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研究生:張佑聖
研究生(外文):Yu-Sheng Chang
論文名稱:變質量平滑粒子動力法在降雨逕流系統之模擬
論文名稱(外文):Modeling rainfall-runoff processes using smoothed particle hydrodynamics with mass-varied particles
指導教授:張倉榮張倉榮引用關係
指導教授(外文):Tsang-Jung Chang
口試委員:許銘熙陳明志柳文成葉克家
口試委員(外文):Ming-Hsi HsuMing-Jyh ChernWen-Cheng LiuKeh-Chia Yeh
口試日期:2017-05-27
學位類別:博士
校院名稱:國立臺灣大學
系所名稱:生物環境系統工程學研究所
學門:工程學門
學類:土木工程學類
論文種類:學術論文
論文出版年:2017
畢業學年度:105
語文別:英文
論文頁數:96
中文關鍵詞:平滑粒子動力法淺水波方程式無網格方法變質量粒子降雨逕流過程入滲下水道
外文關鍵詞:Smoothed particle hydrodynamicsShallow water equationsMeshless methodmass-varied particleRainfall-runoff processInfiltrationSewer
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平滑粒子動力法(Smoothed Particle Hydrodynamics, SPH)為一種拉格朗日(Lagrange)觀點下的無網格粒子數值模擬方法。相較於傳統固定式網格數值方法,SPH在處理自由液面與大形變流場問題上已被許多研究證實具有優勢。近期SPH被運用在求解淺水波方程式(shallow water equations, SWEs)去模擬或是處理各種水力學問題,而發展出SPH-SWEs模式。然而,由於SWEs維度上的限制,目前SPH-SWEs無法處理計算維度外流體增減的問題,如側流、降雨和入滲等現象。因此本研究發展變質量平滑粒子動力法,藉由流體粒子資量的增加或減少來模擬額外維度上流體的增減,並將此新模式運用於研究降雨逕流過程中相關之問題上。
為了測試此新發展之變質量SPH法,研究中選取了三個具有代表性的案例進行模擬。模擬案例包含一維平坦傾斜渠道上的均勻降雨、一維三坡度渠道上不同降雨延時之非均勻降雨與二維複雜地形上之均勻降雨。模擬結果顯示本研究所發展之模式可以不使用源項函數以及增加粒子數來處理增加的流體質量,且在降雨經流過程中所出現的水躍、乾溼床移動邊界流及超臨界/亞臨界/跨流等現象上,皆有不錯的吻合程度,因而證明了此變質量SPH法的穩健與可靠性。
本研究中亦將此變質量SPH法在運用於處理入滲的問題上,結合霍頓(Horton)入滲方程進行求解,模擬逕流時流體減少之現象。另外,本研究亦嘗試以兩分量壓力近似法(Two-component Pressure Approach, TPA)建立一維SPH下水道模式。藉由TPA可使得SPH能以單一控制方程式來同時模擬滿管壓力流與自由夜面流的流況。而透過變質量流體粒子的運用,可將降雨逕流模式(增質量)、入滲模式(減質量)與下水道模式(質量傳輸)結合成一套SPH淹水模式。
Smoothed Particle Hydrodynamics (SPH) is a kind of meshless particle numerical method with Lagrangian concept. Compared to traditional grid-based method, SPH is proved by many researches that it has advantages on dealing with free surface and large deformation problems. Recently SPH has been implemented on solving shallow water equations (SWEs) for simulating or handling the hydrodynamic problems and SPH-SWEs has been developed accordingly. However, due to the limitation of dimensions of SWEs, SPH-SWEs still cannot process the problems of fluid inflow /outflow beyond the computational domain, e.g. lateral flow, rainfall, and infiltration, etc. Thus this research constructs the mass-varied SPH model which uses the mass variation of fluid particle to simulate the fluid inflow/outflow on the external dimension and also applies this new model on investigating the rainfall-runoff processes.
