Natural convection flow of Cu–Water nanofluid in horizontal cylindrical annuli with inner triangular cylinder using lattice Boltzmann method

In this paper the lattice Boltzmann method is used to investigate the effect of nanoparticles on natural convection heat transfer in two-dimensional horizontal annulus. The study consists of an annular-shape enclosure, which is created between a heated triangular inner cylinder and a circular outer cylinder. The inner and outer surface temperatures were set as hot (T_h) and cold temperatures (T_c), respectively and assumed to be isotherms. The effect of nanoparticle volume fraction to the enhancement of heat transfer was examined at different Rayleigh numbers. Furthermore, the effect of vertical, horizontal, and diagonal eccentricities at various locations is examined at Ra = 10⁴. The result is presented in the form of streamlines, isotherms, and local and average Nusselt number. Results show that the Nusselt number and the maximum stream functions increase by augmentation of solid volume fraction. Average Nusselt number increases when the inner cylinder moves downward, but it decreases, when the location of inner cylinder changes horizontally.     

Combined convection heat transfer for thermally developing aiding flow in an inclined circular cylinder with constant heat flux

Experiments have been conducted to study the local and average heat transfer by mixed convection for hydrodynamically fully developed, thermally developing and thermally fully developed laminar air flow in an inclined circular cylinder. The experimental setup consists of aluminum cylinder as test section with 30 mm inside diameter and 900 mm heated length (L/D = 30), is subjected to a constant wall heat flux boundary condition. The investigation covers Reynolds number range from 400 to 1600, heat flux is varied from 70 W/m² to 400 W/m² and cylinder angles of inclination including 30°, 45° and 60°. The hydrodynamically fully developed condition has been achieved by using aluminum entrance section pipes (calming sections) having the same inside diameter as test section pipe but with variable lengths. The entrance sections included two long calming sections, one with length of 180 cm (L/D = 60), another one with length of 240 cm (L/D = 80) and two short calming sections with lengths of 60 cm (L/D = 20), 120 cm (L/D = 40). The results present the surface temperature distribution along the cylinder length, the local and average Nusselt number distribution with the dimensionless axial distance Z⁺. For all entrance sections, the results showed an increase in the Nusselt number values as the heat flux increases and as the angle of cylinder inclination moves from θ = 60° inclined cylinder to θ = 0° horizontal cylinder. The mixed convection regime has been bounded by the convenient selection of Re number range and the heat flux range, so that the obtained Richardson numbers (Ri) is varied approximately from 0.13 to 7.125. The average Nusselt numbers have been correlated with the (Rayleigh numbers/Reynolds numbers) in empirical correlations.     

Lattice Boltzmann simulation of natural convection heat transfer in eccentric annulus

In this article, the natural convection flow in eccentric annulus is simulated numerically by Lattice Boltzmann Model (LBM) based on double-population approach. A numerical strategy presents for dealing with curved boundaries of second order accuracy for both velocity and temperature fields. The effect of vertical, horizontal, and diagonal eccentricity at various locations is examined at Ra = 10⁴ and σ = 2. Velocity and temperature distributions as well as Nusselt number are obtained. The results show that the average Nusselt number increases when the inner cylinder moves downward regardless of the radial position. The validation with previous studies shows that double-population approach can evaluate the velocity and temperature fields in curved boundaries with a good accuracy.     

Investigation of natural circulation in cavities with uniform heat generation for different Prandtl number fluids

