臺灣博碩士論文加值系統

Abstract
A new immersed-boundary method for simulating flows over or inside complex geometries is presented. The present scheme is based on the direct forcing concept, where the boundary condition can be implemented on Eulerian grid directly. The
numerical integration is based on a second-order fractional step method under the staggered grid spatial framework. Based on the direct momentum forcing on the Eulerian grids, a new force formulation imposes the desired velocity distribution V which ensures the satisfaction of the no-slip boundary condition in the intermediate time step without adopting the Lagrangian markers. This forcing procedure involves an interpolation scheme since the immersed boundary in general does not coincide with the grid point. There are three different interpolation schemes adopted. Numerical experiments show that the stability limit is not altered by the proposed techniques and the second order or close to second accurate solutions are obtained. Four different test problems are simulated using present technique(rotating ring flow, rotating concentric rings, lid-driven cavity and flows over a stationary cylinder), and the results are compared with previous experimental and numerical results. The numerical evidences show the accuracy and capability of the proposed methods
for solving complex geometry flow problems. For the IBM schemes adopted, extrapolation scheme A produces relatively better results, while the difference of the interpolation schemes B and C is marginal. However, the advantage of the
interpolation scheme is that its capability to represent the flowfield accurately on both sides of the interface boundary. Therefore, for this kind of applications, scheme
C is preferred.

Contents
1 Introduction 1
1.1 Introduction . . . . . . . . . . . . . . . . .. . . . . . 1
1.2 Paper Survey . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Objective and Motivations . . . . . . . . . . . . . . . . . . . . . . . 8
2 The methodology of immersed boundary technique 9
2.1 Mathematical Formulations . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3 Numerical Algorithm . . . . . . . . . . . . . . . . . . . . . . . 10
2.3.1 Discretization of the Transport Equations . . . . . . . . . . . . . . . . . . . . . . . 11
2.3.2 Final Discretised Form of The Governing Equation . . . . . . . . . . . . . . . . . . . . . . . . 15
2.4 The forcing strategies . . . . . . . . . . . . . . . . . . . . . . . 17
2.4.1 Identi‾cation of forcing points . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.4.2 Evaluation for the forcing points . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.5 The full solution procedure . . . . . . . . . . . . . . . . . . . . . . . 23
2.6 Figures . . . . . . . . . . . . . . . . . . . . . . . . 24
3 Numerical Results 31
3.1 Flow induced by a rotating ring . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.1.1 Steady state for a rotating ring . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.1.2 Unsteady state for a rotating ring . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.2 Flow induced by two concentric rotating rings . . . . . . . . . . . . . . . . . . . . . . . . . .34

3.3 Lid-driven cavity . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.4 The flow past a cylinder . . . . . . . . . . . . . . . . . . . . . . . . 36
4 Conclusion and future work 40
4.1 The conclusion . . . . . . . . . . . . . . . . . . . . . . . 40
4.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . 41
5 Figures & Tables 42

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