# Fluid Dynamics

Geophysical and astrophysical fluids, complex fluids

### Wave radiation (and leaky waves) in GFD

In the tropical atmosphere, the upward propagating gravity waves reach the stratosphere and never return. We found that such leakage effects are significant in the atmosphere - a large atmospheric event that spans the full depth of the troposphere (a "mode one" event) can dissipate in less than two days due to the wave leakage. In both the atmosphere and ocean these scenarios are ubiquitous. We discovered a new mathematical approach to study such problems, which includes finding a physically relevant functional basis for the GFD problem at hand. This PhD project involves investigating various GFD problems and evaluating the effect of leakage in them. This would result in novel understanding of GFD problems and have implications to data analysis. Informal enquiries can be made to Lyuba Chumakova (Lyuba.Chumakova@ed.ac.uk).

### Statistical Mechanics for the Brewing Industry

Brewing is perhaps the oldest example of industrial biotechnology. However, despite continued efforts to scientifically understand the complex processes involved in the transformation of grain into beer, many aspects of brewing remain somewhat of an art. The central player in the industry is yeast, the cells of which are many orders of magnitude smaller than the brewing vessels in which it does its work. Brewers require fine control over the behaviour of yeast, needing it to remain suspended in the liquid whilst it produces alcohol, and then for it to sediment out at the end of the process so that it may be easily removed. Much of the current expertise is a product of hundreds of years of incremental, trial-and-error improvement, with little, if any, input from predictive science. Clearly there is a need for an accurate, efficient and rational modelling approach that would remove some of the guess-and-test elements and would allow the testing of various aspects such as vessel geometries and agitation on the behaviour of the yeast. The multiscale nature of the problem (with interesting and important effects ranging from the size of yeast cells all the way up to that of the vessels) prohibits the use of standard continuum models, such as the Navier-Stokes equations. In addition, the unimaginably large number of cells and vast range of timescales involved prevents the use of full-scale numerical simulations. This project will use a statistical mechanics approach, which couples the microscopic and macroscopic dynamics, based on dynamic density functional theory (DDFT) to study flocculation (clumping) and sedimentation of yeast in brewing vessels, as well as other similar industrial problems. This project will continue an existing collaboration with WEST Brewing, based in Glasgow. Informal enquiries can be made to Ben Goddard (b.goddard@ed.ac.uk).

### Ocean Modelling with Mimetic Finite Element Methods

Mimetic finite element methods are a type of numerical discretisation for partial differential equations which obey discrete analogues of continuous properties of partial differential operators -- for example that the curl of a gradient should be zero, or that the divergence of a curl should be zero. The ability of a numerical method to represent these mathematical properties can often be related to the ability of the method to represent physical balances in discretisations of equations, such as the important physical balances which are present in the equations which govern ocean dynamics. This project will study the application of mimetic finite element methods to the numerical simulation of problems in ocean dynamics, with particular emphasis on problems which are relevant for the large-scale ocean. Informal enquiries can be made to James Maddison (j.r.maddison@ed.ac.uk).

### Energy backscatter by ocean mesoscale eddies

While there are large scale transport processes within the ocean, the ocean is a highly turbulent fluid, and contains a vigorous field of eddies. These eddies are both formed through instabilities of the mean flow, but can also feed back and modify the mean flows which generate them. This can include energy "backscatter" processes, which can lead to local strengthening of the mean. This project will study simplified models of ocean turbulence, with particular emphasis on the backscatter of energy by mesoscale eddies, and how these processes may be interpreted in terms of coarsening of high resolution data. Informal enquiries can be made to James Maddison (j.r.maddison@ed.ac.uk).

