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Publication# Invariant solutions underlying small- and large-scale structures in turbulent boundary layers

Abstract

Flows of gases and liquids interacting with solid objects are often turbulent within a thin boundary layer. As energy dissipation and momentum transfer are dominated by the boundary layer dynamics, many engineering applications can benefit from an improved understanding of physical mechanisms underlying wall-bounded turbulence. Turbulence is often treated as a random stochastic process. The existence of recognisable flow structures with spatial and temporal coherence emerging within turbulent fluctuations however suggests a deterministic description in terms of interacting coherent flow structures. In transitional flows, coherent structures have been related to non-chaotic steady and time-periodic invariant solutions of the Navier-Stokes equations suggesting a description of turbulence as a chaotic walk through a forest of invariant solutions in the system's state space.

The aim of this thesis is to transfer this dynamical systems picture from transitional flows to turbulent boundary layers and to make progress towards describing fully developed wall-bounded turbulence in terms of invariant solutions of the flow equations. We construct invariant solutions underlying two types of important coherent structures in a parallel boundary layer. First, we identify travelling wave solutions of the fully nonlinear Navier-Stokes equations that capture universal small-scale coherent structures in the near-wall region. The travelling waves are asymptotically self-similar and scale in inner units when the Reynolds number approaches infinity. Together with theoretical arguments, the existence of the self-similar solutions suggests all state-space structures supporting turbulence may become self-similar and a dynamical systems description of near-wall turbulence at infinite Reynolds numbers may be possible.

Second, we describe coherent structures spanning the entire turbulent boundary layer. These so-called large-scale motions carry most of the turbulent kinetic energy and physically emerge within a background of small-scale fluctuations. Using spatial filtering approaches, we show that large-scale motions can be isolated from small-scale fluctuations. This allows us to associate large-scale coherent structures with exact solutions of filtered Navier-Stokes equations. We specifically construct several travelling waves and periodic orbits capturing self-sustained large-scale motions at friction Reynolds numbers beyond 1000. We thereby report the first invariant solutions capturing large-scale coherent structures in a boundary layer flow.

While individual invariant solutions successfully capture specific features of turbulence, large sets of invariant solutions and especially of periodic orbits are believed to provide the foundation for a quantitative and predictive description of turbulent flows in terms of invariant solutions. To allow for the construction of sufficiently complete libraries, we propose a novel adjoint-based variational method for finding periodic orbits of spatio-temporally chaotic systems.

Most numerical results were obtained using channelflow 2.0 (channelflow.ch). This open-source software was developed and published by a research team that includes the author of this thesis.

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Ontological neighbourhood

Boundary layer

In physics and fluid mechanics, a boundary layer is the thin layer of fluid in the immediate vicinity of a bounding surface formed by the fluid flowing along the surface. The fluid's interaction with the wall induces a no-slip boundary condition (zero velocity at the wall). The flow velocity then monotonically increases above the surface until it returns to the bulk flow velocity. The thin layer consisting of fluid whose velocity has not yet returned to the bulk flow velocity is called the velocity boundary layer.

Turbulence

In fluid dynamics, turbulence or turbulent flow is fluid motion characterized by chaotic changes in pressure and flow velocity. It is in contrast to a laminar flow, which occurs when a fluid flows in parallel layers, with no disruption between those layers. Turbulence is commonly observed in everyday phenomena such as surf, fast flowing rivers, billowing storm clouds, or smoke from a chimney, and most fluid flows occurring in nature or created in engineering applications are turbulent.

Boundary layer thickness

This page describes some of the parameters used to characterize the thickness and shape of boundary layers formed by fluid flowing along a solid surface. The defining characteristic of boundary layer flow is that at the solid walls, the fluid's velocity is reduced to zero. The boundary layer refers to the thin transition layer between the wall and the bulk fluid flow. The boundary layer concept was originally developed by Ludwig Prandtl and is broadly classified into two types, bounded and unbounded.

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