臺灣博碩士論文加值系統

在光滑粒子水動力法(SPH)中,如果要計算固體受力必須沿著固體邊界進行應力積分。
但是由於SPH是由基於拉格朗日方法因此在計算應力時程序繁複困難,主要是因為需計算固體複雜邊界的法線方向與切線方向,因此本研究引入直接施力沈浸邊界法(DFIB)來計算SPH方法中的固體受力,藉此進行流固耦合的數值計算。相對於原來SPH方法需要進行邊界的應力積分,DFIB方法則只要把固體區域內的虛擬力量積分即可的固體的受力,不需要計算邊界的法線與切線方向,程序上較為簡潔易執行。為了驗證這個SPH-DFIB 方法,固定圓柱在一移動上板引致的穴流以及一移動圓柱在一封閉空間引起的流場用來 作為驗證例子。SPH-DFIB方法計算圓柱受力結果與其他方法或文獻相比具有一致性,因此可證明SPH-DFIB方法在計算固體受力的正確性。另外本SPH-DFIB方法也用於在一自由液面下的移動圓柱所引起的波浪與流場。計算結果顯示,流場型態可依圓柱附近的渦漩變化分為三個模態。同時可發現KC數對於圓柱受力有較為明顯的影響。

The common method to predict the hydrodynamic loading in Smoothed Particle Hydrodynamics (SPH) is the calculation of the surface integral. Because of the Lagrangian nature of the SPH, enforcing the non-slip boundary condition is a challenging task. In addition, calculation of the hydrodynamic force requires information of normal vector which is difficult to be obtained for moving body and complex geometry. To study the solid-fluid interaction and to calculate the hydrodynamic force, the direct forcing immersed boundary method (DFIB) is implemented to the SPH scheme. Instead of surface integration, DFIB method calculate the hydrodynamic force using volume integration. Flow past a cylinder in a lid-driven cavity and an oscillating cylinder in an enclosure at low Keulegan-Carpenter (KC) number are used as validation cases and the results obtained by the proposed SPH-DFIB method are compared with the benchmark results.The comparisons show that the proposed SPH-DFIB method is able to predict hydrodynamic loadings on a cylinder properly. The proposed method is applied to simulate an oscillating cylinder beneath a free surface in a quiescent fluid. Simulations are performed at various KC numbers and depth ratios. The flow patterns are divided into three modes according to the characteristic of wake around the cylinder, the number of vortices and vortex migration patterns. The results demonstrate that the effect of KC number is considerably larger than the depth of submergence on the in-line force. On the contrary, the transverse force and free surface elevation are more affected by depth of submergence, in particular at a low KC number.

Chinese Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . i
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . iv
Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
Nomenclatures ... . . . . . . . . . . . . . . . . . . . . . . . . . viii
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . xii
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . xii
1 INTRODUCTION 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Literature review. . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Synopsis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2 MATHEMATICAL FORMULAE AND NUMERICAL MODEL 8
2.1 Governing equations. . . . . . . . . . . . . . . . . . . . . . . 8
2.2 Numerical method . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3 Detail of computations . . . . . . . . . . . . . . . . . . . . . 17
3 RESULTS AND DISCUSSION 19
3.1 Validation of SPH-DFIB model . . . . . . . . . . . . . . . . . . 19
3.2 Oscillating cylinder beneath a free surface . . . . . . . . . . 23
4 CONCLUSIONS AND FUTURE WORKS 29
4.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.2 Future works . . . . . . . . . . . . . . . . . . . . . . . . . . 31

Antuono, M., Colagrossi, A. & Marrone, S. 2012 Numerical diffusive terms in weakly-compressible SPH schemes. Computer Physics Communications 183 (12), 2570–2580.

Beeman, D. 1976 Some multistep methods for use in molecular dynamics calculations. Journal of Computational Physics 20 (2), 130–139.

Chern, M. J., Kuan, Y. H., Nugroho, G., Lu, G. T & Horng, T. L. 2014 Direct-forcing immersed boundary modeling of vortex-induced vibration of a circular cylinder. Journal of Wind Engineering and Industrial Aerodynamics 134, 109–121.

