Non-darcy natural convection from a heated vertical plate in saturated porous media with mass transfer

The nonsimilar and self-similar flows for the steady natural convection over a vertical heated surface in a saturated porous medium with mass transfer using non-Darcy model have been studied under boundary layer approximations. The differential equations governing both the nonsimilar and self-similar flows have been solved numerically using an implicit finite-difference scheme developed by Keller. The results indicate that both heat transfer and velocity field are appreciably affected by the modified Grashof number and mass transfer except that the effect of the modified Grashof number on the heat transfer for large suction is very small. It is found that the maximum velocity occurs at the wall and it increases as the modified Grashof number or injection increases.     

A numerical study of three-dimensional laminar mixed convection past an open cavity

A numerical study has been carried out to analyze the effects of mixed convective flow over a three-dimensional cavity that lies at the bottom of a horizontal channel. The vertical walls of the cavity are isothermal and all other walls are adiabatic. The cavity is assumed to be cubic in geometry and the flow is laminar and incompressible. A direct numerical simulation is undertaken to investigate the flow structure, the heat transfer characteristics and the complex interaction between the induced stream flow at ambient temperature and the buoyancy-induced flow from the heated wall over a wide range of the Grashof number (10³–10⁶) and two Reynolds numbers Re = 100 and 1000. The computed thermal and flow fields are displayed and discussed in terms of the velocity fields, streamlines, the temperature distribution and the averaged Nusselt number at the heated and cooled walls. It is found that the flow becomes stable at moderate Grashof number and exhibit a three-dimensional structure, while for both high Reynolds and Grashof numbers the mixed convection effects come into play, push the recirculating zone further upstream and the flow becomes unsteady with Kelvin–Helmholtz instabilities at the shear layer.     

The effect of the top and bottom wall temperatures on the laminar natural convection in an air-filled square cavity

The effect of the top and bottom wall temperatures on the natural convection heat transfer characteristics in an air-filled square cavity driven by a difference in the vertical wall temperatures was investigated by measuring the temperature distributions along the heated vertical wall and visualizing the flow patterns in the cavity. The experiments were performed at a horizontal Grashof number of 1.9 × 10⁸. Increasing the top wall temperature resulted in a separated flow region on the top wall, which caused a secondary flow between the separated flow and the boundary layer on the heated vertical wall. This secondary flow had a significant effect on the heat transfer in this region. Changes in the top and bottom wall temperatures changed the temperature gradient and the average temperature of the air outside the thermal boundary layers in the cavity. The local heat transfer along much of the heated vertical wall could be correlated by Nu = C · Ra^0.32, but the constant C increased when the average of the top and bottom wall temperatures increased.     

Comparison of Mixed Convection in a Square Cavity with an Oscillating versus a Constant Velocity Wall

This article presents a numerical investigation of unsteady laminar mixed convection heat transfer in a two-dimensional square cavity. The cavity is configured such that one of the vertical walls is cooled and slides either with a constant speed or with a sinusoidal oscillation. A portion of the opposite stationery wall is heated by a constant temperature heat source while, the remaining walls of the cavity are thermally insulated. Different configurations of sliding wall movement and a series of Richardson numbers and Strouhal numbers are tested. The results indicate that the direction and magnitude of the sliding wall velocity affect the heat transfer rate. At low Richardson numbers, the average heat transfer rate for the cavity with an oscillating wall is found to be lower compared to that for the cavity with a constant velocity wall. In addition, at a fixed Richardson number, as the Strouhal number decreases the oscillation frequency of average Nusselt number on the vertical walls decreases; however, the oscillation amplitude of average Nusselt number increases.     

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.     

Magnetoconvection in an enclosure with partially active vertical walls

Magnetoconvection of an electrically conducting fluid in a square cavity with partially thermally active vertical walls is investigated numerically. The active part of the left side wall is at a higher temperature than the active part of the right side wall. The top, bottom and the inactive parts of the side walls are thermally inactive. Nine different combinations of the relative positions of the active zones are considered. The governing equations are discretized by the control volume method with QUICK scheme and solved numerically by SIMPLE algorithm for the pressure–velocity coupling together with underrelaxation technique. The results are obtained for Grashof numbers between 10⁴ and 10⁶, Hartmann numbers between 0 and 100 and Prandtl numbers 0.054–2.05. The heat transfer characteristics are presented in the form of streamlines and isotherms. The heat transfer rate is maximum for the middle–middle thermally active locations while it is poor for the top–bottom thermally active locations. The average Nusselt number decreases with an increase of Hartmann number and increases with an increase of Grashof number. For sufficiently large magnetic field Ha = 100 the convective mode of heat transfer is converted into conductive mode in the low region of Grashof number than in the high region.     

