MOMENTUM TRANSPORT MECHANISM FOR WATER FLOW OVERPOROUS MEDIA

By Christopher Y. Choi1 and Peter M. Waller2

ABSTRACT: The momentum transport phenomena at the interface of the porous medium and fluid have beennu~erically invest~gated. The single domain approach is used with matching boundary conditions; that is, the~nnkman-Forchhe~mer-extended Darcy equation is used for the present study. Five typical porous media foundm natural and engmeered systems are selected in order to cover a wide range of the Darcy number (6.25 X1O-4 :s Da :S 5.90 X 10- 11

). In addition, six different Reynolds numbers (10 :s R :s 1,000) are tested for eachcase. When Da > 10-7

, the results showed the importance of viscous shear in the channel fluid. The viscousshear propagates across the interface into the porous medium and forms a transition region of disturbed flow inthe porous medium. The depth of penetration is only dependent on the Darcy number of the porous mediumrather than the Reynolds number and the shape of velocity profile. In the vicinity of the interface, it is clearthat Darcy's law is inappropriate to describe flow in a permeable wall fracture or flow over porous media.

INTRODUCTION

Background

In recent years, the slip boundary conditions between fluidsand porous media have been investigated in conjunction withpractical applications such as ground-water hydrology andmass transfer in packed beds. Typical examples of fluid flow,chemical and thermal interaction between a saturated porouslayer and a fluid layer, are encountered in the surface waterand soil system and in ground-water flow between the frac­tures and porous blocks system.

A few studies have concentrated on fluid/porous mediumsystems in conjunction with food and wood drying processes,geophysics, and engineering applications. Oliveira and Hagh­ighi (1994) studied conjugate heat and mass transfer during aconvective drying process of a wood block. Continuity at theapproaching air/wood interface was assumed. Nield and Bejan(1992) devoted a section of their book on convection in porousmedia to the interface conditions, and Prasad (1991) incor­porated an extensive review of this subject. Although matchingboundary conditions at the porous medium and fluid interfacewere proposed and used by some researchers, the conditionsat the fluid and porous medium interface have not been rig­orously verified.

Berkowitz (1989) discussed the existence of a boundarylayer at the fractured wall (a porous medium) interface. Ob­viously, the boundary layer region in the porous medium isaccounted for by the macroscopic shear term, and Darcy's lawis not compatible with the existence of this transitional regionin the porous medium. He demonstrated that neglecting thepresence of such a transition zone can lead to errors of up to20% in estimates of fracture hydraulic conductivities. Chellamet al. (1993) used slip boundary conditions to obtain an ap­proximate solution to the two-dimensional (2D) Navier-Stokesequations for steady flow in a channel bounded by one porouswall subject to uniform suction. They adopted a statistical slipboundary condition at the fluid and porous interface. The ef­fects of fluid slip at the porous boundary on the axial and

'Asst. Prof., Dept. of Agric. and Biosys. Engrg., Univ. of Arizona,Tucson, AZ 85721.

'Asst. Prof., Dept. of Agric. and Biosys. Engrg., Univ. of Arizona,Tucson, AZ.

Note. Associate Editor: Hilary I. Inyang. Discussion open until January1, 1998. To extend the closing date one month, a written request mustbe filed with the ASCE Manager of Journals. The manuscript for thispaper was submitted for review and possible publication on August 12,1996. This paper is part of the Journal of Environmental Engineering,Vol. 123, No.8, August, 1997. ©ASCE, ISSN 0733-9372/97/0008­0792-0799/$4.00 + $.50 per page. Paper No. 13855.

792/ JOURNAL OF ENVIRONMENTAL ENGINEERING / AUGUST 1997

transverse components of fluid velocity, axial pressure drop,and mass transfer were investigated.

It is very difficult to obtain exact solutions in closed formfor a set of 2D coupled elliptic partial differential equations(that is, continuity, Navier-Stokes, and energy and/or chemicaltransport equations) for a fluid/porous system. Fortunately, afew theoretical studies were performed for the onset of naturalconvection in an enclosure (Chen and Chen 1989, 1992). Us­ing the linear stability theory, Chen and Chen (1989) demon­strated the existence of a critical thickness; that is, they ob­served that porous-layer-dominated convection was suddenlytransformed to fluid-layer-dominated convection as the thick­ness ratio increases. The linear stability analyses accuratelypredict the onset of natural convection as well as the size ofnatural convection cells. Later they performed a series of ex­periments to verify their theoretical results (Chen and Chen1992).

In an effort to validate the matching boundary conditions,Choi and Kim (1994) numerically investigated the onset ofconvection when a porous layer underlying a fluid layer washeated from below. At the interface, the matching boundaryconditions of the continuity of longitudinal and transverse ve­locities, pressure, deviatoric normal and shear stresses, tem­perature, and heat flux were implemented. The numerical re­sults were in excellent agreement with the previous reportbased on the linear stability theory reported by Chen and Chen(1989, 1992). In particular, an abrupt change in convectiveflow patterns was accurately predicted, presenting the precip­itous drop of the critical Rayleigh number with an increasingdepth ratio and a rapid change of the wave number. Choi andKim further showed a dramatic change in the number of cellsin the supercritical Raleigh number regime. Overall, the studyproved that the matching interface boundary conditions alongwith the numerical scheme could handle the sensitive Benardconvection problem.

