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  • 學位論文

線性水波通過潛沒平板之解析研究

Analytical Study on Linear Water Waves With a Submerged Flat Plate

指導教授 : 黃良雄
共同指導教授 : 林孟郁

摘要


過去,對於潛沒式水平薄板與波浪交互作用之解析研究,大多以勢流理論的基礎下進行探討,而忽略流體的黏滯性與固體邊界交互作用所造成的影響,然而不同的渦流強度導致流場對於固體作用會有相當大的影響,使得以勢流理論分析流場與真實流場有較大出入。 本研究將範圍設定為線性水波,假設水波為不可壓縮流體,且具有週期性與線性條件,探討線性水波下潛沒式薄板與水波交互作用。首先本文應用線性理論及赫姆霍兹速度拆解定理,將水波速度向量分解為一個無旋向量場和一個螺線向量場,亦即速度場可以分成非旋性部分以及旋性部分,並配合適當之邊界條件及控制程式以解之。 非旋性流場部分以勢流理論下建立可分為兩種,一為波浪通過固定薄板之非旋性散射波解析,另一種為波浪通過薄板且板子會震動之非旋性輻射波解析。本文將此二種解析與Yip and Chwang(1997)之反射與透射曲線作比較確認吻合,證明本研究之數值程式無誤。在本研究中發現,非旋性流場下散射波對於薄板上有造成微小力矩,本研究利用此力矩與薄板上之受力及垂向速度,推求薄板震動擺度,利用此擺動角度將非旋性散射波與非旋性輻射波做連結,並設定外在力矩條件,定義若外在力矩無法抵抗非旋性散射波造成之力矩,此擺動角度之連結將會啟動造成薄板震盪。 旋性流場部分也分為兩種。一為以史托克邊界層理論,建立薄板上下與底床上之邊界層流。邊界層之計算須配合勢流理論將流場拆解為四區,因此在薄板左右端點處,會因為上下兩區皆求得同一點位之渦度,導致薄板在左右端點之渦度需相加,利用此相加之渦度值作為渦度擴散方程式之渦度點源,此為第二種旋性流場部分。   根據結果顯示,勢流理論下,非旋性散射波與非旋性輻射波之連結是相當重要,根據波形與速度場比較,可發現非旋性輻射波下之速度場反而比非旋性散射波之速度場強度來得小,只在薄板兩端有較明顯的變化,且於非旋性輻射波下求得之力矩較非旋性散射波來的小,比較非旋性輻射波與非旋性散射波下之透射率,非旋性輻射波下之透射率相對較小,可證明震動薄板的確適用作為防波結構物,在薄板長與波長比為0.75、1.75、2.75時,非旋性散射波與非旋性輻射波的反射率會最接近,當薄板長度與波長比為1個波長的倍數增加時,非旋性輻射波薄板透射率反而比非旋性散射波來得大,因此欲設計震動薄板作為防波結構物時,須注意避開一倍波長的板長。 旋性流場部分,研究結果顯示初始渦度於薄板端點擴散之現象,其渦度值於薄板端點非常大,端點處之渦度強度與流場內其他區域比較量階相差甚大,因此若以數值模式求解渦度場之變化,於端點處需額外處理,以避免不穩定現象發生,且因渦度擴散導致薄板端點所受之剪應力非常大。在線性條件下,薄板承受之力矩主要由非旋性流場主導,若採用非線性條件計算,旋性流場提供之速度會造成非線性條件下之力矩遠大於線性條件下之力矩,可知非線性條件下薄板承受力矩主要由旋性流場下渦度擴散導致。在運動學的探討上,非旋性流場造成之影響較大,而在動力學上旋性流場則是影響之主因。根據本研究對於運動學與動力學上之探討應能於潛沒式薄板之實驗或是工程上之潛堤設計能有所助益。

並列摘要


In the past, the analytic study of the interaction between submerged horizontal plates and waves is mostly based on the theory of potential flow, while ignoring the effects of fluid viscosity and solid boundary interactions. However, different vortex strength leads to a considerable influence, which makes the potential flow field and the real flow field have a big difference. In the assumption that the fluid is incompressible, periodic and linear. By these conditions, the interaction between the submerged plate and the water wave is discussed. In the present study, the linear theory and the Helmholtz decomposition theorem are used to decompose the velocity field into a irrotational vector field and a solenoidal vector field, and with the appropriate boundary conditions to solve it. First, the irrotational flow field is divided into two types, the irrotational, scattering wave, the irrotational radiation wave which makes the board vibrated. In the present study, the reflection and transmission curves of Yip and Chwang (1997) are consistent with the solutions, which proves that the numerical scheme is correct. It is found that the scattering wave causes a small moment on the thin plate. Using the force and the normal velocity of the thin plate to calculate the angular amplitude. By this swing angle, the scattering wave is connected with the radiation wave. If the external torque cannot resist the torque caused by scattering wave, the plate will start pitching. Second, the rotational flow field is also divided into two types. Using the Stoke boundary layer theory to establish boundary layer flow. To calculate the boundary layer, must disassemble the flow field into four zones; therefore, at the left and right end points there are the vorticity, obtained on the plate and also below the sheet. Add up these two vorticity to be vorticity source, has to go with vorticity diffusion equation to analyze vorticity in the wave field, which is the second part of rotational flow. According to the results, it is very important to connect the scattering wave with radiation wave. By the wave height and the velocity field, it can be found that the velocity field of radiation wave is smaller than scattering wave. The torque obtained by the radiation wave is also smaller than that of the scattering wave. The transmission coefficient under the scattering wave is bigger than another one. It is proved that the vibration plate is suitable as breakwater. When the sheet length to the wavelength ratio is 0.75, 1.75 and 2.75, the reflection coefficient of two kinds of waves is the closest. While the sheet length is multiple of 1 wavelength, the transmission coefficient of the radiation plate is larger than that of the scattering wave; therefore, it should be avoided designing a vibrating plate with sheet length is multiple of 1 wavelength. The results show that the initial vorticity spreads at the end of the thin plate, and at the end of the thin plate the vorticity is very large. The vorticity strength of the end point is very different from the other regions of the flow field; therefore, using numerical model to solve the vorticity field will occurrence of the unstable phenomenon. Also the shear stress caused by the vorticity diffusion is very large. Under the linear condition, the moment of the sheet is mainly driven by the irrotational flow field. If the nonlinear condition is used, the velocity of the rotational flow field will cause the torque under the nonlinear condition to be far greater than the moment under the linear condition. The moment under the nonlinear condition is mainly caused by the vorticity diffusion in the flow field. In the discussion of kinematics, the effect of irrotational flow field is larger, while in the dynamic rotational flow field is the main reason. According to the present study, the study of kinematics and dynamics should be helpful in the design of submerged breakwater.

參考文獻


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