Accuracy Improvement of PLIC-VOF Volume-Tracking Method Using the Equation of Surface Normal Vector

ABSTRACT

The PLIC/SN method that combines the second-order volume tracking method (PLIC-VOF) with the equation of surface normal (SN) vector was recently proposed (M. Sun, “Volume Tracking of Subgrid Particles,” *International Journal for Numerical Methods in Fluids*, Vol. 66, No. 12, 2011, pp. 1530-1554). The method is able to track the motion of a subgrid particle, but the accuracy is not as good as expected on high resolution grids for vortical flows. In this paper, a simple unsplit multidimensional advection algorithm is coupled with the equation of SN vector. The advection algorithm is formulated as the finite volume method, so that it can be used readily for both structured and unstructured grids while maintaining the exact mass conservation. The new method improves the accuracy significantly for high resolution grids. In the well-known test of the time-resolved vortex problem of *T* = 2, the circular interface is resolved with an accuracy better than ever using the equation of SN vector.

Cite this paper

M. Sun, "Accuracy Improvement of PLIC-VOF Volume-Tracking Method Using the Equation of Surface Normal Vector,"*Advances in Pure Mathematics*, Vol. 3 No. 1, 2013, pp. 219-225. doi: 10.4236/apm.2013.31A031.

M. Sun, "Accuracy Improvement of PLIC-VOF Volume-Tracking Method Using the Equation of Surface Normal Vector,"

References

[1] C. W. Hirt and B. D. Nichols, “Volume of Fluid (VOF) Method for the Dynamics of Free Boundaries,” Journal of Computational Physics, Vol. 39, No. 1, 1981, pp. 201-225. doi:10.1016/0021-9991(81)90145-5

[2] W. J. Rider and D. B. Kothe, “Reconstructing Volume Tracking,” Journal of Computational Physics, Vol. 141, No. 2, 1998, pp. 112-152. doi:10.1006/jcph.1998.5906

[3] R. Scardovelli and S. Zaleski, “Direct Numerical Simulation of Free-Surface and Interfacial Flow,” Annual Review of Fluid Mechanics, Vol. 31, 1999, pp. 567-603. doi:10.1146/annurev.fluid.31.1.567

[4] W. F. Noh and P. Woodward, “SLIC (Simple Line Interface Calculation),” In: A. I. van der Vooren and P. J. Zandbergen, Eds., Lecture Notes in Physics, Springer, New York, 1976, p. 330.

[5] M. Rudman, “Volume-Tracking Methods for Interfacial Flow Calculations,” International Journal for Numerical Methods in Fluids, Vol. 24, No. 7, 1997, pp. 671-691. doi:10.1002/(SICI)1097-0363(19970415)24:7<671::AID-FLD508>3.0.CO;2-9

[6] D. L. Youngs, “An Interface Tracking Method for a 3D Eulerian Hydrodynamics Code,” Technical Report 44/92/ 35, AWRE, 1984.

[7] E. G. Puckett, “A Volume of Fluid Interface Tracking Algorithm with Applications to Computing Shock Wave Rarefraction,” Proceedings of the 4th International Symposium on Computational Fluid Dynamics, 1991.

[8] R. Scardovelli and S. Zaleski, “Interface Reconstruction with Least-Square Fit and Split Eulerian-Lagrangian Advection,” International Journal for Numerical Methods in Fluids, Vol. 41, No. 3, 2003, pp. 251-274. doi:10.1002/fld.431

[9] J. López, J. Hernández, P. Gómez and F. Faura, “A Volume of Fluid Method Based on Multidimensional Advection and Spline Interface Reconstruction,” Journal of Computational Physics, Vol. 195, No. 2, 2004, pp. 718- 742. doi:10.1016/j.jcp.2003.10.030

[10] J. E. Pilliod Jr. and E. G. Puckett, “Second-Order Accurate Volume-of-Fluid Algorithms for Tracking Material Interfaces,” Journal of Computational Physics, Vol. 199, No. 2, 2004, pp. 465-502. doi:10.1016/j.jcp.2003.12.023

[11] M. M. Francois and B. K. Swartz, “Interface Curvature via Volume Fractions, Heights, and Mean Values on Nonuniform Rectangular Grids,” Journal of Computational Physics, Vol. 229, No. 3, 2010, pp. 527-540. doi:10.1016/j.jcp.2009.10.022

[12] M. Sussman and E. G. Puckett, “A Coupled Level Set and Volume-of-Fluid Method for Computing 3D and Axisymmetric Incompressible Two-Phase Flows,” Journal of Computational Physics, Vol. 162, No. 2, 2000, pp. 301- 337. doi:10.1006/jcph.2000.6537

[13] M. Raessi, J. Mostaghimi and M. Bussmann, “Advecting Normal Vectors: A New Method for Calculating Interface Normals and Curvatures When Modeling Two-Phase Flows,” Journal of Computational Physics, Vol. 226, No. 1, 2007, pp. 774-794. doi:10.1016/j.jcp.2007.04.023

[14] D. J. E. Harvie and D. F. Fletcher, “A New Volume of Fluid Advection Algorithm: The Defined Donating Region Scheme,” International Journal for Numerical Me- thods in Fluids, Vol. 35, No. 2, 2001, pp. 151-172. doi:10.1002/1097-0363(20010130)35:2<151::AID-FLD87>3.0.CO;2-4

