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

[A.T.]

No it can not accelerate. The mass flowrate entering the control volume always goes to $0$ as $v_x \to w$.

You think there is no relative wind at the boat when Vx = W? Think again, what about Vy?

 

[erobz]

You think there is no relative wind at the boat when Vx = W? Think again, what about Vy?

What $v_y$, the wind is coming in horizontal (purely in the $x$ direction). Are you trying to say that $w$ now has a vertical component $w_y$ as well?

 

[A.T.]

What $v_y$, the wind is coming in horizontal (purely in the $x$ direction). Are you trying to say that $w$ now has a vertical component $w_y$ as well?

You have a moving control volume. The mass flowrate entering the control volume depends on the motion of the airmass relative to the control volume, not relative to the ground.

 

[erobz]

You have a moving control volume. The mass flowrate entering the control volume depends on the motion of the airmass relative to the control volume, not relative to the ground.

The amount of mass entering and exiting the control volume per unit time is frame independent.

 

[erobz]

If the cart has velocity $v_x = w$, none of the flow mass can enter the control volume. it is chasing it at the same speed! You are telling me that I can run at the same speed as a car in front of me and end up inside it. This is just absurd.

 

[A.T.]

The amount of mass entering and exiting the control volume per unit time is frame independent.

Yes, but you compute it based on the velocity of the air relative to the volume, not relative to some arbitrary object like the ground, as you did.

 

[A.T.]

If the cart has velocity $v_x = w$, none of the flow mass can enter the control volume.

See post #36.

 

[erobz]

Yes, but you compute it based on the velocity of the air relative to the volume, not relative to some arbitrary object like the ground, as you did.

What are you talking about? They are the same, what part of frame independent are you not understanding?

The mass flow rate entering the control volume is $\dot m = \rho A ( w - v_x )$. Period.

 

[A.T.]

The mass flow rate entering the control volume is $\dot m = \rho A ( w - v )$.

It's a 2D situation (your first model in post #20), so W and V are 2D vectors.
Is the (W - V) vector zero when Vx = Wx?

 

[erobz]

If I am running with speed $w$ behind a car at speed $w$ holding a bag of rocks, I can't simply smash out the rear window by simply letting a rock go...I must throw the rock with a speed $v$ relative to me. The only way the rock can get into the car is if it has speed $w+v$ relative to the ground.

 

[erobz]

It's a 2D situation (your first model in post #20), so W and V are 2D vectors.
Is the (W - V) vector zero when Vx = Wx?

$w$ is a vector in the $x$ direction. The wind is blowing in the $x$ direction...only in the $x$ direction. Just like it is in #29. You pick a direction of the wind then you try to move at some angle $\theta$ relative to it and examine the "Momentum Equation" in each direction (component basis)

 

[A.T.]

$w$ is a vector in the $x$ direction. The wind is blowing in the $x$ direction...only in the $x$ direction. Just like it is in #29.

You didn't answer my question:
Is the vector (W - V) zero when Vx = Wx?

 

[erobz]

Is the vector (W - V) zero when Vx = Wx?

Firstly $w_x$ makes it seem like $w$ has multiple directions! It doesn't its just plain $w$ in post#20

But Yes. By definition the $w - v_x = 0$, when $w = v_x$.

 

[A.T.]

But Yes. by definition the $w - v_x = 0$, when $w = v_x$.

That wasn't my question. I'm asking about the 2D vector (W - V) with W = (w, 0) and V = (v_x, v_y).

 

[erobz]

Well, its not clear what you are asking. Are you asking if $v_y$ is 0 when $v_x = w$?

The answer to that is no. $v_y = v_x \tan \theta = w \tan \theta$

 

[A.T.]

The mass flow rate entering the control volume is $\dot m = \rho A ( w - v_x )$

You have edited your post and made it wrong. Why would the mass flow rate entering the control volume depend on the motion of the control volume along X only, if the control volume moves along Y as well.

 

[erobz]

You have edited your post and made it wrong. Why would the mass flow rate entering the control volume depend on the motion of the control volume along X only, if the control volume moves along Y as well.

because that is how its getting in! It can only enter from the direction of the wind.

 

[A.T.]

because that is how its getting in! It can only enter from the direction of the wind.

