Drag on circular cylinders with porous outer layers in turbulent cross-flow

The effects of free-stream turbulence intensity and porosity on the drag on cylinders with porous outer layers in cross-flow was investigated experimentally. This work is motivated by the need to better model spotting—a forest fire propagation mechanism in which burning branches and other debris (termed firebrands) are transported away from the main fire by the prevailing wind and ignite new fires. Multiple levels of background turbulence were studied by using no grid, passive grids, and an active grid to generate turbulence intensities of 0.4%, 1.7%, 2.7% and 12.4%. The porous char-layer on the outer surface of firebrands was mimicked by wrapping wire meshes (of 10, 20 and 40 pores per cylinder diameter or PPD) around the cylinders, each to three different layer-thickness fractions ( 1/16 , 1/8 and 1/4 ) of the cylinder’s outer diameter. The drag on one smooth cylinder and nine cylinders with porous outer layers was measured for Reynolds numbers in the range 7000–17 000, at the aforementioned four turbulence intensities. The results showed that (i) the free-stream turbulence intensity and PPD of the wire meshes affect the Reynolds number dependence of the drag coefficient; (ii) the drag coefficient increases with free-stream turbulence intensity when it is relatively low (0.4%–2.7%), then decreases at high intensities (12.4%), and this decrease is more pronounced as the PPD increases; (iii) the drag coefficient increases with the thickness of the porous layer, and asymptotes to an effectively constant value after a critical thickness of about 1/8 of the cylinder’s diameter; and iv) the drag coefficient exhibits a non-monotonic dependence on the PPD of the wire mesh. These results demonstrate the importance of accounting for free-stream turbulence intensity and the parameters that characterize the porous outer layers of cylinders when modeling the drag on firebrands, or in other applications in which there is cross flow over cylinders with porous outer surfaces.

of 0.4%, 1.7%, 2.7% and 12.4%.The porous char-layer on the outer surface of firebrands was mimicked by wrapping wire meshes (of 10, 20 and 40 pores per cylinder diameter or PPD) around the cylinders, each to three different layer-thickness fractions ( 1 /16, 1 /8 and 1 /4) of the cylinder's outer diameter.The drag on one smooth cylinder and nine cylinders with porous outer layers was measured for Reynolds numbers in the range 7000-17 000, at the aforementioned four turbulence intensities.The results showed that (i) the free-stream turbulence intensity and PPD of the wire meshes affect the Reynolds number dependence of the drag coefficient; (ii) the drag coefficient increases with free-stream turbulence intensity when it is relatively low (0.4%-2.7%), then decreases at high intensities (12.4%), and this decrease is more pronounced as the PPD increases; (iii) the drag coefficient increases with the thickness of the porous layer, and asymptotes to an effectively constant value after a critical Original Content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence.Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.

Introduction
Forest fires have increased in frequency and intensity over the past 10-15 years, with significant adverse impacts on humans, flora and fauna (Adams 2013).For example, in 2017, wildfires in Portugal and California resulted in the deaths of 66 and 44 people, respectively (Aon Benfield 2018).In 2023, over 6600 wildfires were reported across Canada (Canadian Interagency Forest Fire Centre 2023), engulfing 18 million hectares (Natural Resources Canada 2023), shattering the prior (1989) record of 7.6 million hectares, and placing roughly 200 000 people under evacuation orders.Given these and the many other events related to damaging and catastrophic forest fires, (i) enhancing our understanding of the mechanisms that lead to their propagation, and (ii) improving the techniques that are used to predict their spread are more pertinent than ever.Viegas (1998) grouped forest fires into three main categories based on the ways they propagate: ground fires, which usually burn without flame, and propagate in the organic layer above the soil; surface fires, in which flaming combustion occurs in vegetation at or near the surface; and crown fires, which extend to the upper layers of the foliage, and may consume groups of trees and achieve a state of sustained propagation to the surrounding foliage.
Crown fires can also spread by spotting, a propagation mechanism in which burning leaves, branches, and/or other debris, termed firebrands, are ejected and entrained into prevailing winds and transported hundreds of meters away from the main fire, igniting new fires (Anthenien et al 2006).In large-scale forest fires, spotting is often the dominant propagation mechanism, and the most difficult to control (Koo et al 2010).
The propagation of forest fires by spotting has been investigated via field observations (Ohmiya and Iwami 2000), laboratory experiments (Tarifa et al 1965, Manzello et al 2011), and numerical simulations (Yin et al 2003, Anthenien et al 2006, Sardoy et al 2007, Oliveira et al 2014).Tarifa et al (1965) conducted wind tunnel experiments to measure the forces and lifetimes of burning cylindrical and spherical objects representing firebrands.Based on these measurements and a simplified two-dimensional mathematical model, they numerically estimated the maximum spotting distance of firebrands.More recently, Oliveira et al (2014) used three-dimensional numerical simulations to determine the effects of initial orientation, oscillations, and spin of wind-borne cylindrical firebrands on their trajectories.Because modeling of forest-fire propagation by spotting involves simulating the trajectories of firebrands, the motivation for the present research is to improve the predictions of the atmospheric transport of firebrands by providing more relevant data on the their aerodynamics.
Up to now, the investigations reported in the published literature have not accounted for two key aspects that could have a significant impact on the drag and lift experienced by wind-borne firebrands, and thus influence their trajectories: the free-stream turbulence in the atmosphere, and the porous char layer that forms on burning firebrands.The atmosphere is typically in a highly turbulent state, in which the turbulence intensities near the ground range from 10%-20% in non-urban areas, and 20%-35% in urban areas; the height of the atmospheric boundary layer over the ground ranges from 400-800 m; and the integral length scale of the turbulence increases with increasing height, ranging from 10 m near the ground to 200 m in the direction of the prevailing winds (Counihan 1975).The characteristics of the char layer formed on burning wood have been investigated by Ragland et al (1991).They found that this char layer has a porosity of 0.8-0.9, and a density of 10%-20% of that of the dry wood.
In the experimental investigations mentioned above, the firebrands were modeled as cylinders, disks or spheres, and the drag and lift on them were measured.In the present work, the drag on non-burning cylinders with porous outer layers in cross-flows was investigated experimentally, with the focus of this work being the determination of the combined effects of the (i) turbulence intensity of the free stream, and (ii) porous outer layer of char on the drag coefficient of firebrands.Specifically, the effects of Reynolds number, free-stream turbulence, porous outer-layer thickness, and porous outer-layer porosity on the drag on cylinders were explicitly studied to better model firebrands, which are transported in a turbulent atmosphere, and which have porous outer layers of varying depths on account of the char that results from their burning.This work was undertaken in a wind tunnel, fitted with passive and active grids that allowed the level of free-stream turbulence to be varied and employed cylinders with porous outer layers of varying porosity and depth, to emulate firebrands.
Having provided an introduction, the remainder of this paper is organized as follows.A brief review of the literature pertaining to the drag on circular cylinders, including the effect of free-stream turbulence and porous outer layers, is presented in section 2. The apparatus and instrumentation are described in section 3, and the experimental conditions described in section 4. The results are presented and discussed in section 5.A further discussion relating the measured drag coefficients to the flow in the wake of the cylinders studied in the prior section will be given in section 6.Finally, concluding remarks are given in section 7.

