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.

First Simulation Stability Results

In broad terms, if the friction-buffer idea is not "complete", it might not be really worth it to get excited about. I could be "wrong" in a certain sense, but only in the deeper more detailed engineering. So I'm trying to open the book on that, and here I will talk about (working name) wobble stability. This is as opposed to axial stability, so let me first give a roadmap of the context to put us in the right place.

Roadmap of Concepts

On the basic claim that flow dividers will reduce drag by the amount already quantified - I don't really have any doubts. Lately I have been using AI to re-do the numbers, and it comes to the same conclusion.

Beyond "flow dividers reduce friction" observation, probably the next most important observation is that you obviously should use inflatable dividers. If not, you create the need for large struts in a space they won't fit. There are pitfalls here, and I have many times had wrong-headed ideas on this. No, they can not be made of saran wrap. Each stage needs to hold an incremental amount of pressure out of the centrifugal pressure (which is much less than 1 bar), which is significant for large sizes. Their material needs are still self-evidently much less _even in total_ than the inner hull where people stand on. Then, _that hull_ is still much lower-need that a space habitat rotating in vacuum. The second-order consequence of this is that you have to walk the pressure profile up to the taper point, which is the subject of quite a few posts here.

These considerations of pressure profile between the friction-buffers not only reinforce the need for stages to hold a pressure difference (mPa range), but also the need for bleed air. You need to continuously have air entering each stage so that the connection point sees a constant flow of air out. This maintains the shape of each stage, which is required to a fairly tight tolerance in terms of fraction of its overall dimensions.

Now since we've arrived at the subject of the connection point, it's clear that the connection point needs to serve some purpose to maintain axial stability. Thus, what I called buttressing, described in a prior post.

As an addendum to that concept, some form of _contact_ is needed to hold axial position (the axial stability problem) because fluid dynamics is not doing you any favors there. It will... dampen movements. But eventually the air will move out of the way, up to the point of allowing contact. Still, there is not any particular _driving_ force we expect from the fluid. Things need to be held in place in some sense, for external air currents things like that.

What happens if contact occurs? Good question! This goes into the unsolved problems list. I might call this the rebound problem. It could depend on surface roughness, it could depend on a lot. Clearly important for safety and needs some results. Any physical experiments will probably hit this relatively fast. You know, by accident.

Returning from that digression, we will label axis of x, y, and z. The axial stability problem is the problem of stability in the z-axis. Now we get to our destination here - what about stability in the (x, y) plane?

That's a big question - and probably the _biggest_ question for the overall viability. Almost everything in this blog hinges on a satisfying resolution to this problem, which I will call the wedge stability. I have code for numerical investigation here, written mostly by AI.

Displacement Formalization

To clarify what it is we even mean to ask, we start with a cross-sectional view. At the center you have the hull (floor where people stand). It is rotating, and it has stages surrounding it rotating at various intermediary speeds.

Now the perturbation should be pretty clear. I find it much easier conceptually to move an arbitrary friction-buffer stage, as opposed to the hull or the outer stage. We can just assume the outer stage is stationary for simplicity and keep the hull stationary too, which is relatively massive making this a fine approximation. So here is a displacement of one stage into the negative x-axis.

In simulations I always displace one stage in the positive x-axis to start. I don't know why I drew it like this. Anyway, this shows the (x,y) axis and a starting displacement, the displacement is just to get the simulation going. Eventually, ultimately, we want it to be stable, so we want the displacement to go away.

What are the concerns? As we get into the wedge effect fundamentals, we can make a prediction.

The Wedge Effect

The wedge effect is what allows rotors of journal-bearing systems to levitate on a thin film of oil for some designs. These days, something called "tilt-pad bearings" makes it not relevant for many large machines. Nonetheless, there is plenty of literature out there on the subject, and in the simulations I have, I'm just re-producing the same integral the literature has.

This key integral measures the "squeeze" as a rotor with an offset pushes fluid into the point of lowest clearance. Thinking of it, still, mostly in the terms the literature does, you have gravity pulling the rotor down, and a fluid force vector that balances gravity which it must do exactly.

The biggest conceptual difficulty I have with this is ignoring the torque imbalance. You can see that the center of pressure must exert a force that is directly "up". That creates a torque because it is not acting in the line of gravity. This... doesn't do anything though. Because the rotor is being spun by a motor. The imbalance might make a spin a little bit slower, but that's it. There's a different equation in the system to account for the changes in angular speed, and that should be covered there. In any case, it's true that the net fluid force does not act in the direction of the center of the circle. Two things are going on:

• pressure buildup causes a large force in radial direction
• depending on the channel width, shear force increases or decreases

These factors are accounted for in the equations used. The shear force comes fairly directly from the separation distance, the retarding torque comes from a fairly simple integral given the offset from center of the rotor. The pressure buildup requires integrating to get the pressures in the first place, and then summing those via another integral, so pretty much a double-integral. I think it's most helpful to look at the pressure over the angles (making a full circle) before the _2nd_ integral so we know what we're dealing with. You can get these from running the script in the git repository:

python sim_250.py

The pressure at any given point pushes the rotor in the direction of its center. But after integrating over all the points, that will push the rotor at some angle that does not align with the offset. At the extreme offset, all of the pressure buildup is right at the choke point, so it _almost_ aligns with the direction of offset, and pushes back to restore its original position. But at the other extreme - a tiny offset, the push is almost entirely to the _side_.