Think again. If there was no wind at all, but the control volume was moving along X and Y, would there be air entering the control volume? From which direction?

 

[erobz]

Think again. If there was no wind at all, but the control volume was moving along X and Y, would there be air entering the control volume? From which direction?

Thats counterproductive to your argument. Scooping mass from in front of $y$direction would impart momentum on the cart opposite its direction of motion! You are trying to use the circular reasoning that the wind can drive the cart into the wind and harness more forward momentum.

 

[A.T.]

You are trying to use the circular reasoning that the wind can drive the cart into the wind and harness more forward momentum.

Nope. The no wind case is just there to demonstrate that your approach for mass flow rate into a moving control volume makes no sense.

So again: If there was no wind at all, but the control volume was moving along X and Y, would there be air entering the control volume? From which direction?

 

[erobz]

Nope. The no wind case is just there to demonstrate that your approach for mass flow rate into a moving control volume makes no sense.

So again: If there was no wind at all, but the control volume was moving along X and Y, would there be air entering the control volume? From which direction?

This is where we are at... If there is no wind, and you give a cart with a sail an initial push it will accelerate away from you!!! Forget about it. You win.

 

[A.T.]

If there is no wind, and you give a cart with a sail an initial push it will accelerate away from you!!!

Nope. Look at the vectors. If true_wind = 0 then apparent_wind = -boat_velocity. So even a infinite lift/drag ratio won't produce a forward_force. The sail_force can be at most 90° from the apparent_wind.

 

[A.T.]

If I am running with speed $w$ behind a car at speed $w$ holding a bag of rocks, I can't simply smash out the rear window by simply letting a rock go..

If the car additionally has a velocity component perpendicular to w (like the boat in your model below) then you can smash a side window by just letting the rocks go.

Or alternatively the car passengers can use a surface to deflect your rocks, and get some propulsion from that (for this the sail in your model needs to be turned by 90° clockwise)

 

[A.T.]

The amount of mass entering and exiting the control volume per unit time is frame independent.

It is frame independent exactly because, no matter which frame you use for analysis, the mass flow rate entering the control volume is computed based on the velocity of the air relative to the control volume. Not relative to your picked frame, like the ground.

You must know this, because you do it for the x-component:

The mass flow rate entering the control volume is $\dot m = \rho A ( w - v_x )$.

The $( w - v_x )$ term in your formula is the x-component: of the flow relative to the control volume. Now you just have to do the same for y-component of the relative flow, which is $( 0 - v_y )$.

So when $v_x = w$ the relative flow is $\begin{pmatrix} 0 \\ -v_y \end{pmatrix}$, therefore you have to rotate the vane in your sketch by 90° clockwise to create propulsion in this situation.

 

[erobz]

It is frame independent exactly because, no matter which frame you use for analysis, the mass flow rate entering the control volume is computed based on the velocity of the air relative to the control volume. Not relative to your picked frame, like the ground.

You must know this, because you do it for the x-component:

The $( w - v_x )$ term in your formula is the x-component: of the flow relative to the control volume. Now you just have to do the same for y-component of the relative flow, which is $( 0 - v_y )$.

So when $v_x = w$ the relative flow is $\begin{pmatrix} 0 \\ -v_y \end{pmatrix}$, therefore you have to rotate the vane in your sketch by 90° clockwise to create propulsion in this situation.

You cannot propel something initially and expect a chain reaction to occur. Momentum entering the vane as $-v_y$ would counter the momentum it was gaining from changing the direction of $w$. The very momentum that is driving it.

In still air you cannot push something with a sail and expect it to accelerate away from you. If I turned the vane as you wish, having it scoop air, and gave it a shove, it would scoop mass from the $y$ direction and sent it to $-x$ direction, thus providing an impulse, which would scoop more mass, and give another impulse, ad infinitum. It just doesn't work like that.

Another thing...my analysis has no mass to "scoop" in the $y$ direction. It is effectively a fluid jet impinging on a sail in the vacuum of space. The analysis provides the highest possible rate of momentum transfer imaginable for a fluid transferring momentum to a sail.

 

[A.T.]

You cannot propel something initially and expect a chain reaction to occur. Momentum entering the vane as $-v_y$ would counter the momentum it was gaining from changing the direction of $w$. The very momentum that is driving it.

