It is well known that when wind turbines are deployed in large arrays, their efficiency decreases due to complex interactions among themselves and with the atmospheric boundary layer ͑ABL͒. For wind farms whose length exceeds the height of the ABL by over an order of magnitude, a "fully developed" flow regime can be established. In this asymptotic regime, changes in the streamwise direction can be neglected and the relevant exchanges occur in the vertical direction. Such a fully developed wind-turbine array boundary layer ͑WTABL͒ has not been studied systematically before. A suite of large eddy simulations ͑LES͒, in which wind turbines are modeled using the classical "drag disk" concept, is performed for various wind-turbine arrangements, turbine loading factors, and surface roughness values. The results are used to quantify the vertical transport of momentum and kinetic energy across the boundary layer. It is shown that the vertical fluxes of kinetic energy are of the same order of magnitude as the power extracted by the forces modeling the wind turbines. In the fully developed WTABL, the kinetic energy extracted by the wind turbines is transported into the wind-turbine region by vertical fluxes associated with turbulence. The results are also used to develop improved models for effective roughness length scales experienced by the ABL. The effective roughness scale is often used to model wind-turbine arrays in simulations of atmospheric dynamics at larger ͑regional and global͒ scales. The results from the LES are compared to several existing models for effective roughness lengths. Based on the observed trends, a modified model is proposed, showing improvement in the predicted effective roughness length.

▪ Abstract  Relationships between small and large scales of motion in turbulent flows are of much interest in large-eddy simulation of turbulence, in which small scales are not explicitly resolved and must be modeled. This paper reviews models that are based on scale-invariance properties of high-Reynolds-number turbulence in the inertial range. The review starts with the Smagorinsky model, but the focus is on dynamic and similarity subgrid models and on evaluating how well these models reproduce the true impact of the small scales on large-scale physics and how they perform in numerical simulations. Various criteria to evaluate the model performance are discussed, including the so-called a posteriori and a priori studies based on direct numerical simulation and experimental data. Issues are addressed mainly in the context of canonical, incompressible flows, but extensions to scalar-transport, compressible, and reacting flows are also mentioned. Other recent modeling approaches are briefly introduced.

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A simple model is presented for the energy-cascading process in the inertial range that fits remarkably well the entire spectrum of scaling exponents for the dissipation field in fully developed turbulence. The scheme is a special case of weighted curdling and its one-dimensional version is a simple generalized two-scale Cantor set with equal scales but unequal weights (with ratio -f ). This set displays all the measured multifractal properties of one-dimensional sections of the dissipation field.PACS numbers: 47.25.-cThe statistical and scaling properties of the field of turbulent energy dissipation have been the subject of a number of studies/"^ and their proper description and modeling are vital for the understanding of fully developed turbulence. To explain the intermittent character of the rate of dissipation e of turbulent kinetic energy, Kolmogorov^ formulated his third hypothesis invoking some statistical independence in the cascading process, which led to the log-normal model for e. This model presents internal inconsistencies and was later shown to be only one of many possibilities.^'"*'^ Mandelbrot' introduced a fractal model for 6, and distinguished between absolute and weighted curdling. The former became well known as the p model, ^ where the flux of energy is transferred to only a fixed fraction /3 of the eddies of smaller scales. This model, as well as the log-normal one, are contradicted by more recent experiments^ concerning scaling exponents of velocity structure functions of high order. To correct this, Frisch and Parisi^ proposed the multifractal model, where singularities of e of different strengths are distributed on interwoven sets of diff'erent fractal dimension. We have shown elsewhere^ that this description is consistent with experiments and that it provides a unifying framework for describing a variety of observations in fully developed turbulence. Benzi et al.^ showed that multifractal distributions can be obtained if (5 is made a random variable in the p model. Another possibility is the weighted curdling introduced by Mandelbrot.^ We will show that the simplest case of nonabsolute curdling describes (at a certain level of description) amazingly well the observed multifractal behavior of 6, and argue that from a metric point of view, fully developed turbulence and the proposed model (p model) are the same within experimental accuracy. As a detailed test of the p model, we compare its predictions with the entire spectrum of generalized dimensions,^^ and (equivalently) the singularity spectrum (the so-called f-a curve'0, both of which have been measured by us^ for the energy-dissipation field in several turbulent flows (see Figs. 1 and 2).Let Er be the dissipation that occurs in a domain of size r. It is convenient to think of Er as the flux of kinetic energy^ from eddies of size r to eddies of smaller size-this flux being actually dissipation when these eddies are of the order of the Kolmogorov length scale rj. This is the classical view of the eddy cascade in the inertial range, and ...

The intermittency of the rate of turbulent energy dissipation ε is investigated experimentally, with special emphasis on its scale-similar facets. This is done using a general formulation in terms of multifractals, and by interpreting measurements in that light. The concept of multiplicative processes in turbulence is (heuristically) shown to lead to multifractal distributions, whose formalism is described in some detail. To prepare proper ground for the interpretation of experimental results, a variety of cascade models is reviewed and their physical contents are analysed qualitatively. Point-probe measurements of ε are made in several laboratory flows and in the atmospheric surface layer, using Taylor's frozen-flow hypothesis. The multifractal spectrum f(α) of ε is measured using different averaging techniques, and the results are shown to be in essential agreement among themselves and with our earlier ones. Also, long data sets obtained in two laboratory flows are used to obtain the latent part of the f(α) curve, confirming Mandelbrot's idea that it can in principle be obtained from linear cuts through a three-dimensional distribution. The tails of distributions of box-averaged dissipation are found to be of the square-root exponential type, and the implications of this finding for the f(α) distribution are discussed. A comparison of the results to a variety of cascade models shows that binomial models give the simplest possible mechanism that reproduces most of the observations. Generalizations to multinomial models are discussed.