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J. Wind Eng. Ind. Aerodyn. 100 (2012) 77–90
Contents lists available at SciVerse ScienceDirect
Journal of Wind Engineering
and Industrial Aerodynamics
journal homepage: www.elsevier.com/locate/jweia
On the stochastic nature of compact debris flight
A. Karimpour, N.B. Kaye n
Glenn Department of Civil Engineering, Clemson University, Clemson, SC 29634, USA
a r t i c l e i n f o
abstract
Article history:
Received 22 June 2011
Received in revised form
1 November 2011
Accepted 3 November 2011
Available online 22 December 2011
The stochastic nature of debris flight is investigated through a series of Monte Carlo simulations based
on the debris flight equations for compact debris presented by Holmes (2004). Any given debris flight
situation presents a number of uncertainties such as the size of the piece of debris and the time-varying
turbulent wind flow. Current debris flight models are largely deterministic and do not account for such
uncertainty in input parameters. The simulations presented model the flight of a single spherical
particle whose diameter is given by a probability distribution function, driven by a turbulent wind with
velocity fluctuations appropriate to the atmospheric boundary layer. The model predicts the mean and
standard deviation of the particle flight distance and impact kinetic energy. Results show that
introducing uncertainty in particle diameter, horizontal turbulence intensity, or vertical turbulence
intensity leads to larger mean values for flight distance and impact kinetic energy, compared to the
condition where there is no variability in input parameters. Introducing input parameter variability also
leads to variability in flight distance and impact kinetic energy that is quantified in this study. While
the simulations presented do not realistically characterize the complex flow within an urban canopy,
the results provide significant physical insight into the influence of particle size variability and
turbulence on the mean and standard deviation of the flight distance and impact kinetic energy.
& 2011 Elsevier Ltd. All rights reserved.
Keywords:
Windborne debris flight
Monte Carlo simulation
Stochastic
Compact debris
1. Introduction
Windborne debris penetrating a building’s envelope can result
in significant damage, varying from simple broken windows and
resulting rain inundation to total destruction of the building.
Damage of the former kind occurred at the Hyatt hotel in downtown New Orleans during Hurricane Katrina. A post-Katrina
assessment found that pea gravel, most likely from the roof of
the adjacent Amoco building, broke 75% of the windows on the
north face of the hotel (Kareem and Bashor, 2006). Because of this
substantial damage, hotel operations were restricted to significantly reduced capacities during subsequent repairs. Total
damages were initially estimated at $100M (Bergen, 2005).
Debris penetration of the building envelope can also result in
the entire collapse of a structure. Once windows are broken, the
wind raises the internal pressure within the building. This
internal pressure increase will increase the net uplift on the
building’s roof, potentially leading to roof separation. If the roof
is actually integrated into the structural bracing of the building,
roof separation can cause a complete collapse of the building.
Post-storm forensic investigations (Sparks, 1998) have found a
number of such structural failures in buildings with large open
n
Corresponding author. Tel.: þ1 864 656 5941.
E-mail address: [email protected] (N.B. Kaye).
0167-6105/$ - see front matter & 2011 Elsevier Ltd. All rights reserved.
doi:10.1016/j.jweia.2011.11.001
internal spaces and un-reinforced walls (e.g., large churches and
big box stores).
The risk posed by windborne debris is not restricted to large
commercial structures; family dwellings are also at risk.
A forensic investigation of 466 houses following Hurricane
Andrew (Sparks et al., 1994) found that 64% had at least one
broken window, most likely due to debris penetration, whereas
only 2% of walls sustained moderate to severe damage resulting
from wind pressures. This suggests that debris impact caused
significantly more damage to housing than structural failures due
to wind pressures. Therefore, a fuller understanding of debris
flight is essential to quantifying the risk of property damage and
personal injury during severe storms.
Existing compact debris flight models are based on Newtonian
mechanics. They consist of equations of motion for a particular
particle in which the forces acting on the particle are the
gravitational body force, drag, and lift forces. Such a model was
presented by Tachikawa (1983, 1988) who was the first to derive
non-dimensional equations for the trajectories of wind born
debris. Tachikawa showed that the dimensionless parameter
!
ra U 2 A
3ra U 2
K¼
¼
ð1Þ
2mg
4rp gd
controls the nature of the flight. The parameters U, m, A, and ra
are the wind speed, particle mass, particle cross sectional area,
and air density, respectively. The term in brackets is the
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A. Karimpour, N.B. Kaye / J. Wind Eng. Ind. Aerodyn. 100 (2012) 77–90
Tachikawa number for a sphere written in terms of the particle
density QUOTE and diameter QUOTE . For large K, debris flight is
relatively flat with high horizontal velocity, whereas for low K the
flight path is more vertical. Tachikawa’s contribution was recognized by having this parameter named after him (Holmes et al.,
2006). Early work on debris flight was largely focused on establishing appropriate realistic impact criteria for standard missiles
to be used in testing the impact resistance of structural cladding
and windows. Therefore, models were deterministic and focused
on fixed inputs and steady wind speeds.
Simplified versions of the Tachikawa equations in two dimensions ignoring lift forces were presented by Holmes (2004).
Equations were derived for the vertical (z) and horizontal (x)
accelerations in terms of the horizontal U and vertical W wind
speeds and the horizontal u and vertical w components of the
particle velocity
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
d x
r CDA
ðUuÞ ðUuÞ2 þðWwÞ2
¼ a
2
2m
dt
ð2Þ
and
2
d z
r CDA
ðWvÞ
¼ a
2m
dt 2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðUuÞ2 þ ðWwÞ2 g:
ð3Þ
where CD is the drag coefficient. Note that the mean vertical
velocity is W ¼ 0 and the mean horizontal velocity is denoted by
U. Holmes (2004) presented two test cases to demonstrate the
behavior of compact debris, a piece of pea gravel appropriate for
use as ballast on a built up roof (a roof consisting of successive
layer of roofing felts laminated together with bitumen and
commonly covered with gravel), and a slightly larger wooden
sphere. The equations were solved numerically to calculate the
flight distance and impact kinetic energy. Additionally, eight
simulations were run with a time-varying velocity with turbulence characteristics appropriate for atmospheric flows. The
turbulence was found to have minimal impact on the mean flight
distance.
