Post-windstorm damage surveys have consistently identified windborne debris as an important cause of building envelope failures, resulting in substantial economic losses from structural repairs, reduced building functionality, and business closure. Along with debris flight studies in tornadic wind fields characterized by rotational flow patterns (Maruyama, 2011;Baker andSterling, 2017, 2018;Bourriez, 2020;Huo et al., 2020;Liu et al., 2021aLiu et al., , 2021bLiu et al., , 2021c)), forensic investigations have also identified debris damage to building façades in urban environments during extreme straight-line wind events, including hurricanes. Post-hurricane damage assessments have documented extensive damage to glass cladding of high-rise buildings during Hurricane Alicia in 1983 in Houston (Kareem, 1985), predominantly attributed to windborne debris missile impacts. Similar glass failures due to windborne debris were recorded after Hurricane Andrew in 1992 (Behr and Minor, 1994). Glass and cladding damage was documented in New Orleans in surveys after Hurricane Katrina in 2005 (Kareem and Bashor, 2006). Following these field investigations, the research community has made significant efforts in windborne debris trajectory studies in straight-line winds, pursuing both experimental (Tachikawa, 1983(Tachikawa, , 1988;;Lin et al., 2007;Kordi and Kopp, 2011;Moghim et al., 2015) and numerical approaches (Holmes, 2004;Baker, 2007;Richards et al., 2008;Kaye, 2015), and subsequently conducting building-level and community-level risk assessments of windborne debris damage (Abdelhady et al., 2019(Abdelhady et al., , 2022;;Angeles and Kijewski-Correa, 2022;Dong et al., 2023;Lyu et al., 2023;Xiong et al., 2025).
Debris research in straight-line wind fields was pioneered several decades ago by Tachikawa (1983Tachikawa ( , 1988)). Subsequent advancements led to the classification of windborne debris based on aerodynamic characteristics (Wills et al., 2002), complemented by numerical models and wind tunnel-derived empirical expressions for trajectory estimation of each debris category (compact, plate and rod). Numerical trajectory simulations for debris in straight-line winds rely on Newton's second law and are based on quasi-steady assumptions for aerodynamic loads, where the aerodynamic forces are computed through the relative wind velocity and predetermined aerodynamic coefficients. In the current literature, most of the existing numerical studies on modeling debris trajectory rely on either uniform wind without turbulence or with low turbulence. For instance, Holmes (2004) proposed two-dimensional flight equations for spherical debris in non-turbulent horizontal flow that can be solved numerically and extended these equations by incorporating horizontal and vertical wind fluctuations using turbulence spectral densities from von Kármán (1948) and Busch and Panofsky (1968). To demonstrate the method, Holmes presented two numerical case studies in uniform unobstructed flow, varying the spherical debris diameter and material, and employing a constant drag coefficient. Debris trajectories were calculated both with and without turbulence effects. In the turbulent case, constant turbulence intensity and mean wind were applied along the trajectory path, neglecting the spatial variations of the flow with height. Baker also provided an analytical solution to two-dimensional compact and plate-type debris flight equations with and without considering atmospheric turbulence and solved for uniform non-turbulent and uniform turbulent wind conditions in Baker (2007). This study only considers the wind temporal variations but not the vertical wind shear. Moghim and Caracoglia (2012a) used a uniform non-turbulent wind field for a compact windborne debris trajectory model, and in another study (Moghim and Caracoglia, 2012b) they superimposed an upward vertical gust on the uniform wind field. Most recently, Li et al. (2024) considered both the spatial and temporal variations of the flow, conducting numerical stochastic simulations of the wind field and spherical debris flight across multiple scenarios. This study has laid the foundation for future experimental validations.
Compared to the numerical studies, there exist limited experimental investigations studying debris trajectory in turbulent conditions (Kordi and Kopp, 2011;Karimpour and Kaye, 2012;Moghim et al., 2015). For example, Moghim et al. (2015) used a single high-speed camera to experimentally investigate spherical and cubic compact debris trajectories in smooth flow and roughly uniform turbulent flow. Karimpour and Kaye (2012) experimentally validated results from a stochastic version of Holmes' flight model (Holmes, 2004) of compact debris flight in unobstructed flow with relatively low turbulence intensity. Kordi and Kopp conducted experimental studies on plate-type debris flight from a building's rooftop under turbulent wind conditions, examining multiple configurations varying in wind angle, debris position, and layout of surrounding buildings (Kordi and Kopp, 2011). None of these experimental studies on compact debris have simultaneously scaled down the debris characteristics and the real-world turbulent wind field. Furthermore, previous studies did not have sufficient debris samples for a reliable statistical estimation. Most recently, Li et al. (2025) conducted wind tunnel experiments to study spherical debris trajectories in turbulent conditions. Conventional wind tunnels at large scales suffer from low-frequency turbulence deficits, while those with active turbulence generation devices can produce realistic full-spectrum turbulence. The authors developed a method to use results from conventional wind tunnel conditions to estimate debris behavior under highly turbulent conditions that cannot be physically generated in these facilities. Experimental results from this study are incorporated here as a baseline for validating the numerical simulation framework.
