Natural Convection in Porous Media: a Numerical Study of Brinkman Model

Natural convection in porous media: a numerical study of Brinkman model

R. Jecl?>, L. Skergef\ E. Petresin^ ^ Faculty of Civil Engineering, University ofMaribor, 2000 Maribor,

Slovenia EMail: [email protected] Faculty of Mechanical Engineering, University ofMaribor, 2000

Maribor, Slovenia EMail: [email protected]

Abstract

The problem of natural convection in porous media is being investigated utilising a Boundary Domain Integral Method. The Brinkman equation (Brinkman- extended Darcy formulation with inertia! term included), which allows the no-slip boundary condition to be satisfied, is used as the starting momentum equation. With the inclusion of the continuity and energy equations the obtained set of governed non-linear partial differential equations allows the transport phenomena of natural convection in homogeneous isotropic non-deformable porous media, saturated with single phase Newtonian fluid, to be conveniently studied. The purpose of this work is to present flow and heat transfer characteristics of the fluid: a) in a vertical porous cavity with vertical walls maintained at different constant temperature where the top and the bottom walls are adiabatic, as well as b) in a horizontal porous layer heated from below.

1 Introduction

Natural convection is one of the most frequently studied transport phenomena in porous media. Due to the general complexity of this phenomena, our studies are based on simplified mathematical model where we assume that:

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- porous media is a material consisting of a solid and a fluid phases. The solid phase is homogeneous, isotropic and non-deformable while the fluid is single phase, Newtonian and the fluid density is taken not to depend on pressure variations, but only on variations of temperature,

- porous media isflui dsaturated , - the solid and the fluid phase are in thermal equilibrium (7 = 7} =7^) and consequently the thermal behaviour of the porous media is described only by a single equation in terms of the average temperature T. When the temperature of the saturating fluid phase in a porous media is not uniform, some flows induced by buoyancy effects may occur. Commonly called free or natural convection movements, these flows depend on density differences due to temperature gradients and boundary conditions (Bear [1]).

2 Governing Equations

The system of partial differential equations governing the phenomena of natural convection in porous media represents the basic conservation balances of mass, momentum and energy and consist of: - continuity equation

• momentum equation

1 dv, 1

Darcy Law Brinkman extension

- energy equation

<5T dv T _ &T d [ „ dT

where v- isfiltratio nvelocity ; ^, /?, y stands for porosity, fluid density, kinematic viscosity of fluid,respectively ; K is permeability of porous media, dPjdXi pressure gradient in the flow direction and s.j is the strain rate tensor

defined as s^ = 1/2 (dv^fffxj +dvj/dx}. The kinematic viscosity is partitioned into constant and perturbated parts so that y = y + / ; T stands for temperature and O is thermal diffiisivity of the porous media defined as a^ -^/pCp ,

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where ^ is heat conductivity of porous media /^ = (1 - ^)A +^A with 4 being heat conductivity of the solid part. The thermal diffiisivity a^ is also partitioned into constant and perturbated parts, i.e., a^ = a^ + a^ . The coefficient

$) is so called heat capacity ratio, where p is density of fluid and solid phase and c^ is specific heat capacity.

Brinkman extension, in our case written in a way to allow a changeable viscosity (as our attempt in the future is to deal also with non-Newtonian saturating fluids), expresses the viscous resistance or viscous drag force exerted by the solid phase on the flowing fluid at their contact surfaces. With the

Brinkman equation we are able to satisfy the no-slip boundary conditions on an impermeable surfaces that bounds the porous media. It is important to stress out that the Brinkman equation is essentially an interpolation scheme between the Navier-Stokes and Darcy equation. It is well known that in the limit when the porosity approaches unity and consequently the permeability tends towards infinity, the Brinkman equation transforms into the classical Navier-Stokes equation for a pure fluid, meanwhile for the permeability converging to zero, the Brinkman term becomes negligible and the Darcy law is being recovered. With above formulated set of conservative equations named modified Navier-

Stokes equations for porous media, we can solve in principal any transport phenomena in porous media if the appropriate hydrodynamic and thermal boundary conditions are precisely defined.

3 Velocity - Vorticity Formulation

Governing equations may be written in different formulations as velocity- pressure, vorticity-stream function, velochy-vorticity, penalty formulations, ect. We have chosen to use the Velocity -Vorticity Formulation (WF) on account of its advantages when dealing with the Boundary Element Method (BEM). This formulation is well suited also for the Boundary Domain Integral Method

(BDDM) which is an extension of the BEM. With the vorticity vector o)j = e^ dv^/dXj representing the curl of the velocity field, the computational scheme is partitioned into its kinematic and kinetic part so that the continuity and momentum equations are replaced by the equations of kinematics and kinetics (Skerget [2]). Applying the curl operator directly to the vorticity defined above, using the continuity equation and with addition of the relaxation parameter a , the kinematic can be formulated in the form of the parabolized kinematic equation written for the plane case of calculation as

^v, 1

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The kinetics is governed by the vorticity transport equation obtained as a curl of the momentum equation which reads:

