Flow Instability and Transition

Module aims

Study of laminar-turbulent transition represents a major branch of modern Fluid Dynamics. The transition takes place due to the fact that at a sufficiently large value of the Reynolds number the flow becomes unstable, in the sense that small perturbations, introduced in the laminar flow due to, say, acoustic noise or free-stream turbulence, are no longer damped by the viscous effects. Instead they start to grow in time, leading to the transition to turbulent state of the flow.  In this lecture course modern mathematical techniques that are used to predict the conditions when the laminar-turbulent transition takes place will be discussed. 

Learning outcomes

Knowledge and understanding On successfully completing this course unit, you will be able to: •    Predict the stability/instability of various flows •    Analyse flows through data-driven flow analysis techniques such as POD, SPOD, and DMD Skills and other attributes On successfully completing this course unit, you will be able to: Intellectual skills •    Apply the knowledge of the laminar-turbulent transition to various practical flows, including aerodynamic flows •    Predict which modes of instability can be expected in different mechanical systems Practical skills •    Predict stability of various mechanical systems (including fluid flows) that are described by partial linear and non-linear differential equations •    Apply data-driven techniques such as POD, SPOD, DMD to analyse the given flow Transferable skills Will be able to use the knowledge in Aerospace Industry, where the delay of the transition is the key element in improving the passenger aircraft design AHEP Learning Outcomes: SM7M, SM8M, EA5m, EA6m, EA7m, D9M, EL11M, P9m 

Module syllabus

1. Introduction (1hr)

2. Basic dynamical systems theory - Jacobian linearisation, phase portrait, bifurcation theory (2hrs)

3. Linear stability for inviscid parallel shear flows -- normal-mode solution, Rayleigh equation, Inflection point theorem, Broken profile analysis (1.5hrs)

5. Linear stability of viscous parallel shear flows – Orr-Sommerfeld-Squire equation, Squire’s theorem, eigenspectra, neutral stability curve, spatial stability theory (1.5hrs)

6. Non-modal theory – Reynolds-Orr equation, the solution to initial value problem, non-orthogonality of eigenvectors, non-normal operators, Algebraic instability, Introduction of optimisation and singular value decomposition, optimal transient growth (time), lift-up/Orr mechanism, resolvent analysis (frequency domain) (4hrs)

7. Spatio-temporal development of instability – Ginzburg-Landau equation, absolute/convective instability, parallel flow (mixing layer), global instability (relating local absolute instability), global transient growth and resolvent (relating to convective instability) (4hrs)

8. Introduction to nonlinear stability theory – weakly nonlinear stability (1hr)

9. Data-driven techniques – POD, SPOD, DMD, Koopman mode analysis, operator-driven vs data-driven stability analysis (3hrs)

10. Applications (2hrs): 1) transition (boundary layer transition and 2D and 3D wake transition)and 2) coherent structures in turbulent flows.

Teaching methods

The module will be delivered primarily through large-class lectures introducing the key concepts and methods, supported by a variety of delivery methods combining the traditional and the technological.  The content is presented via a combination of slides, whiteboard and visualizer.Learning will be reinforced through tutorial question sheets, featuring analytical tasks representative of those carried out by practising engineers. 

Assessment type Assessment description Weighting Pass mark
Examination Closed-book written examination 100% 50%

You will receive feedback on examinations in the form of an examination feedback report on the performance of the entire cohort.

You will receive feedback on your performance whilst undertaking tutorial exercises, during which you will also receive instruction on the correct solution to tutorial problems.

Further individual feedback will be available to you on request via this module’s online feedback forum, through staff office hours and discussions with tutors.


2 hour written examination in the Summer term (100%)

Module leaders

Dr Yongyun Hwang