# Application of Different Turbulence Models Simulating Wind Flow in Complex Terrain: A Case Study for the WindForS Test Site

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## Abstract

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## 1. Introduction

- The most generally applicable approach is to use is the large-eddy simulation (LES). However, the maximum grid resolution is still strongly limited because of the available computer resources. Possibilities, as well as limitations of this modelling approach are given in Chow et al. [2] and Churchfield et al. [3].
- Another option is the detached-eddy simulation (DES) using unsteady Reynolds averaged Navier-Stokes equations (URANS) in combination with two-equation turbulence models in the vicinity of the walls of the computational domain and LES elsewhere [4]. For DES, the computational time is significantly reduced in comparison with LES. As a result of the necessity of a transient simulation and subsequent time averaging for a significant period of time, the computational effort and storage requirements still tend to be too high for practical applications even using massively parallel computers.
- Two-equation turbulence models mostly in combination with wall models, for example, standard k-ε or RNG k-ε models, offer great numerical stability combined with a relatively low demand on computational resources. Their application for the computation of wind flow in complex terrain with strong velocity gradients has shown to give acceptable results despite the assumption of an isotropic turbulence [5,6,7]. However, for many applications, it is stated that the turbulence intensity is not accurately predicted for the k-ε turbulence model and that flow separation is under-predicted. Kim and Patel [8] found that the RNG k-ε model was superior for simulating wind flow in complex terrain, especially for separating flow conditions. The RNG k-ε model also had been successfully used in real complex terrain by Abdi and Bitsuamlak [9].
- A Reynolds stress model (RSM) promises a more accurate description of the anisotropic turbulence in wind flow. However, the computational effort is increased in comparison with the two equation turbulence models and it is numerically not as stable.

_{0}in the logarithmic wall function [10]. Another possibility is the use of canopy models, introducing source terms in the momentum and turbulence equations as first suggested by Svensson and Häggkvist [11]. Similar approaches, all in combination with the standard k-ε model, have been adopted by Liu et al. [12] and Green [13]. Shaw and Schumann [14] set up a test case of a homogeneous forested area, which commonly is used for the verification of these canopy models. Lopes et al. [15] compared the different approaches using the data of Shaw and Schumann [14] and devised a further canopy model. The use of a RSM for canopy flow was first described by Wilson and Shaw [16]. Ayotte et al. [17] established a model aiming at the simulation of neutrally stratified flow in heterogeneous landscapes and compared the simulation results with a flat terrain dataset. Ayotte et al. [17] split the viscous dissipation into a contribution of the spectral eddy cascade, as well as a foliage contribution and the implementation of the RSM is based on the transport equation for the dissipation of the turbulence kinetic energy of the spectral eddy cascade. Dimitris and Panayotis [18] used a similar approach to Ayotte et al. [17]. However, the contribution of the vegetation is considered as source term directly in the transport equation for the total dissipation of turbulent kinetic energy. Dimitris and Panayotis [18] compared the simulation results with measurements from laboratory channels with aquatic vegetation. An application of these kinds of RSM capturing canopy effects for the micro-siting in complex terrain is not known to the authors, indicating that there is a strong need for validation.

## 2. Computational Model

#### 2.1. Continuity and Momentum Equation

_{0}= 288.15 K, p

_{0}= 100,000 Pa, R

_{d}= 287.05 J/(kg K), and g = 9.81 m/s

^{2}.

_{W,i}in the momentum equation (Equation (2)):

_{D}is the constant drag coefficient set to 0.30, and |u| is the magnitude of the velocity vector.

_{C,i}in Equation (2) are defined as follows:

^{−5}s

^{−1}.

_{t}:

_{t}can be expressed by the turbulent kinetic energy k and the dissipation of the turbulent kinetic energy ε assuming isotropic turbulence:

#### 2.2. Energy Equation

#### 2.3. Two Equation Turbulence Models

#### 2.3.1. Standard k-ε Turbulence Model

#### 2.3.2. RNG k-ε Turbulence Model

#### 2.3.3. Canopy Model Source Terms for Two Equation Turbulence Models

#### 2.4. A Second-Order Turbulence Closure

#### Reynolds Stress Turbulence Model (RSM)

_{ij}is the production term:

_{ij,1}and Φ

_{ij,2}are described according to Rotta [28] and Launder et al. [27], respectively:

## 3. Model Verification Using a Homogeneous Canopy Test Case

## 4. Model Setup for the WindForS Test Site

#### 4.1. Initial Conditions for the Parent Domain

#### 4.2. Boundary Conditions for Nested Domains

#### 4.3. Boundary Conditions at the Ground for Parent and Nested Domains

_{0}is set to 0.02 m and 2.67 m for un-forested and urban areas, respectively. For the forested areas, a z

_{0}value of 0.02 m in combination with the canopy models is applied. An LAI of five representing deciduous forest with the LAD distribution according to Figure 5 and a forest height of 20 m is used for the canopy model.

## 5. Results for the WindForS Test Site

#### 5.1. Qualitative Comparison of Simulation Results

#### 5.2. Validation of the Model by Means of Measurement Data

#### 5.2.1. Description of Measurements

#### Multipurpose Airborne Sensor Carrier (MASC)

#### Meteorological Mast

#### 5.2.2. Comparison of Simulation Results and UAV Measurements

#### 5.2.3. Comparison of Simulation Results and Measurements from a Meteorological Wind Mast

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**(

**a**) Normalized longitudinal velocity, original mesh; (

**b**) normalized longitudinal velocity, refined mesh; (

**c**) normalized turbulent kinetic energy, refined mesh; (

**d**) normalized Reynolds stress, refined mesh.

**Figure 3.**Nested domain 4 × 4 with vertical evaluation lines V1–V5 and horizontal evaluation lines 25 m (blue), 75 m (red), and 98 m (orange) above ground, as well as met mast position.

