Momentum Jump Condition at the Boundary Between ...

Journal of Porous Media, 1(3),201-217 (1998)

Momentum Jump Condition at the Boundary Between a Porous Medium and a Homogeneous Fluid: Inertial Effects J. Alberto Ochoa-Tapia l and Stephen

Whitake~

1Division de Ciencias Biisicas e Ingenieria, Universidad Aut6noma Metropolitana-Iztapalapa, Apartado Postal 55-5534, 09340 Mexico D.P., Mexico 2Department of Chemical Engineering and Material Science, University of California at Davis, Davis, California 95616 Received September 3, 1996; Accepted January 28, 1997

ABSTRACT The momentum transfer condition that applies at the boundary between a porous medium and a homogeneous fluid when inertial effects are important is developed as a jump condition based on the nonlocal form of the volumeaveraged momentum equation. Outside the boundary region, this nonlocal form reduces to the classic transport equations, i.e., the Forchheimer equation with the Brinkman correction and the Navier-Stokes equations. The structure of the theory is comparable to that used to develop jump conditions at phase interfaces; thus, experimental measurements are required to determine the two coefficients that appear in the jump condition. The first of these coefficients is associated with an excess viscous stress, whereas the second is related to an excess inertial stress. The theory indicates that both of these coefficients are of order 1.

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Ochoa-Tapia and Whitaker

202

NOMENClATURE

area of the 13-0" interface contained within the averaging volume "V, m2 A'l(t) portion of s'L(t) located in the T] region, m2 A..,(t) portion of 3'L(t) located in the W region, m2 AW'l(t) A'l..,(t), area of the W-T] boundary contained within the large-scale volume, "V oo(t), m 2 3'L(t) bounding surface of the large-scale volume s'L(t), m 2 excess Brinkman stress, N/m2 (B) bounding curve for the area AW'l(t) C Forchheimer correction tensor for the 13 Fi3w phase in the homogeneous W region gravity vector, m/s2 K~~' (I + F[3w), inverse of the generalized Darcy's law permeability tensor that includes the Forchheimer correction, m ~2 unit tensor Darcy's law permeability tensor for the 13 phase in the homogeneous W region, m 2 characteristic length associated with the Lv velocity, m characteristic length associated with the gradient of the velocity, m M second-order tensor that maps E~~(V[3t onto v[3 n, outwardly directed unit normal vector to the curve C that is tangent to the W-T] boundary unit normal vector directed from the 13 phase toward the 0" phase outwardly directed unit normal vector for 0'1 the T] region contained "V oo(t) outwardly directed unit normal vector for the W region contained in "V oo(t) D'lw, unit normal vector directed from the W region toward the T] region pressure in the 13 phase, N/m2 P[3 spatial deviation pressure,p[3 - (P[3)[3, N/m2 p[3 (p[3)i3 intrinsic volume-averaged pressure N/m2 intrinsic volume-averaged pressure in the (p[3)~ homogeneous T] region, N/m2 intrinsic volume-averaged pressure in the homogeneous W region, N/m2 excess inertial stress, N/m2 (R) radius of a spherical averaging volume, m excess bulk stress, N/m2 excess surface stress, N/m volume of the 13 phase contained in the averaging volume "V, m3

V'l(t)

v[3 v[3 (V[3) (V[3)[3 (V[3)'l

x

volume of the 'l1 region contained in "V oo(t), m 3 volume of the W region contained in "V oo(t), m 3 averaging volume, m3 volume of a body in the volume-averaged sense, m2 velocity in the 13 phase, mls v[3 - (V[3)[3, spatial deviation velocity, mls superficial volume-averaged velocity, m/s intrinsic volume-averaged velocity, m/s superficial volume-averaged velocity in the homogeneous T] region, m/s superficial volume-averaged velocity in the homogeneous W region, mls speed of displacement of the W-T] boundary, m/s position vector locating the centroid of the averaging volume, m position vector locating points in the 13 phase relative to the centroid of the averaging volume, m

Greek symbols 131 adjustable coefficient associated with the excess Brinkman and bulk stresses 132 adjustable coefficient associated with the excess inertial stress 8 thickness of the boundary region, m E[3 porosity or volume fraction of the 13 phase E[3w porosity or volume fraction of the 13 phase in the homogeneous W region 'A unit tangent vector to the W-T] boundary J.L[3 viscosity of the 13 phase, Ns/m2 Pi3 mass density of the 13 phase, kg/m3 P[3s excess surface mass density, kg/m 2 Subscripts s surface vector or surface tensor 13 quantity associated with the 13 phase T] quantity associated with the homogeneous T] region W quantity associated with the homogeneous W region wT] quantity associated with the W-T] boundary 00 quantity associated with a large-scale volume Superscripts 13 intrinsic average

203

Momentum Jump Condition

INTRODUCTION In this study, we are interested in developing the momentum jump condition that applies at the boundary between the wand 11 regions illustrated in Fig. 1. The governing point equations and the boundary condition that are valid "everywhere" in the W-l1 system are given by in the \3 phase

V· vJ3 = 0,

C';; +

PJ3

VJ3 .VVJ3 ) = -VPJ3

(1)

+

P~

+ /.LJ3 V2v 13' in the \3 phase B.C.

vJ3

= 0,

at the \3-..

