Convection In Square Cavity Research Articles

Moment-based boundary conditions for lattice Boltzmann simulations of natural convection in cavities

We study a multiple relaxation time lattice Boltzmann model for natural convection with moment-based boundary conditions. The unknown primary variables of the algorithm at a boundary are found by imposing conditions directly upon hydrodynamic moments, which are then translated into conditions for the discrete velocity distribution functions. The method is formulated so that it is consistent with the second order implementation of the discrete velocity Boltzmann equations for fluid flow and temperature. Natural convection in square cavities is studied for Rayleigh numbers ranging from 103 to 108. An excellent agreement with benchmark data is observed and the flow fields are shown to converge with second order accuracy.

Numerical Study of Natural Convection in Square Cavity with Inner Bodies Using Finite Element Method

A numerical study of heat transfer problem by natural convection of a fluid inside a square cavity with two inner bodies is presented. This subject is of great interest in the engineering area, mainly in applications involving development of heat exchangers and cooling or heating systems of bodies by natural convection mechanism. Two cases have been studied. The inner bodies are square in case 1 and circular in case 2. In both cases, the bodies are solid and thermally conductive, the cavity lower and upper horizontal surfaces are isothermal with high temperature Th and low temperature Tc, respectively. Both vertical surfaces are adiabatic. A FORTRAN code using Finite Element Method (FEM) is developed to simulate the problem and solve the governing equations. The distributions of stream function, ψ, dimensionless temperature, θ, and vorticity, ω, are determined. Heat transfer is evaluated by analyzing the behavior of the average Nusselt number. The Grashof number and thermal diffusivity ratio are considered in range from 2 × 104 to 105 and from 0.1 to 100, respectively. The fluid is air with Prandtl number fixed in 0.733.

A three dimensional lattice model for thermal compressible flow on standard lattices

A three-dimensional double distribution function thermal lattice Boltzmann model has been developed for simulation of thermal compressible flows in the low Mach number limit. Both the flow field and energy conservation equation are solved by LB approach. A higher order density distribution function on standard lattices is used to solve the flow field, while an energy distribution function is employed to compute the temperature field. The equation of state of thermal perfect gas is recovered by higher order Hermite polynomial expansions in Navier–Stokes–Fourier equations. The equilibrium distribution functions of D3Q15, D3Q19 and D3Q27 lattices are obtained from the Hermite expansion. They exhibit slight differences originating in differences in the discrete lattice symmetries. The correction terms in LB models for third order derivation are added using an external force in orthogonal polynomials form. Present models are successfully assessed considering several test cases, namely the thermal Couette flow, Rayleigh–Bénard convection, natural convection in square cavity and a spherical explosion in a 3D enclosed box. The numerical results are in good agreement with both analytical solution and results given by previous authors.

Role of various moving walls on energy transfer rates via heat flow visualization during mixed convection in square cavities

Mixed convection in closed cavities are important for various processing industries especially those associated with conservation of energy. Finite element based simulations are carried out for two cases based on the motion of the horizontal wall(s) (cases 1a–1d) or vertical wall(s) (cases 2a–2c). Heat flow distribution within the cavity enclosed by isothermally hot bottom wall, cold side walls and insulated top wall is analyzed for various fluids with Prandtl number, Pr = 0.026 and 7.2, Reynolds number, Re = 10–100 and Grashof number, Gr = 103–105. The direction of motion of wall(s) plays a significant role on the fluid flow field at Pr = 0.026, Gr = 103 and Re = 10 due to dominant forced convection for both horizontally (cases 1a–1d) or vertically (cases 2a–2c) moving wall(s). At Pr = 7.2, Gr = 105 and Re = 100, multiple convective heatline cells are observed for cases 1a–1d. It is found that, the strength of fluid or heatline circulation cells is less at Re = 100 compared to Re = 10 for cases 2a–2c due to weak buoyancy force at high Re. Energy transfer rates are assessed via local and average Nusselt numbers for cases 1a–1d and 2a–2c.

Magneto-natural convection in square cavities with a source-sink pair on different walls

Magnetohydrodynamic natural convection fluid flow and heat transfer in a square enclosure with a pair of source and sink on its walls, filled with liquid Gallium fluid with Prandtl number of 0.02 has been investigated numerically. The heat source and heat sink are maintained at a constant temperature T_{h} and T_{c} , respectively with T_{h} > T_{c} . By variation of relative location of the heat source and sink on the walls of the enclosure, five different cases are generated. The governing equations written in terms of the primitive variables are solved numerically using the finite volume method and the SIMPLER algorithm. Using the developed code, a parametric study is performed, and the effects of the Rayleigh number, the Hartman number, and the locations of the source and sink on the fluid flow and heat transfer inside the enclosure are investigated. The results show that the flow and temperature distributions inside the enclosure are affected by the strength of the magnetic field, the Rayleigh number, and the relative location of the heat source and sink. The magnetic field decreases the rate of heat transfer, suppresses the convection heat transfer, and tends to slows down the flow velocity in the cavity. Moreover in some cases the magnetic field changes the flow pattern inside the enclosure.

