Abstract The Lagrangian smoothed particle hydrodynamics (SPH) method is employed toobtain a meso-/micro-scopic pore-scale insight into the transverse flow across the randomlyaligned fibrous porous media in a 2D domain. Fluid is driven by an external body force, and asquare domain with periodic boundary conditions imposed at both the streamwise and trans-verse flow direction is assumed. The porous matrix is established by randomly embedding acertain number of fibers in the square domain. Fibers are represented by position-fixed SPHparticles, which exert viscous forces upon, and contribute to the density variations of, thenearby fluid particles. An additional repulsive force, similar in form to the 12-6 Lennard-Jonespotential between atoms, is introduced to consider the no-penetrating restraint prescribed bythe solid pore structure. This force is initiated from the fixed solid material particle and mayact on its neighboring moving fluid particles. Fluid flow is visualized by plotting the localvelocity vector field; the meandering fluid flow around the porous microstructures alwaysfollow the paths of least resistance. The simulated steady-state flow field is further used tocalculate the macroscopic permeability. The dimensionless permeability (normalized by thesquared characteristic dimension of the fiber cross section) exhibits an exponential depen-dence on the porosity within the intermediate porosity range, and the derived dimensionlesspermeability—porosity relation is found to have only minor dependence on either the relativearrangement condition among fibers or the fiber cross section (shape or area).
F. Jiang (B)Electrochemical Engine Center (ECEC), Department of Mechanical and Nuclear Engineering,The Pennsylvania State University, University Park, PA 16802, USAe-mail: [email protected]
A. C. M. SousaDepartamento de Engenharia Mecânica, Universidade de Aveiro, Campus Universitário de Santiago,3810-193 Aveiro, Portugal
A. C. M. SousaDepartment of Mechanical Engineering, University of New Brunswick, Fredericton, NB, Canada,E3B 5A3
Liquid composite molding (LCM) operations, such as resin transfer molding (RTM) andstructural reaction injection molding (SRIM), which are being widely used to manufacturepolymer composite components in the civil, aerospace, automotive, and defense industries(Potter 1997), involve fluid flow in fibrous porous media. The understanding of the physicsof the fluid flow in the fiber-reinforced porous preform is a requirement for the design ofthe mold and the choice of the operating procedure which is crutial to the quality of thefinal product. To this goal, CFD numerical modeling can effectively reduce the experimentalparametric investigation effort needed for the optimization of the industrial process. Thus, itcan play an important role in practice. Eulerian-based CFD modeling of fluid flow in porousmedia is in general conducted by using Darcy’s law that can be expressed in the followingform (Phelan and Wise 1996)
〈v〉 = −Kµ
∇ P, (1)
where 〈v〉 and P are the volume averaged (superficial) velocity and the pressure of the fluid(bold text denotes vector or tensor), respectively; K is the permeability tensor of the porousmedium; and µ is the dynamic viscosity of the fluid.
Darcy’s law is a macroscopic-phenomenological model, which is largely based on expe-rimental observations. In this law, the complex interactions between fluid and microscopicporous structures are all lumped in a macroscopic physical quantity—the permeability ten-sor, K. Numerical models based on Darcy’s law have, at least, two shortcomings: (a) theoutcome of the modeling effort is largely dependent on the pre-specified permeability tensor,therefore, inaccurate specification of the permeability tensor leads to erroneous results; and(b) the numerical results cannot convey the effects at pore-level, because the model neglectsthe microscopic information of the pore structures.
As the permeability tensor is of paramount importance to the fluid flow in porous media,numerous research efforts (Phelan and Wise 1996; Sangani and Acrivos 1982; Katz andThompson 1986; Bear 1972) were conducted to establish theoretical relations between thepermeability and other characteristic material properties with the purpose of avoiding time-consuming physical experiments. These theoretical models are generally based on the ana-lysis of the microscopic porous structure of the medium and the subsequent extraction ofthe desired macroscopic information. The complicated, varied pore structures in the fiber-reinforced porous preform present a major challenge to theoretical or semi-empirical modelssince they are all constructed on the basis of artificially simple, regular pores.