Natural convection in enclosures with uniform heat generation and isothermal side walls is studied here. For the rectangular enclosure, two-dimensional conservation equations are solved using SIMPLE algorithm. Parametric studies are conducted to examine the effects of orientation of the cavity, fluid properties (Pr number), and aspect ratio for Rayleigh numbers up to 10⁶. For a horizontal square cavity, the flow becomes periodically oscillating at Ra = 5 × 10⁴ and chaotic at Ra = 8 × 10⁵. With a slight increase in the inclination angle, the oscillations die and for inclination angles greater than 15⁰, the flow attain a steady state over a range of Ra. It is found that for tall cavities (aspect ratio > 1), the steady-state solution is obtained for all values of Ra considered here. However, for wide cavities (aspect ratio < 1), an oscillatory flow regime is observed. The maximum temperature within the cavity is calculated for the range of Ra, aspect ratio and Pr number. Correlations for the maximum cavity temperature is presented here. The values of critical Rayleigh number at which the convection sets in the rectangular cavity are also studied and two distinct criteria are determined to evaluate the critical Rayleigh number. Further, a three-dimensional simulation is performed for a cubic cavity. It is found that the steady state solutions are obtained for all Rayleigh number, except at Ra = 10⁶. This is in contrast to the predictions for a two-dimensional square cavity, which has an oscillatory zone from Ra = 5 × 10⁴ onwards.     

Transient natural convective heat transfer of a low-Prandtl-number fluid inside a horizontal circular cylinder with an inner coaxial triangular cylinder

A numerical study of transient natural convection of liquid gallium (Pr = 0.023) from a horizontal triangular cylinder to its coaxial cylindrical enclosure is performed. The aspect ratio is fixed at 2 and two positions of the inner triangular cylinder are considered. The development of the convective flow and heat transfer is shown via the time histories of the average Nusselt number over the outer circular wall for various Grashof numbers. Temporal phases of the flow development are identified as: initializing, developing, transitioning, and steady/quasi-steady state or oscillating. Typical flow patterns and temperature distributions at these phases are presented by means of streamlines and isotherms, respectively. Pitchfork bifurcation is present for both positions of the inner triangular cylinder when Gr ? 5 × 10⁴. The time-averaged Nusselt number over the outer circular cylinder, the flow development time, and the onset time of pitchfork bifurcation are predicted and scaled with the Grashof number. It is found that the time-averaged Nusselt number is apparently increased by horizontally placing the top side of the inner triangular cylinder for Gr ? 1 × 10⁵.     

Lattice Boltzmann simulation of natural convection heat transfer in an elliptical-triangular annulus

A numerical study for steady-state, laminar natural convection in a horizontal annulus between a heated triangular inner cylinder and cold elliptical outer cylinder was investigated using lattice Boltzmann method. Both inner and outer surfaces are maintained at the constant temperature and air is the working fluid. Study is carried out for Rayleigh numbers ranging from 1.0 × 10³ to 5.0 × 10⁵. The effects of different aspect ratios and elliptical cylinder orientation were studied at different Rayleigh numbers. The local and average Nusselt numbers and percentage of increment heat transfer rate were presented. The average Nusselt number was correlated. The results show that by decreasing the value of aspect ratio and/or increasing the Rayleigh number, the Nusselt number increases. Also the heat transfer rate increases when the ellipse positioned vertically.     

Bénard convection from a circular cylinder in a packed bed

Bénard convection around a circular heated cylinder embedded in a packed bed of spheres is studied numerically. The Forchheimer–Brinkman–extended Darcy momentum model with the Local Thermal Non-Equilibrium energy model is used in the mathematical formulation for the porous layer. The governing parameters considered are the Rayleigh number (10³ ≤ Ra ≤ 5 × 10⁷) and the thermal conductivity ratio (0.1 ≤ k_r ≤ 10,000). The structural properties of the packed bed are kept constant as: cylinder-to-particle diameter ratio D/d = 20 and porosity ε = 0.5, while the Prandtl number is fixed at Pr = 0.71. It is found that the presence of the porous medium suppresses significantly the strong free convection produced in the empty enclosure, and reduces considerably the high intensity of the pair of vortices generated behind the cylinder. Also, the results show that the porous medium can play the role of insulator or enhancer of heat transfer from the heat source, depending mainly on their thermal conductivities regardless of the Rayleigh number.     

