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Particle Fall through the atmosphere

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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

- 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 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/

Medium and small pyroclasts

Fast-falling Large Pyroclasts

D = 1mm

D = 1µm

.01 cm/s

10 m/s

Laminar flow;

RN = 10-2

Turbulent flow;

RN = 106

RN = 20

RN = 40

RN = 104

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…

Particle Reynolds number, Rep:

ratio of inertial force to viscous force per unit mass

Rep = Vtd / v

Vt = particle terminalfall velocity; d = particle diameter; v = fluid kinematicviscosity

Rep:

> 500 turbulent

1-500 transitional

<1 laminar

Drag force:

(i) viscous drag(friction between the fluid and the particle surface)

(ii) form drag(inertial force caused by the acceleration of fluid around the particle as it falls)

From Sparks et al. [1997]

particle accelerates due to gravity

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.

- For large particles (Rep > 500) – inertial forces dominate:

- For small particles(Rep< 1)
-viscous forces dominate:

d = particle diameter

ρp = particle density

ρf = fluid density

g = acceleration due to gravity

Cd = dimensionless drag coefficient

ρp = particle density

g = acceleration due to gravity

d = particle diameter

v = kinematic viscosity

Schneider et al., 1999, J Geophys Res 104 4037-4050

Mean particle size at ~330 km from MSH (Ritzville, WA) was 20 microns; Vt ~0.2-0.4 ms-1

100 micron diameter particle has Vt of ~4-7 ms-1

Environmental parameters determined from the radiosonde sounding taken 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