臺灣博碩士論文加值系統

有限體積法模擬不可壓縮之內外流場
學 生：余重信
指導教授：林三益
中文摘要
本文主要是發展一個數值方法來解不可壓縮的那唯爾-史托克方程式(Navier-Stokes Equation)。在數值方法上，方程式的對流項是使用三階的有限體積法計算，黏滯項為二階的有限體積法。關於非定常流場則是採用二階的克朗克-尼克森(Crank-Nicolson)時間積分。人工壓縮因子法主要是將不可壓縮的那唯爾-史托克方程式由橢圓方程式轉成雙曲線方程式。為了加速數值方法的收斂性，本文將介紹比較三種隱式的DDADI時間積分及一種顯式的朗奇-庫達法的時間積分(Runge-Kutta time Integration)。數值方法的結果顯示空間上的準確度約為2.6階，時間上的準確度約為1.7階。為了節省電腦的運算時間，本文將採用平行計算，而平行計算的結果顯示當計算區塊有適當的分割時，大多數的物理問題都能有不錯的收斂性。本文加入了代數紊流模型來模擬高雷諾數的潛體流場。關於加速物體的流場則是採用動態格點法來加以解決，幾何守恆律的部份在本文也有加以考慮。
流體流經一個圓球的流場是本文研究的重點，該流場的尾流結構依照雷諾數的不同大至上可分為五個範圍，包括了定常流且無分離點發生、定常且為軸對稱分離流、定常非軸對稱分離流，非定常流且有一個尾流主頻，非定常流但是有兩個尾流主頻。本文研究的雷諾數範圍在20到400之間，其結果跟實驗數據比對十分的接近。雷諾數400的流場的現象則由三維流線加以討論。本文的另一個研究的重點在彎管的流場，四種不同幾何外型的彎管流場及其渦流的強度在本文將加以討論，而管內的壓力降及分離流的範圍也將加以比較。本文也將討論人工壓縮因子法在自然對流流場的應用，所討論的物理問題是三維的封閉立方體場，不同的奴塞爾值(Nusselt number)及壁面溫度分佈將加以討論。

Finite Volume Scheme for Incompressible Flows and its Application to
Internal and External Flows
Student : Zhong-Xin Yu
Adviser : San-Yih Lin
ABSTRACT
A numerical method to solve the incompressible Navier-Stokes equations is presented in this thesis. The method uses a third-order upwind finite volume scheme to discretize the convective terms and second-order finite volume method to discretize the viscous terms. For the unsteady flows, the second-order Crank-Nicolson method is used for time integration. The artificial compressibility method is used to solve the incompressible Navier-Stokes equations by a hyperbolic-dominated system of equations. To enhance the convergent rate of the full method, three types of DDADI schemes are introduced and compared in this thesis. An explicit Runge-Kutta method and three implicit DDADI methods, IM1, IM2, and IM3, are then introduced. The numerical results show the whole scheme is about 2.6th order in space and 1.7 order in time. To save the CPU time costing, multizone computations and MPI techniques are investigated here. For a suitable zone-communication structure, the method obtains good convergent rate for most physical problems. The algebra turbulence model is applied in the method to simulate flows over a submerged blunt body under high Reynolds numbers. Dynamic grid method is used to simulate flows over a moving body. The geometric conservation low is taken care.
Flow over a sphere is especially interesting in this theme. The wakes structures of flows over a sphere under various Reynolds numbers are roughly classified into five regions. They include no separation occurring in the wake, steady axis symmetry vortex structure in the wake, steady asymmetry vortex structure in the wake, unsteady with single dominated vortex shedding frequency, and two major frequencies existing in the vortex shedding phenomenon. The detailed flow structures under Reynolds number from 20 to 400 are performed and compared with the experimental and numerical data. The evolutions of the vortex shedding under Re=400 are demonstrated by using 3-D instance streamlines. The pipe flow is another important physical problem. Four different geometric configurations are selected to study the vortex structures and strength of the secondary flows in pipes. Comparisons of pressure drop and separation region are performed. Finally, the natural convection problem is selected to exam the abilities of the artificial compressibility method. The natural convection in a cube with different wall temperature is performed. The temperatures distributions and Nusselt numbers are computed. The numerical results show the method work well. The overall results show that the IM1 and IM2 schemes have good convergent rates for both steady and unsteady problems and perform better than the IM3 and explicit Runge-Kutta schemes do.