From here, you can think of either a ratio of Fr/Ft, or the angle at which the force acts. Since we are free from gravity, this is the only force acting. The friction-buffer in practice has very little mass (even compared to the air), so the force accelerates mass that I just get from the mass of air in the stage's annular space (I know, this has problems). The repository has some graphs of these, but I will not include them here because it is already over-crowded with images.

In any case, just from the basic wedge effect equations from literature we reveal the core concern - given a small displacement we get pure Ft. If you look at a graph where the stage is displaced in positive x-axis, the force acts in positive y-axis. This is _strange_. Right away, it is neither stabilizing or destabilizing. It will cause acceleration, but orthogonal to the displacement. But of course, as it accelerates, it moves, and the force moves with it. Still mostly in the Ft direction, we can logically predict the force to "chase" the rotor around the circle, and here is where the worry begins. It might likely speed up in this case. However, at some point, Fr will grow in magnitude and it is possible to imagine a stable promenade around the circle.

Results

As given, which the most-correct type of numbers, we have an unstable system. This gives the center-line of various stages, and they spiral out. Eventually they hit, and the simulation ends with a violation of geometry.

Were there ways to make it stable? Kind of, yeah. Going back to the wedge effect itself, journal-bearings levitate on a film of oil precisely _because_ of a displacement. So what if we add a steady-state displacement? This is enforced by boundary conditions - meaning the outer stage would be held in place against the motor spinning the null. This is somewhat of a novel concept, that the math leads us to.

Results from those:

This did lead to a stable simulation. I pre-selected x-axis values, but did not so for y-axis values. And if you look back at the wedge effect, again, it doesn't act directly along the line of displacement, so the y-axis values are actually finding their new stable positions. Cool!

Unfortunately, I had to use a fairly extreme displacement. It's not obvious, but this _could_ undermine the entire idea by increasing the drag in the choke point. But these are very imperfect simulations, let's move on.

One thing the AI told me repeatedly, is that there's no dampening added to this simulation in particular. So it could be a forgone conclusion that this is always unstable, with no connection to the physics. This is a hard problem that still needs noodling on. However, I did experiment by adding a "fudge factor" of the Fr and Ft ratios.

Seeing this is kind of comforting... but also disconcerting. I am fairly darned sure that there is some envelope where this can work with constant "wobbles" shown here, but where the stages do not rest at center-line. If reality falls in this regime (which I still don't know) that means that, simultaneously, you CAN use the scheme in the unsupported Russian doll configuration and it will work, but you very well might not WANT to, because it is going to cause persistent movement and you would rather it not.

Take-Aways So Far

In the repo, where most of the content is written by AI, there are some limitations of these simulations outlined. And they are a LOT. Firstly, like oil journal-bearings, we are only even attempting to do laminar simulations. Well, if flow was laminar, the friction-buffers would make no sense to begin with. We are extremely very obviously concerned with turbulent. And that is going to give wildly different results as the channel width expands. So this is very absurdly limited.

Other flaws in the methodology are almost too numerous to count. The simulation does an integral, but in that neglects all momentum of the fluid in the discretized volumes of integration. So we tried very hard to just track the variables for each stage, but tracking variables for each element of fluid in each stage would be more appropriate... just more variables. This would also allow somewhat faithful dampening which are very likely to change the result. Even with these changes, however, I'm not convinced that we can obtain full center-line stability.

But possibly the biggest single flaw, which is unique to this problem, I think is treating any friction divider as a rigid structure to begin with. Even with inflation making giving it shape, it will be fundamentally deform-able. It's just a sheet with a fluid on each side, with tension in the sheet. That is going to give wildly different behavior from the simulation and this is the effect that I am most interested in. As wedge effect literature focuses on pressure buildup as you approach the point of narrowest clearance, I'm not convinced that happens at all in our case, because the sheets can move. Locally.

Overall, I remain cautiously optimistic, but have also accepted that there is likely going to be some space for an engineered solution for stability - for instance, a ring of separators inside of each stage. That can have a negligible impact on drag reduction, and also keep the stages from center-line divergence. The persistent wedge is endlessly fascinating, and shows how more understanding will probably open up the space for more elegant engineered solutions, and I am realizing how deeply of an not-understood problem this is that I'm looking at.

