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On the permeabili ty of unidirectional fibrous media: A parallel computationalapproachChahid K. GhaddarComputerAided ProcessEngineering, Inc., Suite 220, 888 WorcesterStreet, Wellesley,Massachusetts 2181(Received 6 April 1995; accepted 7 July 1995)The problems of viscous and inertial flows through unidirectional fibrous porous media areaddressed using an entirely parallel computational approach. The pertinent partial differentialequations, derived from homogenization theory, are solved by a parallel finite element method inconjunction with Monte Carlo techniques to predict the statistical permeability coefficient. Anip-element method, which mitigates the frequent geometry-induced numerical difficulties, whileproviding both accurate approximations for the permeability coefficient, and rigorous errorestimates, is also presented. The seepage permeability coefficient is determined for a wide range offiber concentration. It is shown to deviate markedly at low porosities from the behavior predicted byearlier cell models, while exhibiting generally good agreement at high, and moderate porosities withthe cell models, and with the limited available experimental and analytical results. Limited butillustrative inertial flow results at moderate Reynolds numbers are also presented for both regularand random arrays. For regular arrays, the flow is found to be unsteady for Reynolds numbersgreater than approximately 150 at which traveling waves characterized by distinct periods andamplitudes are observed. Some modest discrepancy is found in comparison with available datawhich is attributed to the unsteady effects and other numerical issues. For random arrays, severalconfiguration permeability values are calculated and compared satisfactorily against the Erguncorrelation. 0 I995 American Institute of Physics.

1. INTRODUCTIONTransport phenomena in fibrous porous media play acritical role in many man-made and natural processes. Ex-amples include: aerosols filtration,Y2 fluidization3 convec-tive heat exchangers,4 and cell mechanics5P6ust to name afew. Needless to say, a detailed microscopic analysis of thephenomena is plagued by many difficulties such as the sta-tistical nature of the microstructure, the presence of a largerange of disparate length scales, and nonlinearity amongmany others issues. From an engineering view point, how-ever, it usually suffices to predict the macroscopic permeablecharacter of the media rather than the intrinsically compli-cated details of the phenomena.In this paper we consider the permeability behavior ofone class of fibrous porous media in which the flow is strictlytransverse to the axes of the fibers. Such media which can beaccurateIy modeled by two-dimensional (2D) random arraysof cylinders, appear to have accumulated fewer dedicated

efforts to understand their permeable character as opposed torandom beds of spheres for example. In general, the efforts(reviewed below) have either involved artificial assumptionsto simplify the problem to the extent an analytical solutionwas possible, or have lacked the flexibility to perform alarge-scale numerical studies and address nonlinear effects.As a result, rather limited theoretical results are available andare typically short on statistical analysis.The permeability behavior of 2D random arrays of cyl-inders is addressed in this work using an entirely parallelcomputational approach, based on first principles andcoupled with proper statistical analysis. In particular, thepresent strategy departs from earlier efforts in several as-pects: First, by exploiting the well-known benefits associated

with parallel processing, it is possible to perform better sta-tistical treatment and address more ambitious problems whilekeeping the intensive computational requirements within rea-son. Second, me geometry-induced numerical difficultieswhich are typical problems facing numerical treatment, havebeen treated, rather than avoided, with a nip-element methodwhich enjoys rigorous approximation properties. Third, non-linear effects due to moderate Reynolds-number flows whichare not amenable to analytical analysis have been considered.In the following we briefly review previous studies for bothcreeping and inertial flows and then describe the organizationof the current paper.

A. Creeping flow regimeThe permeability in the viscous flow regime, is governedby the well-established7b Darcys law

where (u) is the superficial fluid velocity in the direction ofthe imposed external pressure drop Ap across the thicknessL of the medium, K is the permeability coefficient (in generaltensorial) which depends only on the bore geometry for suf-ficiently small Reynolds numbers, and ,u is the fluid viscos-ity. Darcys equation has been the point of departure formany earlier capillary models aimed at correlating the pres-sure drop-flow rate relation, of which the well-knownCarman-Kozeny equation has often been used. The lattercorrelation- (referred to hereafter simply as the Kozenyequation) in its simplest form provides the following expres-sion for K:

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K= -, k (2)

where k is the Kozeny constant supposedly independent ofthe porosity E, and rh is the mean hydraulic radius defined asthe ratio of free vol ume to the wetted area, which is, for afibrous medium of fiber diameter d, equal to(3)

There is a large volume of data for beds consisting of avariety of spherical and nonspherical particles which indicatethat a value of k-5.0 independent of shape and porosity inthe range 0.26=a(z4)+b(u)m, (4)

extend classical Darcys law by introducing a second nonlin-ear term to account for inertial losses. The coefficients a andb are constants that depen d on the porosity and pore charac-teristics, and m is typically 2. Many attempts have beenmade to accurately define the parameters in (4) to fit thebody of experimental data as compiled by Scheideggerg andDavidson et a2. The most widely accepted relation for prac-tical engineering applications is the Ergun relation2* given interms of a modified friction factor, f , and modified Rey-nolds number, Re I as follows:150f =Re + 1.75,

whereflVP &,

1Re=Re -l--Eand Vp is a nondimensional pressure drop defined as

v,=.A!zTL/d p(u)

(5)

(6)

(7)

(8)where p is the fluid densi ty.C. Organization of paper

This paper is organized into four sections as follows: inthe next section we present the mathematical formulations ofboth the Stokes and inertial flow problems which arefounded upon the periodic supercell model and homogeniza-tion theory. A nip-element method designed to mitigate thesubsequent numerical treatment is also presented. In SectionIII we discuss the pa.ra.llelnumerical treatment of the formu-lations. Results are presented in Section IV which in-

2564 Phys. Fluids, Vol. 7, No. 11, November 199 5 Chahid K. Ghaddar

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d

Yl

l33. 1. Periodic supercell [0,X] containing N cylindrical inclusions ofunity diameterd, with centers ocated at yi , = l,.. . N.eludes three subsections: In Section IV A we present twodemonstrative examples using the nip-element method whichalso serve as a benchmark for the implementation. In SectionIV B we present creeping flow permeability results and com-pare against the predictions of earlier cell models as well asavailable experimental measurements and analytical results.Finally, in Section IV C, we present inertial porous-mediaflow results for regular and random arrays for selected con-centrations and Reynolds numbers along with a comparisonwith the limited available data.II. PROBLEM FORMULATION

We consider a fluid-fiber structure composed of a peri-odically replicated supercell of size X, y E [ O,h] X [0,X]ClR2, consisting of two phases: an incompressible fluidphase of constant viscosity p= 1, and density p; a dispersedphase of N immovabl e co-oriented cylindrical obstacles ofdiameter d = 1, centers yi E {Y}~= (yi , . . . , ylv), and volumet-ric concentration, c= 1 - E, as depicted in Fig. 1. A fixedimposed global pressure gradient, Vp = - 1 jt , [where it isa unit vector in the direction of the y, axis (see Fig. l)], isresponsible for a net transverse fluid motion parallel to they1 axis. [Note the choice (d= l,@= l,Vp= - 1 y^i) rendersp the only variable parameter which may be, equally well,interpreted as a nondimensional control parameter.]A. Viscous flows

The fluid motion governing equations in the supercellare deduced from periodic homogenization theory.2990 Herethe velocity vector u(y)=(u1(yi,y2),u2(y1,y2)) and thepressure, p = p (y), satisfy the steady Stokes equations,

-p+dp=*,i, in a, for i=1,2,d2Ui

djjdyj $Vi (9)

dUi-z=O, in a,. iwhere R is the region of the supercell occupied by the fluid,and Sij is the Kronecker delta. The velocity must vanish onall obstacles boundaries and both the velocity and pressuremust be X-periodic in both y t and y2. It is further requiredfor uniqueness that Japdy= 0.It is important to note that although the Stokes equationsdo not permit a solution for flow, past an isolated cylinder inan infinite domam3t our supercell, however, thanks to thefinite domain and periodic boundary conditions, constitutes awell-posed problem except possibly in the limit of diminish-ing concentration values which are not the focus of thiswork.It is advantageous to work with the variational formula-tion of (9)-(10) since it is required for the ensuing finiteelement analysis. This is based on the well-known varia-tional principles2

u=arg m=hEzJfdv>, (11)where the functional Jo(v) (related to the excess dissipa-tion) takes the form

