dynamics of falling snow. motions of falling objects are generally complex even regularly-shaped...
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Dynamics of falling
snow
Motions of falling objects are generally complex
even regularly-shaped objects such as coins and cards flutter (oscillate from side to side), tumble, and drift sideways
Demonstation: business card change of initial orientation from 0°to 90°is enough to cause big change in fall behavior
Business card held horizontally and vertically
Origin of complex motions
Qualitative description dates back to Maxwell (1853)
Cause: torque created by combination of gravity and lift forces
Forces acting on different points on a body produce several moment arms
Result is complex descending motions
Observations: falling of circular disks through a viscous fluid medium
Stereoscopic cameras used to record the trajectories of disks falling through fluid
disk material: steel and lead
diameter d = 5.1 to 18.0 mm
thickness t = 0.076 to 1.63 mm
fluid: water-glycerol mixture
There are 5 parameters
Disk diameter d, thickness t, density ρ, fluid density ρf , and kinematic viscosity ν.
reduce to 3 dimensionless ratios
moment of inertia I*
Reynolds number Re
aspect ratio t/d
Disks are thin: omit effect of aspect ratio
Dimensionless moment of inertia is
=
Reynolds number is
Re =
where U is mean velocity of falling disk
Definitions of I* and Re
Four distinct motions observed and mapped on “phase diagram”
1) Steady falling
2) Periodic oscillation
3) Chaotic
4) Tumbling
Steady falling
Periodic oscillation
Chaotic
Tumbling with drift
Field et al.
Letters refer to details in Field et al.
I*
Re
“(The disks’) ultimate location of the bottom
of a large container could never be
predicted.”Today we recognize this as a hallmark of
chaos.
Transition to chaotic behavior with
increasing Re and I*
Deterministic chaos
Extreme sensitivity to initial conditions
Example: ski hill with moguls
AHS 260 Day 38 Spring 2013 12
Final positions of skis are very sensitive to their initial positions!
Since there is uncertainty in the exact initial position, the final position is unpredictable.
Ski positions at bottom of hill are much farther apart than at top of hill
Paths of seven skis released from slightly different positions
Ski tracks become widely spaced!
All ski tracks are closely bunched at first
Zhong et al. found a spiral motion as well
planar zigzagtransitional
spiral
Zhong et al.
interactions between disk and induced vorticesplay a significant role
Vortices shed from zigzaggin
g disk
Wake forms
helicoidal shape
Vorticity is produced and fed into wake, scrolling into a
roll
Can fall behavior be modeled numerically?
In principle, we could solve Navier-Stokes equations in 3D
both daunting and unrealistic at higher Re
Computational approaches limited to simple shapes and 2D so far
What about snow crystals or snowflakes (aggregates)?
Casual observation in light winds shows motion is not just downward
snowflake trajectories vary in direction, and vary relative to each other
sideways motion easy to see, but not full swing or spiral
spin around vertical axis occasionally seen
tumbling very occasionally seen
Some due to eddy motions of air, but some due to unstable mode of descent of particle itself
Why study snowflake fall motions?
Computation of aggregation of snow crystals requires knowledge of
fall velocities (well-studied)
fall trajectories (few studies)
Work of Kajikawa on unrimed and rimed crystals, and “early” snowflakes
Importance of fall behavior for cloud physics
Collection efficiency depends on motion of both collector and collected particle
horizontal motions of collector would
• increase sweep volume• allow collisions between
particles with same fallspeed
Other applications
Flutter of cloud particles influences radiation transfer in clouds differential reflectivity signals
Research of M. Kajikawatime interval
= 0.01 s
Melted crystal
Results for unrimed plate-like crystals
Unstable motion first appears as flutter (like (a))
Next swinging (b) appears, followed by spiral
No mention made of chaotic regime
Tumbling did not appear
Latter two points call for further observations
From Field et al.
Fig. 5 Kajikawa 1992
Dendritic crystals Fall with a stable pattern over larger range
of Re than simple plates
may be due to internal ventilation of crystals, seen in model experiments
Motion of dendritic crystals
SD of vertical velocity < 3% of fall velocity, but SD of
horizontal velocity up to 20% of fall velocity
variation in horizontal velocity likely has significant effect on aggregation
Standard deviation of VH (cm s-1)
VH (cm s-
1)
Rimed crystals
Similar to diagram for unrimed crystals
Fig. 2 Kajikawa 1997
Tracking software
I experimented with Tracker, a video analysis tool for physics lab experiments
Allows tracking of objects and analysis and modeling of their motion
Position, velocity and acceleration overlays can be made
autotracking
Problem with snowflakes
Snowflakes fall at ~150 cm s-1
At 30 fps, snowflake falls 5 cm between frames
This is 5x its size, so it’s blurred
Need high-speed camera (problem: cost, for now)
Also need distance scale
relative horizontal vs. vertical motion may be enough to start with
Eddy motions also important we can assume that mean wind motion does not
affect collision process
however fluctuations in wind velocity because of turbulence may enhance collisions
How can turbulence increase collision rate?
by changing the
relative velocities
spatial distribution
collision and coalescence efficiencies between crystals
Even with good measurements….
We cannot characterize motion of each snowflake
can measure average horizontal displacement and velocity, and oscillation frequency
Thus we still need parameterization in numerical cloud or mesoscale models
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