Stretching and folding of Lagrangian coherent structure in cavity flows

• Main Authors: Ching Chang, 張敬
• Other Authors: 伍次寅
• Format: Others
• Language: zh-TW
• Published: 2010
• Online Access: http://ndltd.ncl.edu.tw/handle/72979371872477767914

碩士 === 臺灣大學 === 機械工程學研究所 === 98 === 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.