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Research Papers: Multiphase Flows

Level Set, Phase-Field, and Immersed Boundary Methods for Two-Phase Fluid Flows

[+] Author and Article Information
Haobo Hua

Department of Mathematics,
Korea University,
Seoul 136-713, China;
Department of Mathematics and Physics,
Zhengzhou Institute of Aeronautical
Industry Management,
Zhengzhou 450015, China
e-mail: huahbo@korea.ac.kr

Jaemin Shin

e-mail: zmshin@korea.ac.kr

Junseok Kim

e-mail: cfdkim@korea.ac.kr
Department of Mathematics,
Korea University,
Seoul 136-713, China

1Corresponding author.

Contributed by the Fluids Engineering Division of ASME for publication in the JOURNAL OF FLUIDS ENGINEERING. Manuscript received June 7, 2013; final manuscript received October 8, 2013; published online November 22, 2013. Assoc. Editor: Samuel Paolucci.

J. Fluids Eng 136(2), 021301 (Nov 22, 2013) (14 pages) Paper No: FE-13-1360; doi: 10.1115/1.4025658 History: Received June 07, 2013; Revised October 08, 2013

In this paper, we review and compare the level set, phase-field, and immersed boundary methods for incompressible two-phase flows. The models are based on modified Navier–Stokes and interface evolution equations. We present the basic concepts behind these approaches and discuss the advantages and disadvantages of each method. We also present numerical solutions of the three methods and perform characteristic numerical experiments for two-phase fluid flows.

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Figures

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Fig. 1

Schematic of a two-phase domain

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Fig. 2

(a) Zero contour of the signed distance function φ and (b) surface plot of φ with the zero contour

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Fig. 3

(a) Zero contour of the order parameter φ and (b) surface plot of φ with the zero contour

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Fig. 4

Concentration profile across the diffused interface region with the thickness ξ

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Fig. 5

Immersed boundary configuration X(s,t) for representing the interface Γ

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Fig. 6

(a) Smoothed Heaviside function Hα(φ) for LSM, (b) (1+φ)/2 for PFM, and (c) Indicator function H for IBM.

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Fig. 7

(a) Velocities and pressure near the cell Ωij and (b) Lagrangian points Xl in the computational domain Ω.

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Fig. 8

Smoothed δ functions for LSM and PFM with h = 1/32, α = 3h, and ɛ = 6h/[22 tanh-1(0.9)]

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Fig. 9

(a) Regularized delta functions and (b) Four-point δ functions.

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Fig. 10

Numerical pressure field p for (a) LSM, (b) PFM, and (c) IBM

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Fig. 11

Schematic illustration of droplet deformation under shear flow

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Fig. 12

Deformation number D as a function of time t comparing with results (solid lines) in Ref. [72]

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Fig. 13

(a) Interfaces of the three methods and (b) Deformation numbers as a function of time.

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Fig. 14

Evolution of the droplet using different interfacial thickness parameter (a) α = 0.5h, (b) α = h, and (c) α = 2h in LSM. The contours are drawn at times t = 0, 2.5, and 4.5 from top to bottom. Contour levels are -2h, -h, 0, h, and 2h.

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Fig. 15

Evolution of the droplet using different Peclet number (a) Pe = 0.1/ɛ, (b) Pe = 1/ɛ, and (c) Pe = 10/ɛ in PFM. The contours are drawn at times t = 0, 2.5, and 4.5 from top to bottom. Contour levels are -0.9, -0.45, 0, 0.45, and 0.9.

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Fig. 16

Evolution of the droplet in IBM (a) without and (b) with deletion and addition procedures. The marker points are drawn at times t = 0, 2.5, and 4.5 from top to bottom.

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Fig. 17

Variation of the number of marker points as a function of time in case of Fig. 16(b)

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Fig. 18

Comparison of the falling droplet for the three methods with optimal parameters at time t = 0, 2.5, and 4.5

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