本研究使用多鬆弛係數晶格波茲曼方法模擬層流與紊流於渠道與方管之週期性山坡流場，於渠道之週期性山坡流場，計算之雷諾數為25,50,75,100,2800與5600，而在方管之週期性山坡流場中，計算之雷諾數為25,50,75,100,2800。運用之模型方面，在渠道之週期性山坡流場中運用了D3Q19模型，而方管之週期性山坡流場中則使用了D3Q27模型。渠道與方管之週期性山坡流場的層流結果皆表現出了回流區之分離點隨著雷諾數上升而更早出現、回流區之重新附著點隨著雷諾數上升而更晚出現的現象，另外在雷諾數為75與100時，方管之週期性山坡流場出現了第二個回流區。而在紊流方面，渠道之週期性山坡流場在雷諾數為2800時與Brueuer等人所提供的結果相符，但在雷諾數為5600時則有些許不同。另外在方管之週期性山坡流場於雷諾數為2800的結果中表現出了比渠道之週期性山坡流場更快的主流場方向速度與更劇烈的震盪。而在方管之週期性山坡流場的結果中我們可以觀察到湍流動能雷諾應力之間的分佈與一些由牆面所帶來的現象。
在網格使用方面皆為直角座標下之均勻網格。於邊界之處理方式則根據不同的邊界條件採用了週期性邊界、Half-way反彈格式與根據反彈格式做線性內插的BFL格式。此外，本研究以高速圖形顯示卡運算並採用overlapping的方式，以達到加速運算的效果。

The multiple relaxation time lattice Boltzmann method is adopted in the present
work to simulate the laminar and turbulent flow over the periodic hill in the channel
(Reh =25,50,75,100,2800,5600) and duct (Reh =25,50,75,100,2800).The D3Q19 model is
adopted in the periodic hill in the channel, and for the periodic hill in the duct, the D3Q27
model is adopted. Both the laminar flow over the periodic hill in the channel and duct show
that separation locations appear early, and the reattachment locations appear later with the
Reynolds number increases in the range of present Reynolds numbers. Furthermore, the laminar
flow over the periodic hill in the duct has a second recirculation region when Reynolds numbers
= 75 and 100. The turbulent flow over the periodic hill in the channel shows good agreement
with the benchmark solution when Reh = 2800 but a slight difference in the case of Reh = 5600.
Moreover, the turbulent flow over the periodic hill in the duct shows the complex flow field,
which has a faster velocity in the streamwise direction and a more violent oscillation than the
case of the periodic hill in the channel in Reh = 2800. Moreover, we can observe the proportion
of turbulent kinetic energy between Reynolds stresses and some phenomena caused by the wall
in the cases of the periodic hill in the duct. The uniform mesh in the cartesian coordinates
system is used in the present work. The periodic boundary, the halfway bounce-back scheme,
and the BFL scheme are adopted to deal with different boundary conditions. Furthermore, the
multi-GPU cluster and overlapping strategies are used to accelerate the calculation.

1_Introduction-----------------------------------------------------1
1.1_Introduction-------------------------------------------------1
1.2_Literature survey--------------------------------------------2
1.2.1_Theroy of lattice Boltzmann method---------------------2
1.2.2_Flow over periodic hills-------------------------------2
1.2.3_Boundary conditions------------------------------------3
1.2.4_GPU implementation-------------------------------------5
1.3_Motivation---------------------------------------------------5
2_Methodology------------------------------------------------------7
2.1_The Boltzmann equation---------------------------------------7
2.1.1_Basic assumptions-------------------------------------7
2.1.2_The BGK approximation---------------------------------8
2.1.3_The low-Mach-number approximation--------------------10
2.2_Discretization of Boltzmann equation------------------------11
2.2.1_Discretization of time-------------------------------11
2.2.2_Discretization of phase space------------------------12
3_Numerical algorithm---------------------------------------------15
3.1_Multiple-relaxation-time lattice Boltzmann method-----------15
3.2_Forcing term------------------------------------------------20
3.3_Boundary conditions-----------------------------------------21
3.3.1_Halfway bounce-back boundary condition---------------21
3.3.2_BFL scheme-------------------------------------------22
3.4_GPU implementation------------------------------------------24
4_Numerical results and discussion--------------------------------27
4.1_The geometry of periodic hill-------------------------------27
4.2_The laminar flow over periodic hill in the channel----------29
4.3_The laminar flow over periodic hill in the duct-------------29
4.4_The turbulent flow over periodic hill in the channel--------33
4.4.1_Z plus----------------------------------------------33
4.4.2_Turbulence kinetic energy---------------------------34
4.4.3_Variables in main flow direction compare with
benchmark solution----------------------------------38
4.5_The turbulent flow over periodic hill in duct---------------45
4.5.1_Comparison of the periodic hill in channel and duct-45
4.5.2_Comparison of the periodic hill in duct with
difference location---------------------------------45
5_Conclusions-----------------------------------------------------62
Bibliography------------------------------------------------------63

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