Estimation of the aerodynamic tortuosity of woven/wire screens

Estimation of the Aerodynamic Tortuosity of Woven/Wire Screens1 F.-J. Granados-Ortiza,∗, J. Ortega-Casanovab, A. Lopez-Martineza,c and U.S. Mahabaleshward 2 aDepartment of Engineering, University of Almer´ıa, Almer´ıa, Spain3 bDepartment of Mechanical, Thermal and Fluid Engineering, University of M´alaga4 C/ Dr Ortiz Ramos s/n, 29071 M´alaga, Spain5 cCIAIMBITAL Research Centre, University of Almer´ıa, Almer´ıa, Spain6 dDepartment of Mathematics, University of Davangere, Karnataka, India7 ∗Corresponding author: [email protected]8 Abstract9 The use of wire/woven screens (WSs) is frequent in applications such as particle or insect-proof10 screen in home/greenhouse/farm with natural ventilation. Although this passive element has11 been studied for decades, most previous works have focused on relating the airflow performance12 only to porosity. However, most recent investigations have demonstrated that other pore-related13 parameters such as constriction factor and tortuosity are necessary for the characterisation of14 screens. Tortuosity of WSs is a parameter that has been broadly estimated in the literature,15 whose calculations to date are not physics-based and yield a constant value without dependence16 on airflow velocity. The present investigation proposes a novel method to calculate tortuosity of17 WS. The new approach uses the flow potential flow theory to estimate realistic curvatures of the18 streamlines around the inclined threads of the WS. The calculated tortuosity has been made also19 velocity-dependent, because its value changes for Reynolds numbers below 200, generally. The20 accurate estimation of tortuosity is a very important contribution to the field, because it is a21 missing link to develop a universal model to estimate pressure drop for any WS performance. This22 calculation has been added to AeroScreen software, which allows to obtain porosity, constriction23 and tortuosity from geometry data.24 Keywords: woven screen, tortuosity, porosity, aerodynamic characterisation, potential flow25 theory26 1. INTRODUCTION27 Woven/Wire screens (WSs) are present in different productive sectors and have different ap-28 plications. It is common to use this type of screen in engineering applications to filter particles29 Preprint submitted to Elsevier November 24, 2023 or unwanted elements [1], as protection system in turbines [2], in fluid mixing processes [3], or to30 modify or control the level of turbulence of airflows [4]. However, amongst the different applica-31 tions in engineering, can be outlined their use as protection method par excellence to prevent the32 entry of particles or insects in natural ventilation. They are used in homes to protect humans from33 insects that transmit serious diseases such as malaria [5], in farms to prevent the passage of insects34 that can transmit diseases to animals [6] or in greenhouses [7] to prevent the entry of insects that35 seriously affect crops. Unfortunately, screens at ventilation openings drastically reduce the natural36 ventilation capacity of these [8], reducing the energy of the airflow when entering through vents [9].37 This reduction in the ventilation capacity has the consequence of increasing the temperature and38 humidity inside the building/greenhouse [10], which can be a serious drawback at certain times of39 the year. The effect of screens on airflow turbulence has been analysed in many previous works40 in the wind engineering literature. For instance, in [11] the turbulence management by means41 of different screen geometries is studied computationally via CFD simulations. Also, to mimic42 atmospheric turbulence conditions has been achieved in wind tunnels by screen grids, as seen in43 [12], and turbulence and wake management in wind turbine studies has been studied in [13].44 WS consists of two sets of intertwining weft and warp threads, perpendicular to each other (see45 Figure 1), thus forming a porous structure. This interlaced shape is complex and makes it difficult46 to perform a correct characterisation of screens. For this reason, in previous studies we developed47 methods to improve the characterisation of these woven structures geometrically. First, from a48 two-dimensional point of view [14], where it was developed a methodology to calculate (from digital49 microscope image processing for the identification of the vertices of pores) the separation of the50 threads in the x (weft) and y (warp) direction, Lpx and Lpy, the diameter of the threads in the x51 and y directions, Dhx and Dhy, and the two-dimensional porosity ϕ. Second, in [15] Alvarez and co-52 workers made an approximation to the three-dimensional area of the pore, being the first attempt in53 the literature. However, despite this method was suggested for improving the estimation of three-54 dimensional porosity, in reality this approach was based on planar properties. Thus, in [16] we55 introduced a mathematical method to computationally reconstruct the three-dimensional structure56 of WS meshes and calculate the exact three-dimensional porosity from two-dimensional parameters57 and the thickness, which was the first time a volumetric porosity was calculated analytically for58 WSs. We suggest the reader to see this work for better understanding on the geometric aspects of59 the WSs and how such parameters are related to each other. Finally, we recently suggested a more60 2 advanced method to calculate structural three-dimensional pore properties by a semi-analytical61 mathematical method [17]. This approach allows us to calculate the volumetric porosity and the62 