臺灣博碩士論文加值系統

泥砂運輸與人類的生活息息相關，例如橋墩沖刷、水質估計等等，故泥砂運輸的研究是一直以來都是一個很重要的議題。泥砂顆粒在水中除了隨著水流方向運動之外，也會因為受到紊流的影響而向周圍不規則擴散，此外，其運動的行為可以馬可夫鏈(Markov chain)來近似，因此本研究將泥砂顆粒的運動視為一個隨機過程。本文以力學原理結合序率方法(Stochastic method)來模擬泥砂顆粒在水中的運動軌跡，亦即增強隨機微分方程中的物理性質，使之更貼近自然情形。
為模擬泥砂顆粒的運動行為，本文以朗之萬方程(Langevin equation)為原型所推導出的隨機擴散粒子追蹤模型(Stochastic Diffusion Particle Tracking Model)呈現顆粒運動因紊流而造成的不確定性。其中，隨機擴散粒子追蹤模型主要包含兩種基本元素：平均漂移項(Mean drift term)，即為顆粒隨著水流方向運動；紊流項(Turbulence term)，即顆粒受到紊流作用而有不規則的運動，也稱為布朗運動(Brownian motion)，係利用維納過程(Wiener process)來模擬。
本研究利用隨機擴散粒子追蹤模型來模擬在一般流況下泥砂顆粒的運動軌跡， 分別使用兩種隨機擴散粒子追蹤模型: 單顆粒粒子追蹤模型(One-particle Particle Tracking Model)和雙顆粒粒子追蹤模型(Two-particle Particle Tracking Model)去模擬。其中，雙顆粒粒子追蹤模型比單顆粒粒子追蹤模型多考慮了顆粒在距離相近時候的變化，因為大尺度的渦流(Large scale turbulence)的關係可能使彼此相近的顆粒具有相似的隨機運動。另外，以巨觀的角度去觀察顆粒整體的運動，可以計算出水中的泥砂濃度，且因為泥砂顆粒受到紊流擾動的影響，使得泥砂的濃度也具有不確定的變化。因此本文呈現顆粒軌跡和泥砂濃度的平均值和標準差來表示泥砂顆粒在水中的不確定性。本研究首先和實驗資料比對單顆粒和雙顆粒粒子模型所估計的濃度以驗證模型的可行性，最後使用此模型分別探討層流流場中和紊流流場中顆粒隨機運動的情形，結果顯示在紊流流場中顆粒的隨機運動比較明顯，因此在高雷諾數(Reynolds number)的流場中估計泥砂濃度時，應考慮漩渦對泥砂顆粒所造成的隨機變化，並給予濃度變動範圍較為恰當。此外，泥砂顆粒運動具有馬可夫特性也在本文中證實。然而，如本文結果所顯示，泥沙顆粒的移動距離卻不是和時間呈線性的正比關係，並不符合菲克擴散(Fickian diffusion)。泥砂顆粒具有再懸浮的現象可能導致泥砂擴散為反常擴散(non Fickian diffusion or anomalous diffusion)。

Sediment transport is an important issue for human. It is closely related to human society, such as bridge scour and water quality. A sediment particle in flow not only follows the flow direction, but also diffuses through the surrounding water due to turbulence. Markov chain is used to approach the movement of sediment particles. From this perspective, particle movement is regarded as a stochastic process in our study; moreover, the proposed models simulate particle trajectories based on stochastic methodologies and physical mechanisms, underscoring mechanics in the stochastic differential equation.
To simulate sediment particle movement, the stochastic diffusion particle tracking model (SD-PTM) has been derived from the Langevin equation, which is able to show the random characteristics of sediment movement. SD-PTM has two basic elements, the mean drift term and the turbulence term. One of the particle characteristics, the mean drift term, is that particles follow the flow direction; another one is called the turbulence term that describes random behaviors caused by turbulence diffusion. This movement is known as Brownian motion. In general, the diffusion movement is modeled by the Wiener process.
