# zbMATH — the first resource for mathematics

PICIN: a particle-in-cell solver for incompressible free surface flows with two-way fluid-solid coupling. (English) Zbl 1323.35129
This paper develops a new numerical approach for solutions to the Navier-Stokes equations for free surface flows involving two-way fluid-solid interaction in arbitrary domains. The details of how the free surface boundary, domain boundaries and two-way fluid-solid coupling are handled by a hybrid Eulerian-Lagrangian framework which is based on the full particle particle-in-cell method. And two-way fluid-structure interaction is handled by the distributed Lagrange multiplier approach proposed by N. A. Patankar et al. in [Int. J. Multiphase Flow 26, No. 9, 1509–1524 (2000; Zbl 1137.76712)]. Some numerical results are given. Finally, a comparison of this numerical approach with state-of-the-art numerical results of other researchers is made for each of the test cases presented.
Reviewer: Cheng He (Beijing)

##### MSC:
 35Q30 Navier-Stokes equations 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 76M28 Particle methods and lattice-gas methods 76D05 Navier-Stokes equations for incompressible viscous fluids 65M75 Probabilistic methods, particle methods, etc. for initial value and initial-boundary value problems involving PDEs 35R35 Free boundary problems for PDEs
PICIN
Full Text:
##### References:
 [1] D. Adalsteinsson and J. A. Sethian, The fast construction of extension velocities in level set methods, J. Comput. Phys, 148 (1999), p. 2–22. · Zbl 0919.65074 [2] R. Ando, N. Thurey, and R. Tsuruno, Preserving fluid sheets with adaptively sampled anisotropic particles, IEEE Trans. Visualization Comput. Graphics, 18 (2012), pp. 1202–1214. [3] E. Balaras, Modeling complex boundaries using an external force field on fixed Cartesian grids in large-eddy simulations, Comput. & Fluids, 33 (2004), pp. 375–404. · Zbl 1088.76018 [4] B. Bouscasse, A. Colagrossi, S. Marrone, and A. Souto-Iglesias, Viscous flow past a circular cylinder below a free surface, in Proceedings of the 33rd ASME International Conference on Ocean, Offshore and Arctic Engineering, San Fransisco, CA, 2014, OMAE2014-24488. [5] J. U. Brackbill and H. M Ruppel, FLIP: A method for adaptively zoned, particle-in-cell calculations of fluid flows in two dimensions, J. Comput. Phys., 65 (1986), pp. 314–343. · Zbl 0592.76090 [6] B. Buchner, Green Water on Ship-Type Offshore Structures, Ph.D. thesis, Delft University of Technology, Delft, The Netherlands, 2002. [7] Q. Chen, D. M. Kelly, J. Zang, and A. Dimakopoulos, PICIN: A Particle-In-Cell Solver for Incompressible Free Surface Flows with Two-Way Fluid-Solid Coupling. Part 2: Validation, in preparation. · Zbl 1323.35129 [8] A. J. Chorin, Numerical solution of the Navier–Stokes equations, Math. Comput., 22 (1968), pp. 745–762. · Zbl 0198.50103 [9] A. J. Chorin and J. E. Marsden, A Mathematical Introduction to Fluid Mechanics, Springer-Verlag, New York, 1979. · Zbl 0417.76002 [10] A. Colagrossi, B. Bouscasse, M. Antuono, and S. Marrone, Particle packing algorithm for SPH schemes, Comput. Phys. Comm., 183 (2012), pp. 1641–1653. · Zbl 1307.65140 [11] F. Gibou, R. P. Fedkiw, L.