本研究以有限體積法作為模擬的基礎，使用ANSYS FLUENT 中的連體表面力（Continuum Surface Force, CSF）模型來模擬黏性液滴的自由振動行為，旨在評估CSF 模型的可行性和精確性。由於液滴複雜的形狀振動行為根據線性理論，在空間上可以透過Legendre 模態將液滴分解成各個模態的線性組合，每一個模態都以特定的頻率振動，從而分析並探討液滴形狀振動的行為。研究的主要流程如下: 首先，使用ANSYS FLUENT 進行模擬，模擬完成後獲取每個單元體積上的數據。我們知道在每一個時間點上，液面形狀都有所不同，透過模擬後的數據擷取出在體積分率不為0 或1 的帶狀區域，透過最小二乘法可以找出最接近真實液面的曲線，並與ANSYS FLUENT 模擬後體積分率等位線圖進行邊界上的擬合，擬合完成後，則將其透過Legendre 模態分解並保留低階模態，再使用快速傅立葉轉換(FFT) 將其時間域上的訊號轉換成頻率域上的訊號，就可以計算出低階模態中，每個模態振動頻率的值，以及得到黏性液滴振幅的衰減變化。由於自然的振動頻率可由Rayleigh 計算得出小振幅的理論解，因此我們可以將經由FFT 計算後的頻率與Rayleigh 所計算的理論解進行比對，以判斷結果是否合理，並藉此評估連體表面力模型在模擬黏性液滴振動行為的可行性。
在研究過程中，我們發現連體表面力模型雖然是可行的方法，但存在一些缺點，由於表面張力與曲率有關，在網格較小時，例如模型對稱軸的兩極，曲率的估算誤差可能會導致錯誤的判斷，使得附近出現孤立的衛星液滴(satellite droplets)。儘管存在一些挑戰，結果顯示，我們還是成功以低階的Legendre 模態分解，對連體表面力模型取代黏性液滴上之表面張力作用的作法進行評估，並驗證了連體表面力模型在低模態振動的可靠性!

This study employs the finite volume method as the basis for simulation, using the Continuum Surface Force (CSF) model in ANSYS FLUENT to simulate the shape osillation of viscous droplets. The aim is to evaluate the feasibility and accuracy of the CSF model. Similar to Rayleigh's linear theory of small amplitude approximation, the complicated shape oscillations of the droplet can be decomposed spatially into a linear combination of Legendre modes. Each mode oscillates at a specific frequency. Superposition of all Legendre modes allows us to extricate the droplet's shape oscillation behaviors. The main process of the study is as follows: First, simulations are performed using ANSYS FLUENT with the CSF model enabled. After the simulations are completed, data of the volume fraction of each cell is obtained. Since the interface must lie on the regions where the volume fraction is not 0 or 1, the method of least squares is used to determine the interface that best approximates the actual liquid surface. The interface is then decomposed into various Legendre modes. Fast Fourier Transform (FFT) is applied to the time history of each modes, allowing us to calculate the vibration frequency of each mode and obtaining the amplitude decay rate of each mode of the shape oscillations of the viscous droplet. By comparing the vibration frequencies thus obtained with Rayleigh's frequency, we can assess the feasibility of the Continuum Surface Force model in simulating the vibration behavior of viscous droplets.
We found that the Continuum Surface Force (CSF) model is a viable method for replacing the surface tension in Eulerian description, though with some drawbacks of spurious satellite droplets. Despite these challenges, the results show that we successfully used low-order Legendre mode decomposition to justify the validity of the CSF model in replacing the surface tension effect of viscous droplets.

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