Lagrangian Coherent Structures
Lagrangian Coherent Structures
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DOI：
10.1002/9783527639748.ch3
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发表时间：
2011-11
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影响因子：
--
通讯作者：
S. Shadden
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作者：
S. Shadden
越来越多的证据表明，通过考虑特殊的物质表面，可以有效地研究流体平流，这在这里被称为拉格朗日相干结构（LCS）。1)这些材料表面的特殊之处在于其独特的吸引或排斥性质。值得注意的是，LCS通常是局部最强烈地吸引或排斥流中的材料表面，并且因此对流拓扑结构具有强烈的影响。事实上，通过了解它们的演变，人们往往可以揭示复杂的层流，甚至湍流，流体输送的机制在显着的细节。从概念上讲，LCS可以从动力系统的角度，或从更物理基础的流体力学的角度来接近。从前者开始，我们注意到流体平流由方程_xmx 0 ; t 0 ; tmx; tmx 3：1描述，其中，umx; tmx是流体的速度场，xmx 0 ; t 0 ; tmx描述流体的运动（轨迹），2）在时间t0从位置x 0开始;我们假设名义上的体积保持流，r = u % 0。一般来说，由于流体的运动是混沌的，揭示显著的流动特征有助于我们理解流动是如何组织的。事实上，研究流体力学中的连贯或组织结构肯定是有趣的，只要我们有1）在这里，我们使用LCS来表示单数和复数形式。2)流体元素的概念是一个理想化的概念，但在流体力学建模中被广泛使用，例如，Navier-Stokes方程及其解都是基于这一假设。所得的流量数据通常不会被LCS计算的扩散修正，因为LCS的目标是平流，这实际上是通常考虑的长度和时间尺度上的主要传输模式。然而，对于许多应用，感兴趣的是近似由流体输送的物质的平流（例如，气泡，气溶胶，悬浮液和乳液）;在这种情况下，动力学（方程。（3.1））可以适当地增加[5]，或者流体的LCS可以被视为平流物质的近似LCS，并且在适当的条件下，它们的相关性可以严格地建立[6，7]，也可以参见[8，9]。
Mounting evidence suggests that fluid advection can effectively be studied by considering special material surfaces, which are referred to here as Lagrangian coherent structures (LCSs). 1) What makes these material surfaces special is their distinguished attracting or repelling nature. Notably, LCS are often locally the most strongly attracting or repelling material surfaces in the flow, and as such have a strong influence on the flow topology. In fact, by understanding their evolution, one can often reveal mechanisms that underly complex laminar, and even turbulent, fluid transport in conspicuous detail. Conceptually, LCS can be approached from the dynamical systems perspective, or from a more physically based fluid mechanics perspective. Starting from the former, we note that fluid advection is described by the equation _ xðx 0 ; t 0 ; tÞ ¼ uðx; tÞ ð 3:1Þ where uðx; tÞ is the velocity field of a fluid and xðx 0 ; t 0 ; tÞ describes the motion (trajectory) of a fluid element, or equivalently a material point, 2) starting at position x 0 at time t 0 ; we assume nominally volume-preserving flow, r Á u % 0. Since the motion of fluid is, generally speaking, chaotic, revealing salient flow features helps us understand how the flow is organized. Indeed, the study of coherent or organizing structures in fluid mechanics has surely been of interest for as long as we have 1) Herein we use LCS to abbreviate both singular and plural forms. 2) The concept of a fluid element is an idealization – but one that is overwhelmingly used in modeling fluid mechanics, for example, the Navier-Stokes equation and resulting solution are based on this assumption. Resulting flow data is typically not amended by diffusion for LCS computations, as LCS targets advection, which indeed is the primary mode of transport over the length and time scales typically considered. For many applications though, interest lies in the advection of matter that is approximately transported by the fluid (e.g., bubbles, aerosols, suspensions, and emulsions); in such cases, the dynamics (Eq. (3.1)) can be appropriately augmented [5], or LCS of the fluid can be seen as approximate LCS for the advected matter, and under appropriate conditions, their relevancy can be established rigorously [6, 7], see also [8,9].