CONTENTS
ABSTRACT ………………………………………………………………….…………i
CONTENTS……………………………………………………………………………iii
LIST OF TABLES……………………………………………………………………..vi
LIST OF FIGURES……………………………………………………………..…….vii
NOMENCLATURE……………………………………………………..………...….xiv
CHAPTER
I INTRODUCTION……………………………………………………...1
1.1 Formulations of Governing Equations…………………………………………...1
1.2Numerical Schemes………………………………………………………………2
1.3Multizone and Parallel Computations……………………………………………3
1.4Dynamic Grid…………………………………………………………………….4
1.5Turbulence Models……………………………………………………………….5
1.6Flow over a Sphere……………………………………………………………….5
1.7Natural convection………………………………………………………………..7
1.8Contents and Organizations………………………………………………………7
II NUMERICAL FORMULATIONS………………………………………9
2.1 Governing Equations and Artificial Compressibility……………………………..9
2.2 Space Discretization: Finite-Volume Formulation……………………………….10
2.2.1 Treatment of convection terms……………………………………………10
2.2.2 Treatment of viscous terms……………………………………………….13
2.3 Time Integrations: Runge-Kutta time integration and DDADI Algorithm………14
2.3.1 Steady-state Formulation………………………………………………….14
2.3.2 Unsteady formulation……………………………………………………..20
2.4 Implicit Residue Smoothing Method…………………………………………….20
2.5 Boundary Condition………………………………………………………..……21
2.6 Multizone Technique and Boundary Condition at Interface……………….….....22
2.7 Parallel Computation of Mpi………………………………………………..…...23
2.8 Dynamic Grid Formulations…………………………………………………..…24
2.9 Turbulence Model…………………………………………………………….…27
III SCHEME VALIDATIONS………………………………………….…29
3.1 Vortx-Decay Flow in a Cube……………………………………………………29
3.2 Flow over a Sphere……………………………………………………………...30
3.2.1 Convergence rate………………………………………………………….31
3.2.2 Multizone and parallel computations……………………………………...32
3.3 Driven Cavity Flow under Re=100………………………………………………33
3.4 Internal Pipe Flow under Re=1000………………………………………………36
3.5 Algebra Turbulence Model Testing………………………………………………38
3.6 Conclusions………………………………………………………………………39
IV SIMULATION OF INTERNAL AND EXTERNAL FLOWS…………40
4.1 Introduction………………………………………………………………………40
4.2 Three-Dimensional Driven Cavity flow………………………………………….40
4.3 Flow Over a Sphere………………………………………………………………41
4.3.1 Introduction………………………………………………………………...41
4.3.2 Steady axis symmetry flow without separation…………………………….43
4.3.3 Steady axis symmetric flow with separation……………………………….44
4.3.4 Steady axis asymmetric flow…………………………………………….…44
4.3.5 Time periodic flow…………………………………………………………45
4.4 Flow in a pipe……………………………………………………………………49
4.4.1 Flow field under Re=1000……………………………………………….49
4.4.2 Flow field under Re=400…………………………………………………52
4.5 Flow over a Submerged Blunt Body……………………………………………..52
4.6 Application of Dynamic Grid…………………………………………………….53
4.6.1 Steady-state flow of a submerged blunt body under uniform speed……...53
4.6.2 Accelerating submerge blunt body………………………………………..54
4.7 Conclusion………………………………………………………………………..55
V NATURAL CONVECTION PROBLEM……………………………….57
5.1 Governing Equations…………………………………………………………….57
5.2 Natural Convection in a Cube…………………………………………………..59
5.2.1 Linear wall temperature distribution problem (LTD problem)…………...59
5.2.2 Adiabatic wall temperature problem……………………………….……..60
5.3 Conclusion………………………………………………………………………62
VI CONCLUSIONS AND FURTHER WORKS………………………….63
6.1 Conclusions……………………………………………………………………..63
6.2 Future Work……………………………………………………………………..64
REFERENCE………………………………………………………………..66
TABLES……………………………………………………………………..74
FIGURES……………………………………………………………………76
VITA………………………………………………………………………..187
Publication List……………………………………………………………..188

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