Direct integration of non-axisymmetric Gaussian wind-turbine wake including yaw and wind-veer effects

The performance of a wind farm is significantly influenced by turbine–wake interactions. These interactions are typically quantified for each turbine either by measuring its nacelle wind speed or by evaluating its rotor-averaged wind speed using numerical methods that involve a set of discrete points across the rotor disc. Although various point distributions exist in the literature, we introduce two analytical expressions for integrating non-axisymmetric Gaussian wakes, which account for wake stretching and shearing resulting from upstream turbine yaw and wind veer. The analytical solutions correspond to modelling the target turbine as a circular actuator disc and as an equivalent rectangular actuator disc. The derived expressions are versatile, accommodating any offset and hub-height difference between the wake source (upstream turbine) and the target turbine. Verification against numerical evaluations of the rotor-averaged deficit using 2000 averaging points at various downstream locations from the wake source demonstrates excellent agreement for both analytical solutions at small/moderate veer effects, whereas only the equivalent rectangular-disc solution was accurate under extreme veer conditions. In terms of computational cost compared to vectorised numerical averaging using 16 averaging points, both analytical solutions are computationally efficient with the circular-disc solution being approximately 15 % slower and the rectangular-disc solution being approximately 10 % faster. Furthermore, the analytical expressions are shown to be compatible with multiple wake superposition models and are differentiable, providing a foundation for deriving analytical gradients which can be advantageous for optimisation-based applications.