boiling Flows, Int. J. Multiphase Flow,vol. 24, pp. 387410, 1998.

2. A. Esmaeeli and G. Tryggvason, Computations of Film Boiling. Part I: NumericalMethod, Int. J. Heat Mass Transfer, vol. 47, pp. 54515461, 2004.

3. C. W. Hirt and B. D. Nichols, Volume of Fluid (VOF) Method for the Dynamics of FreeBoundaries, J. Comput. Phys., vol. 39, pp. 201225, 1981.

4. W. J. Rider and D. B. Kothe, Reconstructing Volume Tracking, J. Comput. Phys.,

vol. 141, pp. 112152, 1998.



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5. S. W. J. Welch and J. Wilson, A Volume of Fluid Based Method for Fluid Flows withPhase Change, J. Comput. Phys., vol. 160, pp. 662682, 2000.

6. M. Sussman, P. Smereka, and S. Osher, A Level Set Approach for Computing Solutionsto Incompressible Two-Phase Flow, J. Comput. Phys., vol. 114, pp. 146159, 1994.

7. Y. C. Chang, T. Y. Hou, B. Merriman, and S. Osher, A Level Set Formulation of EulerianInterface Capturing Methods for Incompressible Fluid Flows, J. Comput. Phys., vol. 124,449464, 1996.

8. M. Sussman and E. Fatemi, An Efficient Interface Preserving Level Set Re-distancingAlgorithm and Its Application to Interfacial Incompressible Flow, SIAM J. Sci. Comput.,vol. 20, pp. 11651191, 2000.

9. D. Peng, B. Merriman, S. Osher, H. K. Zhao, and M. Kang, A PDE-Based Fast LocalLevel Set Method, J. Comput. Phys., vol. 155, pp. 410438, 1999.

10. G. Son and N. Hur, A Coupled Level Set and Volume-of-Fluid Method for theBouyancy-Driven Motion of Fluid Particles, Numer. Heat Transfer B, vol. 42, pp. 523542, 2002.

11. G. Son, Efficient Implementation of A Coupled Level Set and Volume-of-Fluid Method

for Three-Dimensional Incompressible Two-Phase Flows, Numer. Heat Transfer B, vol.43, pp. 549565, 2003.

12. M. J. Ni, S. Komori, and N. B. Morley, Direct Simulation of Falling Droplet in a ClosedChannel, Int. J. Heat Mass Transfer, vol. 49, pp. 366376, 2006.

13. Y. F. Yap, J. C. Chai, K. C. Toh, and T. N. Wong, Modeling the Flows of Two Immis-cible Fluids in a Three-Dimensional Square Channel Using the Level-Set Method, Numer.Heat Transfer B, vol. 49, pp. 893904, 2006.

14. Y. F. Yap, J. C. Chai, T. N. Wong, K. C. Toh, and H. Y. Zhang, A GlobalMass Correction Scheme for the Level-Set Method, Numer. Heat Transfer B, vol. 50,pp. 455472, 2006.

15. G. Son, A Level Set Method for Incompressible Two-Fluid Flows with Immersed Solid

Boundaries, Numer. Heat Transfer B, vol. 47, pp. 473489, 2005.16. G. Son and N. Hur, A Level Set Formulation for Incompressible Two-Phase Flows on

Nonorthogonal Grids, Numer. Heat Transfer B, vol. 48, pp. 303316, 2005.17. Y. Suh and G. Son, A Level-Set Method for Simulation of a Thermal Inkjet Process,

Numer. Heat Transfer B, vol. 54, pp. 138156, 2008.

18. S. Chen, B. Merriman, S. Osher, and P. Smereka, A Simple Level Set Method for SolvingStefan Problems, J. Comput. Phys., vol. 135, pp. 829, 1997.

19. H. Zhang, L. L. Zheng, V. Prasad, and T. Y. Hou, A Curvilinear Level Set Formulationfor Highly Deformable Free Surface Problems with Application to Solidification, Numer.Heat Transfer B, vol. 34, pp. 130, 1998.

20. L. L. Zheng and H. Zhang, An Adaptive Level Set Method for Moving-BoundaryProblems: Application to Droplet Spreading and Solidification, Numer. Heat TransferB, vol. 37, pp. 437454, 2000.

