First Glimmers of Experimental Confirmation

I want to be clear that I am very bad at building things, and am humbled by the process of setting up a real experiment. A whole lot goes wrong with doing the things in practice. But I am still happy enough with how things are going to share this attempted 3-divider setup:

You can see staggered heights and radii. In my ideal world, I would have just put these carefully-measured plastic crafts into a bucket, one inside the other, and see the result I'm looking for. It hasn't quite gone so well, but I still think I'm tracking towards the data I'm looking for in the bigger picture.

Phases of Experimentation

There's a lot to think about for putting together the water-medium bucket-scale simulation of flow dividers. I usually don't really think about those things until it goes wrong when I try and actually spin it.

For the first overall geometry, I tried to do a floater-sinker combination on an open cylinder. So nothing holding the cylinder in place, just some things attached that keeps it vertical. This always went wrong because it's fundamentally not stable - it was interesting to measure significant decrease in drum speed, reflecting increased drag from the wobbling state. But even before I got far into this observation, I observed that my first choice for floater material dissolved in water. That went so badly it was almost funny.

Moving on, clearly it was time to experiment with some "spacer" approaches. I tried to Jerry-rig this with a few household items, such as toothpicks. These items were not up to the task. The one exception of note was ping-pong balls. I could believe those would work to hold a divider in place and not add too much extra friction. However, those things are maybe 1/100th the density of water, and I had more practical problem that the divider didn't extend out of the water, vertically, far enough. So with enough speed these just flew out. But a taller divider might still work. Partially sinking the balls was a cute idea, but if it works, it requires less-sticky materials than what I have.

Still not realizing exactly the extent that stickiness (general friction coefficient) is playing, I tried some cardboard spacers. In retrospect, why did I bother? It gave me some good numbers on what happens when something substantially increases the friction the motor is seeing.

This all led to the current approach, which is to cut a circle for the top and bottom of the cylinder that has a hole in the center just large enough to let the metal shaft through. This seemed to address the wobbling issues I was seeing. But the first time, I incorrectly made the thing too short.

In the meantime, the drum itself had a tremendous amount of wobble because I was just bad at drilling the holes. After getting some help with re-doing that, the reference speeds showed dramatic improvements and the fluid movements were a lot more gentle.

Finally, to what was hopefully my last critical oversight, after building the 3 color drums, I included a spacer on the bottom for the drums to "sit" on. I did not realize that they didn't need to sit at all. After a certain speed, there we get a significant lifting force. If you look at what I just wrote on the negative pressure designs, this is already foretold. Fluid forces will push the ends in if the dividers are open to the fluid. Well the connection I'm using is definitely not water-tight so we can be pretty sure this is happening. Because only 1 of the 2 cylinder's end are under water, this... pushes it up. There was nothing to stop its movement up, so the "hat" I created for them would all start flying off and everything went off the rails.

Pretty Sure Working just a Little Bit Now

To set expectations, I have ran a few references with water and no water in the bucket. This gives a sense of the fluid-dynamic specific contribution to the torque. The speed moving the drum (due to the bearings and stuff) was quite significantly less than the motor free spinning the shaft.

For adding a single divider, I have a best-case scenario of adding about 4 RPM speed. This is if the divider itself doesn't add additional friction (which it will) or generally cause chaos as all the other experiments have.

Running all 3 dividers nested in each other (before the hats flew off), it was hard to tell what was going on but speed measurements seemed to be at around parity with the reference. This was encouraging. I did another go with a single divider, and again, before the lifting problem it seemed to be touching parity or maybe sometimes just a little better. It was unclear when the lifting problem was becoming terminal so it's hard to say what range the effect was expected to be measurable.

So onto this configuration, where I might have corrected enough things for it to not be terrible.

