An SPH modeling of bubble rising and coalescing in three dimensions. (English) Zbl 1423.76378
Summary: The numerical simulation of bubbly flows is challenging due to the unstable multiphase interfaces with large density ratios and viscous ratios. The multiphase Smoothed Particle Hydrodynamics (SPH) method is applied in this paper to simulate the phenomena of bubbles rising and coalescing in three dimensions. Firstly, the multiphase SPH model is introduced in detail, including the derivation of the discretized governing equations based on the principle of virtual work, the viscous force, the multiphase interface treatment, the time-stepping scheme, the boundary implementation, etc. Considering the expensive computational cost in three-dimensional (3-D) SPH simulations, the effects of the scale of the computational domain and the density ratio on the multiphase interface are numerically investigated in order to decrease the amount of calculation. Afterwards, several cases of single bubbles rising through viscous fluids are tested and the SPH results are validated by both the experimental data and other numerical results in the literature. Furthermore, the phenomena of bubbles coalescing in both vertical and horizontal directions are simulated and the results agree well with the experimental data. It is found that the background pressure in the equation of state is essential to keep the multiphase interface smooth and stable when the Bond number is relatively small. The fair agreements between the results of SPH and other reference results demonstrate that the present multiphase SPH model is robust and stable enough to accurately simulate the dynamic phenomena of rising bubbles in different conditions, which can give a reference for engineering applications.
References:
| [1] | Grenier, N.; Le Touze, D.; Colagrossi, A.; Antuono, M.; Colicchio, G., Viscous bubbly flows simulation with an interface SPH model, Ocean Eng., 69, 88-102 (2013) |
| [2] | Frising, T.; Noik, C.; Dalmazzone, C., The liquid/liquid sedimentation process: from droplet coalescence to technologically enhanced water/oil emulsion gravity separators: a review, J. Dispersion Sci. Technol., 27, 1035-1057 (2006) |
| [3] | Clift, R.; Grace, J. R.; Weber, M. E., Bubbles, Drops and Particles (1978), Academic Press |
| [4] | Annaland, M. V.; Deen, N. G.; Kuipers, J. A.M., Numerical simulation of gas bubbles behaviour using a three-dimensional volume of fluid method, Chem. Eng. Sci., 60, 2999-3011 (2005) |
| [5] | Sussman, M.; Smereka, P.; Osher, S., A level set approach for computing solutions to incompressible two-phase flow, J. Comput. Phys., 114, 146-159 (1994) · Zbl 0808.76077 |
| [6] | Croce, R.; Griebel, M.; Schweitzer, M. A., Numerical simulation of bubble and droplet deformation by a level set approach with surface tension in three dimensions, Internat. J. Numer. Methods Fluids, 62, 963-993 (2010) · Zbl 1423.76455 |
| [7] | Hua, J.; Stene, J. F.; Lin, P., Numerical simulation of 3D bubbles rising in viscous liquids using a front tracking method, J. Comput. Phys., 227, 3358-3382 (2008) · Zbl 1329.76262 |
| [8] | Annaland, M. V.; Dijkhuizen, W.; Deen, N. G.; Kuipers, J. A.M., Numerical simulation of behavior of gas bubbles using a 3-D front-tracking method, AIChE J., 52, 99-110 (2006) |
| [9] | Hua, J.; Lou, J., Numerical simulation of bubble rising in viscous liquid, J. Comput. Phys., 222, 769-795 (2007) · Zbl 1158.76404 |
| [10] | Yu, Z.; Fan, L. S., An interaction potential based lattice Boltzmann method with adaptive mesh refinement (AMR) for two-phase flow simulation, J. Comput. Phys., 228, 6456-6478 (2009) · Zbl 1261.76048 |
| [11] | Yu, Z.; Yang, H.; Fan, L.-S., Numerical simulation of bubble interactions using an adaptive lattice Boltzmann method, Chem. Eng. Sci., 66, 3441-3451 (2011) |
