本研究主要發展一種數值方法模擬計算多相流體與顆粒交互作用之問題。此計算方法利用兩種顆粒碰撞模型與壓力修正法耦合來計算顆粒與顆粒和顆粒與固體邊界之間在流體內之碰撞力與力矩。對於在單相流體內的碰撞問題利用彈性碰撞法來計算顆粒於流體內的運動，而較在二相流體內的顆粒運動則使用離散元素法。在模擬流體與顆粒交互作用之流場現象中，利用直接力量浸入邊界法與流體體積法來計算在自由液面流場內流體與顆粒之交互作用。在本研究中相較於傳統的沉浸邊界法，流體對於固體的作用力直接從流場計算得到；此外對於流體與固體之間的體積分率函數，本研究提出二種較簡單的體積分率函數來定義出流體與固體邊界的體積分率並利用高斯散度定理將面積分轉換成體積分來計算流體與固體之間的作用力以增加計算效率。
本研究使用直接力量浸入邊界壓力修正法配合顆粒碰撞模型在簡單的結構性卡式格點系統下來求解複雜的顆粒流流場現象及動力分析，並使用Transpose-Free Quasi-Minimal Residual (TFQMR)及平行運算來增加收斂及求解速度。在求解各種物理問題中驗證了此計算方法的效率及準確性，並可成功模擬出顆粒流在不同顆粒濃度的動力特性及物理現象。

A direct-forcing immersed boundary pressure correction method is developed to simulate the fluid-particle interaction problems. Two particle collision models are applied to compute the force and torque from particle to particle and particle to solid boundary collisions in the fluid. An elastic collision force method is used to calculate the motion of submersed particle for the particle collision in single-fluid flow and a discrete element method (DEM) is applied for the complex motion of particle in two-fluid flow. For the simulation of the fluid-particle interaction problems, a direct-forcing immersed boundary (IB) method and volume of fluid (VOF) method are used to compute the fluid-particle interaction in the free surface flow. Unlike the IB method proposed by Peskin, the hydrodynamic force is computed directly from the fluid flow, and two simple volume fraction functions are presented to define the interface volume fraction between the fluid and solid; and further, the Gauss’s divergence theorem is applied to derive the hydrodynamic force from the volume integral of the particle instead of the particle surface and to improve the computing efficiency.
In this thesis, the direct-forcing immersed boundary pressure correction method coupled with the particle collision model is used to solve the complex granular flows and its dynamics in the structure Cartesian grid system.　To enhance the convergence　and computation time, the Transpose-Free Quasi-Minimal Residual (TFQMR) method and parallel computation are also applied. The efficiency and accuracy of the presented numerical scheme are validated by the various physical problems and successfully simulate the dynamics and the physical phenomena in the different concentration granular flows.

[1] Peskin, C. S., 1972, "Flow patterns around heart valves - Numerical method," J. Comput. Phys., 10(2), pp. 252-271.
[2] Fadlun, E. A., Verzicco, R., Orlandi, P., and Mohd-Yusof, J., 2000, "Combined immersed-boundary finite-difference methods for three-dimensional complex flow simulations," J. Comput. Phys., 161(1), pp. 35-60.
[3] Feng, Z. G., and Michaelides, E. E., 2005, "Proteus: a direct forcing method in the simulations of particulate flows," J. Comput. Phys., 202(1), pp. 20-51.
[4] Uhlmann, M., 2005, "An immersed boundary method with direct forcing for the simulation of particulate flows," J. Comput. Phys., 209(2), pp. 448-476.
[5] Lin, S. Y., Chin, Y. H., Hu, J. J., and Chen, Y. C., 2011, "A pressure correction method for fluid-particle interaction flow: Direct-forcing method and sedimentation flow," Int. J. Numer. Meth. Fl., 67(12), pp. 1771-1798.
[6] Hirt, C. W., and Nichols, B. D., 1981, "Volume of fluid (VOF) Method for the dynamics of free boundaries," J. Comput. Phys., 39(1), pp. 201-225.
