Down wind faster than the wind

[H'ethetheth]

Treadmills are not fans.

You are of course correct; treadmills are indeed not fans. The claim is however not that they are; the claim is that the wind is created by the motion of the treadmill.

Look, you can keep being indignant about telling us what you think wind is, but I assure you, it would help us understand where you're coming from. It's not a trick question.

Also, you have a strange and relatively verbose way of saying "no".

 

[ThinAirDesigns]

You are of course correct; treadmills are indeed not fans. The claim is however not that they are; the claim is that the wind is created by the motion of the treadmill..

Tsig, here's a claim for you:

If I simply increased the size of my treadmill enough, I could set you in an easy chair on the porch of a house on a hot summer day and with the treadmill set at 10mph, you would be telling us how sweet the tea was and how nice the breeze was.

The flag on your porch column would stand proud, the wind chimes would be a 'clanging and the shrubs and trees would be swaying -- all because of a treadmill that "can't make wind". And of course you would be arguing strenuously with us that it's REAL wind and would not believe me when I told you that it was all just a treadmill.

JB

 

[spork]

Tsig, here's a claim for you:

If I simply increased the size of my treadmill enough, I could set you in an easy chair on the porch of a house on a hot summer day and with the treadmill set at 10mph...

And he'd still be a troll.

 

[Clive]

Necessary and Sufficient Conditions?

I've been following this thread for a little while now after first learning about these carts via the debate that happened over at "Good Math" at scienceblogs. The various videos posted by spork and others, plus the evidence that ice boats can easily achieve a speed faster than the wind (in the true wind direction) means I am pretty much convinced now the whole deal of accelerating from rest and eventually achieving faster than wind speed, directly downwind, can be done (despite being a doubter to start with).

However, I'd like to have a better grasp on what conditions are necessary and sufficient for the whole deal to be achieved (if those can be nailed down). By "whole deal", I mean accelerating from rest in the same direction as the wind and eventually reaching a stable speed in excess of that of the wind, and so on.

Questions:

1. Assume we have a cart that accelerates from an initial condition where it is moving at exactly wind speed (as per the starting point for many of the treadmill videos). Is this cart guaranteed to also be able to accelerate from rest (relative to the ground) and eventually achieve wind speed (and then obviously also exceed it)?

2. Can the conditions when a cart will NOT be able to achieve the goal be clearly laid out?

3. How about "carts" in mediums other than air? How sure can we be that a cart can be built to go "directly down stream faster than the stream" if the stream is water flowing smoothly along a level canal (with a smooth concrete floor, and the cart is running along that floor, etc.)? How about a cart to do the same trick in a flow of magma?

I've tried (at least a little) to get my head around the mathematics of all this. But even with a reasonably strong background in that area it still looks to me like it might be something that is just too complex to have any easy answers. I'd be very happy though if someone else with more skill, knowledge or practical experience in this area can do better and spell out at least some of conditions/mathematics. Or is it always going to come down to someone needing build and test an actual machine, or at least simulate the whole thing in a computer (assuming appropriately powerful software and all other relevant data is available to do that)?

 

[Michael C]

1. Assume we have a cart that accelerates from an initial condition where it is moving at exactly wind speed (as per the starting point for many of the treadmill videos). Is this cart guaranteed to also be able to accelerate from rest (relative to the ground) and eventually achieve wind speed (and then obviously also exceed it)?

I'd give a qualified "yes" here. It would certainly be possible to invent a cart that does one thing at wind speed, but something else below it. If we restrict ourselves to simple designs such as Spork's cart, where there is only one fixed gear ratio, they should all be able to accelerate from below wind speed to above it. Starting from rest may be problematic: since the cart has only one gear ratio, which is optimised for a speed greater than that of the wind, starting from rest is like starting your car in top gear. Maybe wheels will spin, maybe there won't be enough power to get the thing going. If the cart is light and efficient (as is Spork's), starting from rest shouldn't be a problem. The cart will go on accelerating until equilibrium of forces is reached, at which point it will coast at a steady speed.

