# Modelling Uncertainty in t-RANS Simulations of Thermally Stratified Forest Canopy Flows for Wind Energy Studies

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## Abstract

**:**

## 1. Introduction

## 2. Validation Data

#### 2.1. Wind Speed Data

#### 2.2. Canopy Height Data

## 3. CFD Modelling

#### 3.1. Model Equations

_{x}, which is the product of the canopy drag, C

_{d}, and the Leaf Area Density, A(z). In WM, this loss coefficient can be set to a constant value or can vary with height. As no data relating to the vertical structure of the canopy were available, a constant value was used for all simulations. The specific values used for each simulation will be given in the appropriate section. Note that the WM implementation of the drag force and turbulence source are based on a definition of a drag force including a factor ½. In [14] the factor of ½ is omitted in the definition of this parameter. As a consequence, the loss coefficient in [14] is to be interpreted as half the loss coefficient in WindModeller.

#### 3.2. Boundary Conditions

_{ref}, and the height above ground level at which it occurs Z

_{ref}along with the surface roughness ${z}_{0}$. From these user-defined criteria, WM then calculates a value of ${u}_{*}$ using a form of the log law as shown in Equation (33):

#### 3.3. Domain Description

_{0}= 0.04 m which is what would be expected for a site containing low grass [24]. Values of 0.1 m and 0.001 m were also tried, however the impact on results was negligible. This is due to the fact that much of the fetch along the 255° direction is occupied by forestry and thus the surface roughness itself will have a reduced role in dictating the wind characteristics.

#### 3.4. Mesh Sensitivity

_{x}= 0.05 m

^{−1}was used for all simulations along with U

_{ref}= 6.5 m/s at Z

_{ref}= 40 m.

_{ref}= 6.5 m/s.

## 4. Neutral Simulations

#### 4.1. Process

- Reference height, Z
_{ref} - Reference velocity, U
_{ref} - Canopy loss coefficient, L
_{x}: Variable h_{c} - Canopy loss coefficient, L
_{x}: Constant h_{c}

_{c}’ is used, simulations have been conducted using the canopy height data discussed in Section 2.2. When the term ‘Constant h

_{c}’ is used, simulations have been conducted using a constant canopy height for the forested area. The results of this analysis are presented in the following section.

#### 4.2. Results

#### 4.2.1. Reference Height, Z_{ref} and Reference Velocity, U_{ref}

_{ref}and U

_{ref}are used by WM to calculate the value of ${U}_{*}$ and also to define the inlet velocity profile. The simulations summarised in Table 5 and in Table 6 were conducted in order to assess the sensitivity of the model to the prescribed value of Z

_{ref}and U

_{ref}respectively. The default WM value for the canopy loss coefficient, 0.05 m

^{−1}, has been used for all simulations. The results of these simulations are also displayed in Figure 11 where they are compared to the validation dataset. The target neutral range is highlighted in green. In all tabular results, the adjusted parameter is highlighted in bold for clarity.

_{ref}and U

_{ref}. In Figure 11 we see the locus of results in this section indicated as a purple oversized data point, the simulated value of α is in line with the observed value for the neutral events whilst the values of TI are significantly lower. Due to the insensitivity of the model to the prescribed values, it is not possible to correct this discrepancy by adjusting Z

_{ref}or U

_{ref}. This confirms a lack of sensitivity to a change in Reynold’s number when operating at high Reynolds number values in the absence of stability effects or significant separation due to complex terrain downstream of the obstruction.

#### 4.2.2. Canopy Loss Coefficient, L_{x}: Variable h_{c}

_{x}was assessed using the simulations summarised in Table 7. The canopy height was allowed to vary as described by the canopy height data outlined in Section 2.2. The CFD outputs for α and TI at the meteorological mast location are summarised in Figure 12 where they are compared to the validation data.

^{−1}as used in simulation No. 22.