Heat and fluid flow characteristics inside differentially heated square enclosures with single and multiple sliding walls

Fluid flow and heat transfer characteristics of differentially heated lid driven cavities are numerically modeled and analyzed in the present study. One‐, two‐, and four‐sided lid driven cavity configurations are considered with the vertical walls being maintained at different temperatures and the horizontal walls being thermally insulated. Eight different cavity configurations are considered depending on the direction of wall motion. The Prandtl number Pr is taken to be 0.7, the Grashof number is taken to be 10⁴, while two values for the Richardson number Ri are considered, 0.1 and 10. It is found that both the Richardson number and the cavity configuration affect the heat and fluid flow characteristics in the cavity. It is concluded that for Ri=0.1, a four‐sided driven cavity configuration with all walls rotating in the same direction would triple the value of the average Nusselt number at the cold wall when compared to a one‐sided driven cavity configuration. However, for Ri=10, the cavity configuration has minimal effect and all eight cases result in an average Nusselt number value at the cold wall ranging between 1.3 and 1.9. © 2009 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley InterScience ( www.interscience. wiley.com ). DOI 10.1002/htj.20264     

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.     

Analysis of the flow and heat transfer characteristics for assisting incompressible laminar flow past an open cavity

A numerical study has been carried out to analyze the effects of mixed convective assisting flow past three-dimensional open cavity over a wide range of Reynolds (100–1000) and Richardson (0.001–10) numbers. The vertical walls in the inflow and outflow sides are isothermal while all other walls are adiabatic. The cavity is assumed to be cubic in geometry and the flow is laminar. A direct numerical simulation is undertaken to investigate the flow structure, the heat transfer characteristics and the complex interaction between the induced stream flow at ambient temperature and the buoyancy-induced flow from the heated wall. It is found that the flow becomes stable at moderate Grashof number and exhibits a three-dimensional structure, while for high Richardson number the mixed convection effects come into play and push the recirculating zone further upstream and the flow may becomes unstable.     

Investigation of a large top wall temperature on the natural convection plume along a heated vertical wall in a square cavity

The characteristics of the laminar natural convection in an air-filled square cavity heated and cooled on the side walls was studied for cases where the temperature of the top wall was significantly larger than the heated vertical wall. Experiments were performed for a horizontal Grashof number of 1.3 × 10⁸, and non-dimensional top wall temperatures from 1.4 to 2.3. The results show that the plume formed on the heated vertical wall separated from this wall before reaching the top wall. As a result, three different regions were observed in the cavity: a stratified core region, a buoyant plume region, and a highly stratified region above the plume after it had separated from the vertical wall. The highly stratified region above the plume became larger and more stable with an increase of the top wall temperature, stabilizing the motion of the plume across the cavity. The similarity solutions developed by Kulkarni et al. [A.K. Kulkarni, H.R. Jacobs, J.J. Hwang, Similarity solution for natural convection flow over an isothermal vertical wall immersed in thermally stratified medium, Int. J. Heat Mass Transfer 30 (1987) 691–698] to characterize the natural convection heat transfer along an isothermal single vertical plate did not agree with the results for the current measurements; however, the non-similarity model of Chen and Eichhorn [C.C. Chen, R. Eichhorn, Natural convection from a vertical surface to thermally stratified fluid, J. Heat Transfer 98 (1976) 446–451] was in good agreement over most of the wall. There were some discrepancies in the temperature distributions and the heat transfer characteristics, especially at y/H ? 0.8 due to the separated flow in this region.     

