Lasiecka’s Research Targets Fluid Control Strategies for Aerodynamics, Renewable Energy, Engineering and Beyond.

NSF project seeks to suppress turbulence and flutter for wide-ranging applications

Dr. Irena Lasiecka, distinguished university professor and chair in the department of  Mathematical Science, was awarded $340, 000 as the PI [co-PI distinguished university Dr. R. Triggiani] from the Division of Mathematical Sciences of the National Science Foundation for the three-year research project entitled: PDE Control of 3D Fluids, Flow/Fluid-Structure interactions: Finite dimensional strategies for flutter/turbulence suppression.

The goal of this mathematical research project is to identify and construct physically implementable control strategies for (i) suppression of turbulence, which is fluid motion characterized by chaotic changes in pressure and flow velocity, and (ii) control of flutter, which is sustained oscillation that may occur when a nonlinear body is subject to surrounding inviscid flow. It results in a periodic-like instability which often causes structure failure due to the fatigue of the material. Turbulence caused the collapse of the faultily designed NASA Helio in July 2003. Torsion flutter caused the collapse of the faultily designed Tacoma Narrow bridge in November 1940, a few months after it was open to traffic. Heart arteries may be damaged when interacting with pressured fluids.

The results are anticipated to inform several important application areas, including aerodynamic design, and renewable energy systems in which harvesting flutter is a goal. The project involves collaboration with theoretical engineers (at Oak Ridge National Laboratory, Duke University, etc.) to identify the correct model to analyze mathematically. It will train math graduate students through their research involvement in the mathematical analysis and corresponding computations of the research project. While rooted in real-life phenomena with interdisciplinary cross-fertilization, this multi-task project identifies several open problems within mathematical areas - particularly partial differential equations, analysis and mathematical control theory. The project studies questions in control theory and corresponding control strategies with focus on physical phenomena governed by three-dimensional (3D) fluids, flow-structure or fluids-structure interactions which induce strong instabilities in their dynamics (bridges, buildings, airfoils).

The systems are governed by Euler or Navier Stokes equations coupled at the interface with the system of dynamic elasticity to model structures. Instability in fluid dynamics, and how to cope with it, is one of the central issues in the field. The main goal is to understand the phenomenon of flutter-suppression and turbulence-suppression by means of finite dimensional control strategies. The latter restriction, while physically attractive as being practically implementable, is a major challenge to achieve from the mathematical viewpoint. Uniform stabilization of the 3D Navier-Stokes equations by means of a boundary-based, localized (arbitrary small support) feedback control, which moreover is finite dimensional, was an open problem in the international literature for about 20 years. It was resolved in the affirmative - even with the additional constraint of requiring only-tangential-like controls) by the PIs with their first PhD student at the UofM in a lengthy 2021-article in the high-profile e Archive of Rational Mechanics and Analysis. Solution had to overcome numerous conceptual and technical obstacles, some of independent interest. This result is the breakthrough that opens the door to additional investigations on different fluids, such as the Boussinesq system, which models heat transfer in a viscous incompressible heat conducting fluid.

Within the main goal of establishing flutter-suppression and turbulence- suppression by means of finite dimensional control strategies, the specific objectives of the project are to achieve: (i) finite dimensional attracting sets to capture flutter of the oscillating structure; (ii) finite dimensional boundary feed- back stabilization of 3D fluids, and (iii) control theory for 3D fluid-structures with moving interface. The investigation of flutter will require the theory of non-dissipative dynamical systems and their attractors, along with microlocal- pseudo-differential analysis. In this project, investigation of turbulence in 3D will require Besov spaces of tight indices (to replace traditional Sobolev spaces, which fail) and a new maximal regularity (R-boundedness theory) for the novel setting of boundary-feedback partial differential equations. Quasilinear theory and sharp estimates for the hyperbolic Dirichlet-Neumann map will be essential tools in controlling bodies moving in a fluid, where global existence is still an open problem.

The project draws on past work of the PI’s, as documented by a series of papers and research monographs in the area of mathematical control theory of systems governed by partial differential equations. These including: two volumes (1,100 pages) published in 2000 by Cambridge University Press, Encyclopedia of Mathematics and its Applications [which were awarded the rare MathScienceNet Featured Reviews]; a CBMS-NSF-SIAM Lecture Notes on Control Theory of Coupled PDE’s, published by SIAM in 2002; AMS [American Mathematical Society] 2008 volume in the series Memoires -AMS, on long time behavior of second order Evolutions with nonlinear damping. A Springer Ver- lag Monograph in Mathematics [joint with I. Chueshov] (764-pages, published in 2010) in the area of Wellposedness and Longtime behavior of Von Karman system; a Oberwolfach Seminar, Mathematical Theory in Fluid Structure Interactions, published by Birkhauser in 2018.

In addition, during their PDA leave, Spring 2022, the PIs were appointed at the Mathematical Sciences Research Institute, University of California, Berkeley, Irena Lasiecka as a Eisenbud Professor, funded by the Sloan’s Foundation; and Roberto Triggiani as a Research Professor awarded by MSRI. This was a research intense semester of high international profile on Mathematical Theory in Fluid Dynamics, where the PIs gave invited lectures. It provided additional impetus which has influenced new directions of the research proposed. This topic was  also  the object  of  the  four  lectures that each PI delivered as principal lecturer at the  Summer  School  Semester,  Fluid under Control , held at the Mathematics Institute, Czech Academy of Sciences, in Prague, August 2021, with an 80-page chapter per author to appear in the forthcoming Birkhauser volume. Participation of PhD students is a critical aspect of the broader impacts-aspect of this research project.

For more information on this project, contact Lasiecka at lasiecka@memphis.edu.