Obviously, the modeling of fluid flow in fibrous porous media directly from the microsco-pic pore structure level provides refined fluid flow information and it does not resort to Darcy’slaw. Moreover, the simulated fluid field is amenable to the determination of K as well. Overthe last few decades, rapid advances in computer capabilities and computational algorithmsenabled this kind of modeling work. For example, the lattice-gas-automaton (LGA) or latticeBoltzmann method (LBM) (Chen and Doolen 1998), based upon a micro- or meso-scopicmodel of kinetic formulations or the Boltzmann equation, respectively, can bridge the gap bet-ween microscopic structures and macroscopic phenomena. These models have the advantageof allowing parallelism in a straightforward manner for large-scale numerical calculations,
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Transverse Flow in Randomly Aligned Fibrous Porous Media
and they have the attractive feature of non-slip bounce-back solid boundary treatment forthe simulation of fluid flow in porous media. The simulations based on LGA (Chen et al.1991; Koponen et al. 1997) and LBM (Koponen et al. 1998; Cancelliere et al. 1990; Spaidand Phelan 1997) have demonstrated that Darcy’s law can be reproduced by these methods.The simulated porous media in the work by Koponen et al. (1998) describe three dimensio-nal random fiber webs that closely resemble fibrous sheets, such as paper and non-wovenfabrics. The computed permeability of these webs presents an exponential dependence onthe porosity over a large range of porosity and is in good agreement with experimental data(Ingmanson et al. 1959; Jackson and James 1986). Spaid and Phelan (1997) investigated theresin injection process encountered in RTM. The obtained cell permeability for transverseflow through regularly arranged porous tows of circular or elliptical cross-section agrees wellwith the semi-analytical solution. Instead of LBM or LGA, in the present work another re-latively recent numerical technique—smoothed particle hydrodynamics (SPH), is employedto get a mesoscopic/microscopic insight into the transverse fluid flow in randomly alignedfibrous porous media.
SPH is a meshless particle based Lagrangian fluid dynamics simulation technique, inwhich the fluid is represented by a collection of pseudo-particles. These elements or particlesare initially distributed with a specified density distribution and evolve in time according tothe fluid conservation equations (e.g., mass, momentum). Flow properties are determined byan interpolation or smoothing of the nearby particle distribution using a special weightingfunction—the smoothing kernel. This technique was first proposed by Gingold and Mo-naghan (1977) and Lucy (1977) in the context of astrophysical modeling. Through majordevelopments over the past ∼30 years, SPH has been quickly approaching its mature stage.Although its accuracy, stability and adaptivity need further improvement, the SPH methodand its extensions have been successfully implemented in a broad spectrum of problems, asdescribed, e.g., in Jiang et al. 2006 and Jiang and Sousa 2006.
SPH has two main characteristics—meshless and Lagrangian-based—which make it moreadvantageous than the traditional Eulerian-based numerical techniques in what concernsthe following aspects: (a) ease of handling complex free surface and material interface;(b) relatively simple computer codes and ease of machine parallelization; (c) zero numeri-cal diffusion (or artificial viscosity) due to the absence of the convective-advection terms;and (d) particular suitability to tackle problems dealing with multi-physics. In contrast toLBM or LGA, SPH is a meshless particle-based method and offers more freedom in dea-ling with complicated geometries. Application of SPH to fluid flow in porous media has thepotential of providing a mesoscopic/microscopic pore-scale insight into the relevant physics(Zhu et al. 1999).
In recent work, Jiang et al. (2007) constructed a mesoscale SPH model for the fluid flow inisotropic porous media. The porous structure was resolved in a mesoscopic-level by randomlyassigning a certain number of SPH particles to fixed locations. Particularly, a repulsive force,similar in form to the 12-6 Lennard-Jones potential between atoms, was set in place to mimicthe no-penetrating restraint of the interface between fluid and solid pore structure. This forceis initiated from the fixed porous material particle and may act on its neighboring movingfluid particles. In this way, the fluid was directed to pass through the porous structure inphysically realistic paths. The macroscopic Darcy’s law was confirmed to be valid only inthe creeping flow regime. The derived relationship of permeability versus porosity comparedwell to some existing numerical results/experimental data, which demonstrates that the SPHmodel constructed is able to capture the essential features of the fluid flow in porous media.
