The design of wind turbines and wind farms can be improved by increasing the accuracy of the inflow models representing the atmospheric boundary layer. In this work we employ one-dimensional Reynolds-averaged Navier–Stokes (RANS) simulations of the idealized atmospheric boundary layer (ABL), using turbulence closures with a length-scale limiter. These models can represent the mean effects of surface roughness, Coriolis force, limited ABL depth, and neutral and stable atmospheric conditions using four input parameters: the roughness length, the Coriolis parameter, a maximum turbulence length, and the geostrophic wind speed. We find a new model-based Rossby similarity, which reduces the four input parameters to two Rossby numbers with different length scales. In addition, we extend the limited-length-scale turbulence models to treat the mean effect of unstable stratification in steady-state simulations. The original and extended turbulence models are compared with historical measurements of meteorological quantities and profiles of the atmospheric boundary layer for different atmospheric stabilities.

Wind turbines operate in the turbulent atmospheric boundary layer (ABL) but are designed with simplified inflow conditions that represent analytic wind profiles of the atmospheric surface layer (ASL). The ASL corresponds to roughly the first 10 % of the ABL, typically less than 100 m, while the tip heights of modern wind turbines are now sometimes beyond 200 m. Hence, there is a need for inflow models that represent the entire ABL in order to improve the design of wind turbines and wind farms. Such a model should be simple enough to efficiently improve the chain of design tools used by the wind energy industry.

The ABL is complex and changes continuously over time. Idealized, steady-state models can represent long-term-averaged velocity and turbulence profiles of the real ABL, including the effects of Coriolis, atmospheric stability, capping inversion, homogeneous surface roughness and flat terrain; here we exclude the effects of flow inhomogeneity and nonstationarity, which are typically considered by mesoscale and three-dimensional time-varying models.
In this work, we investigate idealized ABL models that are based on one-dimensional Reynolds-averaged Navier–Stokes (RANS) equations, where the only spatial dimension is the height above ground. The output of the model can be used as inflow conditions for three-dimensional RANS simulations of complex terrain

The limited-length-scale turbulence closures of

The article is structured as follows. The background and theory of the idealized ABL are discussed in Sect.

We model the mean steady-state flow in an idealized ABL. Here idealized refers to flow over homogeneous and flat terrain under barotropic conditions such that the geostrophic wind does not vary with height. This flow can be described by the incompressible RANS equations for momentum, where the contribution from the molecular viscosity is neglected due to the high Reynolds number:

The eddy viscosity,

Comparison of analytic and numerical solutions of existing idealized ABL models using

Eddy viscosity closures for the idealized ABL.

The two limited-length-scale turbulence closures discussed in Sect.

One can generically parameterize the turbulence length scale

In order to account for the increase in turbulence length scale in the surface layer under unstable conditions, we add a buoyancy source term

The one-dimensional numerical simulations are performed with EllipSys1D

The limited-length-scale

From the two-equation

The flow is driven by a constant pressure gradient using a prescribed constant geostrophic wind speed. The initial wind speed is set to the geostrophic wind speed at all heights. During the solving procedure, the ABL depth grows from the ground until convergence is achieved, which occurs when the growth rate of the ABL depth is negligible because a balance between the prescribed pressure gradient, the Coriolis forces, and the turbulence stresses is obtained. The flow that we are solving is relatively stiff, and we choose to include the transient terms using a second-order three-level implicit method with a large time step that is set as

The turbulence model constants of the

A grid refinement study of the numerical setup is performed for the limited-length-scale

Grid refinement study of the one-dimensional RANS simulation using the limited-length-scale

The numerical solution of the original limited-length-scale turbulence closures of

Figure

Rossby number similarity of the original turbulence closures.

Considering the non-neutral ABL with Coriolis effects but ignoring the strength of capping inversion (entrainment), in the micrometeorological literature the

Rossby number similarity of the turbulence closures extended to unstable surface layer conditions.

The extended limited-length-scale mixing-length model (Fig.

One could choose to use the friction velocity at the surface,

The Rossby number similarity can be employed to generate a library of ABL profiles for a range of

The obtained Rossby number similarity can only be achieved for a grid-independent numerical setup, as we have shown in Sect.

The ABL depth

Normalized boundary layer depth

We employ the Rossby similarity from Sect.

The geostrophic drag law (GDL) is a widely used relation in boundary layer meteorology and wind resource assessment

Reproduced from

Figure

The fitted

Reproduced from

Figure

ASL validation cases. Fitted

ASL measurements of

Figure

Table

It should be noted that the validation presented in Fig.

ABL validation cases based on

In Case 6 from

Since

Figure

ABL measurements from

From the measurements during Case 9 it was observed that the WRF-modeled ABL depth grew from 300 m to nearly 1200 m, which indicates that the conditions were largely transient; such nonstationary conditions are difficult for a RANS model. More unstable cases are necessary to further validate the extended model, including measurements of turbulence quantities such as the (total) turbulence intensity. It is possible to use validation cases based on turbulence-resolving methods, such as large-eddy simulations, in future work.

The idealized ABL was simulated with a one-dimensional RANS solver, using two different turbulence closures: a limited mixing-length model and a limited-length-scale

The limited-length-scale turbulence closures can represent the effects of stable and neutral stratification but cannot model unstable conditions.
We have proposed simple extensions to overcome this issue, without adding a temperature equation

The application of the one-dimensional RANS simulations to generate inflow profiles for three-dimensional RANS simulations is not performed here and it should be investigated in future work. Ongoing and future work also includes the incorporation of the effect of the capping-inversion strength to accommodate entrainment at the ABL top (softening the ABL lid, one might say); this can be considered as an introduction of an additional length scale. In addition, the effects of length-scale limitation and neglecting the buoyancy force in the momentum equation need to be quantified for three-dimensional RANS simulations of complex terrain and wind farms.

The analytic solution of

The analytic solution of

A GDL can be derived in the form of Eq. (

The cross-isobar angle (angle between the geostrophic wind direction and surface wind direction) can be written as a function of the geostrophic drag coefficient

In this article, we have shown a Rossby similarity of two limited-length-scale turbulence closures using the geostrophic wind speed,

The Rossby similarity based on

It should be noted that

Geostrophic drag coefficient simulated by the limited-length-scale

Rossby number similarity of the limited-length-scale

The numerical results are generated with proprietary software, although the data presented can be made available by contacting the corresponding author.

MPVDL performed the simulations, obtained the model-based Rossby number similarity for the

The authors declare that they have no conflict of interest.

This paper was edited by Johan Meyers and reviewed by Javier Sanz Rodrigo and one anonymous referee.