New approaches to diagnosing exact recurrent flows embedded in two-dimensional turbulence

Redfern, Edward Matthew

Abstract

Turbulent flow is often viewed as a trajectory wandering through a high dimensional phase-space, momentarily visiting the neighbourhood of unstable invariant solutions, e.g. unstable periodic orbits, travelling waves and equilibria. Periodic orbit theory allows for the prediction of statistical quantities of the flow if unstable periodic orbits can be identified. Recurrent flow analysis finds invariant solutions, in particular periodic orbits, by first locating near recurrent episodes of the state vector, usually in a numerical solution in the form of spatial Fourier coefficients. Using a Newton- GMRES-hookstep method (NGh), we attempt to converge these near recurrences to exactly periodic solutions of the governing equations. However, a near recurrence using this method is not sufficient condition for convergence. Additionally, as the Reynolds number increases, solution instability tends to increase, and converging solutions becomes more difficult. We will study turbulence within doubly periodic two-dimensional Kolmogorov flow in a square box.
A new measure of recurrence is introduced that measures the phase space direction of a turbulent trajectory. This new measure is shown to detect exact recurrence in the flow using known solutions. We combine the direction measure and distance measure using weightings to bias towards one or the other. The direction measure is then used as a filter in NGh to predict whether a trial solution will converge, resulting in a 3 times speed up. We will discuss how an appropriate set of phases in the form φ3 1,2 = ϕk1 + ϕk2 −ϕk arise from the governing equations and are invariant with respect to translation. ϕki are the individual Fourier phases such that wavevectors satisfy the triad condition k1 + k2 = k. We form a recurrence residual using dynamical phases, along with their amplitudes, to find invariant solutions. We show that this residual provides a more effective measure of near recurrence and will therefore pass better conditioned candidate solutions to the Newton method. We discuss the effectiveness of a dynamical phase residual compared to the existing method and other symmetry reduction techniques. Moreover, we show that dynamical phases allow us to unlock interesting new solutions of the Navier-Stokes equations. We are able to present a prediction of turbulence using periodic orbits that: provides an excellent prediction of the regular dynamics and most accurately predicts intermediate and extreme dynamics of all UPO studies to date.

Citation

Redfern, E. M. (2024). New approaches to diagnosing exact recurrent flows embedded in two-dimensional turbulence. (Thesis). Keele University. Retrieved from https://keele-repository.worktribe.com/output/775657

• Thesis Type: Thesis
• Deposit Date: Mar 15, 2024
• Publicly Available Date: Mar 15, 2024
• Public URL: https://keele-repository.worktribe.com/output/775657
• Award Date: 2024-03