Parameter-free numerical method for modeling thermal convection in square cavities in a wide range of Rayleigh numbers

Some computational results for the two and three-dimensional Davis benchmark are presented. This benchmark represents thermal convection in a square (cubical) cavity with vertical active walls in a wide range of Rayleigh numbers (104to 1014), which covers both laminar and highly turbulent flows. A turbulence model with parameters that depend on a Rayleigh number and require adjustment is usually used to describe turbulent flows. An alternative is Direct Numerical Simulation (DNS) methods, but they demand extremely large computational grids. Recently there has been an increasing interest in DNS methods with incomplete resolution, which are able to provide in some cases acceptable results without resolving Kolmogorov scales. On the basis of such an approach the so-called parameter-free computational techniques have been developed. These methods cover wide range of Rayleigh numbers and allow computing various integral properties of heat transport on relatively coarse computational grids. In this paper, a new numerical method based on the CABARET scheme is proposed for solving Navier-Stokes equations with Boussinesq approximation. This turbulent model-free technique includes no additional parameters and has a second-order approximation scheme in time and space on uniform and non-uniform computational grids with minimal computational stencil. Testing of the technique on the Davis benchmark and the sequence of refined grids shows that the method allows one to compute integral heat fluxes with a high degree of accuracy both for laminar and highly turbulent flows. For the Rayleigh numbers up to 1014, a several percent accuracy has been achieved on an extremely coarse grid consisting of 20×20 cells refined toward boundary. There is no a definite and comprehensive explanation of this computational phenomenon. Cautious optimism exists regarding the perspectives of using the new method of thermal convection computations for low Prandtl numbers typical of liquid metals.

Natural convection in square cavity with heat generating baffles

A numerical study of natural convection in an air filled differentially heated square cavity containing two vertical thin heat generating baffles is presented. The baffles are kept centrally separated by a distance d in the cavity and assumed to generate heat uniformly. A variety of thermal boundary conditions are imposed on the vertical walls while the horizontal walls are considered to be insulated. The Grashof number and Prandtl number are fixed at 107 and 0.71, respectively. A steady state results for the vorticity–stream function formulation are presented in the form of streamline and isotherm plots and average Nusselt numbers. The effects of various thermal boundary conditions on the vertical walls and different position of the baffles on heat transfer and fluid flow are investigated.

Numerical simulations of the effect of an isotropic heat field on the entropy generation due to natural convection in a square cavity

Entropy generation associated with laminar natural convection in an infinite square cavity, subjected to an isotropic heat field with different intensities; was numerically investigated for different values of Rayleigh number. The numerical work was carried out using, an in-house CFD code written in FORTRAN, which discretizes non-dimensional forms of the governing equations using the finite volume method and solves the resulting system of equations using Gauss-Seidal method utilizing a TDMA algorithm. Proper code validation was undertaken in order to establish the entropy generation calculations. It was found that the increase in the isotropic heat field intensity resulted in a corresponding exponential increase of the entropy augmentation number, and promoted high values of Bejan number within the flow. The entropy generation due to heat transfer was approximately one order of magnitude higher than the entropy generation due to fluid friction. The spatial uniformity of the Bejan number was more sensitive to the change in Rayleigh number than to the heat field intensity. The thermodynamic penalty of the isotropic heat field is shown by means of global integrals of the entropy source terms over the entire flow domain.

Entropy generation and natural convection in square cavities with wavy walls

The aim of the present work is to study the entropy generation in the natural convection process in square cavities with hot wavy walls through numerical simulations for different undulations and Rayleigh numbers, while keeping the Prandtl number constant. The results show that the hot wall geometry affects notably the heat transfer rate in the cavity. It has been found in the present numerical study that the mean Nusselt number in the case of heat transfer in a cavity with wavy walls is lower, as compared to heat transfer in a cavity without undulations. Based on the obtained dimensionless velocity and temperature values, the distributions of the local entropy generation due to heat transfer and fluid friction, the local Bejan number, and the local entropy generation are determined and plotted for different undulations and Rayleigh numbers. The study is performed for Rayleigh numbers 103 < Ra < 105, irreversibility coefficients 10−4 < φ < 10−2, and Prandtl numbers Pr = 0.71. The total entropy generation is found to increase with increasing undulation number.

