## MAS411 Topics in Advanced Fluid Mechanics

 Semester 1, 2019/20 20 Credits Lecturer: Prof Koji Ohkitani Home page 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
• 1D model equations
• Vortex sheet problem
• Vortex patch problem
• 3D Euler equations
• 3D Navier-Stokes equations

## 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 in time, 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 and analysis of blowup of the CLM equation.
They will learn alternative (more general) ways of writing down 3D Euler equations. This impulse formalism is 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 patch 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%]. Format: 4 questions from 5. 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.
• Burgers equations We will describe two proofs of Cole-Hopf linearisation for the Burgers equation and its traveling-wave solution.
• CLM equation We study exact solution of this baby vorticity equation and see how blowup shows up.
• 3D Euler equations We describe impulse formalism by which we express known first integrals (Kelvin circulation, Cauchy invariant etc.) in an unified manner.
• 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 variant.
• Vortex patch problem We study how to describe the Kirchhoff elliptic vortex respond to the external linear strain.
• Vortex sheet problem We reproduce the well-known dispersion relationship (growth rates) of Kelvin-Helmholtz problem using the Birkhoff-Rott equation.