The same set of equations was analyzed by Baker (2007) for
more general cases of compact debris flight. Baker wrote the
equations in non-dimensional form though in this case the
non-dimensionalization resulted in a parameter O ¼1/K. Though
there are no general analytic solutions to (2) and (3), Baker showed
that after sufficient flight time a steady solution can be found in
which the particle will travel horizontally at the mean wind speed
and vertically at its terminal velocity. Baker also investigated the role
of turbulence and found that it had a negligible effect on the flight
outcomes; however, this is likely because the simulations were run at
the gust wind speed rather than the mean wind speed.
The issue of how to most appropriately define the ‘‘mean wind
speed’’ is complex. Previous studies (e.g. Baker, 2007) have argued
that the appropriate speed is the gust wind speed as debris
typically is launched during strong gusts and the resulting flight
time is short compared to the averaging time needed to get a
steady mean (typically of the order of 10 min). However, there are
circumstances in which debris is launched continuously, rather
than just during peak gusts. For example, it is possible, during
severe storms, to get continuous scour of gravel from a roof (Kind,
1986). In this case, although the individual flight times are still
short, the mean velocity for the event will be the ten minute
mean. Some debris will travel during peak gusts while others will
be transported by lower velocity winds. For the purposes of this
study the term ‘mean velocity’ will represent the ten minute wind
speed and turbulent fluctuations will be superimposed onto this
mean. Interestingly, it will be shown that, for a large enough
sample size of particles being launched into a turbulent wind field
with mean velocity as defined, the mean flight distance is given
by the flight distance of a particle transported at a velocity
analogous to the gust speed (see Section 4.1 below).
A further complicating factor is the appropriate parameterization of the atmospheric boundary layer. Again previous researchers have ignored boundary layer profiles using the same
argument that the particles are transported by gusts and that
the velocity profiles for such gusts are more uniform. There is also
the added complexity of how to account for the complexities of
the urban canopy flow, which is dominated by flow around
buildings, wakes, and regions of high vertical and transverse
shear. Rather than attempt to parameterize this complexity, this
study is focused on quantifying the influence of turbulent fluctuations and particle size variability of the mean and standard
deviation of the flight outcomes. As such, the velocity profile is
taken to be uniform. Note that this represents a worst case as
inclusion of a boundary layer profile would result in lower wind
speeds near the ground reducing the horizontal velocity of the
object, and therefore its flight distance and impact kinetic energy.
Further, ignoring the canopy flow means that the mean vertical
velocity is equal to zero.
Debris flight models are becoming increasingly sophisticated
at accounting for turbulence in the atmospheric boundary layer
(Holmes, 2004), lift forces (Baker, 2007), plate like debris (Lin
et al., 2007), three dimensional motions including rotation
(Richards et al., 2008), and lift off criteria (Wills et al., 2002).
With the exceptions of Holmes (2004) and Baker (2007), who
each briefly discuss the role of atmospheric turbulence, the debris
flight models are deterministic. These models have constant
parameter inputs and solve for the flight of a single object.
However, debris flight is not a deterministic phenomenon.
Take the simple case of the flight of roof gravel blown off a built
up roof. The individual particle size is not known and is best
described by some probability distribution function. Vertical and
horizontal turbulent fluctuations in the wind velocity will also
lead to variations in the flight distance and impact kinetic energy.
Further, the particle shape will not be uniform. Therefore,
compact debris flight is a stochastic process in which there is
statistical uncertainty and variability for a range of input parameters that results in uncertainty and variability in the flight
outcomes, taken to be the horizontal distance and impact kinetic
energy.
The stochastic nature of debris flight is also observed in other
cases. For example the impact location of a shingle blown off a
house is extremely sensitive to the launch angle and roof location,
initial wind speed and angle, and the building wake (Kordi and
Kopp, 2011). Given the uncertainty in these conditions, and the
sensitivity of the flight distance to these parameters, the debris
field is best described in terms of a probability distribution
function with the function parameters related to the statistical
properties of the input conditions.
The goal of this paper is to investigate the stochastic nature of
compact debris flight and the role of input uncertainty in the
classical debris flight model. The impact of uncertainty is investigated through a series of Monte Carlo simulations based on the
debris flight equations of Holmes (2004) (Eqs. (2) and (3)), and
analysis of the steady solutions of the flight equations based on
the work of Baker (2007). In particular, the effect of statistical
distributions of particle diameter, horizontal turbulence intensity,
and vertical turbulence intensity on the flight distance and impact
kinetic energy are investigated. The authors selected roof gravel
flight as a test case as it is a pressing problem, and it is possible to
make a reasonable estimate of the statistical properties of the
particles. Further, the equations are solved in dimensional form as
the statistical properties of the aggregate depend on the mean
particle size. That is, there is no universal coefficient of variation
for particle size. As with previous studies of compact debris
A. Karimpour, N.B. Kaye / J. Wind Eng. Ind. Aerodyn. 100 (2012) 77–90
2. Monte Carlo simulations
A series of Monte Carlo simulations were run to examine the
influence of particle size (d) uncertainty and turbulent velocity
fluctuations, parameterized in terms of turbulence intensities in
the horizontal (Iu) and vertical (Iw) directions, on the flight
distance and impact kinetic energy of roof gravel. The simulations
were run using the compact debris flight equations of Holmes
(2004). The particles were assumed to be spherical and the drag
coefficient was assumed to be constant throughout the flight.
Simulations were run varying d with Iu and Iw equal to zero,
keeping d constant while varying one of Iu and Iw, keeping one
parameter constant and varying the other two, and varying all
three parameters at the same time. For each simulation, a random
set of input conditions (particle size and time varying wind field)
was generated for a large number of cases and each case was
solved numerically using MATLAB. For each case results were
calculated for initial release heights of 1, 2, 5, 10, 20, and 50 m.
Before running the simulations, it is important to establish the
appropriate statistical properties of the input conditions (particle
size distributions and turbulence properties), and to calculate the
number of simulations per test case required to ensure that the
flight outcome statistics are accurately captured.
2.1. Gravel size distribution
The exact distribution function appropriate for gravel diameter
d is not known. For gravel, the size range is roughly based on
the largest and smallest sieves used to categorize the stone.
Considering some general distribution function as shown schematically in Fig. 1(a), it can be seen that a small sampled portion
will have an approximately linear distribution. However, a uniform distribution also gives a good first approximation, see
Fig. 1(b), and requires no data on the statistical properties of
the gravel source or manufacturing processes, only the largest and
smallest stone size. The equivalent cumulative distribution function would be a straight line between the upper and lower limits.
