# Simulation of Snow Drift and the Effects of Snow  Particles on Wind

Authors: Jie Zhang; Ning Huang
Journal: Modelling and Simulation in Engineering
 (2008)

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## Abstract

Coupled equations between wind and saltating particles are presented for a stable wind blowing over an infinite plane bed and the equations are solved for a simplified particle-bed impact process. The calculated results show that the saltating snow particles strongly affect the velocity distribution of the wind, causing a deviation from a logarithmically distributed wind velocity profile. The average height and length of saltating snow particle trajectories exponentially increase as the friction velocity increases; the ejected snow number flux and the streamwise snow transport rate also increase as the friction velocity increases.


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## Body

## 1. Introduction

Blowing snow occurs in many parts of the world and can
originate from a variety of sources, including ice cover, mountain glaciers,
and avalanches [1]. It also contributes to natural disasters
such as ice-snow floods, avalanches, and mudslides [2] that
negatively impact local environments and economies and sometimes cause the loss
of human life.In China, blowing snow is common throughout the provinces of Qinghai,
Tibet, Gansu, Sichuan, and Inner Mongolia. Snow frequently accumulates on
highways in these regions, causing considerable traffic delays, and the
diminished visibility due to blowing snow results in hazardous driving
conditions [3].Wind-snow interactions have been studied as far back as the early
twentieth century, when the characteristics of blowing snow were investigated
in Antarctica [4]. Later,
many other studies examined the movement and trajectory of snow particles via
wind tunnel experiments, high-speed photography, and theoretical and numerical
analyses (e.g., [5–8]). In China, extensive wind tunnel simulations of wind-snow
interactions have been conducted with the aim of developing a qualitative approach
to minimize the
destructive effects of blowing snow [3].Several recent
theoretical analyses of saltating snow particles (such as [2, 9–11]) do not address how such particles
affect the wind velocity distribution, thus their conclusions may have limited practical use,
given that saltating particles have long been known to appreciably change the
speed of the wind that drives them [6, 7]. This paper will present coupled equations
between wind and saltating particles for a stable wind blowing over an infinite
plane bed, and the equations will be solved for a simplified particle-bed
impact process.


## 2. Force Analysis of a Snow Particle in Wind

High-speed
photographs of snow particles in a wind tunnel [12] revealed several
distinct types of motion that arise at different wind speeds. As wind speed
becomes progressively higher, the following types of particle motion appear:
vibrating in place, rolling along the snow surface, slipping along the snow
surface, saltating (bouncing) along the snow surface, and suspension in air while
moving along with the wind. When saltation is present, it is the predominant
mode of particle movement [12] and so it is our main concern.Surface
grains with a dendritic shape tend to be broken by impacting grains, thus
grains become small and more elastic in time and the bulk density increases [13], so people usually assumed the drifting snow grains to be spherical in
their research [14–16]). Figure 1
depicts the forces (not to scale) exerted on a spherical snow particle in a
region, where wind flows in the positive x direction and where y is the particle height above the bed.
DP and ρP are the snow particle diameter and density, respectively.Figure 1
Forces acting on a saltating particle.ρ and g are the air density and gravitational
acceleration, respectively; and  forces acting on the particle are (1) weight W; (2) aerodynamic drag FD; (3) aerodynamic lift FL; (4) Magnus lift (due to particle spin) FM; (5) electrical force FE.The weight may be expressed as(1)W=16πDP3(ρP−ρ)g.The wind speed,u,
is greater than the speed of the saltating particle with respect to the snow
surface, so the drag force on the particle acts in the positive x direction and can be expressed as [17, 18]
(2)FD=18CDρπDP2Vr2,
where CD is the drag coefficient and Vr is the velocity of the particle relative to
the air. Vr is given by
(3)Vr=[(dxdt−u)2+(dydt)2]1/2.For
a spherical particle, the drag coefficient is given by the empirical formula
[19]
(4)CD=24Re+61+(Re)1/2+0.4,
where Re is the Reynolds number, which may be expressed as
(5)Re=DpνVr,
where ν is the kinematic viscosity of air.Experiments have
shown that the forcesFL, FM,
and FE are typically two orders of magnitude smaller
than W and FD [17], so we will not
consider them.