(12)Here Z={(ui .u2). E (H~,(fl),H~,(R))~diuv=O}, andH,&(n) is the space of all X-doubly periodic functionswhich vanish on 65l, and for which both the function andderivative are square-integrable over 0.In order to arrive at the standard velocity-pressure weakform, the constrained maximization (11) is transformed intoan unconstrained saddle problem by enlarging the velocityspace to include all functions (UlPU2) inWW>~~,W)), and introducing a Lagrange multiplier- the pressure, p - to impose the incompressibility con-straint. Taking the tirst variation of the resulting Lagrangian,we obtain the weak form: Find CU1442,P)E (H~#(n>,H~,(n>,L~,o(~)) such that

dy- I,$pdy= lau,dy,

V(Ul a21 E e&w2, (13)

IdUj- q - dy=O,Cl dYi vq -%,oWL

where Lip(a) is the space of all h-doubly periodic func-tions q(y) which are square-integrable over 0 (note thatcandidate pressures need not be continuous), and for whichSdy= 0.The transverse conjiguration permeability,K= ~({y}~,c,h), is defined through Darcys relation (1) andtakes the form 1K(Ul)= T;ZI nu,dy.By equating the first variation of the functional (12) to zeroand using (1 ), K can be expressed as

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1K=-T A- (16)which is related to the Helmholtz stationary dissipationprinciple.33-35 It is worthy to note that although we considerthe isotropic permeabili ty, the extremizing property (16) ex-tends to the full permeability tensor following the develop-ment in Nir et a1.36The permeability, of course, depends on the particularinclusions centers {y}N. By presuming the latter are randomvariables, {Y}N, prescribed according to a joint probabilitydensity function (JPDF), ftyJN( { Y}~, c , A), renders the per-meability a random variable whose average, K,(c,A) (inde-pendent of {YIN) is then given byK,(C,A)I Co,AlxCoAIP({Y}N,C,h)f~y)N({Y}N,c,A)dY.(17)The effective permeability, K, , is representative of a randomperiodic fibrous media whose microstructure can be ad-equately predicted by f{vI,. Nevertheless, it is plausible tobelieve that as X--+00, the periodic media will resemble atruly random media (provided the macroscale L%-A) and,hence, their ,corresponding statistical properties are appli-cable to each other. More precisely, the limiting processlimb,, K,(C,A), or realistically, K,(c, A> A) for someh(c), assumes a well-defined value which, for all practicalpurposes, representative of true random media. This has beendemonstrated numerically for an analogous problem of de-termining the effective conduct ivity in thermal composites24where A(c) is defined such that for XaR(c), (i) K,(c,A) nolonger changes appreciably, and (ii) the standard deviation ofthe random variable K=K({Y}~,c,A), uK(c,A), is suffi-ciently small.In order to compute (17) one needs to know the inclu-sion JPDFffvlN which is typically unknown. Here the inclu-sion JPDF is assumed to be isotropic and homogeneousbased on a random acceptance-rejection sequential additionprocess73824which is believed to be relevant to manyphysical systems. [Note that a particular configuration, ingeneral, may be anisotropic, but, on the average, the proper-ties are expected to be isotropic (scalar).]

B. Inertial flowsIn the event of important convective inertial effects, thefluid motion in the periodic supercell is governed by the(possibly unsteady) Navier-Stokes equations,

p[~+Uj~]-~+~=fi li, in f1, for i=l,:;8)

(19)subject to the no-slip condition at the obstacles boundaries,and the A-periodicity of both the velocity and pressure inboth yt and y2 directions. It should be pointed out that thesupercell equations are actually rationalized, rather than

rigorously derived using periodic homogenization theory ashas been possible for the Stokes problem. Here the nonlinearconvective term precludes complete decoupling between thetensorial components of the homogenized, and supercell sub-problems [save for the special case of weak inertial effects3(~4 1 )] which poses a series of difficulties for the subse-quent solution. The adopted model, widely accepted in theliterature,25V26s justified by a control-volume type argu-ment in which the obvious contribution of the homogenizedproblem to the supercell problem is the linearized pressureappearing locally as p = - I yt . Furthermore, the interesthere is not in the details of the tlow field of the macroscaleproblem but rather the bulk permeabil ity relating the pres-sure drop to the how rate, which is precisely what we calcu-late from the supercell problem The latter should be, due tospatial averaging, less sensitive to the flow field details.The weak form of equations (18) and (19) takes the fol-lowing form. Find (ut ,uZ,p) E (H~#(n),H~,(n),~~,(n)).such that

-

I

di pdy= nu ,dy,tidYi I v(~, ,u2j E (H&W>)2T(20)I dUi- q - dy=O,IL aYi vq E L;,,(sz).

Turning to the permeability expression, we adopt the ba-sic definition provided by Darcys equat ion (1) which in-volves no modeling assumptions. The transverse configura-tion permeability, equation (15), remains valid, althoughtemporal averaging is required in unsteady situations,K=; Jo*{U,(t))dt= $ ;~oj-~U~(t)dydt. (22)

where T is the period of the unsteady flow. The permeability,K, however, will also depend on the flow Reynolds number,Re,

(23)where the last equality follows from the choice(d= I,pu= 1), and equation (15).Last, we show that the permeability can only decreasefor nonzero Reynolds numbers. In fact, we have from (20)

PEJ uldy,R

where we have used the incompressibility condition (19).Due to A-periodicity of the velocity, the convective term,SnUiUj (duildyj) dy, vanishes. Indeed

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P) FIG. 3. The excised nip region, 22 Upper bound (left); lower bound (right).

FIG. 2. Porous-media nip-element strategies : (aj nip-region enlargement(upper bound); (b) nip-region blockage (lower bound).

azci 1uillj- dy=iaYj s an UjU~ .njdS= 0 ,05)where ds is a differential element of &I defined by the outerunit normal n. Here we again use (19) followed by applica-tion of the divergence theorem. Using (25), averaging (24)over one period, T, and recognizing that the period averageof Sn$(dufldt) dy vanishes, we obtain

u*dydt. (26)From (22) and (26) we conclude that

(27)

which is identical to the temporally-averaged (16) despitethe presence of convective contributions. Now, given that uis not the Stokes solution as defined by (13) and (14), andhence the maximizing property (16) is lost, it follows thatJo(u) and hence, K can only decrease relative to the Stokespermeability.C. A nip-element method

The computation of the configuration permeability is fre-quently plagued by the presence of excessively close pairs ofinclusions forming very small gaps-nip regions. This, at best,results in stringent mesh generation, ill-conditioned systemsand poor load balancing. At worst, numerical treatment maysimply not be possible and the particular realizations must bediscarded. The nip-element method is a computational arti-fice proposed in order to significantly mitigate these difficul-ties at the expense of introducing small bounding errors. Themethod is discussed below.1. viscolls flows

The (Stokes) permeability, K, enjoys the variational prin-ciple (16) which facilitates rigorous construction of nip-region-based lower and upper bounds. To generate an upperbound, K[JB , we simply enlarge the nip region as shown inFig. 2(a), thus allowing for a larger flux of fluid across thegap than would actually pass. To obtain a lower bound,KLB, we simply replace the nip region with a blockage as

shown in Fig. 2(b), preventing flow across the gap. Thesemodels will not change the weak variational statement of theoriginal problem as given by equations (13) and (14), exceptfor replacing the original domain s1 by the new modifieddomain, fi, defined as g=n\@for the lower bound, and asc= $1U L8 for the upper bound, where 58 s the excised nipregion as defined by Fig. 3 for the two different nips. The nipregion is defined by two geometrical parameters: the inter-particle spacing, a; the nip height, /?, above the center lineconnecting the two forming inclusions. Thus, any two inclu-sions with a center-to-center distance less than 1 + (Y, areconsidered forming a nip, where LY, is a specified criticalvalue.The proofs for the claimed hierarchy of the bounds,KLB< KC KUB, are based on variational embedding argu-ments, in which a maximizing solution on a smaller domainis extended (by zero, due to the no slip boundary condition)to construct an admissible but non-maximizing candidate ona larger domain. To translate this statement mathematicallylet un be the correct Stokes solution for the bare (no nip)problem defined over the domain 61; uq be the correctStokes solution for the nip-augmented problem defined over5Y and Olg be the zero field extending over the region .LV.For the upper-bound permeability, K~, we can write [usingequations (11) and (16)]:

