A complete set of macroscopic two-equation turbulence model equations has been established for analyzing turbulent flow and heat transfer within porous media. The volume-averaged transport equations for the mass, momentum, energy, turbulence kinetic energy and its dissipation rate were derived by spatially averaging the Reynolds-averaged set of the governing equations. The additional terms representing production and dissipation of turbulence kinetic energy are modeled introducing two unknown model constants, which are determined from a numerical experiment using a spatially periodic array. In order to investigate the validity of the present macroscopic turbulence model, a macroscopically unidirectional turbulent flow through an infinite array of square rods is considered from both micro- and macroscopic-views. It has been found that the stream-wise variations of the turbulence kinetic energy and its dissipation rate predicted by the present macroscopic turbulence model agree well with those obtained from a large scale microscopic computation over an entire field of saturated porous medium.
A volume averaging theory (VAT) established in the field of fluid saturated porous media has been successfully exploited to derive a general set of bioheat transfer equations for blood flows and its surrounding biological tissue. A closed set of macroscopic governing equations for both velocity and temperature fields in intra-and extra-vascular phases has been established, for the first time, using the theory of anisotropic porous media. Firstly, two individual macroscopic energy equations are derived for the blood flow and its surrounding tissue under the thermal non-equilibrium condition. The blood perfusion term is identified and modeled in consideration of the transvascular flow in the extravascular region, while the dispersion and interfacial heat transfer terms are modeled according to conventional porous media treatments. It is shown that the resulting two-energy equation model reduces to Pennes model, Wulff model and their modifications, under appropriate conditions. Subsequently, the two-energy equation model has been extended to the three-energy equation version, in order to account for the countercurrent heat transfer between closely spaced arteries and veins in the circulatory system and its effect on the peripheral heat transfer. This general form of three-energy equation model naturally reduces to the energy equations for the tissue, proposed by Chato, Keller and Seiler. Controversial issues on blood perfusion, dispersion and interfacial heat transfer coefficient are discussed in a rigorous mathematical manner.
A numerical experiment at a pore scale using a full set of Navier-Stokes and energy equations has been conducted to simulate laminar fluid flow and heat transfer through an anisotropic porous medium. A collection of square rods placed in an infinite two-dimensional space has been proposed as a numerical model of microscopic porous structure. The degree of anisotropy was varied by changing the transverse center-to-center distance with the longitudinal center-to-center distance being fixed. Extensive calculations were carried out for various sets of the macroscopic flow angle, Reynolds number and degree of anisotropy. The numerical results thus obtained were integrated over a space to determine the permeability tensor, Forchheimer tensor and directional interfacial heat transfer coefficient. It has been found that the principal axes of the permeability tensor (which controls the viscous drag in the low Reynolds number range) differ significantly from those of the Forchheimer tensor (which controls the form drag in the high Reynolds number range), The study also reveals that the variation of the directional interfacial heat transfer coefficient with respect to the macroscopic flow angle is analogous to that of the directional permeability. Simple subscale model equations for the permeability tensor, Forchheimer tensor and directional Nusselt number have been proposed for possible applications of VAT to investigate flow and heat transfer within complex heat and fluid flow equipment consisting of small scale elements.
Thermal dispersion in convective flow in porous media has been numerically investigated using a two-dimensional periodic model of porous structure. A macroscopically uniform flow is assumed to pass through a collection of square rods placed regularly in an infinite space, where a macroscopically linear temperature gradient is imposed perpendicularly to the flow direction. Due to the periodicity of the model, only one structural unit is taken for a calculation domain to resolve an entire domain of porous medium. Continuity, Navier–Stokes and energy equations are solved numerically to describe the microscopic velocity and temperature fields at a pore scale. The numerical results thus obtained are integrated over a unit structure to evaluate the thermal dispersion and the molecular diffusion due to tortuosity. The resulting correlation for a high-Peclet-number range agrees well with available experimental data.
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