As the first step toward a better understanding of the com­plex nature of chemical and thermal diffusion and advectionbetween a saturated porous medium and a fluid layer, the pres­ent study focuses on the momentum transport phenomenaacross the interface of the porous medium and fluid. Fig. 1schematically presents the effect of viscous shear in the chan­nel fluid as the shear propagates across the interface into theporous medium and forms a transition region of disturbed flowin the porous medium. The depth of the transition region willbe greatly influenced by the type of porous medium. Darcy'slaw is clearly not applicable in that region of the porous me­dium since no macroscopic shear term is associated with themomentum equation. The "disturbed" flow in the porous re­gion may significantly influence chemical and/or thermaltransport processes in the porous region near the interface.

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FIG. 2. Coordinate Systems and Boundary Conditions

FIG. 1. Effect of Viscous Shear In a Doubly Layered Channel(1,2)

1V'v =0; v'Vv =-- Vp + VfV~ + g

Pf

The conservation equations for the porous region are based ona general flow model, which includes the effects of flow inertiaas well as friction caused by macroscopic shear. The gener­alized flow model was semiempirically derived and is knownas the Brinkman-Forchheimer-extended Darcy model (Vafaiand Tien 1981). The governing equations for the porous layerare

u =0, v =0 at y =0 and 0 S x S L (5d)

u =0, v =0 at x =0 and 0 s y S hp (5a)

u =uo, v =0 at x =0 and hp S Y S H (5b)

au- =0, v =0 at x =L and 0 S Y S H (5c)ax

1 2 Vf Fv'Vv = -p; Vp + veffV V - Kv - VK Ivlv + g (4)

The appropriate boundary conditions for the present problemare

V'V =0 (3)

au- =0, v =0 at y =H and 0 S x S L (5e)ay

In addition, the following matching conditions have to be sat­isfied at the interface of the porous/fluid layer:

u 1.>-/1; =u IY-h;' vl.>-/l; =vl.>-/l;; p 1.>-/1; =pl.>-/l; (6a,b)

j..l.eff (:~ + ::) 1.>-/1; = j..l.f (:; + ::) I.>-/It (6c)

These conditions express the continuity of longitudinal andtransverse velocities, pressure, and normal and shear stresses.The condition in (6c) represents the matching shear stress,which is an extension of the condition used by Neale andNader (1974) for flow that is not parallel to the porous/fluidinterface.

Some investigators prefer to use Beavers and Josef's slipvelocity condition (1969) at the interface with the Darcy flowmodel, while others prefer boundary conditions with the gen­eral flow model because it enables them to incorporate thesingle-domain approach. This single-domain approach is alsoeffective for studying the motion of the fluid in the region thatis partially filled with a porous medium and partially filledwith a fluid. For this reason, the single-domain approach wasadopted by Choi and Kim (1994).

Beavers and coworkers (1967, 1970) estimated that the ef­fective viscosity J,L.ff is approximately 0.01 J,Lf for porous me­dium permeabilities in the range 10- 11 to 10-9 m2

• Lundgren(1972) determined j..l..jf theoretically for a moderately dense bedof stationary uniform spheres. His theoretical results werequalitatively consistent with those of Beavers and his cowork­ers (1967, 1970). In addition, he predicted that the value ofj..l..jf can be larger than that of j..l.f for a very coarse porousmedium; that is, j..l.tJf ;z: j..l.f. For simplicity, however, many re­searchers assumed j..l.'ff equal to J,Lf (Neale and Nader 1974;

JOURNAL OF ENVIRONMENTAL ENGINEERING / AUGUST 1997/793

while the overlying fluid layer has the fluid viscosity j..l.f. It isassumed that the flow is steady, laminar, incompressible, iso­thermal, and 2D. In addition, the thermophysical properties ofthe fluid and the porous matrix are assumed to be constant,and the fluid saturated porous medium is considered homo­geneous and isotropic. The conservation equations for massand momentum in the fluid region are

EXIT

FLUID REGION

INLET

· ;....,.. .' Iturbed zone..............................................'I;;. !!!~.~~~~

'-P..,ltratIon DepthDiffusion and CorMctlon In '"- Media

Applications

The present study may contribute to understanding the fun­damentals of the fluid flow at the fluid and porous media in­terface. An improved assessment of flow velocity should beof direct interest to the evaluation of pollutant migration,where the existence of depository solids may significantly af­fect the rate of contaminant transport. In addition, this studycan have practical engineering applications in high-level nu­clear waste disposal (Rasmuson and Neretnieks 1986; Tsanget at. 1988). The accurate numerical estimation of the frictionalforce at the interface due to the fluid flow on the porous mediamay be valuable in predicting soil erosion. In petroleum res­ervoir engineering practice, it is important to predict fluid flowoccurring in adjacent regions of fractured open channels. Athermal and/or chemical transport equation can be readily cou­pled with the momentum equation and interface boundary con­ditions introduced in this study. Therefore, the present studycan also be applied to engineering problems concerningpacked-bed thermal storage systems, fibrous granular insula­tion, and porous journal bearing.