Literature review
The goal in this section is not the presentation of an exhaustive review of the literature pertaining to flows over a circular cylinder; instead, it is a review of publications directly related to the drag on circular cylinders (section 2.1), the effect of free-stream turbulence (section 2.2), and the effect of porous outer layers (section 2.3).

Drag on circular cylinders
The drag on stationary smooth circular cylinders in cross-flow has been the subject of numerous investigations over the past century, and it has now been clearly demonstrated that the flow around circular cylinders varies with the Reynolds number (Re D ≡ ρU∞D µ ) (von Kármán 1911, Roshko 1954, Lienhard 1966).The dimensionless parameter used for quantifying the drag on an object in cross-flow is the drag coefficient (C D ), which is defined as: where F drag is the drag force on an object, ρ is the fluid density, U ∞ is the mean upstream velocity of the flow, and A proj is the projected frontal area of the object.This drag can be divided into two categories: viscous drag, caused by the friction on the surface of the object; and form drag, caused by the difference between the pressure forces on the upstream and downstream faces of the object.
One of the first to publish measurements of C D on circular cylinders was Wieselsberger (1921).In his study, the drag coefficient was measured over a wide range of Reynolds numbers.For Re D ≲ 10 3 , he observed that C D decreases with increasing Re D .In the range 10 3 ≲ Re D ≲ 1.5 × 10 5 , C D is relatively invariant with respect to changes in Re D .In the range 1.5 × 10 5 ≲ Re D ≲ 3.5 × 10 5 , the boundary layer (on the portion of the cylinder surface with attached flow) undergoes transition to turbulence, and the flow separation point on the cylinder moves downstream, from approximately 80-140 degrees (from the front stagnation point).This significantly and fairly abruptly reduces the form drag, and therefore C D -a phenomenon that is sometimes referred to as the drag crisis.The Reynolds number at which this occurs is referred to as the critical Reynolds number Re D,crit ; it is often defined (Bearman and Morel 1983) as that at which the C D drops to 0.8, and is generally around Re D ∼ 3 × 10 5 .
In many published experimental investigations, the cylinders are assumed to be infinitely long, and the flow over them quasi-two-dimensional.However, the aspect ratio of the cylinder (i.e. the ratio of its length to its diameter, L/D) can have an important effect on the drag coefficient and its dependence on the Reynolds number.In the results published by Wieselsberger (1921), the aspect ratios were high enough (280 and above) to be considered infinite.However, in most laboratories, it is difficult to achieve such high aspect ratios.Zdravkovich et al (1989) summarized the effect of the aspect ratio on C D using data from multiple investigations and concluded that even a cylinder with an aspect ratio of 20 exhibits a significantly lower C D than that of an infinitely-long cylinder.
In addition to the effect of finite cylinder aspect ratios, the study of cylinders in wind tunnel experiments can also be affected by tunnel-wall-interference and flow-blockage effects if the ratio of the cylinder diameter to the test section height (or width) is not negligible.This causes the measured values of C D to be overestimated.Fortunately, correlations exist to correct for flow-blockage blockage effects.To this end, Allen and Vincenti (1944) published correlations that apply to a variety of shapes and also account for compressibility in the flow at high Mach numbers (M).Dalton (1971) modified this approach specifically for cylinders in cross-flow, when compressibility effects are negligible (M < 0.2), and obtained the following result: where C ′ D is the measured drag coefficient, C D is the corrected one, D is the cylinder diameter and H is the wind tunnel height for the case of a horizontal cylinder.
The drag on circular cylinders can also be influenced by the end conditions.When a cylinder has one or both ends free in the flow, three-dimensional flow along the span is induced (Zdravkovich 1997b).In the case of a cylinder that spans the cross-section of a wind tunnel, the tunnel-wall boundary-layers separate around the cylinder, creating a 'horseshoe swirl system' (Zdravkovich 1997b).By adding plates at each end of the cylinder, the boundary layers become very thin, and three-dimensional effects are minimized (Stansby 1974).There have been numerous investigations concerning the optimal shape and size for end plates.Kubo et al (1989), using circular end plates with different diameters, found that the ratio of the end-plate diameter to the cylinder diameter had to be at least 8 for the effects on C D to be minimal.Stansby (1974) found that square end plates with sides seven times the cylinder diameter in length rendered the flow almost completely two-dimensional along its axis.He also found that the addition of end plates reduces the base pressure along the cylinder (i.e. the pressure 180 • from the front stagnation point), therefore increasing C D .
Finally, another parameter that may influence C D on circular cylinders is the free-stream turbulence intensity (T i ), which is defined as: where u rms is the root-mean-square value of the fluctuating component of the velocity, and U ∞ is the mean free-stream velocity.The free-stream turbulence intensity in the above-mentioned studies was in the range of 0.1% ⩽ T i ⩽ 0.7%, however the effects of these levels of turbulence on C D have not been fully investigated.Moreover, in many publications on the drag on circular cylinders, T i is not reported.
Because of the aforementioned effects of the aspect ratio, wind-tunnel blockage, end conditions, and free-stream turbulence intensity, particular care must be taken when comparing published values for C D .The work of Norberg (1987) and Delany and Sorensen (1953), being undertaken at Reynolds numbers at the lower and upper ranges, respectively, of those used in the present experiments, and with similar end conditions, will serve as a useful comparison with results of the present work (to be discussed in section 5.1).It must, however, be noted that the aspect ratios of the cylinders used in their experiments, 80 and 28, respectively, are higher than those in this project (L/D = 12).The results of Okamoto and Yagita (1973), for cylinders with similar L/D as the present work, and with experiments at Reynolds numbers at the upper end of the range of Re D studied, may also be useful for comparison; however, they have different cylinder end conditions.Fage and Warsap (1929) were among the first researchers to observe that increasing the freestream turbulence intensity accelerated the transition to turbulence in the boundary layer over the cylinder, decreased Re D,crit , and shifted the C D versus Re D curves to the left.This trend has also been observed in the works of Arie et al (1981) and Cheung and Melbourne (1983).

