As wind farms continue to grow in size, mesoscale effects such as blockage and gravity waves become increasingly important. Allaerts & Meyers (J. Fluid Mech., vol. 862, 2019, pp. 990–1028) proposed an atmospheric perturbation model (APM) that can simulate the interaction of wind farms and the atmospheric boundary layer while keeping computational costs low. The model resolves the mesoscale flow, and couples to a wake model to estimate the turbine inflow velocities at the microscale. This study presents a new way of coupling the mesoscale APM to a wake model, based on matching the velocity between the models throughout the farm. This method performs well, but requires good estimates of the turbine-level velocity fields by the wake model. Additionally, we investigate the mesoscale effects of a large wind farm, and find that aside from the turbine forces and increased turbulence levels, the dispersive stresses due to subgrid flow heterogeneity also play an important role at the entrance of the farm, and contribute to the global blockage effect. By using the wake model coupling, we can explicitly incorporate these stresses in the model. The resulting APM is validated using 27 prior large-eddy simulations of a large wind farm under different atmospheric conditions. The APM and large-eddy simulation results are compared on both mesoscale and turbine scale, and on turbine power output. The APM captures the overall effects that gravity waves have on wind farm power production, and significantly outperforms standard wake models.

We propose an efficient method to reconstruct the turbulent flow field in a neutrally stratified atmospheric boundary layer using large-eddy simulation (LES) and a series of lidar measurements. The reconstruction is formulated as a strong four-dimensional variational data assimilation problem, which involves optimizing two competing terms that contribute in the objective functional. The first term is a likelihood term, while the second contains the initial background distribution of turbulent velocity fluctuations and works as a regularization term. However, computing and storing the full background covariance tensor in turbulent flows is time consuming and resource intensive. In the current work, we investigate the possibility of replacing the complex background tensor by simple analytical approximations based on spectral tensors such as the Hunt–Graham–Wilson (HGW) model (Boundary-Layer Meteorol., vol. 85, 1997, pp. 35–52) or the Mann model (J. Fluid Mech., vol. 273, 1994, pp. 141–168). Afterwards, the problem is solved using a quasi-Newton algorithm and preconditioned to enhance the convergence rate. We test the method using virtual lidar measurements collected on a fine reference LES. Results show a super-linear convergence rate of the optimization algorithm to a local minimum and very good agreement between virtual lidar measurements and reconstruction in the scanning region. Furthermore, we demonstrate that incorporating the Saffman energy spectrum ($E(k) \sim k^2$ where E is the energy spectrum and k is the magnitude of the wavenumber vector) at low wavenumbers into the Mann spectral tensor yields a longer streamwise correlation length, resulting in reduced reconstruction error when compared with the Batchelor spectrum ($E(k) \sim k^4$). Finally, we observe that using the HGW model or Mann model with a Saffman spectrum yields similar results.

We investigate the reconstruction of a turbulent flow field in the atmospheric boundary layer from a time series of lidar measurements, using large-eddy simulations (LES) and a four-dimensional variational data assimilation algorithm. This leads to an optimisation problem in which the error between measurements and simulations is minimised over an observation time horizon. We also consider reconstruction based on a Taylor's frozen turbulence (TFT) model as a point of comparison. To evaluate the approach, we construct a series of virtual lidar measurements from a fine-grid LES of a pressure-driven boundary layer. The reconstruction uses LES on a coarser mesh and smaller domain, and results are compared to the fine-grid reference. Two lidar scanning modes are considered: a classical plan-position-indicator mode, which swipes the lidar beam in a horizontal plane, and a three-dimensional pattern that is based on a Lissajous curve. We find that normalised errors lie between $15\,\%$ and $25\,\%$ (error variance normalised by background variance) in the scanning region, and increase to $100\,\%$ over a distance that is comparable to the correlation length scale outside this scanning region. Moreover, LES outperforms TFT by 30 %–70 % depending on scanning mode and location.

