OPEN ACCESS
This paper presents Direct Numerical Simulation (DNS) of the falling motion of single and multiple deformable drops in a vertical channel. A systematic study of the wall effect on the motion of single drop is performed for Eötvös number (0.5£Eo£5), Morton number (10−3£M£10-8), and confinement ratio CR = 2. Second, the gravity-driven motion of multiple drops and their interactions are studied in a periodic vertical channel for CR = 4. These simulations are performed using a multiple marker level-set methodology, integrated in a finite-volume framework on a collocated unstructured grid. Each droplet is described by a level-set function, which allows capturing multiple interfaces in the same control volume, avoiding the numerical merging of the droplets. Numerical algorithms for fluid motion and interface capturing have been developed in the context of the finite-volume and level-set methodology, surface tension is modeled by means of the continuous surface force approach, and the pressure-velocity coupling is solved using a fractional-step projection method. DNS of single drop shows that they migrate to the symmetry axis of the channel when the Reynolds number is low, following a monotonic approach or damped oscillations according to the dimensionless parameters. If Eötvös number increases, stronger oscillations around the symmetry axis are observed. Simulations of multiple drops show that the collision of two drops follows the drafting-kissing tumbling (DKT) phenomenon. Deformable drops do not collide with the wall, whereas DKT phenomenon in the droplet swarm leads to the formation of groups which move through the center of the channel.
conservative level-set method, DNS, drops, interface capturing, multiphase flow, multiple marker, surface tension, vertical channel
[1] Tryggvason, G., Dabiri, S., Abouhasanzadeh, B. & Lu, J., Multiscale considerations in direct numerical simulations of multiphase flows. Physics of Fluids, 25, 031302, 2013. https://doi.org/10.1063/1.4793543
[2] Tryggvason, G., Bunner, B., Esmaeeli, A., Juric, D., Al-Rawahi, N., Tauber, W., Han, J., Nas, S. & Jan, Y-J., A front-tracking method for the computations of multiphase flow. Journal of Computational Physics, 169, pp. 708–759, 2001. https://doi.org/10.1006/jcph.2001.6726
[3] Fakhari, A. & Rahimian, M.H., Simulation of falling droplet by the lattice Botzmann method. Communications in Nonlinear Science and Numerical Simulation, 14, pp. 3046–3055, 2009.https://doi.org/10.1016/j.cnsns.2008.10.017
[4] Balcázar, N., Lehmkhul, O., Rigola, J. & Oliva, A., A multiple marker level-set method for simulation of deformable fluid particles. International Journal of Multiphase Flow, 74, pp. 125–142, 2015.https://doi.org/10.1016/j.ijmultiphaseflow.2015.04.009
[5] Balcázar, N., Jofre, L., Lehmkhul, O., Castro, J. & Rigola, J., A finite-volume/levelset method for simulating two-phase flows on unstructured grids. International Journal of Multiphase Flow, 64, pp. 55–72, 2014. https://doi.org/10.1016/j.ijmultiphaseflow.2014.04.008
[6] Balcázar, N., Rigola, J., Castro, J. & Oliva, A., A level-set model for thermocapillary motion of deformable fluid particles. International Journal of Heat and Fluid Flow, 62, pp. 324–343, 2016.https://doi.org/10.1016/j.ijheatfluidflow.2016.09.015
[7] Balcázar, N., Lemhkuhl, O., Jofre, L. & Oliva, A., Level-set simulations of buoyancydriven motion of single and multiple bubbles. International Journal of Heat and Fluid Flow, 56, pp. 91–107, 2015.https://doi.org/10.1016/j.ijheatfluidflow.2015.07.004
