Work

Below are a selection of non-journal type, or unpublished, works.  For a full list of journal articles, please see my Publications page or visit my Google Scholar page.

I am the founder and owner of openpipeflow.org, a free and fast simulation code for the study of pipe flow, particularly as a dynamical system.  Non-problem-specific codes for dynamical systems include A Jacobian-free Newton-Krylov (JFNK) code (MATLAB / FORTRAN90) for finding solutions of high-dimensional nonlinear equations – or low-dimensional if you like!

I have also written a double pendulum android app .  With two double-pendula side-by-side, it is specifically designed to show how initially similar states diverge – the chaotic system exhibits sensitivity to initial conditions (SIC). It also shows how the chaotic paths follow particular repeating patterns, periodic orbits.  

The videos below are about the link between chaotic dynamics and periodic orbits – read a description of the mathematics behind the videos.

Double-pendulum app

The double pendulum is a chaotic system.  The app is designed to help see the exponential rate of divergence between almost identical initial conditions

Periodic orbits

Periodic orbits in pipe flow - projection of the dynamics on to three axes: Input energy, Dissipation, Energy of the flow.

Periodic orbits form the 'backbone' of turbulent pipe flow.  Budanur, Short, Farazmand & Cvitanovic  (2017)

Turbulent pipe flow

Blue indicates regions of slow flow, 'streaks', close to the wall.  They are associated with streamwise-aligned vortices, in yellow. 

A periodic orbit in pipe flow

This spatially-localised solution to the equations governing flow in a pipe is periodic in time.  It looks similar to localised 'puffs' of turbulence observed in experiments.

Earth's magnetic field

Radial-component of Earth's magnetic field at the core-mantle boundary:  Blue - intensity of field lines going into the core; Red - field lines coming out.

To be compared with simulation to right/below.

Simulations of the geodynamo

The 'geodynamo' describes the process that generates Earth's magnetic field, which involves flow in the liquid iron outer core.

Matching of flux patches at high-latitudes associated with temperature variations at the base of the mantle.  Gubbins, Willis and Sreenivasan (2007).