To validate this novel mass-varied SPH model, three benchmark case studies are adopted to conduct numerical simulations, including uniform rainfall over a 1D flat sloping channel, nonuniform rain falling over a 1D three-slope channel with different rainfall durations, and uniform rainfall over a 2D plot with complex topography. The simulated results indicate that the proposed treatment can avoid the necessity of a source term function of mass variation, and no additional particles are needed for the increase of mass. Rainfall-runoff processes can be well captured in the presence of hydraulic jumps, dry/wet bed flows, and supercritical/subcritical/transcritical flows. The proposed treatment using mass-varied particles was proven robust and reliable for modeling rainfall-runoff processes.
In this thesis the mass-varied SPH model is also utilized on solving the infiltration problems associated with Horton formula to simulate the fluid decrease during the runoff process. In addition, this research tries to develop a one-dimensional SPH sewer model with two-component pressure approach (TPA). With this approach, SPH can simulate the full-pipe pressure flows and free surface flows at the same time with single governing equation. Afterwards in future study, by means of mass-varied fluid particles, we can integrate the rainfall-runoff model (mass addition), infiltration model (mass reduction) and sewer model (mass transfer) into a SPH flood model.
摘 要 II
Abstract II
Table of Contents IV
List of Figures VI
List of Tables VIII
Chapter 1 Introduction 1
1.1 Objectives 1
1.2 Research method and SPH 2
1.3 Thesis structure 6
Chapter 2 Literature reviews 7
2.1 SPH theory 7
2.1.1 Wall boundary condition 7
2.1.2 Inflow and outflow boundary 8
2.1.3 SPH formula modification 9
2.1.4 Particle searching 9
2.1.5 Error analysis 10
2.2 SPH application on shallow water equations and SPH-SWEs model 10
2.3 Rainfall-runoff processes and models 11
2.4 Infiltration 12
2.5 Drainage systems and sewers 13
Chapter 3 Theories and Methods 16
3.1 Introduction of SPH 16
3.2 Central concept and fundamental formulations 16
3.3 Particle approximation 18
3.4 Derivation of fundamental formulations 19
3.4.1 First derivative 20
3.4.2 Second derivative 21
3.5 The choices of kernel functions 22
Chapter 4 SPH for shallow water equations and numerical techniques 25
4.1 Shallow water equations and SPH-SWEs 25
4.2 Water depth/cross-section wetted area evolution 28
4.3 Using mass-varied particles and the modified smoothing length updating formulation 31
4.4 Discretization of the momentum equation 36
4.5 Evolution of bed gradient term and friction term 37
4.6 Velocity correction in momentum equation for mass variation and extreme small water depth 39
4.7 Artificial viscosity and stabilization term 40
4.8 Time stepping and integration 41
4.9 Nearest neighboring particle searching (NNPS) 42
4.9.1 All-pair search 43
4.9.2 Linked-list 43
4.10 Wall boundary conditions 44
4.10.1 The Ghost particles method 45
4.10.2 The simplified MVBP method 46
4.10.3 Periodic boundary condition 49
4.11 Open boundary conditions 49
4.11.1 Method of specified time interval for 1D Q-A form of SWEs 50
4.11.2 Riemann invariants for 2D u-dw form of SWEs 51
4.12 Green-Ampt Infiltration module 54
4.13 1D SPH sewer module 55
4.14 Calculation process 58
Chapter 5 Model validations and applications 66
5.1 Uniform rainfall over a 1D flat channel 67
5.1.1 Convergence analysis of the particle number 67
5.1.2 Numerical accuracies of discharge, water depth, velocity and Froude number 68
5.2 Nonuniform rainfall over a 1D three-slope channel 69
5.3 Uniform rainfall over a 2D plot with complex topography 70
5.3.1 Convergence analysis of the particle number 71
5.3.2 Numerical accuracies of discharge, water depth, velocity and Froude number 72
5.4 Green-Ampt infiltration 73
5.5 1D pipe flow with SPH-SWEs 74
Chapter 6 Conclusions 87
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