本論文應用多圖形顯示卡叢集以單鬆弛時間與熵晶格波茲曼法計算方管紊流。利用多圖形顯示卡之平行計算的優勢，模擬時間可被縮短，因此紊流模擬可更快速達到完全發展流，且可進行更密網格設定的模擬。雖於2013年Kang和Hassan的研究已指出使用晶格波茲曼法模擬方管紊流須使用D3Q27模型，但本論文仍使用單鬆弛時間與熵晶格波茲曼法搭配不同模型D3Q15、D3Q19、D3Q27，來進一步探討模型對方管層流Reτ = 20、紊流Reτ = 360模擬的影響。另外，本論文亦應用Smagorinsky的subgrid-scale model與Van Direct damping function進行紊流Reτ = 360流場的大渦數值模擬。同時，使用540 × 90 × 90、552 × 138 × 138、1080 × 180 × 180三套網格模擬紊流Reτ = 360，以測試Reτ與網格間最佳比例。然後，利用本研究所提出的Reτ與網格間最佳比例，模擬紊流Reτ = 600的流場。在物理觀點的探討，本論文對紊流Reτ = 360中紊流結構有所研究。最後部分為本論文之多圖形顯示卡叢集的計算平效率。

In this research, Poiseuille turbulent square duct flow simulations with single-relaxation time lattice Boltzmann method (SRT LBM) and entropic lattice Boltzmann method (ELBM) on multi-GPU are studied. Taking the advantage of GPU, computation times can be shortened, thus fully developed turbulent flow is achieved easily and a fine mesh is able to be employed. Although the LBM with D3Q27 model is needed essentially for turbulent duct flow explored by Kang and Hassan in 2013, SRT LBM and ELBM with different models (i.e.D3Q15, D3Q19 and D3Q27) are still adopted to investigate the influence of D3Q15 and D3Q19 in laminar duct flow at friction Reynolds number Reτ = 20 and in turbulent duct flow at Reτ = 360. To simulate turbulent duct flow with LES model, Smagorinsky subgrid-scale model with Van Direct damping function is utilized. And, the effect of simulations with and without LES model are discussed in detail. Meanwhile, turbulent duct flow simulations at Reτ= 360 in different grid sizes of 540 × 90 × 90, 552 × 138 × 138 and 1080 × 180 × 180 are conducted as grid sensitivity test to obtain optimal mesh size related Reτ. Then, turbulent duct flow simulations with D3Q27 SRT LBM at Reτ = 600 are demonstrated. From physical phenomena viewpoint, turbulent structure is investigated in the turbulent duct flow at Reτ = 360. In addition, parallel performance of multi-GPU cluster is also discussed.

1. Introduction 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Literature survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.1 Turbulent Poiseuille flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.2 Large eddy simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.3 Theory of lattice Boltzmann method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.4 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2.5 GPU implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3 Motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2. Methodology 11
2.1 The Boltzmann equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 The BGK approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3 The low-Mach-number approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.4 Discretization of the Boltzmann equation . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.4.1 Discretization of time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.4.2 Discretization of phase space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.5 The Chapman-Enskog expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.6 Large Eddy Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.6.1 The filtering operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.6.2 The filtered Navier-Stokes equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.6.3 The lattice Boltzmann Subgrid-scale model . . . . . . . . . . . . . . . . . . . . . . . 20
2.7 Entropic lattice Blotzmann method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3. Numerical algorithm 26
3.1 Simulation procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.2 Boundary condition implementations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.3 The external forcing term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.4 GPU implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.5 Two dimensional domain decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4. Numerical results and discussion 36
4.1 Laminar Poiseuille square duct flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.2 Turbulent Poiseuille square duct flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.2.1 Comparisons of LBM with different models . . . . . . . . . . . . . . . . . . . . . . . 38
4.2.2 Effect of LES model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.2.3 Grid sensitivity test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.2.4 Turbulent Poiseuille duct flow at 〖Re〗_τ = 600 . . . . . . . . . . . . . . . . . . . . . . . 42
4.2.5 Turbulence structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.3 Parallel performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

5. Conclusions and future work 81

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