The present work is a further extension of the study conducted by Jiang et al. (2007) andits primary aim is twofold. The first goal is to construct a convenient but effective numerical
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F. Jiang, A. C. M. Sousa
model for the simulation of fluid flow in porous media of complicated pore structures. Thesecond goal is to visualize the microscale transverse flow in randomly aligned fibrous po-rous media and to explore the effects of microscopic pore structures on a macroscopic flowproperty: the permeability. The solid matrix of porous medium is formed by randomly positio-ning a certain number of fibers within a two-dimensional square domain. Each porous systemonly encompasses fibers of a fixed cross-section (shape and area). Different pore-structureshave fibers of different cross-sectional shape (circular or square) or different sectional area.By varying the number of the embedded fibers, the porosity of the porous system is ad-justed. The relative arrangement condition between fibers is also taken as a participatingfactor to investigate the dependence of the permeability on the porous microstructures. Twolimiting situations are considered: (a) the embedded fibers are permitted to overlap with nopreconditioning, and (b) the embedded fibers are isolated from each other.
2 Modeling Strategy
2.1 SPH Formulation and Methodology
SPH formulation is directly based on the resolution of the macroscopic governing equationsof fluid flow. Derivation of the SPH formulation from traditional Eulerian-based equationsis routine; therefore, in this article is only briefly described. For a basic understanding ofthe derivation process, one may be referred to previous work, such as, Morris et al. 1997;Zhu et al. 1999; and Jiang et al. 2006.
In SPH, the continuous flow at time t is represented by a collection of N particles locatedat position rn(t) and moving with velocity vn(t), i = 1, 2, . . .. . ., N . The ‘smoothed’ valueof any field quantity q(r, t) at a space point r is a weighted sum of all contributions from theneighboring particles
〈q (r, t)〉 =N∑
j=1
m j
ρ(r j
)q(r j , t
)w
(∣∣r − r j∣∣ , h
), (2)
where m j and ρ(r j ) denote the mass and density of particle j , respectively. The w(|r|, h)
is the weight or smoothing function with h being the smoothing length. In the present SPHimplementation, the high-order 3-splines quintic kernel (Liu et al. 2003) is used and thesmoothing length h is equal to
√2�S, with �S being the average separation of the SPH
particles. In the SPH formulation the gradient of q (r, t) is determined as:
〈∇q (r, t)〉 =N∑
j=1
m j
ρ j
[q
(r j , t
) − q (r, t)]∇w
(∣∣r − r j∣∣ , h
). (3)
Equation 3 is already symmetrized (Jiang et al. 2006), and for the second order viscousdiffusive term, the SPH theory gives the following formulation (Jiang et al. 2006; Morris etal. 1997)
1
ρ∇ · (µ∇q) =
∑
j
m j
ρnρ j
(µn + µ j
)qnj
1∣∣rnj∣∣∂wnj
∂rn, (4)
where qnj = qn − q j , rnj = rn − rj and wnj = w(|rn − r j |, h).
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Transverse Flow in Randomly Aligned Fibrous Porous Media
In terms of Eq. 3, the ‘smoothed’ version of the continuity equation is
dρn
dt=
∑
j
m jvnj · rnj∣∣rnj
∣∣∂wnj
∂rn, (5)
The summation is over all neighboring particles j with exception of particle n itself andvnj = vn − v j .
The transformation of the Navier–Stokes equations into their SPH version is relativelyinvolved, because it requires the symmetrization of the pressure terms (Jiang et al. 2006) tosatisfy momentum conservation, and the adequate treatment of the viscous terms. In SPH,the symmetrization version of pressure gradient can be formulated as:
1
ρn∇ pn =
∑
j
m j
(pn
ρ2n
+ p j
ρ2j
)rnj∣∣rnj
∣∣∂wnj
∂rn, (6)
From Eqs. 4 and 6, the standard SPH form of N-S equation is derived (Morris et al. 1997)
dvn
dt= −
∑
j
m j
(pn
ρ2n
+ p j
ρ2j
)rnj∣∣rnj
∣∣∂wnj
∂rn+
∑
j
m j(µn + µ j
)vnj
ρnρ j
(1∣∣rnj
∣∣∂wnj
∂rn
)+ FB,
(7)
where FB denotes an external body force. In this work, the following SPH expression for themomentum equations is used.
