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MODULE TITLE	Waves, Instabilities and Turbulence	CREDIT VALUE	15
MODULE CODE	MTHM030	MODULE CONVENER	Prof Andrew Hillier (Coordinator)

DURATION: TERM	1	2	3
DURATION: WEEKS	0	11	0

Number of Students Taking Module (anticipated)	13

DESCRIPTION - summary of the module content

Waves and turbulence are both ubiquitous phenomena in fluid flows, and both are often associated with the instability of simpler flows. They arise in many important theoretical and practical applications, ranging from engineering to meteorology and astrophysics. This module will extend your ability to formulate fluid flow problems in terms of partial differential equations, and develop a range of mathematical techniques for analysing the fluid behaviour. The module will emphasise physical interpretation, as well as mathematical technique. Computer-based practical exercises will help you to visualise fluid phenomena and develop your understanding.

Pre-requisite modules: MTH1003 Mathematical Modelling and MTH3007 Fluid Dynamics, or equivalent.

AIMS - intentions of the module

The aim of this module is to develop a range of applied mathematics techniques for analysing the behaviour of waves, instabilities, and turbulence in fluid flows, and to apply the techniques to understand a variety of physical processes in fluids. The material will build on the third-year module MTH3007 on viscous fluids. You will develop an appreciation of the richness and complexity of fluid phenomena, as well as the wide range of situations in which these phenomena arise.

INTENDED LEARNING OUTCOMES (ILOs) (see assessment section below for how ILOs will be assessed)

On successful completion of this module, you should be able to:

Module Specific Skills and Knowledge:

1 Explain key mathematical techniques that can be used to analyse waves, instabilities, and turbulence in fluid flows; 2 Apply those techniques to specific problems and give physical interpretations of the results;

Discipline Specific Skills and Knowledge:

3 Formulate physical problems mathematically;

4 Interpret mathematical solutions in terms of physical processes;

5 Use numerical simulations to investigate physical processes;

Personal and Key Transferable / Employment Skills and Knowledge:

6 Critically analyse practical problems and questions and express them in mathematical terms;

7 Manage time effectively and prioritise activities.

SYLLABUS PLAN - summary of the structure and academic content of the module

- Waves: examples; different physical mechanisms for waves. Linearisation; dispersion relation. Phase and group velocity. WKB theory; ray tracing; reflection, refraction. Momentum and energy transport by waves. Critical layers. Wave, mean-flow interaction;

-Instabilities: normal mode analysis. Examples, including Kelvin-Helmholtz instability, centrifugal instability, parallel shear flows, Rayleigh-Benard convection. Rayleigh's inflection point criterion, Fjortoft's theorem, Howard's semi-circle theorem;

- Turbulence: Reynolds number, inertial range, dimensional arguments, scaling arguments, and eddy viscosity. Three-dimensional turbulence: conserved quantities, energy cascade; -5/3 spectrum. Two-dimensional turbulence: energy and enstrophy cascades, -3 spectrum. Intermittency and other corrections to the simplest theories.

LEARNING AND TEACHING

LEARNING ACTIVITIES AND TEACHING METHODS (given in hours of study time)

Scheduled Learning & Teaching Activities	33.00	Guided Independent Study	117.00	Placement / Study Abroad	0.00

DETAILS OF LEARNING ACTIVITIES AND TEACHING METHODS

Category	Hours of study time	Description
Scheduled Learning and Teaching Activities	28	Lectures
Scheduled Learning and Teaching Activities	5	Examples classes
Guided Independent Study	25	Computer-based practical experiments
Guided Independent Study	25	Coursework
Guided Independent Study	67	Reading, revision, preparation

ASSESSMENT

FORMATIVE ASSESSMENT - for feedback and development purposes; does not count towards module grade

Form of Assessment	Size of Assessment (e.g. duration/length)	ILOs Assessed	Feedback Method
Three Problem Sheets	Approx. 10 hours each	1-4 & 6-7	Examples classes, comments on each script, comments uploaded to ELE, model solutions uploaded to ELE

SUMMATIVE ASSESSMENT (% of credit)

Coursework	40	Written Exams	60	Practical Exams	0

DETAILS OF SUMMATIVE ASSESSMENT

Form of Assessment	% of Credit	Size of Assessment (e.g. duration/length)	ILOs Assessed	Feedback Method
Written Exam, Closed Book	60	2 hours (Summer)	1-4 & 6	Written/Verbal on request
One problem sheet	20	Approx. 25 hours	1-4 & 6-7	Examples classes, comments on each script, comments uploaded to ELE, model solutions uploaded to ELE
One assessed computer based exercise	20	Approx. 10 hours	1-7	Examples classes, comments on each script

DETAILS OF RE-ASSESSMENT (where required by referral or deferral)

Original Form of Assessment	Form of Re-assessment	ILOs Re-assessed	Time Scale for Re-assessment
Written Exam*	Written Exam (60%)	1-4 & 6	August Ref/Def Period
One Problem Sheet*	One Problem sheet (20%)	1-4 & 6-7	August Ref/Def Period
One Computer Based Exercise *	One Computer Based Exercise (20%)	1-7	August Ref/Def Period

RE-ASSESSMENT NOTES

Deferrals: Reassessment will be by coursework and/or written exam in the deferred element only. For deferred candidates, the module mark will be uncapped.

Referrals: Reassessment will be by a single written exam worth 100% of the module only. As it is a referral, the mark will be capped at 50%.

RESOURCES

INDICATIVE LEARNING RESOURCES - The following list is offered as an indication of the type & level of
information that you are expected to consult. Further guidance will be provided by the Module Convener

ELE: http://vle.exeter.ac.uk

Web based and Electronic Resources:

Computer codes will be provided via ELE as the basis for the practical exercises.

Reading list for this module:

Type	Author	Title	Publisher	Year	ISBN	Search
Set	Lighthill, J.	Waves in Fluids	Cambridge University Press	2001	ISBN: 0521010454	[Library]
Set	Drazin, P.G.	Introduction to Hydrodynamic Stability	Cambridge University Press	2002	0521009650	[Library]
Set	Drazin, P.G. and Reid, W.H.	Hydrodynamic Stability	Cambridge University Press	1981	0521289807	[Library]
Set	Salmon, R.	Lectures on Geophysical Fluid Dynamics	Oxford University Press	1998	0195108086	[Library]

CREDIT VALUE	15	ECTS VALUE	7.5

PRE-REQUISITE MODULES	MTH3007, MTH1003
CO-REQUISITE MODULES

NQF LEVEL (FHEQ)	7	AVAILABLE AS DISTANCE LEARNING	No
ORIGIN DATE	Tuesday 10 July 2018	LAST REVISION DATE	Thursday 26 January 2023

KEY WORDS SEARCH	Fluid Dynamics

MTHM030 - Waves, Instabilities and Turbulence (2023)