A numerical study of laminar free convection heat transfer between inner sphere and outer vertical cylinder

The effects of height and radius ratio with a Newtonian fluid have been investigated numerically to determine heat transfer by natural convection between the sphere and vertical cylinder with isothermal boundary conditions. The inner sphere and outer vertical cylinder were heated and cooled in a steady change of temperature. Calculations were carried out systematically for a range of the Rayleigh numbers to determine the average Nusslet numbers which are affected by the geometric ratio parameters (HR and RR) on the flow and temperature fields. The governing equations, in terms of vorticity, stream function and temperature are expressed in a spherical polar coordinate system. Results of the parametric study conducted further reveal that the heat and flow fields are primarily dependent on the Rayleigh number and height and radius ratio, for a Prandtl number of 0.7, with the Rayleigh number ranging from 10³ to 10⁶, and the height and radius ratio varying from 1.2 to 5.0. Above all, the specification of different convective configurations has a significant effect on the average heat transfer rate across the composite annulus gap.     

EFFECTS OF THERMAL BOUNDARY CONDITION ON BUOYANCY DRIVEN TRANSITIONAL AIR FLOW IN A VERTICAL CYLINDER HEATED FROM BELOW

A three-dimensional unsteady numerical computation was carried out here to investigate the effects of the thermal boundary condition on the convection flow in a vertical, bottom heated cylinder containing air. The thermal condition at the sidewall of the cylinder is assumed to be thermally well insulated or perfectly conducting. Results were obtained for air in a cylinder of finite aspect ratio (Gamma = 2) for various thermal Rayleigh numbers. The predicted results indicate that the flow in the sidewall insulated cylinder is highly asymmetric even at steady state and contains multicellular vortices. The flow formation processes leading to the above structures are relatively complicated. In the early transient two axisymmetric circular vortex rolls, one on top of another, appear. Then the rolls merge asymmetrically. In the late stage the flow deflection by the cylinder top and bottom results in a very complex flow. In the cylinder with a perfectly conducting sidewall the transition from a steady to a time dependent flow is subcritical. However, in the cylinder with an insulated sidewall the flow transition is supercritical.     

A numerical investigation of transient natural convection heat transfer of aqueous nanofluids in a horizontal concentric annulus

A numerical study of transient natural convection heat transfer of aqueous nanofluids in a horizontal annulus between two coaxial cylinders is presented. The effective thermophysical properties of water in the presence of copper oxide nanoparticles with four different volume fractions are predicted using existing models, in which the effects of the Brownian motion of nanoparticles are taken into consideration. The predicted development of convective flow and heat transfer of nanofluids is presented by means of the average Nusselt number over the outer cylinder. The flow development time towards a steady state and the time-averaged Nusselt number are predicted and scaled with Rayleigh number. It is shown that at constant Rayleigh numbers, the time-averaged Nusselt number is gradually lowered as the volume fraction of nanoparticles is increased. In addition, the time-averaged Nusselt number will be overestimated if the Brownian motion effects are not considered.     

Prandtl number effect on transition of free-convective flows in a wide-gap horizontal annulus

Transition phenomena of flow patterns in natural convection in an annulus with a small-diameter inner cylinder are numerically investigated for the fluids of 0.01≤Pr≤1. At 0.01≤Pr≤0.2, the solution is unique, and several kinds of multicellular flow patterns are observed at high Rayleigh numbers. At 0.3≤Pr≤1, however, both of the steady unicellular and bicellular flows exist above a certain critical Rayleigh number. The critical Rayleigh number is increased, as Pr increases. The overall Nusselt number of the bicellular flow is little than that of the unicellular flow.     

Dual free-convective flows in a horizontal annulus with a constant heat flux wall

Natural convection in a horizontal annulus with a constant heat flux wall is investigated for the fluids of 0.2?Pr?1. The outer cylinder is kept at a constant temperature, and the inner cylinder is heated with a constant heat flux. By using a numerical approach in solving the unsteady governing equations of flow and temperature fields, it is shown that dual steady solutions exist above a critical Rayleigh number.     