dvn
dt= −
∑
j
m j
(pn
ρ2n
+ p j
ρ2j
+ RGηnj
)rnj∣∣rnj
∣∣∂wnj
∂rn
+∑
j
m j(µn + µ j
)vnj
ρnρ j
(1∣∣rnj
∣∣∂wnj
∂rn
)+ FB +
∑f , (8)
Equation 8 was proposed by Jiang et al. 2007, and it differs from the standard SPH N-Sequation in what concerns two terms on the RHS of this equation, namely: (a) RGη
nj , anartificial pressure, which is used to restrain the tensile instability (Monaghan 2000); (b)
∑f ,
an additional force, which is introduced with the purpose of correctly mimicking the no-penetrating restraint prescribed by the solid pore structure. The term RGη
nj is calculated asfollows:
Gnj = wnj/w (�S, h), (9)
R =
⎧⎪⎨
⎪⎩
ϕ1 |pn |/ρ2n if pn < 0
ϕ1∣∣p j
∣∣/ρ2j if p j < 0
ϕ2
(|pn |/ρ2
n + ∣∣p j∣∣/ρ2
j
)if pn > 0 and p j > 0
(10)
The factors ϕ1 and ϕ2 are taken to be 0.2 and 0.05, respectively. The exponential η is dependenton the smoothing kernel as η ≥ w(0, h)/w(�S, h); 3.0 is adopted for η in this work. Therepulsive force f (per unit mass) is initiated from a porous material particle and is appliedto the neighboring fluid particles within a distance less than a threshold value r0. In theseconditions, for most of the fluid particles,
∑f = 0. f is assumed (Jiang et al. 2007) to be of
the 12-6 Lennard-Jones pair potential form, namely
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F. Jiang, A. C. M. Sousa
f (r) = φFB
[(r0
r
)12 −(r0
r
)6]
rr
(r < r0), (11)
Determination of the constants: φ and r0 were discussed in detail in Jiang et al. (2007); thevalues r0 = 0.6�S and φ=15.0 were found to be suitable for the present SPH implementation.
A quasi-compressible method is used to comply with the incompressibility condition forthe fluid, which relies upon an artificial equation of state,
p = p0
[(ρ
ρ0
)γ
− 1.0
], (12)
where p0 is the magnitude of the pressure for the material state corresponding to the referencedensity ρ0. An artificially large value is specified for γ , γ = 7.0, to satisfy the weak-compressibility of the fluid. The value of the reference pressure p0 determines an “artificial”sound speed cs , which controls the permitted “compressibility” of the incompressible fluid.The values of p0 and cs are taken in accordance with the following relation
c2s = γ p0
ρ0= max
(0.01U 2, 0.01FBL ,
0.01Uµ
ρ0 L
), (13)
where U and L are the characteristic (or maximum) fluid velocity and the characteristic lengthof the geometry, respectively; Eq. 13 ensures the density variation of the fluid be below alower threshold value of approximately 1.0%.
The positions of SPH particles are determined directly from the flow field using
drn
dt= vn . (14)
A direct numerical integration algorithm is used to perform the integration of Eqs. 5, 8, and14. The time step �t is adaptive and determined by the CFL condition (Morris et al. 1997)
�t = min
⎛
⎝ 0.125h
maxn
(∼ 100.0U, |vn |) , 0.25
√h
|dvn/dt | ,0.25h
cs,n
⎞
⎠ . (15)
On the RHS of Eq. 15, the quantity ∼100.0U is introduced in the denominator of the firstconstraint to restrict it to a small value, so that the calculation advances mostly in a fixedsmall time step. In this way the accuracy of the integration procedure can be enhanced andthe calculation stability is also improved.
2.2 Microscale Pore Structures
The porous matrix is established by randomly packing a certain number of fibers. Only thetransverse flow across the fibers is simulated in the present work. Typical 2D porous micro-structures are illustrated in Fig. 1. The cross-sectional shape of the fibers is either circular(Fig. 1a) or square (Fig. 1b), and the fibers can be either overlapped with no preconditioning(Fig. 1a, b) or be isolated from each other (Fig. 1c). Each porous system is formed by packingfibers of fixed cross-section (shape and area). Different porous systems have fibers of dif-ferent cross-sectional shape, different cross-sectional area or different relative arrangementcondition.
