Jensen (Park) Wake Deficit behind a Wind Turbine

A turbine takes energy out of the wind, so the air behind it is slower, and a second turbine standing in that wake makes less power. This relation gives the velocity deficit $\delta$, the fraction of the free wind lost, at a distance $x$ downwind. It is the simple model N. O. Jensen published at Risø in 1983 and Katic, Højstrup and Jensen used for wind farms in 1986. The wake is a cone with a flat profile that starts as wide as the rotor and widens linearly with distance. It is part momentum theory and part fitted constant: the first factor is physical, while the wake decay constant $k$ is fitted.

At the rotor the wind is slowed by $1 - \sqrt{1 - C_t}$, where $C_t$ is the thrust coefficient of the turbine, between 0 and 1. That is twice the induction from momentum theory: at the Betz optimum $C_t$ is 8/9 and the deficit at the rotor is 2/3, so a third of the free wind is left. Moving downwind, the deficit is divided by $(1 + 2kx/D)^{2}$: the wake radius grows by $k$ for each unit of distance, and the same loss of momentum is spread over a bigger area. $D$ is the rotor diameter, so $x/D$ counts rotor diameters. Ma and colleagues (2022) use $k = 0.075$ over land and 0.04 over the sea, but other authors use somewhat different values, so treat $k$ as a fitted constant.

For $C_t = 0.8$, $k = 0.075$ and five rotor diameters downwind (500 m behind a 100 m rotor), the deficit is $(1 - \sqrt{0.2})/(1 + 2 \times 0.075 \times 5)^{2} = 0.552786/3.0625 = 18.050$ %. Since power goes as the cube of the speed, a turbine standing there makes only about 55 % of the power of one in the free wind. Over the sea, with $k = 0.04$ and seven diameters, the deficit is $0.552786/2.4336 = 22.715$ %. Run backwards, an 18.05 % deficit at five diameters with $k = 0.075$ gives $C_t = 0.8$.

The model ignores the turbulence the wake adds and the way wakes of several turbines overlap; wind farm tools add those. The sources are Jensen (1983), Risø-M-2411; Katic, Højstrup and Jensen (1986); and Ma, Archer and Vasel-Be-Hagh (2022), Wind Energy Science, which gives the deficit in this form.