Taper-nested With Slightly More Space

Starting out, let's walk though a graph of the style given in a prior post. The question we are asking is "where can we terminate sheet number N", because they don't come to center-line. Somewhat arbitrarily, a relative speed of 3 m/s is allowed, as a seal must be established over that relative speed. The two series of air-relative and tube-relative give the maximum possible radius where the sheet will be at 3 m/s relative to the tube or to air.

In this case I am using a 500 meter radius tube, because I'm also interested in exploring those larger sizes here.

An easy observation is that, on the sides, relatively few sheets are allowed to have a relatively large opening. Because these are relatively few in number, I've draw in a "design" line where I'll simply use less than the maximum for the edges. I believe this will help both sanity (for now) and possibly logistics in practice.

The meaning of taper-nested is that the sheets come in towards center-line as much as needed. This graph means to quantify. Using these specific numbers (with 32 sheets, chosen somewhat arbitrarily), I want to give a better scaled illustration of what a "larger" tube of 500 meter radius might look like in practice, as a cross-section. You should note that the access opening here is more like 20 meters in radius which is somewhat less than the opening for the sheets around the center. For scale:

Going from big to small, with this being an end-on view:

• Grey outer circle: 500 meter radius of actual tube itself
• Green circle: the center opening of the outer-most friction-buffer
• Yellow circle: the center opening of the middle friction-buffer, the smallest center opening of the center openings
• White center: the access opening for moving people, goods, and air, in and out of the tube

This helps to illustrate the R^2 factor, as an increase in linear dimension of 2 is a difference in area of 4. So it certainly seems worth it to use any extra buffer space possible, as opposed to having a long access tube where the access opening is.

Aside: there is a weirdness here, which I need to give in specific terms:

• Sheet 16
• moving +3 m/s relative to the tube
• moving -3 m/s relative to the outside (stationary) air
• The actual hull of the tube itself, buttressed against sheet 16
• moving 3+3 = 6 m/s relative to the stationary outside air

This is a oddball kind of situation. At first look, it seems to me the most reasonable thing would be add 2 additional sheets around the access opening that divide the relative speed between the tube and outside air. This would only cover the distance about 20 meters to 40 meters in radius. Dreamily, I wonder if these might be retractable for cases where traffic is higher or large items have to be moved in or out.

Wrapping up this thought, I want to give an accurate sketch of the 500 meter case, at last.

The somewhat incredible density is why I'm always talking about a 250 meter reference, instead of this, which is twice as large and has twice as many sheets. This is more material-intensive, and less space-efficient. Nonetheless, I must acknowledge value in human factors of slower rotation (thus Coriolis forces) and and more wide-open interior space.

I also needed this to articulate the side view of taper-nested in mostly the standard sense but just adding some very detailed-oriented tweaks.

The Bleed System

The diagram above makes it very hard to believe that the sheets could resist any external force. Again, these will be inflated like a balloon. Each sheet hits either a rotating structure or an externally-connected static structure at an angle near to 90 degrees, maybe 80 or 85 degrees. This means that it lacks any kind of backstop, or buttressing, for the flow dividers as I suggested in another post.

In other posts I have mentioned another design element that I assume is present in basically all my writings. In essentially any case you need a way to put additional air into the sheets to keep and outward pressure to help maintain the shape. The rotation itself does this to some degree, but probably not enough, and not for the entire shape. These prior pictures here mention a small flow of air. Let me redo that same thing, but with the 500 meter reference in mind, and more granularity, and introducing a new thing - soft segmenting of the shape.

Again, return to the fact that there is no surface for the friction-buffers to "push" against to maintain position, at least in the axial direction. What's a way around that? Well you can just puff air into one end of the friction-buffer. Because of the small channel size, this is likely enough on its own. This is a big construction, so we probably going to want at least a handful of these in every sheet, add some sensing, and control systems to respond if needed.

But there's one more way we can do better. I added dotted lines to suggest that we can introduce additional very suggestive flow dividers between two of the sheets. These would not be a tight seal (they have no need to be), but would just lightly reduce the amount of air flow your valving or pumping would need to move to affect the shape at a particular place.

Bleed System Design Space

Similar to the previous pictures I've drawn on this, the above illustration implies that the air bleed system is a network of valves. These are passive devices that let air flow from an inside sheet to the next out-most stage depending on how far they open. Going all the way to the tube itself, pressure is always higher in the more inward stages, because they are rotating faster.

Even if the valves are passive, they may or may not have any controls applied to them. Without computer control, they are simply holes in the sheet... probably holes of carefully selected areas. These would simply be holes that let some air through, taking advantage of the pumping that is already happening by the main rotor. This air can is then used to maintain the inflation of the sheets, and the air ultimately leaks out the sides. The reason this has started to interest me is that I've realized that this could be relevant for early experiments. As those experiments are mainly going to be concerned with wedge effect and wobble, shape-keeping is likely to play into that. Sacrificing a teensy tiny amount of efficiency could mitigate potentially larger problems of contact and failure.

