**1. Introduction**

Problems of transport phenomena in porous media have been widely investigated over the last few decades, mainly because of several important applications, which could be found in industry and environment, e.g., building insulation systems, dispersion of contaminants through water saturated soil, protection of groundwater resources, combustion technology.

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Buoyancy driven flows in porous enclosures have been simulated using different mathematical models and numerical techniques. Most commonly used mathematical model of governing momentum equation is the Darcy's law, which is valid for the laminar flow regime (Re < 10), where the velocities are low and the viscous forces are predominant over inertia forces [1]. Extensions of the governing momentum equations have been made by analogy with the Navier-Stokes equations with addition of Brinkman term in order to consider the viscous diffusion and Forchheimer term to study the inertia effects on the free convection [2].

Another possible extension of BEM is the boundary domain integral method (BDIM), which enables solving of strong nonlinear problems, where the domain-based effects are dominant, typical for the examples of the diffusion-convection problems [30–32]. The numerical algorithm solves the velocity-vorticity formulation of the Navier-Stokes equations, which separate the numerical scheme into the kinematic and kinetic computational parts. Consequently, the pressure is removed from the field functions conservation equations, and the calculation of the boundary pressure values is eliminated. Further advantage of BDIM is efficient dealing with boundary conditions on the solid boundaries in case of solving the vorticity equation. The vorticity is calculated explicitly from the kinematic computational part without using any approximate formulae. Using the subdomain technique [33], the problem of fully populated system matrices and corresponding memory requirements can be importantly reduced. A very stable and accurate numerical description of coupled diffusion-convection problems follows the use of Green's functions of the appropriate linear differential operators instead of upwinding schemes of different orders, as this is the case in other domain-type numerical techniques, which also eliminate the oscillations in the numerical solutions. A wavelet compression method for a single-domain BEM in 2D was introduced in [34]. In this chapter presented numerical algorithm is based on the combination of single- and subdomain BEM. The main advantage of the used method is that it enables an accurate prediction of the vorticity fields, which are in general defined as a curl of the velocity field. The vorticity is generated on the walls of the domain and influences the development of the flow field and furthermore the heat transfer. The single-domain BEM is used to solve the kinematics equation. The method is based on the fast multipole algorithm (FMM), which was introduced by Greengard and Rokhlin [35] for particle simulations and was later used for a wide variety of problems, e.g., for acceleration of the boundary integral Laplace equation by [36] and for coupling with BEM for the boundary matrices by [37]. The sub-domain BEM is used to solve the equations of the diffusion-advection type. A mesh of entire domain is made, where the integral equations are written for each of the sub-domains separately ([38–41]). Functions are discretized using the continuous quadratic boundary elements, whereas flux is discretized using discontinuous linear boundary elements, which enable to avoid flux definition problems in corners and edges. An over-determined system of linear equations is obtained, which is solved by a least squares manner.

Simulation of Natural Convection in Porous Media by Boundary Element Method

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A numerical approach based on the BEM has been used to solve a problem of buoyancy driven flows in porous media domain, saturated with pure fluid or nanofluid. The mathematical model is based on the Navier-Stokes equations, which are averaged over the representative elementary volume and rewritten into the velocity-vorticity formulation. The influence of several governing parameters, e.g., Rayleigh number, Darcy number, and volume fraction of

The most general mathematical model for the transport phenomena in porous media is based on the volume-averaged Navier-Stokes equations, which are primarily written on the

nanoparticles, on the heat transfer and fluid flow characteristics is analyzed.

**2. Mathematical model**

**2.1. Governing equations**

Problems of natural convection in porous media were studied intensively in last few decades, mainly for the cases of two-dimensional geometries. Two types of geometries are commonly investigated: porous enclosures where temperature gradient is imposed horizontally [3–7] or vertically [8–11]. Studies considering three-dimensional geometries are rare and are usually confined on using a simplified mathematical model, e.g., Darcy model or to conditions of heating from below [12–18]. Researches considering three-dimensional cavities with the condition of heating from the side were published in [19–21].

Recently, several researchers have been investigated buoyant flow in porous media domains saturated with nanofluids [22–24]. Nanoscale particles are often added to working fluids in order to enhance heat transfer or cooling processes. A comprehensive review of the studies considering convection heat transfer in porous media saturated with nanofluid was published in [25].

