兩非平行板間之二維週期性暫態黏性流動

The Periodic Two-Dimensional Radial Flow of Viscous Fluid Between Two Inclined Plane Walls with an Oscillating Source

臧淞(Sung Tsang)

指導教授 ： 孫珍理

國立臺灣大學/工學院/機械工程學研究所/碩士(2013年)

https://doi.org/10.6342/NTU.2013.00049

摘要

本研究針對兩非平行板間，具黏性及不可壓縮性之二維週期性暫態黏性流動問題，既半週期為噴嘴流，另半週期為擴張流，利用similarity transformation，由物理模型 (Navier Stokes equations) 推導出一非線性ordinary differential equation (ODE) 及壓力方程式，再利用elliptic function求解流場結構，並透過值級分析探討壓力梯度與速度分布間之可能性關係。在後續分析部分，藉著類比AC電路概念得到流場流阻抗Zf，並利用流阻抗比評估暫態擴散器之流動特性。當中改變之參數有驅動源震盪角頻率、入出口長度比h/R、半角及Reynolds number Re。另一方面，我們利用壓降及體積流率Q，計算所需之平均功率，並探討平均功率隨各參數而改變之關係。由值級分析，得知當無因次中心速度F0夠大，且震盪角頻率夠小時，擴散器之壓力降將為負，這意味著無法在給定任一弦波壓降的情況下求解速度場，只能由已知速度反向求解壓力。由流阻抗Zf的觀點，我們得知當半角及Reynolds number Re增加時，流阻抗比具有兩種不同之變化趨勢。首先當震盪角頻率較低時，流阻抗比先隨半角或Reynolds number Re增加而下降，直到一最佳點後，流阻抗比才隨半角或Reynolds number Re的增加而上升；反之，在高震盪角頻率情況下，流阻抗比隨半角或Reynolds number增加而下降。接著當入出口長度比h/R上升或震盪角頻率上升時，流場將轉而由非穩態項所主導，且隨著非穩態項逐漸變強，流阻抗比將隨入出口長度比h/R上升或震盪角頻率上升而上升。此外，在改變入出口長度比h/R及震盪角頻率的過程中，我們還發現流阻抗比具有兩個平坦之區域，分別為當Womersley number (Wo) 小於0.1時的準穩態區及Womersley number (Wo) 大於10的非穩態區。另一方面，對於流場平均功率來說，我們發現在低震盪角頻率情況下，平均功率先隨半角增加而下降，直到一最佳點後，平均功率才隨半角增加而上升；反之在高震盪角頻率情況下，平均功率隨半角增加而上升。但是不管是在低或高震盪角頻率，若改變Reynolds number Re時，平均功率則是隨Reynolds number Re增加而上升。而對於入出口長度比h/R及震盪角頻率來說，擴散器之壓降則分別隨入出口長度比h/R上升而下降及隨震盪角頻率上升而上升，故平均功率隨入出口長度比h/R上升而下降及隨震盪角頻率上升而上升。

關鍵字

擴散器 ； 噴嘴 ； 震盪源 ； 流阻抗 ； 徑向週期性流動 ； 非平行板 ； 相似變換 ； 魏爾斯特拉斯橢圓函數

並列摘要

In this thesis, we employ similarity transformation and elliptic functions to solve the two dimensional radial flow of viscous fluid between two inclined plane walls with an oscillating source. Before the solving process, we carry out a scaling analysis to identify the allowable pressure gradient and maximum velocity. Moreover, we use the AC-circuit analogy to obtain the flow impedance Zf. By determining the ratio of flow impedance in opposite flowing direction, physical characteristics of the flow field are investigated. We also calculate the average power from instantaneous pressure drop Δp and flow rate Q. Influences of angular frequency w, slendness h/R, half angle a and Reynolds number Re in the ratio of flow impedance and average power are studied.
Through scaling analysis, we find that the pressure gradient across a diffuser must be positive when the angular frequency w is sufficiently small while the non-dimensional central velocity F0 is sufficiently large. As a result, there is a upper maximum of pressure drop Δp above which velocity solution cannot be found and we can only calculate pressure drop Δp for a given velocity oscillation. At low angular frequency w, there exists an optimal Reynolds number Re and half angle a that minimize the ratio of flow impedance. For Re < Reopt or a < aopt, the ratio of flow impedance decreases with Reynolds number Re or half angle a. For Re > Reopt or a > aopt, the ratio of flow impedance increases with Reynolds number Re or half angle a. In contrast, the ratio of flow impedance increases monotonically with both increasing half angle a and Reynolds number Re at high angular frequency w. As slendness h/R and angular frequency w augment (or Womersley number Wo), the effect of unsteady term amplifies. This leads to an increase in the ratio of flow impedance with both slendness h/R and angular frequency w. The influence of Womersley number Wo in ratio of flow impedance can be categorized into three regions: quasi steady region (Wo < 0.1), transition region (0.1 ≤ Wo ≤ 10) and unsteady region (Wo > 10). On the other hand, at low angular frequency w, there exists an optimal half angle a that minimize the average power. For a < aopt, the average power decreases with half angle a. For a > aopt, the average power increases with half angle a. In contrast, the average power increases monotonically with increasing half angle a at high angular frequency w. Moreover, for Reynolds number Re, the average power increases monotonically with increasing Reynolds number Re either high angular frequency w or low angular frequency w. As slendness h/R and angular frequency w augment (or Womersley number Wo), pressure drop Δp decreases with increasing slendness h/R while increases with increasing angular frequency w. This leads to a decrease in the average power with increasing slendness h/R and an increase in the average power with increasing angular frequency w.

並列關鍵字

diffuser ； nozzle ； oscillating source ； flow impedance ； periodic radial flow ； inclined plane walls ； similarity transformation ； Weierstrassian elliptic function

參考文獻

[2] I. Horschler, W. Schroder, and M. Meinke, "On the assumption of steadiness of nasal cavity flow," Journal of Biomechanics, vol. 43, pp. 1081-1085, 2010.

[4] C. A. Taylor and J. Humphrey, "Open problems in computational vascular biomechanics: hemodynamics and arterial wall mechanics," Computer Methods in Applied Mechanics and Engineering, vol. 198, pp. 3514-3523, 2009.

[6] F. Amirouche, Y. Zhou, and T. Johnson, "Current micropump technologies and their biomedical applications," Microsystem Technologies, vol. 15, pp. 647-666, 2009.

[7] A. Olsson, G. Stemme, and E. Stemme, "A valve-less planar fluid pump with two pump chambers," Sensors and Actuators A: Physical, vol. 47, pp. 549-556, 1995.

[8] A. Olsson, G. r. Stemme, and E. Stemme, "Diffuser-element design investigation for valve-less pumps," Sensors and Actuators A: Physical, vol. 57, pp. 137-143, 1996.