本研究發展一直接力量沉浸邊界壓力修正法，藉由數值求解那威爾-史托克方程式得到流體與顆粒在流場中的交互作用。此方法採用二階迎風有限體積法求解對流項和二階中央差分法求解黏滯項。此方法採用交錯網格防止棋盤式壓力場的發生，並預測一個速度場，然後在質量守恆的條件下獲得更新後的壓力。
對於適當雷諾數，一顆球正向碰撞牆的過程中，在球的後方將引起一渦環，在撞擊結束後，其後方的渦流將會流過球並和牆上產生的渦流發展成一個複雜的渦流環系統。本研究主要是發展一個直接力量沉浸邊界壓力修正法去模擬一顆球撞擊牆後的渦流。我們使用直接力量沉浸邊界法來截取顆粒在流體中的運動。對比於傳統的沉浸邊界法，此方法在使用上相對簡單和有效率。在計算流場作用於球體上的力量，藉由格林定理將面積法轉換成體積分，搭配直接力量沉浸邊界法中的體積容率求解。在球體等速接近牆的研究中，Cox 和 Brenner將經典的潤滑公式中的雷諾數以理論形式延長到50。所以本論文在球等速接近牆的問題中，採用Re=3-50的區間。我們提出一個明確的力量方程式來描述一顆球從靜止落下到很接近牆的力量趨勢。此力量方程式藉由最小平方法去擬合數值結果求得。對於球在重力影響下的自由落體撞擊牆的模擬過程中，我們也提出一個明確的力量方程式，其中包含了半穩定狀態的黏滯阻力和不穩定狀態的附加質量力量和歷史力量。本研究提出的方程式符合低雷諾數理論。

A direct-forcing immersed-boundary (IB) pressure correction method is developed to solve Navier–Stokes equations and investigate the flow fields of fluid–particle interaction problems. The proposed method uses a second-order upwind finite-volume scheme for convective terms, a second-order central finite-volume scheme for viscous terms, as well as a staggered grid to prevent the occurrence of a checkerboard pressure field. The method is also used to predict a velocity field and to obtain updated pressure under mass conservation.
For moderate Reynolds numbers, a sphere colliding with a wall in the normal direction will result in a trailing recirculating wake that threads over the sphere after impact and develops into a complex vortex-ring system as it interacts with the vorticity generated at the wall. To capture particle motion in a flow, the IB method is a popular choice for mapping particle motion into force density along a solid surface.

A direct-forcing IB pressure correction method is used to numerically investigate the total hydrodynamic force for a fully immersed solid sphere in normal approach to a plane wall. For constant approaches, Cox and Brenner’s (1967) classic asymptotic formula in the lubrication regime is extended to cover particle Reynolds number Re= < 50. An explicit force formula is proposed by fitting the current simulation results. We also consider the gravity-induced descent of an immersed sphere toward a wall, as well as propose an equation for motion containing a quasi-steady viscous drag, two unsteady component-added mass forces, and a history force. Possible effects from liquid inertia at finite-Re and wall at small gaps are described by least-square fitting of the simulation results. The proposed formula agrees with existing low-Re theories.

REFERENCES
1. Abdo, M., Mateescu, D., 2005. Low-Reynolds Number Aerodynamics of Airfoils at Incidence., 43rd AIAA Aerospace Sciences Meeting and Exhibit 10-13 January.
2. Ambari, A., Gauthiermanuel, B., Guyon, E., 1984. Wall Effects on a Sphere Translating at Constant Velocity. J Fluid Mech 149, 235-253.
3. Ardekani, A.M., Rangel, R.H., 2008. Numerical investigation of particle-particle and particle-wall collisions in a viscous fluid. J Fluid Mech 596, 437-466.
4. Brennen, C.E., 2005. Fundamentals of multiphase flow. New York: Cambridge University Press.
5. Brenner, H., 1961. The Slow Motion of a Sphere through a Viscous Fluid Towards a Plane Surface. Chem Eng Sci 16, 242-251.
6. Clift, R., Grace, J. R., Webber, M. E., 1978. Bubbles, drops, and particles. New York: Academic Press.
7. Cooley, M.D.A., Oneill, M.E., 1969. On Slow Motion Generated in a Viscous Fluid by Approach of a Sphere to a Plane Wall or Stationary Sphere. Mathematika 16, 37-49.
8. Coutanceau, M., Bouard, R., 1977. Experimental-Determination of Main Features of Viscous-Flow in Wake of a Circular-Cylinder in Uniform Translation .1. Steady Flow. J Fluid Mech 79, 231.
9. Cox, R.G., Brenner, H., 1967. Slow Motion of a Sphere through a Viscous Fluid Towards a Plane Surface .PartII. Small gap Widths Including Inertial Effects. Chem Eng Sci 22, 1753-1777.
10. Dance, S.L., Maxey, M.R., 2003. Incorporation of lubrication effects into the force-coupling method for particulate two-phase flow. J Comput Phys 189, 212-238.
