Undergrad DDWFTTW: Looking for the least confusing explanation

[particlezoo]

I have tried to find the least confusing (or rather, the most 'deconfusing') explanation for why it is possible for a cart with a prop to travel directly downwind faster than the wind. So far this is what I've come up with:

Initial discussion

At first glance, a cart traveling down faster than the wind would experience a headwind. The drag of the headwind would slow down the cart. However, if that cart also has spinning blades, the back side of the blades could run into the air, such that the pressure on the blades is from behind, not in front. So this would have the effect of applying a forward pressure on the vehicle which would force the cart, prop, bearings, etc. to move forward.

Because the bearings hold the axle, the increased translational velocity of bearings relative to the ground would cause the wheels and axles to spin. So at the expense of some of the gained translational kinetic energy of the cart and bearings, the wheels and axles will gain rotational kinetic energy in addition to their own translational kinetic energy. Because the prop is connected to the axle via a belt, the prop must then recover rotational kinetic energy at the expense of the rotational and translational kinetic energies of the rest of the cart (as well at the expense of some of the prop's own translational kinetic energy). This gain of rotational kinetic energy by the prop must overcome the loss of rotational kinetic energy due to the drag by the air that it slices through.

Revealing the hidden assumption

When we say the cart is traveling directly downwind faster than the wind, we are saying that both the cart and the wind have a speed, but one is greater - relative to something. Strictly speaking, the norm of the velocity difference between the cart and this something is greater than the norm of the velocity difference between the air and this something.

In the initial discussion, we choose this something to be our ground. Then we say the cart is traveling directly down the wind faster than the wind as a result of work done on it (relative to the ground) and that kinetic energy of the wind (relative to the ground) is extracted upon by the cart to provide this work.

Forces or work, which is more important?

Ignoring relativity, the forces do not change with the frame of reference, but the displacements do. Therefore the work done on or done by the air, ground, cart, the parts, etc. is dependent on the inertial frame of the independent observer. So if we choose an observer independent of the ground, then work may be done by the air on the cart, or vice versa, and work may be done by the ground on the cart, or vice versa.

However, if by displacements we meant displacements relative to the initial rest frame of each material entity and not displacements relative to an arbitrary inertial observer, then given a reasonable assumption that the ground has a very large inertia relative to the cart and the air, the ground may be seen as receiving no work upon it, according to its unchanging rest frame. The cart may be receiving a force according to its frame, but the rate at which work is done on the cart it is ill defined in this frame unless an initial rest frame for the cart can be chosen. If this could be done for both the cart and the air, then one may say the air does work on the cart (or vice versa), depending on the initial "rest" frames for the air and the cart. Lacking specification of these initial rest frames, then the transfer of energy between the air and the cart and between the ground and the cart is frame-dependent.

So it is not so much that the wind does work on the cart, or the ground does work on the cart, or whatever combination one so desires, since that is the arbitrary result of choosing a frame of reference. In the end, we are dealing with differences in velocity and their norms (i.e. speeds relative to ground), and therefore differences in momentum and their derivatives with time (i.e. forces). Therefore, the deciding factor that makes DDWFTTW possible are the forces on the bodies and not work done between them.

If one insists to use an work-based or power-based explanation, they should clearly specify the frame of reference in which the kinetic energies are defined, and if they are going to analyze the problem from different frames of reference, they should make it clear that kinetic energies and powers are not the same between different frames of reference. If they jump between different frames of reference when thinking about the DDWFTTW in terms of kinetic energies and powers, they will likely confuse themselves if they are not careful.
Kevin M.

 

[rcgldr]

The easiest frame of reference is probably the cart's frame of reference. Assume the cart is not accelerating, so that its frame of reference is inertial. When the car is moving fast enough to experience an apparent headwind (or zero wind) from its frame of reference, the ground is moving faster. There are two sets of Newton third law pairs: the wheels exert a forwards force onto the ground coexistent with the ground exerting a backwards force onto the wheels which in turn drive the propeller which exerts a backwards force onto the air coexistent with the air exerting a forwards force onto the propeller. Since the ground is moving faster than the apparent wind, the cart exploits the situation using an effective reduction in gearing from wheels to propeller, which divides the speed and multiplies the force. The propeller generates a greater force but at a lesser speed, exploiting the fact that the propeller interacts with the slower moving apparent headwind. There are losses in the system (aerodynamic drag, rolling resistance, drivetrain losses, ... ) but a DDWFTTW vechicle named "Blackbird" was efficient enough to achieve a downwind speed about 2.8 times the true wind speed (about 28 mph in a 10 mph wind).

