Vortex-based spatiotemporal characterization of nonlinear flows
Abstract
Although the ubiquity of vortices in nature has been recognized by artists for over seven centuries, it was the work of artist and scientist Leonardo da Vinci that provided the monumental transition from an aesthetic form to a scientific tool. DaVinci used vortices to describe the motions he observed in air currents, flowing water and blood flow in the human heart. Five centuries later, the Navier-Stokes equations allow us to recreate the swirling motions of fluid observed in nature. Computational fluid dynamic (CFD) simulations have provided a lens through which to study the role of vortices in a wide variety of modern day applications. The research summarized below represents an effort to look through this lens and bring into focus the practical use of vortices in describing nonlinear flows. Vortex-based spatiotemporal characterizations are obtained using two specific mathematical tools: vortex core lines (VCL) and proper orthogonal decomposition (POD). By applying these tools, we find that vortices continue to provide new insights in the realm of biofluids, urban flows and the phase space of dynamical systems. The insights we have gained are described in this thesis. Our primary focus is on biofluids. Specifically, we seek to gain new insights into the connection between vortices and vascular diseases in order to provide more effective methods for clinical diagnosis and treatment. We highlight several applications in which VCL and POD are used to characterize the flow conditions in a heart pump, identify stenosis in carotid arteries and validate numerical models against PIV-based experimental data. Next, we quantify the spatial complexity and temporal stability of hemodynamics generated by a database of 210 patient-specific aneurysm geometries. Visual classifications of the hemodynamics are compared to the automated, quantitative classifications. The quantities characterizing the hemodynamics are then compared to clinical data to determine conditions that are most conducive to rupture. Flows that form multiple vortices and undergo large-scale structural changes over the cardiac cycle are found to pose the most significant risk to patients. Concepts from dynamical systems are then applied to explain the formation of large-scale vortical flow structures in cerebral aneurysms. This is done by investigating the role of critical points along vortex core lines. We provide evidence that critical points are created and destroyed in saddle-node bifurcations during the cardiac cycle and that these bifurcations are responsible for changing the large-scale flow structure inside the aneurysm. Uncovering and understanding these mechanisms is the first step towards individualized treatments designed to suppress the creation of specific blood flow patterns that are known to present a risk of rupture. A simple differential dynamical system is used to illustrate the dynamical systems related concepts. Two examples illustrating the use of vortex-based methods in other domains are highlighted at the end of this work. The first example uses realistic CFD modeling of air flow through subway tunnels and stations to study the spread of accidental or planned release of airborne chemical or biological contaminants. Quantities from the vortex-based characterizations are shown to provide clear signatures that correlate to the dispersion and transport of pollutants though the stations. The second example examines swirling flow structures in the phase space of dynamical systems. Descriptions of vortices and their properties are extended to higher dimensions within the special class of differential dynamical systems.
- Publication:
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Ph.D. Thesis
- Pub Date:
- 2013
- Bibcode:
- 2013PhDT.......190B
- Keywords:
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- Physics, General;Physics, Fluid and Plasma