[8] Olsson, E. & Kreiss, G., A conservative level set method for two phase flow. Journal of Computational Physics, 210, pp. 225–246, 2005. https://doi.org/10.1016/j.jcp.2005.04.007
[9] Sussman, M., Smereka, P. & Osher, S. A level set approach for computing solutions to incompressible two-phase flow. Journal of Computational Physics, 144, pp. 146–159, 1994.https://doi.org/10.1006/jcph.1994.1155
[10] Hirt, C. & Nichols, B., Volume of fluid (VOF) method for the dynamics of free boundary. Journal of Computational Physics, 39, pp. 201–225, 1981. https://doi.org/10.1016/0021-9991(81)90145-5
[11] Balcázar, N., Lehmkhul, O., Jofre, L., Rigola, J. & Oliva, A., A coupled volume-offluid/levelset method for simulation of two-phase flows on unstructured meshes. Computers and Fluids, 124, pp. 12–29, 2016. https://doi.org/10.1016/j.compfluid.2015.10.005
[12] Coyajee, E. & Boersma, J.B, Numerical simulation of drop impact on a liquid-liquid interface with a multiple marker front-capturing method. Journal of Computational Physics, 228, pp. 4444–4467, 2009.https://doi.org/10.1016/j.jcp.2009.03.014
[13] Sussman, M. & Puckett, E.G., A coupled level set and volume-of-fluid method for computing 3D and axisymmetric incompressible two-phase flows. Journal of Computational Physics, 162, pp. 301–337, 2000.https://doi.org/10.1006/jcph.2000.6537
[14] Clift, R., Grace, J.R. & Weber, M.E., Bubbles, drops and particles. Academic Press, New York, 1978.
[15] Moore, D.W., The rise of a gas bubble in a viscous fluid. Journal of Fluid Mechanics,6, pp. 113–130, 1959.https://doi.org/10.1017/s0022112059000520
[16] Taylor, T.D. & Acrivos, A., On the deformation and drag of falling viscous drop at low Reynolds number. Journal of Fluid Mechanics, 18, pp. 466–476, 1964. https://doi.org/10.1017/s0022112064000349
[17] Brignell, A.S., The deformation of a liquid drop at small Reynolds number. Quarterly Journal of Mechanics and Applied Mathematics, 26, pp. 99–107, 1973. https://doi.org/10.1093/qjmam/26.1.99
[18] Han, J. & Tryggvason, G., Secondary breakup of axisymmetric liquid drops I. Acceleration by a constant body force. Physics of Fluids, 11(12), pp. 3650–3667, 1999. https://doi.org/10.1063/1.870229
[19] Mortazavi, S. & Tryggvason, G., A numerical study of the motion of drops in Poiseuille flow. Part 1. Lateral migration of one drop. Journal of Fluid Mechanics, 41, pp. 325–350, 2000.https://doi.org/10.1017/s0022112099008204
[20] Amiri, M. & Mortazavi, S., Three-dimensional numerical simulation of sediment ing drops inside a vertical channel. International Journal of Multiphase Flow, 56, pp. 40–53, 2013.https://doi.org/10.1016/j.ijmultiphaseflow.2013.05.007
[21] Brackbill, J.U., Kothe, D.B. & Zemach, C., A continuum method for modeling surface tension. Journal of Computational Physics, 100, pp. 335–354, 1992. https://doi.org/10.1016/0021-9991(92)90240-y
[22] Gottlieb, S. & Shu, C.W., Total variation dimishing Runge-Kutta schemes. Mathematics of Computations, 67, pp. 73–85, 1998.https://doi.org/10.1090/s0025-5718-98-00913-2
[23] Chorin, A.J., Numerical solution of the Navier-Stokes equations. Mathematics of Computation, 22, pp. 745–762, 1968.https://doi.org/10.1090/s0025-5718-1968-0242392-2
[24] Mohamed-Kassim, Z. & Longmire, E.K., Drop impact on a liquid–liquid interface. Physics of Fluids, 15, pp. 3263–3273, 2003.https://doi.org/10.1063/1.1609993
[25] Feng, J., Hu, H.H. & Joseph, D.D., Direct simulation of initial value problems for the motion of solid bodies in a Newtonian fluid Part 1. Sedimentation. Journal of Fluid Mechanics, 261, pp. 95–134, 1994.https://doi.org/10.1017/s0022112094000285