Dimensionless numbers
Bagnold Number
(Iverson, 1997)
- volume fraction of granular
solids, mass density of solids, characteristic grain diameter, shear
strain rate, dynamic viscosity of pore fluid with suspended fine sediment
(Iverson and Denlinger, 2001)
- υ* = maximum possible grain concentration
- grain collision vs. viscous
stresses in steady, uniform shear flows
- Nb < ~40 indicated macroviscous regime, bulk normal and shear stresses
are both proportional to the shear rate
- Nb > ~450 (or 200 – Iverson’s
definition), indicate a collision-dominated flow regime, bulk normal and
shear stresses are both proportional to shear rate squared
Darcy Number
(Iverson, 1997)- k = hydraulic permeability
- interaction stress vs. solid
inertial stress
- tendency for pore fluid
pressure developed between moving grains to buffer grain interactions
Fluidization Number
(Iverson and Denlinger, 2001)- ratio of fluidization scale and
velocity scale
- Nf << 1 probably applies in
most flows and thereby support the validity of simplified equations of
motion (Iverson and Denlinger, 2001)
Friction Number
(Iverson, 1997)
- N = number of grains above
grain surface
- ratio of frictional stresses
(shear stress by sustained grain contacts) to viscous shear stress (NBag/NSav)
- like the Bingham number, but
characterizes stresses borne by distinct solid and fluid phases. (Bingham
characterizes stresses in a one-phase material that exhibits both
viscosity and strength)
- frictional dominates viscous forces for Nf > 2000 (Iverson, 1997); but Nf > 100 for Parsons, 2001.
Froude Number
- velocity, gravity, thickness
- proportional to inertial force /
gravitational force, can be used to quantify the type of flow.
- < 1.0, the flow is subcritical (tranquil)
- = 1.0, the flow is critical.
- > 1.0, the flow is
supercritical and would be characterized as rapid flowing.
- dynamic similitude can be attained if,
as thickness is scaled down by lamba, velocity
is scaled down by lambda^0.5.
Mass Number
- volume fractions and densities
of solid and fluid
- ratio of solid inertia to fluid
inertia in the mixture
- quantifies effects of
volumetric grain concentration
- >1 implies that momentum
transport by solid grains may dominate
Quasi-Reynolds Number
(Iverson and Denlinger, 2001)
- dynamic scaling factor
analogous to Reynolds # in Newtonian fluid mechanics
- usually over 10^6
(Iverson, 1997)
Pore Pressure Number
(Iverson and Denlinger, 2001)- pore pressure diffusivity, runout length, gravity, height
- represents the timescale for downslope flow divided by the timescale for pore
pressure diffusion normal to the flow direction
- small values (<<1) indicate
that pore pressure advection and diffusion operate on different
timescales, which justifies decoupling into separate equations that can be
solved sequentially
Savage Number
(Iverson, 1997)
(Iverson and Denlinger, 2001)- shear strain rate, density
solids, characteristic diameter, number of grains above slip surface,
density solid, density fluid, gravitational acceleration, angle of
internal friction
- grain collision stresses to
gravitational grain contact (frictional) stresses
- if Ns > 0.1 at typical
depths H, grain collision stresses may affect flow dynamics significantly
(Savage and Hutter, 1989)
References
Iverson, R.M., 1997, The physics of debris flows, Reviews of Geophysics, 35(3): 245-296.
Iverson, R.M., and Denlinger, R.P., 2001, Flow of variably fluidized granular masses across
three-dimensional terrain, 1. Coulomb mixture theory, Journal of Geophysical Research, 106(B1): 537-552.