Effect of free-stream turbulence
It is not clear how T i affects C D in the subcritical range, as there is wide scatter in the published results.For example, a study by Norberg (1987), in which Re D = 3000 and 8000, exhibited an increase in C D as T i was increased from 0.1% to 1.4%.Ko and Graf (1972), who studied the drag over a cylinder for 1300 ⩽ Re D ⩽ 8000 and 1.3% ⩽ T i ⩽ 21.3%, found that C D initially decreased with increasing T i , reaching a minimum at T i ≈ 4%; and C D then increased.And Arie et al (1981), undertaking experiments in the ranges of 7900 ⩽ Re D ⩽ 54 000 and 1% ⩽ T i ⩽ 15.8%, found that C D decreased with increasing T i .
The effect of the ratio of the longitudinal integral length scale (ℓ x ) of the background turbulence to the diameter is also unclear.Arie et al (1981) observed that C D decreases with decreasing ℓ x /D when ℓ x /D < 1, and is unaffected when ℓ x /D > 1. Ko and Graf (1972) found no clear trend between C D and ℓ x /D.Perhaps the best summary of the effects of ℓ x /D can be found in the work of Bearman and Morel (1983), who stated: 'As far as the length scale is concerned, increasing its ratio to characteristic body dimension sometimes increases the turbulence effects, sometimes it attenuates them, and in a number of cases it has been found to have no effect at all over a wide range of length scales.' To provide some physical explanations of the aforementioned observations, Bearman and Morel (1983) identified three mechanisms by which the free-stream turbulence can affect the flow over bluff bodies: (i) transition to turbulence at lower values of the Reynolds number, (ii) enhanced mixing and entrainment, and (iii) distortion of the turbulence itself by the mean flow.A possible reason for the scatter in published results is that more than one of these mechanisms may occur at the same or different locations in the flow.Lastly, it bears noting that the effect of background turbulence of the flow over cylinders and other bluff bodies has also been extensively studied in the context of its effect on the heat transfer (Giedt 1951, Smith and Kuethe 1966, Torii and Yang 1993, Lee et al 1994, Scholten and Murray 1998, Sanitjai and Goldstein 2001, Sak et al 2007, Son et al 2010, Quintino 2012).

Effect of porous outer layers
Until now, studies of solid cylinders with porous outer layers have principally focused on applications in flow control and vibration damping (Bhattacharyya and Singh 2011), aerodynamic noise reduction (Sueki et al 2010, Liu et al 2014, Geyer 2020, Sharma et al 2023) or heat transfer (Sobera et al 2003).The few investigations of the drag on solid cylinders with porous outer layers have been mostly numerical in nature (e.g.Naito and Fukagata 2012), with more experimental research in this area being required (Liu et al 2014).Bhattacharyya and Singh (2011) studied the flow over a solid cylinder with a porous outer layer in cross-flow using numerical simulations, and found that the porous layer reduced the drag, when compared to a solid cylinder with no porous outer layer.In another numerical study, Bruneau and Mortazavi (2006) observed that C D either increased or decreased depending on the porosity of the outer layer.Ozkan et al (2012) investigated the wakes of cylinders surrounded by a permeable cylinder in shallow water flows.They found that the effect of the porous outer cylinders was to decrease the turbulent kinetic energy and Reynolds shear stresses in their wakes.
In an attempt to induce drag reductions on the flow over bluff bodies, Klausmann and Ruck (2017) experimentally studied the flow over cylinders with a symmetrically-applied porous outer layer or coating on a part of the leeward side of the cylinder (with coating angle between 40 and 160 degrees).They chose a partial porous coating because preliminary tests on fully coated cylinders exhibited an increase in C D .They observed that the partial porous layer increased the base pressure on the leeward side of the cylinder, leading to a decrease in C D of up to 13%, when compared to a smooth cylinder.Watanabe et al (1989Watanabe et al ( , 1991) ) conducted wind tunnel experiments on solid cylinders surrounded by different (porous) fabrics.They observed that the presence of such fabrics caused the separation point to move forward, towards the front stagnation point on the cylinder.Though their study did not include drag measurements, such a reduction of the separation angle would cause an increase in C D , due to an increase in form drag (Bearman and Morel 1983).

