Commun. Comput. Phys., 11 (2012), pp. 1347-1371.


Numerical Simulation of Free Surface by an Area-Preserving Level Set Method

Tony W. H. Sheu 1*, C. H. Yu 2

1 Department of Engineering Science and Ocean Engineering, National Taiwan University, No. 1, Sec. 4, Roosevelt Road, Taipei, Taiwan; Taida Institute of Mathematical Sciences (TIMS), National Taiwan University; and Center for Quantum Science and Engineering (CQSE), National Taiwan University.
2 Department of Engineering Science and Ocean Engineering, National Taiwan University, No. 1, Sec. 4, Roosevelt Road, Taipei, Taiwan.

Received 15 May 2010; Accepted (in revised version) 15 May 2011
Available online 30 November 2011
doi:10.4208/cicp.120510.150511s

Abstract

We apply in this study an area preserving level set method to simulate gas/water interface flow. For the sake of accuracy, the spatial derivative terms in the equations of motion for an incompressible fluid flow are approximated by the fifth-order accurate upwinding combined compact difference (UCCD) scheme. This scheme development employs two coupled equations to calculate the first- and second-order derivative terms in the momentum equations. For accurately predicting the level set value, the interface tracking scheme is also developed to minimize phase error of the first-order derivative term shown in the pure advection equation. For the purpose of retaining the long-term accurate Hamiltonian in the advection equation for the level set function, the time derivative term is discretized by the sixth-order accurate symplectic Runge-Kutta scheme. Also, to keep as a distance function for ensuring the front having a finite thickness for all time, the re-initialization equation is used. For the verification of the optimized UCCD scheme for the pure advection equation, two benchmark problems have been chosen to investigate in this study. The level set method with excellent area conservation property proposed for capturing the interface in incompressible fluid flows is also verified by solving the dam-break, Rayleigh-Taylor instability, two-bubble rising in water, and droplet falling problems.

AMS subject classifications: 65E05, 76T10

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Key words: Level set method, phase error, upwinding combined compact scheme, Hamiltonian, symplectic Runge-Kutta, area conservation property.

*Corresponding author.
Email: twhsheu@ntu.edu.tw (T. W. H. Sheu)
 

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