本文在弱可壓縮壓力方程式的基礎下，提出了沉浸式邊界法和壓力修正法相結合的方法以處理流固耦合問題，其中包含固定邊界與移動邊界問題。本數值方法為完全顯式並使用在交錯笛卡爾網格上，並使用多核圖形顯示卡進行平行計算。 本篇使用幾個基準問題以驗證當前工作中所提出的方法的準確性，包括流場通過不對稱放置的圓柱體、於靜止水箱中移動的圓柱與等流速的流場通過球體的問題。 並驗證所提出的方法可獲得與基準解決方案相符的結果。 此外，本篇也將演示出弱可壓縮壓力方程式中數值參數(包括普朗特數與馬赫數)對模擬結果所產生的影響。

Within the framework of the general pressure equation, both fixed and moving boundary problems are handled in the present thesis. A combination of the immersed boundary method and the pressure correction approach is proposed to deal with the fluid-structure interaction problems. The computation is fully explicit for implementing on multi-GPUs with staggered Cartesian mesh. Several benchmarking problems are simulated to validate the proposed method accuracy in the present work, including flow over an asymmetrically placed cylinder, moving cylinder, and flow over a sphere. The proposed method can obtain results compatible with the benchmark solutions. Furthermore, the influence of the numerical parameters in the general pressure equation will also be demonstrated.

1 Introduction 1
1.1 Literature Survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Objectives and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Numerical Methods 7
2.1 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Mathematical Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3 Forcing Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.4 Calculation of lift and drag forces . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.5 Full Solution Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3 Numerical Results 18
3.1 Poiseuille channel flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.2 Flow past an asymmetrically placed cylinder . . . . . . . . . . . . . . . . . . . . 19
3.3 In-line oscillating cylinder in a fluid at rest . . . . . . . . . . . . . . . . . . . . . 22
3.4 Uniform flow past a still sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4 Conclusion 38
5 Future Work 40

[1] H.-B. Deng, Y.-Q. Xu, D.-D. Chen, H. Dai, J.Wu, and F.-B. Tian, “On numerical modeling
of animal swimming and flight,” Computational Mechanics, vol. 52, no. 6, pp. 1221–1242,
2013.
[2] X. Wu, X. Zhang, X. Tian, X. Li, and W. Lu, “A review on fluid dynamics of flapping
foils,” Ocean Engineering, vol. 195, p. 106712, 2020.
[3] T. A.Ward, C. J. Fearday, E. Salami, and N. Binti Soin, “A bibliometric review of progress
in micro air vehicle research,” International Journal of Micro Air Vehicles, vol. 9, no. 2,
pp. 146–165, 2017.
[4] J. Young, J. C. Lai, and M. F. Platzer, “A review of progress and challenges in flapping
foil power generation,” Progress in aerospace sciences, vol. 67, pp. 2–28, 2014.
[5] W. Kim and H. Choi, “Immersed boundary methods for fluid-structure interaction: A
review,” International Journal of Heat and Fluid Flow, vol. 75, pp. 301–309, 2019.
[6] B. E. Griffith and N. A. Patankar, “Immersed methods for fluid–structure interaction,”
Annual review of fluid mechanics, vol. 52, pp. 421–448, 2020.
[7] C. S. Peskin, “Flow patterns around heart valves: A numerical method,” Journal of
Computational Physics, vol. 10, no. 2, pp. 252–271, 1972.
[8] D. Goldstein, R. Handler, and L. Sirovich, “Modeling a no-slip flow boundary with an
external force field,” Journal of computational physics, vol. 105, no. 2, pp. 354–366, 1993.
[9] J. Mohd-Yusof, “Combined immersed-boundary/b-spline methods for simulations of flow
in complex geometries,” Center for turbulence research annual research briefs, vol. 161,
no. 1, pp. 317–327, 1997.
[10] E. Fadlun, R. Verzicco, P. Orlandi, and J. Mohd-Yusof, “Combined immersed-boundary
finite-difference methods for three-dimensional complex flow simulations,” Journal of
computational physics, vol. 161, no. 1, pp. 35–60, 2000.