The aim of this study is to simulate sediment particle trajectories under the normal flow condition by the SD-PTMs, one-particle PTM and two-particle PTM. The difference between the single particle model and the paired particle model is that the paired particle model accounts for large eddy turbulence. In other words, the paired particles may have similar random movement if the locations of particles are in the immediate vicinity of each other. Besides, to observe assemblage of particles’ motion in the macroscopic manner, the sediment concentrations can be estimated. Moreover, sediment concentrations involve the property of uncertainty on account of sediment particles’ stochastic trajectories. Therefore, to demonstrate such uncertainty of sediment particles, the ensemble means and ensemble standard deviations of sediment trajectory as well as concentrations are presented in the study respectively. The proposed models are validated against experimental data by ensemble mean velocity and sediment concentrations. Moreover, this study also discussed the random movement of sediment particles under various flow conditions, laminar cavity flow and fully developed turbulent open channel flow. Results show that the random movement of sediment particles is significant in turbulent flow. Thus, it is appropriate to consider the fluctuation of sediment concentrations under high Reynolds number flow conditions. Besides, the Markovian property of the PTMs is validated in our study. However, the variance of particle displacement and time are not a linear proportion as the result. Resuspension of sediment particles may cause particle movement to be anomalous diffusion.

口試委員會審定書 #
中文摘要 ii
ABSTRACT iv
CONTENTS vi
LIST OF FIGURES ix
LIST OF TABLES xi
Chapter 1 Introduction 1
1.1 Problem statement 3
1.2 Research Hypotheses 4
1.3 Objectives of Study 7
1.4 Overview of Thesis 7
Chapter 2 Literature Review 8
2.1 Stochastic Methods 8
2.1.1 The Eulerian model 9
2.1.2 The Lagrangian model 10
2.2 Pickup Probability 11
2.3 Turbulent diffusion and dispersion 15
2.4 Summary 17
Chapter 3 Stochastic Theories 18
3.1 Markov Process 18
3.2 Brownian Motion 19
3.3 Stochastic Diffusion Process 21
3.4 Numerical Approximation for Stochastic Differential Equations 23
3.5 Summary 27
Chapter 4 Development of Stochastic Particle Tracking Model of Suspended Sediment Transport 28
4.1 Introduction 28
4.2 Model Assumptions 29
4.3 Model Development 30
4.3.1 Stochastic Diffusion Model – One-Particle Particle Tracking Model 30
4.3.2 Stochastic Diffusion Model – Two-Particle Particle Tracking Model 33
4.4 Determination of Hydraulic Parameters in Open Channel Flow 36
4.4.1 Velocity Profile 36
4.4.2 Particle Settling Velocity 39
4.4.3 Diffusion Coefficient 41
4.4.4 Re-suspension Mechanism 46
4.5 Simulation Results 48
4.6 Summary and Conclusions 54
Chapter 5 Application of The Stochastic Particle Tracking Model 56
5.1 Introduction 56
5.2 Case study of validating with experimental data 57
5.3 Case study of particle movement under two-dimensional laminar flow conditions 64
5.4 Case study of particle movement under fully developed uniform channel flow 68
5.5 Summary and Conclusions 74
Chapter 6 Summary and Recommendations 77
6.1 Summary and Conclusions 77
6.2 Recommendations for Future Research 78
REFERENCE 79
APPENDIX 85

[1]Absi, R., Marchandon, S., and Lavarde, M. (2011, April). Turbulent diffusion of suspended particles: analysis of the turbulent Schmidt number. In Defect and diffusion forum (Vol. 312, pp. 794-799). Trans Tech Publications.
[2]Argall, R., Sanders, B. F., and Poon, Y. K. (2004). Random-Walk Suspended Sediment Transport and Settling Model. In Estuarine and Coastal Modeling. Eighth International Conference.
[3]Bolin, D. (2009). Computationally efficient methods in spatial statistics: Applications in environmental modeling. Licentiate Theses in Mathematical Sciences, 2009(4).
[4]Borgas, M. S., & Sawford, B. L. (1994). A family of stochastic models for two-particle dispersion in isotropic homogeneous stationary turbulence. Journal of fluid mechanics, 279, 69-99.
[5]Di Nucci, C., & Spena, A. R. (2011). Universal probability distributions of turbulence in open channel flows. Journal of Hydraulic Research, 49(5), 702-702.
[6]Bose, S. K., and Dey, S. (2013). Sediment Entrainment Probability and Threshold of Sediment Suspension: Exponential-Based Approach. Journal of Hydraulic Engineering ASCE, 139(10), 1099-1106.