21. G. Son and V. K. Dhir, Numerical Simulation of Film Boiling near Critical Pressures witha Level Set Method, J. Heat Transfer, vol. 120, pp. 183192, 1998.

22. G. Son and V. K. Dhir, A Level Set Method for Analysis of Film Boiling on an ImmersedSolid Surface, Numer. Heat Transfer B, vol. 52, pp. 153177, 2007.

23. F. Gibou, L. Chen, D. Nguyen, and S. Banerjee, A Level Set Based Sharp InterfaceMethod for the Multiphase Incompressible Navier-Stokes Equations with Phase Change,J. Comput. Phys., vol. 222, pp. 536555, 2007.

24. J. Wu, V. K. Dhir, and J. Qian, Numerical Simulation of Subcoalled Nucleate Boiling byCoupling Level-Set Method with Moving-Mesh Method, Numer. Heat Transfer B, vol. 51,pp. 535563, 2007.



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25. J. A. Sethian, Level Set Methods and Fast Marching Methods, 2nd ed., CambridgeUniversity Press, New York, 1999.

26. S. Osher and R. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces,Springer-Verlag, New York, 2003.

27. R. N. Bracewell, The Fourier Transform and Its Applications, 3d ed., pp. 6177,McGraw-Hill, Boston, 2000.

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APPENDIX: NUMERICAL TEST

Here, a 2-D test problem is solved to find the error incurred by smoothing theHeaviside and Dirac delta functions. The test problem considered is a circular inter-face of radius R 0.1 located centrally at xc;yc 0:5; 0:5 in a square computa-

tional domain of unit dimension. The circular region is filled with fluid 1 and therest of the domain is filled with fluid 2. The test is performed on grid resolution ofN2 with number of control volumes N 16, 32, 64, 128, and 256.

The true values of the Heaviside and Dirac delta functions are calculated by ageometric procedure, wherein for each CV in the domain, the volume (area in 2D)occupied by fluid 1 is calculated geometrically and the cell center volume fraction=Heaviside function, Hi,j, is determined using Eq. (3). Furthermore, the cell centerDirac delta function is obtained from Eq. (7), where the surface area (curve lengthin 2D) of the interface DSi,j is calculated geometrically in all the partially filled CVs.

For the calculation of the smoothed functions, the exact level set fieldis obtained at the centroid of all the CVs in the domain as /i;j Rffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffi

xi;j xc2 yi;j yc

2q

. These values are used to calculate smoothed Heaviside

function He,i,j [(Eq. 5)] and the smoothed delta function de,i,j [(Eq. 8)], where the

commonly used [621] value of 2e 3Dx is taken for the width of the diffusedinterface.

Two comparisons are made to evaluate the effect of using the smoothedfunctions. First, a grid point-by-point comparison of the true=sharp and smoothenedfunctions is evaluated by the normalized L2 error expressed as

EL2

ffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiPi;j Hi;j He;i;j 2

q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPi;j H

2i;j

q EL2

ffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiPi;j DSi;j de;i;jDV 2

q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPi;j DS

2i;j

qSecond, a global measure of error in the calculation of the total volume and surfacearea of the interface in the domain is made, using the smoothed functions He insteadof HP in Eq. (3) and de instead of d in Eq. (7), respectively. The global error isexpressed as

EV pR2

Pi;j He;i;jDVi;j

pR2EA

2pR P

i;j dei;jDVi;j

2pR



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Table 1 shows that the both the errors for both the functions are reduced

substantially with grid refinement, and they are reduced to less than 5% on the finestgrid size.

Table 1. Variation of error due to smoothening of Heaviside and Dirac delta functions with grid

refinement

Percentage error due to

Number of grid pointsSmoothed Heaviside function (%) Smoothed Dirac delta function (%)

N2 EL2 EV EL2 EA

162 35.21 10.24 11.14 0.53

322 17.64 2.80 8.91 0.11

642 8.83 0.71 6.84 0.16

1282 4.47 0.17 4.41 0.12

2562 2.23 0.04 3.16 0.04


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Gada and Sharma Level Set