| [12] | Yu, Z.; Yang, H.; Fan, L. S., Numerical simulation of bubble interactions using an adaptive lattice Boltzmann method, Chem. Eng. Sci., 66, 3441-3451 (2011) |
| [13] | Amaya-Bower, L.; Lee, T., Single bubble rising dynamics for moderate Reynolds number using Lattice Boltzmann Method, Comput. & Fluids, 39, 1191-1207 (2010) · Zbl 1242.76264 |
| [14] | Liu, M.; Liu, G., Smoothed particle hydrodynamics (SPH): an overview and recent developments, Arch. Comput. Methods Eng., 17, 25-76 (2010) · Zbl 1348.76117 |
| [15] | Liu, G. R.; Liu, M., Smoothed Particle Hydrodynamics: A Meshfree Particle Method (2003), World Scientific · Zbl 1046.76001 |
| [16] | Gingold, R. A.; Monaghan, J. J., Smoothed particle hydrodynamics-theory and application to non-spherical stars, Mon. Not. R. Astron. Soc., 181, 375-389 (1977) · Zbl 0421.76032 |
| [17] | Lucy, L. B., A numerical approach to the testing of the fission hypothesis, Astron. J., 82, 1013-1024 (1977) |
| [18] | Monaghan, J. J., Simulating free surface flows with SPH, J. Comput. Phys., 110, 399-406 (1994) · Zbl 0794.76073 |
| [19] | Marrone, S.; Antuono, M.; Colagrossi, A.; Colicchio, G.; Le Touzé, D.; Graziani, G., \( \delta \)-SPH model for simulating violent impact flows, Comput. Methods Appl. Mech. Engrg., 200, 1526-1542 (2011) · Zbl 1228.76116 |
| [20] | Barcarolo, D. A., Improvement of the precision and the efficiency of the SPH method: theoretical and numerical study (2013), (Ph.D. thesis of Ecole Centrale de Nantes) |
| [21] | Colagrossi, A.; Landrini, M., Numerical simulation of interfacial flows by smoothed particle hydrodynamics, J. Comput. Phys., 191, 448-475 (2003) · Zbl 1028.76039 |
| [22] | Antuono, M.; Colagrossi, A.; Marrone, S.; Lugni, C., Propagation of gravity waves through an SPH scheme with numerical diffusive terms, Comput. Phys. Comm., 182, 866-877 (2011) · Zbl 1333.76054 |
| [23] | Bouscasse, B.; Colagrossi, A.; Marrone, S.; Antuono, M., Nonlinear water wave interaction with floating bodies in SPH, J. Fluids Struct., 42, 112-129 (2013) |
| [24] | Cao, X. Y.; Ming, F. R.; Zhang, A. M., Sloshing in a rectangular tank based on SPH simulation, Appl. Ocean Res., 47, 241-254 (2014) |
| [25] | Hu, X.; Adams, N., A multi-phase SPH method for macroscopic and mesoscopic flows, J. Comput. Phys., 213, 844-861 (2006) · Zbl 1136.76419 |
| [26] | Hu, X.; Adams, N., An incompressible multi-phase SPH method, J. Comput. Phys., 227, 264-278 (2007) · Zbl 1126.76045 |
| [27] | Grenier, N.; Antuono, M.; Colagrossi, A.; Le Touzé, D.; Alessandrini, B., An Hamiltonian interface SPH formulation for multi-fluid and free surface flows, J. Comput. Phys., 228, 8380-8393 (2009) · Zbl 1333.76056 |
| [28] | Monaghan, J. J.; Rafiee, A., A simple SPH algorithm for multi-fluid flow with high density ratios, Internat. J. Numer. Methods Fluids, 71, 537-561 (2013) · Zbl 1430.76400 |
| [29] | Bhaga, D.; Weber, M., Bubbles in viscous liquids: shapes, wakes and velocities, J. Fluid Mech., 105, 61-85 (1981) |
| [31] | Duineveld, P., Bouncing and coalescence of bubble pairs rising at high Reynolds number in pure water or aqueous surfactant solutions, (Fascination of Fluid Dynamics (1998), Springer), 409-439 · Zbl 0922.76016 |
| [32] | Das, A. K.; Das, P. K., Bubble evolution and necking at a submerged orifice for the complete range of orifice tilt, AIChE J., 59, 630-642 (2013) |
| [33] | Das, A. K.; Das, P. K., Bubble evolution through a submerged orifice using smoothed particle hydrodynamics: effect of different thermophysical properties, Ind. Eng. Chem. Res., 48, 8726-8735 (2009) |
| [34] | Das, A. K.; Das, P. K., Bubble evolution through submerged orifice using smoothed particle hydrodynamics: basic formulation and model validation, Chem. Eng. Sci., 64, 2281-2290 (2009) |
| [35] | Das, A. K.; Das, P. K., Incorporation of diffuse interface in smoothed particle hydrodynamics: Implementation of the scheme and case studies, Internat. J. Numer. Methods Fluids, 67, 671-699 (2011) · Zbl 1329.76320 |
