The high computational demand of large-eddy simulations (LESs) remains the biggest obstacle for a wider applicability of the method in the field of wind energy. Recent progress of GPU-based (graphics processing unit) lattice Boltzmann frameworks provides significant performance gains alleviating such constraints. The presented work investigates the potential of LES of wind turbine wakes using the cumulant lattice Boltzmann method (CLBM). The wind turbine is represented by the actuator line model (ALM). The implementation is validated and discussed by means of a code-to-code comparison to an established finite-volume Navier–Stokes solver. To this end, the ALM is subjected to both laminar and turbulent inflow while a standard Smagorinsky sub-grid-scale model is employed in the two numerical approaches. The resulting wake characteristics are discussed in terms of the first- and second-order statistics as well the spectra of the turbulence kinetic energy. The near-wake characteristics in laminar inflow are shown to match closely with differences of less than 3 % in the wake deficit. Larger discrepancies are found in the far wake and relate to differences in the point of the laminar-turbulent transition of the wake. In line with other studies, these differences can be attributed to the different orders of accuracy of the two methods. Consistently better agreement is found in turbulent inflow due to the lower impact of the numerical scheme on the wake transition. In summary, the study outlines the feasibility of wind turbine simulations using the CLBM and further validates the presented set-up. Furthermore, it highlights the computational potential of GPU-based LBM implementations for wind energy applications. For the presented cases, near-real-time performance was achieved using a single, off-the-shelf GPU on a local workstation.

Large-eddy simulations (LESs) can provide valuable insights into the aerodynamic interaction of wind turbines. In comparison to modelling approaches with lower fidelity, LESs allow for the investigation of aerodynamic effects that are directly associated with the transient nature of highly turbulent flows as found in the atmospheric boundary layer (ABL). Resolving the transient large energy-containing turbulent structures does, however, come at a high computational cost that is far beyond, for instance, that of Reynolds-averaged approaches

Despite the growing capacities of modern high-performance-computing (HPC) clusters, computational power remains the biggest bottleneck for such large-scale LES applications. Over the last 3 decades the lattice Boltzmann method (LBM) has evolved into a viable alternative to classical computational fluid dynamics (CFD) approaches with significantly increased computational performance

One method of special importance for the modelling of wind turbines in LES is the actuator line model (ALM). The ALM, as well as other actuator-type models, couple a CFD simulation to an extension of the blade element momentum (BEM) method. Using the locally sampled flow velocity, body forces of a blade element are computed using empirically determined lift and drag coefficients of the referring aerofoil section. These are then again applied in the domain of the CFD simulation

The objective of this paper is to analyse the wake of a single wind turbine simulated with the ALM and the cumulant lattice Boltzmann method (CLBM), a recently developed high-fidelity collision operator that is particularly suited for high-Reynolds-number flows

To the authors' knowledge, this study constitutes the first application of the CLBM to wind turbine wake simulations. Moreover, application-oriented studies of the utilized parameterized version of this collision operator (as further outlined in Sect.

The remainder of the paper is organized as follows: Sect.

In the following we provide a brief description of the LBM. This comprises a description of the governing equations as well as more specific topics relevant for the presented studies, such as sub-grid-scale (SGS) modelling and the implementation of the ALM. For a more detailed description of the fundamentals, the interested reader is referred to the work by

The LBM solves the kinetic Boltzmann equation, i.e. the transport equation of particle distribution functions (PDFs)

Schematic of three-dimensional velocity lattices. Coordinate-normal planes marked in yellow. Each vector refers to a discrete velocity

Macroscopic quantities can generally be obtained from the raw velocity moments of the PDFs:

In summary, the simplicity of the LBM leads to a straightforward explicit algorithm. Numerically, it is realized by decomposing and rearranging
Eqs. (

Due to poor numerical stability of the original LBGK model, various alternative approaches have been presented. These mostly relate to the class of multiple-relaxation-time (MRT) models; see for instance

