This paper details the background to the WakeBlaster model: a purpose-built, parabolic three-dimensional RANS solver, developed by ProPlanEn. WakeBlaster is a field model, rather than a single turbine model; it therefore eliminates the need for an empirical wake superposition model. It belongs to a class of very fast (a few core seconds, per flow case) mid-fidelity models, which are designed for industrial application in wind farm design, operation, and control.

The domain is a three-dimensional structured grid, a node spacing of a tenth of a rotor diameter, by default. WakeBlaster uses eddy viscosity turbulence closure, which is parameterized by the local shear, time-lagged turbulence development, and stability corrections for ambient shear and turbulence decay. The model prescribes a profile at the end of the near wake, and the spatial variation of ambient flow, by using output from an external flow model.

In wind farms, wind turbines located downstream of other turbines will experience wake losses. Wind farm development and assessment processes require multiple iterations of configurations, as well as fast project turnaround.

A good understanding of how wake loss works can give a company the competitive edge, while an unexpected systematic performance loss can eliminate the expected profit from a project, or even from an entire project pipeline. Given the importance of wake losses, it may appear contradictory that many in the industry still use analytical single turbine wake models. Using single turbine wake models means that the wake from each turbine is propagated independently, wake expansion is not impacted by neighbouring wakes, and multiple wake deficits are superimposed using an empirical wake superposition model. Single wake models are based on an approach suggested 40 years ago, by

The longevity of the single wake model approach also speaks for the quality and practical usefulness of these early models. However, in order to provide accuracy for the full range of wind farms (e.g. large wind farms, closely cross-spaced farms, low hub height wind farms, wind farms with stable conditions, or offshore wind farms), an increasing number of empirical corrections had to be made, and parameters added, informed by new experimental data from wind farms, scale experiments, or higher-fidelity models – see, for example,

The increased computational power and scalability available today allows higher-fidelity wake models to be used in the iterative process of wind farm design. These models widen the operational envelope, include more physics, and reduce model uncertainties in non-standard situations. The theory behind one such model is presented in this paper: a 3D RANS (Reynolds-averaged Navier–Stokes) wind farm wake model, WakeBlaster.

In order to gain a more detailed understanding of wake losses in a wind energy research context, two groups of 3D RANS codes have been developed. The models are referred to as “field models”, to distinguish them from the single turbine models by

The first group of 3D RANS codes consists of parabolic solvers, using the thin shear layer approximation; see

The second group of 3D RANS field models, the elliptic solvers, is more widespread. Elliptic solvers are generally more powerful, and they iterate equations numerically, in order to allow information to be transported in all directions; this makes them more expensive computationally (by several orders of magnitude). These models use a

The WakeBlaster model developed by ProPlanEn by

The fundamental equations and assumptions for this solver are shown in the Sect.

The WakeBlaster wind farm simulator is based on a Reynolds-averaged Navier–Stokes (RANS) set of equations, which is used to solve the propagation of wake dissipation through the farm domain, in Cartesian 3-dimensional coordinates. In order to account for the fluctuation term of the velocity vector, it uses eddy viscosity turbulence closure, where the eddy viscosity is calculated from the combined wake and ambient wind speed shear profiles.

The wake model uses RANS equations for momentum conservation and mass flow conservation to calculate the three components of wind velocity in the axial, lateral, and vertical directions. Cartesian 3-dimensional vectors are used for displacement

The Reynolds-averaged momentum and mass conservation equation can be expressed in two dimensions, for either a free jet or a wake submerged in an incompressible fluid, as given by

The following simplifying assumptions are applied by Abramovich for a stationary free wake, expanding into an infinite region:

The effect of molecular viscosity is small

Flow pressure gradients can be neglected in most cases

The flow is stationary with respect to the mean velocities

Fluctuations along the flow change much slower than in the transversal direction

After substituting the continuity equation and applying the simplifying assumptions,

The ambient wind field is determined by an external flow model, and it determines the inflow conditions and spatial variations over a site. The turbine is represented by its hub height, diameter, and other readily available and measured characteristics.

