# A Method of Accelerating the Convergence of Computational Fluid Dynamics for Micro-Siting Wind Mapping

## Abstract

**:**

## 1. Introduction

_{ijkl}refers to the probability density function that exhibits the frequency of occurrence of the j-th wind speed bin in the i-th wind direction sector and the k-th atmospheric stability interval in the l-th wind turbine location, while P

_{ijkl}refers to the power output (kW). To calculate the AEP from Equation (1) after determining the wind turbine layout in the wind farm area, f

_{ijkl}and P

_{ijkl}should be identified first.

_{ijkl}is given by the power curve of the wind turbine, and f

_{ijkl}can be expressed by the Weibull distribution function, which is a probability distribution function of wind speed. f

_{ijkm}can be evaluated by long-term correction after the measurement data have been extrapolated up to the wind turbine hub height at the installation location (l = m) of the meteorological tower. Either way, the results of a simulation of a mesoscale numerical weather prediction can be downscaled [6]. However, to determine f

_{ijkl}at an arbitrary location within the wind farm area, the meteorological correlation between f

_{ijkm}and f

_{ijkl}should be evaluated by wind flow modeling, which process is called micro-siting.

_{ijkl}at the arbitrary location can be calculated (as presented in the following equation) by applying the acceleration or deceleration ratio of the wind speed V predicted in the numerical modeling and f

_{ijkm}at the meteorological tower’s location.

## 2. Methods and Data

#### 2.1. Method of Generating the Initial Conditions

#### 2.2. Wind Mapping Steps for Convergence Acceleration

- Step 1:
- CFD simulation is performed by applying a conventional initial condition for the four wind direction sectors, i.e., N, NE, E, and SE, thereby, obtaining the converged solution.
- Step 2:
- The mirrored IC is applied with regard to the four wind direction sectors, i.e., S, SW, W, and SW, which have 180° symmetry with the above wind directions respectively, to perform the CFD simulation and thereby, obtain the convergence solution.
- Step 3:
- Either the composed IC or the shifted IC is applied with regard to the other eight wind directions, i.e., NNE, ENE, ESE, SSE, SSW, SWS, NWN, and NNW, to conduct the CFD simulation and thereby, obtain the convergence solution.

#### 2.3. Verification of the Convergence Acceleration Effect

_{geo}) in the upper top part of the atmospheric boundary layer were set to 5 m/s, 10 m/s, and 15 m/s. As reference information, the Hundhammerfjellet wind farm is located along the ridge in the central part of the peninsula extending SSW to NNE in the center of the domain, as shown in Figure 2. Assuming that the maximum altitude above sea level (236 meters) of the Hundhammerfjellet ridge is set to a characteristic length, the corresponding Reynolds numbers become 0.8 × 10

^{8}, 1.6 × 10

^{8}, and 2.4 × 10

^{8}, respectively.

## 3. Results and Discussion

#### 3.1. Reynolds Number Invariance of the Wind Field

_{geo}. Furthermore, a low wind speed region was observed below the wind speed ratio S < 0.2 on the lee side due to the rapid downslope after a speed-up of more than S > 1.0 at the Hundhammerfjellet ridge in the central part of the computational domain. Figure 3b shows the difference between case S

_{15}at a geostrophic wind speed of 15 m/s and case S

_{05}at 5 m/s, that is, dS = S

_{15}− S

_{05}. The region which revealed the largest difference in the domain was the deceleration area of wind speed due to the downslope and this effect produced a long transport pattern along the downwind direction.

#### 3.2. Evaluation of New Initial Conditions

#### 3.2.1. Error Analysis of New Initial Conditions

#### 3.2.2. Sensitivity Analysis of Directional Interval

#### 3.3. Convergence Acceleration by the New Initial Conditions

#### 3.3.1. Convergence Acceleration of the Individual CFD Simulation

#### 3.3.2. Convergence Acceleration of the Overall CFD Simulations

- Step 1:
- For the four wind direction sectors, i.e., N, NE, E, and SE, the ABL IC took 7167 s until convergence.
- Step 2:
- For the four symmetric wind direction sectors, i.e., S, SW, W, and NW, the converged solutions of the previous four sectors of N, NE, E, and SE were used to generate the mirrored IC, and took 3603 s until convergence.
- Step 3:
- For the remaining eight wind direction sectors, i.e., NNE, ENE, ESE, SSE, SSW, SWS, NWN, and NNW, the converged solutions of the previous eight sectors were used to generate either the vector composed IC (e.g., NNE from N+NE) or the shifted IC (e.g., SSW from either S or SSW), and took 6719 s until convergence.

