近二十年來，晶格玻茲曼方法（Lattice Boltzmann Method）憑藉其平行效率與邊界處理上的優勢逐漸成為替代傳統納維-斯托克斯求解器（Navier-Stokes Solvers）來建模與模擬紊性熱流場之強有力工具。雖然晶格玻茲曼模型在靜態條件下已經獲得大量關注，但其在旋轉狀態下的研究卻還處於初級階段。因此本文基於旋轉坐標首度提出了一種新的晶格玻茲曼模型以用來對旋轉系統之紊性熱流場進行大渦模擬（Large Eddy Simulation）。大渦模擬之次格點模型為改善剪力的司馬格林斯基模型（Shear-Improved Smagorinsky Model）。由於此模型中的應變率可以直接透過非平衡態分佈函數就地算出，因此其整過計算過程完全局部化。為了驗證以上提出之方法，本文藉其模擬了壓差驅動且具跨向旋轉與熱傳之紊性平行板流。基於摩擦速度與平行板半高之雷諾數（Reτ）固定為194而基於摩擦速度與平行板全高之旋轉數（Roτ）則從0變化至3.0。工作流體為空氣，其普朗特數（Pr）為0.71。計算結果包括平均速度、雷諾應力、均方根紊動速度、平均溫度、均方根紊動溫度以及紊性熱流密度。透過與前人直接模擬（Direct Numerical Simulation）之數據比較可以發現本研究結果與前人數據具有較高的一致性，這也驗證了本研究方法用於模擬旋轉紊性熱流場之可行性。
旋轉紊性內流場在很多工業應用中都扮演者非常重要的角色，如燃氣渦輪機、旋轉電極、化學反應器、分離器等。然而人們對這些設備中許多具挑戰性的物理現象，如速度峰值在正方形管道中偏向壓力壁（Pressure Wall）而在平行板中則偏向吸力壁（Suction Wall）等，並未完全了解。因此本文基於先前新方法對跨向旋轉方管之紊性全展流進行首次模擬研究。Reτ與Roτ分別固定為150與2.5而通道寬高比（AR）從1變化至6而後∞（平行板）。透過與前人直接模擬的數據進行比較，本方法在模擬具旋轉離心力與科氏力之紊性正方形管流（AR = 1）的可行性進一步得到驗證。隨著AR的增加，在方形管中首次發現存在著一個臨界AR = 4，當寬高比低於此值時平均主流峰值偏向壓力壁而低於此值時其偏向吸力壁。此一臨界AR值也出現在表征管道中心紊流狀態之各向異性不變量圖（Anisotropic Invariant Map）中。藉助宏觀統御方程，本文從根本上闡明了平均主流峰值偏轉之物理機制以及臨界AR出現之原因。此外在所有的管流中，埃克曼層傳輸（Ekman Layer Transport）在空間和時間上皆持續，且對核心區之影響隨著AR的增加而減弱。為了進一步解釋流場對熱傳的影響，本文對以上AR = 1、4與∞管道在Pr = 0.71時進行了熱傳的研究。加熱方式為上下壁等溫而側壁絕熱。研究結果發現對所有方管其二次流被兩個逆向旋轉的側壁渦旋所主導，此對渦旋顯著地促進了壓力壁兩角落附近的熱傳。相較於側壁渦旋，普朗特第二類二次流（Prandtl’s Secondary Flow of the Second Kind）對熱傳的貢獻則較小。關於熱傳的一個新發現是平均溫度分佈較純熱傳導結果之偏差在AR < 4時因埃克曼運動為負而在AR > 4時沿管道高度方向大部分區域為正。另外在所有參數中雷諾應力分量<u+v+>與熱傳之相關性最高，但該相關性會隨著AR的減少而下降。
晶格玻茲曼方法的重要缺點之一就是均勻網格限制。如此一來，為了在高雷諾數下獲得精確的結果，需要對網格進行全局加密以解析到最小的流力尺度，這就意味著計算成本的驟升。為解決此問題，本文進一步發展了多區域格點加密技術從而使晶格玻茲曼方法能夠在多重解析度格點上模擬三維流場和熱傳。該方法在同時包括外力與能量源項情況下使用一種三維縮放算法與二維雙三次插值來解決粗細格點間非平衡態函數的耦合問題。 隨後本文用新提出的加密方法模擬了三個基準算例，即三維通道強制對流、立方凹穴自然對流以及紊性平行板流，並將計算結果與前人數據進行比較，發現二者具有較好的一致性，這表明當前加密算法可準確模擬三維熱流場問題。

Due to its advantages in parallel efficiency and wall treatment, the lattice Boltzmann method (LBM) has emerged as a powerful alternative to traditional Navier-Stokes solvers for modeling and simulating turbulent thermal flows in recent two decades. Although increasing attention has been paid to stationary lattice Boltzmann models, the development of rotational ones is still at its very early stage. In this study, LBM in a rotating frame of reference is firstly proposed for large eddy simulation of turbulent flows and heat transfer with system rotation. The subgrid scale closure is modeled by a shear-improved Smagorinsky model. Since the strain rates are also locally determined by the non-equilibrium part of the distribution function, the calculation process is entirely local. To validate the approach, the pressure-driven turbulent channel flow with spanwise rotation and heat transfer is simulated. The Reynolds number (Reτ) characterized by the friction velocity and channel half height is fixed at 194 whereas the rotation number (Roτ) in terms of the friction velocity and channel height ranges from 0 to 3.0. A working fluid of air with a Prandtl number of 0.71 is chosen. Computed results are demonstrated in terms of mean velocity, Reynolds stress, root mean square (RMS) velocity fluctuations, mean temperature, RMS temperature fluctuations, and turbulent heat flux. Good agreements are attained between the present predictions and previous direct numerical simulation (DNS) data, which confirms the applicability of the proposed method for computation of rotating turbulent flows and heat transfer.
