TRR 181 Seminar "Local Weak-Coupling Approach to Model Order Reduction of the Sabra Shell Model of Turbulence"

The TRR 181 seminar is held by Denny Gohlke (Universität Hamburg, PhD in M4) on May 23, 11 am at Universität Hamburg, Bundesstr. 53, room 22/23.

Abstract

The major challenge in turbulence modeling is to tackle the closure problem arising from the nonlinear term of the Navier–Stokes equations and to construct a reduced-order model, a closure scheme, for a set of variables of interest ranging from large-scales up to a certain cutoff scale. The effect of unresolved small-scale processes has to be parametrised by use of physically and mathematically consistent assumptions about the dynamics of turbulence fluctuations. Much effort has been made over the past decades to develop such closure schemes with the aim of predicting the energy spectrum correctly: From the concept of eddy viscosity where the needed assumption of scale separation is not justified near the cutoff scale to stochastic backscatter schemes up to the sophisticated statistical approaches like two-point closures. Most of these closure schemes found also their use in large eddy simulations of geophysical turbulent flows.

A new model order reduction technique deducible from the Mori-Zwanzig theory of statistical mechanics has been recently proposed by Wouters & Lucarini [J. Stat. Phys. (2013) 151 (5): 850-860]: The framework of Mori-Zwanzig provides exact but implicit expressions for the effect of the unresolved small-scale processes on the projected dynamics of the resolved scales. Assuming a weak coupling between the resolved and unresolved part of a dynamical system, Wouters & Lucarini used a second-order expansion for the projection operators, and derived a generalized Langevin equation containing a deterministic (eddy viscosity), stochastic (backscatter), and a non-Markovian memory term. These expressions are determined explicitly depending on the Sinai-Ruelle-Bowen measures of the unperturbed dynamics of the uncoupled parts.

In this seminar, I will give a small overview on the turbulence closure problem and closure schemes, and will present the results obtained by the application of this Wouters-Lucarini reduction technique to the Sabra shell model of turbulence which corresponds to a discretised low-dimensional version of the Navier-Stokes equations in Fourier space. The focus will be on the energy spectrum and the intermittent energy fluxes: The parametrised shell system will be compared with the original system in dependency of the cutoff shell demonstrating the importance of the memory term in absence of a clear scale separation. I will also sketch the difficulties in the computation of the memory term. Discussions about possible applications of this weak-coupling approach to model order reduction of more complex turbulence systems are welcome.