Rayleigh–Bénard convection in a supercritical fluid along its critical isochore in a shallow cavity

We perform a numerical investigation of the Rayleigh–Bénard convection in supercritical nitrogen in a shallow enclosure with an aspect ratio of 4. The transient and steady-state fluid behaviors over a wide range of initial distances to the critical point along the critical isochore are obtained and analyzed in response to modest homogeneous bottom heating. On account of the fluid layer being extremely thin, density stratification is notably excluded from consideration herein, which leads to the dominating role of the Rayleigh criterion in the onset of convection. Following the Boussinesq approximation, we find the power law scaling relationships over five decades of the Rayleigh number (Ra) for various transient quantities including the exponential growth rate of the mean enstrophy in the cavity and the characteristic times of the development of convective motion. The correlation of the Nusselt number versus the Rayleigh number shows asymptotic features at the two ends of the Ra spectrum, which incidentally correspond to different convection patterns. Under the regime of high Ra, the heat transfer through the fluid cavity is enhanced by the turbulent bursts of thermal plumes from the boundary layers. On the other hand, under the regime of low Ra, it is the orderly multicellular flow that moves heat from the bottom of the layer to the top, which includes a transition from a four-cell structure to a six-cell structure with decreasing Ra.     

Thermal lattice-BGK model based on large-eddy simulation of turbulent natural convection due to internal heat generation

To simulate turbulent convection at high Rayleigh number (Ra), we propose a new thermal lattice-BGK (LBGK) model based on large eddy simulation (LES). Two-dimensional numerical simulations of natural convection with internal heat generation in a square cavity were performed at Ra from 10⁶ to 10¹³ with Prandtl numbers (Pr) at 0.25 and 0.60. Simulation results indicate that our model is fit to simulate high Ra flow for its better numerical stability. At Ra = 10¹³, a global turbulent has occurred. With a further increase in Ra, the flow will arrive in a fully turbulence regime. The Nusselt–Rayleigh relationship is also discussed.     

Effects of thermal boundary conditions on natural convection flows within a square cavity

A numerical study to investigate the steady laminar natural convection flow in a square cavity with uniformly and non-uniformly heated bottom wall, and adiabatic top wall maintaining constant temperature of cold vertical walls has been performed. A penalty finite element method with bi-quadratic rectangular elements has been used to solve the governing mass, momentum and energy equations. The numerical procedure adopted in the present study yields consistent performance over a wide range of parameters (Rayleigh number Ra, 10³ ⩽ Ra ⩽ 10⁵ and Prandtl number Pr, 0.7 ⩽ Pr ⩽ 10) with respect to continuous and discontinuous Dirichlet boundary conditions. Non-uniform heating of the bottom wall produces greater heat transfer rates at the center of the bottom wall than the uniform heating case for all Rayleigh numbers; however, average Nusselt numbers show overall lower heat transfer rates for the non-uniform heating case. Critical Rayleigh numbers for conduction dominant heat transfer cases have been obtained and for convection dominated regimes, power law correlations between average Nusselt number and Rayleigh numbers are presented.     

Entropy generation in natural convection in a symmetrically and uniformly heated vertical channel

In this study numerical predictions of local and global entropy generation rates in natural convection in air in a vertical channel symmetrically heated at uniform heat flux are reported. Results of entropy generation analysis are obtained by solving the entropy generation equation based on the velocity and temperature data. The analyzed regime is two-dimensional, laminar and steady state. The numerical procedure expands an existing computer code on natural convection in vertical channels. Results in terms of fields and profiles of local entropy generation, for various Rayleigh number, Ra, and aspect ratio values, L/b, are given. The distributions of local values show different behaviours for the different Ra values. A correlation between global entropy generation rates, Rayleigh number and aspect ratio is proposed in the ranges 10³ ⩽ Ra ⩽ 10⁶ and 5 ⩽ L/b ⩽ 20.     

