二維有限解析法明渠水理與輸砂模式之研發與應用

Abstract

對於不規則天然渠道之流場及底床沖淤而言，均存在沿程縱方向以及橫斷面方向之變化，如選擇水深平均二維模式，除可模擬縱向長距離之流場特性外，亦可得到橫斷面方向之水理與底床沖淤變化之資訊。本研究之目的即在發展一水深平均二維水理及輸砂數值模式，其主要特點有：(1)考慮天然河道不規則之邊壁條件，建立非正交貼壁座標系統，推導位於該座標系統上之水深平均水理及輸砂控制方程式，並根據流場幾何特性，利用不同格點產生法建構計算格點；(2)針對台灣河川坡陡流急之特性，水理部分考慮移流作用的影響，採用顯式有限解析法進行水理控制方程式之離散化，此數值方法主要係用以求解雙曲線型偏微分方程式，因此適用於一般常見之明渠流；(3)輸砂模式採用與水理模式非耦合演算的計算方式，並藉由修正各粒徑沈滓起動剪應力，以考慮非均勻沉滓間相互的影響；此外，考慮懸浮載傳輸所具有之移流特性，將懸浮載與河床載分開計算，探討原應用於具垂直維度模式之底床附近河床載與懸浮載間沈滓交換速率的估算方式，並參考van Rijn對懸浮載濃度剖面之研究成果提出適用於水深平均模式之沈滓交換速率估算方法，稱之為積分法，期能更精確地模擬渠底沖淤機制。 本研究將所發展之模式應用至多種定床流場以及動床渠道之沖淤模擬，並引用試驗資料來加以驗證。就水理模式部分，分析之案例包括：定量流迴水、變量流洪水波傳遞、移流作用顯著且存在分離流與環流之滯洪區流場及丁壩流場、以及超臨界流與亞臨界流同時存在或交替發生且自由水面具有大曲率變化之潰壩流場及水躍現象等。就輸砂模式部分，首先檢驗積分法估算沈滓交換速率之正確性，藉由一清水沖刷案例，模擬其沈滓濃度剖面發展的過程，所得結果與試驗資料相近，證實了積分法配合適當的濃度剖面可以彌補模式在水深平均過程中所失去之垂直方向沈滓沈降與擴散作用的影響。此外，為進一步瞭解輸砂模式對於渠道沖淤及河床質粒徑組成改變之模擬能力，分別就均勻與非均勻沈滓條件下之渠道沖刷與淤積案例，以及更複雜之沖淤交替案例進行模擬與比較分析。根據數模結果，顯式有限解析法適用於移流作用顯著之明渠流場之模擬，而積分法考慮了水深方向沈滓濃度分佈對於河床載與懸浮載間沈滓交換速率之影響，因而改進了水深平均模式本質上之限制並提升預測渠道沖淤之精度。未來可針對台灣各河川中具有足夠現場資料者，進行模式內各輸砂經驗式之參數檢定、沖淤模擬及必要之改良，以增進本模式之實用價值。

For irregular natural channels, there exist longitudinal and lateral variations in flow field and channel bed evolution. A depth-averaged horizontal 2-D model has the capability of simulating the long-distanced longitudinal variations of flow characteristics, as well as the lateral variations of channel cross sections. The purpose of the study is to develop a depth-averaged horizontal 2-D hydraulic and sediment transport model. The main special features of the model include: (1) Considering the irregular cross-sectional geometry of natural channel, a non-orthogonal body-fitted coordinates system is constructed, and the depth-averaged hydraulic and sediment transport equations are derived on the coordinates. The computational grids are then generated by one of the grid generation methods, which is suitable for the flow geometry characteristics. (2) Aiming at the steep channel slope and high flow velocity of Taiwan’s rivers, the hydraulic model considers the dominating convection effects. The explicit finite analytic method is adopted to discretize the governing equations of flow. This numerical method is suitable for the hyperbolic-type partial differential equation; thus it is expected to be applicable to the general open channel flows. (3) The sediment model is uncoupled with the hydraulic model. The incipient shear stress for individual sediment particle is modified to account for the interaction among nonuniform sediments. In view of the strong convection character of suspended load, the suspended load and bed-load transport are separately computed. In addition, the method for estimating the sediment exchange rate near bed between suspended load and bed load in the vertical 2-D models is examined to see whether it is still applicable to depth-averaged horizontal 2-D models. By utilizing the sediment concentration profile developed by van Rijn, a new approach, which is suitable for use in the depth-averaged models and named integral approach, for estimating the sediment exchange rate near the channel bed is proposed to make more accurate prediction of channel scour or deposition mechanism. In this study, the proposed model is applied to various fixed-bed flow and mobile-bed aggradation/degradation simulations. Experimental data are used to verify the applicability and accuracy of the model. For the hydraulic model, case studies include: steady backwater flow, unsteady floodwater flow, flows with strong convection and with separation and circulation in a storage basin and in a flume with Groyne, and dam-break and hydraulic jump flows with sub-critical and supercritical regimes occurring simultaneously and sequentially, and with strong curvature of the free surface. For the sediment transport model, the accuracy of the integral approach for estimating the sediment exchange rate is first verified by a scouring experiment with clear water entering in the inlet. The predicted sediment concentration profiles are compared with the experimental measurements and generally good agreements are obtained. This fact shows that the integral approach with appropriate concentration profiles could compensate for the missing mechanism of vertical settling and diffusion of sediment in the depth-averaged model. In addition, for further understanding of its capability for predicting the channel bed evolution and the change of composition of bed material, the model is applied to simulate the cases with aggrading and degrading beds, respectively, with uniform or nonuniform sediment, and the more complex cases with aggrading and degrading bed alternatively. Comparisons against the experimental data show that the explicit finite analytic method is applicable to the open channel flows with strong convection, and the newly-proposed integral approach, which considers the non-equilibrium sediment concentration profiles on the exchange rate between suspended load and bed load, improves the inherent limitation of the depth-averaged model and increases the accuracy of predicting the channel bed evolution. In future, on the basis of the available field data of Taiwan’s rivers, the model parameters can be calibrated, and numerical simulations and necessary modifications of the model can proceed to enhance the model’s practical value.