Effect of power-law fluid behavior on momentum and heat transfer characteristics of an inclined square cylinder in steady flow regime

The governing equations describing the momentum and heat transfer phenomena of power-law non-Newtonian fluids over a heated square cylinder at 45° of incidence in the two-dimensional (2-D) steady flow regime are solved numerically. Extensive results on the detailed structure of the flow and temperature fields as well as on the gross engineering parameters are presented over the following ranges of conditions: 0.2 ? n ? 1; 0.1 ? Re ? 40 and 0.7 ? Pr ? 100. At low Reynolds numbers, the flow remains attached to the surface of the cylinder. This seems to occur for all values of power-law index, at least up to about Re = 1. On the other hand, twin standing vortices were seen to form at Re = 10 for all values of power-law index considered herein. The influence of the Reynolds number and power-law index is delineated on the detailed structure of the flow field (streamlines), wake characteristics and surface pressure distribution as well as on the value of drag coefficients. Similarly, the effect of Prandtl number is studied on forced convective heat transfer for the two commonly encountered boundary conditions, namely, constant temperature or constant heat flux prescribed on the surface of the cylinder. Using the computed numerical results, simple heat transfer correlations are obtained in terms of the Nusselt number as a function of the pertinent governing parameters thereby enabling the prediction of the rate of heat transfer between the fluid and the immersed cylinder. In addition, variation of the local Nusselt number on the surface of the inclined of square cylinder and representative isotherm plots are also presented to elucidate the effect of Reynolds number, Prandtl number and power-law index on the heat transfer phenomenon.     

Jet impingement cooling of a constant heat flux horizontal surface in a confined porous medium: Mixed convection regime

In the present study, numerical investigation of jet impingement cooling of a constant heat flux horizontal surface immersed in a confined porous channel is performed under mixed convection conditions with the limitation of the Darcy model. The results are presented in the mixed convection regime with wide ranges of the governing parameters: Péclet number (1 ? Pe ? 1000), Rayleigh number (10 ? Ra ? 100), half jet width (0.1 ? D ? 1.0), and the distance between the jet and the heated portion (0.1 ? H ? 1.0). It is found that the average Nusselt number increases with increase in either Rayleigh number or jet width for high values of Péclet number. The average Nusselt number also increases with decrease in the distance between the jet and the heated portion. The correlation for Nu_avg in the forced convection regime is suggested. It is shown that mixed convection mode can cause minimum average Nusselt number unfavorably due to counteraction of jet flow against buoyancy driven flow. Hence, careful consideration must be given while designing a system of jet impingement cooling through porous medium.     

Jet impingement cooling of a horizontal surface in an unconfined porous medium: Mixed convection regime

Two-dimensional slot jet impingement cooling of an isothermal horizontal surface immersed in an unconfined porous medium is simulated numerically to gain insight into thermal characteristics under mixed convection conditions with the limitation of the Darcy model. The jet direction is considered to be perpendicular from the top to the horizontal heated element; therefore, the jet flow and the buoyancy driven flow are in opposite directions. The results are presented in the mixed convection regime with wide ranges of the governing parameters: Péclet number (1 ? Pe ? 1000), Rayleigh number (10 ? Ra ? 100), half jet width (0.1 ? D ? 0.5), and the distance between the jet and the heated portion (0.1 ? H ? 1.0). It is found that the average Nusselt number increases with increase in either Rayleigh number or jet width for high values of Péclet number. The average Nusselt number also increases with decrease in the distance between the jet and the heated portion. It is shown that mixed convection mode can cause minimum average Nusselt number at two values of Péclet number and a maximum average Nusselt number occurs in between theses two Péclet numbers at higher Rayleigh number due to counteraction of jet flow against buoyancy driven flow. Hence careful consideration must be given while designing a system of jet impingement cooling through porous medium.     

Effects of moving lid direction on MHD mixed convection in a linearly heated cavity

Effects of moving lid-direction on MHD mixed convection in a cavity with the bottom wall being linearly heated are analyzed using a numerical technique. Vertical walls of the enclosure are adiabatic and the sliding wall at the top has constant temperature. The lid moves in the negative and positive x-direction. Finite volume method has been used to solve the governing equations. Results are presented for different values of Hartmann number (0 ? Ha ? 30), Reynolds number (100 ? Re ? 1000) and Grashof number (10⁴ ? Gr ? 10⁶). It is found that direction of lid is more effective on heat transfer and fluid flow in the case of mixed convection than it is the case in forced convection. Heat transfer is also decreased with increasing of magnetic field for all studied parameters.     