[15] M. Sun, “Volume Tracking of Subgrid Particles,” International Journal for Numerical Methods in Fluids, Vol. 66, No. 12, 2011, pp. 1530-1554. doi:10.1002/fld.2331

[16] D. Igra and M. Sun, “Shock-Water Column Interaction, from Initial Impact to Fragmentation Onset,” AIAA Journal, Vol. 48, No. 12, 2010, pp. 2763-2771. doi:10.2514/1.44901

[17] D. J. E. Harvie and D. F. Fletcher, “A New Volume of Fluid Advection Algorithm: The Stream Scheme,” Journal of Computational Physics, Vol. 162, No. 1, 2000, pp. 1-32. doi:10.1006/jcph.2000.6510

[1] C. W. Hirt and B. D. Nichols, “Volume of Fluid (VOF) Method for the Dynamics of Free Boundaries,” Journal of Computational Physics, Vol. 39, No. 1, 1981, pp. 201-225. doi:10.1016/0021-9991(81)90145-5

[2] W. J. Rider and D. B. Kothe, “Reconstructing Volume Tracking,” Journal of Computational Physics, Vol. 141, No. 2, 1998, pp. 112-152. doi:10.1006/jcph.1998.5906

[3] R. Scardovelli and S. Zaleski, “Direct Numerical Simulation of Free-Surface and Interfacial Flow,” Annual Review of Fluid Mechanics, Vol. 31, 1999, pp. 567-603. doi:10.1146/annurev.fluid.31.1.567

[4] W. F. Noh and P. Woodward, “SLIC (Simple Line Interface Calculation),” In: A. I. van der Vooren and P. J. Zandbergen, Eds., Lecture Notes in Physics, Springer, New York, 1976, p. 330.

[5] M. Rudman, “Volume-Tracking Methods for Interfacial Flow Calculations,” International Journal for Numerical Methods in Fluids, Vol. 24, No. 7, 1997, pp. 671-691. doi:10.1002/(SICI)1097-0363(19970415)24:7<671::AID-FLD508>3.0.CO;2-9

[6] D. L. Youngs, “An Interface Tracking Method for a 3D Eulerian Hydrodynamics Code,” Technical Report 44/92/ 35, AWRE, 1984.

[7] E. G. Puckett, “A Volume of Fluid Interface Tracking Algorithm with Applications to Computing Shock Wave Rarefraction,” Proceedings of the 4th International Symposium on Computational Fluid Dynamics, 1991.

[8] R. Scardovelli and S. Zaleski, “Interface Reconstruction with Least-Square Fit and Split Eulerian-Lagrangian Advection,” International Journal for Numerical Methods in Fluids, Vol. 41, No. 3, 2003, pp. 251-274. doi:10.1002/fld.431

[9] J. López, J. Hernández, P. Gómez and F. Faura, “A Volume of Fluid Method Based on Multidimensional Advection and Spline Interface Reconstruction,” Journal of Computational Physics, Vol. 195, No. 2, 2004, pp. 718- 742. doi:10.1016/j.jcp.2003.10.030

[10] J. E. Pilliod Jr. and E. G. Puckett, “Second-Order Accurate Volume-of-Fluid Algorithms for Tracking Material Interfaces,” Journal of Computational Physics, Vol. 199, No. 2, 2004, pp. 465-502. doi:10.1016/j.jcp.2003.12.023

[11] M. M. Francois and B. K. Swartz, “Interface Curvature via Volume Fractions, Heights, and Mean Values on Nonuniform Rectangular Grids,” Journal of Computational Physics, Vol. 229, No. 3, 2010, pp. 527-540. doi:10.1016/j.jcp.2009.10.022

[12] M. Sussman and E. G. Puckett, “A Coupled Level Set and Volume-of-Fluid Method for Computing 3D and Axisymmetric Incompressible Two-Phase Flows,” Journal of Computational Physics, Vol. 162, No. 2, 2000, pp. 301- 337. doi:10.1006/jcph.2000.6537

[13] M. Raessi, J. Mostaghimi and M. Bussmann, “Advecting Normal Vectors: A New Method for Calculating Interface Normals and Curvatures When Modeling Two-Phase Flows,” Journal of Computational Physics, Vol. 226, No. 1, 2007, pp. 774-794. doi:10.1016/j.jcp.2007.04.023

[14] D. J. E. Harvie and D. F. Fletcher, “A New Volume of Fluid Advection Algorithm: The Defined Donating Region Scheme,” International Journal for Numerical Me- thods in Fluids, Vol. 35, No. 2, 2001, pp. 151-172. doi:10.1002/1097-0363(20010130)35:2<151::AID-FLD87>3.0.CO;2-4

[15] M. Sun, “Volume Tracking of Subgrid Particles,” International Journal for Numerical Methods in Fluids, Vol. 66, No. 12, 2011, pp. 1530-1554. doi:10.1002/fld.2331

[16] D. Igra and M. Sun, “Shock-Water Column Interaction, from Initial Impact to Fragmentation Onset,” AIAA Journal, Vol. 48, No. 12, 2010, pp. 2763-2771. doi:10.2514/1.44901

[17] D. J. E. Harvie and D. F. Fletcher, “A New Volume of Fluid Advection Algorithm: The Stream Scheme,” Journal of Computational Physics, Vol. 162, No. 1, 2000, pp. 1-32. doi:10.1006/jcph.2000.6510