constriction factor, which is a complex measure of how the cross-section of pores is constricted63 thus affecting the flow past screens. This is different to the tortuosity parameter, whose accurate64 calculation is the objective of the present paper, and which relates to the average elongation of65 streamlines in comparison to a straight streamline [18]. Figure 1: Example of wire/woven screen (WS). a) Microscope image of a plain square WS for a greenhouse, and b) 3D computational model of the WS. 66 One of the main interests in characterising WSs is to estimate the performance of flow-past-67 screens, as part of an aerodynamic characterisation (which includes wind aerodynamic loads [19]).68 Many investigations have been published in the literature, but there are limitations in most studies69 due to incomplete modelling. These studies are related to certain geometric characteristics, but70 two-dimensional geometric parameters are the standard (pore lengths, wire diameters or two-71 dimensional porosity parameters), but WSs are three-dimensional, as they have thickness. Different72 authors have obtained empirical models that attempt to estimate the aerodynamic properties73 such as pressure drop coefficient of WSs from a Reynolds number based on the diameter of the74 threads and two-dimensional porosity [20,21,22,23], although their accuracy is doubtful due to75 considerable modelling errors. These have been later used, for instance, to classify insect-proof76 screens [24]. Lu et al. [25] obtained a model to estimate the permeability of fabrics, whose77 discharge coefficient was calculated from the two-dimensional porosity. Various authors have also78 related the discharge coefficient to a Reynolds number based on the wetted perimeter of the orifice79 [26,27,28]. From these models, it is possible to estimate the natural ventilation capacity, and80 to integrate these models into greenhouse energy balance studies [29,30]. This emphasises the81 importance of having the best characterisation of the properties of the screens, in order to aim82 at better predictive models and simulations. When performing simulations, the characteristics83 of the WSs must be manually input to Computational Fluid Dynamic (CFD) softwares. CFD84 3 simulations allow, amongst many other applications, to study the natural ventilation patterns85 in buildings/greenhouses, and the microclimate conditioning. In these simulations, WSs are a86 boundary condition set as a thin porous surface, onto which the properties of the real screen are87 input [31,32,33,34,35]. To perform direct simulations of the pores for ventilation estimations88 within the large computational domain is not practical, as the pore holes have negligible size in89 comparison to the full room/greenhouse. For instance, in [31,36] the pressure jump (∆P) due90 to the presence of porous screens with square pores has been analysed via CFD and by including91 a model for the pressure jump, since an appropriate model does not require to model details92 of the geometry of any screen in simulations. It is thus of top importance to perform a realistic93 characterisation of the properties of screens for a reliable input in simulations (to avoid propagation94 of large errors).95 As aforementioned, pressure drop caused by a porous medium can be modelled. Specifically, it96 is well-known in mechanical sciences that can be modelled by the modified Darcy’s equation [37].97 When the flow passes through a WS, a pressure drop is produced. This is expressed as a function98 of the velocity of the air passing through the WS according to ∆P=a1U2+a2U[8,38,39]. In99 this equation, a1and a2are two modelling coefficients that depend on two important mechanical100 characteristics of the mesh: the permeability of the porous medium (Kp), which depends on the101 geometry of the porous medium; and the inertial factor (Y), which depends on the nature of the102 porous media [40,41].103 Despite of their extensive investigation over the years, these two parameters are still a matter of104 controversy, due to there is not a universal model for them. Researchers have developed empirical105 models over the years which do not share the same parameters. For example, for porous media,106 Nield and Bejan [41] presented two models that permit to estimate permeability (Kp) based on107 two-dimensional porosity and the diameter of the threads/wires. In addition, they estimated the108 inertial factor (Y), based on the diameter of the threads/wires and the diameter of the pores.109 Miguel et al. [42] developed models for these parameters based only on two-dimensional porosity.110 Much more recently, Lopez et al. [40] developed more advanced models that allow to estimate Kp 111 from the two-dimensional porosity and the diameter of the threads; and Yfrom the diameter of112 the threads and the diameter of the inner circumference of the pore. Wind tunnel measurements113 and CFD numerical modelling has been also a support in other investigations to achieve better114 characterisations [43] or to identify key parameters (pressure loss coefficient, drag coefficient and115 4 Reynolds number) in the flow-past-screen behaviour and in the exploration of scaling laws in wind116 tunnel experiments [44].117 The most recent trends in the mechanical/aerodynamic characterisation of screens are oriented118 to the estimation of new parameters that have been ignored in the classic literature. These param-119 eters are the constriction factor and tortuosity, as exposed in the pioneering work developed by120 Berg [45,46]. Although both parameters have been confused by some previous authors, they are121 