This is seen to be approximately the case for graded gravel based
on ASTM D 1863-05 (ASTM, 2005; see Fig. 2). For the simulations
presented in this paper, d is taken to be uniformly distributed
between the maximum and minimum values appropriate for roof
gravel. The mean and standard deviation are therefore given by
d¼
dmax þ dmin
2
and
sd ¼
dmax dmin
pffiffiffiffiffiffi ,
12
ð4Þ
respectively (Harris and Stocker, 1998). The basic test case was a
uniform particle size distribution with dmin ¼4.75 mm and
dmax ¼9.5 mm. However, simulations were also run for wider
and narrower particle size ranges to explore the effect of input
standard deviation on both the mean and variance in the flight
outcomes.
Finer Than Sieve Specified (%)
(Holmes, 2004; Baker, 2007), the study is restricted to two
dimensions as the lift forces on compact debris are negligible
and therefore transverse motions can be ignored.
While it would, in theory, be possible to solve the nondimensional form of the equations, as per Baker (2007), this
would require non-dimensionalizing the input probability
distributions. Considering multiple uncertainties would lead to
complex probability distributions for the controlling parameter
(O). Such an approach would not result in universal conclusions
that are the primary benefit of using dimensionless equations as
the complex non-dimensional probability distributions would be
specific to a given problem.
The remainder of the paper is structured as follows. Section 2
describes the model developed to investigate the effect of atmospheric turbulence and particle size uncertainty on the flight path.
Section 3 presents results from numerical solutions of the debris
flight equations for different atmospheric turbulence intensities,
and appropriate distributions for gravel size. In Section 4, various
analytical approaches for estimating the impact of input parameter uncertainty on flight outcomes are presented. The results of
a series of wind tunnel particle flight tests are shown in Section 5
demonstrating the validity of the stochastic modeling approach.
The significance of the results and a more general discussion of
the stochastic nature of debris flight are presented in Section 6,
along with conclusions.
79
100
90
80
70
60
50
40
30
20
10
0
0
5
10
15
20
Particle size (mm)
25
30
Fig. 2. Cumulative distribution of typical graded gravel particle diameters based
on ASTM D 1863-05 (ASTM, 2005). Dashed line—size 6, solid line—size 7.
Fig. 1. (a) Generic distribution function for a particular variable such as gravel size. (b) Sampled variable (for example, due to sieving of gravel) showing uniform
distribution approximation.
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A. Karimpour, N.B. Kaye / J. Wind Eng. Ind. Aerodyn. 100 (2012) 77–90
2.2. Turbulent wind field
The turbulent wind field was generated using the technique
proposed by Holmes (1978) in which the wind field is created in
frequency space with wave amplitudes deterministically sized
based on the appropriate power spectra for horizontal and
vertical fluctuations, and the phase angle randomly generated
using a uniform distribution between 0 and 2p. The spectra
employed are the same as those used by Holmes (2004), namely
the von Karman form (von Karman, 1948) in the horizontal:
nSu ðnÞ
s2u
¼
4ðnlu =UÞ
½1 þ 70:8ðnlu =UÞ2 5=6
ð5Þ
and
nSw ðnÞ
s2w
¼
2:15ðnz=UÞ
1 þ 11:16ðnz=UÞ5=3
ð6Þ
in vertical (Busch and Panofsky, 1968). The frequency is denoted
by n, su and sw are the standard deviations of the horizontal and
vertical velocity fluctuations, lu is a length scale for the turbulence, and U is the mean horizontal velocity.
The spectra in (5) and (6) are Eulerian, and therefore only
strictly applicable at a particular point in space, whereas the
particles are moving. An alternative would be to use Lagrangian
spectra, however this poses a different set of challenges. Firstly,
the particles, particularly early in their flight, do not move with
the wind but are rather accelerating toward the wind speed, so
the fluctuations they are exposed to are neither Eulerian nor
Lagrangian. Secondly, Lagrangian spectra are not as readily available (Holmes, 2004). Ideally, a full turbulence simulation of the
atmospheric boundary layer would be conducted for each flight
instance giving velocity values at each point in space and time.
However, this approach is exceptionally computationally expensive. Rather, the study was conducted using the spectra in (5) and
(6) as a first order estimate of the turbulence felt by the particles
during flight. This approach has two advantages. First, it allows
direct comparison of this study’s results with those of Holmes
(2004), and second, it allows a computationally efficient model to
be run that will illustrate the role of turbulent fluctuations on the
mean and variance of compact debris flight outcomes. For the
simulations run herein, lu ¼100 m and the base case intensities
were Iu ¼20%, and Iw ¼12% to match those of Holmes (2004).
Sample horizontal and vertical wind velocities are shown in Fig. 3.
Holmes (2004) also considered the correlation between the
vertical and horizontal velocity fluctuations. The random phase
angles used to generate the horizontal fluctuations were used to
generate vertical fluctuations and a small fraction of this velocity
time series was added to the original vertical velocity time series
to achieve an appropriate correlation coefficient. In this study the
correlation coefficient was not considered for the bulk of the
simulations, and no correction was made to the vertical velocity
time series. Instead, a separate series of simulations were run to
investigate the role of the correlation coefficient on the flight
outcomes. In these separate simulations, the correlation coefficient was the only parameter varied.
2.3. Required number of simulations
A series of simulations were run in order to establish the
number of individual cases required to accurately characterize the
statistical properties of the flight outcomes. A set of simulations
were run with d ¼ 7:13 mm, sd ¼1.37 mm, U ¼ 18:9m=s, Iu ¼20%,
and Iw ¼12% and a release height of 20 m. The simulations were
repeated 5 times each with 5, 10, 50, 100, 500, 1000, 5000, and
10000 runs per simulation. Fig. 4 shows the mean flight distance
Fig. 3. Sample velocity time sequences in the horizontal (upper line) and vertical
(lower line) directions.
Fig. 4. Plot of mean flight distance calculated based on different numbers of runs
per simulation. Simulation based on d ¼ 7.13 mm, sd ¼1.37 mm, u ¼18.9 m/s,
Iu ¼ 20%, and Iw ¼12% and a release height of 20 m.
for each set plotted against the number of runs per set. As one
would expect, as the number of runs per simulation increased, the
variation in the mean flight distance decreased. When 10,000
simulations were run, the variability in the mean flight distance
was less than 1%. This was regarded as a reasonable variation and
so each simulation set was run for 10,000 individual cases.
