Crosswind problem (pgs. 34-35, Thinking Physics, 3rd edition)

[erobz]

I still can't figure out what all the confusion is surrounding $\beta$, perhaps I'm miss interpreting what it "looks like", but I think all of that not an issue. If the outflow is exiting at a constant angle $\beta$ (imagine there is a pipe sticking out of the control volume at angle $\beta$ relative to vertical in the diagram, it checks out; No bounds on velocity for some angle in this idealism. Any real body faces significant opposition to that proposition via drag. In theoretically idealistic cases I have to cast out my prior doubt of this.

 

[Gleb1964]

One question, may be I missed before. Do you have control volume stationary at ground frame or it is moving with the carriage? If it is moving, the pipe sticking out moving volume would not blow the flow in direction of the pipe pointing.

 

[A.T.]

I still can't figure out what all the confusion is surrounding $\beta$,

It's not clear why $\beta$ is defined between two velocity vectors taken from two different frames of reference. It's not wrong, just odd.

If the outflow is exiting at a constant angle $\beta$ (imagine there is a pipe sticking out of the control volume at angle $\beta$ relative to vertical in the diagram,

You didn't specify the frame in which you measure the outflow direction. If the thing in your diagram is a pipe, then the outflow in the pipe-frame will indeed be $\beta$ relative to vertical. But in the ground-frame the outflow in will not will be $\beta$ relative to vertical (unless v = 0 or $\beta$ = 45°).

More importantly, the inflow side of you pipe is not aligned with the inflow in the pipe-frame (as it should be for efficient operation), instead it is aligned with the inflow in the ground-frame (which seems like an arbitrary choice that makes no sense).

 

[erobz]

One question, may be I missed before. Do you have control volume stationary at ground frame or it is moving with the carriage? If it is moving, the pipe sticking out moving volume would not blow the flow in direction of the pipe pointing.

I don't have it blowing the flow in the direction of the pipe in the inertial frame. This should be apparent from the mathematics surrounding the momentum transfers. The Reynolds Transport Theorem is valid in the inertial frame only. So every momentum flow is converted to the inertial frame in the equations.

For instance, I don't say the momentum outflow in the $\boldsymbol{u_x}$ is simply:
$$ - \dot m \sqrt{ ( w+y)^2 + v_x^2 }\sin \beta $$

I say it is in the inertial frame:

$$ \dot m \left( v_x - \sqrt{ ( w+y)^2 + v_x^2 }\sin \beta \right) $$

I'm not drawing the "true outflow flow momentum vector" in the inertial frame...It's just a diagram.

 

[A.T.]

With the optimal vane orientation (bisecting apparent wind angle in the moving frame), the deflected wind is moving in opposite direction to the boat speed (parallel to the boat speed, but in opposite direction). The air leaving the control volume in direction opposite to the boat speed.
Assuming that the vane angle is adjusted to the optimal at any speed, that means that at any speed the air is deflected in direction opposite to the boat moving direction. And that is valid both in the ground frame and in the boat frame.

Are you referring to the tacking upwind case here? Because compared to that, in the case of tacking downwind with an VMG > windspeed, the roles of water and air are swapped. So what you wrote about air deflection by the sail here, applies to the water deflection by the keel there.

 

[erobz]

More importantly, the inflow side of you pipe is not aligned with the inflow in the pipe-frame (as it should be for efficient operation), instead it is aligned with the inflow in the ground-frame (which seems like an arbitrary choice that makes no sense).

This is irrelevant, infinite velocity is still obtained even with some theoretical inefficiency you claim exists. Its like you are trying to talk me out of it now! I was of the opinion that you were blowing smoke before, and now that I derive the EOM's it see that its not quackery. Take your win, otherwise write your own equations and show me, because I'm tired of playing these games.

 

[A.T.]

This is irrelevant, infinite velocity is still obtained even with some theoretical inefficiency you claim exists.

In this upwind case you considered here, the air will enter the inlet of the pipe for any v, even if you keep it vertical for all v. So you can get away with this.

But for the downwind case (which was the actual starting point of the debate) the range of relative inflow direction if far greater (for v = 0 to inf). So such a fixed pipe model is problematic, because it might have the inlet pointing the wrong way (more than 90° off the relative inflow). This could create a false-limit, while an actual sail can always be oriented in the optimal direction.