Natural convection heat transfer between inner sphere and outer vertically eccentric cylinder

The effects of eccentricity and geometric configuration with a Newtonian fluid have been investigated numerically to determine heat transfer by natural convection between the sphere and vertical cylinder with isothermal boundary conditions. The inner sphere and outer vertical cylinder were heated and cooled in a steady change of temperature. Calculations were carried out systematically for a range of the Rayleigh numbers to determine the average Nusselt numbers which are affected by the geometric ratio (HR:RR) and eccentricity (ε) parameters on the flow and temperature fields. The governing equations, in terms of vorticity, stream function and temperature are expressed in a spherical polar coordinate system. Results of the parametric study conducted further reveal that the heat and flow fields are primarily dependent on the Rayleigh number, eccentricity and geometric configuration, for a Prandtl number of 0.7, with the Rayleigh number ranging from 10³ to 10⁶, the three eccentricities and two geometric configurations. Above all, the specification of different convective configurations has a significant effect on the average heat transfer rate across the composite annulus gap.     

TWO-DIMENSIONAL AND UNSTEADY NATURAL CONVECTION IN A HORIZONTAL ENCLOSURE WITH A SQUARE BODY

A two-dimensional solution for unsteady natural convection in an enclosure with a square body is obtained using an accurate and efficient Chevyshev spectral collocation method. A spectral multidomain methodology is used to handle a square body located at the center of the computational domain. The physical model considered here is that a square body is located at the center between the bottom hot and top cold walls. To see the effects of the presence of a body on natural convection between the hot and cold walls, we considered the cases that the body maintains the adiabatic and isothermal thermal boundary conditions for different Rayleigh numbers varying in the range of 103 to 106. When the Rayleigh number is small, the flow and temperature distribution between the hot and cold walls shows a symmetrical and steady pattern. At the intermediate Rayleigh number, the fluid flow and temperature fields maintain the steady state but change their shape to the nonsymmetrical pattern. When the Rayleigh number is high, the flow and temperature fields become time dependent, and their time-averaged shapes approach the symmetric pattern again. The Rayleigh number for the fluid flow and temperature fields to become nonsymmetrical and time dependent depends on the thermal boundary conditions of a body. The variation of time- and surface-averaged Nusselt numbers on the hot and cold walls and at the body surfaces for different Rayleigh numbers and thermal boundary conditions are also presented to show the overall heat transfer characteristics in the system.     

Numerical investigation of laminar natural convective heat transfer from a horizontal triangular cylinder to its concentric cylindrical enclosure

A numerical simulation was conducted to investigate the steady laminar natural convective heat transfer for air within the horizontal annulus between a heated triangular cylinder and its circular cylindrical enclosure. The Boussinesq approximation was applied to model the buoyancy-driven effect and the governing equations were solved using the finite volume method. Four different Rayleigh numbers and four different radius ratios were considered, and four different inclination angles for the inner triangular cylinder were investigated as well. The computed flow and temperature fields were demonstrated in the form of streamlines and isotherms. Variations of the maximum stream function and the local and average Nusselt numbers were displayed as functions of the above-mentioned parameters. Correlations of the average Nusselt number were proposed based on curve fitting. At constant radius ratio, inclination angles of the inner triangular cylinder are found to have negligible effects on the average Nusselt number.     

Numerical investigation of natural convection phenomena in a uniformly heated circular cylinder immersed in square enclosure filled with air at different vertical locations

In this model, a numerical study of two dimensional steady natural convection is performed for a uniform heat source applied on the inner circular cylinder in a square air (Pr = 0.7) filled enclosure in which all boundaries are assumed to be isothermal (at a constant low temperature). The developed mathematical model is governed by the coupled equations of continuity, momentum and energy and is solved by finite volume method. The effects of vertical cylinder locations and Rayleigh numbers on fluid flow and heat transfer performance are investigated. Rayleigh number is varied from 10³ to 10⁶ and the location of the inner cylinder is changed vertically along the centerline of the enclosure from − 0.25 L to 0.25 L upward and downward, respectively. It is found that at small Rayleigh numbers does not have much influence on the flow field while at high Rayleigh numbers have considerable effect on the flow pattern. In addition, the numerical solutions yield a two cellular flow field between the inner cylinder and the enclosure. Also, the total average Nusselt number behaves nonlinearly as a function of locations. Results are presented in terms of the streamlines, isotherms, local and average Nusselt numbers. Detailed results of the numerical has been compared with literature ones, and it gives a reliable agreement.     