#### 4.2.3. Canopy Loss Coefficient, L_{x}: Constant h_{c}

_{x}when using a constant rather than a variable canopy height. Firstly, the canopy height was set to 11 m which is the average of the canopy height data summarised in Figure 8. The simulations conducted using this height are summarised in Table 8.

#### 4.3. Discussion

_{x}is used and simulation No. 38 where the value of L

_{x}has been tuned. In Figure 14, these simulated profiles are presented along with the average profiles of all neutral events in the validation dataset.

_{x}, is the product of the canopy drag, C

_{d}, and the Leaf Area Density (LAD), A(z). A value of C

_{d}= 0.15 has been suggested by [25] as being appropriate for a variety of forest canopy types. This would indicate that the average LAD for the Vaudeville forest is approximately 4.6 m

^{−1}if use the value of L

_{x}from No. 38.

^{−1}were suggested, values of 0.5–3 m

^{−1}were more common.

^{−1}for the Vaudeville forest is high but realistic. Given that we are considering a mixed canopy and that the validation dataset relates to the summer months, this value is plausible.

## 5. Stable Simulations

#### 5.1. Process

#### 5.2. Results

#### 5.3. Discussion

## 6. Unstable Simulations

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

Symbol | Definition | Units |

$A\left(z\right)$ | Leaf area density at height z | m^{−1} |

${C}_{d}$ | Canopy drag coefficient | dimensionless |

${C}_{\mu},{\alpha}_{3},{\beta}_{3}$ | Turbulence model constants for SST model | dimensionless |

${C}_{\epsilon 3},\text{}{C}_{\epsilon 4},{\beta}_{d},{\beta}_{p},$ | Turbulence model constants specific to forest canopy model | dimensionless |

${C}_{P}$ | Fluid specific heat capacity at constant pressure | J/(kg K) |

${F}_{1},\text{}{F}_{2}$ | Wall distance functions in SST model | dimensionless |

${F}_{F}$ | Forestry switch | dimensionless |

${F}_{B,i}$ | Buoyancy force per unit volume in the i-direction | kg/(m^{2} s^{2}) |

${F}_{Cor,i}$ | Coriolis force per unit volume in the i-direction | kg/(m^{2} s^{2}) |

${F}_{D,i}$ | Drag force per unit volume in the i-direction | kg/(m^{2} s^{2}) |

${F}_{U}$ | Momentum flux | N/m^{2} |

$f$ | Coriolis parameter | s^{−1} |

$g$ | Gravity acceleration | m/s^{2} |

${h}_{s}$ | Equivelent sand grain roughness | m |

$k$ | Turbulence kinetic energy | m^{2}/s^{2} |

$p$ | Pressure | Pa |

${P}_{k}$ | Shear turbulence production per unit volume | kg/(m s^{3}) |

${P}_{kB}$ | Buoyancy turbulence production per unit volume | kg/(m s^{3}) |

${P}_{\omega B}$ | Buoyancy production term for eddy frequency, per unit volume | kg/(m^{3} s^{2}) |

${S}_{\epsilon}$ | Turbulence dissipation source per unit volume | kg/(m s^{4}) |

${S}_{k}$ | Turbulence kinetic energy production from forestry drag, per unit volume | kg/(m s^{3}) |

${S}_{\omega}$ | Eddy frequency production from forestry drag, per unit volume | kg/(m^{3} s^{2}) |