In rough-wall boundary layers, wall-parallel non-homogeneous mean-flow solutions exist that lead to so-called dispersive velocity components and dispersive stresses. They play a significant role in the mean-flow momentum balance near the wall, but typically disappear in the outer layer. A theoretical framework is presented to study the decay of dispersive motions in the outer layer. To this end, the problem is formulated in Fourier space, and a set of governing ordinary differential equations per mode in wavenumber space is derived by linearizing the Reynolds-averaged Navier–Stokes equations around a constant background velocity. With further simplifications, analytically tractable solutions are found consisting of linear combinations of $\exp (-kz)$ and $\exp (-Kz)$, with $z$ the wall distance, $k$ the magnitude of the horizontal wavevector $\boldsymbol{k}$, and where $K(\boldsymbol{k},Re)$ is a function of $\boldsymbol{k}$ and the Reynolds number $Re$. Moreover, for $k\rightarrow \infty$ or $k_{1}\rightarrow 0$ (with $k_{1}$ the stream-wise wavenumber), $K\rightarrow k$ is found, in which case solutions consist of a linear combination of $\exp (-kz)$ and $z\exp (-kz)$, and are independent of the Reynolds number. These analytical relations are compared in the limit of $k_{1}=0$ to the rough boundary layer experiments by Vanderwel & Ganapathisubramani (J. Fluid Mech., vol. 774, 2015, R2) and are in reasonable agreement for $\ell _{k}/\unicode[STIX]{x1D6FF}\leqslant 0.5$, with $\unicode[STIX]{x1D6FF}$ the boundary-layer thickness and $\ell _{k}=2\unicode[STIX]{x03C0}/k$.

Flow blockage by large wind farms leads to an upward displacement of the boundary layer, which may excite atmospheric gravity waves in the free atmosphere and on the interface between the boundary layer and the free atmosphere. In the current study, we assess the sensitivity of wind-farm gravity-wave excitation to important dimensionless groups and investigate the feedback of gravity-wave-induced pressure fields to wind-farm energy extraction. The sensitivity analysis is performed using a fast boundary-layer model that is developed to this end. It is based on a three-layer representation of the atmosphere in an idealised barotropic environment, and is coupled with an analytical wake model to account for turbine wake interactions. We first validate the model in two-dimensional mode with data from previous large-eddy simulations of ‘infinitely’ wide wind farms, and then use the model to investigate the sensitivity of wind-farm-induced gravity waves to atmospheric state and wind-farm configuration. We find that the inversion layer induces flow physics similar to shallow-water flow and that the corresponding Froude number plays a crucial role. Gravity-wave excitation is maximal at a critical Froude number equal to one, but the feedback on energy extraction is highest when the Froude number is slightly below one due to a trade-off between amplitude and upstream impact of gravity waves. The effect of surface friction and internal gravity waves is to reduce the flow perturbation and the related power loss by dissipating or dispersing perturbation energy. With respect to the wind-farm configuration, we find that gravity-wave-induced power loss increases with wind-farm size and turbine height. Moreover, we find that gravity-wave effects are small for very wide or very long wind farms and attain a maximum at a width-to-depth ratio of approximately $3/2$.

Recent experiments have revealed the existence of very long streamwise features, denoted as very-large-scale motions (VLSMs), in the thermally neutral atmospheric boundary layer (ABL) (Hutchins et al., Boundary-Layer Meteorol., vol. 145(2), 2012, pp. 273–306). The aim of our study is to elaborate the role of these large-scale anisotropic patterns in wind-energy harvesting with special emphasis on the organization of turbulent fields around wind turbines. To this end, we perform large-eddy simulation (LES) of a turbine row operating under neutral conditions. The ABL data are produced separately in a very long domain of $240\unicode[STIX]{x1D6FF}$, where $\unicode[STIX]{x1D6FF}$ is the ABL thickness, to ensure a realistic representation for very large scales of $O(10\unicode[STIX]{x1D6FF})$. VLSMs are extracted from the LES database using a cutoff at streamwise wavelength $\unicode[STIX]{x1D706}_{x}=5\unicode[STIX]{x1D6FF}$, or $\unicode[STIX]{x1D706}_{x}=50D$ in terms of turbine diameter. Reynolds averaging of low-pass filtered fields shows that the interaction of VLSMs and turbines produce very-long-wavelength motions in the wake region, which contain approximately $20\,\%$ of the resolved Reynolds shear stress, and $30\,\%$ of the resolved streamwise kinetic energy in the shear layers. To further elucidate these statistics, we conduct a geometrical analysis using conditional averaging based on large-scale low- and high-velocity events. The conditional eddies provide evidence for very long (${\sim}10\unicode[STIX]{x1D6FF}$) and wide (${\sim}\unicode[STIX]{x1D6FF}$) streak–roller structures around the turbine row. Although all of these eddies share the same streak–roller topology, there are remarkable modifications in the morphology of the conditional eddies whose cores are located sideways to the turbines. In these cases, the turbine row pushes the whole low- or high-momentum streak aside, and prevails as a sharp boundary to the low–high-momentum streak pair. In this process, accompanying rollers remain relatively unaffected. This creates a two-way flux towards the turbine row. These observations provide some insights about the high lateral spreading observed in the large-scale Reynolds stress fields.

