風浪生成機制的直接數值模擬

Abstract

利用直接數值模擬的方法，建立一個空氣與水的耦合紊流模式，並以此耦合紊流模式探討風浪的生成機為本論文的研究重點。其中風浪的生成機制可因風速的不同而不同，我們僅以低風速下產生的風浪為研究範疇。在考慮低風速的情況下，水面上得到的平均風應力約為0.089 dyn cm-2 ，依此平均風應力而得到的空氣和水的摩擦速度，分別約為8.6 cm s-1 和0.3 cm s-1。由於此空氣與水的耦合紊流模式，在空氣與水的介面上，同時滿足速度與應力連續的條件，所以這個耦合模式可以同時捕捉到空氣與水體的運動和它們之間的交互作用。研究顯示，發展最快之波浪的波長與實驗的量測結果很接近，波長約為8~12公分。而且，在波浪生成之後，因波浪成長速率的不同，可分為線性與指數成長兩階段，也與理論和觀測的結果相同。但是，受限於在空氣與水的介面處使用了線性的邊界條件，因此當波浪的梯度大於0.01時，即無法再利用此耦合模式繼續進行模擬。模擬的時間間隔，約為波浪開始生成之後70秒內的發展過程。波浪生成之後，我們分析了波浪對空氣與水體中紊流場的影響；也執行了一些敏感性測試，包括水體中的紊流場、表面張力和空氣的高度。藉由與理論的結果比較，在線性的成長階段，我們的波浪成長率只有在較高的空氣高度的算例中，與Phillips (1957) 的理論預測較一致；在指數的成長階段，有些的波浪成長率與Belcher & Hunt (1993)的理論預測、Plant (1982)所統計的實驗觀測和一些數值模擬的結果一致。但是，有些波浪的成長率則較前人的研究結果大2~3倍。雖然在量的比較上與前人的結果有些許的差異，但是在風浪生成機制的定性條件上，和Phillips (1957)與Belcher & Hunt (1993)所提的機制是符合的。風的能量能夠傳輸至波浪的主要因素：在線性的波浪成長階段，如Phillips (1957)所提的機制一樣，來自紊流所引起的壓力擾動；在指數的波浪成長階段，如Belcher & Hunt (1993)所提的機制一樣，來自波浪所引起的壓力擾動與波浪之間所形成的形狀阻力。

An air-water coupled model is developed to investigate wind-wave generation processes at low wind speed where the surface wind stress is about and the associated surface friction velocities of the air and the water are and , respectively. The air-water coupled model satisfies continuity of velocity and stress at the interface simultaneously, and hence can capture the interaction between air and water motions. Our simulations show that the wavelength of the fastest growing waves agrees with laboratory measurements and the wave growth consists of linear and exponential growth stages as suggested by theoretical and experimental studies. Constrained by the linearization of the interfacial boundary conditions, we perform simulations only for a short time period, about 70s; the maximum wave slope of our simulated waves is and the associated wave age is , which is a slow moving wave. The effects of waves on turbulence statistics above and below the interface are examined. Sensitivity tests are carried out to investigate the effects of turbulence in the water, surface tension, and the numerical depth of the air domain. The growth rates of the simulated waves are compared to Phillips’ (1957) theory for linear growth and to Plant’s (1982) experimental data and previous simulation results for exponential growth. In the exponential growth stage, some of the simulated wave growth rates are comparable to previous studies, but some are about 2~3 times larger than previous studies. In the linear growth stage, the simulated wave growth rates are sensitive to the numerical depth of the air domain, and are comparable to Phillips’ prediction only for the larger air domain. In qualitative agreement with the theories proposed by Phillips (1957) and Belcher and Hunt (1993) for slow moving waves, the mechanisms for the energy transfer from wind to waves in our simulations are mainly from turbulence-induced pressure fluctuations in the linear growth stage and due to the in-phase relationship between wave slope and wave-induced pressure fluctuations in the exponential growth stage, respectively.