clearly different, as outlined by Berg. The constriction factor is a parameter that quantifies how122 the constricting and expanding nature of a pore leads to variations in flow velocity [46] due to the123 conservation of mass (Navier-Stokes continuity equation). The origin of the tortuosity parameter124 can be found in the semi-empirical investigations developed by Kozeny [47,48] and Carman [49],125 who observed that linking microscopic fluid velocity to Darcy’s velocity in porous media involves126 the scaling with a factor, namely tortuosity τ. This scaling was later studied to find a proper127 modelling related to the fluid flow characteristics. Analogies with electric conductance and fluid128 flow were intended [50], to finally conclude that microscopic hydraulic conductance could be a129 good descriptor of fluid flow in porous media. Thus, (aerodynamic or hydraulic) tortuosity of a130 medium can be defined as the deviation from the straight pathline of a microscopic flow, which can131 be identified by the changes in length of streamlines [51,46]. Tortuosity is not a very popular term132 in fluid flow past screens, but it definitely is in general porous media literature [52,53,54,55,56],133 and of course in analogous electric conductance studies [45,46,57]. Whilst the constriction factor134 of woven screens has been explored in detail by the authors of the present paper (see our recent135 work in [17]), the accurate estimation of tortuosity of WS has been still unexplored.136 Regarding the types of tortuosity, previous literature has defined mainly three different groups:137 geometric, hydraulic and diffusional tortuosity [18]. Geometric tortuosity depends only on the138 geometry of the porous medium, whilst hydraulic tortuosity depends on other aspects such as fluid139 flow velocity (or mass-flow rate) through the porous medium, since the calculation is based on140 a ratio of path lengths of fluid flow (e.g. streamlines) over a straight path. On the other hand,141 diffusional tortuosity provides a measure on how diffusion in terms of capillarity evolves over a142 porous medium [18]. As aforementioned, analogies between electric and hydraulic tortuosity exist143 [50], and some authors even compared these two types of tortuosity in porous solids studies (aiming144 to use it in sedimentary rock studies) [58], showing that electric tortuosity (obtained by a measure145 of all current “streamlines” across each node along the medium) is lower than hydraulic tortuosity146 5 when compared for different porosities. Other types of tortuosity can be defined similarly to the147 abovementioned types, as thermal or acoustic tortuosity [18].148 In terms of how to calculate the path lines, there is some freedom as seen in the literature. As149 stated in [59], where several ways to estimate tortuosity are reviewed, to determine the tortuosity150 value is challenging. This is so mainly due to the difficulties in measuring the path lines, which151 are difficult to simulate and not measurable experimentally (in general).152 There are interesting methods in the literature to estimate tortuosity, specially when the porous153 medium has a complex porous structure, as for instance granular media. Amongst these methods,154 one can point out the popular Waterfall Algorithm [60], which intends to search for the shortest155 possible path throughout granular beds consisting of spherical particles. This method is appropri-156 ate for such porous medium and provides the lowest possible tortuosity, as there is no curvature by157 “adherence” once the granular object is surpassed so that the paths (homologous to streamlines158 in our work) are shortened. The estimation of the lowest possible tortuosity can be good as initial159 guess, but for WSs better options must be explored, as the structure is not composed by high160 density paths with cascade-like collisions but large size pores in which streamlines adapt to the161 thread surface due to Coanda effect. In [60], the Waterfall Algorithm was also compared to other162 methods such as the A-Star Algorithm, Path Searching Algorithm, Random Walk technique, and163 Path Tracking Method [61].164 In [59] it is shown that the Lattice Boltzmann Method (LBM) can be used to compute flow165 velocity, thus hydraulic tortuosity can be approximated by the ratio of the average magnitude of166 the intrinsic velocity over the entire volume and the velocity volumetric average along the flow167 direction, as originally introduced in [18]. Nevertheless, despite this is interesting, the calculation168 is not much different to a standard CFD simulation based on Finite Volume Methods (FVM),169 from which streamlines elongation can be directly measured. Finally, the only work related to170 calculation of tortuosity of WSs in the literature is Wang et al. [62], where a geometric-like171 tortuosity (not dependent on flow velocity) is calculated. In their work, streamlines are highly172 simplified by considering that their curvature is identical to the radii of the threads except at the173 central area (pore). Across the pore area the streamlines are considered as straight streamlines of174 the length of the thickness of the screen. This can be, hence, notably improved.175 The research gap addressed in the present manuscript is about providing a trustworthy method-176 ology to estimate tortuosity of the flow across woven screens by means of a physics-based method177 6 supported by the potential flow theory. Although there are many works in the literature that men-178 tioned tortuosity as an influential factor in pressure drop in screens (see, e.g. [63,64,65,66,67]),179 