Fazendeiro, L.A.M.; (2011) Unstable periodic orbits in turbulent hydrodynamics. Doctoral thesis , UCL (University College London).

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Abstract

In this work we describe a novel parallel space-time algorithm for the computation of periodic solutions of the driven, incompressible Navier-Stokes equations in the turbulent regime. Efforts to apply the machinery of dynamical systems theory to fluid turbulence depend on the ability to accurately and reliably compute such unstable periodic orbits (UPOs). These UPOs can be used to construct the dynamical zeta function of the system, from which very accurate turbulent averages of observables can be extracted from first principles, thus circumventing the inherently statistical description of fluid turbulence. In order to identify these orbits we use a space-time variational principle, first introduced in 2004. This approach has not, to the best of our knowledge, been used before on dynamical systems of high dimension because of the formidable storage and computation required. In this thesis we describe the utilization of petascale high performance computation to the problem of applying this space-time algorithm to hydrodynamic turbulence. The lattice-Boltzmann method is used to simulate the Navier-Stokes equations, due to its locality, and is implemented in a fully-parallel software package using the Message Passing Interface. This implementation, called HYPO4D, was successfully deployed on a large variety of platforms both in the UK and the US with an extremely good scalability to tens of thousands of computing cores. Based on this fluid solver other routines were developed, for the systematic location of suitable candidate spacetime minima and their numerical relaxation, using the gradient descent and conjugate gradient algorithms. Following this methodology, several UPOs are identified in homogeneous turbulence driven by an Arnold-Beltrami-Childress force field in three spatial dimensions, at Reynolds numbers corresponding to weakly-turbulent flow. We characterize the transition to turbulence in the ABC flow and the periodic orbits computed, for a flow with Re = 371, after the transients have died down. The work concludes with a discussion of the potential for this approach to become a new paradigm in the study of driven dissipative dynamical systems.

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