K-A Jn(Ua)

1=pn(un>+Jdo)

=KUBT 08)where the last inequality follows because the constructedfield un U 018, although admissible, is not the correct Stokessolution over the domain 8 and thus will not maximize thefunctional Jdv) . The proof for the lower-bound permeabil-ity, KI.B , is analogous and is presented in Ghaddar.39We remark that sharper bounds can also be constructedin an analogous approach to the effective conductivity prob-lem presented in Cruz, Ghaddar and Patera. For example,to obtain a sharper lower bound, we can assume a one di-mensional Poiseuille flow in the nip gap region in which the

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candidate velocity profiles are reduced to parallel parabolicones. .The lower bound will obtain physically because themodel implies an infinite viscosity in the direction normal tothe flow, and mathematically because of the additional re-striction on the functional space in which we search for asolution. The mathematical treatment is, of course, morecomplicated with the presence of the Lagrange multiplier, thepressure. The Poiseuille flow assumption, in fact, is veryclose to reality in the nip regions. Therefore, we expect ourcrude models to be rather sharp. This is because of the sec-ond order dependence of the velocity, and hence, the perme-ability on the inter-particle spacing. This has been observednumerically and presented in Sec. IV A.2. Extension to inertial flows

Unfortunately, the Stokes bounds are no longer rigorousfor the Navier-Stokes problem due to the loss of thefunctional-maximizing property of the permeability, equa-tion (16). However, it is expected that, the bounds will stillwork since the tlow is expected to be effectively inertia-freein the narrow. nip regions even when the global Reynoldsnumber may be large. Sensitivity derivatives, could be used(e.g., with respect to say, the interparticle spacing) to assessthe merits of this naive approach.III. PARALLEL, NUMERICAL TREATMENT

The numerical solution of the supercell problem encom-passes four main tasks: domain partitioning; mesh genera-tion; finite element solution; parallel implementation. Thesetasks are discussed below.A. Domain partitioning

A tensor-product partitioning strategy is employed to di-vide the original supercell domain into P smaller non-overlapping Subdomains or subcells, where P is the targetnumber of processors. The procedure, presented in detail inGhaddar,39 is accomplished by first, overlaying an initialstructured grid upon the square supercell domain. The gridlines are then individually repositioned to eliminate any ex-isting geometrical degeneracies. Next, the necessary geo-metrical and topological information associated with the in-dividual subcells (identified with the grid bricks) areobtained. Last, the bare supercell is augmented with the de-sired type of nip elements, and the required modifications areperformed.The tensor-product partitioning procedure is illustratedin Fig. 4 for an original realization containing 50 inclusionsat a concentration of 0.5, which is partitioned into P= 16subcells. Generally, the resulting subcells will comprisesmaller disconnected domains. This fact, however, will notsacrifice the structured topology associated with the initialgrid which has a significant impact on reducing the commu-nication overhead in the parallel computing environment andhence. boosting the performance.

B. Mesh generationMesh generation is performed in parallel and is based on

triangular finite elements which provide greater geom&ric

original domain

nip-augmented partitioned domainFIG. 4. Example of a tensor-product artition of a realization containing 50inclusions at 0.5 concentration nto lower-bound nip-augmented16 subcells.

flexibility for decomposition of complex domains as opposedto quadrilateral elements. In this work we employed aVoronoi-based triangulator, MSHPTG,4* for its demonstratedrobustness, minimal input information, and satisfactoryresult.24 MSHPTG requires on input a boundary grid and thecorresponding topological description. It honors the bound-ary grid density in the generation of interior elements, whichmakes it suitable for achieving a desired mesh density distri-bution in a fully automated algorithm by simply controllingthe boundary grid.The main problem is then reduced to how to discretizethe boundary curves. The boundary resolution which ispropagated smoothly to the interior, must therefore respectregions of potentially high gradients if discretization errorsare not to dominate or even destroy the numerical solution.These high gradients are expected to occur at regions of clus-tered inclusions. The procedure used for gridding the bound-aries is presented in detail in Ghaddar.39 It makes use of theso-called minimal distance as a guide in the node distribution


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FIG. 5. Finite element mesh generated n parallel for a realization of 17inclusions at 0.55 concentration,partitioned nto four subcells.Here 11 nipsare present.The mesh s composedof 4243 triangles.

process. The minimal distance between a point y and a sur-face S, denoted as d&y,S) is defined bydHi9.S)=m4y-xl, (29)

XESwhere [y-xl is the usual Euclidean distance between twopoints. This distance is used in conjunction with the follow-ing distribution functior? for meshing of arc and line seg-ments making up the boundaries of the subcells,

1h(y) = I -1mldH(y,S) + l/h,,, . r (30)where h is the mesh spacing at point y; h,,, is a nominalmesh spacing parameter which is defaulted to, for suffi-ciently large dH ; nz is a refinement parameter that controlsthe near-cylinder density; and r is a global control refinementparameter typically set to unity. In effect equation (30) hon-ors the relative closeness of neighboring inclusions whendistributing the nodes along the inclusion surface. The reso-lution will be denser in clustered regions and lighter in well-conditioned regions. Figure 5 shows the finite element meshfor a realization of 17 inclusions at 0.55 concentration parti-tioned into four subcells.

C. Finite element treatmentThere are two basic ingredients in the finite elementanalysis: discretization of the weak variational form; solutionof the resulting discrete system of equations. For the firsttask we use isoparametric second order triangular finite ele-ments for accurate boundary representation and greatermeshing flexibility. For the second task, we adopt iterativesolution algor ithms since they require both dramatically lessmemory and significantly fewer operations than direct meth-ods, and they are more readily and efficiently parallelized.In the following we discuss the Stokes and the Navier-Stokes problems, respectively. We note that, for purpose of

simplicity, parallel implications associated with the iterativesolution algorithms are deferred until Sec. Ill D.

1. The Stokes problemDiscretization. The space-restricted statement of theweak variational equations (13) and (14) takes the followingform: Finduh E [V,(fL)] andph E P,(a) suchthatuiUh,V)-b(Ph,V)=(cSil,V),v4V,W)12, (31)-biq,ufJ=O, vqe Phin), (32)

where a(.,.) is the standard continuous Laplacian operator,dub v)=I V.vV-u,dCl; iwn

b (. , .) is the bilinear form,b(q,v) = s qV -vdR; (34)n

(.,.) is the inner product,(q.u) = /gpvdO:

and the subspaces V,, and(35)

Ph are defined as

where lok reads: restricted to finite element k. The choice ofthe approximation spaces for both the velocity and pressureis made such that the inf-sup condition4 is satisfied. In par-ticular, the choice is designed on the use of the43aylor-Hood PI-P1 quadratic and linear triangular ele-ments pair that are known to honor the inf-sup condition.Proceeding with the discretization we arrive at the fol-lowing linear system:

[Q, -It* I;]( ;;}=(By, (38)

where A is the discrete Laplacian operator,A =ij (39)

Dr=D,,i is the derivative operator corresponding to thebilinear form b,Dmni z s ji,,~ don;- I i40)

where 4, and (6, are the nodal basis associated with thelinear and quadratic meshes, respectively; B = Bij is the massmatrix defined by(41)

and (1) is a vector of unity entries. It should be pointed outthat the system matrix in (38) as constructed, does not incor-Phys. Fluids, Vol . 7, No. 11, November 1995 Chahid K. Ghaddar 2569