Experimental investigation of mixed convection heat transfer for thermally developing flow in a horizontal circular cylinder

An experimental investigation is presented on mixed (free and forced) convection to study the local and average heat transfer for hydrodynamically fully developed, thermally developing and thermally fully developed laminar air flow in a horizontal circular cylinder. The experimental setup consists of aluminum cylinder as test section with 30 mm inside diameter and 900 mm heated length (L/D = 30), is subjected to a constant wall heat flux boundary condition. The investigation covers Reynolds number range from 400 to 1600, the heat flux varied from 60 W/m² to 400 W/m² and with cylinder inclination angle of θ = 0° (horizontal). The hydrodynamically fully developed condition is achieved by using an aluminum entrance section pipes (calming sections) having the same inside diameter as test section pipe but with variable lengths. The entrance sections included two long calming sections, one with length of 180 cm (L/D = 60), another one with length of 240 cm (L/D = 80) and two short calming sections with lengths 60 cm (L/D = 20), 120 cm (L/D = 40). The surface temperature variation along the cylinder surface, the local and average Nusselt number variation with the dimensionless axial distance Z⁺ were presented. For all entrance sections, it was found an increase in the Nusselt number values as the heat flux increases. It was concluded that the free convection effects tended to decrease the heat transfer results at low Re while to increase the heat transfer results for high Re. The combined convection regime could be bounded by a suitable selection of Re number ranges and the heat flux ranges. The obtained Richardson numbers (Ri) range varied approximately from 0.13 to 7.125. The average Nusselt numbers were correlated with the (Rayleigh numbers/Reynolds numbers). The proposed correlation has been compared with available literature and showed satisfactory agreement.     

Natural convective flow in an inclined lid-driven enclosure with a heated thin plate in the middle

Natural convection heat transfer from a heated thin plate located in the middle of a lid-driven inclined square enclosure has been analyzed numerically. Left and right of the cavity are adiabatic, the two horizontal walls have constant temperature lower than the plate’s temperature. The study is formulated in terms of the vorticity-stream function procedure and numerical solution was performed using a fully higher-order compact (FHOC) finite difference scheme on the 9-point 2D stencil. Air was chosen as a working fluid (Pr = 0.71). Two cases are considered depending on the position of heated thin plate (Case I, horizontal position; Case II, vertical position). Governing parameters, which are effective on flow field and temperature distribution, are Rayleigh number values (Ra) ranging from 10³ to 10⁵ and inclination angles γ (0° ? γ < 360°). The fluid flow, heat transfer and heat transport characteristics were illustrated by streamlines, isotherms and Nusselt number (Nu). It is found that fluid flow and temperature fields strongly depend on Rayleigh numbers and inclination angles. Further, for the vertical located position of thin plate heat transfer becomes more enhanced with lower γ at various Rayleigh numbers.     

Endwall heat transfer and fluid flow around an inclined short cylinder

Measurements of endwall heat transfer and flow field around a short single cylinder have been performed to examine the influence of cylinder inclination at low Reynolds number (Re_D = 1.0 × 10⁴). Both ends of the cylinder are attached to the endwalls and the length-to-diameter ratio of the cylinder varies from 2.7 to 4, depending on the inclination angle. Endwall heat transfer contours (obtained from transient liquid crystals) and endwall flow visualization results consistently indicate that the interaction between the horseshoe vortices around the cylinder and the wakes shed from the cylinder varies with the inclination. Spanwise pressure gradient induced by the inclination causes: (i) skewing of the upstream main flow as it approaches the cylinder; (ii) formation of a jet-like flow immediately downstream of the cylinder, followed by its impingement onto the endwall; and (iii) skewed separation line along the cylinder span from the cylinder axis.     

