# Mathematics

## ECMM730 - Waves, Instabilities and Turbulence (2015)

MODULE TITLE CREDIT VALUE Waves, Instabilities and Turbulence 15 ECMM730 Prof Beth Wingate (Coordinator)
DURATION: TERM 1 2 3
DURATION: WEEKS 11
 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.

Prerequisite module: ECM3707 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 ECM3707 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.
Personal and Key Transferable / Employment Skills and Knowledge:
5 critically analyse practical problems and questions and express them in mathematical terms;
6 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 Guided Independent Study Placement / Study Abroad 33 117 0
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 20 Computer based practical experiments Guided independent study 40 Coursework Guided independent study 57 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
Two computer based exercises Approx. 10 hours each 1-6 Examples classes, written comments on scripts

SUMMATIVE ASSESSMENT (% of credit)
 Coursework Written Exams Practical Exams 20 80 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 80 2 hours 1-5 Oral, on request
Three problem sheets 15 Approx. 10 hours each 1-6 Examples classes, comments on each script, comments uploaded to ELE, model solutions uploaded to ELE.
One assessed computer based exercise 5 Approx. 10 hours 1-6 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
As above Written exam 1-6 August Ref/Def period

RE-ASSESSMENT NOTES

Referred and deferred assessment will be by examination. For referrals, only the examination will count, a mark of 40% being awarded if the examination is passed. For deferrals, candidates will be awarded the higher of the deferred examination mark or the deferred examination mark combined with the original coursework mark.

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.