應用熵晶格波茲曼法(ELBM)與大渦數值模擬(LES)分析方管內紊流流場為目前論文之目標。相較於LBGK，ELBM能改善數值的穩定性。另外，在LES方面，以Smargorinsky 模型作為subgrid-scale 模型來模擬紊流流場。當使用D3Q19-LBM時進行紊流模擬，會發現在角落會額外出現非物理的小vortex。我們將依據Kang的方式，以D3Q27-LBM來進行模擬，以解決此問題的發生。在模擬紊流流場之前，我們先利用條件RE_tau = 20、方管內之壓力驅動流，並以數值解與解析解相比較，驗證ELBM能否替代Navier-Stokes方程式。利用RE_tau = 300作為條件，模擬方管內之紊流流場。所得結果顯示，ELBM能夠捕捉紊流之特性。然而，目前的模擬結果仍需改進，尤其在近牆區域，所得之模擬結果尚未相當精準。

In this study, the entropic Lattice Boltzmann method (ELBM) and the Large eddy simulation (LES) will be applied to simulate the 3D turbulent duct flow. Comparing of the LBGK, the ELBM can improve the numerical stability. Also, the LES with Smargorinsky subgrid-scale model is adopted to simulate the turbulent duct flow. When utilizing the D3Q19 model, the non-physical small counter-rotating vortex on the duct corner were observed when simulating turbulent duct flow. Due to this reason, we will follow Kang et al. [18] adopting D3Q27 model within ELBM framework to simulate turbulent duct flow. In the beginning, we simulate the laminar Poiseuille flow at frictional Reynolds number Re = 20 and compare with the analytic solution to validate that whether ELBM is capable to recover the Navier-Stokes equation. Second, we simulate Re = 300 for 3D turbulent Poiseuille flow in a square duct. Results show that our simulation can capture correctly the turbulent quantities compared with existing measurement and data of Direct Numerical Simulation
(DNS). However, the present grid resolution is not adequate to fully capture the near wall turbulent structure.

1 Introduction 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . 1
1.2 Literature survey. . . . . . . . . . . . . . . . . . . 3
1.2.1 Lattice Boltzmann method . . . . . . . . . . . . . . 3
1.2.2 Entropic lattice Boltzmann models . . . . . . .. . . 3
1.2.3 Boundary conditions . . . . . . . . . . . . . . . . 4
1.2.4 Large eddy simulations . . . . . . . . . . . . . . . 5
1.2.5 Turbulent Poiseuille flow . . . . . . .. . . . . . . 6
1.3 Objective . . . . . . . . . . . . . . . . . . . . . . 7
2 Methodology . . . . . . . . . . . . . . . . . . . . . . 8
2.1 The Boltzmann equation . . . . . . . . . . . . . . . . 8
2.2 The BGK and the low-Mach-number approximations . .. . 10
2.2.1 The BGK approximation . . . . . . . . . . . . . . . 10
2.2.2 The low-Mach-number approximation . . . . . . . . . 12
2.3 Discretization of the BGK equation . . . . . . . . . 13
2.3.1 Spatial discretization . . . . . . . . . . . . . . 13
2.3.2 Temporal discretization . . . . . . . . . . . . . . 15
2.3.3 The chapman-Enskog Expansion . . . . . . . . . . . 17
2.4 Entropic lattice Blotzmann method . . . . . . . . . . 17
2.5 Large Eddy Simulation . . . . . . . . . . . . . . . . 19
2.5.1 The filtering operation . . . . . . . . . . . . . 19
2.5.2 The filtered Navier-Stokes equations . . . . . . . 20
3 Numerical algorithm . . . . . . . . . . . . . . . . . .22
3.1 Simulation procedure . . . . . . . . . . . . . . . . 22
3.1.1 The entropic condition . . . . . . . . . . . . . . 24
3.2 The external forcing term . . . . . . . . . . . . . . 24
3.3 Boundary conditions . . . . . . . . . . . . . . . . . 24
3.4 The lattice Boltzmann Subgrid-scale model . . . . . . 27
3.5 Parallel algorithm . . . . .. . . . . . . . . . . . . 30
4 Numerical results . . . . .. . . . . . . . . . . . . . .32
4.1 3D laminar Poiseuille flow . . . . . . . .. . . . . . 32
4.2 3D turbulent Poiseuille flow . . . . . . . . . . . . 33
5 Conclusions 50

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