The second largest Lyapunov exponent and transition to chaos of natural convection in a rectangular cavity

In this study we have proposed an accurate and simple method to evaluate the Lyapunov spectrum. The method is suitable for any discretization method that finally expresses a governing equation system in the form of an ordinary differential equation system. The method was applied to evaluate up to the second largest Lyapunov exponents for natural convection in a rectangular cavity with heated and cooled side walls. The main results are as follows: (1) the largest and second largest Lyapunov exponents can be evaluated without any parameters that affects the exponents. (2) The second largest Lyapunov exponent makes it possible to classify quantitatively thermal convection fields into five regimes against the Rayleigh number and to clarify the transition route from steady state to chaos by identifying the first and second Hopf bifurcations. (3) The fluctuation in thermal convection fields just over the critical Rayleigh number at which Hopf bifurcation occurs can be quantitatively explained by using normalized Lyapunov vectors, associated with the computation of the Lyapunov exponents, just under the critical point.     

Double-diffusive Marangoni convection in a rectangular cavity: Transition to chaos

Transition to chaos in double-diffusive Marangoni convection in a rectangular cavity with horizontal temperature and concentration gradients is considered. Attention is restricted to the special case when the resultant thermal and solutal Marangoni effects are equal and opposing. Direct numerical simulation is used and some techniques from nonlinear dynamics are adopted to identify the different dynamic regimes. It is found that the supercritical solution branch takes a quasi-periodicity and phase locking route to chaos while the subcritical branch follows the Ruelle–Takens–Newhouse scenario. Transient intermittency in the supercritical branch is observed and physical instability mechanisms of the subcritical branch are identified.     

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…     

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

~nonSpectral finite difference schemes are developedmainly to analyze partial differential equationsnumerically with boundary conditions andlor initial...din..,[i.2]. Its main feature lies in the fact that noerror is introduced in decomposing the original set ofpartial differential equations into spectral components.Thus, given a system of decomposed simultaneousdifferential equations, supplemented with decomposedboundary I initial conditions, mathematical features tothe original padal differe…     

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.     

Computation of a turbulent natural convection in a rectangular cavity with the elliptic-blending second-moment closure

A numerical study of a turbulent natural convection in an enclosure with the elliptic-blending second-moment closure (EBM) is presented. The primary emphasis of the study is placed on an investigation of the accuracy and numerical stability of the elliptic-blending second-moment closure for the turbulent natural convection flow. The turbulent heat fluxes in this model are treated by the general gradient diffusion hypothesis (GGDH). The model is applied to the prediction of a natural convection in a rectangular cavity and the computed results are compared with the experimental data commonly used for a validation of the turbulence models. The results are also compared with those by the two-layer model, the SST model, the V2-f model and the second-moment differential stress and flux model. It is shown that the elliptic blending model predicts as good as or better than the existing models for the mean velocity and turbulent quantities although this model employs a simpler GGDH for treating the turbulent heat fluxes.     

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。     

Natural convection and entropy generation in a square cavity

A natural convection in a square cavity finds considerable interest in thermal engineering applications. However, the use of entropy generation concept enables to identify the optimum conditions for its practical application. Consequently, in the present study, natural convection in a square cavity with differential top and bottom wall temperatures is investigated. A numerical scheme using the control volume approach is introduced when discretizing the governing flow and energy equations. The study is extended to include the analysis of the entropy in the cavity. It is found that the local rise of temperature occurs at the right bottom of the cavity due to vertical circulation developed in the cavity. The entropy generation amplifies when circulation along the x-axis increases and, the entropy generation becomes minimum for a particular Rayleigh number. © 1998 John Wiley & Sons, Ltd.     