Solid-liquid separation processes in water treatment use across-flow membrane filtration technique that involves a pres­surized fluid flow parallel to the porous membranes; that is,parallel plate membrane modules consisting of a channelbounded by two porous walls are used for filtration (Chellamet al. 1992; Belfort and Nagata 1985). Similarly, membranefiltration to separate plasma and cellular components fromwhole blood (plasmapheresis) is often carried out in a filtrationcell with a porous wall (Castino et al. 1978). In both cases,the equations and boundary conditions introduced in this studycan be directly applied. In addition, many other practical ap­plications can be found in engineering, biological, and envi­ronmental problems.

FORMULATiON

Mathematical Formulation

The coordinate system and the corresponding physical con­figuration are shown in Fig. 2. The thickness of the f1uid­saturated porous layer hp is O.5H. Several different aspect ra­tios (=LlH) were carefully tested for the present study. Due tothe parabolic nature of the problem, the location of the exitboundary does not influence the upstream flow field except atthe end region. Throughout the present study, therefore, theaspect ratio was chosen as 5, based on a series of numericalexperiments. The porous layer has the effective viscosity J,L'ff'

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Ooms et al. 1970) regardless of porous medium permeabilities;likewise, IJ-eff is simply assumed to be the same as IJ-f in thepresent paper, and the validity of this assumption is discussed.

Numerical Simulation

For the present calculations, the continuum approach withone set of conservation equations, that is, (3) and (4), wasadopted instead of using a two-domain approach where fluidand porous layers are treated separately with coupling condi­tions at the interface. This continuum approach is more effi­cient because both the porous and the fluid layers can be mod­eled as a single domain governed by one set of equations. Inaddition, the corresponding solution satisfies the continuity ofthe velocity, stress, temperature, and the heat flux across theporous/fluid interface without an involved iterative procedurefor matching the interface conditions as described in (6). Asmentioned earlier, this approach was extensively tested andvalidated by Choi and Kim (1994).

Let us introduce a set of dimensionless coordinates, veloc­ities, stream function 'l', and vorticity n as

x y u v I/! wHX =-, Y =-, U :::: -, y:::: -, 'I' =-, and .0 = - (7)

H H u" Uo uoH u"

was used to handle abrupt variations in the thermophysicalproperties across the interface. This ensures the continuity ofconvective and diffusive fluxes across the interface withoutrequiring the use of an excessively fine grid structure.

A control-volume-based finite difference method was usedto solve the system of partial differential equations for thevorticity and stream function [Patankar 1980). The main loopsof the discretized partial differential equations were vectorized,using the extrapolated Jacobi scheme. This iteration scheme isbased on a double cyclic routine, which translates into a sweepof only half of the grid points at each iteration step.

Additional calculations were carried out to evaluate the ef­fects of the porous material and the depth of momentum pen­etration on the skin friction at the interface. The results for theskin friction were presented as a dimensionless friction coef­ficient Cf defined as

IJ-f au Iay int 2 aul

Cf = pfuol2 =R aY Int (13)

Note that the coefficient is proportional to the shear stress atthe interface.

Note that some of the important dimensionless numbers are

where the dimensional stream function and vorticity are de­fined as

where K and £ = permeability and porosity, respectively. Thesource term in the momentum equation S is negligible in thefluid region as Da- 1 and A become zero, and the equation isessentially the same as the Navier-Stokes equation for New­tonian fluids. On the other hand, as the Darcy number de­creases in the porous region, the equation becomes the onebased on the Darcy model. The harmonic mean formulation

s =__1 .0 + ..!.. [u.!!- (_1) _y.!!- (_1)]R·Da R aY Da ax Da

+ (u_aA _ y_aA) YU2 + yZaY ax

+ A (u ayuz+ yZ _ Y ay""'u-=z-+-y--:Z) _ AYU 2 + vznay ax

(11)

(14)

Stability and Accuracy of Numerical Scheme

RESULTS AND ANALYSIS

To ensure convergence and stability, underrelaxation wasused. In general, the relaxation parameters were set at 0.7 forall variables. When the Darcy number decreases, the programbecame increasingly unstable, and the relaxation parameterswere reduced to less than 0.1 to ensure convergence. For eachvariable, the numerical integration was performed until the fol­lowing convergence criterion was satisfied:

l4>n-4>

nli.1 i.1 < 10-8

max 4>n -itJ

where 4>i.1 stands for n and 'I' at a node (i, j); and n denotesthe iteration number. After a series of numerical experiments,82 X 82 and 164 X 82 nonuniform grids were selected andused depending on the aspect ratio of the calculation domain.In particular, very fine grids were located near the fluid andporous medium interface. Further refinement of the grid sizeshowed less than 1% difference in the converged results. Asan additional check for the convergence, the mass balancewas calculated at the inlet and exit for each run. It was foundthat the differences between influx and outflow of the fluidwere less than 1% for all converged cases. The present FOR­TRAN program was further vectorized so that it was usedefficiently when processed on a CONVEX C4620 minisuper­computer.

