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Abstract The filtered lifting line theory is an analytical approach used to solve the equations of flow subjected to body forces with a Gaussian distribution, such as used in the actuator line model. In the original formulation, the changes in chord length along the blade were assumed to be small. This assumption can lead to errors in the induced velocities predicted by the theory compared to full solutions of the equations. In this work, we revisit the original derivation and provide a more general formulation that can account for significant changes in chord along the blade. The revised formulation can be applied to wings with significant changes in chord along the span, such as wind turbine blades. 

We introduce an analytical model that describes the vertical structure of Ekman boundary layer flows coupled to the Monin-Obukhov Similarity Theory (MOST) surface layer repre- sentation, which is valid for conventionally neutral (CNBL) and stable (SBL) atmospheric conditions. The model is based on a self-similar profile of horizontal stress for both CNBL and SBL flows that merges the classic 3/2 power law profile with a MOST-consistent stress profile in the surface layer. The velocity profiles are then obtained from the Ekman momentum balance equation. The same stress model is used to derive a new self-consistent Geostrophic Drag Law (GDL). We determine the ABL height (h) using an equilibrium boundary layer height model and parameterize the surface heat flux for quasi-steady SBL flows as a function of a prescribed surface temperature cooling rate. The ABL height and GDL equations can then be solved together to obtain the friction velocity (u∗) and the cross-isobaric angle (α0) as a function of known input parameters such as the Geostrophic wind speed and surface roughness (z0). We show that the model predictions agree well with simulation data from the literature and newly generated Large Eddy Simulations (LES). These results indicate that the proposed model provides an efficient and relatively accurate self-consistent approach for predicting the mean wind velocity distribution in CNBL and SBL flows. 

The dynamics of the turbulent atmospheric boundary layer play a fundamental role in wind farm energy production, governing the velocity field that enters the farm as well as the turbulent mixing that regenerates energy for extraction at downstream rows. Understanding the dynamic interactions among turbines, wind farms, and the atmospheric boundary layer can therefore be beneficial in improving the efficiency of wind farm control approaches. Anticipated increases in the sizes of new wind farms to meet renewable energy targets will increase the importance of exploiting this understanding to advance wind farm control capabilities. This review discusses approaches for modeling and estimation of the wind farm flow field that have exploited such knowledge in closed-loop control, to varying degrees. We focus on power tracking as an example application that will be of critical importance as wind farms transition into their anticipated role as major suppliers of electricity. The discussion highlights the benefits of including the dynamics of the flow field in control and points to critical shortcomings of the current approaches. Expected final online publication date for the Annual Review of Control, Robotics, and Autonomous Systems, Volume 5 is May 2022. Please see http://www.annualreviews.org/page/journal/pubdates for revised estimates. 

Motivated by the need for compact descriptions of the evolution of non-classical wakes behind yawed wind turbines, we develop an analytical model to predict the shape of curled wakes. Interest in such modelling arises due to the potential of wake steering as a strategy for mitigating power reduction and unsteady loading of downstream turbines in wind farms. We first estimate the distribution of the shed vorticity at the wake edge due to both yaw offset and rotating blades. By considering the wake edge as an ideally thin vortex sheet, we describe its evolution in time moving with the flow. Vortex sheet equations are solved using a power series expansion method, and an approximate solution for the wake shape is obtained. The vortex sheet time evolution is then mapped into a spatial evolution by using a convection velocity. Apart from the wake shape, the lateral deflection of the wake including ground effects is modelled. Our results show that there exists a universal solution for the shape of curled wakes if suitable dimensionless variables are employed. For the case of turbulent boundary layer inflow, the decay of vortex sheet circulation due to turbulent diffusion is included. Finally, we modify the Gaussian wake model by incorporating the predicted shape and deflection of the curled wake, so that we can calculate the wake profiles behind yawed turbines. Model predictions are validated against large-eddy simulations and laboratory experiments for turbines with various operating conditions. 

A quantitative understanding of the dominant mechanisms that govern the generation and decay of the counter-rotating vortex pair (CVP) produced by yawed wind turbines is needed to fully realize the potential of yawing for wind farm power maximization and regulation. Observations from large eddy simulations (LES) of yawed wind turbines in the turbulent atmospheric boundary layer and concepts from the aircraft trailing vortex literature inform a model for the shed vorticity and circulation. The model is formed through analytical integration of simplified forms of the vorticity transport equation. Based on an eddy viscosity approach, it uses the boundary-layer friction velocity as the velocity scale and the width of the vorticity distribution itself as the length scale. As with the widely used Jensen model for wake deficit evolution in wind farms, our analytical expressions do not require costly numerical integration of differential equations. The predicted downstream decay of maximum vorticity and total circulation agree well with LES results. We also show that the vorticity length scale grows linearly with downstream distance and find several power laws for the decay of maximum vorticity. These results support the notion that the decay of the CVP is dominated by gradual cancellation of the vorticity at the line of symmetry of the wake through cross-diffusion.