[HN Gopher] The mystery of the small dimensionless number with a...
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The mystery of the small dimensionless number with a big effect
Author : pseudolus
Score : 29 points
Date : 2021-12-27 12:06 UTC (1 days ago)
(HTM) web link (phys.org)
(TXT) w3m dump (phys.org)
| kevinpet wrote:
| Mach number is an odd example of a dimensionless number,
| especially with the example of measuring it in kph or mph (as
| opposed to being independent of altitude or temperature).
| oakwhiz wrote:
| Funny, I was thinking the exact opposite. The Mach number has
| two velocities that cancel out, and velocity tends to be an
| easy concept to grasp, making it a textbook example of a
| dimensionless number. The part where Mach numbers are converted
| back to velocities requires you to input knowledge on the speed
| of sound, but there isn't a single answer for that unless you
| know the composition of the fluid, which is dependent on
| chemical composition, altitude and temperature. If you take a
| guess for that number, it's a unit conversion error rather than
| a variation in the dimensionless quantity itself.
| ianai wrote:
| It's a mathematics of chaos paper/number:
|
| "Two competing effects determine the vertical motion and
| concentration of particles in this region--gravity pulling them
| down to the ground and turbulent air that generates drag forces
| that can lift them up. Researchers often quantify these competing
| effects by a non-dimensional settling number, Sv, which is the
| ratio between how fast the particles settle in the absence of
| turbulence and the characteristic speed of the turbulent air flow
| near the surface. The conventional wisdom is that when Sv is very
| large, the effects of turbulent winds on the particle motion can
| be ignored, while when Sv is very small, the effects of
| gravitational settling can be ignored."
|
| "our numerical simulations revealed something very surprising;
| gravitational settling strongly affected the particle
| concentration profiles in a turbulent boundary layer even when Sv
| was very small."
| motohagiography wrote:
| Reading this as a complete layman, I'm interpreting that these
| non-dimensional numbers are like a natural constant, not unlike
| pi, e, or feigenbaums constant that appears as a proportion in
| a relationship between types of objects (line to a circle,
| recursion and compounding, period of a discrete function to a
| continuous one etc), but it is the artifact of a locally
| defined or observed dynamic.
|
| As though the "non-dimensional" fruit constant would be arise
| from a quantity of apple as measured by a unit defined as the
| average weight of a sample of oranges. If the orange average
| yields a Real number, small changes in either quantity have
| random effects and probably makes the comparison of apples and
| oranges appear chaotic, and some derivative in the relationship
| between them could yield an irrational constant.
|
| Their example of R0 for viral spread and Mach2 always being
| twice the speed of sound are ways of enumerating a phenomenon
| based on some consistent physically observed measure (e.g.
| speed of sound in our atmosphere), and then applying an
| abstraction or counterfactual to the number (2x, half, etc),
| and you get irrational artifacts as a result. That is, you can
| use these non-dimensional numbers to reason effectively about
| physical phenomena, but not to compute precise simulations of
| them them in reasonable resource/timeframes.
|
| I'd wonder if the precision of a given non-dimensional
| measurement and the complexity class of the algorithm required
| to compute it past a certain point of precision have a fast
| heuristic for determining it. Again, layman with language and
| not the tools or background, but coupled with an emerging
| popular interest in analog computing (optical chips, AI,
| fractals, synthesis, quantum etc) it just piques the
| imagination about what in math is signal, and what may be
| noise.
| MattPalmer1086 wrote:
| If you read further, it says that although Sv is small, there
| are always places where the other effects are smaller, and so
| it dominates. No matter how small Sv is, as long as it's not
| zero, this will always be true.
| [deleted]
| patcassidy2000 wrote:
| The fine structure constant is a bigger mystery in my opinion,
| though physics has alot of unexplained constants right now.
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