The porous region is assumed to be saturated with water at300°K. Five typical porous media found in natural and engi­neered systems were selected for numerical simulations andare listed in Table 1. Based on the type of the medium in theporous region, each case of the present analysis is abbreviatedas (1) FM for foam metal; (2) BS for 6 mm porcelain ber!

(8)

(9)

(10)

(12)FeH

and A:::: VKR =uoH, Da = Kz'v H

where

al/! ill/! GV auu :::: -, 11:::: --, and w :::: - - -

ay ax ax ay

Then the dimensionless form of (3) and (4) becomes

a2'1' a2'1'-+-=-.0ilX 2 ay2

a'l' an _ iI'I' an :::: ..!.. (il2n + aZn) +aY ax ax ilY R ax2 ay2 s

TABLE 1. Important Dimensionless Parameters

Materal Abbreviation E K Da A(1 ) (2) (3) (4) (5) (6)

Foam metal (Vafai and Tien 1981) FM 0.98 1.00 X 10-· 6.25 X 10-4 7.09 X 10+0

6 mm porcelain Berl saddle (Carman 1938) BS 0.75 2.00 X 10-7 1.25 X 10-' 2.39 X 10+1

3 mm glass beads (Choi and Kulacki 1992) GB 0.39 9.46 X 10-· 5.16 X 10-· 2.92 X 10+2

Unconsolidated sand (Muskat 1937) US 0.40 6.5 X 10- 11 4.06 X 10-- 3.37 X 10+3

Silt loam GE 3 (van Genuchten 1980) SL 0.40 5.90 X 10- 14 3.69 X 10- 11 1.12 X 1O+S

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___ -:-- =:= -- Alp-5(12aI2) :=-- f------ ----------1-- Alp-IO(I64aI2)=

1\

0cJO.0.080.070.060.05

0.04

Cr0.03

0.02

.......r-- Be 500

-~r--~_dl--I---""'--r-'-~'/I-.--••- +..- -..- -+..-..- -..-.-+..-..- - -+ :--~ ---j---+i""ooo.--+--f----l--+---!------'--~--_-.,.,•• I :

XFIG. 4. Effects of Aspect Ratio on C,

enced by the exit boundary for both aspect ratios (5 and 10)as shown in Fig. 4. However, the upstream values of C, forX < 4 are not influenced by the exit boundary, regardless ofthe aspect ratio. This is mainly due to the parabolic natureof the problem. Other variables show similar end effects. Ex­cept for the results in the exit region, therefore, the resultsof the present calculations will remain the same at any aspectratio.

The velocity profiles in the channel are illustrated in Fig. 5,in which I indicates the nodal number of the nonuniform gridin the X-direction, and the corresponding locations in the com­putational domain are (1) X = 0 at I = I; (2) X = 0.0444 at I=5; (3) X = 0.1671 at I = 10; (4) X = 0.5359 at I = 20; (5)X = 2.2745 at! = 50; and (6) X = 3.8176 at! = 70, respectively.As shown in Fig. 5(a), the velocity profile develops from auniform distribution at the inlet to the parabolic shape beyondthe entrance region. The penetration of velocity profiles intothe foam metal (FM) is pronounced. It is apparent that theflow is nearly fully developed when X> 2.0, and therefore thevelocity profile remains the same after I = 50 at R = 10. Whenthe berl saddle (BS) is used for the porous medium [Fig. 5(b)),the depth of the transitional zone is visibly reduced. It is alsoobserved that the penetration depth at I = 5 is greater thanthose at downstream because of the sudden expansion of thefluid into the porous medium near the inlet boundary. Such apenetration is immediately damped in the porous region andthe depth of penetration remains almost the same beyond theentrance region. When the glass beads (GB) were used [Fig.5(c)) as a porous medium, there was a slight penetration ofthe velocity profile into the porous region. As the Reynoldsnumber increases, the length of the entrance region is expectedto be longer. For R = 1,000, Fig. 5(d) shows that the velocitydistribution is continuously developing in the entire calculationdomain. The overshoot of the velocity profiles near the inter­face even at I = 20 is observed due to the extended entranceregion. It is apparent that the depth of penetration is hardlyinfluenced by the increase of the Reynolds number, and furtherdetails are discussed later. The prediction of the transition zonein the porous region is not possible without slip boundary con­ditions at the interface, as described in Fig. 6.