## 3. Equations of Motion and Boundary Conditions

To analyze the saltation of snow particles, we must
consider (1) the acceleration of the particles by the wind, (2) the force on
the wind due to saltating particles, (3) the collision of particles with the
snow surface. For a stable wind blowing over an infinite plane bed, Ungar and
Haff [17], Werner [20], and so on applied a modified Navier-Stokes
equation:
(6)Fx+ddy(ρk2y2(dudy)2)=0,
where Fx is the horizontal force per unit volume on the
wind due to the presence of saltating particles, k is von Karman's constant, y is the height above the snow surface, and u is the wind speed in the X-direction.The two dominant forces acting on a snow particle can be
decomposed intox and y components to yield the equations of motion
for its two-dimensional trajectory:
(7)mpd2xdt2=fx=−12CDρVr(dxdt−u)π(Dp2)2,mpd2ydt2=fy=−mg−12CDρVr(dydt)π(Dp2)2,
where x and y are the horizontal and vertical coordinates, respectively, g is
the acceleration due to gravity, mp and Dp are the snow particle mass and diameter, and fx, fy are the x and y components of the net force experienced by the
particle.During the process of saltation, each particle will cross
the same heighty twice, once upward
and once downward. Let σ↑(y) and σ↓(y)
be equal to the number density of upward and downward particles,
respectively, as a function of height. Similarly, let fx↑ and fx↓
be equal to the horizontal force on a single upward and downward
particle. By Newton's third law, per unit volume, the horizontal force exerted on the wind must be
equal and opposite to the horizontal force exerted on the particles:
(8)Fx=−[σ↑(y)fx↑+σ↓(y)fx↓].
This equation can be rewritten as
(9)Fx=−s[fx↑y˙↑(y)−fx↓y˙↓(y)],
where s is the number of
particles ejected per unit area of bed per unit time (assuming all particles
are ejected at the same initial speed), y˙↑(y) and y˙↓(y) are the vertical speed of a single upward and
downward particle, respectively.When a saltating
grain approaches the snow bed with a velocityVi, the number of ejected grains with is given
by [17]
(10)S(V→0,V→i)=AVi2Dpgδ3(V→0−BVi3Dpge→y),
where Vi is the magnitude of V→i, e→y is the unit vector in the positive y
direction, V⇀0 is the lift-off velocity of a vertically
ejected grain, A and B are dimensionless constants. Obviously,
(10) means that a saltating grain impacts the bed with a speed V→i, then the number of ejected grains with a same speed BVi3/(Dpg)e⇀y is AVi2/(Dpg).For (6), the
effective bed roughness is Dp/30, giving us the boundary condition y=Dp/30, u=0.Assuming all
grains are ejected at the same speed, (7) are solved using the boundary
condition
(11)u∗=lim(kydudy).


## 4. Computational Method

According to Bagnold [21], the transport of grains is limited, for a given wind speed, by
an “equilibrium mechanism” whereby the number of grains per unit time ejected
into the saltation layer is equal to that falling onto the bed. Thus, for each
grain that impacts the bed, another grain enters the saltation layer. So in
(10), the value of AVi2/(Dpg) is unity.Based on observations at air temperatures less than −6°C, Zhonglong and Yuan
[1] classified surface snow particles into three types: new snow (diameter
less than 1.0 mm,
average density 0.06 g/cm), fine
snow (diameter less than 0.5 mm,
average density less than 0.18 g/cm),
and old fine snow (diameter from 0.5 mm to 1.0 mm, average density 0.23 g/cm). The experimental results with
which we compare our theoretical results are limited to fine snow.Zhonglong [3] identified a threshold wind speed for fine snow particles, at a height of 1.85 m, of 3.65 m/s; below this speed, the number of saltating
grains is too small to cause a measurable decrease in wind speed. From this
result, the particle lift-off speed V0 can be calculated and then used to solve (7) to find Vi.
Then, by solving (6) and (7), we can obtain the wind profile and the
snow particle trajectory.To reach a numerical solution,(1)
input values
fors and for du/dy at y=Dp/30 and then solve (6),(2)
substitute the result from (6) into (7) to
calculate the particle trajectory and find the value of Vi,(3)
substitute the results from (7) into (9) to calculate Fx(y) and then substitute Fx(y) into (6),(4)
repeat the process 1–3 until the wind
profile calculated in step (3) adequately approaches the value calculated in
step (1),(5)
if the precision condition|Vi−Dpg/A|<ε (ε is a positive number with a sufficient small
value) is not satisfied, then the input values must be adjusted accordingly,
and  (6), (7), and (9) solved
iteratively until the condition is satisfied.