The waked wind field is set up by creating a two-dimensional flow plane, which forms a cross section along the

The grid spacing is set by default to 0.1 D (rotor diameter). In the vertical direction, the grid starts at the ground

Axial-momentum theory prescribes pressure building up in the induction zone upstream of any wind turbine or wind farm and pressure recovery in the near wake downstream of the rotor. The momentum that each of the turbines extracts in the process is the wind-speed-dependent thrust coefficient, as a function of the idealized incident wind speed,

In the model, the momentum deficit is injected at the end of the near wake (which is assumed to be at 2 D downstream of the rotor) of each turbine, and it is distributed over an expanded rotor area, using the blunt bell-shaped wind speed deficit profile from

The flow plane is propagated according to Eq. (

At each half-step of the solving process, the horizontal and vertical velocities,

In practice, due to the assumption of incompressibility, this formulation will lead to a local velocity shear, resulting in non-zero lateral and vertical velocities that are infinitely far from the source of shear. In reality this would not be the case, due to the compressibility of air. Therefore, in order to account for the effect of compressibility, a spatial damping term is introduced so that

In the lateral direction, the physical boundary conditions are that

The key term controlling the rate of wake dissipation is eddy viscosity. Eddy viscosity has dimensions of length squared over time, and it can be represented by multiplying a length scale of the shear layer by a velocity scale of the flow field.

WakeBlaster calculates eddy viscosity from the shear profile of axial velocity in the

Create a combined flow plane by multiplying the ambient surface layer wind speed profile by the waked flow plane velocity

For each point, identify the local minimum and maximum velocity. For a point located at

In each of the two directions, the component of eddy viscosity is calculated as

The overall eddy viscosity is the calculated as

Calculation of the vertical component of eddy viscosity by finding the points of minimum and maximum velocity within a given height range.

For a logarithmic wind speed profile in the vertical direction with no lateral variation, this method leads to an eddy viscosity that is proportional to the height above ground.

The eddy viscosity, as so far described in Sect.

The “fixed” model obeys a first-order lag equation:

The “turbulence-dependent” model gives a larger lag distance when the eddy viscosity and turbulence are low, and it obeys the following equation:

When simulating atmospheric conditions that are not neutral, the calculation of eddy viscosity is modified. This modification uses the Monin–Obukhov length,

Furthermore, according to

WakeBlaster calculates the power output using power curve input from the user. In order to calculate accurate power, corresponding to the variant wind speed across the rotor, a rotor-equivalent wind speed (

A general directional variability of the wind within each flow case is included in a standard power curve. A rotor yaw angle can be set per turbine, to consider in the power calculation a known average directional misalignment with the rotor plane. A model to modify the power curve for site-specific directional variability over the rotor, for example changes with height or for specific meteorological conditions, is not included in the model.

WakeBlaster uses IEC methods in

In this section, the grid dependence and sensitivity is analysed, and an estimate of the numerical uncertainty is thereby provided. Computational performance for large wind farms is verified, and offshore wind farm model predictions are inspected graphically for plausibility.

The model uses a structured grid, in terrain-following coordinates. The grid resolution is scaled with a length scale characterizing the specific flow – the rotor diameter. The grid is equally spaced in all directions, and no stretching, compression, or nesting is applied to any part of the domain. The minimalist design is computationally efficient, and it avoids potential numerical errors – at grid interfaces which do not match, for example.

The solver is designed for a single purpose: to model the impact of wind turbines on the underlying flow and the consequential wind farm wake losses. A wind turbine's wake scales with its rotor diameter and its height above ground. In order to match the dominant scale in the flow for each wind farm, the grid resolution is fixed at 0.1 D; it thus scales with the rotor diameter.

Analysis of the sensitivity of model results to changes in grid resolution verifies that the results are not sensitive to grid resolution over the expected range of application. Challenges could arise – for example, when using an average resolution in wind farms with mixed turbine diameters and turbines mounted at low hub heights. In an annual energy calculation, the overall wake loss is composed of several thousand individual flow cases. Wake loss model errors are commonly estimated to be in the range of 10 %–20 %, relative to the average annual wake loss. Numerical errors should be 1 order of magnitude lower. Ignoring error compensation between flow cases, an error of 1 %–2 % (relative to the wind speed difference for an individual flow case) is acceptable.

The grid dependency study was carried out for the following scenario: a single turbine, with ambient wind speed perpendicular to the rotor plane. Ambient conditions were a wind speed of 8 m s

Numerical error due to the grid spacing, based on the difference in wind speed from the hypothetical zero grid spacing value calculated by Richardson extrapolation. The scenario assessed has an ambient wind speed of 8 m s

The sensitivity was tested in a flow case with a strong wake, and the results are presented in Fig.

The numerical error, due to grid spacing for an operational range of up to just above 0.1 D, is below 1 %. At a coarser resolution the model can no longer resolve the structure of the flow sufficiently. The current choice of grid resolution (0.1 D) represents a reasonable compromise between computational efficiency and model accuracy.