Further calculations with the unconsolidated sand (US) andthe silt loam (SL) show virtually no penetration of velocityprofile into the porous region; that is, transitional regions areunmeasurably small. In addition, it is extremely difficult toobtain converged solutions when the Darcy number becomesextremely small (Da = 3.69 X 10- 11 for SL). The solutionsfor SL are obtained with an excessive number of the iterations(up to 10' iterations) because of unusually small relaxationparameters. In this case, the solutions always diverge when therelaxation parameter is greater than 0.02.

Velocity profiles in 0 s Y s 0.45 for FM and BS are de­picted in Fig. 6 in an effort to explore the developing velocityprofiles in the porous region. Substantial differences in the

Ed. Jlealo.11oo .... «)(a)

1 RqI..llee lb)

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

(b)

=~,

1

1

0.7

0.9

0.5o

0.6

0.8

(c)

saddle; (3) GB for 3 mm glass beads; (4) US for unconsoli­dated sand; and (5) SL for silt loam GE3. The selection ofporous media is based on the magnitude of the Darcy numbersince the number plays a significant role in predicting the tran­sition zone. The present study therefore covers a wide rangeof the Darcy number, that is, 3.69 X 10- 11 S Da S 6.25 X10-4

• Six different Reynolds numbers were tested for eachcase; R = 10, 50, 100, 200, 500, and 1,000.

The results generally showed a developing flow pattern nearthe inlet region. For the low Reynolds numbers (R = 10 and50), the velocity quickly became fully developed regardless ofthe type of porous medium. Significant influence of the inletand exit was observed for all cases. As an example, Fig. 3(a)shows streamlines of the flow at R = 10 for GB. The corre­sponding velocity vectors are presented in Figs. 3(b and c). Itis interesting to observe upward velocity vectors at the inletnear the interface [Fig. 3(b)). Such an overshoot of the velocityprofile at the short distance of the entrance was due to thesudden frictional force on the initial slug flow at the interfacesurface. Away from the immediate entrance region, the max­imum velocity occurred at the top free surface as expected.Many researchers previously observed this phenomenon, andHadim (1994) recently summarized it. In the exit region,slightly downward streamlines and velocity vectors areshown near the interface (X = 0.5) in Figs. 3(a and c). Suchan end effect at the interface is pronounced as the frictioncoefficient, and the velocity components are calculated anddepicted. The frictional coefficients, for example, are influ-

0.8 -

0.9 -

0.7

0.6

0.54.5 4.55 4.6 4.65 4.7 4.75 4.8 4.85 4.9 4.95 0.5

FIG. 3. Effects of Aspect Ratio on c,: (a) Streamlines of Flowat R = 10 for GB; (b) Inlet Velocity Vectors; (c) exit Velocity Vec­tors

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0.4 r--f---I---I--+---I---+--I---I--I

0.3 1---1----1---1--+--+----+-+---1--1

1.2 1.4 1.6

i,

!If(b) i 1=1 5 10! '0I / ~i ~

! 'I ~,

- Fluid Region,

II ~I

~~.,-/V I

:

IA --- iV

! !I

i!,

, i

Porous RegionI I I

i I,iI

o-0.2 0 0.2 0.4 0.6 0.6

0.9

0.8

tl 0.7

'!~.:l 0.8

I 0.5Y

0.4

0.3

0.2

0.1

1.2 1.4 1.6

Porous Region

0.2 0.4 0.6 0.8

(a) I I I : ;1= 5 II~Jl0.9 -+-'l---~I' -','-+-,!--iH--t+I-'=fJfA'l~~i----l

i I i/_~"

0.8 I I ) I If ~0.7 f- Fluid Region Ii V J

iA~0.6 , ~/

V ~0.5 r

0.1

i,

0.2 r--+-+--+--+--+----+-+--+!--I

y

u u

1.2 1.4

(d) I I

I70

I- Fluid Region F'<l 50

I 0 <]-20

.,&:;~::::: 10-

I=-- 5

1=1

iI

I Porous RegionI

I,

I io-0.2 0 0.2 0.4 0.6 0.8

0.9

0.8

0.7

0.6

Y 0.5

0.4

0.3

0.2

0.1

1.2 1.4 1.6

I •

i i1=1 'oJ(c) 5i I i

i i ! i I; \

i i I I Rw

II

I ! ~v 'fl::J! !

- Fluid Region

c/ I

I

~ __.1 II-'

~:~'--.• ,..-/ I I

j~~-~ i I

I

-f----II

c--- ----r-- !

IPorous Region

i i , I1-----i !

Ii !!-- ,

I I 1 Ii

i io-0.2 0 0.2 0.4 0.6 0.8

0.9

0.8

0.7

0.8

Y 0.5

0.4

0.3

0.2

0.1

u uFIG. 5. Velocity Profiles for (a) FM, R = 10; (b) BS, R = 10; (c) GB, R = 10; and (d) GB, R = 1,000

magnitude of velocity are observed when the Darcy numberdecreases. Additional calculations for GB, US, and SL shownegligible velocity components in the porous region except thetransitional area near the interface for GB. For both FM andBS, the horizontal velocity profiles penetrate close to the im­permeable wall, and therefore the viscous and convectiveterms in the momentum equation as well as the no-slip con­dition at the bottom wall become important for these highlyporous structures. Like the fluid region as shown in Figs. 5(aand b), fully developed velocity profiles in the porous mediumwere observed after approximately X > 2. The viscous term

becomes less important as the porosity becomes small (Choiand Kulacki 1993).

