Maximum Acceleration Due to Buoyancy

[arkie87]

If you have a one cubic meter streamlined-shape filled with air submerged under water on earth, what is the maximum acceleration possible from the object? A simple calculation using Newton's law shows that the acceleration can be rho_water/rho_air*g, but this fails to take into account the fact that water has to be accelerated under the object to take its place.

However, it is unclear to me whether this water rushing in underneath can accelerate faster than g. While the water directly below the rocket must accelerate at 1 g, the surrounding water and above would be accelerating significantly slower than 1 g, and the weight of this fluid might be able to accelerate the fluid following the object to faster than 1 g, since its weight is much larger than the weight of the fluid following the object.

A related thought experiment is that since water in a tank of pressurized air can be accelerated through an infinitely small nozzle at a g proportional to the pressure, one can achieve the same pressure with a tall rocket-shaped air-filled object as long as the rocket is sufficiently tall. As long as the rocket diameter is much smaller than the container diameter, the velocity in the container will be negligible. 

Thoughts? Am i making a mistake somewhere?

[natsirt721]

This is sort of a meaningless question, and I'll try to explain why.

Buoyant force is based on a pressure gradient, and drag is based on velocity, ergo the maximum acceleration will be when you first let the object go. If the pressure gradient is extreme enough (read: if your object is tall) then this force could be pretty significant.  I wouldn't consider the air inside the object as a fluid, and instead consider the entire thing a solid body with a given density.

Example: 1 meter hollow aluminum cube, wall thickness 1cm = ~200kg. Force on a side = Area * rho * g * d, where d is m from the surface.  Force on the top = 1m2 * 1000kg/m3* 9.81m/s2 * h, force on the bottom = 1m2 * 1000 kg/m2 * 9.81 m/s2 * h+1 (cube is 1m tall so the bottom is 1m lower than the top). Subtract those to get the delta, h cancels, force delta is 9810N up. Gravity is 9.81m/s2 * 200kg = 1960N down. Net force is 7850N up, which for a 200kg object provides an instantaneous acceleration of a = F/m ~= 39 m/s2.  Of course, this is only valid for the exact instant you release the cube, as soon as it starts to move you get drag effects, and the cube will probably start to rotate and the force delta will decrease.  You can increase this by increasing the height of the body, and by reducing its mass. 

So, maximum acceleration is about 4 gees, but you're only going to do better than a gee for, say a second or so. At that point, your velocity through the fluid produces enough drag that the forces are balanced and you ascend at a constant velocity. This velocity is largely dependent of the ballistic properties of the object, and I won't attempt to derive the value for the example above.

[arkie87]

3 minutes ago, natsirt721 said:

This is sort of a meaningless question, and I'll try to explain why.

Buoyant force is based on a pressure gradient, and drag is based on velocity, ergo the maximum acceleration will be when you first let the object go. If the pressure gradient is extreme enough (read: if your object is tall) then this force could be pretty significant.  I wouldn't consider the air inside the object as a fluid, and instead consider the entire thing a solid body with a given density.

Example: 1 meter hollow aluminum cube, wall thickness 1cm = ~200kg. Force on a side = Area * rho * g * d, where d is m from the surface.  Force on the top = 1m2 * 1000kg/m3* 9.81m/s2 * h, force on the bottom = 1m2 * 1000 kg/m2 * 9.81 m/s2 * h+1 (cube is 1m tall so the bottom is 1m lower than the top). Subtract those to get the delta, h cancels, force delta is 9810N up. Gravity is 9.81m/s2 * 200kg = 1960N down. Net force is 7850N up, which for a 200kg object provides an instantaneous acceleration of a = F/m ~= 39 m/s2.  Of course, this is only valid for the exact instant you release the cube, as soon as it starts to move you get drag effects, and the cube will probably start to rotate and the force delta will decrease.  You can increase this by increasing the height of the body, and by reducing its mass. 

So, maximum acceleration is about 4 gees, but you're only going to do better than a gee for, say a second or so. At that point, your velocity through the fluid produces enough drag that the forces are balanced and you ascend at a constant velocity. This velocity is largely dependent of the ballistic properties of the object, and I won't attempt to derive the value for the example above.

sorry to say it, but you didnt really answer my question (or get the point of it). 

[natsirt721]

Could you clarify the point? There were a few questions in there, and I tried to answer the title question as best I could.

Most of the fluid that fills the space beneath the object as it ascends doesn't come from below, it comes in from the sides. Some fluid travels upwards with the object, as friction between the fluid and the object imparts some momentum on the fluid, but the fluid is always moving slower than the object, and compared to the size of the object the total volume moving upwards is significantly less than the displaced volume.  