2. Can the conditions when a cart will NOT be able to achieve the goal be clearly laid out?

Well, there are all sorts of carts that won't do this. If, once more, we restrict ourselves to the design of Spork's cart, the conditions are relatively simple. See links to mathematical analyses lower down.

3. How about "carts" in mediums other than air? How sure can we be that a cart can be built to go "directly down stream faster than the stream" if the stream is water flowing smoothly along a level canal (with a smooth concrete floor, and the cart is running along that floor, etc.)? How about a cart to do the same trick in a flow of magma?

If we can build a cart that has, so to speak, one foot in one medium and one in the other, we're well on our way. In each particular case, we have to see what techniques we can use, and how we can reduce losses due to drag and friction. Water and magma are similar to air: we need something like a propeller that will move through the water or magma at a certain speed which is slower than that of the cart relative to the ground, and we need to make sure that the rest of the cart offers as little resistance as possible to the water or magma. We also need wheels that won't slip on the ground.

I've tried (at least a little) to get my head around the mathematics of all this. But even with a reasonably strong background in that area it still looks to me like it might be something that is just too complex to have any easy answers. I'd be very happy though if someone else with more skill, knowledge or practical experience in this area can do better and spell out at least some of conditions/mathematics. Or is it always going to come down to someone needing build and test an actual machine, or at least simulate the whole thing in a computer (assuming appropriately powerful software and all other relevant data is available to do that)?

Thabiguy has already posted some good analyses. Look at http://www.internationalskeptics.com/forums/showpost.php?p=4192317&postcount=57 and http://www.internationalskeptics.com/forums/showpost.php?p=4218525&postcount=937

 

[spork]

Clive,

You bit off more than I can chew in a single post, so I'm going to try and hit the high spots, and hopefully you can tell me where you want to dig deeper.

First, the math really doesn't have to be terribly complicated. In aero we frequently use models that are perfectly rigorous, without getting into unneccessary detail. The way we model a wing for example is to characterise it's lift to drag ratio, its span, aspect ratio, and overall coefficient of lift. This will tell us most of what we need to know about a wing for most of our purposes without having to solve the simultaneous Navier-Stokes equations using computational fluid dynamics codes. We can take a similar approach with the prop cart - and delve deeper only if desired.

There are really only a couple of imporant parameters needed to characterise the cart: advance ratio and overall efficiency (if we neglect scale).

Whether we are talking about pulling a yo-yo by the string, or putting this cart ( http://www.putfile.com/pic/2794071 ) in a tailwind, or analysing the prop-cart, the advance ratio tells us whether the cart will go downwind faster than the wind, or it will go upwind - and how fast.

The advance ratio is simply the theoretical ratio of speed the cart moves through the air (or string in the case of the yo-yo) divided by the speed it moves over the ground. For the prop cart this is the distance the prop would theoretically advance in a single rotation divided by the distance the wheels would travel in the same single rotation of the prop. It is determined by the prop pitch and the gearing (including wheel diameter).

If the advance ratio is less than 1.0 we have a cart that will go downwind faster than the wind. If the ratio is greater than 1.0 the cart will go upwind. As the advance ratio approaches 1.0 from either side the theoretical speed of the cart will go to greater multiples of the wind speed (upwind or downwind). At 1.0 the answer is undefined (it would be infinitely as fast as the wind both upwind and downwind at the same time - doesn't make much sense).

However, moving the advance ratio towards 1.0 is exactly like putting your car in high gear. You go faster, but you have less guts to climb a hill (or overcome real-world losses). This is where the overall efficiency of the cart comes in. We lump prop efficiency, transmission losses, aero drag, and rolling resistance all into this parameter. The more efficient the vehicle, the closer we can make its advance ratio to 1.0, and therefore the faster it can go - either upwind or downwind.

Now onto the math...

There are a number of ways to approach the math without getting overly complex. I think the most obvious approaches involve energy balance, or force-vector diagram. I find the force vector diagram more compelling, but both certainly have their place. In the case of the force vector diagram it's far easier if we unwind the path of the prop tips and look at this path in two dimensions. From there we can easily show how it winds up to produce the DDWFTTW cart.

