Study information

Waves, Instabilities and Turbulence - 2024 entry

MODULE TITLEWaves, Instabilities and Turbulence CREDIT VALUE15
MODULE CODEMTHM030 MODULE CONVENERProf 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 Guided Independent Study 117 Placement / Study Abroad 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 12 Coursework
Guided Independent Study 80 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 30 Written Exams 70 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 70 2 hours 1-4 & 6 Written/Verbal on request
One problem sheet 10 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 (70%) 1-4 & 6 Referral/deferral period
One Problem Sheet One Problem sheet (10%) 1-4 & 6-7 Referral/deferral period
One Computer Based Exercise One Computer Based Exercise (20%) 1-7 Referral/deferral 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

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 Edition Publisher Year ISBN
Set Lighthill, J. Waves in Fluids Cambridge University Press 2001 ISBN: 0521010454
Set Drazin, P.G. Introduction to Hydrodynamic Stability Cambridge University Press 2002 0521009650
Set Drazin, P.G. and Reid, W.H. Hydrodynamic Stability Cambridge University Press 1981 0521289807
Set Salmon, R. Lectures on Geophysical Fluid Dynamics Oxford University Press 1998 0195108086
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 12th March 2024 LAST REVISION DATE Friday 15th March 2024
KEY WORDS SEARCH Fluid Dynamics

Please note that all modules are subject to change, please get in touch if you have any questions about this module.