結合沉浸邊界速度修正法之半古典橢圓統計格子波茲曼流場模擬

Hydrodynamic Flow Simulation Using Semiclassical Lattice Boltzmann-Elliposidal Statistical Method with Immersed Boundary Velocity Correction Method

蔡承翰(Cheng-Han Tsai)

指導教授 ： 楊照彥

國立臺灣大學/工學院/應用力學研究所/碩士(2014年)

https://doi.org/10.6342/NTU.2014.00566

摘要

在連體模型中，主要的統御方程式是透過歐拉或納維爾方程式來求解。然而在氣體稀薄程度提升時，將不再適用，此時方程式為波茲曼方程式。在相關的研究中，對波茲曼方程式作空間及速度離散，即可推導出格子波茲曼法 (Lattice Boltzmann Method, LBM)。
在本文中，根據Uehling-Uhlenbeck Boltzmann-BGK方程（Uehling-Uhlenbeck Boltzmann Bhatnagar-Gross-Krook Equation）及橢圓統計BGK方程（Ellipsoidal Statistical BGK equation, ESBGK）所發展出的半古典橢圓統計格子波茲曼法，透過這個方法，我們可以有效利用其主導的分布函數來計算出所需之宏觀量。利用此法以及D2Q9速度格子模型為基礎，模擬在三種不同粒子統計下，流體流過使用沉浸邊界速度修正法(Immersed Boundary Velocity Correction Method, IBVCM)的圓柱所產生的渦漩、渦度、壓力以及阻力係數之比較。由模擬結果顯示，ESBGK模型之間三種不同的粒子統計差異(Bose-Einstein統計、Fermi-Dirac統計和Maxwell-Boltzmann統計)，並且隨著雷諾數的增加，流線出現了馮卡門漩渦現象。除此之外，還可運用半古典格子波茲曼可還原成古典格子波茲曼法的特性(MB統計)，與其相關研究作驗證。當圓柱在流道中持續的移動，此時模擬比較而觀察出尾流渦漩在三種統計中有明顯不同的差異結果。

關鍵字

半古典格子波茲曼方法 ； 半古典橢圓統計格子波茲曼法 ； D2Q9模型 ； 圓柱繞流 ； 沉浸邊界速度修正法

並列摘要

Modeling gases in the continuum level is traditionally achieved with macroscopic level by using Euler or Navier-Stokes equations. However, as the degree of rarefaction of a gas increses, the governing equation becomes Boltzmann Equation. The Lattice Boltzmann method is derived by discretizing Boltzmann equation in physical and velocity space.
In the study, the development of a semiclassical lattice Boltzmann–Ellipsoidal Statistical method is based on the Uehling-Uhlenbeck Boltzmann-BGK equation. According to the method, we can effectively link its dominant distribution function to calculate the quantities of macroscopic properties. Here, we present simulations of the flow over cylinder for several Reynolds numbers based on D2Q9 lattice model and the semiclassical lattice Boltzmann–Ellipsoidal Statistical method. In this work, the Immerse Boundary Velocity Correction method (IBVCM) has been used to model the boundary of the cylinder. We compare the results of vortices, pressure, drag coefficient for different particle statistics : Bose-Einstein, Fermi-Dirac and Maxwell-Boltzmann statistics. By studying the streamlines, we observed von Karman vortex street phenomenon as the Reynolds number increases. In addition, the movement of the cylinder boundary is not stationary in the flow channel by taking advantage of the boundary using IBVCM. We observed the wake and found differences in the results of the three statistics.

並列關鍵字

Semiclassical lattice Boltzmann method ； Semiclassical lattice Boltzmann–Ellipsoidal Statistical method ； D2Q9 lattice model ； flow over circular cylinder ； Immersed Boundary Velocity Correction Method

參考文獻

32 洪立昕 半古典晶格波滋曼方法，國立台灣大學工學院應用力學所博士論文，台北（2011）。

1 Bohr, N. I. On the constitution of atoms and molecules. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 26, 1-25 (1913).

2 Heisenberg, W. Uber den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik. Zeitschrift fur Physik 43, 172-198 (1927).

3 Einstein, A. Quantentheorie des einatomigen idealen Gases. Akademie der Wissenshaften, in Kommission bei W. de Gruyter, (1924).

4 Dirac, P. A. On the theory of quantum mechanics. Proceedings of the Royal Society A 112, 661-677 (1926).