Computational modeling and experimental validation of the air permeability of woven structures on the basis of simulation of jet systems

Abstract

The paper presents computational fluid dynamics-based numerical simulation of the through-thickness air permeability of woven structures, applying the theory of jet systems. The flow through the interstices between the warp and weft threads is modeled as an “in-corridor”-ordered jet system, formed by nine jets, issuing from nine pores of the woven structure. Fifteen cases were simulated and three different turbulence models were applied in the simulation: k-ɛ, k-ω and Reynolds stress model. The five simulated woven structures were manufactured and their air permeability was measured experimentally. The performed validation of the numerical results with the experimental values of the air permeability showed very good correlation with the experimental results. The analysis and the verification showed that the method can be applied for further investigation not only of the woven fabrics’ air permeability, but also for investigation of the flow after a textile barrier of a woven type.

Keywords

air permeability numerical simulation computational fluid dynamics jet systems woven structures

The air permeability of woven structures can be easily measured when the fabrics exist. However, in the production of woven textiles with prescribed air permeability, computational fluid dynamics (CFD) modeling can be an effective tool during the design stage. The CFD-based simulation enables reliable assessment of the fabric permeability to be done, avoiding manufacturing of sets of samples and their experimental investigation.

Air permeability is mainly influenced by the porosity of the textile structures. In the particular case of woven fabrics the porosity is related to constructional and geometric characteristics, such as type of threads (filaments or spun yarns), pattern (single, double or complex structures), warp and weft density, thickness, etc. It determines the thermal insulation efficiency, thermal comfort of apparel, precision of the filter media, barrier fabric performance, drying abilities, etc.

There are several published studies, which deal with the problem of air permeability of woven fabrics. Many of them are based on established relationships between permeability of inter-yarn interstices and fabric structures: the Hagen–Poiseuille formula together with Darcy law was used by Kulichenko and Langenhove¹ to develop an equation for theoretical analysis of the dependence of the air permeability on the fabric structure. The Hagen–Poiseuille law was applied by Xu and Wang² as well, but the modeling was limited to the case of woven structures made of monofilaments. In addition, statistical methods³ and neuro nets⁴ were used to describe the dependence between the parameters of the woven structures and their air permeability. CFD codes were used as tools for simulation of through-thickness permeability. A commercial CFD software package⁵ was used to assess the fabric filtration ability in function of the shape of the particles, while Grouve et al.⁶ developed specialized CFD code for simulation of the permeability in the transversal direction.

A particular part of the reported studies is related to investigation of the porosity and description of the interrelation between porosity and structural parameters.^7–9 A geometrical model, which is based on linear density, weave factor and relative fabric density on one hand, and the porosity, on the other, was presented by Dubrovski.¹⁰ Although several authors considered the pores as having cylindrical shape with a constant diameter over all their length,^1,2,11,12 the pore size and shape, as well as the pore distribution, are completely uneven.^13,14 However, the application of CFD for air-permeability simulation requires the microstructure of the pores and their distribution in the woven structure to be described as close as possible to the real situation.¹⁵ Recent successful attempts in this direction are the prediction of the air permeability of plain woven fabrics with constant warp density and changeable weft settings on the basis of volume porosity¹⁶ and the study on high-density woven structures and their pore morphology.¹⁷

In our previous work, a new method for simulation of the air permeability of woven structures was introduced and tested.¹⁸ The method is based on an approximation of the air flow, passing through the woven fabric, as an “in-corridor”-ordered jet system. Different geometries (circular and square shape of the pores) and configurations (3 × 3 and 5 × 5 jets), as well as types of grids (coarse, fine and hybrid), were studied. The method, however, was tested with experimental data for jet systems, but not with real textile structures.

In the present study the through-thickness air permeability of five woven structures, used for the production of apparel and interior textiles, is investigated numerically. The woven samples are produced with different warp and weft densities and patterns so as to obtain structures from very open to tight ones. The air flow in the through-thickness direction of the sample is treated as a 3 × 3 “in-corridor”-ordered jet system, which includes four warp and four weft threads from each sample and nine pores all together. The modeling is based on Reynolds averaged Navier–Stokes (RANS) equations. Three different turbulence models are applied: k-ɛ, k-ω and Reynolds stress model (RSM). The numerical results are assessed on the basic of comparison with experimentally measured air permeability of the studied woven samples.

