fluid saturated porous cavityBEM for natural convection in non-Newtonian
' Faculty of Civil Engineering, University of Maribor, SloveniaR. Jecl', L. Skerget2&L E. PetreSin'
Faculty of Mechanical Engineering, University of Maribor, Slovenia
Abstract
porous cavity will be presented.representing the flow and heat transfer characteristics of the fluid within thesaturate the vertical porous cavity heated from the side. Numerical resultsadequate for many non-Newtonian fluids is considered, representing the fluid thatkinetic part. To evaluate the presented approach the Carreau model which isformulation, therefore the computation scheme is partitioned into kinematic andgoverning equations are transformed by the velocity-vorticity variablesorder to analyse the effects of the available non-Nebtonian viscosity. Theunsteady incompressible Newtonian fluid flow in porous medium is extended inBoundary Domain Integral Method (BDIM) for the numerical solution ofNewtonian fluids in porous medium domain. To solve the stated problem theMethod (BEM) in the analysis of the flow transport phenomena of non-The main purpose of this work is to present the use of the Boundary Element
1 Introduction
phenomena in porous medium, and is therefore used also in our test case.conditions. Natural convection is one of the most frequently studied transportdepend on density differences due to temperature gradients and boundarymay occur. These flows, commonly called free or natural convection movements,in a porous medium is not uniform, certain flows induced by buoyancy effectsrepresenting this relationship. When the temperature of the saturating fluid phaseshear rate. A few parametric viscosity models are available in the literaturebehaviour characterised by a non-linear relationship between shear stress andMany fluids encountered in engineering applications exhibit non-Newtonian
different temperature. The left wall is isothermally heated and the right one isinsulated boundary conditions while the vertical walls are maintained at auniform throughout the cavity. The top and the bottom walls are subjected toparticles, while the porosity and permeability of the medium are assumed to beBoussinesq approximation. Furthermore, the solid phase is made of sphericalconstant except for the density variation, which is handled according to theproperties of the fluid and the solid phases of the porous medium are taken to bevariations, but only on variations of the temperature. The thermophysicalwhile the fluid is single phase and its density is taken not to depend on pressureporous medium. The solid phase is homogeneous, isotropic and non-deformable,cavity is filled with a material consisting of a solid and a fluid phases calledsolution domain is a two-dimensional square cavity with a side length D . TheThe configuration described in the present investigation is shown in Fig. 1. The
Y
isothermally cooled.
Lporous medium
Pf ffflflflfflflf IFig. 1: Geometry and boundary conditions for the porous cavity.
findings and analytical solutions for some rather simplified cases (Bear andnumerous empirical transport coefficients that are supplemented by experimentaland a fluid phases. The averaging method, however, requires information onthat irrespective of its position in porous medium, it always contain both a solidthe suitable representative elementary volume, which has to be determined such,transformed to the macroscopic level using the volume-average technique over
The point governing equations for mass, momentum and energy are
of mass is the continuity equation:The obtained macroscopic equation representing the basic conservation balancementioned assumptions, relating the properties of the discussed porous medium.Bachmat [l]).Therefore, it is of great importance to take into account all above-
a xi
av i- 0 ,--
momentum:The momentum equation represents the basic conservation balance ofwhere vi is volume-averaged velocity, and xi the i -th coordinate.
