Non-Darcy free convection in a thermally stratified porous medium along a vertical plate with variable heat flux

Free convection along an impermeable vertical plate embedded in a thermally stratified, fluid-saturated porous medium is analyzed. The wall heat flux is varied in a power-law form. The non-Darcian effects, such as solid-boundary viscous resistances, high-flow-rate inertia forces, near wall nonuniform porosity distribution and thermal dispersion, have been considered in the present study. Due to the variation of porosity in the near wall region, the stagnant thermal conductivity also varies accordingly. The nonsimilar system of transformed equations is solved with Keller's Box method. It is shown that the thermal stratification effect and the higher value of the exponent m can increase the local Nusselt number. Also the non-Darcian and thermal dispersion effects significantly influence the velocity and temperature profiles and local Nusselt number.     

Mixed convection heat transfer from a horizontal plate with variable surface heat flux in a porous medium

In this article nonsimilarity solution for mixed convection from a horizontal surface in a saturated porous medium was obtained for the case of variable surface heat flux. The entire mixed convection regime, ranging from pure forced convection to pure free convection, is considered by introducing a single nonsimilarity parameter. Heat transfer results are predicted by employing four different flow models, namely, Darcy's law, the Ergun model, and the Brinkman-Forchheimer-extended Darcy model with constant and variable porosity. The variable porosity effect is approximated by an exponential function. Effects of transverse thermal dispersion are taken into consideration in the energy equation, along with variable stagnant thermal conductivities. The formulation of the present problem shows that the flow and heat transfer characteristics depend on five parameters, that is, the power in the variation of surface heat flux, the nonsimilarity mixed-convection parameter, the inertia effect parameter, the boundary effect parameter, and the ratio of thermal conductivity of the fluid phase to that of the solid phase. Numerical results for the local Nusselt number variations, based on the various flow models, are presented for the entire mixed convection regime. The impacts␣of different governing parameters on the heat transfer results are thoroughly investigated. Received on 7 August 1997     

Convective Instability in Slip Flow in a Vertical Circular Porous Microchannel

The paper represents an analysis of convective instability in a vertical cylindrical porous microchannel performed using the Galerkin method. The dependence of the critical Rayleigh number on the Darcy, Knudsen, and Prandtl numbers, as well as on the ratio of the thermal conductivities of the fluid and the wall, was obtained. It was shown that a decrease in permeability of the porous medium (in other words, increase in its porosity) causes an increase in flow stability. This effect is substantially nonlinear. Under the condition Da?>?0.1, the effect of the porosity on the critical Rayleigh number practically vanishes. Strengthening of the slippage effects leads to an increase in the instability of the entire system. The slippage effect on the critical Rayleigh number is nonlinear. The level of nonlinearity depends on the Prandtl number. With an increase in the Prandtl number, the effect of slippage on the onset of convection weakens. With an increase in the ratio of the thermal conductivities of the fluid and the wall, the influence of the Prandtl number decreases. At high values of the Prandtl numbers (Pr?>?10), its influence practically vanishes.

     

Effects of thermal dispersion and stratification on non-darcy mixed convection from a vertical plate in a porous medium

A boundary layer analysis is presented for the mixed convection from a vertical plate embedded in a porous medium. The effects of thermal dispersion and stratification on the flow and temperature fields are investigated. The conservation equations that govern the problem are reduced to a system of nonlinear ordinary differential equations. The resulting equations were solved on the basis of the local similarity approach. Received on 12 February 1998     

Effect of thermal dispersion on free convection in a fluid saturated porous medium

The present article considers a numerical study of thermal dispersion effect on the non-Darcy natural convection over a vertical flat plate in a fluid saturated porous medium. Forchheimer extension is considered in the flow equations. The coefficient of thermal diffusivity has been assumed to be the sum of molecular diffusivity and the dispersion thermal diffusivity due to mechanical dispersion. The non-dimensional governing equations are solved by the finite element method (FEM) with a Newton–Raphson solver. Numerical results for the details of the stream function, velocity and temperature contours and profiles as well as heat transfer rates in terms of Nusselt number are obtained. The study shows that the increase in thermal dispersion coefficient of the porous medium results in more heat energy to disperse away in the normal direction to the wall. This induces more fluid to flow along the wall, enhancing the heat transfer coefficient particularly near the wall.     

