What fun could you have with a fountain design in artificial gravity?

[DaveC426913]

TL;DR

If you were gong to construct a fountain inside a hollow-rotating asteroid colony, what could you do to highlight the weird effects?

Backstory (TL;DR):

I've been searching for an allegory/symbolism for a journey to - and creation of - a colony inside an asteroid.​

​

I played around with Greco-Roman legends, notably Aeneus' escape from Troy and journey by sea to a new land where he created what would become Rome.​

​

But my ever-lovin' suggested Xanadu from Kubla Khan - a paradise that talks a lot about domes and rooves and sunless seas, and it's too much to pass up.​

​

"Where Alph, the sacred river, ran Through caverns measureless to man Down to a sunless sea."​

"So twice five miles of fertile ground With walls and towers were girdled round"​

​

One prominent symbol in the poem is a fountain.​

A mighty fountain momently was forced:​

Amid whose swift half-intermitted burst​

Huge fragments vaulted like rebounding hail,​

Or chaffy grain beneath the thresher’s flail:​

And mid these dancing rocks at once and ever​

It flung up momently the sacred river.​

So, how might you creatively engineer a fountain in a rotating inside-out colony to show off its peculiar physics?

Could you take advantage of the Coriolis Force, or otherwise differential forces and make some sort of periodic clock?

I guess, in its simplest form, a single stream from a fountain that went in circles would trace out an ellipse (of as-yet undetermined eccentricity) on the ground around it - one side a very short parabolic arc, the other a long arc.


Specs:
- assume an inner surface gravity of 1/3g
- asteroid of 10km in diameter, perhaps 15km along its axis.
This produces a rotation rate of .24rpm.

 

[DaveC426913]

Hm. I am no good at these calcs, and I don't trust my chatbot, but it is indicating that sufficiently interesting phenomena may be hard to produce and/or be too subtle to be really cool.

After all, the very nature of a large rotating enclosure is partly to minimize these very Coriolis Effects.

eg. if I wanted the arcs of the spinward jet(s) to return to the base of the fountain, it seems to take on the order of 30 seconds or so - a half rotation or more - and hundreds of metres high - which is far too long for a stream of water - or even drops of water - to remain anywhere near collimated.

 

[DaveC426913]

Trying to work this out for myself using this calculator

For a 20m tall fountain, shooting 45 degree jets all around at 15.79mph, I get a deflection of les than 5m on a ring that's more than 50m in diameter.

I'll try some other params, but this is not looking good.

Here are the numbers I used:

 

[Ibix]

A single straight-up stream would land off-center, so there must be an angle at which a diagonal stream would loop back and land at the launch point.

Working on the inertial frame where the cylinder doesn't translate, let the base of the fountain be at $x,y=0,0$ at $t=0$, with the rotation acis of the cylinder at $x,y=0,R$. At later times the fountain base position is $x=R\sin(\omega t)$, $y=R-R\cos(\omega t)$. Let its muzzle velocity be $v$ and make angle $\theta$ from the vertical. A water droplet launched at $t=0$ has position in the inertial frame $x=\omega R t + v\sin(\theta) t$, $y=v\cos(\theta)t$. Insisting that the droplet lands back at the fountain yields$$\begin{eqnarray}
R\sin(\omega t)&=&R\omega t+v\sin(\theta) t\\
R-R\cos(\omega t)&=&v\cos(\theta) t
\end{eqnarray}$$There are three parameters there (launch speed, launch angle and flight time) and only two constraints, so you should be able to choose your loop size. Unfortunately it's numerical solutions only, but approximating 1 and 2 to first non-vanishing order we get$$\begin{eqnarray*}
-R\frac 16(\omega t)^3&=&v\sin(\theta) t\\
R\frac 12(\omega t)^2&=&v\cos(\theta)t
\end{eqnarray*}$$Dividing the two yields$$-\frac 13\omega t=\tan(\theta)$$and squaring and adding yields$$v^2=R^2\omega^4t^2\left(\frac 1{36}\omega^2t^2 + \frac 14\right)$$which means that you can pick any one of flight time, nozzle angle (negative here meaning that it needs to be tilted away from vertical against the direction of rotation) and nozzle velocity, and get approximate values for the other two (note that the fourth order polynomial connecting $v$ and $t$ can be solved for $t$ by realising that it's a quadratic in $t^2$). The peak height of the loop will be the maximum radial distance between the curve of the cylinder and the chord path the water follows, which is $R-R\cos(\omega t/2)\approx \frac 18R(\omega t)^2$.

So specify your nozzle velocity, which will give you $t$ and hence $\theta$. See what kind of flight time and peak height you get for your parameters. Then, if it looks plausible, you can use equations 1 and 2 to refine the details (Excel solver should be able to handle it given the approximate analytical parameters).