Apparatus and instrumentation
The experiments described herein were conducted in a 0.407 × 0.407m 2 cross-section, 4.5-mlong, open-circuit wind tunnel, located in the Aerodynamics Laboratory at McGill University.The tunnel was originally built and operated at the Sibley School of Mechanical and Aerospace Engineering at Cornell University.Multiple investigations have been carried out in this wind tunnel over the last four decades, including, but hardly limited to, the works in Warhaft (1980Warhaft ( , 1981Warhaft ( , 1984)), Sirivat and Warhaft (1983), Mydlarski and Warhaft (1996), Ayyalasomayajula and Warhaft (2006), Gylfason and Warhaft (2009).The interested reader can refer to these works for more details about the tunnel.The principal changes made to the wind tunnel in its new location are (i) its installation as a horizontal wind tunnel (in comparison to a vertical installation at Cornell University), and (ii) the purchase of a new 10 hp AC motor (Baldor EM3714T-5), variable frequency drive (ABB ACH550-UH) and blower (Cincinnati HBDI-240) for the tunnel.The wind tunnel's flow conditioning and test sections remain unchanged.
Four different levels of free-stream turbulence intensity were investigated in this study.The first level corresponded to the residual background turbulence in the tunnel with no grid installed.The three other levels of turbulence intensities were generated by: (i) a 1" passive grid, (ii) a 2" passive grid, and (iii) an active grid.The active grid is the same one as that used by Mydlarski and Warhaft (1996).However, the operation of the active grid in the present work was improved over that in Mydlarski and Warhaft (1996), being now operated in double-random mode (as opposed to single-random mode operation in Mydlarski and Warhaft (1996)-see Poorte and Biesheuvel (2002) and Mydlarski (2017) for additional discussions on modes of operation of active grids).The different operation of the active grid was achieved by the construction of a new active-grid controller using one Arduino Uno as a master controller, and fourteen Elegoo Arduino Nanos as slave Arduinos, each of which controlled a stepper motor driving each of the fourteen rotating bars of the active grid.See Cohen (2019) for the details of the controller and the operating parameters of the active grid.
Measurements of the flow in the wind tunnel were made by way of hot-wire anemometry, using a DANTEC 55M10 constant temperature anemometer and a 5 µm tungsten hot-wire sensor.The hot-wire length was approximately 1 mm, resulting in a length-to-diameter ratio of 200.The drag force on the cylinder was measured by way of a Kineoptics WTB 3.0 wind tunnel balance, capable of measuring lift, drag, and pitching moment, using three load cells (Honeywell, Model 31), with a range of ±250 g.To calibrate the wind tunnel balance, a screw was installed in the bracket on top of the balance, to which a nylon string was hooked and strung over a low-friction pulley.Gauge blocks, carefully weighed to a precision of ±0.01 g, were hung from the string to provide loads of 0-100 g, transmitting a static horizontal force to the balance.The resulting load-cell voltages were recorded, and a calibration curve was fitted to the data for each load cell.It was found during preliminary tests that the load cells produced the most repeatable results when in compression.Thus, all three load cells were pre-compressed by amounts that ensured they would remain in compression for the full range of drag forces encountered in this work (10g ≲ F drag ≲ 70g).Further details pertaining to the wind tunnel balance and its calibration can be found in Cohen (2019).
The output signals of the hot-wire anemometer and the load cells from the wind tunnel balance were digitized using a 16-bit National Instrument PCI 6143 data acquisition card and a BNC 2110 connector block, and recorded using custom-made LabVIEW programs.When measuring the velocity fluctuations, a Krohn-Hite 3382 filter was used to (i) high-pass filter the signal from the hot-wire at 0.1 Hz, (ii) low-pass filter the signal at the Kolmogorov frequency, and (iii) amplify the filtered signal to minimize the discretization error.
For the data obtained from the hot-wire anemometer, the mean value of the signal (which was not high-pass filtered) was computed with a data set of 1.024 × 10 4 samples recorded at a rate of 400 Hz, for a total sampling time of 0.4 min, such that the mean velocity converged.The filtered signal (used to calculate velocity fluctuations) consisted of a time series of 4.096 × 10 4 samples recorded at a rate of 400 Hz, for a total sampling time of 1.7 min, such that all statistical moments of the fluctuations up to fourth order were converged.Data sets for the calculation of spectra consisted of 1.6384 × 10 6 samples recorded at twice the Kolmogorov frequency (which varied for each case).
When measuring the drag at mean velocities of 6, 8 and 10 ms −1 , the load-cell data from the wind tunnel balance consisted of sets of 4.096 × 10 4 samples, recorded at a rate of 400 Hz, for a total sampling time of 1.7 min to ensure statistical convergence of the data.When the mean velocity was 4 ms −1 , 1.2288 × 10 5 samples were recorded at a sampling rate of 400 Hz, for a total sampling time of 5.1 min (This longer sampling time was required, given the lower vortex shedding frequency at this lowest velocity, and the corresponding longer period of oscillation of the flow in the cylinder's wake.)Before a typical series of tests, the mean free-stream velocity was measured with the hotwire at the tunnel centerline, upstream of the cylinder.It was then removed and the voltages from the wind tunnel balance load cells were recorded.The hot-wire was then used again to take a final velocity measurement at the tunnel centerline.Using the average of the two velocity measurements, and the average of the drag forces measured with the three load cells, the drag coefficient was computed, after applying the method of Dalton (1971) to correct for blockage effects.
Some experiments were undertaken with a smooth aluminum cylinder, of 1" (0.0254 m) diameter and 22" (0.559 m) total length, to (i) benchmark our measurements, and (ii) serve as a basis for comparison for the measurements undertaken with cylinders with porous outer layers.However, the majority of the experiments were done with the cylinders with porous outer layers, which were made by wrapping different wire meshes around solid aluminum cylinders; the outer diameter of the porous layer, in all cases, was 1 ± 0.05" (0.0254 ± 0.00127 m).To achieve porous layer thicknesses (PLTs) of 1 /4" (0.0064 m), 1 /8" (0.0032 m), and 1 /16" (0.0016 m), the upper 14" (0.356 m) lengths of the solid cylinders were machined to diameters of 1 /2" (0.0127 m), 3 /4" (0.0191 m), and 7 /8" (0.0222 m), respectively.Three different plainweave square steel wire meshes, purchased from McMaster-Carr (9219T176, 9219T199 and 9219T956), were used.Their properties are summarized in table 1.These properties can be used to estimate other related properties of the wire meshes and the porous outer layers they were used to make employing expressions proposed by Zhao et al (2013), for example.
The cylinders with outer porous layers were constructed by (i) cutting the wire mesh to the appropriate length and width, (ii) gluing a thin strip of the mesh to the solid aluminum cylinders, (iii) tightly wrapping the mesh around the cylinder until the desired diameter was reached, and then iv) gluing the outermost layer of the mesh to the one below it, at about the same circumferential location at which the other end of the mesh was glued to the cylinder.Full details of the procedure for fabrication of the cylinders with porous outer layers, along with gluing and wrapping jigs that were designed to aid in their construction, are given in Cohen (2019).A total of nine cylinders with porous outer layers were made, corresponding to the three different wire meshes and three different porous-layer thicknesses mentioned above.Photographs of three of the cylinders with porous outer layers are presented in figure 1.
To minimize end effects on the flow over the cylinder, rectangular end plates were installed near the top and bottom of the cylinder adjacent to the walls of the wind tunnel.Following the recommendations of Stansby (1974), the end-plate dimensions were 7D cyl wide in the crossstream direction, and 8D cyl long in the stream-wise direction, with the longitudinal axis of the cylinder located 3.5D cyl from their upstream edge, and 4.5D cyl from their downstream edge.The corners of the end plates were rounded to minimize the formation of tip vortices.
The end plates were placed outside the boundary layers on the wind-tunnel walls, 2" (0.0508 m) from the bottom wall and 2.75" (0.0699 m) from the top wall.(Note that the top wall of the wind-tunnel test section diverges to ensure a constant mean velocity in the tunnel.)Following the design of Klausmann and Ruck (2017), 1.66" (0.0421 m) diameter PVC pipes (schedule 80, 1 1 /4" nominal) were installed between the end plates and the tunnel walls to prevent flow over the cylinder in these regions.The resulting length of the cylinder between the two end plates (i.e. the length of the cylinder exposed to air flow in the wind tunnel) was 11.75" (0.298 m).
Each cylinder was mounted on the wind tunnel balance (which was installed below the wind tunnel) by way of an aluminum block fastened to the bottom of the cylinder.When installed in the wind tunnel, the 'seam' on the outermost porous layer was aligned with the rear stagnation point of the cylinder to minimize any effect it might have on the flow over the cylinder.The aluminum block was fastened to a bracket at the top of the wind tunnel balance.A schematic of the wind tunnel and a photograph of a cylinder mounted in the tunnel in this manner are presented in figure 2.
An uncertainty analysis of the velocity and force measurements in this work is given in Cohen ( 2019) and summarized herein.The uncertainty analysis of the hot-wire anemometry measurements, which accounted for uncertainty in the (i) calibration equipment (a TSI model 1128B calibration jet in conjunction with a MKS 220D pressure transducer), (ii) curve-fit to the calibration data, (iii) variations in ambient air temperature, and (iv) data acquisition board, resulted in a total, relative uncertainty of ±1.3%.(The uncertainty arising from changes in air pressure, humidity and angular position of the hot-wire probe were assumed to be negligible.)The uncertainty in the force measurements was attributed to uncertainty in the (i) load cells, (ii) calibration weights, (iii) data acquisition board, and (iv) curve-fit to the calibration data.The total, relative uncertainty in the force measurements arising from the aforementioned sources was found to be ±0.59%.The relative uncertainty in calculated quantities were as follows.Given the measured maximum relative uncertainty in the cylinder's diameter (±0.60%) and length (±0.15%), and uncertainty in the density and kinematic viscosity of the air being solely attributed to the uncertainty in the measurements of the ambient air temperature (±0.32%), the total relative uncertainties in the measured Reynolds number, drag coefficient, and turbulence intensity were estimated to be ±1.5%,±2.1% and ±1.9%, respectively.The repeatability of our measurements will be addressed in section 5.

