## Spring 2016 Seminars and Abstracts

### Why Are Owls So Quiet And What Have They Got To Do With Wind Turbines

**Date**: Monday 26 January, 2pm, (ARTS 01.02)

**Speaker**: Lorna Ayton (DAMTP, University of Cambridge)

**Abstract**: Owls are well-known to be almost silent as they fly. Several features of their wings, including a porous, flexible trailing edge, have been proposed as key reasons why. In this talk I will discuss how a poroelastic (i.e. both porous and flexible) extension to a finite rigid plate can reduce sound scattering, and how this design feature could be implemented to reduce the noise generated by wind turbines. The effects of the length, porosity, and flexibility of the extension are discussed in an attempt to identify the optimal poroelastic extension for noise reduction. Analytical results are obtained using the Wiener-Hopf method.

### School Colloquium

Divergent Series: From Thomas Bayes' Bewilderment to Today's Resurgence Via The Rainbow

**Date**: Wednesday 3rd February, **4pm**, (LT3)

**Speaker**: Michael Berry (University of Bristol)

**Abstract**: Following the discovery by Bayes in 1747 that Stirling’s series for the factorial is divergent, the study of asymptotic series has today reached the stage of enabling summation of the divergent tails of many series with an accuracy far beyond that of the smallest term. Several of these advances sprang from developments of Airy’s theory of waves near optical caustics such as the rainbow. Key understandings by Euler, Stokes, Dingle and Écalle unify the different series corresponding to different parameter domains, culminating in the concept of resurgence: quantifying the way in which the low orders of such series reappear in the high orders.

### Vortex Dynamics in a Superfluid

**Date**: Monday 15th February, 2pm, (ARTS 2.02)

**Speaker**: Cecilia Rorai (University of Cambridge)

**Abstract**: Quantized vortices in superfluids are mobile and interacting topological defects. Two non-parallel quantized vortices annihilate or propagate in two dimensions, whereas they can reconnect in three dimensions. The nature of vortex dynamics is quantum mechanical, involving the atomically thin vortex cores, but it also influences the large scale dynamics of quantum turbulence, causing a tangle of quantum vortices to evolve in time, and eventually decay.

I will present results on vortex dipoles and vortex reconnection obtained by integrating the Gross-Pitaevskii equation and compare them to some experimental data.

### Some Aspects of Vortices in Light- and Matter Waves

**Date**: Monday 22nd February, 2pm, (ARTS 01.02)

**Speaker**: Fabian Maucher (Durham University)

**Abstract**: The first part of the talk will present recent work on nonlinear light propagation in the presence of competing local and nonlocal nonlinearities. Such system could be realized in a gas of thermal alkaline atoms. Apart from spatial soliton formation, the different length scales of the nonlocality can give rise to filamentation and subsequent self-organised lattice formation in the beam profile, akin to the superfluid-supersolid phase transition in Bose-Einstein condensates (BECs). The particular role of optical vorticity in the process of the pattern formation will be emphasized.

The second part will focus on exciting knotted vortex lines in BECs. We suggest using a light field containing a knotted vortex line as probe field of a Raman-pulse that drives a coherent two-photon Raman transition of three-level atoms with Lambda-level configuration. We elaborate on experimental feasibility as well as on subsequent dynamics of the matter wave.

### Probing Cosmic Superfluids

**Date**: Monday 29th February, 2pm, (ARTS 01.02)

**Speaker**: Nils Andersson (University of Southampton)

**Abstract**: Neutron stars are the exotic remnants left over after the supernova explosions in which massive stars end their lives. They are associated with a range of phenomena observed in radio, X-rays, gamma-rays and hopefully soon gravitational waves as well. Because of the extreme densities of these stars, where more than the mass of our Sun is compressed inside a 10 kilometre radius, they represent exotic physics that cannot be tested in the laboratory. The state of matter in these systems is hotly debated, but it is generally accepted that they are cold enough to contain superfluid components. There is strong observational support for this idea, but we are still far away from having truly quantitative models. In this talk I will describe the nature of the problem, summarize the key physics and outline the computational modelling required to make progress.

