槽流中拉格朗日相關結構的拉伸與摺疊效應

Abstract

對一個廣義的非穩態流場(general unsteady flow)，其內部隨著時間的變動，會發展出一些獨特的結構，稱為相關結構(coherent structure)。相關結構在一般流體力學慣用的歐拉(Eulerian)描述下的物理量，如速度場、壓力場、渦度等等，缺乏一個明確的定義標準，因此會面臨定義上的困難。在動態系統理論中，則提供了有力的工具，在狀態空間中的鞍點具有兩種獨特的流形：穩定與不穩定流形(stable & unstable manifolds)，在穩定流形上的系統軌跡會互相靠近，而其附近的軌跡會互相發散；不穩定流形上的軌跡會互相發散，其附近的軌跡互相靠近。我們在流場中放入粒子，然後追蹤其在流場中的軌跡，藉由上述穩定與不穩定流形將軌跡區隔成不同動態區域的特性，可將其近似為相關結構的邊界，進而觀察出流場中的相關結構。因為利用粒子追蹤的觀點，所以稱為拉格朗日相關結構(Lagrangian coherent structure, LCS)。 本文採用有限時間李亞普諾夫指數(finite-time Lyapunov exponent, FTLE)法，計算凹槽內部流場(cavity flow)中每個粒子在一段有限時間[0,T]的軌跡，並求得每條軌跡在此段時間內，與附近軌跡之間彼此的最大形變量，取對數(logarithm)後為FTLE值 σ_0^T，並以 σ_0^T的局部最大值做為近似的LCS邊界。流場資料的取得採用雙時步(dual time step)虛擬壓縮性(pseudo-compressibility)，配合有限體積(finite volume)法來求解二維納維爾－史托克斯方程組(2D Navier-Stokes equations)。並在流場中加入染料，觀察在其中混合傳輸的效果，並與FTLE法近似得到的LCS邊界做比較。得到的結果可作為研究流場中動態行為，如混合、傳輸等物理現象的依據。

For a general unsteady flow, there are some unique geometry patterns that developed by the flow field as time evolution. These special structures are called “coherent structures”. In the fluid mechanics, the properties like velocity, pressure, vorticity etc., are described from the Eulerian viewpoint. The lack of an unambiguous value to define which flow region is coherent makes it difficult to locate the coherent structure using the Eulerian properties. The dynamical systems theory can provide some useful concepts to approximate the coherent structure in flows: the saddle point and its stable and unstable manifolds in the state space. For the stable manifold, it converses the trajectories on it and repels the trajectories nearby; for the unstable one, it disperses the trajectories on it and attracts the trajectories nearby. The stable and unstable manifolds play crucial roles of the boundaries that divide distinct dynamical regions. With the above ideas, we can calculate the particles’ trajectory in flows and use the stable and unstable manifolds as the approximation of the coherent structure boundaries. Due to the particle-base of description, the coherent structures are also called “Lagrangian coherent structures” (LCS). Our research using the “finite-time Lyapunov exponent method” applies on the cavity flow. We calculate each particle’s trajectories in the flow field over the time interval [0, T] and obtain each trajectory’s maximum dispersion with its nearby trajectories. After taking logarithm on the maximum dispersion, we get the finite-time Lyapunov exponent (FTLE) σ_0^T and plot the its contour over the domain. The local maximum values of the FTLE could be approximate as the LCS boundaries. The velocity data of the cavity flow is obtained by the CFD solver, which uses the finite volume method to solve the 2D Navier-Stokes equations with the dual time steps and the pseudo-compressibility techniques. We also put some virtual dyes in flow to observe their evolution with flow field and compare the distribution patterns with the LCS boundaries from the FTLE approximation. The result provides some clues to the mixing phenomenon and particle transport in the unsteady flows.