Understanding transition to turbulence in fluids: computing hidden flows and beyond
Alvar Messeguer Departament de Física, Universitat Politècnica de Catalunya alvaro.meseguer(at)upc.edu
Lloc: sala de graus antiga de la Facultat de Física de la UB Predicting when and how a fluid in laminar motion may become turbulent is one of the most challenging (and yet unsolved) fundamental problems of classical physics. Current computational power allows for the accurate numerical simulation of fluid flows by means of CFD (Computational Fluid Dynamics) software packages. The majority of CFD codes currently used by scientists and engineers are essentially time evolution solvers (also termed as time-steppers)
of the fluid that provide time integrations of the Navier-Stokes initial value problem. Therefore, classical CFD codes can only reproduce stable flows, regardless the nature of their dynamics: steady, time-periodic, almost-periodic or chaotic. In other words, current standard CFD time-stepping codes can only identify local attractors. Recent studies [2,1] have clearly evidenced that the destabilization of many open flows (such as the ones appearing in pipes or channels) may be related to the presence of unstable Navier-Stokes solutions close to the stable base flow. Forecasting hydrodynamic instabilities in many fluid flows therefore requires the detection of the presence of these repelling Navier-Stokes solutions and their eventual computation. These exact Navier-Stokes flows are hidden since they are unstable and cannot be approached by standard CFD time-stepping codes. Numerical computation of these types of unstable flows is an even more challenging task, where new mathematical concepts such edge state dynamics and top-notch numerical weaponry such as Newton-Krylov-Poincare solvers are required. These techniques allow for the computation of unstable steady flows, as well as travelling waves, or modulated travelling pulses. In this talk we will describe different strategies and techniques used to identify these hidden or unstable Navier-Stokes flows (from steady profiles, to travelling waves or relative periodic orbits) responsible for the transition. We will also explain how to track their linear stability when the parameters of the problem are varied so that bifurcations can be anticipated. Whereas these techniques have been essentially developed within the framework of fluid dynamics, they can easily be adapted to explore the phase space of other deterministic nonlinear PDE arising in physics.  Deguchi, K., Meseguer, A. and Mellibovsky, F., Phys. Rev. Lett. , 112, 184502 (2014).
 Mellibovsky, F., Meseguer, A., Schneider, T. M. and Eckhardt, B., Phys. Rev. Lett. ,103, 054502 (2009).
 Meseguer, A.,Mellibovsky, F., Avila, M. and Marques, F. Phys. Rev. E, 80, 046315 (2009).