臺灣大學: 工程科學及海洋工程學研究所許文翰游景皓Yu, Ching-HaoChing-HaoYu2013-03-272018-06-282013-03-272018-06-282010http://ntur.lib.ntu.edu.tw//handle/246246/252457本論文對於純量方程與動量方程式中之對流項，提供一數值方法以獲得較好的相位誤差。在多項流的問題中，本論文利用等位函數法與守恆型態的等位函數法來捕捉自由液面。此外，對於複雜外型的處理，使用了沉侵邊界法來模擬。最後，固體與流體之間的運動使用了等位函數/沉侵邊界法來計算。 1、具上風效應之合併緊緻差分法： 本論文發展一具有上風效應高階準確之差分法，此方法主要概念是合併了二個(或者更多)緊緻差分格式，以獲得更高階的準確性。此外，我們更嚴謹的分析此方法之消散與耗散之行為，對於計算的問題都有良好的結果。 2、等位函數法： 本論文應用等位函數法模擬自由液面流。在處理動量方程式中採用具有上風合併緊緻差分法。這個方法利用二條方程式，同時處理一階微分、二階微分 。在等位函數方程式中，也是利用此合併緊緻差分法來求解。時間方面之處理則採用六階具有辛算子之落吉-庫塔方法求解。在計算完等位函數方程式後，為了隨時保持平滑區間一定之寬度，使用了重距離化方程式來解決。最後利用本方法模擬了潰壩流、瑞德-泰勒、氣泡浮升與水滴落下等問題。 3、守恆型態之等位函數法： 同樣的，處理動量方程式中採用具有上風效應之合併緊緻差分法。時間方面之處理也是採用六階落吉-庫塔方法求解。此外，守恆型態的等位函數法在第二步的算則中，採用同時具有壓縮項與人工黏滯項之方程式。最後利用此方法來求解潰壩流、瑞德-泰勒、氣泡浮升與水滴落下等問題。 4、 沉侵邊界法： 為了能夠求解複雜外型的流動問題，本論文提出了沉浸邊界方法。沉浸邊界方法的概念是在動量方程式中加入一虛擬源項，以其能達到滿足物體非滑移之邊界條件。本論文提出之沉浸邊界方法不同於經由代數來達到插值的目的，而是經由求解微分方程式來達成，故較容易擴展至三維之情況。本方法探討流體經過圓柱之問題，測試後得知，本論文之沉浸邊界方法，與前人之研究結果相當吻合。 5、 等位函數/沉侵邊界法： 最後，本論文將等位函數法搭配沉浸邊界法來求解流體-固體間之運動。藉由合併此兩種方法，模擬了一潰壩流流經半圓柱與半圓柱在靜止水中來回移動之情形。Abstract In this dissertation, I propose two schemes which accommodate a better dispersion relation for the convective terms shown in the transport equation, such as in the level set equation and momentum equations. For the multi-phase tracking problems, the free surface has been tracked by the level set method (LS) and the conservative level set method (CLS). In addition, the immersed boundary method (IBM) is developed to simulate mechanical systems in which structures may interact with the fluid flows. The interaction between free surface flow and structure is also investigated using a level set/immersed boundary coupled method (LS-IBM). (I) Minimized phase error upwinding combined compact difference schemes: All of schemes are proposed to enhance the convective stability by virtue of the increased dispersive accuracy and they have been rigorously developed through the dispersion and dissipation analyses. To verify the proposed method, several problems will be investigated. The results with good rates of convergence are demonstrated for all the investigated problems. (II) Level set method: I apply a level set method to simulate gas/water interface flow. For the sake of accuracy, the spatial derivative terms in the equations of motion for an incompressible fluid flow are approximated by the higher-order accurate minimized phase error upwinding combined compact difference (UCCD) scheme. This scheme development employs two (or three) combined equations to calculate the first- and second-order derivative terms (and third-order derivative terms). For accurately predicting the level set value, the interface tracking scheme is also developed to preserve the theoretical phase error of the first-order derivative term shown in the level set equation. For the purpose of retaining the longtime accurate Hamiltonian in the advection equation for the level set function, the time derivative term is discretized by the sixth-order accurate symplectic Runge-Kutta scheme. Also, to keep as a distance function for ensuring the front having a finite thickness for all time, the reinitialization equation is used. For the verification of the proposed scheme for the pure advection equation, several benchmark problems have been chosen in this dissertation. The level set method with excellent area preservation property proposed for capturing the interface in incompressible fluid flows is also verified by solving the dambreak, Rayleigh-Taylor instability, two-bubble rising in water, and droplet falling in water problems. (III) Conservative level set method: A two-step conservative level set method is proposed to simulate the gas/water two-phase flow. For the sake of accuracy, the spatial derivative terms in the equations of motion for an incompressible fluid flow are approximated by the upwinding combined compact scheme. For accurately predicting the level set function, the advection scheme with minimized phase error advection scheme is developed to preserve the theoretical phase error for the first-order derivative terms shown in the pure advection equation cast in conservative form. For the purpose of retaining its long-time accurate Casimir functionals and Hamiltonians in the transport equation for the level set function, the time derivative term is discretized by the sixth-order accurate symplectic Runge-Kutta scheme. To resolve contact discontinuity oscillations near interface, nonlinear compression flux term and artificial damping term are properly added to the second-step equation in the conservative level set method. For the verification of the proposed phase-minimized scheme applied in non-staggered grids for solving the incompressible flow equations, several benchmark problems have been chosen in this dissertation. The conservative level set method with area-preserving property proposed for capturing the interface in incompressible fluid flows is verified by solving the dam-break, Rayleigh-Taylor instability, bubble rising in water, and droplet falling in water problems. Good agreement with the referenced solutions is demonstrated in all the investigated problems. (IV) Immersed boundary method: I use the immersed boundary method to solve the flow equations in irregular and timevarying domains. The artificial momentum forcing term applied at certain points in cells containing both of the fluid and solid allows an imposition of velocity condition to account for the solid body motion. We develop in this study a differential-based interpolation scheme which can be easily extended to perform the three-dimensional simulation. The results obtained from the proposed immersed boundary method agree well with other numerical and experimental results for the chosen benchmark problems. The accuracy and fidelity of the Immersed boundary (IB) flow solver developed to predict flows with irregular boundaries are therefore demonstrated. (V) Level set/immersed boundary method: A level set/immersed boundary coupled method combines the level set method and the immersed boundary method. This method implemented in Navier-Stokes solver allows simulation of interaction between the fluid flow with free surface and bodies. A solution algorithm is proposed to prescribe the exact forcing points near the solid boundaries for providing an accurate numerical solution. The discretized linear system of the Poisson pressure equation is solved using the divergence-free-condition (DFC) compensated flow solver. The predicted results are in good agreement with numerical simulations.12211388 bytesapplication/pdfen-US等位函數法沈浸邊界法複雜外型自由液面level set methodimmersed boundary methodcomplex domainfree surfacethesishttp://ntur.lib.ntu.edu.tw/bitstream/246246/252457/1/ntu-99-D95525002-1.pdf