Reynolds-averaged Navier–Stokes (RANS) simulations of wind turbine wakes are usually conducted with two-equation turbulence models based on the Boussinesq hypothesis; these are simple and robust but lack the capability of predicting various turbulence phenomena. Using the explicit algebraic Reynolds stress model (EARSM) of

As wind farms increase in size and number of turbines, increasingly more attention should be given to the study of wind turbine wakes, as they can account for a relatively large power production decrease. Simple engineering models, such as the classic

A wind turbine wake is a complex three-dimensional swirling flow, and its development is governed by turbulent mixing, which is strongly influenced by density stratification in the atmospheric boundary layer (ABL) and the interaction between the ABL and the wake itself. To model a more physically correct wind farm flow we therefore solve the Navier–Stokes equations, which in essence are a reformulation of Newton's second law, along with conservation of mass. The process of discretizing and solving these equations on computers is known as computational fluid dynamics (CFD). Unfortunately, the Reynolds number of atmospheric flows is so large

Linear EVMs based on the Boussinesq hypothesis (e.g., mixing length,

It appears that only

The EARSM framework of

Section

The turbulence models utilized in this paper assume incompressible, non-stratified flow; no system rotation (no Coriolis or centrifugal contributions); isotropic dissipation of TKE; and high Reynolds number flow.

The Boussinesq hypothesis is used to obtain the Reynolds stresses

To simplify the Boussinesq hypothesis, Eq. (

Finally, it can be noted that the time derivative is retained in the transport equations, Eqs. (

The

In the freestream of a neutral ASL,

Table

Model constants for the

The EARS model of

Using the complete two-dimensional tensor basis for the anisotropy tensor

The solution procedure is more thoroughly described in Sect.

Model constants for the WJ-EARSM as recommended by

The 3D model is derived in an analogous way to the 2D model

Unfortunately, there does not exist an analytical solution for

One can show that the 3D model reduces to the 2D model in two-dimensional mean flows, where

The flow cases are simulated with EllipSys3D, which is a finite-volume CFD solver developed and described in detail by

In the

A segregated solver (i.e., solving the

Segregated solver procedure for WJ-EARSM in EllipSys3D. Note that

The 1D version of EllipSys3D, EllipSys1D

For both EllipSys1D and EllipSys3D, the procedure of calling the WJ-EARSM is as such:

Use the most recent solution of momentum and turbulence transport equations to calculate the normalized strain rate and rotation rate tensors,

Calculate tensors and invariants.

Calculate

Calculate coefficients,

Calculate anisotropy tensor,

Calculate TKE shear production,

Calculate

As is clear from the previous section, the expressions in the WJ-EARSM are considerably longer compared to the ones of the

Even before running the verification cases, we consider the general class of “simple shear flows” (a.k.a. 1D parallel flows), where the normalized strain rate and rotation rate tensors are

Analytical off-diagonal anisotropy in simple shear flow.

Homogeneous shear flow (see review by

Mesh and the prescribed velocity profile in EllipSys1D for the homogeneous-shear-flow case.

In principle, homogeneous shear flow is an unsteady 0D case. The turbulence evolves identically at all positions in space; hence there is actually no need for a spatial discretization as shown in Fig.

The simulation parameters used are as follows.

Simulation of homogeneous shear flow with WJ-EARSM (full line) and the model's analytical asymptotic values (dashed lines) with different sets of constants. The simulation data are extracted at

A nice feature of homogeneous shear flow for verification purposes is that it evolves to an asymptotic state and that an analytic solution exists for this state; e.g., the asymptotic production-to-dissipation ratio can be derived to be

Analytic, asymptotic homogeneous shear flow formulas. In homogeneous shear flow,

In 1D parallel flows,

Half-channel EllipSys1D setup. The sketch to the right shows the convention of the coordinate system used for rough-wall simulations.

The second reference flow is the fully developed, steady-state half-channel flow, a.k.a. pressure-driven boundary layer (PDBL) flow, and can be solved in 1D using a grid as sketched in Fig.

The input parameters for the simulations are as follows:

The region in the lower part of the domain is known as the “log layer”, which is characterized by equilibrium turbulence, i.e.,

Analytical values in the log layer of the half-channel flow.

