What the characteristic force is

Dimensional analysis has an interesting result in fluid mechanics: the quantity $chi=eta^2/rho$ has the units of force, where $eta$ is the usual dynamic viscosity. The Reynolds number can then be thought of as
$$text{Re} = F_text{Re}/chi,$$
where the Reynolds force $F_text{Re}$ is geometry-dependent but can generally be written as $eta~u~ell$ or perhaps more suggestively $rho~v~ell~nu$, suggesting some product of mass-flow and momentum-diffusion.

The attractiveness is that one might get a microscopic, geometry-independent characterization of turbulence. Laminar flow in some sense displays forces less than $chi$ and turbulent flow displays forces greater than it. This is at least folklore in biophysics, where single-celled organisms often create forces on the pico-Newton scale less than $chi$ with cilia and flagella. It suggests that a large colony of bacteria can surprisingly team up to create turbulent flows if they can all align and work collectively, even if each individual one does not have the force to do it: this has been observed experimentally.

In addition some organisms need to concentrate their efforts, apparently to exert a higher force over a shorter time to grab their prey. (One inconvenient thing about laminar flow is that when you are trying to approach something you often push it away from you, as many people may have experienced while cooking or so and trying to get some little particle out of a cup/bowl/pot of water and just approaching the thing slowly, it seems to creatively evade your spoon.)

We recently got a question here on PSE which asked whether astronauts could use their breath in zero gravity to propel themselves to a wall, the difference between laminar and turbulent flow regimes is that laminar flow is reversible and the astronaut would have to first induce some sort of rotation so as to breathe in a different direction than he/she breathes out; for turbulent flow this would not be necessary, it just matters that the speed of inhalation is slower than the speed of exhalation.

My confusion around it

My problem here is that the Reynolds force defined above is still quite geometry-dependent and it is not completely clear whether, say, it is proper for me to estimate the force of the breath in say a Newton or centi-Newton range, observe that that is greater than the characteristic force by several orders of magnitude, and conclude that we are in a turbulent regime. It’s in other words not clear to me from dimensional analysis alone why the characteristic force is an important quantity, or why the Reynolds force is an actual force (or a scale for actual forces) occurring somewhere in the system.

Is there a derivation of this sort of compare-a-force-to-$chi$ reasoning from mechanical principles and say the Navier-Stokes equations, rather than from scaling principles and dimensional analysis?