A numerical study of mixed convection in a two-sided lid-driven tall cavity containing a heated triangular block for non-Newtonian power-law fluids

The study of hydrodynamics and thermal characteristics inside a lid-driven cavity has been one of the most captivating problems in computational fluid dynamics. In this numerical work, the mixed convection phenomenon inside a two-dimensional, tall lid-driven cavity with top and bottom lids moving in opposite directions, +x and –x, respectively, has been explored for non-Newtonian power-law fluids. The cavity contains a uniformly heated equilateral triangular obstacle at its geometric center.  Numerical experimentation is performed for a range of flow governing parameters, such as aspect ratio (0.25, 0.5, and 0.75), Prandtl number (1, 50, and 100) Richardson number (0.1, 1, and 10), power-law index (0.6–1.4) and Grashof number of 10⁴. The physical perceptions of the cavity are explained by using streamline and isotherm contours. The fluid movement is limited adjacent to the moving wall concerning the Richardson number at the lower Prandtl number. With a rise in the aspect ratio of the cavity, the flow-pattern becomes more dispersed inside the cavity. Heat transfer enhancement is observed at a lower aspect ratio equal to 0.25.     

辐射侧加热腔内自然对流传热特性研究

为了明确辐射侧加热封闭方腔内半透明流体的自然对流传热现象及规律,采用有限体积法进行数值模拟研究,分析了瑞利数和光学厚度对流场、温度场以及传热特性的影响。结果表明：与传统侧壁加热腔内自然对流相比,辐射侧加热腔内等温线和流场分布规律不一致;随着瑞利数和光学厚度增加,涡心由中心位置沿直线向辐射入射侧斜上方偏移;随着瑞利数增加,等温线变得更均匀;随着光学厚度增加,等温线变密,努塞尔数Nu与瑞利数RaT的标度律指数减小,当光学厚度增加到一定时标度律不再变化,此时传热标度律与传统恒壁温侧加热腔内自然对流相当,满足Nu~Ra0.29T。     

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.     

Lattice Boltzmann simulation of natural convection in an open ended cavity

Natural convection in an open ended cavity is simulated using Lattice Boltzmann Method (LBM). The paper is intended to address the physics of flow and heat transfer in open end cavities and close end slots. The flow is induced into the cavity by buoyancy force due to a heated vertical wall. Also, the paper demonstrated that open boundary conditions used at the opening of the cavity is reliable, where the predicted results are similar to conventional CFD method (finite volume method, FVM) predictions. Prandtl number (Pr) is fixed to 0.71 (air) while Rayleigh number (Ra) and aspect ratio (A) of the cavity are changed in the range of 10⁴–10⁶ and of 0.5–10, respectively. It is found that the rate of heat transfer deceases asymptotically as the aspect ratio increases and may reach conduction limit for large aspect ratio. The flow evaluation in the cavity starts with recirculation inside the cavity, as the time proceeds the flow inside the cavity communicates with the ambient.     

Numerical simulation of turbulent natural convection of nanofluid with thermal radiation inside a tall enclosure under the influence of magnetohydrodynamic

In the present study, the natural convective heat transfer in the turbulent flow of water/CuO nanofluid with volumetric radiation and magnetic field inside a tall enclosure has been numerically investigated. The thermophysical properties of nanofluid have been considered variable with temperature and the effects of Brownian motion of nanoparticles have been considered. The main objective of this work is an investigation of the effect of using water/CuO nanofluid and presence of magnetic field on turbulent natural convection in three types of enclosures (vertical, inclined, and horizontal) by considering the volumetric radiation. The governing equations on turbulent flow domain under the influence of the magnetic field and by considering the combination of volumetric radiation and natural convection have been solved by a coupled algorithm. For validating the present research, a comparison has been carried out with the laminar natural convection flow under the influence of the magnetic field and radiation effects and also, the natural turbulent convection flow of previous studies and a proper coincidence has been achieved. The results indicated that by increasing volume fraction and Hartmann number the average Nusselt number enhances and reduces, respectively. By adding 1% CuO nanoparticles to the base fluid, heat transfer improves from 10.59% to 17.05%. However, by increasing the volume fraction from 1% to 4%, heat transfer improves from 1.35% to 4.90%. By increasing Hartmann number from 0 to 600, heat transfer reduces from 9.29% to 22.07%. Also, the results show that the ratio of deviation angle of the enclosure to the horizontal surface has considerable effects on heat transfer performance. Therefore, in similar conditions, the inclined enclosure with a deviation angle of 45° compared to the vertical and horizontal enclosure has better thermal performance.     