Experimental conditions
Although the wind tunnel in which the present work was undertaken has been used in multiple investigations over the last four decades, this was the first one in its new location (and orientation).A full characterization of the flow in the wind tunnel in its new location, without and with the three turbulence generating grids (and without any cylinders being installed) is given in Cohen (2019) and Blais (2021).The results are consistent with prior experiments undertaken in this wind tunnel Warhaft (1980Warhaft ( , 1981Warhaft ( , 1984)), Sirivat and Warhaft (1983), Mydlarski and Warhaft (1996), Ayyalasomayajula and Warhaft (2006), Gylfason and Warhaft (2009).A summary of the flow conditions in which the present experiments were undertaken is provided below.
Properties of the flow (including the turbulence intensities, integral length scales, and other turbulence statistics) were measured in the wind tunnel (without the cylinder) at x = 2.65 m (see figure 2(a)), just downstream of where the cylinder would be located in the wind tunnel (x cyl = 2.57 m).The results are summarized in table 2 for nominal free-stream velocities of 4, 6, 8 and 10 ms −1 .As expected, the turbulence generated with the active grid exhibits significantly higher values of the turbulence intensity.
Moreover, the uniformity of the flow between the two cylinder end plates was measured (without the cylinder being installed in the tunnel, and with the holes in the endplates being sealed).Mean velocities were measured over this 11.75" (0.298 m) span, at the four nominal free-stream velocities (4, 6, 8 and 10 ms −1 ) and found to deviate from the average by at most 2.9%, 4.8%, 6.1% and 2.1% for the different levels of background turbulence generated in the tunnel with no grid, the passive 1" grid, the passive 2" grid and the active grid, respectively.

Results
This section is devoted to summarizing the results of experiments quantifying the drag on cylinders, with and without porous outer layers.To this end, the first subsection presents the drag coefficient for a smooth cylinder at different Reynolds numbers, and compares them to values from the literature, for the purpose of validating our approach.The second subsection quantifies the drag coefficient for a smooth cylinder in flows of different free-stream turbulence intensities.In the third and final subsection, the drag on cylinders with porous outer layers, including the effects of the Reynolds number, free-stream turbulence intensity, porous layer thickness, and pores per (cylinder) diameter (PPD) of the wire meshes (used for creating the porous layer), are summarized.

Drag on smooth cylinders: validation experiments
To benchmark the experimental apparatus, the drag coefficient on a smooth cylinder was determined using measurements in the wind tunnel with no grid installed (T i = 0.4%), at nominal mean velocities of 4, 6, 8 and 10 ms −1 (7000 ≲ Re D ≲ 17 000).This range of mean velocity was selected to obtain Reynolds numbers similar to those experienced by cylindrical firebrands being transported in the atmosphere.The results are plotted in figure 3 and also tabulated in table 3.As mentioned in section 2.1, the drag coefficients for circular cylinders in cross-flow depend upon the aspect ratio, cylinder end conditions, blockage effects, and free-stream turbulence intensity.The present results are in general agreement with prior results obtained under similar conditions.Specifically, the data agree with those of Delany and Sorensen (1953) (±5%), who had similar cylinder end conditions as the present work, though with a higher aspect ratio (L/D = 28, compared to 12 for this work).Since the drag coefficient increases with aspect ratio, this may explain why their results are 3-5% higher than the current results.The present results are also in agreement with those of Okamoto and Yagita (1973), whose cylinders had a similar aspect ratio, but different end conditions (one free end and one end plate, as opposed to the two end plates used in this work).The results of Wieselsberger (1921) and Norberg (1987) are significantly higher than the current results, likely due to the substantially higher aspect ratios (>280 and 80, respectively) in their experiments, as well as the lack of a correction for blockage effects, which cause the drag coefficient to decrease.Note, however, that the drag coefficient determined in the current experiments increased slightly with Reynolds number-a trend also observed by both Wieselsberger (1921) and Norberg (1987) in the range 3000 ≲ Re D ≲ 20 000.From these results, we can conclude that our measurements of C D for smooth cylinders in flows with low levels of background turbulence are consistent with prior measurements.

Drag on smooth cylinders: effects of free-stream turbulence
The drag coefficient for a smooth cylinder was further determined in flows with three different grids installed in the wind tunnel, for the same ranges of mean velocity, and compared with results at T i = 0.4%.The results are summarized in figure 4. At relatively low turbulence intensities (T i = 0.4%, 1.7% and 2.7%), the drag coefficient increases with increasing turbulence intensity (i.e.C D,0.4% ≲ C D,1.7% ≲ C D,2.7% ).However, at high turbulence intensity (T i = 12.4%), the drag coefficient is reduced, attaining similar values as those for T i = 0.4%.As will be demonstrated in the next section, the same trend was also observed for the cylinders with porous outer layers, i.e. the drag coefficient increases with turbulence intensity at relatively low turbulence intensities, then decreases at high turbulence intensity.A possible explanation for this behavior will be presented later.
To determine the repeatability of the results, experiments for obtaining the drag coefficient for a smooth cylinder at the four nominal mean velocities studied herein (4, 6, 8 and 10 ms −1 ), which correspond to nominal cylinder Reynolds numbers (Re D ) of 7 × 10 3 , 10 × 10 3 , 13 × 10 3 and 17 × 10 3 , were repeated for the following cases: (i) no grid (T i = 0.4%), 5 runs; (ii) 1" passive grid (T i = 1.7%), 4 runs; and (iii) 2" passive grid (T i = 2.7%), 4 runs.The results of the repeatability tests are presented in figure 5, and the mean values (C D ) and standard deviations (σ CD ) are summarized in table 4. The results showed excellent repeatability in all cases, with a standard deviation of the drag coefficient of 0.01 − 0.03, and a maximum relative uncertainty of 3.1%, where the relative uncertainty is taken as the standard deviation divided by the value of the drag coefficient.This is slightly higher than the relative uncertainty of 2.1% presented in section 3, likely because this value was computed by combining relative uncertainties of different measured parameters.

Drag on cylinders with porous outer layers
Nine cylinders with porous outer layers were used in these experiments.They correspond to the various combinations of cylinders with (i) three different values of outer-layer porosity  ) and standard deviations (σ CD ) of the measured drag coefficient for repeated runs for a smooth cylinder for the cases of no grid (T i = 0.4%), a 1" passive grid (T i = 1.7%), and a 2" passive grid (T i = 2.7%).

