Treatment of turbulent heat fluxes with the elliptic-blending second-moment closure for turbulent natural convection flows

In this paper a comparative study on the treatment of the turbulent heat fluxes with the elliptic-blending second-moment closure for natural convection flows is presented. Three different cases for treating the turbulent heat fluxes are considered. Those are the generalized gradient diffusion hypothesis (GGDH), the algebraic flux model (AFM) and the differential flux model (DFM). These models are implemented in a computer code especially designed for an evaluation of turbulent models. Calculations are performed for turbulent natural convection flows in an 1:5 rectangular cavity (Ra = 4.3 × 10¹⁰) and in a square cavity with conducting top and bottom walls (Ra = 1.58 × 10⁹). The calculated results are compared with the available experimental data. The results show that the GGDH, AFM and DFM models produce sufficiently accurate solutions for the turbulent natural convection in an 1:5 rectangular cavity where the strength of the thermal stratification is weak in a central region of the cavity. However, the GGDH model produces very erroneous solutions for the turbulent natural convection in a square cavity with conducting walls where the Rayleigh number is relatively small and the thermally stratified region is dominant. The AFM and DFM produce very accurate solutions for both cases without invoking any numerical problems.     

Entropy generation of turbulent double-diffusive natural convection in a rectangle cavity

Turbulent double-diffusive natural convection is of fundamental interest and practical importance. In the present work we investigate systematically the effects of thermal Rayleigh number (Ra), ratio of buoyancy forces (N) and aspect ratio (A) on entropy generation of turbulent double-diffusive natural convection in a rectangle cavity. Several conclusions are obtained: (1) The total entropy generation number (S_total) increases with Ra, and the relative total entropy generation rates are nearly insensitive to Ra when Ra ≤ 10⁹; (2) Since N > 1, S_total increases quickly and linearly with N and the relative total entropy generation rate due to diffusive irreversibility becomes the dominant irreversibility; and (3) S_total increases nearly linearly with A. The relative total entropy generation rate due to diffusive and thermal irreversibilities both are monotonic decreasing functions against A while that due to viscous irreversibility is a monotonic increasing function with A. More important, through the present work we observe a new phenomenon named as “spatial self-copy” in such convectional flow. The “spatial self-copy” phenomenon implies that large-scale regular patterns may emerge through small-scale irregular and stochastic distributions. But it is still an open question required further investigation to reveal the physical meanings hidden behind it.     

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.     

Computation of turbulent natural convection with the elliptic-blending differential and algebraic flux models

A computation of turbulent natural convection in enclosures with the elliptic-blending based differential and algebraic flux models is presented. The primary emphasis of the study is placed on an investigation of accuracy of the treatment of turbulent heat fluxes with the elliptic-blending second-moment closure for the turbulent natural convection flows. The turbulent heat fluxes are treated by the elliptic-blending based algebraic and differential flux models. The proposed models are applied to the prediction of turbulent natural convections in a 1:5 rectangular cavity and in a square cavity with conducting top and bottom walls. It is shown that both the elliptic-blending based models predict well the mean velocity and temperature, thereby the wall shear stress and Nusselt number. It is also shown that the elliptic-blending based algebraic flux model produces solutions which are as accurate as those by the differential flux model.     

A spectral finite difference analysis of natural convection in a rectangular equilateral triangle cavity

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Entropy generation in turbulent natural convection due to internal heat generation

In this study numerical predictions of entropy generation in turbulent natural convection due to internal heat generation in a square cavity are reported for the first time. Results of entropy generation analysis are obtained by solving the entropy generation equation. The values of velocity and temperature, which are the inputs of the entropy generation equation, are obtained by an improved thermal lattice-BGK model proposed in this paper. The analyzed range is wide, varying from the steady laminar symmetric state to the fully turbulent state. Distributions of entropy generation numbers, for various Rayleigh numbers, Prandtl numbers, and Eckert numbers, are given.     

Study of turbulent natural convection coupled with thermal radiation in a vertical cavity having a large form factor

In this study, a numerical simulation study of turbulent natural convection coupled with thermal radiation in a vertical cavity differentially heated and filled with air assumed as a transparent fluid was carried out. The cavity has a variable form factor which can reach large values. The vertical walls are subjected to constant temperatures (T_c and T_f), whereas the horizontal walls are assumed adiabatic. The flow inside the cavity is turbulent and turbulence was modeled by using the $K?\epsilon$ model, and to take into account of the radiative transfer, the discrete ordinate model (DO) was introduced. To solve the different equations, Ansys‐Fluent software based on the finite volume method was used. Some numerical results obtained for the Rayleigh number value of 10¹¹ have been validated by some existing results in the theory. It is found that the thermal radiation has a significant influence on the flow structure and temperature variation where the flow becomes reinforced. It accelerates the airflow inside the cavity and gives the formation of significant velocity and temperature gradients along the walls of the cavity. Taking into account of the surface, thermal radiation is essential in the correct evaluation of temperature in the cavity.     