1)2(E;~

CONCLUSIONS In this study, we have developed a generalized volume-averaged form of the Navier-Stokes equations that is valid everywhere in a system composed of a porous medium and a homogeneous fluid. This generalized momentum equation can be used to construct a momentum flux jump condition that contains two parameters of order 1 that must be determined by comparison with experiment. In this development, all surface momentum transport has been neglected, and we have required that the volume average velocity be a continuous function of position in the boundary region between the porous medium and the homogeneous fluid.

APPENDIX

+ 1) (72)

For the process illustrated in Fig. 5, we summarize the boundary conditions as

We begin the development of the momentum flux jump condition, or the surface momentum transport equation, with the "generalized" volume-averaged momentum equation in the following form:

Momentum Jump Condition

213

a(p~E;1(V~» + t'7. [ ( -2( )( ) + -1(- - ) at v p~ E~ v~ v~ E~ V~V~

neous wand TJ regions are given by the following equations. w region:

+ E;1(V~V~)ex)] + p~E ;1VE~· (E ;2(V~)(V~) + E;1(V~V~) + E;1(V~V~)ex) = V ·(-I(p~)~ + JL~E;1V(V~» + JL~E;1(V In E~)2(V~) - JL~ct>~

a(p~vw)

-a-t- + (AI)

= V· Tw

To develop the jump condition associated with this equation, we need to simplify the nomenclature by means of the following definitions:

TJ region:

v = E;1(V~)

(A2a)

U = E;1(V~V~)

+

E;1(v~v~)ex

U = E;1VE~· (E;2(V~)(V~)

+

+

(A2b) E;1(V~V~)

V·(p~vwvw)

+

+

V·(p~Uw)

p~g - JL~ct>~w

(A6)

(A7)

We made use of the following conditions that apply to the quantities defined by Eqs. (A2): Uw =U1]=O

(A8a)

E;1(v~v~)ex)

(A2c)

b", = b1] = 0

(A8b)

b = E;1(V In E~)2(V~)

(A2d)

U1] = 0

(A8c)

T = -I(p~)~

(A2e)

ct>~1] = 0

(A8d)

+

JL~E;1V(V~)

With the use of these definitions, we obtain a more attractive generalized momentum equation that takes the following form:

The integral of Eq. (A6) is given by

f

Vw (')

a(p~v)

-- +

at

= V· T

V·(p~vv)

+

p~g

+

+

V·(p~U)

+

p~u

JL~b - JL~ct>~

a(p~vw) - dV

at

+

L

n",·(p~vwv",)

dA

A w (,)

(A3)

Now let "V oo(t) be a volume bound by a surface .stloo(t), which has a speed of displacement (Slattery, 1990), given by V· n, where n is the outwardly directed unit normal vector associated with the surface .stloo(t). This volume is illustrated in Fig. 4, and in terms of "V oo(t), the "integral statement of" Eq. (A3) is given by

:r t

IVoo(t)

p~v dV +

r

)v

V·(p~U) dV and one can use the general transport theorem (Whitaker, 1981) to express the first term in this momentum balance as

oo (')

The portion of "V oo(t) that lies in the w region will be designated by Vit), whereas the part that lies in the TJ region will be identified by V1](t). It follows that (AS)

The forms of Eq. (A3) that are valid in the homoge-

(AlO) We need to emphasize that the speed of displacement of .stloo(t) is V· n, and that the velocity is given by v = (v~)~ = Vw in the homogeneous w region and by v = v ~ = v1] in the homogeneous TJ region. In addition, we

214

Ochoa-Tapia and Whitaker

need to point out that the speed of displacement of AWl)(t) is zero, i.e., W· OWl) = O. Use of Eq. (AlO) in Eq. (A9) leads to the slightly simplified form, given by

surface momentum in the momentum jump condition will assure us that the total time rate of change of momentum will be given correctly by the solution of Eqs. (A6) and (A7). Equation (A13) is the preferred representation of an excess function; however, sometimes it is convenient to use the alternate form given by the following. Excess surface momentum:

(;'\14)

(All)

When we subtract Eqs. (All) and (A12) from Eq. (A4), and make use of either Eq. (A13) or (A14), we obtain the integral momentum jump condition given by

The analogous form for the TJ region is given by

:t f

Pj3vl) dV

+

V"(/)

1

ol)W • (Pi3v l)vl) dA

A"w

(A12) in which vl) = vi3 in the homogeneous TJ region. To develop the momentum jump condition, we subtract Eqs. (All) and (A12) from the integral of the generalized result given by Eq. (A4). Before doing that we need to define the excess surface momentum by the following relation. Excess surface momentum: d dt

r

Pj3v dV =

)1'00(1)

+

dd t

f

V"(/)

!!:. dt

f

Pj3vl) dV

Pj3v dV

Vw(/)

+

i +1

=

OW· (T - Tw) dA

Aw(/)

~·(T-Tl)dA

A"(/)

W

1

d Pj3sVs dV dt A~(/)

(A13)

We note that the term on the left-hand side represents all of the momentum contained in "V oo(t). The first two terms on the right-hand side represent the momentum contained in "V oo(t), as determined by Eqs. (A6) and (A7), and because these equations are not valid in the boundary region, the first two terms on the right-hand side of Eq. (A13) will not necessarily be equal to the term on the left-hand side. Inclusion of the excess

+

f

VJI)

1-1j3(