[11] M.-C. Lai and C. S. Peskin, “An immersed boundary method with formal second-order
accuracy and reduced numerical viscosity,” Journal of computational Physics, vol. 160,
no. 2, pp. 705–719, 2000.
[12] J. Kim, D. Kim, and H. Choi, “An immersed-boundary finite-volume method for
simulations of flow in complex geometries,” Journal of computational physics, vol. 171,
no. 1, pp. 132–150, 2001.
[13] J. Yang and E. Balaras, “An embedded-boundary formulation for large-eddy simulation of
turbulent flows interacting with moving boundaries,” Journal of Computational Physics,
vol. 215, no. 1, pp. 12–40, 2006.
[14] S.-W. Su, M.-C. Lai, and C.-A. Lin, “An immersed boundary technique for simulating
complex flows with rigid boundary,” Computers & Fluids, vol. 36, no. 2, pp. 313–324,
2007.
[15] J.-I. Choi, R. C. Oberoi, J. R. Edwards, and J. A. Rosati, “An immersed boundary method
for complex incompressible flows,” Journal of Computational Physics, vol. 224, no. 2,
pp. 757–784, 2007.
[16] A. Mark and B. G. van Wachem, “Derivation and validation of a novel implicit secondorder
accurate immersed boundary method,” Journal of Computational Physics, vol. 227,
no. 13, pp. 6660–6680, 2008.
[17] C.-C. Liao, Y.-W. Chang, C.-A. Lin, and J. McDonough, “Simulating flows with moving
rigid boundary using immersed-boundary method,” Computers & Fluids, vol. 39, no. 1,
pp. 152–167, 2010.
[18] H. Luo, H. Dai, P. J. F. de Sousa, and B. Yin, “On the numerical oscillation of the directforcing
immersed-boundary method for moving boundaries,” Computers & Fluids, vol. 56,
pp. 61–76, 2012.
[19] C. Liu and C. Hu, “An efficient immersed boundary treatment for complex moving object,”
Journal of Computational Physics, vol. 274, pp. 654–680, 2014.
[20] D. M. Martins, D. M. Albuquerque, and J. C. Pereira, “Continuity constrained leastsquares
interpolation for sfo suppression in immersed boundary methods,” Journal of
Computational Physics, vol. 336, pp. 608–626, 2017.
[21] S. Takeuchi, H. Fukuoka, J. Gu, and T. Kajishima, “Interaction problem between fluid
and membrane by a consistent direct discretisation approach,” Journal of Computational
Physics, vol. 371, pp. 1018–1042, 2018.
[22] J. Yang, “Sharp interface direct forcing immersed boundary methods: a summary of some
algorithms and applications,” Journal of Hydrodynamics, vol. 28, no. 5, pp. 713–730, 2016.
[23] J. Lee, J. Kim, H. Choi, and K.-S. Yang, “Sources of spurious force oscillations from an
immersed boundary method for moving-body problems,” Journal of computational physics,
vol. 230, no. 7, pp. 2677–2695, 2011.
[24] L. A. Miller and C. S. Peskin, “When vortices stick: an aerodynamic transition in tiny
insect flight,” Journal of Experimental Biology, vol. 207, no. 17, pp. 3073–3088, 2004.
[25] D. Kim and H. Choi, “Immersed boundary method for flow around an arbitrarily moving
body,” Journal of Computational Physics, vol. 212, no. 2, pp. 662–680, 2006.
[26] A. J. Chorin, “A numerical method for solving incompressible viscous flow problems,”
Journal of Computational Physics, vol. 2, no. 1, pp. 12–26, 1967.
[27] R. Peyret, “Unsteady evolution of a horizontal jet in a stratified fluid,” Journal of Fluid
Mechanics, vol. 78, no. 1, p. 49–63, 1976.
