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Mess with your minds: A wind powered craft going faster than a tail wind speed.
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IanB:

--- Quote from: electrodacus on December 21, 2021, 06:38:24 pm ---I do not see how else you can call the pressure differential created by the propeller other than a form of energy storage.
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A propeller does not create a pressure differential. A propeller moves ("propels") air. It creates thrust by the law of conservation of momentum, by accelerating the air that goes through it. The air leaving a propeller has a lower pressure because it is moving. You can see this described here:

https://www.quora.com/Why-is-the-static-pressure-before-a-fan-higher-than-after-the-fan

And illustrated here by experiment:

https://youtu.be/f2QfVJe7yEg?t=198

Given that there is no "pressure bubble" behind the propeller, there is no energy storage there to push back on the propeller. Once the air has gone through the propeller it has done its work, there is no more work to give.
electrodacus:

--- Quote from: fourfathom on December 21, 2021, 06:52:42 pm ---The applicable equation is some_factor * air density * area * (wind speed - ground speed)^3 

Since ground speed = 0, the equation becomes:
some_factor * air density * area * (wind speed)^3

The difference is that your equation assumes that the vehicle has no contact with the ground, and behaves something like a balloon.  This is certainly not the case with the DDWFTTW vehicles we ae discussing.

--- End quote ---

That will be the correct equation for something that is not moving relative to ground like a wind turbine. For a wind turbine that is exactly the equation just that you add the turbine efficiency.
If you install the wind turbine on a vehicle and then drive direct down wind then equation of the power output for that wind turbine will again contain (wind speed - vehicle speed) since the wind speed relative to that moving wind turbine will be lower the faster the vehicle moves.
electrodacus:

--- Quote from: IanB on December 21, 2021, 07:00:24 pm ---
A propeller does not create a pressure differential. A propeller moves ("propels") air. It creates thrust by the law of conservation of momentum, by accelerating the air that goes through it. The air leaving a propeller has a lower pressure because it is moving. You can see this described here:

https://www.quora.com/Why-is-the-static-pressure-before-a-fan-higher-than-after-the-fan

And illustrated here by experiment:

https://youtu.be/f2QfVJe7yEg?t=198

Given that there is no "pressure bubble" behind the propeller, there is no energy storage there to push back on the propeller. Once the air has gone through the propeller it has done its work, there is no more work to give.

--- End quote ---

Sorry but you are just ignoring the fact that air is a compressible fluid.
Are you saying this graph below is completely wrong ?
https://en.wikipedia.org/wiki/Axial_fan_design
IanB:

--- Quote from: electrodacus on December 21, 2021, 06:07:52 pm ---I say that available wind power is defined by this equation  0.5 * air density * area * (wind speed - vehicle speed)^3
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The equation  0.5 * (air density) * (area) * (wind speed)^3  gives the total flow of kinetic energy from wind moving through a particular cross section of area, and is applicable to wind turbine design as the air is moving through the turbine. A turbine could never extract more power than this from the wind (it will always be less than this). Note that this equation only works if the wind moves through the area of interest, like the disk area of a turbine.

Since a sail is a solid barrier, the wind does not, and cannot, move through the sail. Therefore, this equation cannot be applied to sails.

Another way of looking at this is to observe that the equation is (kinetic energy of wind per unit mass) x (mass flow of wind moving through control area). Since the mass flow of wind through the sail is zero, the energy flow must necessarily be zero.

When working with sails, you have to use force vectors and force balances.
Kleinstein:

--- Quote from: electrodacus on December 21, 2021, 06:07:52 pm ---I say that available wind power is defined by this equation  0.5 * air density * area * (wind speed - vehicle speed)^3

--- End quote ---
I say this is wrong, and the PDF on wind power you linked also says something different:
They get the wind power from the change in kinetic energy for the air passing though the "turbine" area. The same way of calculations,  when slowing down the air to the vehicle speed this would be 0.5 * air density * area * (wind speed^2 - vehicle speed^2)*(wind speed - vehicle speed)
 = 0.5 * air density * area * (wind speed - vehicle speed)^2*(wind speed + vehicle speed).
With a simple sail the power would be from drag force time vehicle speed and thus within the approximations in these equations:
P_sail =  0.5 * air density * area * (wind speed - vehicle speed)^2* vehicle speed

With the active driven prop there can be even more power available, as the air may slow down even more. This is especially true for a speed close to or higher than the speed of the wind. This is because this way the wind can be slowed down to a speed below the vehicle speed.
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