Abstract
摘要:本計畫針對分散懸浮系統沉降行為,作一系統性研究。直接簡單的現象學理論(如下述之PNK理論)一直受引用來解釋這個效應,但由於其假說與推導有多方面的限制與簡化,因此配合顆粒-顆粒、顆粒-流體、顆粒-壁面作用之理論發展並從事精確的數值計算,以瞭解現象學理論的適用性,進一步解釋物理機制進而修正推廣現象學理論之適用範圍為本計畫之目的、乃至啟發雙相藕合模式(two-way coupling)之發展以應用於更複雜之流況是長期努力之目標。
我們在流體中考慮多個固體運動效應,不但流體運動方程式不再全域適用,且理論推導及數值分析亦變得複雜許多。我們將根據數學分析、實驗、數值模擬之三方面相互配合,研究粒子於液體中運動之基本力學性質及物理機制,進一步推導出相關理論公式或半經驗公式,以利科學及工程問題之有效應用。本項研究本質上涉及非線性偏微分方程組的二相流問題,因此結合數學、流體力學、數值計算三方面的專家,將使對此現象有更深入的瞭解。為使計算工作順利進行,我們選用本研究群參與U Houston 團隊TW Pan (潘從輝) 與R Glowinski共同發展之分散式拉氏乘數/虛擬區域法(Distributed Lagrangian Multiplier/Fictitious Domain, DLM/FD)[9-12, 28-30],計畫擬分三階段先後或同時進行下列子題研究:
第一年:單分散懸浮系統沉降行為。
計算結果將與PNK模型作一比較,進一步修正推廣模型之適用性。我們初步數據顯示,在單分散系統中,粒子沉澱速度大致符合該理論模型,但存在擾動,因沉澱過程中,懸浮粒子運動、粒子與流體間彼此作用以及管壁之影響,均與上述參數之改變,沉澱時間之更迭,而顯現出不同之顆粒行為與流場結構(流場局部對流與整體對流),值得吾人進一步探究。
第二、三年:雙分散懸浮系統沉降行為。
其次在輕、重粒子等雙/多分散系統,特別是在沉澱與懸浮粒子交界面上,發現與PNK理論間存在一些偏移或相當顯著的差異,此現象在文獻中未曾被提及,吾人將藉由對單分散系統中顆粒沉澱行為之瞭解,解釋PNK模型於粒子交界面所產生之偏移,進一步修正推廣PNK模型甚或提出適用性廣的新模型。本研究將進一步擴展至多分散懸浮系統。
第四至五年:三維之分散懸浮系統計算。
開發及應用三維之分散式拉格朗日乘數/虛擬區域法,以更準確模擬真實之物理現象,並驗諸實驗數據,進行上述諸因素之影響的探討,據以修正PNK模型,比較二、三維分散懸浮系統沉降行為之差異。為簡便有效應用於更複雜之實際系統,本研究亦提供一絕佳的機會來檢視更一般的固液二相流系統的雙向偶合模式問題。
Abstract: This proposal is aimed at theoretical and computational study of the Boycott effect in mono-disperse and bi-disperse suspensions under the effect of gravitational force. In the past, the phenomenological theory such as the PNK model (proposed by Ponder, Nakamura and Kuroda) is often used to explain the simple but actually complicated settling behaviors of solid particles in an inclined vessel. The PNK model was derived based on simple physical laws and assumptions without resorting to accurate solution of the equations of motion.
As the research involves a set of nonlinear partial differential equations in fluid mechanics, we will bring together experts in mathematics, fluid dynamics and numerical computation for a profounder understanding of the phenomena. On the numerical side, we plan to provide a more comprehensive study of the effect by solving the Navier-Stokes equations using the method of distributed Lagrange multiplier-based fictitious domain formulation. The method has been mainly developed by Pan and Glowinski [9–12], [28-30], and the PI of this project (CC Chang) had also an opportunity to participate in the development of its 3D version [29]. The numerical results provide useful data for parametric studies of the influence of the important factors such as the particle density, solid fraction of light and heavy particles, tilted angle of the vessel, geometric size and wall effect. The more ambitious goal of this investigation is: to examine the existing models of two-way coupling behaviors between solids and fluids. In a term of five years, we propose to investigate
(1) Settling behaviors of mono-dispersive suspensions
In the past, the PNK model is shown to explain well the settling of mono-dispersive suspensions in some cases. Numerical results are used to check the validity of the PNK model and for possibly improving and generalizing the PNK by incorporating the effects of important parameters such as the inclined angle of the vessel, number density of the particles as well as interactions of particle-particle, particle-fluid, particle-wall boundary.
(2) Settling behaviors of bi-dispersive and multi-dispersive suspensions
This phase is to address the same problems as in Phase 1, but extends the research into bi-dispersive and multi-dispersive suspensions. This phase of study may lead to a possible proposal of a new model that goes beyond the PNK.
(3) Settling behaviors of 3D dispersive suspensions As the realistic experiments are performed in 3D, it is necessary to develop the method of DLM/FD so that it is suitable for applications to study 3D settling behaviors of dispersive suspensions. To identify the crucial factors that distinguish the 2D and 3D Boycott effect is the main focus of the research in this phase.
(4) Applications to modeling relevant two-phase flow. An important application is modeling of cloud and precipitation at the levels of particle-particle interactions. In particular, we will examine two-way coupling of particle-fluid systems. Direct simulations provide an excellent opportunity to examine the existing models of two-way coupling behaviors between solids and fluids.
Keyword(s)
(1)單/雙分散懸浮系統
(2)沉降行為
(3) Boycott效應
(4) PNK模型
(5)分散式拉氏乘數/虛擬區域法(拉格朗日乘數法/虛擬區域法)
(1) Dispersive suspension
(2) Settling behavior
(3) Boycott effect
(4) PNK model (Ponder
Nakamura & Kuroda)
(5) Method of distributed Lagrange multiplier/ fictitious domain
(DLM/FD)