RE: [Algorithms] Underwater Physics

[Graham R. <gr...@ar...>]

Arrrgggggg! I can always count on this list to point out my brain farts!
And there's always a brain fart. Kim will also enjoy reading my
conclusions below.

Charles' reply reminded me that I don't really know that much about low
Reynold's number flow, beyond a project on creeping flow (Re < 1) when I
was a young undergrad lad. I've dealt mostly with airplane aerodynamics
and flight dynamics, and flows for Reynold's numbers above something
like 100,000 (below this is kind of black art land for airplane
aerodynamicists). And I don't recall seeing previously exactly where the
hell the linear drag model actually comes from. But every time I get in
a discussion about drag calcs for games, it comes up. Its time to know.

So, I looked back to some old reference material. The quadratic
relationship between drag and speed (when Reynold's number is fixed)
arises out of a pure dimensional analysis relating force to velocity, a
characteristic length, viscosity, and density. Reynold's number is a
result of this dimensional analysis. The quadratic relationship is
ALWAYS valid, at ALL Reynold's numbers, as long as drag coefficient is
defined in the classical sense.

So, where does the linear drag model come from? It arises due to the
behavior of that drag coefficient, CD. The Reynold's number is embedded
inside CD, as classically defined. CD is basically Reynold's number to
some power. Reynold's number is a linear function of speed, so if there
is a Reynold's number regime for which the exponent of Reynold's number
is on the order of -1, the drag force equation will have a linear speed
term in the denominator that cancels out one of the speed terms in the
numerator, thus leaving us with drag being a linear function of speed.
It looks something like this:

Drag =3D 0.5 * rho * area * speed * speed * CD

CD =3D (Re) ^ a =3D (speed * length * rho / mu) ^ a

So,

Drag =3D 0.5 * rho * area * speed * speed * [(speed * length * rho / mu) =
^
a]

The a varies, basically determines the shape of the CD(Re) curve,
and....for some Reynold's number ranges....becomes negative! Which, as
above makes Drag a linear function of speed.

Where does a become negative? Depends on the shape of the object.
Remember the CD vs. Reynold's number curve shown in the PDF file I
referenced in my earlier post? Look there and see that CD for a sphere
has an inverse relationship with Re when Re < 5000. Therefore, for
spherical objects, drag is a linear function of speed for Re < 5000, and
a quadratic function of speed for Re > 5000. Kind of makes sense, and
you'd probably have guessed that without all the analysis just due to
the dramatic change in the curve.

http://www.physics.ubc.ca/~waltham/scione/C_D.pdf

So, to summarize. Both drag forces models are accurate at all Reynold's
numbers, as long as you use the correct "CD". If you want to use a CD
that is constant always, then for some Re below a certain value
(dependent on shape) you will have to transition from a quadratic to a
linear drag model. But be aware that the CD for the linear model will be
very different from the CD for the quadratic model, and will change
discontinuously when you transition between drag models. Otherwise the
drag force will discontinuously change at the transition point and that
could cause bad things to happen (you can't just use CD =3D 1.0 for all
Re's). If you choose CD to be the classical CD, and vary it using a
table lookup or function curve, then you should be using the quadratic
model (or you'll have a units problem).

Gosh, that was probably an overkill analysis. But, at least I now have a
better story when the subject comes up again!

Graham

> -----Original Message-----
> From: gda...@li...=20
> [mailto:gda...@li...] On=20
> Behalf Of Charles Bloom
> Sent: Thursday, May 27, 2004 5:54 PM
> To: gda...@li...
> Subject: RE: [Algorithms] Underwater Physics
>=20
>=20
>=20
> In water, you're actually quite often in the linear-speed=20
> drag range.  Note=20
> that it doesn't just apply to "slow" flow, it also applies to=20
> pretty fast=20
> flow on large objects (eg. it's based on a uniteless Reynolds=20
> number).  For=20
> decent water drag you'll want to lerp between linear and squared drag=20
> across some range.
>=20
> > >
> > >For very low speed flow, there is also a drag model that is linear=20
> > >with speed instead of quadratic. A bit cheaper to compute.
> > >