A thermal non-equilibrium approach for 2D natural convection due to lateral heat flux: Square as well as slender enclosure

This paper reports the influence of local thermal non-equilibrium state between solid porous matrix and saturated fluid on natural convection in enclosure. The two dimensional steady flow is induced by constant heat flux on side walls of the enclosure, when both horizontal walls are insulated. The governing equations are solved numerically by ADI method and analytically by using parallel flow assumption valid for slender enclosure. A comparative study has been made between convection in square cavity and the same in slender enclosure. Numerical experiments indicate that in comparison with square cavity, where a sharp decrease of local heat transfer rate for fluid (Nuf) takes place up to a certain small value of interface heat transfer coefficient (H), in slender enclosure a smooth decrease of Nuf has been observed in the entire range of H when conductivity ratio (γ) is very small. However, for relatively high values of γ (e.g., γ=10), Nuf is almost independent of H in both geometries. For a given γ, when the value of H is relatively very high, up to a certain value of Ra the difference between both solid as well as fluid temperature rates is negligible. Corresponding temperature contours of two phases becomes almost identical in magnitude as well as patterns point of view, which indicates that equilibrium state between two phases is achieved. Finally, in equilibrium state maximum heat transfer rate used to take place at a certain value (Ao) of aspect ratio, lying in between 1 and 1.5, and is almost independent of all equilibrium parameters. In case of non-equilibrium state, Ao acts as an increasing function of γ.

Transient Natural Convection in Porous Square Cavity Heated and Cooled on Adjacent Walls

Transient natural convection in a square cavity filled with a porous medium is studied numerically. The cavity is assumed heated from one vertical wall and cooled at the top, while the other walls are kept adiabatic. The governing equations are solved numerically by a finite difference method. The effects of Rayleigh number on the initial transient state up to the steady state are investigated for Rayleigh number ranging from 10 to 2 × 102. The evolutions of flow patterns and temperature distributions were presented for Rayleigh numbers, Ra = 102 and 103. It is observed that the time taken to reach the steady state is longer for low Rayleigh number and shorter for high Rayleigh number.

Entropy generation due to natural convection in discretely heated porous square cavities

Optimization of industrial processes for higher energy efficiency may be effectively carried out based on the thermodynamic approach of entropy generation minimization (EGM). This approach provides the key insights on how the available energy (exergy) is being destroyed during the process and the ways to minimize its destruction. In this study, EGM approach is implemented for the analysis of optimal thermal mixing and temperature uniformity due to natural convection in square cavities filled with porous medium for the material processing applications. Effect of the permeability of the porous medium and the role of distributed heating in enhancing the thermal mixing, temperature uniformity and minimization of entropy generation is analyzed. It is found that at lower Darcy number ( Da), the thermal mixing is low and the heat transfer irreversibility dominates the total entropy generation. In contrast, thermal mixing is improved due to enhanced convection at higher Da. Friction irreversibility is found to dominate the total entropy generation for higher Prandtl number ( Pr) fluids at higher Da, whereas the heat transfer irreversibility dominates the total entropy generation for lower Pr fluids. Based on EGM analysis, it is established that larger thermal mixing at high Darcy number may not be always recommended as the total entropy production is quite large at high Darcy number. Overall, it is found that the distributed heating methodology with multiple heat sources may be an efficient strategy for the optimal thermal processing of materials.

Laminar Natural Convection in a Square Cavity with a Partition on the Heated Vertical Wall

The laminar natural convection in an air-filled square cavity with a partition on the heated vertical wall was experimentally investigated. Temperature measurements and flow visualizations were performed for cases with heated and cooled vertical walls (corresponding to a global Grashof number Gr H of approximately 1.4 × 108) and non dimensional top wall temperatures θ T of 0.57 (insulated) to 2.3. Experiments were performed with an aluminum partition with non dimensional height H P /H of 0.0625 and 0.125 attached to the heated vertical wall at y/H = 0.65 and 0.95. The blockage effect and/or the thermal effect of the partition resulted in changes to the temperature and flow fields but were mainly limited to the vicinity of the partition. For the cases with the heated top wall, the change in the height of the partition at y/H = 0.95 resulted in changes to the ambient temperature outside the boundary layer due to the reduction of the size of the recirculating flow in the corner region. The changes in the partition height and the top wall temperature affected the blockage effect of the partition, resulting in the local Nusselt number near the corner region to be affected. The local Nusselt number over most of the heated vertical wall of the partitioned cavity (y/H < 0.7) was correlated to the local Rayleigh number in the form Nu = C · Ra n .