Numerical calculation of turbulent natural convection in a cylindrical transformer enclosure

Numerical calculations of turbulent natural convection in the enclosure of the 20 kVA oil-immersed transformer model are presented. The transformer is modeled as two concentric cylinders with different heights and diameters. The correlating equation between the mean Nusselt number and the Rayleigh number can be obtained for Ra = 10⁸ to 10⁹. The thermal boundary layers are well represented in the temperature distributions along the wall of the transformer model. The flow stratification between the hot and cold walls cannot be seen in the transformer model. The turbulence eddy viscosity has its maximum at the center of the core and its maximum values at the top of the core are slightly larger than those at the bottom of the core. © 1999 Scripta Technica, Heat Transfer Asian Res, 28(6): 429–441, 1999     

Oscillatory natural convection of water-glycerin mixture in a tall rectangular cavity: Transition to the chaos

This paper presents results of a numerical investigation involving bifurcation sequence leading to chaos in natural convection inside a vertically tall rectangular cavity having an aspect ratio equal to 15 and a Prandtl number equal to 125, corresponding to water-glycerin mixture. The flow is characterized by a vertical stratification of the temperature field for Grashof numbers greater than or equal to 200 that is outside the conduction regime and it is stationary monocellular up to a critical value Gr_c = 2800 where a periodic oscillatory regime appears. As Grashof number is increased, a transition from a steady periodic bicellular flow to an oscillatory multicellular flow, with 2 main central cells and 2 secondary cells, occurs. The regime remains periodic until Gr = 3100 where there is a first appearance of the chaotic regime which extends over a narrow interval of the Grashof number delimited by Gr = 3200.     

Conjugate natural convection in a square cavity with finite thickness horizontal walls

The effect of conduction of horizontal walls on natural convection heat transfer in a square cavity is numerically investigated. The vertical walls of the cavity are at different constant temperatures while the outer surfaces of horizontal walls are insulated. A code based on vorticity–stream function is written to solve the governing equations simultaneously over the entire computational domain. The dimensionless wall thickness of cavity is taken as 0.1. The steady state results are obtained for wide ranges of Rayleigh number (10³ < Ra < 10⁶) and thermal conductivity ratio (0 < K < 50). The variation of heat transfer rate through the cavity and horizontal walls with Rayleigh number and conductivity ratio is analyzed. It is found that although the horizontal walls do not directly reduce temperature difference between the vertical walls of cavity, they decrease heat transfer rate across the cavity particularly for high values of Rayleigh number and thermal conductivity ratio. Heatline visualization technique is a useful application for conjugate heat transfer problems as shown in this study.     

Natural convection in an air-filled cavity: Experimental results at large Rayleigh numbers

A large-scale experimental setup is built and instrumented. It consists in a 4 m-high cavity with a horizontal cross-section equal to 0.86 × 1.00 m². Two opposite vertical walls are heated and cooled down; other walls (lateral walls, ceiling and floor) are made of insulating medium covered with a thin and low-emissivity film designed to minimize radiative effects into the cavity. The temperatures of active walls are imposed, constant and equally distributed around the ambient temperature in order to reduce heat losses. The temperature difference between the hot and cold walls is chosen to respect the Boussinesq approximation. Under these assumptions, Rayleigh number values up to 1.2 × 10¹¹ (ΔT = 20 °C) can be obtained. The centre-symmetry is verified on the thermal stratification. Influence of the temperature difference and of wall emissivities on the stratification parameter (dimensionless vertical temperature gradient) is discussed. Velocity measurements allow the velocity field to be obtained and provide information on flows encountered in the cavity. Temperature measurements are also carried out in the whole cavity. In the paper, a complete experimental characterization is provided: airflow inside the cavity is analyzed and the Nusselt number along the hot and the cold wall is presented.     

Double-diffusive turbulent natural convection in a porous square cavity with opposing temperature and concentration gradients

This paper presents results for coupled heat and mass transport under laminar and turbulent flow regimes in porous cavities. Two driving mechanisms are considered to contribute to the overall momentum transport, namely temperature driven and concentration driven mass fluxes. Aiding and opposing flows are considered, where temperature and concentration gradients are either in the same direction or of different sign, respectively. Modeled equations are presented based on the double-decomposition concept, which considers both time fluctuations and spatial deviations about mean values. Turbulent transport is accounted for via a macroscopic version of the k–ε model. Variation of the cavity Nusselt and Sherwood numbers due to changes on N, where N is the ratio of solute to thermal Grashof numbers, is presented. Results indicate that for adding cases, mass and heat transfer across the cavity are enhanced faster than for cases with opposing temperature and concentration gradients. For the conditions here investigated, the use a turbulence model gave results for Nu and Sh that were nearly double when compared with laminar results for the same conditions.     

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.     

