The role of exponentially-small terms in transition to turbulence.
Lead Research Organisation:
University of Cambridge
Department Name: Applied Maths and Theoretical Physics
Abstract
In a conventional asymptotic analysis of high-Reynolds-number flows, only one or two terms of the asymptotic expansions are calculated. It is assumed that the asymptotic expansion converges to the Navier-Stokes solution as the Reynolds number increases. However, over the last decade or so, examples have been furnished where this is not the case, e.g. in the case of high-Reynolds-number unsteady flows that are susceptible to Rayleigh instabilities. The aim of this project is to place the understanding of this phenomenon on a firmer mathematical basis. In particular, it is speculated that this is a Stokes phenomenon arising from the turn-on of exponentially-small terms. The first part of the project will be to demonstrate that this is the case for a one-dimensional model problem that captures the essentials, before extending the analysis to more realistic flows.
Organisations
People |
ORCID iD |
| Stephen Cowley (Primary Supervisor) | |
| Christopher Sear (Student) |
Studentship Projects
| Project Reference | Relationship | Related To | Start | End | Student Name |
|---|---|---|---|---|---|
| EP/N509620/1 | 01/10/2016 | 30/09/2022 | |||
| 1936270 | Studentship | EP/N509620/1 | 01/10/2017 | 30/06/2021 | Christopher Sear |
| Description | By using methods adopted by both my supervisor and some experts on exponential asymptotics, I have made some progress in explaining the small time structure of solutions to the Kuramoto-Sivashinsky equation, which we are using as a toy model to explain the structure of other equations with small parameters - most notably the Navier Stokes equations at high Reynolds number. So far, I am reasonably confident of the picture of the KS solution within the complex x-plane at early times in certain regions, and am now investigating how the solution in these regions impacts the solution on the real x-axis. |
| Exploitation Route | If we can demonstrate a method for fixing terms in exponential asymptotics for the KS equation, it may be replicable for many other problems of physical interest such as the Navier Stokes equations. Asymptotic solutions to such equations are very useful, as they are much easier to derive than exact analytic solutions (which it may not be possible to find at all) but can be less expensive, more reliable and contain more information about the physics of a problem than numerical solutions, and are frequently used in all areas of applied mathematics, especially in fluid dynamics and acoustics. |
| Sectors | Aerospace, Defence and Marine,Energy,Manufacturing, including Industrial Biotechology,Other |
| Description | Presentations on my project to the Waves group in DAMTP (and to other academics working on fluid dynamics) |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | Local |
| Primary Audience | Postgraduate students |
| Results and Impact | I have given three 30-50 minute talks to the Waves group (of which I am a member) at DAMTP, as well as a short 10 minute presentation to researchers in fluid dynamics within the department. The purpose was to share my research with my colleagues, both to get feedback/advice from them and to give myself practice at explaining my research to others. |
| Year(s) Of Engagement Activity | 2018,2019,2020 |