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.