## Autumn 2013 Autumn 2013

### Stability in the Presence of Transport Terms

Date: Monday 30th September, 2pm, (ARTS 3.03)
Speaker: Prof. Arieh Iserles (University of Cambridge)
Abstract: Numerous PDEs are of the form ∂u/∂t + V(x) · u = ℒ(u,x,t), where ℒ is an operator, e.g. convection-diffusion, the forward Kolmogorov equation, the Fokker-Planck equation and the Bolzmann equation. Once this equation, on an arbitrary grid, is discretised using finite differences, in tandem with splitting (e.g. the Strang splitting), the discretisation of transport terms V(x) · u is typically unstable. In this talk we prove that such a discretisation is always stable as long as differentiation is approximated with a skew-symmetric matrix but that skew-symmetry on an arbitrary grid is generically consistent with just first-order approximation. We derive conditions that guarantee high-order skew-symmetric approximation of the first derivative and present a method to construct matrices of this kind which are, in addition, banded.

### Textures and Topology in Liquid Crystals

Date: Monday 7th October, 1pm, (ARTS 2.03)
Speaker: Dr. Gareth P. Alexander (University of Warwick)
Abstract: The textures of liquid crystals have always been central in identifying mesophases and understanding their properties. They are deeply connected with topology, both of the liquid crystal and of the environment that it lives in, so that controlling and describing topological properties provides a key tool in constructing and understanding complex three-dimensional textures. I will describe two applications of topological ideas to studying mesophases: the general characterisation of three-dimensional textures in nematics using the Pontryagin–Thom construction — illustrated through the analysis of torons and the Hopf fibration in confined cholesterics — and the formation of elegant knots and links around colloidal particles representing non-orientable surfaces.

### Discontinuous Galerkin Method for Convection Dominated Partial Differential Equations

Date: Monday 14th October, 2pm, (ARTS 3.03)
Speaker: Chi-Wang Shu (Brown University)
Abstract: Discontinuous Galerkin (DG) method is a finite element method with features from high resolution finite difference and finite volume schemes such as approximate Riemann solvers and nonlinear limiters. It was originally designed for solving hyperbolic conservation laws but has been generalised later to solve higher order convection dominated partial differential equations (PDEs) such as convection diffusion equations and convection dispersion equations. The DG method has been widely applied, in areas such as computational fluid dynamics, computational electromagnetism, and semiconductor device simulations, just to name a few. In this talk we will give a general survey of the DG method, emphasizing its designing principles and main ingredients. We will also describe some of the recent developments in DG methods.

### Hydrodynamics of Rogue Waves

Date: Monday 28th October, 2pm (SCI 3.05)
Speaker: Dr Miguel Onorato (Università degli Studi di Torino)
Abstract: I will discuss how the modulational instability can influence the statistical properties of surface gravity waves. I will show that, using the approach of wave turbulence theory, deviations from Gaussian statistics can be naturally described. In particular, I will discuss the role of bound and free modes in the determination of the statistical properties of the surface elevation. Experimental results will be also compared with theory.

### On the Collision of Sea Breeze Gravity Currents

Date: Monday 4th November, 2pm (ARTS 3.03)
Speaker: Karin van der Wiel (UEA, PhD Student at ENV)
Abstract: Converging and colliding sea/ land breezes can develop over narrow islands, peninsulas or enclosed seas. Sea breezes are an example  of gravity currents, currents forced by horizontal density differences. Gravity currents can be created in the laboratory using so-called 'lock exchange' experiments. Laboratory experiments of colliding sea-breezes/ gravity currents will be presented. Different aspects of these collisions will be discussed, i.e. shape of the collision front, vertical displacements and mixing efficiencies. Finally some theoretical considerations will be done.

### Energy Fluxes in Models of Wind-Driven Ocean Circulation

Date: Monday 18th November, 2pm (ARTS 2.03)
Speaker: Prof. David N. Straub (McGill University)
Abstract: The general circulation of the oceans is forced primarily by large scale, slowly varying wind stress.  How energy moves around between different spatial scales and different classes of flow is an important, but still poorly understood question.  Here we consider energy fluxes in two different dynamical models.  The first assumes quasigeostrophy in a basin setting. The quasigeostrophic approximation filters potentially important exchanges between balanced (low frequency) and unbalanced (high frequency) classes of motion.  Nevertheless, the resulting energetics are interesting in that they differ substantially from the phenomenology of geostrophic turbulence, which has mostly been worked out assuming periodic or unbounded geometries.  In particular, we find that the nonlinear inverse energy cascade familiar from studies of 2-dimensional and geostrophic turbulence does not extend between energy source and sink scales, as is commonly supposed.  Rather, it is part of a "double cascade" in which a linear 'beta' term allows for a forward cascade of energy that is essentially equal and opposite to the nonlinear cascade.  We also consider interactions between slow and fast modes in a primitive equation model channel setting.  Forcing includes steady and high frequency components and we find that near-inertial oscillations excited by the high frequency forcing can have a significant impact on the low frequency (or geostrophic) part of the solution.  At moderate Rossby number, the near-inertial modes act to drain energy from the geostrophic flow.

