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

Timetable

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

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Lectures

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