A 3D VOF model in cylindrical coordinates

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1 A 3D VOF model in cylindrical coordinates Marmar Mehrabadi and Markus Bussmann Department of Mechanical and Industrial Engineering, University of Toronto Recently, volume of fluid (VOF) methods have improved greatly, and they have been widely used to model multiphase phenomena associated with droplets, bubbles and sprays. Nearly all such methods, however, have been developed in Cartesian coordinates. Yet, there are many interesting multiphase flow problems that must be modeled in three-dimensional cylindrical coordinates; examples include the instability of liquid films inside or outside of a rotating cylinder, and the instability of immiscible interfacial flows between rotating concentric cylinders. There are other problems that can be solved on a Cartesian mesh, but for which a cylindrical mesh is a more suitable choice which leads to better accuracy at the same resolution; for example, simulating normal droplet impact onto a solid surface on a Cartesian mesh can produce numerical errors such as the asymmetric spread of a droplet that should remain axisymmetric[1]. To our knowledge, the only VOF model that has been developed in cylindrical coordinates is that of Chen et. al.[2] in which the volume fractions are advected using an algebraic method, i.e. an upwind/downwind scheme is used to calculate flux volumes. Such a VOF model leads to interface diffusion, and to unphysical volume fractions that need to be corrected. Our objective was to develop an accurate two-phase flow model in cylindrical geometry, with a focus on the accurate implementation of three aspects : interface tracking, the modeling of flows with high density ratios, and the modeling of surface tension. Here, we will briefly describe the methodology while focusing on the three above aspects. The incompressible Navier Stokes equations are discretized according to a colocated finite volume scheme. An explicit projection method is employed for time discretization. A one-phase approximation of the Navier- Stokes equations is used to model the two phases. Volume fractions f, are advected using a piecewise linear interface calculation (PLIC) method. In the PLIC method in three-dimensional domains, the interface is approximated by planes in interfacial cells. To define a plane, which can be written in the form of n.x = ρ, the plane normal n, and the plane constant, ρ, need to be determined. The normal is calculated from a height function (HF) which we describe when

2 we discuss the curvature calculation. The plane constant ρ is calculated so that the truncated volume, V truncated, is equal to the volume of the fluid in each cell, V fluid. Instead of solving for ρ directly, an iterative method is used to to find ρ so that V truncated (ρ) = V fluid. A geometrical algorithm has been developed to calculate the V truncated for a given plane of n and ρ, based on the algorithm first developed by Rider and Kothe [3] for determining the linear interface segments in a two dimensional Cartesian mesh. The reconstructed interface together with the velocity field are then used to estimate the volume fractions in the next time step. The volume fraction advection equation is solved using an operator split method. To model flows with high density ratios accurately, the mass (volume fraction) advection and momentum advection should be tightly coupled [4, 5]. Therefore, the Navier-Stokes equations are solved in a fully conservative form, where mass fluxes are calculated from the volume fraction advection algorithm. To model surface tension accurately two aspects are important: the surface tension model and the accuracy of curvature calculation. Here, we focus on the curvature calculation. Recent studies on the various methods of curvature calculation using volume fractions, have shown that Height Function (HF) method and the PROST (parabolic reconstruction of surface tension) method [6] are superior to other methods in terms of the magnitude of error and the order of convergence rate [7, 8]. The PROST method, however, is computationally more expensive, and the implementation of the method is not straightforward [8]. Therefore, the HF method was implemented to calculate curvatures. For a surface of the form R = h(θ, z), curvature, κ, can be derived as κ = 1 R 2 θ + R 2 + R 2 zr 23 (R2 θ RR zz + R 3 R zz 2RR θ R θz R z +RR 2 z R θθ R 2 R 2 z 2R2 θ + RR θθ R 2 ). R in the cell (i,j,k) is estimated by adding the volumes of one of the fluids in a column of cells in the R direction. The derivatives of R are approximated using second order finite difference schemes with R values calculated in a S D 3 3 stencil around cell (i,j,k), where S D is the stencil depth which is determined so that all of the cells in the first and last row are inside or outside of the heavy fluid. Curvature for surfaces of the forms Θ = h(r, z) and Z = h(r, θ) can be calculated similarly. Selection of the direction of the stencil (between R = h(θ, z), Θ = h(r, z), or Z = h(r, θ)) is based on the stencil depth, SD: the direction with the minimum SD leads to the most accurate estimation of curvature. We have completed the development of the model, and are beginning to validate our model. Here, we present the results for one such test. A simple but stringent test case for validating a surface tension calculation is the static drop test, where the pressure jump across the interface, and the magnitude of the induced currents are indicative of the accuracy of the surface tension implementation. The test was performed to compare the curvature

3 calculation using two methods: the HF method and a method based on calculating κ = ˆn from a convolved volume fraction field (CF). A continuum surface force model was employed for both cases. The test parameters correspond to the case tested by Gerlach et. al. [8] at high density ratios. Figure 1 shows the computational mesh. Figure 2 shows the pressure and velocity fields after one time step ( t = 1 5 ). The results show that HF method leads to a much smoother pressure field, and much smaller parasitic velocities. We propose to present the model and the results of a series validation tests, with an emphasis on those aspects that are particular to the cylindrical coordinate system. References [1] S. B. Johnson and J. P. Delplanque, Analyzing the results of droplet impact simulation by volume of fluid methods: Physics or numerical errors?, in ILASS Americas 18th Annual Conference on Liquid Atomization and Spray Systems, Irvine, CA, May 25. [2] L. Chen, S. V. Garimella, J. A. Reizes, and E. Leonardie, The development of a bubble rising in a viscous liquid, Journal of Fluid Mechanics, vol. 387, p. 61, [3] W. J. Rider and D. B. Kothe, Reconstructing volume tracking, Journal of Computational Physics, vol. 141, no. 2, p. 112, [4] M. Rudman, A volume-tracking method for incompressible multifluid flows with large density variations, International Journal for Numerical Methods in Fluids, vol. 28, no. 2, p. 357, [5] M. Bussmann, D. B. Kothe, and J. M. Sicilian, Modeling high density ratio incompressible interfacial flows, in Proceedings of the Sixth ASME/JSME Joint Fluids Engineering Conference, July 14-18, Montreal, Canada, 22. [6] Y. Renardy and M. Renardy, Prost: A parabolic reconstruction of surface tension for the volume-of-fluid method, Journal of Computational Physics, vol. 183, pp , DEC [7] S. J. Cummins, M. M. Francois, and D. B. Kothe, Estimating curvature from volume fractions, Computers and Structures, vol. 83, no. 6-7, p. 425, 25. [8] D. Gerlach, G. Tomar, G. Biswas, and F. Durst, Comparison of volume-of-fluid methods for surface tension-dominant two-phase flows, International Journal of Heat and Mass Transfer, vol. 49, pp , FEB 26.

4 Figure 1: The mesh used to test the static drop. The number of cells per radius is 16.

5 (a) (b) Figure 2: Comparison of pressure contours and velocity vectors for an static drop test using (a) HF method, and (b) CF method, after one time step ( t = 1 5 ). The radius of the drop is 2, the surface tension coefficient is 23.61, and the density ratio is 1 3. The maximum velocity is when using CF method and when using HF method.

Drop Impact Simulation with a Velocity-Dependent Contact Angle

ILASS Americas 19th Annual Conference on Liquid Atomization and Spray Systems, Toronto, Canada, May 2006 Drop Impact Simulation with a Velocity-Dependent Contact Angle S. Afkhami and M. Bussmann Department