If one or both of these approaches are of interest, just say so. I'll be happy to see if I can make them clear.

 

[screensnot]

I'm not sure in what sense we can consider a blimp "fundamentally a ground vehicle".

I see that you have labeled the machine a 'ground vehicle', but I am not sure why.

The initial idea was to go downwind faster than the wind. Nobody said it had to be a 'ground vehicle'.

I guess you'll have to define 'ground vehicle' first, if you are going to place this constraint on us.

If my blimp has wheels that touch the ground, why doesn't that fit your description of 'ground vehicle'?

 

[tsig]

And he'd still be a troll.

If you would just build the device you would not have to call me names.

 

[screensnot]

He did build it, Tsig. Did you not see it?

 

[tsig]

I see that you have labeled the machine a 'ground vehicle', but I am not sure why.

The initial idea was to go downwind faster than the wind. Nobody said it had to be a 'ground vehicle'.

I guess you'll have to define 'ground vehicle' first, if you are going to place this constraint on us.

If my blimp has wheels that touch the ground, why doesn't that fit your description of 'ground vehicle'?

Because we have been talking about a ground vehicle all along.

 

[screensnot]

Because we have been talking about a ground vehicle all along.

We have been talking about a DDWFTTW vehicle. How does a blimp with wheels not fit the description (assuming it goes faster than the wind)?

 

[Thabiguy]

There is no wind on the treadmill.

Treadmills are not fans.

Ice boats and treadmills are red herrings.

Now, if there is wind on every red herring that is not a fan, but no wind on iceboats, which of the following is true, which is false, and which cannot be determined?

a) Some treadmills are not iceboats.
b) Some fans are red herrings.
c) All iceboats are fans.

 

[spork]

D) tsig is still a troll.

 

[Clive]

I'd give a qualified "yes" here. It would certainly be possible to invent a cart that does one thing at wind speed, but something else below it. If we restrict ourselves to simple designs such as Spork's cart, where there is only one fixed gear ratio, they should all be able to accelerate from below wind speed to above it. Starting from rest may be problematic: since the cart has only one gear ratio, which is optimised for a speed greater than that of the wind, starting from rest is like starting your car in top gear. Maybe wheels will spin, maybe there won't be enough power to get the thing going. If the cart is light and efficient (as is Spork's), starting from rest shouldn't be a problem. The cart will go on accelerating until equilibrium of forces is reached, at which point it will coast at a steady speed.

Thanks for your responses Michael (and the links to the earlier posts even though I haven't fully digested those yet).

In my first question I was in fact thinking primarily of a cart using essentially the same design as Spork's. I guess the main thrust (sorry!!) of that was to find out if there is some fairly straightforward reasoning to show that demonstrating acceleration from wind speed (as per the treadmill videos) is in fact all that really needs to be shown to prove the whole thing, starting from rest, etc., is possible, or whether it might still turn out that the very same cart and in the same wind conditions might not actually be able to self-start, or perhaps even that it might self-start but then still not actually reach wind speed after that.

How about a "Spork cart" that can be shown to self-start, and can also be shown accelerate from wind speed. Can we be 100% sure that this is enough to guarantee it can do the whole thing?

As for my last question about a cart that ran in water (instead of air), again I was imagining a "Spork cart", but possibly built from different materials, and with different wheel size, gearing, and prop configuration as needed. It seems clear to me (but only based on intuition again) that as the medium (replacing the air in the original scenario) becomes more viscous and/or dense it also becomes more likely that there is no straightforward variation of the "Spork cart" that will work. Too viscous and you get too much loss through drag, and if it was too dense then you can't get enough traction with the ground (regardless of viscosity)?

 

[humber]

D) tsig is still a troll.

In the same way that Spork is still unable to knockdown the balance argument, even though he has had time to "phone a friend".

 

[Clive]

The advance ratio is simply the theoretical ratio of speed the cart moves through the air (or string in the case of the yo-yo) divided by the speed it moves over the ground. For the prop cart this is the distance the prop would theoretically advance in a single rotation divided by the distance the wheels would travel in the same single rotation of the prop. It is determined by the prop pitch and the gearing (including wheel diameter).

