# Module descriptors - Advanced Computational Methods

The module descriptors for this programme can be found below.

Modules shown are for the current academic year and are subject to change depending on your year of entry.

Please note that the curriculum of this programme is currently being reviewed as part of a College-wide process to introduce a standardised modular structure. As a result, the content and assessment structures of this course may change for your year of entry. We therefore recommend that you check this course page before finalising your application and after submitting it as we will aim to update this page as soon as any changes are ratified by the College.

Find out more about the limited circumstances in which we may need to make changes to or in relation to our courses, the type of changes we may make and how we will tell you about changes we have made.

## 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.

### Assessments

This module presents opportunities for both formative and summative assessment.
You will be formatively assessed through progress tests and tutorial sessions.
You will have additional opportunities to self-assess your learning via tutorial problem sheets.
You will be summatively assessed by a written closed-book examination at the end of the module.

 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.

### Core

• #### Hydrodynamic stability

Drazin, P. G.

2nd ed., Cambridge : Cambridge University Press

• #### Hydrodynamic Stability / [electronic resource] P. G. Drazin, W. H. Reid.

Drazin, P. G.,

2nd ed., Cambridge University Press,

• #### Introduction to hydrodynamic stability /

Drazin, P. G.

Cambridge University Press

• #### Hydrodynamic Instabilities / [electronic resource]

Charru, Francois,

Cambridge University Press

• #### Hydrodynamic instabilities

Charru, FranoĢis.

Cambridge University Press

• #### Boundary-layer theory /

Schlichting, Hermann,

9th edition., Springer,

• #### Stability and transition in shear flows

Schmid, Peter J., author.

Softcover reprint of the original First edition 2001., New York : Springer

• #### Hydrodynamics and nonlinear instabilities

Cambridge University Press

• #### Hydrodynamics and nonlinear instabilities

Cambridge University Press