1. Introduction

Wind energy is one of the leading renewable energy technologies and is key to the renewable energy transition. When wind turbines are placed together in a wind farm, they produce less energy than when placed in isolation. The harvesting of wind energy leads to the formation of wind turbine wakes, a region with reduced wind speed, behind each turbine. These wind turbine wakes affect the performance of downstream turbines in the farm. In addition, large wind farms act as additional resistance to the atmospheric boundary layer (ABL). This reduces the wind speed upstream and inside the farm, which affects the power production of the wind farm compared with the ideal situation where the upstream wind speed is not affected (Nishino & Dunstan Reference Nishino and Dunstan2020). This effect is known as farm blockage (Bleeg et al. Reference Bleeg, Purcell, Ruisi and Traiger2018; Segalini & Dahlberg Reference Segalini and Dahlberg2020) and its importance has only been recognised recently as large wind farms began to operate. Understanding the interactions between wind farms and flow in the ABL flow is one of the grand challenges the wind energy community faces (Veers et al. Reference Veers2019, Reference Veers2022), and is a prime example of a 21st century fluid dynamics problem for which more fundamental insight needs to be developed.

Two main analytical approaches are employed to model wind farm performance, which is crucial to optimise their design. The first approach is based on modelling wind turbine wakes and can be considered a bottom-up or wake modelling approach (Jensen Reference Jensen1983; Bastankhah & Porté-Agel Reference Bastankhah and Porté-Agel2014). Wake models generally work well for the entrance region of the wind farm and are commonly employed to optimise wind farm design. However, field and wind tunnel data comparisons have demonstrated that wake models have limitations in capturing wake–wake and wind farm–ABL interactions.

The second approach is known as the top-down or single-column approach, in which momentum analysis and horizontal averaging are used to estimate the flow inside the wind farm. Seminal works (Newman Reference Newman1977; Frandsen Reference Frandsen1992; Frandsen et al. Reference Frandsen, Barthelmie, Pryor, Rathmann, Larsen, Højstrup and Thøgersen2006) show that including a resistive force at the turbine height and the resistive force at the ground allows for crucial insights into the vertical velocity profile inside the wind farm. Calaf, Meneveau & Meyers (Reference Calaf, Meneveau and Meyers2010) developed an improved top-down model by including an additional wake layer with enhanced turbulence levels. Later works have extended this approach to finite-length wind farms (Meneveau Reference Meneveau2012), and included atmospheric stratification effects (Abkar & Porté-Agel Reference Abkar and Porté-Agel2013; Peña & Rathmann Reference Peña and Rathmann2014; Sescu & Meneveau Reference Sescu and Meneveau2015; Li et al. Reference Li, Liu, Lu and Stevens2022). Although top-down models include the response of an idealised ABL to a large wind farm, this approach cannot account for layout effects.

Efforts at combining top-down and bottom-up approaches include the work of Frandsen et al. (Reference Frandsen, Barthelmie, Pryor, Rathmann, Larsen, Højstrup and Thøgersen2006). This work developed a novel coupling between wake and top-down models, which considers a single column of turbines in a farm. However, this limits the applicability of the model to regular arrays. The coupled wake boundary layer (CWBL) model (Stevens, Gayme & Meneveau Reference Stevens, Gayme and Meneveau2016) widens the applicability of coupled models by introducing a two-way coupling between a wake model and a top-down model. This approach agrees well with large eddy simulations (LES) and field observations and has been extended to general wind farm layouts (Starke et al. Reference Starke, Meneveau, King and Gayme2021).

An important question is whether there is a theoretically derivable maximum power density for large wind farms. For single turbines, the German physicist Albert Betz showed, using the principles of mass and momentum conservation of the airflow through an idealised actuator disc, that no turbine can capture more than $59.3\,\%$ of the kinetic energy in the wind. In reality, most turbines are placed within large wind farm clusters. For wind farms, no fundamental performance limit has been derived yet as the underlying mechanism depends on turbulence. For extended wind farms the primary source of kinetic energy is the flow energy entrained from the geostrophic wind above the farm. Hence, the turbulence entrainment sets a physical limit to power output density that can be obtained with wind farms. This limit should not be confused by the optimal wind turbine spacing, which results from the balance between economical considerations and the wind farm fluid dynamics (Meyers & Meneveau Reference Meyers and Meneveau2012; Stevens et al. Reference Stevens, Hobbs, Ramos and Meneveau2017).

Although deriving a fundamental limit for the performance of wind energy clusters from first principles has remained elusive, various modelling approaches have been explored. As top-down models estimate the energy flux from above they can be used to estimate the maximum wind farm power density (Meneveau Reference Meneveau2019). Luzzatto-Fegiz & Caulfield (Reference Luzzatto-Fegiz and Caulfield2018) developed a two-interface entrainment model for fully developed wind farms to analyse the power output density of wind farms. Their main result is that the wind farm power density is proportional to the rate at which energy is entrained from the ABL. The predicted power output agrees well with field measurements when tailoring the model to reflect current wind farm designs. Furthermore, they showed that the performance of a wind farm can be about an order of magnitude higher in the idealised situation where the boundary layer mixes perfectly with the flow inside the wind farm. Antonini & Caldeira (Reference Antonini and Caldeira2021) showed with mesoscale simulations and model calculations that the maximum achievable power output density of wind farms is primarily determined by the strength of the geostrophic wind that drives the boundary layer flow. Kirby, Nishino & Dunstan (Reference Kirby, Nishino and Dunstan2022) introduced a combined theoretical and computational approach to analyse fluid mechanics processes that determine the optimal performance of extended wind farms. This approach provides new estimates for optimal wind power density and a novel way to study the effective efficiency of extended wind turbine arrays.