Both the resin fluid and the fibers are represented by SPH particles. The positions of theSPH particles used to represent the solid pore matrix are fixed throughout the simulation.
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Transverse Flow in Randomly Aligned Fibrous Porous Media
Fig. 1 Typical 2D porous microstructures formed by randomly packing a number of fibers: (a) fibers are ofcircular cross section with unlimited overlapping between fibers; (b) fibers are of square cross section withunlimited overlapping between fibers; and (c) fibers are of circular cross section and the fibers are isolatedfrom each other. Void space represents the fluid region
Thus, the porosity ε of a porous system is determined as
ε = 1 − number of fixed particles
total number of particles. (16)
Fixed porous material particles contribute to the density variations and exert viscous forcesupon the nearby fluid particles, while their densities are kept constant. The no-penetratingrestraint prescribed by solid pore structure is ensured by activating a repulsive force, asexpressed by Eq. 11, on a fluid particle once it is away from a porous material particle by adistance < r0. With this additional force, the fluid particles are directed to pass by the porestructures in physically acceptable paths (Jiang et al. 2007).
2.3 Other Relevant Conditions
The simulated domain is a two-dimensional square of dimensions 10 mm×10 mm with per-iodic boundary conditions applied in both the streamwise and transverse flow directions.Initially, 100×100 SPH particles are distributed uniformly within the simulated domain(i.e., �x = �y = �S = 0.1 mm) and are all-motionless; the density for all the SPHparticles is specified to be ρ0.
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F. Jiang, A. C. M. Sousa
The fluid flow is driven by a constant body force (FBx) imposed in the streamwise flowdirection. The density, velocity, and position of each fluid particle evolve in time by integratingEqs. 5, 8, and 14 respectively; while all the porous material particles have a fixed positionand constant density (ρ0). The given physical properties of the resin fluid are µ = 0.1 Pa·sand ρ0 = 1000 kg/m3. The applied body force per unit of mass FBx is 10 m/s2 for most of thesimulations; however, when the porosity is very high, a much smaller body force (FBx = 1.0or 0.5 m/s2) is imposed to keep the fluid flow in the creeping flow regime.
3 Results and Discussion
3.1 Flow Visualization
Practical LCM procedure requires sufficient wetting of all the fibers by the resin. An identifiedserious hurdle for LCM process is the so-called preferential flow or edge flow, which iscaused by inhomogeneity of the porous preform or unavoidable clearances existing betweenthe preform edges and mould walls. Preferential flow results in the maldistribution of injectedresin and makes part(s) of the fibrous preform not be saturated or fully saturated, i.e., dryspot(s). To exactly predict the preferential flow so as to optimize the resin injection processis crucial to the quality of the final product. Pore scale SPH model can provide fluid flowinformation at microscopic/mesoscopic pore level and it has the potential of being veryuseful in practice. Figures 2 and 3 present the visualization of the fluid motion across thealigned fibers due to the applied body force (FBx). Preferential flow paths are identified bydepicting the local velocity vectors. The fluid meanders in the porous microstructures andits path has a tortuosity determined by the location of the embedded fibers. The microscopicflows involved in porous systems of different micro-structures do not exhibit substantiallydifferent patterns and the fluid always flows through the path of least resistance among theporous structures.
3.2 Derivation of the Permeability
The permeability is an important macroscopic property for fluid flow in fibrous porous media.In terms of the simulated steady state flow field, the permeability for the transverse flowthrough the fibrous porous medium can be calculated as
k = µ
ρ0 FBx〈u〉, (17)
and this formulation is based on Eq. 1. In order to facilitate the comparison between thepresent results and existing numerical results/experimental data, the permeability k (ofa square domain) must be normalized. The normalization parameter is taken to be the squareof the characteristic dimension, d2, of the embedded fibers; d is defined as the hydrodyna-mic radius of the embedded fiber’s cross section: for fibers of circular cross section, d isthe radius; while for fibers of square cross section, d should be the half side-length of thesquare. It is worth pointing out that due to the finite numerical discretization, the circularfiber cross section cannot be approximated accurately and d is calculated exactly in terms ofthe definition of hydrodynamic radius, instead of simply specifying it equal to the radius ofthe desired circular cross section.
