線性黏滯水流解析方法應用於搖板式造波機

海岸與港灣工程之海工結構設計，是否能達到有效利用的效果，與是否能對波場做正確的分析息息相關；水流黏滯性所造成之影響是分析的其中一項關鍵。然而，以往對於渦流之研究多著墨於數值模擬之運算，對於流場之解析解仍有許多未臻完美之處，又以往對於造波流場之解析研究多引用非黏性流體之假設(如Dalrymple et al.)，忽略黏滯性與固體邊界交互作用所造成的影響。本研究假設水波為不可壓縮之均質流體，並假設週期與線性條件，嘗試建立黏滯水流之解析方法，應用非旋性勢函數、旋性流函數與渦度構成方程組，於研究中提出解析之流程與對應之邊界條件，並將本套方法延伸至搖板式造波機，在微小振幅的狀況下，探討搖板式造波機作用之波場與物理機制。首先，本文應用線性理論將流場拆解為非旋性部分與旋性部分，並給定適當之邊界條件與控制方程式；然而方程式中包含兩個變數，無法於一次計算過程求解，因此本研究提出兩階段解析流程，利用初始程序解析非旋姓、旋性兩部分只包含單一變數之邊界值問題，再使用非旋性部份、旋性部分之解析解交互迭代得到完整之解答。其次，本文提出應用渦度-流函數方程式(vorticity-stream function equation)做為渦度計算之邊界條件，配合擴散方程式(diffusion equation)解析渦度於造波流場之分佈，主要原因為各邊界可利用之條件已於非旋性勢函數與旋性流函數計算時使用完畢，並無多餘之邊界條件可供渦度計算之用，因此本研究引用渦度-流函數方程式做為渦度計算之邊界條件，成功解析流場內之渦度分佈。解析結果顯示渦度量值沿水平方向減少，影響範圍侷限於造波板附近，同時渦度分佈出現於自由液面與底床部份，有效彌平造波板端點渦度不連續之弔詭；又因將流體之黏滯性納入考慮，所以造波流場靠近邊界處會出現邊界層流，使得速度分佈異於以往之非旋性造波流場。比較黏滯性造波流場與非旋性造波流場可發現，整體流場趨勢變化不大，靠近造波板處會因黏滯性造成渦度之分佈，於底床部份則是因為邊界層流之因素，造成速度無滑移之條件，遠離造波板處則是與非旋性流場相似。由於線性黏滯水流解析方法具有解析黏致水流流場之能力，並將解釋相關物理量，因此本研究對於有關渦度計算、造波機實驗設計等方面應用，應能有所助益。

關鍵字

二維；線性；週期波；速度勢；旋性流函數；渦度

並列摘要

Optimizing design to harbor and coastal structures for the purpose of working efficiently always depends on accurate analysis and simulation to the wave field. One of the essential analysis is the effect resulted by viscosity. However, most researches focus on numerical simulation instead of analytical solution. Moreover, the approximation of inviscid fluid to the analytical research might neglect the effect of interaction between solid boundary and fluid. In the assumption that the homogeneous fluid is incompressible, priodic and linearity, the small amplitude wave theory is adopted in the present study to investigate the wave field and physical mechanism of flap type wave-maker. First, the present study utilizes linearity to separate wave field into irrotational and rotational part and determines the governing equation and boundary condition. But, the equation could not be solved in one process because of including two variables. Hence, the present study proposes the two processes analytical method. It uses first guess process to solve two parts including only one varible, then applies iteration process to solve the whole problem. Second, the present study ultilizes vorticity-streamfunction equation to be the boundary conditions of vorticity and diffusion equation to analyze vorticity in the wave field. The reason is that there are no boundary conditions for calculation of vorticity. Thus, the present study takes identy of vorticity and streamfunction as the boundary conditions and analyzes the distribution of vorticity in the wave field successfully. The result indicates that the distribution of vorticity might decay along x direction, the effect of vorticity is confined near the wave-maker plate and might appear in the free surface and bottom which make up the questionable problem at endpoint. Furthermore, owing to considering the viscosity, boundary layer flow would appear near the boundary and causes the velocity distribution to be different from irrotaitonal wave field. Comparing viscid wave field with irrotation wave field, the two wave fields almost have the same behavior while vorticity would appear near the plate due to viscosity. At the bottom, viscid wave field has no-slip condition because of boundary layer flow. Due to the correct expression of vorticity distribution in the wave-maker, the present study is helpful in analysis of vorticity and experiments about wave-maker etc.

並列關鍵字

two-dimension ； linearity ； periodic wave ； potential function ； rotational streamfunction ； vorticity

參考文獻

1. Antonopoulos, K.A., and Tzivanidis, C. (1996). “Analytical solution of boundary value problems of heat conduction in composite regions with arbitrary convection boundary conditions.” Acta Mechanica, 118, 65-78.

2. Batchelor, G.K. (1967). “An introduction to fluid dynamics.” Cambridge University Press.

3. Biésel, F. and Suquet, F. (1951). “Les Appareils Générateurs de houle en Laboratoire.” La Huille Blanche Vol. 6 (2), (4), (5).

4. Carslaw, H.S. and Jaeger, J.C. (1959). “CONDUCTION of HEAT IN SOLIDS.” Oxford at the Clarendon Press

6. Dean, R.G., and Dalrymple, A.D. (1984). “Water wave mechanics for engineers and scientists.” Prentice Hall, Inc.

國際替代計量

線性黏滯水流解析方法應用於搖板式造波機

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