Mathematical Model for the Analysis of Wind-Turbine Wakes

The concept of wind farms with clustered wind turbines at a given site seems to offer an attractive means for extracting wind power on a large scale. Techniques for minimizing the effect of upstream wind-turbine wakes on downstream wind turbines are needed to optimize overall performance of the wind-turbine array. A numerical model for prediction of the interaction of the wind turbine with the prevailing wind flow is described. The model is based on a numerical solution of the three-dimensional Navier-Stokes equations for the planetary boundary layer with the hydrostatic approximation. Three different hypothetical wind-turbine configurations are analyzed to demonstrate the utility of this model. Model predictions from the present study compare favorably with the basic characteristics of measured wind-turbine wakes. Nomenclature a = axial interference factor D = turbine blade diameter / = Coriolis parameter k = von Karman constant KM = turbulent diffusion coefficient KM - dKM/dz, gradient of turbulent diffusion coefficient L = Monin-Obukov length P = atmospheric pressure t =time U = characteristic wind speed u = velocity component in x direction ur = velocity at reference height zr u* = friction velocity v = velocity component in y direction w = velocity component in z direction x,y,z = orthogonal Cartesian coordinates Z0 = aerodynamic surface roughness Z = height of the inversion base A = incremental change p = atmospheric density 6,\I/ = dimensionless functions Subscripts