DDWFTTW - Directly Downwind Faster Than The Wind

Hi Rick,

I apologize for the confusion I caused by not being clear, but I'm not a neophyte here. I've washed down the crow with the kool-aid and even before that I knew the prop is not acting a turbine. So if what I write next is not clear on first read, please take as a given that I'm not confused about prop vs. turbine and read it again. I'm influenced by Puritan writers but don't have their skill; pay particular attention to my commas which I use when I break up a thought because I am trying to qualify it - sort of like the wheels (which are the turbine buckets in the cart's reality - I told you I understood) and the prop fighting each other for a net gain. It should not take more than half a dozen reads to understand; from what I have seen of your posts you'll get it on the first or second go.

The analogy, and it is a very limited analogy, to the turbine I was trying to think about is that the true wind has a given speed before it encounters the prop, and it (the wind) has a lower speed (and a different velocity) after it encounters the prop because it has applied force to, and given energy and/or momentum to, the cart. I.e. I am looking at the cart and its sails/blades/prop as a black box and as a turbine bucket as another black box. In that limited sense and in that limited sense only, there most definitely is an analogy between the cart black box and the turbine bucket black box in that they both extract momentum and energy from available potentials that exist between the motive fluid and the ground/nozzles/shell. (Phew! I had no idea so much was implicit in my previous post; no wonder it made no sense.)

Given that very limited analogy, and possibly (probably) stretching it too far, there is a principle in impulse turbines that the theoretical upper limit on efficiency comes from reducing the incoming fluid velocity by half with the bucket moving at half the incoming fluid velocity. I was trying to figure out if there is an analogous property inherent in the DDWFTTW cart design. I just realized it's probably in the Drela papers and I missed it so I'll go read them again. It's probably the L/D limit from Marchaj applied to the prop/cart/wind geometry.

 

... at that terminal speed.

I assume you mean from at the beginning i.e. startup when the prop foils are stalled. As you approach terminal speed two things happen:

1) the blade efficiency drops as the angle of attack (AOA) drops.

2) the cart overall efficiency drops both with the blade efficiency and with the increasing losses. If the overall efficiency always improved with increasing cart speed there might be no upper limit.

Maybe efficiency isn't the right term for the blade, but in the blade-apparent wind frame the prop can no longer deliver increased momentum to the cart as all the air being turned is used to overcome losses at the terminal speed, and increasing cart speed increases losses with which the prop can't keep up as it approaches feathering (luffing in cloth sailing terms i.e. low AOA).

Actually, maybe that is what I am getting at: the optimum design has the blade efficiency reaching a maximum at cart terminal speed and so blade efficiency never falls off. As you say it would seem that VP should get you most of the way there.

 

That's where I begin to wonder where the energy comes from if you don't have an engine supplying it. You reach wind speed, you engage friction driven wheels to a propeller that only puts out 85% of the energy put into it and
A: You keep speeding up, faster and faster.
B: you slow down
I'm thinking B.

 

Ooh! Ooh! Let me! Let me do this one!

You can go back through this thread, this has been covered at length. I'll try to summarize.

The energy comes from the velocity differential between the air and the ground. The prop blades are the cart's connection to the air, and the wheels are cart's connection to the ground.

Code:

_____High_Energy_Source_(Wind)____
            |
            v
      +-----------+
      |   Cart    |----->Work Extracted
      +-----------+
            |
            |
____________v____________
     Low Energy Sink (Ground)

As the cart velocity increases, its losses (prop and cart drag, rolling friction) increase as its ability to extract work (power) from the wind decreases. When the power extracted balances the losses, the cart has achieved maximum velocity.


If you really do have your thermodynamicist hat on, that should be enough to convince you why it works. The rest is in the details explaining how it works.


Wheel-Prop Energy balance (Thrust & drag):

The wheels need only provide enough energy to the prop to move the cart to leeward through (i.e. wrt) the true wind. The prop is moving at (Cartspeed-TrueWindspeed) wrt the air; the wheels are moving at Cartspeed wrt the ground. The energy to travel to leeward at true windspeed over the ground is already in the wind (ignoring losses), even if there is no prop.


Wheel-Prop Torque (Force) balance (drag):

The wheels need only provide enough force (torque) to balance the drag force of the prop in the plane of the prop rotation, in the wind that the prop sees.


The wheel-prop connection is "merely" clever gearing to make the intersection of any true wind streamline with a cart surface - specifically the prop blades - have a component that moves upwind relative to the true wind.

Basically, here is the least complicated diagram that seals it for me:

Code:

 Cart
 Velocity
 Vector 
 Wrt
 Ground
      ^<--___
Blade |      ---___
  |   |            ---___ Blade Velocity Vector Wrt Ground
  v   |                  ---___
  \   |                        ---___
   \  <-------------------------------O
    \      Blade Velocity Vector wrt Cart
   ^ \
   |
   |
   True
   Wind
   Velocity
   Vector

The prop pitch and the gearing can all be adjusted by design to make the qualitative relationships above work; the point is that the prop blade, moving along the big diagonal vector wrt the ground, is able to push back against the true wind even though the cart is moving faster than the true wind.

