3Distance 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
5Particle FalloutAfter 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).
6Reynolds Number ReReynolds 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, andturbulent where Re > 500.
7Medium and small pyroclasts Fast-falling Large Pyroclasts RN =dvt/Medium and small pyroclastsFast-falling Large PyroclastsD = 1mmD = 1µm.01 cm/s10 m/sLaminar flow;RN = 10-2Turbulent flow;RN = 106RN = 20RN = 40RN = 104Fluid 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…
8Conventional Wisdom: Particle Settling Particle Reynolds number, Rep:ratio of inertial force to viscous force per unit massRep = Vtd / vVt = particle terminal fall velocity; d = particle diameter; v = fluid kinematic viscosityRep :> 500 turbulent1-500 transitional<1 laminarDrag 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. particle accelerates due to gravity
9Larger 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.
10Particle Terminal Fall Velocity For large particles (Rep > 500) – inertial forces dominate:For small particles (Rep < 1)- viscous forces dominate:d = particle diameterρp = particle densityρf = fluid densityg = acceleration due to gravityCd = dimensionless drag coefficientρp = particle densityg = acceleration due to gravityd = particle diameterv = kinematic viscosity
11Fall of spherical particles in earth’s atmosphere This diagram shows how fast fine spherical ash particles would fall if they were simple particles. In reality we know that most ash this size falls out in less than 24 hours.Schneider et al., 1999, J Geophys Res
12Particle Terminal Fall Velocity Mean particle size at ~330 km from MSH (Ritzville, WA) was 20 microns; Vt ~ ms-1100 micron diameter particle has Vt of ~4-7 ms-1
13Atmospheric Structure Environmental parameters determined from the radiosonde sounding taken at Spokane International Airport at 1800 UTC on 18 May 1980.