### PhD Student Talks

**Date**: Monday 7th March, 2pm, (Queens 1.03)

**Speaker**: Awatif Alhowaity

**Title: **Solidification Caused by Under-Cooling

**Abstract**: Many crude oils contain dissolved waxes that can precipitate out of solution and become deposited on the internal walls oil pipes. The waxy oils are transported through very long pipelines from warm walls to cooler conditions in the pipe. An important phenomenon occurring during the under-cooling of the pipeline is the formation of solid matter inside the pipe. The wax deposition is one of the most serious problems, potentially restricting flow and plugging the pipe. However, the wax deposits begin to form when the temperature is below the wax appearance temperature (WAT). We model the particle’s growth in the oil pipe once the temperature falls below the WAT. We determine the temperature distribution, formulate and solve the self-similar problem of wax particle growth from a single point. A numerical method is used to compute the solution of initial value problem of diffusion and transport of wax towards the particle. The numerical solution is compared to the self-similar solution.

**Speaker**: Tanmay Inamdar

**Title: **Measures and Slaloms

**Abstract**: : Any compact subset of the reals has the following properties:.

1) It is compact,

2) It has a linear order which generates its topology,

3) It has a countable dense set, i.e., it is separable.

A classical result of G. Cantor implies that these properties in fact characterise compact subsets of the reals. This led M. Suslin to ask if separability above can be somewhat weakened to the property of having the countable chain condition, i.e., that every disjoint family of open sets is countable. This question is now known to be independent of ZFC.

General topologists have, however, tried to strengthen Suslin's hypothesis in various ways in order to obtain another characterisation of the compact subsets of the reals. A promising candidate until recently consisted of a strengthening of 'has a linear order' with 'does not map continuously onto an uncountable power of [0,1]', until S. Todorcevic constructed a counterexample in ZFC. We consider a further strengthening of this statement, where 'ccc' is replaced by 'supports a measure'. We show that under an extra axiom, Todorcevic's space can be constructed so as to have a measure, whereas under another (necessarily contradictory) extra axiom, no such space can carry a measure. We also show, without assuming any extra axiom, that such a space can be constructed so as to not support a measure. Various other non-separable compactifications of the natural numbers are constructed.

All of this proceeds by an analysis of the effect of forcing with a measure algebra on certain families of subsets of the natural numbers known as 'slaloms' which were used by Todorcevic in his construction. Joint work with Piotr Borodulin-Nadzieja (Wroclaw).

### Critically Balanced Rotating and Stratified Turbulence

**Date**: Monday 18th April, 2pm, (JSC 3.02)

**Speaker**: Alexander Schekochihin (University of Oxford)

**Abstract**: I will discuss the principle of critical balance in strong turbulence: the idea that, in media where both nonlinear interactions and linear wave propagation are supported, turbulent cascades organise themselves in such a way that the characteristic times associated with linear propagation and nonlinear decorrelation are comparable scale by scale. This idea originates from theories of magnetohydrodynamic and plasma turbulence, but appears to lead to interesting and plausible conclusions when applied in hydrodynamic contexts as well. I will outline a theory of strongly rotating turbulence as an example of the application of the critical balance. If time permits, I will also show how it works in stratified turbulence. The latter has interesting applications to turbulence in the intergalactic medium, which I may or may not have time to cover.

References:

Nazarenko & Schekochihin, JFM 677, 134 (2011)

Zhuravleva et al., Nature 515, 85 (2014)