This study provides a baseline validation of the numerical model based on Newton's second law for spherical debris flight in unobstructed highly turbulent boundary layer wind, without considering source and/or surrounding buildings. The study evaluates the accuracy of numerical models in predicting spherical debris trajectories by comparing the modeled versus experimentally observed debris landing positions under multiple turbulence scenarios. Experiments at the University of Florida boundary layer wind tunnel facility (UF BLWT) were conducted with properly scaled wind field and debris characteristics to reflect real-world conditions. The experiments examined two unobstructed flow scenarios with varying mean wind speeds and turbulence intensiies. These distinct flow conditions were selected to ensure a comprehensive validation of the numerical model across different wind fields, enhancing the robustness of the validation process. The first scenario generates partial turbulence that only matches the high-frequency portion of the spectrum, while the second entails fullspectrum turbulence representing realistic suburban exposure. The measured wind at this facility directly provided the inputs to the numerical simulations across five cases: four with lower longitudinal turbulence intensities (around 10 %) with varying mean wind speeds at the debris release height, and one with high longitudinal turbulence intensity (approximately 23.7 %) at the debris release height. Moreover, previous research on spherical debris trajectories has primarily focused on validating conventional models within the Reynolds number range where the drag coefficient remains constant. Consequently, these studies have not addressed the uncertainties associated with time-dependent aerodynamic drag coefficients that occur in Reynolds number sensitive regions. This study examines how the Reynolds number dependency of the drag coefficient affects debris trajectories. Additionally, an approximate model derived from the validated numerical approach is presented for rapid prediction of debris mean longitudinal travel distance.
The scales and parameters used in the wind tunnel tests and corresponding numerical simulations are outlined in Table 1. To ensure scalability of debris motion from experimental testing to real-world conditions, Froude number similarity was invoked in the wind tunnel, preserving the ratio of inertial forces to gravitational forces. The test cases correspond to the full-scale prototype of 30 mm diameter spherical debris (analogous to typical rooftop gravel), released into an unobstructed airflow from an elevation of 20 m above ground level. Glass beads were selected due to their density properties resembling gravel, and hence the density scaling ratio is 1:1. Under the selected length scale of 1:20 based on the practical constraints of the wind tunnel, the debris release height in the wind tunnel tests was 1 m. The debris release height also served as the reference height for characterizing the wind profile. The wind field parameters were adopted from ASCE standards for suburban terrain; the target mean wind profile was based
where the reference velocity at the reference height of 1 m, the roughness length of 0.015 m, and the zero-plane displacement height of 0.047 m are denoted by U 1 (z r ), z 0 , and d. The target longitudinal turbulence intensity profile was adopted from ASCE/SEI 49-21 (ASCE, 2021):
(2)
The five experimental case studies are presented in Table 2. The facility features a long-fetch, low-speed open circuit tunnel with dimensions of 6 m × 3 m, as illustrated in Fig. 1(a). Throughout the experimental investigation, two distinct turbulent scenarios were studied: partial-spectrum turbulence and full-spectrum turbulence. For the partial turbulence scenario (Cases 1-4), turbulence was passively generated using a configuration of eight vane axial fans (VAF), Irwin spires, and a rapidly reconfigurable and precision-controlled roughness element grid (Terraformer) (Catarelli et al., 2020a(Catarelli et al., , 2020b)). Cases 1 through 4 have different mean wind speeds but similar release height longitudinal turbulence intensity to test the numerical simulation performance at multiple wind speeds. In the full-spectrum turbulence scenario (Case 5), the Irwin spires were removed and an array of 319 actively controlled modular fans, denoted as the Flow Field Modulator (FFM), operated in series with the VAFs to generate large-scale turbulent structures and simulate full-spectrum turbulent flow resembling atmospheric boundary layer wind (Fig. 1(a-b)). A control strategy known as the governing convergence algorithm (GCA) iteratively adjusts the inputs of eight VAFs and 319 modular fans to produce the desired wind field characteristics (Pinyochotiwong et al., 2025).
Prior to conducting debris release tests, wind profiles were measured at the center of the turntable to establish undisturbed baseline flow conditions. These measurements were used to directly provide wind field inputs to the numerical simulations. Wind velocity profiles were documented using 12 Vectoflow probes (shown in Fig. 1(c)) sampling at 850 Hz for the five test cases. These instruments, arranged at 125 mm vertical increments, captured all three velocity components simultaneously. The measurement locations included the debris release position without the release device present. The measured and target wind field metrics for Cases 1 and 5 are shown in Fig. 2. Although the passive mechanism used in Case 1 successfully replicates the target mean wind profile, it cannot reproduce the target longitudinal turbulence intensity due to the low-frequency energy deficit, which is well known in boundary layer wind tunnel facilities. For Case 5, this limitation was overcome by implementing the FFM to introduce additional low-frequency turbulence (and increasing the integral length scales). This enables the examination of the impact of full-spectrum turbulence on experimental debris flight.