Nominal
Nominal No Grid 1" Passive Grid 2" Passive Grid Speed Reynolds No. (quantified by the pores per diameter (PPD) of the mesh used to produce it-see table 1), and (ii) three different PLTs.These experiments were conducted at the four different nominal mean velocities and the four turbulence intensities mentioned earlier.Given that the drag on a smooth cylinder was also measured at the four different mean velocities with four different levels of background turbulence, a total of 160 (= (9 + 1) × 4 × 4) cases were investigated.Note that the actual Reynolds number differed slightly for each case for similar nominal velocities because (i) there was some inevitable minor variability in the values of mean velocity, even though an effort was made to get them as close as possible to the nominal values of 4, 6, 8 and 10 ms −1 ; and (ii) each cylinder with a porous outer layer has a slightly different mean diameter, as previously noted.However, the actual values of Re D are used when plotting the data, even if referred to in the text by their nominal values.
In the following subsections, the results are presented in a way that highlights the effects on the drag coefficient of the (i) Reynolds number, (ii) free-stream turbulence intensity, (iii) PLT, and (iv) PPD.

Effects of Reynolds number.
The effects of the Reynolds number on the drag coefficients for the cylinders with porous outer layers are summarized in figure 6.Firstly, it can be observed that for a turbulence intensity of 0.4%, the drag coefficient increases with Reynolds number, as in the case for a smooth cylinder.On the other hand, at a turbulence intensity of 12.4%, the drag coefficient is relatively independent of Reynolds number.At turbulence intensities of 1.7% and 2.7%, the relationship between C D and Re D appears to be affected by the PPD: at 10 PPD, the drag coefficient is relatively invariant to changes in Reynolds number; while at 20 PPD and 40 PPD, the drag coefficient increases with Reynolds number.
As was mentioned earlier, the results of Wieselsberger (1921) and Norberg (1987) (for smooth cylinders) exhibit a drag coefficient that increases with the Reynolds number in the range 3000 ≲ Re D ≲ 20 000, then remains relatively flat in the range 20 000 ≲ Re D ≲ 10 5 .Also, as was mentioned in section 2.2, experiments by Fage and Warsap (1929) (later validated by other researchers) show that as the free-stream turbulence intensity increases, the critical Reynolds number decreases, causing a shift of the C D versus Re D curve to the left.This shift towards lower Reynolds numbers presumably explains why the slope of C D versus Re D curves tends towards zero with increasing turbulence intensity.
Regarding the aforementioned tendency for C D to become independent of Re D for the cylinders with the (most coarse) 10 PPD porous layers, we hypothesize that is because of their higher surface roughness, which results in higher turbulence intensities near the surface of the cylinder, thus having similar effects on the flow in the boundary layer (over the portion of the cylinder surface on which the flow remains attached) to those caused by free-stream turbulence in flows over smooth cylinders.These observations may indicate that phenomena tending to promote turbulence in the boundary layer (i.e. higher Reynolds numbers, free-streams with high levels of turbulence, porous/rough surfaces of the cylinder) may render the drag coefficient less sensitive to changes in the Reynolds number.These notions will be further developed in the subsections that follow.

Effects of free-stream turbulence.
The effects of the free-stream turbulence intensity on the drag coefficients for cylinders with porous outer layers are summarized in figure 7.As in the case for a smooth cylinder, the drag coefficient increases with increasing turbulence intensity for relatively low turbulence intensities (0.4%, 1.7% and 2.7%), then exhibits a decrease at the highest turbulence intensity (12.4%).It is worth noting that the magnitude of the drop in C D is a function of the roughness (i.e.PPD) of the porous layer: for 10 PPD, C D,0.4% ≲ C D,12.4% ≈ C D,1.7% ≲ C D,2.7% ; for 20 PPD, C D,0.4% ≲ C D,12.4% ≲ C D,1.7% ≲ C D,2.7% ; and for 40 PPD, C D,12.4% ≲ C D,0.4% ≲ C D,1.7% ≲ C D,2.7% .Recall from the discussions in the previous section that C D,12.4% ≈ C D,0.4% ≲ C D,1.7% ≲ C D,2.7% for a smooth cylinder.Thus, the drop in C D at the highest turbulence intensities becomes generally less pronounced as the porous layer roughness increases.
To explain these observations, it is hypothesized that, at the relatively low turbulence intensities (0.4%-2.7%), an increase in the turbulence intensity increases the friction drag on the cylinders without reducing the form drag, causing a net increase in the drag coefficient.A similar result was obtained by Norberg (1987), who observed an increase in the drag coefficient between turbulence intensities of 0.1% and 1.4%.However, as was noted in section 2.1, there is wide scatter in published results on the effect of the turbulence intensity on the drag coefficient in the subcritical range.For high turbulence intensities, the friction drag would continue to increase as in the other cases; however, this effect on the friction drag is 'overpowered' by a decrease in form drag, perhaps due to the displacement of the separation point of the boundary layer along the cylinder, away from the front stagnation point, potentially associated with a transition to turbulent flow in the boundary layer, resulting from the intense free-stream turbulence, and a narrowing of the wake.This phenomenon appears to be less pronounced as the PPD value decreases (i.e.roughness increases) -perhaps due to a concomitant increase in friction drag.

Effects of porous-layer thickness.
The effects of the porous-layer thickness (PLT) on the drag coefficient are quantified in figure 8.Note that in this figure, a porous-layer thickness of zero corresponds to the smooth cylinder.From the results presented in this figure, two principal observations can be made: first, the drag coefficient appears to be a monotonically increasing function of the thickness of the porous layer; and second, the drag coefficient appears to asymptote to a constant value for PLT/D beyond 1 /8D, such that C D,1/8D ≈ C D,1/4D .As was mentioned in section 2.1, Klausmann and Ruck (2017) remarked that experiments on cylinders with porous outer layers exhibited an increase in the drag coefficient compared to smooth cylinders, though they did not publish any results to this effect.The first observation outlined above (C D,Porous > C D,Smooth ) is in agreement with these findings, as well as the simulations of Naito and Fukagata (2012).However, the present results contradict the numerical simulations by Bruneau and Mortazavi (2006) and Bhattacharyya and Singh (2011), which showed a decrease in the drag coefficient from that for a smooth cylinder to one with a porous outer layer.A possible explanation for this discrepancy may lie with the uncertainties in the models used to approximate the flow (assumed to be laminar) in the porous layer.For example, Bhattacharyya and Singh (2011) discretize the volume-averaged Darcy-Forchheimer equation to describe the laminar flow in the porous layer, without explicitly resolving the full details of the flow in the interstices of the porous layer, which would be prohibitively expensive, and effectively impossible due to the difficulty in specifying the exact topology of the porous media.
The observation that the drag coefficient increases with increasing PLT and asymptotes to a fixed value may be explained as follows: as the thickness of the porous layer is increased, more fluid is able to flow within the porous outer layer, causing increased drag, and therefore higher C D .However, past a certain thickness, the air flow does not penetrate the full porous layer, all the way to the solid inner cylinder and a further increase of the PLT has no effect.In these experiments, this critical thickness of the porous layer (PLT crit ) appears to occur somewhere in the range 1 /16 ⩽ PLT crit /D ⩽ 1 /8, which corresponds to values of PLT crit /D between 6% and 12%.
Figure 9. Effects of the pores per diameter (PPD) on the drag coefficient for cylinders with porous outer layers.Zero PPD corresponds to the smooth cylinder.