| [36] | Szewc, K.; Pozorski, J.; Minier, J. P., Simulations of single bubbles rising through viscous liquids using smoothed particle hydrodynamics, Int. J. Multiph. Flow, 50, 98-105 (2013) |
| [38] | Adami, S.; Hu, X.; Adams, N., A new surface-tension formulation for multi-phase SPH using a reproducing divergence approximation, J. Comput. Phys., 229, 5011-5021 (2010) · Zbl 1346.76161 |
| [39] | Morris, J. P.; Fox, P. J.; Zhu, Y., Modeling low Reynolds number incompressible flows using SPH, J. Comput. Phys., 136, 214-226 (1997) · Zbl 0889.76066 |
| [40] | Brackbill, J.; Kothe, D. B.; Zemach, C., A continuum method for modeling surface tension, J. Comput. Phys., 100, 335-354 (1992) · Zbl 0775.76110 |
| [41] | Marrone, S.; Colagrossi, A.; Antuono, M.; Colicchio, G.; Graziani, G., An accurate SPH modeling of viscous flows around bodies at low and moderate Reynolds numbers, J. Comput. Phys., 245, 456-475 (2013) · Zbl 1349.76715 |
| [42] | Colagrossi, A.; Bouscasse, B.; Antuono, M.; Marrone, S., Particle packing algorithm for SPH schemes, Comput. Phys. Commun., 183, 1641-1653 (2012) · Zbl 1307.65140 |
| [43] | Molteni, D.; Colagrossi, A., A simple procedure to improve the pressure evaluation in hydrodynamic context using the SPH, Comput. Phys. Commun., 180, 861-872 (2009) · Zbl 1198.76108 |
| [44] | Hongbin, J.; Xin, D., On criterions for smoothed particle hydrodynamics kernels in stable field, J. Comput. Phys., 202, 699-709 (2005) · Zbl 1061.76067 |
| [45] | Colagrossi, A.; Antuono, M.; Le Touzé, D., Theoretical considerations on the free-surface role in the smoothed-particle-hydrodynamics model, Phys. Rev. E, 79, 056701 (2009) |
| [46] | Colagrossi, A.; Antuono, M.; Souto-Iglesias, A.; Le Touzé, D., Theoretical analysis and numerical verification of the consistency of viscous smoothed-particle-hydrodynamics formulations in simulating free-surface flows, Phys. Rev. E, 84, 026705 (2011) |
| [47] | Monaghan, J.; Gingold, R., Shock simulation by the particle method SPH, J. Comput. Phys., 52, 374-389 (1983) · Zbl 0572.76059 |
| [48] | Morris, J. P., Simulating surface tension with smoothed particle hydrodynamics, Internat. J. Numer. Methods Fluids, 33, 333-353 (2000) · Zbl 0985.76072 |
| [49] | Breinlinger, T.; Polfer, P.; Hashibon, A.; Kraft, T., Surface tension and wetting effects with smoothed particle hydrodynamics, J. Comput. Phys., 243, 14-27 (2013) · Zbl 1349.76666 |
| [50] | Zainali, A.; Tofighi, N.; Shadloo, M.; Yildiz, M., Numerical investigation of Newtonian and non-Newtonian multiphase flows using ISPH method, Comput. Methods Appl. Mech. Engrg., 254, 99-113 (2013) · Zbl 1297.76137 |
| [51] | Adami, S.; Hu, X.; Adams, N., A generalized wall boundary condition for smoothed particle hydrodynamics, J. Comput. Phys., 231, 21, 7057-7075 (2012) |
| [52] | Chen, Z.; Zong, Z.; Liu, M.; Zou, L.; Li, H.; Shu, C., An SPH model for multiphase flows with complex interfaces and large density differences, J. Comput. Phys., 283, 169-188 (2015) · Zbl 1351.76231 |
| [53] | Hysing, S.; Turek, S.; Kuzmin, D.; Parolini, N.; Burman, E.; Ganesan, S.; Tobiska, L., Quantitative benchmark computations of two-dimensional bubble dynamics, Internat. J. Numer. Methods Fluids, 60, 1259-1288 (2009) · Zbl 1273.76276 |
| [54] | Grace, J., Shapes and velocities of bubbles rising in infinite liquids, Trans. Inst. Chem. Eng., 51, 116-120 (1973) |
| [55] | Chen, R.; Tian, W.; Su, G.; Qiu, S.; Ishiwatari, Y.; Oka, Y., Numerical investigation on coalescence of bubble pairs rising in a stagnant liquid, Chem. Eng. Sci., 66, 5055-5063 (2011) |
| [56] | Marrone, S., Enhanced SPH modeling of free-surface flows with large deformations (2012), Università di Roma Sapienza, (Ph.D. thesis) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
