A numerical study of the steady forced convection heat transfer from an unconfined circular cylinder

Forced convection heat transfer from an unconfined circular cylinder in the steady cross-flow regime has been studied using a finite volume method (FVM) implemented on a Cartesian grid system in the range as 10 ≤ Re ≤ 45 and 0.7 ≤ Pr ≤ 400. The numerical results are used to develop simple correlations for Nusselt number as a function of the pertinent dimensionless variables. In addition to average Nusselt number, the effects of Re, Pr and thermal boundary conditions on the temperature field near the cylinder and on the local Nusselt number distributions have also been presented to provide further physical insights into the nature of the flow. The rate of heat transfer increases with an increase in the Reynolds and/or Prandtl numbers. The uniform heat flux condition always shows higher value of heat transfer coefficient than the constant wall temperature at the surface of the cylinder for the same Reynolds and Prandtl numbers. The maximum difference between the two values is around 15–20%.     

Experimental investigation of mixed convection heat transfer in a horizontal and inclined rectangular channel

 Mixed convection heat transfer in rectangular channels has been investigated experimentally under various operating conditions. The lower surface of the channel is subjected to a uniform heat flux, sidewalls are insulated and adiabatic, and the upper surface is exposed to the surrounding fluid. Experiments were conducted for Pr=0.7, aspect ratios AR=5 and 10, inclination angles 0° ≤ θ ≤ 30°, Reynolds numbers 50 ≤ Re ≤ 1000, and modified Grashof numbers Gr*=7.0 × 10⁵ to 4.0 × 10⁷. From the parametric study, local Nusselt number distributions were obtained and effects of channel inclination, surface heat flux and Reynolds number on the onset of instability were investigated. Results related to the buoyancy affected secondary flow and the onset of instability have been discussed. Some of the results obtained from the experimental measurements are also compared with the literature, and a good agreement was observed. The onset of instability was found to move upstream for increasing Grashof number and increasing aspect ratio. On the other hand, onset of instability was delayed for increasing Reynolds number and increasing inclination angle. Received on 19 March 2001 / Published online: 29 November 2001     

Three dimensional unsteady convection and mass transfer flow through porous medium

The problem of three dimensional unsteady convection flow through a porous medium, with effect of mass transfer bounded by an infinite vertical porous plate is discussed, when the suction at the plate is transverse sinusoidal and the plate temperature oscillates in time about a constant mean. Assuming the free stream velocity to be uniform, approximate solutions are obtained for the flow field, the temperature field, the skin-friction and the rate of heat transfer. The dependence of solution on Pr (Prandtl number), Gr (Grashof number based on temperature), Gc (modified Grashof number based on concentration difference), Sc (Schimdt number), the frequency and the permeability parameter is also investigated.     

Direct numerical simulation of turbulent flow and heat transfer in a square duct with natural convection

In this paper, a direct numerical simulation of a fully developed turbulent flow and heat transfer are studied in a square duct with an imposed temperature difference between the vertical walls and the perfectly insulated horizontal walls. The natural convection is considered on the cross section in the duct. The numerical scheme employs a time-splitting method to integrate the three dimensional incompressible Navier-Stokes equation. The unsteady flow field was simulated at a Reynolds number of 400 based on the Mean friction velocity and the hydraulic diameter (Re _m = 6200), while the Prandtl number (Pr) is assumed 0.71. Four different Grashof numbers (Gr = 10⁴, 10⁵, 10⁶ and 10⁷) are considered. The results show that the secondary flow and turbulent characteristics are not affected obviously at lower Grashof number (Gr ≤ 10⁵) cases, while for the higher Grashof number cases, natural convection has an important effect, but the mean flow and mean temperature at the cross section are also affected strongly by Reynolds stresses. Compared with the laminar heat transfer at the same Grashof number, the intensity of the combined heat transfer is somewhat decreased.     

