The difference in surface roughness between land and sea, and the terrain complexities, lead to spatially heterogeneous atmospheric conditions, and therefore affect the propagation and dynamics of wind‐turbine and wind‐farm wakes. Currently, these flow heterogeneities and their effects on plant aerodynamics are not modeled in the majority of engineering wake models. In this study, we address this issue by developing a new wake‐merging method capable of superimposing the waked flow on a heterogeneous background velocity field. We couple the proposed wake‐merging method with four different wake models, i.e. the Gaussian, super‐Gaussian, double‐Gaussian and Ishihara model, and we test its performance against LES results, dual‐Doppler radar measurements and SCADA data from the Horns Rev, London Array, and Westermost Rough farms. The standard Jensen model with quadratic superposition is also included. In homogeneous conditions, the new method predicts slightly higher velocity deficits than the linear superposition method. Overall, the distributions of the difference in power ratio between the two wake‐merging methods predictions and observations show a similar mean absolute error (MAE) and interquartile range (IQR) in such conditions. On the other hand, the new wake‐merging method predictions display a lower MAE with a similar IQR in case of a spatially varying background velocity, being overall more accurate than the ones obtained with linear superposition. The most accurate estimates are obtained when the wake‐merging methods are coupled with the double‐Gaussian and Gaussian single‐wake models. In contrast, the Jensen and super‐Gaussian wake models overestimate the velocity deficits for the majority of cases analyzed.

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In the present study, we use large-eddy simulation (LES) to investigate how a capping inversion in combination with a stable free atmosphere influences the flow development and energy extraction in a large finite wind farm with a staggered and aligned layout. In the conventionally neutral boundary layer (CNBL), we find that gravity waves induce an unfavourable pressure gradient in the induction region of the farm which contributes to the upstream blockage, decreasing the available energy for first-row turbines. However, a favourable pressure gradient establishes through the farm in such conditions, which redistributes the energy and enhances wake recovery. These results are compared with a farm operating in the neutral boundary layer (NBL). Here, we find that only hydrodynamic effects induced by the turbines drag play a role, which cause minor pressure perturbations across the domain. For the selected atmospheric conditions, the power losses generated by the upstream blockage are balanced by the enhanced wake recovery promoted by the favourable pressure gradient throughout the farm. Consequently, the staggered farm efficiency in the CNBL is 8.8% higher than in the NBL. We note that this difference in efficiency is slightly enhanced by the 0.5? difference in wind direction at the location of the first-row turbines between the CNBL and NBL cases, which is caused by the presence of flow blockage. Since both simulations are forced with an equal turbulent velocity profile, the variation in performance is solely caused by the different vertical temperature profiles in the main domain. Finally, the staggered layout leads to a slightly stronger flow blockage than the aligned one when both farms operate in the CNBL.

Abstract. Recently, it has been shown that flow blockage in large wind farms may lift up the top of the boundary layer, thereby triggering atmospheric gravity waves in the inversion layer and in the free atmosphere. These waves impose significant pressure gradients in the boundary layer, causing detrimental consequences in terms of a farm's efficiency. In the current study, we investigate the idea of controlling the wind farm in order to mitigate the efficiency drop due to wind-farm-induced gravity waves and blockage. The analysis is performed using a fast boundary layer model which divides the vertical structure of the atmosphere into three layers. The wind-farm drag force is applied over the whole wind-farm area in the lowest layer and is directly proportional to the wind-farm thrust set-point distribution. We implement an optimization model in order to derive the thrust-coefficient distribution, which maximizes the wind-farm energy extraction. We use a continuous adjoint method to efficiently compute gradients for the optimization algorithm, which is based on a quasi-Newton method. Power gains are evaluated with respect to a reference thrust-coefficient distribution based on the Betz–Joukowsky set point. We consider thrust coefficients that can change in space, as well as in time, i.e. considering time-periodic signals. However, in all our optimization results, we find that optimal thrust-coefficient distributions are steady; any time-periodic distribution is less optimal. The (steady) optimal thrust-coefficient distribution is inversely related to the vertical displacement of the boundary layer. Hence, it assumes a sinusoidal behaviour in the streamwise direction in subcritical flow conditions, whereas it becomes a U-shaped curve when the flow is supercritical. The sensitivity of the power gain to the atmospheric state is studied using the developed optimization tool for almost 2000 different atmospheric states. Overall, power gains above 4 % were observed for 77 % of the cases with peaks up to 14 % for weakly stratified atmospheres in critical flow regimes.

