Topological bifurcations in fluid flows with application to boundary layer eruption, vortex pair interactions and exotic wakes

The overall goal of this thesis is to introduce a systematic approach for analysing spatial structures in fluid dynamical problems. The approach describes how to interpret a flow that is already known from experiments or from solving NavierStokes equations analytically or numerically. In other words, we introduce a detailed method for post-processing of velocity or vorticity fields.

A natural starting point for the analysis of a fluid dynamical problem is to make it clear what parameters the problem depends on. Under variation of these external parameters, a flow may undergo some qualitative changes. In this thesis, we analyse such qualitative changes with tools from bifurcation theory.

We mainly focus on describing the flow structure based on the vortices that are present in the fluid. In this context, a qualitative change in the flow occurs when a new vortex is created or destroyed. We apply some well known results from the field of topological fluid dynamics to analyse the changes in the topology of the vorticity field - more specifically, we analyse the creation and destruction of extremal points of vorticity, which we consider to be feature points for vortices.

Furthermore, we ask ourselves how the framework we know from topological fluid dynamics can be extended to deal with topological changes of other quantities describing the structure of a flow. As an example of a different quantity describing the flow structure, we consider the regions of a fluid where the rotation dominates the strain.

These regions are by the Q-criterion defined as vortices [29]. Since the Q-vortex is defined as a region instead of a critical point, the framework we know from topological fluid dynamics can not be directly applied. In this thesis, we extend the framework by continuing the development of a complete bifurcation theory for Q-vortices.

The theory describes the qualitative changes that may occur when we allow several parameters to vary. In addition, we analyse a special bifurcation that occurs only in a flow with a line of symmetry. All our theoretical results are topological in nature, meaning that we do not distinguish between vortex patterns which can be continuously deformed into each other.

The thesis is built around three specific case studies, in which we analyse three classical fluid mechanical problems using bifurcation theory. In all three cases, the dynamics of the vortices are simulated on a computer and we obtain a very accurate description of the flow. The bifurcation theory we develop in this thesis provides a list of possible vortex bifurcation events.

The individual bifurcation events are analysed along the way when they are observed in one of the three case studies. The application of our theoretical results leads to the following main insights into the case studies: Case I. A vortex convected close to a no-slip wall induces a viscous response from the boundary layer.

The boundary layer may organise itself into secondary vortex structures which may erupt from the wall. We study this phenomenon in two dimensions and show that the eruption process can be described as a sequence of four possible bifurcations of vortices defined by the Q-criterion. Furthermore, we show that most of these bifurcation events appear to be very robust to variations in the Reynolds number.

Case II. When several vortices are present in a flow at the same time, they will naturally interact with each other. We study the simplest possible case where two vortices interact in a two dimensional flow. By applying the Q-criterion, we show that two co-rotating vortices merge only if their relative strength ratio is below a certain threshold.

Furthermore, we show that this threshold can be identified as a bifurcation phenomenon itself. Finally, we show that for sufficiently small Reynolds numbers, the so-called core growth model can be used instead of Navier-stokes simulations as a simple, analytically tractable model with low dimension.

Case III. At a sufficiently high Reynolds number, the flow around a stationary cylinder results in the formation of the famous von Kármán vortex street. When the cylinder starts oscillating transverse to the flow direction, the vortex shedding pattern becomes significantly more complex. When the amplitude increases the pattern changes from two single vortices (2S mode) to a pair and a single vortex (P+S mode) being shed per oscillation cycle.

In a two dimensional flow, we study the topological changes of the wake pattern when vortices are defined as extremal points of the vorticity field. We identify four values of the amplitude that define critical stages in the transition from 2S to P+S wake mode.