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Transverse Flow in Randomly Aligned Fibrous Porous Media
Fig. 2 Microscopic flows in a porous system with unlimited overlapping between circular fibers with crosssection of 0.6-mm diameter. (Note: the length scales for the velocity vectors in the two plots are not the sameand resin fluid advances from the left to the right side.) The background of the plot depicts the location of thefibers. (a) ε = 0.4238; (b) ε = 0.6344
The calculated dimensionless permeability for the porous system formed by packingcircular fibers of 0.6 mm diameter cross section with unlimited overlapping between fibersis depicted in Fig. 4, as a function of porosity.
The comparison between the SPH simulation results and existing experimental data(Ingmanson et al. 1959; Jackson and James 1986) indicates the present SPH calculation
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F. Jiang, A. C. M. Sousa
Fig. 3 Flow visualization inrandomly aligned fibrous poroussystems. (Note: the length scalesfor the velocity vectors in thethree plots are not the same andresin fluid flows from the left tothe right side.) The background ofthe plot depicts the location of thefibers. (a) unlimited overlappingbetween fibers of 0.6×0.6 mmsquare cross section, ε = 0.705;(b) unlimited overlappingbetween circular fibers with crosssection of 0.8 mm diameter,ε = 0.7061; (c) isolated circularfibers with cross section of0.6 mm diameter, ε = 0.4661
technique captures the main features of the fluid flow in fibrous porous media. Computatio-nal results derived from other numerical techniques (Koponen et al. 1997; Higdon and Ford1996) are also consistent, at least in trend, with the present SPH simulation predictions. Itshould be noted that there are at least two free parameters present in this comparison. First,different porous systems were studied, namely, the LGA results (Koponen et al. 1997) wereobtained for two-dimensional porous systems with rectangular obstacles either randomlydistributed (depicted by downward open triangles in Fig. 4)) or placed in a regular squarelattice (upward open triangles in Fig. 4); the physical experiments (Ingmanson et al. 1959;
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Transverse Flow in Randomly Aligned Fibrous Porous Media
Fig. 3 continued
Fig. 4 A comparison between the calculated permeability for the porous system formed by circular fi-bers of 0.6 mm diameter cross section (with unlimited overlapping between fibers) and existing numericalresults/experimental data
Jackson and James 1986) were performed with respect to three-dimensional compressed fi-ber mats and fibrous filters; the (Eulerian-based) numerical results (Higdon and Ford 1996)were reported for a three-dimensional porous system of a face-centered-cubic (fcc) array offibers. As a second free parameter for comparison, different physical properties of the fluidwere assumed. For example, in the present study the viscosity of the fluid is specified as,µ = 0.1 Pa s, which may be different from that used by other researchers, since in many
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F. Jiang, A. C. M. Sousa
cases this parameter is not reported. A nearly linear dependence of permeability on viscosity(with fixed porosity) was reported in the work by Koponen et al. (1998).
From Fig. 4, it is clearly seen that when the porosity is smaller than 0.7, the SPH-calculateddimensionless permeability is slightly larger than the FEM result, but is noticeably larger thanthat obtained by LGA technique. Besides the above-described free parameters among dif-ferent numerical efforts, there is another reason present. For the porous systems of unlimitedoverlapping fibers, the embedded fibers may be connected and adjoined to each other, andthey form larger obstacles (as shown in Figs. 2a, b; 3b). This holds true especially when theporosity is low. Therefore, assuming the hydrodynamic radius as the characteristic dimensionof the individual fibers, to a great extent, underestimates the actual characteristic dimensionof the porous system, which leads to larger dimensionless permeability values. Moreover, forporous systems established by randomly distributed inclusions, the randomness can affect theinterior non-homogeneity of the system, which might cause a large variation of the permea-bility. Koponen et al. (1997) stated that the non-homogeneity could increase the permeabilityas much as 50% as compared to a porous system with homogeneous pore structure. Notably,the SPH calculated permeability has large departure from the LGA prediction in very lowporosity range (ε < 0.5). Non-homogeneity in structure has significant influence on the ef-fective porosity when the porosity is approaching the percolation threshold. Slight variationat effective porosity may lead to order of change of the permeability. This additional uncer-tainty makes it extremely difficult to accurately predict the permeability of porous systemwith very low porosity by using any numerical technique discussed here.