Numerical study of turbulent double-diffusive natural convection in a square cavity by LES-based lattice Boltzmann model

Turbulent double-diffusive natural convection in a square cavity represents numerous important problems in practice as well as in fundamental. However up to date the study on it is quite sparse and most previous studies just focus on laminar regime. To the best knowledge of the present authors, only several k–? models were developed to investigate turbulent double-diffusive convection and there is no attempt to use Large Eddy Simulation (LES). In order to deepen our knowledge on turbulent double-diffusive convection in a square cavity, we propose a novel LES-based lattice Boltzmann (LB) model to simulate such turbulent convectional flow. Previous LES-based LB models can be recovered from the present model. We find that the symmetry of the fluid circulation becomes broken since the Rayleigh number Ra = 10⁸, although the asymmetry is more clear when Ra ? 10¹⁰. More important, in the present study we find the power-law relationship among the Nusselt (Nu), the ratio of buoyancy forces (N) and the Rayleigh number (Ra) still exists in turbulent regime. The formula among them can be concluded as Nu = a × (Ra × ∣1 ? N∣)^b + c. The values of parameters a, b and c are given in this work.     

Numerical investigation for natural convection of a nanofluid in an inclined L-shaped cavity in the presence of an inclined magnetic field

The problem of natural convection in an inclined L-shaped enclosure filled with Cu/water nanofluid that operates under differentially heated walls in the presence of an inclined magnetic field is presented in this paper. The fully implicit finite difference method is used to solve the governing equations. A comparison with previously published results in special case of the present study is performed and a very good agreement is found. Heat transfer and fluid flow are examined for parameters of the Hartmann number (0 ≤ Ha ≤ 100), the nanoparticles volume fraction (0% ≤ ϕ ≤ 20%), the cavity inclination angle (0° ≤ ϑ ≤ 300°), the magnetic field inclination angle (0° ≤ γ ≤ 270°), the cavity aspect ratio (0.25 ≤ AR ≤ 0.6) and the Rayleigh number (10³ ≤ Ra ≤ 10⁶). It is found that, the presence of the magnetic field in the fluid region causes a significant reduction in the fluid flow and heat transfer characteristics. Also, a good enhancement in the heat transfer rate can be obtained by adding the copper nanoparticles to the base fluid.     

Lattice Boltzmann simulation of natural convection around a horizontal elliptic cylinder inside a square enclosure

Natural convection between a square outer cylinder and a heated elliptic inner cylinder has been studied numerically. The inner and outer walls are maintained at temperatures T_h and T_c, respectively, with T_h > T_c. Lattice Boltzmann method (LBM) has been used to investigate the hydrodynamic and thermal behaviors of the fluid at various vertical positions of the inner cylinder for different Rayleigh numbers ranging from 10³ to 10⁶. The results show that streamlines, isotherms, and the number, size and formation of the cells strongly depend on the Rayleigh number and the position of inner cylinder. The changes in heat transfer quantities have also been presented.     

On thermal control of devices contained in inclined hemispherical cavities with dome oriented downwards and subjected to transient natural convection

In this study, relationships of Nusselt–Rayleigh–Fourier type are proposed for the case of air-filled hemispherical cavity whose dome is oriented downwards and maintained isothermal. Its disk is subjected to a constant heat flux and inclined at an angle varying between 90° (vertical position) and 180° (disk horizontal with dome oriented downwards). The numerical approach is performed in transient regime by means of the finite volume method for Rayleigh numbers in the range of 10⁴ − 5 × 10⁸. These results are confirmed at steady state by measurements done for some configurations in a previous study for the same Rayleigh and inclination ranges. Otherwise, they complete other surveys considering inclination angles varying between 0° (horizontal cavity with dome oriented upwards) and 90° (vertical cavity) for a wider range of Rayleigh numbers. The correlations allow thermal control of devices submitted to natural convection in hemispherical cavities during the time preceding the steady state after their switch on.     

