臺灣博碩士論文加值系統

This thesis uses immersed interface method to simulate flows with elastic boundary and the unclosed elastic boundary problem.
Unusually, here we develop the new model with finite volume method in staggered grids. It also can achieve equivalent accuracy to finite difference method with less jump conditions. In order to improve the accuracy of pressure, we have to derive a pressure correction equation for the fractional step method. To maintain the uniformity of the intersection locations for a moving boundary problem, we use cubic spline interpolation to reconstruct Lagrangian
markers with constant arc length and Fourier filtering to smooth the curve throughout each time step after moving location. Numerical experiments also show the stability limit can be improved if the distributions of Lagrangian markers are uniform. Several problems are simulated by using the present method.

Abstract ........................................i
List of Figures .................................iv
1 Introduction...........................................1
1.1Introduction .........................................1
1.2 Paper Survey ........................................5
1.3 Objective and Motivations ...........................9
2 The Methodology of Immersed Boundary Technique ........11
2.1 Mathematical Formulations ...........................11
2.2 Governing Equations .................................11
2.3 The Forcing Strategy.................................13
2.3.1 Identification of Forcing Points ..................17
2.4 The Calculation of Force Density ....................18
2.4.1 Spring force and bending force ....................18
2.4.2 curvature force .................................. 20
2.4.3 Spring force and curvature force ..................20
2.5 Evaluation of Boundary Moving Velocity ..............20
2.5.1 Periodic Cubic Spline Interpolation................21
2.5.2 Bilinear Interpolation.............................22
2.5.3 Reconstructing Lagrangian Markers Distribution ....22
2.6 Numerical Algorithm..................................23
2.6.1 Discretization of the Transport Equations..........23
2.6.2 Fractional Step Method.............................27
2.7 The Full Solution Procedure..........................29
3 Numerical Simulations and Results......................37
3.1 The accuracy of the jump correction algorithm........37
3.2 Normal force on an interface[54].....................40
3.3 Tangential force on an interface[54].................41
3.4 Elliptic elastic membrane............................42
3.4.1 Force generated by Spring and Bending force........42
3.4.2 Force generated by Surface tension and curvature...43
3.5 Unclosed elastic membrane with fixed end points......44
3.5.2 A semicircle at the equilibrium state..............45
4 Conclusion.............................................46
4.1 Theconclusion........................................46
4.2 Future work..........................................47
5 Figures................................................48

[1] Lin CH, Lin CA. Simple high-order bounded convection scheme to model discontinuities. AIAA J. 1997;35:563.
[2] Ye T, Mittal R, Udaykumar HS, Shyy W. An accurate Cartesian grid method for viscous incompressible flows with complex immersed boundaries. J. Comp.Phys. 1999;156:209.
[3] Li Z. The Immersed Interface Method - A Numerical Approach for PartialDifferential Equations with Interfaces. Ph.D. thesis University of Washington 1994.
[4] Li Z. A note on immersed interface methods for three dimensional elliptic equations. Comput. Math. Appl. 1996;31:9.
[5] LeVeque RJ, Li Z. The immersed interface method for elliptic equations with discontinuous coefficients and singular sources. SIAM J. Numer. Anal. 1994;31:1019.
[6] LeVeque RJ, Li Z. Immersed interface method for Stokes flow with elastic boundaries or surface tension. SIAM J. Sci. Comput. 1997;18:709.
[7] Andreas Wiegmann, Kenneth P. Bube. The immersed interface method for nonlinear differential equations with discontinuous coefficients and singular sources. SIAM J. Numer. Anal. 1998;35:177.
[8] Andreas Wiegmann, Kenneth P. Bube. The explicit-jump immersed interface method: finite difference methods for PDEs with piecewise smooth solutions.
SIAM J. Numer. Anal. 2000;37:827.
[9] Aaron L. Fogelson, James P. Keener. Immersed interface method for Neumann and related problems in two and three dimensions. SIAM J. Sci. Comput.
2000;22:1630.
