Boundary-Layer Meteorology

, Volume 128, Issue 2, pp 229–254 | Cite as

A Mixed Spectral-Integration Model for Neutral Mean Wind Flow Over Hills

  • Jean-François Corbett
  • Søren Ott
  • Lars Landberg
Original Paper


A linear model for neutral surface-layer flow over orography is presented. The Reynolds-Averaged Navier-Stokes and E\({\varepsilon}\) turbulence closure equations are expressed in a terrain-following coordinate system created from a simple analytical expression in the Fourier domain. The perturbation equations are solved spectrally horizontally and by numerical integration vertically. Non-dimensional solutions are stored in look-up tables for quick re-use. Model results are compared to measurements, as well as other authors’ flow models in three test cases. The model is implemented and tested in two-dimensional space; the equations for a full three-dimensional version are presented.


Complex terrain Mixed spectral-integration Wind-flow model 

List of Symbols


Matrix of coefficients of the system of ODEs


Matrix of coefficients A in the limit \({{k\acute{z} \rightarrow \infty}}\)

\({C_{\mu},C_{\varepsilon 1},C_{\varepsilon 2}}\)

Constants of the \({E-\varepsilon}\) turbulence model


Turbulent kinetic energy (TKE)


Height above ground level


Depth of the boundary layer


Terrain height


The imaginary unit \({\sqrt{-1}}\)


\({\equiv \det \frac{\partial x_i}{\partial \acute{x}_{j}}}\), the Jacobian of the coordinate transformation


Wavenumber in coordinate \({\acute{x}_{i}}\), i = 1, 2


Wavenumber: in the 2D case, we drop the subscript: k = k1


Horizontal wavenumber vector: k = (k1, k2)


Hydrostatic pressure


Expansion parameter


Matrix of source terms in the system of ODEs


ith Component of the velocity: (u1u2u3) = (uvw)


ith Component of the velocity in transformed coordinates (normalized by J)


Cartesian coordinates: (x1x2x3) = (xyz)


Transformed coordinates: \({(\acute{x}_1,\,\acute{x}_2,\,\acute{x}_3) = (\acute{x},\,\acute{y},\,\acute{z})}\)


Transformed horizontal position vector: \({\acute{\bf{x}} = (\acute{x}_1,\,\acute{x}_2)}\)


Matrix of dependent variables in the system of ODEs


“Vertical” coordinate in the terrain-following coordinate system


Aerodynamic roughness length


Dissipation of turbulent kinetic energy


Von Karman constant


Coordinate transformation in dimension i: \({x_i = \acute{x}_i + \lambda_i}\)


Eddy viscosity


Production of turbulent kinetic energy


Density of air

\({\sigma_{E}, \sigma_{\varepsilon}}\)

Constants of the \({E-\varepsilon}\) turbulence model


Vorticity vector


Vertical component of the vorticity in the transformed coordinates

These subscripts, superscripts, and other markers modify the meaning of the quantity \({{\phi}}\):


In transformed coordinates


Fourier-transformed horizontally (along \({\acute{x}}\) and \({\acute{y}}\))


Vertical derivative \({\partial \phi/\partial \acute{z}}\)


Scalar: zero-order (basic flow), first-order perturbation

\({\phi_i^{(0)}, \phi_i^{(1)}}\)

