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.     

Numerical simulation of laminar and turbulent buoyancy-driven flows using a lattice Boltzmann based algorithm

The capability of simulating natural and forced convection has been recently developed and integrated into PowerFLOW, a general purpose CFD solver based on the lattice Boltzmann algorithm. Several benchmark tests have been performed to validate this buoyancy model. Two typical cases of Rayleigh-Bénard convection with the Rayleigh number slightly above (Ra=2000) and below (Ra=1500) the critical Rayleigh number of 1708 were tested to verify the conceptual and algorithmic correctness of the buoyancy model. Then simulations of turbulent natural convection in an enclosed tall cavity with two different Rayleigh numbers, Ra=0.86×10⁶ and Ra=1.43×10⁶, were carried out and found to be in a very good agreement with the experiments of Betts and Bokhari.     

NATURAL CONVECTION IN A TWO-DIMENSIONAL DIFFERENTIALLY HEATED SQUARE ENCLOSURE UNDERGOING ROTATION

A numerical study of two-dimensional natural convection in a rotating and differentially heated square enclosure has been presented by solving the conservation equations of mass, momentum, and energy in a rotating coordinate system using the finite difference method. Considering air to be the fluid medium in the cavity, the results are presented for a wide range of Rayleigh numbers (Ra), Taylor numbers (Ta), and rotational Rayleigh numbers (Ra _w). It is found that a significant enhancement in heat transfer can be achieved due to rotational effects. At a particular Ra, increase in Ta results in an increase in frequency of oscillations of the dynamical variables and also gives rise to formation of a mushroom-shaped plume in the core of the cavity. For constant Ta, an increase in Ra results in formation of thinner thermal boundary layers at the isothermal walls and stable thermal stratification in the core of the cavity. The stratification becomes unstable when the Ta     

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.     

Internal gravity wave resonance of thermal convection fields in a square cavity with heat‐flux vibration

In this paper, the thermal convection field and its resonance phenomena in a square cavity with sinusoidal heat‐flux vibration were numerically investigated. As the angular velocity ω is changed, the thermal convection field at Pr = 0.71,Ra = 10⁶ is found to be classified into 5 regions. In particular, the field has the local maximum relative amplitude of midplane Nusselt's number at ω_c = 350, which corresponds to the angular velocity of internal gravity wave ω_r estimated by a theoretical equation proposed by Thorpe. This shows that the local enhancement is induced by internal gravity wave resonance. Such correspondence is observed for Ra ≥ 10⁵,Ra ≥ 10⁶ for Pr = 0.71, 7.1, respectively. For these ranges of Ra we propose a correlation equation, a function of Pr and Gr only, to estimate the resonant angular velocity. © 2006 Wiley Periodicals, Inc. Heat Trans Asian Res, 35(5): 309–322, 2006; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/htj.20122     

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.     

Constructal design of vertical multiscale triangular fins in natural convection

Constructal design of vertical multiscale triangular fins in natural convection is investigated in this paper. The design consists of two parts. The first part is for single-scale triangular fins. The objective in the first design is to reach to the highest heat transfer density from the fins for three fin angles (15°, 30°, and 45°). The single-scale fins are placed in a horizontal array and considered as isothermal fins. The degrees of freedom are the fin angle, and the fin-to-fin spacing. The constraint is the fin height. The second part is for multiscale fins where small fins are placed between the large fins which are optimized in the first part. In the second part, the angles of the large and small scales fins are kept constant at (15°). The optimal fin-to-fin spacing which is obtained in the first part is considered a constraint in the second part. The Rayleigh numbers in this design are (Ra = 10³, 10⁴, and 10⁵). The two-dimensional mass, momentum, and energy equations for natural convection are solved with the finite volume method. The results show that there is a benefit of placing the small-scale fins where the percentage increase in the heat transfer density is (10.22%) at (Ra = 10³), and (50.6%) at (Ra = 10⁵) due to existence of the small fins between the large fins.     

