Analytical wind turbine wake models are widely used to predict the wake velocity deficit. In these models, the wake growth rate is a key parameter specified mainly with empirical formulations. In this study, a new physics-based model is proposed and validated to predict the wake expansion downstream of a turbine based on the incoming ambient turbulence and turbine operating conditions. The new model utilises Taylor diffusion theory, the Gaussian wake model, turbulent mixing layer theory and the analogy between wind turbine wake expansion and scalar diffusion. These components ensure that the model conserves mass and momentum in the far wake and accounts for the ambient turbulence and turbine-induced turbulence effects on the wake expansion. To account for the turbulence relevant scales that contribute to the wake expansion, the model uses the root-mean-square of the low-pass filtered radial velocity component. A simplified version that only requires the unfiltered velocity standard deviation and turbulence integral scale is also proposed. In addition, a new relation for the near-wake length is derived. The model performance is validated using large-eddy simulation data of a wind turbine wake under neutral atmospheric conditions with a wide range of incoming turbulence levels. The results show that the proposed model yields reasonable predictions of the wake width, maximum velocity deficit and near-wake length. In the case with a relatively low incoming streamwise turbulence intensity of 0.05, the ambient and turbine-induced terms in the model contribute almost equally to the wake width, rendering them both crucial for reasonable wake predictions.

In this study, stereo particle imaging velocimetry measurements are performed at multiple streamwise locations behind a yawed wind turbine to reveal the formation mechanisms of the counter-rotating vortex pair (CVP), and a point vortex transportation (PVT) model is proposed to reproduce the top–down asymmetric kidney-shaped wake (also referred to as a curled wake). Results indicate that the CVP formed behind a yawed wind turbine originates from the complex interactions between the hub vortex and the streamwise components of the blade tip vortices, which is fundamentally different from the case of a yawed drag disk where the hub vortex is absent. Specifically, when the yaw angle exceeds a critical value, a small part of the streamwise vorticity shed from the rotor disk edge switches its sign from negative to positive and subsequently merges with the concentrated hub vortex under mutual induction, creating a patch of positive vorticity; meanwhile, the remaining streamwise vorticity distributed along the rotor edge curls and evolves into another patch of negative vorticity. These two patches of streamwise vorticity essentially constitute the CVP. Based on the physics learnt from the experiments, the non-uniform cross-stream velocity fields are first reconstructed by a cloud of point vortices distributed along the rotor edge and a hub vortex located in the rotor centre, and subsequently used to numerically solve a simplified transportation–diffusion equation of the wake velocity deficit, which altogether constitute the PVT model. This physics-based reduced-order model is the first model capable of accurately reproducing the wake deformation behind a yawed wind turbine.

Analytical wind turbine wake models and wake superposition methods are prevailing tools widely adopted by the wind energy community to predict the power production of wind farms. However, none of the existing wake superposition methods conserve the streamwise momentum. In this study, a novel wake superposition method capable of conserving the total momentum deficit in the streamwise direction is derived theoretically, and its performance is validated with both particle imaging velocimetry measurements and large-eddy simulation results. Detailed inter-method comparisons show that the novel wake superposition method outperforms all the existing methods by delivering an accurate prediction of the power production and the centreline wake velocity deficit, with a typical error of less than 5 % (excluding the near-wake region). Additionally, the momentum-conserving wake superposition method is extended to combine the transverse velocities induced by yawed wind turbines, and the secondary wake steering effect crucial to the power optimization in active wake control is well reproduced.