MHD natural convection in an enclosure from two semi-circular heaters on the bottom wall

A computational study of the thermal and dynamical behavior of fluid in an enclosure with two isothermal semi-circular heaters is presented. The top wall and the flat surfaces on bottom wall are adiabatic while the vertical walls are kept at lower temperature than the semi-circular heaters. The radius of curvature of the semi-circular surfaces is chosen as one tenth of the cavity wall length. The governing equations are solved by the Galerkin weighted residual finite element method. The effect of magnetic field on the flow is another important parameter in this study. Numerical simulations were performed for several values of Rayleigh number (10³ ? Ra ? 10⁶), Hartmann number (0 ? Ha ? 50) and the distance between two semi-circular heaters (0.2 ? D ? 0.8). In all cases the Prandtl number is taken as 7. It is found that the distance between the semi-circular heaters is the most important parameter affecting the heat and fluid flow fields. In addition, Hartmann number was found to have an adverse affect on heat transfer.     

Numerical study of turbulent slot jet impingement cooling on a semi-circular concave surface

An investigation of the flow field and heat transfer characteristics of a slot turbulent jet impinging on a semi-circular concave surface with uniform heat flux has been carried out numerically in this study. The turbulent governing equations are solved by a control-volume-based finite-difference method with a power-law scheme and the well-known k–ε turbulence model and its associate wall function to describe the turbulent structure. In addition, a body-fitted curvilinear coordinate system is employed to transform the physical domain into a computational domain.Numerical computations have been conducted with variations of jet exit Reynolds number Re_2B (5920 ? Re_2B ? 23,700), dimensionless jet-to-surface distance H/B (0.5 ? H/B ? 12), dimensionless jet width B/D (0.033 ? B/D ? 0.05) and the heat flux q″ (1663 W/m² ? q″ ? 5663 W/m²). The theoretical model developed is validated by comparing the numerical predictions with available experimental data in the literature. The variations of local Nusselt numbers along the semi-circular concave surface decrease monotonically from its maximum value at the stagnation point. The numerical results show that the local Nusselt numbers are reasonably predicted with a maximum discrepancy within 15%. As the Reynolds number fixes, the effect of the impingement distance (H/B) on the average Nusselt (Nu_avg) is not significant except at low H/B = 0.5. This study provides fundamental insight into turbulent slot jet impingement cooling on the semi-circular concave surface.     

Natural convection effects on heat and mass transfer in a curvilinear triangular cavity

Finite element method is used in this study to analyze the effects of buoyancy ratio and Lewis number on heat and mass transfer in a triangular cavity with zig-zag shaped bottom wall. Buoyancy ratio is defined as the ratio of Grashof number of solutal and thermal. Inclined walls of the cavity have lower temperature and concentration according to zig-zag shaped bottom wall. Enclosed space consists mostly of an absorber plate and two inclined glass covers that form a cavity. Both high temperature and high concentrations are applied to bottom corrugated wall. Computations were done for different values of buoyancy ratio (?10 ? Br ? 10), Lewis number (0.1 ? Le ? 20) and thermal Rayleigh number (10⁴ ? Ra_T ? 10⁶). Streamlines, isotherms, iso-concentration, average Nusselt and Sherwood numbers are obtained. It is found that average Nusselt and Sherwood numbers increase by 89.18% and 101.91% respectively as Br increases from ?10 to 20 at Ra_T = 10⁶. Also, average Nusselt decreases by 16.22% and Sherwood numbers increases by 144.84% as Le increases from 0.1 to 20 at this Rayleigh number.     

