Edimilson J. Braga

Turbulent Heat Transfer in an Enclosure With a Horizontal Permeable Plate in the Middle

Turbulent natural convection in a vertical two-dimensional square cavity, isothermally heated from below and cooled at the upper surface, is numerically analyzed using the finite volume method. The enclosure has a thin horizontal porous obstruction, made of a highly porous material and extremely permeable, located at the cavity midheight. Governing equations are written in terms of primitive variables and are recast into a general form. For empty cavities, no discrepancies result for the Nusselt number when laminar and turbulent model solutions are compared for Rayleigh numbers up to 10 7 . Also, in general the porous obstruction decreases the heat transfer across the heated walls showing overall lower Nusselt numbers when compared with those without the porous obstruction. However, the presence of a porous plate in the cavity seems to force an earlier separation from laminar to turbulence model solutions due to higher generation rates of turbulent kinetic energy into the porous matrix.

NUMERICAL SIMULATION OF TURBULENT FLOW IN SMALL-ANGLE DIFFUSERS AND CONTRACTIONS USING A NEW WALL TREATMENT AND A LINEAR HIGH REYNOLDS k–ε MODEL

This work presents numerical prediction for the turbulent flow field confined in a circular duct past a segment of gradually varying cross section. Both expanding and contracting sections are investigated. Equations of boundary-layer type are used and the linear k–ϵ model, in its high Reynolds form, is applied. A new correlation for treating the grid point closest to the wall is proposed. A marching-forward method is employed for sweeping the computational domain. Computations are first performed for developing and fully developed constant-area ducts in order to assess the reliability of the code. Results are then presented for contractions and diffusers, where comparisons with experimental data for air and water are carried out. Turbulence damping in contractions and its enhancement in diffusers are calculated correctly. Further, for contractions with angles of up to 21°, the use of a parabolic solver shows good agreement with experimental values for the mean and statistical quantities. For diffusers, adverse pressure gradient along the flow limits the quality of the predictions as the angle and length of diffuser increase past 5° and 10 duct radii, respectively.

Use of porous-continuum and continuum models for determining the permeability of porous cavities under turbulent free convection

ABSTRACT The aim of this work is to estimate the permeability of porous enclosures for numerical solutions of turbulent natural convection in a square cavity. The motivation is that available permeability correlations were proposed based on force rather than natural convection through permeable media. Although commonly seen as a medium property, permeability is measured with a flow through the permeable structure and, as such, its value may carry a flow type dependency. Here, it is assumed that a fixed amount of a solid conducting material is distributed within the cavity and two mathematical models are used and compared when calculating the cavity Nusselt number. First, a porous-continuum model is considered based on the assumption that the solid and the fluid phases are observed as a single medium, over which volume- and time-averaged transport equations apply. Second, a continuum model is used to solve local momentum and energy equations, in both the solid and void spaces, through a conjugate heat transfer solution. The average Nusselt number at the hot wall obtained from the porous-continuum model for several Darcy numbers are compared with those obtained with the continuum model using up to N = 1,024 obstacles within the cavity. When comparing the two methodologies, this study shows that the average Nusselt number calculated by each approach differs by as much as 32% when the number of obstacles N is increased to 1,024. Based on that, an adjustment on the used correlation for calculating the porous medium permeability is proposed to match the Nusselt numbers calculated with the two models. Results indicate that the use of the new correlation gives results for Nu that differ less than about 4% for the range 4 < N < 1,024.

Turbulent Heat Transfer in a Horizontal Enclosure With a Thin Porous Obstruction in the Middle

Turbulent natural convection in a horizontal two-dimensional square cavity, isothermally heated from below and cooled at the upper surface, is numerically analyzed using the finite volume method and a generalized coordinate system. The enclosure has a thin horizontal porous obstruction located at the cavity mid height. Governing equations are written in terms of primitive variables and are recast into a general from. In general, the porous obstruction decreases the heat transfer across the heated walls showing an overall lower Nusselt numbers when compared with those without the same porous obstruction. However, the presence of a porous obstruction. However, the presence of a porous obstruction in a square cavity seems to force an earlier transition from laminar to turbulent regime due to higher generation rates of turbulent kinetic energy into the porous matrix.© 2003 ASME

NATURAL CONVECTION I N TURBULENT REGIME I N CONCENTRIC AND ECCENTRIC HORIZONTAL ANNULAR REGIONS

Steady laminar and turbulent natu ral convection in two dimensional concentric and eccentric annular cavities, isothermally heated from the inner cylinder and cooled from the outer wall, is numerically analyzed using the finite volume method. Benchmark results for laminar and turbulent flo ws are compared with similar calculations by Cho et al. (1982) (JHT, 104), Kenjeres & Hanjalic (1995) (IJHFF, 16) and the experimental data of Kuehn & Goldstein (1978) (JHT, 100) and McLeod & Bishop (1989) (IJHMT, 32). Governing equations are written in terms of primitive variables and are recast into a general form. For laminar flows, isotherms and streamlines for annuli with the same eccentricity, but located at dif ferent angular positions, are presented for Ra L=10 4 and R i/R o=0.3846, where R o and R i are the cylinder radii and Ra L is the Rayleigh number based on a characteristic length R o - R i. Turbulence flow calculated for Ra L=2.5x10 6 and 1.22x10 7 , for both