There is an even prettier vid here:

http://i47.tinypic.com/m7xf09.gif

[created by Eytee on Wikipedia]

which is the same thing just done with with a sail and a centerboard representing the boat's connection to the two media (air on sail; water on keel) to tap the differential energy between those media. The beauty of that vid is that it shows

1) the intersection (red circle) of the sail with a true wind streamline moving to leeward slower than the true wind, and

2) the true wind (blue circles) hitting the back of the sail at that intersection, while

3) the hull and any fixed point on the entire sail itself are moving to leeward faster than the true wind.

 

The overall efficiency will not get any higher than around 75 to 80%. Hence the maximum possible vehicle speed is somewhere between 4 to 5 times windspeed.

With something that is self-starting and not CVT or variable pitch it will be less than 70%. So possible maximum in this case about 3 times windspeed.


On the matter of the propeller. It drags air in from the surrounding and accelerates it. Have a close look at the table back in #159 (reproduced here) and try to work out what is actually happening. Forget about anything to do with sails. The sail analogy is like thinking about the propellers on an aeroplane being sails. The L/D of a high aspect prop blade can be 100 but this is only one of the factors that influence the overall efficiency of a propeller.

Looking at the last column in the attached table at 10m/s. The air coming into the propeller is 5m/s. It is accelerated to 8m/s behind the prop. From an observer on the ground the air in front of the prop is moving at 5m/s. Behind the prop it is moving at 2m/s.

As an example take 12sq.m of air then the energy in the wind at 5m/s is:
E5 = 1/2 * 1.2 * 12 * 5^2 = 180W
In the slipstream of a 12sq.m propeller that slows the wind speed to 2m/s the energy is:
E2 = 1/2 *1.2 * 12 * 2^2 = 29.8W

Hence the energy extracted from the air is 150.2W.

The propeller can never reverse the airflow over the ground. This means the maximum energy that can be taken from 12sq.m with a wind speed of 5m/s is 180W. If you try to gear the propeller to make the flow negative the cart will go into the wind with propeller becoming a turbine.

Think of the propeller as a propeller not as sails. Propellers pull air in and accelerate it out. Look at the diagram in post #161. You do not think of a boat propeller as a sail or dagger board. It is a propeller. The situation with the propeller on the cart can be analysed in exactly the same way you work out what a propeller on an aeroplane or boat does.

The prop on the cart was designed using JavaProp. This is free software. You can use it to determine what efficiency it possible. Make sure you have the right Re# for the speed and chord of the blades as this has a large influence on efficiency.

Rick

 

Thanks for providing those numbers.

Did the ThinAirDesigns team use JavaProp? Are your numbers based on their design parameters or do you think you picked similar ones?

How do you define efficiency for a prop? Is it energy-based or velocity-ratio-based or something else?

A prop and a sail do the same thing: you say toh-may-to; I say toh-mah-to; please bear with my wooden bridge syndrome (if you ask a stone mason to build you a bridge, you get a stone bridge; ask a welder and you get a steel bridge; ask a carpenter and you get ...) and I will appreciate your facility with prop equations and terminology. I do agree the propeller equations are the way to go for these numbers (thanks again), especially given the varying radii and velocities along the blade, but there is a direct analogy between sails and propellers because they both only accelerate - i.e. change the velocity vector of - the air they encounter. Remove the hull from the vid (http://i47.tinypic.com/m7xf09.gif) and you could be looking down the radius of the cart's prop.

Nothing can literally pull or drag air, a fluid with no tensile strength. By accelerating air a sail or prop can create a low pressure into which higher pressure air from the far field flows (differentially diffuses, actually, if I remember my BS&L), but I understand what you mean by pull or drag.


Thanks again,

 

Something just struck me

Maybe somebody already pointed this out, but ...

I think there are some who believe a cart that could go DUWFTTW (per Bauer, 1969 and Barkla, 1971) but do not believe DDWFTTW.

I just realized that the DDWFTTW cart is also a DUGFTTG (Direct Up-Ground Faster Than The Ground) device. I.e. it goes faster through the wind against the ground than the ground is going against the wind. DUWFTTW is also DDGFTTG.

So if someone knows DUWFTTW is possible, they can't deny DUGFTTG because the mechanics are the same in the differential between two media. From DUGFTTG just swap the meaning of the media (or flip the Black Pearl and your viewing camera over) and you have DDWFTTW.

 

The Thinair guys used JavaProp for their prop.

The little table is an approximation to keep it simple. It is not based on a specific propeller. I have given analysis for a DDWFTTW propeller and turbine for a boat on another thread that has windmill in the title.

