Effect of sinusoidal wavy bottom surface on mixed convection heat transfer in a lid-driven cavity

The current numerical study is conducted to analyze mixed convection heat transfer in lid-driven cavity with a sinusoidal wavy bottom surface. The cavity vertical walls are insulated while the wavy bottom surface is maintained at a uniform temperature higher than the top lid. In addition, the transport equations are solved using the finite element formulation based on the Galerkin method of weighted residuals. The validity of the numerical code used is ascertained by comparing our results with previously published results. The implications of Richardson number, number of wavy surface undulation and amplitude of the wavy surface on the flow structure and heat transfer characteristics are investigated in detail while the Prandtl number is considered equal to unity. The trend of the local heat transfer is found to follow a wavy pattern. The results of this investigation illustrate that the average Nusselt number increases with an increase in both the amplitude of the wavy surface and Reynolds number. Furthermore, optimum heat transfer is achieved when the wavy surface is designated with two undulations while subjected to low Richardson numbers.     

Double-diffusive natural convection in a fluid saturated porous cavity with a freely convecting wall

Double — diffusive natural convection in fluid saturated porous medium has been investigated using a generalised porous medium model. One of the vertical walls of the porous cavity considered is subjected to convective heat and mass transfer conditions. The results show that the flow, heat and mass transfer become sensitive to applied mass transfer coefficient in both the Darcy and non-Darcy flow regimes. It is also observed that the Sherwood number approaches a constant value as the solutal Biot number increases.     

Soret and Dufour effects on free convection along a vertical wavy surface in a fluid saturated Darcy porous medium

In this article, free convection of heat and mass transfer along a vertical wavy surface in a Newtonian fluid saturated Darcy porous medium is studied by considering cross diffusion (namely the Soret and the Dufour effects) in the medium. The vertical wavy wall and the flow governing equations are transformed to a plane geometry case by using a suitable transformation. Then a similarity solution to this problem is presented under the large Darcy–Rayleigh number assumption. The governing partial differential equations are reduced to a set of ordinary differential equations that are integrated using numerical methods to study the nature of the non-dimensional heat and mass transfer coefficients in the medium. The results are presented for a range of the flow governing parameters such as the diffusivity ratio parameter, the buoyancy ratio parameter, the Soret parameter, the Dufour parameter and the amplitude of the wavy surface.     

Numerical study of natural convection in a vertical porous annulus with discrete heating

In this paper natural convection flows in a vertical annulus filled with a fluid-saturated porous medium has been investigated when the inner wall is subject to discrete heating. The outer wall is maintained isothermally at a lower temperature, while the top and bottom walls, and the unheated portions of the inner wall are kept adiabatic. Through the Brinkman-extended Darcy equation, the relative importance of discrete heating on natural convection in the porous annulus is examined. An implicit finite difference method has been used to solve the governing equations of the flow system. The analysis is carried out for a wide range of modified Rayleigh and Darcy numbers for different heat source lengths and locations. It is observed that placing of the heater in lower half of the inner wall rather than placing the heater near the top and bottom portions of the inner wall produces maximum heat transfer. The numerical results reveal that an increase in the radius ratio, modified Rayleigh number and Darcy number increases the heat transfer, while the heat transfer decreases with an increase in the length of the heater. The maximum temperature at the heater surface increases with an increase in the heater length, while it decreases when the modified Rayleigh number and Darcy number increases. Further, we find that the size and location of the heater effects the flow intensity and heat transfer rate in the annular cavity.     

Free convection in a wavy cavity filled with a porous medium

The steady-state free convection inside a cavity made of two horizontal straight walls and two vertical bent-wavy walls and filled with a fluid-saturated porous medium is numerically investigated in the present paper. The wavy walls are assumed to follow a profile of cosine curve. The horizontal walls are kept adiabatic, while the bent-wavy walls are isothermal but kept at different temperatures. The Darcy and energy equations (in non-dimensional stream function and temperature formulation) are solved numerically using the Galerkin Finite Element Method (FEM). Flow and heat transfer characteristics (isothermal, streamlines and local and average Nusselt numbers) are investigated for some values of the Rayleigh number, cavity aspect ratio and surface waviness parameter. The present results are compared with those reported in the open literature for a square cavity with straight walls. It was found that these results are in excellent agreement.     