Heat convection from a cylinder performing steady rotation or rotary oscillation – Part II: Rotary oscillation

This paper deals with the problem of combined (forced and natural) convection from a horizontal cylinder performing oscillating rotary motion in a quiescent fluid of infinite extent. While forced convection is caused by cylinder oscillation, the natural convection is caused by the buoyancy driven flow. The heat transfer process is governed by Rayleigh number, Ra, Reynolds number, Re, and the dimensionless frequency of oscillation, S. The study covers Ra up to 10³, Re up to 400 and S up to 0.8. The results showed that, for the same Ra, the time-averaged rate of heat transfer lies in between two limiting values. The first, is the steady state heat rate due to natural convection from a fixed cylinder and the second is the steady state heat rate from a cylinder rotating steadily at a speed equal to the maximum speed of rotational oscillation. The smaller the value of Re the nearer the time-averaged Nusselt number to that of fixed cylinder at the same Ra and the higher Re the lower the average Nusselt number. The effect of frequency is only limited to changing the amplitude of the fluctuating Nusselt number. Received on 15 December 1997     

Numerical Investigation of the Natural Convection of Water in the Neighborhood of the Density Inversion Point For Grashof Numbers up to 106

The natural convection of fresh water in a square cell is considered at a temperature close to the density inversion temperature for Grashof numbers 2.9 · 10⁴ Gr 10⁶. As a result of the numerical investigation, one steady-state and three self-oscillating regimes are found in addition to the three steady-state flows previously detected earlier and described for low Grashof numbers ( 0 Gr 2 · 10⁵). The basic characteristics of the unsteady flows are analyzed by means of the Fourier method, the fundamental oscillation frequencies are found, and the flow evolution and the variation of the oscillation characteristics with increase in the Gr number are considered.     

Onset of unsteady axi-symmetric laminar natural convection in a vertical cylindrical enclosure heated at the wall

In the present study laminar transition to oscillatory convection of fluids having different Prandtl numbers in a laterally heated vertical cylindrical enclosure for different aspect ratios (melt height to crucible radius) of 2–4 is investigated numerically for 0.01 ≤ Pr ≤ 10. Numerical solution to two-dimensional axisymmetric transient Navier Stokes equations and energy equation were solved by finite volume method using SIMPLE algorithm. Numerical results illustrate that there exists a critical Rayleigh number for each Prandtl number beyond which sustained laminar oscillatory flow sets in. The oscillatory regime was characterised by the oscillation of the average kinetic energy and average thermal energy of the melt. For a given aspect ratio, critical Rayleigh number increases with Pr upto 1 and then flattens. It was observed that for low Prandtl number fluids, Pr < 1.0, critical Rayleigh number is found to increase with increase in aspect ratio while for high Prandtl number fluids, Pr ≥ 1.0, it is found to decrease with increase in aspect ratio. The influence of aspect ratio on the transient behaviour of the melt volume below and above the critical Rayleigh number was studied.     

Transient natural convection of low-Prandtl-number fluids in a differentially heated cavity

The transient convective motion in a two-dimensional square cavity driven by a temperature gradient is analysed. The cavity is filled with a low-Prandtl-number fluid and the vertical walls are maintained at constant but different temperatures, while the horizontal boundaries are adiabatic. A control volume approach with a staggered grid is employed to formulate the finite difference equations. Numerically accurate solutions are obtained for Prandtl numbers of 0·001, 0·005 and 0·01 and for Grashof numbers up to 1 × 10⁷. It was found that the flow field exhibits periodic oscillation at the critical Grashof numbers, which are dependent on the Prandtl number. As the Prandtl number is decreased, the critical Grashof number and the frequency of oscillation decrease. Prior to the oscillatory flow, steady state solutions with an oscillatory transient period were predicted. In addition to the main circulation, four weak circulations were predicted at the corners of the cavity.     