The fluctuations in power output from wind farms display significantly reduced spectra compared to single wind turbines due to power smoothing and averaging. In order to better understand these spectral features and to relate them to properties of turbulent boundary layers, we perform a wind tunnel experiment in which we measure spatio-temporal characteristics of an experimental surrogate of the power output from a micro wind farm with 100 porous disk models. The experimental results show that the frequency spectrum of the total wind farm power follows a power law with a slope between $-5/3$ and $-2$, and up to lower frequencies than seen for any individual turbine model. In agreement with previous studies in the literature, peaks in the spectrum are observed at frequencies corresponding to the mean flow convection time between consecutive turbines. In the current work we interpret the sum of power extraction from an array of turbines as a discrete spatial filtering of a turbulent boundary layer and derive the associated transfer function. We apply it to an existing model for the wavenumber–frequency spectrum of turbulent boundary layers. This approach allows us to verify the individual roles of Doppler shift and broadening of frequencies on the resulting spatially sampled frequency spectrum. Comparison with the wind tunnel data confirms that the approach captures and explains the main features in the spectrum, indicating the crucial role of the interaction between the spatial sampling and the space–time correlations inherently present in the flow. The frequency spectrum of the aggregated power from a wind farm thus depends on both the spectrum of the incoming turbulence and its modulation by the spatial distribution of turbines in the boundary layer flow.

While neutral atmospheric boundary layers are rare over land, they occur frequently over sea. In these cases they are almost always of the conventionally neutral type, in which the neutral boundary layer is capped by a strong inversion layer and a stably stratified atmosphere aloft. In the current study, we use large-eddy simulations (LES) to investigate the interaction between a large wind farm that has a fetch of 15 km and a conventionally neutral boundary layer (CNBL) in typical offshore conditions. At the domain inlet, we consider three different equilibrium CNBLs with heights of approximately 300 m, 500 m and 1000 m that are generated in a separate precursor LES. We find that the height of the inflow boundary layer has a significant impact on the wind farm flow development. First of all, above the farm, an internal boundary layer develops that interacts downwind with the capping inversion for the two lowest CNBL cases. Secondly, the upward displacement of the boundary layer by flow deceleration in the wind farm excites gravity waves in the inversion layer and the free atmosphere above. For the lower CNBL cases, these waves induce significant pressure gradients in the farm (both favourable and unfavourable depending on location and case). A detailed energy budget analysis in the turbine region shows that energy extracted by the wind turbines comes both from flow deceleration and from vertical turbulent entrainment. Though turbulent transport dominates near the end of the farm, flow deceleration remains significant, i.e. up to 35 % of the turbulent flux for the lowest CNBL case. In fact, while the turbulent fluxes are fully developed after eight turbine rows, the mean flow does not reach a stationary regime. A further energy budget analysis over the rest of the CNBL reveals that all energy available at turbine level comes from upwind kinetic energy in the boundary layer. In the lower CNBL cases, the pressure field induced by gravity waves plays an important role in redistributing this energy throughout the farm. Overall, in all cases entrainment at the capping inversion is negligible, and also the work done by the mean background pressure gradient, arising from the geostrophic balance in the free atmosphere, is small.