Hydromagnetic effect on mixed convection in a lid-driven cavity with sinusoidal corrugated bottom surface

The present numerical simulation is conducted to analyze the mixed convection flow and heat transfer in a lid-driven cavity with sinusoidal wavy bottom surface in presence of transverse magnetic field. The enclosure is saturated with electrically conducting fluid. The cavity vertical walls are insulated while the wavy bottom surface is maintained at a uniform temperature higher than the top lid. In addition, the transport equations are solved by using the finite element formulation based on the Galerkin method of weighted residuals. The implications of Reynolds number (Re), Hartmann number (Ha) and number of undulations (λ) on the flow structure and heat transfer characteristics are investigated in detail while, Prandtl number (Pr) and Rayleigh number (Ra) are considered fixed. The trend of the local heat transfer is found to follow a wavy pattern. The results of this investigation illustrate that the average Nusselt number (Nu) at the heated surface increases with an increase of the number of waves as well as the Reynolds number, while decreases with increasing Hartmann number.     

Transient natural convection with temperature-dependent viscosity in a square partially porous cavity having a heat-generating source

A numerical study is performed on the transient natural convection with a temperature-dependent viscosity inside a square partially porous cavity with a local heat-generating and heat-conducting source. The vertical walls of the cavity are kept at constant cooling temperature while the horizontal walls are adiabatic. The discrete heat-conducting and heat-generating source is located on the bottom wall. A porous layer is located under the clear fluid layer. Governing equations formulated in dimensionless stream function, vorticity and temperature variables with corresponding initial and boundary conditions are solved using implicit finite difference schemes of the second order. The control parameters are the Darcy number, Ostrogradsky number, viscosity variation parameter, height of the porous layer, and dimensionless time. The effects of these parameters on the average Nusselt number along the heat source surface, average temperature of the heater, fluid flow rate inside the cavity, as well as on the streamlines and isotherms are analyzed. The results show that porous layer thickness and viscosity of the working fluid are very good control parameters for optimization of the passive cooling system.     

Thermosolutal convection from a discrete heat and solute source in a vertical porous annulus

Double-diffusive convection in a vertical annulus filled with a fluid-saturated porous medium is numerically investigated with the aim to understand the effects of a discrete source of heat and solute on the fluid flow and heat and mass transfer rates. The porous annulus is subject to heat and mass fluxes from a portion of the inner wall, while the outer wall is maintained at uniform temperature and concentration. In the formulation of the problem, the Darcy–Brinkman model is adopted to the fluid flow in the porous annulus. The influence of the main controlling parameters, such as thermal Rayleigh number, Darcy number, Lewis number, buoyancy ratio and radius ratio are investigated on the flow patterns, and heat and mass transfer rates for different locations of the heat and solute source. The numerical results show that the flow structure and the rates of heat and mass transfer strongly depend on the location of the heat and solute source. Further, the buoyancy ratio at which flow transition and flow reversal occur is significantly influenced by the thermal Rayleigh number, Darcy number, Lewis number and the segment location. The average Nusselt and Sherwood numbers increase with an increase in radius ratio, Darcy and thermal Rayleigh numbers. It is found that the location for stronger flow circulation is not associated with higher heat and mass transfer rates in the porous annular cavity.     

The effect of the position of the heated thin porous fin on the laminar natural convection heat transfer in a differentially heated cavity

Laminar natural convection heat transfer in a differentially heated cavity with two thin porous fins attached to the hot wall and bottom insulated surface was studied numerically for various pertinent parameters. Such parameters include Richardson number, Darcy number, thermal conductivity ratio, and location of the porous fin. The left wall of the cavity is assumed to be uniformly heated while the right wall is kept at a lower temperature. In addition, the horizontal walls of the cavity were considered insulated. Furthermore, the governing transport equations within the porous media were written according to the volume-average theory. The governing equations are solved using a finite element formulation based on the Galerkin method of weighted residuals. The results of this investigation showed that the presence of a horizontal porous fin increases the average Nusselt number when compared with the differentially heated cavity for various Richardson numbers and thermal conductivity ratios. However, a vertical porous fin attached to the bottom insulated surface exhibited a lower average Nusselt number than the no-fin case.     

A Thermal Nonequilibrium Approach to Natural Convection in a Square Enclosure Due to the Partially Cooled Sidewalls of the Enclosure

This article presents a numerical investigation of steady non-Darcy natural convection heat transfer in a square cavity filled with a heat-generating porous medium with partial cooling using a local thermal nonequilibrium (LTNE) model. Five different partial cooling boundary conditions and the fully cooled boundary condition are investigated under LTNE and local thermal equilibrium (LTE). The cooling portions of the left and the right sidewalls of the cavity are maintained at temperature T ₀ while the enclosure's top and bottom walls, as well as the inactive parts of its sidewalls, are kept insulated. The simulation results show that the placement order of wall cooling has a significant effect on the flow pattern and heat transfer rate. Compared with the fully cooled wall, the partially cooled wall of the cavity yielded a higher local Nusselt number for both fluid and solid phases. Under the same boundary conditions, the LTNE and LTE models can demonstrate significant differences in flow patterns and temperature fields. The total heat transfer rate increases with both Darcy number and Rayleigh number. Enhancement of interphase heat transfer coefficient (H) reduces the impact of Darcy number on the heat transfer rate of a porous cavity. Also, the total heat transfer rate of the porous medium decreases steadily with thermal conductivity ratio γ and interphase heat transfer coefficient H.     