Yes, the water that fills the gap can accelerate faster than a gee, the acceleration depends on the water pressure and the speed of the body.  If the body is going fast enough (or the fluid pressure is low enough), the fluid won't be able to accelerate fast enough to keep up, and you get cavitation, where a vacuum (or rather, rarified pocket of water vapor) forms between the water and the body itself. 

As for the title question, there is no definitive answer. The actual peak acceleration due to buoyancy depends on several factors, namely:

• The geometry  of the object
• The mass of the object
• The density of the fluid

[arkie87]

59 minutes ago, natsirt721 said:

Could you clarify the point? There were a few questions in there, and I tried to answer the title question as best I could.

Most of the fluid that fills the space beneath the object as it ascends doesn't come from below, it comes in from the sides. Some fluid travels upwards with the object, as friction between the fluid and the object imparts some momentum on the fluid, but the fluid is always moving slower than the object, and compared to the size of the object the total volume moving upwards is significantly less than the displaced volume.  

Yes, the water that fills the gap can accelerate faster than a gee, the acceleration depends on the water pressure and the speed of the body.  If the body is going fast enough (or the fluid pressure is low enough), the fluid won't be able to accelerate fast enough to keep up, and you get cavitation, where a vacuum (or rather, rarified pocket of water vapor) forms between the water and the body itself. 

As for the title question, there is no definitive answer. The actual peak acceleration due to buoyancy depends on several factors, namely:

• The geometry  of the object
• The mass of the object
• The density of the fluid

My main question is *neglecting drag*, what is the maximum acceleration of the body. Theoretically, the limit should be the ratio of the densities of the fluid and the object times g. In reality, drag can be neglected as diameter --> infinity. 

Unless there is cavitation, there needs to be enough volume flow to fill the void left by the object. I dont know how you can say there isnt?

I

[natsirt721]

Perhaps I misunderstood the question. I interpret is thusly:

Take a body, whose density is less than that of water's. Move it below the surface of the ocean, to some arbitrary depth. Release the body. What is the body's maximum acceleration between when it is release and when it reaches the surface?

Or, take a rubber duck, hold it at the bottom of a filled bathtub, and let it go. What is the duck's maximum acceleration between when it is released and when it reaches the surface?

The answer to this question, is that it depends on the geometry of the object, the mass of the object, and the density of the fluid. Using a force balance and tweaking those parameters, with zero velocity (and therefore zero drag) any value of instantaneous acceleration is achievable. One could imagine an extremely tall body with huge surface area but very little mass, submerged in an incredibly dense fluid, which would produce massive accelerations. One could just as easily imagine a massive body such that gravity overcomes the buoyant force (this is known as 'sinking'), or a body with exactly the right parameters such that it neither floats or sinks (neutrally buoyant). The equations are totally continuous, so theoretically any value of acceleration is achievable - you need to constrain the problem more in order to get a definite range, and significantly more if you want a concrete answer.

If this is not the question to which the answer you seek, I'm afraid I don't know what it is you want to know.

  

Edited June 18, 2019 by natsirt721

[arkie87]

8 hours ago, natsirt721 said:

Perhaps I misunderstood the question. I interpret is thusly:

Take a body, whose density is less than that of water's. Move it below the surface of the ocean, to some arbitrary depth. Release the body. What is the body's maximum acceleration between when it is release and when it reaches the surface?

Or, take a rubber duck, hold it at the bottom of a filled bathtub, and let it go. What is the duck's maximum acceleration between when it is released and when it reaches the surface?

The answer to this question, is that it depends on the geometry of the object, the mass of the object, and the density of the fluid. Using a force balance and tweaking those parameters, with zero velocity (and therefore zero drag) any value of instantaneous acceleration is achievable. One could imagine an extremely tall body with huge surface area but very little mass, submerged in an incredibly dense fluid, which would produce massive accelerations. One could just as easily imagine a massive body such that gravity overcomes the buoyant force (this is known as 'sinking'), or a body with exactly the right parameters such that it neither floats or sinks (neutrally buoyant). The equations are totally continuous, so theoretically any value of acceleration is achievable - you need to constrain the problem more in order to get a definite range, and significantly more if you want a concrete answer.

If this is not the question to which the answer you seek, I'm afraid I don't know what it is you want to know.

  

Let's limit the fluid to water, let's limit the acceleration due to gravity to 1 g, and let's limit the object to being filled with air (and zero wall thickness). 