A 2-D numerical study of chaotic flow in a natural convection loop

This paper numerically investigates the nonlinear dynamics of the unstable convection regime of the thermal convection loop, an experimental analogue of the Lorenz model. The lower half of the toroidal loop is heated and maintained at a constant high temperature, while the upper half is cooled at a constant low temperature. Subject to the proper boundary conditions, the system of governing equations is solved using a finite volume method. The numerical simulations are performed for water corresponding to Pr = 5.83 and Rayleigh number varying from 1000 to 150,000. In the case of a loop heated from below and cooled from above, it has been demonstrated theoretically and experimentally in the literature that multiple flow regimes are possible. Numerical results in terms of streamlines, isotherms, and local heat flux distributions along the walls are presented for each flow regime. Although several studies have investigated the chaotic regime of convection loops, there have been no detailed numerical simulations of the dynamics of flow reversals. Fine-scale flow behavior during the transition from one flow direction to another is illustrated by the temporal evolution of temperature distribution, mass flow rate, and local heat flux at selected locations in the system. Issues related to the observed Kelvin–Helmholtz instabilities are discussed.     

MHD natural convection in a laterally and volumetrically heated square cavity

A numerical study is presented of unsteady two-dimensional natural convection of an electrically conducting fluid in a laterally and volumetrically heated square cavity under the influence of a magnetic field. The flow is characterized by the external Rayleigh number, Ra_E, determined from the temperature difference of the side walls, the internal Rayleigh number, Ra_I, determined from the volumetric heat rate, and the Hartmann number, Ha, determined from the strength of the imposed magnetic field. Starting from given values of Ra_E and Ha, for which the flow has a steady unicellular pattern, and gradually increasing the ratio S = Ra_I/Ra_E, oscillatory convective flow may occur. The initial steady unicellular flow for S = 0 may undergo transition to steady or unsteady multicellular flow up to a threshold value, Ra_I,cr, of the internal Rayleigh number depending on Ha. Oscillatory multicellular flow fields were observed for S values up to 100 for the range 10⁵-10⁶ of Ra_E studied. The increase of the ratio S results usually in a transition from steady to unsteady flow but there have also been cases where the increase of S results in an inverse transition from unsteady to steady flow. Moreover, the usual damping effect of increasing Hartmann number is not found to be straightforward connected with the resulting flow patterns in the present flow configuration.     

Transient natural convection in a cylindrical enclosure at high Rayleigh numbers

The transient state of natural convection in a vertical cylindrical enclosure is studied numerically for water at high Rayleigh numbers, extending into values characteristic of the turbulent flow regime. Several two-equation turbulence models are used for this purpose. Heating is provided along the cylindrical surface at a constant heat flux, with the horizontal bounding surfaces being adiabatic and the development of stratification is studied. Such a configuration is very relevant to thermal storage tanks or solar thermal system vessels and the study aims at providing insight into the behavior of the system at the boundary between laminar and turbulent flow so that the appropriate numerical treatment may be adopted in future studies. The main aspect ratio considered is L/D=1 and the Rayleigh number (based on the length L) varies in the range 10¹⁰?Ra?10¹³ for laminar flow and 5×10¹³?Ra?10¹⁵ for turbulent flow, values for which previous data in the literature are all but non-existent. The attainment of a quasi-steady state is achieved after the fluid undergoes an oscillating pattern where secondary flows alternately appear and vanish. These patterns affect the development of stratification in the vessel. Low-Reynolds k-? models predict eventually a relaminarization at large times, but models employing the high-Re form of the k-? model obtain sustained or very slowly decaying turbulence instead. Comparisons are made with experimental results where applicable.     

CFD-based analysis of entropy generation due to mean and fluctuating fields during turbulent natural convection in a square enclosure

The current study aims to numerically investigate the entropy generation during the natural convection flow of air in a square cavity. The governing equations for the conservation of mass, momentum, energy, and turbulence are solved using a control volume-based technique employing the commercial code Fluent. Runs have been performed for both laminar and turbulent flow regimes by varying the Rayleigh number (Ra) from 10³ to 10¹⁰. On the other hand, various viscous distribution coefficients (ϕ = 10⁻⁴, 10⁻³, and 10⁻²) and constant Prandtl number (Pr = 0.71) were considered. Given the conflicting perspectives in the literature regarding the entropy generation under turbulent regimes, more research is needed to better understand the impact that the fluctuating flow has on entropy production. The four terms of entropy generation inherent to turbulent natural convection (entropy generation due to dissipation in the mean and the fluctuating velocity fields in addition to the heat flux due to the mean and the fluctuating temperature) are computed in the present work and compared to calculations based on only mean values of temperature and velocity gradients. It was found that taking into account the fluctuating terms of temperatures and velocities augment the total entropy generation by 10.10%, 14.43%, and, 17.70%, up to 32.60%, respectively, for Ra = 5 × 10⁸, Ra = 10⁹, Ra = 1.58 × 10⁹, and Ra = 10¹⁰. The gain shows the tendency to increase with the Rayleigh number. Thus, the fluctuating terms cannot be neglected particularly for high Rayleigh numbers. Furthermore, unlike entropy production due to the mean flow field, numerical outcomes reveal that the generated irreversibilities due to fluctuating flow are located around the upper hot and the lower cold corners of the heated walls. In addition, a numerical relationship between the first and the second laws of thermodynamics has been derived. A promising result that emerged from this study has shown that the Nusselt number and therefore the first law of thermodynamics is sufficient to estimate the heat part of entropy generation without the necessity of using the second law.     