For extremely large tubes, these are extremely large operations, and if we can do better we would. The problem with passive valving is that it puts the air in a free-jet condition where it loses a LOT of its energy, because it's more than what's needed. The alternative is that you could have pumps go from the outside in, as opposed to inside out. The pressure difference you need to deal with is relatively small, and these are technology-level of computer case fans. But again, this could feed into the control system to prevent sheet-to-sheet contact which could be a very big deal.

The (lack of) Prior Art on Annular (Cylindrical) Flow Dividers

Myself, using old Google Scholar, I previously struggled to find any literature that might hint at the fluid dynamics mechanism I've proposed on this blog, which I tend to call the "friction-buffers". Let me be super clear what this describes:

• Layman:
• A tube in air is rotating, and it is surrounded by cylinders of increasing size. The cylinders rotate at slower speeds, because they are dragged by air movements caused by the tube. The cylinders allow us to spin the tube more easily than if they were not there.
• Kind of fancy:
• Multi-Annulus Taylor–Couette system with freely-moving intermediate cylinders
• Another:
• Segmented annular flow with passive staggered rotation enabled by flow dividers that suppress Taylor Vortices

I want to be abundantly clear about this - I do not want this to be my original idea. That just makes it harder to defend. Over and over again I tell myself that somebody must have come up with this idea before. But if I search I come back empty handed.

Gemini Research Output

That hasn't changed. But what has changed is that we have AI now, so at least I can prove that the AI can't find comparable background literature either. In case anyone was going to ask, although I am a major AI power-user, absolutely none of what's written here is from AI. Here's output from the latest Gemini:

https://gemini.google.com/share/0c90b978750a

Let me highlight the 2 key conclusions

1. It agrees with me. It didn't even take convincing by follow-up prompts, which other AIs have.
2. It can't find background literature either

On the first point, here's an evidentiary quote

The investigation into using intermediate, free-floating concentric cylinders to reduce viscous drag and suppress turbulence in high-speed annular centrifuges confirms the viability of this hydrodynamic control technique as a compelling alternative to vacuum operation.

I want to call out the bold, and really emphasize that this goes way further with the claims than I ever would have. This is probably not useful for Uranium enrichment or any other lab centrifuge applications. If drag matters, you can probably solve it by pulling a vacuum, and if you can reasonably pull a vacuum, that is obviously superior to the method I've proposed. I wouldn't call that "compelling", but if I was in academic I would probably write that anyway... but that's just an academia thing.

In any case, if this could be useful and practical for centrifuges, it's almost guaranteed to work for the much larger and slower case of free-floating tubes in microgravity. This is a full-throated endorsement. All the AIs agree, for whatever that's worth, which isn't much.

And as for (2), I'll just have to show you what it did find.

Rotating Half-Discs Drag Reduction

The best reference by far is:

2021

Reduction of turbulent skin-friction drag by passively rotating discs

https://eprints.whiterose.ac.uk/id/eprint/209042/1/2106.12824v1.pdf

This is abundantly clear that it proposes to have discs on a surface with a fluid moving over it, where half of the disc is kept under a divider. This allows the disc surface to be closer to the velocity of the fluid as opposed to the surface. It is very intuitive how this could work with the discs being rotated passively by the fluid, just like it is intuitive that you could reduce drag by putting a treadmill on it. These are just slightly more practical variations of a passive treadmill (if I am to tell it).

You see half of the discs because the other half is covered

Ok but this is still a bit abstract without a use case. I like this sketch, because it looks like an airplane wing.

Airplane wing maybe, my own interpretation

Could this method improve the fuel economy of an airplane? Yeah, that's totally physically possible. This idea does share many properties of the friction-buffers proposed in this blog.

• Introduce a flow divider (in my case) or a skin friction attacher (their case), which is a "sheet" in all cases
• Allow that sheet to move passively, meaning, it is moved by the flow itself
• There is an expectation that the flow becomes less turbulent, and the energy lost due to viscous forces decreases

So almost-check, check, and check. The other difference we might point out is the entire geometry is different - a surface vs. annular flow.

It looks like this paper also covered the same thing

2013

Turbulent drag reduction through oscillating discs

https://www.researchgate.net/publication/258796176_Turbulent_drag_reduction_through_rotating_discs

However, that was much harder to follow because none of the pictures made it entirely clear what it was showing.

Other Close Misses

This paper describes a passive mechanism for drag reduction in Taylor-Couette flow. Seems promising!

2024

Research on the Sealing Performance of Segmented Annular Seals Based on Fluid–Solid–Thermal Coupling Model

https://discovery.researcher.life/article/research-on-the-sealing-performance-of-segmented-annular-seals-based-on-fluid-solid-thermal-coupling-model/aabc017bc11133f9a41c1f81a3ec4b33

However, that is very clear that it optimized using a groove design. And looking at the pictures further, it might not even be talking about the general topic at all.