normalised density-temperature variation function taken asgradient in the flow direction, p the fluid density, g i gravity. F is the
kinematic viscosity, K permeability of porous medium, a P /ax j pressure
where $J is porosity, v f the fluid kinematic viscosity, veff the effective
coefficient of the fluid. The coefficient K is independent of the nature of thedensity at temperature T , and ,l?,being the thermal volume expansionF = (p-pn)/po= -&(T - T o ) ,with p. denoting the reference fluid mass
the hydraulic radius theory of Charman-Kozeny leads to the relationshipfor the case of a simple geometry. For example, in the case of beds of particlesquantity. It is possible to calculate K in terms of geometrical parameters, at leastorder tensor, but for the case of an isotropic medium it can be taken as a scalarfluid but it depends on the geometry of the medium. In general, it is a second-
values (Chen and Hsiao [2]).for K may be replaced by 180, and several other authors determined similarexperimental investigations Ergun has found that the constant 150 in expressionK =di@3/150(1-@y , where d, is the solid particle diameter. With his
effective viscosity veffdepends on the geometry of the porous medium. It mayforce exerted by the solid phase on tie flowing fluid at their contact surfaces. Theterm or Brinkman extension that expresses the viscous resistance or viscous dragpure fluid (Nield and Bejan [ 3 ] ) .The Laplace term is commonly called Brinkmananalogous to the Laplacian term that appears in the Navier-Stokes equations forconsists of two viscous terms. The first is the usual Darcy term, and the second is
The momentum equation (2), commonly known as Brinkman equation
denoting viscosity ratio, is introduced:have a different value than the fluid viscosity v f , therefore parameter A
Navier-Stokes equation as K +- and to the Darcy equation as K +0 .Darcy equations. Namely, the Brinkman equation reduces to a form of theequation is essentially an interpolation scheme between the Navier-Stokes andindeed required. The Brinkman equation has a parameter K , therefore thisin a porous medium and in an adjacent viscous fluid, then the Laplacian term iscompare flows in porous medium with those in pure fluids, or to match solutionsis important to satisfy the no-slip boundary condition, when one wishes toconfirmed that for many practical purposes there is no need to include it. But if ithave disagreed on weather to include the Laplacian term or not, and it has beenvalue of porosity a reasonable approximation is A = 1.I n several articles authorsvalue is often approximated by A = l/@(Jecl et a l . [4]) but in cases with a high(Bear and Bachmat [l]). Since h depends on the geometry of the medium, itsh the expression A = l/@ T * , where T * is called tortuosity of the mediumA detailed averaging process for an isotropic medium yields for the parameter
model may be applied for the relationship between the stress tensor o, or shearneglecting the elastic properties of the fluid, the simple viscous constitutivetheir simplicity and applicability. Focusing on incompressible fluids andare still based on the use of generalised Newtonian fluid models (GNF) due todomain. Many practical simulations of process fluid flows in complex geometries
Now we introduce the non-Newtonian fluid that saturates our porous medium
given by the following constitutive hypothesis z, = 2p(j)i., , where the shear
rate dependent, p =p ( j ). Obviously, the viscous stress tensor zy for GNF isKronecker function and p is dynamic viscosity which is assumed to be strainstress z, and shear rate i, in the form CT,= - p i j j+ 2 p (y).2,, where S, is
models are available in the literature, for example the “power law”, “the Carreau-shear rate being i.,= l/2 (a v i / d x + iil v /d xi ) . A few parametric viscosity
strain rate or magnitude of the shear strain rate tensor is y = (2.2, S,,)%, and the
paper is the Carreau model:Yasuda model”, the “ E h s model”, while the constitutive model applied in this
momentum equation is divided into two parts and the equation (2) is now:into constant and perturbed parts so that v f =Vf+Ff the Brinkman extension in
the momentum equation (2) by v f = p/p . If the viscosity is further partitioned
constant. The dynamic viscosity p is related with the kinematic viscosity vf inwhere p. and p, are the zero and infinite shear rate viscosities, and x a time
Energy equation represents the basic conservation balance of energy:
/ l e = Am+ Ad . The stagnant component based on the experimental findings istwo constituents, a stagnant component A,,, and a dispersion component Ad aseffective thermal conductivity emerges as a combination of the conductivities ofespecially when As/Af >> 1 . Therefore another possibility to compute theBejan [3]). Unfortunately, the mixing rule could give rise to considerable errorwhere Af and As are the fluid and the solid thermal conductivity (Nield and
conventionally approximated with the classic mixing rule Ae =@A, + (l -@)As,porous medium. The effective thermal conductivity of a porous medium can betemperature, and iz, represents the effective thermal conductivity of the saturatedsolid and the fluid specific heats at constant pressure respectively, T stands forwhere p, and p are the solid and fluid densities respectively, c, and cf the
and 6= !L, whereas the dispersion conductivity is4
J
with B = 1.25[?l%" = O . + ~ ~ ] ~ ,
vector is 1 v 1 =Jm'2swith v, and v y being the filtration velocity in x and y
will be similarly as the kinematic viscosity, partitioned into constant andThe thermal diffusivity of the porous medium is defined as a p = /%?/pcf , and
directions respectively, and af the thermal diffusivity of the fluid (Amiri [5]).