Wall-bounded rotating turbulent flows have played a pivotal role in many industrial applications like gas turbines, rotating-disk electrodes, chemical reactors, separators, etc. However, some challenging physical phenomena in these devices like mean velocity peak shift, towards the pressure wall in square ducts versus the suction wall in channels, have never been thoroughly understood. Hence, the numerical investigations of fully developed turbulent thermal flow in spanwise rotating rectangular ducts are subsequently performed by the newly proposed method for the first time. Reτ and Roτ are fixed at 150 and 2.5, respectively, whereas the duct aspect ratio (AR) varies from 1 to 6 and then ∞ (channel). By comparing the present results with previous DNS data, the capability of the rotating model for predicting turbulent flows in a square duct (AR = 1) subject to rotation induced centrifugal and Coriolis forces are proved. With increasing aspect ratio, it is further observed that there exists a critical AR = 4, beyond or below which the peak of the mean velocity profile is skewed to the suction or pressure side. The same critical value of AR also appears in the anisotropic invariant map indicated by the turbulence state in the duct center. Based on the macroscopic equations, the physical mechanism of mean velocity peak shift and critical value of AR are radically clarified. Furthermore, the Ekman layer transport is persistent temporally and spatially in all duct flows and its influence on the core region decreases with increasing AR. To reveal the effect of fluid flow on heat transfer, scalar transport in rectangular ducts is calculated for AR = 1, 4, and ∞ at Pr = 0.71. Constant high and low temperature are imposed on the top (pressure) and bottom (suction) wall, respectively, whereas the sidewalls are assumed to be adiabatic. The simulation results show that the dominance of two large counter-rotating sidewall vortices in the mean secondary flows of small AR ducts significantly contributes to the heat transfer near the two pressure wall corners. In contrast, the Prandtl’s secondary flow of the second kind demonstrate less contribution to scalar transport. The new finding is that the mean temperature deviation from conduction solution displays negative value in the entire duct height for AR < 4 due to the Ekman motion whereas positive results in the most portions of the duct height for AR > 4. Additionally, the Reynolds stress component <u+v+> correlates with the scalar transport among all flow parameters. However, its relevance becomes relatively lower for smaller values of AR.