Theoretical background

The current study is based on systematic experimental research on jet systems^19–21 and experimental investigation of the pores’ morphology and distribution in woven structures,¹⁴ as well as a numerically tested method for CFD simulation of the air permeability, based on jet systems’ theory.¹⁸

Jet systems

The flow resulting from a jet system, is a complex three-dimensional (3D) flow. It is formed due to the interaction of turbulent flows (jets), issuing from openings (holes) with circular, square or rectangular shape of the cross-section. The single jets can be²⁰

in-line ordered (in a system of at least three jets);

in-corridor ordered (in a system of at least nine jets); or

chess-board ordered jets (in a system of at least seven jets).

Figure 1 shows an in-corridor ordered jets system with nine square openings, with a length of 2b₀. Basic parameters of the jet system are the steps S_y and S_z between the single free jets. The figure illustrates the development of the flow in the transversal direction of the openings and the gradually intermixing of the single jets into a resulting flow field.

Figure 1.

In-corridor ordered jet system.

The jets issue with initial velocity u₀ and temperature Т_0. The jets’ interaction starts after the initial region I, within the mixing region II, where the velocity field is characterized by U_min (minimal velocity) and U_m (maximal velocity) along a cross-section. The further development of the flow downstream transforms the velocity field into one typical for a single jet – region III.

The woven fabrics have strongly defined geometry of repeated alternation of threads and voids in the direction of both warp and weft threads. Therefore, the flow, passing in the transversal direction (through-thickness permeability), can be simulated as an “in-corridor”-ordered system of jets.

It was experimentally proven by Stankov²⁰ that in the case of an “in-corridor”-ordered jet system, every single jet is representative of each of them, if the jet is surrounded by eight other jets from the system. All surrounding jets influence the “central jet” and play the role of “boundary conditions” that influence the flow development from the “central jet”. Therefore, the common approach, used in modeling of free jets, namely to simulate quarter of a jet, is not appropriate for jet systems.

Figure 2 illustrates the approximation of the woven structure as a jet system and its virtual geometry.

Figure 2.

Jet system approach in modeling of the woven structure.

The following parameters are used for this approximation:

average diameter of warp yarns d_wp;

average diameter of weft yarns d_wf;

equivalent pore size $2 b_{0} = \sqrt{S_{m}}$ , where S_m is the mean pore area.

On this basis the main parameters of the jet system (see Figure 1) can be determined as

S_{y} = 2 b_{0} + d_{wp}

S_{z} = 2 b_{0} + d_{wf}

Shape of the pores

The shape of the pores in the case of monofilaments is closer to rectangular or square. In the case of multifilaments and staple fiber yarns, the pores have a shape closer to circular: the “corners” of the interstices become round due to the friction between the warp and weft threads.¹⁴

It was found in our previous work¹⁸ that there is no difference either in the numerical results, or in the required central processing unit (CPU) time in the simulation of the air flow, between jet systems with circular and square openings. Therefore, the choice of the shape of the pores would depend on the personal preference of the researcher and on the type of the grid used, which has to assure “high quality” of the control volumes, convergence promptness, etc.²¹

Grid type

Three types of grids, namely coarse, fine and hybrid grids, were tested in our preliminary work.¹⁸ It was found that the fine grid gives best results and it was selected for the present investigation as well.

Characteristics of the studied woven structures

The samples were manufactured especially for the purpose of the present study so as to cover a wide interval of volume porosity of their structure. The samples were woven, using 100% cotton staple fiber yarns and a Sulzer projectile loom. The basic characteristics of the used ring spun yarns are shown in Table 1.

Table 1.

Main characteristics of the ring spun yarns

Sample no	Linear density, tex	Twist, m^–1	Type
1	20	209	Ring, CO
2	28	157	Ring, CO
3	30	155	Ring, CO
4	36	152	Ring, CO

The gray fabrics were washed in a bath with 3 g/l 25% water solution of NaON as a finishing process and the final fabrics were experimentally investigated. Table 2 summarizes the parameters of the samples. The arrangement is done in accordance with volume porosity (sorted from the highest to the lowest value).

Table 2.

Parameters of the investigated woven structures

Sample	Warp density, [ends/dm	Weft density, picks/dm	Linear density warp, tex	Linear density weft, tex	Thickness, mm	Weave	Weight, g/m²	Volume porosity, %	Mean pore area S_m, mm²
1	176	124	28	28	0.45	plain	89	51.62	0.255
2	284	294	20	20	0.39	plain	133	29.07	0.038
3	234	254	30	30	0.47	plain	157	25.94	0.031
4	383	338	28	28	0.68	twill 2/1	185	10.51	0.014
5	386	286	30	36	0.71	twill 3/1	202	10.11	0.023

Microscopic pictures of the woven structures are shown in Figure 3. The pictures clearly demonstrate the difference in samples’ porosity and the tube-like structure of the interstices.

