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Article Is Structured As Follows Wind Energ. Sci., 5, 355–374, 2020 https://doi.org/10.5194/wes-5-355-2020 © Author(s) 2020. This work is distributed under the Creative Commons Attribution 4.0 License. Rossby number similarity of an atmospheric RANS model using limited-length-scale turbulence closures extended to unstable stratification Maarten Paul van der Laan, Mark Kelly, Rogier Floors, and Alfredo Peña DTU Wind Energy, Technical University of Denmark, Risø Campus, Frederiksborgvej 399, 4000 Roskilde, Denmark Correspondence: Maarten Paul van der Laan ([email protected]) Received: 9 October 2019 – Discussion started: 21 October 2019 Revised: 17 January 2020 – Accepted: 20 February 2020 – Published: 26 March 2020 Abstract. 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 turbu- lence 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. 1 Introduction nonstationarity, which are typically considered by mesoscale and three-dimensional time-varying models. In this work, Wind turbines operate in the turbulent atmospheric boundary we investigate idealized ABL models that are based on layer (ABL) but are designed with simplified inflow condi- one-dimensional Reynolds-averaged Navier–Stokes (RANS) tions that represent analytic wind profiles of the atmospheric equations, where the only spatial dimension is the height surface layer (ASL). The ASL corresponds to roughly the above ground. The output of the model can be used as first 10 % of the ABL, typically less than 100 m, while the inflow conditions for three-dimensional RANS simulations tip heights of modern wind turbines are now sometimes be- of complex terrain (Koblitz et al., 2015) and wind farms yond 200 m. Hence, there is a need for inflow models that (van der Laan and Sørensen, 2017b). Turbulence is mod- represent the entire ABL in order to improve the design of eled here by two limited-length-scale turbulence closures, wind turbines and wind farms. Such a model should be sim- the mixing-length model of Blackadar(1962) and the two- ple enough to efficiently improve the chain of design tools equation k–" model of Apsley and Castro(1997). These used by the wind energy industry. turbulence models can simulate one-dimensional stable and The ABL is complex and changes continuously over neutral ABLs without the necessity of a temperature equa- time. Idealized, steady-state models can represent long-term- tion and a momentum source term of buoyancy. In other averaged velocity and turbulence profiles of the real ABL, words, all temperature effects are represented by the turbu- including the effects of Coriolis, atmospheric stability, cap- lence model. The limited-length-scale turbulence models de- ping inversion, homogeneous surface roughness and flat ter- pend on four parameters: the roughness length, the Coriolis rain; here we exclude the effects of flow inhomogeneity and parameter, the geostrophic wind speed, and a chosen maxi- Published by Copernicus Publications on behalf of the European Academy of Wind Energy e.V. 356 M. P. van der Laan et al.: Rossby number similarity of an atmospheric RANS model mum turbulence length scale that is related to the ABL depth. We show that the normalized profiles of wind speed, wind di- DU d dU D f (V − V ) C νT D 0; rection and turbulence quantities are only dependent on two Dt c G dz dz dimensionless parameters that represent the ratio of the in- DV d dV ertial to the Coriolis force, based on two different length D −f (U − U ) C νT D 0; (1) Dt c G dz dz scales: the roughness length and the maximum turbulence length scale. These dimensionless parameters are Rossby where U and V are the mean horizontal velocity compo- numbers. The Rossby number based on the roughness length nents, UG and VG are the corresponding mean geostrophic is known as the surface Rossby number, as introduced by velocities, fc D 2sin(λ) is the Coriolis parameter with Lettau(1959), while the Rossby number based on the max- as Earth’s angular velocity and λ as the latitude, and z is the imum turbulence length is a new dimensionless parameter. height above ground. In addition, the Reynolds stresses u0w0 The obtained model-based Rossby number similarity is used and v0w0 are modeled by the linear stress–strain relation- to validate a range of simulations with historical measure- 0 0 0 0 ship of Boussinesq(1897): u w D −νT dU=dz and v w D ments of the geostrophic drag coefficient and cross-isobar an- −νT dV =dz, where νT is the eddy viscosity. The boundary gle. In addition, we show that both RANS models’ solutions conditions for U and V are U D V D 0 at z D z0 and U D UG are bounded by two analytic solutions of the idealized ABL. and V D VG at z ! 1, where z0 is the roughness length. The limited-length-scale turbulence closures of Blackadar Note that it is possible to write the two momentum equations (1962) and Apsley and Castro(1997) can model the effect as a single ordinary differential equation: of stable and neutral stability but cannot model the unstable atmosphere. We propose simple extensions to solve this is- d dW νT D if W; (2) sue and validate the results of the extended k–" model with dz dz c measurements of wind speed and wind direction profiles. The model extensions lead to a third Rossby number, where where W ≡ (U −UG)Ci(V −VG) is a complex variable and the length scale is based on the Obukhov length. The lim- i2 D −1. ited mixing-length model is not considered in the compari- The eddy viscosity, νT , needs to be modeled in order to son with measurements because we are mainly interested in close the system of equations. The eddy viscosity can be the limited-length-scale k–" model. The k–" model is more written as νT D u∗`, where u∗ and ` represent turbulence applicable to wind energy applications because it can also velocity and turbulence length scales. For a constant eddy provide an estimate of the turbulence intensity, which is not viscosity, the equations can be solved analytically, and the available from the limited mixing-length model of Blackadar solution is known as the Ekman spiral (Ekman, 1905), which (1962). The limited mixing-length model is applied here to includes the wind direction change with height due to Corio- show that the same model-based Rossby number similarity lis effects. The Ekman spiral can also be considered a laminar as obtained for the k–" model is recovered. solution, since one can neglect the turbulence in the momen- The article is structured as follows. The background and tum equations and set the molecular viscosity to determine theory of the idealized ABL are discussed in Sect.2. Ex- the rate of mixing. For an eddy viscosity that increases lin- tensions to unstable surface layer stratification are presented early with height, the equations can also be solved analyti- in Sect.3. Section4 presents the methodology of the one- cally, as introduced by Ellison(1956) and discussed by Kr- dimensional RANS simulations. The model-based Rossby ishna(1980) and Constantin and Johnson(2019). The two similarity is illustrated in Sect.5. The simulation results of analytic solutions are provided in AppendixA. One can re- the limited-length-scale k–" model including the extension late the analytic solution of Ellison(1956) to the (neutral) to unstable conditions are compared with measurements in ASL (z zi), while the Ekman spiral is more valid for alti- Sect.6. tudes around the ABL depth zi. Neither of the two analytic solutions result in a realistic representation of the entire (ide- 2 Background and theory – idealized models of alized) ABL. A combination of both a linear eddy viscosity ∼ the ABL for z zi and a constant eddy viscosity for z zi should provide a more realistic solution. For example, the eddy vis- We model the mean steady-state flow in an idealized ABL. cosity could have the form νT D κu∗0zexp(−z=h), where Here idealized refers to flow over homogeneous and flat ter- νT increases linearly with height for z h as expected in rain under barotropic conditions such that the geostrophic the surface layer; then it reaches a maximum value at z D h, wind does not vary with height. This flow can be described by and decreases to zero for z > h. Note that u∗0 is the friction the incompressible RANS equations for momentum, where velocity near the surface. Constantin and Johnson(2019) de- the contribution from the molecular viscosity is neglected rived a number of solutions for a variable eddy viscosity, al- due to the high Reynolds number: though an explicit solution for the entire idealized ABL with a realistic eddy viscosity (in the previously mentioned form) has not been found yet.
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