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.     

Impingement heat transfer of a single thermal plume on the upper wall

We present numerical calculations of the generation, growth and impingement of a thermal plume in a two-dimensional buoyancy induced flow. Numerical values are obtained for the aspect ratios H/W=1/4, 3/8, 1/2, the Grashof numbers Gr=10⁴, 10⁵, and the Prandtl number Pr=170. Impingement heat transfer on the upper wall is evaluated at various times. Numerical results show that before a thermal plume impinges on the upper heated wall, the thermal conduction layer, which is the stable stratification, near the upper wall becomes thinner and the local heat transfer peaks. The local Nusselt number approaches the steady condition after the impingement of a thermal plume. Additionally, under certain conditions the stream function takes a symmetrical form of two ellipses.     

Double-Diffusive Natural Convection in a Mixture-Filled Cavity with Walls' Opposite Temperatures and Concentrations

ABSTRACT

This paper deals with natural convection flows evolving inside an ended and differentially heated cavity, which is filled either with an air or an air–CO₂ mixture. The investigation was conducted through the laminar regime to analyze buoyancy ratio changes' effect on heat and mass transfers both in aiding and opposing flows. The thermal Rayleigh number was varied from 10³ to 10⁷. Streamlines, isotherms, iso-concentrations, and local and average Nusselt and Sherwood numbers are provided to demonstrate the convective flow induced. The governing equations are solved by finite volume method using SIMPLEC algorithm to handle the pressure–velocity coupling. The buoyancy ratio effect on dynamic, thermal, and mass fields is noteworthy, exhibiting both the competition between thermosolutal forces and fields' stratification. From the results, it turned out that, in general, when the buoyancy ratio is: (1) positive, thermosolutal buoyancy forces are cooperative, (2) nil, solutal buoyancy forces are weak and the flow is merely thermoconvective, (3) negative and greater than ?1, buoyancy effects are competing and thermal convection dominates, (4) ?1, buoyancy effects are canceled and heat and mass transfers are driven only by diffusion, and (5) less than ?1, buoyancy forces compete with a dominant solutal convection.     

Characteristics of vortex flow in a low speed air jet impinging onto a heated disk in a vertical cylindrical chamber

An experiment combining flow visualization and transient temperature measurement is carried out to investigate the characteristics of the mixed convective vortex flow resulting from a low speed air jet impinging onto a heated horizontal circular disk confined in a vertical adiabatic cylindrical chamber. Attention is focused on the conditions leading to the onset of the inertia and buoyancy driven vortex rolls and the effects of governing nondimensional groups on the steady and time dependent vortex flow. More specifically, experiments are conducted for the jet Reynolds number varied from 0 to 1082 and Rayleigh number from 0 to 18,790 for two different injection pipes. The results show that typically the steady vortex flow in the processing chamber consists of two inertia-driven and one buoyancy-driven circular vortex rolls. The secondary inertia-driven roll only appears at high jet Reynolds numbers. At low buoyancy-to-inertia ratio Gr/Re_j² the vortex rolls are steady and axisymmetric. But at certain high Gr/Re_j² the vortex flow becomes unstable and the vortex rolls are somewhat deformed. Besides, new vortex rolls can be induced by the additional thermal rising from the heated disk and the splitting of the primary inertia-driven roll. The temporal characteristics of the time periodic vortex flows are examined in detail. In the region dominated by the new rolls the flow oscillates significantly. Finally, empirical equations are proposed to correlate the oscillation frequency of the time periodic flow, and the size and location of the vortex rolls. Furthermore, the conditions for the onset of the buoyancy driven rolls are given. A flow regime map is provided to delineate the temporal state of the vortex flow.     

Laminar natural convection in inclined open shallow cavities

Laminar steady state natural convection in inclined shallow cavities has been numerically studied. The side facing the opening is heated by a constant heat flux, sides perpendicular to the heated side are insulated and the opening is in contact with a fluid at constant temperature and pressure. Equations of mass, momentum and energy are solved using constant properties and Boussinesq approximation and assuming an approximate boundary conditions at the opening. Isotherms and streamlines are produced, heat and mass transfer is calculated for Rayleigh numbers from 10³ to 10¹⁰, cavity aspect ratio A=H/L from 1 to 0.125. The results show that flow and heat transfer are governed by Rayleigh number, aspect ratio and the inclination. Heat transfer approaches asymptotic values at Rayleigh numbers independent of the aspect ratio. The asymptotic values are close to that for a flat plate with constant heat flux. The effect of elongation of open cavities is to delay this asymptotic behavior. It is also found that the inclination angle of the heated plate is an important parameter affecting volumetric flow rate and the heat transfer.     

