Wind-turbine wakes over two-dimensional (2D) hills with different slope gradients are systematically investigated using large-eddy simulation with wind turbine parameterized as actuator disk model and hilly terrain modeled by immersed boundary method. The chosen hill models represent typical hilly terrains with and without flow recirculation in the wake of the hills. The flow characteristics of wind-turbine wakes [including mean velocity, wake-center trajectory, turbulence statistics, and mean kinetic energy (MKE) budgets] and the power performance are analyzed, and the related flow mechanisms are elucidated in our study. It is found that the velocity deficit in turbine wakes cannot be acceptably represented by the Gaussian model in the wake of the steep hill until at a further distance. It is also found that the assumption that the wake-center trajectory maintains a nearly constant elevation downwind of the hilltop proposed by Shamsoddin and Porté-Agel [“Wind turbine wakes over hills,” J. Fluid Mech. 855, 671–702 (2018)] may not be applicable in particular for the steep hill cases. Furthermore, the hilltop is the optimal location for turbine placement because the turbine harvests more wind energy due to the speed-up effect and suffers less fatigue loading due to the lower turbulence levels. Both the turbulence levels and the magnitude of vertical turbulent flux are found to drop below those of the flat ground case on the windward side of the hills, and they also decrease within the hill wake region compared with the no-turbine cases. A detailed analysis of MKE budgets reveals that the budgets of pressure transport and mean convection are mainly responsible for balancing the MKE in turbine wakes over hilly terrain.

A novel analytical model is proposed and validated in this paper to predict the velocity deficit in the wake downwind of a wind turbine. The model is derived by employing mass and momentum conservation and assuming a cosine-shaped distribution for the velocity deficit. In this model, a modified wake growth rate rather than a constant one is chosen to take into account the effects of the ambient turbulence and the mechanical turbulence generated. The model was tested against field observations, wind-tunnel measurements in different thrust operations and high-resolution large-eddy simulations (LES) for two aerodynamic roughness lengths. It was found that the normalized velocity deficit predicted by the proposed model shows good agreement with experimental and numerical data in terms of shape and magnitude in the far wake region ( $x/d_0>3$ ). Based on the proposed model, predictions from multiple views and at different locations are demonstrated to show the spatial distribution of streamwise velocity downwind of a wind turbine. The result shows that the model is suitable for predicting streamwise velocity fields and thus could provide some references for the selection of wind turbine spacing.

Research on turbulent wakes under pressure gradients has gradually attracted more attention due to its fundamental and applied significance. This study aims to propose new analytical solutions [including a new full solution (FS) and a new asymptotic solution (AS)] for two-dimensional and three-dimensional wakes. The model is developed based on the principle of the conservation of mass and momentum and the assumption that the velocity deficit in wakes follows the cosine distribution. The new asymptotic form is obtained by adding a high-order term to the streamwise momentum deficit equation. It is found that the new FS and the new AS are in excellent agreement with the experimental and numerical results. The new AS exhibits excellent convergence toward the FS, whereas significant discrepancies are observed between the original AS and the FS, particularly under adverse-pressure-gradient conditions. A comprehensive comparison between the FS and the ASs, as well as an in-depth analysis of the behavior of the FS, is conducted by varying the base flow inputs with different strengths and widths. The results further confirm a much better convergence performance of the new AS than that of the original one. It is also found that the FS exhibits instability characteristics when the maximum velocity deficit exceeds a critical threshold value. Compared to the relatively complicated FS and the original AS with lower accuracy, the new AS has the advantage of being a simple analytical form while retaining good predictive performance, which allows easy and accurate inspection of the wake development under pressure gradients. The mathematical model of wake flows presented in this paper is beneficial for scientific and engineering communities.