Discretization Techniques of Height Function Method for Greater Increased Accuracy of Mass Conservation



A height function method has been used to solve the shape of free surfaces in incompressible viscous flows for hydrodynamics. Three proposed discretization techniques for the height function method are developed with particular attention to the law of mass conservation. The concept of the proposed techniques is to place a control volume on the most appropriate location in any staggered grid system. First, the proposed techniques and the conventional technique are verified with a simple problem whose exact solution is known. Then, all numerical techniques are examined with a more complicated problem to investigate their accuracy. The simulated results of the proposed techniques are compared to those of conventional technique. Finally, it is concluded that (1) the proposed techniques will give better results than the conventional technique if the grid resolution is sufficiently fine, (2) the first proposed technique gives poorer results than the other proposed techniques, and (3) the second proposed technique gives better results than the third proposed technique, but the third proposed technique is easier to apply due to its explicit form of the equation.



Finite volume method, free surface, height function method, hydrodynamic, two-phase flow

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R Scardovelli and S Zaleski. Direct numerical simulation of free-surface and interfacial flow. Annu. Rev. Fluid Mech. 1999; 31, 567-603.

H Lan and Z Zhang. Comparative study of the free-surface boundary condition in two-dimensional finite-difference elastic wave field simulation. J. Geophys. Eng. 2011; 8, 275-86.

FH Harlow and JE Welch. Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface. Phys. Fluid. 1965; 8, 2182-9.

J Lin, Y Kamotani and S Ostrach. An experimental study of free surface deformation in oscillatory thermocapillary flow. Acta Astronaut. 1995; 35, 525-36.

T Hazuku, T Takamasa and K Okamoto. Simultaneous measuring system for free surface and liquid velocity distributions using PIV and LFD. Exp. Therm. Fluid. Sci. 2003; 27, 677-87.

JM Campin, A Adcroft, C Hill and J Marshall. Conservation of properties in a free-surface model. Ocean. Model. 2004; 6, 221-44.

S Smolentsev and R Miraghaie. Study of a free surface in open-channel water flows in the regime from “Weak” to “Strong” turbulence. Int. J. Multiphas. Flow 2005; 31, 921-39.

L Lu, Y Li and B Teng. Numerical simulation of turbulent free surface flow over obstruction. J. Hydrodyn. 2008; 20, 414-23.

F Murzyn and H Chanson. Free-surface fluctuations in hydraulic jumps: Experimental observations. Exp. Therm. Fluid. Sci. 2009; 33, 1055-64.

SH Sadathosseini, SM Mousaviraad, B Firoozabadi and G Ahmadi. Numerical simulation of free-surface waves and wave induced separation. Sci. Iran. 2008; 15, 323-31.

S Mayer, A Garapon and LS Sorensen. A fractional step method for unsteady free-surface flow with applications to non-linear wave dynamics. Int. J. Numer. Meth. Fluid. 1998; 28, 293-315.

E Aulisa, S Manservisi and R Scardovelli. A surface marker algorithm coupled to an area-preserving marker redistribution method for three-dimensional interface tracking. J. Comput. Phys. 2004; 197, 555-84.

PE Raad, S Chen and DB Johnson. The introduction of the micro cells to treat pressure in free surface fluid flow problems. J. Fluid. Eng. 1995; 117, 683-90.

BD Nichols and CW Hirt. Calculating three-dimensional free surface flows in the vicinity of submerged and exposed structures. J. Comput. Phys. 1971; 12, 234-46.

S McKee, MF Tome, VG Ferreira, JA Cuminato, A Castelo, FS Sousa and N Mangiavacchi. The MAC method. Comput. Fluid. 2008; 37, 907-30.

CW Hirt and BD Nichols. Volume of fluid (VOF) method for the dynamics of free boundaries. J. Comput. Phys. 1981; 39, 201-25.

P Liovic, M Rudman and JL Liow. Numerical modelling of free surface flows in metallurgical vessels. In: Proceedings of the 2nd International Conference on CFD in the Minerals and Process Industries CSIRO, Melbourne, Australia, 1999, p. 255-60.

SJ Cummins, MM Francois and DB Kothe. Estimating curvature from volume fractions. Comput. Struct. 2005; 83, 425-34.

S Afkhami and M Bussmann. Height function-based contact angles for VOF simulations of contact line phenomena. In: Proceedings of the 5th International Conference on Scientific Computing and Applications, Banff Alberta, Canada, 2006.

S Afkhami and M Bussmann. Height functions for applying contact angles to 2D VOF simulations. Int. J. Numer. Meth. Fluid. 2008; 57, 453-72.

PA Ferdowsi and M Bussmann. Second-order accurate normals from height functions. J. Comput. Phys. 2008; 227, 9293-302.

J López, C Zanzi, P Gómez, R Zamora, F Faura and J Hernández. An improved height function technique for computing interface curvature from volume fractions. Comput. Meth. Appl. Mech. Eng. 2009; 198, 2555-64.

JH Ferziger and M Perić. Computational Methods for Fluid Dynamics. 3rd ed. Springer-Verlag Berlin Heidelberg New York, Germany, 2002, p. 381-2.

S Afkhami and M Bussmann. Height Functions for applying contact angles to 3D VOF simulations. Int. J. Numer. Meth. Fluid. 2009; 61, 827-47.


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