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A bounded porous box saturated with Newtonian fluid and subjected to a sinusoidal temperature gradient has various practical applications, such as solar energy storage, groundwater remediation, food processing, and chemical reactors. We address the generalization of the classical Rayleigh–Bénard convection problem in a horizontal fluid layer in an infinitely large domain heated from below to a finite three-dimensional box. We also look into a more intricate form of the modulated Rayleigh–Bénard problem in which the temperature at the bottom boundary varies sinusoidally. The Rayleigh number quantifies the non-sinusoidal part of the temperature gradient, while the amplitude and frequency of modulation describe the sinusoidal one. The critical Rayleigh number is determined using linear and nonlinear stability analyses; for the latter, the energy method is used. There is a possibility of subcritical instabilities, as evidenced by the energy stability estimates being lower than the linear ones. Furthermore, eigenvalues are obtained as a function of aspect ratios, modulation amplitude, and frequency for varying Darcy numbers. Modulation amplitude more significantly triggers a change in flow patterns at the onset of convection compared to the effect of other parameters. Considering water-saturated porous media made up of different materials, we report the critical temperature difference between lower and upper surfaces required for the onset of convection. In addition, a comparison between such a temperature difference obtained from linear theory and the energy method is also provided in the same manner. It is observed that subharmonic instability occurs for all considered porous media packed densely or sparsely. [ABSTRACT FROM AUTHOR]

A unified model for the analysis of two-dimensional Brinkman–Bénard/Rayleigh–Bénard/ Darcy–Bénard convection in cylindrical and rectangular enclosures ( C E / R E ) saturated by a Newtonian liquid is presented by adopting the local thermal non-equilibrium ( L T N E ) model for the heat transfer between fluid and solid phases. The actual thermophysical properties of water and porous media are used. The range of permissible values for all the parameters is calculated and used in the analysis. The result of the local thermal equilibrium ( L T E ) model is obtained as a particular case of the L T N E model through the use of asymptotic analyses. The critical value of the Rayleigh number at which the entropy generates in the system is reported in the study. The analytical expression for the number of Bénard cells formed in the system at the onset of convection as a function of the aspect ratio, S o , and parameters appearing in the problem is obtained. For a given value of S o it was found that in comparison with the case of L T E , more number of cells manifest in the case of L T N E . Likewise, smaller cells form in the D B C problem when compared with the corresponding problem of B B C . In the case of R B C , fewer cells form when compared to that in the case of B B C and D B C . The above findings are true in both C E and R E . In other words, the presence of a porous medium results in the production of less entropy in the system, or a more significant number of cells represents the case of less entropy production in the system. For small and finite S o , the appearance of the first cell differs in the C E and R E problems. [ABSTRACT FROM AUTHOR]

An analytical methodology is developed to study the pollutant dispersion in a turbulent wind flow over a two-dimensional hill with a small slope. As in a typical boundary layer problem, the flow domain is divided into an inner and an outer region: the inviscid outer region is further subdivided into an upper and a middle layer while the viscous inner region is subdivided into a shear stress and an inner surface layer. Based on the Reynolds-averaged Navier-Stokes equations and the continuity equations, closed form analytical solutions of the stream functions and velocities are readily obtained for all regions in the domain. The velocity information is then imported into the diffusion equation, and the pollutant concentration distribution is readily solved. For reasons of turbulent shear, a variational method with adjustments to the streamline coordinate system is used to obtain an accurate solution of the pollutant concentration. Results show that when the source is located in the upper layer, the concentrations decrease with distance along the upwind side of the hill and tend to reach a constant value rapidly near the hilltop. Similar results are observed when the source is located in the middle layer. However, due to the reduction of wind speed in the middle layer, the concentrations become saturated at a later upslope position as compared to the source in the upper layer. This methodology is shown to be able to provide a quick and accurate estimate of local pollutant patterns and can be applied to any flow field provided that the streamlines can be specified through the velocities. DOI: 10.1061/(ASCE)0733-9399(2006)132:1(99) CE Database subject headings: Air pollution; Topography; Wind; Dispersion; Slopes.

In this article we investigate the fluid flow through a thin fracture modelled as a fluid-saturated porous medium. We assume that the fracture has constrictions and that the flow is governed by the prescribed pressure drop between the edges of the fracture. The problem is described by the Darcy-Lapwood-Brinkman model acknowledging the Brinkman extension of the Darcy law as well as the flow inertia. Using asymptotic analysis with respect to the thickness of the fracture, we derive the explicit higher-order approximation for the velocity distribution. We make an error analysis to comment on the order of accuracy of the method used and also to provide rigorous justification for the model. [ABSTRACT FROM AUTHOR]