Current research projects
- Unstable periodic orbits in homogeneous, isotropic turbulence
Collaborators: Shigeo Kida (retired, see Researchgate); Genta Kawahara (Osaka University, Japan); Tatsuya Yasuda (Nagoya Institute of Technology); Alberto Vela-Martín (Universidad Politécnica de Madrid).
Related publications: V3, VKK, KKV, VKY, VVKY, VVK.
Key words: turbulence, periodic orbit theory, scientific computing.
Abstract: In the first project, we computed unstable periodic solutions in Kida-Pelz flow and investigated their statistical and dynamical properties as well as their dependence on the Reynolds number. The ultimate goal is to compute periodic orbits which reproduce Kolmogorov statistics in the inertial range. Since the number of degrees of freedom this requires is too high for our current numerical methods and hardware, we resorted to LES modelling. In LES of box turbulence we identified a highly unstable periodic orbits that reproduces a small but significant Kolmogorov spectrum.
- Scaling and dynamics in the Kuramoto-Sivashinsky equation
Collaborators: Kazumasa Takeuchi (Tokyo Institute of Technology, Japan); Chris Chow, Hendrick de Haan.
Related publications: V4.
Key words: Yakhot's conjecture, scaling laws, scientific computing. Abstract: There is a long-standing conjecture due to Yakhot that the long-time, large-scale behaviour of the Kuramoto-Sivashinsky equation falls into the Kardar-Parisi-Zhang universality class, in which we find mostly stochastic models describing, for instance, ballistic deposition. There is no conclusive theoretical evidence in favour of this conjecture and numerical evidence is hard to produce. One must perform a great number of very accurate simulations on very large domains in order to obtain reliable statistics. In this project, we are trying to relate scaling exponents obtained from numerical experiments to those predicted by KPZ theory and dynamic renormalization, with the verification or rejection of the conjecture as the ultimate goal.
- Modelling collective motion of cells
Collaborators: Luciano Buono, Mitchell Kovacic, Eryn Frawley, Hendrick de Haan; Raluca Eftimie (University of Dundee, UK).
Related publications: bBEKV.
Key words: mathematical modelling, nonlocal PDEs, collective motion.
Abstract: Mitchell and Eryn studied models of animal aggregation in one and two spatial dimensions, respectively. These models are complicated because they contain nonlocal terms, modelling the processing of information by individuals in a finite spatial range aroud them. While these models were mostly meant to apply to large mammals, we now think that they might be particularly useful for decribing a certain type of cell motility in the high density limit. If we manage to tune a nonlocal PDE model to describe the collective motion of these cells, we could answer various questions about the self-organisation and dynamics without resorting to massive agent-based simulations.
- Global bifurcations in subcritical shear flow
Collaborators: Genta Kawahara, Susumu Goto (Osaka University, Japan); Behzad Nikzad.
Related publications: VK, VKM, pV, VGo.
Key words: shear turbulence, subcritical transition, global bifurcations, scientific computing.
Abstract: Since the seminar work by Nagata in the early 1980s, there has been ample interest in the mathematical description of subcritical transition to turbulence in shear flows. The edge state hypothesis asserts that certain flows with a simple spatial structure mediate between the laminar and turbulent states of shear flow. We are trying to unravel the global connections (homoclinic or heteroclinic) of such states, going on the idea that subcritical turbulence is a random-ish walk between many, possibly infinitely many, invariant solutions such as equilibria, periodic orbits and traveling waves. To this end we use qualitatively correct low-order models as well as full-fledged direct numerical simulations.
- Dynamical analysis of mean field cortex models
Collaborators: David Liley, Federico Frascoli (Swinburne University of Technology, Melbourne, Australia);
Ingo Bojak (University of Reading, UK); Loukia Tsakanikas, Kevin Green, Laura Green.
Related publications: VL, FVBL, pFDVBL, bLBDVFF, GV, VG.
Key words: EEG modelling, mean-field model, bifurcation analysis, pattern formation.
Abstract: In this project, we study a mean-field (PDE) model of human cortical activity. We have performed extensive simulations and bifurcation analysis of the model and several low-order reductions. Some of the questions we address concern the generation of the alpha rhythm, the effect of anaesthetic agents and the effect of external noise.