To estimate the penetration of velocity profile quantitatively,the depth of penetration Ypt is defined as being the distancemeasured from the interface wall into the porous region atwhich the velocity V is within 0.1 % of Vo (i.e., V = 0.001 Vo ).

The concept of Ypt is similar to a transition region thicknessintroduced by Neale and Nadar (1974). For both FM and BS,Ypt is located near the bottom wall except in the entrance re­gion as shown in Figs. 5(a and b). On the other hand Ypt islocated at the fluid/porous media interface (i.e., X = 0.5) for

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, "rrr(b)

i 1// 10

II

i~

iiI \, 20

! \"0 lIT 70

IiI

I

(a) I;:;---1 __-------t

- fIV VI/ ~O

/ \0& 70

I

I / )

~!

x

5

5

4.5

4.543.5

3.5 4

3

x2.5 3

2.5

2

2

1.5

1.5

0.5

0.5

FIG. 7. Depth of Penetration YptforGB

(a)- -_.f- --I--- - .._~- f---

,\

:':::Ile-lOOII

~ J.~

10

oo

0.5 c---r-..__.-.....--,...--..,..-,-...--..,.--...,---.--.....,...--.0.495 fr-...-+-~~__"."'--.......+I......·.,.I----TI...·-· ...._~'--....··_.....·""'_........._.--.....~""'_.....;""--'10.49 -- --.. Re. 10, so, 100, 1000

0.485

D.48 1--+--+--+--+---+---+---1---+---\---1Y 0.475

pt 0.47 I---l---l---t---t---t---t---t---t---t---;

0.465 I---l---j---t--+---t---t---t---t---t---;

0.48 I---l---j---t--+---t---t-~-t---t---t---I

0.455 (----l---j--+----f---t---t---t---t----t---I

0.45 t...... ....

a

illustrate this, horizontal and vertical velocity components atthe interface for GB are depicted in Fig. 8. The vertical ve­locity remains zero except in the inlet and exit regions [Fig.8(b)J. On the other hand, the horizontal velocity componentVI," constantly decreases except in the exit region as the ve­locity profile develops when R =500 and 1,000. When thevelocity is fully developed [e.g., R = 10 and X > 1.0 as shownin Fig. 5(c)], Vi., appears to be constant. In all cases, thechanges in Yp , are negligible regardless of the conditions ofvelocity profile in the fluid region and at the interface. In sum­mary, the depth of penetration is governed by the dampingforce of the porous medium rather than by the velocity profilein the fluid region, the Reynolds number, and the horizontalvelocity of the fluid at the interface. It should be noted thatthe damping force is the sum of the viscous forces exerted bythe individual particles on the penetrating fluid. It also shouldbe noted that additional calculations at Da close to that of GBshow similar results.

The interface horizontal velocity is plotted in Fig. 9(a) whenR = 100. As expected, Vi., decreases as the Darcy numberdecreases. The change in Vi., becomes very small when theDarcy number is less than that of the glass beads (i.e., Da <

0.05

0.04

0.03

U1nt

0.02

0.01

0.01

0.010.005

0.005

u

o

o

o-0.005

o-0.005

0.45

0.4

0.35

0.3

0.25

Y0.2

0.15

0.1

0.05

0.45

0.4

0.35

0.3

0.25

Y0.2

0.15

0.1

0.05

FIG. 8. Velocity Components along Interface (Y=0.5): (a) Uln,

and (b) v,nt for GB

JOURNAL OF ENVIRONMENTAL ENGINEERING I AUGUST 1997/797

(b)

\.-------~

i'Ra - 10, 100. SOO. lad 1000

uFIG. 6. Velocity Profiles in Porous Region for <a) FM; and (b)BSwhenR=10

US and SL; that is, no measurable penetration due to the vis­cous stress from the fluid part is observed. For GB, Yp , isdepicted in Fig. 7. The effects of the entrance and exit bound­ary conditions are visible. Surprisingly, Yp , remains almost thesame at various Reynolds numbers and distances in the X­direction except in the entrance and exit regions. In Fig. 3, aspresented earlier, the decrease of C/ is observed for both R =500 and 1,000. In addition, developing velocity profiles in thefluid region are observed in Figs. 5(c and d). Evidently, thecontinuous changes in the horizontal velocity component Vare also expected along the fluid/porous media interface. To

0.1

0.05

a

V1nl -'l.05

-'l.1

-'l.15

-'l.2a 0.5 1.5 2 2.5

x3 3.5 4 4.5 5

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0.6 .---.,....---,--.......,~--r--"'"'T'"----,.--.,..--...,.....---r----.