Mixed convective transport in a lid-driven cavity containing a nanofluid and a rotating circular cylinder at the center

The mixed convective transport of Cu-H₂O nanofluid in a differentially heated and lid-driven square enclosure in the presence of a rotating circular cylinder is investigated numerically. The top wall of the enclosure is sliding from left to right at a uniform speed while all other walls are stationary. A thermally insulated circular cylinder is placed centrally within the enclosure. The cylinder can rotate about its centroidal axis. The top and bottom walls are kept isothermal at different temperatures while the side walls are assumed adiabatic. Simulations are performed for, Richardson number 1 ≤ Ri ≤ 10, dimensionless rotational speed 0 ≤ Ω ≤ 5 and nanoparticle concentration 0 ≤ ϕ ≤ 0.20 keeping the Grashof number fixed as Gr = 10⁴. The flow and thermal fields are analyzed through streamline and isotherm plots for various Ω and Ri. Furthermore, the drag coefficient of the moving lid and Nusselt number of the hot wall are also computed to understand the effects of Ω and Ri on them. It is observed that the heat transfer greatly depends on the rotational speed of the cylinder, mixed convective strength and the nanoparticle concentration.     

Numerical predictions for stable buoyant regimes within a square cavity containing a heated horizontal cylinder

Buoyancy-induced flow regimes are investigated numerically for the basic case of a horizontal cylinder centred into a long co-axial square-sectioned cavity. In the frame of the 2D assumption, the threshold for the occurrence of time-dependent behaviour is explored. Stable symmetric and non-symmetric steady-state solutions, as well as unsteady regimes are observed, depending on the Rayleigh number, Ra, and the aspect ratio of the cavity, d. Four d-values are considered (d = 0.2, 0.4, 0.6, and 0.8). Heat transfer results are correlated by a single equation covering the full subcritical region.     

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.     

Numerical simulation of natural convection heat transfer in a porous cavity heated from below using a non-Darcian and thermal non-equilibrium model

The present paper investigates the numerical simulation of steady laminar incompressible natural convection heat transfer in an enclosed cavity that is filled with a fluid-saturated porous medium. The bottom wall is subjected to a relatively higher temperature than the top wall while the vertical walls are considered to be insulated. The flow field is modeled upon incorporating different non-Darcian effects, such as the convective term, Brinkman effect and Forchhiemer quadratic inertial effect. Moreover the two-equation model is used to separately account for the local fluid and solid temperatures. The numerical solution is obtained through the application of the finite volume method. The appraisals of the sought objectives are performed upon identifying key dimensionless groups of parameters. These dimensionless groups along with their operating domains are: Rayleigh number 1 ⩽ Ra ⩽ 400, Darcy number 10⁻⁴ ⩽ Da ⩽ 10⁻³, effective fluid-to-solid thermal conductivity ratio 0.1 ⩽ κ ⩽ 1.0, and the modified Biot number 1 ⩽ χ ⩽ 100. The non-Darcian effects are first examined over a broad range of Rayleigh number. Next, the implications of the group of parameters on the flow circulation intensity, local thermal non-equilibrium (LTNE) and average Nusselt number are highlighted and pertinent observations are documented.     

Numerical studies on laminar natural convection inside inclined cylinders of unity aspect ratio

The effect of cylinder inclination on thermal buoyancy induced flows and internal natural convective heat transfer is explored using CFD simulations. The cylinder’s top and bottom surfaces were maintained at different temperatures while the curved surface was adiabatic. The aspect ratio (length/diameter) of the cylinder was unity and the Prandtl number of the fluid was fixed at 0.71. The Rayleigh number of the confined fluid was varied from 10³ to 3.1 × 10⁴ by changing the specified end wall temperatures. The critical Rayleigh number was estimated to be 3800 for the vertical cylinder. Relaxing the convergence criterion caused false hysteresis in the converged results for the vertical cylinder. Typical natural convective fluid flow and temperature patterns obtained under laminar flow conditions are illustrated for various inclinations ranging from 0° to 180°. Flow visualization studies revealed complex three-dimensional patterns. Different thermal–hydrodynamic regimes were identified and were classified in terms of Rayleigh number and angle of inclination. Empirical correlations for the Nusselt number and maximum velocities in the domain as a function of the inclination angle and Rayleigh number are developed.     

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.