Advanced Computational Methods 

for 

Turbulent Incompressible Flow

Course structure: 6 credits (1-2-0-6)

Assignments: 50%

• Based on topics discussed in class, intended to aid understanding.

• Will be primarily numerical and require coding.

Term project: 50%

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)

No tests / exams

Office Hours:

Wednesdays: 10am -- 11am

Thursdays:  9:45am -- 10:45am

Location: #308, Transit Building

Lecture 1: 

• 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:

1. Reynolds transport theorem

2. Buckingham's Pi theorem

3. Hans G. Hornung, Dimensional Analysis, Chapters 1-3

Lecture 2:

• The compressible equations in 3D: vector form, Einstein tensor notation; criteria for incompressibility

• The Boussinesq approximation following Spiegel and Veronis, and its validity

Reading:

1. Spiegel and Veronis, Astrophys. J. (1959), On the Boussinesq Approximation for a Compressible Fluid

2. D. J. Tritton, Physical Fluid Dynamics, Chapter 14 Appendix: The Boussinesq Equation

3. Landau and Lifshitz, Fluid Mechanics, Chapter 1, #10

Lecture 3:

• The Boussinesq energy equations for lapsing and non-lapsing ambients

• Nondimensionalisation and nondimensional parameters

• Example: Rayleigh-Benard convection

Reading:

1. Leo Kadanoff, Turbulent Heat Flow: Structures and Scaling, Physics Today (2001)

2. J. B. S. Haldane, On Being the Right Size, in Possible Worlds (1927)

3. Hans G. Hornung, Dimensional Analysis, Chapters 1-3 

Lecture 4:

• Rotating Boussinesq-Navier-Stokes equations

• Geostrophic balance and the Rossby number

• Taylor Proudman Theorem

• Shallow water approximation and the shallow water equations

Reading:

1. G. Batchelor, Fluid Dynamics, Chapter 3.2

2. Geoffrey Vallis, Atmospheric and Oceanic Fluid Dynamics, Chapter 3

Video:

Record Player Fluid Dynamics: A Taylor Column Experiment

Lecture 5:

1. 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

2. Numerical solutions of PDEs: The diffusion equation

   • Relation to the random walk (sketch)

   • Discretisation in space and time

Assignment: Download here

Lecture 6: (24 / 1 / 2023)

1. Well-posed and ill-posed differential equations

2. Consistency, stability and convergence of numerical schemes; the Lax equivalence theorem

3. von Neumann error analysis for the first-order timestepping scheme for the diffusion equation (Lec 5, Assignment 1)

4. Modus Operandi for a PDE: discretisation in space + higher-order timestepping scheme

5. 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)

Lecture 7: (27 / 1 / 2023)

1. Analysis of linear PDEs: the residual term in the error equation

2. Faliure of the Lax equivalence theorem in Nonliner PDEs

3. Equivalent differential equation for a scheme

4. Tutorial on Assignment 1

Assignment 2: Download here

Lecture 8: (1 / 2 / 2023)

1. Semi-discrete and fully discrete numerical schemes: general forms

2. Operator calculus and its use in deriving the order of numerical schemes

3. The polynomial a(z) for a given numerical scheme

4. Example: diffusion equation with forward-time and central-space discretisation

Lecture 9: (3 / 2 / 2023)

1. Stability of semi-discrete and fully discrete numerical schemes using a(z)

2. Example: Crank Nicolson method for the diffusion equation

Lecture 10: (7 / 2 / 2023)

1. Discrete and semi-discrete Fourier transforms

2. Basis sets and function interpolation: spectral accuracy

3. Energy in the wave equation:

   • integration by the Euler method

   • integration by the first-order upwind method

Reading: Arieh Iserles (see above)

The Cooley-Tukey algorithm

Lecture 11: (10 / 2 / 2023)

1. Stiff systems of ODEs,

2. Linear stability and A-stability of numerical ODE schemes,

Reading: Arieh Iserles, Chapter 4.

Assignment 3: download here

Lecture 12: (28 / 2 / 2023)

1. Recap of Lecs 1-11.

2. Chorin's operator splitting for the Navier-Stokes equations

3. Orszag & Patterson's pseudospectral method

References:

• Chorin's paper: here

• Orszag's paper: here

Lecture 13: (3 / 3 / 2023)

Tutorial on FFTs for Assignments 3/4

Lecture 14: (10 / 3 / 2023)

1. The Poisson equation: basic properties, ellipticity

2. The tri-diagonal matrix algorithm (TDMA / Thomas) for the Poisson equation in 1D.

3. Jacobi and Gauss-Siedel methods iterative methods for solution of the Poisson equation; successive over-relaxation

References: 

• Suhas V. Patankar, Chapter 4.2-7 (TDMA)

Lecture 15: (14 / 3 / 2023)

1. The spectral properties of iterative methods: slow convergence at small wavelengths

2. The Multigrid algorithm: O(NlogN)  and O(N) 

References: 

• Gilbert Strang lecture notes on the multigrid algorithm

Assignment 5/6: The Poisson Equation

Lecture 16: (17  / 3 / 2023)

1. Chaos and the logistic map

2. Transition to turbulence, Landau's route through successive bifurcations

References:

• P. A. Davidson, Turbulence, Chapter 3

Lecture 17: (21 / 3 / 2023)

1. Kolmogorov's theory of fully developed turbulence

2. Different ways of averaging turbulent flows: ensemble averages, time averages, volume averages

References:

• P. A. Davidson, Turbulence, Chapter 3

Lecture 18: (24 / 3 / 2023)

1. Turbulence quantities: velocity autocorrelation, second order structure-functions, energy spectra

References:

• P. A. Davidson, Turbulence, Chapter 3

Lecture 19: (5 / 4 / 2023)

1. Dealiasing of signals in (pseudo)spectral analysis

2. Parallelisation: The need for parallelisation of code

3. Shared and distributed memory paradigms for parallel computing

4. 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