(a)0.4 II-'\\~-....,-- +-/-t-

P

-

M

-- ----. +--- ----1---+--+----1

0.2 -'=BS

o ~G~B~=+:;::;~~:::~=t::::::t:::4:::::t=:::1

-0.2 ~. S and ~~----- -- -.- - --

-0.4 r-+--t--+---+---j---+-....,--+---t---I

x

/~J....~U=S_ _+_--. ~~

543

0.1 _

0.08 p",,,,,,,,-0.06 "-_--'-__~__~_--'-___'___~_--'-__-'-___'____--J

o 2

o~~: .-\~-Ql!..._---- ----- ---- -- ---- ....._-- ----. -------- ---..0.16 ~

0.14 ~"'-

0.12 ""--""'"........... BS, "'_Cr

x

Dint

~8 ~7 ~6 ~s ~4 ~3

10 10 10 10 10 10

Da

(c)

r------------------------~ 0.16

0.14

0.12

0.10

0.08

0.06

0.04

0.02

0.00

0.11

0.10

Cf

0.09

0.08

0.07 -1110

FIG. 9. With R =100: (a) U",,; (b) Cli and (c) U,n,and C,at X= 4

10-6), and eventually the velocity becomes nearly zero as

shown for US and SL. On the other hand, the velocity gradientat the interface increases as the Darcy number decreases, sincethere is no penetration of fluid and damping force in the po­rous region. As Da becomes extremely small, the velocity gra­dient eventually approaches that of the fluid flow over theimpermeable wall. Accordingly, the viscous shear stress as­ymptotically increases as Da decreases. Values of Cf are nearlyindependent of the Darcy number when Da is less than it isfor GB, as shown in Fig. 9(b). Fig. 9(c) shows the values of~nI and Cf at X = 4. The figure effectively depicts the asymp­totic decrease of Uifll and increase of Cf as Da decreases. WhenDa = 10-7 or less, the changes in Uin, and Cf are negligible,

7981 JOURNAL OF ENVIRONMENTAL ENGINEERING 1AUGUST 1997

that is, the results asymptotically approach those of an imper­meable solid for the porous part.

For simplicity, the present investigation is based on the as­sumption that fJ..ff = fJ.f. As mentioned earlier, however, theeffective viscosity is a function of the permeability K. Unfor­tunately, there is no theoretical or experimental relationshiPsthat is valid for the wide range of K used in this study (10- 5

m2 < K :S 10-6 m2). The present "numerical" experiments

[see Table I and Fig. 9(c)] generally indicate that the viscousterm (and other non-Darcy terms, of course) in the momentumequation becomes less important for averge and dense porousmedia. Consequently, the value of the effective viscosity is notsignificant. As the porosity approaches I, the effective viscos-

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ity becomes the fluid viscosity. Thus, the assumption J.L-u = J.LImay be reasonable for very coarse porous media. It should benoted that the significance of the effective viscosity should befurther investigated.

CONCLUSIONS AND RECOMMENDATIONS

This study demonstrated that the present matching boundaryconditions along with the Brinkman-Forchheimer-extendedmomentum equation could predict the transition region of thevelocity profile at the interface. The effects of viscous shearstress at the interface are particularly important for coarse po­rous structures such as highly porous fissured rock, glassbeads, and foam metals. However, the penetration depth is notmeasurable when Da < 10-7

; that is, the slip boundary con­ditions at the interface play a negligible role. It is interestingto observe that the depth of penetration is only dependent onthe Darcy number of the porous medium rather than the flowspeed and the shape of the velocity profile in the fluid region.In the vicinity of the interface, it is clear that Darcy's lawalone is inappropriate to describe flow in a permeable wallfracture and flow over a porous medium. It is recommendedthat the role of each non-Darcy term (i.e., the Brinkman andthe Forchheimer terms) should be separately examined at var­ious R. Experimental study at high R will further verify thevalidity of the present boundary conditions and the numericalscheme.

ACKNOWLEDGMENT

This study was funded in part by a grant from the Agricultural Ex­periment Station that we gratefully acknowledge. Computing facilitieswere provided by the University of Arizona.

APPENDIX I. REFERENCES

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Beavers, G. S., Sparrow, E. M., and Magnuson, R. A. (1970). "Parallelflow in a channel and a bounding porous medium." J. Basic Engrg.,92,843-848.

Belfort, G., and Nagata. N. (1985). "Fluid mechanics and cross-flowfiltration: Some thoughts." Desalination, 53(1), 57-79.

Berkowitz, B. (1989). "Boundary conditions along permeable fracturewalls: Influence on flow and conductivity." Water Resour. Res., 25(8),1919-1922.

Carman, P. C. (1938). "The determination of the specific surface of pow­ers." Int. J. Soc. Chem. Ind., 57, 225-234.

Castino, F., Friedman, L. I., Soloman, B. A.• Colton, C. K.• and Lysaght,M. J. (1978). "The filtration of plasma from whole blood: A novelapproach to clinical detoxication." Artificial kidney, artificial liver, andartificial cells, T. M. S. Chang. ed.• Plenum Press, New York, N.Y.259-266.

