濃密顆粒流本構模型之新觀點與其於斜坡流之理論解

New Aspects of Constitutive Model for Dense Granular Flows and Their Analytic Solutions to Inclined Flows

李庚霖(Keng-Lin Lee)

指導教授 ： 楊馥菱

國立臺灣大學/工學院/機械工程學研究所/博士(2016年)

https://doi.org/10.6342/NTU201603685

摘要

濃密乾顆粒流廣泛出現在穀物流、土石流、砂流，乃至於公路上車 流、工業輸送帶上的物流。由於其固有的相變複雜性，鮮少有解析模型能清楚地解釋巨觀現象背後的物理機制，以及提供便於工程應用的預測公式。除此之外，既存連續體本構模型(constitutive model)均侷限 在某種流況或流場。因此，本論文提出新的解析觀點來解釋及預測乾 顆粒斜坡流的複雜現象，並進一步提出更完備的本構模型。

首先，本文以顆粒間的關聯運動來解釋與描述有限質量斜坡流的沉 積行為。由關聯長度的本購模型(Ertas and Halsey 2002)出發，在忽略慣性影響以及淺層流的假設下，可求得流動厚度與關聯長度的動態漸近解(asymptotic solution)，並比較兩者在斜坡上的分佈，決定何時運 動的顆粒關聯結構體能有效地將底部摩擦阻力傳遍流體層，進而減速 至流動靜止。結果顯示沉積現象可被一運動波方程式描述。此方程式可用來建立流體軌跡與沉積長度的預測公式，與文獻中的實驗結果一致。
本文接下來研究連續質量斜坡流的動態行為與深度速度分佈，本構模型採用廣為接受的µ(I)流變律。由完整的動量方程式出發，將慣性項的影響視為一高階微擾，修正一階無慣性影響的解。結果顯示，慣性效應會在流體接近前端時產生一個向下游的推動力，弱化壓力梯度使流層厚度平滑，並增強底部黏滯阻力，使深度速度分佈從上游的Bagnold轉變為下游的Plug的型態。根據漸進分析的結果，本文建立一新的厚度分布解，改善了於深度平均模型(depth-averaged model)中假設特定深度速度模式的解。類似的分析被進一步應用到推導有限質量斜坡流的模型，並且使用關聯長度的概念來定義模型的有效範圍。

本文最後在Ginzburg-Landau相變理論框架中建立一新的本構模型，以慣性數I(Inertial number)為有序參數(order parameter)。參考非線性系統中pitchfork bifurcation理論去建立 Landau 自由能函數，再藉由量岡分析與物理推論去確立其中係數。此模型能清楚地顯示顆粒動量傳輸特性，與非局部現象背後的能量守恆關係式。在斜坡流的應用上，此模型能定量預測受遲滯與非局部行為影響的固液相變現象，並引導出一無因次參數描述厚層流與淺層流的速度分布轉變。此外，著名的Pouliquen平均流律在厚層流中以解析方式重現，與模型中固有的動量傳輸特性密切相關。本論文最後也提出幾項能改進此模型的方向，以期應用到更複雜的流場。

關鍵字

濃密顆粒流 ； 土石流 ； 漸近分析 ； Ginzburg-Landau 相變理論

並列摘要

Dense granular flow is ubiquitous in geophysical events and industrial applications. Owing to its multi-phasic nature, a unified constitutive description is still challenging and an analytical solution that reveals underlying mechanisms for macroscopic phenomenon is nearly impossible. Hence, this the- sis attempts asymptotic analysis to seek analytic solution to gravity-driven surface inclined flows in new aspects regarding different flow conditions or constitutive models. We also attempt a novel constitutive model which can complement the existing models to reproduce all the important flow features as the first in the literature.

The first problem studies finite propagation of finite mass with the correlation- length stress model of Ertas and Halsey (2002) and seek asymptotic solution under inertia-free and shallowness assumptions and no-slip basal condition. Solutions are found for both the non-uniform evolving flow height and cor- relation length which are compared to determine when basal resistance transmitted through the length is effective to arrest a whole layer. The predicted bulk dynamics agrees quantitatively to the literature measurements and our approach further provides a microscopic phase transition mechanism and reveals a kinematic-wave property of deposition process.

We also solve the front dynamics and velocity depth profile in a steady continuous flow down a rough incline using the µ(I) rheological model in full momentum equation. Analytic solution is sought by treating inertial effect as a small perturbation to a typical non-inertial solution. The solution shows that flow inertia weakens streamwise pressure gradient and strengthens basal resistance to result in a milder surface profile and facilitate a transition from upstream Bagnold to downstream plug flow profile. An approximate solution is further constructed from these two limiting solutions to improve the prediction from a depth-averaged model with prescribed Bagnold or plug flow. Similar asymptotic analysis is further applied to flow with finite mass and the correlation length model is used to estimate the model valid range.

Finally, a novel constitutive model is formulated using the concept of Ginzburg-Landau phase transition model and the inertial number I as an order parameter. In particular, a free energy potential is formulated by extending a subcritical bifurcation theory to describe granular hysteresis. The series expansion coefficient and exponents in the model are determined uniquely via physical scaling arguments so that the final form can capture all the flow features that are captured only partially by existing models. Further novelty of our approach is to demonstrate energy conservation among external shear work and internal energy modes. Finally, a dimensionless parameter that measures the significance of stress nonlocality is discovered to reflect the transition between creeping, Bagnold, and plug flows when flow conditions are changed.

並列關鍵字

Dense granular flows ； Debris flow ； Asymptotic analysis ； Ginzbur-Landau phase transition theory.

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