Study information

# Waves, Instabilities and Turbulence - 2023 entry

MODULE TITLE CREDIT VALUE Waves, Instabilities and Turbulence 15 MTHM030 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 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 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 Written Exams Practical Exams 40 60 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
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

Web based and Electronic Resources:

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