Radiative and Thermal Dispersion Effects on Non-Darcy Natural Convection with Lateral Mass Flux for Non-Newtonian Fluid from a Vertical Flat Plate in a Saturated Porous Medium

The problem of natural convective heat transfer for a non-Newtonian fluid from an impermeable vertical plate embedded in a fluid-saturated porous medium has been analyzed. Non-Darcian, radiative and thermal dispersion effects have been considered in the present analysis. The governing boundary layer equations and boundary conditions are cast into a dimensionless form and simplified by using a similarity transformation. The resulting system of equations is solved by using a double shooting Runge–Kutta method. The effect of viscosity index n, the conduction–radiation parameter R, the non-Darcy parameter Gr*, the thermal dispersion parameter Ds and the suction/injection parameter f_w on the fluid velocities, temperatures and the local Nusselt number are discussed.     

Numerical Simulation of Particle Dispersion in a Spatially Developing Mixing Layer

Although there have been several numerical studies on particle dispersion in mixing layers, most of them have been conducted for temporally evolving mixing layers. In this study, numerical simulations of a spatially developing mixing layer are performed to investigate particle dispersion under various conditions. The full compressible Navier--Stokes equations are solved with a high-order compact finite difference scheme, along with high-order time-integration. Accurate non-reflecting boundary conditions for the fluid flow are used, and several methods for introducing particles into the computational domain are tested. The particles are traced using a Lagrangian approach assuming one-way coupling between the continuous and the dispersed phases. The study focuses on the roles of the large-scale vortex structures in particle dispersion at low, medium and high Stokes numbers, which highlights the important effects of interacting vortex structures in nearby regions in the spatially developing mixing layer. The effects of particles with randomly distributed sizes (or Stokes numbers) are also investigated. Both instantaneous flow fields and statistical quantities are analyzed, which reveals essential features of particle dispersion in spatially developing free shear flows, which are different from those observed in temporally developing flows. The inclusion of the gravity not only modifies the overall dispersion patterns, but also enhances stream-crossing by particles. Received 7 June 2001 and accepted 19 February 2002     

Thermal Dispersion Effects on Combined Convection in Non-Newtonian Fluids Along a Nonisothermal Vertical Plate in a Porous Medium

The method of non-similarity solution is used to study the influence of thermal dispersion on combined convection from vertical surfaces in a porous medium saturated with a power-law type non-Newtonian fluid. The coefficient of thermal diffusivity has been assumed to be the sum of molecular diffusivity and the dispersion thermal diffusivity due to mechanical dispersion. The transformed conservation laws are solved numerically for the case of variable surface heat flux conditions. Results for the details of the velocity and temperature fields as well as the Nusselt number have been presented.     

Uncertainty Analysis of Radionuclide Transport in a Fractured Coastal Aquifer with Geothermal Effects

Groundwater flow and radionuclide transport at the Milrow underground nuclear test site on Amchitka Island are modeled using two-dimensional numerical simulations. A multi-parameter uncertainty analysis is adapted and used to address the effects of uncertainties associated with the definition of the modeled processes and the values of the parameters governing these processes. In particular, we focus on the effects on radioactive transport of uncertainties associated with conduction and convection of heat relative to the uncertainties associated with other flow and transport parameters. These include recharge, hydraulic conductivity, fracture porosity, dispersivity and strength of matrix diffusion. The flow model is conceptualized to address the problem of density-driven flow under conditions of variable salinity and geothermal gradient. The conceptual transport model simulates the advection–dispersion process, the diffusion process from the high-velocity fractures into the porous matrix blocks, and radioactive decay.For this case study, the uncertainty of the recharge-conductivity ratio contributes the most to the output uncertainty (standard deviation of mass flux across the seafloor). The location of the freshwater–saltwater transition zone changes dramatically as this ratio changes with the thickness of the freshwater lens and the location of the seepage face changing as well. In the context of radionuclide transport from the nuclear test cavity that is located in the area where the transition zone is uncertain, travel times of radionuclide mass from the cavity to the seepage face along the seafloor are significantly impacted. The variation in transition zone location changes the velocity magnitude at the cavity location by a large factor (probably an order of magnitude). When this effect is combined with porosity and matrix diffusion uncertainty, the uncertainty of transport results becomes large. Although thermal parameters have an effect on the solution of the flow problem and also on travel times of radionuclides, the effect is relatively small compared to other flow and transport parameters.     