only Wang et al. [62] dared to provide an approximation to tortuosity, overcoming other vague180 estimations such as the one related to porosity only by Carman [68]. Other authors, even sug-181 gested that tortuosity can be considered constant and equal to unity for wired/woven screens [63],182 which is an inaccurate estimation since each mesh and Reynolds number has a different tortuosity,183 which (in addition to other parameters) finally affects to the way that pressure drop takes place184 [64,62,46], being pressure drop different for each mesh even at the same flow velocities [40]. The185 following sentence summarises the current state-of-the-art in wire/woven screens: despite these186 screens have been studied for decades, their characterisation is still dull and inaccurate, with con-187 troversy amongst publications (many publications in the literature omit a proper characterisation188 of screens such as constriction factor or tortuosity or they are based on 2D properties only). One189 of the aspects that lead to this scenario is that the mathematical modelling of wire/woven screens190 is not easy (as we pointed out in previous publications [16,17]). Actually, in [17], we suggested191 that researchers may have discarded to include mathematically complex parameters such as the192 constriction factor (whose rationale can be extended to the present manuscript on tortuosity) be-193 cause they have no tools or resources to obtain this parameter. The present work aims to go a step194 beyond and provides a physics-based approach to estimate the tortuosity of a wire screen. This195 parameter is added to the existing AeroScreen software, developed by the authors to democratise196 the use of our approaches in the characterisation of screens.197 This manuscript is structured as follows. Section 2provides a brief explanation of the method-198 ology and motivation of this work. Section 3shows the mathematical details of the proposed199 methods to estimate the tortuosity of WSs. These methodologies, as well as CFD simulations as200 support, are validated with data in Section 4. In Section 5tortuosity of signature WSs is investi-201 gated, including analytic calculations and CFD simulations. Finally, conclusions drawn from the202 present investigation are given in Section 6.203 2. METHODOLOGY & MOTIVATION204 As aforementioned, the recent advances made in Berg [46] and Wang et al. [62] are leading to205 the use of porosity, constriction factor and tortuosity as the most influential parameters to fully206 characterise screens. Whilst volumetric porosity and constriction factor have been analysed by the207 7 authors in previous publications [16,17], a reliable estimation of tortuosity is still missing in the208 literature, since only two broad approximations are available to date [62,68]. The methodology209 in Wang et al. [62] consisted on assuming that the streamlines are curved exactly of length equals210 to half of the circumference of the diameter of the thread for those streamlines over the thread211 surfaces, and assuming the rest with no curvature. However, this is not realistic; firstly, because212 the threads are not horizontal but inclined, thus the airflow is not passing through rounded shapes.213 And secondly, because it is well known in fluid dynamics that, in the absence of flow separation,214 the flow streamlines adapt to the shape of the solid body gently. Thus, since in WSs tortuosity215 can be either estimated as geometric (dependent on geometry only and focused on the shortest216 possible lengths) or hydraulic tortuosity (by considering the effective path lengths) [69], Wang et217 al. is an approximation to a geometric tortuosity, which is not accurate. For this reason, two218 methods are suggested to improve the estimation of tortuosity, which will be also universal to any219 plain square/rectangular woven screen.220 The first method consists of applying a correction to the estimation of tortuosity suggested by221 Wang et al. [62]. Whereas they considered that the flow is passing through a round horizontal222 cylinder (thread), we propose to correct the shape of the thread according to the real inclination223 in the interlaced geometry. That is to say, the shape is not fully round, but elliptic, committing a224 non-negligible error by considering it as fully circular. The streamlines around the threads will be225 considered as the arc of the elliptic cross-sectional shape.226 The second method attempts to go a step beyond and calculate the aerodynamic or hydraulic227 tortuosity. By means of the potential flow theory from fluid dynamics, a more realistic interpre-228 tation of the streamlines around the threads will be considered. As in the first suggested method,229 the inclination of the threads will have a role, and the airflow passes through elliptic (a specific230 oval shape) objects that represent the 2D cross-section of the threads/wires in the direction of231 the airflow stream. This elliptic shape will be reconstructed by the potential theory as a joint232 source and sink of intensity mlocated at certain distances −aand +a, respectively, to recreate233 virtually the oval object dependent on the inclination. The streamfunction will finally allow to234 obtain the mathematical expression of the streamlines, whose length can be calculated by integra-235 tion. The mount of threads due to the interlaced shapes will be also modelled by the presence of236 two cylinders in tandem. Thus, this novel approach will then provide streamline lengths that do237 not rely on simplifications but actual fluid mechanic analytical and formal expressions. A further238 