11. Davis, R.H., Serayssol, J.M., Hinch, E.J., 1986. The Elastohydrodynamic Collision of two Spheres. J Fluid Mech 163, 479-497.
12. Dennis, S.C.R., Chang, G.Z., 1970. Numerical Solutions for Steady Flow Past a Circular Cylinder at Reynolds Numbers up to 100. J Fluid Mech 42, 471-489.
13 Dutsch, H., Durst, F., Becker, S. and Lienhart H., 1998. Low-Reynolds-number flow around an oscillating circular cylinder at low Keulegan-Carpenter numbers, J. Fluid Mech.360, 249-271.
14. Eames, I., Dalziel, S.B., 2000. Dust resuspension by the flow around an impacting sphere. J Fluid Mech 403, 305-328.
15. Fadlun, E.A., Verzicco, R., Orlandi, P., Mohd-Yusof, J., 2000. Combined immersed-boundary finite-difference methods for three-dimensional complex flow simulations. J Comput Phys 161, 35-60.
16. Feng, Z.G., Michaelides, E.E., 2004. The immersed boundary-lattice Boltzmann method for solving fluid-particles interaction problems. J Comput Phys 195, 602-628.
17. Feng, Z.G., Michaelides, E.E., 2005. Proteus: a direct forcing method in the simulations of particulate flows. J Comput Phys 202, 20-51.
18. Fornberg, B., 1980. A Numerical Study of Steady Viscous-Flow Past a Circular-Cylinder. J Fluid Mech 98, 819-855.
19. Fornberg, B., 1988. Steady Viscous-Flow Past a Sphere at High Reynolds-Numbers. J Fluid Mech 190, 471-489.
20. Ghia, U., Ghia, K.N., Shin, C.T., 1982. High-Re Solutions for Incompressible-Flow Using the Navier Stokes Equations and a Multigrid Method. J Comput Phys 48, 387-411.
21. Glowinski, R., Pan, T.W., Hesla, T.I., Joseph, D.D., 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, 363-426.
22 Gondret, P., Lance, M. & Petit, L. 2002. Bouncing motion of spherical particles in fluids. Phys. Fluids 14, 643–652.
23. He, X.Y., Doolen, G., 1997. Lattice Boltzmann method on curvilinear coordinates system: Flow around a circular cylinder. J Comput Phys 134, 306-315.
24. Hofler, K., Schwarzer, S., 2000. Navier-Stokes simulation with constraint forces: Finite-difference method for particle-laden flows and complex geometries. Phys Rev E 61, 7146-7160.
25. Hou, S.L., Zou, Q., Chen, S.Y., Doolen, G., Cogley, A.C., 1995. Simulation of Cavity Flow by the Lattice Boltzmann Method. J Comput Phys 118, 329-347.
26 Hu, H.H., 1996. Direct simulation of flows of solid-liquid mixtures. Int J Multiphas Flow 22, 335-352.
27. Ingber, M.S., 1990. Dynamic Simulation of the Hydrodynamic Interaction among Immersed Particles in Stokes-Flow. Int J Numer Meth Fl 10, 791-809.
28. Issa, R.I., 1986. Solution of the Implicitly Discretized Fluid-Flow Equations by Operator-Splitting. J Comput Phys 62, 40-65.
29. Jiang, Y., Chen, C.P., Tucker, P.K., 1991. Multigrid Solution of Unsteady Navier-Stokes Equations Using a Pressure Method. Numer Heat Tr a-Appl 20, 81-93.
30. Johnson, A.A., Tezduyar, T.E., 1996. Simulation of multiple spheres falling in a liquid-filled tube. Comput Method Appl M 134, 351-373.
31. Johnson, A.A., Tezduyar, T.E., 1999. Advanced mesh generation and update methods for 3D flow simulations. Comput Mech 23, 130-143.
32. Johnson, T.A., Patel, V.C., 1999. Flow past a sphere up to a Reynolds number of 300. J Fluid Mech 378, 19-70.
33. Joseph, G.G., Zenit, R., Hunt, M.L., Rosenwinkel, A.M., 2001. Particle-wall collisions in a viscous fluid. J Fluid Mech 433, 329-346.
34. Kim, J., Kim, D., Choi, H., 2001. An immersed-boundary finite-volume method for simulations of flow in complex geometries. J Comput Phys 171, 132-150.
35. Kunz, P.J., Kroo, I., 2002. Analysis and design of airfoils for use at ultra-low Reynolds numbers. Progr Astronaut Aero 195, 35-60.
36. Ladd, A.J.C., 1994a. Numerical Simulations of Particulate Suspensions Via a Discretized Boltzmann-Equation .Part1. Theoretical Foundation. J Fluid Mech 271, 285-309.
37. Ladd, A.J.C., 1994b. Numerical Simulations of Particulate Suspensions Via a Discretized Boltzmann-Equation .Part2. Numerical Results. J Fluid Mech 271, 311-339.
