平滑粒子動力法在淺水波方程式與微粒濃度場之應用

SPH on shallow water equations and particulate matter concentration distribution

張高華(Kao-Hua Chang)

指導教授 ： 張倉榮

國立臺灣大學/生物資源暨農學院/生物環境系統工程學研究所/博士(2012年)

https://doi.org/10.6342/NTU.2012.10260

摘要

平滑粒子動力法(smoothed particle hydrodynamics，簡稱SPH)為一種拉格郎君觀點的無網格數值方法。SPH特性是以粒子呈現物理空間，任何物理量皆是以內插方式計算。目前SPH被許多研究案例證實其在處理大形變問題上有著令人讚賞的結果，特別是含有自由液面的流況問題。近期SPH被應用於求解淺水波方程式(shallow water equations，簡稱SWE)去模擬潰壩流況，但皆屬於封閉邊界問題；Vacondio等更進一步提出特徵邊界法去處理含有開放邊界的明渠流問題。由於特徵邊界法是以黎曼不變數建立出入流的邊界條件，而黎曼不變數僅存在於矩形與三角形斷面渠道。為了讓SPH-SWE模組能夠適用於非矩形斷面渠道，本研究引用特定時間間距法，其為求解特徵方程式去建立無論是矩形或非矩形渠道案例的出入流邊界條件。此外，目前SPH-SWE模組皆是求解水深與流速的相依變數，此僅能描述定型渠道流況，而無法呈現受非定型渠道的渠寬變化影響之流況，故本研究是求解斷面積與流量的相依變數。為了測試本研究所建立的新SPH-SWE模組，選取三個具有代表性案例進行模擬。模擬案例包含非均一底床坡度、不同出入流邊界條件、矩形與梯形斷面渠道、及定型與非定型渠道等情境以供測試。經模擬與解析及量測結果比較，顯示兩者趨勢頗為吻合，說明了新SPH-SWE模組能夠處理明渠流問題。
另外，因為SPH實為一種內插法，本研究將SPH應用於室內懸浮微粒的濃度場計算，在此稱作為權重內插法。選擇一含有入流口與初流口的單間房間為模擬區域，並於入流口連續施放微粒進行流場與粒子軌跡的二相流模擬。濃度計算方面，除了權重內插法以外，還有目前普遍被使用的樣本體積法，目的為比較兩者之差異。最後，在利用權重內插法計算粒子濃度時，兩種粒子搜尋法被用來比較計算效率，即all-pair與linked-list搜尋法。結果顯示，linked-list搜尋法是較有效率，而且當濃度觀測點數超過O(104)時，all-pair與linked-list搜尋法所耗費CPU時間之比值會趨於定值，約為28。

關鍵字

平滑粒子動力法 ； 淺水波方程式 ； 特定時間間距法 ； 開放邊界 ； 非矩形與非定型渠道 ； 懸浮微粒 ； All-pair與Linked-list搜尋法

並列摘要

Smoothed particle hydrodynamics (SPH) is a Lagrangian meshless method. In SPH, particles are used to present physical domains and any physical quantity is computed by interpolating. Many research cases have been proven that SPH can deal with the problems of large deformation. Recently, SPH is applied to solve the shallow water equations (SWE) to simulate dam-break flow with closed boundaries. Vacondio et al. further proposed the characteristic boundary method to predict open channel flows with open boundaries. In the characteristic boundary method, the Riemann invariants are used to establish in/out-flow boundary conditions. However, the Riemann invariants are formulated only for the cases of rectangular and triangular channels. For the purpose that SPH-SWE can simulate non-rectangular channel flows, the newly SPH-SWE approach with the specified time interval method is proposed in this research. Thus, the in/out-flow boundary conditions are set up in non-rectangular channels by solving the characteristic equations in the newly SPH-SWE approach. On the other hand, the dependent variables of water depth and water velocity are solved in the traditional SPH-SWE. To reflect the effect of variable channel width on flows in the non-prismatic channels, the dependent variables of wetted cross-section area and water discharge are solved herein. Three benchmark study cases are used to validate the ability of the newly SPH-SWE approach on non-rectangular and non-prismatic channel flows. The study cases include non-uniform bed slope, various combinations of in/out-flow boundary conditions, rectangular and non-rectangular, and prismatic and non-prismatic etc. In comparison with analytic and measured results, the simulated results show good predictions. It can be demonstrated that the newly SPH-SWE approach is capable of simulating open channel flows.
Another part of the research, SPH is utilized to compute the indoor particulate matter (PM) concentrations (called the kernel method). A single room with the inlet and outlet is chosen as the simulation domain and PM is injected continuously at the inlet. The simulations of wind flow field and particle trajectory are executed. Two methods, i.e. the kernel method and the sampling volume method, are applied for the computation of PM concentrations. To make a comparison between the two methods, it can be found that the kernel method is more efficient with the same accuracy. Finally, the all-pair and linked-list search algorithms are used in the kernel method. In the view of the efficiency, the linked-list search algorithm is more efficient. It can be detected that the ratio of required CPU time between the all-pair and linked-list search algorithms is 28 as the number of observed concentration points exceeds O(104).

並列關鍵字

Smoothed particle hydrodynamics ； Shallow water equations ； Method of specified time interval ； Open boundaries ； Non-rectangular and non-prismatic channel ； Particulate matter ； All-pair and Linked-list search algorithm

參考文獻

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