Jeff's Lunchbreak

There's an interesting topic that stirs quite a bit of debate in certain circles these days - whether or not a wind powered cart can travel directly downwind faster than the wind, itself. This concept is generally known by one of two acronyms, Down Wind Faster Than The Wind, DWFTTW, or the slightly longer Directly Down Wind Faster Than The Wind, DDWFTTW.

The cart in the concept has a propeller connected to wheels through a driveshaft and transmission. You might intuitively think that this is impossible. I know I did. It sounds too much like a perpetual motion machine, with the wheels powering the propeller which pushes the wheels. So, I thought I would set out to prove it impossible with a few free body diagrams, but now that I've studied the diagrams, I find myself thinking that it might actually work.

Background

First, to get a bit of background on this concept, here are some of the websites of its main proponents:

• Faster Than the Wind Team's Official Blog
• Downwind Faster Than the Wind

To get an idea of just how much passion this concept evokes, take a look at some of the discussion threads arguing over it:

• JREF Discussion Forum, Thread 1
• JREF Discussion Forum, Thread 2 (this one's still active)
• Mythbusters Fan Club Forum
• SkepticBlog
• Good Math, Bad Math

As one last link before getting into my own discussion, the Faster Than the Wind Team claims to have built and tested a car that demonstrates the concept. The North American Land Sailing Association (NALSA) witnessed and authenticated the event. While some might think that would be enough to convince doubters, many remain skeptical of the claims and suspect foul play or incompetence (such as not running directly downwind or measuring windspeed incorrectly in the more generous accusations).

• NALSA's Direct Downwind Record Attempt Page

Here's some video of the record claiming run.

Main Issues

There are two big discussions in this debate, whether a DWFTTW car is even possible, and second, whether the widely circulated 'Treadmill Experiment' is useful in demonstrating its validity. Since the first claim is more interesting, I'll address that first.

Is DWFTTW Possible?

To address this, I drew up some free body diagrams. All the diagrams are shown in an assumed steady state condition. For simplicity, the vehicles are all simplified as just a single wheel, a prop/turbine, and a transmission connecting them.

Let's start with something that we know works, an upwind vehicle.

In this case, V_wind will be greater than V_ground. The prop/turbine will be acting as a turbine, so it will be creating drag. The wheel will be driving the vehicle, so F_ground will be in the forward direction. Since the vehicle is in equilibrium:
F_ground = Drag

To calculate the power from the turbine and the power from the wheel:

P_turbine = Drag * V_wind
P_wheel = F_ground * V_ground

Since Drag = F_ground, if V_wind > V_ground, then P_turbine > P_wheel. That's what it needs to be to overcome transmission losses and the vehicle wind drag that I didn't account for. So, the diagram and analysis agree with what we expect from reality.

Next, let's move on to a case that we know doesn't work, a vehicle with no wind.

Once again, to maintain equilibrium:
F_ground = Thrust

Since I labeled the forces slightly differently, here are the power equations:

P_prop = Thrust * V_wind
P_wheel = F_ground * V_ground

Since Thrust = F_ground and V_wind = V_ground, then P_prop = P_wheel. That doesn't work, since transmission losses will sap the energy out of that system, as will the air drag. I think it should be obvious enough that if you reverse the drive direction (i.e. a turbine powering the wheels), that it still comes out to P_turbine = P_wheel, which doesn't work. So again, the diagram and analysis agree with what we expect from reality.

Now, let's move on to the DDWFTTW case.

Once again, to maintain equilibrium:
F_ground = Thrust

The power equations are the same as the previous case:

P_prop = Thrust * V_wind
P_wheel = F_ground * V_ground

Since Thrust = F_ground, if V_ground > V_wind, then P_wheel > P_prop. Like the first case, that's what it needs to be to overcome transmission losses and the vehicle wind drag.

So, it seems counter-intuitive, but unless I've made a mistake somewhere, it looks like it should work. Maybe there is something to what the propenents have been saying, that the differential velocities are the source of energy, and why this isn't a perpetual motion machine. As the diagrams show, if there's no wind at all, then the vehicle doesn't run.

The Treadmill Experiment

An early proof of concept experiment that made its rounds on the Internet was to put one of these carts on a treadmill and see what happened. The video is included in the links I gave at the start of this entry, but I'll embed it here to make it easier for you.

The treadmill experiment prompted two big questions - is it representative of a cart moving downwind over the ground, and does it demonstrate the validity of the DWFTTW concept?

The answer to the first question is a clear yes. The treadmill is an equivalent reference frame. That's how wind tunnels work – it's all about relative velocities. If a treadmill is moving at a steady 10 mph in still air, it's the same as the ground being stationary with a steady 10 mph wind.

Consider this. The Earth's surface is not stationary itself. Given a circumference of approximately 25,000 miles, and a rotation period of 24 hours, the ground is moving at just over 1000 mph at the equator (and that's ignoring the Earth's motion around the Sun, the Sun's motion around the Milky Way, and the Milky Way's motion about the local galactic cluster). In other words, the Earth could be considered a giant treadmill. But we can safely neglect that if we use a frame of reference that moves along with the earth. It's the same thing with the treadmill. As long as all the relative velocities are equivalent, then your reference frames are equivalent.

But, did the experiments in the video demonstrate the validity of DWFTTW? I think the answer is yes to that as well, but I also have an idea for another experiment.

My initial skeptical thought was that by physically holding the cart stationary on the treadmill before releasing it, they were storing energy by spinning up the propeller on the cart. When they released the cart, it would surge forward using that stored energy. It's just like a toy helicopter where you pull a string to make it take off. There's nothing surprising about that.