### A Review of Nonlinear Eigenvalue Problems

Date: Monday 25th November, 2pm (ARTS 3.03)
Speaker: Prof. Francoise Tisseur (Manchester University)
Abstract: In its most general form, the nonlinear eigenvalue problem is to find scalars $\lambda$ (eigenvalues) and nonzero vectors $\mathbf{x}$ and $\mathbf{y}$ (right and left eigenvectors) satisfying $N(\lambda) \mathbf{x}=0$ and $\mathbf{y}^* N(\lambda) = 0$, where $N : \mathbb{C} \rightarrow \mathbb{C}^{n\times n}$ is an analytic function. In practice, the matrix elements are most often polynomial, rational or exponential functions of $\lambda$ or a combination of these. These problems underpin many areas of computational science and engineering. They can be difficult to solve due to large problem size, ill conditioning (which means that the problem is very sensitive to perturbations and hence hard to solve accurately), or simply a lack of good numerical methods. My aim is to discuss applications and to review the recent research directions in this area.

### Robust Error Estimates for Stabilised Finite Element Methods Applied to High Reynolds Number Flows

Date: Monday 2nd December, 2pm (EFRY 1.34)
Speaker: Prof. Erik Burman (UCL)
Abstract: The computation of solutions to the equations of fluid flow at high Reynolds flow is a challenging problem. Indeed at high Reynolds number the appearance of layers in the solution makes standard conservative numerical methods such as the Galerkin method unstable and some resolution dependent artificial viscosity must be added. Even if numerical stability can be ensured, it is not clear what quantities can be reliably computed due to the lack of stability of the underlying physical problem, even in the linear case, unless the velocity field is Lipschitz continuous. In this talk we will discuss what we can provably compute for some different models of fluid mechanics. Starting with the one dimensional Burgers' equation we show how the E-condition of Oleinik can be used to prove a priori and a posteriori error estimates that are robust with respect to the Reynolds number. We then widen the scope and discuss how these results may be generalised to scalar convection--diffusion equations in two or three dimensions or the Navier-Stokes' equations in two space dimensions. To lift the standard Lipschitz hypothesis on the velocity field in higher dimension than one we propose a scale separation assumption under which the bulk of the energy is transported in smooth large scales with high Reynolds number and (arbitrarily) strong fluctuations are concentrated to small scales whose relative Reynolds number is order unity. Under this assumption and using the stability properties of the stabilised finite element methods we prove robust error estimates also for these cases.

### Dispersion in the Large-Deviation Regime

Date: Monday 9th December, 2pm (ARTS 3.07)
Speaker: Prof. Jacques Vanneste (University of Edinburgh)
Abstract: The dispersion of a passive scalar in a fluid through the combined action of advection and molecular diffusion can often be described as a diffusive process, with an effective diffusivity that is enhanced compared to the molecular value. This description fails to capture the tails of the scalar concentration in initial-value problems, however. This talk addresses this issue and shows how the theory of large deviation can be applied to capture the concentration tails by solving a family of eigenvalue problems. Two types of flows are considered: shear flows and cellular flows. In both cases, large deviation is shown to generalise classical results (Taylor dispersion for shear flows, homogenisation results for cellular flows). Explicit asymptotic results are obtained in the limit of large Péclet number corresponding to small molecular diffusivity. The implications of the results for the problem of front propagation in reacting flows are also discussed.

## Spring 2014 Spring 2014

### Nonstandard Bifurcation Phenomena in PDEs with Variable Exponenta in PDa

Date: Monday 20th January, 3pm (ARTS 2.03)
Speaker: Prof. Vicentiu Radulescu (Mathematics Institute of the Romanian Academy)
Abstract: We establish some bifurcation properties for nonlinear elliptic PDEs with variable exponent and we point out the differences with respect to the standard case corresponding to the Laplace operator. The mathematical treatment combines variational and topological arguments, and the study is inspired by models in non-Newtonian fluids and image processing.

### Color to Grayscale

Date: Monday 27th January, 2pm, (SCI 3.05)
Speaker: Dr. Hans Rivertz (Sør-Trøndelag University College)
Abstract: The simplest Color to grayscale transforms are weighted averages of the color channels. Luminance is one of these methods. The color contrast is often lost in the luminance version. I will talk about some methods that will try to represent color contrast as greyscale contrast. If there are something like the "best" color to grayscale transformation. The search for that method is still going on.