Evolution to chaotic mixed convection in a multiple ventilated cavity

The characteristics of transition from laminar to chaotic mixed convection in a two-dimensional multiple ventilated cavity is analyzed in this paper. The horizontal air streams enter the cavity from the two inflow-openings near the top of both vertical walls, while the outflow openings are near the bottoms of both vertical walls. The results obtained for a range of the Richardson number, Ri, from 0.01 to 5 at Pr = 0.71, the Reynolds number, Re, from 1000 to 2500 and the inlet flow angle, φ, based on 0°, 20°, 45° and 70°. The results show that, as Ri increases, the solution may exhibit a change from steady-state to periodic oscillation, and then to non-periodic oscillatory state. However, the flow inside the cavity becomes steady-state again as Richardson number increases further. The results also show that the effect of inlet flow angle on the oscillations of mixed convection is evident, the configuration with φ = 0° is the most unstable among the four values of φ. The non-periodic oscillatory solution at Re = 2500 is studied by means of phase portraits, correlation dimension, Kolmogorov entropy and Lyapunov exponents to detect chaos. The phase portraits show the evolution of the attractor from a stable fixed point to a limited cycle to chaos, and finally, to a stable fixed point again, and the correlation dimension, Kolmogorov entropy and the largest Lyapunov numbers all show that the behavior of mixed convection in this dynamical system lies on a low-dimensional chaotic attractor according to the non-periodic oscillatory solution.     

Effects of diameter ratio of adiabatic circular cylinder and tilt angle on natural convection from a square open tilted cavity

A numerical analysis is carried out to study the performance of natural convection inside a square open tilted cavity filled with air. An adiabatic circular solid cylinder is placed at the center of the cavity and the sidewall in front of the breathing space is heated by a constant heat flux. The top and bottom walls are kept at the ambient constant temperature. Two‐dimensional forms of Navier–Stokes equations along with the energy equations are solved using the Galerkin finite element method. Results are obtained for a range of Grashof numbers from 10³ to 10⁶ at Pr = 0.71 while the tilt angle varies from 0 to 45° and the diameter ratio of the cylinder is considered to be 0.2, 0.3, and 0.4 with constant physical properties. The parametric studies for a wide range of cylinder diameter ratios and cavity tilt angles show significant features of the present problem in terms of stream functions and temperature profiles. The computational results indicate that the heat transfer coefficient is strongly influenced by the above governing parameters. It is also found that the average Nusselt number decreases when the diameter ratio increases. © 2012 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley Online Library ( wileyonlinelibrary.com/journal/htj ). DOI 10.1002/htj.21001     

Energy and entropy analysis in a square cavity with protruding body: effects of protruding body aspect ratio

In the present study, the simulation of mixed convection in a square cavity with protruding body having different aspect ratios is carried out. The governing flow and energy equations are solved numerically using a control volume approach. Air is used as fluid in the cavity while steel is considered as protruding body. To investigate the heat transfer characteristics due to different aspect ratios of the protruding body, Stanton number (St) variation with the aspect ratio is considered. The entropy analysis is carried out to determine the irreversibility generated in the cavity for different aspect ratios. The heat transfer to irreversibility ratio is determined for each aspect ratio. It is found that the aspect ratio influences the heat transfer characteristics and the irreversibility generated in the cavity, in which case, the heat transfer increases at high aspect ratio while the irreversibility reduces. Moreover, heat transfer to irreversibility ratio improves considerably at an aspect ratio of 3. Copyright © 2002 John Wiley & Sons, Ltd.     

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.     

A numerical investigation of transient natural convection heat transfer of aqueous nanofluids in a differentially heated square cavity

Transient natural convection heat transfer of aqueous nanofluids in a differentially heated square cavity is investigated numerically. The effective thermal conductivity and dynamic viscosity of nanofluids are predicted by using the proposed models that take the contribution of Brownian motion of nanoparticles into account. Three different Rayleigh numbers and five different volume fractions of nanoparticles are considered. The development of natural convection is presented through the evolutions of the average Nusselt number along the cold side wall. The predicted flow development times and time-averaged Nusselt numbers are scaled with Rayleigh number. In addition, the time-averaged Nusselt numbers are presented in terms of volume fraction of nanoparticles. It is shown that at constant Rayleigh numbers, the time-averaged Nusselt number is lowered with increasing volume fraction of nanoparticles.