The time histories and distributions of longitudinal wind speed at release height for Cases 1-5 are presented in Fig. 3 and Fig. 4. For Cases 1-4 (Fig. 4(a-d)), statistical properties including standard deviation, skewness, and kurtosis are similar across the various mean wind speeds. When comparing Cases 1 and 5 (Fig. 4(a) and (e)), the standard deviation almost doubles when the low-frequency components in the turbulence spectrum are recovered. The added low-frequency energy for Case 5 is shown in Fig. 3.
Reliable debris release and tracking were critical in minimizing uncertainty in measuring the trajectory of every debris release trial. Release of debris from a constant predefined position was accomplished using a customized debris release device designed and fabricated by UF BLWT staff, placed at the center of the turntable (Figs. 5 and 7). A column of debris (glass beads) was fed to the release point at the top of the device, and a precision actuator controlled a rod to slowly push individual beads into the flow at a constant rate. The device was designed and anchored to prevent wind-induced vibration at the release point.
Debris trajectories were recorded using a high-resolution stereo vision system consisting of two Baumer VLXT-240M.I cameras equipped with Navitar lenses with 12 -mm focal length. The dual-camera stereo system operated in synchronization with the debris release device, capturing debris flight in continuous 1-minute recording intervals. The actuated piston released a single debris particle per second, while the high-speed cameras captured the motion at 100 frames per second. Postprocessing was conducted on the image dataset by employing a customized computer vision technique including camera calibration, debris tracking, and the projection of debris from pixel format to realworld 3D space (Anarak et al., 2024).
The reliable detection and tracking of transparent spherical glass debris between consecutive frames presented multiple significant technical challenges. This obstacle was addressed through two enhancement strategies: implementation of a specialized tracking algorithm and optimization of environmental contrast conditions. The interior wind tunnel test section surfaces were covered with matte black paper to create maximum visual contrast. Optical visibility was further enhanced using two flicker-free Aputure LS 1200d Pro lights, equipped with narrow reflectors to obtain maximum light reflection from the glass particles. Camera and lighting positions are illustrated in Fig. 6; Fig. 7 presents the trajectory tracking results during debris flight for a single debris sample. The cameras were located in the wall of the wind tunnel at approximately the location where the label 'Test Section' appears in Fig. 1(a).
This section reviews the equation of motion based on Newton's second law for three-dimensional spherical debris trajectory in straightline winds. Previous experimental investigations have indicated that this equation of motion is valid for predicting debris flight trajectories in non-turbulent or low turbulence conditions (Karimpour and Kaye, 2012). In this equation of motion, the quasi-steady aerodynamic load is applied to the debris due to its relatively small size. For a non-spinning spherical debris particle moving in straight-line wind, the aerodynamic force component can be simplified by disregarding the Magnus effect (Baker, 2007) and associated lift forces, thereby reducing the analysis to primarily consider drag-induced forces. Therefore, after initial flight perturbations dissipate, the aerodynamic forces on spherical debris can be characterized using a single drag coefficient (Baker and Sterling, 2017). The quasi-steady state drag coefficient represents aerodynamic responses under controlled, quasi-steady conditions where flow patterns have fully developed and stabilized around the object at a specific Reynolds number. Within this quasi-steady flow, the motion of the debris particle in direction i (i = 1, 2, 3 represents longitudinal, lateral, and vertical direction motion) can be formulated as:
where the displacement and wind velocity vectors are represented by x and U, and F a and F g represent the aerodynamic and gravitational force vectors, respectively. The gravitational acceleration vector is (0,0,-g), where g represents gravitational acceleration. In this equation, t denotes time; m is the debris mass; ρ a is air density; A refers to the projected frontal area of the spherical debris; and C d is the quasi-steady state drag coefficient. Fig. 8 presents a schematic diagram of the forces acting on the debris along with the governing equation of debris flight.
The equation of motion described in Section 3.1 has been widely applied to predict spherical debris trajectories in turbulent flow fields, but its experimental validation for highly turbulent conditions has not been reported in the literature. Our preliminary work indicates that debris trajectories in turbulent flow are primarily influenced by lowfrequency wind components, where velocity changes occur gradually, creating quasi-steady flow conditions over discrete time intervals (Li et al., 2024). This preliminary study provides a basis for estimating debris trajectories during turbulent straight-line wind conditions by allowing aerodynamic forces on compact debris to be computed using quasi-steady coefficients at each time step. High-frequency turbulence effects on debris trajectories can be reasonably ignored, as they introduce minor distortions to flight paths without significantly influencing the main trajectory.