Effect of aspect ratio on natural convection in power-law liquids from a heated horizontal elliptic cylinder

The influence of aspect ratio and shear-dependent viscosity on free convection heat transfer from a horizontal heated elliptic cylinder in power-law fluids has been investigated. In particular, the coupled momentum and energy equations have been solved numerically over the following ranges of conditions: Grashof number, 10 ? Gr ? 10⁵; Prandtl number, 0.72 ? Pr ? 100; power-law index, 0.3 ? n ? 1.5 and aspect ratio, 0.2 ? E ? 5. The new extensive results demonstrate the influence of the Grashof number (Gr), Prandtl number (Pr), power-law index (n) and aspect ratio (E) on the macroscopic heat and momentum transfer characteristics like local and average values of Nusselt number (Nu) and drag coefficients (C_D). Further insights are developed by examining the structure of the flow and temperature fields adjacent to the cylinder. Broadly speaking, all else being equal, shear- thinning fluid behaviour promotes heat transfer whereas shear-thickening viscosity has a deleterious effect on it with reference to that in Newtonian fluids. Also, the rate of heat transfer gradually increases as the cylinder shape passes from blunt to slender with respect to the direction of gravity. Finally, the present numerical values of the Nusselt number are correlated using a simple analytical form which facilitates interpolation of the present results for the intermediate values of the governing parameters. The paper is concluded by presenting detailed comparisons with the previous numerical and experimental results available in the literature, especially in Newtonian fluids.     

Effects of Reynolds and Prandtl numbers on heat transfer from a square cylinder in the unsteady flow regime

The effects of the Reynolds and Prandtl numbers on the rate of heat transfer from a square cylinder are investigated numerically in the unsteady two-dimensional periodic flow regime, for the range of conditions 60 ? Re ? 160 and 0.7 ? Pr ? 50 (the maximum value of Peclet number being 4000). A semi-explicit finite volume method has been used on a non-uniform collocated grid arrangement to solve the governing equations. Using the present numerical results, simple heat transfer correlations are obtained for the constant temperature and constant heat flux conditions on the solid square cylinder. In addition, the variation of the time averaged local Nusselt number on the each face of the obstacle and representative isotherm plots are presented to elucidate the role of Prandtl number on heat transfer in the unsteady flow regime.     

Numerical study of the heat sink with un-uniform fin width designs

The thermal performances of the heat sink with un-uniform fin width designs with an impingement cooling were investigated numerically. The governing equations are discretized by using a control-volume-based finite-difference method with a power-law scheme on an orthogonal non-uniform staggered grid. The coupling of the velocity and the pressure terms of momentum equations are solved by the SIMPLEC algorithm. The well-known k ? ε two-equations turbulence model is employed to describe the turbulent structure and behavior. The parameters include the five Reynolds number (Re = 5000–25000), three fin heights (H = 35, 40, 45 mm), and five fin width designs (Type-1–Type-5). The objective of this study is to examine the effects of the fin shape of the heat sink on the thermal performance. The results show that the Nusselt number increases with the Reynolds number. The increment of the Nusselt number decreases gradually with the increasing Reynolds number. Furthermore, the effects of fin dimensions on the Nusselt number at high Reynolds numbers are more significant than that at low Reynolds numbers. It is also found that there is potential for optimizing the un-uniform fin width design.     

Laminar natural convection in a square cavity: Low Prandtl numbers and large density differences

Steady natural convection at low Prandtl numbers caused by large density differences in a square cavity heated through the side walls is investigated numerically and theoretically. An appropriate dimensionless parameter characterizing the density differences of the working fluid is identified by the Gay-Lussac number. The Boussinesq assumption is achieved when the Gay-Lussac number tends to zero. The Nusselt number is derived for the ranges in Rayleigh number 10 ? Ra ? 10⁸, in Prandtl number 0.0071 ? Pr ? 7.1 and in Gay-Lussac number 0 ? Ga < 2. The effects of the Rayleigh, Prandtl and Gay-Lussac numbers on the Nusselt number are discussed on physical grounds by means of a scale analysis. Finally, based on physical arguments, a heat transfer correlation is proposed, valid for all Prandtl and Gay-Lussac number ranges addressed.     