Despite significant stability improvements, several fundamental deficiencies of MRT models render the approach unsuitable for high-Reynolds-number flows as required for studies of wind turbines in the ABL. Referring to the seminal paper by

A simple and widely adopted choice in the CLBM is to set all relaxation rates of higher-order cumulants to 1, commonly referred to as the AllOne cumulant. In this case, higher-order cumulants are instantly relaxed towards the reference equilibrium. This unconditionally damps all higher-order perturbations, providing an inherently stable solution and thereby an extremely robust numerical framework. Numerous studies have shown that the AllOne CLBM can be readily applied to high-Reynolds-number flows

From a theoretical point of view the parameterized CLBM can arguably be seen as one of the most advanced collision models today, in terms of both accuracy and stability. Nevertheless, the complexity of the collision model makes it more computationally demanding in terms compared to SRT and MRT models. Furthermore, the CLBM is only defined on the D3Q27 velocity lattice as opposed to SRT and MRT models that typically employ D3Q19 lattices. Consequently, it also requires more memory. In addition to the aforementioned theoretical considerations, we therefore provide a pre-study on the suitability of other collision models for this application in Sect.

For the sake of simplicity as well as numerical efficiency and accuracy, implementations of the LBM are commonly nondimensionalized. Physical units are therefore rescaled to non-dimensional lattice units (hereafter indexed

In this study we employ the LBM for an incompressible problem. As in the majority of applications, this implies that compressibility effects are assumed to have negligible effects on the flow physics of interests. The Mach number is thus merely required to be small, yet does not necessarily have to comply with the physically correct value of the problem. For incompressible flows it therefore commonly reduces to a somewhat free parameter affecting numerical accuracy in the incompressible limit

Early on, LES approaches were used in LBM frameworks

Using a standard constant Smagorinsky model, the eddy viscosity can be determined locally using the well-known formulation

A crucial characteristic of the CLBM is the model's inherent numerical stability. As opposed to many other collision models, it does not require the stabilizing features of explicit turbulence models, even for under-resolved highly turbulent flows. The stabilizing characteristic of the original AllOne cumulant approach appears rather obvious as it unconditionally resets all higher-order cumulants in each time step. The fourth-order accuracy of the parameterized approach, however, relies on the temporal memory of these higher-order cumulants. Therefore,

The lattice Boltzmann actuator line implementation used in this study closely follows the original description in NS frameworks as presented by

Differences between ALM implementations in NS and LBM frameworks are obviously small. The latter can be expected given that the link between the model itself and the flow solver is simply made by exchanging information of velocity and body forces. Lastly, it is worth mentioning that the locality of all subroutines of the ALM allows for a perfect parallelization. The model is therefore efficiently parallelized on the GPU, in line with the general architecture of the utilized LBM solver (see Sect.

In light of the code-to-code comparison, the simulations in both frameworks were set up in the most similar manner possible. This refers to the grid, the boundary conditions, and the implementation of the ALM. Nevertheless, certain differences remain unavoidable due to the inherently different numerical approaches. Further details, as well as the set-up in general, will be given in the following.

The LBM simulations are performed using the GPU-accelerated Efficient Lattice Boltzmann Environment “elbe”

Symmetry boundary conditions (zero gradient with no penetration) are applied at the lateral boundaries of the domain, referring to a simple bounceback with reversed tangential components

As a NS reference we consult the multipurpose flow solver EllipSys3D developed at the Technical University of Denmark (DTU) by

The governing equations are formulated in a collocated finite-volume approach. Diffusive and convective terms are discretized using second-order central differences and a blend of third-order QUICK (10 %) and fourth-order central differences (90 %), respectively. The blended scheme for the convective term was shown to provide sufficient numerical stability while keeping numerical diffusion to a minimum

Symmetry conditions are applied at the lateral boundaries, equivalently to the LB set-up. The outlet boundary condition prescribes a zero velocity gradient.

For the evaluation of the ALM we choose one of the most prominent test cases in this context, i.e. the simulation of the NREL 5 MW reference turbine

Schematic of the case set-up outlining the dimensions of the computational domain, position of the turbine, and definition of coordinates.

