Windborne debris poses a threat to building envelopes ranging from low-rise residential housing to tall buildings in an urban setting with glass facades/cladding systems (Minor, 1994;Gurley and Masters, 2011;Jain, 2015). Building envelope damage can lead to subsequent water intrusion, extensive interior damage, and additional debris further damaging the structure and potentially resulting in a risk to occupant safety (Pita et al., 2016;Johnson et al., 2018;Wei et al., 2024a). The resulting loss of building functionality (occupant dislocation and business interruption/closures) may last for extended periods, and hence compromise community resilience (Wei et al., 2024b), underscoring the importance of reducing the vulnerability of infrastructure to windborne debris.
The damage risk for building envelopes due to windborne debris depends on the debris type, flight initiation (e.g., Kordi and Kopp, 2011;Kakimpa et al., 2011), flight trajectory (e.g., Holmes, 2004;Baker, 2007), and impact mechanism (e.g., Fernandez et al., 2010;Masters et al., 2010;Zhang et al., 2013). Among these factors that determine the risk of building envelopes, debris flight initiation and trajectory are sensitive to turbulent wind field. Hence, accurate modeling of debris impact risk depends on an accurate understanding of the turbulent wind field around buildings. The local wind field in the rooftop region significantly affects initial conditions of debris flight, while the turbulent wake wind field directly influences debris flight trajectory and contributes to uncertainties in final debris impact/landing location and momentum. Existing studies have taken the route of simplifying approaches to model this process. For example, initial location and velocity are either arbitrarily assumed as random variables (Ai et al., 2023;Lyu et al., 2023) or based on simple parametric models on roof top flows (Dong et al., 2023). For the wake flow, only the temporally constant mean wind field based on Reynolds-averaged Navier-Stokes equations (RANS) simulation (Ai et al., 2023;Lyu et al., 2023) and spatially fully correlated wind fluctuations (Dong et al., 2023) are considered in the existing numerical studies of debris flight J o u r n a l P r e -p r o o f simulation. The simplifications in existing studies may not fully capture complex debris flight behavior in the spatiotemporally varying turbulent wake flow. Since these simplifications encompass multiple aspects of the flow, the relative influence of individual simplifications on simulation accuracy is not revealed.
A recent review of windborne debris simulation studies (Zhao et al., 2021) reveals the common use of simple unobstructed open terrain flow conditions without considering the influence of surrounding buildings on the wind field (i.e., open flow conditions). Several studies have investigated flight behavior in the mean wind field without fluctuations (e.g., Lin et al., 2006;Holmes et al., 2006), but fewer studies have incorporated the effects of turbulence. Holmes (2004) and Baker (2007) briefly discussed the effects of turbulence on the flight trajectory of compact and sheet debris. They found that turbulence can produce significant variability in individual trajectories but may have little effect on average trajectories. Karimpour and Kaye (2012) studied the stochastic nature of windborne debris, where the effects of vertical and along-wind wind fluctuations on flight distance and impact kinetic energy were investigated with a uniform twodimensional background flow. With the authors noting the high computational cost of simulating wind fluctuations along the trajectories, the abovementioned studies assumed that all the spatial points undergo an identical fluctuation following a Gaussian distribution with a target spectrum. Moghim andCaracoglia (2012a and2012b) simulated a uniform (spatially identical) upward vertical gust with short duration and investigated its influence on the trajectory of compact debris as well as the impact risk for a proximate tall building. Moghim and Caracoglia (2014) extended this to a more complex turbulent wind field, where the Gaussian turbulence at discrete points on the "inlet boundary" are first simulated with prescribed cross-spectrum and then propagated through the field using Taylor's frozen turbulence hypothesis to determine wind speed J o u r n a l P r e -p r o o f at the time stepping instantaneous location of debris in flight. Some of these numerical simulations have been validated in the wind tunnel tests (Karimpour and Kaye, 2012;Moghim et al., 2015).
