Open Journal of Civil Engineering, 2013, 3, 13-17
http://dx.doi.org/10.4236/ojce.2013.33B003 Published Online September 2013 (http://www.scirp.org/journal/ojce)
Similarity Study on Snowdri ft Win d Tunnel Test
Weihua Wang1,2, Haili Liao1, Mingshui Li1, Hanjie Huang2
1School of Civil Engineering, Southwest Jiaotong University, Chengdu, China
2China Aerodynamics Research and Development Center, Mianyang, China
Received July 2013
ABSTRACT
The model for snowdrift wind tunnel test needs to be similar with the prototype. Based on detailed analysis in aspects of
geometry, kinematics and dynamics, the major similarity parameters that need to be satisfied are gained. The contradic-
tion between the Reynolds number and Froude number as well as the problem of time scale is introduced, and the selec-
tions of the model parameters are specified. Lastly, an example of snowdrift wind tunnel test by adoption of quartz sand
as the model of snow grains is presented. The flow field and the snow distributions on a typical stepped roof were in-
vestigated. The results show that the flow filed characters are in good agreement with the field observations, and the
stepped roof snow depth distributions are basically consistent with the observation results.
Keywords: Simi larity Parameters; Wind Tunnel Test; Snowdrift; Roof
1. Introduction
Snowdrift forms the unbalanced distribution on roof un-
der the wind actions, leading to excessively great snow
load on local roof, eventually resulting in collapse of the
building [1]. Such incidents of structure damage by snow
load are not rarely seen. Especially in recent years, with
changes of global climate, wind and snow disasters be-
come more and more frequent, exerting significantly ne-
gative impact on people’s daily life, thus snow disaster
prediction and prevention has drawn increasingly more
attentions.
Wind-induced snow drifts include creep, saltation and
suspension, as shown in Figure 1. Wind-induced snow
drift research is mainly conducted through field observa-
tion, wind tunnel (or water flume) test and numerical
simulation, with each having their own advantages and
drawbacks. Wind tunnel test, due to its accessibility for
easy control and systematic study, is regarded as an im-
portant research method. Snowdrift wind tunnel simula-
tion needs to keep similarities between the model and
prototype in geometry, kinematics, and dynamics.
2. Geometric Similarities
Geometric similarity requir es equivalence of the ratios of
geometric dimensions to characteristic dimensions be-
tween the model and prototype,
( )( )
mp
lL lL=
(1)
where l is linear dimension, L is a characteristic length,
and subscripts m and p refer to model and prototype re-
spectively.
According to the similarity criterion, all the geometric
dimensions in the physical model shall meet Equation (1),
including the terrain and surrounding buildings in the test
domain. The geometric dimensions of snow grains, how-
ever, would introduce difficulties in simulations. For ex-
ample, for simulation of medium-sized snow grains in a
diameter of 0.5 mm, in case of the geometric scale being
1/50, the diameter of the model snow grain needs to be
10 μm. Obviously, there is difficulty in operations on
such small particles in the lab; and the excessively small
size of particles would lead to unduly great threshold
velocity. Fortunately, abandonment of this parameter
would not bring in significant impact [3]. The diameters
of snow grain model for wind tunnel test generally can
select a range of 0.05 - 0.2 mm.
Figure 1. Sketch of saltation, suspension and creep trajecto-
ries [2].

W. H. WANG ET AL.
14
3. Kinematic Similarities
Kinematic similarity requires equivalence of the ratio s of
flow speeds at various points between the model and pro-
totype,
( )( )
mp
uU uU=
(2)
where u is the flow speed and U is the characteristic ve-
locity.
Kinematic similarity implies the reproduction of the
atmospheric turbulent structures. The flow field velocity
profile at the atmospheric boundary layer conforms to the
logarithmic law, with the incoming flow mainly simu-
lated with Jensen number. Owen [4] pointed out that wi-
thin the flow field having particle movement, saltation
particles continuously absorbed energy from the air flow,
leading to the decrease of flow speed, this was equivalent
to increase the effective aerodynamic roughness-length
(z0’) of the field. The flow field velocity profile outside
the saltation layer conforms to the following:
*2
*
2
( )ln
ugz
uz C
u
κ