2.4. Numerical technique
For each simulation set, 10,000 particle diameters and wind
velocity time series were generated. These were used as inputs to
a MATLAB code that solved for the flight path using a 4th order
Runge–Kutta scheme with a time step of 0.02 s. For each case the
flight path was tracked over time for 10 s of flight time.
Cubic spline interpolation was then used to find the horizontal
displacement and particle kinetic energy at vertical distances of 1,
2, 5, 10, 20, and 50 m below the release height. This data was then
used to calculate the mean and standard deviation of the flight
A. Karimpour, N.B. Kaye / J. Wind Eng. Ind. Aerodyn. 100 (2012) 77–90
distance and impact kinetic energy at each height. The numerical
scheme was tested for the deterministic case presented in Holmes
(2004) of a 8 mm piece of gravel released from a height of 10 m
with a constant horizontal wind speed of 20 m/s. The resulting
flight distance and impact kinetic energy were the same as those
presented by Holmes (2004).
In all cases, the mean horizontal velocity is assumed to be
uniform with height. This is a reasonable assumption for short
flight times such as those typical for debris being blown off
buildings, as the flight time is significantly less than the averaging
time required to establish a smooth mean boundary layer profile.
This assumption may be inappropriate for debris flight that has
longer flight times, such as ember flight from wild fires in which
embers can be lofted high into the atmosphere. Ignoring mean
velocity variation with height is a conservative assumption in that
it leads to a worse case (longer flight distance) (Karimpour and
Kaye, 2010).
3. Simulation results
Although the simulations were run in dimensional form, the
results are largely presented in non-dimensional form. Inputs and
flight outcomes are scaled on the inputs and outcomes of the
base case of a gravel particle with density rp ¼2000 kg/m3 and
diameter d ¼ 7:13 mm released into a wind with horizontal
velocity U ¼ 18:9 m=s. The velocity was calculated based on the
model of Wills et al. (2002) and the diameter is based on data in
Table 1 of ASTM D 1863-05 (ASTM, 2005). For all results
presented below, only the variation about these mean values is
changed, and the mean properties are always held constant.
The flight outcomes are scaled on the flight outcomes of the
mean case. The mean flight distance and mean kinetic energy are
denoted by
X¼
x
xðd ¼ d,Iu ¼ Iw ¼ 0Þ
and
KE ¼
ke
keðd ¼ d,Iu ¼ Iw ¼ 0Þ
81
simulation gave flight distance and impact kinetic energy for six
different release heights, making a total of 810,000 individual
simulations for 4.9 million simulated flights.
Not all the data is presented below, but rather summary data
that quantifies the role of input uncertainty on outcome uncertainty. Results quantifying the change in mean flight distance and
kinetic energy due to variability in input conditions are given in
Section 3.1. In Section 3.2 the variability in outcomes due to input
variability is discussed. The change in outcomes is also dependent
on the flight distance, so the variation in mean outcomes at
different heights is presented in Section 3.3. Results illustrating
the impact of the correlation between the vertical and horizontal
turbulence fluctuations are shown in Section 3.4.
3.1. Effect of input parameter uncertainty on mean flight outcomes
In this section we present data on X¼X(CVd,Iu,Iw) and
KE ¼KE(CVd,Iu,Iw). Although simulations were run for six different
release heights, only results for a vertical drop of 50 m are
reported. The effect of release height is discussed in Section 3.3.
Plots of dimensionless flight distance X and dimensionless impact
kinetic energy KE versus CVd, Iu, and Iw are shown in Fig. 5.
ð7Þ
where xðd ¼ d,Iu ¼ Iw ¼ 0Þ and keðd ¼ d,Iu ¼ Iw ¼ 0Þ represent the
flight distance and impact kinetic energy calculated using the
mean particle diameter and no turbulence whereas x and ke are
the mean flight distance and impact kinetic energy based on a
10,000 run simulation varying at least one parameter. Values of X
or KE greater than one indicate that the mean flight distance or
kinetic energy is greater than that for the deterministic case. The
standard deviation of the diameter is denoted as sd, and is
reported as a coefficient of variation (CV):
CV d ¼
sd
d
:
ð8Þ
Turbulence is reported in terms of the turbulence intensities Iu
and Iw. Outcome variations are reported in terms of their
coefficients of variation:
CV x ¼
sx
x
and
CV ke ¼
ske
ke
:
ð9Þ
The mean and variance in the outcomes ðX,KE,CV x ,CV ke Þ are
functions of the mean and variance of the inputs ðd,U,CV d ,Iu ,Iw Þ
and the release height.
A total of 71 different cases were run with 10,000 simulations
per case. The cases included simulations in which only one
parameter was varied (9 varying diameter, 8 varying Iu, and
9 varying Iw), two were varied and one held constant (8 with
Iw ¼0%, 8 with Iu ¼ 0%, and 12 with the diameter held constant),
and cases where all three parameters were allowed to vary
(17 cases). A further 100,000 simulations were run in which the
correlation between the vertical and horizontal turbulence fluctuations was varied while keeping the diameter constant. Each
Fig. 5. Effect of input parameter variation on (a) mean flight distance and
(b) mean impact kinetic energy.
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A. Karimpour, N.B. Kaye / J. Wind Eng. Ind. Aerodyn. 100 (2012) 77–90
The results in this figure are based on simulations in which only
one parameter is varied at a time.
By introducing an uncertainty into the system, the dimensionless flight distance X increases. This implies that ignoring uncertainty in the input parameters will lead to an underestimation of
the mean flight distance. The mean flight distance increases as the
square of the input coefficient of variation (CVd,Iu,Iw). See Section
4.1 for an analysis of this behavior. The impact of variation in
particle diameter is the most significant for a given CV. For typical
real world values (CVd ¼0.2, Iu ¼20%, and Iw ¼12%) the mean flight
distance increases by 2% due to size variation, 1% due to
horizontal turbulent fluctuations, and less that 0.5% due to
vertical turbulent fluctuations. The effect of varying all three
parameters at once is discussed in Section 3.3.
Direct comparison with Holmes (2004) indicates that the
relative increase in flight distance when both horizontal and
vertical turbulences are included is approximately 1.5% averaged
over 10,000 simulations compared with 0.5% found by Holmes
(2004) based on 8 simulations. A larger number of simulations by
Holmes would likely resolve this discrepancy (see Fig. 4).
A similar trend is observed for the mean impact kinetic energy.