Chellam, S., and Wiesner, M. R. (1993). "Slip flow through porous mediawith permeable boundaries: implications for the dimensional scaling ofpacked beds." Water Envir. Res., 65(6), 744-749.

Chellam, S., Wiesner, M. R., and Dawson, C. (1992). "Slip at a uniformlyporous boundary: effect on fluid flow and mass transfer." J. Engrg.Mathematics, 46(4), 481-492.

Chen, F., and Chen, C. F. (1989). "Experimental investigation of con­vective stability in a superposed fluid and porous layer when heatedfrom below." J. Fluid Mech., 207(1), 311-321.

Chen, F., and Chen, C. F. (1992). "Convection in a superposed fluid andporous layers." J. Fluid Mech., 234(1), 97-119.

Choi, C. Y., and Kim, S. J. (1994). "Modeling of boundary conditionsat the soil and water interface." Proc., ASAE Int. Winter Meeting, Am.Soc. Agric. Engrs., St. Joseph. Mich.

Choi, C. Y., and Kulacki, F. A. (1992). "Mixed convection in a verticalporous annulus." J. Heat Transfer, 114(1),143-151.

Choi, C. Y., and Kulacki, F. A. (1993). "Non-Darcian effects on mixedconvection in a vertical porous annulus." J. Heat Transfer, 115(2),506-510.

Hadim, A. (1994). "Forced convection in a porous channel with localizedheat sources." J. Heat Transfer, 116(2),465-472.

Lundgren, T. S. (1972). "Slow flow through stationary random beds andsuspensions of spheres." J. Fluid Mech., 51(1), 273-299.

Muskat, M. (1937). The flow of homogeneous fluids through porous me­dia. McGraw-Hili, New York, N.Y.

Neale, G., and Nadar, W. (1974). "Practical significance of Brinkmanextension of Darcy's law: coupled parallel flows within a channel anda boundary porous medium." Can. J. Chern. Engrg., 52, 472-478.

Nield, D. A., and Bejan, A. (1992). Convection in porous media.Springer-Verlag New York, Inc., New York, N.Y.

Oliviera, L. S., and Haghighi, K. (1994). "Finite element analysis ofconjugate heat and mass transfer during drying of porous media."Proc., ASAE Winter Annu. Meeting, Am. Soc. Agric. Engrs., St. Joseph,Mich.

Ooms, G., Mijnlieff, P. F., and Beckers, H. L. (1970). "Frictional forcesexerted by a frictional fluid on a permeable particles with particularreference to polymer coils." J. Chemical Phys., 53(11),4123-4130.

Patankar, S. (1980). Numerical heat transfer and fluid flow. Hemisphere,New York, N.Y.

Prasad, V. (1991). "Convective flow interaction and heat transfer betweenfluid and porous layers." Convective heat and mass transfer in porousmedia, S. Kakac et aI., eds., Kluwer Academic, Dordrecht, The Neth­erlands, 173-224.

Rasmuson, A., and Neretnieks, I. (1986). "Radionuclide transport in fastchannels in crystalline rock." Water Resour. Res., 22, 1247-1256.

Tsang, Y. W., Tsang, C. F., Neretnieks, I., and Moreno, L. (1988). "Flowand tracer transport in fractured media: A variable aperture channelmodel and its properties." Water Resour. Res., 24, 2049-2060.

Vafai, K., and Tien, C. L. (1981). "Boundary and inertial effects on flowand heat transfer in porous media." Int. J. Heat and Mass Transfer,24, 195-203.

van Genuchten, M. T. (1980). "A closed-form equation for predicting thehydraulic conductivity of unsaturated soils." Soil Sci. Soc. Am. J., 44.892-899.

APPENDIX II. NOTATION

The following symbols are used in this paper:

Cf = friction coefficient;Da = Darcy number;

F = Forchheimer number;g = gravity constant (m/s2

);

H = total height of fluid and porous layers (m);hf = height of fluid region (m);hp = height of porous region (m);K = permeability (m2

);

L = horizontal length of channel (m);p = pressure (Pa);R = Reynolds number;S = source term;

U, V = dimensionless horizontal and vertical velocities;u, v = horizontal and vertical velocities (m/s);

v = velocity vector [u, v] (m/s);X, Y = dimensionless horizontal and vertical coordinates;x, y = horizontal and vertical coordinates (m);

E = porosity;A inertia parameter;J.L = dynamic viscosity (N' s/m2

);

v = kinematic viscosity (m2/s);p density (kg/m3

);

'I' = dimensionless stream function;IjJ = stream function (m%);.0 = dimensionless vorticity; andw = vorticity (S-I).

Subscripts

eff = effective thermophysical properties;f = fluid;

int = interface;o = inlet; andp = porous medium.

JOURNAL OF ENVIRONMENTAL ENGINEERING / AUGUST 1997/799

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