 

[DaveC426913]

Yeah, unfortunately, the net result is that the landing point (or ring) is merely unilaterally shifted antispinward a few metres. Visually, it is no different than if there were a mild breeze.

Quite anti-climatic.

I was hoping for some delightful effect. Maybe even a precession.

 

[DaveC426913]

I suspect I may have to incorporate a gyroscope, or some other way to create a precession.

That's unfortunate. It would have been cool to have a fountain do something weird and wonderful.

 

[sbrothy]

An interesting question, albeit beyond my math skills. Although, I seem to remember that even in Arthur C. Clarke's Raman ship the Coriolis force was noticable:

``Everything OK, Skipper,'' he reported. ``We're just passing the halfway mark. Joe, Will, any problems?''

``Same here,'' added Myron, ``But watch out for the Coriolis force. It's starting to build up.''

So Mercer had already noticed. When he let go of the rungs, he had a distinct tendency to drift off to the right. He knew perfectly well that this was merely the effect of Rama's spin, but it seemed as if some mysterious force was gently pushing him away from the ladder.

--- http://lichen.phys.uregina.ca/eclectic/rama.html

But yeah, with the various pseudoforces involved something artful should be possible I'd imagine.

 

[DaveC426913]

An interesting question, albeit beyond my math skills.

It's beyond mine too, but I found this calculator, which is exactly what I need. Inputs the params of the station, then the fiddle with the launch trajectory and it visualizes the arc for me.

 

[DaveC426913]

I have upped the gravity to 0.5g (0.3rpm), and I reduced the station diameter to 2.5km.

So the colonists build a tower 110m tall.
The water spouts should probably not be faster than 15mph. 20mph seems to be terminal velocity of a water stream. Any faster and it breaks into droplets anyway.

They can make a fountain whose spinward streams carry them back to the base of the fountain and whose antispinward streams fall 71m away:

Not sure if I'm going to do much better than that with this simplest of fountains.

I'm still open to other ideas. Maybe I'll go with an artificially-precessing fountain, which could be more modest - maybe 10 or 20m tall, but it just rotates in sync with the station. Less cool.

 

[sbrothy]

Ah, I see. I like this laconic comment:

The Coriolis effect (named for some guy) [...]

Still, I'm fatigued. My laptop died and I had to build a new one out of parts lying around. Thank god I have all this electronic equipment. And I've been fighting with the webpage all evening. So even using a calculator is beyond me now! It's time for a cartoon and then going to bed.

EDIT: But yeah, nice picture you're "painting".

 

[sbrothy]

The term Euler force was new to me but I knew it as the tangential pseudoforce. So I guess I learned even more today. Typical physicsforums!

EDIT: Explore apparent forces in rotating systems focusing on Coriolis and Euler forces and their impact on motion in non-inertial frames of reference (www.ai-futureschool.com).

 

[DaveC426913]

Actually, I suppose this might be more visually impactful. Curve and tilt the tower by about 15 degrees, so the entire fountain is askew:

 

[sbrothy]

Actually, I suppose this might be more visually impactful. Tilt the tower by about 15 degrees, so the entire fountain is askew:
View attachment 371663

Exotic, certainly! I'd imagine it would be akin to sea-sickness living with these forces on a space station or similar. I'm not sure a skewed fountain would help though!

 

[sbrothy]

TL;DR: If you were gong to construct a fountain inside a hollow-rotating asteroid colony, what could you do to highlight the weird effects?

Backstory (TL;DR):

I've been searching for an allegory/symbolism for a journey to - and creation of - a colony inside an asteroid.​

​

I played around with Greco-Roman legends, notably Aeneus' escape from Troy and journey by sea to a new land where he created what would become Rome.​

​

But my ever-lovin' suggested Xanadu from Kubla Khan - a paradise that talks a lot about domes and rooves and sunless seas, and it's too much to pass up.​

​

"Where Alph, the sacred river, ran Through caverns measureless to man Down to a sunless sea."​

"So twice five miles of fertile ground With walls and towers were girdled round"​

​

One prominent symbol in the poem is a fountain.​

A mighty fountain momently was forced:​

Amid whose swift half-intermitted burst​

Huge fragments vaulted like rebounding hail,​

Or chaffy grain beneath the thresher’s flail:​

And mid these dancing rocks at once and ever​

It flung up momently the sacred river.​

So, how might you creatively engineer a fountain in a rotating inside-out colony to show off its peculiar physics?

Could you take advantage of the Coriolis Force, or otherwise differential forces and make some sort of periodic clock?

I guess, in its simplest form, a single stream from a fountain that went in circles would trace out an ellipse (of as-yet undetermined eccentricity) on the ground around it - one side a very short parabolic arc, the other a long arc.


