Based on topics discussed in class, intended to aid understanding.
Will be primarily numerical and require coding.
Reproduce a journal article, chosen in consultation with your advisor and the course instructor, relevant to your research.
Will require implementing a numerical solver for the Navier-Stokes equations
Will require the use of plotting libraries (e.g. matplotlib)
Wednesdays: 10am -- 11am
Thursdays: 9:45am -- 10:45am
Location: #308, Transit Building
Introduction to the instructor, teaching assistants and students, course structure, course aims
The equations of the dynamics of a compressible fluid: mass, momentum, energy
Equations of sound
Slides: here.
Reading:
Reynolds transport theorem
Buckingham's Pi theorem
Hans G. Hornung, Dimensional Analysis, Chapters 1-3
The compressible equations in 3D: vector form, Einstein tensor notation; criteria for incompressibility
The Boussinesq approximation following Spiegel and Veronis, and its validity
Reading:
Spiegel and Veronis, Astrophys. J. (1959), On the Boussinesq Approximation for a Compressible Fluid
D. J. Tritton, Physical Fluid Dynamics, Chapter 14 Appendix: The Boussinesq Equation
Landau and Lifshitz, Fluid Mechanics, Chapter 1, #10
The Boussinesq energy equations for lapsing and non-lapsing ambients
Nondimensionalisation and nondimensional parameters
Example: Rayleigh-Benard convection
Reading:
Leo Kadanoff, Turbulent Heat Flow: Structures and Scaling, Physics Today (2001)
J. B. S. Haldane, On Being the Right Size, in Possible Worlds (1927)
Hans G. Hornung, Dimensional Analysis, Chapters 1-3
Rotating Boussinesq-Navier-Stokes equations
Geostrophic balance and the Rossby number
Taylor Proudman Theorem
Shallow water approximation and the shallow water equations
Reading:
G. Batchelor, Fluid Dynamics, Chapter 3.2
Geoffrey Vallis, Atmospheric and Oceanic Fluid Dynamics, Chapter 3
Video:
Record Player Fluid Dynamics: A Taylor Column Experiment
Shallow water equations (continued): (see Vallis above)
Stretching and Relative height conservation
Potential vorticity conservation in the shallow water equations
The analogy between shallow water and compressible equations
Numerical solutions of PDEs: The diffusion equation
Relation to the random walk (sketch)
Discretisation in space and time
Assignment: Download here
Well-posed and ill-posed differential equations
Consistency, stability and convergence of numerical schemes; the Lax equivalence theorem
von Neumann error analysis for the first-order timestepping scheme for the diffusion equation (Lec 5, Assignment 1)
Modus Operandi for a PDE: discretisation in space + higher-order timestepping scheme
Implicit and explicit schemes; Crank--Nicolson method for the diffusion equation
Reading:
Arieh Iserles, Chapter 13 "The Diffusion Equation" (Note: rigorously mathematical; will take some work to read / understand; I will try to find a more engineering-style reference)
Analysis of linear PDEs: the residual term in the error equation
Faliure of the Lax equivalence theorem in Nonliner PDEs
Equivalent differential equation for a scheme
Tutorial on Assignment 1
Assignment 2: Download here
Semi-discrete and fully discrete numerical schemes: general forms
Operator calculus and its use in deriving the order of numerical schemes
The polynomial a(z) for a given numerical scheme
Example: diffusion equation with forward-time and central-space discretisation
Stability of semi-discrete and fully discrete numerical schemes using a(z)
Example: Crank Nicolson method for the diffusion equation
Discrete and semi-discrete Fourier transforms
Basis sets and function interpolation: spectral accuracy
Energy in the wave equation:
integration by the Euler method
integration by the first-order upwind method
Reading: Arieh Iserles (see above)
Stiff systems of ODEs,
Linear stability and A-stability of numerical ODE schemes,
Reading: Arieh Iserles, Chapter 4.
Assignment 3: download here
Recap of Lecs 1-11.
Chorin's operator splitting for the Navier-Stokes equations
Orszag & Patterson's pseudospectral method
References:
Tutorial on FFTs for Assignments 3/4
The Poisson equation: basic properties, ellipticity
The tri-diagonal matrix algorithm (TDMA / Thomas) for the Poisson equation in 1D.
Jacobi and Gauss-Siedel methods iterative methods for solution of the Poisson equation; successive over-relaxation
References:
The spectral properties of iterative methods: slow convergence at small wavelengths
The Multigrid algorithm: O(NlogN) and O(N)
References:
Gilbert Strang lecture notes on the multigrid algorithm
Assignment 5/6: The Poisson Equation
Chaos and the logistic map
Transition to turbulence, Landau's route through successive bifurcations
References:
P. A. Davidson, Turbulence, Chapter 3
Kolmogorov's theory of fully developed turbulence
Different ways of averaging turbulent flows: ensemble averages, time averages, volume averages
References:
P. A. Davidson, Turbulence, Chapter 3
Turbulence quantities: velocity autocorrelation, second order structure-functions, energy spectra
References:
P. A. Davidson, Turbulence, Chapter 3
Dealiasing of signals in (pseudo)spectral analysis
Parallelisation: The need for parallelisation of code
Shared and distributed memory paradigms for parallel computing
Message-passing for distributed memory: send/recv, blocking/non-blocking commands
References:
Good reference for MPI with examples: https://rookiehpc.org/mpi/index.html
Some exercises: https://www.dartmouth.edu/~rc/classes/intro_mpi/print_pages.shtml