Hydrodynamic Stability Theory

Co-ordinators:Dr R. Hewitt, Mathematics Department, OWENS, (Maths Blg 18.07 x 55918)

email: hewitt@ma.man.ac.uk


 
Assessment
Credits
Contact hours
% Exam
% Coursework
Level
10
24h lectures, 8h tutorials
80
20
M

General Description

Many fluid flows of practical importance are unstable in the sense that small disturbances superimposed on the basic mean flow can amplify and significantly distort the basic state.In this course we investigate the hydrodynamic stability of a variety of fluid ranging from thin layers heated from below, to the flow between rotating cylinders, and shear and boundary layer flows.

Aims

The aim of this course is to look at various topics in hydrodynamic stability theory and introduce students to some of the classical as well as more modern ideas and techniques. 

Objectives

By the end of this course students should be able to (i) derive linearised stability equations for a given basic state, (ii) perform a normal-mode analysis leading to an eigenvalue problems, (iii) use the ideas of weakly non-linear stability theory in simple flows, (iv) appreciate the different physical mechanisms leading to instability in fluid flows.

Course Content

(1)Basic concepts of stability theory, stability, instability, normal modes,marginal stability, neutral curves, temporal/spatial growth. 

(2)Rayleigh-Benard instability. Navier-Stokes equations and formulation of the linearised stability problem. Boundary conditions for free-free, rigid-free and rigid-rigid problems. Exact solution of the free-free boundary problem, and discussion of the marginal stability properties. Cell patterns. Experimental observations. 

(3)Shear Flow boundary layer instability. Stability of parallel flows. Inviscid stability ttheory and properties of Rayleigh equation. Inflexion point criteria, Fjortoft's theorem. Howard's semi-circle theorem. Viscous/Tollmien-Schlichting instability. Orr-Sommerfeld equation. Energy methods. Parallel flow approximation and application to boundary layers. Gaster Transformation. Discussion of enmethod, Brief discussion of non-parallel stability theories, and High Reynolds number asymptotic approach. Absolute instabilities. 

(4)Introduction to nonlinear stability theory.The Stuart-Landau equation. Local bifurcation theory:Saddle-node, Pitchfork, Hopf and transcritical bifurcations.Structural (topological) stability.The Ginsberg-Landau equation and modulation.

(5)Benjamin–Feir instability.

(6)Time – dependent flows – Mathieu’s equation and the parametric pendulum.

Associated Information

 

This course will be assessed via a single 2-hour closed bookexamination paper (80%). Coursework counting 20% will be used in the assessment of this module.The examination will take place immediately after the Easter break.

Suggested Books:

`Hydrodynamic Stability', by P.G. Drazin & W. Reid , C.U.P. 

`Laminar Boundary Layers', (ed. L. Rosenhead), Dover paperback.Chap IX by J.T. Stuart. 

`Hydrodynamic and Hydromagnetic Stability', S. Chandrasekhar, Dover paperback.