= +


(3)
where u* is the shear velocity, ĸ is the Von Karman con-
stant, and C is a constant.
Kinematic similarity implies that the restitute coeffi-
cients of the model and the prototype should be kept
consistent. However, studies show that the restitute coef-
ficient has no significant impact on the particle move-
ment [5]. Therefore, this parameter is generally ignored
in wind tunnel tests. Besides, th e atmospheric turbulen ce
intensity and turbulence length scale should also be sim-
ilar in theory, but it can be relaxed in wind tunnel tests
when the time average results are concerned.
4. Dynamic Similarities
Dynamic similarity is a very important and complicated
condition for the simulation of physical models. Snow-
drift is a gas-solid two-phas e flo w, with very complicated
mutual coupling mechanism. Dimensional analysis show-
ed that there are dozens of relevant parameters for simu-
lation [6]. Clearly, it’s impossible to satisfy all the para-
meters, but only the major parameters need to be taken
into consideration. This section presents the derivation of
the similarity parameters through the analysis of particle
motion and transport mechanism.
4.1. Equations of Motion
Assuming snow particle is approximately spherical with
a diameter of d, thus its motion Equation can be ex-
pressed as
3 323
6 686
ppD rr
d
dd Cdd
dt
π πππ
ρρρ ρ = −−+
uguug R
(4)
where ρ is the fluid density, ρ
p is the particle density, CD
is the drag coefficient, g is the acceleration vector, u is
the particle speed vector, and ur is the particle speed rel-
ative to fluid.
Gravity, drag and buoyancy are the most important
forces for an aeolian grain than other forces. So the Equ-
ation (4) can red uce to
31
14
Drr
pp
dC
dt d
ρρ
ρρ

=−−



ug uu
(5)
Assuming
ˆU=uu
;
ˆ
t tT=
, where T is a character-
ristic time, and
T LU=
, thus nondimensional Equation
(5) is
2
ˆ3ˆˆ
1
ˆ4
Drr
pp
dL L
C
dt d U
ρρ
ρρ

= −−



ug uu
(6)
Assuming
( )
2
1
1
p
Lg U ρρ Π= −
,
1
Π
is the
Froude number, g is the gravity acceleration, and
2
34
Dp
LC d ρρ
Π=
. Assuming
f
w
is the terminal ve-
locity of particle, then equilibrium of drag, gravitational
force and buoyancy yields
2
2
3
14
f
D
pp
w
LgL C
dU
U
ρρ
ρρ
 
−=
 
 

(7)
and
( )
2
21 f
UwΠ=Π
(8)
Defining nondimensional parameter
3f
UwΠ=
, then
2
Π is the combination of
1
Π
and
3
Π
.
4.2. Mass Tra nsport
Wind-induced snow transport is mainly achieved by sal-
tation. Saltation o ccurs within the lower layer about 2cm
high above the surface [7]. Provided that a typical salta-
tion particle ascending horizontal speed is 0, the average
horizontal landing speed is u2, the average saltation
length is l (as shown in Figure 1), and Q (kg/m/s) is the
mass transport rate, then the shearing force of wind act-
ing on the saltation transport is
τ
s=Qu2/l. Assuming the
total shearing force of wind is
τ , then the shearing force
of wind acting on the surface bed is
τ
b =
τ - τ
s. Owen [4 ]
has suggested that the shearing force of wind acting on
the bed should be kept on threshold level, i.e.
τ
b =
τ
th. As
if
τ
b >
τ
th, more particles would be inspired to make mo-
tion, leading to
τ
s increases and
τ
b decreases; otherwise,
if
τ
b <
τ
th, more particles tend to cease motion, so
τ
s de-
creases and
τ
b increases. It looks like that there exists a
self-balancing mechanism in mass transport.
So based on above analysis there is
( )
22
2 **th th
Qu luu ττ ρ =−= −
(9)
Provided the average ascending vertical velocity of

W. H. WANG ET AL.
15
particle is v1, then there is u2/l = g/v1, and v1 is propor-
tional to friction velocity [8], i.e. v1 ≈
α u*, where α is a
coefficient. Substituting those relationships into Equation
(9) yields
2
*
32
**
1
th
u Qg
uu
α
ρ

= −


(10)
Equation (10) shows that nondimensional mass trans-
port rate is related to the ratio of shear velocity and thre-
shold shear velocity. Assuming
4**th
uuΠ=
, the para-
meter
4
Π
can be replaced as
5th
UUΠ=
, where Uth is
the threshold reference velocity.
In summary, the dynamic similarity parameters mainly
include
( )( )
22
mp
11
pp
Lg ULg U ρρ ρρ

