濃密乾顆粒流之連體數值計算方法及流場應力邊界條件

濃密顆粒流之連續體模型近年來受到廣泛的研究，但在實際流場預測的應用上仍有許多限制深具挑戰性，主要的原因來自於不完整的本固模型與未知的邊界條件。所建立的運動方程式為非線性，所需的數值解一直到近年才有所突破，究其困難之根本，在於顆粒材料的雙相特性，顆粒間的摩擦接觸讓顆粒體在近穩態時有承受剪切力、類似固態的行為，但卻能隨著微觀上顆粒碰撞行為的發展轉成似液態的特性。針對諸多顆粒流未解的議題，本論文在近年來廣泛的接受局部(local)μ(I)流變本固模型上，發展有限體積法(FVM)之數值模型，並首度引入考量非碰撞傳遞機制的非局部模型修正。其中，針對μ(I)本固模型之黏塑性特性會造成數值計算上的歧異行為，我們採用正規化法(Regularization technique)處理。所發展的數值模型能成功重現不同幾何之顆粒流場中的實驗即離散元素計算的結果，包括穩態傾斜流中Bagnold至sub-Bagnold速度場隨著流場福祿數（Froude number）及非局部特性的轉換、於傾斜滑道上崩塌之流動、有無重力之平板剪切流中流場隨非局部特性由線性轉成Ｓ型的轉變，都首度在連續體模擬中重現。另一方面，在暫態以及穩態的離散模擬中發現，在一定體積中量測之側牆摩擦係數μw 並非常數且會隨著流動深度而遞減。此種現象在顆粒崩塌的離散模擬中，歸咎於一個無因次的轉動因子Ω所造成，其定義為顆粒的轉動速度以及移動速度的比值。但此假設尚未受到實驗的證明由於量測上的困難，於是本文提出了一個新的演算法用以量測顆粒崩塌實驗中，靠近側牆之顆粒的轉動速度。此實驗量測的角速度量值將用於檢測一個於文獻中提出的摩擦係數遞減的模型，進而再與另一個文獻中所提出的邊界條件模型進行比較。為了連結兩個不同模型的關係，我們發現角速度(ω)會與顆粒溫度(T)有線性關係，接著根據無因次分析發現，轉動因子Ω與一個新的無因次參數√(T/(p/ρ))有關。此微觀的參數對邊界條件的影響，將為邊界條件模型的發展開啟新的視野。

關鍵字

乾顆粒流；連續體模擬；正規化法；非局部模型；顆粒轉動；有效摩擦係數

並列摘要

A generic continuum rheology model for dense granular material has been widely investigated but its success in practical applications is still far from satisfactory due to unsettled issues in the rheology model and the flow boundary condition. The resulting governing equations are often nonlinear and require numerical solutions but a feasible continuum solver is not available until very recently [59]. The main difficulty results from the dual nature of a granular material that it possess yield strength as a solid in quasi-static state but transits to flow as a non-Newtonian fluid in agitated state. Hence, the first part of this thesis sets out to fill the gaps by developing a continuum solver (finite volume method with pressure implicit of splitting operator scheme) for a recently proposed local μ(I) rheology model in which we further incorporate non-local effect on transport mechanism for the very first time in literature. In particular, we propose a regularization scheme to handle the tricky viscoplastic behavior of μ(I) rheology so that new flow phenomenon can be predicted over a wider range of bulk Forde number in the configurations. The unique features include a non-Bagnold velocity profile in shallow surface flows on mild slope and the non-linear velocity profile in simple shear cell flows. The other half of the thesis studies how to assign the bulk wall friction coefficient for a Coulomb-type stress assignment in both laboratory experiments and discrete element (DE) simulations. Using finite mass avalanche as the benchmark problem, a promoted degree of grain rotation relative to translation, characterized by a non-dimensional rotation index Ω , has been identified as a reduction mechanism in DE simulated avalanche events [103]. In search of experimental evidence, we develop a novel image processing algorithm to measure angular velocity vector of individual sphere from the high-speed digital images at the flow boundary. The measured angular velocities are employed in a friction degradation model for μw/f with f being pure sliding friction coefficient between grains and flow boundary, which is further compared to the other literature model in terms of granular temperature T [5]. To correlate the two models, we discover a linear relation between the measured angular speed ω and T for the first time in the literature which links a bulk continuum property to particle-level dynamics. Furthermore, we conducted dimension analysis to further reveal a monotonic decay of the rotation index Ω with √(T/(p/ρ)) where p and ρ are confining pressure and grain intrinsic density. The finding shall shed light on how the microscopic mechanism for friction degradation correlates to bulk continuum properties. More importantly, it provides an explicit relation to solve the closure problem in continuum flow modeling when a flow-dependent boundary condition model is desired.

並列關鍵字

dry granular flow ； continuum simulation ； regularization technique ； non-local rheology ； particle rotation ； effective friction coefficient

參考文獻

[1] Ancey, C. & Evesque, P. 2000 Frictional-collisional regime for granular suspension flows down an inclined channel. Phys. Rev. E 62, 8349–8360.

[2] Ancey, C. 2001 Dry flows down an inclined channel: Experimental investigations on the frictional-collision regimes. Phys. Rev. E 65, 011304.

[3] Artoni, R., Santomaso, A. C., Massimiliano, G. & Canu, P. 2012 Scaling laws for the slip velocity in dense granular flows. Phys. Rev. E 108, 238002.

[4] Artoni, R. & Santomaso, A. 2014 Effective wall slip in chutes and channels: experiments and discrete element simulations. Granular Matter 16, 377–382.

[5] Artoni, R. & Richard, P. 2015 Effective wall friction in wall-bounded 3d dense granular flows. Phys. Rev. Lett. 115, 158001.

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濃密乾顆粒流之連體數值計算方法及流場應力邊界條件

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