Upwind turbines, which have rotors in front of the towers, have been predominant throughout the decades of commercial wind turbine history. However, downwind turbines, which have rotors behind the towers, are gathering attention due to their technical and economic advantages regarding modern/future wind turbines. Downwind rotors generally have negative rotor tilt to avoid collision of the blade and the tower. One of the advantages of downwind turbines is performance in complex terrains. The negatively tilted rotors of downwind turbines are favorable for upflow wind in complex terrain [
1]. Furthermore, the yaw measurement devices in front of the rotor and the nacelle also contribute to yaw control accuracy [
2]. Downwind turbines can be advantageous in floating offshore wind turbine (FOWT) applications, so the share of the downwind rotor is much higher in FOWTs [
3,
4,
5] compared to the rest of the market. New FOWT concepts are now appearing, with some downwind rotors [
6,
7] inclining rearward a couple of degrees due to the rotor thrust, causing the rotor–wind misalignment to become smaller in complex terrain conditions. Rotor position and rotor coning further contribute to the stability of floating turbines with small yaw stiffness and damping [
8,
9]. Downwind turbines are also considered to be advantageous for large-scale wind turbines, mainly due to their compatibility for lighter, more elastic blades and load mitigation by appropriately coned rotors [
10].
The most essential drawback of downwind turbines is the tower shadow effect, which generates impulsive loads and infrasound when the blades pass through the wake of the tower [
11]. Many design load cases are defined by combinations of wind turbine conditions, as well as various wind and marine conditions based on international design standards, such as IEC61400-1 [
12] and IEC61400-3-1 [
13]. All of the aerodynamics, the elasticities of the structure, the controls, and the hydrodynamics strongly affect the loads and performances of the wind turbines. The blade-element momentum (BEM) method is commonly used in analyses, with some modifications or extensions for three-dimensional and dynamic effects [
14,
15], as it is accurate enough with low cost analysis. Therefore, development of the tower shadow model for BEM is the most important technical challenge in the design and analysis of downwind wind turbines. Previous studies considered the variable loads of downwind turbines by tower shadow effects, demonstrating applications of the tower wake wind speed profile while ignoring the interaction between the rotor and the tower. Matiz-Chicacausa and Lopez [
16] conducted the analysis of the tower shadow effects by the actuator line model, which showed good agreement with computational fluid dynamics (CFD). Wang and Coton [
17] developed a high-resolution tower shadow model, which showed good agreement with an experiment except for high angle of attack conditions. Zahle et al. [
18] conducted a two-dimensional CFD experiment for tower shadow effects on three different tower configurations. Zhao et al. [
19] compared upwind and downwind rotors with two different rotor speed conditions using CFD. Van der Male et al. [
20] showed that the tower shadow of downwind turbine strongly affects the fatigue damages of downwind turbine by the aeroelastic simulation with Madsen’s wake profile model. The aerodynamic interaction with between the rotor and the tower and dynamic effects are ignored in the study. Yoshida and Kiyoki [
21] developed the load equivalent tower shadow modeling method for the BEM of downwind turbines. It defines a bell-shaped wind speed profile of the tower wake by its three parameters, i.e., depth and width of the tower wake profile and the defined point, which could be adapted to the load history through the tower wake using the wind turbine CFD. This was a considerable finding, as it considered the aerodynamic interaction between the rotor and the tower, which could not be taken into account in previous models, thereby providing realistic load fluctuation in the tower shadow. This technique was practically used in the design and analysis of commercial downwind turbines, such as SUBARU 80/20 [
22], later Hitachi 2 MW, and Hitachi 5 MW [
23]. This method is still useful for analysis but not for practical design applications, as it needs CFD for each condition to identify the parameters. In addition, this model does not consider the dynamic tower shadow effects. Munduate et al. [
24] developed a dynamic tower shadow model, which considers dynamic stall effect. Although it ignores the mutual interaction between the rotor and the tower, it shows good agreement with wind tunnel tests with a 1.0 m rotor model, particularly in the context of asymmetry between the entrance and exit of the tower wake. However, the model still demonstrated two problems: (1) it did not express the load increase before the entrance of the tower wake, thereby affecting the fatigue, and (2) it uses an empirical tower wake model to determine the wind speed profile behind the tower.
Considering these situations, a dynamic tower shadow modeling method is developed herein for the BEM calculation of downwind turbines. Munduate’s model is modified and extended to solve problems (1) and (2), as mentioned above. Furthermore, the scale effect of the model is also discussed in this study.