本研究中，使用沉浸邊界法做流固耦合的問題分析。在探討兩球的不同幾何擺放位置對兩球沉降的(牽引(drafting)、接觸(kissing)、翻滾(Tumbling))DKT 現象的影響中，擺放方式分為Setup-A:大球置於標準球上方；Setup-B:相等尺寸之兩標準球:Setup-C:大球置於標準球下方。在Setup-C 的兩球擺放位置為大球置於標準球下方時，隨初始間距上升，兩球的接觸時間延後，最後轉變為(牽引(drafting)到
分離(separate))DS 現象。本研究探討初始間距對DS 現象的影響，產生DS 現象之最小初始間距隨球徑上升而下降，牽引時間在此最小初始間距時亦隨球徑增加而減少。
在模擬複雜幾何結構之流固耦合問題上，本研究使用標準鑲嵌語言法(Standard Tessellation Language (STL))格式的三角形表面網格，呈現幾何物體之外形。不同幾何物體之STL 格式三角形表面網格檔案由CAD 建立，STL 檔案中的物體表面之法向量與三點位置向量被應用於結構網格上判斷固液交界面。研究中利用球面公式驗證STL 格式之三角形表面網格之固液交界判斷之結果。並且在以一旋轉橢圓體驗證下平行運算與流速之結果。網格測試，由雷諾數為100 時，不同網格
密度下之流場通過ㄧ固定球體之情形，驗證流場。最後模擬在靜止流場中一個風扇低速旋轉的結果(雷諾數為100)。

In the present study, the problems of solid-fluid interaction using the immersed boundary strategy are investigated. The DKT phenomenon of two sedimenting spheres is presented. The DS phenomenon is observed in the case that the larger sphere is at the bottom. The duration of drafting term in the smallest initial gap in DS phenomenon is investigated at different diameter ratio in the case. On the other hand, in order to simulate the complex-shape object, the method of finding the boundary with the triangular facet surface is added into the numerical
method. The triangular facet surface of the object is performed in Standard Tessellation Language (STL) format. In the STL format, the normal vector and three positions of points are recorded in x, y, z-directions. The STL files for the different objects are designed by using CAD. The characteristic of the STL format triangular facet surface is used in the identification of the points around and inside the solid object. The forcing points, decided by using the STL-format triangular facets, is tested in the sphere case and the location of the forcing points are matched with the location found by employing the equation of sphere surface. The parallel computing and altering position of shape are used and validated with the case of rotating ellipsoid. Then, the case of the viscous flow past a sphere is presented. Finally, the case of a rotating turbine with a static fluid domain is performed.

Abstract i
Contents ii
List of Figures iv
List of Tables vi
1 Introduction 1
1.1 Introduction 1
1.2 Literature Survey 2
1.3 Objectives and Motivations 6
2 Numerical Methods 8
2.1 Immersed-Boundary Method 8
2.1.1 Mathematical Formulation 9
2.1.2 Numerical Scheme 10
2.1.3 Forcing Strategies 11
2.2 Treatment of immersed boundary with triangular facet surface 13
2.2.1 Establishment of bounding boxes 14
2.2.2 Determinations of points around the interface boundary domain 15
2.2.3 Determinations of solid domain 16
2.2.4 Determinations of forcing points 16
2.2.5 Determinations of interpolation 17
2.3 Determinations of lift and drag forces 18
2.4 Complete solution procedure 19
3 Numerical Results 35
3.1 Sedimentation of spheres 35
3.1.1 Collision between particles 36
3.1.2 code validation 37
3.1.3 collision between two spheres 38
3.1.4 sedimentation of two spheres 38
3.1.5 Different initial positions of two sedimenting spheres 39
3.1.6 DKT and DS phenomenon of two sedimenting spheres 40
3.2 Complex structure with triangular facet surface 41
3.2.1 Comparison of rotating sphere 42
3.2.2 Test of rotating ellipsoid 42
3.3 Flows past a sphere 43
3.4 Rotating turbine in static fluid 44
4 Conclusion 70

[1] C.S. Peskin. 1972. Flow patterns around heart valves: a numerical method. J.

Comp. Phys.10 : 252-271.


[2] C.S. Peskin. 1982. The fluid dynamics of heart valves: Experimental, theritiacal and computational methods. ANNU. REV. FLUID. MECH.14 : 235-259.

[3] T. Ye, R. Mittal, H.S. Udaykumar, W. Shyy. 1999. An accurate Cartesian grid method for viscous incompressible flows with complex immersed boundaries. J. Comput. Phys. 156 : 209-240.

