均質化理論應用於剛性孔隙介質流之研究

Homogenization Theory Applied to Rigid Porous Media Flow

吳映昕(Ying-Hsin Wu)

指導教授 ： 劉格非

國立臺灣大學/工學院/土木工程學研究所/博士(2013年)

https://doi.org/10.6342/NTU.2013.00844

摘要

本研究運用均質化理論分析剛性孔隙介質流的問題，包含：(a)非飽和土壤內非穩態的滲流問題；(b)水平方向週期振動造成乾砂堆崩落的分析。
均質化理論的主要分析工具為多尺度微擾法和體積積分平均法。首先定義問題中微觀空間尺度(或稱小尺度)與巨觀空間尺度(大尺度)，再以微觀尺度取一代表控制體積(簡稱REV)。利用多尺度微擾法將問題展開後，從微觀尺度開始求解REV的運動。接著利用體積積分平均法得到REV的平均運動後，搭配巨觀尺度的邊界條件，即可求出巨觀尺度下的運動。
非飽和土壤內之非穩態滲流問題中，微觀尺度係為孔隙之特徵尺度，巨觀尺度則是整塊土壤特徵尺寸。土壤內的顆粒假設為剛性體且無凝聚性。假設孔隙內水的流速緩慢，即孔隙水流運動之雷諾數的數量級為 。利用no-slip邊界條件與自由液面動力邊界條件可推導出微觀尺度的運動。對微觀尺度取體積積分得到REV之含水量與孔隙內壓力水頭和流速之關係，此關係驗證為Richards'方程式(Richards, 1931)。
乾砂堆受水平振動作用的問題中，我們分析砂堆開始發生崩落時振動頻率與振幅間的關係。假設振動為一維正弦函數型式的週期性振動。砂堆為乾燥不含任何液體。微觀尺度令為砂堆中孔隙的尺度；巨觀尺度則為砂堆之大小。假設砂子顆粒為鋼性體且無凝聚性。假設孔隙內空氣運動之雷諾數的數量級大於 。利用均質化裡論分析可得孔隙內空氣運動對於砂子顆粒的影響，再配合砂子顆粒之力平衡與力矩平衡，分析得到砂子受振動影響而開始運動時振動頻率和振幅的關係。最後，利用室內振動台實驗來驗證解析理論分析結果，從結果發現在小振幅時理論分析會高估砂堆崩落的頻率；在大振幅時理論與實驗結果相吻合。

關鍵字

均質化理論 ； 多尺度微擾法 ； 體積積分平均法 ； 剛性孔隙介質 ； 非穩態滲流 ； 砂堆崩落啟動

並列摘要

Using Homogenization theory and experiments, we study the two problems of the motions of fluid through rigid porous media, including (a) unsteady seepage flow in unsaturated and static soil; (b) initiation of collapse of sandpile under horizontal oscillatory vibration.

The key tool of homogenization theory is multiple-scale perturbation method with ensemble average. Before application of homogenization theory, we need to define the microscopic characteristic length scale (or small scale) and macroscopic scale (or large scale), respectively. Based on the microscale the representative elementary volume (REV for short) is defined. Using multiscale perturbation method we start the analysis from solving the REV problem on microscale. With ensemble average the REV averaged motion can be obtained. By macroscopic boundary conditions, the macroscopic motion is derived eventually .

As for the seepage problem, the characteristic length for microscale is set to be equal to pore-size, and total bulk-size for macroscale. The REV in the soil is set to be one order larger than pore-size. Solids in the REV are assumed to be rigid and cohesionless. The liquid velocity in the pore is assumed to be slow, and corresponding Reynolds number is about . By no-slip boundary condition on solid boundary in REV, we could obtain the microscopic flow conditions. Using spatial ensemble average under the microscopic scale, we obtain the relation between water content, pressure head and velocities on macroscale. This macroscopic averaged equation is validated to be equal to Richards’ equation (Richards, 1931).

As for the problem of sandpile under horizontal vibration, we study the relation between vibrating frequency and amplitude for the initiation of sandpile collapse. The vibration form is an one-dimensional sinusoidal function. The sandpile is composed of dry cohesionless and rigid sand grains without any liquid inside. The microscopic length scale is assumed to be equal to pore-size, and sanpile-size for macroscopic scale. The Reynolds number of the air flow in the pore is assumed to be greater than . Using homogenization theory we derive the force exerted on solid boundary by air motion in the pores. According to equilibrium of forces and moments, we obtained the instability condition of a sand grain and the relation between vibrating frequency and amplitude for initiation of sandpile collapse. Finally, using shaking table the experiments were conducted to be verified with the result by analytic analysis. Comparing with experimental data, the theoretical analysis gives good prediction for motion in large amplitude, but overestimate for motions in small amplitude.

並列關鍵字

homogenization theory ； multiple-scale perturbation method ； spatial ensemble average ； rigid porous media ； unsteady seepage flow ； initiation of sandpile collapse

參考文獻

尹孝元、黃清哲、連惠邦、李秉乾、周天穎、王晉倫 (2006). 自動化土石流觀測系統之發展及應用. 中華水土保持學報, 37(2):91-100.

水土保持局 (2009) 98年土石流年報, 行政院農業委員會水土保持局. (pp. 135)

Seed, H. B., Lysmer, J., & Martin, P. P. (1976). Pore-water pressure changes during soil liquefaction. Journal of the geotechnical engineering division, 102(4), pp. 323-346.

周憲德 & 張藝耀 (2004). 斜坡堆積圓球即圓柱顆粒受水平振動時之傾角分析. 中國土木水利工程學刊, 16(1), pp. 137-146.

Ancey, C. (2007). Plasticity and geophysical flows: A review. Journal of Non-Newtonian Fluid Mechanics, 142(1), pp. 4-35.