Natural convection cooling process from two identical porous-covering electronic components

In the present work, we focus to study numerically the natural convection cooling process from two identical electronic components located on the bottom wall of a two-dimensional cavity. Each electronic component is covered by a porous medium with finite thickness. The conservation equations governing the problem are discretized using the finite volume method and the SIMPLER algorithm is used to handle the nonlinear character of conservation equations. Calculations were carried out for the following control parameters: the porous/fluid thermal conductivity ratio (1 ≤ Rk_P1 ≤ 100), the Darcy and Rayleigh numbers (10⁻¹ ≤ Da ≤ 10⁻⁶, 10³ ≤ Ra ≤ 10⁶), the first porous-cover thickness (0.05 ≤ e₁ ≤ 0.3), and the separation distance between components (0.2 ≤ S ≤ 1) to highlight their influence on the cooling process. The results show that under specific values of the Darcy and Rayleigh numbers and in the limiting case of a high value of the porous/fluid thermal conductivity ratio (Rk_P1 = 100), a decrease in the maximum components temperature, up to 95%, is observed by increasing the porous-cover thickness from 0.05 to 0.3. In addition, by increasing the permeability, the Rayleigh number or the separation distance, an improvement in the cooling process of the two components greater than or equal to 22% is achieved.     

Unsteady natural convection within a differentially heated enclosure of sinusoidal corrugated side walls

Numerically investigation of natural convection within a differentially heated modified square enclosure with sinusoidally corrugated side walls has been performed for different values of Rayleigh number. The fluid inside the enclosure considered is air and is quiescent, initially. The top and bottom surfaces are flat and considered as adiabatic. Results reveal three main stages: an initial stage, a transitory or oscillatory stage and a steady stage for the development of natural convection flow inside the corrugated cavity. The numerical scheme is based on the finite element method adapted to triangular non-uniform mesh element by a non-linear parametric solution algorithm. Investigation has been performed for the Rayleigh number, Ra ranging from 10⁵ to 10⁸ with variation of corrugation amplitude and frequency. Constant physical properties for the fluid medium have been assumed except for the density where Boussinesq’s approximation has been considered. Results have been presented in terms of the isotherms, streamlines, temperature plots, average Nusselt numbers, traveling waves and thermal boundary layer thickness plots, temperature and velocity profiles. The effects of sudden differential heating and its consequent transient behavior on fluid flow and heat transfer characteristics have been observed for the range of governing parameters. The present results show that the transient phenomena are greatly influenced by the variation of the Rayleigh number with corrugation amplitude and frequency.     

Natural convection in a square cavity filled with a porous medium: Effects of various thermal boundary conditions

Natural convection flows in a square cavity filled with a porous matrix has been studied numerically using penalty finite element method for uniformly and non-uniformly heated bottom wall, and adiabatic top wall maintaining constant temperature of cold vertical walls. Darcy–Forchheimer model is used to simulate the momentum transfer in the porous medium. The numerical procedure is adopted in the present study yields consistent performance over a wide range of parameters (Rayleigh number Ra, 10³ ⩽ Ra ⩽ 10⁶, Darcy number Da, 10⁻⁵ ⩽ Da ⩽ 10⁻³, and Prandtl number Pr, 0.71 ⩽ Pr ⩽ 10) with respect to continuous and discontinuous thermal boundary conditions. Numerical results are presented in terms of stream functions, temperature profiles and Nusselt numbers. Non-uniform heating of the bottom wall produces greater heat transfer rate at the center of the bottom wall than uniform heating case for all Rayleigh numbers but average Nusselt number shows overall lower heat transfer rate for non-uniform heating case. It has been found that the heat transfer is primarily due to conduction for Da ⩽ 10⁻⁵ irrespective of Ra and Pr. The conductive heat transfer regime as a function of Ra has also been reported for Da ⩾ 10⁻⁴. Critical Rayleigh numbers for conduction dominant heat transfer cases have been obtained and for convection dominated regimes the power law correlations between average Nusselt number and Rayleigh numbers are presented.     

Conjugate turbulent natural convection in the roof enclosure of a heavy construction building during winter

This paper presents the results of a study of conjugate turbulent natural convection inside a building attic in the shape of a rectangular enclosure bounded by realistic walls made from composite construction materials under winter day boundary conditions. The effects of cavity aspect ratio, Rayleigh number (Ra), depth of the external concrete beam, and external wall construction materials on the flow and heat transfer characteristics were the main focus of the investigation. The Shear stress transport k–ω turbulence model is implemented to calculate air-flow velocities and temperatures in a steady, turbulent, two-dimensional conjugate natural convection heat transfer inside an attic. The governing equations were solved by employing the line-by-line tri-diagonal matrix algorithm (TDMA) control volume method. For Ra ranging from 10⁷ to 10¹⁰, steady-state results of the streamline and temperature contours in addition to local and mean Nusselt numbers at all surfaces of the cavity were obtained. The results show that the values of Ra, attic aspect ratio and the composite wall materials have significant effect on the temperature and stream function contours within the enclosure, and the heat flux out of the room through the enclosure.     

