Project IV (MATH4072) 2022-23

Hydrodynamic Instabilities

Laura Currie

Description

A turbulent flow resulting from instability in Taylor–Couette flow with counter-rotating cylinders as studied in Crowley et al. 2020 .

Understanding the stability of solutions to the equations that govern fluid dynamics is key to understanding why we observe some of these solutions in nature, and not others. In many astrophysical and geophysical contexts, simple flows are often unstable and transition to turbulence. In order to be observed, a flow must be stable to the small perturbations that will inevitably be present in a complex system.

There are many known examples of hydrodynamic instabilities that lead to interesting behaviour in fluid systems. For example, Kelvin Helmholtz instability (as observed in clouds formations on Earth and on Jupter and Saturn), centrifugal instabilities (resulting from the effects of rotation or curavture), baroclinic instability (responsible for atmospheric weather systems), plus many more. You will have already met the basic concept of instability in Fluid Mechanics III, but this project will allow you to investigate more deeply this rich area of fluid dynamics.

To begin the project we will read broadly to gather an awareness of the wide range of instabilities occurring in fluid dynamics. We will also review the Navier-Stokes equations and the key features of linear stability analysis which can be applied to these equations. Following this, each student will be encouraged to examine further the linear stability of their chosen system, exploring natural extensions. Other/further potential avenues of study could include looking at nonlinear stability theory and/or numerical solutions to the equations.

Prerequisites

Fluid Mechanics III is required. Some basic knowledge of numerical computations in Python (or similar) may be useful, but not essential.

Resources

  • Textbook by Drazin & Reid (classic textbook on Hydrodynamic Stability)
  • Textbook by Vallis (textbook containing lots more than fluid instabilties but has nice sections focussing on instabitlities in atmospheric and oceanic fluid dynamics)
  • Textbook by Chandrasekhar (classic textbook for linear stability problems)
  • Paper by Davey (1962) (one of many research articles examining hydrodynamic stabilities. This one studies the growth of vortices in flow between rotating cylinders)

For more information email: Laura Currie