## 5. Analysis of Results

For snow-wind
coupling, the particle trajectory heights and lengths will increase with the
friction velocity (Figure2(a)).(a) Saltation trajectories at different friction velocities. (b) (i) Photograph of
particle trajectories. (ii) Trace of particle motion on the internal of each
spot is 3.7 m/s.
(a)(b)Figures3(a) and 3(b) compare
the coupled and uncoupled particle trajectories at two different friction
velocities.(a) A comparison of coupled and uncoupled particle trajectories foru*=0.56 m/s. (b) A comparison of coupled and
uncoupled particle trajectories for u*=0.24 m/s.
(a)(b)The particle trajectory is strongly influenced by
coupling. Whenu*=0.56 m/s, the coupled trajectory height exceeds
the uncoupled height by 48%; when u*=0.24 m/s, the coupled trajectory height exceeds
the uncoupled height by 20%.Kikuchi [6],
Maeno et al. [7], and  Zhonglong [3] observed that snow-wind coupling
causes the vertical wind velocity profile to deviate from a logarithmic
distribution. In Figure 4, our calculated results illustrate this deviation at
different friction velocities; the effect tends to be more pronounced at higher
wind speeds when the ejected particle rate s is greater.Figure 4
Calculated wind profiles with drifting fine snow.Figures5(a) and 5(b) show our calculated
results for coupled and uncoupled wind profiles based on the experimental data
of  Zhonglong [3]. Our coupled profile is qualitatively consistent with Wang's
result.(a) A comparison of the coupled and uncoupled wind velocity profiles. (b) The
experimentally determined wind profile of Zhonglong [3].
(a)(b)Figure6
compares our calculations of trajectory height versus friction velocity with
Wang's experimental result (grain diameter = 0.5 mm, mean density = 0.9 g/cm3).Figure 6
Trajectory height versus friction velocity.According to Zhonglong [3], the average
height of saltating snow particles increases exponentially with friction
velocity. Figure 6 shows a numerical fit of our calculated values, which is given
by(12)H=α1exp(u∗β1)+λ1,
where
(13)α1=0.00527m,β1=0.19m/s,λ1=−0.0032m.Figures7(a) and 7(b)
compare our calculated values of particle
trajectory length versus friction velocity with the experimental result of
Maeno et al.  [8], showing qualitative agreement. A numerical fit for our values is
given by
(14)L=α2exp(u∗β2)+λ2,
where
(15)α2=0.083m,β2=0.19m/s,λ2=0.00052m.(a) Our results for trajectory length versus friction
velocity. (b) The experimental result of  Maeno et al. [8].
(a)(b)We have used our
numerical model to calculates, the
number of ejected particles per unit area of bed per unit time, based on the
experimental results of  Zhonglong [3]. Figure 8 shows these values as a function
of friction velocity, an exponential relationship that is given by
(16)N=α3exp(u∗β)+λ3,
where α3=4127.1m−2s−1, β3=0.15m/s, and λ3=−16541.4m−2s−1.Figure 8
Ejected grain number versus friction velocity.Zhonglong [3] proposed, based on experimental results, that
the saltation transport rate varies as the cube of friction velocity. Figure 9
shows our calculated values for Q, the transport rate per unit width of bed,
which is linear to the cube of friction velocity for u*3 greater than 0.2 m3/s3.
Our result is consistent with that of  Dziji [5], validating our theoretical
model in this respect.Figure 9
Saltation transport rate versus cube of friction velocity.


## 6. Conclusion

In this paper, coupled equations
between wind and saltating snow particles are presented for a stable wind
blowing over an infinite plane bed, and the equations are solved for a
simplified particle-bed impact process.(1)
Snow-wind coupling results in
considerably different particle trajectories when compared to a model in which
no coupling exists.(2)
The average height and length of saltating snow particles
increase exponentially with friction velocity.(3)
Our model provides the number of
particles ejected from a unit area of bed per unit time, given experimental
data for particle trajectory height and friction velocity.(4)
The transport rate of saltating
particles is linear to the cube of friction velocity, in agreement with
Kobayashi [5].



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*Source: 408075-2009-03-05.xml*