Just running with the middle divider present (blue). Down at the bottom, there is a spacer both under and above the bottom circle of the drum. I obtained some numbers, and they're not much:

V	Reference	With blue divider
6	135.8	135.4
7	159.4	160
8	183.5	184
9	207.1	207.6
10	232	233

But we are now hitting around 1 of the 4 RPM. I'm using an automotive tachometer, and because it's doing a timing measurement, I actually believe it is accurate to the 0.1 RPM. The larger problem is the physical speed stability, and it tends to get worse before the dividers break apart. Otherwise it is >1 RPM roughly, in good conditions. So this still might be better to call it a statistical suggestion of a speedup.

I need to go higher voltage & speed, but still have a problem with the hat flying off... right around the 10 V mark, which is also the speed the expected 4 RPM speedup would apply to.

This is still extremely underwhelming, but these experiments have seen a gradual improvement from the dividers making the speed drop, to reaching parity with the reference, to now, maybe just dipping into the improvement territory. Still lots of build quality issues to address. Once I get a much more satisfying speedup, and generally ability to spin the thing faster, that will open up a number of theoretical questions to testing.

Broad Strokes of a Physical Test Plan

Simulations I presented in last post are grasping at something that can not really, realistically, be achieved. Even if I were to put in a turbulent model, account for momentum correctly, it would be very difficult to allow deformations of the flow-dividers.

I believe the answer to this is physical experiments, which are already fairly common in the adjacent research space. As I've gotten further into the topic, I've realized that the Russian doll type velocity staging is weirder than I originally thought, and actually non-trivial in its implementation. The core arguments hold, but the lack of similar applications on Earth leave us with a surprisingly empty engineering space, in terms of background literature. So the next logical step is to start experiments.

Reynolds Number Ranges

Before we even add flow dividers, we are going to pretend that we are doing basic Taylor–Couette flow, which is just a rotating drum inside a larger cylinder. In all numbers I'm giving here, I will not do anything fancier than that.

For the table, I am going to select (describe) a particular physical thing, like a bucket. I know what size bucket I can buy, so I will start with an outer radius, $r_o$ from the available product, I can potentially get. Define the gap $g$ as the difference between the outer radius $r_o$ and the inner radius $r_i$.

$$ g = r_o - r_i $$

The gap-based Reynolds number $\mathrm{Re}$ uses the relative tangential speed $\Delta U$, gap $g$, and kinematic viscosity $\nu$.

$$ \mathrm{Re} = \frac{\Delta U \, g}{\nu} $$

Angular speed $\omega$ is the tangential speed $\Delta U$ divided by the inner radius $r_i$.

$$ \omega = \frac{\Delta U}{r_i} $$

Rotation rate in revolutions per minute (rpm) is angular speed $\omega$ converted from radians per second.

$$ \mathrm{rpm} = \frac{\omega}{2\pi}\,60 = \frac{\Delta U}{2\pi r_i}\,60 $$

Solving for $\Delta U$ gives tangential speed from rpm and $r_i$.

$$ \Delta U = \frac{2\pi r_i\,\mathrm{rpm}}{60} $$

A simple turbulent wall-drag scaling relates available shaft power $P$ to steady-state speed $\Delta U$ using density $\rho$, friction factor $C_f$, inner radius $r_i$, and active length $L$.

$$ P \approx \pi \,\rho\, C_f \, r_i \, L \, (\Delta U)^3 $$

A smooth-turbulent closure for the friction factor uses $C_f$ as a function of Reynolds number $\mathrm{Re}$.

$$ C_f \approx 0.079\,\mathrm{Re}^{-0.25} $$

The power model is coupled to the flow state through the same Reynolds definition $\mathrm{Re}=\Delta U g/\nu$.

$$ \mathrm{Re} = \frac{\Delta U \, g}{\nu} $$

This is all a little scatter-shot, but it gives enough background to fairly simply fill in the remaining columns after we have selected some bounding inputs from the hardware store. Those inputs are:

• Outer radius $r_o$ set by the given container we have available or the maximum extent we're willing to build at that moment
• Length, L, also constrained by container. In most cases, by the vertical dimension.
• Available power, P, this is set by the motor we expect to use.