Heat Transfer in Cavities Having a Fixed Amount of Solid Material

This work compares two different approaches for obtaining numerical solutions for laminar natural convection within a square cavity, which is filled by a fixed amount of a solid conducting material. The first model considered, namely, porous-continuum model, is based on the assumption that the solid and the fluid phases are seen as the same medium, over which volume-averaged transport equations apply. Secondly, a continuum model is considered to solve the momentum equations for the fluid phase that would resemble a conjugate heat transfer problem in both the solid and the void space. In the continuum model, the solid phase is composed of square obstacles, equally spaced within the cavity. In both models, governing equations are numerically solved using the finite volume method. The average Nusselt number at the hot wall, obtained from the porous-continuum model, for several Darcy numbers, are compared with those obtained with the second approach, namely the continuum model, with different number of obstacles. When comparing the two methodologies, this study shows that the average Nusselt number calculated for each approach for the same Ram differs between each other and that this discrepancy increases as the Darcy number decreases, in the porous-continuum model, or the number of blocks increases and their size decreases, in the continuum model. A correlation is suggested to modify the macroscopic thermal expansion coefficient in order to match the average Nusselt numbers calculated by the two models for Ram = const = 104 and Da ranging from 1.2060×10−4 to 1.Copyright

Laminar Free Convection in Inclined Enclosures Filled With a Fluid Saturated Porous Medium

Detailed numerical computations for steady-state laminar natural convection within in oblique cavities totally filled with a fluid saturated porous medium is numerically analyzed using the finite volume method in a generalized coordinate system. The inclined walls are maintained at constant but different temperatures, while the horizontal walls are kept insulated. Governing equations are written in terms of primitive variables and are recast into a general form. Flow and heat transfer characteristics, (streamlines, isotherms and average Nusselt number), are investigated for Rayleigh number ranging from 103 to 104 and inclined angles ranging from 0° to 45°. In general, present results show good agreement with previous works. Analyses of important environmental and engineering flows can benefit from the derivations herein and, ultimately, it is expected that additional research on this new subject be stimulated by the work here presented.© 2004 ASME

Computation of Turbulent Free Convection in Oblique Porous Enclosures Using a Macroscopic Two-Equation Model

Computations for turbulent natural convection within an inclined cavity totally filled with a fluid saturated porous medium are presented. The finite volume method in a generalized coordinate system is applied. The inclined walls are maintained at constant but different temperatures, while the horizontal walls are kept insulated. Governing equations are written in terms of primitive variables and are recast into a general form, Flow and heat transfer characteristics, (streamlines, isotherms and average Nusselt number), are investigated for a wide range of values of Rayleigh number and inclined angle. The turbulent model used is the standard k-e model with a wall function. In this work, the turbulence model is first switched off and the laminar branch of the solution is found. Subsequently, the turbulence model is included so that the solution merges to the laminar branch for a reducing Ram . This convergence of results as Ram decreases can be seen as an estimate of the so-called relaminarization phenomenon. Present solutions are compared with published results and the influence of the inclination angle on Racr is analyzed. For Ram greater than around 104 , both laminar and turbulent flow solutions deviate, indicating that such critical value for Ram was reached.Copyright

Turbulent Natural Convection in Horizontal Composite Cavities

Turbulent natural convection in a two-dimensional horizontal composite square cavity, isothermally heated at the left side and cooled from the opposing surface, is numerically analyzed using the finite volume method. The composite square cavity is formed by three distinct regions, namely, clear, porous and solid region. Accordingly, the development of a numerical tool able to treat all these regions as one computational domain is of advantage for engineering design of thermal systems. Governing equations are written in terms of primitive variables and are recast into a general form. It was found that the fluid begins to permeate the porous medium for values of Ra greater than 106 . Nusselt number values show that for the range of Ra analyzed there are no significant variation between the laminar and turbulent model solution..Copyright

Turbulent Natural Convection in Enclosures With Clear Fluid and Completely Filled With Porous Material

Steady laminar and turbulent natural convection in a two-dimensional square cavity, isothermally heated from the left side and cooled from the opposing side, is numerically analyzed using the finite volume method. Benchmark results for laminar and turbulent flows are compared with similar numerical solutions in the literature. The cases of clear and porous media are considered. Governing equations are written in terms of primitive variables and are recast into a general form. The effects of Rayleigh number on flow pattern and energy transport are investigated for Ra ranging from 103 to 1010 for clear media and 101 to 106 for porous media. The turbulence model used was the standard k–e along with the wall function approach. All results presented herein showed reasonable agreement with calculations presented in the literature. Critical values for the Rayleigh number for the onset of turbulence are suggested. The main objective of this work is to validate a numerical tool for simulating turbulent natural convection in both clear and porous media.Copyright