Imposing the free-slip condition with a continuous forcing immersed boundary method. (English) Zbl 1351.76111
Summary: The numerical simulation of spherical and ellipsoidal bubbles in purified fluids requires the imposition of the free-slip boundary condition at the bubble surface. This paper describes a numerical method for the implementation of free-slip boundary conditions in the context of immersed boundary methods. In contrast to other numerical approaches for multiphase flows, the realization is not straightforward. The reason is that the immersed boundary method treats the liquid as well as the gas phase as a field of constant density and viscosity with a fictitious fluid inside the bubble. The motion of the disperse phase is computed explicitly by solving the momentum balance for each of its elements and is coupled to the continuous phase via additional source terms in the Navier-Stokes equations. The paper starts with illustrating that an ad hoc method is unsuccessful. On this basis, a new method is proposed employing appropriate direct forcing at the bubble surface. A central finding is that with common ratios between the step size of the grid and the bubble diameter, curvature terms need to be accounted for to obtain satisfactory results. The new method is first developed for spherical objects and then extended to generally curved interfaces. This is done by introducing a local coordinate system which approximates the surface in the vicinity of a Lagrangian marker with the help of the two principal curvatures of the surface at this point. The numerical scheme is then validated for spherical and ellipsoidal objects with or without prescribed constant angular velocity. It is shown that the proposed method achieves similar convergence behavior as the method for no-slip boundaries. The results are compared to analytical solutions for creeping flow around a sphere and to numerical reference data obtained on a body-fitted grid. The numerical tests confirm the excellent performance of the proposed method.
MSC:
| 76M12 | Finite volume methods applied to problems in fluid mechanics |
| 65M08 | Finite volume methods for initial value and initial-boundary value problems involving PDEs |
| 76T10 | Liquid-gas two-phase flows, bubbly flows |
References:
| [1] | Aland, S.; Schwarz, S.; Fröhlich, J.; Voigt, A., Modeling and numerical approximations for bubbles in liquid metal, Eur. Phys. J. Spec. Top., 220, 185-194 (2013) |
| [2] | Allen, M. P.; Tildesley, D. J., Computer Simulation of Liquids (1989), Oxford University Press · Zbl 0703.68099 |
| [3] | Alves, S. S.; Orvalho, S. P.; Vasconcelos, J. M.T., Effect of bubble contamination on rise velocity and mass transfer, Chem. Eng. Sci., 60, 1-9 (2005) |
| [4] | Anderson, E.; Bai, Z.; Bischof, C.; Blackford, S.; Demmel, J.; Dongarra, J.; Du Croz, J.; Greenbaum, A.; Hammarling, S.; McKenney, A.; Sorensen, D., LAPACK Users’ Guide (1987), Society for Industrial and Applied Mathematics · Zbl 0934.65030 |
| [5] | Balaras, E., Modeling complex boundaries using an external force field on fixed Cartesian grids in large-eddy simulations, Comput. Fluids, 33, 375-404 (2004) · Zbl 1088.76018 |
| [6] | Borazjani, I.; Ge, L.; Sotiropoulos, F., Curvilinear immersed boundary method for simulating fluid structure interaction with complex 3D rigid bodies, J. Comput. Phys., 227, 7587-7620 (2008) · Zbl 1213.76129 |
| [7] | Chan-Braun, C.; García-Villalba, M.; Uhlmann, M., Force and torque acting on particles in a transitionally rough open-channel flow, J. Fluid Mech., 684, 441 (2011) · Zbl 1241.76393 |
| [8] | Clift, R.; Grace, J. R.; Weber, M. E., Bubbles, Drops, and Particles (1978), Dover Publications |
| [9] | Crowe, C., Multiphase Flow Handbook (2006), CRC Press · Zbl 1098.76001 |
