I am interested in turbulence and dynamical systems theory. The hope that dynamical systems theory could shed light on turbulence goes back some sixty years to Hopf. But it has proven difficult to translate Hopf's insights into concrete results, due to the very high (in principle infinite) dimensionality of the Navier-Stokes equations.
However, there have been some very exciting developments in recent years, triggered by advances in numerical methods and computer power, and by insightful research in the physics of wall-bounded flows. It is now possible to treat direct numerical simulations of Navier-Stokes directly as high-dimensional dynamical systems, for example, to find their equilibria and compute their linear stability.
My own research is an effort to analyze turbulent dynamics in terms of numerically exact solutions of the Navier-Stokes equations: equilibria, traveling waves, and periodic orbits. My current work focuses on plane Couette flow above the onset of turbulence, in preparation for applying the same ideas to dynamics of the turbulent boundary layer.
I am also interested in numerical analysis, computational fluid dynamics, and practical matters of scientific computing, such as developing flexible research software, and getting the most out of Linux boxes. I have focused these interests on the development of Channelflow, a set of high-level software tools and libraries for research in turbulence in channel geometries. Channelflow opens new ground in flexibility and ease-of-use in computational fluid dynamics. Give it a try!
The Julia programming language is the future of scientific computing. Julia is a revolutionary new open-source language with the high-level, dynamic, general-purpose feel of Python, the numerical focus, syntax, and libraries of Matlab, the execution speed of C, and the metaprogramming sophistication of Lisp. I am leading an NSF-funded project to incorporate Julia into the scientific computing curriculum at UNH. The first stage is incorporating Julia into Math 753/853 Numerical Methods.