- 6v" - + — -,< .\ (5) ^ ^ ^

The new variable appears in vorticity equation, namely T^ = t/fi which is so called modified vorticity time step, introduced only as a necessary mathematical step allowing us to use the WF on our momentum equation. The last term of eqn. (5) represent a contribution from non-linear material properties defined as

dx) dy dx ' dx dy By

For the same reason as with vorticity kinetics we introduce new modified temperature time step also into energy equation ty = f/cr that permits us to write energy equation in the form

4 Boundary Domain Integral Equations

The obtained set of partial differential equations (4), (5), (6) have to be, using weighted residual techniques in combination with appropriate fundamental solutions - Green functions and completed with suitable boundary conditions, transformed into boundary domain integral equations. Integral representation can be derived by using integral representation of a parabolic diffusion-convective equation, rendering boundary domain integral equations for the plane (2D): - kinematics

/9 ' - eg J a-j-dZl + plvj^ u

describing the time dependent transport of velocity field v,, where w* is an elliptic modified Helmholtz fundamental solution which takes into account the effects of geometry, time step and material properties (Jecl [3]), - vorticity kinetics

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describing the time dependent transport of vorticity field CD in an integral formulation, where U* =y(/>u and u is the elliptic diffusion-convective fundamental solution of steady diffusion-convective PDE with firstorde r reaction term (Skerget [2]), considering the effects of geometry, material properties, time step and velocity v, which has to be decomposed into an average constant vector and perturbation vector as v^ = v) + v," , - heat energy kinetics

(9)

"P a

describing the time dependent transport of temperature in an integral formulation, where U* = au and %* is once again the elliptic diffusion-convective fundamental solution of steady diffiision-convective PDE with reaction term.

5 Discretized Boundary Domain Integral Equations

For the numerical approximate solution of field functions (velocity, vorticity, temperature), the corresponding boundary domain integral representation is written in a discretized manner in which the integrals over the boundary and domain are approximated by a sum of integrals over E individual boundary elements and C internal cells (Hribersek [4]). The discretized equation for kinematic is written for all boundary nodes resulting in matrix system

(10)

to be solved for unknown boundary velocity components or their normal derivatives, while the computation of internal domain velocity components, if needed, is done afterwards in an explicit manner, point by point.

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The discretized equations for vorticity kinetic and for heat energy kinetic should be written for all boundary and internal nodes resulting in a matrix systems for vorticity kinetic

yielding the solution of the unknown boundary vorticity flux values and unknown domain vorticity values. For the heat energy kinetic we can write

(12)

to determine unknown boundary temperature flux or boundary temperature values and temperature internal domain values. Here [H\ \G\ [/>,], [B] are matrices composed of integrals, representing the integration, taken over individual boundary elements and over the internal cells. The system of discretized equations is solved by coupling kinetic and kinematic equations and subdomain technique is used. In our case each subdomain consist of four discontinuous 3-node quadratic boundary elements and one continuous 9-node corner continuous quadratic internal cell.

6 Test examples

To check validity of proposed numerical procedure we will discus the problem of natural convection in a vertical porous cavity and in a porous layer. Unlike in cavities heated from the side, where fluid motion is present as soon as an infinitesimally small temperature difference is applied across the layer, in layers heated from below convection is possible only if the temperature difference exceeds a critical threshold value. Furthermore, as the temperature difference increases, in layers heated from the side the flow remains unicellular, whereas in layers heated from below the flow evolves as a collection of Benard cells whose number increases discretely.

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6.1 Natural convection in a vertical porous cavity

The description of the physical problem is shown in Fig. 1. and represent a two- dimensional, vertical cavityfille dwit h an isotropic, homogeneous, fluid-saturated porous media whose one vertical wall is isothermally heated, the other one is isothermally cooled, and the horizontal walls are adiabatic.

s dT l&y = 0 2

i- porous TC = -05 H media

dT[dy =0 D

Figure 1.: Geometry and boundary conditions for porous cavity

Whenever considering Brinkman equation we have to deal with so called Darcy number, appearing as the ratio between the permeability and the characteristic length multiplied with the viscosity ratio which is in our case equal to the reciprocity value of porosity. We must stress out that, with the use of BDIM the Darcy number is not explicitly derived as we are not employing the non- dimensionalisation of governing equations which is the common procedure used with other numerical methods. We will use that parameter only from the reasons of comparison with the published results, noting that the permeability itself completely defines the porous media when BDIM is used. Thus, the governing parameters for the presented problem are: - porosity 0,

- permeability of porous media K, defined in the terms of so called modified Darcy number as Da = K/fiD*

- aspect ratio A = H/D, - modified (porous) Rayleigh number Ra* = gfiKDhT/y dp ,

- specific heat ratio & = + p^ c^ fpc^ (1 - $), where D, H , A7 are the width of the cavity, the height of the cavity and the temperature difference between hot and cold walls, respectively. Parameter J3 is the isobaric coefficient of thermal expansion of a fluid.