Natural convection in a partially porous cavity: Roads to chaos

Abstract

A numerical study of a periodic and chaotic behavior of natural convection in a square cavity, with a porous layer is presented. The cavity under study consists of two opposite vertical walls of which the lower half walls are hot (hold high temperature) and the upper half walls are cold (hold low temperature), whereas the horizontal walls are adiabatic. The porous medium is located in the lower part of the cavity. The natural heat transfer and the Darcy Brinkman equations are solved by using the finite volume method and the TDMA. The results show that thermal natural convection evolves towards a deterministic chaos by following Curry York scenario.     

Transient and non-Darcian effects on natural convection flow in a vertical channel partially filled with porous medium: Analysis with Forchheimer–Brinkman extended Darcy model

The present study addresses the transient as well as non-Darcian effects on laminar natural convection flow in a vertical channel partially filled with porous medium. Forchheimer–Brinkman extended Darcy model is assumed to simulate momentum transfer within the porous medium. Two regions are coupled by equating the velocity and shear stress in the case of momentum equation while matching of the temperature and heat flux is taken for thermal energy equation. Approximate solutions are obtained using perturbation technique. Variations in velocity field with Darcy number, Grashof number, kinematic viscosity ratio, distance of interface and variations in temperature distribution with thermal conductivity ratio, distance of interface are obtained and depicted graphically. The skin-friction and rate of heat transfer at the channel walls are also derived and the numerical values for various physical parameters are tabulated.     

Combined convection flow in triangular wavy chamber filled with water–CuO nanofluid: Effect of viscosity models

This work is focused on the numerical modeling of steady laminar combined convection flow in a vertical triangular wavy enclosure filled with water–CuO nanofluid. The left and right vertical walls of the cavity take the form of a triangular wavy pattern. The bottom and top horizontal walls are mechanically driven. The lower and upper surfaces move to the right and left direction at the same constant speed respectively. They maintain constant temperature lower than both vertical walls. Two different nanofluid models namely, the Brinkman model and the Pak and Cho correlation are employed. The developed equations are given in terms of the Navier Stokes and the energy equation and are non-dimensionalized and then solved numerically subject to appropriate boundary conditions by the Galerkin's finite-element method. Comparisons with published work are performed and found to be in good agreement. A parametric study is conducted and a selective set of graphical results is presented. The effects of the Reynolds number, Richardson number and the nanoparticles volume fraction on the flow and heat transfer characteristics in the cavity are displayed to compare the predictions obtained by the two different nanofluid models. Heat transfer enhancement can be obtained significantly due to the presence of nanoparticles. The rate of heat transfer is accentuated moderately by falling the Richardson number and rising the Reynolds number as well as the solid volume fraction.     

Combined convection heat transfer in a porous lid-driven enclosure due to heater with finite length

A numerical work is performed to analyze combined convection heat transfer and fluid flow in a partially heated porous lid-driven enclosure. The top wall of enclosure moves from left to right with constant velocity and temperature. Heater with finite length is located on the fixed wall where its center of location changes along the walls. The finite volume-based finite-difference method is applied for numerical experiments. Parameters effective on flow and thermal fields are Richardson number, Darcy number, center of heater and heater length. The results are shown that the best heat transfer is formed when the heater is located on the left vertical wall.     

Effect of Darcy,fluid rayleigh and heat generation parameters on natural convection in a porous square enclosure: A Brinkman-extended Darcy model

A Pressure-velocity solution for natural convection for fluid saturated heat generating porous medium in a square enclosure is analysed by finite element method. The numerical solutions obtained for wide range of fluid Rayleigh number, Ra_f, Darcy number, Da, and heat generating number, Q_d. The justification for taking these non-dimensional parameters independently is to establish the effect of individual parameters on flow patterns. It has been observed that peak temperature occurs at the top central part and weaker velocity prevails near the vertical walls of the enclosure due to the heat generation parameter alone. On comparison, the modified Rayleigh number used by the earlier investigators[4,6], can not explain explicitly the effect of heat generation parameter on natural convection within an enclosure having differentially heated vertical walls. At higher Darcy number, the peak temperature and peak velocity are comparatively more, resulting in better enhancement of heat transfer rate.     

