k-epsilon-Sk

An extended $k-\varepsilon$ model, called $k-\varepsilon-S_k$, integrated with OpenFOAM for simulation of turbine wakes and power losses in wind farms, developed by Zehtabiyan-Rezaie and Abkar (2024) at the Fluid Mechanics and Turbulence research group at Aarhus University, Denmark.

Description

We introduce an extended version of the standard $k-\varepsilon$ model within the OpenFOAM framework, with a primary focus on its application in the context of wind-farm simulations. The standard $k-\varepsilon$ model, while widely used, is known to exhibit limitations in accurately representing turbulence characteristics within the wake region of turbines. This often results in an overestimation of turbulence intensity and, consequently, an inaccurate prediction of the power losses in wind farms. To address this challenge, $k-\varepsilon-S_k$ model incorporates an additional term in the turbulent kinetic energy (TKE) transport equation, which accounts for the influence of turbine forces. This modification is based on a rigorous analytical approach derived from fundamental physical principles. For a comprehensive understanding of $k-\varepsilon-S_k$ model and its application in wind-farm simulation, additional information can be found in this publication. This approach, originated from physical principles and derived from an analytical approach, can be similarly applied to the TKE equation in other widely used empirical models (e.g., $k-\varepsilon$ and $k-\omega$ families).

Target platform

The code has been rigorously tested and verified to be fully compatible with OpenFOAM v-2112, ensuring its smooth integration and reliable performance with this specific release.

How to set the model

1- Download the source code.

2- To enable the momentum-source calculator of the actuator-disk model without rotation (ADM-NR) in OpenFOAM to calculate the source terms based on the disk-averaged velocity, follow these instructions:

$\bullet$ Go to your work directory via the following command:

cd $WM_PROJECT_USER_DIR

$\bullet$ Copy the folder ADM_NR_diskBased to your work directory, and compile the new library with the following command

wmake

3- Copy the folder testCase to your run directory. To initiate the simulation, adjust number of numberOfSubdomains in system/decomposeParDict, and execute the following command:

.//Allrun

If you use $ck = 0$ in the kSource's implemented in fvOptions file as scalarCodedSource, the additional term in the turbulent kinetic energy equation is deactivated, and the model transforms to the standard $k-\varepsilon$.

4- You will find the results of the calculated velocity, thrust, and power in the disk folders in the postProcessing directory. You can also use the lineSample and Surfaces features in the system directory to extract data over lines and surfaces.

5- Note that for simulation of a new wind farm with different operating conditions of turbines, you need to:

$\bullet$ Update the system/blockMeshDict and system/topoSetDict based on the layout of the wind farm under study.

$\bullet$ Update the disks' information in constant/fvOptions based on the layout of the wind farm under study and the operating conditions of the turbines.

$\bullet$ Update the the operating conditions of the turbines in kSource's implemented in fvOptions file as scalarCodedSource.

Evaluation of $k-\varepsilon-S_k$ model's performance

Normalized power of turbines in a six-turbine case under full-wake conditions $\theta_w = 270^\circ$, predicted by the standard and extended $k-\varepsilon$ models against large-eddy simulations from the study of Eidi et al. (2021). The inter-turbine spacing is 7 rotor diameters and the turbines have a rotor diameter, hub height, and an induction factor of 80 m, 70 m, and 0.25, respectively. The inflow velocity and inflow turbulence intensity at the hub height are 8 m/s and 5.8%, respectively.

The performance of the $k-\varepsilon-S_k$ model has been further assessed through extensive testing against data from wind-tunnel measurements and large-eddy simulations for three validation cases with different layouts under full and partial wake conditions, a $10 \times 3$ array of turbines, and the Horns rev 1 Offshore Wind Farm in the study of Zehtabiyan-Rezaie and Abkar (2024).

How to cite

Please, cite this library as:

@article{ZEHTABIYANREZAIE_RENE2024,
title = {An extended k−ɛ model for wake-flow simulation of wind farms},
author = {Navid Zehtabiyan-Rezaie and Mahdi Abkar},
journal = {Renewable Energy},
volume = {222},
pages = {119904},
year = {2024},
doi = {https://doi.org/10.1016/j.renene.2023.119904},
url = {https://www.sciencedirect.com/science/article/pii/S0960148123018190}
}