More confusion about the faster-than-wind Blackbird

Question

There's been plenty of discussion on here about the Blackbird as featured in this video. I get that it harvests energy from the relative velocity of air to ground. But I still can't wrap my head around it in the reference frame of the vehicle.

In that frame, the wind speed is already zero or backwards. Let's say we're at wind speed, so it's zero. So we have to give the air some net backward momentum. But the vehicle is still speeding up in this frame, hence gaining energy, so the energy of the air has to decrease to compensate.

If the air were a rigid body, we couldn't increase its momentum without increasing its energy. But since it's a fluid, we can decrease the pressure of a fluid element while increasing its average velocity, and if the pressure decrease is enough, its total energy decreases. Voila, we get both thrust and energy from the air. No problem there.

But my issue is that, if you just look at the propeller-air system, it's indistinguishable from a fan running in a room, because the free field is stationary in this frame. We know a room fan takes energy to run -- sure, part of that is just due to internal friction, but I have to think it would run much easier in vacuum, from which I deduce that the fan is transferring energy to the air, not vice versa. And if this airflow is no different from that of the Blackbird, the air would also be gaining energy in the latter case.

So where's the gap in my reasoning?

Answer 1

I deduce that the fan is transferring energy to the air, not vice versa.

That depends on your frame of reference. In the frame where the fan is at rest, that is correct. In a frame where the fan is moving forward, it is not. There the fan is slowing the air.

The direction of energy transfer depends on the frame you are examining. In the case of the blackbird, both the ground energy transfer and the air energy transfer are affected by this choice of frame, but the net energy transfer will not be affected.

Answer 2

Here is how I make sense of that machine. Your mileage may vary and will probably be lower in California (this is an old mechanical engineering joke, laughter is optional):

Let's take the case where the wind is blowing fast from the west and the vehicle is pointing north. the operator swivels the head bearing of the windmill to face the blades westward into the wind and the blades start to turn. The vehicle is now slowly moving north, propelled by the work harvested by the windmill. Because it is moving slowly, the direction of the relative wind is still almost straight to the west, and you can then analyze the windmill using all the usual techniques, which contain no mysteries.

As the vehicle picks up speed, the relative wind gains a northward component and the operator adjusts the swivel angle of the blade disc to face it head-on. She measures a greater wind velocity because it is now the vector sum of her motion northward and the wind's westward direction. More velocity means more harvestable work in the wind, and the vehicle speeds up heading north.

She then trims the blade disc to match the relative wind and gets still more speed. The blades are now spinning like sixty and you now begin modeling the blade disc as if it were a set of autorotating helicopter blades that generate lift as the relative wind flows through the blade disc, only in this case the blades spin sideways and the lift force is expressed as thrust which propels the vehicle northward.

Now you can use the equations for helicopter blades (or actually, autogyro blades) to get the relationship between the speed and direction of the relative wind and the lift force generated by the blade disc.

This way of visualizing the windmill-powered vehicle makes it easier (for me, anyway) to keep track of what's happening.

Answer 3

Go back to sailboats, which have a centerboard, or to iceboats, which have skates, both of which are constrained to move along a straight line. The sail is at an angle to that. The lift vector of the sail is nearly but not perfectly perpendicular to that straight line.

So it's like squeezing a slippery wedge or pumpkin seed. There's no limit to how fast it can go, except for friction, and the travel line can be pointed in almost any direction except directly down or up-wind.

Now let the sailboat or iceboat travel in a helix around a watery cylinder aligned with the wind. You have that car. The pitch of the propeller acts as the sail's angle, and the gearing to the wheels acts like the centerboard, controlling how far forward it can go for each rotation of the "sail".