However, the mean impact kinetic energy is much more sensitive
to variability in the particle diameter CVd and almost totally
insensitive to variations of Iu and Iw. Again, the mean impact
kinetic energy is greater than the impact kinetic energy of the
particle with mean diameter. As with the flight distance results,
the increase in KE scales on the square of the input CV (see
Section 4.1). For the same typical case described in the previous
paragraph, KE increases by 12% due to size variation, and by less
than 0.5% due to horizontal and vertical turbulent fluctuations.
3.2. Effect of input parameter uncertainty on output results
distribution
In this section we present data on CV x ¼ CV x ðCV d ,Iu ,Iw Þ and
CV ke ¼ CV ke ðCV d ,Iu ,Iw Þ. Again data is only presented for a release
height of 50 m and generally only for simulations in which one
parameter is varied at a time. Clearly, introducing variability into
the input conditions will result in variability in the flight distance
and impact kinetic energy. This can be seen in Figs. 6 and 7, which
Fig. 6. Histograms of particle flight distance for different input variability. Top left—CVd ¼ 0.2, Iu ¼ Iw ¼0, top right—Iu ¼20%, CVd ¼ Iw ¼0, bottom left—CVd ¼ Iu ¼0, and
Iw ¼12%, and bottom right—CVd ¼ 0.2, Iu ¼20%, and Iw ¼ 12%.
A. Karimpour, N.B. Kaye / J. Wind Eng. Ind. Aerodyn. 100 (2012) 77–90
83
Fig. 7. Histograms of particle impact kinetic energy for different input variability. Top left—CVd ¼0.2, Iu ¼ Iw ¼0, top right—Iu ¼ 20% , CVd ¼ Iw ¼ 0, bottom left—CVd ¼ Iu ¼0,
and Iw ¼ 12%, and bottom right CVd ¼ 0.2, Iu ¼20%, and Iw ¼ 12%.
show histograms of flight distance (Fig. 6) and impact kinetic
energy (Fig. 7) when the inputs are varied.
A number of points are worth noting from these figures. First,
the outcome distributions resulting from vertical turbulent
fluctuations exhibit the least spread. Second, the distributions
could not reasonably be described in terms of the most common
probability distributions. In particular, the outcomes due to
turbulence exhibit a distinct double peak. Finally, running
10,000 flights per case provides accurate predictions of the
outcome mean and standard deviations (Fig. 4) and provides
relatively smooth outcome histograms (Figs. 6 and 7).
The data from each of these histograms, along with data from
14 other simulations, were used to calculate the standard deviation of the flight distance and impact kinetic energy, shown in
Fig. 8. This figure shows log–log scale plots of output CV as a
function of input CV. For all cases the data falls on a straight line
indicating a power law relationship between the input CV and the
outcome CV. For both flight distance and impact kinetic energy,
their CV scales linearly with the particle diameter CV and with
the square of the turbulence intensities. See Section 4.2 for an
analysis of this behavior.
For typical real world values (CVd ¼0.2, Iu ¼20%, and Iw ¼12%),
variation in flight distance CVx ¼0.14 for particle diameter variation, CVx ¼0.10 for horizontal turbulence, and CVx ¼0.04 for
vertical turbulence. The impact kinetic energy variation is
comparable when considering turbulence (CVke ¼0.08 due to Iu
and CVke ¼0.03 due to Iw). However, variation due to particle size
variation is significantly greater (CVke ¼ 0.56). For a 10 m release
height the coefficient of variation in flight distance due to
horizontal and vertical turbulences is CVx ¼0.15 compared
to CVx ¼0.105 reported by Holmes (2004). Again, this minor
discrepancy is attributable to the small number of simulations
run by Holmes.
3.3. Effect of release height and multiple parameter variation
Sections 3.1 and 3.2 discussed the role of individual parameter
variation on outcome variation. All cases considered had a release
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A. Karimpour, N.B. Kaye / J. Wind Eng. Ind. Aerodyn. 100 (2012) 77–90
Fig. 8. Effect of input parameter variation on (a) flight distance CV and (b) impact kinetic energy CV.
height of 50 m and mean input values of d ¼ 7:13 mm,
U ¼ 18:9m=s, and W ¼ 0 m=s. This section considers the role of
release height on the mean and standard deviation of flight
distance and kinetic energy, as well as the impact of varying
more than one parameter at a time. Results are presented for
release heights of H¼1, 2, 5, 10, 20, and 50 m with the same mean
input conditions. For all results presented, the variability/uncertainty in the input conditions is fixed with CVd ¼ 0.2, Iu ¼20%, and
Iw ¼12%. Fig. 9(a) and (b) shows the change in mean flight
distance (X) and impact kinetic energy (KE) for different release
heights when varying the input parameters one at a time and
when varying all three together. The behavior is quite different for
the two outcomes.
The relative change in the mean flight distance (X) decreases
with increasing release height from 5% (when varying all three
inputs) at 1 m to 3% for a release height of 50 m. Further,
X decreases with increasing release height regardless of which
parameter is varied though the decline is more significant when
only varying the particle size compared to horizontal and vertical
turbulences. The sum of the percentage increases due to individual parameter variations is slightly more than the percentage
increase when varying all three parameters at the same time. For
example, for a release height of 20 m, the sum of the percentage
increases in X when varying one parameter at a time is 3.8%
whereas the percentage increase when varying all three at once
was only 3.3%.
In contrast, the relative change in the mean impact kinetic
energy (KE) increases with increasing release height from 10%
(when varying all three inputs) at 1 m to 14% for a release height
of 50 m. However, this is entirely due to variations in the particle
size. For the simulations in which the particle size was kept
constant, KE decreased with increasing height. The total variation
in mean impact kinetic energy is dominated by the particle size
variation. Interestingly, and in contrast to mean flight distance,
the sum of the percentage increases due to individual parameter
variations is slightly less than the percentage increase when
varying all three parameters at the same time. Again, taking the
example of a release height of 20 m, the sum of the percentage
increases in KE when varying one parameter at a time is 11%
whereas the percentage increase when varying all three at once is
almost 12%. While it is interesting to note the difference in
behavior between flight distance and kinetic energy increases,
the actual differences are insignificant.