### A Generalized Model for Optimal Transport of Images Including Dissipation and Density Modulation

**Date**: Monday 16th May, 2pm, (SCI 0.31)

**Speaker**: Carola-Bibiane Schönlieb (Cambridge)

**Abstract**: In this talk I will present a new model in which the optimal transport and the metamorphosis perspectives are combined. For a pair of given input images geodesic paths in the space of images are defined as minimizers of a resulting path energy. To this end, the underlying Riemannian metric measures the rate of transport cost and the rate of viscous dissipation. Furthermore, the model is capable to deal with strongly varying image contrast and explicitly allows for sources and sinks in the transport equations which are incorporated in the metric related to the metamorphosis approach by Trouv'e and Younes. In the non-viscous case with source term existence of geodesic paths is proven in the space of measures. The proposed model is explored on the range from merely optimal transport to strongly dissipative dynamics. For this model a robust and effective variational time discretization of geodesic paths is proposed. This requires to minimize a discrete path energy consisting of a sum of consecutive image matching functionals. These functionals are defined on corresponding pairs of intensity functions and on associated pairwise matching deformations. Existence of time discrete geodesics is demonstrated. Furthermore, a finite element implementation is proposed and applied to instructive test cases and to real images. In the non-viscous case this is compared to the algorithm proposed by Benamou and Brenier including a discretization of the source term. Finally, the model is generalized to define discrete weighted barycentres with applications to textures and objects. This is joint work with Jan Maas, Martin Rumpf and Stefan Simon.

### Multiwavelets and Outlier Detection for Troubled-Cell Indication

**AMENDED Date: **Tuesday 24th May, **3pm,** (SCI 0.31)

**Speaker**: Thea Vuik (Delft University of Technology)

**Abstract**: In general, solutions of nonlinear hyperbolic PDEs contain shocks or develop discontinuities. One option for improving the numerical treatment of the spurious oscillations that occur near these artifacts is through the application of a limiter. The cells where such treatment is necessary are referred to as troubled cells, which are selected using a troubled-cell indicator. These indicators perform well as long as a suitable, problem-dependent parameter is chosen. The optimal parameter is chosen such that the minimal number of troubled cells is detected and the resulting approximation is free of spurious oscillations. In general, many tests are required to obtain this optimal parameter for each problem.

In this presentation, I will introduce a new indication technique based on the multiwavelet decomposition of the approximation. I will show that the multiwavelet coefficients at the highest level can be used to detect discontinuities in the (derivatives of the) DG approximation. In addition, we will see that the sudden increase or decrease of the indicator value with respect to the neighboring values is important for detection. Indication basically reduces to detecting outliers, which is done using Tukey's boxplot approach. We provide an algorithm that can be applied to various troubled-cell indication variables. Using this technique, the problem-dependent parameter that the original indicator requires, is no longer necessary, as the parameter will be chosen automatically.

### Solitary Waves on a Ferrofluid Jet

**Date: Wednesday **8th June, **1:30pm,** (SCI 0.31)

**Speaker**: Dag Nilsson (Lund University)

**Abstract**: We consider a current carrying rod surrounded by a ferromagnetic fluid in the presence of a magnetic field. In such a setup on can consider waves on the ferrofluid jet, and in particular solitary waves. In this talk I will present some new results regarding existence of solitary waves on a ferrofluid jet. As in the case of water waves, the governing equations for this problem can be written as a free boundary value problem with nonlinear boundary conditions. Instead of working with these equations directly, we use a spatial dynamics approach and formulate the problem as an evolution equation. This equation is then studied by using the center manifold theorem. These techniques have previously been applied successfully in order to find solitary water waves. This talk is based on a joint work with Professor Mark Groves from Saarland University.

## Autumn 2015 Seminars and Abstracts

### Sparse Grid Discontinuous Galerkin Methods for High-Dimensional Elliptic Equations

**Date**: Thursday 10th September, 2pm, (SCI 3.05)

**Speaker**: Yingda Cheng (Michigan State University)

**Abstract**: In this talk, we develop an interior penalty method on sparse grid for computations of high-dimensional elliptic equations. Using a hierarchical basis representation, we construct a sparse finite element approximation space, reducing the degree of freedom from the standard $O(h^{-d})$ to $O(h^{-1}|\log_2 h|^{d-1})$ for $d$-dimensional problems, where $h$ is the uniform mesh size in each dimension. Compared to the traditional full grid approaches, the accuracy of the numerical approximation of this method is only slightly deteriorated by a factor of $|\log_2 h|^{d-1}$ in the energy norm. Error estimates are provided and confirmed by numerical tests in multi-dimensions.

