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

The natural convection boundary layer adjacent to an inclined plate subject to sudden cooling boundary condition has been studied. It is found 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. A scaling relation for the onset of instability of the boundary layer is achieved. The scaling relations have been developed by equating important terms of the governing equations based on the development of the boundary layer with time. The flow adjacent to the plate can be classified broadly into a conductive, a stable convective or an unstable convective regime determined by the Rayleigh number. Proper scales have been established to quantify the flow properties in each of these flow regimes. An appropriate identification of the time when the instability may set in is discussed. A numerical verification of the time for the onset of instability is also presented in this study. Different flow regimes based on the stability of the boundary layer have been discussed with numerical results.

Natural convection in attic-shaped spaces subject to sudden and ramp heating boundary conditions

In this study, a discussion of the fluid dynamics in the attic space is reported, focusing on its transient response to sudden and linear changes of temperature along the two inclined walls. The transient behaviour of an attic space is relevant to our daily life. The instantaneous and non-instantaneous (ramp) heating boundary condition is applied on the sloping walls of the attic space. A theoretical understanding of the transient behaviour of the flow in the enclosure is performed through scaling analysis. A proper identification of the timescales, the velocity and the thickness relevant to the flow that develops inside the cavity makes it possible to predict theoretically the basic flow features that will survive once the thermal flow in the enclosure reaches a steady state. A time scale for the heating-up of the whole cavity together with the heat transfer scales through the inclined walls has also been obtained through scaling analysis. All scales are verified by the numerical simulations.

Natural convection in attics subject to instantaneous and ramp cooling boundary conditions

A fundamental study of the fluid dynamics inside an attic shaped triangular enclosure with cold upper walls and adiabatic horizontal bottom wall is reported in this study. The transient behaviour of the attic fluid which is relevant to our daily life is examined based on a scaling analysis. The transient phenomenon begins with the instantaneous cooling and the cooling with linear decreases of temperature up to some specific time (ramp time) and then maintain constant of the upper sloped walls. It is shown that both inclined walls develop a thermal boundary layer whose thicknesses increase towards steady-state or quasi-steady values. A proper identification of the timescales, the velocity and the thickness relevant to the flow that develops inside the cavity makes it possible to predict theoretically the basic flow features that will survive once the thermal flow in the enclosure reaches a steady state. A time scale for the cooling-down of the whole cavity together with the heat transfer scales through the inclined walls has also been obtained through scaling analysis. All scales are verified by the numerical simulations.

Natural convection and heat transfer in attics subject to periodic thermal forcing

The heat transfer through the attics of buildings under realistic thermal forcing has been considered in this study. A periodic temperature boundary condition is applied on the sloping walls of the attic to show the basic flow features in the attic space over diurnal cycles. The numerical results reveal that, during the daytime heating stage, the flow in the attic space is stratified; whereas at the night-time cooling stage, the flow becomes unstable. A symmetrical solution is seen for relatively low Rayleigh numbers. However, as the Ra gradually increases, a transition occurs at a critical value of Ra. Above this critical value, an asymmetrical solution exhibiting a pitchfork bifurcation arises at the night-time. It is also found that the calculated heat transfer rate at the night-time cooling stage is much higher than that during the daytime heating stage.

Radiation effects on natural convection laminar flow from a horizontal circular cylinder

The effect of radiation on natural convection flow from an isothermal circular cylinder has been investigated numerically in this study. The governing boundary layer equations of motion are transformed into a non-dimensional form and the resulting nonlinear systems of partial differential equations are reduced to convenient boundary layer equations, which are then solved numerically by two distinct efficient methods namely: (i) implicit finite differencemethod or the Keller-Box Method (KBM) and (ii) Straight Forward Finite Difference Method (SFFD). Numerical results are presented by velocity and temperature distribution of the fluid as well as heat transfer characteristics, namely the shearing stress and the local heat transfer rate in terms of the local skin-friction coefficient and the local Nusselt number for a wide range of surface heating parameter and radiation-conduction parameter. Due to the effects of the radiation the skin-friction coefficients as well as the rate of heat transfer increased and consequently the momentum and thermal boundary layer thickness enhanced.

The effect of temperature dependent viscosity on MHD natural convection flow from an isothermal sphere

Laminar magnetohydrodynamic (MHD) natural convection flow from an isothermal sphere immersed in a fluid with viscosity proportional to linear function of temperature has been studied. The governing boundary layer equations are transformed into a non-dimensional form and the resulting nonlinear system of partial differential equations are reduced to convenient form which are solved numerically by two very efficient methods, namely, (i) Implicit finite difference method together with Keller box scheme and (ii) Direct numerical scheme. Numerical results are presented by velocity and temperature distribution, streamlines and isotherms of the fluid as well as heat transfer characteristics, namely the local skin-friction coefficients and the local heat transfer rate for a wide range of magnetohydrodynamic paramagnet and viscosity-variation parameter.