Laminar Flow and Heat Transfer Phenomena Across a Confined Semicircular Bluff Body at Low Reynolds Numbers

The present study is concerned with the simulation of incompressible Newtonian fluid flow and heat transfer over a long semicircular bluff body in a channel at low Reynolds numbers. In particular, wall effects on the forced convection from a (heated) semicircular cylinder confined in a horizontal channel are investigated for Reynolds number = 1–40 and blockage ratio = 16.67–50% for air as the working fluid. Flow and thermal fields are found steady for the preceding range of settings. The onset of flow separation increases as the wall confinement increases. The size of the recirculation zone downstream of a semicircular cylinder is seen to increase almost linearly with Reynolds number for a fixed blockage ratio, but it decreases with increasing blockage ratio for a fixed Reynolds number. As expected, total drag coefficient and its components decrease with increasing value of Reynolds number. However, with increasing blockage ratio, the values of these drag coefficients increase. On the basis of equal projected area, the total drag coefficient for the present flow system is found to be greater than the corresponding drag in the case of the unconfined semicircular cylinder. Similarly, the overall drag in the case of a confined semicircular cylinder is found to be greater than that of a confined circular cylinder for the appropriate range of dimensionless control parameters. The maximum augmentation in heat transfer for blockage ratios of 25% and 50% is found to be approximately 16% and 51% with respect to the corresponding value at the blockage ratio of 16.67% at Reynolds number = 1. Finally, the correlations of wake length, drag coefficient, and average Nusselt number are obtained.     

Flow visualization of natural convection in a vertical channel with asymmetric heating

A dynamical study of the flow in an asymmetrically heated vertical plane channel has been conducted experimentally. Experiments were carried out in water for three aspect ratios and for a range of modified Rayleigh numbers corresponding to the boundary layer flow regime. The flow dynamics were characterized by means of visualization techniques based on laser tomography using discrete and continuous tracers. Flow visualizations were carried out in the plane of symmetry of the channel along its entire height. The investigations focused more specifically on the influence of the aspect ratio and the modified Rayleigh number on the flow structure both in steady-state regime and during the transitional phase occurring just after the start of the heating. An upward boundary layer flow is found near the heated wall, accompanied by a reverse flow developing on the opposite side from the top open-end of the channel. In steady state, the reverse flow takes the form of an elongated eight-shaped structure with two main recirculation cells. The length of the upper cell of the eight-shape structure decreases with increasing aspect ratio. For a fixed aspect ratio, the increase in modified Rayleigh number results in a decrease in the penetration of the reverse flow. During the transient the flow structure is shown to evolve from a single cell to a final eight-shaped structure.     

Isotherms in a horizontal differentially heated cavity at intermediate Rayleigh numbers

Rayleigh-Benard convection in a horizontal, differentially heated, high aspect ratio fluid layer is considered. Experiments with three Rayleigh numbers (13900, 34800 and 51800) corresponding to different flow regimes have been performed. The temperature field in the fluid has been recorded in the form of interferometric projections, the fringe patterns representing isotherms. For steady and quasi-steady flow, the three dimensional temperature field has been obtained using an iterative algebraic reconstruction technique. Results show the formation of longitudinal rolls in the cavity at a Rayleigh number of 13900. The flow is unsteady at a Rayleigh number of 34800, but a strong indication of the formation of cubic rolls is seen. At a Rayleigh number of 51800, the flow field is completely unsteady with no discernible pattern. The local ray-averaged Nusselt numbers at each of the heated and cooled walls have been determined at the two lower Rayleigh numbers. The systematic variation of Nusselt number over each of the solid surfaces confirms the flow models proposed at these Rayleigh numbers. The average Nusselt number qualitatively matches the published correlations.     

Natural convection from a confined horizontal cylinder: the optimum distance between the confining walls

The laminar natural convection from an isothermal horizontal cylinder confined between vertical walls, at low Rayleigh numbers, is investigated by theoretical, experimental and numerical methods. The height of the walls is kept constant, however, their distance is changed to study its effect on the rate of the heat transfer. Results are incorporated into a single equation which gives the Nusselt number as a function of the ratio of the wall distance to cylinder diameter, t/D, and the Rayleigh number. There is an optimum distance between the walls for which heat transfer is maximum.     

Numerical investigation of transient natural convection in a vertical channel-chimney system symmetrically heated at uniform heat flux

In the present numerical investigation, a transient numerical analysis for natural convection in air, between two vertical parallel plates (channel), heated at uniform heat flux, with adiabatic parallel plates downstream (chimney), is carried out by means of the finite volume method. The analyzed transient problem is two-dimensional and laminar. The computational domain is made up of the channel-chimney system, and two reservoirs, placed upstream the channel and downstream the chimney. The reservoirs are important because they simulate the thermal and fluid dynamic behaviors far away from the inflow and outflow regions. Results are presented in terms of wall temperature and air velocity profiles. They are given at different Rayleigh number and expansion ratios (chimney gap/channel gap) for a fixed channel aspect ratio (channel height/channel gap) equal to 10 and extension ratio (channel-chimney height/channel height) equal to 2.0. Wall temperature profiles over a period show the presence of overshoots and undershoots. The comparison among the maximum wall temperatures shows that the simple channel is the most critical configuration at steady state condition, but it is the best configuration during the transient heating at the first overshoot. As indicated by the temperature profiles, average Nusselt number profiles over a period of consideration show minimum and maximum values and oscillations before the steady state. Stream function fields allow to observe the development of fluid dynamic structures inside the channel-chimney system, particularly how and when the cold inflow is present and develops.     

