**3. Subgrid-scale modeling**

2 Computational and Numerical Simulations

turbulent boundary-layer flow with a Reynolds number of order *Re* ∼ 108 or higher. The ABL flow has a huge continuous range of turbulent eddy scales, ranging from the integral scale, on the order of *L* ∼ *O*(1 km), down to the Kolmogorov viscous dissipation scale *η* ∼ *O*(1 mm). Prediction of ABL flow is complicated by the often strong temporal and spatial variability of the land-surface characteristics (e.g., surface temperature and aerodynamic roughness). Moreover, land surfaces are often characterized by complex topography, which is in many cases multifractal [79], as well as spatial heterogeneity of aerodynamic roughness and temperature associated with different land-cover types. This leads to highly non-linear

Accurate modeling turbulent transport of momentum and scalars in ABL is of great importance to forecast weather, climate, air pollution, wind loads on structures, and wind energy resources. Of special relevance are the seminal works of Deardorff, who first performed actual LESs of channel flow [23] and ABL flow [24]. In the last decades, LES has become a powerful tool to study turbulent transport and mixing in the ABL. Numerical simulations have been used to investigate the impact of different surface types (homogeneous, heterogeneous, flat, complex topography) on turbulent fluxes of momentum and scalars, such as temperature, water vapor and pollutants [e.g., 1, 2, 11, 25, 48, 55, 72, 87, 96, 99]. Recently, LES studies of the interaction between ABL turbulence and wind turbines, and the interference effects among wind turbines have been carried out, in order to understand the impact of wind farms on local meteorology as well as to optimize the design

However, there are still some open issues that need to be addressed in order to make LES a more accurate tool for turbulence simulations. The main weakness of LES is associated with our limited ability to accurately account for the dynamics that are not explicitly resolved in the simulations (because they occur at scales smaller than the grid size). Here, we present a summary of our recent efforts to improve subgrid-scale parameterizations and, thus, to make

Along with laminar-turbulent transition, turbulence parameterization constitutes the most critical part of flow modeling [68, 80]. In high-Reynolds-number turbulent flows computational limitations impose the choice of a grid size <sup>∆</sup>*grid* substantially larger than the smallest scale of motion. In LES, the separation of scales between resolved and unresolved scales is achieved by filtering (with a spatial filter of characteristic size <sup>∆</sup> <sup>∆</sup>*grid*) the equations describing the transport of mass, momentum and scalar quantities. In particular, the filtered equations (using the Boussinesq approximation) governing continuity,

= 0 , (1)

+ *f* 

*<sup>i</sup>* , (2)

, (3)

*<sup>∂</sup>*2*<sup>u</sup><sup>i</sup> ∂xj∂xj*

interactions between the complexity of the land surfaces and the ABL flow.

(turbine siting) of wind energy projects [e.g., 16, 36, 37, 59, 78, 97].

LES a more reliable tool to study turbulent flows.

conservation of momentum and scalar transport are

*∂ui <sup>∂</sup><sup>t</sup>* <sup>+</sup>

> *∂θ <sup>∂</sup><sup>t</sup>* <sup>+</sup> *<sup>u</sup><sup>i</sup>*

*∂ui ∂xi*

*<sup>∂</sup><sup>u</sup>iu<sup>j</sup> ∂xj*

> *∂θ ∂xi*

<sup>=</sup> <sup>−</sup> *<sup>∂</sup><sup>p</sup> ∂xi*

<sup>=</sup> <sup>−</sup> *<sup>∂</sup>qi ∂xi* + *κ*

<sup>−</sup> *∂τij ∂xj* + *ν*

> *∂*2*θ ∂xi∂xi*

**2. LES governing equations**

This section provides a brief overview of standard eddy-viscosity/diffusivity models. On the basis of some mathematical and physical constraints, we developed a new nonlinear approach.
