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 The development of robust and efficient numerical algorithms for both steady-state and transient simulations of the Stokes equations and Navier-Stokes equations for two-dimensional and three-dimensional incompressible viscous flow is an active research field. A meshless method based on the multiquadrics (MQ) method has been developed to solve the 2D and 3D Stokes flows and Navier-Stokes equations in velocity-vorticity formulation. Numerical results are also reported using the method of fundamental solutions (MFS) in order to compare its performance with the MQ method. The method of fundamental solutions (MFS) based on the Stokeslet is successfully implemented for the numerical solution of Stokes flow problems. The MFS does not require a discretized interior domain and boundary integration to obtain the solutions for the flow variables. We implemented the MQ method to solve the Stokes equations and Navier-Stokes equations for two and three-dimensional flow problems. The method employed a coupled numerical solution algorithm by combining the boundary equations along with the governing equations to form a single global matrix for all the field variables. The computation of the velocity and the vorticity variables are completed by satisfying the continuity equation for the velocity field and the solenoidal constraint for the vorticity field. The multiquadrics method is found to be an efficient scheme for low Reynolds number flows, which has been demonstrated in this study. Two-dimensional and three-dimensional flow solutions for Stokes equations in a circular cavity, square cavity and cubic cavity are established and compared with available benchmark solutions by using both the MFS and MQ algorithms.
 誌謝 i摘要 iiAbstract iiiTable of Contents ivList of Tables viiList of Figures viiiSymbols xvChapter 1 Introduction 11.1 Mathematical modeling 11.2 Numerical methods 21.3 Meshless methods 41.4 Objectives of the present work 61.5 Organization of the dissertation 7Chapter 2 Literature Review 82.1 Method of fundamental solutions 82.2 Multiquadrics method 10 2.2.1 Stokes flow 10 2.2.2 Navier –Stokes equations 13Chapter 3 Governing equations 183.1 Navier-Stokes equations in the primitive variable form 183.2 Navier-Stokes equations in the velocity-vorticity form 203.3 Stokes flow equations in the velocity-vorticity form 223.4 Stokeslet equations in the primitive variables form 243.5 Initial and boundary conditions 28Chapter 4 Numerical solutions using multiquadrics method and method of fundamental solutions 304.1 Numerical solutions of 2D and 3D Stokes flow using MFS 304.2 Numerical solutions of Stokes flow using MQ 33 4.2.1 2D Stokes flow using MQ 33 4.2.2 3D Stokes flow using MQ 354.3 Numerical solutions of Navier-Stokes equations using MQ 39 4.3.1 2D Navier-Stokes equations using MQ 39 4.3.2 3D Navier-Stokes equations using MQ 43Chapter 5 Results and discussions of method of fundamental solutions for Stokes flows 555.1 MFS and Stokeslet solutions 555.2 2D Stokes flow in a circular cavity 555.3 2D Stokes flow in a square cavity 635.4 2D Stokes flow in a cavity with wave-shaped bottom 695.5 3D Stokes flow in a cubical cavity 73Chapter 6 Results and discussions of multiquadrics method for Stokes flows and Navier-Stokes equations 806.1 MQ solutions for 2D Stokes flows 80 6.1.1 2D results of Stokes flow in a square cavity using MQ 80 6.1.2 2D results of Stokes flow in a circular cavity using MQ 826.2 MQ solutions for 3D Stokes flow in a cubic cavity 856.3 MQ solutions for Navier-Stokes equations 94 6.3.1 MQ solutions for 2D Navier-Stokes equations 94 6.3.2 MQ solutions for 3D Navier-Stokes equations 966.4 Effect of Space parameter c 109Chapter 7 Conclusions 1227.1 Main conclusions 1227.2 Limitations and scope for further research126References 129Appendix 1:Multiquadrics with radial basis function 135A.1.1 Meshless numerical methods 135A.1.2 Radial basis function 137A.1.3 Illustration of the MQ method 141Appendix 2: Fundamental solutions using Stokeslet 152Appendix 3: List of publications 158 國圖紙本論文     top 簡易查詢 | 進階查詢 | 熱門排行 | 我的研究室   