| [10] | Cuenot, B.; Magnaudet, J.; Spennato, B., The effects of slightly soluble surfactants on the flow around a spherical bubble, J. Fluid Mech., 339, 25-35 (1997) · Zbl 0892.76011 |
| [11] | Diebel, J., Representing attitude: Euler angles, unit quaternions, and rotation vectors, Matrix, 58, 15-16 (2006) |
| [12] | Duineveld, P. C., The rise velocity and shape of bubbles in pure water at high Reynolds number, J. Fluid Mech., 292, 325-332 (1995) |
| [13] | Bel Fdhila, R.; Duineveld, P. C., The effect of surfactant on the rise of a spherical bubble at high Reynolds and Peclet numbers, Phys. Fluids, 8, 310 (1996) · Zbl 1027.76673 |
| [14] | Ferzinger, J. H.; Peric, M., Computational Methods for Fluid Dynamics (2002), Springer-Verlag · Zbl 0998.76001 |
| [15] | Francois, M.; Uzgoren, E.; Jackson, J.; Shyy, W., Multigrid computations with the immersed boundary technique for multiphase flows, Int. J. Numer. Methods Heat Fluid Flow, 14, 98-115 (2004) · Zbl 1064.76081 |
| [16] | Fröhlich, J.; Schwarz, S.; Heitkam, S.; Santarelli, C.; Zhang, C.; Vogt, T.; Boden, S.; Andruszkiewicz, A.; Eckert, K.; Odenbach, S., Influence of magnetic fields on the behavior of bubbles in liquid metals, Eur. Phys. J. Spec. Top., 220, 167-183 (2013) |
| [17] | Hadamard, J. S., Mouvement permanent lent d’une sphere liquide et visqueuse dans un liquide visqueux, C. R. Acad. Sci., 152, 1735 (1911) · JFM 42.0813.01 |
| [18] | Happel, J.; Brenner, H., Low Reynolds Number Hydrodynamics (1983), Springer: Springer Netherlands |
| [19] | Harlow, F. H.; Welch, J. E., Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface, Phys. Fluids, 8, 2182-2189 (1965) · Zbl 1180.76043 |
| [20] | Heitkam, S.; Drenckhan, W.; Fröhlich, J., Packing spheres tightly: influence of mechanical stability on close-packed sphere structures, Phys. Rev. Lett., 108, 148302 (2012) |
| [21] | 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 |
| [22] | Kajishima, T.; Takiguchi, S., Interaction between particle clusters and particle-induced turbulence, Int. J. Heat Fluid Flow, 23, 639-646 (2002) |
| [23] | Kajishima, T.; Takiguchi, S.; Hamasaki, H.; Miyake, Y., Turbulence structure of particle-laden flow in a vertical plane channel due to vortex shedding, JSME Int. J. Ser. B, 44, 526-535 (2001) |
| [24] | Kempe, T., A numerical method for interface-resolving simulations of particle-laden flows with collisions (2013), Institute of Fluid Mechanics, TU Dresden, TUDpress Verlag der Wissenschaften GmbH, PhD thesis |
| [25] | Kempe, T.; Fröhlich, J., Collision modelling for the interface-resolved simulation of spherical particles in viscous fluids, J. Fluid Mech., 709, 445-489 (2012) · Zbl 1275.76206 |
| [26] | Kempe, T.; Fröhlich, J., An improved immersed boundary method with direct forcing for the simulation of particle laden flows, J. Comput. Phys., 231, 3663-3684 (2012) · Zbl 1402.76143 |
| [27] | Kempe, T.; Vowinckel, B.; Fröhlich, J., On the relevance of collision modeling for interface-resolving simulations of sediment transport in open channel flow, Int. J. Multiph. Flow, 58, 214-235 (2014) |
| [28] | Kidanemariam, A.; Chan-Braun, C.; Doychev, T.; Uhlmann, M., Direct numerical simulation of horizontal open channel flow with finite-size, heavy particles at low solid volume fraction, New J. Phys., 15, 025031 (2013) |
| [29] | Kim, D.; Choi, H., Immersed boundary method for flow around an arbitrarily moving body, J. Comput. Phys., 212, 662-680 (2006) · Zbl 1161.76520 |
| [30] | Kim, J.; Kim, D.; Choi, H., An immersed-boundary finite-volume method for simulations of flow in complex geometries, J. Comput. Phys., 171, 132-150 (2001) · Zbl 1057.76039 |
| [31] | Kim, S.; Karrila, S. J., Microhydrodynamics (2005), Dover Publications |
| [32] | Kühnel, W.; Hunt, B., Differential Geometry: Curves-Surfaces-Manifolds (2005), American Mathematical Society · Zbl 1082.53001 |