Looking further into papers on Taylor-Couette flow is an exercise in madness. One takes two cylinders (tubes) and places them next to each other and has them spin.

I'm tempted to believe that somebody wouldn't write a paper on this friction-divider concept. Let me explain why in 2 scenarios:

• Do not include the effect of oscillations with movement of the cylinders (global stability problem), you've effectively made an undergrad-level problem, not worthy of CFD or of a paper. It's too easy and the effect is obvious
• Include the movement of cylinders, you've added mechanical boundary movement to your CFD at which point you've made a problem that's too hard and give up

So basically, to get progress, I, or you (the reader) need to do it ourselves.

CFD or Experiment

Experiment.

This brings me back to my prior dichotomy. It's either "CFD is useless" or "CFD is impossible". Also, looking towards the specific application, the predictable objection should be a tiny bit more nuanced than "you can't divide the flow in half". Problems you're most likely to hit probably won't occur until you're at a high number of sheets. Not just 1. I'm thinking 4+ sheets to really get some value out of it.

I have some material to write on global oscillations, and maybe it's correct in its approach. But there's no natural confidence in it. Actually doing the experiment, showing the friction-buffers stay in place, and demonstrating a performance gain, would get miles and miles further confirmation that eigenvalues from the coefficients that you came up with on your own. Garbage in, garbage out.

Numbers on Staggering, and new Buttressing Idea

 Qualitative drawings of the friction-buffers sometimes miss an important details, and I believe this is true of the final illustration of my last post. Here, I hope to simply drive this train to thought to its final conclusion in broad terms.

Specific Numerical Methods for the "Ruler" Shape

Doing this took a spreadsheet (intent is that it is publicly visible).

Let's start out writing down some criteria. I was inconsistently using an incorrect criteria that "radius of sheet termination is where its velocity is within +/- 3 m/s from its neighbor". This is not correct, so I'll write the correct criteria in a bullet point:

• constraint: All sheets at all point must move +/- 3 m/s relative to all its neighbors, at all points.
• consequence: As you widen the opening of one sheet, the relative velocity of its revealed neighbors must be +/- 3 m/s
• for some sheets: this is the same as being within 3 m/s of its neighbor
• for other sheets: they must divide up the flow between sheet numbers far away from its position

These are stranger than they look, and give the reason for the "ruler" shape I outlined before. However, I only did that qualitatively so I'll just do the same in a quantitative since here. You can refer to the spreadsheet for numbers, but for completeness I need to give the algorithm because it's not as simple as an equation. Start with the scenarios of interest:

• 125 meter radius, 8 sheets
• 250 meter radius, 16 sheets
• 500 meter radius, 32 sheets

These are chosen because they satisfy the 2^N number, and also because they're close enough to previously published numbers such as to be presumed acceptable in terms of power consumption. Now, for each scenario, for each sheet:

• Calculate the statically-obvious values for each sheet (1-8, 1-16, 1-32)
• The radius of the sheet in the middle cross section
• The angular velocity of the sheet
• For the 0th and Nth entry you can enter in the tube's angular velocity and 0 angular velocity
• Go to the N/2th sheet
• This sheet will have the 0th and Nth sheet as it's revealed neighbors, so those are entered as the 2 relevant neighbor angular velocities
• Divide each remaining numerical segment into 2, and fill in revealed neighbors for its apparent neighbor
• Example: the N/4th sheet has 0th and N/2th neighbor as neighbors
• Repeat this process to fill in the rest of all the neighbor angular velocities
• From those revealed neighbor angular velocities, find the point that the neighbors are moving 3 m/s relative to each other, and that's how wide the sheet's hole can be 

This is very weird, but it still produces the same "ruler" shape. Here, I report the height from the floor which corresponds well to the illustrations from my last post.

R=125 meter scenario

R=250 meter scenario

R=500 meter scenario

In all these cases you still get the "ruler" shape... however, even for the smallest "ticks" on the ruler, there is still significant height from the floor, about half-way. This just seems to be how the math falls out, you can reason out why if you try hard.

Material Reduction from Staggering

Comparing to the taper-nested reference, we use less material, because the only real observation here is that some openings for some sheets can be made larger (thus, using less material). So let's slap an improvement number on those.

		Sheet area
R	sheets	no staggering (km3)	staggering (km3)	improvement
125	8	2.850	2.916	2.25%
250	16	21.213	21.707	2.27%
500	32	162.888	166.682	2.28%

It's as well-established prior observation on this blog (almost from the first post) that you have to use more dividers (and thus divider area) as you go to bigger radii. I didn't adjust for this, meaning that 500 meters gives bigger livable area in addition to needing more supporting material, so keep that in mind.

The "no staggering" case presumes that all sheets have a hole which matches the size of the access hole exactly. Then there are a number of route geometry work to sum up all the areas of all the sheets, and that's what I did for both the "no staggering" case and with the data shown in the plots.