$t$ | Time | s |

$\mathrm{TI}$ | Turbulence intensity | dimensionless |

$\left|U\right|$ | Modulus of the windspeed | m/s |

$\overline{U}$ | 10 min mean wind speed | m/s |

$U\left(z\right)$ | Velocity at reference height z | m/s |

${U}_{i,j}$ | Wind speed in the i-direction, j-direction | m/s |

${U}_{i,geo}$ | Geostrophic wind speed in the i-direction | m/s |

${U}_{40,80}$ | 10 min mean wind speed at 40 m, 80 m | m/s |

${u}_{*}$ | Friction velocity | m/s |

${x}_{i}$ | Spatial coordinate in i-direction | m |

${y}_{i}$ | Spatial coordinate in i-direction | m |

$\alpha $ | Shear exponent factor | dimensionless |

$\beta $ | Thermal expansion coefficient | K^{−1} |

$\sigma $,$\text{}{\sigma}_{\theta}$,$\text{}{\sigma}_{k},\text{}{\sigma}_{\omega 2},{\sigma}_{\omega 3}$ | Turbulent Prandtl number for momentum, temperature, $k$ and $\omega $ | dimensionless |

${\sigma}_{u}$ | Standard deviation of wind speed over 10 min, sampled at a rate of 1 Hz | m/s |

$\epsilon $ | Turbulence disspation rate | m^{2}/s^{3} |

$\theta $ | Potential temperature | K |

$\kappa $ | Von Karmen constant | dimensionless |

$\lambda $ | Fluid conductivity | W/(m K) |

$\mu $ | Fluid viscosity | kg/(m s) |

${\mu}_{T}$ | Eddy viscosity | kg/(m s) |

$\rho $ | Fluid density | kg/m^{−3} |

$\omega $ | Turbulence eddy frequency | s^{−1} |

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**Figure 1.**Location of the Vaudeville meteorological mast is indicated by the red marker. [Picture credit: www.maps.google.com].

**Figure 2.**Institut National de l’Information Géographique et Forestière (IGN) map of the Vaudeville region. The meteorological mast location (46°26′58″ N, 05°35′02″ E) is marked with by the red X circumscribed by a red circle. Turbine locations are indicated by a red inverted Y [8].

**Figure 3.**Aerial photograph of the Vaudeville site showing the 250–260° direction sector. Distances to the forest edge are indicated by the red arrows. [Picture credit: www.maps.google.com].

**Figure 4.**A summary of the available data showing the number of observations and the maximum recorded irradiance level for each month. Month 1 relates to January 2010. The yellow shading identifies the months selected for analysis.

**Figure 5.**Observed wind shear and turbulence intensity at the Vaudeville site for the 250–260° direction sectors for July and August 2010. The red lines indicated the applied neutral threshold values. Turbulence intensity values are calculated at 80 m.

**Figure 6.**Neutral thresholds are applied to the selected data. Points in the sector with the green background are considered to be neutral, blue are stable and red unstable. Profiles for the oversized data points in each of these sectors are given in Figure 7. Only events with wind speeds of >3 m/s at 40 m are displayed in this figure.

**Figure 7.**Sample profiles for the oversized data points in Figure 6.

**Figure 8.**The height distribution of trees for the region outlined in Figure 8.

**Figure 9.**(

**a**) Mesh zones created by WM. The red dot in (

**b**) indicates the meteorological mast location. The same domain is detailed in (

**a**,

**b**).

**Figure 11.**The locus of the results of Simulations 1–11 are represented by the purple oversized data point.

**Figure 12.**The results of Simulations 12–22 are represented by the purple oversized data points. The reference numbers shown correspond to the simulation numbers given in Table 7.

**Figure 14.**Graphs showing the simulated normalised velocity (

**a**) and turbulence intensity (

**b**) profiles at the meteorological mast location for simulations No. 4 & No. 38. The field data points represent the average value at that height for all neutral events whilst the horizontal bars indicate the range of recorded values at each height in terms of 2 × Standard Deviation.

**Figure 15.**The results of Simulations 48–53 are represented by the blue oversized data points. The reference numbers shown correspond to the simulation numbers given in Table 12.

**Figure 16.**Graphs showing the simulated normalised velocity (

**a**) and turbulence intensity (

**b**) profiles at the meteorological mast location for simulation No. 51. The field data points represent the average value at that height for all stable events whilst the horizontal bars indicate the range of recorded values at each height in terms of 2 × Standard Deviation.