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porate the associated boundary conditions of X-periodicity, the old values. The uniqueness condition for the pressure isno slip inclusions walls and the zero average of the pressure effected by imposing the zero average requirement on ph.vector. In the context of iterative solver, these boundary con- Finally we remark that no special treatment is required in theditions are imposed as additional steps to the iterative algo- presence of nip elements other than enforcing the no sliprithm (described below). condition for the velocity along the nip edges.The dissipation-related expression for Kh correspondingto (16) is given by 2. The Navier-Stokes problemDiscretization. The spatial discretization of the varia-tional, unsteady, incompressible, Navier-Stokes equations(20) and (21). takes the form (for clarity we drop the sub-script h)

1Kh=--Tx- (42)

Due to the functional dependence of Kh on the Laplacianoperator A, Kh iS expected to inherit the convergence prop-erties of A (see Wang and Fix44), that is, third order conver-gence with second order finite elements. This is verified inSection IV Al.Solution strategy. The solution is based on the iterativeUzawa algorithm,45~46which decouples the pressure and ve-locity (via application of block Gaussian elimination to thesystem matrix) into two equations,

2sphEtz, -D,IA-BaiI 9 (43)AUi,h= D,Tph+ BSi,,

where S is defined by2

S=C DiA-D I .i=l

(44)

(45)Equation (43) is first solved for the pressure by a nestedconjugate gradient iteration48,49 ollowed by two elliptic so-lutions for the velocity components. (Unfortunately, no pre-conditioning was used.)The A-periodicity of the velocity and pressure is dis-cretely imposed on the iterate by identifying appropriatepairs of nodes as the same node along periodic edges,50.51that is, summing the contributions and writing the sum over

-+C(U)Ui =-Aui+DTp+BSil,I- DiUi=O , (47)

for i= 1,2, where C(u) = C,(U) is the skew-symmetric dis-crete convection operator:(48)

Temporal discretization is carried out by treating the vis-cous term implicitly using first order backward Euler integra-tion scheme, and the nonlinear convection term explicitly bythe third order Adams-Bashforth. This leads to the followingsemi-implicit time-stepping scheme:I 1+ Atmte. B u;+-D;pfpn+n

=&Buy-pi a,C(U-q)u~-q+s~i*I (49)q=fJ-Di,;+l=O, (5ojwhere At is the time step at iteration ~1. The coefficientsLYE re defined by an algebraic functions of the three last timesteps,

~2At-1(Atn-1+Atn-2)+6Atn(2Atn-*fAt-2)+4Atn2Atn-1(Atn-+Atn-2) (51)

1 6At(At-+At-*)+4Atn2*=-E I At-1&-2 1 (52)6APAt- +4Atn2 1At,,- I+ Atn-2)AtTL-2 63)

For the case of a constant At, the coefficients aq reduce tothe classical constants documented with third order Adams-Bashforth, namely LYE s , a1 = 2 and ~ys 2.

The merit of this temporal scheme mixing is twofold:first by treating the nonlinear term explicitly, the equationsbecome Iinear at every time step which would significantlysimplify the solution; and second, the implicit treatment ofthe diffusion operator alleviates the stringent time step re-2570 Phys. Fluids, Vol. 7, No. 11, November 1995

quirement associated with the stability of the diffusion op-erator. The time step is then governed solely by the maxi-mum eigenvalue of the convection operator which must lieinside the stability region of the explicit scheme to honor thewell-known Courant condition.39 A value of 0.5 for the latterhas proved a stable limit for the moderate range of Reynoldsnumbers considered.The temporally-averaged numerical permeability followsfrom the dissipation related expression, (42):

(54)or, equivalently, from the Darcys definition:

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1 TKh?c Buldt, (59where T is the period of flow oscillations. The sum in (55) isover the entire length of the global vector Bul , where B isthe system mass matrix [equation (41)]. The two expressionsshould be equivalent which provides a useful check for theimplementation. The temporal averaging in unsteady situa-tions is effected by averaging the permeability over a fulltime period.

Solution strategy. One possible solution strategy wouldbe to apply the Uzawa algorithm to (49)-(50) at each timestep. While this is an accurate method, it suffers from thevery ill-conditioned pressure-solve operator; in addition, theexpensive nested Helmholtz inverse solution makes this ap-proach very inefficient. Instead a recent operator splittingmethod is used which breaks the original system into a se-quence of three decoupled subproblems integrated in a newintermediate variable5253 as follows:

&b.d vertex

2fw+= D;p-l- -&, Buy-px q=o a,C(un-q)u;-q+ Bail,

(56)E(P n+l_p)=-& D&f, (57)

0 r.

1 ,oo:: 1 - I

Illrri,&;+l+; &nB-lD;(p+-p), (58)where ti is the intermediate velocity field,H = [A + (p/A tj B] is the Helmholtz operator, and E is theconsistent Poisson operator given by

0 r I 0 L IFIG. 6. Definitions of essential parallel constructs.

2E= C DiB-DT. (59)I=1

It is important to note that this first order split formulationincurs additional temporal errors of order At. However, ifsteady time-independent solutions are reached, these errorsvanish completely. Furthermore, this splitting scheme is per-formed on the well-posed discrete original equations basedupon a consistent approximation spaces and thus requires noartificial or special treatment of boundary conditions, whichplagues the more classical fractional splitting method basedon the continuous original equations.34

where Iok] represents the area of element ank. Note that thissubstitution also eliminates the iterations requirement for thelast step (58). The error incurred due to this approximationvanishes completely if a steady state time-independent solu-tion is obtained, as is evident from equation (46). For thecase of an unsteady solution, a small damping effect wasobserved when using the lumped mass matrix (see SectionIV c 1).D. Parallel considerations

The first and third steps are solved by the standard con-jugate gradient iterations. On the other hand, the pressuresolution requires a nested conjugate gradient iteration similarto the Uzawa used for the Stokes problem. The mass matrix,however, is very well-conditioned and when pre-conditionedby the diagonal lumped mass matrix, the number of innerconjugate gradient iterations is reduced to order unity, whichsignificantly mitigates the otherwise ser ious drawback of thealgorithm. To circumvent this inner solve completely we re-place the mass matrix, B, by the diagonal lumped mass ma-trix ji defined elementally as

In this section we describe briefly some of the aspects ofthe parallel implementation on a distributed-memory,MIMD, message-passing architecture, here the Intel iPSC/860 32-node hypercube. In particular, we do not intend todiscuss the broader context of parallel processing,46,55,56 utrather to outline briefly the parallel constructs necessary forperforming the communication- involved tasks, as well as theparallel performance measures of the implementation.1. Parallel constructs

%=~ace(BkjlfikI (60)

The parallel constructs are best described by referring tothe schematic shown in Fig. 6 in which a supercell is parti-tioned into, here, four subcells. The relevant information as-sociated with each subcell is then loaded onto the assignedprocessor. This information includes the geometrical and to-

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pological descriptions which are needed for the subsequentmeshing; pointers that identify common boundaries withother processors along with other associated information forthe required exchange of data. The subcells are subsequentlytriangulated followed by a global checking to ensure the in-tegrity of the entire supercell mesh.The data structures are based on the now standard tech-nique of elemental evaluation of operators in which no as-sembling of the system operator is required.6*47 nstead, thequadratures are performed only to assemble elemental matri-ces. All the iterates are stored locally based on the individualgrids of the subcells. For example, in Fig. 6, the solutionvectors are split into four locally defined data structures. Atthe end of the solution stage, each subcell will hold the cor-rect information based on its mesh, and hence the solution iscompletely known over the entire supercell.The back bone of the Uzawa and the splitting scheme isthe iterative conjugate gradient algorithm. The computationalcomplexity of the latter involves two main operations:matrix-vector product and vector-vector inner product. Thelatter is performed locally on each processor followed by aglobal log2P gather-scatter operation which sums from allthe processors and redistributes the total sum back. Theformer is effected using the elemental evaluation techniqueby carrying out the multiplications locally on each processor.This is followed by a direct stiffness summation, DSS, pro-cedure which sums the local values at the elemental localnodes associated with a global node and redistributes thetotal sum back to the elemental local nodes (see Fig. 6 fordefinitions of local and global nodes). By virtue of the par-titioning procedure and also due to periodicity, certain globalnodes will appear on more than one (and as many as four)processors. This will require in addition to the local DSS oneach processor parallel communication (exchange of data)among processors either to attain the correct global valueslocally, or to impose periodic boundary conditions. It be-.comes critical for parallel efficiency to minimize the numberof messages (sends/receives) and to maximize their lengths.For this purpose, boundary nodes are classified and groupedinto either vertices if shared between more than two proces-sors, or edges if shared between strictly two processors. TheParallel part of the DSS (in addition to the serial part per-formed locally on each processor), is then to sum the contri-butions from all local nodes associated with edges and ver-tices and redistribute the correct sum back to the contributinglocal nodes. (For details on the implementation seeGhaddar?9)

