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

[erobz]

I would agree even with unrealistic "jet wind" concept, asking consider infinity long sail instead.

I can begin with that. So you want a single jet, infinitely long planar sail oriented relative to the cart such that the force is maximized on the cart in the direction of motion?

 

[A.T.]

How do you want me to orient\select geometry for the vane such that the craft will outrun the jet(s) in the direction of the jet for some track angle $\theta$?

You can orient the sail like in the image below:

w : wind relative to the ground
w' : wind relative to the boat
v: boat velocity relative to the ground

2) The direction of the impinging fluid jet(s) is fixed to the right $\rightarrow^+$

This sounds wrong. The direction of fluid inflow onto the sail (w') is not fixed. It changes with v:

w' = w - v

See blue arrows above.

4) The frame which we consider the forces is the inertial frame.

The force by the air on the sail is frame invariant.

 

[erobz]

You can orient the sail like in the image below:

https://www.physicsforums.com/attachments/322853
w : wind relative to the ground
w' : wind relative to the boat
v: boat velocity relative to the groundThis sounds wrong. The direction of fluid inflow onto the sail (w') is not fixed. It changes with v:

w' = w - v

See blue arrows above.The force by the air on the sail is frame invariant.

So it looks like we can't agree...can't say I'm surprised. The wind caries momentum into the sail,and is carried out in the direction the sail deflects it. This relative wind you keep trying to invoke for the analysis doesn't exist in the inertial frame. You are making up momentum that doesn't exist in the inertial frame. I'm doing the analysis in the inertial frame (a frame fixed to the ground)

 

[A.T.]

So it looks like we can't agree...can't say I'm surprised.

Fortunately real world evidence agrees with me.

The wind caries momentum into the sail,and is carried out in the direction the sail deflects it. This relative wind you keep trying to invoke for the analysis doesn't exist in the inertial frame. You are making up momentum that doesn't exist in the inertial frame.

1) We have empirical data on lift/drag ratios, which are all based on the relative wind. Those are way more realistic than the simplifying assumptions about the air deflection that you have to make.

2) The way you yourself keep drawing the flow (as if the vane was fixed) can only refer to the relative wind, The deflection in the ground frame, where both: the air and the vane are moving looks different.

I'm doing the analysis in the inertial frame (a frame fixed to the ground)

You can do that, but keep in mind: The mass flow rate into your moving control volume depends on the wind relative to the control volume. No matter how much you try to ignore it, the varaying relative wind still comes in.

 

[erobz]

2) The way you yourself keep drawing the flow (as if the vane was fixed) can only refer to the relative wind, The deflection in the ground frame, where both: the air and the vane are moving looks different.

How I draw it, or how it looks is not important. What is important is properly accounting for the change in momentum of the flow w.r.t the stationary inertial frame.

You can do that, but keep in mind: The mass flow rate into your moving control volume depends on the wind relative to the control volume. No matter how much you try to ignore it, the varaying relative wind still comes in.

Yes. Thank you.

 

[Gleb1964]

.. This relative wind you keep trying to invoke for the analysis doesn't exist in the inertial frame. You are making up momentum that doesn't exist in the inertial frame. I'm doing the analysis in the inertial frame (a frame fixed to the ground)

Not quite agree. You can consider inertial frame with the boat at any speed. Been sailing a bit, the apparent (relative) wind is a very real thing, especially when you are sitting in the boat.

 

[erobz]

Not quite agree. You can consider inertial frame with the boat at any speed. Been sailing a bit, the apparent (relative) wind is a very real thing, especially when you are sitting in the boat.

You are in a non-inertial frame when you are on the boat accelerating. Let's just talk about how we do Newtonian Mechanics. If you see something is off with the physics I will present, say something. That is what should be the main focus here.

 

[Gleb1964]

At any moment of acceleration it is possible to induce an instant inertial frame to the boat. That inertial frame is good for understand the correct change of the air momentum.
I would even not complicate the task, but just assume the boat moving with constant speed. Make a solve with the constant boat speed. It is always possible to assume that boat has some resistance, keeping it speed constant. Forgot about how boat is getting to the speed. Make it simple.