http://en.wikipedia.org/wiki/Blackbird_(land_yacht)

From the carts frame of reference, energy is extracted by slowing down the speed of the Earth's surface (with respect to the cart). The Earth is massive so the change in speed is extremely tiny. The extracted energy, minus losses, is added to the apparent headwind by the propeller.

- - -

From an inertial frame of reference tied to the surface of the Earth at a moment in time, the cart's propeller slows down the true wind, extracting energy from the true wind, and from that inertial frame of reference, after losses, adds some energy to the Earth by speeding up the movement of the Earth's surface by a very tiny amount.

 

[CWatters]

The two simplest (if incomplete) explanations I have heard include..

a) The lift produced by a wing is greater than drag. Lift is what ultimately pulls the cart along and drag is what slows it down.
or
b) It's just like a yacht tacking, but in 3D.

 

[A.T.]

The two simplest (if incomplete) explanations I have heard include..

a) The lift produced by a wing is greater than drag. Lift is what ultimately pulls the cart along and drag is what slows it down.

That is indeed incomplete / not specific enough.

b) It's just like a yacht tacking, but in 3D.

That is actually complete, but understanding tacking involves a). Once you understand how yachts tack downwind with velocity made good greater than windspeed, DDWFTTW is almost trivial:

 

[A.T.]

If one insists to use an work-based or power-based explanation, they should clearly specify the frame of reference in which the kinetic energies are defined, and if they are going to analyze the problem from different frames of reference, they should make it clear that kinetic energies and powers are not the same between different frames of reference. If they jump between different frames of reference when thinking about the DDWFTTW in terms of kinetic energies and powers, they will likely confuse themselves if they are not careful.

Some powers are frame independent:
- power transmitted in a shaft : angular velocity * torque (across inertial frames)
- power dissipated / generated / consumed at the interface of two moving bodies (based on their relative velocity)

See page 12 for a short power based explanation:
http://www.aapt.org/physicsteam/2013/upload/E3-1-7-solutions.pdf

Same thing, just more detailed:
http://orbit.dtu.dk/fedora/objects/orbit:55484/datastreams/file_3748519/content

 

[rcgldr]

Once you understand how yachts tack downwind with velocity made good greater than windspeed, DDWFTTW is almost trivial ...

I never liked the tacking analogy. A DDWFTTW cart could use any efficient thrust generating mechanism driven by the wheels. Propellers are used because of efficiency, not because of a similarity to tacking.

 

[A.T.]

I never liked the tacking analogy. A DDWFTTW cart could use any efficient thrust generating mechanism ...

Correct, but "...any efficient thrust generating mechanism..." is too abstract for many. Tacking is proven to work in practice, showing that we actually can be efficient enough working in air.

 

[CWatters]

Has anyone investigated if its necessary to have the wheels or water prop power the rotor?

It's known that some racing sailing boats or land yachts can go down wind faster than the wind by tacking. So in theory you could arrange for two of those (on opposite tacks) to tow a light weight barge. The barge could go directly down wind while the two boats tack back and forth in front of it. Ok so there are some engineering challenges but the principle would seem to be ok.

You can argue that not all of the rig is going directly down wind but not all of the cart discussed above does either.

 

[rcgldr]

Has anyone investigated if its necessary to have the wheels or water prop power the rotor?

It's known that some racing sailing boats or land yachts can go down wind faster than the wind by tacking. So in theory you could arrange for two of those (on opposite tacks) to tow a light weight barge. The barge could go directly down wind while the two boats tack back and forth in front of it. Ok so there are some engineering challenges but the principle would seem to be ok.

You can argue that not all of the rig is going directly down wind but not all of the cart discussed above does either.

This is the reason I don't like the tacking analogy for a DDWFTTW cart. With the pair of land yachts, the ground is opposing the apparent crosswind experienced by each land yacht, which is a function of the true wind times sin(heading with respect to wind), and independent of the land yacht's speed.