Half-channel simulation results (full lines) and analytical log-layer solutions (dashed lines).

The flow profiles with the analytical log-layer values are plotted in Fig.

The square duct geometry and boundary conditions are shown in Fig.

Square duct geometry and a cross-section showing the secondary corner flows. Sketch made with inspiration from

Although, the fully developed square duct flow might appear as a two-dimensional problem, it in fact features a full three-dimensional flow field, due to the secondary corner flows, also sketched in Fig.

Streamwise

In this paper, we simulate a fully turbulent square duct flow (high Reynolds number) with rough-wall BCs; hence the flow is independent of

Parameters used for simulation of square duct flow in EllipSys3D. The brackets

Simulation parameters for the single-wake V80 case. Note that the LES uses neutral PDBL for the atmospheric model.

Streamwise velocity

The streamwise and vertical velocities (

The 3D WJ-EARSM performed similarly to the 2D WJ-EARSM, although with a slightly weaker secondary flow, which can be seen in Fig.

Flow domain

This concludes the verification studies, where the WJ-EARSM has been seen to give expected results for three canonical flows. Furthermore, the last case of square duct flow clearly demonstrates that EARS models are able to predict physical phenomena that two-equation models based on the linear Boussinesq hypothesis cannot do.

This section concerns the application of the EARS model to a single-wind-turbine wake. The numerical CFD setup is similar to that used in many previous RANS studies

The case simulated is similar to the case used by

The neutral ASL inflow profile

WJ-EARSM dependence on

Inflow profiles for the single-wake V80 case.

In the freestream, the neutral ASL profiles, Eq. (

In the neutral ASL, we have

The inflow profiles of the LES and RANS simulations are shown in Fig.

Eigendecomposition of the Reynolds stress tensors can be used to describe the turbulence state through its three real eigenvalues, which describe the fluctuations in the three orthogonal, principal directions. Several techniques (e.g., eigenvalue map, invariant map, barycentric map, and Lumley triangle) combine the eigenvalues and visualize them with two-dimensional maps; in Fig.

RGB-colored barycentric triangle, with RANS/LES inflow data shaded by height; white lines at right mark (

Streamwise velocity

Wake data in the form of velocity and TI contours at hub height and profiles at three downstream positions are shown in Figs.

Streamwise velocity

Overall the wake velocity contours in Fig.

Disk-averaged streamwise velocity and TI for the single-wake V80 case.

Considering the wake profiles in Fig.

Normal stresses at hub height for the single-wake V80 case.

Lateral

Lastly, we want to emphasize that the turbulence model is not solely responsible for the wake results: the same turbulence model applied for the same case but with different codes/solvers can yield significantly different results, as can be seen in the comparison of the

For more insights on the wake mixing and turbulence, we now turn to the second-order statistics of turbulence, namely the individual Reynolds stress components. The normal components are shown in Fig.

Transport of

Simulation parameters for the aligned-row case from TotalControl. The AD scaling and 1D mom'm control methods are described in detail by

To conclude, we see some advantages but also disadvantages with using WJ-EARSM over the

In the inflow section, Sect.

RGB-colored turbulence componentiality at hub height for the single-wake V80 case.

To test the WJ-EARSM in a wind farm scenario, we simulate the lower row in the TotalControl rot90 reference wind farm, which consists of eight aligned wind turbines with 5

A

Streamwise velocity contour at hub height for the aligned-row TotalControl case.

TI contour at hub height for the aligned-row TotalControl case.

The velocity contours in Fig.

The turbulence intensity contours in Fig.

Streamwise velocity and turbulence intensity at the axial line going through the AD centers for the aligned-row TotalControl case.

In conclusion, the WJ-EARSM appears numerically stable and well behaved (e.g., no monotonic decreasing velocity deficit or other unphysical effects) for interacting wakes, and as in the single-wake case, there are both some improvements and some less desirable effects of the model over the

This paper documents and explains our implementation and application of an EARSM

Three canonical flow cases – homogeneous shear flow, half-channel flow, and square duct flow – were used to verify the implementation of the model and also showcased some of the advantages with an EARSM over traditional linear EVMs, namely the prediction of freestream turbulence anisotropy and secondary flow phenomena. All three cases have either analytical asymptotes or DNS data to compare against and are easy to set up, which makes them ideal for verification purposes.

