Mechanics of Sand Transport - Dune Worlds: How Windblown Sand Shapes Planetary Landscapes - page 38

Fig. 4.4 The drag coefficient of
a sphere as a function of
Reynolds number. At left,
viscous forces dominate, and the
drag coefficient varies as (24/Re)
and yields Stokes' law. Over
*100\Re\100,000, C d is fairly
constant (changing the shape
dramatically changes C d by only
a factor of 2), but C d drops
sharply (the 'drag crisis') at a
roughness/turbulence-dependent
Re value of *100,000

Thus L = 1/2SC l q f U 2 and D = 1/2SC d q f U 2 , with S the
particle area (=pd 2 /4 for a sphere). The parameter group
1/2q f U 2 is called the dynamic pressure, and is equal to the
kinetic energy per unit volume of the airflow.
If we consider a particle away from others and away
from the planet surface, it will be exposed to weight and lift/
drag only. In the absence of wind, it will be accelerated
downwards by its weight until it is descending relative to
the fluid at a speed Vs (terminal velocity) where the drag
balances the weight (since the particle is falling vertically,
the drag is upwards). Equating the weight and drag we find:
D ¼ 1 =ð SC d r f U 2 ¼ 1 =ð pd 2 C d q f V s ¼ W
¼ p = 6

approached for small dust grains in the thin Mars atmosphere,
but for everything else the fluid behaves as a continuum.
The second parameter is the Mach number, the ratio of
fluid velocity to the sound speed in the medium. The 'sound
barrier' is the phenomenon where the drag coefficient rises
sharply (and the lift can change) as an aircraft approaches
the speed of sound. This was actually recognized (Lorenz
2006) in 1740 by an astute artilleryman, Benjamin Robins,
whose cannonballs did not go exactly where he expected; he
also recognized the role of spin, hence the Robins-Magnus
effect.

For all the situations we'll consider, the Mach

number is small and the flow is subsonic.
But the last parameter, the Reynolds number Re (named
after Osborne Reynolds, a British physicist), is what causes
complications. This is the ratio of inertial forces to viscous
forces, and defines whether a flow is laminar or turbulent. In
laminar flow, the fluid follows streamlines which stay
constant in time. In turbulent flow, streamlines can be
drawn to define the average flow path, but individual parcels
of fluid wander and swirl across these average lines, causing
much more vigorous mixing. Reynolds performed a series
of experiments in glass-walled pipes to determine what
caused flow to become either laminar or turbulent and found
that

and thus

Þ d 3

ð q p q f

ð 4 : 1 Þ

C d q f 0 : 5

V s ¼ 4 =ð d q p q f

All else being equal, the terminal velocity varies as the
square root of diameter.
The above is all pretty much 17th-century Newton stuff,
and looks straightforward because all the complexity is
hidden in the innocent-looking C l and C d lift and drag
coefficients. These are functions of shape (which is why car
and aeroplane manufacturers spend lots of time with wind
tunnels and computer simulations) and also on three
dimensionless parameters which define the way the fluid
behaves, whether it is gas or liquid.
We can dismiss a couple of these rather quickly. First is
the Knudsen number, the ratio of the mean free path of air
molecules to the particle size—for everything we'll discuss
this number is very small and fluids behave as a continuum.
For satellites in the upper atmosphere, the mean free path is
large and air molecules behave as unconnected billiard balls
travelling in basically straight lines ('free molecular flow'—
how Newton imagined things, where the drag coefficient C d
is exactly 2.0). The borderline between these regimes is

the criterion related to the ratio of the following

parameters:

Re ¼ udq f = l ð 4 : 2 Þ

where u is velocity, d is particle dimension, q f is the fluid
density, and l is the fluid (dynamic) viscosity. From many
laboratory experiments in over a century of work, the drag
coefficient of various shapes as a function of this quantity is
very well known (Fig. 4.4 ). In sedimentology, the Shields
curve is essentially a redrawing of this drag coefficient
relation, but with slightly different axes.