Fig. 9(c) and (d) shows the coefficient of variation for the flight
distance and impact kinetic energy, respectively. The outcome CV
trends are similar to those observed for the increase in mean
outcomes. The flight distance CV decreases with increasing height
for all three input parameters. The most rapid decrease in CV is
due to horizontal turbulence fluctuations. At a release height of
1 m, CVx ¼0.19, which reduces to CVx ¼0.10 for a 50 m release
height. This reduction is due to the increased flight time. The
further the particle falls, the longer the flight time. Therefore, the
probability that the flight distance is significantly influenced by a
short intense gust is less for longer flight times. That is, the
probability that the wind speed averaged over the flight time is
significantly different from the mean wind speed (say the 10 min
wind speed) decreases with increasing flight time (i.e. with
increasing averaging time).
As with the mean impact kinetic energy, CVke increases with
increasing release height. Again, CVke is dominated by variation in
the particle diameter with turbulence playing only a minor role.
For example, at a release height of 20 m, CVke ¼0.52 when varying
all three parameters and CVke ¼0.518 when varying only the
particle size. A full discussion of the influence of varying multiple
inputs on outcome CV is left to Section 4.3.
3.4. Impact of turbulent fluctuation correlation
As stated in Section 2.2, no attempt was made to manipulate
the velocity fields so that there was a correlation between the
vertical and horizontal turbulences. In fact, with the exception of
Fig. 9, only one turbulence component was turned on at a time.
This section presents results of a series of simulations in which
the correlation coefficient
P
r ¼ u0 w0
puw
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð10Þ
P
P ffi
u02 w02
was systematically varied. The terms u0 and w0 represent the
instantaneous fluctuations about the mean of the horizontal and
vertical velocities. In the atmospheric boundary layer there is a
small negative correlation between these fluctuations, which
physically represents a downward turbulent transport of
momentum.
Simulations were run with d ¼ 7:13 mm, U ¼ 18:9 m=s,
W ¼ 0 m=s, CVd ¼0, Iu ¼20%, and Iw ¼12% and a release height of
50 m. For each simulation, the vertical and horizontal turbulence
fluctuations were generated in the manner described in Section
2.2. Then, following the approach of Holmes (2004), a second set
of vertical fluctuations was developed using the vertical power
spectra and the random phase angles used in generating the
A. Karimpour, N.B. Kaye / J. Wind Eng. Ind. Aerodyn. 100 (2012) 77–90
85
Fig. 9. Flight outcome change as a function of release height for vertical turbulence (triangles), horizontal turbulence (squares), particle size (diamonds), and all three
(crosses). (a) Mean flight distance, (b) mean impact kinetic energy, (c) flight distance CV, and (d) impact kinetic energy CV.
horizontal fluctuations. A small fraction of this second set of
vertical fluctuations was added to the first set and ruw was
calculated. No attempt was made to fine tune this process to give
a particular value of ruw. Rather, 100,000 simulations were run
and the data sets were placed in groups with different ranges of
ruw. The groups were 0.1 wide and were centered at ruw ¼ .05,
.15, .25, .35, and .45. Each group contained at least 9,000
simulations, which were then analyzed to calculate the flight
outcome mean and standard deviation. Plots of the variation in
the mean (X,KE) and coefficient of variation (CVx,CVke) of flight
distance and impact kinetic energy against ruw are given in
Fig. 10. Each point represents all the flights in a particular group
and is plotted at the center of the group.
Clearly the correlation coefficient has a negligible impact on
the mean flight distance and kinetic energy. Over the range of ruw
investigated the mean flight distance varied by less than 0.5% and
the impact kinetic energy by less than 0.25%. However, the
coefficient of variation did vary though with a larger variation
in CVx (from 0.08 to 0.12) than in CVke (from 0.09 to 0.105).
Interestingly the trend in CV is not consistent between the two
cases. A peak in CVke occurs for ruw ¼ 0.35 and decreases
as ruw approaches zero. In contrast CVx increases as ruw
approaches zero.
The trend in CVx can be understood in terms of the nature of
the fluctuations. If the particle is influenced by an upward gust,
positive w0 , then for values of ruw o0 there is likely to be a
corresponding gust in the negative x direction. That is, an upward
gust that will tend to increase flight time will be accompanied by
a negative horizontal gust that will reduce the flight distance. For
negative w0 the opposite is true, that is, a down gust that will tend
to shorten the flight will be accompanied by a positive horizontal
gust that will tend to extend the flight distance. As u0 and w0
become uncorrelated (ruw ¼0) the probability that an upward
gust is accompanied by a positive horizontal gust, and vice versa,
increases, leading to a greater probability of longer and shorter
flight distances. Therefore, the more negatively correlated u0 and
w0 the less variation would be expected in the flight distance. The
probability of a particularly long flight is no greater than that of a
particularly short flight as a result of any correlation. Hence the
mean flight distance is virtually unchanged regardless of ruw.
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A. Karimpour, N.B. Kaye / J. Wind Eng. Ind. Aerodyn. 100 (2012) 77–90
Fig. 10. Mean (top row) and standard deviation (bottom row) of flight distance (left column) and kinetic energy (right column) as a function of the correlation coefficient ruv.
4. Analysis of outcome variation
As shown above, uncertainty and variability in the particle
diameter, horizontal wind velocity, and vertical wind velocity
increases the mean flight distance and impact kinetic energy. For
both these outcome parameters, both the mean and standard
deviation of the outcome are altered by input uncertainty. It is the
goal of this section to present approximate analytical expressions
for the outcome changes and to discuss the limitations of
analytical approaches to this problem.
The impact of variability in the particle diameter on mean
flight distance and kinetic energy in the absence of turbulence is
considered. Writing the diameter of an individual particle as
xðd þ DdÞ xðdÞ þ x0 ðdÞDd þ
x00 ðdÞ
ðDdÞ2 þOðDdÞ3 :
2
ð12Þ
Varying d randomly N times between dmin and dmax will give N
estimates for flight distance. Ignoring higher order terms, the
mean value of these N estimated flight distances is
xðd þ DdÞ N
1X
x00 ðdÞ
ðDdÞ2i
xðdÞ þ x0 ðdÞðDdÞi þ
N 1
2
ð13Þ
Dividing by the mean flight distance, taking the limit as N-N,
and noting that the average of Dd¼ 0 leads to
xðdÞ
1þ
x00 ðdÞ
2xðdÞ
s2d :
ð15Þ
(where H is the release height) and
3HU
x00 ðdÞ ¼ pffiffiffi 5=2
4 ad
X ¼ 1þ
ð16Þ
ð14Þ
To estimate the value of x00 ðdÞ=2xðdÞ we turn to Baker (2007)
who showed that compact debris, which is allowed to fall long
enough, will travel horizontally at the mean wind velocity and
3
CV 2d :
8
ð17Þ
The analysis for the impact kinetic energy results in a slightly
more complicated expression
ð11Þ
and denoting the flight distance as x ¼x(d), a Taylor series
expansion about the mean particle diameter can be used to
approximate the flight distance of an individual particle. That is
xðdÞ
HU
xðdÞ ¼ pffiffiffiffiffiffi
ad
where a ¼4rpg/raCD. Therefore, the mean flight distance is given
by
4.1. Impact of input variability on mean outcome
d ¼ d þ Dd
vertically at its terminal velocity. Making the simplifying assumption that the time taken to adjust to this flight is small compared
to the total flight time, then xðdÞ and x00 ðdÞ are given by
KE ¼
keðdÞ
keðdÞ
1þ
3ðu2 þ 2adÞ
ðu2 þ adÞ
CV 2d :
ð18Þ
These approximate expressions for the change in mean flight
distance (17) and impact kinetic energy (18) are compared to the
results of the simulations in Fig. 11.
