Modelling the effects of boundary proximity on a tidal rotor using the actuator line method
Authors: Edwards Huw , Chen Xiaosheng , Vogel Christopher , Willden Richard 
Published: September 2023

For a turbine in unconstrained flow, the maximum power coefficient (CP ) is limited to 16/27 of the undisturbedkinetic energy flux through the rotor area according to work attributed to Betz, Lanchester, and Joukowsky [1].This maximum, often referred to as the Betz limit, occurs when the rotor presents the optimum resistance tothe incoming flow, imparting enough force to generate power without overly choking the flow through the rotor.However, in the context of tidal stream energy, the flow is often constrained by the seabed and the free surfacechanging the balance of optimal rotor resistance. Thus, the limit for maximum power extraction is modifiedto the form first presented by Garrett and Cummins, CP = (16/27)(1 − B)−2 [2]. The factor B, known as theblockage ratio, represents the fraction of the channel cross-section occupied by the rotor swept area, and allowsrotors operating in confined conditions to theoretically exceed the Betz limit. Subsequent theoretical work byNishino and Willden extended this model to demonstrate that constructive interference effects between closelyspaced turbines in a co-planar fence can allow efficiency increases above the Betz limit, even in an infinitely widechannel where the global blockage ratio is negligible [3]. This phenomenon, attributed to the concept of localblockage, defined as the ratio of rotor swept area to the surrounding flow passage area, has been observed byseveral studies both experimentally, and numerically using actuator disk and blade element momentum methods[4, 5, 6]. However, neither the experimental nor the numerical studies provided detail on unsteady loading effectsstemming from azimuthal variations in the flow field caused by anisotropy in the local blockage, such as whenthe rotor is not centred in the flow passage. In this study, a single tidal rotor is simulated using the actuatorline model embedded in a Reynolds-Averaged Navier-Stokes solver with varying degrees of anisotropic blockageimposed by proximity to a non-deformable upper boundary. The investigation is carried out in the contextof the Supergen ORE Unsteady Tidal Turbine Benchmarking Project using a 1.6 m rotor in a computationaldomain equivalent to the towing tank dimensions at the Qinetiq Haslar facility in which the turbine was testedexperimentally [7]. The discrete blade representation and unsteady nature of the actuator line method allowsinvestigation of variations in loads and the importance of boundary proximity by extracting the spanwise loaddistributions and local flow parameters at different positions around the azimuth. The effects of two differenttip-loss correction models on the spanwise force distributions and overall loads are investigated. Analysis of theintegrated rotor loads shows potential for an increase in the maximum CP of ∼1% with changing proximity tothe upper boundary without any detriment to the power-to-thrust ratio. However, anisotropy in the local flowpassage can result in azimuthally varying blade forces, introducing an additional source of fatigue loading tothe rotor and drive train.