# MAGIC014: Hydrodynamic Stability Theory

## Course details

### Semester

Spring 2008
Monday, January 21st to Friday, March 14th; Monday, April 28th to Friday, May 16th

### Hours

Live lecture hours
20
Recorded lecture hours
0
Total advised study hours
0

Tuesdays
12:05 - 12:55
Thursdays
09:05 - 09:55

## Description

This is offered as a core course for Applied.

### Prerequisites

No prerequisites information is available yet.

### Syllabus

1. Introduction (2 lectures)
• Derivation of the Navier-Stokes equations
• Boundary conditions
• Non-dimensionalisation
• Additional forces and equations: Coriolis force, buoyancy
• Boussinesq approximation
2. Basics of stability theory (2 lectures)
• Swift-Hohenberg equation as a model
• Linear stability. Dispersion relation.
• Marginal stability curve.
• Weakly nonlinear theory.
• Normal form for pitchfork bifurcation
• Global stability
3. Rayleigh-Benard convection (4 lectures)
• Basic state. Linear theory. Normal modes.
• Marginal stability curve.
• Weakly nonlinear theory. Modified perturbation theory.
• Global stability for two-dimensional solutions
• Truncation: the Lorenz equations
4. Double-diffusive convection (2 lectures)
• Rotating convection, plane layer and spherical geometry
• Linear theory: real and complex eigenvalues,
• Takens-Bogdanov point.
• Taylor-Proudman theorem.
• Thermosolutal convection.
• Linear theory: real and complex eigenvalues.
• Salt fingers.
5. Instabilities of parallel flows (6 lectures)
• Instabilities of invicid shear flows. Linear theory.
• Squire's theorem. Rayleigh's equation.
• Plane Couette flow.
• Rayleigh's inflexion point criterion.
• Howard's semi-circle theorem.
• Examples: Kelvin-Helmholtz, bounded shear layer.
• Role of stratification. Role of viscosity, global stability.
• Shear flow instabilities of viscous fluids.
• Orr-Sommerfeld equation.
• Examples: plane Couette flow, plane Poiseuille flow, pipe flow, Taylor-Couette flow.
• Problems with normal mode analysis.
• Pseudo-spectrum and non-normality.
• Absolute and convective instabilities.
• Finite domain effects.
6. Introduction to pattern formation (3 lectures)
• Stripes, squares and hexagons. Weakly nonlinear theory.
• Mode interactions.
• Oscillatory patterns: standing and travelling waves.
• Long-wave instabilities of patterns: Eckhaus, Benjamin-Feir, etc.
7. Introduction to the transition to turbulence (1 lectures)
• Supercritical vs subcritical bifurcations.
• Fully developed turbulence. Turbulent cascade.
• Energy spectrum. Isotropic turbulence.
• Mean plus fluctuations, Reynolds stress.
• Closures: eddy viscosity, Subgrid-scale modelling, similarity models

## Lecturers

• AR

### Professor Alastair Rucklidge

University
University of Leeds
Role
Main contact
• RH

### Dr Rainer Hollerbach

University
University of Leeds
• DH

### Professor David Hughes

University
University of Leeds
• CJ

### Professor Chris Jones

University
University of Leeds
• ST

### Professor Steven Tobias

University
University of Leeds

## Bibliography

No bibliography has been specified for this course.

## Assessment

### Attention needed

Assessment information will be available nearer the time.

## Files

Only consortium members have access to these files.