In theory, a DDWFTTW cart could be made with no parts that move perpendicular to the true wind. For example, the wheels could drive an enclosed paddle wheel (axis parallel to wheel axis) with vents located front (intake) and back (output) so that the vented paddle wheel generates thrust (similar to a vane pump).

 

[A.T.]

Has anyone investigated if its necessary to have the wheels or water prop power the rotor?

Of course it is necessary. It would violate energy conservation, if the rotor would produce steady thrust in a relative headwind without power input via the shaft.

It's known that some racing sailing boats or land yachts can go down wind faster than the wind by tacking. So in theory you could arrange for two of those (on opposite tacks) to tow a light weight barge. The barge could go directly down wind while the two boats tack back and forth in front of it.

Yes, and the lateral movement of the yachts would be powered by their keels, just like the rotation of the propeller is powered by the wheels. Note that the aerodynamic force on the yachts opposes their lateral movement, just like the aerodynamic torque at the prop opposes its rotation.

 

[A.T.]

...the ground is opposing the apparent crosswind...

I guess you mean that the surface provides the force that drives the yachts laterally (perpendicular to true wind). Just like the DDWFTTW wheels provide the torque, that drives the propeller blades laterally.

But you can make it even simpler:

In the surface frame the interaction with the surface (via keel / wheels) is just a constraint, which does no work, but merely constrains the airfoils (yacht sails / prop blades ) to a certain path.

 

[rcgldr]

I guess you mean that the surface provides the force that drives the yachts laterally (perpendicular to true wind). Just like the DDWFTTW wheels provide the torque, that drives the propeller blades laterally.

Ignoring drag, the force from the surface does no work because the force is perpendicular to the path of the land yacht. As I posted before, it opposes the apparent crosswind which = true wind times sin(heading with respect to true wind), allowing the apparent crosswind to be diverted by the sail to produce thrust to drive the land yacht forward.

In a DDWFTTW cart, the ground force is in the opposite direction of the path of the cart, performing "negative" work on the cart to drive the propeller, which combined with the tailwind, performs "positive" work on the cart.

 

[A.T.]

Ignoring drag, the force from the surface does no work because the force is perpendicular to the path of the land yacht. As I posted before, it opposes the apparent crosswind which = true wind times sin(heading with respect to true wind), allowing the apparent crosswind to be diverted by the sail to produce thrust to drive the land yacht forward.

In a DDWFTTW cart, the ground force is in the opposite direction of the path of the cart, performing "negative" work on the cart to drive the propeller, which combined with the tailwind, performs "positive" work on the cart.

You really should take the advice from the OP:

If one insists to use an work-based or power-based explanation, they should clearly specify the frame of reference in which the kinetic energies are defined, and if they are going to analyze the problem from different frames of reference, they should make it clear that kinetic energies and powers are not the same between different frames of reference. If they jump between different frames of reference when thinking about the DDWFTTW in terms of kinetic energies and powers, they will likely confuse themselves if they are not careful.

 

[rcgldr]

since the force is perpendicular to the path of the land yacht from any inertial frame of reference, the work done is zero.

 

[A.T.]

since the force is perpendicular to the path of the land yacht from any inertial frame of reference, the work done is zero.

No, the path is frame dependent, and so is the work done by a force acting along that path.

The key point I was making was that in the land yacht case, the surface force is perpendicular to the velocity of the land yacht (ignoring drag), while in the DDWFTTW case, the surface force is in the opposite direction of the velocity of a DDWFTTW cart

The tacking analogy doesn't have one tacking yacht, but two connected yachts:

It's known that some racing sailing boats or land yachts can go down wind faster than the wind by tacking. So in theory you could arrange for two of those (on opposite tacks) to tow a light weight barge. The barge could go directly down wind while the two boats tack back and forth in front of it.

The net force from the surface on the whole contraption points backwards, just like for the DDWFTTW cart.

 

[rcgldr]

No, the path is frame dependent, and so is the work done by a force acting along that path.

It doesn't matter that the path is frame dependent, only that the force is always perpendicular to the path (perpendicular to the land yacht's velocity), regardless of the frame of reference. It's similar to the situation where a centripetal force never performs any work, regardless of the frame of reference.