Temporal convergence of the turbulent kinetic energy

As a starting point we compare the results obtained with the CLBM to the NS reference in uniform laminar inflow. The simplicity of the case eliminates various uncertainties associated with more complex yet possibly more realistic inflow conditions. Also, it becomes more straightforward to analyse the effect of the numerical scheme on the downstream evolution of the wake and particularly the onset of turbulence as recently discussed by

In both solvers we apply the constant Smagorinsky model outlined in Sect.

As for the ALM, the blades in all cases are discretized by 64 points. The smearing width is set to 0.078125

Results of the simulations for the time-averaged tangential and normal force components of all cases are given in Fig.

Mean tangential force

Firstly, we compare the time-averaged cross-stream velocity profiles, given in Fig.

Cross-stream profiles of the mean stream-wise velocity

Relative difference (

It can be seen that the two numerical approaches are in good agreement in the near wake of the turbine. Up until

In the near-wake region discussed here, viscous effects usually only play a minor role. Among others, this is shown in a small wake recovery. Also, the rotational velocity does not change significantly. The wake is thus mostly governed by the inviscid flow solution

Further downstream (

Contour plots of the instantaneous stream-wise velocity

More quantitatively, the breakdown of the wake can be observed by means of a drastic increase in the turbulence intensity Ti as depicted in Fig.

Cross-stream profiles of the turbulence intensity

Stream-wise evolution of the turbulence intensity Ti at

The mechanism of the transition of wind turbine wakes has been extensively described based on ALM simulations; see, e.g.

As a last aspect we analyse the one-point turbulence kinetic energy spectra. The spectra shown in Fig.

One-point turbulent kinetic energy spectra in the near (

The energy content in the near wake (

At

Further downstream at

Rendering of the instantaneous contours of the

Laminar inflow cases allow for a good comparison of fundamental numerical aspects as discussed in Sect.

At the inlet we prescribe homogeneous isotropic turbulence (HIT) based on the von Kármán energy spectrum. The three-dimensional field of velocity fluctuations is generated based on the method developed by

Figure

Stream-wise evolution of the turbulence intensity Ti without ALM. Each data point

Figure

One-point turbulent kinetic energy spectra at the turbine position (

Analogously to Sect.

Cross-stream profiles of the mean stream-wise velocity

Relative difference (

Profiles of the turbulence intensity are shown in Fig.

Cross-stream profiles of the turbulence intensity Ti of the CLBM and NS references in turbulent inflow. For the legend, see Fig.

In the case discussed here the transition is dominated by instabilities introduced by the ambient turbulence. As opposed to the transition in laminar inflow, the impact of the dissipative characteristics of the numerical scheme here appears to be subordinate, if not negligible. Without imposed turbulence, perturbations triggering the transition grow within the wake itself starting from infinitesimal magnitudes as outlined in Sect.

The spectra of the turbulent kinetic energy at three different downstream positions are provided in Fig.

One-point turbulent kinetic energy spectra in the near wake (

Lastly, we shall comment on the small differences in the ambient turbulence shown earlier. Based on the above elaborations one might expect a more notable impact on the wake characteristics. With regards to this we refer to the study by

A further aspect of the CLBM to be discussed is the impact of the limiter of the third-order cumulants described in Eq. (

In the code-to-code comparison the limiter was practically switched off for the sake of comparison. Hence, numerical stability was also solely provided by the Smagorinsky model. Motivated by the lack of experience with the use of the limiter, we provide a brief investigation of the characteristics of the wake in comparison to the case with the Smagorinsky model used in Sect.

Contour plots of the mean stream-wise velocity and turbulence intensity are shown in Fig.