In addition to straight-line wind fields, existing studies have considered vortex wind fields such as tornadoes. Noting the simplifications from neglected turbulence in many existing studies (e.g., Baker and Sterling, 2017;Abdelhady et al., 2021), computational fluid dynamics (CFD) based on large eddy simulation (LES) has been used to include the tornado turbulence in debris flight computation (e.g., Maruyama, 2011;Huo et al., 2020;Liu et al., 2021a;Liu et al., 2021b).
In addition, Liu et al. (2021c) behavior to these wind features will inform the design of debris flight tracking wind tunnel tests and building façade debris vulnerability modeling efforts that address more complex urban wind fields around mid/high-rise buildings with the presence of local vortices and wake regions.
The simulation methodology of this study is schematically shown in Fig. 1. For statistical analysis of debris flight characteristics requiring input of wind speed along the debris flight trajectory, the premise is to simulate time histories of longitudinal wind speed over a fixed spatial grid forming a vertical plane parallel to the horizontal mean wind direction. With the simulated wind field, the two-dimensional debris flight trajectories are computed by releasing 𝑁 𝐷𝑅 debris at random time steps. This debris releasing process is repeated 𝑁 𝑊𝐺 times, resulting in a total of 𝑁 𝐷𝑅 × 𝑁 𝑊𝐺 simulated debris flight trajectories for a reliable statistical estimate of the debris flight characteristics. The selection of proper values of 𝑁 𝐷𝑅 and 𝑁 𝑊𝐺 is discussed later in this paper.
The next two sections describe the debris flight model employed in this study and the stochastic wind field simulation approach to incorporate increasingly realistic spatial correlation features concurrent with Gaussian and then non-Gaussian probability content. The result analysis and implications for wind tunnel testing are subsequent, followed by the concluding remarks and future directions.
J o u r n a l P r e -p r o o f 2 DEBRIS FLIGHT MODEL This study considers two-dimensional flight trajectory of spherical debris. The adoption of spherical debris allows for elimination of complex uncertainties from the time-varying aerodynamic lift/drag as in debris of irregular shapes. The two-dimensional simplification of the debris flight is justified by the open flow environment investigated in this study. The governing J o u r n a l P r e -p r o o f equation of the debris flight (see Fig. 2), based on the quasi-steady aerodynamic load, can be expressed as (Holmes, 2004):
where x and z are the displacements in along-wind and vertical directions; t is the time; m is the mass of the debris calculated as 𝑚 = 4 3 𝜋𝑟 3 𝜌 (r is the debris radius and 𝜌 is the debris density); g is the gravitational acceleration; 𝜌 𝑎 is the density of air; A is the projected frontal area for a spherical debris 𝐴 = 𝜋𝑟 2 ; 𝐶 𝑑 is the drag coefficient; the values of these parameters are listed in Table 1, and some of them are determined by the practical considerations for the planned wind tunnel tests where the flight trajectories of scaled debris will be tracked using high-speed cameras; , which can be substituted into Eq. ( 1) to derive:
J o u r n a l P r e -p r o o f Table 1. Parameters for simulating debris flight trajectories Parameters Values Debris radius r 1.5 cm Debris density 𝜌 2.5 g/cm 3 Debris mass m 35.3 g Debris release height 20 m Wind speed at release height 32.2 m/s Drag coefficient 𝐶 𝑑 0.5 Gravitational acceleration g 9.8 m/s 2 Air density 𝜌 𝑎 1.225 kg/m 3 3 SIMULATION OF TURBULENT WIND FIELD This section discusses the simulation of the turbulent wind field as the input to the debris flight model. Multiple schemes are employed to model the degree of correlation in the along-wind turbulence component among the spatially separated grid points, as well as the probabilistic turbulence properties. Table 2 describes the content and sequencing of six different combinations of spatial correlation and probabilistic turbulence properties used in this study, together with their J o u r n a l P r e -p r o o f implementation in debris flight model (more details in the subsections of each wind field model).
