Application of Sum-of-squares of Polynomials Technique in Fluid Dynamics

Lead Research Organisation: Imperial College London
Department Name: Dept of Aeronautics


This PhD project will feature two primary objectives. Firstly, we aim to improve the UODESys toolbox. In
particular, we aim to improve the efficiency of the toolbox and obtain tighter bounds on the X term, which
appears in 2 and hence in the uncertain dynamical system. It is proposed to investigate different methods of
bounding the X term.
One of the main shortcomings of UODESys is that inefficient nested loops are used to implement certain
computations. We will replace nested loops with more efficient vectorised code wherever possible, in order to
speed up the derivation of the uncertain system.
The second primary objective of this research is to apply the SOS optimisation technique to specific fluid
flows. This will be done by first deriving a finite-dimensional (truncated) Galerkin approximation to the NSEs.
The finite Galerkin basis ui will be chosen based on physical considerations. Namely, the finite-dimensional
model should capture all the physical characteristics of the actual fluid flow. The resulting ODE system can
then be analysed. We will focus our efforts on determining the maximum Reynolds number for which a steady
solution of the ODE system is stable using a combination of Lyapunov stability theory and SOS optimisation,
thus enabling us to obtain a stability limit on the Reynolds number that is higher than the energy stability limit.
In addition, for various flows, we would like to derive rigorous bounds on flow characteristics in the turbulent
regime. If required, in both of these applications UODESys will be used to derive a reduced and uncertain ODE
system to reduce computational complexity.


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Studentship Projects

Project Reference Relationship Related To Start End Student Name
EP/N509486/1 01/10/2016 30/09/2021
2092930 Studentship EP/N509486/1 01/10/2017 30/09/2021 Mayur Venkatram Lakshmi
Description Many physical and non-physical phenomena can be mathematically modelled with dynamical systems. Dynamical systems are ubiquitous in all areas of applied mathematics. The analysis of chaotic dynamical systems presents many challenges, and this PhD project is focused on a theoretical and numerical study of such systems. Unstable periodic orbits (UPOs) are particularly interesting features of chaotic systems, as the set of all UPOs effectively forms a skeleton around which the chaotic behaviour develops. A new technique of searching for unstable periodic orbits (UPOs) with Sum-of-squares (SOS) optimisation has been developed. Our new technique allows one to find UPOs in cases where existing methods may fail, and has the potential to become a powerful new tool in the numerical analysis of dynamical systems. The development of our new technique has led to a publication in the SIAM Journal on Applied Dynamical Systems, in which we present the results of applying our new technique to a nine-dimensional dynamical system. This system is a model for a fluid flow between two flat plates, and chaotic solutions correspond to the physical phenomena of turbulence. In this publication we also report two new theoretical results. The work carried out in this PhD project links together two different areas of applied mathematics -- convex optimisation and dynamical systems theory. In the process of developing our new UPO localisation method, we have also developed the capability to determine highly accurate bounds on the long-time average of dynamical characteristics. Our method contrasts with traditional methods of computing long-time averages, which rely on time-consuming and expensive direct numerical simulations. Future work is focused on further improvement of our UPO localisation technique, in terms of efficacy and efficiency. Furthermore, we aim to apply our technique to dynamical systems modelling phenomena other than fluid flows, such as dynamical systems in the field of Orbital Mechanics. Another line of investigation is to adapt our techniques to leverage the characteristic features of the dynamical system which is to be analysed.
Exploitation Route Our new technique of localising Unstable Periodic Orbits (UPOs) could have far-reaching consequences. In particular, UPOs discovered using our new technique can be used to understand the behaviour of dynamical systems which exhibit spatio-temporal chaos. Applications to systems modelling fluid flows will elucidate the physical mechanisms related to turbulence. Our work can thus be used to help drive advances in numerous fields such as Physics, Aeronautical Engineering and Mechanical Engineering. Other fields where our new techniques can potentially be applied include Financial Mathematics and Operations Management. Part of the future work plan in this PhD project is to investigate potential applications in the aforementioned subject areas. In particular, the work carried out in this PhD project is directly applicable to the theory and practical implementation of Optimal Control, as knowledge of UPOs allows one to design control strategies to stabilise desired dynamics or suppress undesired ones. The full scope of applications remains to be seen. However, the work carried out in this PhD project has the potential to tackle a variety of long-standing problems in both academia and industry.
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