[4] R.J. LeVeque, Z. Li. 1994. The immersed interface method for elliptic equations with discontinuous coefficients and singular sources. SIAM J. Numer. Anal. 156
: 1019-1044.


[5] R.J. LeVeque, Z. Li. 1997. The immersed interface method for Stokes flow with elastic boundaries or surface tension. Siam J. Dci. Comput. 18: 709-735.

[6] D. Calhoun. 2002. A Cartesian grid method for solving the two-dimensional streamfunction-vorticity equations in irresgular regions. J. Comp. Phys. 176 :
231-275.


[7] Z. Li, M.C. Lai. 2001. The ommersed interface method for the Navier-Stokes equations with singular forces. J. Comp. Phys.171 : 822-842.





ii



[8] M.C. Lai, C.S. Peskin. 2000. An immersed boundary method with formal second order accuracy and reduced numerical viscosity. J. Comp. Phy. 160
: 705-719.


[9] Z. Li. 2003. An overview of the immersed interface method and its applications.

Twaiwanese J. Math 7 : 1.


[10] L.E. Silva, A. Silveira-Neto, J.J.R. Damasceno. 2003. Numerical simulation of two-dimensional flow over a circular cylinder using the immersed boundary method. J. Comp. Phys. 189 : 351-370.

[11] D. Glodstein, R. Handler, L. Sirovich. 1993. Modeling a no-slip flow with an external force field. J. Comp. Phys. 105 : 354-366.

[12] D. Glodstein, R. Handler, L. Sirovich. 1995. Direct numerical simulation of turlent flow over a modeled riblet covered surface. J. Fluid Mech. 302 : 333-
376.


[13] E.M. Saiki, S. Biringen. 1996. Numerical simulation of a cylinder in uniform flow : application of a virtual boundary method. J. COmp. Phys. 123 : 450-465.

[14] J. Mohd-Yusof. 1997. Combined immersed boundary/B-Spline method for simulationsof flows in complex geometries in complex geometries. CTR annual research briefs NASA Ames/Stanford University.

[15] R. Verzicco, J. Mohd-Yusof, P.Orlandi, D. Haworth. 2000. LES in complex geometries using boundary body forces. AIAA Journal 38 : 27-433.

[16] E.A. Fadlum, R. Verzicco, P. Orlandi, J. Mohd-Yusof. 2000. Combined immersed boundary methods for three dimensional complex flow simulations. J. Comp. Phys. 161 : 35-60.

[17] E. Balaras. 2004. Modeling complex boundaries using an external force field on fixed Cartesian grids in large-eddy simulations. Comput. Fluids 33 : 375-404.



[18] Y.H. Tseng, J.H. Ferziger. 2003. A ghost-cell immersed boundary boundary method for flow in complex geometry. J. Comp. Phys. 192 : 593-623.

[19] M. Tyagi, S. Acharya. 2005. Large eddy simulation of turbulent flows in complex and moving rigid geometries using the immersed boundary method. J. Numer. Meth. Fluids 48 : 691-722.

[20] H.S. Udaykumar, R. MIttal, W. Shyy. 1999. Computation of solid-liquid phase fronts in the sharp interface limit on fixed grids. J. Comp. Phys. 153 : 535-574.

[21] H.S. Udaykumar, R. MIttal, P. Rampunggoon, A. Khanna. 2001. A sharp interface Cartesian grid method for simulating flows with complex moving boundaries. J. Comp. Phys. 174 : 345-380.

[22] S. Marella, S. Krishnan, H. Liu, H.S. Udaykumar. 2005. Sharp interface Cartesian grid method I: An easily implemented technique for 3D moving boundary computations. J. Comp. Phys. 210 : 1-31.

[23] P.H. Chang, C.C. Liao , H.W. Hsu , S.H. Liu and C.A. Lin. 2014. Simulations of laminar and turbulent flows over periodic hills with immersed boundary method. Comput. Fluids 92 : 233-243.

[24] S.W. Su, M.C. Lai, C.A. Lin. 2007. A simple immersed boundary technique for simulating amplex flows with rigid boundary. Comput. Fluids 36 : 313-324.

[25] J. Kim, D. Kim, H. Choi. 2001. An immersed boundary finite volume method for simulations of flow in complex geometries. J. Comp. Phys. 171 : 132-150.

[26] D. Kim, H. Choi. 2006. Immersed boundary method for flow around an arbitrarily moving body. J. Comp. Phys. 212 : 662-680.

