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.