[7]Cai, S. T. "Settling of Sediment Particles in Quiescent Water -(1) The Effect of Concentration on Fall Velocity. Journal of Physics, Vol. 12, No. 5, 1965, pp. 402-408. (in Chinese)
[8]Cellino, M. (1998). Experimental study of suspension flow in open channels. PhD thesisDépt. de Génie Civil, Ecole Polytechnique Fédérale de Lausanne
[9]Cheng, N. S., and Chiew, Y. M. (1998). Pickup probability for sediment entrainment. Journal of Hydraulic Engineering ASCE, 124(2), 232-235.
[10]Cheng, N. S., and Chiew, Y. M. (1999). Analysis of initiation of sediment suspension from bed load. Journal of Hydraulic Engineering ASCE, 125(8), 855-861.
[11]Chien, N. and Wan, Z. (1999). Mechanics of Sediment Transport, Translated under the guidance of Mcnown, J. S., ASCE Press, Reston, VA.
[12]Chou, Y. J. (in press). Fully Developed Uniform Channel Flow Simulate by Large eddy Simulation. Retrieved from Multi-Scale flow Physics & Computation Lab
[13]Davidson L (2015). Fluid Mechanics, Turbulent Flow and Turbulence Modeling. Goteborg, Sweden: Chalmers University of Technology.
[14]Coleman, N. L. 1970. Flume studies of the sediment transfer coefficient, Water Resources Res., 6(3), 801-809.
[15]Diamant, H., Cui, B., Lin, B., and Rice, S. A. (2005). Correlated particle dynamics in concentrated quasi-two-dimensional suspensions. Journal of Physics: Condensed Matter, 17(49), S4047.
[16]Dimou, K. N., & Adams, E. E. (1993). A random-walk, particle tracking model for well-mixed estuaries and coastal waters. Estuarine, Coastal and Shelf Science, 37(1), 99-110.
[17]Durbin, P. A. (1980). A stochastic model of two-particle dispersion and concentration fluctuations in homogeneous turbulence. Journal of Fluid Mechanics, 100(02), 279-302.
[18]Einstein, H. A. (1942). Formula for the transportation of bed load. Trans. Am. Soc. Civ. Eng., 107, 561–597.
[19]Einstein, H. A. (1950). The bed-load function for sediment transportation in open channel flows. Technical Bulletin No. 1026, U.S. Dept. of Agriculture, Soil Conservation Service, Washington, DC.
[20]Elder, J. W. (1959). The dispersion of marked fluid in turbulent shear flow. Journal of fluid mechanics, 5(04), 544-560.
[21]Fisher, H.B.; List, EJ.; Koh, R.C.Y.; Imberger, J.; Brooks, N.H. (1979): Mixing in inland and coastal waters. Academic Press, New York.
[22]French, R.H., (1985). Open-Channel Hydraulics. McGraw-Hill, New York, NY, 705 pp.
[23]Gardiner, C.W., (1985). Handbook of Stochastic Models, 2nd ed. Springer, Heidelberg.
[24]Ghia, U. K. N. G., Ghia, K. N., and Shin, C. T. (1982). High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method. Journal of computational physics, 48(3), 387-411.
[25]Heemink, A. W. (1990). Stochastic modelling of dispersion in shallow water. Stochastic hydrology and hydraulics, 4(2), 161-174.
[26]Higham, D. J. (2001). An algorithmic introduction to numerical simulation of stochastic differential equations. SIAM review, 43(3), 525-546.
[27]Keefer, T. N. (1971). The relation of turbulence to diffusion in open-channel flows (No. 72-206). US Geological Survey.
[28]Kirmse, D. W. (1964). A Monte Carlo study of turbulent diffusion, PhD Dissertation, Iowa State University of Science and Technology, Ames, Iowa.
[29]Lee, H., and Balachandar, S. (2012). Critical shear stress for incipient motion of a particle on a rough bed. Journal of Geophysical Research: Earth Surface, 117(F1).
[30]Man C. (2007). Stochastic Modeling of Suspended Sediment Transport in Regular and Extreme Flow Environments. Ph.D. Dissertation, State University of New York at Buffalo, Buffalo, NY
[31]Man C. and Tsai C. (2007), A stochastic partial differential equation based model for suspended sediment transport in surface water flows. Journal of Engineering Mechanics ASCE, Vol. 133, No. 4, pp. 422–430
[32]Muste, M., Yu, K., Fujita, I., & Ettema, R. (2009). Two-phase flow insights into open-channel flows with suspended particles of different densities. Environmental fluid mechanics, 9(2), 161-186.