**Table 1.**Meteorological sensors present on the Vaudeville meteorological mast. Instrumentation model numbers are given in parenthesis where available. Sonic anemometer were orientated into the prevailing wind from the south west and thus were not affected by tower shadow for the director sector considered, shown in Figure 3.

Height (m) | Sensor 1 | Sensor 2 | Sensor 3 |
---|---|---|---|

80 | Temperature sensor (PT 100, SKS Sensors, Vantaa, Finland) | 3D Sonic anemometer (Metek USA-1) | Cup Anemometer (Thies First class, Thies, Göttingen, Germany) |

70 | Wind vane (Thies compact) | - | - |

60 | Temperature sensor (PT 100) | 3D Sonic anemometer (Metek USA-1) | - |

40 | Temperature sensor (PT 100) | 3D Sonic anemometer (Metek USA-1) | - |

10 | Temperature sensor (PT 100) | 3D Sonic anemometer (Metek USA-1) | - |

3 | Temperature & Humidity (CS215) | Pyranometer (CMP6, Kipp & Zonen, Delft, The Neterlands) | - |

1 | Pluviometer | - | - |

-1 | Temperature sensor (PT 100) | - | - |

**Table 2.**Time and date at which each of the profiles in Figure 7 were recorded.

Stability Class | Time & Date |
---|---|

Stable | 19:40 13 July 2010 |

Neutral | 23:40 17 August 2010 |

Unstable | 12:00 10 August 2010 |

**Table 3.**Modelling constants used for the canopy model [14].

Constant | Value |
---|---|

${\beta}_{p}$ | 0.17 |

${\beta}_{d}$ | 3.37 |

${C}_{\epsilon 4}$ | 0.9 |

${C}_{\epsilon 5}$ | 0.9 |

Mesh | Maximum Cell Size | Control Volumes | Nodes | CPU Time | |
---|---|---|---|---|---|

Hz | Vt | ||||

Coarse | 100 m | 100 m | 87,696 | 93,478 | 5 min |

Medium | 20 m | 50 m | 2,149,056 | 2,215,626 | 60 min |

Fine | 10 m | 25 m | 13,418,460 | 13,638,322 | 480 min |

**Table 5.**Summary of simulations run to investigate the sensitivity of the CFD model to the prescribed value of Z

_{ref}. The adjusted parameter is in italics.

Simulation No. | CFD Settings | CFD Output | |||||
---|---|---|---|---|---|---|---|

Z_{ref} (m) | U_{ref} (m/s) | L_{x} (m^{−1}) | C_{μ} | h_{c} (m) | α | TI | |

1 | 40 | 6.5 | 0.05 | 0.09 | Variable | 0.415 | 0.142 |

2 | 60 | 6.5 | 0.05 | 0.09 | Variable | 0.415 | 0.142 |

3 | 80 | 6.5 | 0.05 | 0.09 | Variable | 0.415 | 0.142 |

4 | 100 | 6.5 | 0.05 | 0.09 | Variable | 0.415 | 0.142 |

5 | 500 | 6.5 | 0.05 | 0.09 | Variable | 0.415 | 0.142 |

**Table 6.**Summary of simulations run to investigate the sensitivity of the CFD model to the prescribed value of U

_{ref}. The adjusted parameter is in italics.

Simulation No. | CFD Settings | CFD Output | |||||
---|---|---|---|---|---|---|---|

Z_{ref} (m) | U_{ref} (m/s) | L_{x} (m^{−1}) | C_{μ} | h_{c} (m) | α | TI | |

6 | 100 | 5 | 0.05 | 0.09 | Variable | 0.418 | 0.141 |

7 | 100 | 5.5 | 0.05 | 0.09 | Variable | 0.418 | 0.141 |

8 | 100 | 6 | 0.05 | 0.09 | Variable | 0.417 | 0.141 |

9 | 100 | 7 | 0.05 | 0.09 | Variable | 0.417 | 0.141 |

10 | 100 | 13 | 0.05 | 0.09 | Variable | 0.418 | 0.142 |

11 | 100 | 20 | 0.05 | 0.09 | Variable | 0.418 | 0.142 |

**Table 7.**Summary of simulations run to investigate the sensitivity of the CFD model to the prescribed value of L

_{x}with a variable canopy height. The adjusted parameter is in italics.

