區域近似特解法求解不可壓縮納維爾-史托克斯方程組

Localized method of approximate particular solution for solving incompressible Navier-Stokes equations

林宗毅(Chung-Yi Lin)

指導教授 ： 楊德良

國立臺灣大學/工學院/土木工程學研究所/博士(2015年)

https://doi.org/10.6342/NTU.2015.02850

摘要

區域近似特解法是近年正在發展中的無網格數值方法。本論文第一章先介紹了區域近似特解法，從特解法到近似特解法的發展歷程。接著引入區域化技巧、徑向基底方程式用正規化技巧、利用改良十字形選點法選擇區計算域，使區域近似特解法比近似特解法能更有效率的求解大尺度電腦運算或即時模擬問題。本研究也詳細提供了適用於多維度多種運算子與等式、方程式之區域近似特解法通用推導過程。目的是要使用區域近似特解法求解多維度不可壓縮納維爾-史托克斯方程組。

為了用丘林的投影法求解不可壓縮納維爾-史托克斯方程組，本論文先在第二章與第三章對波以松方程式和伯格斯方程組做了一系列的測試。第二章用波以松方程式展示了區域近似特解法是否使用了足夠的總點數、合理之區域計算域、有效的正規化技巧之差別。第三章提供了一個利用柯爾-霍夫轉換式作數值近似求解柏格斯方程組之技巧，但此方法在求解微分邊界所遇到的困難仍待未來研究進一步解決。本研究推導了多維度通用的柯爾-霍夫轉換式，包含了對必要條件之數值應用推導式。柯爾-霍夫轉換式將控制方程式轉換為單一變數擴散方程式，其初始條件需要以奇異值分解法求之，而多維度問題則需要最小平方法求之。第三章提供了四個實驗以展示本研究提出之數值模式對於使用不同總點數、不同區域點數、不同時間步間隔、不規則計算域與非結構性佈點之應用能力。

介紹完波以森分成式與柏格斯方程組後，第四章與第五章集中在本研究之主要題目:求解多維度納維爾-史托克斯方程組。本研究利用區域近似特解法對丘林的投影法做數值處理來求解納維爾-史托克斯方程組，兩個章節求解了板驅動空穴流與後向階梯流問題。第四章之二維數值試驗提供了和文獻或不同方法比較之詳盡數據，而第五章則試圖以比第四章更有效率方法求解三維數值試驗。

本研究亦提出在不同計算遇幾何形狀、不同時間解離方法、不同佈點法、不同控制方程式與不同流場特性下的形狀參數之最適範圍。所有實驗的形狀參數之最適範圍皆維持在特定範圍內，５∼８５有可能提供精準解，而２０∼３０則在不同問題皆能夠穩定地求出精準解。

最後，本研究驗證了區域近似特解法求解多維度不可壓縮納維爾-史托克斯方程組織能力，並提出在未來研究可行之改進本數值模式之方法。

關鍵字

區域近似特解法 ； 改良十字形選點法 ； 徑向基底函數 ； 形狀參數 ； 柏格斯方程組 ； 柯爾-霍夫轉換式 ； 投影法 ； 不可壓縮納維爾－史托克斯方程組

並列摘要

The localized method of approximate particular solutions (LMAPS) is a developing meshless numerical method. The LMAPS as introduced in Chapter 1, is developed from the method of particular solutions (MPS) and the method of approximate particular solutions (MAPS). Localization technique, along with normalization technique for radial basis functions and modified cross-shaped selection for local influence area allows the LMAPS to be more efficient than the MAPS for large scale computations or real time simulations. This research also provides generalized derivation applicable for various operators or equations in detail. The goal of this research is to use the LMAPS to solve multi-dimensional incompressible Navier-Stokes equations.

In order to apply Chorin’s approach for solving incompressible Navier-Stokes equations, Poisson equation and Burgers equations have been tested prior to incompressible Navier-Stokes equations, respectively in Chapter 2 and Chapter 3. Chapter 2 uses Poisson equation to demonstrate the difference of applying the LMAPS with or without certain amount of global points, reasonable local influence area, or sufficient normalization technique. While in Chapter 3, an alternative numerical approach for obtaining solutions of Burgers equations was provided, via Cole-Hopf transformation, although it still has difficulties in dealing problems with Neumann boundary conditions. A generalized derivation of Cole-Hopf transformation is also demonstrated, including the derivation for numerical implementations of the essential conditions. Applying Cole-Hopf transformation transforms the governing equations into diffusion equation, it requires singular value decomposition (SVD) to solve the initial conditions, and least squares method (LSM) for solving multi-dimensional problems. Four experiments have been carried out to demonstrate capabilities of the proposed scheme with different number of global points, different number of local points, different time interval, irregular domain, and unstructured point distribution.

After introducing Poisson equation and Burgers equations, Chapter 4 and 5 focus on the main task of solving multi-dimensional incompressible Navier-Stokes equations. The numerical solutions of incompressible Navier-Stokes equations is solved by the LMAPS with implementation of Chorin’s projection method, both chapter provide experiments on lid-driven cavity flow and backward-facing step flow. The experiments of two-dimensional viscous flow in Chapter 4 involve close investigations in matching the details with the results in literature or by other numerical methods, while Chapter 5 tries to solve three-dimensional problems with more efficiency and less consuming comparing with the experiments of Chapter 4.

The optimal range for shape parameter has been determined while applying the proposed scheme to cases with different domain geometry, different temporal discretization, different point distribution, different governing equations, and different flow characteristics. This research proves the proposed scheme is capable of finding the same stable optimal range of shape parameter for all experiments given, can possibly give some accurate solutions, and can get accurate solution stably for different experiments.

Finally, this research verifies the capability of the LMAPS to be able to solve multi-dimensional incompressible Navier-Stokes equations, and some possible approach for improving the proposed scheme is mentioned in future works.

並列關鍵字

The localized method of approximate particular solutions ； modified cross-shaped selection ； radial basis function ； shape parameter ； Burgers equations ； Cole-Hopf transformation ； projection method ； incompressible Navier-Stokes equations

參考文獻

Research Background

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