 

[erobz]

This is the diagram for your suggested infinite planar sail. My goal is to optimize the angle of the sail w.r.t. the cart $\beta$ such that the component of the force $F$ ( shown in purple ) in the direction of motion is maximized. I.e. I'm hoping to maximize $F \cos \gamma$. Seeing the equations I'm developing I don't suspect to have a prayer of getting there in general, but I'll see what I can come up with.

If you don't like it, shoot it down before I go further.

 

[erobz]

At any moment of acceleration it is possible to induce an instant inertial frame to the boat. That inertial frame is good for understand the correct change of the air momentum.
I would even not complicate the task, but just assume the boat moving with constant speed. Make a solve with the constant boat speed. It is always possible to assume that boat has some resistance, keeping it speed constant. Forgot about how boat is getting to the speed. Make it simple.

Unfortunately it can't work like that. I must form some first order differential equation and look at the limit of said equation as $\frac{dv}{dt} \to 0$.

 

[Gleb1964]

The orientation of the sail is not correct at #149.

 

[erobz]

The orientation of the sail is not correct at #149.

The sail has no orientation in that diagram. Its orientation relative to the cart is characterized by the parameter $\beta$ which I hope to first optimize as a function of $v_x$. I.e the expectation is that there is a little man on the boat that is turning the sail to maximize the component of the applied force $F$ in the direction of motion over the entire duration of the analysis.

 

[Gleb1964]

The optimal sail would be bisecting between apparent wind direction and boat direction, the force on the sail is perpendicular to the sail plane. That mean the optimal force direction is known for every boat speed.

 

[erobz]

The optimal sail would be bisecting between apparent wind direction and boat direction, the force on the sail is perpendicular to the sail plane. That mean the optimal force direction is known for every boat speed.

If the sail were fixed at the angle $\beta$(making it a constant), that you say it is ( whatever it is), with the force $F$ perpendicular to the sail that implies that $\frac{F_y}{F_x}$ or $\frac{F_x}{F_y}$ is a constant w.r.t. to velocity.

The problem with that is:

$$ F_x = \dot m \left( w \sin ( \beta + \theta ) + v_x - w \right) $$

$$F_y = \dot m \left( v_y - w \cos ( \beta + \theta ) \right) $$

If you sub $v_y = v_x \tan \theta$ it should be apparent by inspection that isn't the case as:

$$ \frac{F_x}{F_y} = \frac{w \sin ( \beta + \theta ) + v_x - w}{ v_x \tan \theta - w \cos ( \beta + \theta ) } = f(v_x)$$

There is no one that wants it to be a constant more than I do right now, as the solution to that problem is straight forward...but its not. @A.T. already made a point of this in an earlier post.

 

[Gleb1964]

Here is illustration of sail boat diagram.
The boat is descending downwind faster than wind, but you can see, that the sail line is moving slower than wind. That makes possible for wind to make a force on the sail, which has a component (not illustrated) keeping or accelerating the boat speed.

Do you have questions to this diagram?

 

[erobz]

Here is illustration of sail boat diagram.
The boat is descending downwind faster than wind, but you can see, that the sail line is moving slower than wind. That makes possible for wind to make a force on the sail, which has a component (not illustrated) keeping or accelerating the boat speed.

Do you have questions to this diagram?

View attachment 322863

Its not analysis. Quit patronizing me with "vectors" that supposedly reduce everything I just went over to nothing. I am applying Newtons Laws of motion, as instructed to do so by the Fluid Mechanics textbook sitting in front me. I don't accept..."but look at randomly drawn vectors", so stop wasting your time. If you are a physicist, you should be much more inclined to hear what Newton has to say about, or figure out how what I'm saying is incorrect in an agreed upon framework of Classical Physics i.e. Newtonian Mechanics and tell me about it.

 

[Gleb1964]

Your flow analysis suppose to come to the same. What way do you defining the outgoing flow?

 

[erobz]

Your flow analysis suppose to come to the same. What way do you defining the outgoing flow?

Look at the diagram. It shows everything I have planned.

This equation represents Newtons Second in Fluid Mechanics under the assumption of constant cross-sectional properties:

$$ \sum F = \frac{d}{dt} \int_{cv} \rho v ~d V\llap{-} + \sum_{cs} \dot m_o v_o - \sum_{cs} \dot m_i v_i $$

 

[A.T.]

