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## ECMM730 - Waves, Instabilities and Turbulence (2018)

MODULE TITLE | Waves, Instabilities and Turbulence | CREDIT VALUE | 15 |
---|---|---|---|

MODULE CODE | ECMM730 | MODULE CONVENER | Prof Beth Wingate (Coordinator) |

DURATION: TERM | 1 | 2 | 3 |
---|---|---|---|

DURATION: WEEKS | 11 |

Number of Students Taking Module (anticipated) | 20 |
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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 modules: ECM3707 equivalent. Also, students should have a sound knowledge of Vector Calculus, as provided for example by ECM2706/MTH2004. Students will also need to have a sound knowledge of either Matlab or Python programming and should have taken MTH2005 or MTH3039 or MTHM703 or ECM2704 or ECM3739 or ECMM703 or equivalent.**

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.

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.

- 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.

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

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 |

Coursework | 20 | Written Exams | 80 | Practical Exams | 0 |
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Form of Assessment | % of Credit | Size of Assessment (e.g. duration/length) | ILOs Assessed | Feedback Method |
---|---|---|---|---|

Written exam, closed book | 80 | 2 hours - Summer Exam Period | 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 |

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 |

Referred and deferred assessment will be by examination. For referrals, only the examination will count, a mark of 50% 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.

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 | Edition | Publisher | Year | ISBN | Search |
---|---|---|---|---|---|---|---|

Set | Salmon, R | Lectures on Geophysical Fluid Dynamics | Oxford University Press | 1998 | 0195108086 | [Library] | |

Set | Drazin, P.G., and Reid, W.H | Hydrodynamic Stability | Cambridge University Press | 1981 | 0521289807 | [Library] | |

Set | Drazin, P.G | Introduction to Hydrodynamic Stability | Cambridge University Press | 2002 | 0521009650 | [Library] | |

Set | Lighthill, J | Waves in Fluids | Cambridge University Press | 2001 | ISBN: 0521010454 | [Library] |

CREDIT VALUE | 15 | ECTS VALUE | 7.5 |
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PRE-REQUISITE MODULES | ECM3707 |
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CO-REQUISITE MODULES |

NQF LEVEL (FHEQ) | 7 | AVAILABLE AS DISTANCE LEARNING | No |
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ORIGIN DATE | Thursday 06 July 2017 | LAST REVISION DATE | Thursday 28 February 2019 |

KEY WORDS SEARCH | Fluid dynamics |
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