8 velocity-corrected version will be studied by including a correction function to this estimation of239 tortuosity.240 3. MATHEMATICAL METHODS TO ESTIMATE TORTUOSITY OF WOVEN WIRE241 SCREENS242 The tortuosity parameter is related to the deviation of the flow streamlines when passing243 through the screen thickness [46]. As mentioned above, the estimation of tortuosity is one of the244 three most relevant parameters in the analysis of flow-past-screens. In this section, methods to245 estimate this parameter will be described, starting from the existing simplified approach in Wang246 et al. [62], and proposing two improvements to this method.247 3.1. Simplifications in the estimation of tortuosity due to flow across a woven wire screen248 To calculate or estimate the value of tortuosity in wire screens, one has to bear in mind first how249 these screens are designed. Wire screens consist of interlaced threads forming a woven structure,250 as represented in Figure 2. The interlacing of the threads may vary depending on the diameter251 of the threads. For instance, it is very common to see screens that are fully symmetric: the252 diameter of the xand y-threads (meaning by xand ythe direction of the threads) is the same,253 the spacing between these threads is the same too, and the thickness of the screen is two times254 the diameter. However, the geometry may get a bit more complex if the scenario is the opposite255 (different diameter of threads, rectangular pore cross-section, thickness larger than the sum of the256 xand y-thread diameters) [16].257 Figure 2: Example of wire woven mesh formed by threads/wires of the same diameter in all directions and square projected pore. Image from the CAD repository: [70]. 9 This theoretical approach was one of the most relevant applications in the development of aviation364 in the mid 20th century [72,73].365 The present problem under study can be divided into several parts in order to use the potential366 flow equations. Threads can be considered as cylindrical shapes, whose 2D potential flow can be367 modelled by means of a free stream and a source term. The source term represents the radial368 motion of fluid particles per unit length. Likewise, when the threads have certain inclination, the369 flow cannot be approximated as a simple source term, but a combination of a source term and a370 sink, because the cross-section of the cylinder with a plane in the direction of the free stream is not371 round but elliptic. Thus, it is more accurate to model this shape as a Rankine oval. For Rankine372 ovals, the potential function in cartesian coordinates is expressed as:373 ϕro(z,x) = Uz +m 2πlogp(z+a)2+x2−m 2πlogp(z−a)2+x2,(16) where Uis the free stream velocity, and mis the intensity of the sink (m < 0) or source (m > 0)374 located at z=−aand z=apositions, respectively (see Figure 6). By means of Equations375 (14)-(15), the velocity field due to velocity potential is obtained as376 uro(z,x) = ∂ϕ ∂z =U+m 2π z+a (z+a)2+x2−m 2π z−a (z−a)2+x2,(17) 377 vro(z,x) = ∂ϕ ∂x =m 2π x (z+a)2+x2−m 2π x (z−a)2+x2,(18) and the streamfunction:378 ψro(z,x) = Ux +m 2πtan−1x z+a−m 2πtan−1x z−a.(19) As the objective is to mimic the exact shape of the elliptic cross-section (a particularisation of the379 oval shape) of the wires/threads, the major axis (position of the stagnation points) and the height380 must be first obtained. Since the stagnation points located at z=Aand z=−Aare those with381 uro = 0 (the major axis is then 2A), one has just to solve uro(±A,0) = 0, which leads to:382 A=±rm a πU +a2.(20) Similarly, the height Bof the ellipse can be obtained at the position with vro = 0 and the stream-383 function ψro(z= 0, x =B) = 0, leading to the following equation:384 m aU 1 2−1 πtan−1B a−B a= 0,(21) 16 which must be solved by an iterative method. To find the solution to this equation, the initial guess385 is of high importance for a robust search. We suggest the initial guess m0= 10Band a0= 2A/3386 to solve the system of the two non-linear equations described above, so that the algorithm starts387 with a realistic initial value. Figure 6: Streamlines over a Rankine oval (blue lines) in the z-x plane with free stream velocity U. Only streamlines with x > 0 and outside the oval are shown. The red markers are the positions of the source (z=−a) and sink (z=a). 388 At this stage, the next step is to identify the elliptic geometry to be reproduced by the potential389 flow, in order to solve the system of non-linear equations formed by (20) and (21). The elliptic390 geometry depends on the inclination of each thread, as well as other geometric inputs. To this391 aim, the workflow depicted in Figure 7is followed. The process starts with the solution of the392 non-linear equations that model the interlacing of the threads, which are solved according to the393 geometric data of the WS (diameter of wires, spacing between wires, thickness of the screen and394 configuration 1 or 2 [16]). Once these equations are solved, the inclination angles are known, since395 the entire WS shape has been now reconstructed. Then, Equation (10) can be used to obtain396 the values of Aand Bthat identify the dimensions of the cross-section, and with this data and397 the value of the free stream velocity U, the sink/source intensity mand their position acan be398 obtained by solving the system of non-linear equations formed by Equation (20) and (21). With399 all this information, the streamlines around the elliptic shape can be generated.400 The determination of the streamlines length is also challenging. Streamlines are obtained from401 the streamfunction, since they represent lines of constant value of the streamfunction. Therefore,402 they are extracted from ψro(z,x) = k, with ka constant value of ψro. Nevertheless, because of403 the streamlines are implicit functions, it is not possible to obtain the streamlines lengths from the404 17 𝐷𝐷ℎ𝑥𝑥 𝐷𝐷ℎ𝑦𝑦 𝐿𝐿𝑝𝑝𝑥𝑥 𝐿𝐿𝑝𝑝𝑦𝑦 𝑒𝑒 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 Solution to the non-linear system of eqs. 