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Simulations of gas-liquid compressible-incompressible systems using SPH. (English) Zbl 1411.76140
Summary: Gas bubbles immersed in a liquid and flowing through a large pressure gradient undergo volumetric deformation in addition to possible deviatoric deformation. While the high density liquid phase can be assumed to be an incompressible fluid, the gas phase needs to be modeled as a compressible fluid for such bubble flow problems. The Rayleigh-Plesset (RP) equation describes such a bubble undergoing volumetric deformation due to changes in pressure in the ambient incompressible fluid in the presence of capillary force at its boundary, assuming axisymmetric dynamics. We propose a compressible-incompressible coupling of smoothed particle hydrodynamics (SPH) and validate this coupling against the RP model in two dimensions. This study complements the SPH simulations of a different class of compressible-incompressible systems where an outer compressible phase affects the dynamics of an inner incompressible phase. For different density ratios, a sinusoidal pressure variation is applied to the ambient incompressible liquid and the response of the bubble in terms of volumetric deformation is observed and compared with the solutions of the axisymmetric RP equation.
##### MSC:
 76M28 Particle methods and lattice-gas methods 65M75 Probabilistic methods, particle methods, etc. for initial value and initial-boundary value problems involving PDEs 76T10 Liquid-gas two-phase flows, bubbly flows
BiCGstab
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##### References:
 [1] Adami, S.; Hu, X.; Adams, N., A new surface-tension formulation for multi-phase SPH using a reproducing divergence approximation, J Comput Phys, 229, 13, 5011-5021, (2010) · Zbl 1346.76161 [2] Avulapati, M. M.; Ravikrishna, R. V., An experimental study on effervescent atomization of bio-oil fuels, Atomization Sprays, 22, 8, 663-685, (2012) [3] Billaud, M.; Gallice, G.; Nkonga, B., A simple stabilized finite element method for solving two phase compressible-incompressible interface flows, Comput Method Appl Mech Eng, 200, 9, 1272-1290, (2011) · Zbl 1225.76191 [4] Chaniotis, A.; Poulikakos, D.; Koumoutsakos, P., Remeshed smoothed particle hydrodynamics for the simulation of viscous and heat conducting flows, J Comput Phys, 182, 1, 67-90, (2002) · Zbl 1048.76046 [5] Colagrossi, A.; Landrini, M., Numerical simulation of interfacial flows by smoothed particle hydrodynamics, J Comput Phys, 191, 2, 448-475, (2003) · Zbl 1028.76039 [6] Cummins, S. J.; Rudman, M., An SPH projection method, J Comput Phys, 152, 2, 584-607, (1999) · Zbl 0954.76074 [7] Dehnen, W.; Aly, H., Improving convergence in smoothed particle hydrodynamics simulations without pairing instability, Mon Not R Astron Soc, 425, 2, 1068-1082, (2012) [8] Ferrari, A.; Munz, C. D.; Weigand, B., A high order sharp-interface method with local time stepping for compressible multiphase flows, Commun Comput Phys, 9, 1, 205, (2011) · Zbl 1284.76265 [9] Gadgil, H. P.; Raghunandan, B. N., Some features of spray breakup in effervescent atomizers, Exp Fluids, 50, 2, 329-338, (2011) [10] 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 [11] Hauke, G.; Hughes, T., A unified approach to compressible and incompressible flows, Comput Method Appl Mech Eng, 113, 3, 389-395, (1994) · Zbl 0845.76040 [12] Hauke, G.; Hughes, T. J., A comparative study of different sets of variables for solving compressible and incompressible flows, Comput Method Appl Mech Eng, 153, 1, 1-44, (1998) · Zbl 0957.76028 [13] Jedelsky, J.; Jicha, M.; Slama, J.; Otahal, J., Development of an effervescent atomizer for industrial burners, Energy Fuels, 23, 12, 6121-6130, (2009) [14] Lin, H.; Storey, B. D.; Szeri, A. J., Inertially driven inhomogeneities in violently collapsing bubbles: the validity of the Rayleigh-Plesset equation, J Fluid Mech, 452, 145-162, (2002) · Zbl 1074.76050 [15] Lind, S.; Stansby, P.; Rogers, B. D., Incompressible-compressible flows with a transient discontinuous interface using smoothed particle hydrodynamics (SPH), J Comput Phys, 309, 129-147, (2016) · Zbl 1351.76241 [16] Liu, T.; Khoo, B.; Wang, C., The ghost fluid method for compressible gas-water simulation, J Comput Phys, 204, 1, 193-221, (2005) · Zbl 1190.76160 [17] Loebker, D. W.; Empie, H. L., High mass flowrate effervescent spraying of a high viscosity Newtonian liquid, Proceedings of the 10th Annual Conference on Liquid Atomization and Spray Systems. Atlanta, Georgia: the Institute, 253-257, (1997) [18] Monaghan, J. J., Smoothed particle hydrodynamics, Annu Rev Astron Astrophys, 30, 543-574, (1992) [19] Morris, J. P., Simulating surface tension with smoothed particle hydrodynamics, Int J Numer Methods Fluids, 33, 3, 333-353, (2000) · Zbl 0985.76072 [20] Morris, J. P.; Fox, P. J.; Zhu, Y., Modeling low Reynolds number incompressible flows using SPH, J Comput Phys, 136, 1, 214-226, (1997) · Zbl 0889.76066 [21] Nair, P.; Pöschel, T., Dynamic capillary phenomena using incompressible SPH, Chem Eng Sci, 176, 192-204, (2018) [22] Nair, P.; Tomar, G., An improved free surface modeling for incompressible SPH, Comput Fluids, 102, 304-314, (2014) · Zbl 1391.76626 [23] Nair, P.; Tomar, G., Volume conservation issues in incompressible smoothed particle hydrodynamics, J Comput Phys, 297, 689-699, (2015) · Zbl 1349.76721 [24] Panton, R. L., Incompressible flow, (2006), John Wiley & Sons [25] Pesch, L.; van der Vegt, J. J., A discontinuous Galerkin finite element discretization of the Euler equations for compressible and incompressible fluids, J Comput Phys, 227, 11, 5426-5446, (2008) · Zbl 1144.76033 [26] Plesset, M., The dynamics of cavitation bubbles, J Appl Mech, 16, 277-282, (1949) [27] Rayleigh, L., Viii. on the pressure developed in a liquid during the collapse of a spherical cavity, Lond, Edinburgh, Dublin Philos Mag J Sci, 34, 200, 94-98, (1917) · JFM 46.1274.01 [28] Rezavand, M.; Taeibi-Rahni, M.; Rauch, W., An ISPH scheme for numerical simulation of multiphase flows with complex interfaces and high density ratios, Comput Math Appl, (2018) [29] Sleijpen, G. L.; Van der Vorst, H. A.; Fokkema, D. R., BiCGstab (l) and other hybrid bi-CG methods, Numer Algorithms, 7, 1, 75-109, (1994) · Zbl 0810.65027 [30] Sovani, S.; Sojka, P.; Lefebvre, A., Effervescent atomization, Prog Energy Combust Sci, 27, 4, 483-521, (2001) [31] Violeau, D.; Leroy, A., On the maximum time step in weakly compressible sph, J Comput Phys, 256, 388-415, (2014) · Zbl 1349.76745 [32] Wendland, H., Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree, Adv Comput Math, 4, 1, 389-396, (1995) · Zbl 0838.41014 [33] Whitlow, J.; Lefebvre, A. H., Effervescent atomizer operation and spray characteristics, Atomization Sprays, 3, 2, (1993) [34] Xu, R.; Stansby, P.; Laurence, D., Accuracy and stability in incompressible SPH (ISPH) based on the projection method and a new approach, J Comput Phys, 228, 18, 6703-6725, (2009) · Zbl 1261.76047 [35] Zhang, M., Simulation of surface tension in 2D and 3D with smoothed particle hydrodynamics method, J Comput Phys, 229, 19, 7238-7259, (2010) · Zbl 1426.76623
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