Simulation No. | CFD Settings | CFD Output | |||||
---|---|---|---|---|---|---|---|

Z_{ref} (m) | U_{ref} (m/s) | L_{x} (m^{−1}) | C_{μ} | h_{c} (m) | α | TI | |

12 | 100 | 6.5 | 0.001 | 0.09 | Variable | 0.223 | 0.095 |

13 | 100 | 6.5 | 0.01 | 0.09 | Variable | 0.373 | 0.129 |

14 | 100 | 6.5 | 0.02 | 0.09 | Variable | 0.397 | 0.135 |

15 | 100 | 6.5 | 0.03 | 0.09 | Variable | 0.405 | 0.138 |

16 | 100 | 6.5 | 0.04 | 0.09 | Variable | 0.411 | 0.140 |

17 | 100 | 6.5 | 0.045 | 0.09 | Variable | 0.413 | 0.141 |

18 | 100 | 6.5 | 0.06 | 0.09 | Variable | 0.420 | 0.144 |

19 | 100 | 6.5 | 0.07 | 0.09 | Variable | 0.423 | 0.145 |

20 | 100 | 6.5 | 0.08 | 0.09 | Variable | 0.426 | 0.146 |

21 | 100 | 6.5 | 0.09 | 0.09 | Variable | 0.430 | 0.148 |

22 | 100 | 6.5 | 0.5 | 0.09 | Variable | 0.484 | 0.169 |

**Table 8.**Summary of simulations run to investigate the sensitivity of the CFD model to the prescribed value of L

_{x}with a constant canopy height of 11 m. The adjusted parameter is in italics.

Simulation No. | CFD Settings | CFD Output | |||||
---|---|---|---|---|---|---|---|

Z_{ref} (m) | U_{ref} (m/s) | L_{x} (m^{−1}) | C_{μ} | h_{c} (m) | α | TI | |

23 | 100 | 6.5 | 0.02 | 0.09 | 11 | 0.360 | 0.130 |

24 | 100 | 6.5 | 0.03 | 0.09 | 11 | 0.360 | 0.130 |

25 | 100 | 6.5 | 0.04 | 0.09 | 11 | 0.363 | 0.130 |

26 | 100 | 6.5 | 0.05 | 0.09 | 11 | 0.365 | 0.131 |

27 | 100 | 6.5 | 0.06 | 0.09 | 11 | 0.368 | 0.133 |

28 | 100 | 6.5 | 0.09 | 0.09 | 11 | 0.374 | 0.136 |

29 | 100 | 6.5 | 0.12 | 0.09 | 11 | 0.379 | 0.138 |

30 | 100 | 6.5 | 0.15 | 0.09 | 11 | 0.383 | 0.141 |

31 | 100 | 6.5 | 0.2 | 0.09 | 11 | 0.389 | 0.144 |

32 | 100 | 6.5 | 0.3 | 0.09 | 11 | 0.397 | 0.148 |

33 | 100 | 6.5 | 0.4 | 0.09 | 11 | 0.404 | 0.151 |

34 | 100 | 6.5 | 0.6 | 0.09 | 11 | 0.412 | 0.156 |

35 | 100 | 6.5 | 0.7 | 0.09 | 11 | 0.415 | 0.158 |

36 | 100 | 6.5 | 0.8 | 0.09 | 11 | 0.414 | 0.158 |

**Table 9.**Summary of simulations run to investigate the sensitivity of the CFD model to the prescribed value of L

_{x}with a constant canopy height of 20 m. The adjusted parameter is in italics.

