本篇論文主要使用不同種類顯式數值演算法在圖形顯示卡上進行複雜流場的模擬研究，其中包含DTAC、PM、LBM、GPE、EDAC等方法。這項研究主要有四個目的；第一個是將PM方法從CPU平台轉移至GPU平台進行模擬，本文使用多重網格方法並且搭配粗網格聚合之技術來進行收斂加速。從模擬結果可以得到最佳的多重網格為採用到8^3的粗網格以及最佳粗網格聚合的網格數為32^3，並且搭配粗網格聚合方法可以得到2.3至2.6倍的加速。

第二個目標為探討各顯式數值方法之計算效率，藉由模擬紊流腔體拉板流場與黏彈流體自然對流，比較各顯式方法對於不同剛性流場的優劣。結果表明，在Nvidia Tesla V100顯示卡上，我們所使用的PM演算法搭配多重網格方法比原先在CPU平台的PM方法加速了1000倍，也比弱可壓縮方法(GPE)加速了3倍。相反的，當我們針對剛性較大的黏彈流體自然對流進行模擬，發現GPE方法因為避免求解泊松方程式以及有著較好的平行效率，得到了比PM方法快2.2倍的加速效果。

第三個目的為使用GPE方法來模擬FENE-P流體的自然對流流場。結果表明，低彈性力影響(Wi=10)時，接近壁面的最大水平以及垂直速度能觀察到些微的增加，相反的，高彈性力影響(Wi=100,L^2_{max}=500)時，水平速度以及垂直速度明顯下降，導致壁面熱傳降低。針對不同$Ra$數，可以觀察到彈性力的影響在高Ra數時變的不明顯。此外我們也利用尺度分析推導出一個關係公式來精準捕捉黏彈流體在自然對流的熱傳行為。最後發現在Ra=10^4時，觀察到12%的熱傳降低效果。

第四個目標為利用弱可壓縮流場模擬兩相流場，我們藉由使用GPE與EDAC方法來探討在壓力方程式中的對流項對模擬兩相流場的影響。從水壩潰堤以及注水模擬結果可以發現，當有對流項引入的壓力方程式(EDAC)，擁有較好的計算穩定性，也從模擬案例可以觀察到弱可壓縮流方法在掌握這種高剛性流場以及大尺度網格的高效率求解能力。

In the present study, several explicit numerical methods are investigated on Newtonian and non-Newtonian flows using the graphics processing unit (GPU) cluster. Due to the nature of the parallel structure of GPU, the explicit method is suitable to adopt for the computational fluid dynamics (CFD). Previous studies developed several numerical algorithms, including the projection method with conjugate gradient smoother on CPU, the dual-time artificial compressibility (DTAC) method on GPU, the lattice Boltzmann method (LBM) on GPU, and the weakly compprssible method on GPU. The projection method is migrated to the GPU platform with multigrid technology in the present study. However, the advantage of each method is not clear. Hence, the methods are examined by simulating the different stiff systems to judge their performance.

\noindent First, a GPU-enabled numerical procedure based on the projection method is developed for simulating incompressible turbulent flows. The pressure Poisson equation is efficiently solved using the V-cycle geometric multigrid method. Additionally, the coarse grid aggregation (CGA) technique enhances the multigrid level of multi-GPU simulations, resulting in significant performance improvements. The validity of the proposed method is confirmed through direct numerical simulations of the turbulent lid-driven cavity (LDC) flows at a Reynolds number of 3200. The computed mean, and turbulence quantities closely match the available measured data, validating the accuracy of the approach.
For the cubic cavity under consideration, the optimized minimum grid sizes for multigrid and CGA are determined to be $8^3$ and $32^3$, respectively. An additional speedup of approximately 2.3 to 2.6 is achieved by employing CGA. In terms of performance based on turbulent LDC simulations, the current implementation demonstrates compatibility with the LBM while also being three times faster than the general pressure equation scheme. The superior performance of the GPU implementation over CPU is further highlighted, with a remarkable one thousandfold speedup observed between the Nvidia Tesla V100 and a single core of the Intel I7-6900K (8 cores). Specifically, the performance of one Tesla V100 is found to be equivalent to 125 I7-6900K CPUs. On the other hand, a 2.2 times speedup is achieved using the GPE method over the multigrid-based projection method solver for the natural convection flow with FENE-P fluid.