The solutions of the problems of transport phenomena in porous media have been obtained using different numerical methods, where the most commonly used methods are the finite element method (FEM), the finite difference method (FDM), and the finite volume method (FVM). As an alternative to others, in engineering practice widely used methods, the BEM was developed mainly because it was very efficient for solving potential problems of fluid mechanics (inviscid fluid flow, heat conduction, etc.), where the mathematical transformation of the governing set of partial differential equation results in boundary integral equations only. To rewrite the partial differential equation into an equivalent integral representation, the known fundamental solutions of the differential operator [26] and the Green's theorem are used. The discretized system contains only a fully populated system of integrals over boundary elements, which represent the main advantage over the volume-based methods.

When dealing with nonhomogenous and nonlinear problems, e.g., diffusion-convection problems, the domain integrals occur in the integral representation as well, which demands the extension of the classical BEM in order to additionally deal the problem within the domain. The main issue in this case is the evaluation of the domain matrices, which are full and unsymmetrical and require a lot of storage space. Several techniques have been developed in order to eliminate the domain integrals or transform them into the boundary integrals. One of the possibilities is the dual reciprocity boundary element method (DRBEM), which transforms domain integrals into a finite series of boundary integrals [27–29]. The nonhomogenous term is expanded in terms of radial basis functions. Since the discretization of the domain is represented only by grid points and the discretization of the geometry and fields on the boundary is piecewise polygonal, the DRBEM is still more flexible and efficient against other numerical methods, e.g., FDM.

Another possible extension of BEM is the boundary domain integral method (BDIM), which enables solving of strong nonlinear problems, where the domain-based effects are dominant, typical for the examples of the diffusion-convection problems [30–32]. The numerical algorithm solves the velocity-vorticity formulation of the Navier-Stokes equations, which separate the numerical scheme into the kinematic and kinetic computational parts. Consequently, the pressure is removed from the field functions conservation equations, and the calculation of the boundary pressure values is eliminated. Further advantage of BDIM is efficient dealing with boundary conditions on the solid boundaries in case of solving the vorticity equation. The vorticity is calculated explicitly from the kinematic computational part without using any approximate formulae. Using the subdomain technique [33], the problem of fully populated system matrices and corresponding memory requirements can be importantly reduced. A very stable and accurate numerical description of coupled diffusion-convection problems follows the use of Green's functions of the appropriate linear differential operators instead of upwinding schemes of different orders, as this is the case in other domain-type numerical techniques, which also eliminate the oscillations in the numerical solutions. A wavelet compression method for a single-domain BEM in 2D was introduced in [34].

In this chapter presented numerical algorithm is based on the combination of single- and subdomain BEM. The main advantage of the used method is that it enables an accurate prediction of the vorticity fields, which are in general defined as a curl of the velocity field. The vorticity is generated on the walls of the domain and influences the development of the flow field and furthermore the heat transfer. The single-domain BEM is used to solve the kinematics equation. The method is based on the fast multipole algorithm (FMM), which was introduced by Greengard and Rokhlin [35] for particle simulations and was later used for a wide variety of problems, e.g., for acceleration of the boundary integral Laplace equation by [36] and for coupling with BEM for the boundary matrices by [37]. The sub-domain BEM is used to solve the equations of the diffusion-advection type. A mesh of entire domain is made, where the integral equations are written for each of the sub-domains separately ([38–41]). Functions are discretized using the continuous quadratic boundary elements, whereas flux is discretized using discontinuous linear boundary elements, which enable to avoid flux definition problems in corners and edges. An over-determined system of linear equations is obtained, which is solved by a least squares manner.

A numerical approach based on the BEM has been used to solve a problem of buoyancy driven flows in porous media domain, saturated with pure fluid or nanofluid. The mathematical model is based on the Navier-Stokes equations, which are averaged over the representative elementary volume and rewritten into the velocity-vorticity formulation. The influence of several governing parameters, e.g., Rayleigh number, Darcy number, and volume fraction of nanoparticles, on the heat transfer and fluid flow characteristics is analyzed.