Melting of nano-enhanced phase change material in a cavity heated sinusoidal from below: Numerical study using lattice Boltzmann method

Natural convection and melting of ice as a phase change material dispersed with copper nanoparticles are numerically investigated. Square cavity filled with nano-mixture (Cu−ice) subjected to sinusoidal temperature distributions from the hot bottom boundary. The phase change process and heat transfer are formulated and solved using the enthalpy-based lattice Boltzmann method. Home-built numerical code is developed and validated. The effect of Rayleigh number (Ra = 10⁴, 10⁵, and 10⁶) and copper nanoparticle concentration (ϕ = 0%, 1%, 3%, and 5%) on the flow characteristics and thermal performance of NePCM during the melting process is examined. According to the numerical results, the melting and charging times decrease by increasing the Rayleigh number. It is also observed that increasing the volume fraction of nanoparticle decrease melting time by up to 10%.     

Heat transfer enhancement in cubical enclosures with vertical fins

Natural convection of air in a cubical enclosure with a thick partition fitted vertically on the hot wall is numerically investigated for Rayleigh numbers of 10³–10⁶. A three dimensional convective circulation is generated, in which the cold flow sweeps the fin faces and the hot wall, with low flow blockage. The combined contributions of these faces cause heat transfer enhancements over 40% at high Rayleigh numbers and thermal conductivity ratios (R_k). These enhancements significantly exceed the ones obtained with horizontal fins. Even low R_k values cause heat transfer enhancements, except at Ra = 10⁴.     

NATURAL CONVECTION AND ENTROPY GENERATION FROM A CYLINDER WITH HIGH CONDUCTIVITY FINS

The problem of laminar natural convection from a horizontal cylinder with multiple equally spaced high conductivity fins on its outer surface was investigated numerically. The effect of several combinations of number of fins and fin height on the average effective Nusselt number was studied over a wide range of Rayleigh numbers. The results showed that there was an optimal combination of number of fins and fin height for maximum heat transfer from the cylinder for a given value of Rayleigh number. A high number of short fins slightly decreased the heat transfer from the cylinder. The calculated velocity and temperature profiles also were used to study the total entropy generation. The total entropy production was dominated by entropy generation due to thermal effects. The exception was at Ra _D = 10³ and a large cylinder diameter where entropy generation was dominated by entropy generation due to viscous effects. This information can be used to access the changes in the thermodynamic efficiency due to the addition of fins to enhance the natural convection heat transfer from a horizontal cylinder.     

Computation of a turbulent Rayleigh–Benard convection with the elliptic-blending second-moment closure

This communication reports briefly on the computational results of a turbulent Rayleigh–Benard convection with the elliptic-blending second-moment closure (EBM). The primary emphasis of the study is placed on an investigation of accuracy and numerical stability of the elliptic-blending second-moment closure for the turbulent Rayleigh–Benard convection. The turbulent heat fluxes in this study are treated by the algebraic flux model where the molecular dissipation rate of turbulent heat flux is included. The model is applied to the prediction of the turbulent Rayleigh–Benard convection for Rayleigh numbers ranging from Ra = 2 × 10⁶ to Ra = 10⁹, and the computed results are compared with the previous experimental correlations, T-RANS and LES results. The predicted cell-averaged Nusselt number follows the correlation by Peng et al. [S.H. Peng, K. Hanjalic, L. Davidson, Large-eddy simulation and deduced scaling analysis of Rayleigh–Benard convection up to Ra = 10⁹, J. Turbulence 7 (2006) 1–29.] (Nu = 0.162Ra^0.286) in the ‘soft’ convective turbulence region (2 × 10⁶ ≤ Ra ≤ 4 × 10⁷) and it follows the experimental correlation by Niemela et al. [J.J. Niemela, L. Skrbek, K.R., Sreenivasan, R.J. Donnelly, Turbulent convection at very high Rayleigh numbers, Nature 404 (2000) 837–840.] (Nu = 0.124Ra^0.309) in the ‘hard’ convective turbulence region (10⁸ ≤ Ra ≤ 10⁹) within 5% accuracy. This result shows that the elliptic-blending second-moment closure with an algebraic flux model predicts very accurately the Rayleigh–Benard convection.     