To systematically analyze the influence of the spatial correlation of turbulence, Section 3.1 begins with the two extremes of no correlation and full spatial correlation to establish debris flight behavior boundaries (here the "full correlation" does not have a time delay). Considering that the key factor is the fluctuation correlation between the points along the debris flight trajectory (with coordinate variation in both vertical and along-wind direction), the cases of "no vertical and full along-wind correlation" and "full vertical and no along-wind correlation" are equivalent to the case of "no vertical and no along-wind correlation", and hence are not included in Table 2. After the two extreme cases, more realistic scenarios of partial vertical correlation and frozen turbulencebased along-wind propagation (i.e., full correlation with time delay) are investigated. Wind fluctuations in Section 3.1 are assumed to follow the Gaussian distribution, while non-Gaussian wind turbulence will be introduced later in Section 3.2.
For all simulations considered, the mean wind speed 𝑈 ̅ (𝑧) along the elevation z follows the logarithmic law (ASCE/SEI 49-12):
𝑈 𝑟 is the reference wind speed at reference height 𝑧 𝑟 ; 𝑈 𝑟 = 32.2 m/s is selected in this study with 𝑧 𝑟 = 20 m (selected based on debris releasing height for a typical mid-rise building); the debris release location is also at the reference height 𝑧 𝑟 ; 𝑧 0 = 0.3 m is the roughness length for suburban terrain; d = 0.94 m is the displacement height.
The vertical profile of turbulence intensity 𝐼 𝑢 (𝑧) is modeled following (ASCE/SEI 49-12):
The wind turbulence follows the von Karman spectrum:
J o u r n a l P r e -p r o o f
]} 5 6
(5)
where 𝜎 𝑠 = 𝑈 ̅ (𝑧)𝐼 𝑢 (𝑧) is the standard deviation of the wind fluctuation; 𝐿 𝑢 (𝑧) is the turbulence integral length scale varying with elevation, which is calculated as (ASCE/SEI 7-16):
𝐿 𝑢 (𝑧) = 97.54( 𝑧 10
) 1/3 (6) This study models open flow conditions (no proximity buildings) and assumes zero mean vertical wind speed, i.e., 𝑊 ̅ =0. Furthermore, it is expected that compared to along-wind turbulence the influence of vertical turbulence on debris flight behavior is relatively weak due to: (1) vertical turbulence intensity is usually smaller than along-wind turbulence in open flow (Hui et al., 2009;He et al., 2020), and (2) vertical turbulence usually has fewer low-frequency components in the spectrum (Dyrbye and Hansen, 1996;Moghim and Caracoglia, 2014) and hence smaller spatial scales, indicating that its effect on debris motion can be more easily canceled out during the flight.
In addition, currently there is a lack a standard accepted model for the correlation between vertical and along-wind fluctuation (Benowitz and Deodatis, 2015;Liu et al., 2023). Hence, vertical turbulence is not considered in this study but will be considered in follow up work that use particle image velocimetry (PIV) measurements as the baseline wind field.
J o u r n a l P r e -p r o o f (a) Time-domain result (b) Frequency-domain result Figure 4. No spatial correlation: simulated standard Gaussian white noise 𝑢 𝑤 (𝑡)
When wind fluctuations at all spatial points on the grid are fully correlated, a time history at a single grid point (say z = zr and x = 0) is generated. That time history is then translated and dilated to impart the appropriate mean and turbulence intensity at each height on the grid. For a given height z, the time history of wind speed at every horizontal grid point x is identical and without J o u r n a l P r e -p r o o f time lag. Under this description, the wind speed experienced by the windborne debris at any location and time 𝑈(𝑥, 𝑧, 𝑡) can be calculated as (see Fig. 5):
where 𝑢 𝑠 (𝑡) is a unit-variance wind fluctuation with prescribed power spectrum density (PSD)
𝑆(𝜔) based on Eq. ( 5). For this case, 𝑈 ̅ (𝑧) in Eq. ( 5) is set to be 𝑈 𝑟 while the integral length scale is 𝐿 𝑢 (𝑧 𝑟 ), essentially assuming all the spatial points in the wind field undergo scaled fluctuation at reference elevation 𝑧 𝑟 (this is also the debris release location). The spectral representation method (SRM) (Deodatis, 1996) is employed to simulate the wind fluctuation 𝑢 𝑠 (𝑡) with target spectrum 𝑆(𝜔). A sample of simulated unit-variance wind fluctuation 𝑢 𝑠 (𝑡) is shown in Fig. 6.