[7] Pilliod, J. E., and Puckett, E. G., 2004, "Second-order accurate volume-of-fluid algorithms for tracking material interfaces," J. Comput. Phys., 199(2), pp. 465-502.
[8] Gueyffier, D., Li, J., Nadim, A., Scardovelli, R., and Zaleski, S., 1999, "Volume-of-fluid interface tracking with smoothed surface stress methods for three-dimensional flows," J. Comput. Phys., 152(2), pp. 423-456.
[9] Lin, S. Y., Chin, Y. H., Wu, C. M., Lin, J. F., and Chen, Y. C., 2012, "A pressure correction-volume of fluid method for simulation of two-phase flows," Int. J. Numer. Meth. Fl., 68(2), pp. 181-195.
[10] Osher, S., and Sethian, J. A., 1988, "Fronts propagating with curvature-dependent Speed - algorithms based on Hamilton-Jacobi formulations," J. Comput. Phys., 79(1), pp. 12-49.
[11] Ye, T., Shyy, W., and Chung, J. N., 2001, "A fixed-grid, sharp-interface method for bubble dynamics and phase change," J. Comput. Phys., 174(2), pp. 781-815.
[12] Uzgoren, E., Sim, J., and Shyy, W., 2009, "Marker-based, 3-D adaptive cartesian grid method for multiphase flow around irregular geometries," Commun. Comput. Phys., 5(1), pp. 1-41.
[13] Monaghan, J. J., 1992, "Smoothed particle hydrodynamics," Annu. Rev. Astron. Astr., 30, pp. 543-574.
[14] Monaghan, J. J., 1994, "Simulating free-surface flows with SPH," J. Comput. Phys., 110(2), pp. 399-406.
[15] Bombardelli, F. A., Hirt, C. W., and Garcia, M. H., 2001, "Computations of curved free surface water flow on spiral concentrators - Discussion," J. Hydraul. Eng.-ASCE, 127(7), pp. 629-631.
[16] Osher, S., and Chakravarthy, S., 1984, "High-resolution schemes and the entropy condition," SIAM J. Numer. Anal., 21(5), pp. 955-984.
[17] Lin, S. Y., and Yu, Z. X., 2002, "Vortex structure and strength of secondary flows in model aortic arches," Int. J. Numer. Meth. Fl., 40(3-4), pp. 379-389.
[18] Lin, S. Y., and Chin, Y. S., 1995, "Comparison of higher resolution Euler schemes for aeroacoustic computations," AIAA J., 33(2), pp. 237-245.
[19] Hoomans, B. P. B., Kuipers, J. A. M., Briels, W. J., and vanSwaaij, W. P. M., 1996, "Discrete particle simulation of bubble and slug formation in a two-dimensional gas-fluidised bed: A hard-sphere approach," Chem. Eng. Sci., 51(1), pp. 99-118.
[20] Deen, N. G., Annaland, M. V., Van der Hoef, M. A., and Kuipers, J. A. M., 2007, "Review of discrete particle modeling of fluidized beds," Chem. Eng. Sci., 62(1-2), pp. 28-44.
[21] Cundall, P. A., and Strack, O. D. L., 1979, "Discrete numerical-model for granular assemblies," Geotechnique, 29(1), pp. 47-65.
[22] Tsuji, Y., Kawaguchi, T., and Tanaka, T., 1993, "Discrete particle simulation of 2-dimensional fluidized-bed," Powder Technol., 77(1), pp. 79-87.
[23] Zhang, X., and Vu-Quoc, L., 2002, "Modeling the dependence of the coefficient of restitution on the impact velocity in elasto-plastic collisions," Int. J. Impact. Eng., 27(3), pp. 317-341.
[24] Stevens, A. B., and Hrenya, C. M., 2005, "Comparison of soft-sphere models to measurements of collision properties during normal impacts," Powder Technol., 154(2-3), pp. 99-109.
