| Summary: | 博士 === 國立臺灣大學 === 應用力學研究所 === 101 === 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.
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