一維不恆定流分析之可變阻力係數研究

Abstract

在計算水力學研究中，模擬管流暫態現象（水錘）與計算全流域河川不恆定流中洪水波運移情況這兩個領域，雖含有一些相異點，但事實上是十分相近的，因為它們有相似的連續方程式與運動方程式，為一組二應變數及二自變數的雙曲線型非線性偏微分方程式。可以轉換為特徵方程式與以特徵線解法直接求解。 本研究以賴經都教授所創的兩個1維不恆定流水理模式，進行(1)紊流條件與層流條件下的管流暫態現象分析及(2)淡水河全流域河川不恆定流分析，這兩個水理模式的共同特點是都只有一個可以調整參數也就是阻力係數。本研究說明層流條件下的管流暫態現象及淡水河全流域河川不恆定流分析過程中阻力係數不是定值常數，只有考慮實際流況符合物理意義條件下的可變阻力係數，在計算時才有較好的的準確度。 在管流暫態現象分析方面，本研究重新檢視賴經都博士1961年完成的重要文獻” A study of waterhammer including effect of hydraulic losses”，首先重建數值模式、逐步分析計算過程使用指定時間間隔內差法及提高計算準確度使用的外差法導致錯誤產生原因、重新提出特徵方程式有限差分修正計算式，解決原模式數值計算時會產生震動突波問題。本研究重新設計可維持上游邊界在固定壓力（水頭高度）條件下，進行低雷諾數紊流與層流條件的水錘實驗。本研究以3種不同可變阻力係數計算方法評估數值模式阻力項，並與上述實驗結果比較，證明可變阻力係數在非紊流條件下管流暫態現象（水錘）分析的重要性。 在全流域河川不恆定流研究方面，考慮自然界多數河川流況均為紊流條件，若河川渠道水位變化劇烈，河槽阻力係數如為單一常數，將無法同時滿足平時低水位與洪水時高水位計算準確度需求，本研究提出複式河道的河槽阻力係數值應隨水位高程變化而不同，而決定阻力係數變化範圍的最低限值與最高限值（範圍內假設為線性變化）可依據平時低水位及颱洪時期高水位觀測記錄來率定。本研究說明此種考慮水位與河槽阻力係數變化關係，已成功應用於於台灣淡水河全流域，包含狹隘經人為整治複式河槽的基隆河，寬廣複式河道的淡水河、大漢溪、新店溪及人工開鑿的二重疏洪道。 本論文在水錘研究以實驗室量測壓力的變化值，明渠流研究則以淡水河流域各水位站的觀測水位值，來證明可變阻力係數在數值模式之重要性與效率性。各個數值計算，均與前人研究之數值解及實驗結果比較，本研究另模擬現實淡水河流域颱洪問題，來說明全流域河川不恆定流模式之適用性。

The flow of open channel and that of closed-conduit or pipe have many things in common, but also have a number of dissimilarities. The former is the flow under free surface, any disturbance generated is transmitted by the gravity wave, the cross sections of the channel containing the flow are often quite irregular, and so forth. The latter is the flow under pressure, any disturbance generated is transmitted by the elasticity wave, the cross sections of the conduit are usually uniform, and so forth. Regardless of such dissimilarities, both flows can be analyzed in quite a similar way. For example, the governing equations can be derived from the principles of continuity and of motion, the equations thus derived are of hyperbolic-type partial differential equations, amenable for numerical solution by the method of characteristics, from which it is possible to build powerful and useful unsteady flow models. These two types of flow models saw amazingly similar paths of development and progress in the past. They both started from simple channel configuration and basic parameter involvements. They then advanced step by step to sophisticated model forms of today, with the development of flow resistance coefficients in particular. In this study, two 1-D unsteady flow models, one each from the aforementioned two flow regimes, transient flow or water-hammer in a circular pipe/pipe-system and the flood flow in a tide-affected river basin, were selected for intensive investigation as to their similarities and dissimilarities, their paralleling patterns of model advancements, and augmented treatments and applications of variable flow-resistance coefficients in the model. In regards to the research of transient pipe flow, Dr. Chintu Lai’s dissertation, “A study of waterhammer including effect of hydraulic losses”, published in 1961, which was condensed and recast as: Streeter and Lai (1963), ”Water-Hammer Analysis Including Fluid Friction”. Based on Dr. Lai’s dissertation, this study first recreated his original computer program, then using this program to verify the accuracy and reliability for many important parts of the original work, and also to analyze some discrepancies that were present in some parts of the work dealing with low velocity flow (i.e., laminar flow). A modified form of the finite difference expression for the characteristics equations was also proposed in the study to deal with the problem of numerical oscillation existed in the original program. This study attempted an improved system that would enable the upstream boundary condition to sustain a constant pressure head, while conducting the water hammer experiment of low Reynolds number turbulent flow and of transient laminar flow. To this end, the treatment of distinct friction terms in a numerical model with three different variable flow-resistance coefficients showed the importance and efficiency of using variable flow-resistance coefficient in water hammer analysis. A brief review of some earlier numerical models using a constant resistance coefficient in single, regular reaches of waterways showed their improvements in two aspects: i) the improvement toward more complex channel geometries and reach configurations, and ii) the improvement toward use of variable flow-resistance coefficients. Consideration of the above two aspects, together with the relationship between stage and flow-resistance coefficient, a basin-wide flood flow model with variable flow-resistance coefficients has been successfully applied to Tamsui River system. In this study, a 1-D unsteady open-channel flow model was organized using the multimode method of characteristics of the second kind (MMOC-Ⅱ). In the research of pipe flows, the pressure profile was measured in the lab, and in the research of open channel flows, the observed stages in different stage stations of Tamsui River system were used to prove the importance and efficiency of variable flow-resistance coefficient in a numerical model. All calculated data in this study were compared with the data obtained from the previous experiments. For field calibration, verification, and application of the complex-compound channel model, i.e., the basin-wide channel flow model, the data from some actual typhoon-flood events occurred in the Tamsui River basin were utilized.