[25] Zhang, S., Kuwabara, S., Suzuki, T., Kawano, Y., Morita, K., and Fukuda, K., 2009, "Simulation of solid-fluid mixture flow using moving particle methods," J. Comput. Phys., 228(7), pp. 2552-2565.
[26] Raad, P. E., and Bidoae, R., 2005, "The three-dimensional Eulerian-Lagrangian marker and micro cell method for the simulation of free surface flows," J. Comput. Phys., 203(2), pp. 668-699.
[27] Gómez-Gesteira, M., and Dalrymple, R. A., 2004, "Using a three-dimensional smoothed particle hydrodynamics method for wave impact on a tall structure," J. Waterw. Port. C-ASCE, 130(2), pp. 63-69.
[28] Silvester, T. B., Cleary, P.W., "Wave-structure interaction using smoothed particle hydrodynamics," Proc. The 5th International Conference on CFD in the Process Industries.
[29] Kleefsman, K. M. T., Fekken, G., Veldman, A. E. P., Iwanowski, B., and Buchner, B., 2005, "A volume-of-fluid based simulation method for wave impact problems," J. Comput. Phys., 206(1), pp. 363-393.
[30] Shieh, C. L., Ting, C. H., and Pan, H. W., 2008, "Impulsive force of debris flow on a curved dam," Int. J. Sediment. Res., 23(2), pp. 149-158.
[31] Ting, C.H., 2009, "Impulsive force resulted from debris flow acting on the sabo dam," Ph.D, National Cheng Kung University.
[32] Jiang, Y., Chen, C. P., and Tucker, P. K., 1991, "Multigrid solution of unsteady Navier-Stokes equations using a pressure method," Numer. Heat Tr. A-Appl., 20(1), pp. 81-93.
[33] Issa, R. I., 1986, "Solution of the Implicitly Discretized Fluid-Flow Equations by Operator-Splitting," J. Comput. Phys., 62(1), pp. 40-65.
[34] Lin, S. Y., Chen, Yi. Cheng, 2013, "A pressure correction-volume of fluid method for simulations of fluid–particle interaction and impact problems," Int. J. Multiphase Flow, 49, p. 18.
[35] Pan, K. L., and Yin, G. C., 2012, "Parallel strategies of front-tracking method for simulation of multiphase flows," Comput. Fluids., 67, pp. 123-129.
[36] Amritkar, A., Tafti, D., Liu, R., Kufrin, R., and Chapman, B., 2012, "OpenMP parallelism for fluid and fluid-particulate systems," Parallel Comput., 38(9), pp. 501-517.
[37] Hoeflinger, J., Alavilli, P., Jackson, T., and Kuhn, B., 2001, "Producing scalable performance with OpenMP: Experiments with two CFD applications," Parallel Comput., 27(4), pp. 391-413.
[38] Hoeflinger, J., Kuhn, B., Nagel, W., Petersen, P., Rajic, H., Shah, S., Vetter, J., Voss, M., and Woo, R., 2001, "An integrated performance visualizer for MPI/OpenMP programs," OpenMP Shared Memory Parallel Programming, Proceedings, 2104, pp. 40-52.
[39] Norden, M., Holmgren, S., and Thune, M., 2006, "OpenMP versus MPI for PDE solvers based on regular sparse numerical operators," Future Gener. Comp. Sy., 22(1-2), pp. 194-203.
[40] Tafit, D., 2001, "GenIDLEST – a scalable parallel computational tool for simulating complex turbulent flows," Proc. In Proceedings of the ASME Fluids Engineering Division, 256, pp. 347-356.
[41] Freund, R. W., 1993, "A transpose-free quasi-minimal residual algorithm for non-Hermitian linear-systems," SIAM J. Sci. Comput., 14(2), pp. 470-482.
[42] Armfield, S. W., 1991, "Finite-difference solutions of the Navier-Stokes equations on staggered and non-staggered grids," Comput. Fluids, 20(1), pp. 1-17.