These are the numerical inputs for rows in the literal table below. However, you might note that a motor doesn't just have a power. You also need to get it such that it provides the correct speed. The approach I'm taking (assuming will be taken) is that for a given experiment, from this data, we basically find out how fast the motor needs to go. Then that feeds into what kind of motor we get. This likely requires some gearing, and later experiments might swap out gearing as needed.

Experiment	Outer R (m)	Length (m)	Power	$\mathrm{<u>\$\backslash Delta\; U\$\; (m/s)</u>}$	Re	drum rpm
Bucket-water	0.140	0.30	100 W	7.40	4.15e5	842
Pool-water	1.524	1.10	1 kW	4.69	1.41e6	36.6
Backyard-air	1.524	1.10	1 kW	42.3	8.46e5	330
Lake-water	10.0	15.0	15 kW	2.91	5.83e6	3.48
Hangar-air	10.0	15.0	15 kW	26.3	3.50e6	31.4
Space hab 250m	—	—	—	49.5	6.60e6	1.89

I've put simple names on the experiment scales. The first row comes from what kind of 5 gallon bucket you can get from the hardware store. The second row comes from some basic searching on what kind of above-ground pool (low quality would be sufficient) I can buy.

Then the power numbers are partly speculation, and another part, what motor would have a cost commensurate to the cost of the other stuff in the experiment.

Air has a convenience factor for experiment scaling - it ups you to a Reynolds number that you wouldn't otherwise counter except at a much larger scale. Compare lake-water to the space habitat and you get the point. This lake-level experiment would provide an appropriate level of validation before you went and launched something into orbit for real... at least in some senses.

The biggest drawback of water is that it is incomprehensible, and air is the ultimate objective. So it makes sense to run the experiment with air as the medium.

This leaves the big question of "how" you would conduct such an experiment. And that's something I have a few ideas on.

Driving Shaft and Half Scale

Return to the basic thing that we need. I like to illustrate with simple conical pinched ends. And in case there was any doubt, flow-dividers go inside other flow dividers. Dotted lines are to mark what wouldn't be seen from the outside.

This isn't very practical. Once you finish constructing one of the layers, you will have de-construct it to ever take it apart again. So I fully anticipate a half-scale kind of experiment where you would lob off one of the two end tapers. A shaft in the middle would be applying torque in any case, which I'll illustrate here.

You also have to hold them in place, the axial stability problem isn't really particularly interesting academically, so it would be better to isolate that factor and just investigate the wedge-effect type stability. Here is where another property of water is helpful. You can use the half-scale setup to also helpfully hand-wave the axial stability. My proposal for this is to add floaties to all of the flow dividers. These floaties would be circular (very thing donuts), made with Great Stuff or something similar.

Lately, I have been racking my brain on whether or not this can be a valid setup. Like, if it fails, would it be failing due to a reason that is meaningful? I think so, but it seems important to articulate why. As I've done many times here, you have to start from the access opening, and work your way out for each stage. As a result of this "walking", each stage is expected to hold some amount of pressure. This should still work starting from the bottom opening.

My challenge is to consider whether this can be compatible with the idea of adding floaties for the half-scale experiment. After all, the air above the water has an effectively constant pressure, so this would seem to violate the pressure differential on each stage. But not necessarily so. As these are rotating, water behaves as you would expect, with the surface demonstrating a slope. The little bit of rise-up of water on the inside should maintain the pressure differential.

Oh, things can go wrong with this. The rise-up could knock over some of the divider or the floatie, and that would be a failure. Or it could spill over. In all of these cases, however, it should be a fairly obvious failure mechanism. With this mental picture, I feel relatively good about the theory for moving forward with this solution for water experiments.

Air experiments have a different challenge. Because we do not live in micro-gravity, we would need a new solution. I believe that would not be the half-scale experiment details here. Instead, you would likely add a circular Helium bladder to keep each stage up. Doing things in air should technically require keeping both end tapers in place. That sure sounds hard, but it's a problem for another day.

Objective

So, what would we expect to get from this? The theory, however imperfect, does give us some ideas. Firstly, we want to replicate the instability that we predict. If we can't... that would be very notable. Astonishingly, I still don't really have an answer here. So they'll collide or not and I don't know.

