利用多鬆弛時間適應性網格加密晶格波茲曼法模型來直接數值模擬平板渠道紊流(Reτ = 180, 395, 640)。更進一步地，平板渠道紊流具壁面噴流(Reτ =150, v0/uτ = 0.05) 和 (Reτ = 250, v0/uτ = 0.05, 0.1, 0.16, 0.26)第一次用多鬆弛時間晶格波茲曼法來模擬。為了用直接數值方法來擷取近壁面邊界層的流場特性，一個利用高∆+階層加密程序被建立，用來捕捉近壁面流場快速變化的特性。對於目前實現的平板紊流而言，三個以壁面為單位∆+網格大小，大約是1, 2, 4。對於進階的平板紊流壁面具噴流(Reτ =150, v0/uτ = 0.05)而言，其三個∆+網格大小，大約是0.853, 1.705, and 3.409。對於另一個的平板紊流壁面具噴流的模擬(Reτ =250, v0/uτ =0.05,0.1, 0.16, 0.26)，其三個∆+網格大小，大約是0.781, 1.563, and 3.125。由於晶格波茲曼法的數值顯示特性，此數值程序在圖形顯示卡叢集上執行以及在統一計算架構上實現。強標度律(strong scaling)顯示了現行模擬極佳的可擴展性。整體而言，純粹平板紊流(Reτ = 180, 395, 640) 在D3Q19和D3Q27模型下，模擬結果顯示幾乎沒有不同。和標準答案相比下，前述具有適應性網格的模擬，產生令人滿意的紊流量。更進一步，紊流能量收支也進行了討論。

Direct numerical simulations (DNS) of wall-bounded turbulent channel flows(Reτ =180, 395, 640) are conducted by utilizing multi-relaxation-time(MRT) adaptive mesh-refinement Lattice Boltzmann method(LBM) models. Furthermore, wall-bounded turbulent channel flows with wall transpiration(Reτ =150, v0/uτ = 0.05) and (Reτ = 250, v0/uτ = 0.05, 0.1, 0.16, 0.26) are also simulated with MRT LBM for the first time. In order to catch the flow property of the near-wall boundary layer with direct numerical simulations, a hierarchy refinement procedure that utilizes high ∆+ is built for capturing fast variation of near-wall flow properties. For the current implementation of wall-bounded turbulent channel flows, the three grid spacings in the wall unit are (∆+) ∼ 1, 2, and 4. For the advanced wall-bounded turbulent channel flows with wall transpiration (Reτ =150, v0/uτ =0.05), the three grid spacings are ∆+ ∼ 0.853, 1.705, and 3.409. For another wall-bounded turbulent channel flows with wall transpiration (Reτ =250,v0/uτ =0.05,
0.1, 0.16, 0.26), the three grid spacings are ∆+ ∼ 0.781, 1.563, and 3.125. Due to the Lattice Boltzmann numerical explicit property, the numerical procedure is conducted on the graphics processing unit(GPU) cluster and implemented on the compute unified device architecture(CUDA) framework. Strong scaling shows
excellent scalability of the current simulation. Overall, the present pure channel flows simulated results (Reτ =180, 395, and 640) with D3Q19 and D3Q27 models demonstrate slight differences. So, the advanced channel flows with wall transpiration(Reτ =150,v0/uτ =0.05) and (Reτ =250,v0/uτ =0.05, 0.1, 0.16, 0.26) are performed with D3Q19 model only. With adaptive mesh refinement, the precedent simulations produce satisfactory turbulent quantities when compared with the benchmark solutions. Furthermore, budgets of the turbulence kinetic energy are also discussed.

1 Introduction 1
1.1 The DNS channel flow with wall transpiration . . . . . . . . . . . . . 1
1.2 Turbulent DNS channel flow . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 LBM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.4 AMR of LBM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.5 GPU . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.6 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2 Methodology 7
2.1 Mathematical formulation . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Mesh refinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3 Energy k budget and Reynolds stress equation . . . . . . . . . . . . . 14
2.4 GPU implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3 Numerical results and discussion 17
3.1 Turbulent DNS channel flow without wall transpiration . . . . . . . . 17
3.1.1 Geometry and computational domain . . . . . . . . . . . . . . 17
3.1.2 Turbulent quantities . . . . . . . . . . . . . . . . . . . . . . . 20
3.2 Turbulent DNS channel flow with wall transpiration . . . . . . . . . . 31
3.2.1 Geometry and computational domain . . . . . . . . . . . . . . 31
3.2.2 Turbulent quantities . . . . . . . . . . . . . . . . . . . . . . . 35
3.2.3 Reynold stress equation . . . . . . . . . . . . . . . . . . . . . 39
iii
3.2.4 GPU performance . . . . . . . . . . . . . . . . . . . . . . . . . 43
4 Conclusions 46
4.1 Turbulent DNS channel flow without wall transpiration . . . . . . . . 46
4.2 Turbulent DNS channel flow with wall transpiration . . . . . . . . . . 47
4.3 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

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