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Research Activities

[Emerging Research Directions]

  • Optimal Control of Free-Boundary Problems in Fluid Mechanics, with applications, e.g., to aquatic propulsion
  • Computational Functional Analysis, in particular, the use of inverse methods to saturate estimates
  • Inverse Problems for Flows in Porous Media

    • [Main Research Themes (past & present)]

    • Optimal Control of Vortex Systems -- This research represents one of the first systematic applications of the modern theory of optimal control to systems of point vortices. Such systems arise as solutions of the 2D Euler equation, and are mathematically tractable alternatives to solutions of the Navier--Stokes equation. The techniques employed in our investigations range from rigorous mathematical analysis, allowing us to prove theorems concerning the behavior of some model problems, to large--scale computational studies of actual flows. We successfully applied linear feedback control strategies to stabilize a family of point vortex models of a laminar wake flow. Mathematical analysis of this problem allowed us to prove the existence of a center manifold in the controlled system, whose presence was also confirmed by numerical computations. Results of this long--term research effort are synthesized in a recent invited review paper.

    • Adjoint-Based Optimization in Fluid Mechanics -- The goal of this research is development and implementation of efficient computational techniques for solution of a range of optimization (inverse) problems in fluid mechanics and related fields. A key ingredient of our approaches is the use of a suitably--defined adjoint system which is employed to determine the gradient of the cost functional. Our early work in this area focused on the optimal control of a flow past a circular cylinder which is a classical problem in hydrodynamics. Subsequent investigations involved development of regularization strategies for adjoint--based optimization which, in addition to accelerating convergence of the method, make it also possible to apply such techniques to a broader range of problems. Our recent work concerned optimal control of problems in moving domains, such as thermo--fluid phenomena with phase changes. The motivation for these investigations comes from the industry, where our Industrial Partner (General Motors of Canada, Ltd.) is interested in applying such techniques to optimization of advanced joining processes, such as MIG welding. With these applications in mind, we developed novel computational algorithms based on the shape--differential and ``noncylindrical'' calculus. These methods were validated and applied to study some model problems, and were also extended these results to establish an optimization framework for a high-fidelity multiphysics model of a welding process.

    • Fundamental Properties of Bluff Body Wake Flows -- This research effort seeks to characterize, both theoretically and mathematically, certain fundamental aspects of flows past obstacles. In addition to their independent interest, these results are also important to our work on the two problems above. Our early investigations of this subject focused on careful characterization of the drag force in controlled and uncontrolled wake flows. Another related contribution addressed the question of efficient computation of hydrodynamic forces in open flows based on the velocity and vorticity fields alone. In yet another investigation we constructed a new family of solutions of the Euler equation with some interesting control--theoretic properties as a generalization of the celebrated ``Foppl model''.

    • Estimation of Turbulent Flows -- This research dealt with the problem of estimating the state of a turbulent flow based on some incomplete and noisy measurements. The specific question addressed in this work concerned the irreducible set of measurements needed to fully reconstruct such a turbulent flow.

    • Geometrical Statistics in 2D Turbulent Flows -- In this investigation, which was a part of a broader effort focused on fundamental properties of turbulent flows, we sought to provide an understanding of the connection between universality of small scales and nonhomogeneity of large scale motions in two-dimensional turbulence. In our studies we employed the concept of geometrical statistics introduced earlier to study certain characteristics of Fourier and wavelet filtering of turbulent flows.

    [Current Research Collaborations]

    • Flavio Dickstein (Departamento de Matematica Aplicada, Universidade Federal do Rio de Janeiro, Brazil)
    • Alan Elcrat (Department of Mathematics and Statistics, Wichita State University, USA)
    • Angelo Iollo (Institut de Mathématiques, Université Bordeaux I, France)
    • Nicholas Kevlahan (Department of Mathematics and Statistics, McMaster University, Canada)
    • Tony Murphy (The Commonwealth Scientific and Industrial Research Organisation, Australia)
    • Andrew Nowakowski (Department of Mechanical Engineering, The University of Sheffield, UK)
    • Koji Ohkitani (Department of Applied Mathematics, The University of Sheffield, UK)
    • Luca Zannetti (Aerospace Engineering Department, Politecnico di Torino, Italy)

    [Research Funding]