-T. Cheng, and M. Kang, A second-order-accurate symmetric discretization of the Poisson equation on irregular domains, J. Comput. Phys., 176 (2002), pp. 205–227. · Zbl 0996.65108 [12] F. H. Harlow, A Machine Calculation Method for Hydrodynamic Problems, Technical report LAMS-1956, Los Alamos Scientific Laboratory, Los Alamos, NM, 1955. [13] F. H. Harlow, The particle-in-cell computing method for fluid dynamics, in Methods in Computational Physics, B. Alder, ed., Academic Press, New York, 1964, pp. 319–343. [14] F. H. Harlow and J. E. Welch, Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface, Phys. Fluids, 8 (1965), pp. 2182–2189. · Zbl 1180.76043 [15] J. P. Hughes and D. I. Graham, Comparison of incompressible and weakly-compressible SPH models for free-surface water flows, J. Hydraulic Res., 48 (2010), pp. 105–117. [16] C. B. Jiang, J. Chen, H. S. Tang, and Y. Z. Cheng, Hydrodynamic processes on a beach: Wave breaking, up-rush and backwash, Commun. Nonlinear Sci. Numer. Simul., 16 (2011), pp. 3126–3139. · Zbl 1419.76480 [17] S. Koshizuka, A. Nobe, and Y. Oka, Numerical analysis of breaking waves using the moving particle semi-implicit method, Internat. J. Numer. Methods Fluids, 26 (1998), pp. 751–769. · Zbl 0928.76086 [18] P. Lin and P. L. F. Liu, A numerical study of breaking waves in the surf zone, J. Fluid Mech., 359 (1998), pp. 239–264. · Zbl 0916.76009 [19] S. J. Lind, R. Xu, P. K. Stansby, and B. D. Rogers, Incompressible smoothed particle hydrodynamics for free-surface flows: A generalised diffusion-based algorithm for stability and validations for impulsive flows and propagating waves, J. Comput. Phys., 231 (2012), pp. 1499–1523. · Zbl 1286.76118 [20] G. Markham and M. V. Proctor, C.E.G.B. report, Tech. report TRPD/L/0063/M82, CEGB, London, 1983. [21] S. Marrone, M. Antuono, A. Colagrossi, G. Colicchio, D. Le Touze, and G. Graziani, $$δ$$-SPH model for simulating violent impact flows, Comput. Methods Appl. Mech. Engrg., 200 (2011), pp. 1526–1542. · Zbl 1228.76116 [22] J. J. Monaghan, SPH without a tensile instability, J. Comput. Phys., 159 (2000), pp. 290–311. · Zbl 0980.76065 [23] J. J. Monaghan, Smoothed particle hydrodynamics and its diverse applications, Annu. Rev. Fluid Mech., 44 (2012), pp. 323–346. · Zbl 1361.76019 [24] J. J. Monaghan and J. C. Lattanzio, A refined method for astrophysical problems, Astronomy Astrophys., 149 (1985), pp. 135–143. · Zbl 0622.76054 [25] J. P. Morris, P. J. Fox, and Y. Zhu, Modeling low Reynolds number incompressible flows using SPH, J. Comput. Phys., 136 (1997), pp. 214–226. · Zbl 0889.76066 [26] Y. Ng, C. Min, and F. Gibou, An efficient fluid-solid coupling algorithm for single-phase flows, J. Comput. Phys., 228 (2009), pp. 8807–8829. · Zbl 1245.76019 [27] B. D. Nichols and C. W. Hirt, Improved free surface boundary conditions for numerical incompressible-flow simulations, J. Comput. Phys., 8 (1971), pp. 434–448. · Zbl 0227.76048 [28] W. F. Noh, CEL: A time-dependent, two-space-dimensional, coupled Eulerian-Lagrange code, in Methods in Computational Physics, B. Alder, ed., Academic Press, New York, 1964, pp. 117–179. [29] N. A. Patankar, A formulation for fast computations of rigid particulate flows, in Center for Turbulence Research Annual Research Briefs 2001, Stanford University, Stanford, CA, 2001, pp. 185–196. [30] N. A. Patankar, P. Singh, D. D. Joseph, R. Glowinski, and T.-W. Pan, A new formulation of the distributed Lagrange multiplier/ fictitious domain method for particulate flows, Int. J. Multiphase Flow, 26 (2000), pp. 1509–1524. · Zbl 1137.76712 [31] W. H. Press, S. A. Teulowsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in FORTRAN, The Art of Scientific Computing, 2nd ed., Cambridge University Press, Cambridge, UK, 1994. [32] A. Ralston, Runge–Kutta methods with minimum error bound, Math. Comp., 16 (1962), pp. 431–437. · Zbl 0105.31903 [33] J. A. Sethian, Level Set Methods and Fast Marching Methods, 9th ed., Cambridge University Press, Cambridge, UK, 2008. · Zbl 0994.65090 [34] H. Zhao, A fast sweeping method for Eikonal equations, Math. Comp., 74 (2004), pp. 603–627. · Zbl 1070.65113 [35] Y Zhu and R Bridson, Animating sand as a fluid, in Proceedings of ACM SIGGRAPH 2005, ACM, New York, 2005, pp. 965–972.
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.