Wind Farm Wakes: The Betz Limit and the Jensen Model

Every turbine steals wind from the one behind it

A wind turbine does not just harvest energy from the air passing through its rotor -- it slows that air down and leaves a turbulent, lower-speed wake trailing downwind, often stretching ten rotor diameters or more before it fully re-mixes with the free stream. Any turbine sited inside that wake sees less energy in the wind reaching it, and because power scales with the cube of wind speed, even a modest speed deficit costs a large chunk of output. Wind farm layout is, at its core, the problem of packing turbines close enough to use the land or seabed efficiently while keeping wake losses tolerable.

A single turbine's available power follows the standard wind-power formula, with the power coefficient Cp expressing how much of that available power the rotor actually converts to shaft power:

P = 1/2 · rho · A · v^3 · Cp
rho = air density, A = rotor swept area, v = wind speed, Cp = power coefficient

The Betz limit: 59.3%, and no higher

In 1919 Albert Betz showed that no turbine, however cleverly designed, can convert more than 16/27 (about 59.3%) of the kinetic energy in the wind passing through its swept area into shaft power. The argument is pure momentum conservation: a turbine that extracted all the wind's kinetic energy would have to bring the air to a dead stop right behind the rotor, which would block any more air from flowing through at all. The optimum turns out to be slowing the wind to exactly one third of its free-stream speed by the time it reaches the rotor plane -- extract more and you choke the flow, extract less and you leave energy on the table. Real turbines reach roughly 75 to 85 percent of the Betz limit once gearbox, generator and blade-aerodynamic losses are included, putting practical Cp around 0.45.

The Jensen (Park) wake model

The simplest and still most widely used engineering wake model, published by Niels Jensen in 1983 and often called the Park model, treats the wake as a cone that expands linearly with downwind distance and carries a uniform velocity deficit inside it. The deficit shrinks with distance because the wake mixes with surrounding air and the cone's cross-section grows:

v_wake(x) = v_0 · [1 - (1 - sqrt(1 - Ct)) / (1 + 2 · k · x / D)^2]
x = downwind distance, D = rotor diameter, Ct = thrust coefficient, k = wake decay constant

The wake decay constant k is typically around 0.075 offshore, where the sea surface is smooth and turbulence that would otherwise re-mix the wake faster is low, and closer to 0.04 to 0.1 onshore depending on terrain roughness. This single difference is a big part of why offshore wakes travel further and offshore farms need wider turbine spacing than the terrain alone would suggest.

Multiple wakes and layout trade-offs

When a downwind turbine sits inside two or more overlapping wakes, the standard engineering approach sums the velocity deficits in quadrature (root-sum-square) rather than simply adding them, since the losses do not stack linearly. Layout optimisation then becomes a real trade-off against cost: spacing turbines 8 to 10 rotor diameters apart in the prevailing wind direction all but eliminates wake losses but multiplies the land or seabed area, cabling and access-road cost per megawatt; packing them at 3 to 5 diameters saves on infrastructure but can cost 10 to 20 percent of the farm's annual output to wake losses. Real farms compromise with wider spacing along the dominant wind direction and tighter spacing across it, and increasingly use active wake steering -- yawing upwind turbines a few degrees off the wind to deflect their wake sideways, away from the turbine directly behind them.

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