arXiv 1006.53502010-06-12physics.flu-dyn

二维剪切流转捩模型

A two-dimensional model of shear-flow transition

Norman R. Lebovitz

备注：: Appears also on http://www.cs.uchicago.edu/~lebovitz

摘要：

我们探讨了一个二维动力学系统建模过渡剪切流，试图了解的“边缘”状态的性质。后者是相空间中的不变集，它将层流态的吸引域B分成两部分，这两部分通过再层化轨道的性质而彼此区分。该模型由R参数化，R代表雷诺数。原点是所有R值的稳定平衡点，代表层流。该系统具有四个发生质变的临界参数值：R_{sn}、R_h、R_{bh}和R_ \infty。如果R < R_{sn} but for R >R_{sn}还有两个平衡点X_{lb}和X_{ub}，则原点是全局稳定的。其中，X_{lb}对于R > R_{sn}的所有值都是不稳定的，而X_{ub}对于R < R_{bh}是稳定的，因此具有其自身的吸引域D。当R = R_h时，发生同宿分岔，同时形成同宿环和边态。当R_h < R < R_{bh}时，X_{lb}的稳定流形边缘态构成了\partial B的一部分，即原点吸引域的边界。\partial B的另一部分是包围D的周期轨道P。P和D随着R的增加而缩小。当R = R_{bh}时，存在一个“向后Hopf”分岔，在此分岔处X_{ub}失去稳定性，P和D消失.对于R_{bh} < R < R_ \infty，边缘是“纯”的，因为它是唯一位于原点吸引盆之外的相空间结构。随着R的增加，点X_{lb}逐渐后退到更大的距离，在R = R_ \infty处有一个奇点，在那里它变成无穷大。当R > R_ \infty X_{lb}重新出现时，边缘态消失，几何结构有利于从层流态永久转变，随着R值的增加，这种情况越来越明显。

英文摘要：

We explore a two-dimensional dynamical system modeling transition in shear flows to try to understand the nature of an 'edge' state. The latter is an invariant set in phase space separating the basin of attraction B of the laminar state into two parts distinguished from one another by the nature of relaminarizing orbits. The model is parametrized by R, a stand-in for Reynolds number. The origin is a stable equilibrium point for all values of R and represents the laminar flow. The system possesses four critical parameter values at which qualitative changes take place, R_{sn}, R_h, R_{bh} and R_\infty. The origin is globally stable if R < R_{sn} but for R > R_{sn} has two further equilibrium points, X_{lb} and X_{ub} . Of these X_{lb} is unstable for all values of R > R_{sn} whereas X_{ub} is stable for R < R_{bh} and therefore possesses its own basin of attraction D. At R = R_h a homoclinic bifurcation takes place with the simultaneous formation of a homoclinic loop and an edge state. For R_h < R < R_{bh} the edge state, which is the stable manifold of X_{lb}, forms part of \partial B, the boundary of the basin of attraction of the origin. The other part of \partial B is a periodic orbit P bounding D. P and D shrink with increasing R. At R = R_{bh} there is a 'backwards Hopf' bifurcation at which X_{ub} loses its stability and P and D disappear. For R_{bh} < R < R_\infty the edge is 'pure' in the sense that it is the only phase space structure that lies outside the basin of attraction of the origin. As R increases the point X_{lb} recedes to progressively greater distances, with a singularity at R = R_\infty where it becomes infinite. For R > R_\infty X_{lb} has reappeared, the edge state has disappeared, and the geometrical structure favors permanent transition from the laminar state, increasingly so for increasing values of R.

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摘要：

英文摘要：

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