Variability of the rotor aspect ratio is one of the inherent characteristics of vertical-axis wind turbines (VAWTs) which differentiates them especially from the more conventional horizontal-axis wind turbines. In this study, we intend to investigate the effect of rotor aspect ratio on VAWT wakes. In particular, we aim to find out whether a common behaviour exists in the mean flow field of such wakes. In order to do so, we first design and perform a set of numerical experiments (using our already validated large-eddy simulation framework) to obtain the mean flow field of the wakes of three VAWTs of different aspect ratio (2, 1 and ${\textstyle \frac{1}{4}}$) and the same thrust coefficient ($C_{T}=0.8$). After observing the obvious differences in these three wakes, by using the classical momentum integral and the concept of momentum diameter, we come up with an appropriate normalization length scale $D_{eq}=\sqrt{(4/\unicode[STIX]{x03C0})DH}$, where $D$ is the rotor diameter and $H$ is the rotor height. By normalizing the lengths (both streamwise and lateral) involved in the mean velocity profiles by $D_{eq}$, we obtain a remarkable collapse of the wake profiles for the three aspect ratios. As a corollary, cross-sections of wakes of turbines with different aspect ratios eventually converge to a circular shape – not an elliptical one, for example, as one might presume intuitively. This result influences the modelling of VAWT far wakes and, in turn, has implications on the optimal configuration of VAWT farms.

Understanding and predicting the behaviour of wind turbine wake flows over hills is important for optimal design of wind-farm configurations on topography. In this study, we present an analytical modelling framework together with large-eddy simulation (LES) results to investigate turbine wakes over two-dimensional hills. The analytical model consists of two steps. In the first step, we deal with the effect of the pressure gradient on the wake evolution; and in the second step, we consider the effect of the hill-induced streamline distortion on the wake. This model enables us to obtain the wake recovery rate, the mean velocity and velocity deficit profiles and the wake trajectory in the presence of the hill. Moreover, we perform LES to test our model and also to obtain new complementary insight about such flows. Especially, we take advantage of the LES data to perform a special analysis of the behaviour of the wake on the leeward side of the hill. It is found that the mainly favourable pressure gradient on the windward side of the hill accelerates the wake recovery and the adverse pressure gradient on the leeward side decelerates it. The wake trajectory for a hill of the same height as the turbine’s hub height is found to closely follow the hill profile on the windward side, but it maintains an almost constant elevation (a horizontal line) downstream of the hilltop. The trajectory of the wake on the leeward side is also studied for a limiting case of an escarpment, and it is shown that an internal boundary layer forms on the plateau which leads to an upward displacement of the wake centre. Finally, a parametric study of the position of the turbine with respect to the hill is performed to further elucidate the effect of the hill-induced pressure gradient on the wind turbine wake recovery.

Turbulent axisymmetric wakes under pressure gradient have received little attention in the literature, in spite of their fundamental and practical importance, for example, in the case of wind turbine wakes over topography. In this paper, we develop an analytical framework to analyse turbulent axisymmetric wakes under different pressure gradient conditions. Specifically, we develop a model to predict how an arbitrary imposed pressure gradient perturbs the evolution of the zero-pressure-gradient wake. The starting point of the model is the basic mean conservation of the streamwise momentum equation. We take advantage of the self-similarity of the wake velocity deficit and the assumption that the ratio of the maximum velocity deficit to the wake width is independent of the pressure gradient; such an assumption is supported experimentally for planar wakes, and numerically for axisymmetric wakes in this study. Furthermore, an asymptotic solution for the problem is also derived. The problem is considered for both an axisymmetric strain and a planar strain. The inputs to the model are the imposed pressure gradient and the wake width in the zero-pressure-gradient case. To validate the model results, a set of large-eddy simulations (LES) are performed. Comparing the evolution of the maximum velocity deficit and the wake width, the model results and the LES data show good agreement. Similarly to planar wakes, it is observed that the axisymmetric wake recovers faster in the favourable pressure gradient compared with the adverse one.