3.3 The Dependence of Permeability on Porous Micro- Structures
A series of SPH calculations with respect to different porous systems of different fibercross-sectional shapes (square or circular), different fiber cross-sectional areas and differentrelative arrangement conditions among fibers was conducted to examine the dependence ofthe permeability on the porous microstructures.
Figure 5 displays the non-dimensional permeability for the porous systems of two differentfibers cross-sectional shapes (circular and square). The observed trend of the dimensionlesspermeability versus porosity is very similar for both cases, which indicates the fiber crosssectional shape has little effect upon the dimensionless permeability.
Fig. 5 Dependence of the permeability on the fiber cross sectional shape
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Transverse Flow in Randomly Aligned Fibrous Porous Media
Fig. 6 Dependence of the permeability on the fiber cross sectional areas
There are some distinct features displayed by the permeability—porosity relation shown inFig. 5. First, the permeability has a sharp increase as expected when the porosity ε approachesone (k is infinite for ε = 1). Second, the permeability seems to follow an exponentialdependence on the porosity in the intermediate porosity range (Koponen et al. 1998; Jianget al. 2007), 0.5 ≤ ε ≤ 0.8. A regression fit to the calculated data yields the followingexponential relationship
ln(k/d2) = −3.13 + 9.43ε. (18)
The correlation coefficient between the simulated points and the exponentially fitted curve is0.99. Third, the SPH simulations indicate the percolation threshold of the investigated porousmedia is smaller than 0.4. A value of 0.33 for the percolation threshold was used (Koponenet al. 1997) for a 2D porous system consisting of randomly distributed rectangular obstacles.
Figure 6 shows the dimensionless permeability for the porous systems of different fibercross sectional areas. For the porous systems consisting of 0.6 mm or 0.8 mm diameter circularfibers, the dimensionless permeability varies with the porosity in an almost identical way;however, some deviations from the two systems are apparent, when the porosity is outsidethe intermediate range, for the porous system of 1.0-mm diameter circular fibers. Since thecomputational domain is just a 10×10 mm square, the embedded fibers of 1.0-mm diametercross section, most likely, will destroy the x–y plane isotropism of the system, which mayyield the above-mentioned computed data deviations. Therefore it is reasonable to infer thatthe fiber cross-sectional area also has a little influence on the dimensionless permeability.
The fiber relative arrangement, either with unlimited overlapping of the fibers or full isola-tion from each other, is considered to examine its effect on the non-dimensional permeability.As displayed in Fig. 3c, even if no-interaction is pre-specified as the relative arrangementcondition between fibers, neighboring fibers in the numerical porous system may touch eachother, which is largely due to the finite SPH particle resolution. However, this relative arran-gement condition does avoid the occurrence of fiber overlapping. Relevant computed dataare depicted in Fig. 7. The fiber relative arrangement condition also has no significant effecton the dimensionless permeability.
A simplified dimensional analysis suggests the permeability k of a porous medium is afunction of the effective porosity εeff (≤ ε), flow path tortuosity τ , and the specific surface
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F. Jiang, A. C. M. Sousa
Fig. 7 Dependence of the permeability on the fiber relative arrangement conditions (circular fibers of 0.6 mmdiameter cross section)
area s—the ratio of the fiber wetted area to the solid volume. k has the units of a squaredlength; τ and εeff are dimensionless; and s has the units of the inverse of length. Hence,k should be related to εeff , τ and s with a functional of εeff and τ being divided by thesquared s. Kozeny capillary theory indicates the permeability k is directly proportional to thecubic εeff , and inversely proportional to the τ square as well as s square, i.e., k ∝ ε3
eff/τ2s2.
More information about Kozeny theory is provided in, for instance, Koponen et al. 1997 andJiang et al. 2007.
By comparing the porous microstructures shown in Fig. 2a and 3c, it can be noted theporous medium consisting of isolated fibers effectively avoids the formation of the so-calledoccluded pores (pores that are encircled by porous material) and dead-end pores. Almost allthe physically existing pores in the porous system shown by Fig. 3c are conducting pores andcontribute to the fluid flow (εeff ≈ ε), which should lead to an increase of the permeability.However, due to the isolation between fibers, the specific surface area also increases, whichcauses a decrease of the permeability; as expected, the fiber relative arrangement conditionshould also have some influence on the flow path tortuosity τ . All these factors, when com-bined, lead to the overall effect of near-independence of the permeability—porosity relationfrom different fiber relative arrangement conditions of the porous systems.
