Transactions on Modelling and Simulation vol 24, © 1999 WIT Press, www.witpress.com, ISSN 1743-355X 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: Transactions on Modelling and Simulation vol 24, © 1999 WIT Press, www.witpress.com, ISSN 1743-355X 146 Boundary Elements - 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 is fluid saturated, - 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 <?v/v, Darcy Law Brinkman extension - energy equation <5T dv T _ &T d [ „ dT where v- is filtration velocity; ^, /?, 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 , Transactions on Modelling and Simulation vol 24, © 1999 WIT Press, www.witpress.com, ISSN 1743-355X Boundary Elements 147 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 <?V, do* dXdX a dt ** dx Transactions on Modelling and Simulation vol 24, © 1999 WIT Press, www.witpress.com, ISSN 1743-355X 148 Boundary Elements 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 6T 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 <Kl 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 Transactions on Modelling and Simulation vol 24, © 1999 WIT Press, www.witpress.com, ISSN 1743-355X Boundary Elements 149 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 first order 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. Transactions on Modelling and Simulation vol 24, © 1999 WIT Press, www.witpress.com, ISSN 1743-355X 150 B oundary Elements 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