Chern, M. J., Shiu, W. C. & Horng, T. L. 2013 Immersed boundary modeling for interaction of oscillatory flow with cylinder array under effects of flow direction and cylinder arrangement. Journal of Fluids and Structures 43, 325–346.

Colagrossi, A., Antuono, M. & Le Touze, D. 2009 Theoretical considerations on the free-surface role in the smoothed-particle-hydrodynamics model. Physical Review E - Statistical, Nonlinear, and Soft Matter Physics 79 (5), 1–13.

Colagrossi, A. & Landrini, M. 2003 Numerical simulation of interfacial flows by smoothed particle hydrodynamics. Journal of Computational Physics 191 (2),
448–475.

Crespo, A. J. C.,Gomez G. M.,Dalrymple, R. A. 2007 Boundary conditions generated by dynamic particles in SPH methods. Computers, Materials and Continua 5 (3), 173–184.

Dutch, H., Durst F., Becker, S. & Lienhart, H. 1998 Oscillating Circular Cylinder At Low KeuleganCarpenter numbers. Journal of Fluid Mechanics 360, 249-271.

Fadlun, E. A., Verzicco, R., Orlandi, P. & Mohd-Yusof, J. 2000 Combined Immersed-Boundary Finite-Difference Methods for Three-Dimensional Complex Flow Simulations. Journal of Computational Physics 161 (1), 35–60.

Gomez-Gesteira, M., Rogers, B. D., Crespo, Alejandro J. C., Dalrymple, R. a., Narayanaswamy, M. & Domınguez, Jose M. 2012 SPHysics deveopment of a free-surface fluid solverPart 1: Theory and formulations. Computers & Geosciences 48 (January 2016), 289–299.

Lam, K. M. & Dai, G. Q. 2002 Formation of vortex street and vortex pair from a circular cylinder oscillating in water. Experimental Thermal and Fluid Science 26 (8),901–915.

Lin, J.-C. & Rockwell, D. 1999 Horizontal oscillations of a cylinder beneath a free surface: vortex formation and loading. Journal of Fluid Mechanics 389, 1–26.

Marrone, S. 2011 Enhanced SPH modeling of free-surface flows with large deformations. PhD thesis, University of Rome.

Mayer, S., Garapon, A. & Srensen, L.S. 1998 Free-Surface Flow With Applications To Non-Linear Wave Dynamics. International Journal for Numerical Methods in Fluids 315 (April 1997), 293–315.

Molteni, D. & Colagrossi, A. 2009 A simple procedure to improve the pressure evaluation in hydrodynamic context using the SPH. Computer Physics Communications 180 (6), 861–872.

Monaghan, J. J. & Gingold, R. A. 1983 Shock simulation by the particle method SPH. Journal of Computational Physics 52 (2), 374–389.

Monaghan, J. J. & Kos, A 1999 Solitary Waves on a Cretan Beach. Journal of Waterway, Port, Coastal, and Ocean Engineering 125 (3), 145–154.

Moussa, B. B. & Vila, J. P. 2000 Convergence of SPH Method for Scalar Non-linear Conservation Laws. SIAM Journal on Numerical Analysis 37 (3), 863–887.

Noor, D.Z., Chern, M.J. & Horng, T.L. 2009 An immersed boundary method to solve fluid solid interaction problems. Computational Mechanics 44, 447–453.

Peskin, C.S. 1972 Flow patterns around heart valves: A numerical method. Journal of Computational Physics 10 (2), 252–271.

Schwaiger, H. F. 2008 An implicit corrected SPH formulation for thermal diffusion with linear free surface boundary conditions. International Journal for Numerical Methods in Engineering 75 (6), 647–671.

Shen, L., Chan, E. S. & Lin, P. 2009 Calculation of hydrodynamic forces acting on a submerged moving object using immersed boundary method. Computers and Fluids 38 (3), 691–702.

Wendland, H. 1995 Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree. Advances in Computational Mathematics 4 (1), 389–396.