Accurate prediction of the spatial evolution of turbulent wake flows under pressure gradient conditions is required in some engineering applications such as the design of high-lift devices and wind farms over topography. In this paper, we aim to develop an analytical model to predict the evolution of a turbulent planar wake under an arbitrary pressure gradient condition. The model is based on the cross-stream integration of the streamwise momentum equation and uses the self-similarity of the mean flow. We have also made an experimentally supported assumption that the ratio of the maximum velocity deficit to the wake width is independent of the imposed pressure gradient. The asymptotic response of the wake to the pressure gradient is also investigated. After its derivation, the model is successfully validated against experimental data by comparing the evolution of the wake width and maximum velocity deficit. The inputs of the model are the imposed pressure gradient and the wake width under zero pressure gradient. The model does not require any parameter tuning and is deemed to be practical, computationally fast, accurate enough, and therefore useful for the scientific and engineering communities.

This work is dedicated to systematically studying and predicting the wake characteristics of a yawed wind turbine immersed in a turbulent boundary layer. To achieve this goal, wind tunnel experiments were performed to characterize the wake of a horizontal-axis wind turbine model. A high-resolution stereoscopic particle image velocimetry system was used to measure the three velocity components in the turbine wake under different yaw angles and tip-speed ratios. Moreover, power and thrust measurements were carried out to analyse the performance of the wind turbine. These detailed wind tunnel measurements were then used to perform a budget study of the continuity and Reynolds-averaged Navier–Stokes equations for the wake of a yawed turbine. This theoretical analysis revealed some notable features of the wakes of yawed turbines, such as the asymmetric distribution of the wake skew angle with respect to the wake centre. Under highly yawed conditions, the formation of a counter-rotating vortex pair in the wake cross-section as well as the vertical displacement of the wake centre were shown and analysed. Finally, this study enabled us to develop general governing equations upon which a simple and computationally inexpensive analytical model was built. The proposed model aims at predicting the wake deflection and the far-wake velocity distribution for yawed turbines. Comparisons of model predictions with the wind tunnel measurements show that this simple model can acceptably predict the velocity distribution in the far wake of a yawed turbine. Apart from the ability of the model to predict wake flows in yawed conditions, it can provide valuable physical insight on the behaviour of turbine wakes in this complex situation.

An experimental investigation was conducted to study the wall boundary condition for large-eddy simulation (LES) of a turbulent boundary layer at Rθ = 3500. Most boundary condition formulations for LES require the specification of the instantaneous filtered wall shear stress field based upon the filtered velocity field at the closest grid point above the wall. Three conventional boundary conditions are tested using simultaneously obtained filtered wall shear stress and streamwise and wall-normal velocities, at locations nominally within the log region of the flow. This was done using arrays of hot-film sensors and X-wire probes. The results indicate that models based on streamwise velocity perform better than those using the wall-normal velocity, but overall significant discrepancies were found for all three models. A new model is proposed which gives better agreement with the shear stress measured at the wall. The new model is also based on the streamwise velocity but is formulated so as to be consistent with ‘outer-flow’ scaling similarity of the streamwise velocity spectra. It is therefore expected to be more generally applicable over a larger range of Reynolds numbers at any first-grid position within the log region of the boundary layer.

A scale-dependent dynamic subgrid-scale model for large-eddy simulation of turbulent flows is proposed. Unlike the traditional dynamic model, it does not rely on the assumption that the model coefficient is scale invariant. The model is based on a second test-filtering operation which allows us to determine from the simulation how the coefficient varies with scale. The scale-dependent model is tested in simulations of a neutral atmospheric boundary layer. In this application, near the ground the grid scale is by necessity comparable to the local integral scale (of the order of the distance to the wall). With the grid scale and/or the test-filter scale being outside the inertial range, scale invariance is broken. The results are compared with those from (a) the traditional Smagorinsky model that requires specification of the coefficient and of a wall damping function, and (b) the standard dynamic model that assumes scale invariance of the coefficient. In the near-surface region the traditional Smagorinsky and standard dynamic models are too dissipative and not dissipative enough, respectively. Simulations with the scale-dependent dynamic model yield the expected trends of the coefficient as a function of scale and give improved predictions of velocity spectra at different heights from the ground. Consistent with the improved dissipation characteristics, the scale-dependent model also yields improved mean velocity profiles.