Effects of porosity of the porous outer layer.
The effects of the porosity of the outer layer on the drag coefficient (which are quantified by studying the dependence of C D on the PPD of the mesh used in the fabrication of the cylinders' porous outer layers) are summarized in figure 9. Note that in this figure, zero PPD corresponds to the smooth cylinder.For almost all cases at relatively low turbulent intensities (T i = 0.4%, 1.7% and 2.7%), the drag coefficient decreases from porous layers with 10 PPD to those with 20 PPD, and then increases for porous layers with 40 PPD, such that C D,20PPD ≲ C D,10PPD ≲ C D,40PPD .At the highest turbulent intensity (T i = 12.4%), the drag coefficient increases with decreasing PPD, such that To advance a physical explanation of these effects, further studies, with additional values of T i and PPD, are required.In this context, flow visualization, or near-cylinder velocity measurements (by particle image velocimetry, for example), could also be very useful, especially if such non-monotonic evolutions of C D can be related to a trade-off between increased friction drag due to an increasingly rough porous outer layers, and decreased form drag if/when the separation point moves towards the cylinder's rear stagnation point.

Further discussion
The main objective of this research was to investigate the effects of Reynolds number, freestream turbulence, porous outer-layer thickness, and outer-layer pore-size on the drag on cylinders.To complement the main results of this work, which were described in section 5, we present in this section a brief discussion of the flow in the turbulent wakes for some representative cases of the cylinders studied herein, to provide additional qualitative insights into the wakes downstream of cylinders with porous outer layers.
The effects of the porous layer thickness on the mean and rms longitudinal velocity profiles in the wake are first depicted in figures 10(a) and (b), respectively, for a representative case (10 PPD, T i = 0.4%).With respect to the mean velocity, one observes that the velocity deficit in the wake increases with increasing thickness of the porous layer.This observation is consistent with the general trend of increasing drag coefficient with increasing porous layer thickness, observed in figure 8, since the momentum deficit can be linked to the drag force on the cylinder 1 .In these results, we observe the tendency towards asymptotic velocity profiles for PLT/D ⩾ 1/8, as also observed in the measurements of the drag coefficient.
Root-mean-square velocity profiles are shown in figure 10(b) for the same (10 PPD, T i = 0.4%) case.Of particular interest are the observed bimodal rms velocity profiles for the PLT/D = 1/8 and 1/4 PLT, which are notably different from the unimodal ones observed for the smooth cylinder and that with the thinnest porous outer later (PLT/D = 1/16).This result it consistent with the work of Ozkan et al (2012), who studied the wakes of cylinders with porous outer layers in shallow flow.For smooth cylinders, it has been observed that the rms velocity profile is initially bimodal in the near wake (x/D ≲ 10), then becomes unimodal, then tends again to a bimodal profile in the far wake (x/D ≳ 100) (Antonia andMi 1998, Tang et al 2016).In addition, they showed that the rms velocities decreased in magnitude as x/D increases for smooth cylinders.However, in a comparison of wakes downstream of (smooth, solid) cylinders with those downstream of screens Antonia and Mi (1998), demonstrated that (i) the u rms profiles of screens increased in magnitude as x/D increased, and (ii) approached the self-similar state of (far) wakes of solid cylinders at smaller downstream distances than did the u rms profiles in the wakes of solid cylinders.These observations would imply that the 1 We note that the estimates of drag in section 5 are more reliable than those inferred from the velocity profiles measured downstream of the cylinders with porous outer layers, for two related reasons.Firstly, to measured the drag force on a cylinder as accurately as possible requires the use of end-plates, as discussed in section 3, to ensure the effects of wall boundary layers on the flow over the cylinder are minimized.However, in the region of the wake downstream of the end plates, the flow is no longer two-dimensional, on account of the said end plates.Secondly, a force balance measures the drag force averaged over the surface of the cylinder exposed to the flow, whereas such velocity profiles are only measured in one plane normal to the cylinder's axis.These are important effects that must be considered when studying the velocity field downstream of a cylinder that is also connected to a force balance.wakes of cylinders with thicker porous outer layers follow a similar trend to those of wakes of screens-perhaps due the fact their porous natures reduce the 'effective diameter' of the cylinders, and hence accelerate their evolution to a self-similar state, by achieving large values of (effective) x/D sooner.Moreover Ozgoren (2006), has demonstrated how the separation points of the flow over cylinders of different cross-sections can impact their wakes.Thus the effects of the porous outer layers on the separation of the boundary layer that were hypothesized in section 5 may play a similar role in the evolution of the wakes studied herein.
Figure 11 plots the one-dimensional power spectral densities (i.e.power spectra) of the longitudinal velocity fluctuations in the wake of the cylinders for the four levels of turbulence intensity of the free-stream flow for the 10 PPD, PLT/D = 1/8 case.Note that the spectra are plotted dimensionally, to demonstrate the collapse of the data at small scales for all cases.Were these spectra plotted non-dimensionally using Kolmogorov non-dimensionalizations  (Kolmogorov 1941, Tennekes and Lumley 1972, Pope 2000), such a result would be expected.However, the observed collapse at small scales when plotted dimensionally indicates that the small-scale structure of the turbulent wakes is not excessively affected by the free-stream turbulence (for the ranges of parameters studied in this work).Figure 11(a) depicts spectra measured along the centerline of the wake (y/D = 0), whereas figure 11(b) depicts spectra measured at an off-axis location (y/D = 1.25).One first remarks that the spectra for three lowest turbulence intensities collapse over all scales, whereas the spectra for the case with the highest free-stream turbulence intensity (T i = 12.4%, generated by the active grid) diverges from the other three spectra at low frequencies / large scales.This indicates that the effects of the background turbulence on the wake dynamics are limited to the largest scales, and that the effect of a turbulent background flow of moderately high intensity is to increase the turbulent kinetic energy in the wake, even at its centerline, which would be the location in the wake least affected by the free-stream flow, given that it is the location farthest from the free stream.The power spectra measured away the centerline (at y/D = 1.25, and therefore closer to the freesteam) are similar to those measured at the centerline, with one notable difference.The spectra for three lowest turbulence intensities exhibit a clear spike in the spectrum at approximately 75-80 Hz, which roughly corresponds to a Strouhal number of St = f D/U ∞ ≈ 0.2, indicating that vortex shedding is present in these three cases (consistent with expectations for the wake of a cylinder at Re = 17000, (Lienhard 1966)).However, it bears noting that the signature spike associated with vortex shedding is absent for the case with the highest free-stream turbulence intensity.In this case, the action of the moderately-intense free-stream turbulence eliminates the periodic shedding of vortices (for the given flow parameters in this case).
Lastly, probability density functions (PDFs) of the longitudinal velocity fluctuations at the centerline of the wake are shown in figure 12 for the 20 PPD, PLT/D = 1/8 case, for the four different turbulence intensities of the free-stream.Figure 12(a) plots the PDFs in linearlinear form, in which the PDFs are found to be nearly Gaussian.Figure 12(b) plots the same four PDFs in log-linear form, to be able to clearly depict the tails of the PDFs.The PDFs for the three lowest turbulence intensities are similar and collapse very well.However, as best observed in figure 12(b), the PDF for the case with the highest free-stream turbulence intensity exhibits wider tails, indicting that the effect of the intense background turbulence is to increase the magnitude of the velocity fluctuations in the wake, even at the centerline.