Viscous dissipation effects on natural convection from a vertical plate with uniform surface heat flux placed in a thermally stratified media

In the present study we investigate the effect of viscous dissipation on natural convection from a vertical plate placed in a thermally stratified environment. The reduced equations are integrated by employing the implicit finite difference scheme of Keller box method and obtained the effect of heat due to viscous dissipation on the local skin friction and local Nusselt number at various stratification levels, for fluids having Prandtl numbers of 10, 50, and 100. Solutions are also obtained using the perturbation technique for small values of viscous dissipation parameters $\xi$ and compared to the finite difference solutions for 0 · $\xi$ · 1. Effect of viscous dissipation and temperature stratification are also shown on the velocity and temperature distributions in the boundary layer region.

Natural convection flow with combined buoyancy effects due to thermal and mass diffusion in a thermally stratified media

We present here a numerical study of laminar doubly diffusive free convection flows adjacent to a vertical surface in a stable thermally stratified medium. The governing equations of mass, momentum, energy and species are non-dimensionalized. These equations have been solved by using an implicit finite difference method and local non-similarity method. The results show many interesting aspects of complex interaction of the two buoyant mechanisms that have been shown in both the tabular as well as graphical form.

Natural convection from a plane vertical porous surface in non-isothermal surroundings

In this paper, laminar natural convection flow from a permeable and isothermal vertical surface placed in non-isothermal surroundings is considered. Introducing appropriate transformations into the boundary layer equations governing the flow derives non-similar boundary layer equations. Results of both the analytical and numerical solutions are then presented in the form of skin-friction and Nusselt number. Numerical solutions of the transformed non-similar boundary layer equations are obtained by three distinct solution methods, (i) the perturbation solutions for small � (ii) the asymptotic solution for large � (iii) the implicit finite difference method for all � where � is the transpiration parameter. Perturbation solutions for small and large values of � are compared with the finite difference solutions for different values of pertinent parameters, namely, the Prandtl number Pr, and the ambient temperature gradient n.

A review of natural convection and heat transfer in attic-shaped space

This paper presents a review of studies on natural convection heat transfer in the triangular enclosure namely, in attic-shaped space. Much research activity has been devoted to this topic over the last three decades with a view to providing thermal comfort to the occupants in attic-shaped buildings and to minimising the energy costs associated with heating and air-conditioning. Two basic thermal boundary conditions of attic are considered to represent hot and cold climates or day and night time. This paper also reports on a significant number of studies which have been performed recently on other topics related to the attic space, for example, attics subject to localized heating and attics filled with porous media.

On the natural convection boundary layer adjacent to an inclined flat plate subject to ramp heating

An investigation of the natural convection boundary layer adjacent to an inclined semi-infinite plate subject to a temperature boundary condition which follows a ramp function up until some specified time and then remains constant is reported. The development of the flow from start-up to a steadystate has been described based on scaling analyses and verified by numerical simulations. Attention in this study has been given to fluids having a Prandtl number Pr less than unity. The boundary layer flow depends on the comparison of the time at which the ramp heating is completed and the time at which the boundary layer completes its growth. If the ramp time is long compared with the steady state time, the layer reaches a quasi steady mode in which the growth of the layer is governed solely by the thermal balance between convection and conduction. On the other hand, if the ramp is completed before the layer becomes steady; the subsequent growth is governed by the balance between buoyancy and inertia, as for the case of instantaneous heating.

Natural convection in a triangular enclosure subject to periodic thermal forcing

The effects of periodic thermal forcing on the flow field and heat transfer through an attic space are examined numerically in this paper. We consider the case with a fixed aspect ratio of 0.5 and a fixed Grashof number of 1.33×106. The numerical results reveal that, during the daytime, the flow is stratified; whereas at the night-time, the flow becomes unstable. A number of regular plumes and vortices are observed in the contours of isotherms and stream functions respectively. Moreover, the flow appears to be symmetric during the daytime, and becomes asymmetric at the night-time. It is also found that the flow is weaker during the daytime than that at the night-time in the present case, and the calculated heat transfer rate at the night-time is approximately three times greater than the heat transfer rate during the daytime.

Scaling analysis of unsteady natural convection boundary layer for ramp heating

A scaling analysis for the natural convection boundary layer adjacent to an inclined semi-infinite plate subject to a non-instantaneous heating in the form of an imposed wall temperature which increases linearly up to a prescribed steady value over a prescribed time is reported. The development of the flow from start-up to a steady-state has been described based on scaling analyses and verified by numerical simulations. The analysis reveals that, if the period of temperature growth on the wall is sufficiently long, the boundary layer reaches a quasisteady mode before the growth of the temperature is completed. In this mode the thermal boundary layer at first grows in thickness and then contracts with increasing time. However, if the imposed wall temperature growth period is sufficiently short, the boundary layer develops differently, but after the wall temperature growth is completed, the boundary layer develops as though the start up had been instantaneous. The steady state values of the boundary layer for both cases are ultimately the same.

Scaling analysis of unsteady natural convection boundary layer for instantaneous heating

Natural convection thermal boundary layer adjacent to an instantaneous heated inclined flat plate is investigated through a scaling analysis and verified by direct numerical simulations. It is revealed from the analysis that the development of the boundary layer may be characterized by three distinct stages, i.e. a start-up stage, a transitional stage and a steady state stage. These three stages can be clearly identified from the numerical simulations. Major scales including the flow velocity, flow development time, and the thermal and viscous boundary layer thicknesses are established to quantify the flow development at different stages and over a wide range of flow parameters. Details of the scaling analysis are described in this paper.