Simulation No. | CFD Settings | CFD Output | |||||
---|---|---|---|---|---|---|---|

Z_{ref} (m) | U_{ref} (m/s) | L_{x} (m^{−1}) | C_{μ} | h_{c} (m) | α | TI | |

37 | 100 | 6.5 | 0.05 | 0.09 | 20 | 0.458 | 0.154 |

38 | 100 | 6.5 | 0.7 | 0.09 | 20 | 0.462 | 0.176 |

39 | 100 | 6.5 | 0.9 | 0.09 | 20 | 0.465 | 0.179 |

**Table 10.**Summary of simulations run to investigate the sensitivity of the CFD model to the prescribed value of L

_{x}with a constant canopy height of 25 m. The adjusted parameter is in italics.

Simulation No. | CFD Settings | CFD Output | |||||
---|---|---|---|---|---|---|---|

Z_{ref} (m) | U_{ref} (m/s) | L_{x} (m^{−1}) | C_{μ} | h_{c} (m) | α | TI | |

40 | 100 | 6.5 | 0.05 | 0.09 | 25 | 0.544 | 0.193 |

41 | 100 | 6.5 | 0.9 | 0.09 | 25 | 0.570 | 0.238 |

**Table 11.**Summary of simulations run to investigate the sensitivity of the CFD model to the prescribed value of L

_{x}with a constant canopy height of 30 m. The adjusted parameter is in italics.

Simulation No. | CFD Settings | CFD Output | |||||
---|---|---|---|---|---|---|---|

Z_{ref} (m) | U_{ref} (m/s) | L_{x} (m^{−1}) | C_{μ} | h_{c} (m) | α | TI | |

42 | 100 | 6.5 | 0.05 | 0.09 | 30 | 0.572 | 0.174 |

43 | 100 | 6.5 | 0.7 | 0.09 | 30 | 0.514 | 0.193 |

44 | 100 | 6.5 | 0.9 | 0.09 | 30 | 0.515 | 0.197 |

**Table 12.**Summary of simulations run to investigate the sensitivity of the CFD model to the prescribed temperature of the domain floor. The time given is the physical computational time that the simulation required to reach a converged solution. The adjusted parameter is in italics.

Simulation No. | Floor Temperature Difference from Ambient (Kelvin) | CFD Output | ||
---|---|---|---|---|

α | TI | Time (min) | ||

48 | −0.5 | 0.594 | 0.120 | 800 |

49 | −1 | 0.626 | 0.117 | 828 |

50 | −5 | 0.720 | 0.107 | 1088 |

51 | −10 | 0.764 | 0.099 | 2204 |

52 | −25 | 0.833 | 0.092 | 2434 |

53 | −50 | 0.864 | 0.068 | 2574 |

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## Share and Cite

**MDPI and ACS Style**

Desmond, C.J.; Watson, S.; Montavon, C.; Murphy, J. Modelling Uncertainty in t-RANS Simulations of Thermally Stratified Forest Canopy Flows for Wind Energy Studies. *Energies* **2018**, *11*, 1703.
https://doi.org/10.3390/en11071703

**AMA Style**

Desmond CJ, Watson S, Montavon C, Murphy J. Modelling Uncertainty in t-RANS Simulations of Thermally Stratified Forest Canopy Flows for Wind Energy Studies. *Energies*. 2018; 11(7):1703.
https://doi.org/10.3390/en11071703

**Chicago/Turabian Style**

Desmond, Cian J., Simon Watson, Christiane Montavon, and Jimmy Murphy. 2018. "Modelling Uncertainty in t-RANS Simulations of Thermally Stratified Forest Canopy Flows for Wind Energy Studies" *Energies* 11, no. 7: 1703.
https://doi.org/10.3390/en11071703