perturbed parts u p= i T p +Lip . Introducing the heat capacity ratio o :
9 b C : ) + O - d ~ P C s )cT= (7)
P C . f
the heat energy equation (6) can be rewritten in the following form:
accordancewith the problem description, the initial and boundary conditions are:hydrodynamic and thermal boundary conditions are precisely defined. Inprincipal any transport phenomena in porous medium if the appropriatenamed modified Navier-Stokes equations for porous medium, we can solve inWith above formulated set of conservative equations (l), (5) and (S), commonly
v, = VJ =0 for x=O,D and y =O,D
V , = V , = o , F = 0 for t = O
for y=O,D (9)
T = F c = O
T=T, = l for x = 0
for x = D
3 Numerical scheme and procedure
With the vorticity vector mi= eyk&,/h,,representing the curl of the velocitytransformed with the use of the velocity-vorticity (VVF) variables formulation.the obtained modified Navier-Stokes equations (l), (5) and @), are furtherclassical BEM for solving complicated diffusion-convective problems. In BDIM,Boundary Domain Integral Method (BDIM) represents an extension of the
kinematic can be formulated i n the form of the parabolic kinematic equation:continuity equation and with addition of the relaxation parameter a , theApplying the curl operator directly to the vorticity defined above, using theequations will consequently be written for the case of planar geometry only.present work are limited to the two-dimensional case, all the subsequentkinematics and kinetics (Skerget et al. [6]). As the computational results of theso that the continuity and momentum equations are replaced by the equations offield, the computational scheme is partitioned into its kinematic and kinetic part
use the VVF on our momentum equation, the vorticity transport equation isnow:modified vorticity time step,only as a necessary mathematical step allowing us tothe momentum equation (5) . Introducing the new variable T, = t /@, the so calledThe kinetics is governed by the vorticity transport equation obtained as a curl of
properties:The last term of equation (11) represent a contribution from non-linear material
rewrite equation (8) in the form:temperature time step Z, = t / o also into energy equation which permits us toFor the same reason as with vorticity kinetics, we introduce new modified
-+vj-=ap-d T d T - a 2 T +L[z$).ar, axj ax jax j a x j (12)
level second order implicit scheme:field function U (velocity, vorticity, temperature) is approximated by the threeimportance to the success of the BDIM. For all equations, the time derivative of aconvective equation, the accurate integral representation of which is of keyet al. [6]). Each of this equation can be represented with the parabolic diffusion-which the weighted residuals technique of the BDIM has to be applied (SkergetEquations (lo), (11) and (12) represent the leading non-linear set of equations to
(l$+1 ~ 3un+'-4th" + l P
2At (13)
equation is of the form:The velocity equation (10) rewritten as a nonhomogenous parabolic diffusionwith n denoting the time step number.
a----- 3 ' ~ ; 'vi + b = o ,a x j a x j a t (14)
with the following corresponding integral representation:
Parameter c ( { ) denotes the fundamental solution related coefficient depending
the parabolic diffusion fundamental solution:on the position of the source point, r is the boundary of the domain R. U* is
* 1u =--..p(-&),
4 7 z m
following matrix form of kinematic integral equation (15) appears:influence matrices, and incorporation of boundary and initial conditions, theanalytically. After discretization of the computational domain, assembly of allindividual time increment, the time integrals in equation (15) may be evaluatedreference point S. Assuming constant variation of the velocity within thewith z = tn+'- t n and r being the distance fi-om the source point 5 to the
[H]{v,}=[G]{%} + e q [ G ] { w n j } - e , [ D j ] { ~ } + [ B ] { v ~ - ~ } . (17)
convective equation with a constant reaction term (Skerget and Jecl [7]):temperature kinetics are based on a nonhomogeneous elliptic diffusion-The formulations of the integral representation for the vorticity kinetics andintegration, taken over all individual boundary elements and all the internal cells.Here [H], [G], [Di],[B] are matrices composed of integrals, representing the
- a 2 u " , U p u + b = O ,k----a x j a x j a x j (18)
kinetics is obtained:the following resulting integral representation for vorticity and temperaturedefined considering the conservation laws and constitutive hypothesis. Finally,where U is taken as vorticity W and temperature T , respectively and k being
permeability and the characteristic length multiplied by the viscosity ratio:Da (Nield and Bejan [3]).The Darcy number is defined as the ratio between themomentum equation (2), we have to deal with the parameter called Darcy numberconditions given by equation (9). Whenever we consider the Brinkman term indescription of the problem is shown on Fig. l., with the initial and boundaryconvection in a square porous cavity heated from the side is investigated. TheTo check the validity of the proposed numerical procedure the problem of natural
