Flow physics beyond the Betz limit

Thesis

A pivotal theoretical result of the early 20th century demonstrated that the limit of energy extraction for a wind or tidal stream turbine from an unconstrained flow is 16/27 of the upstream kinetic flux passing through the turbine area, known as the ‘Betz Limit’. This result relies on several simplifying assumptions, including inviscid, unconstrained and steady flow. The aim of this thesis is to relax these assumptions for more realistic flow conditions, leading to increased energy extraction efficiencies. The thesis first considers relaxing the unconstrained flow assumption. It has previously been shown that tidal fences consisting of turbines placed side-by-side can make use of constructive interference to raise energy extraction efficiency. The current work shows that the asymptotic performance limit of a multi-layered array is a power coefficient of unity, a 27/16 increase on the Betz limit. Contrary to intuition, the minimisation of mixing losses is not the mechanism by which power extraction increases; instead, wake mixing is found to provide an additional source of static pressure recovery. Building on this physical understanding, a new theoretical model is developed that relaxes the inviscid assumption by considering wake mixing for a single turbine, demonstrating an increase in power coefficient under the same pressure recovery mechanism observed for turbine arrays. New algebraic estimates for the optimal power coefficient are proposed, with the maximum disc performance in an unbounded flow of between 0.71 and 0.81, depending on the mixing conditions. For turbine arrays partially spanning a channel, the leading order steady flow assumption is then relaxed to consider unsteady sinusoidal forcing on the coupling between the channel and array, where the mass flux through the turbine fence can respond dynamically and is governed by the channel head. The coupled framework enables optimisation of the number and spacing of turbines in a channel to maximise power per turbine area as a proxy for revenue per cost. The thesis also presents an analytical two-scale momentum-based method for correcting experimental turbine fence performance data. Unwanted global blockage effects due to domain constraints are separated from the intended constructive interference effects caused by local system blockage. The theoretical work highlights the importance of multi-scale energy extraction and mixing-induced wake pressure recovery by which turbine performance can be boosted. Exploiting the underlying physics may allow for increases in energy capture efficiency to pave the way for next-generation turbines, accelerating the shift to energy decarbonisation.

Authors: Dehtyriov, D

• DOI: 10.5287/ora-8r9bpp709
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: renewable energy; optimisation; fluid mechanics
• Subjects: Computational fluid dynamics; Constrained optimization; Engineering