To model the debris flight trajectory in spatiotemporally varying turbulent wind field, Eq. ( 3) was applied with the assumption of Taylor's frozen turbulence hypothesis (Taylor, 1938) in the longitudinal direction, which characterizes downstream turbulence as a time-shifted representation of upstream turbulence at the inlet boundary (Li et al., 2024). Within this frozen turbulence-based correlation in the longitudinal direction framework, the longitudinal wind speed experienced by debris was approximated as it traverses the flow field:
where μ U1 (x) is the temporal mean of the longitudinal wind speed at location x. Based on the spatial variations of wind speed only in the vertical direction, Eq. ( 4) can be simplified to:
where μ U1 (x 3 ) and I U1 (x 3 ) are the temporal mean and turbulence intensity of the longitudinal wind speed at height x 3 , and the u k value can be derived using:
Based on the established framework in Section 3.2, stochastic numerical simulations of debris trajectory were conducted for Cases 1-5 with the time interval corresponding to the sampling rate of the velocity probes (850 Hz). Further investigation of the numerical simulation integration time step is presented in Appendix A. These simulations exclusively considered variability in the wind field, while maintaining constant values for other inputs including debris size, debris density, release height, and air density. Previous studies usually consider a constant drag coefficient of 0.5, assuming that the Reynolds number is in the range where the drag coefficient remains a constant. However, for this study with lower Reynolds numbers, the drag coefficient is very sensitive to the Reynolds number. The change in the drag coefficient can in turn lead to a large variation in the numerical prediction of debris flight. To this end, numerical simulations were conducted using drag coefficient-Reynolds number relationships from two different references (Hoerner, 1965;Schlichting and Gersten, 2016) that give different drag coefficients under the same Reynolds number and hence result in variations in debris flight statistics. Further discussion on the sensitivity of the obtained flight statistics to the drag coefficient, and the appropriate drag coefficient to be used is provided in Section 4.
Two distinct approaches were implemented for the drag coefficient parameter. The first approach uses a constant drag coefficient method where the initial Reynolds number is calculated based on the mean velocity at the release height, and the corresponding drag coefficient is determined according to this Reynolds number and remains constant throughout the entire simulation. In the second approach, the spherical drag coefficient was determined at each computational time step by calculating the relative velocity of the debris, which was defined as the difference between debris speed and wind speed at debris location. The relative velocity enabled the calculation of the instantaneous Reynolds number, which, through the drag-Reynolds relationship, yielded the corresponding instantaneous drag coefficient value. This coefficient was applied for the current time step to calculate aerodynamic forces, followed by updates to velocity and position vectors. This methodology includes the assumption that flow adjustment around the debris occurs sufficiently rapidly such that the flow field closely approximates the quasi-steady condition at each time step. Regardless of which approach was adopted for the drag coefficient, the simulations were stochastic exclusively in terms of the input wind. The performance of numerical model using the two approaches was then compared.
In the stochastic simulation, each debris particle rests at 1 m above ground with zero initial velocity. The process starts by randomly selecting a time instant from which to begin sampling the measured wind time history at the release height (taken as a random starting point from Fig. 3 for the corresponding case number). This selection determines the initial wind conditions when the debris is released. The initial wind speed associated with the selected start time is used to compute debris state (location, velocity, and acceleration) for the next time step. During flight, the vertical position of the debris at each time step is used to identify the wind velocity at the debris location and calculate the relative velocity necessary for determining its subsequent state. Using wind measurements from the center of the turn table (Section 2.1), the u k (in Eq. ( 5)) time history at the closest measured location to the elevation of the debris is derived. The u k value at the debris location is obtained by downstream propagation of turbulence from the measurement location to the debris position using Eq. ( 5) and Eq. ( 6). Using the u k value, the wind velocity is calculated from Eq. ( 5), which is used to update debris state using Eq. ( 3). Following this procedure, the debris location is continuously updated at each time step until landing on the ground. Since the release time is randomly selected, each debris will experience the wind field from a random window of the measured time histories, which thus ensures the stochasticity in the simulation. The schematic description of the debris trajectory modelling for a particle is shown in Fig. 9, and the numerical simulation workflow is presented in Fig. 10.
In this section, an approximate model to predict the horizontal travel distance of debris was derived based on Eq. ( 3), to facilitate fast estimation of debris behavior in engineering practice. Considering that the debris is released from the release height at time t r , the debris displacement at a given time t is expressed as:
where F is a function relating the displacement to the wind speed at spatial location x(τ) at time step τ, here denoted as U[x(τ), τ]. Consequently, the debris impact location is given by:
where t f is the duration of debris flight. As discussed in previous studies (Dong et al., 2023;Li et al., 2024), the debris flight trajectory primarily depends on the initial conditions, and the effect of turbulence on the
Comparison of numerical simulation estimations and experimental measurements for mean along-wind travel distance after applying the displacement threshold for Cases 1-4.
Case 1 Case 2 Case 3 Case 4 EXP (mm) 521.3 697.4 973.2 1214.9 Hoerner's drag coefficient NCD (mm) 581.7 784.0 1012.0 1260.3 NRD (mm) 597.5 806.3 1039.4 1297.3 Absolute error value between EXP & NCD (mm) 60.4 86.6 38.8 45.4 Absolute error value between EXP & NRD (mm) 76.2 108.9 66.2 82.4 Schlichting & Gersten's drag coefficient NCD (mm) 523.0 695.1 891.6 1110.8 NRD (mm) 538.5 721.0 923.7 1150.7 Absolute error value between EXP & NCD (mm) 1.7 2.3 81.6 104.1 Absolute error value between EXP & NRD (mm) 17.2 23.6 49.5 64.2
Comparison of numerical simulation estimations and experimental measurements for standard deviation of along-wind travel distance after applying the displacement threshold for Cases 1-4.