Contour plots of the mean stream-wise velocity

The presented case study underlines that the impact of the limiter is sufficiently large to arbitrarily tune the scheme's dissipativity over a wide range. Hence, the choice of the limiter in underresolved flows is by no means irrelevant despite the negligible influence on the asymptotic order of accuracy. On the other hand, the limiter conceivably states a measure to achieve implicit LES characteristics with the CLBM. As mentioned earlier, though, this clearly requires a more systematic understanding and subsequent tuning. Without the latter, the use of classical well-documented SGS models might remain more practical. Ultimately, they also provide numerical stability while choices for model parameters can build on well-documented experience.

We initially outlined that the main motivation for the use of the LBM in this context is the method's superior computational performance. Nevertheless, a detailed discussion is not the focus of this paper. For further details on this topic we refer to our previous study

The cumulant lattice Boltzmann method was applied to simulate the wake of a single wind turbine in both laminar and turbulent inflow. The turbine was represented by the actuator line model. The presented model was compared against a well-established finite-volume Navier–Stokes solver. It was shown that the cumulant lattice Boltzmann implementation of the actuator line model yields comparable first- and second-order statistics of the wake. More specifically, a very good agreement of the two numerical approaches was found in the near wake in laminar inflow, with differences amounting to less than 3 % in terms of the wake deficit. Larger discrepancies occurring in the far-wake were attributed to differences in the point of transition. These in turn could be related to the different numerical diffusivities of the schemes, building onto previous similar code-to-code comparisons

An additional case study investigated the impact of the third-order cumulant limiter in laminar inflow. It was shown that the choice of the limiter largely affects the dissipativity of the scheme. Likewise, the tunability of this dampening characteristic clearly shows the potential to be used in a more systematic way and might be exploited as an implicit LES feature. Yet, this requires further fundamental investigations in order to understand and calibrate it or even develop procedures to determine optimal values dynamically. As of now, the use of explicit eddy-viscosity SGS models thus appears more practical despite a small computational overhead.

As for future applications of the lattice Boltzmann method to more realistic wind-power-related flow cases, the following conclusions can be drawn. First and foremost, the presented study underlines the suitability of the cumulant lattice Boltzmann method for the simulation of highly turbulent engineering flows. The crucial advantage over other collision operators is the superior numerical stability of the method. No other collision operator initially tested in this study was found to be sufficiently robust using the given grid resolutions. The tested single- and multiple-relaxation-time models therefore do not appear suitable for LES of entire wind farms where higher spatial resolutions are not feasible and viscosities on the lattice scale consequently small. The advantages of the parameterized cumulant clearly render it a preferable collision model for wind turbine simulations and presumably other atmospheric flows. Application-oriented studies of the model are so far limited to this work and the recent study by

Generally, the choice of collision operator and lattice should consider stability, accuracy, memory demand, and performance. Based on the seminal works by

As for this specific set-up, satisfactory stability could only be achieved using the CLBM despite the use of the Smagorinsky model (for the reference formulations in moment space applied to the SRT and MRT models, see

In addition to stability issues, the isotropy of the D3Q19 lattice was shown to be insufficient. Figure

Usually, stability issues as described above can be remedied by using smaller grid spacings. As we consider the latter unfeasible for the described applications, we refrain from further investigations thereof at this point. Moreover,

Instantaneous velocity contours (

Both EllipSys3D and elbe are proprietary software and not publicly available. All data presented in this study can be made available upon request.

HA developed and implemented the LBM ALM; performed the simulations, post-processing, and data analysis; and drafted the original paper. HOE and SI contributed to the conceptualization of the study, discussion of the results, and revision of the manuscript.

The authors declare that they have no conflict of interest.

This article is part of the special issue “Wind Energy Science Conference 2019”. It is a result of the Wind Energy Science Conference 2019, Cork, Ireland, 17–20 June 2019.

The authors would like to thank Martin Gehrke (TUHH) for the productive collaboration on the implementation and testing of the parameterized cumulant LBM. Also, the many fruitful discussions of the case set-up and results with Niels N. Sørensen (DTU) are highly appreciated.

EllipSys3D simulations were performed on resources provided by the Swedish National Infrastructure for Computing (SNIC) at NSC.

This paper was edited by Alessandro Croce and reviewed by two anonymous referees.