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Applying the modified mild-slope equation to solve the wave field around the cylinder mounted on a conical shoal
|Contributors: ||NTOU:Department of Harbor and River Engineering|
modified mild-slope equation;analytical solution;dual reciprocity boundary element method
|Issue Date: ||2011-06-30T07:55:14Z
|Abstract: ||過去對於圓島地形之解析解求解最早以線性淺水長波方程式為主，後來有緩坡方程式(Mild-Slope Equation, MSE)之解析解，然而其週波數卻也是以長波近似，因此亦為線性淺水長波方程式。為突破此一應用侷限，Liu et al.(2004)以Hunt (1979)所提出對分散關係式之顯級數展開方式求解Homma圓島之緩坡方程式近似解析解。將近似解析解應用範圍推進到中間性水波，然而卻未考慮到修正型緩坡方程式(Modified Mild-Slope Equation, MMSE; Chamberlain and Porter, 1995)之底床延伸項於中間性水波之顯著效應。因此，基於上述因素，本文以Hunt (1979)之級數方式，求解一適用於中間性水波條件且含底床延伸項之修正型緩坡方程式在錐形淺灘上圓島之近似解析解。此外，為驗證該近似解析解之合理性，本文並建立一基於修正型緩坡方程式之DRBEM數值模式。 經與前人研究成果以及本文數值模式之結果驗證比對，本文所推導錐形淺灘圓島之修正型緩坡方程式近似解析解與其有良好之一致性。此外，本文亦探討波浪入射錐形淺灘圓島週邊之無因次波高放大率，發現波高放大率由淺水長波漸增至中間性水波達到最大值，又漸減至深水波條件。當波浪條件為中間性水波時，MMSE與MSE之平均相對差異量於坡度1：7.5，波浪週期75sec時有最大值為2.7%；而深水波入射條件時其最大之平均相對差異量則為5.4%，發生於波浪週期60sec。而最大相對差異量則是以坡度1：3波浪週期60sec時之值12.2%為峰值。|
In the past, the analytical solutions for solving waves passing a circular island were mostly based on linear shallow water equation. Later on, the mild-slope equation (MSE) was used to treat this type of problem. However, the wave numbers used in the MSE analysis were still approximated with the long-wave assumption. For this, Liu et al.(2004) adopted the direct solution of liner dispersion relation by Hunt’s (1979) to approximate the wave numbers in the mild-slope equation. They derived an approximate analytical solution of mild-slope equation for waves passing the Homma’s island. Their analytical solution of mild-slope equation was thus extended to intermediate water wave conditions. However, the effects of bottom extended terms in the modified-mild slope equation (MMSE) (Chamberlain and Porter, 1995) in the intermediate water wave conditions were not considered. Thus, based on the Hunt’s method we derived approximate analytical solutions of modified mild-slope equation for a circular island mounted on a conical shoal. The approximate analytical solutions, included the effects of the bottom extended terms, were applicable in the intermediate water wave conditions. Furthermore, to verify the approximate analytical solution we also developed a DRBEM numerical model based on modified mild-slope equation. Comparisons of our approximate analytical solutions with analytical solutions of MSE and numerical solution of MMSE-DRBEM showed good agreements. We further investigated the amplification factors of wave height at various locations around the island. We found that the wave height amplification first increased as the wave conditions changed from shallow water to the intermediate water condition. The amplification reached its maximum at the intermediated wave condition and then the amplification decreased as the wave approached deep water wave condition. We also found that for the intermediate water condition the maximum relative difference between MMSE and MSE was 2.7% for the wave with a 75 sec period and a bed slope of 1:7.5. For the deep water wave conditions, the maximum mean relative difference was 5.4% for the 60 sec wave period and the maxima relative difference was 12.2% for the wave with 60 sec period, on a bed slope 1:3.
|Appears in Collections:||[河海工程學系] 博碩士論文|
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