Case 1 Case 2 Case 3 Case 4 EXP (mm) 64.4 90.1 92.7 109.7 Hoerner's drag coefficient NCD (mm) 75.4 94.1 120.2 142.7 NRD (mm) 70.3 89.0 114.6 136.5 Absolute error value between EXP & NCD (mm) 11.0 4.0 27.5 33.0 Absolute error value between EXP & NRD (mm) 5.9 1.1 21.9 26.8 Schlichting & Gersten's drag coefficient NCD (mm) 68.5 84.6 107.5 127.6 NRD (mm) 62.5 77.8 100.0 120.1 Absolute error value between EXP & NCD (mm) 4.1 5.5 14.8 17.9 Absolute error value between EXP & NRD (mm) 1.9 12.3 7.3 10.4 trajectory can be estimated based on the turbulence characteristics at the release location instead of using the fully turbulent winds across the space. Meanwhile, the spatial variation of wind speed can be considered using the vertical mean wind profile. For a debris released at a specific time t r , Eq. ( 8) is expressed as follows:
where G is a function relating the displacement to the initial conditions, and U 0i [t r ] is the wind speed at release height and time in direction i.
The expected value for debris travel distance, considering the stochastic nature of the U 0i [t r ] at varying release times, can be obtained using:
The term p(U 0i [t r ]|U 0 [t r ]|) in Eq. ( 10) can be simplified as the distribution of U 0i [t r ], which represents the probability distribution function of wind speed in the ith direction at the release height and time. To approximate the expected travel distance, Eq. ( 9) is expanded about the temporal mean of U 0i [t r ], denoted by μ U0i , using Taylor series expansion:
Substituting this expansion into Eq. ( 10) yields:
where L i
) is the nth derivative of debris travel distance carried by the μ U0i with respect to U 0i . Eq. ( 12) can be expressed as:
where σ U0i , γ 1i and γ 2i represent the standard deviation, skewness, and kurtosis of U 0i [t r ], respectively. Vertical wind profile (spatial variations) of boundary layer wind is used to calculate L i (μ U0i ) and derivative terms in Eq. ( 13) to capture the mean flight distance, while the turbulence effect is modeled using the higher order statistics (i.e., the second or higher order statistics) of wind speed at the release height. Based on Eq. ( 13), incorporating higher-order terms is necessary when modeling turbulent wind fields, as using mean vertical wind profile alone may cause deviation from the travel distance calculated. The effect of higher-order terms is investigated in Section 4.2.
This section provides an overview of the experimental and numerical datasets prior to presenting comparative results. In the wind tunnel experiments, the customized computer vision technique successfully tracked 1825 out of 2067 released beads across five cases, achieving an 88 % success rate of the method and providing a robust dataset for result analysis. The 12 % failure rate corresponds to cases where an individual debris particle was confirmed to be released into the air, but the algorithm failed to track its landing location. When a capture fails, this result is easily identified and removed from the dataset before analysis. The numbers of successful samples for Cases 1-5 are 523, 233, 285, 328, and 456, respectively. Larger debris quantities were employed for Cases 1 and 5 to accurately examine the effect of full and partial turbulence spectrum on debris trajectory statistics. For each tracked debris particle, the complete flight trajectory from release until landing was captured. The numerical simulation involved 100,000 realizations for each of the 5 cases to ensure statistical robustness and enable detailed comparison of trajectory characteristics between experimental and numerical procedures. The sufficiency of sampling in experiments and numerical modeling is presented in Appendix B.
Debris flight distance results from numerical simulations and experiments are presented for Cases 1-4 in Figs. relationship proposed by Schlichting and Gersten (2016). Experimental results are presented in row (c) of Figs. 11-18.
In Fig. 11, the left column panels (a1) and (b1) each present the results from 100,000 numerical simulations, while panel (c1) presents the companion experimental results from 523 debris beads. In panel (a1), the top plot presents a histogram (pink) of the landing distance of the debris (mm) downwind of the origin (release location). The lower portion of panel a1 shows a scatter plot of the landing location from each of the 100,000 simulations, represented as red dots, relative to the release location (indicated by a green triangle). In the lower-right corner, a histogram (green) of the landing locations lateral to the release location is shown. The upper right portion of panel a1 presents the mean, standard deviation, skewness, and kurtosis of the landing locations among the 100,000 samples both downwind of the release location origin (X) and lateral to the release location (Y). This figure structure is consistently applied within all panels in Figs. 1112131415161718. The description of the right column (i.e., panels (a2), (b2), (c2)) will follow an analysis of the results in the left column.
Left column results of Figs. 11-18 reveal that the experimental debris exhibit a higher occurrence of short-flight cases than the numerical simulations. This is presented as a higher magnitude negative skewness in the longitudinal direction for the experimental flight statistics (which can also be visualized by comparing the pink histograms), and more frequent individual occurrences of low flight distance in the experimental scatter plot. This disparity may be attributed to imperfections in the debris release device used in the experiments. Debris may have occasionally been influenced by contact and friction from the release device, introducing conditions absent in numerical simulations.