The wind speed simulation duration of 𝑇 𝑢 = 819.2 s was chosen to capture sufficient low frequency contribution to the wind record interacting with the debris. From Eq. 6 it was determined that at the turbulence integral length scale at the debris release height of 20 m is 122.9 m. At the employed mean wind speed of 32.2 m/s, the simulated 819.2 s record permits the sequential passage of approximately 215 integral length scales.
Through comparing Eq. ( 7) and Fig. 4 with Eq. ( 8) and Fig. 6, it is straightforward that the only difference between the no-correlation and full-correlation scenarios is the PSD of the debrisexperienced fluctuation. The unit-variance wind fluctuation 𝑢 𝑠 (𝑡) has more low-frequency energy and less high-frequency energy compared to that of the standard Gaussian white noise 𝑢 𝑤 (𝑡). The difference of spectral property of 𝑢 𝑠 (𝑡) and 𝑢 𝑤 (𝑡) allows revealing the influence of turbulence frequency distribution on the debris flight, which will be discussed in the beginning of Section 4.1.
J o u r n a l P r e -p r o o f (a) Time-domain result (b) Frequency-domain result (area under the curve is one) Figure 6. Full spatial correlation: simulated unit-variance wind fluctuations 𝑢 𝑠 (𝑡)
The upcoming results section will reveal clear differences in debris flight behaviors between the two extreme scenarios of no and full spatial correlation of turbulence, illustrating the potential importance of partial correlation on debris flight. The vertical spatial correlation of longitudinal J o u r n a l P r e -p r o o f fluctuations has been widely studied, and well-accepted models, such as Davenport's distancedecaying coherence (Davenport, 1961), are available. On the other hand, the along-wind spatial correlation of longitudinal fluctuations is less standardized. Taylor's frozen turbulence hypothesis (Taylor, 1938) is still widely presumed. In frozen turbulence-based correlation, the crosscorrelation coefficient function of longitudinal turbulence between two along-wind separated locations peaks with a value of unity at the time lag determined by the distance between the two separated locations divided by the mean wind speed.
This study maintains a consistent distance-decaying partial correlation scheme in the vertical direction while considering two schemes for modeling along-wind correlation: (1) full correlation without time delay, (2) full correlation with time delay based on frozen turbulence.
More realistic along-wind correlations will be considered in future work using PIV measurements as the baseline wind field.
To generate wind fluctuation with partial correlation in the vertical direction and full correlation in the along-wind direction, this study first generates wind fluctuations at n discrete vertical locations along the inlet boundary from origin (0, 0) to debris release location (0, 𝑧 𝑟 ) with equal spacing ∆𝑧 (see Fig. 7). Then, the debris-experienced wind speed can be conveniently simulated based on the assumption of full correlation and the interpolation criteria (𝑧 ≈ 𝑘∆𝑧):
The unit-variance wind fluctuations for the n locations can be simulated by the SRM using the prescribed power spectrum density matrix (PSDM) 𝑺(𝜔) (Deodatis, 1996):
r n a l P r e -p r o o f 𝑺(𝜔) = [ 𝑆 11 (𝜔) 𝑆 12 (𝜔) … 𝑆 1𝑛 (𝜔) 𝑆 21 (𝜔) 𝑆 22 (𝜔) … … … … … … 𝑆 𝑛1 (𝜔) . . . … 𝑆 𝑛𝑛 (𝜔) ]
where the diagonal term is the auto-spectrum:
𝑆 𝑘 (𝜔) is defined by von Karman spectrum in Eq. ( 5); the off-diagonal term is the cross-spectrum:
𝑆 𝑗𝑘 (𝜔) = √𝑆 𝑗 (𝜔) 𝑆 𝑘 (𝜔)𝛾 𝑗𝑘 (𝜔) with j, k=1, 2, …, n and j ≠ k (12)
where 𝛾 𝑗𝑘 (𝜔) is the Davenport coherence function (Davenport, 1961) with a constant decay factor (a) Time histories of 𝑢 19 (𝑡) (z = 20 m) (b) Time histories of 𝑢 1 (𝑡) (z = 2 m) J o u r n a l P r e -p r o o f (c) Auto-spectrum of 𝑢 19 (𝑡) (d) Auto-spectrum of 𝑢 1 (𝑡) (e) Cross-spectrum of 𝑢 19 (𝑡) and 𝑢 1 (𝑡) Figure 8. Simulated correlated wind fluctuations at two elevations 3.1.
