Laminar and turbulent natural convection in rectangular cavities

"This chapter discusses laminar and turbulent natural convection in rectangular cavities. Natural convection in rectangular two-dimensional cavities has become a standard problem in numerical heat transfer because of its relevance in understanding a number of problems in engineering. Current research identified a number of difficulties with regard to the numerical methods and the turbulence modeling for this class of flows. Obtaining numerical predictions at high Rayleigh numbers proved computationally expensive such that results beyond Ra ∼ 1014 are rarely reported. The chapter discusses a study in which it was found that turbulent computations in square cavities can't be extended beyond Ra ∼ O (1012) despite having developed a code that proved very efficient for the high Ra laminar regime. As the Rayleigh number increased, thin boundary layers began to form next to the vertical walls, and the central region became progressively more stagnant and highly stratified. Results obtained for the high Ra laminar regime were in good agreement with existing studies. Turbulence computations, although of a preliminary nature, indicated that a second moment closure model was capable of predicting the experimentally observed flow features."--Publisher Summary

Modeling of Convection and Macrosegregation through Appropriate Consideration of Multiphase/Multiscale Phenomena during Alloy Solidification

Solidification processes are complex in nature, involving multiple phases and several length scales. The properties of solidified products are dictated by the microstructure, the mactostructure, and various defects present in the casting. These, in turn, are governed by the multiphase transport phenomena Occurring at different length scales. In order to control and improve the quality of cast products, it is important to have a thorough understanding of various physical and physicochemical phenomena Occurring at various length scales. preferably through predictive models and controlled experiments. In this context, the modeling of transport phenomena during alloy solidification has evolved over the last few decades due to the complex multiscale nature of the problem. Despite this, a model accounting for all the important length scales directly is computationally prohibitive. Thus, in the past, single-phase continuum models have often been employed with respect to a single length scale to model solidification processing. However, continuous development in understanding the physics of solidification at various length scales oil one hand and the phenomenal growth of computational power oil the other have allowed researchers to use increasingly complex multiphase/multiscale models in recent. times. These models have allowed greater understanding of the coupled micro/macro nature of the process and have made it possible to predict solute segregation and microstructure evolution at different length scales. In this paper, a brief overview of the current status of modeling of convection and macrosegregation in alloy solidification processing is presented.

Steady mixed convection stagnation-point flow of upper convected Maxwell fluids with magnetic field

The steady MHD mixed convection flow of a viscoelastic fluid in the vicinity of two-dimensional stagnation point with magnetic field has been investigated under the assumption that the fluid obeys the upper-convected Maxwell (UCM) model. Boundary layer theory is used to simplify the equations of motion. induced magnetic field and energy which results in three coupled non-linear ordinary differential equations which are well-posed. These equations have been solved by using finite difference method. The results indicate the reduction in the surface velocity gradient, surface heat transfer and displacement thickness with the increase in the elasticity number. These trends are opposite to those reported in the literature for a second-grade fluid. The surface velocity gradient and heat transfer are enhanced by the magnetic and buoyancy parameters. The surface heat transfer increases with the Prandtl number, but the surface velocity gradient decreases.

Effect of surface radiation on conjugate natural convection in a horizontal annulus driven by inner heat generating solid cylinder

This paper reports a numerical study of the laminar conjugate natural convection heat transfer with and without the interaction of the surface radiation in a horizontal cylindrical annulus formed between an inner heat generating solid circular cylinder and an outer isothermal circular boundary. Numerical solutions are obtained by solving the governing equations with a pressure correction method on a collocated (non-staggered) mesh. Steady-state results are presented for the flow and temperature distributions and Nusselt numbers for the heat generation based Grashof number ranging from 10(7) to 10(10), solid-to-fluid thermal conductivity ratios of 1, 5, 10, 50 and 100, radius ratios of 0.226 and 0.452 and surface emissivities of 0-0.8 with air as the working medium. It is observed that surface radiation reduces the convective heat transfer in the annulus compared to the pure natural convection case and enhances the overall Nusselt number.

Unsteady Natural Convection Flow over a Heated Cylinder Buried in a Fluid Saturated Porous Medium

Transient natural convection flow on a heated cylinder buried in a semi-infinite liquid-saturated porous medium has been studied. The unsteadiness in the problem arises due to the cylinder which is heated (cooled) suddenly and then maintained at that temperature. The coupled partial differential equations governing the flow and heat transfer are cast into stream function-temperature formulation, and the solutions are obtained from the initial time to the time when steady state is reached. The heat transfer is found to change significantly with increasing time in a small time interval immediately after the start of the impulsive change, and steady state is reached after some time. The average Nusselt number is found to increase with Rayleigh number When the surface of the cylinder is suddenly cooled, there is a change in the direction of the heat transfer in a small time interval immediately after the start of the impulsive change in the surface temperature;however when the surface temperature is suddenly increased, no such phenomenon is observed.

The influence of the Coriolis force on flux tubes rising through the solar convection zone

In order to study the effect of the Coriolis force due to solar rotation on rising magnetic flux, the authors consider a flux ring, azimuthally symmetric around the rotation axis, starting from rest at the bottom of the convection zone, and then follow the trajectory of the flux ring as it rises. If it is assumed that the flux ring remains azimuthally symmetric during its ascent, then the problem can be described essentially in terms of two parameters: the value of the initial magnetic field in the ring when it starts, and the effective drag experienced by it. For field strengths at the bottom of the convection zone of order 10,000 G or less, it is found that the Coriolis force plays a dominant role and flux rings starting from low latitudes at the bottom are deflected and emerge at latitudes significantly poleward of sunspot zones.

