1. Introduction
Today, renewable energy sources play an important role in reducing CO
2 emissions, which cause global warming. Many organizations and countries are taking action to reduce CO
2 emissions and finance the development of wind energy technology. According to the Economist (2019), 7% of the total energy was received from solar panels and wind turbines in 2019, and the number is expected to increase up to fivefold by 2040 [
1]. The International Energy Agency (IEA) believes and acknowledges that nuclear energy, wind energy, solar PV energy geothermal power, and hydropower will be the main power sources of the 21st century, as there is a limit on nonrenewable sources as oil, natural gas, and coal, since they last up to 30, 50, and 200 years, respectively [
2]. Wind energy is the cleanest green energy available currently, and researchers are developing efficient wind turbines and testing them with wind tunnels. However, the usage of wind tunnels is expensive and time-consuming. Furthermore, scale effects cannot be properly addressed. Therefore, computational fluid dynamics (CFD) simulations are used for numerical simulations, analysis, and design optimization.
A more realistic flow through the rotors of wind turbines can be simulated using computational fluid dynamics (CFD), which numerically solves Navier–Stokes equations. On the other hand, there are other low-fidelity methods such as blade element momentum (BEM) and lifting line panel method (LLPM); however, they have some limitations when it comes to the simulation of complex three-dimensional turbulent flow, since the lifting line theory is based on potential flow theory, while the BEM is 2D momentum theory, which is highly reliant on two-dimensional airfoil data that need empirical corrections [
3]. There are three main CFD approaches to simulate turbulent flows: (1) large eddy simulation (LES); (2) Reynolds-averaged Navier–Stokes (RANS); (3) the hybrid RANS–LES model (HRLM) [
4,
5]. The RANS model for simulation is acceptable as it can appropriately simulate steady and unsteady simulations, while the LES model is good for resolving large-scale energy containing eddies; thus, LES is more accurate than RANS but computationally much more expensive, while HRLM is more accurate than RANS and computationally cheaper than LES. Wind turbine simulation involves complex flows through its rotors, for which the RANS model currently remains the main CFD solver for HAWT aerodynamics as it is computationally cheap and generates acceptable time-averaged results. However, the HRLM will be increasingly used for complex wind turbine aerodynamic analysis and design optimization, as engineers are designing ever larger wind turbines that require multidisciplinary analysis and design, considering fluid–structure interaction, fatigue, and noise reduction, as the LES is too expensive for these tasks [
6,
7].
Arbitrary hybrid turbulence modeling is the arbitrary combination of the RANS turbulence method and LES in the flow field depending on the required resolution in different locations, which gives more accurate results than RANS while being cheaper than LES. By rescaling the conventional RANS equations through the introduction of the resolution control function Fr into the turbulent viscosity of the RANS turbulence model, the formulation of the new VLES model can be achieved. This resolution control factor is the ratio of sub-grid turbulent stress to the RANS/URANS turbulent stress, which can also roughly represent the ratio of modeled turbulent energy to total turbulent energy. It is responsible for smooth transitioning between RANS/URANS/LES/DNS modes depending on local mesh density in comparison with the turbulence integral and Kolmogorov length scales.
A study was performed on the mechanical responses of the National Renewable Energy Laboratory (NREL) Phase VI wind turbine using commercial application ANSYS, and the results were compared with the traditional blade element momentum (BEM) method [
8]. The study found that the blade element momentum method (BEM) was not accurate, and it was also observed that applicable results could be obtained using BEM, but it was time-consuming and challenging. Furthermore, the CFD and FSI methods either overpredicted or underpredicted the power compared with experimental results.
Zhong et al. [
9] performed CFD simulations using the k-omega SST model with different beta star coefficients for the NREL Phase VI wind turbine. It was found that changing the beta star value to 0.11 from 0.09 improved the results; however, the agreement between experimental values and numerical results were not satisfactory when the wind speed was 15 m/s, as the difference was more than 15%.
Another paper [
10] reported a numerical study on the NREL Phase VI wind turbine using different wind speeds between 5 and 21 m/s and the Spalart–Allmaras (SA) turbulence model. The CFD computed results underpredicted the torque compared with NREL experimental data for all wind speeds. It was concluded that the differences between experimental data and CFD results were due to the limitations of the turbulence model.