But, if you watch the video, once the cart is operating, they aren't holding it against the treadmill, they're holding it back against its own thrust. That, to me, is indicative that the cart wants to run faster than the treadmill. Unfortunately, given the short length of their treadmill, the videos never show the cart reaching a steady state.

My suggestion for a better experiment (aside from the full size human carrying cart) is to build a long treadmill inside a building, and put their cart on that with some guides to keep it from running off track. If it achieved a steady state forward velocity relative to a stationary observer (and was well documented by independent observers), then I'd think most reasonable skeptics would be convinced. Alternatively, the cart could be placed on the treadmill before the treadmill was started in motion, so that it wouldn't be touched by human hands at all once the experiment started. Judging by the Faster Than the Wind Team's human carrying cart, I would think they have the means to carry this out, and all remaining doubts could be put to rest.

Alternate Explanation 1

I think a better way to understand this vehicle, is rather than thinking of it as a ground vehicle powered by the wind, think of it as an aircraft powered by the ground. I have a little thought experiment that might help. Envision the vehicle suspended on some rails, with the prop aligned to propel it down the rails, and with the wheel hanging below on a caster that enables the wheel to face any direction. If you put a conveyor up to the wheel with the conveyor running sideways relative to the vehicle, it's obvious that the conveyor will turn the wheel, which will drive the propeller and push the vehicle down the tracks. Now, if you start rotating the conveyor to more closely align with the tracks, it will continue to drive the wheel. The more closely it aligns with the tracks, the higher the drag load that it will impart, but it will continue driving the wheel.

So, think of the cart as an aircraft that with no other forces acting on it would want to be 'at rest' with zero relative windspeed. But, once the ground starts moving relative to the aircraft, it provides a power source that the aircraft can tap into to propel itself.

Alternate Explanation 2

Consider a cart where one set of wheels is turning a generator used to power another set of wheels driven by motors. This obviously won't work. Since both wheels are moving over the ground at the same speed, if the force at each wheel was of equal magnitude, then the power created by the generating wheel would be equal to the power being used by the driving wheel, which doesn't work when you account for losses.

If a cart with a propeller was moving through still air, then it would be the same thing. That was what I tried to explain with the no wind case. When forces are equal and velocities are equal, then powers are equal.

What makes the downwind case work, is that the wheels and the propeller are operating in two different media at two different speeds. For a propeller, we typically look at the thrust generated for a given power, since that's the way engines operate. It's well understood that for a given power, thrust drops with airspeed. But looking at that a different way, it means that to generate a given thrust, the power requirement goes up with airspeed. So, using the no wind day as a baseline, when the thrust from the prop and the drag from the wheels are the same, the input and output powers are the same (which doesn't work because of losses). Now, if you add a little bit of tail wind, it means the propeller is not travelling through the air as fast. If you maintain it at the same thrust, it means the power requirement goes down. So, now we're getting into a regime where the power generated by the wheels is higher than that required by the prop. With enough of a tail wind, the power differential can get big enough to overcome the inefficiencies and make the system actually work.

Conclusion

So in the end, once I gave this a little thought, I surprised myself. I think my initial gut reaction to this concept was wrong, and that the DDWFTTW proponents are right. That's all part of honest skepticism - knowing when to admit you were wrong and to change your views based on new reasoning and evidence. It's certainly nice to know that the Faster Than the Wind Team is most probably honest, and that the videos probably aren't a hoax. Congratulations to them for their achievement.

Now that I've had my say, and hopefully convinced people that this is possible, here's a good article on it:

• Wired - A Long, Strange Trip Downwind Faster Than the Wind

I'll also add that the nice thing about this question is that it's testable. If I get a chance, I'll build a little cart myself. If enough people do this and test it, it should be confirmed pretty quickly. If you're one of the people that feels really strongly about this, go do a test for yourself.

Added 2011-06-13

I figured it might be fun to throw in a few real numbers to get a feel for how this would work. So, I pulled some numbers out of the air to see how the calculations come out.

I started out with a ground speed of 30 ft/s (~20.5 mph). Assuming 100 pounds of drag on the wheels, this works out to 3000 ft-lb/s of power (~5.5 HP). Assuming a 90% efficient drive train, there's 2700 ft-lb/s going into the prop. Now, the next step requires a little understanding of propellers which I've explained on my static site (Theoretical Max Propeller Efficiency). Assuming a figure of merit of 0.9*, and a propeller diameter of 15', the prop will create 170.5 lbs of static thrust. So, at the state where ground speed matches wind speed, the thrust created by the propeller will be greater than the drag on the wheels - the cart will accelerate forward. And since the cart is at zero relative airspeed at that condition, there's no aerodynamic drag to consider. Also note that there's no stored energy from a flywheel effect in this analysis, so the steady state condition will necessarily be at some speed where the cart is going faster than the wind speed.

You can play around with those numbers if you want to. For the given efficiencies and prop diameter, the break even point where thrust = drag is around 495 lbs (27 HP @ 30 ft/s). If you hold the efficiencies and drag constant, the break even point on thrust & drag occurs for a 6.7 ft diameter propeller. All of these numbers appear to be fairly reasonable, giving me yet further confidence that the Faster Than the Wind Team probably achieved what they stated.

* This figure of merit is a measure of how much of the power is going into accelerating the air. This is a more useful measure than efficiency for low speeds, since by definition, propeller efficiency is equal to zero for static thrust. A figure of merit of 1 is the theoretical limit. The propellers I've designed at work typically achieve figures of merit of 0.92 to 0.94 for static thrust.