Unstable atmospheric conditions are often observed during the daytime over land and for significant periods offshore and are hence relevant for wake studies. A simple

Wind turbine wakes have been studied for decades using many different methodologies, including wind tunnels, field experiments, analytical engineering models and numerical simulations. A review of these methodologies is given by

A sub-category of numerical simulations is the Reynolds-averaged Navier–Stokes (RANS) approach, which is a computational fluid dynamics (CFD) method that solves for the mean fields. This means that no time history of the flow is obtained; however, the computational resources required for RANS are very small compared to higher-fidelity CFD methods, making RANS an attractive option for parametric studies or for isolating various physical effects

The part of the atmosphere closest to the ground, i.e., the atmospheric surface layer (ASL), can be parametrized with the similarity theory of

In all the RANS studies discussed thus far, the simulations suffer from a known imbalance in the

Although there seems to be a general consensus that wakes should recover faster in unstable conditions, field measurements by

The balanced

The wakes are simulated with the incompressible, finite-volume flow solver EllipSys3D

The different components of the RANS simulations will be discussed in the following sections.

Numerous articles have been written about MOST, and a historical review is given by

The above relations are valid for

The roughness length

Parameters of the

Examples of four inflow profiles with identical

Analytical MOST profiles. Combinations of low/high TI and neutral/unstable stability. The rotor area of a NREL5MW turbine is shown. Dashed lines are used for

The eddy viscosity profile,

The eddy viscosity is sometimes expressed as a product of turbulent velocity and length scales (see

Turbulence scales in the freestream. Same labels as in Fig.

A recently developed actuator disk (AD) model by

The main advantage of the Joukowsky AD over the widely used “airfoil AD” (e.g.,

No nacelle or tower is included in our simulations, which have been shown to be a good approximation for

A homogeneous, flat lower surface is assumed for all cases in this paper. The inner part of the mesh surrounding the AD is called the “wake domain” and is shown for a typical case in Fig.

Top and side views of the wake domain, the size of which is

The numerical solution strategy of the incompressible RANS equations in EllipSys3D is thoroughly discussed in other publications

As mentioned in the introduction, the flow variables in an empty domain with MOST inflow can be kept in balance by modifying the

The background eddy viscosity shown in Fig.

Streamwise velocity

TKE budgets of the 2017 model

The cause of the

The buoyant production of TKE is

The “2017 model” is the one utilized by

Disk-averaged streamwise velocity,

The upstream (

A clear distinction between the two parametrizations is seen both in the near-wake (

Another deficiency of the unstable 2017 model is illustrated in Fig.

The faster wake recovery of the unstable cstB model compared to the neutral model (as demonstrated in Fig.

Normalized streamwise velocity

As stated in the introduction,

As shown in Fig.

It was recognized by

A more subtle modification arises recognizing that the

Analytical form of the

When both modifications are used, faster wake recovery for a given

Same as Fig.

Overview of test cases. SWiFT:

The cstB model with the

The numerical setup for each case follows that described in Sect.

A large wake benchmark study was conducted by

The inflow parameters of the SWiFT row in Table

For the unstable SWiFT case, the wake profile was only measured at 3

The unstable SWiFT case. The fixed frame of reference experimental and LES results were digitized from

The unstable NTK41 case, where LES and experimental results were digitized from

RANS can generally not be expected to perform better than a well-performed LES, and if it does, it is likely due to fortunate error cancelations. Therefore, from a theoretical point of view one could argue that the performance of RANS should mainly be assessed with regards to how well it matches the LES results. Both RANS and LES use many of the same idealizations (uniform roughness, flat terrain, homogeneous inflow, etc.), and indeed our RANS results in Fig.

For the SWiFT case, the 2017 model seemingly performs better than the cstB model, but this is probably due to fortuitous model error and/or unaccounted mesoscale effects. Model error was expected because the neutral EllipSys3D RANS simulation in the SWiFT study (surprisingly) did not compare well with experimental results

A Nordtank NTK41 500 kW wind turbine was installed at what is now the Risø campus of the Technical University of Denmark in 1992 and was used for many research studies before its decommissioning in 2021. Among these studies, the NTK41 test case of this paper is based on lidar measurements and LESs conducted by

The unstable V80-Abkar case, where LES results were digitized from

The NTK41 turbine is a stall-regulated wind turbine and is therefore operated at constant rotational speed independent of the inflow wind speed

Figure

Relative to the LES, the velocity deficits shown in Fig.

The unstable V80-Keck case, where LES results were digitized from

This case is based on a LES from

Figure

The last case is based on the LES studies by

The steady-state power, thrust coefficient and rotational speed were not given in the paper, so therefore the steady-state curves from the DTU in-house aeroelastic solver, HAWCStab2, were used. These are similar to the curves shown by

The velocity deficit of the new cstB model compares well with the LES data, especially so for

The unstable NREL5MW case, where LES results were digitized from

We have proposed a simple

Originally developed to account for the general over-diffusive nature of

The cstB model with the modified

Testing the cstB model behavior in more complicated scenarios, i.e., aligned row cases (see Appendix

Earlier studies by

Grid-independence study. “Ref” means reference, i.e., the finest resolution available. Metrics are evaluated at

Based on this small grid study, a domain resolution of 10 cells per diameter and an AD resolution of

In summary, the surface force distributions (unit: N m

The normal blade loading,

Disk average of velocity,

Aligned row of 10 turbines with inflow similar to the Case 5 of

Here,

The normal and tangential loadings for the same case as used in Sect.

Finally, in Fig.

Even though the focus of the present paper is on single wakes, wake simulations will in practice most often involve wake interaction. Contrary to engineering models, there is no need for empirical wake superposition methods in RANS, since the wake interaction automatically results from solving the RANS equations.

The Case 5 from the study of

EllipSys3D and PyWakeEllipSys are proprietary software of DTU. Information about the latter can, however, be freely accessed at

The RANS results were generated with DTU's proprietary software, but the data presented can be made available by contacting the corresponding author. Interested parties are also welcome to hand-digitize the results and use them as reference in other publications.

MB performed the RANS simulations and proposed the modifications to the turbulence model. All authors (MB, MPvdL and MK) contributed to derivation of the new model and article writing.

The contact author has declared that neither they nor their co-authors have any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The authors are grateful to the creators of the LES and lidar results used in the validation section. We would also like to thank the two reviewers for their feedback and suggestions.

This paper was edited by Sandrine Aubrun and reviewed by two anonymous referees.