2. Parallel performanceTurning to the parallel performance, it is generally char-acterized by the following four interrelated quantities: thespeed up, S, ,

Ldc(1)Sp=rtalc(P)where tcalc(n) refers to the calculation time on p2processors;the parallel efficiency, rip,

the load balance, A,, ,(62)

maxpEP pdoAb= mqsp pdof- fidoffid0-f +1= fidof 63)where NFf is the number of degrees of freedom, residing onprocessor p and Ndcf is the average number of degrees offreedom which is equal to the total number of degrees of6-eedom divided by the number of processors. Note Ab& 1,and Ab= 1 is the ideal value. This definition leads directly tothe parallel efficiency in the limit of no communication over-head,

1rl,=z 7 (64where ;ip refers to the no-communication-overhead casewhich is, by definition, larger than rip [relation (64) followsby recognizing that, in the limit of no communication and nononlinear hardware effects, t,& P) scales withmaxp,$$f and t,,[,( 1) scales with the total number of de-grees of freedom]; and last, the overall MFLOPS which canbe estimated by multiplying the typical MFLOPS capabilityof a processor by the number of processqrs and the parallelefficiency. These defined measures arc generally problem-dependent, and may vary significantly from one case to an-other. For a typical calculation with O(50,OOO) total degreesof freedom and good load balance, an estimated Q(80)MFLOPS and a parallel efficiency of ~90% was achievedon 16 processors of the Intel i860.39IV. RESULTS AND DISCUSSIONA. Nip-element examples

The nip-element method offers potentially significantcomputational benefits particularly in conjunction with itera-tive solvers. This is a consequence of the elimination of oth-erwise element-dense nip regions which is directly reflectedin improving the problem conditioning, the reduction of de-grees of freedom, and the improvement of load balancing.These combined effects result in considerable CPU savingsrelative to the original nip-free problem.40 Here we presenttwo examples: the first one is intended to demonstrate therigor of the hierarchy of the Stokes bounds for an interestingsingle-nip configuration, whereas the second is intended totest the method in the presence of important convective ef-fects for a random realization.1. The Stokes permeability bounds

We begin by verifying the third order convergence of thepermeability expected with the second-order isoparametricdiscretization. This also serves as a benchmark for the imple-mentation as well as provides a guide for the expected dis-cretization error as a function of mesh density. In Fig. 7 weplot the absolute error, E, = K~ - K,,,,~ versus the represen-tative mesh spacing ff = hnomlr [see equation (30)] usingboth subparametric and isoparametric implementations. Thetests were carried out for a hexagonal array at concentration


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FIG. 7. Convergence f the permeability with mesh spacing or a hexagonalarray at concentration 0.6. The third order convergence s observed withisoparametric inite elements.

0.6. (shown in Fig. 8) for which ~,,,,~=0.001452 (Sanganiand Acrivos). The calculations were performed on four pro-cessors of the Intel i860 hypercube and consumed 30 to 1000seconds of computation time. (Total degrees of freedomranged from 1400 to 12000.) We observe the third orderconvergence of the permeability with isoparametric ele-ments. Note that the skin effects due to subparametricrepresentation of the boundary obscure the third order con-vergence (see Strang and FLK,~ pp. 105-116 and pp. 192-204).Next we consider the realization shown in Fig. 9 whichhas N=2 cylindrical inclusions at a concentration of 0.1, inwhich the centerline connecting the two inclusions is parallel

FIG. 9. Special realization containing two inclusions at 0.1 concentration,(h=3.963).

to the flow direction. The motivation behind this examplestems from the argument that the flow blockage strategy in-curs counter effects of reduced drag which may obscure theexpected lower bound permeability. This is because theblockage strategy reduces the surface area exposed to thefluid and eliminates full dissipative regions, particularly inthis symmetric flow situation in which the nip region has avery minor flow-passing role. The rigor of the bound-hierarchy proof is verified by the numerical results shown inTable I in which the following quantities are presented: therelative error E, (dependent on the exact solution K),

where K* is either ~~~ or ~~~ ; the relative error E, (i&e-pendent of K),

x 100,where

#T+ KUB+KLB2. ;

(66)

the degrees-of-freedom per velocity component, Ndof u .Indeed, these counter effects only render the bounds verysharp. We remark that these results were obtained for thecase CZ=0.075,/I= 0.1 (recall LY s the inter cylinder spacing;j3 is the nip height above the center line) using a relativelyvery fine grid and stringent incomplete iteration control. [The

TABLE I. Bound results for the Stokespermeability of the symmetric real-ization shown in Figure 9.K* or i? E, -6 Ndof u

FIG. 8. A particular finite el ement mesh comprising 1965 finite triangularelements or a hexagonalarray at concentration0.6.

K 0.7439011 *.* . 13697KLB 0.7439008 4.OE-5 *.a 13214KUB 0.7439019 1.5E-4 I** 13357E 0.7439013 ... 3.4E-5 ***

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PIG. 10. Random realization containing 14 inclusions at 0.3 concentration(A = 6.054) used n the inertial nip-element example of Section IV A 2: (a)original geometry; (b) lower-bound geometry; (c) upper -boundgeometry.

discretization error is expected to be 0( 10m7) based on theconvergence test of Fig. 7. This error is smaller than thereported bound gap.]

2. Inertial porous-media flow exampleAs discussed in Section II, the bound hierarchy is stillexpected to obtain at finite Reynolds numbers. This is dem-onstrated using a random realization containing 14 inclusionsat a concentration of 0.3 shown in Fig. 10(a). Here, p, ischosen to be 10000, at which a moderate flow Reynoldsnumber, Re, is expected to result. The geometries of thelower and upper bound calculations are shown respectivelyin Figs. 10(b) and 10(c) in which four nips are present. TableII presents the bound results for the permeability calcula-tions. In addition to the quantities defined in Table I, is alsopresented: GS, defined as the ratio of the maximum edge-length of all finite elements present to the minimum edge-length of all finite elements present; CPU(p) the (wall clock)processing time in seconds on p processors; Ab the loadimbalance defined in (63).

TABLE II:Bound results for the inertial-flow permeability of the randomarray realization shown n Figure 10 at p= 10000.K* or i? E, ,f?, Re Ndof, CPU(16) GS Ala

K 0.006873 ..+ 1.. 68.7 15508 872510 84.9 2.1KLB 0.006811 0.9 ... 68.1 13743 278324 18.7 1.4KC/B 0.007029 2.3 ... 70.2 13783 245500 14.7 1.4i 0.006920 ... 0.7 69.2 -*- 523824 ... ...

2574 Phys. Fluids, Vol. 7, No. II, November 1995

PIG. 11. Speed contours for the Navier-Stokes calculation at ReynoldsnumberRe = 68.7 in the realization of Fig. 10(a).

