**Large-eddy simulation of turbulent flows with applications to atmospheric boundary layer research Large-eddy simulation of turbulent flows with applications to atmospheric boundary layer research**

Hao Lu Hao Lu\*

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/57051 10.5772/57051

**1. Introduction**

In 1932, the British physicist Sir Horace Lamb, in an address to the British Association for the Advancement of Science, reportedly said, "I am an old man now, and when I die and go to heaven there are two matters on which I hope for enlightenment. One is quantum electrodynamics, and the other is the turbulent motion of fluids. And about the former I am rather optimistic."

Why then is the problem of turbulence so difficult? One reason is that the governing equations of turbulence are nonlinear partial differential equations, and appear to be insoluble. There are only partial proofs for the existence, uniqueness and regularity of solutions. What is more, these proofs correspond to simplified cases. It is not clear whether the equations themselves have some hidden randomness, or just the solutions. And if the latter, is it a consequence of the equations, or a consequence of the initial conditions?

With increased computing power over the last three decades, researchers can numerically solve the governing equations to obtain a complete description of a turbulent flow, where the flow variable (e.g., velocity, temperature, and pressure) is expressed as a function of space and time. The direct numerical simulation (DNS) of turbulence is the most straightforward approach to the solution of turbulent flows; however, DNS of high-Reynolds-number flow like atmospheric boundary layer (ABL) is not possible with today's computer resources. Large-eddy simulation (LES) has been introduced to simulate turbulence since the 1960s [82]. In LES, the large-scale motions of the flow are calculated, while the effects of the smaller scales are modeled through the use of a sub-grid scale (SGS) model. The main advantage of LES over computationally cheaper Reynolds-averaged Navier-Stokes (RANS) is the increased level of detail that LES can deliver [e.g., 80]. While RANS provides "averaged" results, LES can potentially provide the kind of high-resolution spatial and temporal information needed for applications.

The ABL is the lowest part of the atmosphere which is in direct interaction with the Earth's surface and responds to surface forcing with time scales of one hour or less. It is a highly

Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2014 Lu; licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

©2012 Lu, licensee InTech. This is an open access chapter distributed under the terms of the Creative

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

10.5772/57051

http://dx.doi.org/10.5772/57051

represents

appears in the

. (4)

is the

193

where (*<sup>u</sup>*1, *<sup>u</sup>*2, *<sup>u</sup>*<sup>3</sup>)=(*<sup>u</sup>*, *<sup>v</sup>*, *<sup>w</sup>*) are the components of the resolved velocity field, *<sup>θ</sup>*

−→<sup>Ω</sup> = (0, 0, <sup>Ω</sup>). The effects of the sub-grid scales on the evolution of *<sup>u</sup><sup>i</sup>* and *<sup>θ</sup>*

*<sup>τ</sup>ij* <sup>=</sup> *<sup>u</sup>iuj* <sup>−</sup> *<sup>u</sup>iu<sup>j</sup>*, and *qi* <sup>=</sup> *<sup>u</sup>*

SGS models are needed to parameterize *τij* and *qi* as a function of the resolved velocity and

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

Base on the Boussinesq hypothesis [14], a common class of SGS models, eddy-viscosity/diffusivity model, parameterizes the SGS stress' deviatoric part and

*<sup>δ</sup>ijτkk* <sup>=</sup> <sup>−</sup>2*νsgsSij*, and *qi* <sup>=</sup> <sup>−</sup> *<sup>ν</sup>sgs*

viscosity and *Scsgs* is the SGS Schmidt number. Several different models have been used to determine the eddy viscosity. The most common one was introduced by Smagorinsky [82] by assuming a local equilibrium between production and dissipation of SGS kinetic energy.

> *CS*<sup>∆</sup> <sup>2</sup> *S*

Smagorinsky coefficient. In isotropic turbulence, if a cut-off filter is used in the inertial subrange and the filter scale <sup>∆</sup> is equal to the grid size, then *Cs* <sup>≈</sup> 0.17 and *Scsgs* <sup>≈</sup> 0.5 [6, 53]. However, flow anisotropy, particularly the presence of a strong mean shear near

*νsgs* =

SGS stress *τij* and the SGS flux *qi*, respectively. They are defined as

diffusivity, and *f*

scalar fields.

approach.

the SGS flux as

where *<sup>S</sup>ij* <sup>=</sup> <sup>1</sup>

where

 *S* = 2 *∂ui ∂xj*

<sup>2</sup>*<sup>S</sup>ijSij*<sup>1</sup> 2

force would be included as *f*

**3. Subgrid-scale modeling**

and the Coriolis force would be included as *f*

**3.1. Standard eddy-viscosity/diffusivity models**

*<sup>τ</sup>ij* <sup>−</sup> <sup>1</sup> 3

The Smagorinsky model computes the eddy viscosity as

<sup>+</sup> *<sup>∂</sup><sup>u</sup><sup>j</sup> ∂xi*

resolved scalar, *p* is the effective pressure, *ν* is the kinematic viscosity, *κ* is the scalar

 *<sup>i</sup>* = *δi*3*g*

the resolved potential temperature, *<sup>θ</sup>*<sup>0</sup> is the reference temperature, �·�*<sup>H</sup>* denotes a horizontal average, *g* is the gravitational acceleration, *fc* is the Coriolis parameter, *δij* is the Kronecker delta, and *εijk* is the alternating unit tensor. In homogeneous rotating turbulence, the Coriolis

*<sup>i</sup>* is a forcing term. In the stable/unstable ABL flow, the buoyancy force

Large-eddy simulation of turbulent flows with applications to atmospheric boundary layer research

*θ* −�*<sup>θ</sup>* �*H*

*<sup>i</sup>* = −2*εij*3Ω*ju<sup>k</sup>*, and without loss of generality, we would chose

*<sup>i</sup><sup>θ</sup>* <sup>−</sup> *<sup>u</sup><sup>i</sup><sup>θ</sup>*

*Scsgs*

is the resolved (filtered) strain rate tensor, *νsgs* is the SGS eddy

is the strain rate, and *CS* is a non-dimensional parameter called

*∂θ ∂xi*

, (6)

, (5)

*<sup>θ</sup>*<sup>0</sup> <sup>+</sup> *fcεij*3*<sup>u</sup><sup>j</sup>*, where *<sup>θ</sup>*

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

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 LES a more reliable tool to study turbulent flows.