Numerical Study of Natural Convection in an Enclosure with an Internal Heat Source at Higher Rayleigh Numbers

Natural convection is extensively used in cooling of large scale electrical and electronic equipments. This work involves study of flow and heat transfer characteristics in enclosures with partial openings having an internal heat source at higher Rayleigh number (Ra_h > 10⁶). It involves the numerical simulation of 2D steady state natural convection in enclosures of different aspect ratios (H/W = 2 and 3) for five Rayleigh numbers (Ra_h = 10⁷, 10⁸, 10⁹, 10¹⁰, and 10¹¹). Two different configurations have been considered based on the number and position of vents—diagonal side (DS) and two inlets one outlet (2I1O). The time dependent nature of the flow is characterized by performing a Fast Fourier Transform (FFT) analysis of temperature and velocity at a characteristic location in the enclosure. The global parameters considered are the mass flow rate driven through the cavity by the heater and the average Nu defined over the heater surface. It is seen that with increase in Ra_h, flow becomes more fluctuating and moves towards chaotic regime and this transition is quicker at lower H/W. For the given configuration both the global parameters increases with increase in Ra_h and decrease in H/W.     

Heat generation effects in natural convection inside a porous annulus

The interplay between internal heat generation and externally driven natural convection inside a porous medium annulus is studied in detail using numerical methods. The axisymmetric domain is bounded with adiabatic top and bottom walls and differentially heated side walls sustaining steady natural convection of a fluid with Prandtl number, Pr = 5, through a porous matrix of volumetric porosity, ? = 0.4. The generalized momentum equation with Brinkman–Darcy–Forchheimer terms and the local thermal non-equilibrium based two-energy equation model are solved to determine the flow and the temperature distribution. Beyond a critical heat generation value defined using an internal Rayleigh number, Ra_I,cr^?, the convection transits from unicellular to bicellular mode, as the annulus T_max becomes higher than the fixed hot-wall temperature. The Ra_I,cr^? increases proportionately when the permeability based external Rayleigh number Ra_E^? and the solid–fluid thermal conductivity ratio γ are independently increased. A correlation is proposed to predict the overall annulus Nu as a function of Ra_E^?, Ra_I^?, Da and γ. It predicts the results within ± 20% accuracy.     

A critical review of buoyancy-induced flow transitions in horizontal annuli

The main mechanisms of transition of buoyancy-induced flows in the horizontal annulus between circular cylinders are reviewed, based on the available literature. Both experimental and theoretical studies are considered. The different scenarios for the evolution of the flow regimes and temperature patterns are tracked, for increasing values of the Rayleigh number, Ra. The occurrence of various instability and bifurcative phenomena is pointed out, and linked to other relevant parameters, such as the radius ratio R and the Prandtl number, Pr. Although most of the relevant literature is on 2D cases, the effect of the third dimension is considered as far as possible. Studies on the influence of the eccentricity of the inner cylinder on the laminar flow and the thermal asset are also reviewed. Finally, open questions and topics for future research are hinted at.     

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.     

Numerical exploration of 1 + 2 type laminar natural convection in a differentially heated square cavity using Chebyshev spectral collocation method

In this paper, the Chebyshev spectral collocation method is applied to explore the unsteady two dimensional (1 + 2 type) laminar natural convection in a differentially heated square cavity at a Rayleigh number (Ra) of 10⁷. The method has embedded the traditional Chorin's algorithm so as to avoid the trouble of seeking the pressure field in the buoyancy driven wall-jet flow. The sensitivity of the δ− parameter has been numerically investigated. It is found that when the δ value is over 11.6173, numerical instability occurs. Comparing the maximum horizontal velocity component with the existing numerical data obtained by solving the Poisson's equation of pressure field reveals that the Chorin's algorithm should be inapplicable for the solution of the benchmark problem of natural convection at Ra = 10⁷ in thermal science.     