Eq. (17) provides a reasonable approximation for the increase
in mean flight distance for small values of CVd. However, the
approximate theory diverges from the simulation results as CVd
increases beyond 0.1. There are two main reasons for this. First,
only the first three terms of the Taylor series were considered in
the analysis, and as the range of Dd increases the higher order
terms will become more significant. Second, the approximation
that the particle is always in the constant velocity steady state is
poor, particularly for low release heights. This is seen in
Fig. 11(a) where (17) always lies below the data points and, the
lower the release height, the greater the discrepancy between the
data and (17). Similar results are seen in the kinetic energy plot
(Fig. 11(b)), though in this case (18) lies above all the points. Both
(17) and (18) provide reasonable estimates of the increase in
mean outcome for small coefficients of variation and perform
better for higher release heights.
A. Karimpour, N.B. Kaye / J. Wind Eng. Ind. Aerodyn. 100 (2012) 77–90
87
Fig. 11. Effect of particle diameter variation on (a) dimensionless flight distance and (b) impact kinetic energy.
Fig. 12. Plot of flight (a) distance and (b) impact kinetic energy, for U ¼ U
has a slope of 1.
qffiffiffiffiffiffiffiffiffiffiffiffi
1 þ I2u against the mean flight distance for simulation results for randomly generated U. The line
The impact of horizontal turbulence fluctuations on the mean
outcomes is considered. One can write the instantaneous
horizontal velocity as
U ¼ U þ u0
ð19Þ
where u0 is the instantaneous horizontal wind velocity fluctuation. In the drag equation the velocity is squared so one might
expect the appropriate mean velocity (U) to be the square root of
the mean of the square of the individual velocities. That is
qffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
b ¼ U 2 ¼ U 2 þ 2ðUu0 Þ þ u02
ð20Þ
U
or
b ¼U
U
qffiffiffiffiffiffiffiffiffiffiffiffi
1 þ I2u
ð21Þ
The mean flight distance and impact kinetic energy can be
b.
calculated by considering a constant wind velocity equal to U
Flight distance and kinetic energy results were calculated using a
constant wind speed given by (21) and plotted against the results
of the Monte Carlo simulations presented in Fig. 5. These results
are shown in Fig. 12. There is an excellent agreement between the
two mean flight distance predictions and there is a little need to
run large scale Monte Carlo simulations in order to establish
mean outcomes. Eq. (21) illustrates why the simulations of Baker
(2007) showed little effect due to turbulence. The Baker (2007)
simulations were run at the gust wind speed, which is similar to
b , and therefore already accounts for the increased flight
using U
distance due to horizontal turbulence. However, Baker’s approach
does not consider variations about the mean.
4.2. Impact of input variability on outcome variability
Treating the outcome uncertainty using standard error analysis techniques, we can write the flight distance uncertainty due to
particle size variation as
@x
Dx Dd
ð22Þ
@d
ignoring higher order terms. Approximating the flight distance by
(17) and taking the variability to scale on the standard deviation,
one can estimate the variation in flight distance scaled on the
mean flight distance as
CV x 1
CV d :
2
ð23Þ
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A. Karimpour, N.B. Kaye / J. Wind Eng. Ind. Aerodyn. 100 (2012) 77–90
Fig. 13. Effect of particle diameter variation on (a) flight distance variation and (b) impact kinetic energy.
Fig. 14. Plot of the square of standard deviation in (a) flight distance and (b) impact kinetic energy, from simulations varying all parameters against the sum of the squares
of the standard deviation due to variation in individual parameters. The line has a slope of 1 for comparison.
Eq. (23) is compared with the simulation results in Fig. 13(a).
The first order approximation provides a reasonable estimate of
the variability in flight distance for small CVd. However, as CVd
increases, the measured variability increases relative to that
predicted. This is again due to ignoring higher order terms in
the approximation and the simplified analytical solution for the
flight distance.
Similar analysis of the impact kinetic energy variation leads to
CV ke 3CV d
ð24Þ
which is plotted in Fig. 13(b). Again, for small CVd (24) provides a
good approximation for the variation in impact kinetic energy.
The analysis presented above also illustrates why the variability in kinetic energy is so much greater than the variation in
flight distance. In the steady constant velocity flight regime, the
flight path scales on d 1/2 whereas the leading term in the kinetic
energy expression is d3. The significant variation in the impact
kinetic energy is not due to variations in flight speed, but rather
variation in particle mass (m d3). Again, both (23) and (24)
provide reasonable estimates of the outcome coefficient of variation for small input coefficient of variation. They also provide
better predictions for higher release heights due to the steady
flight velocity assumption. Improvements could be made by
considering more terms in the Taylor series. However, the quality
of any analytical approximation is limited by the quality of the
approximate analytical solution for the flight path based on the
flight equations. Therefore, a full understanding of variability can
only be achieved through the use of Monte Carlo simulations.