[43] Choi, S. K., Nam, H. Y., and Cho, M., 1994, "Systematic comparison of finite-volume calculation methods with staggered and nonstaggered grid arrangements," Numer. Heat Tr. B-Fund, 25(2), pp. 205-221.
[44] Choi, S. K., Nam, H. Y., and Cho, M., 1994, "Use of staggered and nonstaggered grid arrangements for incompressible-flow calculations on nonorthogonal grids," Numer. Heat. Tr. B-Fund., 25(2), pp. 193-204.
[45] Melaaen, M. C., 1992, "Calculation of fluid-flows with staggered and nonstaggered curvilinear nonorthogonal grids - the Theory," Numer. Heat Tr. B-Fund., 21(1), pp. 1-19.
[46] Acharya, S., and Moukalled, F. H., 1989, "Improvements to incompressible-flow calculation on a nonstaggered curvilinear grid," Numer. Heat Tr. B-Fund., 15(2), pp. 131-152.
[47] Date, A. W., 1993, "Solution of Navier-Stokes equations on non-staggered grid," Int. J. Heat Mass Tran., 36(7), pp. 1913-1922.
[48] Date, A. W., 1996, "Complete pressure correction algorithm for solution of incompressible Navier-Stokes equations on a nonstaggered grid," Numer. Heat Tr. B-Fund., 29(4), pp. 441-458.
[49] Majumdar, S., 1988, "Role of underrelaxation in momentum interpolation for calculation of flow with nonstaggered grids," Numer. Heat Tr. A-Appl., 13(1), pp. 125-132.
[50] Rhie, C. M., and Chow, W. L., 1983, "Numerical study of the turbulent-flow past an airfoil with trailing edge separation," AIAA J., 21(11), pp. 1525-1532.
[51] Yu, B., Kawaguchi, Y., Tao, W. Q., and Ozoe, H., 2002, "Checkerboard pressure predictions due to the underrelaxation factor and time step size for a nonstaggered grid with momentum interpolation method," Numer. Heat Tr. B-Fund., 41(1), pp. 85-94.
[52] Yu, B., Tao, W. Q., Wei, J. J., Kawaguchi, Y., Tagawa, T., and Ozoe, H., 2002, "Discussion on momentum interpolation method for collocated grids of incompressible flow," Numer. Heat Tr. B-Fund., 42(2), pp. 141-166.
[53] Glowinski, R., Pan, T. W., Hesla, T. I., Joseph, D. D., and Periaux, J., 2001, "A fictitious domain approach to the direct numerical simulation of incompressible viscous flow past moving rigid bodies: Application to particulate flow," J. Comput. Phys., 169(2), pp. 363-426.
[54] Malone, K. F., and Xu, B. H., 2008, "Determination of contact parameters for discrete element method simulations of granular systems," Particuology, 6(6), pp. 521-528.
[55] Hofler, K., and Schwarzer, S., 2000, "Navier-Stokes simulation with constraint forces: Finite-difference method for particle-laden flows and complex geometries," Phys. Rev. E, 61(6), pp. 7146-7160.
[56] Lai, M. C., and Peskin, C. S., 2000, "An immersed boundary method with formal second-order accuracy and reduced numerical viscosity," J. Comput. Phys., 160(2), pp. 705-719.
[57] Lee, C., 2003, "Stability characteristics of the virtual boundary method in three-dimensional applications," J. Comput. Phys., 184(2), pp. 559-591.
[58] White, F. M., 2006, Viscous fluid flow, Third Ed. Mc Graw Hill.
[59] Armaly, B. F., Durst, F., Pereira, J. C. F., and Schonung, B., 1983, "Experimental and theoretical investigation of backward-facing step flow," J. Fluid Mech., 127(Feb), pp. 473-496.
[60] Barton, I. E., 1995, "A numerical study of flow over a confined backward-facing step," Int. J. Numer. Meth Fl., 21(8), pp. 653-665.
[61] Taneda, S., 1956, "Experimental investigation of the wake behind a sphere at low Reynolds numbers," J. Phys. Soc. Jpn., 11(10), pp. 1104-1108.
