G’day!

I’m Eric, a final year PhD student in Applied Mathematics at the University of Sydney. My research with Dr Geoffrey Vasil asks a simple question: What can happen when fluids and solids interact? The answers to this deceptively straightforward question have wide-ranging implications for commercial shipping, the global climate, and perhaps even life beyond Earth. My own work combines mathematical analysis, new computational methods, and laboratory experiments in collaboration with mathematicians, scientists, and engineers around the world. I’ve been extremely lucky to explore such fascinating fields and to work with many wonderful people.

Fluid-solid interactions combine the all complexities of fluid and solid dynamics, posing many challenges. Making progress requires new computational methods that can simultaneously capture the different dynamics of fluids and solids. The core of my work is built on the innovative framework Dedalus [1]. Pioneered by my supervisor Dr Geoffrey Vasil and his collaborators, Dedalus is a powerful code that solves user-specified partial differential equations (PDEs) with efficient spectral methods. Dedalus applies best in class numerical techniques in a unified way, allowing users to rapidly develop codes to investigate scientific problems. Here’s an example movie, showing how Dedalus can simulate fluid flow past an object. It’s simple to try adding things like these blue tracers (or indeed, the ellipse!) to help visualise the beautiful flow fields and eddies of the von-Karman vortex street. To anyone interested in simulating PDEs, I can’t recommend Dedalus enough!

Fluid-solid interactions are often seen as incompatible with spectral methods like Dedalus though. My research counters this view through mathematical and computational advances in “diffuse-domain” techniques [2,3]. (The video above shows the improved volume-penalty method for fluid-solid interactions [2].) Diffuse-domain techniques are a simple and versatile way to model objects in fluid flows. They replace complicated boundary conditions with smoothed equation terms. This allows numerical solvers to simulate complex fluid-solid interactions with straightforward algorithms.

Unfortunately, previous implementations were often inefficient. To understand and improve their accuracy, my work uses multiple-scales matched-asymptotic analysis. Using the elegant differential geometry of the signed-distance function (illustrated in the video), we can derive simple corrections that cancel the leading order errors in arbitrary geometries [2,3]. This video illustrates the construction of the signed distance function—finding the set of points at a given distance from an initial surface (e.g. the green curve). The way the signed distance function encodes the shape of the surface is key to the asymptotic analysis.

The first scientific problem I investigated was the “Dead Water” effect, in which boats experience massive drag in density-stratified waters. First documented by explorer Fridtjof Nansen on his famous 1893-96 polar voyage, he noted that dead water “occurs where a surface layer of fresh water rests upon the salt water of the sea”. This prompted the first scientific investigation of dead water in Walfrid Ekman's PhD thesis (who later became a founding father of modern oceanography). Ekman's insightful work, comprising 744 experiments, eyewitness testimony, and a linear model of boat drag, established the surprising origin of dead water: internal waves generated at the pycnocline separating fresh and salty water steal energy from the boat. With Dedalus and the improved volume penalty method [2], we are able to build on Ekman’s explanation. This simulation shows a boat (half-oval) pushed at a constant force through water with a linear density stratification (top). The boat evolves from the balance of applied force and fluid drag, and the vorticity is shown on the bottom. The black bar is a ‘sponge’ to filter the flow in the periodic domain. The simulations reveal the importance of nonlinear features like eddies that have been neglected in previous analyses. I am currently working on laboratory experiments with Prof. Amin Chabchoub and Dr Zachary Benitez in the School of Engineering at The University of Sydney to further investigate these findings before submitting our first paper on the project.

My second project began during my Geophysical Fluid Dynamics Fellowship at the Woods Hole Oceanographic Institution in 2017. I worked with Dr Claudia Cenedese and Dr Craig McConnochie to investigate how iceberg shapes affect melting. Icebergs are an important source of freshwater to the polar oceans, and come in a wide range of shapes and sizes. How quickly icebergs melt is key to understanding their influence on the global climate. Our laboratory experiments showed that previous models underestimated overall melting, and ignored differences between bottom and side melting. However important questions remained. This video shows one experiment, with a dyed ice block (blue) fixed in a flume recirculating warm (20 ºC) salt water at 3.5 cm/s. Notice the increased melting toward the front (left) of the base. Getting to the bottom of this would require some numerical assistance.