Aspect ratio effect on natural convection in a square enclosure with a sinusoidal active thermal wall using a thermal non-equilibrium model

ABSTRACT

A numerical investigation of the aspect ratio effect on natural convection in a square enclosure is carried out by adopting the local thermal non-equilibrium model. The top and bottom walls of the enclosure are adiabatic, the left vertical wall is partially heated and cooled by the sinusoidal thermal boundary condition, and the right vertical wall is maintained at uniform thermal boundary condition. The results show the value of periodicity parameter increasing. The streamlines vary in different patterns, rotating clockwise and counterclockwise simultaneously when N > 1, and the number of clockwise and counterclockwise rotating cells increases with the increase of N and equals the value of N. The sinusoidal local Nusselt number profiles are observed and the wave amplitude of local Nusselt number decreases with the increase of aspect ratio, and the absolute values of average Nusselt number at left wall of porous cavity reach maximum when Ar = 1. The absolute value of solid-to-fluid temperature differences decreases as the inter-phase heat transfer coefficient (H) increases and it increases as the value of aspect ratio increases. The total heat transfer of porous cavity can be enhanced by increasing the aspect ratio and the thermal conductivity ratio.     

Stability and chaotic characteristics of a wall plume

In this study artificial disturbances were introduced into a wall plume adjacent to a vertical side wall, and its stability and chaotic characteristics were experimentally examined. The main results are: (1) for high and low frequency disturbances the neutral region (a region between stability and instability) is lower and higher values of the modified Grashof number than an analytical neutral curve, respectively. (2) As the Grashof number increases, a chaotic fluctuation of temperature is observed above a critical value. This chaotic region is completely included in the region where the wall plume is unstable. (3) The flow can be classified into four regions based on patterns, a stable region, disturbance-amplification region, chaotic region and non-chaotic region.     

Effect of corrugation frequencies on natural convective heat transfer and flow characteristics in a square enclosure of vee-corrugated vertical walls

A numerical investigation has been carried out on natural convective heat transfer and fluid flow in a square cavity with vee-corrugated vertical surfaces. This study covers the range of corrugation frequency from 1 to 3 and Grashof number from 10³ to 10⁵. The corrugation amplitude has been fixed at 5% of the enclosure height. The vorticity stream function formulation with the control volume based finite element method has been used to analyse the effects of corrugation frequency and Grashof number. The investigation shows that the overall heat transfer through the enclosure increases with the increase of corrugation for low Grashof number; but the trend is reversed for high Grashof number.     

Mixed convective flow of two immiscible viscous fluids in a vertical wavy channel with traveling thermal waves

The problem of fully developed laminar mixed convection flow in a vertical wavy channel filled with two immiscible viscous fluids is studied analytically. Non‐linear equations governing the motion have been solved by linearization technique, wherein the flow is assumed to be in two parts; a mean part and a perturbed part. Exact solutions are obtained for the mean part and a perturbed part is solved using long wave approximation. Separate solutions are matched at the interface using suitable matching conditions. Numerical results are presented graphically for the distribution of velocity and temperature fields for varying physical parameters such as Grashof number, viscosity ratio, width ratio, and conductivity ratio. The effect of these parameters on the physical characteristics such as Nusselt number and skin friction at the walls is studied. It is found that Grashof number, viscosity ratio, width ratio, and conductivity ratio enhance the velocity parallel to the flow direction. Reversal effect is observed on the velocity which is perpendicular to the flow direction. The Nusselt number remains invariant on Grashof number. As the width ratio decreases, the Nusselt number decreases at the right wall and increases at the left wall and reversal effect is observed for variations of conductivity ratio. The skin friction increases at the left wavy wall and decreases at the right wavy wall as the Grashof number increases. © 2011 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley Online Library ( wileyonlinelibrary.com/journal/htj ). DOI 10.1002/htj.20379     

Couple stress effects on the MHD oscillatory flow in a fluid-saturated porous layer

In this paper, the oscillatory flow of hydromagnetic couple stress fluid-saturated porous layer with inhomogeneous wall temperatures is studied. The flow is modeled using the modified Darcy equation. The fluid is subjected to a transverse magnetic field and the velocity slip at the lower plate is taken into deliberation. The governing coupled partial differential equations of the flow are transformed to coupled ordinary differential equations and are solved analytically. The impact of the physical parameters such as the Grashof number, Prandtl number, Darcy number, Hartmann number, and couple stress parameters on velocity profiles, temperature, rate of heat transfer, and skin friction are emphasized. The velocity field increased as either the Grashof number, the Darcy number, the suction/injection parameter, and Prandtl number increased nevertheless reverse growth can be seen by increasing the Hartmann number and the couple stress parameter. The temperature field in the channel increases with increasing the suction/injection parameter and Prandtl number but a conflicting development can be seen with increasing the oscillation amplitude. It is interesting to note that skin friction increases on both channel plates as injection increases on the heated plate.     