It is kind of interesting that there is more improvement for larger sizes, but in absolute terms it is marginal in all cases. This will bring me into the economic argument later.

Decent Illustrations of Staggering with Curvature

With this, I can try again with the prior paradigm to draw the friction buffers. I still don't aim for exact numbers, but wanted to get something representative. As you might have guessed, doing anything other than the N=8 scenario would be tedious, so all drawings would map to the R=125 meter scenario.

This gives us a couple of things we were looking for, with some drawbacks.

• Good:
• Friction buffer sheets legitimately have outward curvature everywhere, without any dramatic conceit like a rigid floor integrated (last post)
• The access window is acceptably small-looking, it's an single circle as opposed to a long tunnel, should flow air & people & goods well
• Bad:
• Space inefficient, as you can see big open spaces inside some sheets, mostly an artifact of striving for an exactly circular shape for the inward curve
• The connection angles are terrible at several connection points... I can barely tell what's happening in some places

How can we clean this up? Look back at the plots, you see that sheets 0 (the hull), 4, and 8 all have to come in almost exactly up to the access window. We have not yet taken advantage of the taper-nested idea, so we will do that. However, we want to do it in a way that maintains the direction of outward curvature and gives nice nearly-right connection angles which is important for the air ingress management system.

But visually, this is the thing that didn't work because of the circles. So we will only partially relax that. We will make the mentioned sheet number, 4 (the hull is already straight), and like the hull, straighten it out. This will have a partially-straight segment (might have to be rigid) but the rest of those layers outside of this area will still work like the other dividers.

Doing this, with an eye to connection angle, we'll also return to the slanted hull shape. Here we go:

This is fanciful. To allow the new straight segment we had to re-introduce a long access tunnel. Tradeoffs:

• Good:
• Compared to last, we have good solid connection angles
• Bad (mostly everything else):
• Re-introduced a long access tunnel
• Even less space-efficient

Practical Considerations Leading to Buttressing Design

At this point, we will lean into rigidity, but with a more clear-eyed vision of where we'll go. But first let's better formalize a requirement instead of the hand-waving "connection angle" mention.

• Requirement: we are able to hold the friction-dividers in place
• Radially:
• wobble can be partially dampened by contact near the end cap
• non-periodic bulk forces, like air currents, micro-gravity forces, can be resisted by the edge of the friction-buffer pushing against something
• Axially:
• here too, bulk forces must be resisted by giving it something to push against

If you look at past sketches, it seems clear that taper-zero could have a problem, since it's balloons pushing against balloons. The case of taper-nested seems like this requirement isn't a problem, but you have to pull up the numbers to see that most sheets have to terminate very near to the access window... which isn't necessarily a problem for radial stabilization, but axially there's nothing to push against. It's not obvious that this is a problem but it could be a problem.

To accomplish the ideal for all the stated goals, I'll double-down on the rigid segments for certain flag-leader sheets, and further sacrifice the circular shape. I'll mark the rigid parts with a bold line. This still uses smooth circular curvature in places, but also demands a custom shape.

Instead of a long access tunnel, we will simply angle the sheet N=4 inward. By keeping it at an angle, we still satisfy both the radial & axial push requirement.

Trade-offs here:

• Good:
• Minimal clutter around access window
• Vastly more space efficient, with the theoretical "triangle" wedge shape starting to emerge
• Able to buttress the friction buffers against movement in all directions
• Bad:
• The rigid segment (only for N=4) must now move at a different speed, this requires a stronger direct mechanical coupling over this moving joint for the mechanical forces involved

On the bad, I want to be clear this isn't just a different speed for the outside air (which is stationary), but is between the inner tube and the outside air. So it is offering a "checkpoint" kind of speed so that the slower sheets can push against it and the faster sheets can push against the hull.

If you refer back to the graphs, you can logically predict that very large sizes will have more checkpoint friction-buffers, all requiring mechanical connection and rigid parts. For extreme sizes, you might require not only several checkpoint sheets, but a tree-like network of these extending out at +45 degree and -45 degree angles, every one at a unique speed.

I'll end it here, but want to add one more claim that the benefits of de-cluttering the access window are likely to be greater at greater sizes. Because at those sizes you are managing more sheets with roughly the same spacing between them. So balancing mechanical thrust on those sheets precisely becomes a much bigger deal. In short, when you get big enough you will want buttressing for the friction-buffer sheets. That seems reasonable and intuitive.

My sketch here is awful, they would all be smooth surfaces in practice. 3-D printed, I'm sure. That's likely overkill for small tubes, but this is at least a path to a scalable design that I can endorse, and that's important.

Wide-Mouth Taper and Group-Based Termination (Staggering)

Here, we will entertain an alternative to the tiny access center-hole at the ends of a rotating tube in a gravity balloon (or generally, artificial gravity integrated into atmosphere). Return to our reference design:

• Radius of ground: 250 meters
• Access hold radius: 15 meters

The 15 meters could be pushed, of course. Because it's not that large to begin with, we may be able to accept a larger relative velocity to get, say, 20 meters radius for this access opening at the cost of slightly faster edge speed. That is 40 meters wide. While, to me, this seems sufficient to supply such a tube, it's not the only option and I want to give proper treatment to the alternative.