### Research Challenges in Applied Mathematics to Wave Power

**Date**: Monday 28th September, 2pm, (SCI 0.31)

**Speaker**: Dr Emiliano Renzi (Loughborough)

**Abstract**: Deep understanding and detailed investigation of ocean waves cannot be achieved without the help of mathematics. In this seminar, I will attempt to show how accurate formulation, solution and analysis of the solution provide a significant physical insight on wave power extraction. My talk will focus on the Oyster wave energy converter developed by Aquamarine Power, which is believed to have generated the highest sustained power output of any wave machine in the world (www.aquamarinepower.com).

### Optimal Error Estimates for Discontinuous Galerkin Methods Based On Upwind-Biased Fluxes for Linear Hyperbolic Equations

**Date**: Monday 5th October, 2pm, (ARTS 01.02)

**Speaker**: Dr Xiong Meng (UEA)

**Abstract**: In this talk, we will analyze discontinuous Galerkin methods using upwind-biased numerical fluxes for time-dependent linear conservation laws.

In one dimension, optimal a priori error estimates of order $k+1$ are obtained for the semi-discrete scheme when piecewise polynomials of degree at most $k$ $(k \ge 0)$ are used. Our analysis is valid for arbitrary nonuniform regular meshes and for both periodic boundary conditions and for initial-boundary value problems. We extend the analysis to the multidimensional case on Cartesian meshes when piecewise tensor product polynomials are used, and to the fully discrete scheme with explicit Runge--Kutta time discretization. Numerical experiments are shown to demonstrate the theoretical results. This is a joint work with Chi-Wang Shu and Boying Wu.

### Complex Solutions of the Navier-Stokes Equations

**Date**: Monday 12th October, 2pm, (ARTS 01.02)

**Speaker**: Prof. Jonathon Mestel (Imperial)

**Abstract**: It is well known that low-Reynolds-number flows ($R_e\ll1$) have unique solutions, but this statement may not be true if complex solutions are permitted.

We begin by considering Stokes series, where a general steady velocity field is expanded as a power series in the Reynolds number. At each order, a linear problem determines the coefficient functions, providing an exact closed form representation of the solution for all Reynolds numbers. However, typically the convergence of this series is limited by singularities in the complex $R_e$ plane.

We employ a generalised Padé approximant technique to continue analytically the solution outside the circle of convergence of the series. This identifies other solutions branches, some of them complex. These new solution branches can be followed as they boldly go where no flow has gone before. Sometimes these complex solution branches coalesce giving rise to real solution branches. It is shown that often, an unforced, nonlinear complex "eigensolution" exists, which implies a formal non-uniqueness, even for small and positive $R_e$.

Extensive reference will be made to Dean flow in a slowly curved pipe, but also to flows between concentric, differentially rotating spheres, and to convection in a slot. In addition, certain fundamental exact solutions are shown to possess extra complex solutions. This is joint work with Florencia Boshier.

### Ellipsoidal Harmonics and their Applications

**Date**: Monday 19 October, 2pm, (ARTS 01.02)

**Speaker**: Prof. Ioannis Chatzigeorgiou (UEA)

**Abstract**: Ellipsoidal harmonics are called the products of the separable solutions of the Laplace equation in an ellipsoidal coordinate system. Use of the separation of variables of the Laplace equation in an ellipsoidal system, results in the so-called Lame equations which accordingly provide the Lame functions and finally the ellipsoidal harmonics. The problem with Lame functions (apart from the limited literature on the subject) is that they cannot be calculated in closed forms. The reason is that their derivation requires the computation of their characteristic values which are given as solutions of characteristic polynomials. Therefore the only way to derive Lame functions for arbitrary degree and order is to apply an efficient numerical procedure. In the seminar the speaker will present such a methodology which is able to determine Lame functions (of the first and the second kind) for indefinitely large degree and order. Having calculated the Lame functions, one is able to apply them to various fields of applied mathematics to tackle practical problems. In this context, formulations and results will be presented for the hydrodynamics of fish-like shapes moving close to a wall or in the centre of a channel. Some results will also be presented for a complete different problem that deals with electroencephalography for brain imaging.