NATURAL CONVECTION ABOVE AN ARRAY OF OPEN CAVITIES HEATED FROM BELOW

Abstract

The effect of buoyancy on the flow and heat transfer that develop between a horizontal cold surface and an infinite two-dimensional array of open cavities heated from below is studied numerically. In earlier investigations the steady-state features of this problem were studied for the case of unbounded flow above the cavities. The resulting flow pattern was found to be symmetrical with respect to the centerlines of the cavities. In the present work it is shown that the symmetry of the flow can be destroyed due to the presence of an upper wall. The evolutionary path to steady-state flow is examined, and sustained oscillatory behavior has been observed in several cases. The solution structure is governed by five parameters, i.e., the geometric parameters A = l'/H', B = h'/H', and C = L'/H', the Rayleigh number Ra = gβ ΔT' H' ³/av, and the Prandtl number Pr = v/α. For a geometry with A = ½z, B = ¼, and C = 1, a complicated solution structure is observed upon increasing the Rayleigh number. For Ra ≤ 4 × 10³, a steady symmetric two-cell pattern is observed. This pattern becomes asymmetric for 4 × 10³ < Ra ≤ 9 × 10⁴, periodic for Ra ≤ 3 × 10⁵ and chaotic above that. The transition to periodic convection occurs at lower Rayleigh numbers with decreasing B.     

Experiments and a model of turbulent exchange flow in a vertical pipe

In this paper we present experimental measurements of buoyancy driven turbulent exchange flow in a vertical pipe (L/d ratios of 9–12). The flow is driven by an unstable density difference across the ends of the pipe, created using brine and distilled water. Away from either end, a fully developed region of turbulence exists with a linear density gradient. Using a mixing length model that accounts for the end effects, we obtain the turbulent scales and flux. The Nusselt number scales like the square root of the Rayleigh number (Nu ∼ Ra^1/2). We give an empirical relation to quantify the end effects and hence calculate the flux of the salt (NaCl) given the aspect ratio of the pipe and the overall density difference across it.     

Conjugate heat transfer in open cavities with a discrete heater at its optimized position

In this article, we determined optimum position of a discrete heater by maximizing the conductance and then studied heat transfer and volume flow rate with the discrete heater at its optimum position in open cavities. Continuity, Navier–Stokes and energy equations are solved by finite difference-control volume numerical method. The relevant governing parameters were: the Rayleigh numbers from 10⁶ to 10¹², the Prandtl number, Pr = 0.7, the cavity aspect ratio, A = H/L from 0.5 to 2, the wall thickness l/L from 0.05 to 0.15, the heater size h/L from 0.15 to 0.6, and the conductivity ratio k_r from 1 to 50. We found that the global conductance is an increasing function of the Rayleigh number, the conductivity ratio, and a decreasing function of the wall thickness. Best thermal performance is obtained by positioning the discrete heater at off center and slightly closer to the bottom. The Nusselt number and the volume flow rate in and out the open cavity are an increasing function of the Rayleigh number and the wall thickness, and a decreasing function of the conductivity ratio. The Nusselt number is a decreasing function of the cavity aspect ratio and the volume flow rate is an increasing function of it.     

Onset of secondary flow and enhancement of heat transfer in horizontal convergent and divergent channels heated from below

Experiments for the onset and development of the buoyancy driven secondary air flow and enhancement of heat transfer in a horizontal convergent and a divergent channel have been carried out. The bottom wall of the channel is horizontal and heated uniformly, while the top wall is insulated and inclined with respect to the horizontal plane so as to create a convergence angle of 3° for the convergent channel, or a divergence angle of 3° for the divergent channel. The aspect ratio (width to height) and the ratio of channel length to height at the entrance of the channel is 6.67 and 15, respectively. The Reynolds number ranges from 200 to 2000, the buoyancy parameter, Gr/Re², from 2.5 to 907 and Pr of the air flow is 0.7. Flow structure inside the channel is visualized by injecting smoke at the inlet flowing along the bottom wall. The onset of secondary flow appearing as transverse instability wave and onset of initial protrusion of the bottom heated layer are identified. Secondary flow structures observed are somewhat different from the case in the parallel-plate channel. This is attributed to the destabilization effect of the deceleration in the divergent channel which results in a much earlier initiation of secondary flow and more pronounced enhancement in the heat transfer, and the stabilization effect of the acceleration in the convergent channel which results in a much later initiation of the secondary flow and less pronounced enhancement in the heat transfer. However, the deceleration flow in the divergent channel and the acceleration in the convergent make the average Nusselt numbers approach the results of the parallel-plate channel. Correlation results for the onset of the secondary flow and enhancement of the heat transfer will be presented and discussed.     