ith Component of vector: zero-order (basic flow), first-order perturbation


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  1. Astrup P, Mikkelsen T, Jensen NO (1997) A fast model for mean and turbulent wind characteristics over terrain with mixed surface roughness. Radiat Prot Dosim 73(1–4): 257–260Google Scholar
  2. Athanassiadou M, Castro IP (2001) Neutral flow over a series of rough hills: a laboratory experiment. Boundary-Layer Meteorol. 101: 1–30CrossRefGoogle Scholar
  3. Ayotte KW, Xu D, Taylor PA (1994) The impact of turbulence closure schemes on predictions of the mixed spectral finite-difference model for flow over topography. Boundary-Layer Meteorol 68: 1–33CrossRefGoogle Scholar
  4. Beljaars ACM, Walmsley JL, Taylor PA (1987) A mixed spectral finite-difference model for neutrally stratified boundary-layer flow over roughness changes and topography. Boundary-Layer Meteorol 38: 273–303CrossRefGoogle Scholar
  5. Castro FA, Palma JMLM, Silva Lopez A (2003) Simulation of the Askervein flow. part 1: Reynolds averaged Navier-Stokes equations (\({k-\varepsilon}\) turbulence model). Boundary-Layer Meteorol. 107: 501–530CrossRefGoogle Scholar
  6. Corbett JF (2006) A novel approach for the fast modelling of mean wind flow over complex orography. PhD thesis, University of Copenhagen, Copenhagen, Denmark. Also available from Risø National Laboratory, Roskilde, Denmark, at
  7. Emeis S, Courtney MS, Højstrup J, Jensen NO (1993) Hjardemål experiment data report. Technical Report Risø–M–2289(EN), Risø National Laboratory, RoskildeGoogle Scholar
  8. Emeis S, Frank HP, Fiedler F (1995) Modification of air-flow over an escarpment – results from the hjardemål experiment. Boundary-Layer Meteorol 74: 131–161CrossRefGoogle Scholar
  9. Hunt JCR, Leibovich S, Richards KJ (1988) Turbulent shear flows over low hills. Quart J Roy Meteorol Soc 114: 1435–1470CrossRefGoogle Scholar
  10. Jackson PS, Hunt JCR (1975) Turbulent wind flow over a low hill. Quart J Roy Meteorol Soc 101: 929–955CrossRefGoogle Scholar
  11. Jørgensen BH, Ott S, Sørensen NN, Mann J, Badger J (2006) Computational methods in wind power meteorology. Technical Report Risø–R–1560(EN), Risø National Laboratory, RoskildeGoogle Scholar
  12. Launder BE, Spalding DB (1974) The numerical computation of turbulent flows Comp Meth Appl Mech Eng 3: 269–289Google Scholar
  13. Mason PJ, King JC (1985) Measurements and predictions of flow and turbulence over an isolated hill of moderate slope. Quart J Roy Meteorol Soc 111: 617–640CrossRefGoogle Scholar
  14. Mason PJ, Sykes RI (1979) Flow over an isolated hill of moderate slope. Quart J Roy Meteorol Soc 105: 383–395CrossRefGoogle Scholar
  15. Press WH, Teukolsky SA, Vetterling WT, Flannery BP (1992) Numerical recipes in Fortran 77, 2nd edn. Cambridge University Press 992 ppGoogle Scholar
  16. Risø National Laboratory: (1999) WAsP Engineering.
  17. Salmon JR, Bowen AJ, Hoff AM, Johnson R, Mickle RE, Taylor PA, Tetzlaff G, Walmsley JL (1988a) The Askervein hill project: mean wind variations at fixed heights above ground. Boundary-Layer Meteorol 43: 247–271CrossRefGoogle Scholar
  18. Salmon JR, Teunissen HW, Mickle RE, Taylor (1988b) The Kettles hill project: field observations, wind-tunnel simulations and numerical model preductions for flow over a low hill. Boundary-Layer Meteorol 43: 309–343CrossRefGoogle Scholar
  19. Sørensen NN (1995) General purpose flow solver applied to flow over hills. PhD thesis, Technical University of Denmark, Kgs. Lyngby. Also available as Technical report Risø–R–827(EN) Risø National Laboratory, Roskilde, DenmarkGoogle Scholar
  20. Sykes RI (1980) An asymptotic theory of incompressible turbulent boundary-layer flow over a small hump. J Fluid Mech 101: 647–670CrossRefGoogle Scholar
  21. Taylor PA, Teunissen HW (1987) Askervein hill project: overview and background data. Boundary-Layer Meteorol 39: 15–39CrossRefGoogle Scholar
  22. Troen I, de Baas A (1986) A spectral diagnostic model for wind flow simulation in complex terrain. Proceedings of the European wind energy association conference & exhibition, Rome, pp 37–41Google Scholar
  23. Troen I, Petersen EL (1989) European wind atlas. Risø National Laboratory, RoskildeGoogle Scholar
  24. Walmsley JL, Taylor PA (1996) Boundary-layer flow over topography: impacts of the Askervein study. Boundary-Layer Meteorol 78: 291–320CrossRefGoogle Scholar
  25. Walmsley JL, Salmon JR, Taylor PA (1982) On the application of a model of boundary-layer flow over low hills to real terrain. Boundary-Layer Meteorol 23: 17–46CrossRefGoogle Scholar
  26. Wolfram Research, Inc. (2005) Mathematica 5.2.
  27. Xu D, Ayotte KW, Taylor PA (1994) Development of a nonlinear mixed spectral finite difference model for turbulent boundary-layer flow over topography. Boundary-Layer Meteorol 70: 341–367CrossRefGoogle Scholar
  28. Xu D, Taylor PA (1992) A nonlinear extension of the mixed spectral finite difference model for neutrally stratified turbulent flow over topography. Boundary-Layer Meteorol 59: 177–186CrossRefGoogle Scholar
  29. Zeman O, Jensen NO (1987) Modification of turbulence characteristics in flow over hills. Quart J Roy Meteorol Soc 113: 55–80CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  • Jean-François Corbett
    • 1
  • Søren Ott
    • 2
  • Lars Landberg
    • 3
  1. 1.Garrad Hassan Denmark ApSCopenhagenDenmark
  2. 2.Wind Energy Department, Risø National LaboratoryTechnical University of Denmark (DTU)RoskildeDenmark
  3. 3.Garrad Hassan and PartnersBristolUK

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