Specs:
- assume an inner surface gravity of 1/3g
- asteroid of 10km in diameter, perhaps 15km along its axis.
This produces a rotation rate of .24rpm.

That poem is from The Ancient Mariner by Coleridge, isn't it?

EDIT: I recognize it from reading Douglas Adams.

 

[Ibix]

I was hoping for some delightful effect.

My calculations suggest that with your original parameters and a 10m/s launch velocity about 3° off vertical you could have a 16m high closed loop (a single stream that returns to its launch point) 60cm wide at its widest point. With your revised parameters, 10° off vertical gives you a closed loop 40m high and about 5m wide.

 

[sbrothy]

My calculations suggest that with your original parameters and a 10m/s launch velocity about 3° off vertical you could have a 16m high closed loop (a single stream that returns to its launch point) 60cm wide at its widest point. With your revised parameters, 10° off vertical gives you a closed loop 40m high and about 5m wide.

Hah. Now we're talking!

 

[DaveC426913]

That poem is from The Ancient Mariner by Coleridge, isn't it?

No. It's Kubla Khan by Coleridge, describing Xanadu.

 

[Ibix]

In Bakerloo did Ali Khan
A stately hippodrome decree
Where Alf the bread deliv'ry man
Collided with a draper's van
While doing sixty three.

 

[DaveC426913]

My calculations suggest that with your original parameters and a 10m/s launch velocity about 3° off vertical you could have a 16m high closed loop (a single stream that returns to its launch point) 60cm wide at its widest point. With your revised parameters, 10° off vertical gives you a closed loop 40m high and about 5m wide.

Sorry, the calculator doesn't bare that out.

I must have some parameters wrong.

 

[Ibix]

Sorry, the calculator doesn't bare that out.
View attachment 371686
I must have some parameters wrong.
View attachment 371685

Possible I messed up the maths - I was working on my phone. I'll check on the laptop later.

 

[Ibix]

No, wait, try 100° instead of 80°. You need to launch sightly anti-spinward (or widdershins if we want to sound more mystical).

 

[sbrothy]

No. It's Kubla Khan by Coleridge, describing Xanadu.

Oh. But it was at least Coleridge? I'll have to read his stuff.

 

[A.T.]

If you were gong to construct a fountain inside a hollow-rotating asteroid colony, what could you do to highlight the weird effects?

A spiraling stream around the rotation axis?

 

[sbrothy]

A spiraling stream around the rotation axis?

I visualized something like that but I didn't have the confidence to suggest it. But I guess anyone could claim that now.

 

[Ibix]

A spiraling stream around the rotation axis?

It suspect a stream might break up due to interaction with increasingly fast-moving air. I thought an interesting variant might be to fill a pool near the axis and the take away the base. Repeat, so you get a periodic stream of large blobs of water. You'd want quite a large pool at the bottom for it to hit - although the vertical velocity could be arbitrarily low it could have a substantial horizontal velocity.

 

[jtbell]

​

"Where Alph, the sacred river, ran Through caverns measureless to man Down to a sunless sea."​

This triggered memories of a sci-fi story that I read in my school days in the 1960s, which quoted that poem. I don't think I knew it was by Coleridge until I encountered it in my 12th-grade English literature class.

Now I'm trying to remember which story and author. Somewhere between the late 1940s to mid 1960s. Not Asimov or Heinlein, I think, but someone of that vintage. Any ideas?

 

[DaveC426913]

No, wait, try 100° instead of 80°. You need to launch sightly anti-spinward (or widdershins if we want to sound more mystical).

That doesn't make sense. An object launched antispinward will gain range , not lose it.

 

[Ibix]

That doesn't make sense. An object launched qntispinward will gain range , not lose it.

Did you try it? I can't get the zoom feature to work on the online calculator on my phone.

 

[sbrothy]

This triggered memories of a sci-fi story that I read in my school days in the 1960s, which quoted that poem. I don't think I knew it was by Coleridge until I encountered it in my 12th-grade English literature class.

Now I'm trying to remember which story and author. Somewhere between the late 1940s to mid 1960s. Not Asimov or Heinlein, I think, but someone of that vintage. Any ideas?

As I mentioned Coleridge is a big part of Dirk Gently's Holistic Detective Agency. Whether that's scifi I'm not entirely sure. Entertaining surely but described more like:

a "thumping good detective-ghost-horror-who dunnit-time travel-romantic-musical-comedy-epic"

That and The Long Dark Tea-Time of the Soul

 

[DaveC426913]

Did you try it?

Thats how i got my two stream fountain diagrams:
- set it to 55 degrees, take a screenshot, - set it to 125 degrees take a screenshot,
- combine.

I can't get the zoom feature to work on the online calculator on my phone.

On my lappie I just click and drag out a zoom box. Doesn't seem to have a mobile equivalent.