Fluid flow and heat transfer characteristics of natural convection in square cavities due to discrete source–sink pairs

Laminar natural convection in a two-dimensional square cavity of side length H due to two and three discrete heat source–sink pairs on the vertical sidewalls was numerically investigated. Main efforts were focused on the size and arrangement effects of the sources and sinks on the fluid flow and heat transfer characteristics. The sizes of sources and sinks were, respectively, H/4 for two sources–sinks pairs and H/6 for three sources–sinks pairs. The arrangement of the sources and sinks changes from the separated to staggered modes, i.e., first separately located on two sidewalls, then alternately located on two sidewalls, and finally alternately located on one sidewall. The fluid flow, heat transfer, and heat transport characteristics were illustrated by streamlines, isotherms and averaged Nusselt number, and heatlines. It was found that the total heat transfer was closely related with the number of eddies in the enclosure. When the sources and sinks were split into smaller segments and/or arranged in a staggered mode, the number of eddies in the enclosure would increase and hence heat transfer was augmented.

Experimental analysis of natural convection in square cavities heated from below with 2D-PIV and holographic interferometry techniques

A numerical and experimental analysis was performed to study the natural convection heat transfer in a square cavity heated from below and cooled by the sidewalls. The enclosure was filled with air ( Pr = 0.71) and a discrete heater was mounted on its lower surface; the effect of three different positions was evaluated. The air temperature distribution and the Nusselt numbers at different Rayleigh numbers on the heated strip were measured by an holographic interferometry thanks to the real-time and the double-exposure technique. The double-exposure technique was performed at steady-state and it was used to obtain the isothermal lines in the cavity at different Rayleigh numbers; while the real-time technique was used to control the presence of the plume oscillations and to determinate the achievement of the steady-state. A 2D particle image velocimetry (PIV) was utilized to measure the velocity fields at the same Rayleigh numbers. In particular we analysed the distribution of the velocity vectors and their modulus inside the cavity. The convective phenomenon was studied and the Nusselt numbers were presented as well as the Rayleigh numbers analysed. Moreover experimental and numerical correlations were determined for each position analysed to connect the Rayleigh numbers with the Nusselt numbers. Measured quantities were compared with the numerical results which were obtained with the finite volume code Fluent 6.2.16.

Natural Convection in a Square Cavity with Spatial Side-Wall Temperature Variation

ABSTRACT Laminar natural convection in a two-dimensional square cavity filled with a pure air (Pr = 0.71) is studied numerically in the present article with nonuniform side-wall temperature. The heated vertical wall is assumed to have spatial sinusoidal temperature variations about a constant mean value, which is higher than the cold side-wall temperature, while the top and the bottom walls are adiabatic. A finite-volume method is used to solve numerically the nondimensional governing equations in the vorticity–stream function formulation. The effects of the amplitude and the wave number of the heated side-wall temperature variation on the natural convection in the cavity are investigated. It is found that the average Nusselt number varies based on the hot-wall temperature. It increases with an increase in the amplitude, while the maximum average Nusselt number occurs at the wave number of k = 0.7 for Rayleigh number range 103 ≤ Ra ≤ 106. It is found that the values of maximum fluid circulation occur at a similar wave number, which produces maximum heat transfer for small values of Ra, while it occurs at higher values of wave number at high Ra.

Numerical simulation of two-dimensional piston effect and natural convection in a square cavity heated from one side

Piston effect and natural convection in two-dimensional square cavity heated from one side are solved numerically in which it is assumed that the temperature of the left-hand-side wall is suddenly increased at t=0. According to the piston effect, the bulk temperature suddenly rises and convection cells are generated near not only hot wall but also cold wall. However, the rise of temperature due to natural convection has no influence on the piston effect, because the difference in time scale between the natural convection and piston effect is very large

Transient free convection in a square cavity filled with a porous medium

Transient free convection in a two-dimensional square cavity filled with a porous medium is numerically studied in this paper. The left vertical wall is suddenly heated to a constant temperature T h, while the right wall is suddenly cooled to a constant temperature T c by equal amount relative to an initially uniform temperature distribution. Both the horizontal walls are adiabatic. The finite volume numerical method is used to solve the non-dimensional governing equations. The results are obtained for the initial transient state up to the steady state, and for Rayleigh number values of 10 2–10 4. It is observed that the average Nusselt number showing an undershoot during the transient period and that the time required to reach the steady state is longer for low Rayleigh number and shorter for high Rayleigh number.

To the influence of uniform magnetic field on thermomagnetic convection in square cavity

The influence of a uniform outer magnetic field on natural convection in square cavity is studied. It is discovered that the angle between the directions of temperature gradient and magnetic field influences the convection structure and, as a result, the intensity of heat flux.

Numerical analysis of laminar free convection in square cavity containing heat generating Kr-85 gas.

Numerical analysis of laminar free convection in square cavity containing heat generating Kr-85 gas.