Finite element based heatline approach to study mixed convection in a porous square cavity with various wall thermal boundary conditions

A penalty finite element method based simulation is performed to analyze the influence of various walls thermal boundary conditions on mixed convection lid driven flows in a square cavity filled with porous medium. The relevant parameters in the present study are Darcy number (Da = 10^?5 ? 10^?3), Grashof number (Gr = 10³ ? 10⁵), Prandtl number (Pr = 0.7–7.2), and Reynolds number (Re = 1–10²). Heatline approach of visualizing heat flow is implemented to gain a complete understanding of complex heat flow patterns. Patterns of heatlines and streamlines are qualitatively similar near the core for convection dominant flow for Da = 10^?3. Symmetric distribution in heatlines, similar to streamlines is observed irrespective of Da at higher Gr in natural convection dominant regime corresponding to smaller values of Re. A single circulation cell in heatlines, similar to streamlines is observed at Da = 10^?3 for forced convection dominance and heatlines are found to emanate from a large portion on the bottom wall illustrating enhanced heat flow for Re = 100. Multiple circulation cells in heatlines are observed at higher Da and Gr for Pr = 0.7 and 7.2. The heat transfer rates along the walls are illustrated by the local Nusselt number distribution based on gradients of heatfunctions. Wavy distribution in heat transfer rates is observed with Da ? 10^?4 for non-uniformly heated walls primarily in natural convection dominant regime. In general, exponential variation of average Nusselt numbers with Grashof number is found except the cases where the side walls are linearly heated. Overall, heatlines are found to be a powerful tool to analyze heat transport within the cavity and also a suitable guideline on explaining the Nusselt number variations.     

Effect of rotating cylinder on heat transfer in a square enclosure filled with nanofluids

Convective heat transfer in a differentially heated square enclosure with an inner rotating cylinder is studied theoretically. The free space between the cylinder and the enclosure walls is filled with water–Ag, water–Cu, water–Al₂O₃ or water–TiO₂ nanofluids. The governing equations are formulated for velocity, pressure and temperature formulation and are modeled in COMSOL, a partial differential equation (PDE) solver based on the Galerkin finite element method (GFEM). The governing parameters considered are the solid volume fraction, 0.0 ? ? ? 0.05, the cylinder radius, 0 ? R ? 0.3 and the angular rotational velocity, ?1000 ? Ω ? 1000. The results are presented to show the effect of these parameters on the heat transfer and fluid flow characteristics. It is found that the strength of the flow circulation is much stronger for a higher nanoparticle concentration, a better thermal conductivity value and a smaller cylinder with a faster, negative rotation. The maximum heat transfer are obtained at a high nanoparticle concentration with a good conductivity value, a slow positive rotation and a moderate cylinder size located in the center of the enclosure.     

Laminar free convection from a horizontal semi-circular cylinder to power-law fluids

Extensive numerical results on the flow and thermal fields are presented for free convection from a semi-circular cylinder (flat base upward) immersed in quiescent power-law fluids for the following ranges of conditions: Grashof number, 10 ? Gr ? 10⁵, Prandtl number, 0.72 ? Pr ? 100, and power-law index, 0.2 ? n ? 1.8. The heat transfer characteristics are analyzed in terms of the isotherm patterns, local and average Nusselt number as functions of the pertinent dimensionless parameters. The flow field is visualized in terms of the streamline patterns adjacent to the surface of the cylinder for a range of values of the Grashof number, Prandtl number and power-law index. A separated flow region forms at as low values of the Prandtl number as Pr = 0.72 for n ? 1 (Newtonian and shear-thickening fluids); whereas for shear-thinning fluids (n < 1), the flow remains attached to the cylinder surface over the range of conditions encompassed here. The bubble size grows with Grashof number and it shrinks with Prandtl number. In order to quantify the deviation from the Newtonian behaviour, the normalized values of average Nusselt number are analyzed as a function of the power-law index. In addition, a correlation is proposed for average Nusselt number as a function of the Grashof number, Prandtl number and power-law index. In general terms, shear-thinning fluid behaviour enhances heat transfer whereas shear-thickening has adverse influence on it.     

Thermo-bioconvection in a suspension of gravitactic micro-organisms in vertical cylinders

We investigate the effect of heating or cooling from below at constant temperature and constant heat flux on the development of gravitactic bioconvection in vertical cylinders with stress free sidewalls. The governing equations are the continuity equation, the Navier–Stokes equations with the Boussinesq approximation, the diffusion equation for the motile micro-organisms and the energy equation. The control volume method is used to solve numerically the complete set of governing equations. The governing parameters are the thermal and bioconvection Rayleigh numbers, the bioconvection Peclet number, the Lewis number, the Schmidt number and the aspect ratio. We found that subcritical bifurcations of bioconvection became supercritical bifurcations when the thermal Rayleigh number Ra_T is different than zero. For Ra_T < 0, i.e. for cooling from below, we have opposing buoyancy forces, the convection is decreased and the critical thermo-bioconvection Rayleigh number is increased with respect to that of bioconvection. For Ra_T > 0, i.e. for heating from below, we have cooperating buoyancy forces, the convection is increased and the critical thermo-bioconvection Rayleigh number is decreased with respect to that of bioconvection. Heating and cooling from below at constant temperature and heat flux modify considerably the pattern formation of the gravitactic bioconvection.     