### Optimal flow control: mathematical models and numerical methods

A key application of mathematical modelling to practical scientific problems arises in the field of fluid flow control, where one wishes to design a physical system in which the flow behaves in some desired or optimal way. Two key challenges result from this approach: (i) it needs to be ensured that the mathematical model accurately reflects the physics of the flow and the desired objective, (ii) computational algorithms must be designed to solve the systems of equations resulting from the model accurately and efficiently. This project aims to address these two challenges for fluid dynamics problems motivated by practical and industrial applications. A possible additional component of the project is to apply this methodology within a high performance computing framework. Informal enquiries can be made to John Pearson (j.pearson@ed.ac.uk). This project is related to the EPSRC Fellowship http://gow.epsrc.ac.uk/NGBOViewGrant.aspx?GrantRef=EP/M018857/1.

### Colliding Black Holes and Neutron Stars

The mysterious gamma-ray bursts are the largest explosions since the big bang. They happen on average once a day and last for only a few seconds. One scenario to explain these cataclysmic events involves the collisions of neutrons stars and black holes. The detailed numerical calculations are done on powerful computers to try to simulate the collisions and then explain these bursts. The project would involve first to get acquainted with the computer code and the astrophysical background, and then to extend the code to include new physical effects and/or explore the parameter space. Please contact Max Ruffert (M.Ruffert@ed.ac.uk) for further details. Examples of previous PhDs can be found here and here.

### Dispersion in random flows

When a constituent, e.g. a pollutant, is released in a fluid flow, it disperses through the combined effect of advection by the flow and molecular diffusion. Some time after the release, the dispersion can often be represented as a simple diffusion process, with a diffusivity, termed effective diffusivity, that is enhanced compared to the molecular diffusivity and accounts for the effect of advection. Mathematical techniques, mostly homogenisation methods, have been developed to compute the effective diffusivity for a range of flows of increasing complexity, from simple time-independent flows to time-dependent random flows that model turbulence. The description in terms of an effective diffusivity is however limited to regions of high concentration of the constituent and fails to describe the low-concentration tails. These turn out to be in important in applications, for instance when the constituent undergoes fast chemical reactions that amplify low concentrations. In recent work with Peter Haynes (Cambridge), we have developed a new asymptotic method that predicts the entire concentration profile, including its tails. The technique relies on large-deviation theory and highlights how the statistics in the tails are controlled by rare, extreme events. It has been applied to simple time-independent flows for which explicit results can be obtained in certain limits. This project will extend the large-deviation approach to random flows, starting with a class of spatially correlated, white in time flows. It will combine the analysis of stochastic partial differential equations with numerical simulations, Monte Carlo sampling in particular. The project could evolve in several directions and analyse, for instance, dispersion in complex geometries or in realistic turbulent flows. Informal enquiries can be made to Jacques Vanneste (j.vanneste@ed.ac.uk).

### The geometry of wave-mean flow interaction

Numerous problems in fluid dynamics involve the separation of a flow between a mean component and fluctuations, often regarded as 'waves'. Examples of this include the separation of the atmospheric flow into its average along latitudes and perturbations, the separation of oceanic flow into a slow time-averaged component and fast surface waves, or the separation of numerically simulated flows into resolved and subgrid parts. The main aim of this separation is to obtain a simplified model for both the mean flow and the waves which accounts for their interactions. It is crucial that this simplified model respect key properties of the parent model such as conservation of energy and momentum and, when relevant, circulation. This places strong constraints on the ways of separating waves from the mean flow, and on the models of wave-mean flow interactions. This project examines how wave-mean flow models that satisfy these constraints can be derived by relying on the geometric foundations of the original fluid equations. A fluid flow can be thought of as a trajectory in an infinite-dimensional space (the space of diffeomorphisms in the simplest case) which minimises a certain action. Applying this viewpoint to the wave-mean flow problem, and using differential geometric tools has proved fruitful in the simple setup of inviscid incompressible fluids. The project will generalise this to consider more complex fluid models that include the effect of density variation and compressibility, among others. The result will be a methodology that can be applied to a broad class of problems in geophysical and astrophysical fluid dynamics. Informal enquiries can be made to Jacques Vanneste (j.vanneste@ed.ac.uk).

### Mathematics applied to the environment

Projects applying mathematics to environmental problems and supervised by James Maddison and Jacques Vanneste are available through the Edinburgh Earth, Ecology and Environment Doctoral Training Partnership.