In very large wind farms, the vertical interaction with the atmospheric boundary layer plays an important role, i.e. the total energy extraction is governed by the vertical transport of kinetic energy from higher regions in the boundary layer towards the turbine level. In the current study, we investigate optimal control of wind-farm boundary layers, considering the individual wind turbines as flow actuators, whose energy extraction can be dynamically regulated in time so as to optimally influence the flow field and the vertical energy transport. To this end, we use large-eddy simulations of a fully developed pressure-driven wind-farm boundary layer in a receding-horizon optimal control framework. For the optimization of the wind-turbine controls, a conjugate-gradient optimization method is used in combination with adjoint large-eddy simulations for the determination of the gradients of the cost functional. In a first control study, wind-farm energy extraction is optimized in an aligned wind farm. Results are accumulated over one hour of operation. We find that the energy extraction is increased by 16 % compared to the uncontrolled reference. This is directly related to an increase of the vertical fluxes of energy towards the wind turbines, and vertical shear stresses increase considerably. A further analysis, decomposing the total stresses into dispersive and Reynolds stresses, shows that the dispersive stresses increase drastically, and that the Reynolds stresses decrease on average, but increase in the wake region, leading to better wake recovery. We further observe also that turbulent dissipation levels in the boundary layer increase, and overall the outer layer of the boundary layer enters into a transient decelerating regime, while the inner layer and the turbine region attain a new statistically steady equilibrium within approximately one wind-farm through-flow time. Two additional optimal control cases study penalization of turbulent dissipation. For the current wind-farm geometry, it is found that the ratio between wind-farm energy extraction and turbulent boundary-layer dissipation remains roughly around 70 %, but can be slightly increased by a few per cent by penalizing the dissipation in the optimization objective. For a pressure-driven boundary layer in equilibrium, we estimate that such a shift can lead to an increase in wind-farm energy extraction of 6 %.

As a generalization of the mass–flux based classical stream tube, the concept of momentum and energy transport tubes is discussed as a flow visualization tool. These transport tubes have the property that no fluxes of momentum or energy exist over their respective tube mantles. As an example application using data from large eddy simulation, such tubes are visualized for the mean-flow structure of turbulent flow in large wind farms, in fully developed wind-turbine-array boundary layers. The three-dimensional organization of energy transport tubes changes considerably when turbine spacings are varied, enabling the visualization of the path taken by the kinetic energy flux that is ultimately available at any given turbine within the array.

Following a method by Meldrum and van der Walt, near-rings of matrix maps are defined for general near-rings, not necessarily with identity. The influence of one-sided identities is discussed. When the base near-ring is integral and planar, the near-ring of matrix maps is shown to be simple. Various types of primitivity of the near-ring of matrix maps are discussed when the base near-ring is planar but not integral. Finally, an open problem concerning bijective matrix maps is solved.

We address the sensitivity of large-eddy simulations (LES) to parametric uncertainty in the subgrid-scale model. More specifically, we investigate the sensitivity of the LES statistical moments of decaying homogeneous isotropic turbulence to the uncertainty in the Smagorinsky model free parameter Cs (i.e. the Smagorinsky constant). Our sensitivity methodology relies on the non-intrusive approach of the generalized Polynomial Chaos (gPC) method; the gPC is a spectral non-statistical numerical method well-suited to representing random processes not restricted to Gaussian fields. The analysis is carried out at Reλ, =, 100 and for different grid resolutions and Cs distributions. Numerical predictions are also compared to direct numerical simulation evidence. We have shown that the different turbulent scales of the LES solution respond differently to the variability in Cs. In particular, the study of the relative turbulent kinetic energy distributions for different Cs distributions indicates that small scales are mainly affected by changes in the subgrid-model parametric uncertainty.