The hierarchy of the bounds is obtained despite the rela-tively important convective contributions, [ Re = 0( 701, seeTable II]. (Of course, this does not rule out the possibility ofcounter examples.) Note that the nip gaps form percolationialves; the flow is effectively inertia-free through thesepassages, even though the global Reynolds number is large.This is made evident in Fig. 11 which contours the speedover the nip-free domain. The clustered inclusions in themiddle almost block the flow entirely, forming a stagnationregion. Finally we remark that, given the relatively moderategeometrical stiffness of the original problem (see Table II),the savings for this particular realization are somewhat mod-erate. (We only obtain a factor of 3 reduction in CPU timefor both the lower bound and upper bound calculations com-pared to the original problem.) More typically, for worsegeometrical distortion situations, the conditioning of the con-sistent Poisson operator [equation (59)], associated with thepressure solve deteriorates rapidly, resulting in dramatic ob-served savings.6. Creeping flow permeability results

The supercell permeability, K,(c,X), is calculated at aparticular h, X0%7, for 27 concentration values ofc=O.O5,0.75 and cj=0.1+j0.02,j=0 ,..., 24. The choice ofA is conjectured to be reasonably largea to reduce spatialaveraging effects and hence, K~(c,&,) is expected to be agood .approximation to the true random media.The nip-element procedure has been used rather exten-sively to overcome the dramatic negative effects of the geo-metrical stiffness on the computation time.40 In order tominimize the bound gap error, however, a stringent cr,,c~~=O.05, has been defined. The worst obtained nondimen-sionaI relative bound gap error [see equation (66)] was forthe highest concentration considered, c = 0.58, and amountedto 0.0520.Turning to the Monte Carlo components of the algo-rithm, for any particular concentration, c, we approximatethe integral (17) asthe average of the lower and upper boundsampIe means,

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R

1 0 . . . . . . . . . . . . . . c.. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . * . . . . . . ..: : : : : : : : ! : :: : : : : : : : : : : : : : : : :: : . : : : : : : : :: : : :~ : :~ : :: : : . . :. . z.ii;,:L ::::::: ;z;:;::k::::> >:::::..::

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . ..A

. . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . . . . ::.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 . . . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . . .. . . . . . . . . . . . . . . ^ . . . . . . . . . . . . . . : . . . . . . . . . . . . . . . I . . . . . . . . . . . . . . . : . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . ^. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . : . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .;. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . .

IO ~~iiii~ii::::~ril i!~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~:~~~~~~~~~~~~~:~~~~~~~~~~~~~~:~~~~~........... ................................ . ......................... .. 3........................:ii:iiiii:i_r--j~iiiiiii~iiiiiiiiiiiiiiiiiiiEiiiiiiiii~iiiiiiiiii~iiiiiiii~iiiiiiiiiiiiiii~: ............. .~. .g.. ...... .. ............ ............... .. ............ .. .............10-1 : ............ ...i:,:.:. ............ . ... ........ .... .... ................ .................. ...... .......... ..... .... ...... ............ .. ...>I.::.................... ......................................................................................... .. ........................................, ................................................ ...................... . ............................................. ................ .... ............. ................. ...............................................) .................. . .............. ........ .................+

... .l.li!:l~iii::::~:::::::::::::::i::::::::::::............... ........... ....................+ ...............lii!iiiiiii:ii:::::::::.:::::~~::~~~~~:~~:::~j .:$: i..llIiiii:lji~lii!~~~~~~~~~~.............. . ............................. ;.. ......... .:::::::::::::::aiiiiiiliijiiiiiiiiiittiiiriiilieiiiil::::::.,,S:Fpiiii,ii,iiiiiiiiiiiiiiii.............-::.::::::::::T:::::::::::::::i:::::::::::::::~:::~:::~:::::::~::,.~y&c.:~:I..:;:;::~.............. .. ............. ............... ............... ...........10:::..:..::::::~:::::::::::::::::::::::::::~:~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~;. : i,o~:-:-::~~~~lii~

0 0.1 0.2 0.3 0.4 0.5 0.6C

FIG. 12. The behavior of permeability as a function of fiber concentration n2D random arrays. The enor bars representone mean-normalizedstandarddeviation plus 5% bound gap error.

- -lT=KLB -+- KVB2 -.

wherek, = j$ Ki ,r=

here N, is the sample size, andK,= ~~({Y)~,c,h~),i= l,...,N,, is either the lower- orupper-bound configuration permeability. Likewise, the vari-ance associated with 2, G, is approximated by the averageof the lower and upper bound sample variances,

;i'2, dB t &B2 9

TABLE III. Mean values and associatedvarianCes or the creeping-flowtransverse ermeability of 2D random arrays of cylinders at the listed inclu-sion concentrations.

(69)

(70)

cr,(ch)- N _ Ii=,--$ (Ki-k*j2. (70

prominent of which is the increased drag due to increasedfluid-solid interaction and the likelihood of percolation-typeflow mechanism. This sharp decrease in the permeability canbe also interpreted in terms of simple channel flow mecha-nism. In the latter, the average velocity is proportional to thesquare of the plate separation; in porous medium, the aver-age nearest neighbor interparticle spacing, 0 (c- ), playsthe role of the channel separation. Predicting the form ofdependence on the concentration is not a trivial task and hasbeen carried out analytically only in the limit of diluteconcentrations.57 It is, interesting, however, to note that theplot suggests an exponential decay with increased concentra-tion of the form

where

C I? (5.20.05 9.721E-01 1.0738-030.675 5.848E-01 1.8328-020.10 3.622E-41 1.2638-020.12 3.337E-01 1.784E-020.14 1.980E-01 1.321E-030.16 1.4388-01 9.891E-040.18 1.257E-01 1.981E-030.20 8.753E-02 7.708E-040.22 6.690E-02 8.792E-040.24 5.259E-02 2.476E-040.26 3.715E-02 1.1358-040.28 3.161E-02 7.885E-050.30 2.424E-02. 8.609E-050.32 1.767E-02 2.142E-050.34 1.516E-02 3.847E-050.36 l.l89E-02 1.069E-050.38 9.943E-03 7.553E-960.40 7.179E-03 3.512E-060.42 6.372E-03 3.117E-060.44 4.964E-03 t 2.911E-060.46 3.363E-03 5.249E-070.48 2.972E-03 1.312E-060.50 1.969E-03 4.03OE-070.52 1.720E-03 4.947E-070.54 l.O99E-03 2.04OE-070.56 8.101E-04 1.908E-070.58 6.428E-04 7.614E-08

The sample size N, is set to 20 which is expected toreasonablycontrol the noise contribution. (For details on thesequential acceptance-rejection sampling procedure seeGhaddar3) For N,=20 realizations, the most time consum-ing calculation (~0.58) requires 42 hours of CPU time,and costs roughly $600 on 16 processors at relatively goodpadIe performance ( VI6-90% and 80 MFLOPS). Thiscalculation is simply prohibitive on a workstation.Figure 12 shows the behavior of the permeability datapoints of the 2D random arrays as a function of the inclusionconcentrations, along with error bars representing one mean-normalized standard deviation plus 5% as a conservative ac-count for the bound gap error. (The data points are listed inTable III.) We first note that the sharp diminishing of thepermeability with increased inclusion concentration is awell-known fact on account of many factors. The most

K,(C)=cq)e-PO=, (72)where the constants a0 and /Ia are determined from aweighted least square fit to the data based on the mean-normalized standard deviation as cya= 1.171 andPO= 12.736. Clearly this simple model is only applicable inthe indicated range of concentrations; in the limitc--+c,,,=O.82 (Berrymat?), the model will not predict azero value of the permeabili ty which is not consistent withthe physics of the problem. (No attempt is made to consoli-date the exponential behavior with any analytical analysis.)Next we evaluate the applicability of the Kozeny modelfor the 2D random arrays over the wide range of concentra-tions. In Fig. 13 we plot calculated values of the Kozenyconstant, k [equation (2)] based on the simulation permeabil-ity data points versus the porosity points 1 - cj . We also plot

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k

IO3 ..f . . 1.............. L . .L .t....... -.L . I .. . :...: . . . . . . . . . > ,.........., < . . . ../.............. % . . . . . . . L . :: .. . . . . . . _ . . . . . . . . . . . . . . . . . . . ,............ z . . . . . . ._,.. _ . . . . ,........................................i.. ...................... .i.. .............................................. *L.. ........ .. ..r........;current .... . ...................................................................................................

.......................................

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TABLE IV. A comparison of the creeping-f-low ransversepermeability of2D r andom arrays. Here N is the number of cylinders, N, is the samplesize,X is t he supercell size, and the % diff. is with respect o the presentdata.c Presentdata Sangani and Yao Howell?