Melting effect and Cattaneo-Christov heat flux in fourth-grade material flow through a Darcy-Forchheimer porous medium

The melting phenomenon in two-dimensional (2D) flow of fourth-grade material over a stretching surface is explored. The flow is created via a stretching surface. A Darcy-Forchheimer (D-F) porous medium is considered in the flow field. The heat transport is examined with the existence of the Cattaneo-Christov (C-C) heat flux. The fourth-grade material is electrically conducting subject to an applied magnetic field. The governing partial differential equations (PDEs) are reduced into ordinary differential equations (ODEs) by appropriate transformations. The solutions are constructed analytically through the optimal homotopy analysis method (OHAM). The fluid velocity, temperature, and skin friction are examined under the effects of various involved parameters. The fluid velocity increases with higher material parameters and velocity ratio parameter while decreases with higher magnetic parameter, porosity parameter, and Forchheimer number. The fluid temperature is reduced with higher melting parameter while boosts against higher Prandtl number, magnetic parameter, and thermal relaxation parameter. Furthermore, the skin friction coefficient decreases against higher melting and velocity ratio parameters while increases against higher material parameters, thermal relaxation parameter, and Forchheimer number.

     

Radiation effects on combined convection over a vertical flat plate embedded in a porous medium of variable porosity

The present paper is concerned with the study of radiation effects on the combined (forced-free) convection flow of an optically dense viscous incompressible fluid over a vertical surface embedded in a fluid saturated porous medium of variable porosity with heat generation or absorption. The effects of radiation heat transfer from a porous wall on convection flow are very important in high temperature processes. The inclusion of radiation effects in the energy equation leads to a highly non-linear partial differential equations which are transformed to a system of ordinary differential equations using non-similarity transformation. These equations are then solved numerically using implicit finite-difference method subject to appropriate boundary and matching conditions. A parametric study of the physical parameters such as the particle diameter-based Reynolds number, the flow based Reynolds number, the Grashof number, the heat generation or absorption co-efficient and radiation parameter is conducted on temperature distribution. The effects of radiation and other physical parameters on the local skin friction and on local Nusselt number are shown graphically. It is interesting to observe that the momentum and thermal boundary layer thickness increases with the radiation and decrease with increase in the Prandtl number.     

Jump Condition for Diffusive and Convective Mass Transfer Between a Porous Medium and a Fluid Involving Adsorption and Chemical Reaction

In this paper, mass transfer at the fluid–porous medium boundaries is studied. The problem considers both diffusive and convective transport, along with adsorption and reaction effects in the porous medium. The result is a mass flux jump condition that is expressed in terms of effective transport coefficients. Such coefficients (a total dispersion tensor and effective reaction and adsorption coefficients) may be computed from the solution of the corresponding closure problem here stated and solved as a function of the Péclet number (Pe), the porosity and a local Thiele modulus. For the case of negligible convective transport (i.e., ), the closure problem reduces to the one recently solved by the authors for diffusion and reaction between a fluid and a microporous medium.     

A numerical study of heat transfer of a porous block with the random porosity model in a channel flow

 In this paper, heat transfer of a hot plate with a porous block in a channel flow is numerically investigated. A porous block is simulated as a fin type heat sink. The random/artificial porosity models are used to generate the distribution of porosity. In fact, the distribution of porosity in porous medium is irregular, thus the random porosity model is more realistic than the constant or variable porosity model to describe the phenomena happening in porous medium. Therefore, the distribution of porosity of porous block obeys the random porosity model, and the factors of mean porosity and standard deviation are taken into consideration. The variations of the porosity and the velocity in porous block are no longer smooth. For obtaining more heat transfer rate, the artificial porosity model is proposed. The heat transfer rates of the several cases derived by the artificial porosity model are better than those of the random porosity model. The thermal performance of porous block is larger than that of solid block as the mean porosity is larger than 0.5. Received on 5 March 2001 / Published online: 29 November 2001     

Simultaneous heat and mass transfer by natural convection from a plate embedded in a porous medium with thermal dispersion effects

The problem of steady, laminar, simultaneous heat and mass transfer by natural convection flow over a vertical permeable plate embedded in a uniform porous medium in the presence of inertia and thermal dispersion effects is investigated for the case of linear variations of both the wall temperature and concentration with the distance along the plate. Appropriate transformations are employed to transform the governing differential equations to a non-similar form. The transformed equations are solved numerically by an efficient implicit, iterative, finite-difference scheme. The obtained results are checked against previously published work on special cases of the problem and are found to be in good agreement. A parametric study illustrating the influence of the porous medium effects, heat generation or absorption, wall suction or injection, concentration to thermal buoyancy ratio, thermal dispersion parameter, and the Schmidt number on the fluid velocity, temperature and concentration as well as the skin-friction coefficient and the Nusselt and Sherwood numbers is conducted. The results of this parametric study are shown graphically and the physical aspects of the problem are highlighted and discussed.     