𝜽𝜽𝒙𝒙 𝜽𝜽𝒚𝒚 𝐿𝐿𝑇𝑇𝑥𝑥 𝐿𝐿𝑇𝑇𝑦𝑦 ℎ𝑥𝑥 ℎ𝑦𝑦 Elliptic cross-section from angles 𝐴𝐴=𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑐𝑐 𝐵𝐵=𝑚𝑚𝑚𝑚𝑚𝑚 𝑧𝑧𝑐𝑐 --------------------- 𝑚𝑚and 𝑚𝑚estimated from solving the system of two Eqs (18) and (19) Streamline lengths from potential flow of the estimated cross-section Geometric inputs Geometric outputs Elliptic cross-section of an inclined thread/wire Figure 7: Workflow to obtain the streamline lengths of the potential flow around a Rankine oval from WS geometric inputs (measurements). The workflow starts with the input of the geometric parameters of the screen (diameter of wires [Dhx,Dhy], horizontal spacing between wires or rectangular pore dimensions [Lpx,Lpy], thickness of the screen eand configuration 1 or 2 [16]), then the system of equations that model the interlacing of threads/wires is solved to obtain the full geometric characterisation of the screen (geometric inputs plus inclination of threads [θx,θy], total length of threads [LT x,LT y], vertical spacing between threads [hx,hy]), as explained in [16,17]. Finally, from the geometric characterisation of the screen, the elliptic cross-section is estimated, and the potential flow theory around the Rankine oval can be applied to get the streamlines lengths around the object. derivative of the parametric coordinates (as shown in Equation (11)) nor from the derivative of an405 explicit function. Thus, the only possibility that we found was to compute the streamline lengths406 from the integral of differential portions of streamline. To this objective, we have estimated the407 length in the z-x plane as the sum of the square terms dS2=dz2+dx2. Then, the square root of408 this squared differential term has been integrated over the zand xdomain to obtain the length of409 each streamline (SK) by:410 SK=Zx x0Z−e/2 e/2 dSK=X jqdz2 K,j +dx2 K,j,(22) with the subscript Kdenoting each streamline from the streamfunction ψro(z,x) = k, and the411 limits of the integration in xwill be determined later according to Figure 10. The process is the412 same for the z-y plane, replacing xby y. It is also important to outline that, for a valid estimation413 of the streamlines, the streamlines inside of the elliptic shape must be deleted correctly, since only414 streamlines around the wires are used in the calculation. In order to avoid these, we have set the415 equation of the ellipse ( z2 A2+x2 B2≤1 or z2 A2+y2 B2≤1) as limiting value for the non-accountable416 streamlines. The results from the integral in (22) were validated with the length of straight lines,417 circumference and ellipse lengths, obtaining a perfect match.418 Once the generation of streamlines for the inclined cylinders has been explained (streamlines S2 419 18 and S3in Figure 4), the final step is the generation of streamlines for the corners of the screen pores420 (streamlines S1in Figure 4). These consist of a thread on top of an orthogonal thread (see Figure421 3or 4). Wang et al. [62] proposed that the flow on these corners has a curvature of the sum of the422 half circumferences of Dhx and Dhy. For this reason, a good approximation to this 3D shape in 2D423 can be assuming that the streamlines have a similar curvature to a free stream flow passing over424 two cylinders in tandem as shown in Figure 8(inclination is not relevant this time, since overlapped425 threads are mostly horizontal at the corners). It is obvious that this is just an approximation, but426 from 3D CFD simulations it has been observed that the curvature of the streamlines has no more427 than a 7% relative error difference, so it is useful as simplification. Unfortunately, this cannot be428 validated with experimental data, since there is no experimental data in the literature regarding429 streamline measurements in screens, but the overall performance of CFD simulations in this work430 has been validated with experimental data as will be shown in Section 4.2, so the conclusions from431 the present work are robust enough. In Figure 9can be seen the trajectory of two streamlines at432 the corner of the gauze nº3 wire screen in [74], where the x-thread (of diameter Dhx) mounts the433 y-thread (of diameter Dhy). The streamline that starts just at the corner can be seen to have a434 curvature around the x-thread perpendicular to the plane of view, and then it is curved to the right435 around the y-thread. This is nearly the same distance as if the two threads are in tandem. The436 length of the streamline in the validated CFD simulation is 2.38E−3 m, and from the method in437 Wang et al.