But beyond that, we should absolutely not quit with unstable behavior, but try some stabilizing approaches. One would be to get some neutral buoyancy balls that match the gap distance, and then just throw them in and see how it goes. They probably won't self-sort, but I would want to see this play out. Predicting the most obvious outcome - we would want to add some sort of brace that holes in a cylindrical shape spacers. Spherical balls won't work for this... maybe at that point we would need wheels. At maybe somewhere around there 3D printing parts will help.

So, starting with the bucket-water experiment, we want stability demonstrated, with or without aids. Probably, ideally, with more than one solution to maintain stability. Then with this, prove some confidence to continue scaling up to larger sizes, with the idea that we can still get stability. Then, ultimately, we can get Reynolds number parity with what we would launch into space, and some well-developed corrections for in-compressible cases.

Scaled Experiment Metrics and Development Pathway

Stability remains an issue - it is one of the core weaknesses of the proposition for the gravity balloon concept. The issue isn't whether the fundamental principle of laminarization will work, or even necessarily that the channel flow physics are suitable for this problem, but the behavior of a complex and fairly high-energy system.

If you took a simplistic picture of the wedge effect, the prediction would be a slight restorative force for each layer of the friction buffers. The exact directionality of this force gets complicated. It gets even more complicated by the non-rigid nature of the buffers. What we should really raise our eyebrows at, however, is what happens when you consider all the numerous sheets in tandem. Loosely connected dynamic systems can be prone to failure, and this situation seems like a candidate for that. But then again, maybe not. The solution becomes fairly non-trivial.

Due to the complexity of the problem, our only best option left is to perform scaled experiments. Ideally, we would like to use a fluid that is more convenient, in that the same flow patterns can be produced at a smaller scale. To do this, we will look at the Reynold's number.

Re = rho v d / mu

We would like something with a high density and a low viscosity (in particular, relative to air). It becomes fairly obvious that water is our best option for this. Also, you can see that we have 2 dimensions over which to flex our abilities - that is the velocity and the distance metric. Now it seems pretty clear that we want the smallest structure that we can spin at workable speeds.

These 2 degrees of freedom would still dictate a massive scale of experiment, if taken to apply to the entire system all at once. By that, you can say that d is the diameter of the outer-most friction buffer (for example), and that the system will be a scaled model in terms of all geometric dimensions. The enormous cost of this nudges us to seek cheaper solutions that might go 80% of the way with 20% of the effort. In a more accurate accounting, I'm looking for something more like 10% of the ultimate discoveries with 0.001% of the effort.

Obviously, we might instead take d to measure the distance between sheets, while also relaxing the requirement for strict geometric similarity. This might be nice in order to make something testable by reducing the number of sheets compared to the gravity balloon reference design. Thus, the overall scale and velocities will be dramatically less, while still demonstrating channel-to-channel interactions with the same flow patterns.

This still won't be sufficient to make an immediately tenable experiment. We'll need to relax something related to the channel flow pattern itself. The obvious candidate is to change the channel width relative to the overall tube diameter. However, I will not count this as an independent variable, because I think that (for most cases) it will fall out of the selection of the number of sheets. In any format you choose, it's likely that the ratio of the overall thickness of the friction buffer region will be about 50% of the tube radius. So greater channel width will follow with fewer sheets.

Our hypothetical experiment has been cut-and-slashed a lot by this point, but we're not finished yet! What is truly the really important point? What would we want to learn from this? I would argue that it is the interaction of multiple friction buffers in a (generally sufficiently) turbulent flow regime. Even if you cut that out, there's still some value because it answers some questions about this broader notion of friction buffers (which can even have other applications). However, we do want to answer questions about the friction buffers used in a gravity balloon regarding their stability. We basically know that the answer will be different for laminar and turbulent (or at least somewhat independent). Let me illustrate my thinking in a sketch.

