Boundary-Layer Meteorology

, Volume 128, Issue 2, pp 229–254

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

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

DOI: 10.1007/s10546-008-9284-z

Cite this article as:
Corbett, J., Ott, S. & Landberg, L. Boundary-Layer Meteorol (2008) 128: 229. doi:10.1007/s10546-008-9284-z

Abstract

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.

Keywords

Complex terrainMixed spectral-integrationWind-flow model

List of Symbols

A

Matrix of coefficients of the system of ODEs

A

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

E

Turbulent kinetic energy (TKE)

h

Height above ground level

hBL

Depth of the boundary layer

ht

Terrain height

i

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

J

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

ki

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

k

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

k

Horizontal wavenumber vector: k = (k1, k2)

p

Hydrostatic pressure

s

Expansion parameter

S

Matrix of source terms in the system of ODEs

ui

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

\({\acute{U}_i}\)

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

xi

Cartesian coordinates: (x1x2x3) = (xyz)

\({\acute{x}_i}\)

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

\({\acute{\bf{x}}}\)

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

X

Matrix of dependent variables in the system of ODEs

\({\acute{z}}\)

“Vertical” coordinate in the terrain-following coordinate system

z0

Aerodynamic roughness length

\({\varepsilon}\)

Dissipation of turbulent kinetic energy

κ

Von Karman constant

λi

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

ν

Eddy viscosity

Π

Production of turbulent kinetic energy

ρ

Density of air

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

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

ω

Vorticity vector

\({\acute{\omega}_3}\)

Vertical component of the vorticity in the transformed coordinates

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

\({\acute{\phi}}\)

In transformed coordinates

\({\hat{\phi}}\)

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

\({\phi'}\)

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

\({\phi_0,\phi_1}\)

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

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