Quit patronizing me with "vectors" that supposedly reduce everything I just went over to nothing.

Vectors indeed simplify the analysis substantially.

I am applying Newtons Laws of motion, as instructed to do so by the Fluid Mechanics textbook sitting in front me.

Just because you have a hammer, doesn't mean every problem is a nail. We don't have to go into the complex details of fluid dynamics, because we have empirical data on achievable lift/drag ratios, which leads to the much simpler vector analysis.

I don't accept..."but look at randomly drawn vectors", so stop wasting your time.

What have vectors done to you, that you hate them so much?

If you are a physicist, you should be much more inclined to hear what Newton has to say about,

Here is a paper by a physics professor in a physics journal, analyzing the same thing and using vectors of course:

High-speed sailing, Wolfgang Püschl 2018 Eur. J. Phys. 39 044002
https://iopscience.iop.org/article/10.1088/1361-6404/aab982

 

[A.T.]

The sail has no orientation in that diagram. Its orientation relative to the cart is characterized by the parameter $\beta$ which I hope to first optimize as a function of $v_x$. I.e the expectation is that there is a little man on the boat that is turning the sail to maximize the component of the applied force $F$ in the direction of motion over the entire duration of the analysis.

OK, but note that:
- the example $\beta$ you have drawn is wrong for maximizing $v$. See post #142.
- you have drawn the relative outgoing flow again, because it is parallel to the sail. This is not how the outgoing flow looks like in the ground frame where the sail is moving.

 

[A.T.]

What way do you defining the outgoing flow?

Look at the diagram. It shows everything I have planned.

This equation represents Newtons Second in Fluid Mechanics under the assumption of constant cross-sectional properties:

$$ \sum F = \frac{d}{dt} \int_{cv} \rho v ~d V\llap{-} + \sum_{cs} \dot m_o v_o - \sum_{cs} \dot m_i v_i $$

But your diagram doesn't have $\dot m_o v_o$ in it. The outgoing flow is labeled $\dot m w$, which is of course wrong, because the flowspeed $w$ (in the ground frame) will change after the interaction with the sail.

So how do you plan to determine that $v_o$ for all possible speeds of the boat?

 

[erobz]

But your diagram doesn't have $\dot m_o v_o$ in it. The outgoing flow is labeled $\dot m w$, which is of course wrong, because the flowspeed $w$ (in the ground frame) will change after the interaction with the sail.

The problem is that you don't understand that I'm adding vectors ( well, technically scalars that are associated with the component vectors)! I wrote out the forces acting on the sail in post #154. For example: The momentum of the incoming flow for $F_x$ has three terms; the first two combined describe the momentum of the outflow (notice the dependency of the outflow momentum on component of cart velocity w.r.t the ground frame). The last term describes the momentum inflow relative to inertial frame.

So how do you plan to determine that $v_o$ for all possible speeds of the boat?

The subscripts $i,o$ refer to inflow, outflow respectively. The problem here, given this statement and the fact that I have done this analysis 5 times over with this notation in this thread means that you haven't actually tried to digest any of it until just now.

 

[erobz]

Vectors indeed simplify the analysis substantially.Just because you have a hammer, doesn't mean every problem is a nail. We don't have to go into the complex details of fluid dynamics, because we have empirical data on achievable lift/drag ratios, which leads to the much simpler vector analysis.What have vectors done to you, that you hate them so much?Here is a paper by a physics professor in a physics journal, analyzing the same thing and using vectors of course:

High-speed sailing, Wolfgang Püschl 2018 Eur. J. Phys. 39 044002
https://iopscience.iop.org/article/10.1088/1361-6404/aab982

Vectors are not physics! They are a small part of the process. They do not in and of themselves encapsulate it, and or complete an analysis.

If we are talking fluid mechanics, we are talking Newtons Second, whether it be through the simplified model im proposing, or the more detailed Navier-Stokes Equations.

 

[erobz]

View attachment 322880

OK, but note that:
- the example $\beta$ you have drawn is wrong for maximizing $v$. See post #142.

I said $\beta$ is a parameter I was proposing to optimize as a function of $v_x$, to maximize $F \cos \gamma$.