Analysis of Natural Convection in an Elliptical Cavity,Using a Dini Expansion

INTRODUCTIONNaturalconvectioninanellipticalcavityheatedfrombelowwastreatednumericallybyM..him.ull],usingaFourierspectralfinitedifferencemethod,anditseffectivenesswasshownasinRef.[2].AlsopossibilityoftheextensiontovarioustypesofboundaryconditionsforthespectralfinitedeferenceschemewasproposedinReL[2-3].HerethespectralfinitedifferenceschemeisbeingextendedtoadoptDimexpansions(akindofBesselexpansions).NUMERICALANAlySISBasicAssumptionsbleatedistransiellttwo--dimensionalnaturallaminarco…     

Conditional moment closure prediction of soot formation in turbulent, nonpremixed ethylene flames

Results obtained from incorporating a semiempirical soot model into a first-order conditional moment closure (CMC) approach to modeling turbulent nonpremixed flames of ethylene and air are presented. Soot formation is determined via the solution of two transport equations for soot mass fraction and particle number density, with acetylene and benzene employed as the incipient species responsible for soot nucleation, and the concentrations of these species calculated using a detailed gas-phase kinetic scheme involving 463 reactions and 70 species. The study focuses on the influence of differential diffusion of soot particles on soot volume fraction predictions. The results of calculations are compared with experimental data for three sooting ethylene flames and, in general, predictions of mixing and temperature fields within the three flames show good agreement with data. Soot volume fraction predictions are found to be in significantly better accord with data when differential diffusion is accounted for in the CMC-based soot model, supporting the importance of such effects in sooting flames, as previously noted by Kronenburg et al. in relation to methane combustion. Overall, the study demonstrates that the CMC-based soot model, when used in conjunction with a model of differential diffusion effects, is capable of accurately predicting soot formation in turbulent nonpremixed ethylene-air flames.     

Numerical study on natural convection in a square enclosure containing a rectangular heated cylinder

In this paper, the natural convection in a square enclosure with a rectangular heated cylinder is investigated via the lattice Boltzmann method. A detailed study is conducted on the effect of the cylinder width and the Rayleigh number on the fluid flow and heat transfer. The flow structures and heat transfer patterns are classified into eight buoyant regimes, i.e., four steady regimes, two periodic regimes, one multiple periodic regime, and one chaos regime, two of which are reported for the first time.     

Study of conjugate natural convection between vertical coaxial rectangular cylinders

Laminar natural convection between two coaxial vertical rectangular cylinders is numerically studied in this work. The outer cylinder is connected with vertical rectangular inlet and outlet pipes. The inner cylinder dissipates volumetric heat. The fluid flow and heat transfer characteristics between the cylinders are analyzed in detail for various Grashof numbers. The heat transfer rates on the individual faces of the inner cylinder are reported. The bottom face of the inner cylinder is found to associate with much higher heat rates than those of the other faces. The average Nusselt number on bottom face is more than 2.5 times of the Nusselt number averaged on all the faces. At a given elevation, local Nusselt number on the inner cylinder faces increases towards cylinder edges. The effect of thermal condition of the walls of outer cylinder, inlet and outlet on the natural convection is analyzed. The thermal condition shows strong qualitative and quantitative impact on the fluid flow and heat transfer. The variation of induced flow rate, dimensionless maximum temperature and average Nusselt numbers with Grashof number is studied. Correlations for dimensionless buoyancy-induced mass flow rate and temperature maximum are presented.     

含加热圆管方腔内自然对流的数值研究

采用数值计算方法对含不同直径圆管以及相同直径圆管位置不同方腔内的层流自然对流进行了研究。以冷热壁面温度差为基准的瑞利数Rn为10^6,以圆管壁面热流密度为基准的Ra为10^8。计算结果表明,当圆管处于方腔中间位置时,随着圆管直径的增大,圆管表面局部努塞尔数呈减小趋势。当圆管直径不变时,由于在不同位置处浮力作用的强弱不同,随着圆管在方腔内位置的改变,方腔内流场结构和温度场分布也会发生变化。整个计算结果可为工程设计提供参考。     

A numerical study on entropy generation induced by turbulent forced convection in curved rectangular ducts with various aspect ratios

The present paper analyzes the entropy generation induced by turbulent forced convection in a curved rectangular duct with external heating by numerical methods. The problem is assumed as steady, three-dimensional and turbulent. The flow features, including the secondary flow motions, the distribution of local entropy generation as well as the overall entropy generation in the whole flow fields, are analyzed. For a baseline case with Re = 20,000, external heat flux q? = 0.112 and aspect ratio γ = 1, the results show the entropy generation induced by the frictional irreversibility concentrates within the regions adjacent to the duct walls, whereas the entropy generation resulted from the heat transfer irreversibility only significantly occurs near the outer wall of the duct where the external heat flux imposed. Except the baseline case, two additional cases with aspect ratio equal to 0.25 and 4 are calculated. Through the comparison of the three aspect-ratio cases, it is seen that the resultant entropy generations in the flow fields for the three cases are all dominated by the frictional irreversibilities. Among the three aspect-ratio cases, the resultant entropy generation is minimal in the γ = 1 case. Accordingly, the case with γ = 1 is concluded to be the optimal aspect ratio under the current flow condition based on the minimal entropy generation principle.