The evolution of loop structures in flux rings within the solar convection zone

Choudhuri and Gilman (1987) considered certain implications of the hypothesis that the magnetic flux within the Sun is generated at the bottom of the convection zone and then rises through it. Taking flux rings symmetric around the rotation axis and using reasonable values of different parameters, they found that the Coriolis force deflects these flux rings into trajectories parallel to the rotation axis so that they emerge at rather high latitudes. This paper looks into the question of whether the action of the Coriolis force is subdued when the initial configuration of the flux ring has non-axisymmetries in the form of loop structures. The results depend dramatically on whether the flux ring with the loops lies completely within the convection zone or whether the lower parts of it are embedded in the stable layers underneath the convection zone. In the first case, the Coriolis force supresses the non-axisymmetric perturbations so that the flux ring tends to remain symmetric and the trajectories are very similar to those of Choudhuri and Gilman (1987). In the second case, however, the lower parts of the flux ring may remain anchored underneath the bottom of the convection zone, but the upper parts of the loops still tend to move parallel to the rotation axis and emerge at high latitudes. Thus the problem of the magnetic flux not being able to come out at the sunspot latitudes still persists after the non-axisymmetries in the flux rings are taken into account.

Unsteady natural convection from a horizontal annulus filled with a porous medium

The unsteady natural convection flow from a horizontal cylindrical annulus filled with a non-Darcy porous medium has been studied. The unsteadiness in the problem arises due to the impulsive change in the wall temperature of the outer cylinder. The Navier–Stokes equations along with the energy equation governing the unsteady natural convection flow have been solved by the finite-volume method. The effect of time variation on the heat transfer is more pronounced only in a small time interval immediately after the start of the impulsive motion and the steady state is reached after certain time. The results show that the annulus completely filled with a porous medium has the best insulating effectiveness. Convection in the horizontal annulus is confined mostly at top and bottom regions. Hence, only these regions should be insulated. In case of annulus partially filled with a porous material, insulating the region near the outer cylinder is more effective than insulating the region near the inner cylinder. The effect of Darcy number on the heat transfer is more pronounced than that of the Grashof number.

Unsteady MHD mixed convection flow over an impulsively stretched permeable vertical surface in a quiescent fluid

The unsteady mixed convection flow of an incompressible laminar electrically conducting fluid over an impulsively stretched permeable vertical surface in an unbounded quiescent fluid in the presence of a transverse magnetic field has been investigated. At the same time, the surface temperature is suddenly increased from the surrounding fluid temperature or a constant heat flux is suddenly imposed on the surface. The problem is formulated in such a way that for small time it is governed by Rayleigh type of equation and for large time by Crane type of equation. The non-linear coupled parabolic partial differential equations governing the unsteady mixed convection flow under boundary layer approximations have been solved analytically by using the homotopy analysis method as well as numerically by an implicit finite difference scheme. The local skin friction coefficient and the local Nusselt number are found to decrease rapidly with time in a small time interval and they tend to steady-state values for t* >= 5. They also increase with the buoyancy force and suction, but decrease with injection rate. The local skin friction coefficient increases with the magnetic field, but the local Nusselt number decreases. There is a smooth transition from the unsteady state to the steady state. (C) 2010 Elsevier Ltd. All rights reserved.

Unsteady mixed convection on the stagnation-point flow adjacent to a vertical plate with a magnetic field

An analysis is performed to study the unsteady combined forced and free convection flow (mixed convection flow) of a viscous incompressible electrically conducting fluid in the vicinity of an axisymmetric stagnation point adjacent to a heated vertical surface. The unsteadiness in the flow and temperature fields is due to the free stream velocity, which varies arbitrarily with time. Both constant wall temperature and constant heat flux conditions are considered in this analysis. By using suitable transformations, the Navier-Stokes and energy equations with four independent variables (x, y, z, t) are reduced to a system of partial differential equations with two independent variables (eta, tau). These transformations also uncouple the momentum and energy equations resulting in a primary axisymmetric flow, in an energy equation dependent on the primary flow and in a buoyancy-induced secondary flow dependent on both primary flow and energy. The resulting system of partial differential equations has been solved numerically by using both implicit finite-difference scheme and differential-difference method. An interesting result is that for a decelerating free stream velocity, flow reversal occurs in the primary flow after certain instant of time and the magnetic field delays or prevents the flow reversal. The surface heat transfer and the surface shear stress in the primary flow increase with the magnetic field, but the surface shear stress in the buoyancy-induced secondary flow decreases. Further the heat transfer increases with the Prandtl number, but the surface shear stress in the secondary flow decreases.

Free convection heat transfer from vertical cylinders and wires

Free convection heat transfer from vertical long cylinders and wires were investigated experimentally. The experiments were designed to check the analytical results and the radius of curvature criteria formulated by the same authors. The results for water, the fluid medium used in the present set of experiments, are in excellent agreement with the theory. The results of Hama, Recesso and Christiaens, Kyte, Madden and Piret, in air are also shown to be in close agreement with proposed correlations.

On the ignition and extinction problems in forced convection systems

In this paper the problem of ignition and extinction has been formulated for the flow of a compressible fluid with Prandtl and Schmidt numbers taken as unity. In particular, the problems of (i) a jet impinging on a wall of combustible material and (ii) the opposed jet diffusion flame have been studied. In the wall jet case, three approximations in the momentum equation namely, (i) potential flow, (ii) viscous flow, (ii) viscous incompressible with k = 1 and (iii) Lees' approximation (taking pressure gradient terms zero) are studied. It is shown that the predictions of the mass flow rates at extinction are not very sensitive to the approximations made in the momentum equation. The effects of varying the wall temperature in the case (i) and the jet temperature in the case (ii) on the extinction speeds have been studied. The effects of varying the activation energy and the free stream oxidant concentration in case (ii), have also been investigated.