[10] Li Z. An overview of the immersed interface method and its applications. Taiwanese J. Math. 2003;7:1.
[11] Calhoun D. A Cartesian grid method for solving the two-dimensional streamfunction-vorticity equations in irregular regions. J. Comp. Phys.2002;176:231.
[12] Li Z, Wang C. A fast finite difference method for solving Navier-Stokes equations of irregular domains. Commun. Math. Sci. 2003;1:180.
[13] Li Z, Lai MC. The immersed interface method for the Navier - Stokes equations
with singular forces. J. Comp. Phys. 2001;171:822.
[14] Le DV, Khoo BC, Peraire J. An immersed interface method for the incompressible Navier Stokes equations in irregular domains. Proceedings of the third MIT conference on computational fluid and solid mechanics. Elsevier
Science 2005.
[15] Li Z. An overview of the immersed interface method and its applications. Taiwanese J. Math. 2003;7:1.
[16] Peskin CS. Flow patterns around heart valves: a numerical method. J. Comp. Phys. 1972;10:252-271.
[17] Peskin CS. The fluid dynamics of heart valves: Experimental, theoritiacal and computational methods. Annual Review of Fluid Mechanics 1982;14:235.
[18] Beyer RP, LeVeque RL. Analysis of a one-dimensional model for the immersed boundary method. SIAM J. Numer. Anal. 1992;29:332.
[19] Lai MC, Peskin CS. An immersed boundary method with formal second order accuracy and reduced numerical viscosity. J. Comp. Phys. 2000;160:705.
[20] Peskin CS, Printz BF. Improved volume conservation in the computation of flows with immersed elastic boundaries. J. Comp. Phys. 1993;105:33.
[21] Cortez R, Minion M. The blob projection method for immersed boundary problems, J. Comp. Phys. 2000;161:428.
[22] Mohd-Yusof J. Combined immersed boundary/B-Spline method for simulations
of flows in complex geometries in complex geometries CTR annual research briefs. NASA Ames/Stanford University 1997.
[23] Verzicco R, Mohd-Yusof J, Orlandi P, DHaworth D. Large eddy simulation in complex geometry configurations using boundary body forces. AIAA J.
2000;38:427.
[24] Fadlun EA, Verzicco R, Orlandi P, Mohd-Yusof J. Combined immersed-boundary finite-difference methods for three dimensional complex flow simulations. J. Comp. Phys. 2000;161: 30.
[25] Goldstein D, Handler R, Sirovich L. Modeling a no-slip flow with an external force field. J. Comp. Phys. 1993;105:354.
[26] Goldstein D, Hadler R, Sirovich L. Direct numerical simulation of turbulent
flow over a modeled riblet covered surface. J. Fluid Mech. 1995;302:333.
[27] Lima E Silva ALF, Silveira-Neto A, Damasceno JJR. Numerical simulation of
two-dimensional flows over a circular cylinder using the immersed boundary method. J. Comp. Phys. 2003;189:351.
[28] Tu C, Peskin CS. Stability and instability in the computation of flows with
moving immersed boundaries, SIAM J. Sta. Comput. 1992;13:70.
[29] John M. Stockie, Brian R. Wetton. Analysis of stiffness in the immersed boundary method and implications for time-stepping schemes, J. Comp. Phys. 1999;154:41.
[30] Roma AR, Peskin CS, M.J. Berger. An adaptive version of the immersed boundary method, J. Comp. Phys.1999;153;509.
[31] Zhu L, Peskin CS. Simulation of a flapping flexible filament in a flowing soap
film by the immersed boundary method, J. Comp. Phys. 2002;179:452.
[32] Kim J, Kim D, Choi H. An immersed-boundary finite-volume method for simulations of flow in complex geometries. J. Comp. Phys. 2001;171:132
[33] Balaras E. Modeling complex boundaries using an external force field on fixed Cartesian grids in large-eddy simulations, Computer & Fluids 2004;33:375.