This section considers turbulence correlation in along-wind direction that is more realistic than full correlation. One simple approach to use Taylor's hypothesis of frozen turbulence (Taylor, 1938), which considers the downstream turbulence as the time-delayed version of upstream turbulence at the inlet boundary (see Fig. 9). Under frozen turbulence, the debris-experienced wind speed is calculated as: where 𝑢 𝑘 (𝑡) is the same as that defined in the previous section.
The SRM simulation method employed for all correlation variations (Section 3.1) result in Gaussian wind fluctuations, which does not align with full-scale and wind tunnel observations, including extreme winds (e.g., Balderrama et al., 2012;Fernández-Cabán and Masters, 2017;Zhao et al., 2019;Gurley et al., 2021;Ojeda-Tuz et al., 2023). Non-Gaussian wind fluctuations may change the debris flight trajectory and will be investigated in this section. As described in Table 2, only the distance-decaying vertical correlation and frozen turbulence-based along-wind correlation are considered, and maintained as:
The approach to simulating 𝑢 𝑘 𝑁 (𝑡) utilizes the translation method (Grigoriu, 1984;Grigoriu, 1998) to impart the desired marginal probability density function (MPDF) and adopts a third order Hermite polynomial probability model (Yang et al., 2013;Yang and Gurley, 2015) to describe the MPDF as a function of desired skewness and kurtosis in the turbulence. In summary, the method employs a static polynomial transform of a SRM simulated Gaussian process to simultaneously achieve the desired PSD and MPDF characteristics. The approach is non-iterative and computationally efficient. Although no new contributions to this method are developed in the current study, it is briefly described in Appendix B for the sake of completeness.
Since the purpose of this section is to determine whether debris flight is sensitive to non-Gaussian turbulence features, a simple approach is employed. A uniform skewness profile with a value of 𝜇 𝑆𝐾 (𝑧) = 1 is assumed, which is relatively extreme in the context of field measurements (e.g., Balderrama et al., 2012;Fernández-Cabán and Masters, 2017;Zhao et al., 2019). The kurtosis is obtained from the empirical relationship 𝜇 𝐾𝑇 (𝑧) = 2.86|𝜇 𝑆𝐾 (𝑧) | 2 + 3.02 = 5.88 from hurricane field measurement (Zhao et al., 2019). The second order characteristic follows the identical spectral and coherence models defined in the previous section. Samples of simulated non-Gaussian wind fluctuations are shown in Fig. 10, together with their skewness and kurtosis as well as the auto-and cross-spectrum.
J o u r n a l P r e -p r o o f
The six combinations of spatial correlation and probability content (see Table 2) are employed individually to compute the flight trajectories of spherical debris using the model described in Section 2. This allows the systematic investigation of the influence of spatial correlation and the non-Gaussian probability on debris flight.