My main question is if the acceleration that you compute from a force balance i.e. (weight of displaced water - weight of object)/mass of object is physical, because the fluid underneath or on the sides of the object must accelerate to follow the object (or else you have cavitation), and this acceleration will "absorb" some of the available pressure drop or pressure gradient.

[MinimumSky5]

@arkie87 Yes, you can calculate the force that the water being dragged behind the object is causing, we call it drag!

You're asking for an equation to calculate the low pressure, turbulent region behind the moving object, but that requires the speed of the object, the objects geometry, surface texture, rotation rate, turbulence of the water, and a supercomputer to calculate it all. 

[Nightside]

3 hours ago, arkie87 said:

let's limit the object to being filled with air (and zero wall thickness). 

Let's call this an air bubble.

I was surprised to find one of the best answers I've ever seen on Quora. https://www.quora.com/How-does-the-acceleration-of-a-bubble-change-as-it-moves-to-the-surface-from-the-bottom, which I have quoted in part below.

To summarize a bit, If you ignore that you have to move water out of the way of the rising bubble, acceleration would equal the proportional difference of densities, where air is 1000x less dense than water, so 1000g!

But since you can't ignore that, it maxes out at 2G!

The biggest difference between your case and a typical bubble is that a bubble can expand as hydrostatic pressure decreases, which actually causes buoyancy to increase. - For a fixed volume, buoyancy would not increase as the object rises.

Quote

Kim Aaron, Has PhD in fluid dynamics from Caltech

Answered May 9, 2016 · Author has 5.5k answers and 9.4m answer views

 

A surprising result is that the initial acceleration of a bubble released from rest in a liquid is 2 g's.  Once the bubble gets moving, viscous drag will become important and the acceleration will fall to almost zero.  As the bubble rises, it expands as the hydrostatic pressure becomes less, so the buoyancy force will increase in proportion to diameter cubed, whereas the drag force will only scale in proportion to diameter squared.  Therefore the terminal velocity of the bubble will gradually increase as it rises.  That assumes it remains spherical, which is only true for small bubbles for which surface tension is strong. 

As the bubble gets larger, it will form a spherical cap shape with a flat horizontal lower surface.  Here are some stills from a video showing two spherical cap bubbles merging into one larger bubble. 

Hydrodynamics of Ionic Liquids in Bubble Columns

---

But why is the initial acceleration 2 g's?

If you computed the buoyancy force on a small spherical air bubble in water, it would be the weight of the water displaced.  And the mass of the air would be about 1000 times smaller.  So why isn't the acceleration equal to 1000 g's?  After all, F = ma.   What gives? 

Well, if you only had to accelerate the air bubble, the acceleration would be about 1000 g's.  But you also have to accelerate the water out of the way to make room for the bubble to pass through the water.  There is a nice result from potential flow around a sphere that allows you to compute the kinetic energy of all the water when a sphere moves through stationary water.  And that kinetic energy turns out to be equal to half the kinetic energy of the sphere if it were solid water.  So the force of buoyancy is equal to the weight of water displaced, but the equivalent inertia of all the water moving around the sphere is equivalent to half the mass of water displaced.  So the acceleration is 2 g's (ignoring the tiny mass of the air itself in the bubble).  This is also referred to as "added mass" or "virtual mass" and it is important for the acceleration of helium filled balloons in air if you care. 

 

Why are you interested in this question? Thinking of launching rockets from the sea floor?

[arkie87]

4 hours ago, MinimumSky5 said:

@arkie87 Yes, you can calculate the force that the water being dragged behind the object is causing, we call it drag!

You're asking for an equation to calculate the low pressure, turbulent region behind the moving object, but that requires the speed of the object, the objects geometry, surface texture, rotation rate, turbulence of the water, and a supercomputer to calculate it all. 

arg... when the object is at rest, there is no drag. turbulence isnt an issue either for obvious reasons (Reynolds=0). sheesh.

12 minutes ago, Nightside said:

Let's call this an air bubble.

I was surprised to find one of the best answers I've ever seen on Quora. https://www.quora.com/How-does-the-acceleration-of-a-bubble-change-as-it-moves-to-the-surface-from-the-bottom, which I have quoted in part below.

To summarize a bit, If you ignore that you have to move water out of the way of the rising bubble, acceleration would equal the proportional difference of densities, where air is 1000x less dense than water, so 1000g!

But since you can't ignore that, it maxes out at 2G!

The biggest difference between your case and a typical bubble is that a bubble can expand as hydrostatic pressure decreases, which actually causes buoyancy to increase. - For a fixed volume, buoyancy would not increase as the object rises.

 

Why are you interested in this question? Thinking of launching rockets from the sea floor?