(let me volunteer that I know I illustrated the transition region poorly)

Basically, we want to probe on the minimum edge of turbulent flow regimes with a multi-sheet friction buffer system. Just opt for an outright change of Reynolds number according to the abilities available for experimentation. This would be the 10% of ultimate knowledge I'm interested in. This would set the stage for everything that may (or may not) come in the future.

The good news - all this slashing of the metrics gets our experiment size way down (the scale is highly sensitive to Reynolds number). And that gives us some wiggle room. With lower Reynolds number, we can play around with more sheets (even up to 10 or 16 as I'm dreamed about), while staying at least in the turbulent regime. With the same general equipment, dial the numbers the other direction, and see about higher Reynolds number channels with fewer sheets.

Once you start chewing on this, something new starts to take form - a general format of the development path. Because as these numbers are dialed back up to the full scale (with more resources), its possible to speculate when many different components of the design will be proven in principle. After that, you can image at what point those components will mature into a representative suite of technologies. For instance, at certain numbers, the intra-sheet flow management will become testable. At another point later on, active controls for maintaining the seals could be strapped on.

So now we've covered 4 (mostly) independent factors. I think this is probably the right way to look at scaling of real, physical, experiments. These can start on a household scale.

Small Balloon-Tubes Systems, a Gauntlet of Wires, and Sucking Sheet Pinches

This is a fairly general brain dump of a collection of topics. I could see them all being posts, but not all of them are likely to become posts, so I want to get them out while the concepts are fresh in my mind.

People-Units 

While working on the math for this stuff, I keep coming back to notions of "magic" numbers. There are very defined numerical parameters that we can spin our own abstract tapestry. What's most unique about this project is what defines those bounding parameters - they almost all come down to human biology. Why is a gravity balloon a certain size? Because humans need a certain pressure, and this combines with the _fundamental_ gravitational constant to produce a tangible number.

All this reminds me of the notion of "god" units, or Planck units. The fundamental units span the full range of physical values. Because of this, you can measure practically any complex quantity as a combination of the fundamental ones - like volume.

People units constitute a rougher and more gray set of fundamental constants. Combining the gravity people need with the air properties they need, you can get the characteristic height of Earth's atmosphere, but there are lots of other ways you can come up with different length units.

Minimum Size for Friction Buffers

Lately on NASA Spaceflight forms, I've seen artificial gravity inside of balloon envelopes come up. This has a rather strange similarity to what I've talked about in this blog. The motivations given for this design are predictable - space stations can continue to be thought of as a nice inertial frame of reference, like the ISS, while adding centrifuges in a limited domain. The basic idea is to take a large Bigelow module and put 2 counter-rotating centrifuges. The two can be spun up at the same time so they have minimal effect on the rest of the station.

The minimal effect principle is an objective very much worth pursuing. For near-term space stations, we will expect many roles to be fulfilled by the station, and external operations can not be compromised for the logistics of a spinning module. In this context, it's hard to imagine that anything other than a fully enclosed centrifuge can make sense.

But where does this lead us? Operationally, I can paint somewhat of a picture. If you moved around in such a centrifuge, vomiting seems inevitable. However, limited time spent for the purpose of maintaining health seems possible if you limit people's activities (and compare to the fact that they'll be feeling sick anyway). But what about drag? For something just a few 10s of meters, it's likely that you would leave the annular space alone between the centrifuge and the balloon wall. But at what size will it make sense to add any friction-reducing buffers? It depends on how much energy you're willing to put in, but it seems simple to compare this to the energy expenditure of other station systems.

That sounds like some pretty low-hanging fruit for developing a practical case for more investigation into this tech. Importantly, some push into this area would raise some obvious experimental pathways to establish the friction buffer sheet stability.

Scaled Experiments 

Stability of the friction buffers is a tough topic, so it makes sense to give up on the analysis and defer to experimental evidence at some point. Fortunately for us, the available fluids helps to make the problem easier for us. Air is a low density and low viscosity fluid. Water an extremely obvious stand-in for scaling based on similar Reynolds numbers.

