含渦流作用之重力水波研究

Study of Gravitational Water Wave with Vortex Effect

林孟郁(Meng-Yu Lin)

指導教授 ： 黃良雄

國立臺灣大學/工學院/土木工程學研究所/博士(2008年)

https://doi.org/10.6342/NTU.2008.02685

摘要

本研究的主旨是建立一數值模式，以研究重力水波通過潛沒物體引致渦流現象之二維流場問題。此數值模式的特色在於同時應用了邊界積分法(boundary integral method) 與渦旋法 (vortex method) 的優點。邊界積分法由於無須在計算域內架構網格，因此可免除網格建立的時間耗費以及處理自由水面邊界的困難。另一方面，由於渦旋法中帶有渦度的質點能集中在具有旋性的區域內，因此在處理尺度不同、大大小小的渦旋現象上具有計算上的優勢。此方法能突破一般求解 Navier-Stokes 方程式之數值方法需在網格系統上求解的限制，因而能更有效地將計算資源集中於自由水面波動及渦流運動的
模擬。

數學模式的建立係應用了 Helmholtz decomposition 將速度場分解為非旋性與旋性兩部份，並建立新的待解變數組。自由水面上的渦流或偶極流奇異點分布 (vortex or dipole distribution) 決定速度場的非旋性部份，而物體邊界上的渦流奇異點分布則決定了進入流體的渦度通量 (vorticity flux)。自由水面上的黏滯性效應及渦度的產生予以忽略。統御邊界奇異點強度的邊界積分式屬於第二類 Fredholm 積分式，故可用疊代法求解，以大幅提高計算效率。速度場的旋性部份是由渦度場控制，並運用渦旋法求解。渦旋法將渦度以離散的Lagrangian 質點來表示：質點跟隨流體而運動，其強度則因黏滯性擴散而改變，並由 PSE (particle strength exchange) 法解之。而紊流的效應則採用在大尺度渦流模擬 (large eddy simulation) 中常用的 SGS 模型 (sub-grid scale model)之渦旋黏度 (eddy viscosity) 來處理。

為了驗證模式在自由水面計算上的準確性，本研究選擇了兩個問題來進行測試。其一是孤立波在等水深的水道中之傳遞，此測試之目的在於檢驗本模式計算非線性水波的能力。另一是水中移動之線渦 (line vortex) 引致水面波動的現象，其目的是測試模式中計算渦流對於水面影響的能力。兩項測試均呈現了計算的準確性。而模式在模擬水波通過矩形潛沒物體產生渦流現象之準確性，則是與他人的實驗結果比較並驗證，而計算的收斂性亦同時檢測。結果顯示本模式的計算成果足以呈現與實驗觀測接近的水面波動與渦流生成及消散等流場現象。

在模式的應用部份，本文中呈現了一系列的水波通過矩形潛沒物體的研究，水波形態包含了週期波與孤立波。在週期性水波通過矩形潛沒物體之流場分析中，文中討論了水波之波長、水深及物體長度對於渦流生成的影響。研究結果發現 Keulegan-Carpenter 數對於評估生成渦流之強度及其影響是關鍵的參數。而在孤立波通過矩形潛沒物體之流場分析中，則著重在探討不同波高生成的渦流尺度，以及此渦流對於孤立波的影響。研究發現由於孤立波並不會產生往返的水流，因此與週期波相比渦流將存在較長的時間。而渦流的強度及其影響，則與入射波之波幅呈正比的關係。

關鍵字

自由液面流 ； 渦流生成與消散 ； 潛沒物體 ； 渦旋法 ； 邊界積分法

並列摘要

The present work develops a 2-D numerical scheme which combines vortex
method and boundary integral method to investigate the interaction of water waves with submerged obstacles. The major advantages of this method are the efficiency on solving the free surface motion and the irrotational flow field by using the boundary integral formulations, and the essentially grid-free nature of the vortex particle method for solving the vorticity field. Using this method we may concentrate the computational resources on the simulation of surface waves and on the evaluation of vorticity which is predominantly confined in compact regions.

The numerical method is accomplished by using a Helmholtz decomposition
which decomposes velocity field into its irrotational and rotational components. The distribution of dipole (vortex) along the free surface determines the irrotational component of flow field, and the distribution of vortex along the solid surface determines the vorticity flux into the fluid. Viscous effects and generation of vorticity on the free surface are neglected. For evaluating the strengths of singularities a boundary integral method is used in which the resulting Fredholm integral equations of the second kind are solved efficiently in both storage and work by iteration. Vorticity generated from the submerged obstacle is convected and diffused in the fluid via a Lagrangian vortex (blob) method, using the particle strength exchange (PSE) method for diffusion, with particle redistribution. A sub-grid eddy viscosity model is used to simulate the turbulent effects.

The inviscid part of the numerical method and the performance in the computation of nonlinear waves are tested using a calculation of solitary wave propagation in a uniform channel. A further test for the simulation of the free surface deformation induced by a moving line vortex in the fluid is performed. In these tests good agreements between numerical and theoretical results are obtained. The full model is verified by simulating periodic and solitary waves travelling over a submerged rectangular obstacle, and the results are compared with laboratory measurements. All comparisons exhibit reasonably well agreement.

Applications to periodic and solitary water waves over a submerged rectangular obstacle are given. For periodic waves, the effects of the incident
wave length, still water depth and the length of the obstacle on the vortex generation and evolution are presented. The results lead us to the conclusion that for periodic water waves the Keulegan-Carpenter number is the key parameter in determining the formation and development of the vortices generated from submerged obstacles. For solitary waves, the effects of the incident wave height are discussed. In contrast with periodic waves, the vortices generated from the submerged obstacle in a solitary wave are preserved in a long period because there is no reversed flow. The strength of vortices and their effects are positively correlated with the incident wave height.

並列關鍵字

Free surface flow ； Vortex generation and dissipation ； Submerged obstacle ； Vortex methods ； Boundary integral methods

參考文獻

propagation in shallow water: a finite element model. Int. J. Numer. Methods

Fluids, 16:447–459, 1993.

G. R. Baker. A test of the method of fink and soh for following vortex-sheet motion. J. Fluid Mech., 100:209, 1980.

G. R. Baker, D. I. Meiron, and S. A. Orszag. Generalized vortex methods for freesurface flow problems. J. Fluid Mech., 123:477–501, 1982.

G. R. Baker, D. I. Meiron, and S. A. Orszag. Generalized vortex methods for free surface flow problems II: radiating waves. J. Sci. Comp., 4(3):237–259, 1989.