NATURAL CONVECTION IN A DIFFERENTIALLY HEATED SQUARE CAVITY WITH INTERNAL HEAT GENERATION

A high-resolution, finite-difference numerical study is reported on natural convection in a square cavity. The vertical sidewatts of the cavity are differentially heated, and a uniform internal heat generation is also present. Two principal parameters are considered, the internal Rayleigh number Ra_I, which represents the strength of the internal heat generation, and the external Rayleigh number Rag, which denotes the effect due to the differential heating of the side walls. The internal Rayleigh number varies in the range 10¹⁰ Ra_I ≤ 10⁷, while the external Rayleigh number is set at Ra_E = 5 x 10⁷ for most computations. As the relative strength of the internal heat generation increases, the flows near the tap portion of the heated sidewall are directed downward. When the effect of the internal heat generation is dominant, the thermal energy leaves the system for the surroundings over the top portion of the heated wall. Only in the bottom pari of the heated wall is heat transfer directed into the system. These numerical solutions are in qualitative agreement with the available experimental measurements.     

Three-dimensional natural convection in an enclosure with a sphere at different vertical locations

Numerical calculations are carried out for the three-dimensional natural convection induced by a temperature difference between a cold outer cubic enclosure and a hot inner sphere. The immersed-boundary method (IBM) to model a sphere based on the finite volume method is used to study a three-dimensional natural convection for different Rayleigh numbers varying in the range of 10³–10⁶. This study investigates the effect of the inner sphere location on the heat transfer and fluid flow. The flow and thermal fields eventually reach the steady state for all Rayleigh numbers regardless of the sphere location. For Rayleigh numbers of 10⁵ and 10⁶, the variation of local Nusselt number of the sphere along the circumferential direction is large, showing the strong three dimensionality of the natural convection in the enclosure unlike to the cases of lower Rayleigh numbers of 10³ and 10⁴. For the highest Rayleigh number, the local peaks of the Nusselt number on the top wall of the enclosure shows the sinusoidal distribution along the circumferential direction. The flow and thermal fields, and the local and surface-averaged Nusselt numbers on the sphere and the enclosure are highlighted in detail.     

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.     

Lattice Boltzmann simulation of natural convection in tall enclosures using water/SiO2 nanofluid

Natural convection in enclosures using water/SiO₂ nanofluid is simulated with Lattice Boltzmann method (LBM). This investigation compared with other numerical methods and found to be in excellent agreement. This study has been carried out for the pertinent parameters in the following ranges: the Rayleigh number of base fluid, Ra = 10³-10⁵, the volumetric fraction of nanoparticles between 0 and 4% and aspect ratio (A) of the enclosure between 0.5 and 2. The thermal conductivity of nanofluids is obtained on basis of experimental data. The comparisons show that the average Nusselt number increases with volume fraction for the whole range of Rayleigh numbers and aspect ratios. Also the effect of nanoparticles on heat transfer augments as the enclosure aspect ratio increases.     

Laminar and turbulent natural convection from a stack of thin hollow horizontal cylinders: A numerical study

Three dimensional natural convection from a stack of thin hollow horizontal cylinders has been investigated numerically in both laminar and turbulent regimes of Rayleigh number (Ra) spanning in the range 10⁴ to 10⁸ and 10¹⁰ to 10¹³, respectively. In the present study, the length to diameter ratio (L/D) of the cylinders is considered in the range 0.5–20. Thin horizontal hollow cylinders of three, six, and ten in numbers are arranged in a triangular manner to form three different types of stacks of same length scale. The full Navier-Stokes equation along with the energy equation are solved to predict the flow pattern and heat loss from the cylinders. The present computational study is able to capture very interesting buoyancy plume structures around the stack of short and long hollow cylinders. The average Nusselt number (Nu) shows a positive dependence on Ra for all L/D. In addition, Nu for a stack of three cylinders is found to be marginally higher than the stack of six cylinders followed by ten cylinders. Empirical correlations are developed for Nu as a function of Ra and L/D, which would be beneficial to academic researchers as well as practicing engineers.