Modelling the effects of boundary proximity on a tidal rotor using the actuator line method

Abstract

For a turbine in unconstrained flow, the maximum power coefficient (CP ) is limited to 16/27 of the undisturbed
kinetic 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 to
the 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 surface
changing the balance of optimal rotor resistance. Thus, the limit for maximum power extraction is modified
to the form first presented by Garrett and Cummins, CP = (16/27)(1 − B)⁻² [2]. The factor B, known as the
blockage ratio, represents the fraction of the channel cross-section occupied by the rotor swept area, and allows
rotors operating in confined conditions to theoretically exceed the Betz limit. Subsequent theoretical work by
Nishino and Willden extended this model to demonstrate that constructive interference effects between closely
spaced turbines in a co-planar fence can allow efficiency increases above the Betz limit, even in an infinitely wide
channel where the global blockage ratio is negligible [3]. This phenomenon, attributed to the concept of local
blockage, defined as the ratio of rotor swept area to the surrounding flow passage area, has been observed by
several 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 effects
stemming from azimuthal variations in the flow field caused by anisotropy in the local blockage, such as when
the rotor is not centred in the flow passage. In this study, a single tidal rotor is simulated using the actuator
line model embedded in a Reynolds-Averaged Navier-Stokes solver with varying degrees of anisotropic blockage
imposed by proximity to a non-deformable upper boundary. The investigation is carried out in the context
of the Supergen ORE Unsteady Tidal Turbine Benchmarking Project using a 1.6 m rotor in a computational
domain equivalent to the towing tank dimensions at the Qinetiq Haslar facility in which the turbine was tested
experimentally [7]. The discrete blade representation and unsteady nature of the actuator line method allows
investigation of variations in loads and the importance of boundary proximity by extracting the spanwise load
distributions and local flow parameters at different positions around the azimuth. The effects of two different
tip-loss correction models on the spanwise force distributions and overall loads are investigated. Analysis of the
integrated rotor loads shows potential for an increase in the maximum CP of ∼1% with changing proximity to
the upper boundary without any detriment to the power-to-thrust ratio. However, anisotropy in the local flow
passage can result in azimuthally varying blade forces, introducing an additional source of fatigue loading to
the rotor and drive train.