Natural convection in porous cavity with sinusoidal bottom wall temperature variation

Numerical study of natural convection in a porous cavity is carried out in the present paper. Natural convection is induced when the bottom wall is heated and the top wall is cooled while the vertical walls are adiabatic. The heated wall is assumed to have spatial sinusoidal temperature variation about a constant mean value which is higher than the cold top wall temperature. The non-dimensional governing equations are derived based on the Darcy model. The effects of the amplitude of the bottom wall temperature variation and the heat source length on the natural convection in the cavity are investigated for Rayleigh number range 20–500. It is found that the average Nusselt number increases when the length of the heat source or the amplitude of the temperature variation increases. It is observed that the heat transfer per unit area of the heat source decreases by increasing the length of the heated segment.     

MHD mixed convection of Cu–water nanofluid in a two-sided lid-driven porous cavity with a partial slip

ABSTRACT

A numerical simulation of magneto-hydrodynamic mixed convection flow and heat transfer of Cu–water nanofluid in a square cavity filled with a Darcian porous medium with a partial slip is numerically investigated. The left and right walls of the cavity are moving up with a constant speed in vertical direction, and the partial slip effect is considered along these walls. The top and bottom walls of the cavity are assumed to be adiabatic. The right vertical wall of the cavity is assumed to be kept at a lower temperature, while the left vertical wall is kept at a higher temperature. The developed equations of the mathematical model are nondimensionalized and then solved numerically subject to appropriate boundary conditions by the finite-volume method. A parametric study is performed and a set of graphical results is presented and discussed to demonstrate interesting features of the solution.     

HYDROMAGNETIC NATURAL CONVECTION FROM AN INCLINED POROUS SQUARE ENCLOSURE WITH HEAT GENERATION

The problem of unsteady, laminar, two-dimensional hydromagnetic natural convection heat transfer in an inclined square enclosure filled with a fluid-saturated porous medium in the presence of a transverse magnetic field and fluid heat generation effects is studied numerically. The walls of the enclosure are maintained at constant temperatures. The flow in the porous region is modeled using the Brinkman-extended Darcy's law to account for the no-slip conditions at the walls. The control volume method is used to solve the governing balance equations for different values of the Darcy number, Hartmann number, and the inclination angle. Favorable comparisons with previously published work are performed. These comparisons confirmed the correctness of the numerical results. The obtained numerical results are presented graphically in terms of streamlines and isotherms as well as velocity and temperature profiles at midsections of the cavity to illustrate interesting features of the solution.     

Non-Darcy natural convection of a non-Newtonian fluid in a porous cavity

A numerical study of non-Darcy natural convection in a porous enclosure saturated with a power-law fluid is presented. Hydrodynamic and heat transfer results are reported for the configuration in which the enclosure is heated from a side-wall while the horizontal walls are insulated. The flow in the porous medium is modeled using the modified Brinkman–Forchheimer-extended Darcy model for power-law fluids, which accounts for both inertia and boundary effects. The results indicate that when the power law index is decreased, the circulation within the enclosure increases leading to a higher Nusselt number and these effects are enhanced as the Darcy number is increased. Consequently as the power law index decreases, the onset of the transitions from Darcy regime to Darcy–Forchheimer–Brinkman regime to asymptotic convection (boundary layer) regime shift to higher corresponding values of the Darcy number. An increase in Rayleigh number produces similar effects as a decrease in power law index.     

Natural convection in an enclosure with localized heating and salting from below

Two-dimensional, double diffusion, natural convection in a rectangular enclosure filled with binary fluid saturating porous media is investigated numerically. Multiple motions are driven by the external temperature and concentration differences imposed across horizontal walls with the simultaneous presence of discrete heat and contaminant sources. The general Brinkman-extended Darcy model is adopted to formulate the fluid flow in the cavity. The fluid, heat and moisture transport through the isotropic porous layer are analyzed using the streamlines, heatlines and masslines, and the heat and mass transfer potentials are also explained by the variations of overall Nusselt and Sherwood numbers. The numerical simulations presented here span a wide range of the main parameters (thermal Rayleigh numbers, strip pitches and Darcy number) in the domain of destabilizing solutal buoyancy forces. It is shown that the heat and mass transfer potential can be promoted or inhibited, depending strongly on the permeability of porous medium, the strip pitch, the thermal and solutal Rayleigh numbers.