| [33] | Lamb, H., Hydrodynamics (1993), Dover Publications · Zbl 0828.01012 |
| [34] | Li, Z.; Ito, K., The Immersed Interface Method: Numerical Solutions of PDEs Involving Interfaces and Irregular Domains (2006), Society for Industrial Mathematics · Zbl 1122.65096 |
| [35] | Loth, E., Numerical approaches for motion of dispersed particles, droplets and bubbles, Prog. Energy Combust. Sci., 26, 3, 161-223 (2000) |
| [36] | Magnaudet, J.; Eames, I., The motion of high-Reynolds-number bubbles in inhomogeneous flows, Annu. Rev. Fluid Mech., 32, 659-708 (2000) · Zbl 0989.76082 |
| [37] | Majumdar, S.; Iaccarino, G.; Durbin, P., RANS solvers with adaptive structured boundary non-conforming grids, 353-366 (2001), Center for Turbulence Research, Stanford University, Annual Research Briefs |
| [38] | Malysa, K.; Krasowska, M.; Krzan, M., Influence of surface active substances on bubble motion and collision with various interfaces, Adv. Colloid Interface Sci., 114, 205-225 (2005) |
| [39] | McLaughlin, J. B., Numerical simulation of bubble motion in water, J. Colloid Interface Sci., 184, 614-625 (1996) |
| [40] | Meier, M.; Yadigaroglu, G.; Smith, B. L., A novel technique for including surface tension in PLIC-VOF methods, Eur. J. Mech. B Fluids, 21, 61-73 (2002) · Zbl 1064.76084 |
| [41] | Michaelides, E. E., Particles, Bubbles & Drops: Their Motion, Heat And Mass Transfer (2006), World Scientific Publishing |
| [42] | Mittal, R.; Dong, H.; Bozkurttas, M.; Najjar, F. M.; Vargas, A.; von Loebbecke, A., A versatile sharp interface immersed boundary method for incompressible flows with complex boundaries, J. Comput. Phys., 227, 4825-4852 (2008) · Zbl 1388.76263 |
| [43] | Mittal, R.; Iaccarino, G., Immersed boundary methods, Annu. Rev. Fluid Mech., 37, 239-261 (2005) · Zbl 1117.76049 |
| [44] | Mohd-Yusof, J., Combined immersed boundary/B-spline method for simulations of flows in complex geometries, 317-327 (1997), Center for Turbulence Research, NASA Ames/Stanford University, Annual Research Briefs |
| [45] | Osher, S.; Fedkiw, R. P., Level set methods: an overview and some recent results, J. Comput. Phys., 169, 463-502 (2001) · Zbl 0988.65093 |
| [46] | Peskin, Ch. S., Numerical analysis of blood flow in the heart, J. Comput. Phys., 25, 220-252 (1977) · Zbl 0403.76100 |
| [47] | Peskin, Ch. S., The immersed boundary method, Acta Numer., 11, 479-517 (2003) · Zbl 1123.74309 |
| [48] | Peters, F.; Els, C., An experimental study on slow and fast bubbles in tap water, Chem. Eng. Sci., 82, 194-199 (2012) |
| [49] | Pöschel, T.; Schwager, T., Computational Granular Dynamics: Models and Algorithms (2005), Springer |
| [50] | Prosperetti, A.; Tryggvason, G., Computational Methods for Multiphase Flow (2011), Cambridge University Press |
| [51] | Redfield, J. A.; Houghton, G., Mass transfer and drag coefficients for single bubbles at Reynolds numbers of 0.02-5000, Chem. Eng. Sci., 20, 131-139 (1965) |
| [52] | Roghair, I.; Lau, Y. M.; Deen, N. G.; Slagter, H. M.; Baltussen, M. W.; Van Sint Annaland, M.; Kuipers, J. A.M., On the drag force of bubbles in bubble swarms at intermediate and high Reynolds numbers, Chem. Eng. Sci., 66, 3204-3211 (2011) |
| [53] | Roma, A. M.; Peskin, C. S.; Berger, M. J., An adaptive version of the immersed boundary method, J. Comput. Phys., 153, 509-534 (1999) · Zbl 0953.76069 |
| [54] | Roman, F.; Napoli, E.; Milici, B.; Armenio, V., An improved immersed boundary method for curvilinear grids, Comput. Fluids, 38, 8, 1510-1527 (2009) · Zbl 1242.76210 |
| [55] | Rybczynski, W., Über die fortschreitende Bewegung einer flüssigen Kugel in einem zähen Medium, Bull. Acad. Sci. Cracovi., 40-46 (1911), (in German) · JFM 42.0811.02 |
| [56] | Santarelli, C.; Fröhlich, J., On the pair correlation function in a bubble swarm, Kerntechnik, 1, 50-51 (2013) |