Mechanical efficiency is the same for a propeller as any other mechanical device. It is the ratio of power out over power in.


A sail below stall and a propeller both make use of lifting foils. Beyond that they vary significantly. The air velocity at the tip of the blade can be 10 times or more than the farfield air velocity relative to the prop. If you are going to design a propeller you must take into account the wide variation in velocity as the radius changes. Sails only have a slight variation in velocity over their span. Sails are closer to a wing than the twisted foil of a propeller.

I am certain you would never try to explain the operation of a boat propeller by making a direct analogy to the rudder. That has about the same significance as comparing an air propeller with a sail.

 

You are wrong there. If you go back through this thread and the one linked here:
http://www.boatdesign.net/forums/boat-design/wind-powered-sail-less-boat-24669.html
you will see many people struggle with inertial frames of reference.

There are videos on YouTube showing DDWFTTW carts climbing up a treadmill. It seems a great number of people argue this is completely different to the cart operating in wind over ground.

 

Firstly the wheels are always engaged to the prop. It is direct mechanical connection. The propeller turns as soon as the wheels turn. The gearing is such that the wheels overcome the torque on the propeller when it is at zero speed.

Look at the table I posted a few posts back and at #159.

Consider the last column where the cart is at 10m/s.

Lets say the drag on the wheels to turn the prop is 15N. So the power going into the shaft of the propeller if the power transfer is 95% efficient is 15 * 10 * .95 = 142.5W.

The propeller has an efficiency of 80% so its power output is 114W. With apparent airspeed of 5m/s it will be producing a force of 114 / 5 = 22.8N.

So the net force to cater for windage and rolling resistance is 22.8 -15 = 7.8N. If the resistance is lower than this it will keep accelerating however it will eventually reach a force and power balance.

Once you get past the point of accepting that it works it is very simple physics

Maybe think of an electrically powered propeller driven cart that is free rolling. Lets say it does 5m/s in no wind. Now if the wind is blowing at 5m/s it is obvious that it will be doing 10m/s without any need to increase the rpm of the propeller.

The fundamental understanding is that:
Power = Force X Velocity

The incoming power comes from a drag force applied at the speed of the cart where the wheels contact the ground while the outgoing power produces the thrust force at the prop working on the air being applied at the apparent wind speed. In steady state there is force and power ballance. As long as there is wind and the gearing is the right way around the cart will be able to go faster than the wind downwind.

 

Sadly, yes. I should have said "So if someone competent knows DUWFTTW is possible, they can't logically deny DUGFTTG ..."

 

The wheels' ground-relative Velocity is high (by comparison to the prop).

The prop's air-relative Velocity is low (by comparison to the wheels).

As Rick says, Power = Force * Velocity.

Trying to distill the answer to your question down to a concise form:

The Power In, at the higher wheel-ground relative Velocity uses a smaller Force, pushing the cart backward, to balance the Power Out at the lower prop-air relative Velocity, which has a larger Force pushing the cart forward.

Even when divided by 85% to make up for losses the smaller force at the wheels can still be smaller than the larger force at the prop, so the cart accelerates. When the forces balance, the cart is at steady state.

The first part of your question ("where the energy comes from") is answered by the "why" at the beginning of my long post above.

 

Actually I would explain them all exactly the same way: F = d(mv)/dt.

The rest (twist, F = INTEGRAL_r( d(m(r)v(r)/(dt.dr) dr ), etc) is an implementation detail.

A little history: my Dad worked for a division of a company that made steam turbines. He would run performance and acceptance tests of installed power plant turbines to an accuracy of a quarter percent in some cases. In a steam turbine, most of the work comes from the HP section (high pressure) where

- the rotor is large
- the blades are short
- dV/dr is zero
- there is little if any twist in the blades

In the LP section, everything is the opposite but it only adds the last few tenths of a percent to the work extracted to get the customer to pay a premium to buy your turbine instead of the next guy's.

So like I said, wooden bridge syndrome: I know it's there but if I don't have to deal with twist, as in a qualitative discussion, then I won't: d(mv)/dt does the job. Even when I get semi-quantitative I'm still only interested in an average L/D for the prop (even though I know it's probably bogus and there is no such thing in a rigorous sense) because ballpark is all I need. The 2.85 VMGDW/Vt suggests a system L/D of somewhere between 2.7 and 4 (I already have +/-25% to start with and you want me to worry about dV/dr?), which puts the prop average L/D at perhaps double that. From what you tell me about the L/D at the tip that is reasonable.

I'm not saying it's not important, in fact your analyses of the flow field are both fascinating and important. And I will get there, I just don't need it yet.

If there is one thing that all these discussions have shown it is that no one explanation works with every individual.

Thanks again.

 

You guys are on XKCD too? When do you have time to build this thing? When do you have time for a job?

 

Until you run out of string.

See also:

http://www.youtube.com/watch?v=CWmeTWPN5o4