−= −

(11)
() ( )
mp
ff
Uw Uw=
(12)
( )( )
mp
th th
UU UU=
(13)
5. Reynolds Number and Time Scale
5.1. Reynolds Number
Wind tunnel test model can be established according to
the above rules, such as studied by Isymouv [9], Sant’
Anna [10], O’rourke [11]. Kind [5] suggested that the
Reynolds number should be taken into account and in-
troduced the Reynolds number of roughness-height, i.e.
u*3/(2gν), where ν is the fluid kinematic viscosity coeffi-
cient. The Reynolds number for natural snowdrift is
about 50. In case that the geometry scale of the model is
1/100, as per the relationship between the velocity and
geometric dimension in Equation (11), the Reynolds
number of roughness-height is 0.05, far less than the
lower limit of formation aerodynamic roughness flow.
Therefore, Kind [5] has suggested the model should sa-
tisfy
3
*
2 30ug ν ≥
(14)
The Equation (14) implies that the friction speed of
wind tunnel test should be above 0.2 m/s, but Kind has
also noted that the requirement may be appropriately
relaxed. With high wind velocity, Equation (11) would
not be satisfied, hereby indicating the classic contradic-
tions between the Reynolds number and the Froude
number. Distortion of the Froude number results in that
the saltation length of particles exceeds the required val-
ues. However, it’s believed that as long as the geometry
dimension of model is greater than the saltation length of
particles, the result error would be not so significant [12 ].
The saltation length of particle is about 10 times of the
saltation height, and the average saltation height of par-
ticle is approximately 1.6u*2/(2 g) [8], then the dimension
of the model should comply with the following
2
*
81ugL 
(15)
which can be satisfied readily in a usual wind tunnel test.
5.2. Time Scale
Abandonment of Froude number brings the uncertainty
of time scale. Kind [2] has ignored the effect of Froude
number, and suggested the time scale as
tU L
or
p
tU L ρρ
(16)
Anno [13] has adopted mass transport rate for refer-
ence. The time scale suggested by Anno [13] is generally
deemed as reasonable, but the mass transport rate needs
to be determined, which is difficult for the prototype. The
expression for Anno’ time scale is
2
s
tQ L ρ
(17)
Iversen [6] introduced similar time scale parameters,
but he didn’t adopt mass transport rate explicitly, but a
function of other variable, which need to be determined
through test. One form of the time scales as suggested by
Iversen [6] is
2
1
th
p
U U tU
gLU L
ρ
ρ

−


(18)
6. Wind Tunnel Test
In the present test, a geometric scale ratio of 1/40 was
used. For the snow grain model, the quartz sand with an
average diameter of 0.2 mm in irregular shapes was
adopted. The effect of angle of repose should also be
taken into account in wind tunnel test. The angle of re-
pose for natural snow grains can exceed 90˚, which is
hardly realized for a model particle in wind tunnel.
However, the angle of repose is important only in the test
involved simulation of steep shape; it generally ensures
approximation as closely as possible [2].
The properties of model particles and the major simi-
larity parameters are listed in Ta ble 1. It can be seen that
the u*t 3/(2g
ν ) and u*/u*t of the model fall within the re-
quired range; while wf/u*t is 2 - 3 times larger than the
prototype value, which is mainly because wf of the model
particle is a little too large. The most significant differ-
ence is the Froude number (taking u* for reference) such
that the model is two orders of magnitudes larger than
the prototype, so the Froude number simulation is dis-
torted.
A series of tests were carried out in XNJD-2 wind
tunnel at Southwest Jiaotong University in China. The
working section of the tunnel is 10 m in length, 1.3 m in
width, and 1.5 m in height; the wind velocity can be