Yes, i didnt call it an air bubble since air bubbles can deform under drag force and also expand as the static pressure changes, as you noted. It's hard to tell a priori what minute details strangers on the internet will latch onto instead of answering the question at hand, so i opted for a volume of air with an infitinitely small, massless, rigid shell instead of a bubble.

Yes, thank you for the quora link. Cant say i understand why the limit is 2g though. The answer also appears to indicate that shape would be a factor (since they solve potential flow around a sphere)  i.e. 2g is the solution for a sphere, but what about a rocket-shaped object?

And no, i'm not considering launching objects from the sea floor. just had a discussion where someone claimed "maximum acceleration due to buoyancy is 1g since 1g acceleration causes buoyancy," but that appears to be incorrect...

[Nightside]

3 hours ago, arkie87 said:

“...maximum acceleration due to buoyancy is 1g since 1g acceleration causes buoyancy,"

This would be the case for an object floating on the surface though! 

 

My experience with fluid dynamics is limited to some basic open channel flow through simple pipe calcs and it gets complicated quickly, because the solutions are very sensitive to shape and velocity, most problems need to be solved iteratively. That is to say I’m at my limit of knowledge on this.

[Scotius]

Ekhm... I'm pretty sure any Navy operating ballistic missile submarines solved this problem decades ago But i'm just as sure they will not be eager to share this knowledge with you without good reason

[arkie87]

27 minutes ago, Nightside said:

This would be the case for an object floating on the surface though! 

 

My experience with fluid dynamics is limited to some basic open channel flow through simple pipe calcs and it gets complicated quickly, because the solutions are very sensitive to shape and velocity, most problems need to be solved iteratively. That is to say I’m at my limit of knowledge on this.

yeah, i've run CFD simulations and i have a phd in mechanical engineering in the fields of fluid flow & heat transfer, but because of the accelerating frame of reference, typical navier stokes equations are not applicable..

[mikegarrison]

18 hours ago, arkie87 said:

My main question is *neglecting drag*

Hey arkie, first you state your question without making this HUGE statement here, then you get kind of snarky about an excellent reply.

This is not a good look for you.

[magnemoe]

18 hours ago, natsirt721 said:

Perhaps I misunderstood the question. I interpret is thusly:

Take a body, whose density is less than that of water's. Move it below the surface of the ocean, to some arbitrary depth. Release the body. What is the body's maximum acceleration between when it is release and when it reaches the surface?

Or, take a rubber duck, hold it at the bottom of a filled bathtub, and let it go. What is the duck's maximum acceleration between when it is released and when it reaches the surface?

The answer to this question, is that it depends on the geometry of the object, the mass of the object, and the density of the fluid. Using a force balance and tweaking those parameters, with zero velocity (and therefore zero drag) any value of instantaneous acceleration is achievable. One could imagine an extremely tall body with huge surface area but very little mass, submerged in an incredibly dense fluid, which would produce massive accelerations. One could just as easily imagine a massive body such that gravity overcomes the buoyant force (this is known as 'sinking'), or a body with exactly the right parameters such that it neither floats or sinks (neutrally buoyant). The equations are totally continuous, so theoretically any value of acceleration is achievable - you need to constrain the problem more in order to get a definite range, and significantly more if you want a concrete answer.

If this is not the question to which the answer you seek, I'm afraid I don't know what it is you want to know.

  

Acceleration will always be highest at the start. (ignoring an compressed object from far underwater)
Low speed gives low drag and high acceleration also think that high pressure will give higher force. 

[arkie87]

12 minutes ago, mikegarrison said:

Hey arkie, first you state your question without making this HUGE statement here, then you get kind of snarky about an excellent reply.

This is not a good look for you.

I asked for maximum acceleration, which occurs the instant the object is let go, where drag is zero. Yeah, i didnt specifically state it, but i went ahead and discussed a lot of other items, which the responder didnt address. Just told me stuff i already knew. It can be very frustrating when you ask a specific and highly technical question, and someone responds with basic physics and doesnt address your question.

4 minutes ago, magnemoe said:

Acceleration will always be highest at the start. (ignoring an compressed object from far underwater)
Low speed gives low drag and high acceleration also think that high pressure will give higher force. 

@mikegarrison for instance, this response above... not sure what do with it...

[magnemoe]

47 minutes ago, Scotius said:

Ekhm... I'm pretty sure any Navy operating ballistic missile submarines solved this problem decades ago But i'm just as sure they will not be eager to share this knowledge with you without good reason

More relevant for torpedoes, missiles are ejected by an gas generator, think steam gun who give it enough speed to get to the surface and into the air while stable, then the rocket ignite. 
Torpedoes don't use buoyancy but need an set trust for an speed. Increase speed and range drops. 