\noindent Then, we simulate the 2D viscoelastic natural convection flow using the general pressure equation with parameters of Rayleigh (Ra) number ranging from $10^4$ to $10^7$, Weissenberg (Wi) number ranging from 1 to 100, and maximum characteristic length ($L_{max}^2$) equal to 10, 100, and 500. The solver is validated using Newtonian natural convection cases at Ra number ranging from $10^3$ to $10^7$. Results indicate that at low elasticity effects ($Wi=10$), the maximum horizontal and vertical velocity slightly increased, leading to higher heat transfer. In contrast, high elasticity effects ($Wi=100$, $L_{max}^2=500$) resulted in a decrease in horizontal and vertical velocity, reducing heat transfer. The presence of polymer generated opposing stress near the wall, leading to a decrease in vertical velocity at $Wi=100$, $L_{max}^2=500$.
From the Ra number effect, increasing the value of Ra resulted in a decrease in the amount of stress generated by the polymer.
Additionally, the mean kinetic energy budget shows that buoyant production is the main source of energy generation, and this process is counterbalanced by pressure diffusion, viscous diffusion, and polymer diffusion.
Moreover, a scaling correlation is derived based on the FENE-P fluid for predicting the heat transfer performance. Although the scaling is simplified to produce order-of-magnitude estimates for quantities of interest, the both HTE and HTR effect are nicely captured at a range of $Wi$, $L^2_{max}$ and $b$.
It was found that viscoelastic rheological behavior had a significant impact on flow intensity and thermal structures, with the highest heat transfer rate reducing by 12$\%$ at $Ra=10^4$, $Wi=100$, $L_{max}^2=500$, which represented a resistance force. These findings provide valuable insights into understanding the behavior of viscoelastic fluids in natural convection.

\noindent Finally, weakly compressible method with various artificial pressure evolution equations have been proposed as alternatives to solving the Poisson equation in order to simulate incompressible fluid flows. Notably, the general pressure equation (GPE) and entropically damped artificial compressibility (EDAC) methods have emerged as novel approaches, differing mainly in the presence of an convective term in the pressure equation. In this study, our focus is to analyze the performance of these two pressure evolution equations on GPUs specifically for simulating two-phase flows. To accurately capture the gas-liquid interface and minimize errors associated with surface tension force calculations, our solver incorporates the coupled level set and phase-field equation method. To validate the solver's interface capturing capability, we conducted simulations of the reversed single vortex and Rayleigh Taylor instability problems, yielding results that closely matched established benchmark solutions. Particularly, the inclusion of the convective term in the pressure evolution equation yielded a more stable numerical simulation, especially in the presence of violent two-phase flow. The obtained results revealed intricate interactions between two liquids, including phenomena such as droplet splashing and the formation of dense bubble clusters. Furthermore, the computational efficiency of the solver was demonstrated through high-resolution simulations capable of handling billion-level grid sizes.

Contents
Abstract i
Contents vi
List of Figures xii
List of Tables xv
1 Introduction 1
1.1 Explicit Schemes for GPU Computing . . . . . . . . . . . . . . . 2
1.2 Projection method with multigrid technology . . . . . . . . . . 3
1.3 Simulations of polymer stress induced viscoelastic natural convection
flow by general pressure equation . . . . . . . . . . . . . . . . . 6
1.3.1 Non-Newtonian fluid . . . . . . . . . . . . . . . . . . . . . 6
1.3.2 Viscoelastic natural convection flow . . . . . . . . . . . . . 8
1.4 Numerical simulations of two-phase flow by weakly compressible method 11
1.4.1 Analysis of pressure evolution equation . . . . . . . . . . 11
1.4.2 Interface-capturing method . . . . . . . . . . . . . . . . . 14
1.5 Objective of the Dissertation . . . . . . . . . . . . . . . . . . . . . . . 15
1.6 Outlines of Dissertation . . . . . . . . . . . . . . . . . . . . . . . . . 17
2 Methodology 18
2.1 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2 Projection method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2.1 Multigrid implementation . . . . . . . . . . . . . . . . . . . . 21
2.3 Weakly compressible method . . . . . . . . . . . . . . . . . . . . . . . 30
2.4 FENE-P model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.5 Interface-Capturing method . . . . . . . . . . . . . . . . . . . . . . . 31
2.6 GPU implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3 Performance efficiency 36
3.1 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.1.1 Laminar lid-driven cavity flow . . . . . . . . . . . . . . . . . . 37
3.1.2 Turbulent lid-driven cavity flow . . . . . . . . . . . . . . . . . 37
3.2 Performance of V-cycle multigrid solver . . . . . . . . . . . . . . . . . 48
3.3 Mass conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.4 Performance comparisons with explicit methods . . . . . . . . . . . . 59
3.5 Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4 Simulation of polymer stress induced viscoelastic natural convection
flow by general pressure equation 64
4.1 Scaling analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.2.1 Natural convection of Newtonian fluid . . . . . . . . . . . . . 67
4.2.2 Natural convection of visco-elastic fluid . . . . . . . . . . . . . 70
4.2.3 Flow structure and maximum velocities . . . . . . . . . . . . . 71
4.2.4 Mean kinetic energy . . . . . . . . . . . . . . . . . . . . . . . 78
4.3 Comparison with literature results . . . . . . . . . . . . . . . . . . . . 81
5 Numerical simulations of two-phase flow by weakly compressible
method 85
5.1 Reversed Single Vortex . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.2 Two-Dimensional Rayleigh Taylor instability . . . . . . . . . . . . . . 86
5.3 Three-Dimensional Bubble rising . . . . . . . . . . . . . . . . . . . . 88
5.4 Dam Break . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
5.5 Water Injection into A Container . . . . . . . . . . . . . . . . . . . . 96
6 Conclusions and recommendations for further research 99
6.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
6.1.1 General pressure equation for Natural convection flow
simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
6.1.2 Weakly compressible method for two-phase flow simulations . 101
6.2 Recommendations for further research . . . . . . . . . . . . . . . . . 102

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