José L. Lage

A general two-equation macroscopic turbulence model for incompressible flow in porous media

Abstract view of the practical and fundamental importance to heat and mass transfer, we present a two-equation turbulence model for incompressible flow within a fluid saturated and rigid porous medium. The derivation consists of time-averaging the general (macroscopic) transport equations and closing the model with the classical eddy diffusivity concept and the Kolmogorov-Prandtl relation. The transport equations for the turbulence kinetic energy (κ) and its dissipation rate (e) are attained from the general momentum equations. Analysis of the κ-e equations proves that for a small permeability medium, small enough to minimize the form drag (Forchheimer term), the effect of a porous matrix is to damp turbulence, as physically expected. For the large permeability case the analysis is inconclusive as the Forchheimer term contribution can be to enhance or to damp turbulence. In addition, the model demonstrates that the only possible solution for steady unidirectional flow is zero macroscopic turbulence kinetic energy. The implications of this conclusion are far reaching. Among them, this conclusion supports the hypothesis of having microscopic turbulence, known to exist at high speed flow, damped by the volume averaging process. Therefore, turbulence models derived directly from the general (macroscopic) equations will inevitably fail to characterize accurately turbulence induced by the porous matrix in a microscopic sense.

The resonance of natural convection in an enclosure heated periodically from the side

Abstract This is a numerical and theoretical investigation of natural convection in a two-dimensional square enclosure with one side cold and isothermal, and the other side heated with pulsating heat flux. It is shown numerically that the buoyancy induced circulation resonates to a certain (single) frequency of the pulsating heat input. The resonance is characterized by maximum fluctuations in the total heat transfer rate through the vertical midplane of the cavity. The numerical experiments cover the Prandtl number range 0.01–7, the heat flux Rayleigh number range 103–109, and the nondimensional frequency range 0– 0.3. It is shown that the critical frequencies determined numerically can be anticipated based on theoretical grounds, by matching the period of the pulsating heat input to the period of the rotation (circulation) of the enclosed fluid. In the last section, the same theoretical argument is used to predict the critical period for natural convection resonance in an enclosed porous medium saturated with fluid.

Application of Fractional Calculus to Fluid Mechanics

We present the application of fractional calculus, or the calculus of arbitrary (noninteger) differentiation, to the solution of time-dependent, viscous-diffusion fluid mechanics problems. Together with the Laplace transform method, the application of fractional calculus to the classical transient viscous-diffusion equation in a semi-infinite space is shown to yield explicit analytical (fractional) solutions for the shear-stress and fluid speed anywhere in the domain

A modified form of the κ-ε model for turbulent flows of an incompressible fluid in porous media

Abstract This work extends the recently published work of Antohe and Lage on the development of a macroscopic two-equation turbulence model for an incompressible flow in porous media. The difference occurs in approximating the Forchheimer term in the time-averaged momentum equation. Unlike the Antohe and Lage’s work, where only the linear terms of the expansion are kept, in the presently proposed model we include the second-order correlation term. This inclusion gives rise to an extra term in the transport momentum equation which, in turn, gives rise to additional terms in the transport equations for the turbulent kinetic energy and the dissipation rate. The additional higher order terms produce correlation coefficients that are used to absorb any departure from the clear flow when expressing the two-equation turbulence model for incompressible flow in a porous medium.

Experimental Determination of Permeability and Inertia Coefficients of Mechanically Compressed Aluminum Porous Matrices

A heat exchanger, using mechanically compressed microporous matrices, is being developed for cooling high power electronics. The thermal efficiency of this new device depends on the hydraulic characteristics (porosity Φ, permeability K, and Forchheimer coefficient c F ) of the matrix inserted in it. These quantities have to be obtained experimentally as predictive models do not exist. Twenty-eight compressed matrices are initially chosen for experimental testing. Based on structural requirements, nine matrices are selected for full hydraulic characterization. The determination of permeability and inertia coefficient of each matrix is performed following a proposed direct methodology based on the curve fitting of the experimental results. This methodology is found to yield more consistent and accurate results than existing methods. The uncertainty of the experimental results is evaluated with a new and general procedure that can be applied to any curve fitting technique. Results indicate that the tested matrices have a unique characteristic, that of a relatively wide porosity range, from 0.3 to 0.7, within a relatively narrow permeability range, from 1.0 x 10 -10 m 2 to 12 X 10 -10 m 2 . The inertia coefficient varies from 0.3 to 0.9. These hydraulic characteristics lead to a microporous heat exchanger performing within requirements.