To ensure data quality and reliability of statistical analyses, an automatic outlier removal function was implemented to eliminate potentially erroneous data from both experimental and numerical datasets. The outlier removal function utilizes MATLAB's "rmoutliers" function, which employs a robust statistical method based on scaled median absolute deviation (MAD). This approach identifies outliers as data points that fall more than three scaled MADs away from the median. The MAD is scaled by a factor of 1.4826 to make it equivalent to the standard deviation for normally distributed data, ensuring that the three-scaled-MAD threshold corresponds to the three-sigma rule used in standard outlier detection. Since the median and MAD are less sensitive to extreme values, the outlier detection process is robust to the influence of the outliers it seeks to identify. The data excluding the outliers are presented in the right column in . Within these presentations, EXP, NCD, and NRD denote results from experiments, numerical simulations with constant drag coefficient, and numerical simulations with Reynolds-dependent drag coefficient, respectively.
As demonstrated in the right columns of Figs. 11-18, the outlier removal function has a negligible effect on the simulated data, as numerical simulations inherently do not suffer from the experimental constraints encountered in the wind tunnel. The comparison of debris longitudinal displacement from left and right columns in Figs. 11-18 demonstrates a significant experimental data loss after applying the outlier removal function for Cases 1 and 2 that incorporate lower wind speed data, creating a visible cut in the longitudinal flight histograms and distribution in the right columns. However, for Cases 3 and 4, there is no considerable loss of data, as evidenced by the relatively unchanged statistics shown in the right columns. The numerical simulation was in reasonable agreement with experimental data in terms of debris flight estimations in the lateral direction, considering both left and right columns. The application of the outlier removal function does not significantly affect the lateral displacement statistics in either experimental or numerical datasets. Therefore, the focus of the remaining section is on the longitudinal debris trajectory, which is presented in more detail in Tables 34.
The improved distributional match between datasets before and after applying the outlier removal is further illustrated in Figs. 19202122. In these figures, box plots are positioned above the histograms for both experimental and numerical longitudinal flight distance datasets, with median values displayed numerically. The results demonstrate that data ranges for both numerical and experimental datasets show significantly better agreement after applying the outlier removal function. The quantile-quantile (Q-Q) plots for the experimental and numerical longitudinal flight distance datasets before and after applying the outlier removal function are presented in Figs. 23242526. In these plots, the xaxis and the y-axis represent experimental and numerical simulation quantiles, respectively. The blue points represent the actual quantilequantile pairs from the data comparison, while the red dashed line indicates the linear relationship that would exist if both datasets followed identical distributions. The threshold values shown in the plots represent the outlier detection boundaries applied to eliminate data points that fall outside these statistical limits. Before outlier removal, the blue data points show substantial deviation from the red reference line, particularly in the lower quantile ranges, indicating poor distributional agreement between experimental and numerical data. This deviation suggests the presence of erroneous measurements that distort the statistical characteristics of the experimental dataset. After applying the outlier removal function, the blue points align much more closely with the red reference line, demonstrating that the two datasets exhibit nearly identical statistical distributions. This improved agreement validates the effectiveness of the outlier detection method.
To understand why the first method (constant drag coefficient) has lower estimations of the mean longitudinal flight distance than the second method (variable drag coefficient), it is essential to examine the drag coefficient behavior during simulation. Sample drag coefficient trends are presented in Figs. 27-28 with results for both a single debris sample trajectory and 100 random samples' trajectories, demonstrating the increasing trend of drag coefficients in the NRD approach compared to the constant NCD drag coefficient value. When using the varying drag coefficient method, the drag coefficient trend generally increases throughout the simulation because as the debris descends to lower heights, the wind velocity generally decreases while the debris accelerates, resulting in smaller relative velocities and correspondingly smaller Reynolds numbers. Since the Reynolds number operates within a sensitive region where the drag coefficient is heavily dependent on Reynolds number variation, the instantaneous drag coefficient tends to increase substantially with decreasing Reynolds number, leading to higher trajectory results. However, the constant drag coefficient approach smooths out unsteady effects that the Reynolds number-dependent model overreacts to.
As can be seen in the captions of Figs. 11-18, when using the NCD approach to simulate the trajectories, Hoerner's drag coefficient is always higher than Schlichting & Gersten's drag coefficient. The difference in drag coefficients explains the observation that Hoerner's drag coefficient overestimates and Schlichting & Gersten's drag coefficient often underestimates the mean flight distance compared to experimental values. For example, for Case 3, the constant drag coefficient used in the NCD approach from Hoerner (1965) is 0.520, while the same approach from Schlichting and Gersten (2016) yields 0.445 as the drag coefficient, resulting in mean longitudinal displacements of 1012 and 891.6 mm, respectively, and the experimental value is 973.2 mm.
Comparison of numerical simulation estimates and experimental measurements for mean along-wind travel distance after applying the displacement threshold for Case 5.
