Re: scaling laws

[John Denker <jsd@MONMOUTH.COM>]

At 10:45 AM 11/29/00 -0500, Joseph Bellina wrote:
> By accident the whole issue of dimensions has taken on a new meaning for
> me based on some work my spouse is doing regarding early use of
> wind-tunnels in this country and England.  One problem was how to scale
> up (pardon me Bob) the wind tunnel measurements to be useful on full
> scale aeroplanes, since flight tests in those days were frequently
> tragic.
> The scaling committee of the National Physical Laboratory at Teddington
> included the best minds in England, including as I recall Lord Rayleigh,
> the director of the Cavendish Laboratory.  Rayleigh wrote articles about
> Dimensional Analysis, in which one could construct groups of variables
> where were dimensionless, and so could be scaled.

I wonder if this is the same committee as is mentioned in
  http://www.eng.man.ac.uk/historic/reynolds/oreync.htm

namely a committee including Reynolds, Kelvin, and Froude as well as
Rayleigh.  This committee (especially Reynolds) produced quite a number of
important papers on dimensional analysis and scaling laws (aka similarity
laws).

All this considerably predates the founding of the NPL (1902) and the
invention of airplanes (1903) -- the committee was concerned with the
safety of _ships_.

> I had not realized that this, apparently side-line in my physics course
> which was used mostly for checking the accuracy of students
> mathematical manipulations, and the physical legitimacy of statements,
> had such a noble history.


Noble, yes indeed!

In an elementary introductory course, there is not much use of scaling
laws.  After all, if there is only one mass, one length, and one time-scale
in the problem, there's not much opportunity to form dimensionless groups.

In contrast, in the real world, there are lots and lots of important
dimensionless groups.

1) Aspect ratio is the grand-daddy of these.  In 1638 Galileo published a
paper "On Two New Sciences".  One of these new sciences was the laws of
motion.  What was the other?  SCALING LAWS!  He discussed how you might go
about scaling-up a cat to the size of a horse.  The aspect ratio of the
bones would have to change.  The spatial dimensionality (D) enters these
relationships in an interesting way.

   Vocabulary note:  The "dimensionality" D is a number (typically 3).
   This is essentially unrelated to the concept of "dimensions" of a
   unit such as velocity (typically meters per second).

2) The Reynolds number (1883) is basically the ratio of inertial forces to
viscous forces in a fluid.  This is covered in chapter 1 of typical
fluid-dynamics books.  It is also the topic of section 41-3 of _The Feynman
Lectures on Physics_ volume II.

3) In condensed-matter physics we have critical phenomena.  Near the
critical point, lots of variables diverge (including heat capacity,
compressibility, correlation lengths, et cetera).  They scale like the
reduced temperature (1-T/Tc) raised to some weird fractional power (a
"critical exponent") such as 0.330 or whatever.  These critical exponents
are all dimensionless.  There are some beautiful and amazing _scaling laws_
which are algebraic equations relating certain critical exponents to
certain others, and to the dimensionality D of the space in which the
phenomenon is occurring.

4) Similarly in nonlinear dynamics, we have the Feigenbaum scaling law,
describing how things diverge as we approach the onset of chaos.

===

The list goes on and on.

Bottom line:  There are LOTS AND LOTS of scaling laws.  Some of them
describe what happens when you "zoom in or zoom out" (changing the length
scale) but some of them are much fancier than that.