To start our goal - this is asking for a greater area at which the tube can interface with the surrounding atmosphere. Similar to cans going from the traditional opening to the "wide mouth" opening.

Wide-Mouth Can

Unlike the cans, however, we will go as big as we can, up to occupying the entire cross-section of the tube. Erase everything I've written about the termination of the friction buffers. We have a tube (like the tube of a toilet paper roll), and we know we need the friction buffers around them, and we know that the tube will be moving too fast relative to the atmosphere if we leave it as-is.

Making the Floor Itself Move

To make this work, we will move the ground itself. Similar to how (in any rotating artificial gravity habitat), you can get lower gravity by climbing a ladder, in our new innovation here, you will get lower gravity by walking onto another segment of the tube that rotates more slowly. Yes, this will require some kind of mechanical, moving, coupling between the floor moving at different speeds.

What else do we need? The moving floor segments will be connected to a friction buffer at the matching angular velocity. This connection happens under the floor, so would not be observable to residents. Now we have a tube that rotates fast in the center, and then rotates at decreasing speed as you go to the edge. For less obvious fluid mechanics reasons, this will not be sufficient either. There is quite considerable pumping you will get from this unless the length over which this happens is really really long. That would be possible, but wouldn't be practical. So to smooth out this flow, just like in the other friction buffer termination design, we have to add dividers and this time it goes into the interior of the tube in a very obvious way to the residents. We have enough to sketch now. This starts from the cross-section perspective I've used in prior posts and illustrates the described setup.

Wide-Mouth Tube with Variable Speed Floors

You need a lot of imagination for human-scale stories happening here. The diameter of the large tube is 500 meters, and a human is less than 2 meters. The relative speed between each segment of floor is the same as the existing reference design, 2.9 m/s, 10.5 kph, or 6.5 mph. Speaking personally, my running speed is 7.5 mph, so physical able humans can physically jump from one to the next (assuming no "video game platformer" gap). Moveable walkways travel at more gentle speeds of 2 mph, but the internet informs me that some go up to 9 mph. So real-world movable walkways could cover this, and one on both sides could easily cover it.

Because of these factors, I assume some minor assistance would be added if the gaps were to be traversed on foot, specifically one or two literal moving walkways. What happens after stepping onto the next segment? You are faced with the flow divider. Could a hole just put be in that? Maybe. There is expected to be some pressure gap between one stage and the next, so if there is a door I expect some door-opening resistance, and if it is left open, I expect some notable airflow that contributes to losses. To avoid travel delays, you would need doors along virtually the entire radius so I could see this being a problem. Reminder - if the pressure difference is too great, you can do the good-old 2-doors and a room in-between trick. This is similar to an airlock, but... just ordinary doors.

How would a vehicle travel from the tube interior to the edge? I'm at a loss. You wouldn't use a railed vehicle (how do you match the tracks on the next shell?). A rotating mechanical device to pick up the vehicle and place it at zero velocity on the next shell sounds expensive, but trivially possible. I think my favorite idea is that each shell has ramps built into it, and the vehicle goes in the circumferential direction and jumps to the next shell? Sounds fun.

I frankly have no idea how truly practical it would be to travel this way, it is an exercise left up to the reader. All I have to offer is the observation that it is an additional option. The above-illustrated design does not lose any utility compared to the other designs seen in taper-nested or others throughout this blog. You can ride a lift to the center-line, and then ride a gondola through the center hole into the microgravity space. This checks all of the boxes of being physical, economical, and practical. I just have no idea whether it is useful. If I'm maintaining a list of canonical ideas, things I accept as being in the real design space, count wide-mouth in!

Reducing Clutter, Group-Based (Staggered) Termination

While I'm iffy about the usefulness or need for this, it is academically useful to me to clarify the remaining work we need to optimize the termination of the friction buffers. As you can see in the diagram, this really is the same thing as the other designs, just with a different curvature to the geometry of each sheet (and a floor added, not relevant here).

Let me describe the obvious issue - at the edge of the access space, the sheets are moving at a relative velocity of 0.3 m/s, or less than 1 mph, all while assumed to have the same spacing between sheets as the rest of the geometry (it must, due to the velocity at the floors). This is much less speed reduction than what we need. We can pull that back for some sheets (make the center hole bigger)... but which sheets specifically?

Re-stating our constraint - the relative speed of each sheet must not be more than about 3 m/s relative to its neighbor. But if you reduced the inner radius of all expect the 1st and last, then the relative speed of the 1st and last becomes too high... as they kind of become neighbors. This leads me a form of grouping, like the markers on a ruler.