A theoretical analysis is presented on the behaviour of the model coefficients for the well-known Smagorinsky model and two variational multi-scale (VMS) variants of the Smagorinsky model. The dependency on two important parameters is addressed, i.e. the ratio of the LES-filter width $\varDelta$ and the Kolmogorov scale $\eta$ on the one hand, and the ratio of the integral length scale $L$ and the LES-filter width $\varDelta$ on the other hand. First of all, it is demonstrated that the model coefficients vary strongly with $\varDelta/\eta$. By evaluating the model coefficients as functions of the subgrid activity $s$ (which expresses the relative contribution of the subgrid-scale model in the total dissipation, and corresponds to a nonlinear transformation of $\varDelta/\eta$), we show that a classical Lilly–Smagorinsky model overestimates the dissipation, even in cases where the dissipation of the subgrid-scale model is dominant. Therefore, generic and easy-to-use modifications to the different models are proposed, which provide close approximations to the models employing ‘exact’ coefficients. For the standard Smagorinsky model, this modified model corresponds to approximating the eddy viscosity $\nu_t$ as $\nu_t\,{=}\,(\nu_{\mbox{\textit{\scriptsize Lilly}}}^2\,{+}\,\nu^2)^{1/2} -\nu$, with $\nu_{\mbox{\textit{\scriptsize Lilly}}}$ the turbulent viscosity obtained by employing Lilly's classical Smagorinsky constant and $\nu$ the laminar viscosity. Similar easy-to-use relations are presented for the variational multi-scale Smagorinsky models. Next to the $\varDelta/\eta$ dependence of the model coefficients, the $L/\varDelta$ behaviour is also elaborated. Although a strong dependence on $L/\varDelta$ is observed for low values of the ratio, we do not advocate the use of $L/\varDelta$-dependent model coefficients. Rather, the asymptotic $L/\varDelta$ independence and the speed of asymptotic convergence are used as a tool to compare the quality of subgrid-scale models (e.g. $L/\varDelta \,{>}\, 10$ is a minimum order of magnitude for the small–small VMS model), and differences are observed between the standard Smagorinsky model and its two VMS variants. Finally, for the VMS models, the influence of the shape of the high-pass filter, used in the variational multi-scale formulation, is investigated. We observed that smooth high-pass filters result in more robust VMS Smagorinsky models.

Ru Schottky barrier diodes (SBD's) were fabricated on the Zn face of n-type ZnO. These diodes were irradiated with 1.8 MeV at fluences ranging from 1 ´ 1013 cm-2 to 2.4 ´ 1014 cm-2. Capacitance and current (I) deep level transient spectroscopy (DLTS) was used to characterise the irradiation induced defects. Capacitance DLTS showed that proton irradiation introduced a level, Ep1, at 0.52 eV below the conduction band at an introduction rate of 13±1 cm-1. A defect with a very similar DLTS signature was also present in low concentrations in unirradiated ZnO. I-DLTS revealed that this proton irradiation introduced a defect with an energy level at (0.036± 0.004) eV below the conduction band. This defect is clearly distinguishable from a defect with a level at (0.033± 0.004) eV below the conduction band that was present in the unirradiated sample. It is speculated that these shallow level defects are related to zinc interstitials or complexes involving them.

The developmental interplay between nature and nurture is discussed, with particular reference to implications for research in developmental psychopathology. The general principles include individual differences in reactivity to the environment, two-way interplay between intraindividual biology and environmental influences, and the need to consider broader social contextual features. Individuals actively process their experiences; they also act on their environment to shape and select their experiences, and individual characteristics change over time. Key findings on genetic effects include their ubiquitous influence, the multifactorial origin of most psychopathology, the involvement of several genes in most mental disorders, some genetic effects operate through dimensional risk features rather than directly on disorder, some genetic effects are dependent on gene–environment correlations and interactions, and genetic effects increase with age. Key findings on environmental effects include their ubiquitous influence, the genetic mediation of some supposed environmental effects, the importance of passive gene–environment correlations, the paucity of evidence regarding environmental effects on lifetime liability to psychopathology, the lack of understanding of environmental effects on the organism, and the importance of nonshared environmental effects. Research strategies to investigate environmental risk mediation include the range of genetically sensitive designs, migration studies, secular trend investigations, studies of nonfamilial environments, and examination of intraindividual change in relation to measured environmental alterations. Proximal processes involved in person–environment interplay are discussed in relation to person–environment interactions and evocative and active person–environment correlations.