Commonly, computational fluid dynamics (CFD) simulations are used to predict wind turbine performances under different wind conditions. These simulations represent numerical solutions of the Navier–Stokes equations, which describe fluid flow. Furthermore, for wind turbine CFD simulations, traditional turbulence models such as Spalart–Allmaras, k-epsilon, k-omega, and k-omega SST are used since they are as computationally effective as RANS simulations. However, engineers increasingly need to study more physical effects, such as fluid–structure interaction (FSI), fatigue, and noise for multidisciplinary design analysis and optimization (MDAO), which requires greater resolution of turbulent eddies. In order to make more accurate predictions, turbulence models such as LES, zonal detached eddy simulation (DES), or even direct numerical simulation (DNS) can be used, but they are very computationally expensive and inflexible. Therefore, in this study, we aimed to develop an arbitrary hybrid turbulence modeling (AHTM) approach based on VLES, which can help design engineers to take advantage of this unique and highly flexible approach to tailor the grid according to the design and resolution requirements in different areas of the flow field over the wind turbine without sacrificing accuracy and efficiency. This paper presents the details of the implementation and careful validation of the AHTM method using the NREL Phase VI wind turbine, in comparison with other existing models, such as RANS and URANS, showing that the VLES is the most accurate among those examined. Furthermore, the results of this study demonstrate that the AHTM approach has the flexibility, efficiency, and accuracy to be integrated with transient and concurrent MDAO tools for engineering design in the wind energy industry. Currently, the AHTM implementation is being integrated with the DAFoam for gradient-based multipoint MDAO using an efficient adjoint solver based on the sparse nonlinear optimizer (SNOPT).
4. Implementation of Arbitrary Hybrid Turbulence Model
The following equation describes the general form of control function
Fr, which is established from the ratio of sub-grid scale turbulence energy to the total turbulence energy:
where
is the cutoff length scale,
is the integral length scale, and
is the Kolmogorov length scale. Moreover,
are the mesh dimensions in different directions, and the laminar kinematic viscosity is υ. According to Equation (6), the resolution control function represents the ratio of the unresolved turbulence energy to the total turbulent energy [
14]. It was subsequently modified to the following form, which adopts the minimum value between 1.0 and the modified Speziale model [
15]:
where
Lc,
Li, and
Lk are the turbulent cutoff length scale, integral length scale, and Kolmogorov length scale, respectively,
, and
are mesh scales in different directions, and
is the laminar viscosity. Recommended values for
and
are
n ,
, and
according to the study of Speziale et. al. [
16]. The model constant
can be calibrated using
, which is the model constant of the turbulence model (in this case, SST k-omega model), and
, which is the typical Smagorinsky LES model constant [
8]. Thus,
. This resolution control function can be used to dampen the turbulent viscosity of the k-omega SST turbulence model as
, where
is the new turbulent viscosity,
is the resolution control function, and
is the turbulent viscosity of the turbulence model [
17].
As discussed before, the VLES formulation can give better results than RANS/URANS models and is cheaper than LES; thus, the model was implemented into OpenFOAM through the modification of the k-omega shear stress transport (kOmegaSST) model since this turbulence model gives more accurate results than other turbulence models for wind turbine simulation. First, applying all the recommended values and model constants into the resolution control function, the following expression could be derived for the resolution control function:
Furthermore, the kOmegaSST turbulence model equations proposed by Menter et al. [
16] were the same as given in Equations (3)–(5), and the turbulence viscosity term of the model was modified as follows for VLES implementation:
Now, the VLES was implemented into the OpenFOAM with source codes in the directory mykOmegaSST, and a library was created from mykOmegaSST and other OpenFOAM directories. First, the makeTurbulenceTransportmodels.C file was modified as shown in
Figure 1.
Then, a folder with the name Make was created in the mykOmegaSST directory, and the files and options source codes were created as shown in
Figure 2 and
Figure 3, respectively.
After that, the kOmegaSST files and base files were copied from OpenFOAM to the mykOmegaSST directory, and the modified version of the kOmegaSST or VLES is shown in
Figure 4.
The code shown above is the resolution control function added to the turbulence viscosity of the kOmegaSST. In addition, the recommended values and model constants need to be added as shown in
Figure 5.
Once, all the necessary files were modified and created, the codes were compiled and a library was created via the
wmake libso command. Furthermore, to run the VLES model for any simulation, the controlDict and turbulenceProperties files need to be changed. Links for all the necessary files to compile the model are included in
Appendix A.