- you have drawn the relative outgoing flow again, because it is parallel to the sail. This is not how the outgoing flow looks like in the ground frame where the sail is moving.

This is not a diagram of the flow throughout all time and space. It is a snapshot capturing the momentum change of the impinging jet immediately before and after impacting the sail at some specific time, location on the infinite planar sail.

 

[A.T.]

If we are talking fluid mechanics, we are talking Newtons Second,...

Sure, but you don't have to go back to Newton's Laws for every fluid mechanics problem. In particular for the aerodynamic forces on airfoils we have tons of experimental data, which is more reliable, than what you get by simplifying the situation, in order to solve it analytically.

 

[erobz]

Sure, but you don't have to go back to Newton's Laws for every fluid mechanics problem. In particular for the aerodynamic forces on airfoils we have tons of experimental data, which is more reliable, than what you get by simplifying the situation, in order to solve it analytically.

I can tell you right now I don't have a prayer of solving it analytically. Numerically...maybe. But that has been my whole point. We have this analysis (which is not the analysis of an airfoil, but is a step in that direction) which appears to be virtually humanly unsolvable in general, and you were telling me to just look at some randomly drawn vectors and animations, while critizing my "vectors" that are used as part of an analysis in a traditional way.

If you are able to site a CFD analysis that specifically proves downwind travel faster than the wind in the idealized case presented here, post it. I mean, if this is such a hot button issue I'm finding it hard to believe there aren't any experts that have created the rather theoretically simple CFD model to demonstrate it.

 

[A.T.]

...you were telling me to just look at some randomly drawn vectors...

No, I'm telling you to look at vectors, which are consistent with real world lift/drag-data for airfoils. This removes the whole messy fluid dynamics from the problem. Even for a numerical approach you can use empirical lift/drag-data, instead of CFD.

If you are able to site a CFD analysis that specifically proves downwind travel faster than the wind in the idealized case presented here, post it

"Sure, it works in reality, but can you show it in CFD?"

 

[erobz]

"Sure, it works in reality, but can you show it in CFD?"

"Reality" is messy and an uncontrolled environment. Show it in the laboratory, or simulation where experimental parameters are mostly controlled\fully controlled.

 

[A.T.]

"Reality" is messy and an uncontrolled environment.

Ice boats can do 4 x windspeed in the downwind direction. This is way more than the variability of the wind.

Show it in the laboratory,

The experimental lift/drag-data for airfoils comes from the lab (wind-tunnel). And and it implies that downwind faster than the wind is possible. Which is consistent with outdoor experimental data.

 

[erobz]

Ice boats can do 4 x windspeed in the downwind direction. This is way more than the variability of the wind.The experimental lift/drag-data for airfoils comes from the lab (wind-tunnel). And and it implies that downwind faster than the wind is possible. Which is consistent with outdoor experimental data.

Color me skeptical, and prove it with CFD.

 

[A.T.]

prove it with CFD.

That is not how physics works. It's an empirical science based on experiments. We know lift/drag-data for airfoils from experiments, so we don't need CFD to "prove" that it is possible.

 

[erobz]

That is not how physics works. It's an empirical science based on experiments. We know lift/drag-data for airfoils from experiments, so we don't need CFD to "prove" that it is possible.

This wasn't about a wing, this is about a vane. There is a difference. You are telling me that the solution for a simple vane should be faster than the wind in the direction of the wind...given the complexity of the equations I will derive I say, prove that with CFD. A thin straight (or curved) sheet of canvas, does not an airfoil make.

 

[A.T.]

This wasn't about a wing, this is about a vane.

It doesn't mater how you call it. All that matters is that you have something that can create much more lift than drag. The rest is trigonometry and vector math.

 

[erobz]

It doesn't mater how you call it. All that matters is that you have something that can create much more lift than drag. The rest is trigonometry and vector math.

I just showed you that is substantially more complicated than that! You are ignoring it and defaulting to your "just add the vectors" non-analysis "analysis".

 

[A.T.]

I just showed you that is substantially more complicated than that!

No. You just showed that one can make it more complicated than it needs to be.

There is no need to go back to Newton's Laws or use CFD here, because the question is not how lift is generated on the fundamental level. The question is how fast the boat can go downwind, given the empirically known performance of airfoils.