4.3. Impact of multiple input variations
The analysis above considers only one input variation at a
time. However, ordinarily more than one input varies. If the
uncertainty in each input parameter is statistically independent
of the others, then the Bienayme formula (Loeve, 1977) can be
used to predict the total standard deviation in the outcomes:
X
s2total ¼
s2i
ð25Þ
where stotal is the standard deviation accounting for the complete
problem and the si are the outcome standard deviations due to
the individual input parameters. To verify that (25) is appropriate
for modeling the stochastic nature of compact debris flight, the
square of the standard deviation when all the input parameters
(i.e., d, Iu, and Iw) were varied ðs2x ÞdIu Iw is plotted versus the sum of
the squares of the standard deviations of calculated mean flight
distance when only one of the input parameters is varied,
ðs2x Þd þ ðs2x ÞIu þðs2x ÞIw (see Fig. 14(a)). In the same way, the
A. Karimpour, N.B. Kaye / J. Wind Eng. Ind. Aerodyn. 100 (2012) 77–90
standard deviation in the impact kinetic energy, ðs2 ÞdIu Iw is
ke
plotted versus ðs2 Þd þ ðs2 ÞIu þ ðs2 ÞIw in Fig. 14(b). In both figures
ke
ke
ke
the Bienayme formula (25) can be seen to accurately predict the
combined standard deviation of the flight distance and impact
kinetic energy based on simulations varying only one parameter
at a time. Fig. 14 also provides additional evidence that 10,000
simulations per case is adequate and leads to a good approximation of the outcome standard deviation.
5. Wind tunnel tests
A series of ball-drop experiments were conducted in the
Clemson University Boundary Layer Wind Tunnel to assess the
validity of the statistical modeling approach presented. A remotely controlled robot hand was used to hold and release the balls
into a turbulent wind field. The flight of the particle was filmed
using a high definition video camera. The horizontal flight
distance was measured for each release. In each test the particle
diameter was kept constant however the mean velocity varied
due to turbulent gusts in the wind tunnel. To characterize the
wind a time series of the wind speed was taken over 100 s. The
time series was broken up into 1 s bursts and the mean velocity of
each burst was calculated. The mean and standard deviation of
the one second burst speed were then calculated. The wind speed
measurements were made using a Dwyer pitot tube connected to
a Setra DATUM 2000 digital manometer and a Measurement
89
Computing data acquisition module. Matlab was used for analyzing the velocity data. For the fan speed used in these tests, the
mean one second wind velocity was U ¼ 6:54 m=s and the standard deviation was su ¼0.41 m/s. A ball with a constant diameter
of d ¼37.8 mm and a density of rp ¼89.9 kg/m3 was dropped from
a height of 0.8 m. A total of 128 experiments were run with the
typical flight time being a little less than one second. Images of
the experimental setup are shown in Fig. 15.
A Monte Carlo simulation with 10,000 runs was conducted
using a fixed particle diameter, a uniform vertical velocity profile,
and constant wind speed during flight. The wind speed was
randomly varied from run to run about a mean of 6.54 m/s
assuming a normal distribution with a standard deviation of
0.41 m/s. Histograms of the experimental and simulated flight
distances are shown in Fig. 16. The measured mean flight distance
was 0.42 m while the simulated mean was 0.43 m. The measured
standard deviation was 0.047 m compared to the simulated
standard deviation of 0.049 m. This represents excellent agreement between the simulations and experiments and demonstrates that the Monte Carlo approach to modeling debris flight
can provide physically meaningful predictions as well as more
theoretical insights described in Sections 3 and 4.
6. Discussion and conclusions
Debris flight models are almost exclusively deterministic.
These models typically assume known fixed input parameters
Robot Hand
Object
Fig. 15. Experimental setup for ball drop experiments showing (a) the robot hand and the upwind surface roughness blocks, and (b) the horizontal scale used to measure
the flight distance.
Fig. 16. Histogram of ball flight distance. Left: measured, Right: simulated.
90
A. Karimpour, N.B. Kaye / J. Wind Eng. Ind. Aerodyn. 100 (2012) 77–90
such as wind velocity and particle size as well as constant
coefficients such as the drag coefficient. However, such determinism is very rarely the case and debris flight modeling can be
improved by accounting for model input uncertainty. The results
of almost 5 million flight simulations presented above indicate
that failure to account for uncertainty in the particle size,
horizontal turbulence intensity, and vertical turbulence intensity
will result in under predictions of the mean flight distance and
mean impact kinetic energy and in no information about the
spatial distribution of the particle impact location or variation in
impact kinetic energy.
Monte Carlo simulations, as described in this paper, provide a
means for quantifying the influence of input uncertainty on the
resulting flight characteristics (mean and variance of flight
distance and impact kinetic energy). Running a series of simulations varying each input parameter (i.e., particle diameter, horizontal turbulence intensity, and vertical turbulence intensity)
separately shows the effect of input parameter variability on the
simulation outcome. Introducing uncertainty in any of particle
diameter, horizontal turbulence intensity, and vertical turbulence
intensity leads to larger mean values for flight distance and
impact kinetic energy compared to the deterministic case. For
values typical of gravel blown off a built up roof and a release
height of 20 m, modeling using a mean particle size and wind
speed under-predicts the mean flight distance by over 3% and
underestimates the mean impact kinetic energy by almost 12%.
Increasing the input variability increases the error in the mean
flight outcomes. This could be particularly important in regions of
high turbulence intensity such as in the wake downwind of a
building. Introducing variability in input parameters also leads to
variability in flight distance and impact kinetic energy. Simulation
results for a 20 m release height show coefficients of variation of
0.2 for flight distance and 0.5 for impact kinetic energy.
A number of analytical approaches to understanding and
quantifying the stochastic nature of debris flight were presented
that explain the broad trends observed in the data. However, such
an analysis is severely limited by the lack of analytic solutions to
the debris flight equations, particularly in the presence of turbulence. A full analysis of the impact of parameter uncertainty and
variability is better realized through the Monte Carlo simulation
approach presented.
There are many other sources of uncertainty, even in the
simple problem of a piece of gravel blowing off a high roof. For
example, the launch angle, location on the roof at which flight
initiation occurs, and particle shape will all vary about some
mean. The wind field is also highly complex within the urban
canopy with the flow being dominated by building wakes and
regions of high shear. A full understanding of the risk associated
with compact debris flight in urban areas will require large scale
computational fluid dynamics simulations, which are currently
prohibitively expensive. The approach presented in this paper
provides a computationally efficient method for quantifying the
effect of certain input uncertainties on the distribution of flight
outcomes. Further work is needed to accurately parameterize the
appropriate statistical description of each input parameter and to
extend this work to consider rod-like and plate-like debris.
Acknowledgments
The authors would like to thank two anonymous reviewers
whose detailed comments have greatly improved the paper.
Dr. Karimpour would also like to acknowledge the support of
the Clemson University Glenn Department of Civil Engineering
for their support through a graduate research and teaching
assistantship.
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