Refining Finite-Time Lyapunov Exponent Ridges and the Challenges of Classifying Them

While more rigorous and sophisticated methods for identifying Lagrangian based coherent structures exist, the finite-time Lyapunov exponent (FTLE) field remains a straightforward and popular method for gaining some insight into transport by complex, time-dependent two-dimensional flows. In light of its enduring appeal, and in support of good practice, we begin by investigating the effects of discretization and noise on two numerical approaches for calculating the FTLE field. A practical method to extract and refine FTLE ridges in two-dimensional flows, which builds on previous methods, is then presented. Seeking to better ascertain the role of a FTLE ridge in flow transport, we adapt an existing classification scheme and provide a thorough treatment of the challenges of classifying the types of deformation represented by a FTLE ridge. As a practical demonstration, the methods are applied to an ocean surface velocity field data set generated by a numerical model. (C) 2015 AIP Publishing LLC.ONR N000141210665Center for Nonlinear Dynamic

Lagrangian Based Methods for Coherent Structure Detection

There has been a proliferation in the development of Lagrangian analytical methods for detecting coherent structures in fluid flow transport, yielding a variety of qualitatively different approaches. We present a review of four approaches and demonstrate the utility of these methods via their application to the same sample analytic model, the canonical double-gyre flow, highlighting the pros and cons of each approach. Two of the methods, the geometric and probabilistic approaches, are well established and require velocity field data over the time interval of interest to identify particularly important material lines and surfaces, and influential regions, respectively. The other two approaches, implementing tools from cluster and braid theory, seek coherent structures based on limited trajectory data, attempting to partition the flow transport into distinct regions. All four of these approaches share the common trait that they are objective methods, meaning that their results do not depend on the frame of reference used. For each method, we also present a number of example applications ranging from blood flow and chemical reactions to ocean and atmospheric flows. (C) 2015 AIP Publishing LLC.ONR N000141210665Center for Nonlinear Dynamic

Detecting and analyzing coherent structures in two-dimensional dynamical systems

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mechanical Engineering, 2013.Some pages landscape orientation. Cataloged from PDF version of thesis.Includes bibliographical references (pages 211-218).The identification of coherent structures enhances the understanding of transport by complex flows such as those found at the ocean surface. The rapidly developing approach of Lagrangian Coherent Structures (LCSs) is based on the identification of codimension-1 structures that are locally the strongest repelling material surfaces in forwards or backwards-time over a given time window. Current theory and methods regarding LCSs are surveyed, and we present a modified algorithm for their detection, highlighting the pros and cons of the modified approach. One beneficial aspect of the modified approach is that it is possible to classify and advect LCSs through the time window. We apply the improved detection scheme as well as the classification to a high quality, validated simulation of ocean surface flow near the Ningaloo reef along the coast of Western Australia. This region is home to the longest fringing reef in Australia, a diverse marine environment, and a growing offshore drilling industry, and understanding the surface flows will enable better informed decisions to be made in this environmentally delicate domain. In addition to applying the LCS techniques, for the first time we account for the impact of surface winds on the LCS field by creating a hybrid current-wind velocity field. While the LCS approach is based on rigorous dynamical systems theory, its reliance on an accurate velocity field restricts potential ocean applications to simulations or regions with surface velocity measurements via systems like HF radar stations. An untapped resource is the data collected from float trajectories. With the goal of eventual application to these data sets, we develop a coherent structure detection algorithm utilizing sparse trajectory data. This new approach is based on the application of tools from braid theory, which produce a simplified perspective of the mixing of two-dimensional systems that enables rapid analysis. As a first application, our braid-based approach is applied to a periodically stirred system.by Michael R. Allshouse.Ph. D

Novel applications of diffusion-driven flow

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Mechanical Engineering, 2010.Cataloged from PDF version of thesis.Includes bibliographical references (p. 71-72).Diffusion-driven flow is the result of a conflict between hydrostatic equilibrium in a density stratified fluid and the no-flux boundary condition that must be obeyed on impermeable boundaries that are sloping with respect to gravity. This conflict results in a boundary layer flow, and in this thesis we present two novel applications of diffusion-driven flow. First, it is demonstrated that diffusion-driven flow can spontaneously propel asymmetric floating objects. Then, it is shown that the properties of diffusion-driven flow in a fissure can be exploited to make reliable measurements of molecular diffusivity.by Michael R. Allshouse.S.M

Diffusion-driven flows due to an obstruction layer

Thesis (S.B.)--Massachusetts Institute of Technology, Dept. of Mechanical Engineering, 2008.Includes bibliographical references (p. 73).With the confirmation of experiments, two new concepts about diffusion-driven flow are discussed and demonstrated. Although Phillips- Wunsch flow has been shown to exist along sloping boundaries, it is shown here that even an obstruction layer with only vertical walls produces a type of diffusion-driven flow. This is demonstrated through the execution of a dye test. Additionally, the influence that impermeable obstructions have on the density evolution of a stratified fluid is developed theoretically and demonstrated experimentally. With the use of two different types of obstruction layers, the theory is shown to accurately predict the density evolution.by Michael R. Allshouse.S.B

Internal wave pressure, velocity, and energy flux from density perturbations

Determination of energy transport is crucial for understanding the energy budget and fluid circulation in density varying fluids such as the ocean and the atmosphere. However, it is rarely possible to determine the energy flux field $\mathbf{J} = p \mathbf{u}$, which requires simultaneous measurements of the pressure and velocity perturbation fields, $p$ and $\mathbf{u}$. We present a method for obtaining the instantaneous $\mathbf{J}(x,z,t)$ from density perturbations alone: a Green's function-based calculation yields $p$, and $\mathbf{u}$ is obtained by integrating the continuity equation and the incompressibility condition. We validate our method with results from Navier-Stokes simulations: the Green's function method is applied to the density perturbation field from the simulations, and the result for $\mathbf{J}$ is found to agree typically to within $1\%$ with $\mathbf{J}$ computed directly using $p$ and $\mathbf{u}$ from the Navier-Stokes simulation. We also apply the Green's function method to density perturbation data from laboratory schlieren measurements of internal waves in a stratified fluid, and the result for $\mathbf{J}$ agrees to within $6\%$ with results from Navier-Stokes simulations. Our method for determining the instantaneous velocity, pressure, and energy flux fields applies to any system described by a linear approximation of the density perturbation field, e.g., to small amplitude lee waves and propagating vertical modes. The method can be applied using our Matlab graphical user interface EnergyFlux