I have two types of things in mind:

    sheet Reynolds number
    true scale model

You could scale the entire system of an artificial gravity tube by selecting an experiment geometry that is exactly similar to it but on a tabletop scale. In practice, however, this leads to sizes or speeds and torque that are just not workable. This could not be a tabletop scale experiment.

Instead, it will make more sense to emulate the separation distance and speed of the friction buffer layers, and see how the multi-sheet stability looks with different kinds of configurations.

Problem with all Center Connections
I misspoke somewhat in my previous post introducing transport of commodities. I had presumed that some commodities could be sent through connections that existed exactly on the axial line. This can not possibly be the case.

It is an easy mistake to mistake. You can simply imagine that cargo moving through the center can move slightly to the side of the axial line itself. The rotation speeds will not be substantial for a great distance beyond this, and the weight itself would not be overly burdensome. The problem comes when you realize that the rotating part... well... rotates. You can't simply move cargo to the size of the connection and move it along, because the line (pipe, wire, etc.) going to the colony rotates. If the cargo stalled inside of the plane that this line rotated in, then it would collide with the line.

This seems impractical in my vision of the economy. It would be far better to keep the center-line of artificial gravity tubes completely empty aside from rails which which cargo is moved along with. The challenges for connecting at a larger radius for power, water, information, etc. are completely solvable. Transit of bulk materials is much trickier, so the center line would need to be reserved for these activities.

Relative Movement of Tubes and Balloon

I must take some time to argue with myself on the subject of how the artificial gravity tubes move relative to the balloon "wall". The most simple solution is that they don't move. Actually, this is quite practical in terms of the inflation physics. Halting the rotation of an asteroid in general isn't a hard problem. With a strong tether, you can dangle a large rock from the equator, slowly releasing it to a large radius, pulled by the rotation of the asteroid. This is a cheap way to expel a great amount of the asteroid's rotation. You may still keep a small amount of rotation to stay sun-synchronous. The inflation process itself also reduces the rotation speed. The only cases where this is not practical are small asteroids. Those will be easier to manage in general, and will probably have rotating joints for electric connections.

So I envision artificial gravity tubes fully tethered to the wall. This will help to keep them suspended in-place inside of colonies with insanely huge scaling. It will also develop a hard electrical connection between the tubes and solar panels that may lie out the surface of the asteroid (or slightly off). Things can be balanced by a tether at the asteroid-sun L2 and L1 points (these are not impractically far away either).

Because of this, I will personally have to abandon the idea of the geosynchronous washer-shaped radiator. It's better to not rotate and tie the tubes to the walls (if sufficiently large).

Pressure Management and End Seals

Friction buffers are not rigid. I mean, they're monstrously huge. Instead, they would maintain their shape by having some positive pressure inside of them. Note that this positive pressure is relative to the next outer-most friction buffer sheet. This constitutes some fluid management constraints. Keep in mind that air pressure changes with the rotational acceleration (like a gravity gradient). Because of this, we can draw a graph of the pressure over an outward line from the axis to somewhere on the surface of the outermost friction buffer.

Are there any complications with this scheme? Of course there are. The ends are pinched, remember? As you get closer to the end-cap, the friction buffer sheets pinch in as well. This means that the acceleration gradient will be more gentle. In the limit case, consider that the outer-most sheet is almost stationary, but the 2nd outermost sheet is rotating very slowly. Going from the outside to the pinch point will be a small change in pressure. On the other end of the spectrum, the air pressure changes a great deal from surface to axial line inside the tube itself.

We would like to equalize all the different pressures around the end seals (this would make it easier to seal, clearly), but this isn't possible due to the pressure demands of the friction buffer layers at their full radial position. The real problem comes at both ends of the tube where we should maintain a negative pressure inside the spaces between the friction buffers. Positive pressures are easy, negative pressures are tricky. I'm not sure exactly how this problem would be solved, but I think there are a lot of tricks to mitigate the challenge.

To be clear, I think this is one of the biggest problems for the viability overall. It probably comes somewhere close to the stability of the buffers in general.