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We have tested our numerical model on several different cases, but here we present only the velocity (Fig. 2.) and temperature (Fig. 3.) fields for modified

Rayleigh number Ra* = 100 and different Darcy numbers: Da = 10'*,10 \ 10~*, 10~* . The effect of an increase in Darcy number, appears namely to be very similar at all Rayleigh numbers, although it is known that the effect of viscous (Brinkman) term becomes more important at high Rayleigh numbers.

a) b)

Figure 2.: Velocityfield sfo r A = 2,f l = 05, A7 = 1 and Ra =100 :

a) Da = W(v,,ax = 0283E + 03); b) Da = 10"(v^ = 0249E + 03);

c) Da = 10 2 ( v^ = 0.479E + 02 ); d) Da = 10 ( v^ = 0^78E + 01).

Figure 3.: Temperaturefield sfo r A = 2, 0 = 05 , AT = 1 and Ra =100:

a) Da = W ( Nu = 3.847 ); b) Da = 10'* ( Nu = 3.833 ); c) Da = l(T*(Nu = 3315); d) Da = 10' (Nu = 2.966).

From results obtained for different Darcy numbers we can clearly observe that the velocity and temperature fields are almost identical for Da = 10

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Da = 10 * but with a further increase in the Darcy number (that means with an

increase in permeability K), the velocity and the temperature fields are starting becomes significantly modified. With the use of Boundary Domain Integral Method for natural convection in

porous cavity we can confirm that the effect of the viscous - Brinkman term is negligible as long as Darcy number is less than Da < 10^ which agrees with the

observations of others that solved the same problem based upon the completely different numerical methods (Lauriat [5]).

6.2 Natural convection in a porous layer heated from bellow

The description of the physical problem is shown in Fig. 4. and represent a two- dimensional, porous layer of finite extend, bounded by two horizontal impermeable walls, where the lover one is heated and the upper one is cooled. The governing parameters for the presented problem are:

- porosity , - permeability of porous media K, defined in the terms of so called modified

Darcy number as Da - K/&H*, - aspect ratio A = D/H,

- fluid Rayleigh number Ha, = gfiH* &T/y a ,

- specific heat ratio cr = ^ + /%c^ /pc^ (1 - (/>), where D, H and A7 are the width and the height of porous layer and the

temperature difference between bottom and top surfaces, respectively.

= 0 porous = 0 H media

Figure 4.: Geometry and boundary conditions for porous layer

The above described problem is, at this moment, in the phase of evaluation and testing (Malmou [6]). We estimate that the results, showing the redistribution of the velocity and temperature fields in porous layer heated from below, will soon be available and will be presented at the conference.

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8 Conclusion

The problem of natural convection in porous media is being investigated utilising a Boundary Domain Integral Method. The Brinkman equation is used as the starting momentum equation. The solution is based on the Velocity-Vorticity Formulation of constitutive equations which allows separation of the computational scheme into its kinematic and kinetic part. Elliptic modified

Helmholtz fundamental solution is used for the kinematic part of fluid motion, while elliptic diffiision-convective fundamental solution is employed for a kinetic one. In the numerical evaluations the subdomain technique is being applied, where each subdomain is being constructed of four discontinuous 3-node quadratic boundary elements and one continuous 9-node corner continuous quadratic internal cell. Two main configurations in which the phenomena of natural convection are expected to exist, namely a porous cavity heated from the side and the porous layer heated from below, are presented and discussed in details.

References

1. J. Bear, Y. Bachmat: Introduction to Modelling of Transport Phenomena in Porous Media, Kluver Academic Publishers, 1991.

2. L. Skerget, M. Hribersek, G. Kuhn: Computational Fluid Dynamics by

Boundary-Domain Integral Method, accepted for publication in Int. J. Num. Meth. Eng., 1999.

3. R. Jecl, L. Skerget: Fluid Transport Process in Porous Media Using Boundary

Domain Integral Method, Proc. of the 4th Eur. Comp. Fluid Dynamics Cow/, eds. K. D. Papailiou,..., Chichester, John Willey & Sons, pp. 1180-1185, 1998.

4. M. Hribersek, L. Skerget: Iterative Methods in Solving Navier-Stokes Equations by the Boundary Element Method, Int. Journal for Num. Meth. in

Engineering, John Willey & Sons, Ltd., pp. 115-139, 1996.

5. G. Lauriat, V. Prasad: Natural Convection in a Vertical Porous Cavity: a Numerical Study for Brinkman-Extended Darcy Formulation, Journal of Heat

Transfer, Vol. 109, ASME, pp. 688-696, 1987.

6. M. Malmou, L. Robillard, E. Bilgen, P. Vasseur: Entrainment Effect of a Moving Thermal Wave on Benard Cells in a Horizontal Porous Layer, Proc. of the 2nd Int. Conf. on Adv. Comp. Meth. in Heat Transfer, eds. L.C. Wrobel, C.A. Brebbia, AJ. Nowak, CMP, Southampton, pp. 442-456, 1992.