Figure 3.

Microscopic pictures of the woven structures: (a) Sample 1; (b) Sample 2; (c) Sample 3; (d) Sample 4; (e) Sample 5.

The thickness of the samples was measured in accordance with EN ISO 5084 and each sample was tested 10 times with a pressure of 10 cN/cm². The weight was measured after finishing: 10 measurements of 10 cm × 10 cm samples. The mean pore area was experimentally determined on the basis of an image analysis of microscopic pictures.¹⁴ One hundred pictures were taken with an Optika DM-15 microscope with a built-in digital camera (max resolution 1600 × 1200 pixels) under four times enlargement.

The five woven structures were tested on the Metefem FF-12 apparatus for air permeability.²² The tested area of the fabric was 10 cm². The suction device assured a constant pressure difference between the two sides of the sample of a 10 mm water column (100 Pa). Ten single measurements were made for each structure and the average value was calculated.

Numerical procedure

Due to the complexity of the woven fabric geometry, the following assumptions were accepted:

every simulated sample was presented as a system of 4 × 4 threads, the interconnection of which gave an “in-corridor”-ordered system of 3 × 3 jet flows;

the diameter of the yarn was accepted to be one and the same along the thread;

as the interweaving of the warp and weft threads had an influence on the thickness, an average value from the experimental measurement of the thickness was accepted as the length of the interstices;²³

the areas of the pores were approximated to an average value S_m for the whole sample (see Table 2);

the shape of the pores was approximated and modeled as square one.

The geometry of each of the woven structures was built using the GAMBIT Fluent pre-processor. The sample was immersed in a pipe-like domain, 3 mm after the domain inlet and 8 mm before the domain outlet, so as to assure the total flow formation after the sample. These dimensions were established on the basis of preliminary simulations, as a compromise between the required CPU time and the necessary flow development.¹⁸

A structured grid of hexahedrons was used, as depicted in Figure 4. Тhe numerical details for the meshing of each of the simulated samples are shown in Table 3.

Figure 4.

Meshing of the computational domain for Sample 2.

Table 3.

Numerical details of the meshing of the samples

Sample	Min volume of a control volume, m³	Max volume of a control volume, m³	Size of a control volume, m	Sample size, mm
1	2.3E-13	2.5E-13	6.2E-05	2.85/2.85/0.45
2	2.7E-13	2.8E-13	6.5E-05	1.75/1.75/0.3
3	2.0E-13	2.0E-13	5.9E-05	1.93/1.93/0.47
4	1.9E-13	2.4E-13	6.0E-05	2.4/2.4/0.68
5	1.3E-13	1.3E-13	5.0E-05	2.58/2.58/0.71

The Fluent 6.3.26 CFD commercial software package was used for the simulations. The RANS-based mathematical model was applied.

The turbulence was modeled by using three different turbulence models: k-ɛ and k-ω from the group of the Eddy Viscosity Models (EVMs) and the RSM.

The k-ɛ turbulence model adds two extra transport equations to the RANS equations for the kinetic energy k and its dissipation rate ɛ.

The k-ω turbulence model is also a two-equation model, where instead of the dissipation ɛ of the kinetic energy k, an equation for the specific dissipation ω is included.

The RSM is a higher level turbulence model, a Second-order Closure. It involves additional transport equations for the individual Reynolds stresses. It allows the Reynolds stresses to be directly predicted, when necessary, for more precise analysis of the flow in the region after the textile barriers.

Fifteen cases altogether were simulated numerically.

The solver was set-up as pressure based and implicit, over a 3D space. The pressure inlet was 200 Pa, while the pressure outlet was 100 Pa, which corresponds to 10 mm water column pressure difference between the two sides of the fabric.²⁴ The surrounding walls of the domain were set as symmetrical boundaries.

Computational results

To prove once again the modeling approach, based on jet systems, the flow field through the samples was simulated as two types of “in-corridor” jet systems: 3 × 3 and 9 × 9 (the k-ɛ turbulence model was applied). The simulation results for the velocity field for Sample 3 are shown in Figure 5 (3 × 3 jet system) and Figure 6 (9 × 9 jet system).

Figure 5.

