Transcript AshfallLect5

Particle Fall through the
atmosphere
Lecture #5
Ashfall Class 2009
Distance d travelled by an object falling for
time t:
Time t taken for an object to fall distance d:
Instantaneous velocity vi of a falling object
after elapsed time t:
Instantaneous velocity vi of a falling object
that has travelled distance d:
Average velocity va of an object that has
been falling for time t (averaged over time):
Average velocity va of a falling object that
has travelled distance d (averaged over
time):
use g = 9.8 m/s² (metres per second squared; which might be thought
of as "metres per second, per second”. Assuming SI units, g is measured
in metres per second squared, so d must be measured in metres, t in
seconds and v in metres per second.
air resistance is neglected--- quite inaccurate after only 5 seconds
Particle Fallout
• After a very short time, ~4 seconds, particles
will reach a terminal velocity in earth's
atmosphere, with their gravitational attraction
to the earth balanced by air resistance. Small
particles have dominant air resistance (fall
slowly) while large particles have dominant
gravity (fall rapidly).
Reynolds Number
Re
• Reynolds number is a dimensionless number
(i.e. it has no units) that is a measure of the
type of flow through a fluid. In the case of
falling particles, this describes the way that air
flows around the particle. There are three
basic types:
• laminar where Re < 0.4,
• intermediate where 0.4 < Re < 500, and
• turbulent where Re > 500.
RN =dvt/
Fast-falling
Large
Pyroclasts
Medium and
small pyroclasts
D=
1mm
D=
1µm
.01
cm/
s
Laminar flow;
RN =
10-2
10
m/s
RN = 20
RN = 40
RN = 104
Turbulent flow;
RN = 106
Fluid dynamics applies dimensionless analysis of fall of spheres in the
atmosphere, which shows that experience with large pyroclasts might not apply
to smaller ones which fall much more slowly…
Conventional Wisdom:
Particle Reynolds
number, Re :
Particle Settling ratio of inertial force
p
to viscous force per
unit mass
Drag force:
(i) viscous drag
(friction between
the fluid and the
particle surface)
Rep = Vtd / v
Vt = particle terminal
fall velocity;
d = particle diameter;
v = fluid kinematic
viscosity
(ii) form drag
(inertial force
caused by the
acceleration of
fluid around the
particle as it falls)
Rep :
> 500 turbulent
1-500 transitional
<1 laminar
From Sparks et al. [1997]
particle accelerates due to gravity
8
Larger pyroclasts,
those >2mm in
diameter, fall in a
turbulent flow
regime (Re> 500)
through the
atmosphere. Small
pyroclasts, <1/16
mm (62 μm or 4 Φ),
fall in laminar flow
regime (Re<0.4).
Intermediate size
particles are
transitional.
Particle Terminal Fall Velocity
• For large particles (Rep > 500) –
inertial forces dominate:


4
d
(

p


f)
g
t 



3 C
d

f


V

For small particles (Rep < 1)
- viscous forces dominate:
2
pgd


Vt 
 18




d = particle diameter
ρp = particle density
ρp = particle density
g = acceleration due to gravity
ρf = fluid density
d = particle diameter
g = acceleration due to gravity
v = kinematic viscosity
Cd = dimensionless drag coefficient
10
Fall of spherical particles in earth’s atmosphere
Schneider et al., 1999, J Geophys Res 104 4037-4050
Particle Terminal Fall Velocity
Mean particle size
at ~330 km from
MSH (Ritzville,
WA) was 20
microns; Vt ~0.20.4 ms-1
100 micron
diameter
particle has
Vt of ~4-7
ms-1
12
Atmospheric Structure
Environmental parameters determined from the radiosonde sounding taken
13
at Spokane International Airport at 1800 UTC on 18 May 1980.
Bonadonna et al., 1998
Bonadonna et al., 1998
Bonadonna et al., 1998
Bonadonna et al., 1998
Bonadonna et al., 1998
Bonadonna et al., 1998
Bonadonna et al., 1998
Bonadonna et al., 1998
Figure 2 Typical stereo-pair taken at 8o tilt angle.
Owen P Mills, MS
thesis, Michigan Tech,
2007
Figure 3. Digital elevation map produced from stereo-pair in Figure 2.
Augustine ash P Izbekov
Ash is NOT
spherical!
Riley et al., 2003
Riley et al., 2003
Rose W I, C M Riley and S Dartevelle, 2003, J Geology, 111:115-124.
Riley et al., 2003
Riley et al., 2003
Riley et al., 2003
Riley et al., 2003
Rose W I, C M Riley and S Dartevelle, 2003, J Geology, 111:115-124.
Rose W I, C M Riley and S Dartevelle, 2003, J Geology, 111:115-124.
Riley et al., 2003