Effect of water density inversion on the plane-parallel flow and heat transfer in a channel of constant width

The results of a numerical study of the effect of cold-water density inversion (Prandtl number Pr=11.59) on the flow and heat transfer in a horizontal plane-parallel channel with isothermal top and bottom walls are presented. The calculations were performed for the Grashof number Gr=3·10⁴, the Reynolds number Re=10, and the channel segment length-to-height ratiol/d=40. The wall temperature was so varied that the temperature difference between the top and bottom walls remained constant. Surgut. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 72–78, January–February, 2000.     

Momentum and heat transfer from an asymmetrically confined circular cylinder in a plane channel

Unsteady momentum and heat transfer from an asymmetrically confined circular cylinder in a plane channel is numerically investigated using FLUENT for the ranges of Reynolds numbers as 10≤Re≤500, of the blockage ratio as 0.1≤β≤0.4, and of the gap ratio as 0.125≤γ≤1 for a constant value of the Prandtl number of 0.744. The transition of the flow from steady to unsteady (characterized by critical Re) is determined as a function of γ and β. The effect of γ on the mean drag and lift coefficients, Strouhal number (St), and Nusselt number (Nu _w ) is studied. Critical Re was found to increase with decreasing γ for all values of β. and St were found to increase with decreasing values of γ for fixed β and Re. The effect of decrease in γ on was found to be negligible for all blockage ratios investigated.     

Non-Darcy buoyancy driven flows in a fluid saturated porous medium: the use of asymptotic computational fluid dynamics (ACFD) approach

Natural convection in a fluid saturated porous medium has been numerically investigated using a generalized non-Darcy approach. The governing equations are solved by using Finite Volume approach. First order upwind scheme is employed for convective formulation and SIMPLE algorithm for pressure velocity coupling. Numerical results are presented to study the influence of parameters such as Rayleigh number (10⁶ ≤Ra ≤10⁸), Darcy number (10⁻⁵ ≤ Da ≤ 10⁻²), porosity (0.4 ≤ ɛ ≤ 0.9) and Prandtl number (0.01 ≤ Pr ≤ 10) on the flow behavior and heat transfer. By combining the method of matched asymptotic expansions with computational fluid dynamics (CFD), so called asymptotic computational fluid dynamics (ACFD) technique has been employed to generate correlation for average Nusselt number. The technique is found to be an attractive option for generating correlation and also in the analysis of natural convection in porous medium over a fairly wide range of parameters with fewer simulations for numerical solutions.     

Convection of a viscous incompressible gas in rectangular regions with inlet and outlet channels

A study is made of two-dimensional problems of thermal convection of a viscous incompressible gas in rectangular regions that have gas inlet and outlet channels in the presence of a temperature difference between the bottom and the top (the bottom is heated). In contrast to the well-studied case of natural convection, when no-slip conditions are specified on all boundaries of the region and motion in the region occurs only through the temperature difference [1–4], the heat transfer in the investigated flows is complicated by the additional influence of the forced convection of the gas due to the motion of gas through the inlet and outlet channels. Flows of such type simulate well the processes that take place in many heat transfer devices and in ventilated and air-conditioned industrial premises. Two formulations of the problem are considered. In the first, the gas flow through the inlet and outlet channels is assumed given, and the solution of the problem is determined by the dimensionless Prandtl, Grashof, and Reynolds numbers. In the second case, this flow rate is not given but determined during the solution of the problem. The motion in the region arises from the difference between the temperatures of the bottom and the top of the region, and the motion, in its turn, causes a flow of gas through the inlet and outlet channels. As in the case of natural convection, the solution of the problem in this case is determined by only two dimensionless numbers — the Grashof and Prandtl numbers. By numerical solution of the boundary-value problems for the equations of heat transfer a study is made of the influence of the characteristic dimensionless numbers on the hydrodynamic and temperature fields and the heat fluxes through the boundaries of the region. The solutions of the problems in the two formulations are compared for different positions of the outlet channels.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 126–131, September–October, 1979.We thank G. I. Petrov for discussing the results.     