You requested mathematical rigor, but when the math contradicts your intuition, you are making excuses why you won't accept it. I'm just asking you to be consistent for both axes x & y.

In still air you cannot push something with a sail and expect it to accelerate away from you.

Correct, but we are not talking about no wind. We are talking about your own scenario with wind $w$ along the positive x-direction. Post #57 explains why it doesn't work without wind relative to the ground.

If I turned the vane as you wish, having it scoop air, and gave it a shove, it would scoop mass from the $y$ direction and sent it to $-x$ direction, thus providing an impulse, which would scoop more mass, and give another impulse, ad infinitum. It just doesn't work like that.

That is exactly how it works. That's why there is no fixed limit on the windspeed multiple a sailcraft can achieve, and this applies to the downwind component as well.

But there are practical limits. As you accelerate, the relative flow comes more and more from the front, so you need increasing high lift/drag ratios to still produce a forward force component (see post #57). The achievable lift/drag ratios for the interactions with the air and the surface are limiting the speed of a sailcraft.

 

[A.T.]

...It is effectively a fluid jet impinging on a sail in the vacuum of space...

Fortunately the real wind is a large moving airmass that extends in all directions, so it doesn't have the limitations of such a thin jet model. Just use a very wide jet instead.

 

[erobz]

That is exactly how it works. That's why there is no fixed limit on the windspeed multiple a sailcraft can achieve, and this applies to the downwind component as well.

Thats exactly how it works my foot. You cannot give a sail craft an initial shove in still air and have it accelerate away. That is absolutely ludicrous!

 

[A.T.]

,,,in still air ,,,

Why do you misrepresent what I am saying. I stated many times, that it doesn't work without wind relative to the ground and explained why:

If true_wind = 0 then apparent_wind = -boat_velocity. So even a infinite lift/drag ratio won't produce a forward_force. The sail_force can be at most 90° from the apparent_wind.

 

[erobz]

Why do you misrepresent what I am saying.

That is exactly how it works.

in reply to;

If I turned the vane as you wish, having it scoop air, and gave it a shove, it would scoop mass from the y direction and sent it to −x direction, thus providing an impulse, which would scoop more mass, and give another impulse, ad infinitum. It just doesn't work like that.

 

[A.T.]

Yes, that is exactly how it works, if there is wind relative to the ground.

 

[erobz]

You say this is so trivial early on, but then give no foundational proof. Your idea of mathematical rigor is " just look at the (arbitrarily drawn) vectors on the diagram"!

That doesn't cut it. Color me unconvinced. Good day to you.

 

[A.T.]

You say this is so trivial early on, but then give no foundational proof. Your idea of mathematical rigor is " just look at the arbitrarily drawn vectors on the diagram"!

What about the vectors is wrong? The only assumption there is a lift/drag ratio of about 5, which is very conservative. Airfoils reach 20 and more.

 

[erobz]

If I gave the boat a shove in still air in the direction of the "true wind" in your diagram, that would produce relative flow over the air foil in the direction of the "apparent wind", that in turn would provide a force that accelerated the boat in the direction of the lift force (increasing its velocity), which would increase the sail force by way of increasing the apparent wind! $\circlearrowright$. So in that case (absence of a drag force on the air foil - a function of the apparent flow velocity) the craft would accelerate away from me indefinitely. Now...add in the drag force. A drag force ( since you are talking about drag/lift ratio ) would grow (from 0) such that its component in the direction of the boats velocity matched the component of the lift force, thus accelerating away from me for a while, approaching moving away from me at a constant velocity.

 

[A.T.]

If I gave the boat a shove in still air in the direction of the "true wind" in your diagram,

In "still air" the "true wind" vector is zero. It doesn't have a direction. Can you draw vector diagram of what you mean?

that would produce relative flow over the air foil in the direction of the "apparent wind",

If true_wind = 0 then apparent_wind = -boat_velocity. Since sail_force cannot be more than 90° from the apparent_wind, it cannot have a positive component in the direction of boat_velocity.

So if true_wind = 0 there cannot be a propulsive forward_force from the sail, no matter how much you push initially. There is nothing in the diagram that suggests it would accelerate without true_wind relative to the ground.