N 6 16 1..N, 20 5-8 .0.1 x 6.865 11.209 .Kccr 0.3622 t- 0.1124 0.41750 + 0.03 0.4475% diff. . 15.3 23.6N 17 16 .N, 20 5-x .0.3 x 6.671 6.472 .i?G 0.02424 +- Q.28E-3 0.02550 t 2.OE-3 0.02425% diff. . . 5.2 0.004N 27 9 1..N, 20 5-8 .0.5 h 6.512 3.760 .kk(T 0.001969 t 6.35E-4 0.00235 + 4.24E-4 0.001925%diff. . 19.3 2.2

0 At the higher porosities, E+ I, the permeability isknown to vary as l/c in 3D beds (Batchelo?). Using thisfact in the Kozeny equation, it is easily shown that theKozeny constant varies also as l/c and the model is not validin that limit. The cell models of Happel and Kuwabara areconsistent with this fact as can be seen in Fig. 13. The be-havior of the 2D random arrays appears to be in agreementwith the cell model s although the dependence on c is likelyto be different. Unfortunately, we do not have sufficient datain that limit to fully demonstrate this behavior.We conclude the discussion on Stokes permeability witha comparison to the few semi-analytical results reported bySangani and YaoZ3 and the theoretical predictions ofHowells*i based on a correction to the Brinkmans model.Referring to the comparison presented in Table IV we com-ment the following.

0.08!----- I --- --I0.075 -0 current

0.07 - x Edwards et alx

K 00.065 x0 06 0 0

03:

I -2-l0 20 40 60 80 100 120 140 160 180Re

PIG. 14. Permeability values of the squarearray at c = 0.2 for several Rey-nolds numbers.

TABL E V. Temporally-averaged permeability results for the square array atthe indicated concentrationsand Reynolds numbers.c P K Re Steady/Unsteady

0.2

0.3

0.4

0.5

0.6

0 7.62OE-2 0 S295.8 6.602E-2 19.5 S635.0 6.141E-2 40.0 S900.0 6.0558-2 54.5 S1632.2 5.752E-2 93.9 S2025.0 5.644E-2 114.3 S2500.0 5.532E-2 138.3 S2809.0 5.346E-2 150.2 u3271.8 5.1648-2 168.4 u0 2.544E-2 0 S894.0 2.221E-2 19.9 S1953.6 2.042E-2 39.9 S5112.2 1.8958-2 96.9 S10261.7 1.6568-2 169.9 U0 9.0098-3 0 S2540.2 7.963E-3 20.2 S5610.0 7.221E-3 40.5 S14932.8 6.404E-3 95.6 S30870.5 5.5878-3 172.5 U0 2.949E-3 0 S7956.6 2.617E-3 20.8 S18198.0 ?.315E-3 42.1 S51256.9 2.055E-3 105.3 S118956.0 1:750E-3 208.2 IJ0 7.424E-4 0 S32869.7 6.742E-4 22.2 S74911.7 5.885E-4 44.1 S212152.4 4.8678-4 103.2 S491120.6 3.878E-4 190.4 u

(Note that in the two articles the radius is assumed unityas opposed to the diameter in the present work. Therefore,their data is renormalized such that the radius is half.) Ingeneral the comparison should be regarded as good even atthe largest difference since the confidence intervals overlap.However, it appears that the present data point at the lowestconcentration c = 0.1 is less accurate than the two otherswhich is evident by the large associated standard deviation ofthe current data point and the good agreement of the othertwo. Now given that our sample size is over two times largerthan the sample size used by Sangani and Yao, one suspectsthat the chosen supercell size of X-7 is perhaps not largeenough to filter out the special effects at the lower concen-trations. On the other hand, the present data point at thehighest concentration, c=O.5, seems to be more accuratethan the value reported by Sangani and Yao given our two-times larger supercell size and sample size in addition to thegood agreement with Howells. (In fact, Sangani and Yaoreport convergence difficulties at higher concentrations andlarger number of cylinders.) It is also interesting to note thatHowellss correction to the Brinkmans model by finding theresistance due to a second cylinder and then averaging overall its possible positions not only stabilizes the model, butalso provides accurate results.


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x Edwards et al

x0.018 -

?

2.5K

2

1.5

0.016 I0 50 100 150 200Re Re

IO c= .5

0X

0X

0

X0

Y ,100 200 :Re

06.5

0X

Re

FIG. 15. Permeability values of the squarearray at severalReynolds numbersand our different concentrations. he 0 s indicate current data; he Xs are datareported by Edwards et al.

C. Inertial porous-media flow results1. Regu/ar arrays

Square array results. For the sake of making a compari-son with the limited available numerical data, we restrict ourselected concentrations and Reynolds numbers mainly to thecomprehensive set reported in Edwards et aLz6 These con-centrations and Reynolds numbers arel Cj E {0.2,0.3,0.4,0.5,0.6},*Re E {0,20,40,95,180).Since our solution strategy does not permit defining Re

a priori, Re is calculated a posteriori from (23), where p isestimated based on the permeability results provided in Ed-wards et a1.26 such that to yield the desired Re.2578 Phys. Fluids, Vol. 7, No. 11, November 1995

Figure 14 shows the current permeability results(marked with OS) for the concentration c=O.2 at approxi-mately the Reynolds numbers listed above and four addi-tional values. (The data points are listed in Table V.) Alsoplotted on the same figure the data reported by Edwardset a1.26 (marked with Xs). The comparison for the remainingset of concentrations 0.3, 0.4, 0.5 and 0.6 is provided in Fig.15. Figures 14 and 15 show, generally, a reasonable, agree-ment between the two calculations. However, there are con-spicuous discrepancies that deserve to be investigated to es-tablish the possible sources of errors and, consequently, therelative reliability of the two calculations.

Referring to Figs. 14 and 15 we first note the relativelylarge discrepancy at the Stokes solution (Re =O). Compar-

Chahid K. Ghaddar


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Re = 150.2 r I

-0.4; I5 IO 1.5 20 25 30 35 40

0.4- I02 Re = 168.4

a1 O-02--0.4 -a0 5 IO 15 20 25 30 35 40

time unit

FIG. 16. History plots of the y,-component of the velocity at Re = 150.2,and Re= 168.4, respectively, or the squarearray at concentration0.2.

ing the two calculations to the exact solutions we find thatour Stokes permeability values are virtually exact, whereasEdwards et aZ. permeability values deviate appreciably by ashigh as 6.2% at c = 0.6. This deviation in the Stokes limit ofEdwards et al. indicates a lack of resolution due possibly tolarge iteration or/and discretization errors. In Edwards et al.a penalty finite element method based on the steady formu-lation of the cell problem is used. No iteration error controlis reported for the Newton iteration scheme. The computa-tional domain is restricted to one fourth of the square cell forthe Stokes problem, and half the cell for the finite Reynolds-

Lumoed Mass Matrix

-4.5 : f : :;o 2l.5 1 36.5 iI 3j.5 L3* 31.5

Exact MW ;s Matrix

46 2i.5 zi 2i.5 38 36.5 ;I 3i.5 i 3A.5time unit

FIG. 17. Time history plots for the yt-component of the velocity atp= 3271.8 using both the lumped mass matrix (upper) and the exact massmatrix (lower) for the square array at concentrationc = 0.2.

25- x 0 current+ + Bergelin et al20-R * Eidsath et al

VP 15I x

10

t

fx

x Edwards et al

0'0 50 100Re

-FIG. 18. Comparisonof dimensionless ressuredrop, VP, versusReynoldsnumber, or the squarearray at concentration0.5 between he experimentaldata of Bergelin et al., the numerical data of Eidsath et al., Edwards et al.,and the current data.

number runs, with symmetry assumptions on both the veloc-ity and pressure. Edwards et al. report using typically 400isoparametric biquadratic square finite el ements in half thecell. In this work a mixed finite element method based on theunsteady formulation is used, along with a stringent errorcontrol imposed on both the velocity and pressure for thesteady runs. (More precisely, the Lz-norm of the residual inboth the velocity and pressure is required to be less than

Re=O Re = 19.9

Re = 96.9 Re = 169.9FIG. 19. Streamlinescontours or tlow through a squarearray of cylindersat 0.3 concentrationat the indicated Reynolds numbers.