Free convection around a horizontal circular cylinder. Adimensional empirical equations

An experimental study involving holographic interferometry has been made of the natural convection from a horizontal cylinder in air, and for an extensive range of surface temperatures covering a range of values Gr.Pr between 3600 and 5600. A number of semiempirical equations were obtained that satisfactorily reproduce the adimensional field of temperatures around the cylinder. Moreover, the functions that provide the local and average Nusselt number values in terms of the corresponding Grashof and Prandtl numbers have also been obtained. Their close correspondence with the contrasted empirical formulas shows that the method employed may be used to obtain variation equations of both temperature and Nusselt magnitudes that involve angular position and are fully compatible with the mean values used in practice.     

Transient combined natural convection-conduction in open-ended vertical concentric annuli

Transient conjugate natural convection heat transfer in open-ended vertical concentric annuli is investigated numerically. The governing equations of an induced laminar flow for a fluid of Pr=0.7 are solved using a finite-difference technique. The heating is achieved by a step change in the temperature of the outer surface of the outer tube while the inner surface of the inner tube is kept adiabatic. The range of Grashof number considered is 500≤Gr^*≤10⁵. The effects of solid-fluid conductivity ratio and diffusivity ratio on the transient induced flow characteristics are presented. Received on 25 March 1998     

Numerical study of transient laminar natural convection heat transfer over a sphere subjected to a constant heat flux

Transient laminar natural convection over a sphere which is subjected to a constant heat flux has been studied numerically for high Grashof numbers (10⁵ ≤ Gr ≤ 10⁹) and a wide range of Prandtl numbers (Pr = 0.02, 0.7, 7, and 100). A plume with a mushroom-shaped cap forms above the sphere and drifts upward continuously with time. The size and the level of temperature of the transient cap and plume stem decrease with increasing Gr and Pr. Flow separation and an associated vortex may appear in the wake of the sphere depending on the magnitude of Gr and Pr. A recirculation vortex which appears and grows until “steady state” is attained was found only for the very high Grashof numbers (10⁵ ≤ Gr ≤ 10⁹) and the lowest Prandtl number considered (Pr = 0.02). The appearance and subsequent disappearance of a vortex was observed for Gr = 10⁹ and Pr = 0.7. Over the lower hemisphere, the thickness of both the hydrodynamic (δ_H) and the thermal (δ_T) boundary layers remain nearly constant and the sphere surface is nearly isothermal. The surface temperature presents a local maximum in the wake of the sphere whenever a vortex is established in the wake of the sphere. The surface pressure recovery in the wake of the sphere increases with decreasing Pr and with increasing Gr. For very small Pr, unlike forced convection, the ratio δ_T/δ_H remains close to unity. The results are in good agreement with experimental data and in excellent agreement with numerical results available in the literature. A correlation has also been presented for the overall Nusselt number as a function of Gr and Pr.     

Mixed convection in a vertical channel with discrete heat sources at the wall

The mixed convection in a vertical plane-parallel channel with two heat sources of finite dimensions located at the wall is analyzed on the basis of a two-dimensional numerical simulation. The effect of the distance between the heat sources on the flow pattern and the temperature field is studied. Calculations are performed on the Grashof and Reynolds number ranges from 0–10⁵ and 0–10, respectively, at a Prandtl number of 0.7. The mathematical model is based on the time-dependent Navier-Stokes equations in the Boussinesq approximation. The problem is solved by the finite element method.     

