MAGIC014: Hydrodynamic Stability Theory

Course details

A core MAGIC course

Semester

Spring 2019
Monday, January 21st to Friday, March 29th

Hours

Live lecture hours
10
Recorded lecture hours
0
Total advised study hours
40

Timetable

Tuesdays
13:05 - 13:55

Description

This is offered as a core course for Applied.

Prerequisites

It will be assumed that students are familiar with the Navier-Stokes equations. Any previous experience of perturbation methods would be an advantage, but is not essential, as the main ideas will be introduced as needed.

Syllabus

0. Some pictures of unstable flows (motivation)
1. Introduction
The idea of instability
(Approximately) parallel shear flows - e.g. pipe flow, boundary layers, channel flows, jets, wakes, mixing layers
Shear layer stability equations - reduction to linear ODEs
2. Inviscid stability theory
Stability theorems - inflexion points, etc.
Piecewise-linear profiles
Critical points - Tollmien's solutions
Emergence of layered structures in the long-wave limit
Matched asymptotic expansions
Second order long-wave theory capturing critical layers
3. Viscous stability theory
Thin viscous layers within inviscid flow
Destabilizing effects of viscosity
An interpretation of the viscous instability mechanism
4. Weakly nonlinear theory
Solvability conditions - when do solutions to forced equations exist?
Higher order expansions in the amplitude parameter.
Multiple-scales theory.
Amplitude equations - supercritical/subcritical bifurcations.
Wave interactions - resonant and nonresonant cases.
5. Absolute and convective instabilities
Upstream and downstream propagation.
Initial value problems.
Saddle point methods.

Lecturer

  • JH

    Prof Jonathan Healey

    University
    Keele University

Bibliography

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Assessment

Description

The assessment for this course will be via a single take-home paper made available at the end of the module, with 2 weeks to complete and submit solutions online. Questions may be of different lengths. The number marks for each question will be indicated. The pass mark will be 50%.

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Files

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Lectures

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