# Vertical axis wind turbine vector diagrams

People are happy that a blade can generate torque in most positions around its circular path because they know that sail boats can manage to sail upwind by "tacking" at an angle to the wind. Everyone accepts that no torque is generated facing directly upwind. People are even more happy that sail boats can sail downwind provided that the wind is traveling faster than the boat. The intuitive understanding breaks down when the blade is travelling downwind and much faster than the wind and yet the mathematicians say that the blade is still generating torque rather than consuming it!

The following diagram shows a top view of this downwind motion scenario. The head wind is produced due to the movement of the blade. The real wind is blowing from bottom to top. When the two winds are vector added to produce the apparent wind, the strength of the wind is less and it is now at a shallow angle to the blade such that the blade is not stalled and will generate lift at right angles to the average direction angle of the apparent wind before and after deflection by the blade. This lift vector can be resolved into two components, one component accelerates the blade forwards and the other component is along the boom connecting the blade to the axis of the turbine.

Why is the lift vector almost at right angles to the apparent wind? The wind is deflected through a small angle by the blade but not very much slowed down because the airfoil has a very low drag. The force to deflect the flow is at right angles to the average angle of the flow before and after deflection. If the deflection angle is small, then this is almost the same as at right angles to the incoming airflow (true angle is tilted backward by a few degrees). The ratio of lift to drag for an airfoil can be between 10 and 100 which is why the drag components do not defeat us until the incoming airflow angle drops below a few degrees. Thus the vertical axis wind turbine generates power at all blade positions except when the blades are nearly aligned with the wind (upwind or downwind).

## Building your own vertical axis wind turbine

Now that you know how it works, you will want to build one. Here are some guesses for experimentation.

### Airfoil: NACA63-4-021

***UPDATE*** I no longer recommend this laminar flow airfoil due to stall at large angles of attack. Better to use a turbulant flow airfoil such as NACA 0021 which is more forgiving at high angles of attack.

Big commercial eggbeater type turbines may use symmetric airfoils such as NACA0015 to avoid a pitching moment, but a home built H-rotor (it looks like an H from a distance) with more rigid blades and booms can possibly use a cambered airfoil such as NACA 4415. This has a flat side for easier attachment to the boom and the vector analysis done shows that using camber or increasing angle of attack will produce more torque on one half of the cycle than the other but possibly more overall (See http://home.inreach.com/integener/ for the reasoning behind this.). Also, even if you want a symmetric profile relative to the wind, the headwind is already actually slightly curved due to the circular path of the blade and so a symmetric profile is not ideal. You will instead need a symmetric profile distorted to fit to a mean camber line which is a pure circular arc of the same diameter as the rotor. Here is a table of the % camber needed for various blade width to diameter ratios.

 Blade chordwidth to rotor diameter ratio (chord/diameter) Needed camber (pure circular arc) when rotating fast (0.5/(chord/diameter))*(1-cos(chord/diameter)) *100%  (arguments to cos in radians) 0.4 10% 0.2 5% 0.1 2.5% 0.05 1.25%

Note that these cambers can be significant and are within the range of conventional cambered airfoils (although standard airfoil camber is not a pure circular arc). I have my doubts that use of an angled blade is beneficial since it is already difficult to keep the angle of attack below stall over the whole cycle and I would need to do a computer integration round the whole cycle to be convinced.

Meanwhile, I recommend to use a fat symmetric profile such as NACA 0021 (*** UPDATE *** NACA63-4-021 may have worse stall characteristics than NACA 0021) , with additional pure circular camber (so that it is symmetric with respect to airflow when rotating fast). The large radius leading edge on the profile will give a wide low-drag bucket between Cl of +0.4 and -0.4 and even when completely stalled and side on to the wind, the large radius leading edge may deflect the wind to give some starting propulsion. When the wind is flowing the wrong way (from sharp edge to blunt edge) again the large "nose" will now contribute drag which is what we want when the machine is starting up and is below a TSR of 1. The fat thickness fraction (21%) gives a stiffer blade than the thinner sections such as NACA4415 (15%).