Magnetic field and internal heat generation effects on the free convection in a rectangular cavity filled with a porous medium

A numerical investigation of the steady magnetohydrodynamics free convection in a rectangular cavity filled with a fluid-saturated porous medium and with internal heat generation has been performed. A uniform magnetic field, inclined at an angle γ with respect to the horizontal plane, is externally imposed. The values of the governing parameters are the inclined angle γ = 0, π/6, π/4 and π/2, Hartmann number Ha = 0, 1, 5, 10 and 50, Rayleigh number Ra = 10, 100, 10³ and 10⁵, and the aspect ratio a = 0.01, 0.2, 0.5 and 1 (square cavity). It is shown that the intensity of the core convection is considerably affected by the considered parameters. It is also found that the local Nusselt number Nu_Y decreases on the bottom wall as γ increases (magnetic field changes its direction from the horizontal to the vertical direction) and vice versa for the top wall of the cavity. The reported results are in good agreement with the available published work in the literature.     

Heat transfer enhancement from protruding heat sources using perforated zone between the heat sources

This study is to experimentally investigate the heat transfer enhancement by perforation in air cooling of two in-line rectangular heat sources module. Two separation distances between the heat sources were investigated at s/L = 0.5 and 1.0. The area between the heat sources in both cases were perforated in aligned arrangement such that the holes open area ratio (β) are of 0, 0.0736, 0.1472 and 0.2944. The dimensionless temperature distribution and the average Nusselt number are considered at different values of Reynolds number (3391 ? Re_L ? 10798) and holes open area ratio. It could be seen that perforation could enhance the heat transfer coefficients and reduce the module temperature significantly. Correlations are obtained for the average Nusselt number utilizing the present measurements within the investigated range of the different parameters.     

Hydro-magnetic combined convection in a lid-driven cavity with sinusoidal boundary conditions on both sidewalls

Mixed convection in a square cavity of sinusoidal boundary temperatures at the sidewalls in the presence of magnetic field is investigated numerically. The horizontal walls of the cavity are adiabatic. The governing equations are solved by finite-volume method. The results are obtained for various combinations of amplitude ratio, phase deviation, Richardson number, and Hartmann number. The heat transfer rate increases with the phase deviation up to ? = π/2 and then it decreases for further increase in the phase deviation. The heat transfer rate increases on increasing the amplitude ratio. The flow behavior and heat transfer rate inside the cavity are strongly affected by the presence of the magnetic field.     

Flow over and forced convection heat transfer in Newtonian fluids from a semi-circular cylinder

Momentum and heat transfer characteristics of a semi-circular cylinder immersed in unconfined flowing Newtonian fluids have been investigated numerically. The governing equations, namely, continuity, Navier–Stokes and energy, have been solved in the steady flow regime over wide ranges of the Reynolds number (0.01 ? Re ? 39.5) and Prandtl number (Pr ? 100). Prior to the investigation of drag and heat transfer phenomena, the critical values of the Reynolds number for wake formation (0.55 < Re^c < 0.6) and for the onset of vortex shedding (39.5 < Re_c < 40) have been identified. The corresponding values of the lift coefficient, drag coefficient, and Strouhal number are also presented. After establishing the limit of the steady flow regime, the influence of the Reynolds number (0.01 ? Re ? 39.5) and Prandtl number (Pr = 0.72, 1, 10, 50 and 100) on the global flow and heat transfer characteristics have been elucidated. Detailed kinematics of the flow is investigated in terms of the streamline and vorticity profiles and the variation of pressure coefficient in the vicinity of the cylinder. The functional dependence of the individual and total drag coefficients on the Reynolds number is explored. The Nusselt number shows an additional dependence on the Prandtl number. In addition, the isotherm profiles, local Nusselt number (Nu_L) and average Nusselt number (Nu) are also presented to analyze the heat transfer characteristic of a semi-circular cylinder in Newtonian media.