Scaling of natural convection of an inclined flat plate: Ramp cooling condition

A scaling analysis is performed for the transient boundary layer established adjacent to an inclined flat plate following a ramp cooling boundary condition. The imposed wall temperature decreases linearly up to a specific value over a specific time. It is revealed that if the ramp time is sufficiently large then the boundary layer reaches quasi-steady mode before the growth of the temperature is finished. However, if the ramp time is shorter then the steady state of the boundary layer may be reached after the growth of the temperature is completed. In this case, the ultimate steady state is the same as if the start up had been instantaneous. Note that the cold boundary layer adjacent to the plate is potentially unstable to Rayleigh–Bénard instability if the Rayleigh number exceeds a certain critical value for this cooling case. The onset of instability may set in at different stages of the boundary layer development. A proper identification of the time when the instability may set in is discussed. A numerical verification of the time for the onset of instability is presented in this study. Different flow regimes based on the stability of the boundary layer have also been discussed with numerical results.     

A numerical investigation on the fluid flow and heat transfer in the confined impinging slot jet in the low Reynolds number region for different channel heights

The numerical solution is obtained for unsteady two-dimensional fluid flow and heat transfer in a confined impinging slot jet using the finite volume method. In order to consider the effect of Reynolds number and height ratio on the flow and temperature fields in the channel, the numerical simulations were performed for different Reynolds numbers of 50–500 and different height ratios of 2–5. The critical Reynolds number, beyond which the flow and thermal fields change their state from steady to unsteady, depends on the Reynolds number and height ratio. The unsteadiness gives a big impact on the flow and temperature fields and as a result the pressure coefficient, skin friction coefficient and Nusselt number in the unsteady region show different characteristics from those in the steady region.     

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.     

Transient natural convective heat transfer from a heated triangular cylinder to its air-filled coaxial cylindrical enclosure

A numerical study of transient natural convection inside an air-filled horizontal cylinder with a coaxial inner triangular cylinder is performed with the Prandtl number and aspect ratio being fixed at 0.72 and 2.0, respectively. The effects of two different positions of the inner triangular cylinder and Grashof number are examined. The predicted development of the convective flow and heat transfer is presented by means of snapshots of streamlines and isotherms. It is shown that the flow development can be representatively classified into several stages based on time histories of the average Nusselt number over the outer circular cylinder. The time-averaged Nusselt number is scaled with Grashof number of power 0.21–0.24 and time duration for attaining a steady/quasi-steady state is also correlated to Grashof number.     

Entropy generation and natural convection in rectangular cavities

This work presents a numerical analysis of entropy generation in rectangular cavities that were submitted to the natural convection process. This natural convection process was caused by temperature differences between the vertical walls of the cavities. Momentum and energy equations were used to solve this problem. These equations were coupled by the Boussinesq approximation. Initially the cavities were submitted to uniform temperature and velocity fields. The hypothesis of perfect insulation was considered for the top and bottom walls of the cavity. Impermeability and non-slip condition in the boundary were assumed for every wall of the cavity. The numerical analysis is performed through a two-dimensional model with the Finite Volume method. The results of the entropy generation obtained to a square cavity were used to validate the numerical model and it presented good concordance with results from other authors. Additionally, an analysis of the entropy generation in rectangular cavities was performed with five aspect ratios, five Rayleigh numbers and four irreversibility coefficients. The results of this work indicate that: (a) the total entropy generation in steady state increases linearly in both cases, the aspect ratio and the irreversibility coefficient, and exponentially with the Rayleigh number; (b) the influence of the aspect ratio on Bejan number is proportional to Rayleigh number and inversely proportional to the irreversibility coefficient; (c) for the same aspect ratio, the entropy generation due to the viscous effects increases with the Rayleigh number and, for a certain Rayleigh number, the entropy generation due to the viscous effects also increases with the aspect ratio.