[33]Oh, J., and Tsai, C. W. (2010). A stochastic jump diffusion particle‐tracking model (SJD‐PTM) for sediment transport in open channel flows. Water Resources Research, 46(10)
[34]Pope, S.B., (2000), Turbulent flows, Cambridge University Press, Cambridge, U.K.; New York, U.S.A.
[35]Reynolds, A. M. (1998). A Lagrangian stochastic model for the trajectories of particle pairs and its application to the prediction of concentration variance within plant canopies. Boundary-Layer Meteorology, 88(3), 467-478.
[36]Risken, H. (1989). The Fokker-Planck Equation. Methods of Solution and Applications, vol. 18 of. Springer Series in Synergetics.
[37]Roberts, P. J., and Webster, D. R. (2002). Turbulent diffusion (pp. 7-47). ASCE Press, Reston, Virginia.
[38]Ross, S. M. (2006). Introduction to probability models, Academic press.
[39]Rouse, H. (1937). Modern conceptions of the mechanics of turbulence, Trans. ASCE, 102, 463-543.
[40]Sadat, S.M., Tokaldani, E., Darby, S. and Shafaie, A. (2009). Fall Velocity of Sediment Particles. 4th International. Conference on Water Resources, Hydraulics and Hydrology. February 24-26, 2009, University of Cambridge, UK.
[41]Sawford, B. L. (1991). Reynolds number effects in Lagrangian stochastic models of turbulent dispersion. Physics of Fluids A: Fluid Dynamics (1989-1993), 3(6), 1577-1586.
[42]Sawford, B. L. and Bokgas, M. S. (1994) On the continuity of stochastic models for the Lagrangian velocity in turbulence. physica D.
[43]Shah, S. H. A. M., Heemink, A. W., and Deleersnijder, E. (2011). Assessing Lagrangian schemes for simulating diffusion on non-flat isopycnal surfaces. Ocean Modelling, 39(3), 351-361.
[44]Shlesinger, M. F., West, B. J., & Klafter, J. (1987). Lévy dynamics of enhanced diffusion: Application to turbulence. Physical Review Letters, 58(11), 1100.
[45]Sharma, S. N., and Patel, H. G. (2010). The Fokker-Planck equation. INTECH Open Access Publisher.
[46]Socolofsky, S. A., & Jirka, G. H. (2005). Special topics in mixing and transport processes in the environment. Engineering—lectures, fifth ed., Coastal and Ocean Engineering Division, Texas A&M University.
[47] Spivakovskaya, D., Heemink, A. W., and Schoenmakers, J. G. M. (2007). Two-particle models for the estimation of the mean and standard deviation of concentrations in coastal waters. Stochastic environmental Research and Risk Assessment, 21(3), 235-251.
[48]Spurk, J.H. and N. Aksel, Fluid mechanics. 2008, Berlin: Springer.
[49]Thomson, D. J. (1990). A stochastic model for the motion of particle pairs in isotropic high-Reynolds-number turbulence, and its application to the problem of concentration variance. Journal of fluid mechanics, 210, 113-153.
[50]Chuanjian, M. A. N., & Jungsun, O. H. (2014). Stochastic particle based models for suspended particle movement in surface flows. International Journal of Sediment Research, 29(2), 195-207.
[51] Tsujimoto T. (2010). Diffusion coefficient of suspended sediment and kinematic eddy viscosity of flow containing suspended load. River Flow 2010 -Dittrich, Koll, Aberle & Geisenhainer (eds), 801-806.
[52]Wu, F.-C., and Chou, Y.-J. (2003). Rolling and lifting probabilities for sediment entrainment. Journal of Hydraulic Engineering ASCE, 129(2), 110-119.
[53]Wu, F. C., & Lin, Y. C. (2002). Pickup probability of sediment under log-normal velocity distribution. Journal of Hydraulic Engineering, 128(4), 438-442.
[54] Yen B. C. (2002), Stochastic inference to sediment and fluvial hydraulics. ASCE Journal of Hydraulic Engineering, Vol. 128, No. 4, pp.365-367