If the advance ratio is less than 1.0 we have a cart that will go downwind faster than the wind. If the ratio is greater than 1.0 the cart will go upwind. As the advance ratio approaches 1.0 from either side the theoretical speed of the cart will go to greater multiples of the wind speed (upwind or downwind). At 1.0 the answer is undefined (it would be infinitely as fast as the wind both upwind and downwind at the same time - doesn't make much sense).

I think I need to ask you to clarify my understanding of "advance ratio". What exactly do you mean when you talk about "the theoretical speed the cart moves through the air"? Is this something like imagining the prop is spinning on its shaft but is also able to "fly" freely along the shaft, and that if the cart was moving at (say) one meter per second across the ground, then the prop (once "stable") would be generating lift and drag in proportions that meant it progressed forward on shaft at A metres per sec (where A is the advance ratio)?

We have found the best advance ratio results in the 1.2 to 1.4 range -- meaning that the wheels travel farther than per prop rotation by that multiple over the pitch of the prop per rotation.

The other niggle in my mind regarding the advance ratio is that back in comment #280 to the original GoodMath thread, "JB" said your carts seemed to work best with an advance ratio in the range 1.2-1.4, but according to what you've just told me that would indicate a cart designed to advance upwind rather than to go DDFTTW. Hmmm... just read JB's comment again and noticed he said the 1.2-1.4 ratio means the distance covered by the wheel divided by something to do with the prop so it seems he has inverted your definition!

Anyway, some clarification on "advance ratio" would help me I think. Thanks.

 

[screensnot]

Anyway, some clarification on "advance ratio" would help me I think. Thanks.

Seems to me that JB and spork are just calculating the ratio differently.

Spork says: AR = Prop / wheel

JB says: AR = wheel / prop

Just take the inverse of JB's number to put it in Spork's terms. Or vice-versa.

 

[CORed]

In the same way that Spork is still unable to knockdown the balance argument, even though he has had time to "phone a friend".

There's no need to knock down something that was never standing in the first place.

 

[humber]

There's no need to knock down something that was never standing in the first place.

Ok, put forward another explanation for how the cart remains on the belt as seen. I will knock that down too.

 

[spork]

I think I need to ask you to clarify my understanding of "advance ratio". What exactly do you mean when you talk about "the theoretical speed the cart moves through the air"?

This part is simply the prop pitch. You can imagine the prop moving through jello without anything holding it back. The distance it would advance in one rotation would be the prop's pitch (which in my description is the numerator of the advance ratio). If you want to think of it analytically... the prop's pitch is given by:

pitch/circumference = tan(tip_angle)

pitch = circumference * tan(tip_angle)

pitch = 2*PI*prop_radius * tan(tip_angle)

Where tip_angle is the angle the tip of the blade makes relative to the "disk" of the blade.

Effectively, the pitch of a propeller is the same thing as the thread-pitch of a screw.

So, in my definition you'd define advance_ratio as the prop-pitch divided by the distance the wheels would roll given a single prop rotation. This of course is determined by the gearing and wheel diameter.

wheel_advance = gearing * 2*PI*wheel_radius

So...
advance_ratio = prop_pitch / wheel_advance

advance_ratio = 2*PI*prop_radius * tan(tip_angle) / (gearing * 2*PI*wheel_radius)


Finally:
advance_ratio = prop_radius * tan(tip_angle) / (gearing * wheel_radius)

Where gearing is defined to be greater than 1.0 if the axle spins faster than the prop-shaft.

The other niggle in my mind regarding the advance ratio is that back in comment #280 to the original GoodMath thread, "JB" said your carts seemed to work best with an advance ratio in the range 1.2-1.4, but according to what you've just told me that would indicate a cart designed to advance upwind rather than to go DDFTTW.

Yeah - that's just because we invented this term to define a key parameter of the cart, and didn't actually agree on whether prop_pitch should be the numerator or denominator. I'll see JB at the office tomorrow and administer the appropriate beating.