應用NewC法建立下水道管網模式

Building A Sewer Network Model Using NewC Scheme

蘇玉豪(Yu-Hao Su)

指導教授 ： 李天浩

國立臺灣大學/工學院/土木工程學研究所/碩士(2007年)

https://doi.org/10.6342/NTU.2007.03351

摘要

本研究第一部份，設計包含管渠與人孔的下水道控制體積，對控制體積內的水積分，得到蓄水體積時變率與穿過控制表面流量的連續方程式。利用人孔中存在自由水面，類似在滿管壓力流時增加普里斯曼槽(Preissmann slot)數值技巧的概念，將滿管壓力流轉變為明渠流，以迪聖凡南方程式計算。採用交錯式網格設計、可穩定計算穿臨界流的NewC法，數值化連續方程式與動量方程式，建立可以同時模擬明渠流況與滿管流況的一維下水道演算模式。設計一維、非穩態下水道未滿管與滿管模擬案例，利用本研究開發的模式模擬，證實無論是滿管流況或未滿管流況均可穩定模擬；同時，模擬結果與SWMM的結果的差異均低於1%。

本研究的第二部份是發展高效率的管網系統求解演算法，將龐大、複雜的下水道管網系統，拆解為多個一維下水道計算。利用Sen and Garg (2002)針對非交錯式差分網格和Preissmann四點法所提出的河系數值三步驟求解技巧，配合NewC法的特殊數值邏輯，建立將下水道管網與明渠河系線性化方程式，拆解為多個一維演算系統和匯流口條件聯立解的三步驟求解演算法。首先，針對匯流處等內部邊界，提出適用於NewC法的管網匯流處理方法，清楚定義符號規約(sign convention)，利用個別支流控制方程式的組合，推導得符合真實情形且正確的匯流處(junction)連續方程式，以及匯流處的水位、流量關係式。求解三個演算步驟分別是：第一步驟利用雙掃法第一掃的原理，將三對角矩陣方程式變為二對角矩陣方程式。第二步驟是利用高斯消去法，消去非邊界狀態變數和邊界狀態變數的聯立關係，得到單一渠道上、下游邊界水位和流量四個狀態變數的聯立方程組；再以匯流口交換機制和邊界條件，建立所有一維渠道邊界狀態變數的聯立方程組，求解各一維渠道的所有邊界狀態變數。第三步驟是由已知的邊界狀態變數，再以雙掃法（因為邊界狀態已變更，第一掃係數亦隨之改變）計算各單一渠道內部點的狀態變數。設計Y型、Loop型管網系統非穩態案例，測試本研究開發的管網求解邏輯；各案例都可以穩定計算、得到合理的演算結果，顯示本研究建立的下水道管網與地表明渠河系機制，可成功應用於不同形式之管網。

本研究第三部份，是結合前述以NewC法、管網演算邏輯所建立的明渠模式和下水道模式，初步採用Villemonte的潛堰堰流公式，比較下水道人孔冒出地面的水位和市街流明渠水位，由兩者的高低，決定下水道和地表市街渠道流透過人孔交換的流量，建立地表市街流與下水道水流耦合演算機制。利用管網、河系演算邏輯，結合地表市街流明渠模式與下水道系統模式，並且以迴圈型市街流渠道配合迴圈型下水道管網，設計非穩態下水道滿管溢流，和市街流透過人孔溢流入下水道的案例。測試顯示，模式皆可以穩定演算市街流明渠系統與下水道系統，結果合理估計雙向交換的流量；證實可以利用相同的河系、管網演算機制，耦合地表明渠流和下水道水流模式。

關鍵字

迪聖凡南方程式 ； NewC法 ； 明渠流 ； 滿管流 ； 下水道 ； 管網 ； 市街流 ； 耦合

並列摘要

The first part of this study develops a one-dimensional storm sewer model with NewC scheme (Kutija and Hewett, 2002). It designs a control volume containing one manhole and all the connected half-length pipes and bases on volume-integrated water volume and the discharges through control surface. Given that free surface exists in manholes and that manholes can be considered as the Preissmann-slot applied to pressurized pipe flow, the de St. Venant Equation can be applied to solve both open channel as well as surcharge flow. The NewC scheme is adopted for it’s ability to simulate trans-critical flow with unconditionally stability. Unsteady case studies demonstrate that the model can stably simulate flows under both full and non-full conditions. Compare its output to that of SWMM, the differences of stages and discharges are all within 1%.

The second part devices an efficient algorithm to disjoin solution of sewer network into solving a collection of 1D sewer pipes and simultaneous equations of junctions. The algorithm of Sen and Garg (2002), which utilized non-staggered grid and Preissmann four point scheme, is adapted to the numeric scheme of NewC. Comparing to the continuity equation of junctions, the sign convention to assemble 1D sewer pipes is defined. The algorithm consists of three parts. (1) Use the first sweep of a double-sweep method to transform the tri-diagonal coefficient matrix into a bi-diagonal one. (2) Apply Gauss Elimination to decouple internal state variables from those at external boundaries or internal boundaries, i.e., junctions. The equations of junctions consist of 2 equations derived from each 1D sewer as well as 1 junction or boundary equation at each end. A total of 4N unknowns for N 1D sewer pipes are solved simultaneously. (3) Employ the double-sweep algorithm to solve for the internal state variables of each 1D sewer pipe with known boundary state variables. A Y-shape and a loop sewer network is designed to test the algorithm. The simulations prove the algorithm is stable and efficient.

The third part of this research is to couple the sewer network flow model with the open channel flow model of urban street network. We adopt Villemonte’s formula for submerged weir flow to compute the interchange discharge or lateral flow between sewer system and street flow. The lateral flow is based on the water levels of manholes and those of street flow. Through designed case studies, the coupled model smoothly and accurately simulates the street and sewer flow.

並列關鍵字

de St ； Venant Equation ； NewC scheme ； Open channel flow ； Surcharge flow ； Sewer Network ； Overland flow ； Coupling.

參考文獻

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