雙分佈函數半古典格子波茲曼法之熱流場模擬

Abstract

本文研究中，我們使用耦合半古典格子波茲曼法於雙分佈函數模型下，來計算模擬熱流動問題。 在此模型下，藉著速度分佈函數來計算速度場；總能分佈函數計算溫度場。由古典格子波茲曼法推導出半古典格子波茲曼法來模擬量子氣體，其方法是利用 Uehling-Uhlenbeck Boltzmann-BGK 方程，經Hermite多項式根據Grad’s moment展開推得，再由Gauss-Hermite積分得到巨觀物理量(數量密度、動量、能量)。此半古典格子波茲曼法可以模擬三種量子氣體效應(Maxwell-Boltzmann統計、Bose-Einstein統計、Fermi-Dirac統計)。 於雙分佈函數下，採用D2Q9格子速度模型探討多種Rayleigh number和不同粒子統計情況，模擬自然對流之封閉方腔流、方形腔體之頂蓋有一恆定驅動速度及Rayleigh-Benard自然對流此三種物理問題。由模擬結果分析可以發現，當Rayleigh Number較小時，主要影響整個流場之熱傳方式為傳導所造成，從流線圖可觀察到渦漩會呈近似圓形；隨著Rayleigh Number增大，在流場內熱交換活動越來越劇烈，主要傳遞熱的方式由原本的傳導，漸漸變成以對流為主，其流線圖中的渦漩會漸漸變成橢圓，根據數值結果與前例可得到驗證。因此，使我們瞭解Rayleigh number與流場變化的關係。

A coupled lattice Boltzmann method is proposed for solving thermal flows in the double-distribution-function framework. In the present, a density distribution function is used to simulate the flow field, while a total energy distribution function is employed to simulate the temperature field. The semiclassical lattice Boltzmann equation is used for describing the flow of fermions and bosons. This method is derived by directly projecting the Uehling-Uhlenbeck Boltzmann-BGK equations onto the tensor Hermite polynomials using Grad’s moment expansion method. By applying Gauss-Hermite quadrature to the moment integration, we have the macroscopic quantities (the number density, number density flux, and energy density), and get the equilibrium distribution function which including Maxwell-Boltzmann, Bose-Einstein, and Fermi-Dirac statistics. Simulations of 2D natural convection flows, such as square cavity, lid driven cavity flow, and Rayleigh-Benard thermal convection, based on the two-dimensional nine-velocity (D2Q9) lattice model for several Rayleigh numbers and different particle statistics are shown. For low Rayleigh number, a vortex appears at the center of the cavity. When the Rayleigh number increases, the vortex gradually becomes elliptic. The numerical results are in good agreement with the previous data. The heat is transferred mainly by conduction at small Rayleigh number and by convection at large Rayleigh number. Therefore, we understand the relation of Rayleigh number and the change of the flow.