Qualitative analysis of a model for boundary effects in the Taylor problem
Qualitative analysis of a model for boundary effects in the Taylor problem
复制标题
泰勒问题边界效应模型的定性分析
DOI：
10.1017/s0305004100056759
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发表时间：
1980
影响因子：
0.8
通讯作者：
D. Schaeffer
中科院分区：
文献类型：
--
作者：
D. Schaeffer
流体力学经典分叉实验中的边界效应一直是最近研究的主题，对我们来说，最值得注意的是Benjamin（2）关于在很短的环形空间中Taylor问题中分叉定常流的工作。他的论文提出了一个基本的概念性问题：主流中细胞的数量，一个整数，如何依赖于环的长度L，一个真实的参数？主流是指随着转速从静止状态逐渐增加而在高雷诺数R下形成的流动，其中L保持恒定（例如L = L*）。本雅明数据的相关部分如图0·1所示。在他的实验的很短的环中，主流只具有四个细胞中的两个。(See第1节关于偏向偶数个细胞。）如果L* > L2，L1 < L* < L2或L* < L1，则主流分别具有4个、2个或2个单元;然而，当L1 < L* < L2时，随着R的增加，流的演变不是平滑的，而是随着R通过图中直线{L = L*}与尖点曲线Γ的三个交点的中间而发生跳跃。(It现在人们普遍认为，流动中细胞的发展是在一定雷诺数范围内发生的;这一点不是这里的问题。）曲线Γ将平面分成两个区域，使得根据（R，L）分别位于区域I或区域II中，存在一个或两个稳定的、实验上可观察到的流动。我们可以将区域II中的两个稳定状态称为2胞或4胞，尽管它们的胞状结构在图中出现的中等雷诺数值下仅部分发展。区域II中的两个流动中的一个或另一个在穿过Γ时失去稳定性，从而当（R，L）如上所述穿过从区域II移动到区域I的边界时，产生跳跃的可能性。一般来说，如果（R，L）沿一条从区域I开始，穿过区域II，再进入区域I的路径准静态地沿着变化，则当且仅当点H位于路径穿过边界（在沿沿着Γ的距离意义上）的Γ的两点之间时，在再进入时会发生跳跃。
Boundary effects in the classical bifurcation experiments of fluid mechanics have been the subject of much recent investigation, most notably for us the work of Benjamin(2) on bifurcating steady flows in the Taylor problem in a very short annulus. His paper broaches a fundamental conceptual issue: how does the number of cells in the primary flow, an integer, depend on the length L of the annulus, a real parameter? By the primary flow we mean the flow which develops at high Reynolds number R as the rotation speed is increased gradually from rest, L being held constant (say L = L*). The relevant part of Benjamin's data is presented in Fig. 0·1. For the very short annuli of his experiment the primary flow possessed either two of four cells only. (See Section 1 concerning the bias towards an even number of cells.) If L* > L2, L1 < L* < L2 or L* < L1 the primary flow has 4, 2 or 2 cells respectively; however when L1 < L* < L2 the evolution of the flow as R is increased is not smooth but suffers a jump as R passes the middle of the three intersections of the line {L = L*} with the cusped curve Γ in the figure. (It is now widely accepted that the development of cells in the flow occurs over a range of Reynolds numbers; this point is not the issue here.) The curve Γ divides the plane into two regions such that there are one or two stable, experimentally observable flows according as (R, L) lies in region I or region II, respectively. We may refer to the two stable states in region II as 2 cell or 4 cell, although their cellular structure is only partially developed at the moderate values of the Reynolds number occurring in the figure. One or the other of the two flows in region II loses stability across Γ, giving rise to the possibility of jumps when, as above, (R, L) crosses the boundary moving from region II to region I. In general if (R, L) is varied quasi-statically along a path which begins in region I, passes through region II, and re-enters region I, a jump will occur on re-entry if and only if the point H lies between the two points of Γ where the path crosses the boundary (between in the sense of distance along Γ).