To obtain a reliable estimate on the statistical properties of simulated debris flight, the following parameters need to be properly selected for the balance of computational accuracy and efficiency: (1) the temporal discretization size, ∆𝑡, for wind field simulation and debris flight computation, (2) the spatial discretization size, ∆𝑧, for wind field simulation, (3) the number of realizations, 𝑁 𝑊𝐺 , for wind field simulation, and (4) the number of debris releases, 𝑁 𝐷𝑅 , for uncertainty quantification of debris flight. Sensitivity analysis is presented in Appendix A to determine appropriate values for these parameters. As a result, the values ∆𝑡 = 0.0125 s, ∆𝑧 = 1 m, 𝑁 𝑊𝐺 = 128, and 𝑁 𝐷𝑅 = 2 15 are adopted for this study. Zero initial velocity of debris is assumed for all scenarios. This study focuses on the statistical properties, i.e., mean, standard deviation (STD), skewness, and kurtosis, of the along-wind flight distances 𝐿 𝑥 that are important for debris J o u r n a l P r e -p r o o f risk analysis. In the following presentations, histograms of debris flight distance are presented and compared among the different experiments in Table 2. The probabilistic distribution of debris flight distance is not normalized to an empirical probability density function but kept as a histogram format to allow easier comparisons among figures.
The simulation results of the two extreme scenarios for no and full spatial correlations are shown in Fig. 11(a) and 11(b), respectively. While the mean value of flight distance is very similar, the standard deviation of the debris flight distance for the full-correlation case is almost 10 times larger than that of the no-correlation case. In addition, the distribution of along-wind flight distance is approximately Gaussian for no spatial correlation of turbulence. When full spatial correlation is introduced, the debris flight distribution becomes slightly non-Gaussian with positive skewness and kurtosis larger than 3. These results also demonstrate the influence of low-frequency fluctuations on computing debris flight, considering the differences between the flat white noise spectrum (Fig. 4b) and frequency-decaying von Karman spectrum (Fig. 6b).
It is known that an actual wind field is neither uncorrelated nor fully correlated, and so this comparison is intended to set the boundaries of correlation influence on debris flight behavior within the context of the selected conditions (release height, open flow, spherical debris, etc.). It can be concluded that the presence of correlation is a significant contributor to simulated debris flight behavior. It remains to be determined how sensitive simulated debris flight is to the layered complexities of correlation that span no-correlation through full correlation, as well as the influence of non-Gaussian turbulence.
J o u r n a l P r e -p r o o f
The influence of vertical correlation is investigated through comparing the result of ( 1) full spatial correlation (Section 3.1.2) and ( 2) distance-decaying vertical correlation and full along-wind correlation (Section 3. 1.3.1). The results in Fig. 12 show that the difference between the two scenarios is very small. To further investigate the underlying mechanism, two hypotheses are proposed here.
Hypothesis A: Spatial correlation is large over the relatively short distance between debris release elevation and the ground (as per the Davenport coherence function). That is, this example contrasts full vertical correlation with very large but not full vertical correlation, and little difference is observed.
Hypothesis B: Debris flight trajectories are mostly sensitive to the turbulence at the early stage of flight, and the local wind field covering the initial portion of the debris flight is highly correlated to the wind fluctuation at the debris release location.
Hypothesis A is tested by conducting simulations of debris released at 100 m elevation, where the turbulence correlation between debris release elevation and the ground is much smaller than the case of a 20 m release elevation. The simulation results (not shown) yield insignificant differences between (1) full spatial correlation and (2) distance-decaying vertical correlation and full along-wind correlation, which disproves Hypothesis A.
To test Hypothesis B, two additional cases, deviating from baseline of distance-decaying vertical correlation and full along-wind correlation, are considered: (1) no turbulence for debris traveling between z = 0 m and z =10 m, (2) no turbulence for debris traveling between z = 10m
and z =20 m. The simulation results are shown in Fig. 13. For the case of no turbulence for lower portion of debris flight, the debris flight characteristics (Fig. 13c and 13d) are very close to the baseline (Fig. 13a and 13b). The debris has proved the importance of turbulence in the initial flight, which supports Hypothesis B. In contrast, the case of no turbulence for upper portion of debris flight (Fig. 13e and 13f) has significantly smaller variation in debris flight distance, which reconfirms the higher importance of turbulence in the initial stage of debris flight. This finding of higher importance of turbulence in the initial stage is also consistent with that reported in the literature (Dong et al., 2023). Additional simulations using the "temporal" half to partition the initial and later flight stage have also been conducted in Appendix C to complement the result in Fig. 13 using "spatial" half.
