My research interests include:
- Fluid Dynamics: turbulence, mixing, biological flows, computational fluid dynamics, hydrodynamic stability
- Biological Fluid Mechanics: the glymphatic system, cerebrospinal fluid flow in perivascular spaces
- Nonlinear Dynamics and Chaos: bifurcations, high-dimensional chaos, exact coherent structures, Lagrangian coherent structures, pattern formation, symmetries, synchronization
- Topological Data Analysis: computational homology, persistent homology
Fluid Mechanics of the Glymphatic System (Current University of Rochester Work)
I study the glymphatic system, which is a waste removal system in the brain that may play an important role in the development of neurodegenerative diseases such as Alzheimer’s disease. Professors Kelley, Shang, Thomas, and I collaborate with Professor Maiken Nedergaard, who discovered this system in 2012, to quantitatively measure and model the glymphatic system. Specifically, this pathway includes a brain-wide network of perivascular spaces (PVSs)—annular tunnels around blood vessels—and cerebrospinal fluid (CSF) flows through these spaces to remove waste from the brain.
We have recently developed techniques for measuring CSF flow through PVSs in living mice. Our results, just published in Nature Communications, demonstrate that CSF flow is driven through PVSs by a novel peristalsis-like mechanism arising from arterial pulsations. We also demonstrate that acute arterial hypertension causes changes in the arterial pulsations which lead to a reduction in net CSF flow. This result may offer a causal mechanism that contributes to the correlation between hypertension and dementia, as recently reported by the SPRINT clinical trial, for example. By developing microscopic and macroscopic models, as well as performing in vivo measurements, our collaboration will characterize many of the factors that reduce waste clearance in the brain via impaired glymphatic function.
Forecasting Turbulence Using Exact Coherent Structures (Georgia Tech Dissertation)
Fluid turbulence is notoriously difficult to forecast despite the fact that it is governed by the deterministic Navier-Stokes equation. However, recent advances suggest that special solutions of the Navier-Stokes equation, often called “exact coherent structures” (ECSs), might provide a framework for forecasting turbulence. Calculating ECSs, however, is no trivial task. Such calculations must be done numerically, so it is very important that the numerical simulation accurately captures the experimental system being studied.
In our work, we study a quasi-two-dimensional flow generated by driving a thin layer of electrolyte (3 mm thick) with electromagnetic forces, approximating a canonical problem known as “Kolmogorov flow.” The complexity of the flow is characterized by the Reynolds number Re. To model these thin-layer flows, our research group derived a new form of 2D Navier-Stokes equation by applying a formal depth-averaging procedure to the 3D Navier-Stokes equation. Using this model combined with accurate modeling of the electromagnetic forces driving the experiment and realistic boundary conditions, we have developed simulations that agree remarkably well with the experiment.
With this close agreement, we next searched for ECSs in the weakly turbulent regime (Re=22.5). By “weakly turbulent,” I mean that the flow is chaotic and irregular but doesn’t show the statistical scaling behaviors often used to define turbulence. Here is an hour long time series of fully-resolved velocity fields showing the full flow field in the experiment, sped up by a factor of 75 (arrows indicate the velocity and the color contour is the vorticity):
By studying several of these time series, we identified instances when we expect that the experiment is dynamically “close” to an ECS. We then calculated that ECS by plugging the snapshot from the experiment directly into the simulation and using a special Newton solver. One such calculation is shown below. The similarity between the snapshot from the experiment and the ECS is quite remarkable!
To demonstrate the utility of ECS, we used one such solution to forecast the experiment. In our article published in Physical Review Letters, we demonstrate that one such ECS allows us to forecast the experiment over a period of about 90 seconds (3.4 correlation times). We believe that similar ECS-based approaches may one day offer valuable improvements for predicting several spatiotemporally complex systems, including the weather and climate.
Characterizing Fluid Flows Using Persistent Homology (Georgia Tech Dissertation)
Spatially-extended systems that exhibit complicated dynamics commonly arise in a wide variety of systems in nature and technology, e.g., turbulent fluid flows, neurons firing in the brain, or pacemaker cells in the heart. Describing, understanding, and predicting the behavior of such systems is a challenge for scientists and engineers. Such systems are particularly challenging because they are high dimensional and may possess global symmetries, which we do not know how to optimally deal with at the current time. We are exploring the use of persistent homology to characterize spatiotemporal dynamics in a canonical fluid mechanics problem known as Rayleigh-Bénard convection.
Persistent homology, which is a branch of algebraic topology, provides a powerful mathematical formalism in which the topological characteristics of a pattern are encoded into a so-called persistence diagram (PD). PDs provide a reduced description of a pattern at any given instant of time, while preserving the dynamical properties of that system. This means that during instants of large changes to the patterns (the midplane temperature field, in this case), there are correspondingly large changes to the points in the PDs, as can be seen in the following movie. Additionally, PDs are inherently symmetry-independent, which means, for example, if the flow field below were rotated 90 degrees, the PDs would still be exactly the same, because a rotated flow field is dynamically equivalent.
Each PD can be thought of as a single point in the high-dimensional state space of persistent homology. So, we can then apply persistent homology a second time to this “point cloud” to generate a PD that contains information about the time evolution of this system. These two applications of persistent homology effectively allow us to reduce a flow field with 125,000 degrees of freedom to 2 dominant points in a PD, which is a tremendous dimensionality reduction. For more information, check out our most recent paper.