Abstract. Recent research suggests that atmospheric gravity waves can affect offshore wind-farm performance. A fast wind-farm boundary layer model has been proposed to simulate the effects of these gravity waves on wind-farm operation by Allaerts and Meyers (2019). The current work extends the applicability of that model to free atmospheres in which wind and stability vary with altitude. We validate the model using reference cases from literature on mountain waves. Analysis of a reference flow shows that internal gravity-wave resonance caused by the atmospheric non-uniformity can prohibit perturbations in the atmospheric boundary layer (ABL) at the wavelengths where it occurs. To determine the overall impact of the vertical variations in the atmospheric conditions on wind-farm operation, we consider 1 year of operation of the Belgian–Dutch wind-farm cluster with the extended model. We find that this impact on individual flow cases is often of the same order of magnitude as the total flow perturbation. In 16.6 % of the analyzed flows, the relative difference in upstream velocity reduction between uniform and non-uniform free atmospheres is more than 30 %. However, this impact is small when averaged over all cases. This suggests that variations in the atmospheric conditions should be taken into account when simulating wind-farm operation in specific atmospheric conditions.

Abstract. Recent research suggests that atmospheric gravity waves can affect off-shore wind farm performance. A fast wind-farm boundary-layer model has been proposed to simulate the effects of these gravity waves on wind-farm operation by Allaerts and Meyers (2019). The current work extends the applicability of that model to free atmospheres in which wind and stability vary with altitude. We validate the model using reference cases from literature on mountain waves. Analysis of two reference flows shows that internal gravity wave resonance caused by the atmospheric non-uniformity can prohibit perturbations in the ABL at the wavelengths where it occurs. To determine the overall impact of the vertical variations in the atmospheric conditions on wind farm operation, we consider one year of operation of the Belgian–Dutch wind-farm cluster with the extended model. We find that this impact on individual flow cases is often of the same order of magnitude as the total flow perturbation. In 16.5 % of the analysed flows, the relative difference in upstream velocity reduction between uniform and non-uniform free atmospheres is more than 30 %. However, this impact is small when averaged over all cases. This suggests that variations in the atmospheric conditions should be taken into account when simulating wind-farm operation in specific atmospheric conditions.

Large-eddy simulations of the atmospheric boundary layer are often performed using pseudo-spectral methods, which adopt a fringe-region approach to introduce inflow boundary conditions. However, we notice that a standard fringe-region technique excites spurious gravity waves when stratified atmospheres are considered, therefore enhancing the amount of energy reflected from the top of the domain and perturbing the velocity and pressure fields downstream. In this work, we develop a new fringe-region method that imposes the inflow conditions without altering the flow downstream. This is achieved by locally damping the convective term in the vertical momentum equation. We first apply the standard and wave-free fringe-region techniques to two-dimensional inviscid-flow simulations subjected to 169 different atmospheric states. A similar study is performed on a three-dimensional domain using a couple of atmospheric states. In all cases, the new fringe-region technique imposes the inflow conditions without altering the surrounding flow. Moreover, we also investigate the performance of two non-reflective upper boundary conditions, that is a Rayleigh damping layer (RDL) and a radiation condition (RC). Results highlight the importance of carefully tuning the RDL to limit the distortion of the numerical solution. Also, we find that the tuned RDL outperforms the RC in all cases. Finally, the tuned RDL together with the wave-free fringe-region method are applied to an LES of a wind farm operating in a conventionally neutral boundary layer, for which we measure a reflectivity of only 0.7%.

Abstract. Recently, it has been shown that flow blockage in large wind farms may lift up the top of the boundary layer, thereby triggering atmospheric gravity waves in the inversion layer and in the free atmosphere. These waves impose significant pressure gradients in the boundary layer causing detrimental consequences in terms of farm's efficiency. In the current study, we investigate the idea of controlling the wind farm in order to mitigate the efficiency drop due to wind-farm induced gravity waves and blockage. The analysis is performed using a fast boundary layer model which divides the vertical structure of the atmosphere into three layers. The wind-farm drag force is applied over the whole wind-farm area in the lowest layer and is directly proportional to the wind-farm thrust set-point distribution. We implement an optimization model in order to derive the thrust-coefficient distribution which maximizes the wind-farm energy extraction. We use a continuous adjoint method to efficiently compute gradients for the optimization algorithm, which is based on a quasi-Newton method. Energy gains are evaluated with respect to a reference thrust-coefficient distribution based on the Betz–Joukowsky set point. We consider thrust coefficients that can change in space, as well as in time, i.e. considering time-periodic signals. However, in all our optimization results, we find that optimal thrust-coefficient distributions are steady; any time-periodic distribution is less optimal. The (steady) optimal thrust-coefficient distribution is inversely related to the vertical displacement of the boundary layer. Hence, it assumes a sinusoidal behaviour in the streamwise direction in subcritical flow conditions, whereas it becomes a U-shaped curve when the flow is supercritical. The sensitivity of the energy gain to the atmospheric state is studied using the developed optimization tool for almost two thousand different atmospheric states. Energy gains of up to 14 % are found for weakly stratified atmospheres in critical flow regimes.