J o u r n a l P r e -p r o o f 4.1.1.2 Influence of along-wind correlation The influence of along-wind correlation is investigated by comparing the result of (1) distancedecaying vertical correlation and full along-wind correlation (Section 3.1.3.1) and (2) distancedecaying vertical correlation and frozen turbulence-based along-wind correlation (Section J o u r n a l P r e -p r o o f 3.1.3.2). The results in Fig. 14 suggest very small differences between the two scenarios. This negligible difference can be attributed to the short time delay calculated in the frozen turbulencebased assumption (Eq. 14) due to the small flight distance and high wind speed. This short time delay (e.g., in the order of 0.25s for the flying debris at x = 6m and z = 10m) is smaller than the large period of low-frequency turbulence (e.g., the passage time of an integral length scale for the wind turbulence at z = 10m is in the order of 4s), and the resulting variation in wind speed is not significant for debris flight. (a) Gaussian turbulence; distance-decaying vertical correlation and full along-wind correlation (b) Gaussian turbulence; distance-decaying vertical correlation and frozen turbulence-based along-wind correlation Figure 14. Influence of horizontal correlation on debris flight
This section investigates influence of high-order turbulence behavior on debris flight by comparing the result of Gaussian (Section 3.1.3.2) and non-Gaussian turbulence (Section 3.2) while maintaining the same spatial correlation and power spectral characteristics in both cases (distancedecaying vertical correlation and frozen turbulence-based along-wind correlation). Fig. 15a and 15b shows that the mean value of the horizontal flight distance remains unchanged, while the J o u r n a l P r e -p r o o f standard deviation slightly increases due to the non-Gaussian turbulence. The most pronounced difference is that the skewness and kurtosis of debris flight distance are much larger compared to the Gaussian counterparts. The probability of exceedance 𝑝(𝐿 𝑥 > 𝜇 𝑀𝑁 + 𝑛𝜇 𝑆𝐷 ) for debris flying beyond n times the standard deviation, 𝜇 𝑆𝐷 , from the mean value, 𝜇 𝑀𝑁 , is shown in Fig. 15c, which shows that a Gaussian simulation underestimates the debris flight distance for the extreme cases (the tail region beyond two standard deviation away from mean). These results demonstrate the potential importance of considering non-Gaussian wind fields in debris risk analysis.
For the sake of clarity, the simulation results of all the investigated scenarios are summarized in Fig. 16. J o u r n a l P r e -p r o o f 5 IMPLICATIONS FOR WIND TUNEL TESTING Noting that the debris shape, size and density, release height, wind speed and spectral model used in this study are selected to emulate the conditions feasible in a boundary layer wind tunnel, the obtained results can effectively inform decisions regarding experimental studies of tracking spherical debris flight. The main implications for wind tunnel testing are summarized in the following.
(1) The significant influence of low-frequency turbulence on debris flight demonstrates the value of introducing active turbulence generation such as active controlled fans (e.g., Catarelli et al., 2020;Li et al., 2021) to address the low-frequency turbulence deficit in conventional wind tunnels that employ only passive turbulence generation mechanisms.
(2) Considering the limited number of debris flight tracking tests in the wind tunnel, debris should be sequentially released to the turbulent flow with a relatively large interval so that enough number of low-frequency turbulence can be covered (i.e., avoid the case that all debris are trapped in one single gust).
(3) Based on the statistics of the debris horizontal flight distance, Lx, the view window of the debris tracking system (e.g., high-speed cameras) under the wind speed considered in this study should cover twice the distance of the debris release elevation in the along-wind direction so that the extreme values of debris landing locations can be captured.
( statistics. (a) Skewness (b) Kurtosis Figure A2. Sensitivity analysis on 𝑁 𝑊𝐺 J o u r n a l P r e -p r o o f (a) Mean (b) Standard deviation (c) Skewness (d) Kurtosis Figure A3. Sensitivity analysis on 𝑁 𝐷𝑅 510 511 (a) Mean (b) Standard deviation J o u r n a l P r e -p r o o f (c) Skewness
) 2 + 𝐴 3 ] 1/3