#### 3.3.3. Analysis of Sensitivity of the Computation Cell Numbers

## 4. Conclusions

- (1)
- Mirrored, composed, and shifted ICs were generated from the converged solutions with regard to the different wind direction sectors by using the geometric similarity or vector composition principle. In the case of the CFD simulations of the 16 wind direction sectors in the Hundhammerfjellet wind farm case, the new ICs showed better convergence performance than that of the conventional ABL IC case, shortening the convergence time by 50%. Compared to the converged solution, the new ICs showed approximation errors of only 4.9%~6.4% of the wind speed ratio MAE and 2.10~2.90 of the wind direction MAE, which are about 34% and 44% lower than those of ABL IC, respectively.
- (2)
- When the method of generating the initial conditions proposed in this study and the conventional initial conditions were mixed appropriately by wind direction sector, the overall CFD convergence time was confirmed to have been reduced by around 36% when employing 1 million computation cells and 16 wind direction sectors. Therefore, the proposed method is expected to make a substantial contribution to shortening the micro-siting project period for the design of a wind farm.
- (3)
- The validity of the Reynolds number invariance of atmospheric wind flow over rough terrain was verified from the Hundhammerfjellet wind farm simulation with regard to the gradient wind speeds of 5 m/s, 10 m/s, and 15 m/s. The relative RMSE of the difference in wind speed between the Reynolds number 0.8 × 10
^{8}and 2.4 × 10^{8}cases was only 0.65% and differences were found on the lee side of the steeply sloped, high terrain.

## Funding

## Conflicts of Interest

## Patent

## References

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**Figure 2.**The computational domain around the Hundhammerfjellet wind farm (triangles: wind turbines, cobalt blue region: sea).

**Figure 4.**Error wind field of the mirrored initial condition for the north-easterly wind case (10 meters above ground level).

**Figure 6.**Comparison of the convergence history of monitoring values of the field variables (normalized by maximum; left graphs) and the residual values of the field variables (right graphs).

**Table 1.**Comparison of approximation error of normalized wind speed difference and wind direction difference by the method of generating the initial condition.

Initial Condition | dV/V_{geo} × 100 (%) | dθ | ||||
---|---|---|---|---|---|---|

MBE | MAE | RMSE | MBE | MAE | RMSE | |

Mirrored | 3.3 | 6.4 | 8.0 | −0.97 | 2.35 | 3.75 |

Composed | 4.3 | 4.9 | 5.9 | 0.17 | 2.12 | 3.17 |

Shifted | −1.9 | 6.1 | 7.9 | −0.49 | 2.90 | 6.29 |

ABL | −5.7 | 8.7 | 1.2 | 3.62 | 4.35 | 7.46 |

**Table 2.**Comparison of the convergence times of the 16 wind direction cases between the conventional initial condition (IC) and the proposed ICs.

Wind Directional Sector | Convergence Time (s) | |||
---|---|---|---|---|

Conventional Method | Proposed Method Using New ICs | |||

ABL IC | Step 1 | Step 2 | Step 3 | |

ABL IC | Mirrored IC | Composed or Shifted IC | ||

N | 1740 | 1740 | ||

NNE | 1837 | 919 | ||

NE | 1980 | 1980 | ||

ENE | 1855 | 927 | ||

E | 1565 | 1565 | ||

ESE | 1591 | 795 | ||

SE | 1883 | 1883 | ||

SSE | 1691 | 845 | ||

S | 1770 | 885 | ||

SSW | 1688 | 844 | ||

SW | 1674 | 1004 | ||

SWS | 1564 | 782 | ||

W | 1471 | 735 | ||

NWN | 1583 | 791 | ||

NW | 1631 | 979 | ||

NNW | 1632 | 816 | ||

Total | 27,150 | 7167 | 3603 | 6719 |

17,489 |

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**MDPI and ACS Style**

Kim, H.-G.
A Method of Accelerating the Convergence of Computational Fluid Dynamics for Micro-Siting Wind Mapping. *Computation* **2019**, *7*, 22.
https://doi.org/10.3390/computation7020022

**AMA Style**

Kim H-G.
A Method of Accelerating the Convergence of Computational Fluid Dynamics for Micro-Siting Wind Mapping. *Computation*. 2019; 7(2):22.
https://doi.org/10.3390/computation7020022

**Chicago/Turabian Style**

Kim, Hyun-Goo.
2019. "A Method of Accelerating the Convergence of Computational Fluid Dynamics for Micro-Siting Wind Mapping" *Computation* 7, no. 2: 22.
https://doi.org/10.3390/computation7020022