W. H. WANG ET AL.
16
Table 1. Particle properties and typical similitude parame-
ters
Parameters Model values Prototype values [3,5,14]
Mean diameter (mm) 0.2 0.1 - 0.5
Density (kg/m3) 2560 500 - 900
Angle of repose (˚) 31 >40
u*t (m/s) 0.149 0.118 - 0.28
wf (m/s) 2.13 0.31 - 0.75
u*t3/(2g
ν ) 13.00 7.0 - 70.0
u*/u*t 1.23 - 1.59 0.66 - 7.93
wf/u* 8.987 - 11.768 0.327 - 3.93
u*2/Lg 0.0171 - 0.0287 0.00043 - 0.00072
wf/u*t 14.2 2.16 - 5
varied between 0 and 19 m/s. Before the wind tunn el test,
a 3 m-long qu artz sand particle bed was paved in front of
the model, with a thickness of 3 cm. A 5 m × 4 m collec-
tion box was laid below the wind tunnel exit to collect
the particles flying out of the wind tunnel, which would
be used for calculation of mass transport rate. Figure 2
presents a snapshot of the test.
The flow field was measured by the KANOMAX
anemometer. Measurement of the wind velocity profiles
included four nominal wind speeds: 4.5 m/s, 5 m/s, 5.5
m/s and 6 m/s, corresponding to 3.87 m/s, 4.14 m/s, 4.60
m/s and 4.99 m/s for reference velocities respectively.
The threshold wind velocity was observed as Uth = 4.14
m/s.
Figure 3 shows the measured velocity profiles. As
shown in the figure, when U > Uth, the aerodynamic
roughness-length increases as the wind speed increases.
The u* obtained by fitting the measurement data of wind
velocities are 0.130, 0.149, 0.183 and 0.211 respectively.
According to Equation (3), z0’ is proportional to u*2.
Figure 4 shows the fitting result of velocities. Th e fitting
relationship is z0’ ≈ 0 .00 316u*2/(2g), and R2 = 0.839, so
based on the field data [8] there is z0’p/ z0’m ≈ 38.
The Iv ersen ’s time scale (tp/tm) is calculated as 113. At
the nominal wind speed of 5.5 m/s the Anno’s time scale
is 153.
Figure 5 shows the nondimensional roof snow depth
distributions (Cs) for a typical stepped building model at
different time intervals, wind speeds and directions. The
figure also presents Tsuchiya’s [15] field observation
results for similar model in Hokkaido of Japan. It can be
seen from the figure that the test results are basically
consistent with the field observation.
7. Conclusions
Through the analysis in aspects of geometry, kinematics
and dynamics, the major similarity parameters that need
Figure 2. Snapshot of the test.
0123456
10-4
10-3
10-2
10-1
100
101
102
u (m/s)
z
(cm)
U=3.87m /s
U=4.14m /s
U=4.60m /s
U=4.99m /s
Figure 3. Velocity profiles.
Figure 4. Fitting velocities.
to be satisfied for snowdrift wind tunnel test model are
gained. As for the contradiction between the Reynolds
number and Froude number, the existing wind tunnel
tests generally place priorities on the Reynolds number
above the Froude number, so the main similarity para-
meters that need to be satisfied for models include Equa-
tions (1), (2), (12), (13) and (14). As for Equation (11),
it’s generally ensured to minimize the difference of the
model and the prototype. Other parameters also include
angle of r e pose, restitute coefficient and time scale.

W. H. WANG ET AL.
17
00.5 11.5 22.5 3
0
0.5
1
1.5
2
x/H
Cs
Tsuchiya－1
Tsuchiya－2
Tsuchiya－3
t = 300s, U = 4.60m/s, β = 0
o
t = 300s, U = 4.99m/s, β = 0
o
t = 300s, U = 5.36m/s, β = 0
o
t = 300s, U = 4.99m/s, β = 15
o
t = 900s, U = 4.99m/s, β = 15
o
Figure 5. Roof snow depth distr i but ions.
It should be pointed out that, the snowdrift-involved
wind tunnel test is still in the development stage, thus
there are many things that need to be improved, such as
the impact of natural environment on the physical fea-
tures (temperature, humidity, and sublimation, etc.) of
snow grains in actuality, which has not been taken into
consideration in wind tunnel simulations.
According to the similarity law, a wind tunnel test by
adoption of quartz sand as the snow grain model was
conducted. The flow field characters were measured and
analyzed, and the roof snow distributions of a typical
stepped building were investigated. The results show that,
with the particle saltation, the velocity profiles outside
the saltation layer agree well with the field observations;
the stepped roof snow depth distributions are basically
consistent with the observation results.
8. Acknowledgements
The project is supported by National Natural Science
Foundation of Chi na ( G rant No. 41 272832) .
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