Simulation of conjugate radiation–forced convection heat transfer in a porous medium using the lattice Boltzmann method

In this paper, a lattice Boltzmann method is employed to simulate the conjugate radiation–forced convection heat transfer in a porous medium. The absorbing, emitting, and scattering phenomena are fully included in the model. The effects of different parameters of a silicon carbide porous medium including porosity, pore size, conduction–radiation ratio, extinction coefficient and kinematic viscosity ratio on the temperature and velocity distributions are investigated. The convergence times of modified and regular LBMs for this problem are 15 s and 94 s, respectively, indicating a considerable reduction in the solution time through using the modified LBM. Further, the thermal plume formed behind the porous cylinder elongates as the porosity and pore size increase. This result reveals that the thermal penetration of the porous cylinder increases with increasing the porosity and pore size. Finally, the mean temperature at the channel output increases by about 22% as the extinction coefficient of fluid increases in the range of 0–0.03.

     

Effects of thermal dispersion and stratification on combined convection on a vertical surface embedded in a porous medium

The effects of thermal dispersion and thermal stratification on mixed convection about a vertical surface in a porous medium are studied. The conservation equations that govern the problem are reduced to a system of nonlinear ordinary differential equations. The resulting equations are solved on the basis of the local similarity approximation. The results indicate that both dispersion and stratification effects have considerable influence on the heat transfer rate.     

Thermal dispersion and viscous dissipation effects on non-darcy mixed convection in a fluid saturated porous medium

Mixed convection flow and heat transfer about an isothermal vertical wall embedded in a fluid saturated porous medium with uniform free stream velocity is considered and the effects of thermal dispersion and viscous dissipation in both aiding and opposing flows are analysed. Similarity solution is not possible due to the inclusion of the viscous dissipation term, series solution is obtained, first and second order effects of dissipation revealed that viscous dissipation lowers the heat transfer rate. Observations also revealed that the thermal dispersion effect enhances the heat transfer rate and the effect of viscous dissipation is observed to increase with increasing values of the dispersion parameter. Received on 21 March 1997     

Free Convection Heat Transfer From A Sphere In A Porous Medium Using A Thermal Non-equilibrium Model

This work studies the free convection heat transfer from a sphere with constant wall temperature embedded in a fluid-saturated porous medium using a thermal non-equilibrium model. The governing equations are transformed into boundary-layer partial differential equations by the coordinate transform, and the obtained governing equations are then solved by the cubic spline collocation method. The temperature distributions for fluid and solid phases are shown for different values of the porosity scaled thermal conductivity ratio, the interphase heat transfer parameter, and the streamwise coordinate. The effects of the porosity scaled thermal conductivity ratio and the interphase heat transfer parameter between solid and fluid phases on the local Nusselt numbers for fluid and solid phases are examined. Results show the local Nusset number for the porous medium can be increased by increasing the porosity scaled thermal conductivity ratio. Moreover, the thermal non-equilibrium effect is more significant for low values of the porosity scaled thermal conductivity ratio or the interphase heat transfer parameter.     

Modes of vortex formation and transition to three-dimensionality in the wake of a freely vibrating cylinder

The present work is aimed to give some insight into the relation between vortex shedding modes and transition to three-dimensionality in the wake of a freely vibrating cylinder by establishing a numerical model and analyzing the relevant results of two- and three-dimensional simulations. The compressible flow past an elastically-mounted cylinder is solved by using the immersed boundary method (IB method). The cylinder is free to vibrate in the transverse direction with zero structure damping. The response of displacement amplitude is studied with the variation of reduced velocity. Whether P+S mode exists in three-dimensional flow and the occurrence of 2P mode is caused by flow transition from two-dimensional to three-dimensional are problems of concern. Both 2P and P+S wake modes are observed in two- and three-dimensional simulations. The numerical results indicate that the flow transition from two-dimensional to three-dimensional is coupled with the cylinder vibration in the synchronization/lock-in regime. The wake formation given by three-dimensional simulations suggests that the P+S mode might exist in reality when the flow is reverted to two-dimensional by vortex induced vibration (VIV) at Re=300–350. When Reynolds number increases to 425, the wake formation undergoes transition to three-dimensionality and 2P mode is observed. The effect of mass ratio on the flow transition to three-dimensionality is studied. The relationship between wake modes and aerodynamic forces is discussed.     

Mixed Convection–Radiation Interaction in Power-Law Fluids Along a Nonisothermal Wedge Embedded in a Porous Medium

A boundary layer analysis has been presented for the interaction of mixed convection with thermal radiation in laminar boundary flow from a vertical wedge in a porous medium saturated with a power-law type non-Newtonian fluid. The fluid considered is a gray medium, and the Rosseland approximation is used to describe the radiative heat flux in the energy equation. The transformed conservation laws are solved numerically for the case of variable surface temperature conditions. Results for the details of the velocity and temperature fields as well as the Nusselt number have been presented.