Club Cycom's blade design tool is able to produce plots of NACA63-4-021 of any chord width and cambered to a circular arc with camber of 0%, 1.25%, 2.5%, 5% and 10%.  Even though the tool is designed for horizontal axis wind turbine design it is possible to vary parameters to get the plot we want. Here is how. Plot section through blade tip and alter lift coefficient until chord is the length that you want it. Adjust angle of attack to get final setting angle of 0 degrees. Ensure that "Draw X as cylinder surface distance" is ticked (true)) and that the airfoil with the right amount of additional circular camber is specified. A Young low drag body 60% laminar flow, 30% thickness could be used for booms and struts. Subscribers will be able to load the scenario darr025.zip to see a plot of NACA63-4-021 pure circular cambered at 2.5% (designated as NACA63-4-021025). *** UPDATE *** use NACA0021025 A screen shot is shown below.

Tip speed ratio will be around 4 or 5 and we don't want the Reynolds number to get too low in low winds so we choose a reasonable chord width.

### Number of blades: 2 or 1

Again we want to keep Reynolds number high for better attached flow. Few blades means larger chord widths which also mean better strength/load ratio.

### Diameter: 3 metres

Guessing that solidity = 0.1 would be good for an H-rotor TSR in range 4 or 5. Solidity=Nc/D where N is number of blades, c is chord width and D is diameter. Centrifugal forces get smaller as the radius is larger so large is good. You will probably need to gear it up to the generator. If the generator is a dynamo, this can be run as motor to assist starting. For a 5 m/s wind and a TSR of 4 the rotation is 127 RPM.

### Height (length of blades): 2 meters

Most designs have a squarish swept area. You can reduce this length if you feel your blades are not light and rigid enough. You may want some struts or wires. Club Cycom's blade design tool will calculate RPM and G-force factors for a given TSR, windspeed and radius. They get very big in high winds.

### Safety

Vertical axis wind trubines can be hard to stop and can't be yawed out of the wind. Consider adding a centrifugally operated parachute or plan for safe auto-destruct in a storm. Note that Cycom has not built this machine. If you build one and survive, please let us know if it worked.

### Test Results

Cycom has not build the above machine. A heavy very rough half size approximation using thin  1 meter rough carved wood blades on a wobbly bicycle wheel failed to self start and even when manually started, it failed to keep going in the light wind available. It seems that it is difficult to reach the high speeds needed to keep the angle of attack below stall. Obviously some fatter profiles, decent scale and care will be needed to make a working machine. At lower TSR some way to modulate the angle of attack by tilting the blade or changing the camber will be needed. Some people use a cam and a tail to vary the blade angle during a cycle. See http://www.windstuffnow.com/main/darrieus_type.htm.

### Closer to Savonius

Choosing something with a much higher solidity such as 1.2 will reduce the TSR and bring us closer to the Savonius rotor. Choosing 3 blades will ensure self-start from any position. A much higher camber will be needed. With 3 blades the chord/diameter ratio is 0.4 so we use a 10% camber
(subscribers can load scenario darr10.zip ). The resulting plot is 23cm chord width and so rotor diameter should be 23/0.4 = 57cm. The vertical black reference line on the plot will be the centreline of the boom.*** UPDATE *** Use NACA002110 A screenshot is shown below.

The operation of the above rotor can be explained in many ways any of which should convince the reader that it will at least rotate.
1. The lift vector explanation give at the start of this page.
2. As a drag machine, the wind drag is asymmetric because of the streamlining being better in one direction that the other.
3. Any wind entering the rotor volume will be deflected by the tail as it leaves the rotor disk. Since the tails all are tangential to the circle of rotation, the air will leave tangentially, and, like a rocket motor, the blade will be pushed in the opposite direction to the leaving jet.

There are some problems with the above 3 bladed rotor at low windspeeds of say 5m/s.

1. The very high solidity of 1.2 with 3 blades means a low optimum tip speed ratio,  perhaps only 1.4. The low RPM (would also mean less generator power for a given weight.
2.  The small chord and low speed means a low reynolds number (Reynolds number for air fluid is Re=68459*V*L where V is velocity in m/s and L is chord length in metres). If windspeed is 5m/s then Re=68459*5*1.4*0.23=110218 and airfoils are not often so efficient below Re=100000.
3. There is 3 times as much material and hence cost in 3 blades