Velocity magnitude visualization, Sample 3, 3 × 3 jet system.

Figure 6.

Velocity magnitude visualization, Sample 3, 9 × 9 jet system.

The comparison between the two figures clearly demonstrates that the velocity fields are similar and the flow development has the same character for the two cases: the velocity of the jets decreases fast after the sample and the resulting flow develops rapidly within the studied domain. As a result it can be concluded that the “3 × 3 case” is fully representative for a jet system with an infinite number of jets.

The flow development in the near region after the woven structure is presented in Figure 7. The velocity contours in a cross-section of the domain at a distance of x = 0.5-1-2-3-4-5 mm downstream of Sample 1 (Figure 7(a)) and Sample 3 (Figure 7(b)) are visualized. The comparison between the velocity contours shows that the flow has higher velocity when the pore area is bigger (Figure 7(a)). At the same time the velocity maximum slows down more quickly when the pore size is smaller, which corresponds to a lower porosity (Figure 7(b)).

Figure 7.

Velocity profiles, m/s: (a) Sample 1; (b) Sample 3, Reynolds stress model.

It can be seen as well that at a distance of x = 2 mm after Sample 1, the maximum velocity decreases by 26%, compared to the near sample region (x = 0.5), while the decrement of the maximum flow velocity after Sample 3 is 46.6%. The result confirms numerically the fact that tightly woven fabrics detain easier the air than woven structures with larger pores. Both graphs show that the flow is fully developed at a distance of 5 mm after the sample and the single jets from the jet system have transformed into a free jet.

The static pressure of the flow is visualized in Figure 8. A comparison is made between Sample 1, with highest volume porosity (51.62%, see Table 2) and the tightest structure – Sample 5 (10.11% volume porosity). The results show that the static pressure in the pores changes logically: it is higher in the case of lower porosity.

Figure 8.

Contours of the static pressure, Pa: (a) Sample 1; (b) Sample 5.

Experimental validation

The obtained numerical results for the flow rate through an area of 10 cm² were compared with the measured values of the air permeability of each sample. The flow rate through a single pore was predicted on the basis of the average velocity of the flow through a single pore and the averaged area of the single pore (see Table 2). The flow rate was calculated for all 15 cases of the CFD simulation, based on the different turbulence models. The results are shown in Figure 9.

Figure 9.

Comparison between numerical and experimental results for the air-permeability: volume flow rate through an area of 10 cm².

The relative error between the numerical and experimental results is calculated and the results are summarized in Table 4.

Table 4.

Relative error between numerical and experimental results

Sample no	Relative error, %
Sample no	k-ɛ	k-ω	RSM
1	2.4	14.0	3.1
2	4.4	3.6	4.6
3	9.8	9.9	9.2
4	12.4	14.0	10.9
5	3.2	3.4	2.4

RSM: Reynolds stress model.

There is evidence that the predicted values of the air permeability differ by a maximum of 10.9% (depending on the turbulent model applied) from the experimental values, which can be assessed as a very good result. Moreover, the error is around 3% for three of the modeled samples.

It can be seen that the highest discrepancy between numerical and experimental results occurs when the k-ω turbulence model was applied. In the case of Sample 1, which corresponds to the woven structure with the largest volume porosity, it is a problem of turbulence modeling, as the numerical results obtained with k-ɛ and RSM turbulence models are very similar to the experimental measurements of the flow rate. The simulation results for the air permeability of Samples 3 and 4 also differ from the experimental values, but while in the case of Sample 3 the three turbulent models gave similar numerical results, in the case of Sample 4 the application of the RSM gave the lowest relative error (see Table 3). The similar values of the relative error, however, show that for Samples 3 and 4 the turbulence modeling could not be a reason for the difference between the numerical and experimental results

Conclusions

Computer simulation of the transversal air permeability of woven structures on the basis of the theory of jet systems is numerically studied and experimentally validated. The numerical results show that the flow after the woven sample can be treated as an “in-corridor” jet system, which allows only four warp and weft threads and nine pores altogether to be simulated. The theoretical results obtained are in a good correlation with the experimental values for the air permeability of the tested woven samples.

The main conclusion from the study is that the presented approach can be successfully applied for further studies on the topic of woven structure permeability. The method used can be successfully applied also for precise flow analysis after the textile barrier, when necessary.

Footnotes

Funding

This work was supported by project BG051PO001/3.3-05-001 “Science and Business”, Operational Program “Development of the Human Resources” of the European Social Fund.

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