In September and October of 2018 I visited Dr Benjamin Favier at the Institute for Research in Nonequilibrium Phenomena (IRPHE) in Aix-Marseille University. Dr Favier had recently simulated melting ice in fresh water using phase-field models. The mathematical techniques I developed during this visit led to the theoretical aspects of my first paper on the improved Volume Penalty Method [2], which was implemented in our paper on emergent melting phenomena in Rayleigh- Benard convection [4].

Visiting IRPHE also led to my collaboration with Dr Louis-Alexandre Couston, a Marie Skłodowska-Curie Postdoctoral Fellow at the British Antarctic Survey and the Department of Applied Mathematics and Theoretical Physics (DAMTP), Cambridge. I visited Dr Couston during September 2019, and extended my methods to simulate melting in salt water. This collaboration resulted in our paper on improved phase-field models of melting in salt water [3], and our investigation of the influence of buoyancy on melting underneath glaciers [5]. The method was finally able to reproduce and explain the observations of my Woods Hole project, culminating in our recent paper “Aspect ratio affects iceberg melting” [6]. This movie helps explain the localised increases in melting in experiments. We plot the temperature field of the simulation (black:freezing, yellow:room temperature). This field shows the cool (red) meltwater being released under the ice (black), which is caught in vortices. By plotting the local melt rate (red line), we see that the melt rate coincides with the generation of vortices, which brings up warm background water to melt the base. This causes significantly increased melt rates over time, and we believe this will also increase the melting of actual icebergs.

After finishing my PhD, I plan to continue my work on fluid-solid interactions by investigating the presence of sub-surface oceans on Saturn’s ice moon Enceladus. Enceladus is the most remote known example of liquid water in the solar system. The origin of this liquid ocean has perplexed scientists. Enceladus is too remote for the sun to support liquid water, and too small for radiogenic heating to maintain an ocean. The remaining hypotheses for heating propose that the gravitational tides induced by Saturn are able to generate heat by continually mixing the ocean. Using the recent spherical geometry capabilities of Dedalus [7], I’ll hopefully be able to answer some of these questions!

References
1. Burns, Keaton J., Geoffrey M. Vasil, Jeffrey S. Oishi, Daniel Lecoanet, and Benjamin P. Brown. “Dedalus: A Flexible Framework for Numerical Simulations with Spectral Methods.” Physical Review Research 2, no. 2 (April 23, 2020): 023068.
2. Hester, Eric W., Vasil, Geoffrey. M. & Burns, Keaton. J. “Improving convergence of volume penalized fluid-solid interactions”. arXiv:1903.11914 (2019). (Accepted at Journal of Computational Physics.)
3.Hester, Eric W., Louis-Alexandre Couston, Benjamin Favier, Keaton J. Burns, and Geoffrey M. Vasil. “Improved Phase-Field Models of Melting and Dissolution in Multi-Component Flows.” Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 476, no. 2242 (October 28, 2020): 20200508.
4. Purseed, J., Favier, B., Duchemin, L. & Hester, E. W. “Bistability in Rayleigh-Benard convection with a melting boundary”. Phys. Rev. Fluids 5, 023501 (2020).
5. Couston, L.-A. Hester, E. W., Favier, B., Taylor, J. R., Holland, P. R. & Jenkins, A. “Ice melting in a turbulent stratified shear flow”. arXiv:2004.09879 (2020). (Accepted at Journal of Fluid Mechanics.)
6. Hester, Eric W., Craig D. McConnochie, Claudia Cenedese, Louis-Alexandre Couston, and Geoffrey Vasil. “Aspect Ratio Affects Iceberg Melting.” ArXiv:2009.10281 (2020). (Submitted to Physical Review Fluids.)
7. Lecoanet, Daniel, Geoffrey M. Vasil, Keaton J. Burns, Benjamin P. Brown, and Jeffrey S. Oishi. “Tensor Calculus in Spherical Coordinates Using Jacobi Polynomials. Part-II: Implementation and Examples.” Journal of Computational Physics: X, March 4, 2019, 100012.