二維Poiseuille 流中探討團簇尺寸分佈與尺度分析

Abstract

以數值模擬來解決學術或工程上問題已經有數十年之久的歷史，其成效也是相當顯著，因此本研究即以此法來研究多顆粒在流道中形成團簇的行為，並加以分析及探討出結論。本文一共分成五個章節，第一章主要說明研究過程中參考的文獻，以及某些文章中整理出較為基本的物理現象，並以這些前人研究的結果為基礎，藉以發展出更深入的研究成果。第二章則闡述在本研究中所使用的數學模型，對於流場時解Navier-Stokes equation；同樣的，固體則解Newton-Euler equation，再以流體對顆粒邊界造成的水動力以及力矩積分式將固液方程耦合起來，最後以變分法與拉氏乘數引入方程後推導出最後的數學模型。本研究所使用的數值方法為分佈式拉格朗日乘數/虛擬區域法(DLM/FD)，因此第三章所講解的即為此方法在空間與時間上的離散，同時也說明在計算不同項時所面臨的問題以及解決方法。當前置作業都介紹完後，第四章則是所有模擬的結果，首先我們考慮 的計算域；顆粒的幾何形狀分為圓柱與橢圓，在圓柱幾何面中考慮三種不同大小的顆粒與密度；橢圓則是一種大小與密度，兩者濃度變化考慮為5%至40%之間，並引用Ding and Aidun, 2006兩位學者定義的參數來計算平均團簇尺寸、數量隨時間的變化，在這之中可觀察到高濃度下平均團簇尺寸在transient時的線性爬升現象，接下來我們改變流道雷諾數和顆粒大小來比較上述兩者對團簇數量分佈的影響，並找到圓柱與橢圓的臨界雷諾數後，再以normalization factor將所有資料點scaled成一趨勢，最後我們也計算團簇的迴轉半徑，來觀察團簇在流體中是以什麼型態存在；長鏈狀或是一整團團簇。第五章中本文將所有重要的結果加以整理成一些結論，首先濃度是最直接影響團簇大小的一個因子，增加濃度就會導致團簇尺寸的增加，接著我們改變雷諾數與顆粒參數，雷諾數變化的結果與Ding和Aidun在三維中的模擬相似，這種相似情況即便在改變濃度後仍然會存在。至於顆粒參數的改變則有著相對的結果，首先增加顆粒數量或改變顆粒大小都會造成顆粒在流場中濃度的改變，所以我們假設數量與顆粒大小皆為濃度的函數，如此一來讓三種顆粒大小在同一濃度下比較，就可得到數量與顆粒大小對團簇數量的影響強烈度，最後得到在團簇平均數量的分佈上數量是一個較強烈的影響參數，另外也以三種顆粒的臨界濃度為依據並計算各別的normalization factor，發現在二維流場，雷諾數為 且顆粒濃度不超過臨界濃度情況下，所有的團簇尺寸分佈有一通用的尺度關係，這個關係不與濃度和顆粒的幾何形狀有關，並與Ding與Aidun在三維、Stokes flow中模擬出的關係式一致(Ding and Aidun, 2006)。為了瞭解眾多顆粒與團簇在流場中的運動，本論文計算團簇的迴轉半徑並藉以觀察團簇在流道中的形態，其結果顯示取對數後，團簇的長度變化趨勢會是一線性方程，除此之外本文也解釋低雷諾數下顆粒的移動軌跡以及最後在流道中間堆積的現象，係因Poiseuille Flow在中間有最大的流動速度，並產生尾流將兩旁的顆粒拉進中線上，最後得到一長串的顆粒堆積在流道中線而兩側顆粒驟減的情形。

Solving industrial and engineering problems via the direct numerical simulation (DNS) has been as the third method of scientific investigation besides the theoretical analysis and laboratory experiments for the past several decades. Especially in the past decade the DNS in Computational Fluid Dynamics becomes a major research tool to analyze two-phase flow problems. The major focus of this thesis is to investigate the cluster size distributions and the scaling properties of particles interacting in two-dimensional Poiseuille flow via the DNS. There are five chapters in this thesis. In Chapter One we first recall some important references and introduce the physical phenomena closely related to the particle motion in fluid flow discussed in these references. Based on these fundamental results and theories, we go further in our investigation on the particle clustering and the scaling properties. The mathematical models for the two-phase flow problems are the Navier-Stokes equations for the incompressible viscous fluid flow and the Newton-Euler equations for the particle motion, which are presented in Chapter Two. Through the integration formulas of the hydrodynamic forces and torque, these two sets of governing equations are coupled in a variational formulation. Then via a distributed Lagrange multiplier term, a fictitious domain formulation has been derived. In this thesis, we have used the distributed Lagrange multiplier/fictitious domain method (DLM/FD) as our numerical method for the DNS. In Chapter Three, we introduce the space and time discretization first. Then we describe the computational problems encountered in the simulation and the algorithms used to solve those problems. The aforementioned Chapters are the foundation for the numerical results discussed in Chapter 4. We have considered two kinds of particle shapes, namely circular cylinder and elliptical cylinder, with the variation concentration between 5% and 40%. The computational domain is a 4 by 1 rectangle. The detail values of parameters are given in Chapter Four. Following the analyzing procedures given in Ding and Aidun (2006), we calculate both the averaged value of cluster sizes at each time step and averaged number n(s) of each cluster size s after the cluster distribution reaches it equilibrium. For the cases of higher concentration, the averaged value of cluster sizes increases linearly in time during the initial transient period. The effects of the Reynolds number and the diameter of particles have been studied. We found that the effect of the Reynolds number is similar to the results in Ding and Aidun (2006). Finally, the critical concentration as defined in Ding and Aidun (2006) has been identified for each case and used in the normalization factor to scale all data points of (n(s),s) where s varies through all obtained cluster sizes. Moreover, we have obtained a universal curve by using curve fitting to the normalized data points. Based on the aforementioned results, we conclude that at lower Re (i.e., ) a universal scaling relation for the cluster size distribution is still held for particle interacting in two-dimensional Poiseuille flow for both particle shapes. In aspect of multiple particles moving in two-dimensional Poiseuille flow, we have observed that (1) the particle trajectories are self-regulated at low Reynolds number, which are totally different from those at higher Reynolds number and (2) the particles aggregate in middle of channel and form the long chains at higher concentration due to the higher flow velocity and the drafting between neighboring particles.