 

[A.T.]

...the force is always perpendicular to the path (perpendicular to the land yacht's velocity), regardless of the frame of reference

Velocity is frame dependent. Qualifying "perpendicular to velocity" as "regardless of the frame of reference" makes no sense.

The lateral forces would be related to the backwards force, but it's not clear if the backwards force could be stated as the source force that ultimately generates thrust,

Your distinction between "related" but not "ultimately generating" is a physically meaningless semantic game.

 

[rcgldr]

the lateral movement of the yachts would be powered by their keels

From the sail boat or land yacht frame of reference, the lateral force from the keel or wheels is part of a Newton third law pair. The lateral force from the keel or wheel is opposed by the lateral force related to diversion of the apparent wind, and the net lateral force is zero (otherwise the sail boat or land yacht would be accelerating laterally). Since the net force from this Newton third law pair of forces is zero, it doesn't matter which frame of reference is used, the net force remains zero.

From the sail boat or land yacht frame of reference, what drives the sail boat or land yacht forward is the diversion of the apparent crosswind in a direction aft of the sail boat or land yacht. Again a Newton third law pair, the sail exerts a backwards force on the apparent crosswind, and the affected air exerts a forward force on the sail. As stated before, the lateral forces cancel. Diversion of the apparent headwind contributes to part of the lateral force and also somewhat reduces the net forward force from the sail (similar to lift induced drag). From a ground frame of reference, a component of the flow diverted from the sail is upwind, which reduces the true wind speed (extracting energy from the true wind).

- - -

As for the two land yacht contraption, an analogy could be made comparing the sails to a rotating propeller on a DDWFTTW cart, but if the propeller was replaced by a paddle wheel like thrust generator, there would be no component of movement perpendicular to the wind.

 

[A.T.]

As for the two land yacht contraption, an analogy could be made comparing the sails to a rotating propeller on a DDWFTTW cart,

Yes, and in the frame of the twin-yacht-vehicle the lateral movement of the sails is powered by the keels, just like the rotation of the propeller blades is powered by the wheels in the DDWFTTW-cart frame.

but if the propeller was replaced by a paddle wheel like thrust generator, there would be no component of movement perpendicular to the wind.

I have no idea why you even find this relevant, but a rotating paddle wheel does have parts moving perpendicular to the driving direction (up / down).

 

[rcgldr]

Yes, and in the frame of the twin-yacht-vehicle the lateral movement of the sails is powered by the keels, just like the rotation of the propeller blades is powered by the wheels in the DDWFTTW-cart frame.

The lateral movement of the sails would be powered by the lateral component of forces from the keels. The true-wind component of forces from the keels would be an upwind force.

As for the two land yacht contraption, an analogy could be made comparing the sails to a rotating propeller on a DDWFTTW cart, but if the propeller was replaced by a paddle wheel like thrust generator, there would be no component of movement perpendicular to the wind.

I have no idea why you even find this relevant, but a rotating paddle wheel does have parts moving perpendicular to the driving direction (up / down).

There would be movement perpendicular to the true wind, but not interaction. The parts motion would include perpendicular motion, but in the idealized vehicle, there would be no net interaction with the two media (air and ground or air and water) during any motion perpendicular to the true wind. For example, the wheels driven by the ground in a conventional DDWFTTW vehicle rotate, but there are no net interactions or forces between wheel and ground that are perpendicular to the true wind. My point was that a idealized thrust generator could be used to interact with the air, so that are no net interactions or forces between thrust generator and air that are perpendicular to the true wind, and that an idealized DDWFTTW vehicle would not require any net interaction or forces between the vehicle and the two media that are perpendicular to the true wind. It's just an issue of efficiency.

The key point of DDWFTTW or DUWFTTW is that tacking is not required.

- - -

Getting back to the least confusing explanation of a a vehicle that interacts with two media (for this case air and ground) with a relative velocity (true wind speed from the ground frame of reference) ... One explanation focuses on the effective gearing used to reduce the speed and increase the force from the faster moving media (ground if moving DDWFTTW, air if moving DUWFTTW) to be applied to the slower moving media, which is used in the articles you linked to that discuss the process from a power perspective, which are similar to Mark Drela's analysis (link below). Another explanation uses tacking as an analogy, which is somewhat different in the case of a single land yacht (which is why I don't like tacking analogy). However, in the case of a two land yacht vehicle, the same effective gearing effect is involved, the net force on the vehicle from the sails that interact with the air is downwind and the net force on the vehicle from the wheels (acting like keels) that interact with the ground is upwind, and from the vehicles frame of reference, the gearing is used to reduce the speed and increase the force from the faster moving media.

