Low dimensional models of the transition to turbulence
The transition from a regularly ordered state of fluid motion to chaotic and turbulent regime has been long observed and progressively understood through the discovery and application of theoretical tools and frameworks. One such lens through which the problem can be attacked confines the dynamics of a turbulent system to a set of ordinary differential equations describing the time evolution of velocity modes, allowing for the application of techniques from dynamical systems theory. In this body of work,this framework shall be applied to develop a series of low dimensional models of the transition to turbulence in plane Couette flow (fluid sheared between two parallel plates of anti-parallel velocity). These models will then be investigated to understand certain features of the transition. Initially, the transition to Newtonian turbulence is investigated, with the results used as a starting point for a study of viscoelastic turbulence. The first Chapter of this thesis will introduce the required theoretical knowledge and contextualise the research outline. The second Chapter will then detail an algorithm for generating a low order model of the transition to turbulence. This process is demonstrated by using the method to create a hierarchy of models for plane Couette flow. In Chapter 3, the thesis will then detail an investigation as to why the lifetime of the turbulent state increases rapidly with the Reynolds number Re, the dimensionless flow parameter quantifying the ratio of inertial to viscous forces, using the models created in Chapter 2. The findings of Chapter 2 and Chapter 3 are then used together to form the basis of a study into effects of viscoelasticity, described by the Oldroyd-B model, on the the dynamics of turbulent flow in Chapter 4.