### Hp-Version Composite Discontinuous Galerkin Methods for PDEs on Complicated Domains

Date: Wednesday 12th February, 2pm, (ARTS 3.03)
Speaker: Prof. Paul Houston (University of Nottingham)
Abstract: In this talk we introduce the hp-version discontinuous Galerkin composite finite element method for the discretization of second-order elliptic partial differential equations. This class of methods allows for the approximation of problems posed on computational domains which may contain a huge number of local geometrical features, or micro-structures. While standard numerical methods can be devised for such problems, the computational effort may be extremely high, as the minimal number of elements needed to represent the underlying domain can be very large. In contrast, the minimal dimension of the underlying composite finite element space is independent of the number of geometric features. The key idea in the construction of this latter class of methods is that the computational domain Ω is no longer resolved by the mesh; instead, the finite element basis (or shape) functions are adapted to the geometric details present in Ω. Here we consider both the a priori and a posteriori error analysis of this class of methods, as well as their application within Schwarz-type domain decomposition pre-conditioners.

### Air Cushioning in Liquid-Solid Impacts

Date: Monday 17th February, 2pm, (SCI 3.05)
Speaker: Dr. Richard Purvis (UEA)
Abstract: This talk will examine air cushioning effects in droplet and solid impacts.  Air cushioning occurs just prior to high-speed impacts when high pressures generated in the air cause free-surface to deform prior to impact taking place.  This changes touchdown behaviour, causes bubble entrapment and can considerably affect the pressures created post-impact.  We will introduce a basic model initial, identifying parameter regimes where viscous or inviscid air effects are key.  For the case where viscosity is dominant, we develop a coupled lubricating air-inviscid water model.  Once this basic model is established we will look at various cases, including three-dimensional impacts (with experimentally validated predictions of bubble sizes), oblique impacts, impact of droplets onto thin water-layers, and the influence of surface tension.  The touchdown behaviour in each case will be considered; some cases see accelerated touchdown, others see a delayed touchdown and, in the case of surface tension, touchdown is prevented altogether(!). Finally we will summarise recent work on compressibility effects.

### Skyrmion Scattering

Date: Monday 3rd March, 2pm, (SCI 3.05)
Speaker: Dr David Foster (University of Kent)
Abstract: The Skyrme model is a model of nuclei which can be understood as a geometric energy functional. I shall introduce the Skyrme model. Then I will discuss how its solutions scatter and show how this can be approximated  by treating them as point particles. I will also show some interesting and possibly unique dynamics.

### Challenges in High Reynolds Number Spectral/hp Flow Simulations

Date: Monday 10th March, 2pm, (SCI 3.05)
Speaker: Prof. Spencer Sherwin (Imperial College London)
Abstract: In this presentation we will outline and investigate the challenges of performing high Reynolds number simulations using spectral/hp element discretisations. The spectral/hp element methods combines the good phase and dispersion properties of spectral methods with the geometric flexibility of finite element methods and can be considered as a high order finite element technique. Whilst these properties are attractive from a numerical discretisation a number of challenges exist in apply these methods to high Reynolds number complex geometry problems. In this presentation we will first outline the challenges of generating appropriate meshes where body conforming near wall prismatic elements are used to capture the boundary layer dynamics. We will then highlight the role of  polynomial de-aliasing and spectral vanishing viscosity to stabilise high Reynolds number transient flows.

### PhD Presentations

Date: Monday 24th March, 2pm, (SCI 3.05)
Speaker: PhD Presentations

### PhD Presentations

Date: Monday 31st March, 2pm, (SCI 3.05)
Speaker: PhD Presentations

### Boundary Behaviour of Viscous Fluids: Influence of Wall Roughness and Friction-Driven Boundary Conditions

Date: Monday 7th April, 2pm, (SCI 3.05)
Speaker: Prof. Dorin Bucur (Université de Savoie)
Abstract: We consider a family of solutions to the evolutionary Navier-Stokes system supplemented with the complete slip boundary conditions on domains with rough boundaries. We give a complete description of the asymptotic limit by means of Γ−convergence arguments, and identify a general class of boundary conditions.
(This is a joint work with E. Feireisl and S. Necasova)

## Summer 2014 Summer 2014

### A Fast Solver for the Navier-Stokes Equations

Date: Monday 28th April, 2pm, (SCI 3.05)
Speaker: Prof. Kees Vuik (Delft University of Technology)
Abstract: After linearization and discretization of the incompressible Navier Stokes equations one has to solve block-structured indefinite linear systems. The successful design of robust, scalable, and efficient preconditioners is intimately connected with an understanding of the structure of the resulting block matrix system. Effective preconditioners are often based on an approximate block decomposition of the discretized incompressible Navier Stokes equations. This requires a careful consideration of the spectral properties of the component block operators and their Schur complement operators. Through this purely algebraic view of preconditioning, a simplified system of block component equations is developed. Inclusion of "physics based" preconditioners of the various parts can lead to effective preconditioners with optimal or nearly optimal convergence rates for academic and industrial problems.