[27] J. Yang, E. Balaras. 2006. An embedded-boundary formulation for large-eddy simulation of turbulent flows interacting with moving boundaries. J. Comp. Phys. 215 : 12-24.



[28] C.C. Liao, Y.W. Chang, C.A. Lin and J.M. McDonough. 2010. Simulating flows with moving rigid boundary using immersed-boundary method. Comput. Fluids 39 : 152-167.

[29] R. Glowinski, T.W. Pan, T.I. Hesla, D.D. Joseph. 2001. J. Periaux, A fictious domain approach to the direct numerical simulation of imcompressible viscous flow past moving rigid bodies: application to particulate flow. J. Comp. Phys.
169 : 363-426.


[30] J. Yang, F. Stern. 2014. A sharp interface direct forcing immersed boundary approach for fully resolved simulations of particulate flows. J. Fluids Eng. 136
: 040904-1.


[31] S. Balay and K. Buschelman and W. D. Gropp and D. Kaushik and

M. G. Knepley and L. C. Mclnnes. 2010. PETSc Web page < http :

//www.mcs.anl.gov/petsc >.


[32] H.W. Hsu, F.N. Hwang, Z.H. Wei, S.H. Lai, C.A. Lin. 2011. A parallel multilevel preconditioned iterative pressure Poisson solver for the large-eddy simulation of turbulent flow inside a duct. Comput. Fluids 45 : 138-146.

[33] M.N Chang. 2013. A parallel multilevel presondidioned iterative pressure Poisson solver for 3D lid-driven cavity. Master thesis, Department of Mechanical Engineering, National Tsing Hua University.

[34] A.F Fortes,D.D. Joseph, T.S Lundgren. 1987. Nolinear mechanics of fluidization of beds of spherical particle. J. Fluid Mech 177 : 407-483.

[35] D.D. Joseph, J. Nelson, H.H. Hu, Y.J. Liu. 1987. Competition btween inertial pressures and normal stresses in the flow induced anisotropy of solid particles. J. Fluid Mech 177 : 407-483.



[36] Feng J, Hu HH, Joseph DD. 1994. Direct simulation of initial value problems for the motion of solid bodies in a Newtonian fluid. Part 1. Sedimentation. J. Fluid Mech 261 : 95-134.

[37] Glowinski R, Pan TW, Hesla TI. 1999. Joseph DD,A distributed Lagrange multiplier/fictitious domain method for particulate flows. Int. J. Multiph. Flow
25 : 755-794.


[38] Uhlmann M. 2005. An immersed boundary method with direct forcing for the simulation of particulate flows. J. Comput. Phys. 209 : 448-476.

[39] Wan D. Turek S. 2006. Direct numerical simulation of particulate flow via multigrid FEM techniques and the fictitious boundarymethod. Int. J. Numer. Methods Fluids 51 : 531-566.

[40] Wang L, Guo ZL, Mi JC. 2014. Drafting, kissing and tumbling process of two particles with different sizes. Comput. Fluids 96 : 20-34.

[41] Chuan-Chieh Liao, Wen-Wei Hsiao, Ting-Yu Lin, Chao-An Lin. 2015.

Simulations of two sedimenting-interacting spheres with different sizes and initial configurations using immersed boundary method. Comput. Mech. 55
: 1191-1200.


[42] Johnson A, Tezduyar TE. 1996. Simulation of multiple spheres falling in a liquid-filled tube, Comput. Methods Appl. Mech. Engng. 134 : 351-373.

[43] Johnson A, Tezduyar TE. 1997. 3D simulation of fluid-particle interactions with the number of particles reaching 100. Comput. Methods Appl. Mech. Engng.
145 : 301-321.


[44] Johnson A, Tezduyar TE. 1999. Advanced mesh generation and update methods for 3D flow simulations. Computational Mech. 23 : 130-143.



[45] Sharma N, Patankar NA. 2005 A fast computation technique for the direct numerical simulation of rigid particulate flows. J. Comput. Phys. 205 : 439-
457.


[46] Apte SV, Martin M, Patankar NA. 2009. A numerical method for fully resolved simulation (FRS) of rigid particleflow interactions in complex flows. J. Comput. Phys. 228 : 2712-2738.

[47] Breugem WP. 2012. A second-order accurate immersed boundary method for fully resolved simulations of particle-laden flows. J. Comput. Phys. 231 : 4469-
4498.