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c= .30.028 r I0.028I.024 0 currentx Edwards et al0.018 s?0.016,,I ( 2 , ox

0 50 100 150 200Re

2.5 -

Re Re

x 10 c = .4ii- (,IO-g98- 07-

K B-5- s?4-

03- X

Re

x109c= .I3

1.52

1.4-0

1.3- x1.2. 0

IE 1.1 -X

I-X0.9 - 0

0.8 - X0.71 00 50 100 150 200

FIG. 20. Permeability valuesof the hexagonalarray at severalReynoldsnumbersand four different concentrations. he 0 s indicate currentdata; he Xs aredata reportedby Edwards et al.

1 X low5 in virtually all the steady runs reported.) The com-plete square array cell was considered with periodic bound-ary conditions for both the Stokes and inertia1 runs, with atypical 1600 second order isoparametric triangular finite el-ements, which is twice the number of elements reported byEdwards et al.The lack of resolution could be the primarily reason forthe discrepancy at the finite Reynolds numbers results aswell. Another important factor is the unsteadiness observedat Re, approximately 150 and greater for which the flowexhibited steady periodic oscillations. The unsteadiness re-sults in larger dissipation and thus lower permeability. Thiseffect is clearly observed by the sudden drop in thetemporally-averaged permeability values for the last twolargest Reynolds numbers at c = 0.2. Note that Edwards et al.2580 Phys. Flui ds, Vol. 7, No. 11, November 1995 Chahid K. Ghaddar

steady approach with imposed symmetry assumption on thevelocity and pressure is inconsistent at the higher Reynoldsnumbers.Figure 16 plots the history cur ves of the yt-componentof the velocity at a selected point in the square array atc=O.2 for the two unsteady calculations at Reynolds num-bers Re= 150.2 and Re= 168.4 respectively. We first notethat the onset of unsteadiness at about Re = 150 is consistentwith limited previous visual studies in regular arrays ofspheres.6LY27At Re = 138.3 no oscillations were observed upto 30 time units.) Second, we also observe the expected in-crease in the amplitude of oscillations at the higher Reynoldsnumber which is also accompanied with a slight increase inthe frequency. Third, the unsteadiness phenomenon in peri-odic a rrays is in the form of traveling waves characterized by


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TABLE VI. Temporally-averagedpermeability results for the hexagonalarray at the i ndicated concentrationsand Reynoids numbers.c P K Rt? Steady/Unsteady

0 2.703E-2 0967.2 2.081E-2 20.10.3 2342.5 1.748E-2 40.99467.3 1.311E-2 124.115550.1 1.102E-2 171.2

0 l.O56E-2 023342.6 8.713E-3 20.40.4 4900.0 7.481E-3 36.626114.6 4.784E-3 124.960172.1 3.442E-3 207.10 4.108E-3 05625.0 3.653B-3 20.50.5 12723.8 3.16OE-3 40.260811.6 2.025E-3 123.1119577.6 1.525E-3 182.30 1.45133-3 015351.2 1.354E-3 20.80.6 37869.2 1.200E-3 45.4126878.4 9.0928-4 115.3226861.7 7.314E-4 165.9

SSSSUSSSSUSSSSUSSSSU

distinct periods and amplitudes, and is fundamentally differ-ent from the periodic wake (vortex shedding) phenomenonbehind bluff cylinders. (We remark that further analysis ofunsteadiness phenomena is beyond the scope of the currentstudy.)The unsteadiness raises an important question concem-ing the effect of the lumped mass matrix employed in placeof the actual mass matrix, for reasons of computational sav-ings. [The lumped mass matrix has virtually no effect onsteady time-independent solutions (see Section III C ).] Toinvestigate this effect, we re-compute the flow for the casep=57.2 (yields Re = 168.4 with the lumped mass method)using the exact mass matrix. In Fig. 17 we plot the timehistory of the yt-component of the velocity resulting fromboth strategies at an identical selected history point. The re-sults indicate that the lumped mass matrix has a small damp-ing effect as we observe the small increase in the oscillationsamplitude when using the exact mass matrix. The exact massmatrix method wiIl thus yield a lower permeability due to thehigher ampli tude oscillations and thus higher dissipation.The actual reduction in the permeability however, was foundto be inappreciable (the relative change is less than 0.33%).This slight improvement in the measurement does not war-rant the 0 (5 > 0 ( 10) more expensive exact matrix method.We next compare our results against the available ex-perimental data of Bergelin et al.,63 and the numerical dataof Eidsath et al.= at c= 0.5. The comparison is shown inFig. 18. We first remark that the data are presented in termsof a dimensionless pressure drop, 6, defined in (8) ratherthan permeability, K, used in the previous comparisons. Thetwo are, of course, related by the following relation:

- 1VP=&. (73)

For the first two Reynolds numbers, the figure suggests thatour data lie closer to the experimental measurements of

Bergelin et a2. than the data reported by Edwards et al. andEidsath et al. Our last two data points happen to be at higherReynolds numbers than the rest of the data which precludesaccurate comparison.We conclude the discussion on square arrays by describ-ing the flow structure at different Reynolds numbers. In Fig.19 we plot the streamlines for the square array at concentra-tion 0.3 for several Reynolds numbers. At finite Reynoldsnumber a pair of symmetric vortices is seen to form in thegaps between cylinders. As Reynolds number is increased,the vortices intensify with their centers pushed apart result-ing in parallel streamlines through the free path. At the high-est Reynolds number the flow is unsteady and the plot rep-resents an instantaneous streamlines configuration. A deeperqualitative picture of the flow pattern with increasing Rey-nolds numbers can be described in conjunction with the vi-sual studies of Dybbs and Edwards. The latter consideredporous media consisting of Plexiglas spheres in hexagonalpacking and Plexiglas rods arranged in a complex, fixedthree dimensional geometry. Although the pore structure isdifferent than our simple square array, there is still a closeparallel in the flow structure between the different structures.The visualization studies showed that at Re = 1 boundarylayers begin to develop near the solid boundaries of thepores. At Re> 10, the boundary layers become more pro-nounced and an inertial core appears. As the Reynoldsnumber is increased, the core flows enlarge in size and theirinfluence become more and more pronounced on the overallflow picture. The first evidence of unsteady tlow is observedat Rem 150, in the form of laminar oscillations in the pores.These oscillations take the form of traveling waves charac-terized by distinct periods and amplitudes.

Hexagonal array results. The next type of regular arraysconsidered, is that of a hexagonal arrangement as shown inFig. 8. Again we restrict our selected concentrations andReynolds numbers mainly to the set reported in Edwardset al. to allow for constructive comparison. These concen-trations and Reynolds numbers are

l cj~{0.3,0.4,0.5,0.6},l Re ~{0.20,40,120,180}.Unlike the square array, the permeability of the hexago-nal array differs along any two orthogonal directions. Here,we compute the permeability along the y t-direction for thehexagonal configuration shown in Fig. 8, which we shalldenote simply as K. (Our yt-direction matches y,-directionwith respect to the Edwards et al. hexagonal cell.) It is im-portant to note here that, due to the nonlinear Navier-Stokesequations, knowledge of the permeability for any two or-thogonal directions can not be used (by simply adding vec-tors) to calculate the permeability along a different obliquedirection. This statement is not true in the case of linearStokes flow in which knowledge of the permeability tensor

Kij will uniquely specify the permeability along any desireddirection.We note that our computational domain is four timeslarger than that considered in Edwards et al. (Our hexagonalcell contains four inclusions whereas the Edwards et al. cellcontains two inclusions but they consider only half of it withsymmetry assumptions.) We have used typically O(3000)

Phys. Flui ds, Vol. 7, No. 11, November 1995 Chahid K. Ghaddar 2581wnloaded 29 Dec 2010 to 210.212.58.168. Redistribution subject to AIP license or copyright; see http://pof.aip.org/about/rights_and_permissions

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0.022-.022- Ox.02 -.02 - 0

n 00.018 -.018 - X0.016 -.016 - 0

Re Re

x103 c= .54.5, I

4

i

X3.5Xtc

02.5 -0

2-

1.50

ox0

Y ,100 200 300

Re

c== .616 x lOA

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