本論文研究了插值補充晶格波茲曼法（ISLBM）和半拉格朗日晶格波茲曼法（SLLBM）在類比複雜流動方面的性能。分析考慮了正交坐標系中的不同插值階數，探討了它們對層流流場和湍流流場的影響。使用不同的插值階數對三維泰勒-格林漩渦進行了模擬，揭示了耗散率的變化。採用六階插值的 ISLBM 與晶格波茲曼法（LBM）的解決方案非常相似，而 SLLBM 則傾向於高估耗散。提高網格解析度可以提高耗散率預測的準確性。在類比壁界湍流通道流時，SLLBM 顯示出不穩定性，而 ISLBM 保持穩定。六階插值法在預測湍流耗散率時表現出卓越的性能，同時產生了合理的計算開銷。此外，論文還探討了曲線座標下的流場模擬，強調了投影對流場模擬結果的影響。變形角度的增大使得精確的動能值和耗散率值的獲取面臨挑戰，過度變形會導致模型的某些部分不穩定。

This thesis investigates the performance of the interpolation-supplemented lattice Boltzmann method (ISLBM) and the semi-Lagrangian lattice Boltzmann method (SLLBM) in simulating complex flows. The analysis considers different interpolation orders in the orthogonal coordinate system, exploring their impact from laminar to turbulent flow fields. Simulations of the three-dimensional Taylor-Green vortex are conducted using various interpolation orders, revealing variations in dissipation rates. The ISLBM with sixth-order interpolation closely resembles the lattice Boltzmann method (LBM) solution, while the SLLBM tends to overestimate dissipation. Increasing the grid resolution improves the accuracy of dissipation rate predictions. In the simulation of wall-bounded turbulent channel flow, the SLLBM exhibits instability, while the ISLBM remains stable. The sixth-order interpolation demonstrates superior performance while predicting turbulence dissipation rate producing reasonable computational overhead. Furthermore, the thesis explores the simulation of flow fields in curvilinear coordinates, highlighting the effect of the projection on the results of the flow fields simulation. Increasing deformation angles make accurate kinetic energy while dissipation rate values are challenging to obtain, with excessive deformation causing instability in certain components of the model.

Abstract i
Acknowledgment ii
List of Figures vii
List of Tables x
1 Introduction 1
1.1 Motivation and objective . . . . . . . . . . . . . . . . . 1
1.2 Literature survey . . . . . . . . . . . . . . . . . . . . 3
1.2.1 Simulation of 3-D turbulent flow with the lattice Boltzmann method in non-square grids . . . . . . . . . . . . . . . . . .3
1.3 Outline of the thesis . . . . . . . . . . . . . . . . . . 5
2 The lattice Boltzmann method 6
2.1 The Boltzmann equation . . . . . . . . . . . . . . . . . .6
2.1.1 H-Theorem . . . . . . . . . . . . . . . . . . . . . . . 7
2.1.2 Bhatnagar-Gross-Krook model . . . . . . . . . . . . . . 8
2.2 The lattice Boltzmann method . . . . . . . . . . . . . . .9
2.2.1 Chapman-Enskog Analysis . . . . . . . . . . . . . . . . 11
2.3 Boundary condition implementations . . . . . . . . . . . .13
2.3.1 Non-equilibrium bounce-back boundary condition . . . . .13
2.3.2 Halfway bounce-back boundary condition . . . . . . . . .14
2.4 Parallel Algorithm . . . . . . . . . . . . . . . . . . . .15
3 Interpolation-based lattice Boltzmann method 19
3.1 Semi-Lagrangian lattice Boltzmann method . . . . . . . . .20
3.2 Interpolation-supplemented lattice Boltzmann method . . . 21
3.3 Defining the relationship between mesh size and lattice size . . . . . . . . . . . . 22
4 Influence of interpolation order 25
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . .25
4.2 Analytical solution in fully developed laminar Poiseuille channel flow . . . . . . 26
4.3 Three-dimensional Taylor-Green vortex . . . . . . . . . . 31
4.4 Turbulent Poiseuille channel flow . . . . . . . . . . . . 36
4.4.1 semi-Lagrangian lattice Boltzmann method . . . . . . . .36
4.4.2 Interpolation-supplemented lattice Boltzmann method . . 40
4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . 48
5 Effect of distortion of interpolation stencils 50
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . .50
5.2 Three-dimensional Taylor-Green vortex . . . . . . . . . . 51
5.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . 57
6 Contributions and future recommendations 58
6.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . 58
6.2 Major contributions . . . . . . . . . . . . . . . . . . . 60
6.3 Future recommendations . . . . . . . . . . . . . . . . . .60
6.3.1 Flow over periodic hills . . . . . . . . . . . . . . . .60
6.3.2 Extend to compressible flows and multi-phase flows . . .61
Appendix A Derivation of analytical solutions 62

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