Mechanics of Sand Transport - Dune Worlds: How Windblown Sand Shapes Planetary Landscapes - page 39

Table 4.1 Reynolds numbers for different conditions. Aerospace vehicles and thrown objects have high Re and viscosity does not matter; for
dust, Re is typically much less than 1 and viscosity dominates. Sands in atmospheres tend to be in between with Re of *100, although sands in
liquids tend to have low Re
Setting Density (kg/m 3 ) Viscosity (lPa-s) Size (mm) Speed (m/s) Re

Sand grain in water 1000 1000 0.2 0.01 2.0E+00

Silt particle in water 1000 1000 0.02 0.001 2.0E-02

Pebble in water 1000 1000 20 0.5 1.0E+04

Dust in air at sea level 1.25 17 0.02 0.1 1.5E-01

Sand grain in air at 5000 m 0.8 16 0.25 10 1.3E+02

Sand grain in air at sea level 1.25 17 0.25 10 1.8E+02

Baseball 1.25 17 70 40 2.1E+05

5.0E+08 a 747 airliner in cruise 0.3 15 1E+05 250

Sand grain on Venus 64 35 0.2 1 3.7E+02

3.1E-04 b Dust grain on Mars (6 mbar) 0.02 13 0.002 0.1

Sand grain on Mars (6 mbar) 0.02 13 0.2 10 3.1E+00

Sand grain on Mars (Tharsis) 0.003 13 0.2 25 1.2E+00

Sand grain in Titan air 5.4 6 0.2 1 1.8E+02

Sand grain in Titan sea 660 2000 0.2 0.1 6.6E+00

Huygens probe at touchdown 5.4 6 1300 5 5.9E+06

a

For jet aircraft, Mach number effects can be important

b

For dust on Mars, Knudsen number effects can be important

In general, low Reynolds numbers mean viscous forces
dominate and the flow is laminar; when inertial forces
overpower the resisting viscous forces, the flow is turbulent.
The borderline between the two is somewhat shape- and
condition-dependent, but if Re [*100, then fluid is tur-
bulent. When we calculate the value of Re for various
conditions on different planetary bodies (Table 4.1 ) we find,
somewhat inconveniently, that many situations of interest
are close to this boundary, so we require some care in
evaluating the drag—neither the Stokes limit nor the large-
Re limit are universally applicable.
Because much of the early quantitative work on particle
transport by fluids was in water, where the Reynolds
number turns out to be low, much of the literature on sed-
imentology is shaped by the low-Re limit of fluid behavior.
This in turn focuses on Stokes' law (named after George
Stokes) which estimates the velocity that results when
frictional (drag) and buoyant forces are exactly balanced by
gravitational force (weight) for a spherical particle falling
vertically through a viscous fluid:

limit of C d = 24/Re is substituted). One can see that when
viscosity dominates, the force depends linearly on flow
speed, not on the square of flow speed (put another way,
because velocity is in the Reynolds number, and the drag
coefficient varies as the reciprocal of Reynolds number, one
of the velocities in Eq. 4.3 gets cancelled out). This com-
plicating difference in flow regime proved a challenge in the
historical development of fluid mechanics, with debate
raging among the foremost mathematicians of the 17th and
18th centuries over whether drag depended on speed or the
square of speed, when in fact 'it depends…'.
So, armed with Eq. 4.3 we can calculate the terminal
velocity of particles (Fig. 4.5 ). While for many situations
Stokes law works, in a few cases it is not applicable and the
more general form of Cd must be used.
In general, the settling velocity in air for particles of silt
and clay (or 'dust') is much smaller than the upward
velocities available from turbulent eddies in wind flow (see
Sect. 3.6 ) which is the explanation for why these particles
can be transported great distances in suspension.

Þ gr 2 9l

u s ¼ 2 q s q f ð ð 4 : 3 Þ

4.3 The Boundary Layer where u s is the settling velocity, q s is the particle density, q f

is the fluid density, g is the acceleration of gravity, r is the
particle radius, and l is the dynamic viscosity of the fluid.
The Stokes law in Eq. 4.3 is valid only for laminar flow
around the particle, so it should be restricted to situations
where Re \ 1 (the algebraically-minded will see that this
equation is equivalent to Eq. 4.1 above, when the low Re

The wind speed near the ground approaches zero (in fact
aerodynamicists refer to a 'no slip' condition wherein the
speed exactly adjacent to a surface must be zero, otherwise
the shear stress would be infinite). The layer above the
surface in which the fluid velocity is 99 % or less of its