The grid resolution in the model scales automatically with the rotor diameter. Neither the grid nor the resolution is a variable which should (under normal circumstances) be adjusted by any user.

WakeBlaster is a medium-fidelity tool, which is typically capable of running each flow case in a few seconds, on the single core of a modern processor. With the default settings (a flow plane resolution of 0.1 D and a domain height of 3 D), the time (in seconds) to run a single flow case (

For example, a typical flow case for Horns Rev – a wind farm with 80 turbines arranged in a grid, with inter-turbine spacing of 7 D – runs in about 5

Layout of the Lillgrund wind farm. The turbine rotor diameter is 93 m with a hub height of 68 m. The turbine spacing is approximately 4.3 D, along the south-west to north-east rows, and 3.3 D along the south-east to north-west rows.

Using a three-dimensional wake model, it is possible to create plots of the three-dimensional waked flow field for the complete wind farm, for a particular flow case. This article presents a visualization of a single flow case from the Lillgrund wind farm, located in the Øresund Strait, between Sweden and Denmark. The Lillgrund wind farm presents a good case study, because the small spacing between turbines (3.3 and 4.3 D, along the two principal rows) leads to large wake effects. The layout is shown in Fig.

Three cross-sectional slices in the

Plots of the axial velocity on the wind farm relative to ambient wind speed for a flow case of 8 m s

These simulations indicate that there is significant interaction between wakes originating from individual turbines, and this supports the assumption that the wakes cannot be modelled independently. The wake interaction leads to a complex wake shape downstream of the wind farm. The low hub height of the wind turbines (68 m), relative to their rotor diameter (93 m), results in significant ground–wake interaction effects. As ambient mixing from below is limited, single turbine wakes become asymmetrical in shape, and the point of greatest deficit drifts downwards to below hub height.

The code is a mid-fidelity code designed to be fast and capable of simulating projects with several thousand turbines, working with limited amount of readily available input data, and be used in an iterative design process. This limits the level of detail that can be included in the submodels.

No direct interaction between the turbines and no description of the axial pressure gradient are included in the model. The induction zones directly upstream and downstream (near wake) of turbines can overlap and interact. This may lead to changes in turbine performance and turbine characteristics, and no attempt has been made to quantify such effects.

A basic representation of the ambient flow is used as input to the model. The wake is modelled as a perturbation of the underlying flow. No attempt has been made to model a two-way interaction with the atmospheric boundary layer.

The model uses the directional speed-ups predicted by a suitable flow model (for example in a RSF/WRG format) to account for spatial variation of the wind resource, for example due to orography, or roughness. Further complex terrain effects, like flow separation, are not considered.

The ambient wind direction is assumed to be constant throughout the wind farm. Therefore in curved flows (due to terrain or due to meteorological factors), downstream wake locations may not be accurate.

The WakeBlaster model undergoes continuous, data-driven improvement, and refined models will be added successively.

This is the first publication to present the theoretical background of WakeBlaster in some detail. WakeBlaster is a recently developed 3D RANS solver that is specialized to simulate the waked flow field on wind farms. The characteristics of this model show the desired performance balance between speed and level of detail.

WakeBlaster calculations are provided as a cloud service and designed for integration in other software packages. WakeBlaster is available from ProPlanEn directly (

PB carried out the following: formal analysis, investigation, methodology, software development, data curation, verification, visualization, and writing. WS carried out the following: conceptualization, funding acquisition, project administration, resources, investigation, supervision, methodology, and writing.

WakeBlaster is a commercial product of ProPlanEn Ltd. Wolfgang Schlez is the founder and sole shareholder of ProPlanEn Ltd. Philip Bradstock was employed by ProPlanEn Ltd at the time of carrying out the model development. He is a director of Bitbloom Ltd, providing services to ProPlanEn Ltd.

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.

A number of companies contributed operational wind farm data to this research, and their support is greatly appreciated. ProPlanEn GmbH processed the Lillgrund test case, as part of IEA task-31: WakeBench. The contributions of Staffan Lindahl and Sascha Schmidt (verification and visualization), Michael Tinning (software development), and Vassilis Kostopoulos (verification, investigation, and methodology) are acknowledged.

This research was co-funded by the UK's innovation agency, Innovate UK (grant no. 132381).

This paper was edited by Jens Nørkær Sørensen and reviewed by Paul van der Laan and one anonymous referee.