[34] Udaykumar HS, Mittal R, Rampunggoon, Khanna A. A sharp interface cartensian grid method for simulating flows with complex moving boundaries. J. Comp. Phys. 2001;174:345
[35] Yang J, Balaras E. An embedded boundary formulation forlarge eddy simulation of turbulent flows interacting with moving boundaries. J. Comp. Phys. 2006;215:12
[36] Sheth KS, Pozrikidis C. Effect of inertia on the deformation of liquis drops in simpl shear flow. Computer & Fluids 1995;24:101
[37] Pozrikidis C. Effect of membrane bending stiffness in the deformation of capsules in simple shear flow. J. Fluid Mech. 2001;440:269
[38] Lee J, Pozrikidis C. Effect of surfactant on the deformation of drops and bubbles in Navier-Stokes flow. Computer & Fluids 2006;35:43
[39] Lai MC, Tseng Yu-Hau , Huang Huaxiong. An immersed boundary method for interfical flows with insoluble surfactant, J. Comp. Phys.(2008);7279-7293
[40] Xu J.-J. , Li Z., Lowengrub J.S., Zhao H.-K., A level-set method for interfacial flows with surfactant, J. Comp. Phys.212 (2006)590-616
[41] Su SW, Lai MC, Lin CA. A immersed boundary technique for simulating complex flows with rigid boundary. Computer & Fluids 2007;36:313.
[42] Kim Y, Peskin CS. Penalty immersed boundary method for an Elastic Boundary with Mass. Phys. Fluids 2007;19:053103.
[43] Kim Y, Peskin CS. 3-D parachute simulation by the penalty immersed boundary method, SIAM J. Sci. Comput. in press.
[44] Li Z, Lai MC. A remark jump condition s for the three-dimensional Navier
- Stokes equations involving immersed moving membrane. App. Mathe. Let.
2001;14;149.
[45] Lee L, LeVeque RJ. An immersed interface method for incompressible Navier- Stokes equations. SIAM J. Sci. Comput. 2003;25:832.
[46] Xu S , Wang ZJ. Systemmatic derivation of jump conditions for the immersed interface method in three dimensional flow simulation. J. Comp. Phys.
2006;216:454.
[47] Xu S , Wang ZJ. An immersed interface method for simulating the interaction of a fluid with moving boundaries. SIAM J. Sci. Comput. 2006;27:1948.
[48] Harlow FH and Welsh JE. Numerical calculation of time-dependent viscous incompressible flow of fluid with a free surface. Phys. Fluids 1965;8:2181.
[49] Choi H, Moin P, Effects of the computational time step on numerical solutions of turbulent flow. J. Comp. Phys. 1993;113:1
[50] Chorine AJ. Numeriacal solution of the Navier-stokes equations. Math. Comp. 1968;22:745.
[51] H. A. Van den Vorst, Sonneveld P. CGSTAB, A more smoothly converging variant of CGS. Technical Report 90-50, Delft University of Technology 1990.
[52] Ghia U, Ghia KN, and Shin CT. High-Re solutions for incompressible flow using the Navier-Stokes Equations and a multi-grid method. J. Comp. Phys. 1982;48:387.
[53] Xu S , Wang ZJ. A 3D immersed interface method for fluid-solid interaction. Comput. Methods Appl. Mech. Engrg.. 2008;197:2068
[54] Lai MC, Tseng Hsiao Chieh. A simple implementation of the immersed interface methods for Stokes flows with singular forces. Computers and luids.2008;37:99.
[55] Le DV, Khoo BC, Peraire J. An immersed interface ethod for viscous
incompressible flow involving rigid and flexible boundaries, J. Comp.
Phys.2006;109:138.
[56] Browm D.L. , Cotez R. , Minion M.L. Accurate projection methods for the incompressible Navier-Stokes equations. J. Comp. Phys.1989;85:257
[57] Le DV . An Immersed Interface Method for Solving Viscous Incompressible Flows Involving Rigid and Flexible Boundaries. Ph.D. thesis.2005