Natural convection heat transfer from a horizontal cylinder embedded in a porous medium

This paper presents the results of an experimental investigation of heat transfer by natural convection from a horizontal cylinder embedded in porous media consisting of randomly packed glass spheres saturated by either water or silicone oil. It is shown that the overall range of the Rayleigh number, Ra, can be divided into two subregions, called ‘low’ and ‘high’, in each of which the Nusselt number, Nu, behaves differently. It is demonstrated that the low Ra region corresponds to Darey flow and the high to Forchheimer flow. Correlation equations for Nu for the Darcy regime are presented that account for viscous dissipation, and others for the Forchheimer regime that involve the first and second Forchheimer coefficients. The variation of properties with temperature and the wall effect on porosity (and consequently on heat transfer) are considered. The paper includes information concerning the resistance to flow in porous media that was obtained in conjunction with the heat transfer study.     

Conjugate natural convection in the roof cavity of heavy construction building during summer

This paper presents the results of a study of conjugate natural convection inside a building attic in the shape of a rectangular enclosure bounded by realistic walls made from composite construction materials under summer day boundary conditions. The effects of cavity aspect ratio, Rayleigh number (Ra), and orientation of the external surfaces on the flow and heat transfer characteristics were the main focus of the investigation. The problem was formulated in terms of the vorticity-stream function procedure, and the governing equations for steady, laminar, two-dimensional conjugate natural convection heat transfer were solved by employing the Alternating Direction Implicit (ADI) control volume method along with under-relaxation factors for temperature, vorticity, and stream functions. For Ra ranging from 10³ to 10¹⁰, steady state results of the streamline and temperature contours in addition to local and mean Nusselt number at all surfaces of the cavity were obtained. The results show that the values of Ra and the aspect ratio have significant effect on the temperature and stream function contours within the enclosure. Another important finding of the study is that heat flux into the room increases with the increase of both the aspect ratio and Rayleigh number.     

Simulation of turbulent natural convection in a porous cylindrical annulus using a macroscopic two-equation model

This work presents numerical computations for laminar and turbulent natural convection within a horizontal cylindrical annulus filled with a fluid saturated porous medium. Computations covered the range 25 < Ra_m < 500 and 3.2 × 10⁻⁴ > Da > 3.2 × 10⁻⁶ and made use of the finite volume method. The inner and outer walls are maintained at constant but different temperatures. The macroscopic k–ε turbulence model with wall function is used to handle turbulent flows in porous media. First, the turbulence model is switched off and the laminar branch of the solution is found when increasing the Rayleigh number, Ra_m. Subsequently, the turbulence model is included and calculations start at high Ra_m, merging to the laminar branch for a reducing Ra_m. This convergence of results as Ra_m decreases can be seen as an estimate of the so-called laminarization phenomenon. Here, a critical Rayleigh number was not identified and results indicated that when the porosity, Prandtl number, conductivity ratio between the fluid and the solid matrix and Ra_m are kept fixed, the lower the Darcy number, the higher is the difference of the average Nusselt number given by the laminar and turbulent models.     

Heat transfer by mixed convection in a vertical flow undergoing transition

Heat transfer, for aiding mixed convection from vertical, uniform flux surfaces and for small forced convection effects, is considered here. Simple relations have been proposed to correlate the new experimental data which were obtained in a flow undergoing transition from a laminar regime toward turbulence. Experiments were performed in air at pressures ranging from 4.4 to about 8 bar. The correlation based on experimental data for laminar flow for Pr = 0.7 has been extended to other Prandtl numbers through numerical integration of the transport equations. It is shown that, for both laminar and turbulent mixed convection, the Nusselt number may be successfully correlated, employing suitable combinations of the corresponding heat transfer correlations for forced and for natural convection. The parameter characterizing the mixed convection effect was found to be different in laminar and turbulent flow. However, in each of these regions, the relevant parameter is proportional to the ratio of the applicable characteristic forced and natural convection velocity scales.     

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.