The effect of the top and bottom wall temperatures on the laminar natural convection in an air-filled square cavity

The effect of the top and bottom wall temperatures on the natural convection heat transfer characteristics in an air-filled square cavity driven by a difference in the vertical wall temperatures was investigated by measuring the temperature distributions along the heated vertical wall and visualizing the flow patterns in the cavity. The experiments were performed at a horizontal Grashof number of 1.9 × 10⁸. Increasing the top wall temperature resulted in a separated flow region on the top wall, which caused a secondary flow between the separated flow and the boundary layer on the heated vertical wall. This secondary flow had a significant effect on the heat transfer in this region. Changes in the top and bottom wall temperatures changed the temperature gradient and the average temperature of the air outside the thermal boundary layers in the cavity. The local heat transfer along much of the heated vertical wall could be correlated by Nu = C · Ra^0.32, but the constant C increased when the average of the top and bottom wall temperatures increased.     

Conjugate natural convection in air filled tube inserted a square cavity

Numerical analyses of fluid flow and heat transfer due to buoyancy forces in a tube inserted square cavity filled with fluid were carried out by using control volume method in this study. The cavity was heated from the left wall and cooled from the right isothermally and horizontal walls were adiabatic. A circular tube filled with air was inserted into the square cavity. The case that the inside and outside of the tube were filled with the same fluid (air) was examined. Varied solid materials were chosen as the tube wall. Results were obtained for different Rayleigh numbers (Ra = 10⁴, 10⁵ and 10⁶), thermal conductivity ratio of the fluid to the tube wall (k = 0.1, 1 and 10) and different location centers of the tube (c (0.25 ≤ x ≤ 0.75, 0.25 ≤ y ≤ 0.75)). Comparison with benchmark solutions of the natural convection in a cavity was performed and numerical results gave an acceptable agreement. It was found that varied location of the tube center can lead to different flow fields and heat transfer intensities which are also affected by the value of Rayleigh number.     

Local turbulent opposing mixed convection heat transfer in inclined flat channel for stably stratified airflow

Local turbulent mixed convection heat transfer in inclined flat channels (?=20-90° from horizontal position) for opposing flows was investigated for the case when only upper wall is heated (under stably stratified flow conditions). Wide ranges of airflow parameters are covered: Re=4 × 10³-4 × 10⁴, Gr_q=1.7 × 10⁸-1.4 × 10¹⁰, pressures; p=0.2; 0.4; 0.6; 0.8 MPa. Based on analysis of local heat transfer data and existing information in the literature three characteristic regions in the buoyancy parameter range investigated were identified: region without buoyancy instabilities, transition region and region with buoyancy instabilities in whole heated section. For the region without buoyancy instabilities correlation for calculation of heat transfer in inclined flat channels was suggested.     

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.     

Buoyancy effects on laminar heat transfer in the thermal entrance region of horizontal rectangular channels with uniform wall heat flux for large prandtl number fluid

The Graetz problem for fully developed laminar flow in horizontal rectangular channels with uniform wall heat flux is extended by including buoyancy effects in the analysis for the case of large Prandtl number fluid. A general formulation valid for all Prandtl numbers is presented and the limiting case of large Prandtl number is approached by a numerical method. The typical developments of temperature profile, wall temperature and secondary flow in the thermal entrance region are presented for the case of square channel γ = 1. Local Nusselt number variations are presented for the aspect ratios γ = 0.2, 0.5, 1, 2 and 5 with Rayleigh number as parameter. Due to entry and secondary flow effects, a minimum Nusselt number occurs at some distance from the entrance, depending on the magnitude of Rayleigh number. This behavior is similar to that observed in the thermal entrance region where the transition from laminar to turbulent flow occurs. The effect of Rayleigh number is seen to decrease the thermal entrance length, and the Graetz solution, neglecting buoyancy effects, is found to be applicable only when Rayleigh number is less than about 10³. A study of the practical implications of large Prandtl number on heat transfer results for hydrodynamically and thermally fully developed case reveals that the present heat transfer results are valid for Prandtl number ranging from order 10 to infinity.     

Natural convection heat transfer in a square enclosure containing water near its density maximum

The steady-state flow structure, temperature and heat transfer in a square enclosure heated and cooled on opposite vertical walls and containing cold water near its density maximum are investigated numerically. Interpretation of the results hinges upon a dimensionless density distribution parameter which fixes the orientation of the hot and cold wall temperatures with respect to the extremum temperature and also serves to characterize the distribution of the buoyancy force in the enclosure. Multicellular flow structures are observed for certain ranges of the density distribution parameter independent of the value of Rayleigh number (10³ ⩽ Ra ⩽10⁶). The effect of the density distribution parameter on cross-cavity heat transfer is found to be substantial and is discussed in the context of the changing flow structure. Comparisons with previous studies in the literature are made.     

