Wind Farm Wake Lab (2D)

Power of one turbine

The kinetic power in the wind through the rotor area A is ½ ρ A v³. A turbine extracts at most the fraction Cp = 4a(1-a)², which peaks at a = 1/3 with 16/27 = 0.593 (the Betz limit). Real rotors reach about 0.45. Between cut-in and rated speed power follows v³, then it is held at the rated value until cut-out.

Jensen (Park) wake

Behind a turbine the wake is a cone that widens at the rate k (about 0.04 offshore, 0.075 onshore). Its speed deficit at distance x is δ = (1 - sqrt(1 - CT)) / (1 + 2 k x / D)². It follows from conservation of mass for a top-hat wake. Wakes from several turbines are combined by sum of squares: u = u0 (1 - sqrt(Σ δi²)).

Array efficiency and yield

Array efficiency is farm power divided by N turbines in free wind. Annual yield uses a Weibull distribution of wind speed, f(v) = (k/c)(v/c)k-1 exp[-(v/c)k], with shape 2 (Rayleigh) here.

Checks

(1) A numerical scan of C_p(a) finds the maximum at a = 1/3 and 16/27. (2) The wake deficit from the model equals the one obtained from a mass-flux balance over the expanding cone. (3) The numerical Weibull mean of v³ equals c³ Γ(1 + 3/k).

Simplifications

Top-hat wakes, constant C_T, no turbulence, terrain or atmospheric stability, no wake meandering. Illustrative turbine data, not any manufacturer's.

What to try

• Align the wind with a row and compare with a 45 degree wind.
• Increase spacing from 4 D to 10 D and watch array efficiency rise.
• Switch from the offshore to the onshore wake decay.