For neutral ASL inflow we show that there is a delicate relationship between the turbulence constants that needs to be fulfilled to ensure a non-developing freestream solution and that it also dictates the amount of freestream turbulence anisotropy. It should be noted that this balance of constants is also important for numerical robustness. Comparing the RANS inflow with reference LES data shows that the WJ-EARSM is capable of predicting similar freestream anisotropy, whereas the turbulence of the

A single-wake case was considered first, and it was observed that the 2D version of the EARSM yielded almost identical results to the 3D version, even when tangential forces were applied on the AD; thus we use the 2D model for the remainder of the paper. It should be noted that the 2D version of the WJ-EARSM is a complete and invariant model for general three-dimensional mean flows. Only the particular dependency of pure three-dimensional effects is simplified, which will have a minor impact on most three-dimensional mean flows of interest. The wake profiles of the EARSM were more top-hat-shaped than the profiles observed in the LES data, which might be related to the underlying weak-equilibrium assumption and limitations in the length-scale-determining

Finally, we simulated a row of eight aligned turbines, where the trends from the single-wake case could also be seen, e.g., the top-hat-shaped profiles and better turbulence intensity prediction with EARSM. There were more uncertainties in the comparison with the LES data in this case because different turbine modeling techniques were used, but the case nevertheless shows that the EARSM also behaves sensibly in cases with wake–wake interaction in the sense that the code still converges and that no unphysical trends (such as monotonic increase in wake deficit throughout the row of turbines) are observed.

In conclusion, the EARSM of

Atmospheric conditions in thermally stable stratification and thermal convection strongly influence the turbulence states, anisotropies, and in particular the vertical mixing in the ASL. This will have a fundamental influence on the wake development and the performance of wind parks. The extension of the EARSM to non-neutral conditions has over recent years been developed by

Earlier studies

Grid study of streamwise velocity (upper row) and TI (lower row) profiles at hub height for the V80 case with the 2D WJ-EARSM using different mesh resolutions.

The

Tested sets of turbulence model constants and derived variables for the single-wake V80 case. For all sets, we use

Inflow profiles for the single-wake V80 case with different sets of model constants.

Equation (

Streamwise velocity

Normal stresses at hub height for the single-wake V80 case with different sets of model constants.

Another consideration is that the roughness length and friction velocity also need to be modified to give the same hub height velocity and turbulence intensity according to Eq. (

Figure

Increasing

As discussed in Sect.

Gaussian-filtered (GF) wake profiles.

The wind direction

For example the hub height LES values for the V80 case gives

Assuming zero mean wind direction and a Gaussian distribution gives a simple model for the wind direction variability:

It should be intuitively clear that such a distribution of wind directions would act to smoothen out the wake profile compared to a simulation with

To obtain

The above model for

Both the linear and quadratic filter width models are used in Gaussian convolutions of the 2D WJ-EARSM wake profiles, and the results are shown in Fig.

EllipSys3D is proprietary software of DTU.

The RANS results were generated with EllipSys3D, 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 a reference in other publications.

MB implemented the EARSM in EllipSys1D/3D, performed the RANS simulations, and wrote the initial article draft. SW provided guidance for the implementation and verification of the EARSM, MPvdL suggested re-tuning the model constants and using a Gaussian filter to compensate for the top-hat-shaped wake profiles, and MK contributed to the discussions regarding inflow turbulence in the neutral ASL. All authors (MB, SW, MPvdL, and MK) contributed to editing and finalizing the paper.

The contact author has declared that none of the authors has any competing interests.

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

The authors would like to thank Mahdi Abkar for sharing the LES data of the V80 case and Ishaan Sood and Johan Meyers for making the TotalControl LES data publicly available in an online repository. We would also like to thank the reviewers for suggestions and comments.

This paper was edited by Roland Schmehl and reviewed by J. Blair Perot and Stefano Letizia.