[67] in the estimation of τ0, this is 2.93E−3 m. From our suggested method, the size of438 the same streamline from the potential flow approximation is 2.2E−3 m, which is much closer to439 the CFD data (a 7% of relative error with respect to the CFD results, opposite to a 23% of relative440 error from Wang et al. approximation). Moreover, in our approach, the intensity of the curvature441 of the streamlines is dependent on the position, whilst in the approach by Wang et al. it is always442 maximum (half of circumference). Thus, the accuracy in the estimation is notably increased. The443 difference between our estimation and the CFD simulation is mainly because in the 3D simulation,444 curvature over the thread below starts almost at the central axis of such thread, thus the streamline445 length is a bit longer than in our estimation, as in Figure 8can be seen that the curvature over the446 second cylinder does not start close to the central axis x= 0. Nevertheless, the approximation is447 still quite good. In addition, although not very dramatic, the inclination of the inclined cylinder448 nearby also contributes to lengthen the streamlines starting at the top of the toroidal part slightly449 (see the second streamline in Figure 9closer to the cylindrical part of the thread above). In any450 19 case, the relative error between our estimation and CFD simulations here discussed shows accuracy,451 despite we offer a universal physics-based low-cost approach approximation that does not require452 any CFD/experimental case-by-case testing.453 Figure 8: Streamlines over two cylinders in tandem (blue lines) in the z-x plane with free stream velocity U. Only streamlines with x > 0 and outside the oval are shown. Figure 9: Two corner streamlines from the CFD simulation of gauze nº3 from [74]. The first streamline (from left to right) starts right at the corner and envelopes both threads. The second streamline starts at a position closer to the inclined part of the upper thread, so that the curvature around the bottom thread is more gentle. Finally, the streamfunction of a potential flow around two cylinders in tandem in the z-x plane454 20 is modelled as:455 ψ2c(z,x) = Ux +UR2x (z−d)2+x2+(R1/R2)2x (z+d)2+x2,(23) with R1and R2the radii of the first and second cylinders (to be substituted by Rhx =Dhx/2456 or Rhy =Dhy/2, depending on which is the one on top and bottom), respectively, and 2dis the457 distance between the centrelines of the cylinders. It is obvious that, in order to keep the cylinders458 pulled up, dmust satisfy:459 d=R1+R2 2.(24) Similarly to the Rankine oval case scenario, the streamlines can be represented for ψ2c(z,x) = k,460 as shown in Figure 8.461 The calculation of the streamline lengths is done following the steps described for the Rankine462 oval but for two cylinders in tandem. Once the streamline lengths are known, the surface-averaged463 integration has to be calculated. For this, the area of influence of each potential flow streamline464 has been selected according to the percentage of each thread over the total projected area of the465 pore. That is to say, it has been selected as the 2D proportional part of each thread area over the466 total pore opening of area that covers (Lpx +Dhy)×(Lpy +Dhx). This is shown in Figure 10. The467 areas are identified as follows, according to the sketch depicted in Figure 10:468 Ax=ξx[(Lpx +Dhy)−2ξy], Ay=ξy[(Lpy +Dhx)−2ξx], Ax,y =Ay,x =ξxξy, Ac= [(Lpx +Dhy)−2ξy][(Lpy +Dhx)−2ξx], (25) where the subscripts xand yrefer to the xand y-thread/wire, respectively; the subscript x,y469 refers to the area where the x-thread is over the y-thread, and the subscript y,x viceversa. The ξx 470 and ξyterms represent the extension of the integration of the streamlines for the xand y-threads,471 respectively, which is calculated as the percentage of the thread over the perpendicular coordinate:472 ξx=Rhx + %txLpy =Rhx +Dhx Dhx +Lpy Lpy, ξy=Rhy + %tyLpx =Rhy +Dhy Dhy +Lpx Lpx, (26) where Rhx =Dhx/2 and Rhy =Dhy/2. Finally, tortuosity is calculated as the ratio τ2=Seff,2/e,473 21 Figure 10: Area of influence of each potential flow streamline. The subscripts refer to the thread (x-thread or y-thread), and Astands for the area across which the surface-average effective streamline length will be estimated. E.g., Axis the area of influence which the streamlines Sx(streamlines around the x-thread) go through, in order to perform the calculation given in Equation (27). The transition from Sxto the non-curved streamlines through Acis quite smooth, since the closer to the central part of the pore, the less curved the streamlines are. where the surface-averaged values of the streamlines length are calculated as474 Seff,2=1 At ZZAx SxdA +ZZAy SydA +ZZAx,y Sx,ydA +ZZAy,x Sy,xdA +ZZAc ScdA!,(27) where Sxand Syare the streamlines around the xand y-threads using the potential flow theory475 around the Rankine ovals (see Equations (16)-(21)); and Sx,y and Sy,x are the streamlines of the476 x-thread over the y-thread and vice versa, obtained according to the potential flow theory of two477 cylinders in tandem (see Equation (23)). Scare non-curved (straight) streamlines of length eat478 the central part of the pore.479 The main advantage of this novel approach is that now tortuosity of WSs is, for the first time,480 based on a physics-based method and not a mere approximation related to their geometry. This481 method to estimate tortuosity has been implemented by code in AeroScreen software [75], to obtain482 this parameter instantly by any practitioner.483 22 2.8% 1.3% 0.75% 0.5% Figure 11: Grid convergence of the 3D CFD simulation of the pressure gradient for different dimensionless grid element sizes of a wire screen with ρt= 9 ×9 wires/inch2(namely gauze nº3 in [74]). Grid made dimensionless with the diameter of threads D. The limit value of the pressure gradient without discretisation error when ds →0 (solid red square) has been predicted by a quadratic extrapolation. 