| [57] | Santarelli, C.; Schwarz, S.; Fröhlich, J., Numerical simulation of light ellipsoidal particles in vertical turbulent channel flow, (8th ICMF International Conference on Multiphase Flow. 8th ICMF International Conference on Multiphase Flow, Jeju, Korea (2013)) |
| [58] | Scardovelli, R.; Zaleski, S., Direct numerical simulation of free-surface and interfacial flow, Annu. Rev. Fluid Mech., 31, 567-603 (1999) |
| [59] | Schwarz, S., An immersed boundary method for particles and bubbles in magnetohydrodynamic flows (2014), Institute of Fluid Mechanics, TU Dresden, TUDpress Verlag der Wissenschaften GmbH, PhD thesis |
| [60] | Schwarz, S.; Fröhlich, J., Representation of deformable bubbles by analytically defined shapes in an immersed boundary method, (Simos, T. E.; Psihoyios, G.; Tsitouras, C.; Anastassi, Z., Proceedings ICNAAM. Proceedings ICNAAM, American Institute of Physics Conference Series, vol. 1479 (2012)), 104-108 |
| [61] | Schwarz, S.; Fröhlich, J., Simulation of a bubble chain in a container of high aspect ratio exposed to a magnetic field, Eur. Phys. J. Spec. Top., 220, 195-205 (2013) |
| [62] | Schwarz, S.; Fröhlich, J., Numerical study of single bubble motion in liquid metal exposed to a longitudinal magnetic field, Int. J. Multiph. Flow, 62, 134-151 (2014) |
| [63] | Schwarz, S.; Kempe, T.; Fröhlich, J., A temporal discretization scheme to compute the motion of light particles in viscous flows by an immersed boundary method, J. Comput. Phys., 281, 591-613 (2015) · Zbl 1351.76223 |
| [64] | Spurk, J. H.; Aksel, N., Fluid Mechanics (2007), Springer · Zbl 1140.76001 |
| [65] | Sridhar, G.; Katz, J., Drag and lift forces on microscopic bubbles entrained by a vortex, Phys. Fluids, 7, 389 (1995) |
| [66] | Sussman, M.; Fatemi, E.; Smereka, P.; Osher, S., An improved level set method for incompressible two-phase flows, Comput. Fluids, 27, 663-680 (1998) · Zbl 0967.76078 |
| [67] | Tang, Y.; Kriebitzsch, S. H.L.; Peters, E. A.J. F.; van der Hoef, M. A.; Kuipers, J. A.M., A methodology for highly accurate results of direct numerical simulations: drag force in dense gas-solid flows at intermediate Reynolds number, Int. J. Multiph. Flow, 62, 73-86 (2014) |
| [68] | Tomiyama, A.; Kataoka, I.; Zun, I.; Sakaguchi, T., Drag Coefficients of Single Bubbles under Normal and Micro Gravity Conditions, JSME Int. J. Ser. B, 41, 472-479 (1998) |
| [69] | 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, J. Comput. Phys., 169, 708-759 (2001) · Zbl 1047.76574 |
| [70] | Tryggvason, G.; Scardovelli, R.; Zaleski, S., Direct Numerical Simulations of Gas-Liquid Multiphase Flows (2011), Cambridge University Press · Zbl 1226.76001 |
| [71] | Uhlmann, M., An immersed boundary method with direct forcing for the simulation of particulate flows, J. Comput. Phys., 209, 448-476 (2005) · Zbl 1138.76398 |
| [72] | Uhlmann, M., Interface-resolved direct numerical simulation of vertical particulate channel flow in the turbulent regime, Phys. Fluids, 20, 053305 (2008) · Zbl 1182.76785 |
| [73] | Unverdi, S. O.; Tryggvason, G., A front-tracking method for viscous, incompressible, multi-fluid flows, J. Comput. Phys., 100, 25-37 (1992) · Zbl 0758.76047 |
| [74] | Vowinckel, B.; Kempe, T.; Fröhlich, J., Particle-resolving simulations of bed-load sediment transport, (8th International Conference on Multiphase Flow. 8th International Conference on Multiphase Flow, Jeju, Korea (2013)) |
| [75] | Vowinckel, B.; Kempe, T.; Fröhlich, J., Fluid-particle interaction in turbulent open channel flow with fully-resolved mobile beds, Adv. Water Resour. (2014) |
| [76] | Vowinckel, B.; Kempe, T.; Fröhlich, J.; Nikora, V. I., Numerical simulation of sediment transport in open channel flow, (Murillo, R., River Flow (2012)), 507-514 |
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