 

[erobz]

In this upwind case you considered here, the air will enter the inlet of the pipe for any v, even if you keep it vertical for all v. So you can get away with this.

But for the downwind case (which was the actual starting point of the debate) the range of relative inflow direction if far greater (for v = 0 to inf). So such a fixed pipe model is problematic, because it might have the inlet pointing the wrong way (more than 90° off the relative inflow). This could create a false-limit, while an actual sail can always be oriented in the optimal direction.

We aren't discussing a sail at this point, we are discussing a "black box" control volume, and even without examining the actual mechanics of the sail, it still works. There is nothing to argue about anymore IMO, unless you think the mathematics is truly abusive.

In reality all of this infinite velocity talk is a ridiculous notion anyhow, even with an always perfectly oriented sail! The question was "can one theoretically sail faster than the wind". I would say "in a world without friction or drag it's not theoretically prohibited nor limited". Calculating the real limit in the real world is not something I'm trying to tackle, that's for certain.

 

[A.T.]

In reality all of this infinite velocity talk is a ridiculous notion anyhow, even with an always perfectly oriented sail! The question was "can one theoretically sail faster than the wind". I would say "in a world without friction or drag it's not theoretically prohibited nor limited".

You are conflating different things here:
- moving at infinite multiples of true wind-speed requires drag to be zero
- moving at more than 1 multiple of true wind-speed, including downwind or upwind component > true wind, works fine with real-world drag

The relationship of velocity limits to drag is described here:
https://www.physicsforums.com/threa...king-physics-3rd-edition.1048870/post-6861400

 

[erobz]

You are conflating different things here:
- moving at infinite multiples of true wind-speed requires drag to be zero
- moving at more than 1 multiple of true wind-speed, including downwind or upwind component > true wind, works fine with real-world drag

The relationship of velocity limits to drag is described here:
https://www.physicsforums.com/threa...king-physics-3rd-edition.1048870/post-6861400

I'm not conflating anything. I made no such claim in that statement about the possibility of moving at more than 1 multiple of wind speed in the real world. I said I'm not trying to calculate the limit in the real world. i.e. the model (without drag - nor general relativity) is not bounded, in reality (with drag - and relativity) its certainly is bounded nowhere "near" infinity . Thats the only claim I confidently make without updating the model.

 

[A.T.]

I said I'm not trying to calculate the limit in the real world. i.e. the model (without drag - nor general relativity) is not bounded, in reality (with drag - and relativity) its certainly is bounded nowhere "near" infinity .

Fair enough. But note that relativity puts a limit on the speed terms of distance / time, not on the ratio of boat-speed to wind-speed. That ratio can go to infinity for a fixed boat-speed, as the wind-speed needed to achieve it goes to zero, because efficiency goes to one.

 

[Gleb1964]

Are you referring to the tacking upwind case here? Because compared to that, in the case of tacking downwind with an VMG > windspeed, the roles of water and air are swapped. So what you wrote about air deflection by the sail here, applies to the water deflection by the keel there.

First I was looking the upwind case only. But I have checked the downwind case as well and my conclusion is still keep holding.
In the case of the optimal vane orientation (meant bisecting the direction of motion and the apparent wind in the frame of carriage) the out flow from control volume pointed ether in direction of motion (downwind) or opposite to motion (upwind). That is valid in both the ground frame and the carriage frame.
For me that is interesting finding, I haven't been aware about it before.

 

[A.T.]

First I was looking the upwind case only. But I have checked the downwind case as well and my conclusion is still keep holding.
In the case of the optimal vane orientation (meant bisecting the direction of motion and the apparent wind in the frame of carriage) the out flow from control volume pointed ether in direction of motion (downwind) or opposite to motion (upwind). That is valid in both the ground frame and the carriage frame.
For me that is interesting finding, I haven't been aware about it before.

I think I now understand what you mean by the optimal vane orientation. If we just take a flat sail, that bisects the angle between negated boat velocity and apparent wind, and assume a perfectly elastic collision with it, then after the collision the flow will be along the negated boat velocity. Yes that's always true.