Case 5 EXP (mm) 562.1 Hoerner's drag coefficient NCD (mm) 583.4 NRD (mm) 594.6 Absolute error value between EXP & NCD (mm) 21.3 Absolute error value between EXP & NRD (mm) 32.5 Schlichting & Gersten's drag coefficient NCD (mm) 525.7 NRD (mm) 534.7 Absolute error value between EXP & NCD (mm) 36.4 Absolute error value between EXP & NRD (mm) 27.4
Comparison of numerical simulation estimates and experimental measurements for standard deviation of along-wind travel distance after applying the displacement threshold for Case 5.
Case 5 EXP (mm) 173.7 Hoerner's drag coefficient NCD (mm) 189.8 NRD (mm) 177.0 Absolute error value between EXP & NCD (mm) 16.1 Absolute error value between EXP & NRD (mm) 3.3 Schlichting & Gersten's drag coefficient NCD (mm) 172.5 NRD (mm) 156.5 Absolute error value between EXP & NCD (mm) 1.2 Absolute error value between EXP & NRD (mm) 17.2 K. Yousefi Anarak, S. Li, R.A. Catarelli et al. Advances in Wind Engineering 2 (2025) 100073
These discrepancies highlight the importance of proper drag coefficient selection in debris flight modeling. In fact, some errors may occur when applying the quasi-steady aerodynamic drag to the transient flow experienced by the debris. Furthermore, when selecting drag coefficients, it is important to recognize that the drag coefficient depends not only on the Reynolds number (which is a function of wind velocity) but also on turbulence intensity and integral length scale, though this information is often unavailable in existing literature. Moreover, values extracted from available drag coefficient-Reynolds number plots, while containing experimental data, interpolate drag coefficient values for Reynolds numbers not experimentally investigated. This interpolation might differ from actual values if experimentally determined. Additionally, it is not definitively established whether all particles used in their studies were perfectly spherical. Therefore, a detailed analysis of debris flight in turbulent conditions requires careful consideration of the drag coefficient under the same flow conditions. Although applying outlier removal was intended to mitigate problems associated with potential contact and friction of particles at the moment of release, the presence of the release device itself could also influence the downstream flow where particles fly. For Case 1, in addition to the release location without the release device mechanism (as described in Section 2.1), wind speed measurements were conducted at five additional locations downstream of the release device with the device in place. These measurements were taken from 0.2 m to 1.0 m downstream of the release location at intervals of 0.2 m. These wake region measurements were used to properly scale the wind speed experienced by the debris in the numerical simulation, which accounted for the changes in mean wind speed and turbulence intensity in the wake region. Taylor's frozen turbulence hypothesis was applied using the closest upstream wind speed measurement location to the debris particle. The numerical simulation was repeated using both NCD and NRD approaches, with results shown in As can be seen in Fig. 29, the results show considerable improvement in estimation of both mean and standard deviation when considering both the wake effect due to the presence of the debris release device and outlier removal, suggesting interference of the device with the ideal open flow condition we intended to test. In ideal open flow conditions, it is expected that numerical simulations will therefore provide good estimations of debris statistics. The results when using Schlichting & Gersten's drag coefficients for simulations are closer to the experimental values, suggesting that the drag coefficient relationships from this reference, when using either NCD or NRD approaches, lead to values closer to the experimental measurements. This again highlights the importance of selecting proper drag coefficient values. The aerodynamic drag essentially depends on the flow conditions including mean wind speed and characteristics of the turbulence such as the turbulence spectrum and integral length scale. This phenomenon can be even more complicated due to the transient flow field experienced by the moving debris, which requires future investigation.
This section examines the numerical method's performance in calculating debris travel distance statistics for Case 5, which incorporates both low-and high-frequency components in the wind (see Fig. 2(c)). Numerical simulations were conducted using both constant and Reynolds-dependent drag coefficients, as illustrated in Fig. 30(a1, b1) and Fig. 31(a1,b1), with subsequent comparison to experimental data presented in Fig. 30(c1) and Fig. 31(c1). To be consistent with the previous section in eliminating the outliers from the results, the same outlier removal function was implemented, generating the results shown in Fig. 30(a2,b2,c2) and Fig. 31(a2,b2,c2). Statistical analysis of the results, detailed in Tables 56, demonstrates the differences between numerical simulation statistics and ground truth experimental data.
The same analysis presented for Cases 1-4 is extended to Case 5, with results shown in Figs. [32][33][34][35][36][37] demonstrate that even before outlier removal, Case 5 exhibits distributional agreement between experimental and numerical simulations for both NCD and NRD approaches. This observation is consistent with Fig. 30(c1), which shows skewness values close to zero for Case 5, unlike Cases 1-4. Figs. 34-35 further confirm this agreement, displaying correspondence between the probability density functions of experimental data and both NCD and NRD approaches before and after outlier removal, suggesting that outlier removal is unnecessary for this dataset. Comparison of histograms between Case 1 and Case 5, for instance, a comparison between Fig. 19 (Case 1) and the left column of Fig. 34 reveals that Case 5 has a larger spread of debris longitudinal distances than Case 1. This larger spread can cover up the presence of failed ejections that are obvious in Case 1 due to its narrower spread of flight distances, making outlier removal effective for Case 1 but unnecessary for Case 5. Fig. 36 illustrates the drag coefficient evolution during flight, showing that the NCD value remains generally lower than the drag coefficients calculated during debris flight when using the NRD approach, which explains the overestimation of mean trajectory compared to NCD approach, as discussed previously.