Conclusions
The overarching objective of the present research is to improve the predictions of the atmospheric transport of burning debris such as branches (termed firebrands) that propagate forest fires, in a process called spotting.To this end, firebrands were modeled as cylinders with porous outer layers, to emulate the layer of char that forms on their outer surface.The present work is a first step in achieving this long-term objective, which involved wind tunnel measurements of drag on both smooth cylinders and cylinders with porous outer layers, and focusing on the effects of (i) Reynolds number, (ii) free-stream turbulence, (iii) porous layer thickness, and (iv) porosity of the porous outer layer.The measurements were undertaken in a wind tunnel, equipped with passive and active (and no) grids, to generate different levels of background turbulence, and a force balance.
For all cases, the porous outer layer resulted in higher drag coefficients when compared to smooth cylinders at the same Reynolds number.It was furthermore demonstrated that for lower free-stream turbulence intensities and porous outer layers made with larger PPD meshes, the drag coefficient of cylinders with porous outer layers generally demonstrated gradual increases with Reynolds number over the range of Re D studied herein, as is the case for smooth cylinders (over the same range of Re D ).However, at higher levels of turbulence intensity, the values of C D became relatively invariant to changes in Re D .The onset of independence of C D from variations in Re D was furthermore accelerated for cylinders with more porous (i.e.lower PPD) outer layers.This tendency to render the drag coefficient less sensitive to changes in the Reynolds number was associated with any changes that could enhance turbulence in the boundary layer on the portion of the cylinder surface over which the flow remains attached (i.e. higher Reynolds numbers, free-streams with high levels of turbulence, porous/rough surfaces of the cylinder).With respect to the magnitude of C D (as opposed to its evolution with Re D ), it was found that C D increased monotonically with porous-layer thickness, and appeared to asymptote to a constant value for porous-layer thicknesses beyond approximately 1 /8D (i.e. for porous-layer thicknesses greater than ≈12% of the cylinder diameter).It was hypothesized that beyond this thickness, the porous outer layer becomes effectively 'infinitely thick,' from the perspective of the fluid flowing over it.The dependence of the drag coefficient on the porosity was more complex and non-monotonic.At the lower three free-stream turbulence intensities, a local minimum in C D was observed for cylinders with 20 PPD porous layers.However, at the highest turbulence intensity, the drag coefficient was maximum for cylinders with 10 PPD porous layers, and decreased monotonically as the PPD increased.
From a fundamental physical perspective, further insight into the phenomena underlying the present experiments could be obtained by flow visualization, near-cylinder velocity measurements (by particle-image velocimetry, e.g.Xia et al 2018or Du et al 2022), or by direct or large-eddy numerical simulations.These may, for example, confirm the hypothesis that the observed drop in drag coefficient at large turbulence intensities results from a displacement of the separation point of the boundary layer towards the rear of the cylinder, potentially being associated with a transition to turbulent flow in the boundary layer over the cylinder surface upstream of the flow separation.From a practical perspective related to the long-term objective of this work, there would be a benefit to undertaking additional experiments studying the drag on objects of other geometries also having porous outer layers.A most logical choice in this context would be to quantify the drag on circular and elliptic disks, to model firebrands originating from burning leaves and/or pieces of bark.In the longer term, this work could be extended to burning wooden cylinders and disks, to be able to quantify the effects of the burning on the drag, surface temperature, and mass/volume-evolution of the firebrands.It is also worth noting that the work presented in this paper and extensions of it would be useful in other applications involving cross-flows over cylinders with porous outer layers made of wire meshes: examples include devices for filtration, catalytic converters, and solar energy collectors.

Appendix. Results in tabular form
The main results of section 5 that are depicted in figures 6-9 are tabulated below, in tables A1-A4.The density and viscosity of the air were assessed at the temperature and pressure during the experiments, which were 19 ± 1 • C and 101.3 kPa ±1.3 kPa, respectively.Note that the momentum thickness (δ 2 ) is related to the drag coefficient as follows: C D = 2 δ2 D (Schlichting 1979).

Figure 1 .
Figure 1.Cylinders with porous outer layers, including end views for three different thicknesses of the porous layer.

Figure 2 .
Figure 2. The wind tunnel with the cylinder.(a) Schematic of the wind tunnel in the horizontal plane (top view, not to scale), indicating the coordinate system used in the test section (the z-coordinate, normal to x, y, points vertically upward, starting at the centerline of the wind tunnel).(b) Mounting of a cylinder in the wind tunnel.

Figure 3 .
Figure 3. Drag coefficient as a function of Reynolds number for a smooth cylinder at T i = 0.4%, including results from other investigations.

Figure 4 .
Figure 4. Effects of the turbulence intensity on the drag coefficient for a smooth cylinder.

Figure 6 .
Figure 6.Effects of the Reynolds number on the drag coefficient for cylinders with porous outer layers.

Figure 7 .
Figure 7. Effects of the turbulence intensity on the drag coefficient for cylinders with porous outer layers.

Figure 8 .
Figure 8. Effects of the porous layer thickness on the drag coefficient for cylinders with porous outer layers.A porous layer thickness of zero corresponds to the smooth cylinder.

Figure 12 .
Figure 12.Probability density functions (PDFs) of the longitudinal velocity fluctuations in the wake of the cylinder with a porous outer layer in flows of varying turbulence intensity for the 20 PPD, PLT/D = 1/8 case.(x/D, y/D, z/D) = (20, 0, 0).Re = 17000.(a) linear-linear coordinates (b) log-linear coordinates (to better depict the tails of the PDF).

Table 1 .
Properties of the steel wire meshes used to make the porous outer layers.

Table 2 .
Flow parameters for the four different levels of background turbulence and four different nominal mean velocities studied herein.All statistics are measured at x = 2.65 m without the cylinder in place.The last two lines specify the coefficient (B) and decay exponent (n) of the background turbulence (⟨u 2 ⟩/U 2 = B(x/M) n ).ρ(τ ) ≡ u(t)u(t+τ )

Table 3 .
Tabular comparison of drag coefficients measured for smooth cylinders under similar conditions (i.e.free-stream turbulence intensities (T i ), aspect ratios (L/D), cylinder end conditions (see text), and blockage effects).An asterisk ( * ) denotes that blockage corrections were not applied.

Table 4 .
Mean (C D

Table A1 .
Data for smooth cylinder.