# Estimating wind velocity behind a wind turbine

According to the Wikipedia page on the Betz Limit, the outgoing wind velocity behind a wind turbine can be as low as 33% of the incoming wind velocity. However, this figure assumes that the turbine is operating at the theoretical Betz limit. Since most turbines operate below this limit, is there a straightforward formula for calculating a more realistic estimate of the wind velocity behind a turbine?

The wind behind an operating turbine is turbulent. So no, there is no straightforward formula for calculating a more realistic estimate of the wind velocity behind a turbine. The existence of such a formula is literally a million-dollar question.

However, calculating the approximate speed of the air isn't too hard: we know the energy in any given mass of air is $\frac{1}{2}mv^2$. So if we know the speed upwind of the turbine, the density of that air, and the amount of electricity generated, we can calculate the energy in the air downwind of the turbine, and thus its speed. Assume 95% efficiency for the turbine mechanics & electrics (for kinetic energy that has been harvested and is being converted to electricity), and you'll be near enough.

The study of the wakes of wind turbines is a rich and complex one, with quite a lot of work going on at the moment

Here are some wakes that nature has kindly visualised for us, at Horns Rev windfarm in Denmark (photo courtesy Vattenfall Wind Power)