Center Opening Widened, Relative Velocity Still OK

This is a strange outcome, but I believe it to be legitimate. This also lifts the concern about traffic jams due to a very long and relatively skinny access tunnel. Yes, we still retain a single choke point, but it's not a choke tube. It opens up the atmosphere in the transition region... somewhat.

Implications of Group-Based (Staggered) Termination for Other Designs

This also helps clarify what the actual minimum clutter is in all designs. Through this sketching, I have learned something new about what is possible with almost all friction buffer termination solutions. For taper-nested, a brief sketch:

Taper-zero with Staggered Termination

For years, I have had a gut feeling that something was still off in how I was drawing these, and I believe this is it. The reason it was hard to find was that it was so non-obvious how to state the maximum relative velocity constraint, because it applies to the revealed neighbor sheet, not just the N+1 sheet. It also makes sheet numbers of 2^N desirable.

Even with the briefest visual survey of the taper-zero-staggered sketch, you intuitively see less clutter going towards the center. This concept of staggering the circular opening of each sheet is going to be assumed in most designs going forward.

Putting Some Numbers on the Taper Options

 See the prior post for the definitions here:

Illustrations of the Friction Buffer Tapers

Here, I just want to give more elaboration on the specific connection points because they were only drawn in the abstract before. To do this, I have a simple spreadsheet which I'll just leave a shared link for. To explain what I'm looking for, we'll go by the designs introduced in the prior post.

• taper-nested:
• The connection points can connect to a point along the triangular ramp which is at the end of the rotating cylinder. There is a constraint of a maximum radius where we can connect it, because connecting at a larger radius would leave us with a large relative velocity. The reason for existence of the sheets is to minimize drag, so we can't do that.
• taper-zero:
• Each connection point (confusingly) has to of a small enough radius so that it shields the sheet inside of it. This, again, results in a maximum radius value, beyond which the sheet inside of it will make contact at the air at too high of velocity.
• Note, however, that the design calls for the innermost sheet to envelope basically the entire ramp. This this brings the connection further in, to a smaller radius, it does not violate the maximum radius constraint.

This note on taper-zero is going to cause some trouble communicating, so I'm not going to plot its numbers directly (because they don't apply to all sheets), so I'll instead have a series "air-acceptable R" which is the radius at which any given sheet is moving slow enough relative to the static air so that it can be exposed directly to the outside micogravity.

One item not exposed here is that the access opening is assumed to be at a radius of 15 meters. That is largely set by the arguments around acceptable speed (the 3 m/s thing), so it shouldn't be surprising that no point in the graph dips below this radius.

The biggest thing I want to observe here is that the points are clustered fairly heavily towards the center (low values of R). This does quite a lot to change one's visual image, which were poorly represented by the earlier hand drawings.

Let's take taper-nested as an example. Start with the first sheet, and it can connect about 3/4ths the way along the ramp, next, maybe 1/3rd. But then numbers 6 through 17 are super close together. The sheets at their full cylinder radius should be a large distance apart, like 10 meters, and you won't have this automatically with the triangle shape. Thus, a key takeaway is that you would have to elongate the access tunnel to increase spacing. Because decreasing spacing increases drag, just like increasing relative velocity does.

So my revised mental image of the taper-nested looks more like this.

I've added an additional detail which is compartmentalization in the tapering region. This would be a leaky flow divider, and to be honest I arrived at this addition after a conversation with ChatGPT, where it was concerned about "pumping" being caused in the tapering region. I think it is still small, because the tapering region keeps velocities as low or lower than what they are at the full radius... but I still see the point that we have 2 axes of eddy creation, and a dumb divider is a cheap solution. We already have some controlled air ingress through the channels, so they would bend in the direction of the access tunnel.

Onto the elephant in the room - that seems inconvenient. This would make it harder to transport goods and people in and out. Yes, but it's an obtuse academic over-simplification. Given this reference design (17 sheets), there is clear advantage to having the first 3 or so sheets hug the tube, but beyond that, spending the material to wrap all the way to the access point is pure waste. The direct solution basically winds up being what I've already named taper-zero.

I do have many more thoughts on taper-zero, and I need to mature the numerical constraints for that design (angles get "weird", but this isn't... a technical statement). I almost might start over from scratch with a new approach that helps conceptualize the overall design space better. However...

Rejection of Hybrid Taper Solutions

I speculated in another design direction before.

Very Preliminary Thoughts on a Hybrid Taper Solution

I now believe this was an unproductive dead end. It does not solve any problem that mattered. Total quantity of material strength needed for the sheets will simply never be an issue, ever. You spend more on the ground itself. Sheet positioning is a much bigger issue. Keeping the fully or mostly tensile constructions (balloons) also has a big economic impact. Control systems for the air ingress mater, means of maintaining the seals matter. I think what that design direction was offering was just worthless. In the end, with new designs I'm still working on there might some some non-uniformity between different sheets, but nothing like different material composition from one to the next.