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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 78 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. 80 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. 82 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 84 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. 86 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 88 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. References ASTM, 2005. Standard Specification for Mineral Aggregate Used in Built-up Roofs, D 1863-05. Baker, C.J., 2007. The debris flight equations. Journal of Wind Engineering and Industrial Aerodynamics 95, 329–353. Bergen, K., 2005. Strategic Hotel Capital Estimates Hyatt Regency New Orleans Losses Due to Storm Damage and Lost Business Could Exceed $100 million. Chicago Tribune, October 1, 2005. Busch, N., Panofsky, H.A., 1968. Recent spectra of atmospheric turbulence. Quarterly Journal of Royal Meteorological Society 94, 132–148. Harris, J.W., Stocker, H., 1998. Handbook of Mathematics and Computational Science. Springer-Verlag, New York. Holmes, J.D., 1978. Computer simulation of multiple, correlated wind records usings the inverse fast Fourier transform. Civil Engineering Transactions, Institution of Engineers, Australia CE20 (1), 67–74. Holmes, J.D., 2004. Trajectories of spheres in strong winds with application to windborne debris. Journal of Wind Engineering and Industrial Aerodynamics 92, 9–22. Holmes, J.D., Baker, C.J., Tamura, Y., 2006. The Tachikawa number: a proposal. Journal of Wind Engineering and Industrial Aerodynamics 94 (1), 41–47. Kareem, A., Bashor, R., 2006. Performance of Glass/Cladding of High-Rise Buildings in Hurricane Katrina. Natural Hazards Modeling Laboratory, University of Notre Dame. /http://www.nd.edu/ nathaz/doc/Katrina_AAWE_9-21.pdfS. Karimpour, A., Kaye, N.B., 2010. The debris flight equations in a model atmospheric boundary layer. In: Proceedings of the A.A.W.E. Comp. Wind Eng. Conference, Chapel Hill, NC, USA, May 23–27, 2010. Kind, R.J., 1986. Measurement in small wind tunnels of wind speeds for gravel blow off from rooftops. Journal of Wind Engineering and Industrial Aerodynamics 23, 223–235. Kordi, B., Kopp, G.A., 2011. Effects of initial conditions on the flight of windborne plate debris. Journal of Wind Engineering and Industrial Aerodynamics 99 (5), 601–614. Lin, N., Holmes, J.D., Letchford, C.W., 2007. Trajectories of windborne debris and applications to impact testing. Journal of Structural Engineering (ASCE) 133 (2), 274–282. Loeve, M., 1977. Probability Theory, Graduate Texts in Mathematics, vol. 45, 4th ed. Springer-Verlag. Richards, P.J., Williams., N., Laing., B., McCarty, M., Pond, M., 2008. Numerical calculation of the three-dimensional motion of windborne debris. Journal of Wind Engineering and Industrial Aerodynamics 96, 2188–2202. Sparks, P.R., 1998. The potential for progressive collapse in buildings damaged by hurricanes and tornadoes. In: Proceedings of the First International Conference on Damaged Structures. Sparks, P.R., Schiff, S.D., Reinhold, T.A., 1994. Wind damage to envelopes of houses and consequent insurance losses. Journal of Wind Engineering and Industrial Aerodynamics 53, 145–155. Tachikawa, M., 1983. Trajectories of flat plates in uniform flow with application to wind-generated missiles. Journal of Wind Engineering and Industrial Aerodynamics 14, 443–453. Tachikawa, M., 1988. A method for estimating the distribution range of trajectories of windborne missiles. Journal of Wind Engineering and Industrial Aerodynamics 29, 175–184. von Karman, T., 1948. Progress in the statistical theory of turbulence. Proceedings of the National Academy of Sciences (USA) 34, 530–539. Wills, J.A.B., Lee, B.E., Wyatt, T.A., 2002. A model of windborne debris damage. Journal of Wind Engineering and Industrial Aerodynamics 90, 555–565.