D a = A +D
(24)
for the porous medium (Nield and Bejan [3]):important governing parameter for the present problem is the Rayleigh number(Jecl et al. [4]), where we have chosen the porosity to be @ = 0.5. The otherIn our case the viscosity ratio is equal to the reciprocity of the porosity A = l/@
10x10 subdomains was used. Time step is At = 0.001, while the convergencebe defined and it has been taken as o = 1. A uniform computational mesh ofnumerical results the heat capacity ratio in the heat energy equation (8) needs toother parameters have been defined earlier. In order to illustrate the typicalwhere A T is the temperature difference between hot and cold walls and all the
x = 0.81 and n = 0.364.be Carreau model with the following parameters: pn= 1.01, p _ = 5.9E - 0 4 ,criterion is determined to be E = 5 X Non-Newtonian model was chosen to
and the time evolution of the temperature field at the same time steps is shown onTime evolution of the velocity field at different time steps is presented in Fig. 2.satisfied, what however is not the case when using the Darcy law.boundary condition on the impermeable walls bounding the porous medium isand Prasad [S]). It also turns out that using the Brinkman extension the no-slip(2), Laplace or Brinkman term, becomes negligible for Da < 0.0001 (LauriatBrinkman momentum equation, the effect of the second viscous term in equationThe complete analysis will likely serve to confirm the fact that, when using thephase of evaluation and testing, therefore the simple test case is presented here.the presented theoretical work and chosen parameters is, at this moment, in thethermal behaviours of the porous cavity are studied. A numerical model based on
The effects of the Darcy number and the Rayleigh number on the flow and
and Ra* = 100, Da = lo-', q3 = 0.5, AT = l for Carreau fluid.Fig. 2: Vector velocity fields at different time steps (0.01,0.3 and 0.7 S)
IJ
and Ra* = 100, Da = lo-', q3 = 0.5 , AT = 1 for Carreau fluid.Fig. 3: Temperature fields at different time steps (0.01, 0.3 and 0.7S)
More test cases with detailed final results will be presented at the conference.
5 Conclusion
quadratic internal cell. The proposed numerical procedure is studied for the casenode quadratic boundary elements, and one continuous 9-node corner continuousis applied, where each subdomain is being constructed of four discontinuous 3-fundamental solution is employed for the kinetic part. The subdomain techniquefor the kinematic part of fluid motion, while elliptic diffusion-convectiveits kinematic and kinetic part. Parabolic diffusion fundamental solution is usedconstitutive equations, which allows separation of the computational scheme intofrom the side. The solution is based on the velocity-vorticity formulation ofconsidered representing the fluid that saturates the vertical porous cavity heatedthe Carreau model, which is adequate for many non-Newtonian fluids, is(BDIM). The Brinkman equation is used as the starting momentum equation, andNewtonian fluid is investigated utilising a Boundary Domain Integral MethodThe problem of natural convection in porous medium saturated with non-
for solving the transport phenomena i n porous medium.different parameters. The results indicated that the BDIM can be efficiently usedcharacteristics of the flow and temperature fields in the cavity are analyzed forof natural convection in square porous cavity heated from the side. The
Acknowledgements
Slovenia under the basic research project number 22-3289 (2001).received from the Ministry of Education, Science and Sport of Republic ofThe first author (R. Jecl) would like to acknowledge the financial support
References
Porous Media, Kluver Academic Publishers: Dordrecht, 1991.[ l ] Bear, J., Bachmat, Y., Introduction to Modelling of Transport Phenomena in
Pop, Elsevier Science: Oxford, pp. 31-56, 1998.(Chapter 2). Transportphenomena in porous media, eds. D.B. Ingham and I.
[2] Chen, CK, Hsiao, SW., Transport phenomena in enclosed porous cavities
York Inc., 1992.[3] Nield, D.A, Bejan, A., Convection in Porous Media, Springer-Verlag: New
Methods in Fluids, 35, pp. 39-54, 2001.transport phenomena in porous media. International Journal for Numerical
[4] Jecl, R., Skerget, L., PetreSin, E., Boundary domain integral method for
Transfer, 43, pp. 3513-3527, 2000.cavity filled with porous medium. International Journal of Heat and Mass
[ 5 ] Amiri, A.M., Analysis of momentum and energy transfer in a lid-driven
Methods in Engineering, 46, pp. 1291-1311, 1999.Boundary Domain Integral Method. International Journal for Numerical
[6] Skerget L., HriberSek, M., Kuhn, G., Computational Fluid Dynamics by
hgham and I. Pop, Elsevier Science: Oxford, (in print).porous medium. Transport phenomena in porous media - Vol 2., eds. D.B.
[7] Skerget L., Jecl, R., Boundary element method for transport phenomena in
Transfer, 109, pp. 688-696, 1987.numerical study for Brinkman-extended Darcy formulation. Journal of Heat
[S] Lauriat G., Prasad V., Natural convection in a vertical porous cavity: a