### Cell-Based Modelling for Wound Contraction and Angiogenesis

**Date:** Monday 26 October, 2pm, (ARTS 01.02)

**Speaker:** Fred Vermolen (TU Delft)

**Abstract:** Wound contraction and angiogenesis are biological processes that often take place during healing of wounds and in tumor development. To model these processes, one distinguishes between different types of models, which are descriptive at several scales, ranging from cellular scale (micro-scale) to the tissue scale (macro-scale). The models are on the macro-scale are based on continuum hypotheses, which means that one sets up and solves partial differential equations with the associated boundary and initial conditions. On the smallest scale one models all kinds of cell phenomena on a molecular level. In this talk, we will consider colonies of cells, which are treated as discrete entities, as well as chemical and mechanical signals that are modelled as sets of partial differential equations. Hence, the current approach is a hybrid one.

The process of angiogenesis, which is the formation of a vascular network in tissues, is often modelled by using principles based on cell densities in a continuum approach or on hybrid cellular-continuum level where one uses cellular automata (in particular cellular Potts) models. In this study, we abandon the lattice needed to model the cell positions in cellular automata modelling and instead, we apply a continuous cell-based approach to simulate three-dimensional angiogenesis. Next to the application of this modelling strategy to angiogenesis, we discuss the application of the formalism to wound contraction.

The talk will describe some of the mathematical issues encountered in these models and further some animations will be shown to illustrate the potential merits of our approaches.

### RANS Based Boundary Layer Transition Modelling Using Laminar Kinetic Energy Concept for Wind Energy Applications

**Date**: Monday 2nd November, 2pm, (ARTS 01.02)

**Speaker**: Dr Zhengzhong Sun (City University London)

**Abstract**: The talk presents a numerical investigation of boundary layer transition on a wind turbine airfoil DU91-W2-250 at chord-based Reynolds number $Re_c = 1\times10^6$ using the RANS-based transition model with laminar kinetic energy concept. The $k_L$–$k_T$–$\omega$ transition model is first validated at the angle of attack of $6.24^\circ$ against wind tunnel measurement in terms of lift and drag coefficients, surface pressure distribution and transition location. Observation of flow field in the vicinity of transition location identifies a separation bubble, which attributes to the laminar-turbulent transition. Due to the low-Reynolds number nature for this transition model, study of the entire transition process is possible and the boundary layer evolution across the transition is analysed. The AoA effect on transition taking place on wind turbine airfoil is finally studied. By increasing the angle of attack from $-3$ to $10$, the transition locations are predicted in close agreement with measurement.

### Lagrangian Modelling of Water Waves

**Date**: Monday 9th November, 2pm, (ARTS 01.02)

**Speaker**: Dr Eugeny Buldakov (UCL)

**Abstract**: The main difficulty of Eulerian numerical solvers for water waves modelling is the changing shape of a computational domain. There are numerous methods of dealing with this difficulty, all of them invariably leading to considerable complication of solvers. The natural solution is using equations of fluid motion in Lagrangian form which -- though in some cases are more complicated than the Eulerian counterparts -- to be solved in the fixed domain of Lagrangian coordinates. The presentation discusses the development and application of a numerical solver which uses a simple finite-difference technique applied directly to Lagrangian equations of fluid motion. Stability of the developed numerical scheme and numerical dispersion relation are analysed and draw-backs of the scheme are discussed. A 2D version of the solver is developed and applied to a range of test cases including violent sloshing, tsunami runup, evolution of breaking wave groups and waves over sheared currents. Comparison with flume experiments demonstrates that the solver is able to model evolution of highly non-linear waves with good accuracy. However numerical dispersion may lead to a considerable phase error. Finally, directions of further development and methods to improve model accuracy are discussed.

### Ensemble Visualization and Uncertainty Characterization Using Generalized Notions of Data Depth

**Date**: Wednesday 11th November, 2pm, (ARTS 01.02)

**Speaker**: Mike Kirby (University of Utah)

**Abstract**: When computational methods or predictive simulations are used to model complex phenomena such as dynamics of physical systems, researchers, analysts and decision makers are not only interested in understanding the data but also interested in understanding the uncertainty present in the data as well. In such situations, using ensembles is a common approach to account for the uncertainty or, in a broader sense, explore the possible outcomes of a model. Visualization as an integral component of data-analysis task can significantly facilitate the communication of the characteristics of an ensemble including uncertainty information. Designing visualization schemes suitable for exploration of ensembles is specifically challenging if the quantities of interest are derived feature-sets such as isocontours or streamlines rather than fields of data.