[mikegarrison]

Just now, arkie87 said:

I asked for maximum acceleration, which occurs the instant the object is let go, where drag is zero. Yeah, i didnt specifically state it, but i went ahead and discussed a lot of other items, which the responder didnt address. Just told me stuff i already knew. It can be very frustrating when you ask a specific and highly technical question, and someone responds with basic physics and doesnt address your question.

And my point was that you didn't really ask your question very clearly. IMO, of course.

People come in here and ask for free help with their problems, then complain when the free help that is freely offered isn't exactly what they were looking for. It's ... annoying when that happens.

9 hours ago, arkie87 said:

My main question is if the acceleration that you compute from a force balance i.e. (weight of displaced water - weight of object)/mass of object is physical, because the fluid underneath or on the sides of the object must accelerate to follow the object (or else you have cavitation), and this acceleration will "absorb" some of the available pressure drop or pressure gradient.

Also, the more I think about it, the more convinced I get that you are saying, "ignoring drag, help me calculate a force that is due to the fluid resisting the motion of the object within it". In other words, you are basically trying to ignore drag and then derive drag from first principles. Isn't this thing that you are looking for basically the same as what we call "form drag" in aero?

[arkie87]

1 minute ago, mikegarrison said:

And my point was that you didn't really ask your question very clearly. IMO, of course.

People come in here and ask for free help with their problems, then complain when the free help that is freely offered isn't exactly what they were looking for. It's ... annoying when that happens.

I agree, my question could have been more clear. But in no way did the original response actually respond to the approximately 2/3 of my question either...

And while you might be upset at people who get annoyed at people who try to help them for free with their problems, it is also equally annoying to be on the other end of it, when you ask for help with a specific problem and someone answers a different question altogether without fully reading your question. Or worse-- they start a whole argument about one of your assumptions when its supposed to be a thought experiment... etc.... if i had a penny for every time that has happened on this forum (and others).... i'd have like 10 pennies (at least)

[mikegarrison]

12 minutes ago, arkie87 said:

@mikegarrison for instance, this response above... not sure what do with it...

Well, since it was replying to another poster, I would say the easiest thing for you to do about it would have been to ignore it.

[arkie87]

Just now, mikegarrison said:

Well, since it was replying to another poster, I would say the easiest thing for you to do about it would have been to ignore it.

lol, my bad in this instance. replies like that, i usually do ignore.

its hard to ignore when they begin attacking your assumptions even though its a thought experiment... 

[mikegarrison]

Anyway, did you notice my question about form drag?

I really am starting to think this entire thing is ill-posed. When you first release the object there is no motion so there is no drag, but because there is no motion there is also no backfilling of water or water moving out of the way of the object. So the whole problem disappears except for the basic question of "what is bouyancy?"

Once motion begins, then the water does need to have to start moving out of the way and filling in behind -- but this is what we call "drag". So....

[natsirt721]

2 minutes ago, mikegarrison said:

I really am starting to think this entire thing is ill-posed.

Yeah I can't really grok the question, you want velocity but no drag and acceleration but not from a standstill? That problem has so many degrees of freedom I can't even begin to name them all. I clearly can't help you further, good day.

[arkie87]

2 minutes ago, mikegarrison said:

Anyway, did you notice my question about form drag?

I really am starting to think this entire thing is ill-posed. When you first release the object there is no motion so there is no drag, but because there is no motion there is also no backfilling of water or water moving out of the way of the object. So the whole problem disappears except for the basic question of "what is bouyancy?"

Once motion begins, then the water does need to have to start moving out of the way and filling in behind -- but this is what we call "drag". So....

There is form drag and viscous drag/skin friction. Viscous drag/skin friction can be zero (which is what i meant by zero drag), but form drag cannot i.e. the thought experiment would be ill-posed or self-contradictory.
 

Just now, natsirt721 said:

Yeah I can't really grok the question, you want velocity but no drag and acceleration but not from a standstill? That problem has so many degrees of freedom I can't even begin to name them all. I clearly can't help you further, good day.

zero velocity... so skin friction is zero. yes, acceleration, and yes from standstill ( i never said not from a standstill)?

[mikegarrison]

OK, so what you're asking is how to calculate form drag in water?

If so, I don't know. I'm an aero guy, and we do funny stuff like ignore the mass of the fluid we're moving through, something the hydrodynamics guys can't do. (But they ignore compressibility, which more than makes up for it.)