The effect of the wake behind the release device is illustrated in Fig. 37. Similar to Case 1, wind speeds downstream of the release device were measured from 0.2 m to 1.0 m at intervals of 0.2 m with the device present, and these measurements were incorporated into the numerical simulation. When wake effects are included in the simulation, the results of both methods approach the experimental values more closely. The findings demonstrate that numerical simulation statistics align with experimental data regardless of whether constant or Reynolds-dependent drag coefficients are employed when using drag coefficients from Hoerner (1965), highlighting the need for proper drag coefficient selection in numerical simulations. Finally, the study experimentally validates the numerical method's application under fullturbulence spectrum conditions with high turbulence intensity.
This section compares mean trajectories estimated using Eq. ( 13) with stochastically simulated results using the validated numerical model incorporating 100,000 realizations for each of Cases 1-5. Higher order terms in Eq. ( 13) were computed using the centered finite differences method, which offers the advantage that for uncertainty order n (greater than 1), approximate solutions for mean debris landing locations can be obtained using n + 1 realizations. For example, with second-order uncertainty, mean longitudinal travel distance can be approximated using only three realizations: 14) where h denotes the step size, selected as 10 % mean longitudinal velocity at the release height in this study. Subsequently L 1 (μ U01 ),
L 1 (μ U01 +h) and L 1 (μ U01 -h) represent the predicted landing locations using mean wind profile, upper-bound wind profile (mean wind profile increased by 10 %) and lower-bound wind profile (mean wind profile decreased by 10 %), respectively. The term σ U01 2 denotes the wind speed variance at the release height. Both the approximate solution and stochastic simulation adopt a varying drag coefficient during simulation and use Hoerner's drag coefficient-Reynolds number plot. Fig. 38 demonstrates the effect of higher order terms up to 4th order, showing that using just the first two terms of Eq. ( 13), provides a sufficiently close approximation of mean debris travel distance compared to stochastic simulations with the maximum percentage error of 2.5 % across all cases. This close match between the estimation from the proposed approximate method and the validated numerical model serves to validate the applicability of the approximate debris trajectory model for turbulent cases. Using this approach, the mean longitudinal travel distance for spherical debris traveling within an unobstructed straightline flow, can be readily estimated using the mean wind speed profile and turbulence intensity at the debris release height.
This study sought to validate the numerical model for spherical debris flight under highly turbulent wind conditions. The application of quasi-steady aerodynamic loads is theoretically justified due to the relatively small size of debris in wind gusts. To further validate the theory, five cases were experimentally investigated and numerically simulated, with four sharing approximately 10 % longitudinal turbulence intensity at the release height but varying mean wind speeds, featuring partially generated turbulence spectra with low frequency deficits. The fifth case matched the first case's mean wind speed but employed a fully generated turbulence spectrum with longitudinal turbulence intensity of 23.7 % at the release height.
Debris flight was tracked via a synchronized camera system and processed using computer vision techniques designed for high-speed tracking in low-contrast environments. Statistical analysis of ground impact locations was conducted after performing numerical stochastic simulations using a Newtonian-based system of equations, employing measured wind velocity profiles as inputs. The discrepancies observed between experimental and numerical results can be attributed to several factors, including the drag coefficient sensitivity to the flow conditions, imperfections in the debris release mechanism, the flow disruption caused by the presence of the release device itself in the wind field, and the limitations of Taylor's frozen turbulence hypothesis. The match between numerical and experimental results improved after excluding outlier debris trajectory data and considering the wake behind the device, suggesting potential measurement inaccuracies due to these reasons. The statistics of debris travel distance using both a constant drag coefficient and a variable drag coefficient in the numerical modeling were in good agreement with those obtained experimentally. The findings demonstrate that the numerical model produces results comparable to experimental observations, validating the applicability of debris flight models in highly turbulent boundary layer winds.
Additionally, an approximate method was developed for determining spherical debris landing positions using mean vertical wind profile and longitudinal turbulence intensity at the debris release height, providing a new approach for the rapid estimation of mean flight distance under highly turbulent open flow conditions. This validation of the numerical model in unobstructed, highly turbulent conditions establishes a foundation for future research examining the model's applicability in obstructed flow scenarios involving a neighboring building.
The authors declare the following financial interests/personal relationships that may be considered as potential competing interests: Yanlin Guo, John W. van de Lindt, Kurtis R. Gurley report that financial support was provided by the National Science Foundation (NSF). Yanlin Guo is a editorial board member of Advances in Wind Engineering, and she was not involved in the editorial review or the decision to publish this article. Kimia Yousefi Anarak reports that financial support was provided by Thornton Tomasetti. If there are other authors, they declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
were critical to the success of this study. The authors would also like to express their sincere gratitude to Todd Atadero, Laboratory Support Engineer at Colorado State University, for his invaluable assistance and insights into the experimental work. Additional support through the Thornton Tomasetti Student Innovation Fellowship is also gratefully acknowledged.
Advances in WindEngineering 2 (2025) 100073