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  • 學位論文

序率泥砂力學: 泥砂濃度變異性,非菲克定律擴散,及沉降速度不確定

Stochastic sediment transport modeling: fluctuation sediment concentrations, Non-Fickian diffusivity, settling velocity uncertainty.

指導教授 : 蔡宛珊

摘要


因河川的渦流,使底床的泥沙侵蝕。在實驗中,可以觀察大尺度的渦流,小尺的渦 流,但通常定律模型都假設在渦流是小尺度之下。因此,再這篇論文探討三個主題: 第一,用菲克定律擴散模型去做模擬泥沙濃度的不確定性。第二,利用力平衡與碰 撞的概念,去模擬泥沙濃度的不確定性,第三,由於前兩者都是蓄率模型,對不確 定分析上的模型,進行改良。 第一個主題著重在菲克定律擴散模型,利用馬可夫練的方法,來簡化原來的模 型,讓計算效率提升,對於大量泥沙顆粒的情況下,可以快速在空間與時間上的平 均濃度,並且可以算出泥沙濃度的不確定性,由於計算效率的提升,可以考慮更多 隨機變數。因此在此論文中,除了本身渦流上的物理不確定,加上不均勻顆粒做為 探討。相對於定率模型,這個模型可以提供更多的資訊,例如:實驗上泥沙的不穩 定性可以被計算。 第二個主題著重在單顆顆粒的運動行為,在越重顆粒的運動行為,一般的擴散運 動並不適合去描述泥砂運動過程。因此,在這個主題創造一個四階精度的流場,並 考慮二維速度的相關性,利用升力,重力,拖曳力,浮力來描述顆粒的行為,除此 之外,建立一個隨機碰撞底床的行為模式,來模擬邊界條件形況。顆粒泥沙跳躍的 高速,以及每次跳躍的距離,都與實驗比較,在泥沙濃度上面,在懸浮載和底床載 的情況下,與實驗和定率模型筆。在這個模式中,可以表現出更多的隨機變數,考 慮更多的影響原因,對於未來泥沙濃度模擬,可以有提供更多資訊。 第三個主題,上面兩個模式,在隨機變數上的分析,都是以蒙地卡羅的方式做為 模擬,但蒙地卡羅要許多時間,另外一個方法,點估計法,只利用三個點去做模擬 的精度並不高。因此提出了兩種不同的改良方法,第一個,利用數值的方法去改良 原先點估計法的方式,利用兩次計算結果來估計誤差,進行判斷是否需要更多模擬 點,第二個方法,利用現有的數值積分方法(赫米特高斯積分),此積分必須要假設 變數為高斯分布,因此這篇主題利用蓄率模型,把原先隨機變數的機率分布,假設 為高斯分布承多項式。此多項式讓高階動差與原先變數相同。此方法可以大幅度減 所需要的計算效率,並且提供一個精準的答案。在這個主題中,不同的函數和不同 的機率分布的結果被探討。兩種方法各有優缺點。關鍵字:泥沙濃度、不確定分 析、菲克定率、序率泥砂運動、底床碰撞

並列摘要


The uncertainty of the sediment transport cannot be neglected according to the turbulent bursting. The study adapted two methods to calculated the uncertainty of sediment transport and one method to improved uncertainty analysis method. The first method (first topic) is based on the advection diffusion equation which is in diffusion region. The uncertain of the movement of the sediment particle can be simulated by stochastic particle tracking method(SPTM). By Markov Chain, SPTM can be simplified and calculated the uncertainty of the sediment concentration efficiency. First, the uncertainty of the spatial and temporal sediment concertation caused by the particle size can be calculated. With a similar particle size, the change of temporal sediment concertation is explicit. Secondly the uncertain of the equilibrium sediment concentration caused by fluctuation of turbulence will be estimated. The result of sediment concertation by proposed model is validated against the experiment data and can better describe sediment concentration than the deterministic model (Rouse profile). The first method is more suitable to described the sediment concentration under condition of suspended load. The second method is focus on the trajectory of the sediment particle in non-diffusion region. The particle is force by lifting, gravity, drag, Buoyancy forces. To calculated the foundation force, the more accuracy flow velocity is created by gram charily expansion which considering four order moment of the fluctuation. The bed condition in the study adapt a rebound process. The particle will collide with stochastic particle alignment. In this (second) topic, firstly, the saltation length and height calculated by the proposed model will be validated by experiment. The saltation length and height in the smooth bed is higher than that in rough bed. Secondly the sediment concentration under suspend and bed load condition is compared with Rouse profile and experiment data. Third the supdiffusion and subdiffusion will be discussed. In the first and second topic, the uncertainty of the sediment concentration is calculated by the Monte Carlo Simulation which is extremely time consuming. In the third topic, two new methods are provided. The results simulated by two methods are compared with Simpson method. The first method is derived from the point estimated method(PEM) which only used two or three points. By increasing the accuracy of the simulated result by PEM, the adequative method is used to increase the simulation points. Secondly, Gram Charlier expansion and Hermit Gauss Quadrature is adapted to calculated by uncertainty in the second method. First method is suitable when the simulation points are few and the distribution of the random variable is known. The second method is suitable when the statistical moment of the random variable is only known. The converge of the second method is faster than the other two methods.Key words: Particle tracking method, Markov chain, higher order flow field, Modified Hermite Gauss integral, Adequative Hong’s method.

參考文獻


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