單及多顆懸浮體在Poiseuille管流內之運動

The Motion of a Single and Multiple Neutrally Buoyant Ellipses in Poiseuille Flow

陳世迪(Shih-Di Chen)

指導教授 ： 張建成

國立臺灣大學/工學院/應用力學研究所/博士(2013年)

https://doi.org/10.6342/NTU.2013.02476

摘要

本文探討中性浮力橢圓柱與橢球體在二維及三維管流內之運動問題，並且使用離散式拉格朗日乘數/虛擬區域法計算並研究不同橢圓柱尺寸比K、長寬比A和雷諾數Re對其運動的影響。研究發現單一橢圓柱在二維平面管流的條件下存在一個臨界雷諾數Rec ~ 3：當Re < Rec時，提高雷諾數會使得橢圓柱的質心平衡位置往管壁移動；當Re > Rec時，提高雷諾數反而會使橢圓柱的質心平衡位置靠近管道中央軸線。此外也發現橢圓柱的尺寸比K以及雷諾數Re會決定其平衡位置、質心移動軌跡及轉動模式：(i)在較大的雷諾數(Re≧550)及橢圓柱尺寸比(K≧0.76)條件下，其質心軌跡維持在管道中央時，橢圓柱的角度不會轉動並且其長軸方向固定與流場方向平行。(ii)在中等的雷諾數(Re = 40~500)及橢圓柱尺寸比 (K = 0.4~0.76)條件下，其質心軌跡在管道中央上下震盪時，橢圓柱的角度會來回擺動。(iii)在較低的雷諾數Re及橢圓柱尺寸比(K ≦ 0.4)條件下，其橢圓柱的質心軌跡在非管道中央的平衡位置輕微上下震盪時，橢圓柱會持續轉動。
當管流內僅有少數的橢圓柱(數目為ND = 4~16)時，在較低的雷諾數(Re ≦100)情況下橢圓柱會分散成數個集團在各自的平衡位置輕微上下震盪，而在較高雷諾數(Re = 1000)情況下橢圓柱會集中到介於管道中央及管壁之間的平衡位置。當管流內有較多的橢圓柱(ND = 36~108)時，則不再有明確的平衡位置，反而會在接近管壁處產生一層無橢圓柱存在的無顆粒層，其厚度yf會受橢圓柱的長寬比A以及雷諾數Re影響。我們也注意到當橢圓柱數目提高後所產生的流變效應，此時橢圓柱的平移速度Up以及流場速度場u的分佈皆會有鈍化現象。當二維管流內同時有橢圓柱以及圓柱存在時，管壁旁的無顆粒層厚度yf會隨著雷諾數Re或是橢圓柱長寬比Ae的增加而提高，也發現較大尺寸比(Ke = 0.2)的橢圓柱會將圓柱推離管道中央。
而在三維管流內的橢球體運動結果可發現橢球體最終會有三個不同區域的平衡位置以及轉動模式，其分別對應到不同的雷諾數範圍：(i)在較低的雷諾數範圍內(20 ≦ Re ≦ 800)，橢球體的平衡位置在S-S平衡位置附近，隨著雷諾數的增加平衡位置會更靠近壁面並且產生震盪，此時橢球體的長軸主要在yz平面上轉動；(ii)在中等的雷諾數範圍內(1000 ≦ Re ≦ 1200)，橢球體的平衡位置大約在req = 0.2處，並且有明顯的震盪，此時橢球體的長軸沒有在固定的平面上轉動；(iii)在較高的雷諾數範圍內(1600 ≦ Re ≦ 2000)，橢球體的平衡位置較靠近圓管z軸，隨著雷諾數的增加平衡位置會更靠近圓管z軸，此時橢球體的長軸固定與圓管的z軸平行，但其短軸仍會在xy平面上轉動。

關鍵字

剛性中性懸浮 ； 橢圓柱 ； 橢球體 ； 顆粒-顆粒相互作用 ； 管流 ； 虛擬區域法 ； 離散式拉格朗日乘數

並列摘要

In this article we investigate the motion of neutrally buoyant elliptical cylinders/ellipsoids in the plane/tube Poiseuille flow. The effects of the size ratio K, the aspect ratio A and the Reynolds number Re on the equilibrium position, the center-of-mass trajectory and the orientation dynamic of the ellipses are investigated and simulated by the distributed Lagrange multiplier/fictitious domain method. It is found for a single elliptical cylinder in Poiseuille flow that there is a critical Reynolds number Rec ~ 3: When Re ≦ Rec, the equilibrium position moves towards the wall as Re is increased; When Re ≧ Rec, the equilibrium position moves closer to the central axis with increasing Re. Moreover, the center-of-mass trajectorie and the orientation dynamic of the elliptical cylinder depend on the size ratio K and the Reynolds number Re: (i) For larger K (≧0.76) and Re (≧550), the center-of-mass trajectory moves to the channel center, the elliptical cylinder becomes stationary in orientation with its major axis parallel to the flow. (ii) For moderate K (≧0.4) and Re (= 40~500), the center-of-mass trajectory is oscillating about the channel center, the elliptical cylinder also exhibits an oscillatory orientation dynamics. (iii) For lower K (≦ 0.4) and Re, the center-of-mass trajectory is oscillating mildly about a certain height not at the channel center, the elliptical cylinder is rotary all the time.
For a few elliptical cylinders (the number of cylinders ND = 4~16), the cylinders may scatter into several groups and fluctuate about each averaged position at lower Re (≦100), and converge to an equilibrium position on each side of the channel center at the higher Re (=1000). When there are a larger number of cylinders (ND = 36~108), there are no longer clearly defined averaged equilibrium positions; instead, there exists a particle-free layer next to each wall where no cylinders are found. The thickness of the particle-free layers yf depend on the aspect ratio A of the elliptical cylinders and the Reynolds number Re. It is also interesting to note the rheological effect by observing the more blunted particle velocity Up and the flow velocity u profiles. For the motion of mixed neutrally buoyant elliptical and circular cylinders in the plane Poiseuille flow, the thickness of the particle-free layers yf increase as Re or the aspect ratio of the elliptical cylinder Ae is increased, and the elliptical cylinders with the larger size ratio (Ke = 0.2) tend to push the circular cylinders away from the channel center..
In our study of the motion of single neutrally buoyant ellipsoid in tube Poiseuille flow, we discover the different equilibrium positions and rotational states depending on the ranges of the Reynolds numbers: (i) At the lower Re range (20 ≦ Re ≦ 800), the equilibrium position of the ellipsoid is near the Segre-Silberberg equilibrium position, and move towards the wall as Re is increased. The ellipsoid majorly rotates with its long axis on the yz-plane. (ii) At the moderate Re range (1000 ≦ Re ≦ 1200), the equilibrium position oscillates about at req = 0.2. The long-axis of the ellipsoid does not rotate on a fixed plane. (iii) At the higher Re range (1600 ≦ Re ≦ 2000), the equilibrium position is near the channel center, and move closer to the channel center as Re is increased. The ellipsoid majorly rotates with its short axis on the xy-plane.

並列關鍵字

Rigid neutrally buoyant ； Elliptical cylinder ； Ellipsoid ； Particle-particle interaction ； Poiseuille flow ； Fictitious domain method ； Distributed Lagrange multipliers

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