[28] A. Gilmanov and F. Sotiropoulos, “A hybrid cartesian/immersed boundary method
for simulating flows with 3d, geometrically complex, moving bodies,” Journal of
Computational Physics, vol. 207, no. 2, pp. 457–492, 2005.
[29] J. R. Clausen, “Entropically damped form of artificial compressibility for explicit
simulation of incompressible flow,” Phys. Rev. E, vol. 87, p. 013309, Jan 2013.
[30] Y. T. Delorme, K. Puri, J. Nordstrom, V. Linders, S. Dong, and S. H. Frankel, “A simple
and efficient incompressible navier–stokes solver for unsteady complex geometry flows on
truncated domains,” Computers & Fluids, vol. 150, pp. 84–94, 2017.
[31] A. Toutant, “General and exact pressure evolution equation,” Physics Letters A, vol. 381,
p. 3739–3742, Nov 2017.
[32] A. Toutant, “Numerical simulations of unsteady viscous incompressible flows using general
pressure equation,” Journal of Computational Physics, vol. 374, pp. 822–842, 2018.
[33] X. Shi and C.-A. Lin, “Simulations of wall bounded turbulent flows using general pressure
equation,” Flow, Turbulence and Combustion, pp. 1–16, 2020.
[34] J.-J. Huang, “Numerical simulation of two-phase incompressible viscous flows using general
pressure equation,” 2020.
[35] F. H. Harlow and J. E. Welch, “Numerical calculation of time-dependent viscous
incompressible flow of fluid with free surface,” The physics of fluids, vol. 8, no. 12, pp. 2182–
2189, 1965.
[36] C.-W. Shu and S. Osher, “Efficient implementation of essentially non-oscillatory shockcapturing
schemes,” Journal of computational physics, vol. 77, no. 2, pp. 439–471, 1988.
[37] M. Sch¨afer, S. Turek, F. Durst, E. Krause, and R. Rannacher, “Benchmark computations
of laminar flow around a cylinder,” in Flow simulation with high-performance computers
II, pp. 547–566, Springer, 1996.
[38] H. D¨utsch, F. Durst, S. Becker, and H. Lienhart, “Low-reynolds-number flow around
an oscillating circular cylinder at low keulegan–carpenter numbers,” Journal of Fluid
Mechanics, vol. 360, pp. 249–271, 1998.
[39] K. Ceylan, A. Altunba¸s, and G. Kelbaliyev, “A new model for estimation of drag force
in the flow of newtonian fluids around rigid or deformable particles,” Powder technology,
vol. 119, no. 2-3, pp. 250–256, 2001.
[40] H. Sakamoto and H. Haniu, “The formation mechanism and shedding frequency of vortices
from a sphere in uniform shear flow,” Journal of Fluid Mechanics, vol. 287, pp. 151–171,
1995.
[41] R. Mittal, J. Wilson, and F. Najjar, “Symmetry properties of the transitional sphere wake,”
AIAA journal, vol. 40, no. 3, pp. 579–582, 2002.
[42] T. Johnson and V. Patel, “Flow past a sphere up to a reynolds number of 300,” Journal
of Fluid Mechanics, vol. 378, pp. 19–70, 1999.
[43] P. Bagchi, M. Ha, and S. Balachandar, “Direct numerical simulation of flow and heat
transfer from a sphere in a uniform cross-flow,” J. Fluids Eng., vol. 123, no. 2, pp. 347–
358, 2001.
[44] S. Marella, S. Krishnan, H. Liu, and H. Udaykumar, “Sharp interface cartesian grid method
i: an easily implemented technique for 3d moving boundary computations,” Journal of
Computational Physics, vol. 210, no. 1, pp. 1–31, 2005.
[45] R. Mittal, H. Dong, M. Bozkurttas, F. Najjar, A. Vargas, and A. Von Loebbecke, “A
versatile sharp interface immersed boundary method for incompressible flows with complex
boundaries,” Journal of computational physics, vol. 227, no. 10, pp. 4825–4852, 2008.