MAS411 Topics in Advanced Fluid Mechanics

 Semester 1, 2021/22 20 Credits Lecturer: Dr Yi Li Timetable Reading List Aims Outcomes Assessment Full Syllabus

This module aims to describe advanced mathematical handling of fluid equations in an easily accessible fashion. A number of topics are treated in connection with the mathematical modelling of formation of the (near-)singular structures with concentrated vorticity in inviscid flows. After discussing prototype problems in one and two dimensions, the three-dimensional flows in terms of vortex dynamics are described. Mathematical tools are explained during the unit in a self-contained manner. Candidates are directed to read key original papers on some topics to deepen their understanding.

There are no prerequisites for this module.
No other modules have this module as a prerequisite.

Outline syllabus

• Fluid dynamical equations revisited
• Vortex sheet problem
• Linear instability
• 3D Euler equations
• 1D model equations
• 3D Navier-Stokes equations

Office hours

2pm-3pm Tuesdays.

Aims

This unit aims to familiarise candidates with advanced mathematical techniques used in fluid mechanics, in particular in vortex dynamics, by working out prototype problems.
The mathematical tools we will learn as we go include: the heat kernel for the diffusion equation, Hilbert transforms, methods of complex functions, Fourier series, Taylor series, transformations by stretched-coordinates and method of linear stability analysis.

Learning outcomes

They will learn how to solve 1D model equations analytically. This includes linearisation and traveling-wave solution of the Burgers equation.
They will learn alternative (more general) ways of writing down 3D Euler equations, including the impulse formalism suited to the variational formulation.
They will learn some exact solutions to 3D Navier-Stokes equations, which mimic characteristic structure observed in turbulence.
They will learn formulation of vortex tube problem under the influence of an external straining field.
They will learn the Kelvin-Helmholtz instability on the basis of the Birkhoff-Rott equation.

20 lectures, no tutorials

Assessment

One formal 2.5-hour written examination [80%]. Students will also be required to complete derivations from approx. 5 papers on a reading list [20%].

Full syllabus

• Fluid dynamical equations revisited We give an introduction and motivations.
• 3D Euler equations: We describe impulse formalism and several first integrals (Kelvin circulation, Cauchy invariant etc.).
• 3D Navier-Stokes equations: We describe how the Burgers vortex emerges in the long-time evolution of the Navier-Stokes equations under a special configuration. We also study its variants.
• Vortex sheet problem: We reproduce the well-known dispersion relationship (growth rates) of Kelvin-Helmholtz problem using the Birkhoff-Rott equation.
• Instability: We introduce linear instability for canonical flows.
• Burgers equations: We describe the Cole-Hopf transformation for the Burgers equation and its traveling-wave solution. We discuss its representation in the Fourier space.