Forced convection in a fluid-saturated porous-medium channel with isothermal or isoflux boundaries

We present a fresh theoretical analysis of fully developed forced convection in a fluid-saturated porous-medium channel bounded by parallel plates, with imposed uniform heat flux or isothermal condition at the plates. As a preliminary step, we obtain an ‘exact’ solution of the Brinkman-Forchheimer extension of Darcys momentum equation for flow in the channel. This uniformly valid solution permits a unified treatment of forced convection heat transfer, provides the means for a deeper explanation of the physical phenomena, and also leads to results which are valid for highly porous materials of current practical importance.

Two Types of Nonlinear Pressure-Drop Versus Flow-Rate Relation Observed for Saturated Porous Media

Previous reports of experiments performed with water (Fand et al., and Kececioglu and Jiang) indicated that beyond the Forchheimer regime the rate of change of the hydrostatic pressure gradient along a porous medium suddenly decreases. This abnormal behavior has been termed transition to turbulence in a porous medium. We investigate the relationship between the hydrostatic pressure gradient of a fluid (air) through a porous medium and the average seepage fluid velocity. Our experimental results, reported here, indicate an increase in the hydrostatic pressure rate beyond a certain transition speed, not a decrease. Physical arguments based on a consideration of internal versus extemal incompressible viscous flow are used to justify this distinct behavior, a consequence of the competition between a form dominated transition and a viscous dominated transition. We establish a criterion for the viscous dominated transition from consideration of the results of three porous media with distinct hydraulic characteristics. A theoretical analysis based on the semivariance model validation principle indicates that the pressure gradient versus fluid speed relation indeed departs from the quadratic Forchheimer-extended Darcy flow model, and can be correlated by a cubic function of fluid speed for the velocity range of our experiments.

Amplitude effect on convection induced by time-periodic horizontal heating

Abstract Heat and momentum transport is investigated theoretically and numerically considering a rectangular enclosure filled with clear fluid or with fully saturated porous medium, under time-periodic horizontal heating. Numerical simulations, of various configurations, indicate that the natural convection activity within the enclosure peaks at several discrete frequencies, with the climax attained at a heating frequency referred to as resonance frequency . A general theory for predicting this resonance frequency is developed from the natural frequency of the flow circulating inside the enclosure. The resonance frequency can be calculated by solving a system of non linear equations, function of the averaged Rayleigh number, the Prandtl number, the enclosure aspect ratio, the heating amplitude, and the Darcy number for the porous medium case. Theoretical predictions agree well with numerical results. It is shown that the convection intensity within the enclosure increases linearly with heating amplitude for a wide range of parameters. Moreover, the flow response to pulsating heat is continuously enhanced as the system becomes more permeable. Time evolution graphs, phase-plane portraits, and streamlines highlight several distinct phases of the periodic heating process.

Numerical study of a low permeability microporous heat sink for cooling phased-array radar systems

Abstract Microporous media is being used to develop an improved forced convection cold plate device for removing waste heat from high frequency phased-array radar apertures. The waste heat, generated by transmit and receive microwave functions mounted in separate electronic modules, is conducted to the surfaces of a thin rectangular enclosure (cold plate) through which coolant flows. The performance of the phased-array radar is known to deteriorate very rapidly when the difference in operating temperatures of identical electronic components within each module increases. The cold plate device investigated here, designed to minimize this temperature difference, consists of a microporous layer placed (brazed) within the cold plate. A theoretical transport model is developed around an extended system of two-dimensional equations obtained by considering a thin enclosure and integrating the original three-dimensional equations along the direction of smaller dimension. Thermo/hydraulic characteristics are obtained through numerical simulations considering a low permeability aluminum alloy porous layer, and air, water and PAO as coolants. A theoretical estimate of the global pressure drop across the cold plate is also obtained and compared with the numerical results. The microporous cold plate provides substantially more uniform operating temperature for identical components in all module housings than a cold plate without porous layer. Results also suggest an increased global heat transfer coefficient reducing the operational (junction) temperature of the electronics for the same waste heat.

Convection induced by inclined thermal and solutal gradients in a shallow horizontal layer of a porous medium

A theoretical examination is made of convection, induced by applied thermal and solutal gradients inclined to the vertical, in a shallow horizontal layer of a saturated porous medium. The horizontal components of these gradients induce a Hadley circulation, which becomes unstable when the vertical components are sufficiently large. A linear stability analysis is carried out, and calculations are made using a low-order Galerkin approximation for the various modes of instability. The orientation of the preferred mode and the other critical quantities are determined for representative parameter values.