Betz law and RotCp


I’m working on the OC3 5MW Hywind ( FAST_v7.00.00a-bjj_AeroDyn_v13.00.00a-bjj_BladedDLLInterface_OC3Hywind )
I’m using the Matlab Simulink Implementation ( (R2009b) ) and I’m running Windows 7 -32bit

I can’t figure out why the RotCp signal coming from FAST can be bigger than Betz law 0.59?
Please find the attached figure.

Best regards
Soeren Christiansen
PhD student
Aalborg university

Hi Soeren,

I just started working with FAST recently and had a similar issue. I had asked about this and see below for the response I got from Jason Jonkman at NREL.

I am by no means an expert on FAST and the reason you are seeing Cp greater than Betz limit could be different. I will leave it for more knowledgeable folks here to confirm that. Hope this helps.


Subin Sethuram

Thank you, Subin. I was just going to paste that response in a post, but you beat me to it. I have nothing more to add.

Best regards,

Hi Jason,

Glad to help out. You guys have helped me quite a bit. Just wanted to do my part. On a separate issue, I was corresponding with Marshall on this thread [url]]. I assume he must be quite busy and hasnt had time to get back on the last question I posted. Would it be possible for you to take a peek if you have time? Thank you.



I just did.

Thank you for the fast reply.

Does this mean that FAST generally calculates the Power and Rotor thrust using WindVxi and not the wind speed normal to the rotor plan?

Dear Soeren,

FAST with AeroDyn does not calculate the rotor power and thrust explicitly. Instead, the aerodynamic loads on each node of the blade are what are calculated explicitly. The aerodynamic loads on each node are not based on WindVxi, but on the wind velocity local to the aerodynamic nodes. It is these loads integrated across the rotor that results in the overall rotor power and torque.

I hope that helps.

Best regards,

Hi all,

We are working on a passive flow control Concept but unfortunately we cannot yet model our concept including
drag. For this reason we tried to visualize the impact of the drag coefficient on the wind turbine power coefficient.
What we did. We took different windfields and with different turbulence intensities and made simulations for airfoils
with and without drag coefficient. We made a simulation for about 600s and at the end we took the mean value of the RotCp power coefficient over time.
Then we made a plot with the mean of the RotCp over different reference wind speeds. Here we came about the same issue that the power coefficient had values exceeding the Betz Limit of 16/27. As reference turbine geometry we take the NREL 5MW Baseline turbine.

When I looked through the instantaneous values for RotCp I found rather huge values of about RotCp = 9-10 for some conditions with low wind speeds Vxi.

Now I took the definition of RotCp = RotSpeedRotTorq/(1/2AirDensSweptAreaWindVxi^3) and changed the instantaneous wind
speed with the reference wind speed by multiplying RotCp*(WindVxi/WindVx_ref)^3
So I got instantaneous values for RotCp all smaller than 16/27.
I don’t know if this is all correct what I did but the values were more compatible to the Betz Limit.
Also we only came about these problems for low reference wind speeds and high turbulence intensities what makes sense
because than a change in Wind speed has higher influence due to the power law ^3.

Any ideas if this approach is correct or do you see any flaws.

Also we would be interested if anyone has experience in simulations without drag coefficient. Does it make a big
difference to neglect it.



Hi Andreas,

The power coefficient has less meaning in turbulence. The main problem is that the average of series of values cubed is greater than the cube of the series average. In equation form,

SUM( V_i^3, i=1,2,…N )/N >= ( SUM( V_i, i=1,2,…N )/N )^3.

(The equals only applies if V_i = constant for all i.) For example, if N = 2 with V_1 = 1 and V_2 = 3, then the equation above yields is ( 1^3 + 3^3 )/2 > ( ( 1 + 3 )/2 )^3, or ( 1 + 27 )/2 > 2^3, or 14 > 8.

In the case of the power coefficient, the averaging is from time averaging (over the simulation length), spatial averaging (over the disk), or both.

The true power available in the wind that a wind turbine could extract should be calculated as 1/2*AirDens times the integral over the swept area of the disk of the local undisturbed wind velocities cubed. However, FAST only estimates the power coefficient (RotCp) using the wind power derived from the cube of hub-height wind speed (WindVxi). In shear and turbulence, this estimate of power will be less than the actual power available, and hence, the RotCp estimated by FAST will be high. Instead of calculating RotCp in turbulence, I suggest you simply use the mechanical or electrical power output.

Drag has little influence in the linear region of the airfoil data (where drag is small), but of course, it does have an influence beyond stall, where drag increases substantially. In general, drag will decrease power and increase thrust. The results you plot in your post confirm this.

Best regards,

Hi Subin,

I have also started working on FAST recently. Is it possible for you to share references on 'Betz law and RotCp?

Dear Shankar,

You can find definitions of the “Betz limit” and “power coefficient” in any wind energy textbook, such as “Wind Energy Explained,” by Manwell, McGowan, and Rogers.

Best regards,

Dear Jason,

Thanks for posting a reply. I hadn’t logged in for a some time now.

Dear Shankar,

Like Jason mentioned in his post, any wind energy texbook should give you an explanation. Personally I liked the expalantion in “Aerodynamics of Wind Turbines” by Martin.O.L.Hansen. This book also gives simple algorithm to implement your own BEM code. Another reference is “Wind Energy Handbook” by Tony Burton et. al.

Best Regards,