The dynamics of coherent structures in the wall region of a turbulent boundary layer

TLDR

The authors derive low‑dimensional ordinary differential equations for the wall‑region turbulent boundary layer by Galerkin projecting the Navier–Stokes equations onto experimentally determined streamwise‑roll eigenfunctions. The resulting equations reproduce intermittent and chaotic roll dynamics that mirror experimentally observed ejection and bursting events, demonstrating that autonomous wall‑region bursts are modulated by outer‑layer pressure and establishing a first coherent connection between low‑dimensional chaos and realistic turbulent flow. Kline et al., 1967.

Abstract

We have modelled the wall region of a turbulent boundary layer by expanding the instantaneous field in so-called empirical eigenfunctions, as permitted by the proper orthogonal decomposition theorem (Lumley 1967, 1981). We truncate the representation to obtain low-dimensional sets of ordinary differential equations, from the Navier–Stokes equations, via Galerkin projection. The experimentally determined eigenfunctions of Herzog (1986) are used; these are in the form of streamwise rolls. Our model equations represent the dynamical behaviour of these rolls. We show that these equations exhibit intermittency, which we analyse using the methods of dynamical systems theory, as well as a chaotic regime. We argue that this behaviour captures major aspects of the ejection and bursting events associated with streamwise vortex pairs which have been observed in experimental work (Kline et al. 1967). We show that although this bursting behaviour is produced autonomously in the wall region, and the structure and duration of the bursts is determined there, the pressure signal from the outer part of the boundary layer triggers the bursts, and determines their average frequency. The analysis and conclusions drawn in this paper appear to be among the first to provide a reasonably coherent link between low-dimensional chaotic dynamics and a realistic turbulent open flow system.