**Figure 4.**(

**a**) Wind rose in the timeframe 2014–2015 at DWD measuring station Stötten (734 m a.s.l., 48.6654° latitude, 9.8655° longitude); (

**b**) unmanned aerial vehicle (UAV) measurement race tracks (© 2009 GeoBasis-DE/BKG, © 2016 Google); (

**c**) parent model; (

**d**) nested domain 10 × 10.

**Figure 5.**(

**a**) Velocities computed with k-ε model on a height of 75 m above ground; (

**b**) velocities computed with RNG k-ε model on a height of 75 m above ground; (

**c**) velocities computed with RSM model on a height of 75 m above ground.

**Figure 6.**(

**a**) Horizontal velocity at a constant distance of 25 m above the ground (grey line); (

**b**) turbulence intensity at a constant distance of 25 m above the ground (grey line).

**Figure 7.**(

**a**) Horizontal velocity at a constant distance of 75 m above the ground (grey line); (

**b**) turbulence intensity at a constant distance of 75 m above the ground (grey line).

**Figure 8.**(

**a**) Horizontal velocity at a constant distance of 98 m above the ground (grey line); (

**b**) turbulence intensity at a constant distance of 98 m above the ground (grey line).

**Figure 9.**(

**a**) Principal Reynolds stresses at a constant distance of 75 m above the ground; (

**b**) shear stress $-\overline{{u}^{\prime}{w}^{\prime}}$ at constant distance of 25 m, 75 m, and 98 m above the ground.

**Figure 10.**(

**a**) Horizontal velocity profile at position V1; (

**b**) horizontal velocity profile at position V2; (

**c**) horizontal velocity profile at position V3; (

**d**) horizontal velocity profile at position V4; (

**e**) horizontal velocity profile at position V5; (

**f**) legend and positions of vertical evaluation lines.

**Figure 11.**(

**a**) Inclination position V1; (

**b**) inclination position V2; (

**c**) inclination position V3; (

**d**) inclination position V4; (

**e**) inclination position V5; (

**f**) legend and location of measuring positions.

C_{μ} = 0.09 | C_{ε1} = 1.44 | C_{ε2} = 1.92 | σ_{k} = 1.0 | σ_{ε} = 1.3 |

C_{μ}_{RNG} = 0.085 | C_{ε2RNG} = 1.68 | σ_{kRNG} = 0.7179 | σ_{ε}_{RNG} = 0.7179 | β_{RNG} = 0.012 |

**Table 3.**Constants for the canopy model considered [12].

β_{p} = 1.0 | β_{d} = 4.0 | C_{ε4} = 1.5 | C_{ε5} = 0.6 |

C_{S} = 0.22 | C_{1} = 1.8 | C_{2} = 0.6 | C_{ε} = 0.18 | C_{ε1} = 1.45 | C_{ε} = 1.90 |

**Table 5.**Root mean square deviation (RMSD) between large-eddy simulation (LES) and Reynolds stress model (RSM) using different internal time scale factors $f$.

RMSD | RSM Factor $\mathit{f}$ = 1 | RSM Factor $\mathit{f}$ = 2 | RSM Factor $\mathit{f}$ = 4 | |||
---|---|---|---|---|---|---|

Comp. Domain | Canopy | Comp. Domain | Canopy | Comp. Domain | Canopy | |

Normalized velocity | 0.0577 | 0.0617 | 0.0485 | 0.0377 | 0.0295 | 0.0316 |

Normalized turbulent kinetic energy | 0.3566 | 0.5490 | 0.2810 | 0.3888 | 0.2423 | 0.2310 |

Normalized Reynolds stress | 0.0841 | 0.1020 | 0.0645 | 0.0424 | 0.0636 | 0.0383 |

Case | Number of Cells | Horizontal Grid Resolution at the Ground | Vertical Grid Resolution at the Ground | Maximum Cell Size |
---|---|---|---|---|

Parent Model | 57.6 Mio. | 30 m | 3.0 m | 100 m |

Nested Domain 10 × 10 | 18.5 Mio. | 24 m | 1.5 m | 48.0 m |

Nested Domain 4 × 4 | 6.86 Mio. | 10 m | 1.0 m | 40.0 m |

Computational Domain | k-ε Model | RNG k-ε Model | RSM |
---|---|---|---|

Nested domain 4 × 4 | 100% | 118% | 207% |

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**MDPI and ACS Style**

Knaus, H.; Hofsäß, M.; Rautenberg, A.; Bange, J.
Application of Different Turbulence Models Simulating Wind Flow in Complex Terrain: A Case Study for the WindForS Test Site. *Computation* **2018**, *6*, 43.
https://doi.org/10.3390/computation6030043

**AMA Style**

Knaus H, Hofsäß M, Rautenberg A, Bange J.
Application of Different Turbulence Models Simulating Wind Flow in Complex Terrain: A Case Study for the WindForS Test Site. *Computation*. 2018; 6(3):43.
https://doi.org/10.3390/computation6030043

**Chicago/Turabian Style**

Knaus, Hermann, Martin Hofsäß, Alexander Rautenberg, and Jens Bange.
2018. "Application of Different Turbulence Models Simulating Wind Flow in Complex Terrain: A Case Study for the WindForS Test Site" *Computation* 6, no. 3: 43.
https://doi.org/10.3390/computation6030043