Re: How much can science tell us about sculling?

On 29 May, 07:17, Carl Douglas <c...@xxxxxxxxxxxxxxxxx> wrote:
Nowadays many massively complex systems, e.g. 3-D fluid flows, even
chemically reacting flows, are completely & repeatedly solved - not only
by nature's own analog computer (aka the real thing) but routinely to
very acceptable accuracy by CFD in industries which wouldn't pay for
useless results.

That Pete can point to the massive complexity of some common problems
underlines not only their difficulty but also how well their underlying
science & maths are understood. Life is, of course, far too short &
computers too inadequate to plot the moves of every molecule in a flow,
but that's not needed.

I'm well aware that CFD is practically useful and works. However, as
you know, in the (rare) case where you have reason to believe there
isn't going to be any turbulence in your real system, you can usually
get away with running a CFD model without any calibration beyond the
standard 'this fluid is water', and probably with quite large cells,
and still get good results. When you expect there to be turbulence,
you start needing more calibration work, and what that covers up is
the fact that the model cannot really cope with turbulence (it can
only see sizes down to one cell, which may well be enough to see quite
a lot of what looks like turbulence, but of course it cannot examine
the similarly complex motions that would in a real flow exist within a
single model cell). And this is not a case of your CFD program being
cheap and not programmed to deal with turbulence, nor is it a case
where it's understood by physicists but they haven't programmed a
computer to work with it yet. Physicists do not claim they understand
what processes really operate in turbulent systems; engineers simply
know that whatever those processes are, they seem to be repeatable
enough that you can get away with fudge factors and the computations
don't go too far wrong.

If you prefer a short version: any time you give me a system with a
large number of interacting objects all obeying the same simple rules,
you may think that system is well understood, or the mathematics is
understood - well, my job is in this area, and I will tell you that
usually it is not.

Usually even with vastly simpler rules than anything in fluid dynamics
we can really only say a little bit; we can either talk about local
structure or sometimes global structure, but any time you ask a
question whose answer depends on both, we will not be able to answer
it. The best we will be able to do is usually to run a lot of
simulations and then say we believe they are probably about right.

Here is a good example. The system is the positive whole numbers. You
start at one number, and move to other numbers in succession, with the
following two rules. If you are at an even number x, move to x/2. If
you are at an odd number x, move to 3x+1.

Very simple system; if you try a few different start points you'll
find you always end up with repeating forever the sequence 4->2->1->4.
So - does this always happen, or is there a start position which never
reaches 1?

Another slightly more complicated system: one point, moving on a
square grid (or three-dimensional cubic lattice, if you have a good
imagination). Every second, it moves to one of the four (six in 3d)
neighbouring lattice points; it chooses the point to move to randomly
(with equal probability, and independent of previous choices). This is
a random walk. The behaviour is by now fairly well understood (though
people still publish papers on it after over 100 years of serious work
on the subject). But now change the definition slightly - the random
walk is not allowed to revisit any lattice point. This is a self-
avoiding random walk; it's for example a model for polymer chains
(obviously, there cannot be two bits of the polymer in the same
physical space). Still seems very simple, looks like you ought to be
able to understand it. But we can do almost nothing with that model
mathematically; what we can do is run a few thousand simulations, see
what properties those tend to have, and say we believe that is the
truth.

By the way, Knuth is a smart guy.

Pete
.

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