Computation of buoyancy-induced flow in a heated rotating cavity with an axial throughflow of cooling air

In the cavity between the co-rotating compressor discs in gas turbine engines, the flow is very complex because of the multiple driving forces including the centrifugal buoyancy force, the Coriolis force and the inertial force. Numerical analysis was carried out in a simple rotating cavity with cooling air axial throughflow and a heated shroud. Efforts were focused upon the flow structure and its variations. The results reveal the non-axisymmetrical flow structures with cyclonic and anti-cyclonic circulations, which slip relative to the rotating cavity in the opposite direction (that is, rotate with a slower speed than the cavity) and the patterns remain unchanged. These structures are not unique, and four types with one, two, three, four pairs of circulations are obtained. For any particular set of conditions, the final structure can depend on the path taken: as axial Reynolds number is increased the number of circulation couples increases, and as Grashof number is increased the number of circulation couples decreases. At high Grashof number, the variation of Nu_av with Gr is consistent with the Rayleigh–Bénard convection.     

Natural convection in right-angle porous trapezoidal enclosure partially cooled from inclined wall

A numerical study is conducted to investigate the steady free convection flow in a two-dimensional right-angle trapezoidal enclosure filled with a fluid-saturated porous medium. The left vertical wall of the cavity is heated; the inclined wall is partially cooled; and the remaining walls are insulated (adiabatic). Three different cases are considered. While in Case I the cooler wall is located adjacent to the top wall, in Case II it is located in the middle inclined wall. In Case III, it is located adjacent to the bottom wall. Flow and heat transfer characteristics are studied for a range of parameters: the Rayleigh number, Ra, 100 ≤ Ra ≤ 1000; and the aspect ration, AR = 0.25, 0.50 and 0.75. Numerical results indicate that there exist significant changes in the flow and temperature fields as compared with those of a differentially heated square porous cavity. These results lead, in particular, to the prediction of a position of minimum heat transfer across the cavity, which is of interest in the thermal insulation of buildings and other areas of technology.     

Laminar Mixed-Convection Heat Transfer in a Lid-Driven Cavity with Modified Heated Wall

In this numerical study, steady laminar mixed-convection heat transfer in a two-dimensional square lid-driven cavity with a modified heated wall is investigated over a range of Richardson numbers, including 0.01, 1, and 10. The heated bottom wall of the cavity is characterized by rectangular, triangular, and sinusoidal wave shapes. The cooled top wall of the cavity is sliding with constant velocity, while the vertical walls are kept stationary and adiabatic. The governing equations are solved using a finite-volume technique. The results are presented in the form of streamlines, isotherms, and Nusselt number plots. The effects of the number of undulations and the amplitude on the flow field and heat transfer are also investigated. The predicted results demonstrate that the heat transfer enhancement is generally observed with the modification of the heated wall, while the improvement is found to be more profound for the case of rectangular wave and at low Richardson number.     

Natural Convection in a Trapezoidal Enclosure Heated from the Side with a Baffle Mounted on Its Upper Inclined Surface

A numerical investigation examined the effects on heat transfer of mounting baffles to the upper inclined surfaces of trapezoidal cavities. Two thermal boundary conditions are considered. In the first, the left, short vertical wall is heated while the right, long vertical wall is cooled (buoyancy assisting mode along the upper inclined surface of the cavity). In the second, the right, long vertical wall is heated while the left, short vertical wall is cooled (buoyancy opposing mode along the upper inclined surface of the cavity). For each boundary condition, computations are performed for three baffle heights, two baffle locations, four Rayleigh number values, and three Prandtl number values. Results are displayed in terms of streamlines, isotherms, and local and average Nusselt number values. For both boundary conditions, predictions reveal a decrease in heat transfer in the presence of baffles, with its rate generally increasing with increasing baffle height and Prandtl number. For a given baffle height, a higher decrease in heat transfer is generally obtained with baffles located close to the short vertical wall. Average Nusselt number correlations for both boundary conditions are presented.