Effects of Heating Location and Size on Natural Convection in Partially Heated Open-Ended Enclosure by Using Lattice Boltzmann Method

Natural convection heat transfer in an square enclosure, consisting of a partially heated west wall with east end open to ambient, is investigated numerically by using an in-house computational fluid dynamics solver based on thermal lattice Boltzmann method. In particular, the influences of Rayleigh number (10³–10⁶), heating location (bottom, middle, and top) on west wall, and dimensionless heating length (0.25–0.75) on momentum and heat transfer characteristics of air are presented and discussed. The streamline patterns show bifurcation at the lowest Rayleigh number for bottom and middle heating, whereas at the highest Rayleigh number, all heating positions yield bifurcation and elongation of flow patterns with a secondary vortex near the lower side of open end. The middle and bottom heating locations show a linear increase in Nusselt number with heater size, whereas inverse dependence is seen for top heating. The maximum heat transfer is observed in the case of middle heating. As expected, average Nusselt number increased with increasing Rayleigh number. Finally, the functional dependence of the average Nusselt number on flow governing parameters (Rayleigh number and heating length) for different heating locations is presented as a simple predictive empirical relationship.     

Ecoulements induits par la force gravifique dans une cavité cylindrique en rotation

A spectral Tau-Chebyshev method is used for the prediction of the motion of a rotating Boussinesq fluid driven by buoyancy from a horizontal temperature gradient. The analysis is made for the axisymmetric regime in annular cavities when the physical parameters vary as: 0≤ Re≤ 2500; Ra = 10², ± 10⁴, 10⁵; 0 ≤ Fr ≤ 25. The transition from the thermal convection regime to the rotation dominated regime results from the competition between the various buoyancy, centrifugal and Coriolis forces. The validity of the asymptotical solutions derived from Hunter (1966) when Fr ⪡ 1 andRe ⪢ 1, is discussed when Re varies.     

Heat and fluid flow characteristics inside differentially heated square enclosures with single and multiple sliding walls

Fluid flow and heat transfer characteristics of differentially heated lid driven cavities are numerically modeled and analyzed in the present study. One‐, two‐, and four‐sided lid driven cavity configurations are considered with the vertical walls being maintained at different temperatures and the horizontal walls being thermally insulated. Eight different cavity configurations are considered depending on the direction of wall motion. The Prandtl number Pr is taken to be 0.7, the Grashof number is taken to be 10⁴, while two values for the Richardson number Ri are considered, 0.1 and 10. It is found that both the Richardson number and the cavity configuration affect the heat and fluid flow characteristics in the cavity. It is concluded that for Ri=0.1, a four‐sided driven cavity configuration with all walls rotating in the same direction would triple the value of the average Nusselt number at the cold wall when compared to a one‐sided driven cavity configuration. However, for Ri=10, the cavity configuration has minimal effect and all eight cases result in an average Nusselt number value at the cold wall ranging between 1.3 and 1.9. © 2009 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley InterScience ( www.interscience. wiley.com ). DOI 10.1002/htj.20264     

Jet impingement cooling of a horizontal surface in an unconfined porous medium: Mixed convection regime

Two-dimensional slot jet impingement cooling of an isothermal horizontal surface immersed in an unconfined porous medium is simulated numerically to gain insight into thermal characteristics under mixed convection conditions with the limitation of the Darcy model. The jet direction is considered to be perpendicular from the top to the horizontal heated element; therefore, the jet flow and the buoyancy driven flow are in opposite directions. The results are presented in the mixed convection regime with wide ranges of the governing parameters: Péclet number (1 ? Pe ? 1000), Rayleigh number (10 ? Ra ? 100), half jet width (0.1 ? D ? 0.5), and the distance between the jet and the heated portion (0.1 ? H ? 1.0). It is found that the average Nusselt number increases with increase in either Rayleigh number or jet width for high values of Péclet number. The average Nusselt number also increases with decrease in the distance between the jet and the heated portion. It is shown that mixed convection mode can cause minimum average Nusselt number at two values of Péclet number and a maximum average Nusselt number occurs in between theses two Péclet numbers at higher Rayleigh number due to counteraction of jet flow against buoyancy driven flow. Hence careful consideration must be given while designing a system of jet impingement cooling through porous medium.