In this talk, I will introduce novel ensemble visualization paradigms that use a class of nonparametric statistical analysis techniques called data depth to derive robust statistical summaries from an ensemble of feature-sets (from scalar or vector fields). This class of visualization techniques is based on the generalization of conventional univariate boxplots. Generalizations of boxplot provide an intuitive yet rigorous approach to studying variability while preserving the main features shared among the members. They also aid in highlighting descriptive information such as the most representative ensemble member (median) and potential outlying members. The nonparametric nature and robustness of data depth analysis and boxplot visualization makes it an advantageous approach to study uncertainty in various applications ranging from image analysis to fluid simulation to weather and climate modeling.

This is joint work with Mahsa Mirzargar and Ross Whitaker

### An Introduction to the Hybridized Discontinuous Galerkin Method

**Date**: Monday 16th November, 2pm, (ARTS 01.02)

**Speaker**: Dr Liangyue Ji

**Abstract**: The hybridized discontinuous Galerkin (HDG) method was first introduced for linear second-order parabolic equations by Prof. Bernardo Cockburn and his collaborators. The main favourable feature of these methods is that their approximate solutions can be expressed in an element-by-element fashion in terms of a numerical trace. In this framework, the globally coupled degrees of freedom, which only involves those of the numerical trace on the element face, is significantly reduced compared to standard DG methods, leading to an efficient implementation. Moreover, the numerical trace containing the super-convergence property allows computation of a new enhanced approximation by a different type of post-processing technique. In this talk, I will discuss this new characterization of the approximated solution given by the HDG method for second-order parabolic equations.

### A Unified Analysis of Algebraic Flux-Correction Schemes

**Date**: Monday 30th November, 2pm, (ARTS 01.02)

**Speaker**: Dr Gabriel Barrenechea (Strathclyde)

**Abstract**: In this talk I will review recent results on the mathematical analysis of algebraic flux-correction (AFC) schemes. The schemes are designed mainly to preserve a discrete version of the maximum principle, and, unlike usual stabilised finite element schemes, AFC schemes do not start from a weak formulation, but rather they only "see" the linear system resulting from the discretisation. This lack of weak formulation is one of the main issues that have prevented from providing a mathematical analysis for this class of schemes. Then, the first step is to to write this scheme in a weak form, and then obtain stability results. Positivity of the scheme is proved under some appropriate conditions on the mesh, and convergence results are proved.

### Control of Falling Liquid Films

**Date**: Monday 7th December, 2pm, (ARTS 01.02)

**Speaker**: Dr Alice Thompson (Imperial)

**Abstract**: The flow of a fluid layer with one interface exposed to the air and the other an inclined planar wall becomes unstable due to inertial effects when the fluid layer is sufficiently thick or the slope sufficiently steep. This free surface flow of a single fluid layer has industrial applications including coating and heat transfer, which benefit from smooth and wavy interfaces respectively. In this talk, I will discuss how the dynamics of the system can be altered by introducing deliberately spatially-varying or time-dependent perturbations via a shaped wall, chemical coatings, or the injection of fluid through the wall. I will focus on the case of fluid injection, and compare the effect on flow dynamics of choosing steady non-zero injection, or using the injection as a responsive control mechanism. I will show that applying steady spatially-periodic injection always leads to non-uniform states, can enable new bifurcations and complicated time-dependent behaviour, and significantly alters the trajectories of particles in the flow. In the second case, I will demonstrate that using injection as a feedback control mechanism based on real-time observations of the interface is remarkably effective, even when combined with localised feedback and actuation. Furthermore, the controls can be used to drive the system towards arbitrary steady states and travelling waves, and the qualitative effects are independent of the details of the flow modelling.

.