Mark Drela's analysis:

http://www.boatdesign.net/forums/attachments/propulsion/28167d1231128492-ddwfttw-directly-downwind-faster-than-wind-ddw2.pdf

 

[Redbelly98]

Here is the simplest explanation I've been able to come up with:

Suppose the cart is moving at the same velocity as the wind. Then there is no drag force. If we can neglect rolling resistance and friction in the axles, then the only horizontal force on the cart is in the forward direction and due to the propeller pushing backward on the air.

Since the force on the cart is in the forward direction, it will accelerate and go faster than the wind.

 

[A.T.]

Suppose the cart is moving at the same velocity as the wind. Then there is no drag force. If we can neglect rolling resistance and friction in the axles, then the only horizontal force on the cart is in the forward direction and due to the propeller pushing backward on the air.

No. Even without those dissipative loses, there is still a drag force at the wheels, which have to turn the propeller against the aerodynamic torque. The key here is the difference in relative speeds (ground-cart vs. air-cart), which allows this drag force to be lower than the propeller thrust.

 

[Redbelly98]

No. Even without those dissipative loses, there is still a drag force at the wheels, which have to turn the propeller against the aerodynamic torque.

Aargh, yes, good point.

 

[A.T.]

I have tried to find the least confusing (or rather, the most 'deconfusing') explanation for why it is possible for a cart with a prop to travel directly downwind faster than the wind.

Rick Cavallaro, the pilot of the Blackbird on the record runs, offers his favorite explanations in this talk:

 

[JR Jonsson]

As a kid i played with a spool and was fachinated abaut the fakt that the spool came closer too me even when it meant that the thred was winded up on the spool. There you have the force aspect in this matter.

Imagine you are in a train with no floor.
Put down a wheel, driven by the moving ground beneth, that with a transmission drives you forward in the train with apropiate equippment.
The moving train itself makes you travel faster then the train, like the wind itself makes you move faster then the wind.

 

[mrspeedybob]

The air and the ground are moving relative to one another, therefore, they have kinetic energy relative to each other.
It is possible to build an apparatus to harness that energy.
Movement of the apparatus parallel to the air/ground movement necessitates certain design considerations, but does not invalidate the possibility of capturing energy from the air/ground movement.
Energy, from whatever source, including that described above, can be used to propel the apparatus.

 

[A.T.]

Explanation by Veritasium at 17:30min:

 

[anorlunda]

Good video. I was one of the ones who said "impossible". It appears that I was wrong.

It is still hard to grasp, but his explanation plus animations makes it easier.

 

[Redbelly98]

Explanation by Veritasium at 17:30min:

I was going to post the same link in one of these old DDWFTTW threads, but you beat me to it.
There is another good, 2-minute explanation starting at 6:45 in the video:

 

[anorlunda]

There's a good lesson in this story. When we evaluate devices looking for impossible PM (Perpetual Motion), there's really only one clue that's important -- conservation of energy (COE). If a machine violates COE, it is impossible. To me and others, when first hearing of this vehicle it seemed to violate energy conservation. But in the video explanation starting at 17:30, the key comes a 19:00 where Derek shows where the energy comes from to make the vehicle go faster than the wind downstream. Case closed, if that's true COE is preserved and the machine is possible.

But the next step is to analyze where the energy goes, and how it is transformed in the various steps. That can be exceedingly complicated. I don't claim to understand all those steps in that vehicle. But the second step is unnecessary to answer the question, "Possible or Impossible?"

Once COE is assured, we can say "possible" without knowing any more details of the design or operation. Conversely, if we know that COE is violated, we can say "impossible" without knowing any more details of the design or operation. That infuriates some advocates of PM machines, who insist that we explain every step in the process in detail before issuing a verdict on the feasibility.

Such is the power of science. Physical laws allow us to state some conclusions with confidence, even without exhaustive analysis of the details.