One drawback of LBM is that uniform-sized meshes are generally required. Consequently, to obtain accurate results at high Reτ, the mesh size has to be adjusted to resolve the smallest scales, which means a surge in computational cost. Therefore, a multi-domain grid refinement technique that enables LBM to work on multi-resolution grid for 3D flow and heat transfer is further developed. The coupling of non-equilibrium distribution functions between coarse and fine grids is handled by a 3D scaling technique, considering both external forces and scalar source terms, and 2D bicubic interpolation scheme. The proposed algorithm is then applied to simulate three benchmark cases, i.e., the forced convection in a 3D channel, natural convection in a cubical cavity, and turbulent channel flow with heat transfer, and its results are thoroughly compared with previous data. The very decent match shows that the present approach is capable of accurate prediction of thermal flow in 3D domains.

Abstract iii
Acknowledgments v
List of Tables x
List of Figures xi
List of Symbols xvii
Chapter 1 Introduction 1
1.1 Preliminary Remarks 1
1.2 Literature Review 3
1.2.1 Lattice Boltzmann Model for Turbulent Flows 3
1.2.2 Turbulent Flow in Rotating Rectangular Ducts 6
1.2.3 Grid Refinement Techniques within LBM Framework 9
1.3 Objectives 12
Chapter 2 Governing Equations 31
2.1 Lattice Boltzmann Equations 31
2.2 Chapman-Enskog Analysis 33
2.3 Boundary Conditions 36
2.3.1 On-Lattice Boundary Treatment 37
2.3.2 Boundary Condition for Curved Walls 38
2.4 Parallel Algorithm 39
Chapter 3 Rotating Lattice Boltzmann Model 43
3.1 Lattice Boltzmann Equations in a Rotating Coordinate 43
3.2 Shear-Improved Subgrid Scale Closure 45
3.3 Rotating Turbulent Channel Flow with Heat Transfer 47
3.3.1 Comparisons of Turbulence Statistics 49
3.3.2 Instantaneous Flow and Temperature Field 52
3.4 Summary 54
Chapter 4 Turbulent Flow in Rotating Rectangular Ducts 70
4.1 Numerical Setup 70
4.2 Validation 71
4.3 Results and Discussion 71
4.3.1 Turbulence Statistics 72
4.3.2 Secondary Flows 74
4.3.3 Anisotropy Invariant Map 75
4.3.4 Instantaneous Flow Structures 76
4.4 Summary 77
Chapter 5 Rotating Turbulent Duct Flow with Scalar Transport 87
5.1 Problem Description 87
5.2 Computational Details 87
5.3 Results and Discussion 87
5.3.1 Validation 88
5.3.2 Turbulence Statistics 88
5.3.3 Analog between Fluid Flow and Scalar Transport 90
5.3.4 Instantaneous Velocity and Scalar Fields 92
5.4 Summary 94
Chapter 6 Multidomain Grid Refinement 104
6.1 Basic Concepts of Grid Refinement 104
6.2 Mapping between Coarse and Fine Grids 104
6.3 Interpolation Schemes 107
6.4 Numerical Validations and Discussion 108
6.4.1 3D Poiseuille Flow with Constant Wall Heat Flux 108
6.4.2 Natural Convection in a Cubical Cavity 110
6.4.3 Turbulent Channel Flow with Heat Transfer 112
6.5 Summary 113
Chapter 7 Conclusions and Future Recommendations 128
7.1 Conclusions 128
7.1.1 Lattice Boltzmann Modeling of Fluid Rotation 128
7.1.2 Rotating Turbulent Duct Flow 128
7.1.3 Multidomain Grid Refinement 129
7.2 Major Contributions 130
7.3 Future Recommendations 131
Appendix A RANS Modeling of Rotating Turbulent Thermal Flow 132
A.1 Governing Equations 132
A.2 Turbulence Models 133
A.2.1 Realizable k-ε Model 134
A.2.2 k-ω SST Model 135
A.2.3 Reynolds Stress Model with Linear Pressure Strain 135
A.3 Near-Wall Treatment 136
A.4 Numerical Methods 137
Appendix B Heat Transfer in Rotating Two-Pass Rhombic Ducts 139
B.1 Introduction 139
B.2 Description of Problem 144
B.3 Computational Procedure 145
B.3.1 Comparison of EVMs and RSM 145
B.3.2 Boundary Conditions 145
B.3.3 Computational Grid Details 146
B.4 Results and Discussion 147
B.5 Conclusions 160
Appendix C Response to the Comments of Oral Defense Committee 175
References 183
Curriculum Vitae 197

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