Unit information: Advanced Fluid Dynamics in 2011/12

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Unit name Advanced Fluid Dynamics
Unit code MATHM0600
Credit points 20
Level of study M/7
Teaching block(s) Teaching Block 1 (weeks 1 - 12)
Unit director Professor. Hogg
Open unit status Not open
Pre-requisites

MATH20100, MATH20900, MATH33200
Co-requisites

None
School/department School of Mathematics
Faculty Faculty of Science

Description including Unit Aims

The behaviour of ordinary fluids like oil, water, or air can be understood on the basis of a single equation, due to Navier and Stokes. The description of fluid motion thus amounts to finding solutions to the Navier-Stokes equation, a mathematical problem of almost infinite variability and often staggering complexity. Fluid mechanics has broken up into a great number of subfields, this course will try to give a more unified view by emphasizing mathematical structures that reappear in different guises in almost all those sub-specialties.

Aims

Understanding the principles governing fluid flow snd the mathematical models used to investigate them.

Syllabus

  1. The governing equation for viscous fluid flow: Conservation of mass; stress and rate of strain tensors; Navier-Stokes equation; dissipation; boundary conditions; vorticity.
  2. Simple fluid flows: Solution to the Navier-Stokes equations for simple geometries; steady flows along pipes and between parallel plates; oscillating flows; impulsively started flows; flows with circular streamlines.
  3. Dynamical similarity and the Reynolds Number: Dimensionless governing equations and the definition of the Reynolds number; interpretation; simple examples.
  4. Flows with negligible inertia: Stokes' equations; corner flows; settling particles; lubrication flows; spreading droplets.
  5. Flows with large Reynolds Numbers: Singular perturbations of the Navier-Stokes equations; boundary layer flows; similarity solutions; production of vorticity; separation; wakes.
  6. Instabilities: Linear instability theory of shear flows and vortex sheets.
  7. Turbulence: Isotropic turbulence; Kolmogorov spectrum and energy cascade; Reynolds averaged equations and closures.

Relation to Other Units

This unit is a continuation of the Level 3 Fluid Dynamics unit and an investiagtion of more advanced topics. This unit is self-contained and it is not necessary to have previously attended Level 3 Fluid Dynamics. However familarity with the key themes and ideas of Level 3 Fluid Dynamics would be advantageous.

Intended Learning Outcomes

After taking this unit, students should:

  • know the basic equations and the underlying concepts
  • realise the importance of the Reynolds number and other non-dimensional parameters
  • know how to set up the appropriate mathematical equations for a given flow problem
  • appreciate the general concepts of stability and scaling

Transferable Skills:

Ability to transfer physical questions into well-defined mathematical problems. Understanding the critical parameters of a problem and developing intuition for the behaviour of a system as a function of these parameters.

Teaching Information

A unit of 30 lectures spread over 12 weeks. Regular homework assignments are set.

Assessment Information

The assessment mark for Advanced Fluid Dynamics is calculated from a 2½-hour written examination in April consisting of FIVE questions. A candidate's best FOUR answers will be used for assessment.

Calculators are NOT permitted, but candidates may bring ALL their written lecture notes to the examination.

Reading and References

  1. L.D. Landau & E.M. Lifshitz Fluid Mechanics, Pergamon 1959
  2. D.J. Acheson, Elementary Fluid Dynamics, Oxford.
  3. T.E. Faber, Fluid dynamics for physicists, Cambridge, 1995
  4. G.K. Batchelor, An introduction to fluid mechanics, Cambridge 1965

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