顆粒崩塌流場之動力行為: 實驗與模擬之研究

Dynamics of Granular Avalanches:An Experimental and Numerical Study

沈立宗(Li-Tsung Sheng)

指導教授 ： 蕭述三

國立中央大學/工學院/機械工程學系/博士(2012年)

摘要

顆粒材質在傾斜面上經重力驅動造成流動的現象是在許多工業上重要的應用或常見於地球物理相關領域。而目前諸多對於此類崩塌流場的學術研究，皆視為一個顆粒流的狀態，即針對一堆粒子所造成崩塌流場的現象去做其分析研究，所以顆粒崩塌流的研究，成為此類研究核心的部份，也是最基礎與最重要的部份。
本論文將針對顆粒崩塌流場的動力行為在實驗與模擬上做為研究。
在實驗部分將提出一個透過表面量測方法來估算當顆粒流體於矩形滑道中加速運動時的體積佔有率之變化。我們將設計一個傾斜角可變動且長150公分、寬5公分的矩形流槽。利用四個不同的傾斜角來做顆粒體加速運動的測試量測，而角度的變化範圍也包含了所使用的顆粒體的內部摩擦角。為了觀察顆粒體運動時自由表面與底層表面加速運動的情形，在滑道上下兩側均架設鏡子，以反射出顆粒運動時的影像。之後再利用PIV(Particleimage velocimetry)去計算顆粒流體的速度與流體厚度，且利用電子天秤去量測流體的質量流率，接下來利用線性內差去計算出流體的體積流率並推算出最後的顆粒體積佔有率。此外我們也將針對此線性內差去做敏感性的測試(Sensitivity analysis)，經過分析過後發現誤差約為±6%，也就是說我們所選用的內差法對於所計算出來的顆粒體積佔有率並沒有太大的影響。而最後所計算出來的顆粒體積佔有率，我們也將透過直接擷取量測的方法去做一個驗證，也證明利用內差法所計算出來的結果是正確的。相較於其他非直接量測的方法，例如：MRI (Magnetic resonance imaging) 或PEPT(Radioactive positron emission particle tracking)，本論文所發展出來的方法是比較符合經濟效益以及安全性的方法。在四個不同傾斜角的實驗測試中所推算出的體積佔有率的變化分佈，可以發現呈現兩種現象。而此分界點也剛好是顆粒體的內部摩擦角。兩個較小的傾斜角測試中，體積佔有率呈現線性遞減。但較大的兩個傾斜角則呈現出凹面曲線(concave)的分佈。
而在模擬部分，本論文也發展一個針對乾顆粒流場的運動行為之二相混合動力模型，此乾顆粒流場之間隙流體將視為空氣。此模型為包含二相的質量守恆及動量守恆方程式的二維動力模型。流體在流動過程中，空氣進出流體表面的物理現象也被考慮於此動力模型，而且空氣進出顆粒流體的量與顆粒流體擠壓或拉伸行為有關。因顆粒體與空氣之密度相差甚大，經過因次分析之後，模型統御方程式也呈現流體運動行為被顆粒體所主導的現象，也導致固氣二相速度差距很小。數個模擬測試結果也展現出此模型理論可對於顆粒流場運動行為做一預測之能力， 第一個測試為模擬一個穩態流場流經一個隆起物。第二個測試則模擬流體在一個平面上，且初始狀態為一個不穩定狀態的流動行為。第三個測試則是在各種不同初始狀態下，模擬壩體崩塌的問題。第四個測試則模擬一團有限質量的顆粒流體於傾斜面崩塌並最後於平面上沉積的流動行為。最後也將模擬結果與實驗結果做對照，皆呈現出非常吻合的結果。期待本論文的研究結果對於未來探討顆粒崩塌流場相關研究有新的參考價值。

關鍵字

顆粒流 ； 崩塌流 ； 顆粒體積佔有率 ； 二相流模型

並列摘要

Rapid flows of granular materials on inclined surfaces are often encountered in engineering applications involving mineral and food processing, bulk materials handling, and they are also found in geophysical situations. In the geophysical context, rock-falls, landslides and snow-slab avalanches set up to large-scale granular materials in motion. Because of the importance of industrial and geophysical applications, granular chute flow down inclines is a popular fundamental research issue of rapid flows.
In this thesis, we detail a method for estimating the flux-averaged solid fraction of a steady granular flows moving down an inclined rectangular chute using velocity measurements from along the perimeter cross-section, combined with knowledge of the mass flow rate through the cross-section. The chute is 5 cm wide and 150 cm long with an adjustable inclination angle. Four inclination angles, from 27◦ to 36◦ at 3◦ intervals are tested. This angle range overlaps the internal friction angle of the glass beads, which are of 4 mm nominal diameter. Two slender mirrors are installed at the top and the bottom of the transparent chute to reflect images of the flow down the chute of the two surfaces. This allows photographic recording of the flow with a PIV imaging system and measurement of the flow depth. The mass flow rate can be calibrated simultaneously by collecting the accumulated mass at the chute exit. A linear interpolation scheme is proposed to interpolate the volume flow rate in each section of the chute. Sensitivity analysis suggests that the relative standard deviation of this scheme is about ±6%, i.e., the resultant solid volume fraction is only moderately dependent on the interpolation scheme for the tested cases. This is further confirmed by a direct intercepting method. Compared to the sophisticated magnetic resonance imaging (MRI) or the radioactive positron emission particle tracking (PEPT) methods, the present method is verified as a cost-effective and nonhazardous alternative for ordinary laboratories. Two distinct groups of streamwise dependence of the solid fractions are found. They are seperated by the inclination angle of the chute which agreed with the internal friction angle. In the experiments by using two smaller inclination angles, the solid fraction ratios are found to be linear functions of the streamwise distance, whilst for the two larger inclination angles, the ratios have a nonlinear concave shape. All decrease with growing downstream distance.
Dry granular flows are also modeled in this thesis by utilizing a set of equations akin to a two-phase mixture system, in which the interstitial fluid is air. The resultant system of equations for a two-dimensional configuration includes two continuity and two momentum balance equations for the two respective constituents. The density variation is described by considering the phenomenon of air entrainment/extrusion at the flow surface, where the entrainment rate is assumed to be dependent on the divergent or convergent behavior of the solid constituent. The density difference between the two constituents is extremely large, so, as a consequence scaling analysis reveals that the flow behavior is dominated by the solid species, yielding small relative velocities between the two constituents. A non-oscillatory central (NOC) scheme with total variation diminishing (TVD) limiters is implemented. Four numerical examples are investigated: the first one considers that a steady flow moves over a bump; the second is related to the flow behaviors on a horizontal plane with an unstable initial condition; the third example is devoted to simulating a dam-break problem with respect to different initial conditions; and the fourth one investigates the behavior of a finite mass of granular material flowing down an inclined plane. Moreover, the simulation results also compare with the experimental results. It is demonstrated that the simulations and experimental observations are in excellent agreement. The key features and the capability of the equations to model the behavior are illustrated in these numerical results.
The works in this thesis are relatively fundamental. However, these issues are important for investigating the dynamic behavior of dry granular avalanches. We wish that the results will bring some new information to the research field and also contribute to related industries.

並列關鍵字

granular flow ； avalanches ； solid volume fraction ； two-phase model

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