4. VALIDATION484 Unfortunately, there is no experimental data of concretely tortuosity (nor streamline lengths)485 of WSs, since, as said in the introduction, only recent works have highlighted this parameter in486 the characterisation of screens. However, we have found both experimental and numerical data of487 the pressure gradient through certain representative WSs, which have been used to validate the488 computations. For these reasons, a total of three WSs from [74] (with density of threads ρt= 6×6,489 9×9, and 14 ×14 wires/inch2) with three different airflow velocities (9 simulations in total) have490 been simulated via CFD, in order to validate a CFD model with the numerical and experimental491 results from the said reference work [74].492 Finally, in order to be confident with the methodology and CFD set-up to be applied in all493 simulations, firstly, a grid convergence study was carried out to select the optimal mesh for all494 computations.495 4.1. CFD grid convergence study496 Four different grids with four different dimensionless element sizes have been simulated (element497 sizes are made dimensionless with the diameter Dof the threads, as it is the same diameter for498 all threads). By using the coarsest one as reference (with size ds4≈0.1), the successive levels of499 23 refinement were done by dividing the face element size by a constant factor of r= 1.3 by following500 the approach in [76]. E.g. the next grid (a finer one) had a discretisation size of ds3=ds4/r,501 and so on. This means that the finest grid has approximately 11.2M of cells. In Figure 11 it is502 illustrated the pressure gradient through the screen ρt= 9 ×9 (namely gauze nº3 in [74]) with503 the different computational grid sizes and an inlet flow velocity of 1.5 m/s. The limit value of504 the pressure gradient without discretisation error when ds →0 has been predicted by a quadratic505 extrapolation in Figure 11, as shown by a solid red square. The percentage values in the plot show506 the relative error of every value with respect to the limit value. Finally, it was selected the medium507 grid size ds2=ds4/r2for our CFD simulations, with approximately 8.7M of cells and with an error508 of just 0.75%. This discretisation error is very low, whilst the computational simulation elapsed509 time is acceptable. In Figure 12 it is shown the definitive computational mesh, which is used to510 solve the Navier-Stokes equations numerically. The figure shows that this optimal mesh around511 the WS fits very well the geometry. The equations are solved numerically in the present study512 by means of the finite volume software ANSYS Fluent, where the velocity-pressure coupling was513 solved by means of the SIMPLE (Semi-Implicit Method for Pressure-Linked Equations) algorithm514 [77]. The simulation was run until numerical convergence is achieved, by establishing residuals515 below 10−4and fluid properties constant when advancing the iterations. Additionally, spatial516 discretisation methods were second-order accurate for pressure and momentum, whereas the least517 square cell-based method was used for gradient discretisation. Since the objective of the present518 manuscript is to measure the elongation of the path lines (streamlines), these are shown in Figure519 13 by means of three different views. The curvature of the streamlines shows agreement with520 the assumptions made in Section 2regarding how streamline curvature decreases gradually when521 moving towards the centre of the pore. The validation of the computational simulation is detailed522 next.523 4.2. Validation of the CFD simulation with experimental and computational data524 In order to reproduce and validate the results presented in this study, the work done in [74] with525 three WS, namely gauze nº1, 3 and 5, were now simulated for three different inlet flow velocities526 (they do not report the Reynolds number but dimensional data throughout the manuscript). To527 be able to match the CFD simulations with the data from [74] makes our simulations specially528 trustworthy, since their work also includes heat transfer (heated wires). Since the authors did529 not provide quantitative information about the used boundary conditions, we have identified from530 24 (a) (b) Figure 12: General view (a) and zoom in (b) of the definitive generated mesh according to the grid convergence analysis. (a) Front view (b) Isometric view (c) Bottom view Figure 13: Pathlines through the geometry coloured by the velocity magnitude. The threads of the WS are shown in light grey. their data that the threads were at a constant temperature of 340K and inlet air at 300K. The531 physical properties of air (density, viscosity, thermal conductivity and specific heat) were configured532 as temperature dependent as the authors did in their work. This lack of information made the533 validation process complicated but, as shown in Figures 14 and 15, our computational data match534 very well their reported experimental and computational results for any of the three tested velocities535 25 of aerodynamic (hydraulic) tortuosity, and thus opens new possibilities to characterise screens652 by manufacturers and to obtain optimal designs in industry parametrically. Another important653 application is CFD simulation of large domains: to test e.g. the effect of a certain screen on654 natural ventilation in a building, the WS is a porous media which is input as boundary condition655 on a 2D surface (either pressure drop or permeability and inertial factor are the inputs, which are656 calculated from porosity, constriction factor and tortuosity).657 7. ACKNOWLEDGMENTS658 The first author acknowledges the Ram´on y Cajal 2021 Excellence Research Grant action659 from the Spanish Ministry of Science and Innovation (FSE/AGENCIA ESTATAL DE INVES-660 TIGACI´ ON), as well as the Andalusian Research, Development and Innovation Plan (PAIDI -661 Junta de Andalucia) postdoctorate fundings. 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