[62] Wu, J. S., and Faeth, G. M., 1993, "Sphere wakes in still surroundings at intermediate Reynolds numbers," AIAA J., 31(8), pp. 1448-1455.
[63] Tomboulides, A. G., 1993, "Direct and large-eddy simulation of wake flows: flow past a sphere," PhD, Princetion University.
[64] Achenbac.E, 1974, "Vortex shedding from spheres," J. Fluid Mech., 62, pp. 209-221.
[65] Fornberg, B., 1988, "Steady viscous- Flow past a sphere at high Reynolds numbers," J. Fluid Mech., 190, pp. 471-489.
[66] Johnson, T. A., and Patel, V. C., 1999, "Flow past a sphere up to a Reynolds number of 300," J. Fluid Mech., 378, pp. 19-70.
[67] Marella, S., Krishnan, S., Liu, H., and Udaykumar, H. S., 2005, "Sharp interface Cartesian grid method I: An easily implemented technique for 3D moving boundary computations," J. Comput. Phys., 210(1), pp. 1-31.
[68] Clift, R., Grace, J.R., Weber, M.E., 1978, Bubbles, Drops and Particles., Acdemic Press: New York.
[69] ten Cate, A., Nieuwstad, C. H., Derksen, J. J., and Van den Akker, H. E. A., 2002, "Particle imaging velocimetry experiments and lattice-Boltzmann simulations on a single sphere settling under gravity," Phys. Fluids, 14(11), pp. 4012-4025.
[70] Vu-Quoc, L., Lesburg, L., and Zhang, X., 2004, "An accurate tangential force-displacement model for granular-flow simulations: Contacting spheres with plastic deformation, force-driven formulation," J. Comput. Phys., 196(1), pp. 298-326.
[71] Vu-Quoc, L., Zhang, X., and Lesburg, L., 2000, "A normal force-displacement model for contacting spheres accounting for plastic deformation: Force-driven formulation," J. Appl. Mech.-T. ASME, 67(2), pp. 363-371.
[72] Johnson, K. L., 1985, Contact mechanics, Cambridge University Press, New York.
[73] Hertz, H., 1882, "Ueber die Berührung fester elastischer Körper," J. Reine Angew. Math., 92, pp. 156-171.
[74] Mindlin R.D., D. H., 1952, "Elastic spheres in contact under varying oblique forces," ASME J. Appl. Mech., 20, pp. 327-344.
[75] Kelecy, F. J., and Pletcher, R. H., 1997, "The development of a free surface capturing approach for multidimensional free surface flows in closed containers," J. Comput. Phys., 138(2), pp. 939-980.
[76] Martin, J. C., Moyce W.J., 1952, "An experimental study of the collapse of liquid columns on a rigid horizontal plane," Phil. Trans. R. Soc. A, 244, pp. 312-324.
[77] Feng, Z. G., and Michaelides, E. E., 2004, "The immersed boundary-lattice Boltzmann method for solving fluid-particles interaction problems," J. Comput. Phys., 195(2), pp. 602-628.
[78] Lin, S. Y., Lin, C. T., Chin, Y. H., and Tai, Y. H., 2011, "A direct-forcing pressure-based lattice Boltzmann method for solving fluid-particle interaction problems," Int. J. Numer. Meth. Fl., 66(5), pp. 648-670.
[79] Tyvand, P. A., and Miloh, T., 1995, "Free-surface flow due to impulsive motion of a submerged circular-cylinder," J. Fluid. Mech., 286, pp. 67-101.
[80] Greenhow, M., and Moyo, S., 1997, "Water entry and exit of horizontal circular cylinders," Phil. Trans. R. Soc. A, 355(1724), pp. 551-563.
[81] Lin, P. Z., 2007, "A fixed-grid model for simulation of a moving body in free surface flows," Comput. Fluids, 36(3), pp. 549-561.
[82] Currie, I. G., 2004, Fundamental mechanics of fluids, Second Ed. Mc Graw Hill.