[48] Charles Hull. 1988. StereoLithography Interface Specification. 3D Systems. Inc.

July 1988


[49] Nastase-Dan Ciobota. 2012. Standard Tessellation Language in Rapid Prototyping Technology. The Scientific Bulletin of VALA HIA University MATERIALS and MECHANICS Nr. 7 (year 10) : 201.

[50] G. Tryggvason , B. Bunner, A. Esmaeeli, D. Juric, N. Al-Rawahi, W. Tauber, J.

Han, S. Nas, Y.-J. Jan. 20001. A Front-Tracking Method for the Computations of Multiphase Flow. J. Comp. Phys. 169 : 708-759.

[51] A. Gilmanov, F. Sotiropoulos, E. Balaras. 2003. A general reconstruction algorithm for simulating flows with 3D complex immerd boundaries on Cartesian grids. J. Comp. Phys. 191 : 660-669.

[52] Anvar Gilmanov, Fotis Sotiropoulos. 2005. A hybrid Cartesian/immersed boundary method for simulating flows with 3D, geometrically complex, moving bodies. J. Comp. Phys. 207 : 457-492.



[53] I. Borazjani, L. Ge, and F. Sotiropoulos. 2008. Curvilinear Immersed Boundary Method for Simulating Fluid Structure Interaction with Complex 3D Rigid Bodies. J Comput Phys. 227 : 75877620.

[54] R. Mittal, H. Dong, M. Bozkurttas, F.M. Najjar, A. Vargas, A. von loebbeck.

2008. A versatile sharp interface immerd boundary method for incompressible flows with complex boundarys. J. Comp. Phys. 227 : 4825-4852.

[55] Borazjani. 2013. Fluidstructure interaction, immersed boundary-finite element method simulations of bio-prosthetic heart valves. Comput.Methods in Appl.Mech.Eng. 257 : 103116.

[56] Gianluca Iaccarino, Roberto Verzicco. 2003. Immersed boundary technique for turbulent flow simulations. Appl. Mech. Rev. 56 : 331-347.

[57] Jianming Yang, Frederick Stern. 2013. Robust and Efficient Setup Procedure for Complex Triangulations in Immersed Boundary Simulations. J. Fluids Eng.
135 : Issue 10


[58] Jung-Il Choi, Roshan C. Oberoi, Jack R. Edwards, Jacky A. Rosati. 2007.

An immersed boundary method for complex incompressible flows. J. Comput. Phys. 224 : 757-784.

[59] H. Choi, P. Moin. 1994. Effects of the computational time step on numerical solutions of turbulent flow. J. Comput. Phys. 113 : 1-4.

[60] J. Kim, P. Moin. 1985. Application of a fractional-step method to incompressible Navier-Stokes equations. J. Comput. Phys. 59 : 308-323.

[61] E. Haines. 1994. Point in polygon strategies. Academic Press Graphics Gems

Series : 24-46.



[62] Iman Borazjani, Liang Ge, and Fotis Sotiropoulos. 2008. Curvilinear Immersed Boundary Method for Simulating Fluid Structure Interaction with Complex 3D Rigid Bodies. J. Comput Phys. 227 : 75877620.

[63] D.V. Le, B.C. Khoo, K.M. Lim. 2008. An implicit-forcing immersed boundary method for simulating viscous flows in irregular domain. Comp. Methods Appl. Engrg. 197 : 2118-2130.

[64] A. ten Cate, C.H. Nieuwstad, J.J. Derksen, H.E. A Van den Akker. 2002.

Particle imaging velocimetry experiments and lattice-Boltzmann simulations on a single sphere settling under gravity. Phys. Fluids 14 : 4012-4025.

[65] T. A. Johnson, V. C. Patel. 1999. Flow past a sphere up to a Reynolds number of 300. Fluid. Mech 378 : 19-70.

[66] P. K. Mohanta, C. A. Lin, R. S. PATIL. 2016. Three dimensional simulations of Wind Turbine flows with Immersed Boundary Method. Undergraduate Thesis,BIRLA INSTITUTE OF TECHNOLOGY AND SCIENCE PILANI, GOA CAMPUS

[67] Z.G. Feng, E.E. Michaelides. 2005. Proteus: a direct forcing method in the simulations of particulate flows. J. Comput. Phys. 202 : 251.

[68] J. Jonkman, S. Butterfield, W. Musial, and G. Scott. 2009. Definition of a

5-MW Reference Wind Turbine for Offshore System Development. Technical

Report NREL/TP-500-38060 February 2009