Heat transfer in a power-law fluid film over a unsteady stretching sheet

In this paper the problem of momentum and heat transfer in a thin liquid film of power-law fluid on an unsteady stretching surface has been studied. Numerical solutions are obtained for some representative values of the unsteadiness parameter S and the power-law index n for a wide range of the generalized Prandtl number, 0.001 ≤ Pr ≤ 1000. Typical temperature and velocity profiles, the dimensionless film thickness, free-surface temperature, and the surface heat fluxes are presented at selected controlling parameters. The results show that increasing the value of n tends to increase the boundary-layer thickness and broadens the temperature distributions. The free-surface temperature of a shear thinning fluid is larger than that of a Newtonian fluid, but the opposite trend is true for a shear thickening fluid. For small generalized Prandtl numbers, the surface heat flux increases with a decrease in n, but the impacts of n on the heat transfer diminish for Pr greater than a moderate value (approximately 1 ≤ Pr ≤ 10, depending on the magnitude of S).     

Combined heat and mass transfer along a vertical moving cylinder with a free stream

The mixed convection flow over a continuous moving vertical slender cylinder under the combined buoyancy effect of thermal and mass diffusion has been studied. Both uniform wall temperature (concentration) and uniform heat (mass) flux cases are included in the analysis. The problem is formulated in such a manner that when the ratio λ(= u _w/(u _w + u _∞), where u _w and u _∞ are the wall and free stream velocities, is zero, the problem reduces to the flow over a stationary cylinder, and when λ = 1 it reduces to the flow over a moving cylinder in an ambient fluid. The partial differential equations governing the flow have been solved numerically using an implicit finite-difference scheme. We have also obtained the solution using a perturbation technique with Shanks transformation. This transformation has been used to increase the range of the validity of the solution. For some particular cases closed form solutions are obtained. The surface skin friction, heat transfer and mass transfer increase with the buoyancy forces. The buoyancy forces cause considerable overshoot in the velocity profiles. The Prandtl number and the Schmidt number strongly affect the surface heat transfer and the mass transfer, respectively. The surface skin friction decreases as the relative velocity between the surface and free stream decreases. Received on 17 May 1999     

Experimental investigation of mixed convection from an array of discrete heat sources at the bottom of a horizontal channel

Mixed convection heat transfer from an array of discrete heat sources inside a rectangular channel has been investigated experimentally under various operating conditions for air. The lower surface of the channel was equipped with 8 × 4 flush-mounted heat sources subjected to uniform heat flux, sidewalls and the upper wall are insulated and adiabatic. The experimental parametric study was made for an aspect ratio of AR = 10, Reynolds numbers 241 Re_Dh 980, and modified Grashof numbers Gr^* = 9.53 × 10⁵ to 1.53 × 10⁷ . From the experimental measurements, surface temperature distributions of the discrete heat sources were obtained and effects of Reynolds and Grashof numbers on these temperatures were investigated. Furthermore, Nusselt number distributions were calculated for different Reynolds and Grashof numbers, with emphasis on changes obtained for different discrete heat source locations. From these results, the buoyancy affected secondary flow and the onset of instability have been discussed. Results show that surface temperatures increase with increasing Grashof number and decrease with increasing Reynolds number. However, with the increase in the buoyancy affected secondary flow and the onset of instability, temperatures level off and even drop as a result of heat transfer enhancement. This outcome can also be observed from the variation of the row-averaged Nusselt number showing an increase towards the exit, especially for low Reynolds numbers.     

Forced convection flow over a microsphere

Forced convection heat transfer characteristics around a microsphere subjected to uniform heat flux boundary condition is numerically investigated in this study. Moderate to high values of Reynolds number and a wide range of Prandtl number are considered. The analysis assumes that the continuity assumption is valid and hence the Navier–Stokes equations are solved for the range of Knudsen number of 0.001 ≤ Kn ≤ 0.1. The appropriate boundary conditions at the surface of the microsphere; the velocity slip and temperature jump are applied. The effect of the flow parameters: Re, Pr and Kn on the velocity and temperature distribution is presented and hence a better control on the boundary layer thickness can be achieved in the microscale level. Furthermore, the effect of the controlling parameters on the delay of flow separation, reduced shear stress, drag coefficient and on the Nusselt number profiles is also presented in the results.