4 Error Estimate
As described in Sect. 2, for the fibrous porous systems of circular fibers (Fig. 1a, c), the fibercircular cross section is numerically approximated by a certain number of square elements(SPH particles). Obviously, due to its finite resolution, the circular interface between fibersand fluid can not be fully reproduced, which yields some error introduced into the finalpermeability data. The magnitude of this error can be estimated by considering an idealizedporous system formed by embedding a 0.6 mm diameter circle in a 1.0×1.0 mm2 domain. Tomeet the situations of interest in the present study, the basic numerical elements are taken assquares of 0.1 mm side-length. In total, 32 square elements are used to represent the circularinclusion. The numerical porous system together with the desired porous system is depicted in
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Transverse Flow in Randomly Aligned Fibrous Porous Media
Fig. 8 Illustration of thedeviation between the numericalporous system and the desiredporous system with a circularinclusion
Table 1 Evaluation of the error caused by finite resolution to describe the circular inclusion
Fig. 8. The parameters numerically obtained for the pore-structure present minor deviationsto the desired values as listed in Table 1. The effect of the departure between the desiredand the numerical data upon the flow path tortuosity τ is assumed practically negligible;therefore τ is not taken into account here. As already mentioned, the porosity that is used inthe relation of dimensionless permeability versus porosity is obtained from Eq. 16. Hence,in order to perform the comparison, the parameters for an equivalent porous system that hasa circular inclusion and has the same porosity as the numerical porous system have to becalculated, and they are also listed in Table 1. Inspection of the d and s data for the numericalporous system and the equivalent system indicates the derived dimensionless permeabilityof numerical porous system gives a good representation of the porous system with a circularinclusion; the error caused by the approximation is ∼2.5%.
For the fibrous porous systems formed by fibers of square cross section (Fig. 1b), the squarenumerical elements (SPH particles) can reproduce the pore-structures accurately. Thus,the computational results for this type of porous systems are free of the above-mentionedapproximation error. In what follows a grid-dependence test is carried out with respect toone case reported to this type of porous system.
For the porous system having the pore-structures shown in Fig. 3a, the Richardsonextrapolation technique was used to examine the dependence of the resulting permeabilityon the particle resolution. The dimensionless permeability as a linear function of 1/(numberof SPH particles) was computed based on the SPH particle resolution of 50×50, 100×100,and 200×200, respectively. The results obtained indicate the SPH calculation for 100×100
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particle resolution gives a dimensionless permeability that deviates by approximately 7.9%from the extrapolated value for an infinite SPH particle number.
5 Summary and Conclusions
The SPH technique has the potential of modeling the micro-flows in porous media fromthe viewpoint of microscopic pore-level. In what concerns the transverse fluid flow acrossrandomly aligned fibrous porous media, the present work constructs a 2-D SPH numericalmodel. The developed model fully utilizes the meshless characteristic of SPH methodologyand thus offers great flexibility in the construction of complicated pore structures.
The proposed modeling approach enables a mesoscopic insight into the fluid flow. Pore-scale preferential flow paths are visualized, the fluid meanders in the porous microstructuresand, as expected, always flows along the paths of least resistance. The macroscopic featuresof the fluid flow are fully captured by this model. For instance, the dimensionless permeability(normalized by the squared characteristic dimension of the fiber cross section) is found toexhibit an exponential dependence on the porosity in the intermediate porosity range.
In addition, this model was used to explore the effect of the micro-structures inside thefibrous porous medium on the dimensionless permeability—porosity relation. Neither therelative arrangement between fibers nor the fiber cross-section (shape or area) has significanteffect on this relation.
Acknowledgments Financial support was received partly from the FCT (Foundation for Science andTechnology, Portugal) research grant POCTI/EME/59728/2004, NSERC (Natural Sciences and EngineeringResearch Council of Canada) Discovery Grant 12875 (ACMS) and the Post-doctoral Fellowship,SFRH/BPD/20273/2004 (FJ).
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