Unstable periodic orbits in weak turbulence

Abstract

We report on a sophisticated numerical study of a 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). For example, the UPOs may be used to construct the dynamical zeta function of the system, from which very accurate turbulent averages of observables may be extracted. Though a number of algorithms for computing such orbits have been proposed and tested, in this paper we focus on a space–time variational principle introduced by Lan and Cvitanović in 2004 [15]. This method has not, to our knowledge, been tested on dynamical systems of high dimension because of the formidable storage and computation required. In this paper, we use petascale computation to apply this algorithm to weak hydrodynamic turbulence. We begin with a brief description and reformulation of the space–time algorithm of Lan and Cvitanović. We then describe how to apply this algorithm to the lattice-Boltzmann method for the solution of the Navier–Stokes equations. In particular, we describe the fully parallel implementation of this algorithm using the Message Passing Interface. This implementation, called HYPO4D, has been successfully deployed on a large variety of platforms both in the UK and the USA and has shown very good scalability to tens of thousands of computing cores. Finally, we describe the application of this implementation to the problem of weak homogeneous turbulence driven by an Arnold–Beltrami–Childress force field in three spatial dimensions, at a Reynolds number of 371. We commence by systematically searching for nearly periodic orbits as candidate solutions from which to begin the relaxation; we then apply the variational algorithm until convergence is obtained. Because the algorithm requires storage of the space–time lattice, even the smallest orbits require resources on the order of tens of thousands of computing cores. Using this approach, two UPOs have been identified and some of their properties have been analysed.