CFD applications in maritime industry, for example hull resistance prediction, involve high Reynolds number flows modelled by the incompressible Reynolds-averaged Navier-Stokes equations. The system of equations is discretized with a cell-centred finite-volume method with colocated variables. After linearization, various SIMPLE-type preconditioners can be applied to solve the discrete system. In this presentation, we discuss their performance for flows with Reynolds number up to 10^9 and cell aspect ratio up to 10^6.

Reference: SIMPLE-type preconditioners for cell-centred, colocated finite volume discretization of incompressible Reynolds-averaged Navier-Stokes equations C.M. Klaij and C. Vuik International Journal for Numerical Methods in Fluids, 71, pp. 830-849, 2013

http://ta.twi.tudelft.nl/nw/users/vuik/papers/Kla12V.pdf

### Coreless Vortices and Topological Interfaces in Spinor Bose-Einstein Condensates

Date: Monday 12 May, 2pm (SCI 0.31)
Speaker: Dr Magnus Borgh (University of Southampton)
Abstract: I will give a brief introduction to the rich phenomenology of topological defects, such as vortices, and textures in atomic spinor Bose-Einstein condensates (BECs), which form when the spin degree of freedom of the atoms is not frozen out by strong magnetic fields. Together with the high control over parameters in experiments with ultracold atoms, this makes spinor BECs excellent candidates as test-beds for studies of complex defect states. I will present two particular examples: I will show how a stable coreless vortex can form a composite topological defect with distinct small- and large-distance topology in a hierarchical core structure. I will also discuss how atoms with spin-degree of freedom can be used to study the interface between regions of different broken symmetry, as occurs in such disparate systems as the boundary between A and B phases of superfluid He-3, and cosmological brane-inflation scenarios.

### Wave Turbulence in Vibrating Elastic Plates

Date: Monday 19 May, 2pm (SCI 0.31)
Speaker: Prof. Sergio Rica (Universidad Adolfo Ibanez, Santiago, Chile)
Abstract: The dynamics of random weakly nonlinear waves is studied in the framework of vibrating thin elastic plates. According to the theory of weak turbulence a nonlinear wave system evolves in long-time creating a slow energy transfer from one mode to another. In collaboration with Gustavo During, Christophe Josserand, we have derived a kinetic equation for the spectral density of the amplitude of the waves. It has been predicted that such a non-equilibrium theory describes the approach to an equilibrium wave spectrum, and describes also an energy cascade, often called the Kolmogorov-Zakharov spectrum. More recently, we have shown that a non stationary inverse cascade could exist in this framework, we show substantial evidence of the existence of an inverse cascade, opening the possibility of self organisation for a larger class of systems. This inverse cascade transports the spectral density of the amplitude of the waves from short up to large scales, in- creasing the distribution of long waves despite the short waves fluctuations. This cascade appears to be self-similar and we show explicitly a tendency to build a long wave coherent structure. The existence of condensation of classical elastic waves is also discussed.

### The Physics of Beer Foaming-Up

Date: Monday 9 June, 3pm (SCI 0.31)
Speaker: Javier Rodríguez-Rodríguez (Carlos III University of Madrid)
Abstract: Besides the obvious interest in the field of recreational physics — which initially attracted our attention to the problem — understanding the formation of foam in a supersaturated carbonated liquid after an impact on the container involves the careful physical description of a number of processes of great interest in bubble dynamics and multiphase flows in general. In order of appearance in this problem: strong pressure wave propagation in bubbly liquids, bubble-bubble interaction in clusters, bubble collapse and break-up, diffusive bubble-liquid mass transfer and the dynamics of bubble-laden plumes. As a matter of fact, very similar physical phenomena are found in the study of oil reservoirs using seismic waves or in the formation of mud volcanoes. In this talk, we will present experimental results showing that the overall foaming-up process can be divided into three well-defined stages dominated by different physical effects and occurring in different time-scales. Namely, cavitation and bubble collapse, diffusion-driven growth of bubble clouds and buoyancy-driven autocatalytic bubble plumes. Besides the experiments, quantitative analyses of these stages is presented.