38. Ladd, A.J.C., Verberg, R., 2001. Lattice-Boltzmann simulations of particle-fluid suspensions. J Stat Phys 104, 1191-1251.
39. Lai, M.C., Peskin, C.S., 2000. An immersed boundary method with formal second-order accuracy and reduced numerical viscosity. J Comput Phys 160, 705-719.
40. Lamb, H., 1932. Hydrodynamics. New York: Dover.
41. Leclair, B.P., Hamielec, A.E., Pruppach.Hr, 1970. A Numerical Study of Drag on a Sphere at Low and Intermediate Reynolds Numbers. J Atmos Sci 27, 308.
42. Lee, C., 2003. Stability characteristics of the virtual boundary method in three-dimensional applications. J Comput Phys 184, 559-591.
43. Lin, S.Y., Chen, Y.C., 2013. A pressure correction-volume of fluid method for simulations of fluid-particle interaction and impact problems. Int J Multiphas Flow 49, 31-48.
44. Lin, S.Y., Chin, Y.H., Hu, J.J., Chen, Y.C., 2011a. A pressure correction method for fluid-particle interaction flow: Direct-forcing method and sedimentation flow. Int J Numer Meth Fl 67, 1771-1798.
45. Lin, S.Y., Lin, C.T., Chin, Y.H., Tai, Y.H., 2011b. A direct-forcing pressure-based lattice Boltzmann method for solving fluid-particle interaction problems. Int J Numer Meth Fl 66, 648-670.
46. Lomholt, S., Stenum, B., Maxey, M.R., 2002. Experimental verification of the force coupling method for particulate flows. Int J Multiphas Flow 28, 225-246.
47. Magnaudet, J., Rivero, M., Fabre, J., 1995. Accelerated Flows Past a Rigid Sphere or a Spherical Bubble .1. Steady Straining Flow. J Fluid Mech 284, 97-135.
48. Marella, S., Krishnan, S., Liu, H., Udaykumar, H.S., 2005. Sharp interface Cartesian grid method I: An easily implemented technique for 3D moving boundary computations. J Comput Phys 210, 1-31.
49. Marston, J.O., Yong, W., Thoroddsen, S.T., 2010. Direct verification of the lubrication force on a sphere travelling through a viscous film upon approach to a solid wall. J Fluid Mech 655, 515-526.
50. Michaelides, E.E., 1997. Review - The transient equation of motion for particles, bubbles, and droplets. J Fluid Eng-T Asme 119, 233-247.
51. Milne-Thomson, L.M.M., 1968. Theoretical Hydrodynamics., 5th ed ed. New York: Dover.
52. Nieuwstadt, F., Keller, H. B., 1973. Viscous flow past circular cylinders. Computers & Fluids. 1, 59-71.
53 Oseen, C. W., 1910. Über die Stokes'sche Formel und über eine verwandte Aufgabe in der Hydrodynamik. Arkiv för Matematik Astronomi och Fysik 6, pp. 1-20.
54. Pan, T.W., Joseph, D.D., Bai, R., Glowinski, R., Sarin, V., 2002. Fluidization of 1204 spheres: simulation and experiment. J Fluid Mech 451, 169-191.
55. Peskin, C.S., 1977. Numerical-Analysis of Blood-Flow in Heart. J Comput Phys 25, 220-252.
56. Pozrikidis, C., 1999. A spectral-element method for particulate Stokes flow. J Comput Phys 156, 360-381.
57. Schouveiler, L., Thompson, M. C., Leweke, T., Hourigan, K., 2009. Vortex Dynamics Associated with the Impact of a Cylinder with a Wall Iutam Bookser 14, 235-242.
58 Swearingen, J. D., Crouch, J. D. & Handler, R. A. 1995. Dynamics and stability of a vortex ring impacting a solid boundary. J. Fluid Mech. 297, 1–28.
59. ten Cate, A., Nieuwstad, C.H., Derksen, J.J., 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, 4012-4025.
60. Thompson, M.C., Hourigan, K., Cheung, A., Leweke, T., 2006. Hydrodynamics of a particle impact on a wall. Appl Math Model 30, 1356-1369.
61. Thompson, M.C., Leweke, T., Hourigan, K., 2007. Sphere-wall collisions: vortex dynamics and stability. J Fluid Mech 575, 121-148.
62. Uhlmann, M., 2005. An immersed boundary method with direct forcing for the simulation of particulate flows. J Comput Phys 209, 448-476.
63 Walker, J. D. A., Smith, C. R., Cerra, A. W. & Doligalski, T. L. 1987. The impact of a vortex ring on a wall. J. Fluid Mech. 181, 99–140.
64. White, F.M., 2006. Viscous Fluid Flow., 3rd ed. ed. McGraw-Hill.
65. Yang, F.L., 2006. Interaction law for a collision between two solid particles in a viscous liquid. California Institute of Technology, USA.
66. Yang, F.L., 2010. A formula for the wall-amplified added mass coefficient for a solid sphere in normal approach to a wall and its application for such motion at low Reynolds number. Phys Fluids 22.