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