Stationary transverse rolls and U-rolls in limiting low Reynolds number mixed convective air flow near the convective threshold in a horizontal flat duct

Combined experimental flow visualization and temperature measurement are carried out in the present study to explore the buoyancy driven vortex flow patterns in a limiting low Reynolds number mixed convective air flow through a bottom heated horizontal flat duct. In Particular, attention is paid to the flow approaching the natural convection limit (Re=0) for Re=1.0 and 2.0 with the Rayleigh number near the critical level for the onset of convection for 1200?Ra?4000. Results from the flow visualization have revealed two unfamiliar vortex flow patterns which were not seen in our earlier study [Int. J. Heat Mass Transfer 44 (4) (2001) 705]. One is characterized by the stable stationary transverse rolls in the duct entry and stable longitudinal rolls in the downstream. Another is in the form of U-rolls. The relations of these two patterns with those reported in the literature from analytical, numerical and experimental studies are discussed. Moreover, stable longitudinal rolls along with nonperiodic traversing waves, and mixed longitudinal and transverse rolls as well as irregular cells which appear in the higher Reynolds number for 3.0?Re?5.0 are also noted here. The temporal and spatial characteristics of the unfamiliar vortex flows are inspected in detail. In addition, the flow formation processes leading to the two unfamiliar vortex flow structures are also examined carefully. During the flow formation we noted merging of longitudinal and transverse rolls to form U-rolls, splitting of rolls into cells and the reverse process of cell integration into rolls, aside from the generation of the longitudinal and transverse rolls. Finally, a flow regime map is provided to delineate various vortex flow structures observed in this study and in the previous study (cf. the above-mentioned reference) driven by the slightly supercritical and subcritical buoyancies for 1.0?Re?5.0.     

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.     

A further investigation of effects of jet-disk separation distance on steady mixed convective vortex flow resulting from a confined impinging air jet

We extend our previous study [J.C. Hsieh, T.F. Lin, Effects of jet-to-disk separation distance on the characteristics of mixed convective vortex flow in an impinging air jet confined in a cylindrical chamber, Int. J. Heat Mass Transfer 48 (2005) 511–525] here to further investigate how the jet-disk separation distance H affects the mixed convective vortex flow resulting from a round air jet impinging onto a heated horizontal circular disk confined in a vertical cylindrical chamber. The experiment is conducted for the jet-disk separation distance varying from 40.0 to 60.0 mm and the jet flow rate is varied from 0 to 12.0 slpm (standard liter per minute) for the jet Reynolds number Re_j ranging from 0 to 1623. The temperature difference between the disk and the air injected into the chamber is varied from 0 to 25.0 °C for the Rayleigh number Ra ranging from 0 to 507,348. The data from the present study for the ratio H/D_j = 4–6 are compared with our previous study for H/D_j = 1–3. The results indicate that the critical jet Reynolds numbers for the onsets of the secondary and tertiary inertia-driven rolls and for the onset of the buoyancy-driven roll vary nonmonotonically with the jet-disk separation distance due to the complicate changes of the vortex flow structure with H. In the steady vortex flow, both the primary inertia-driven roll and the buoyancy-driven roll get larger at increasing jet-disk separation distance before they contact with each other for H/D_j = 1 and 2. But for H/D_j ≧ 3 the primary roll and buoyancy roll do not always grow at increasing H. Finally, empirical correlations are proposed for the critical conditions leading to the onsets of the inertia- and buoyancy-driven vortex rolls.     

MHD natural convection in a laterally and volumetrically heated square cavity

A numerical study is presented of unsteady two-dimensional natural convection of an electrically conducting fluid in a laterally and volumetrically heated square cavity under the influence of a magnetic field. The flow is characterized by the external Rayleigh number, Ra_E, determined from the temperature difference of the side walls, the internal Rayleigh number, Ra_I, determined from the volumetric heat rate, and the Hartmann number, Ha, determined from the strength of the imposed magnetic field. Starting from given values of Ra_E and Ha, for which the flow has a steady unicellular pattern, and gradually increasing the ratio S = Ra_I/Ra_E, oscillatory convective flow may occur. The initial steady unicellular flow for S = 0 may undergo transition to steady or unsteady multicellular flow up to a threshold value, Ra_I,cr, of the internal Rayleigh number depending on Ha. Oscillatory multicellular flow fields were observed for S values up to 100 for the range 10⁵-10⁶ of Ra_E studied. The increase of the ratio S results usually in a transition from steady to unsteady flow but there have also been cases where the increase of S results in an inverse transition from unsteady to steady flow. Moreover, the usual damping effect of increasing Hartmann number is not found to be straightforward connected with the resulting flow patterns in the present flow configuration.     

NATURAL CONVECTION-RADIATION INTERACTIONS IN A CUBE FILLED WITH A NONGRAY GAS

A three-dimensional numerical study was performed on interactions of natural convection and radiation in a cubical enclosure filled with carbon dioxide gas. The enclosure was heated differentially by two opposing vertical walls. Gas radiation was analyzed by the P₁ differential approximation method and the weighted sum of gray gas model. Computations were carried out over a range of the Rayleigh number, Ra, between 10⁵ and 10⁹. The Prandtl number and the overheat ratio were held fixed at 0·68 and 1·0, respectively. Unsteady transitional flows were computed by a direct simulation method, without using any explicit turbulence models. From the predictions, a mean heat transfer correlation has been proposed as Nu = 0·323 Ra^0·342 in the surface/gas radiation mode, where Nu is the time and spatially averaged Nusselt number at the isothermal walls.     

Transient and steady natural convection from a heat source embedded in thermally stratified porous layer

Numerical results are presented for transient and steady natural convection from a heat source buried in a saturated, porous layer. The porous layer is thermally stratified with a negative temperature gradient but its corresponding Rayleigh number is not greater than that of the critical value (Ra_c = 40). The effects of the size of the heat source and its buried depth are also investigated. The results, which include the buoyancy-induced flow patterns and temperature profiles, as well as the heat transfer coefficient in terms of the Nusselt number, are presented for a wide range of Rayleigh number and stratification parameter. The results thus obtained have important implications for applications in geothermal energy and underground disposal of nuclear wastes.     

Flow patterns and heat transfer enhancement in low-Reynolds–Rayleigh-number channel flow

This paper presents an experimental study on buoyancy-induced flow patterns and heat transfer characteristics of airflow through a horizontal rectangular channel. The channel had an aspect ratio of six, and its bottom and sidewalls were heated, whereas the top of the channel was cooled. The experiments were conducted at the Reynolds numbers 40 and Rayleigh numbers ranging from 100 to 4200. The Nusselt number and the temperature distributions on the top surface of the channel were measured simultaneously at different thermal/flow conditions, and the heat transfer characteristics of the channel was evaluated, together with the flow patterns in the channel. The results showed that due to the heated sidewalls, which was an `imperfect' factor comparing with the classic Rayleigh–Bénard channel, the longitudinal vortex rolls can occur at the Rayleigh number Ra=100, starting with number of rolls N=2 and then N=4 as the Ra increases, rather than the N=6 mode for the same channel with `perfect' sidewalls. In the present study, the six-roll mode occurred at Ra=1730 and above, but an initial trigger was required. Otherwise the four-roll mode would continue to be the dominant flow pattern at high Rayleigh numbers. It was demonstrated that significant heat transfer enhancement could be achieved in low Reynolds and Rayleigh number flow if the longitudinal vortex rolls were excited in the channel.     

Non-Darcy effects in buoyancy driven flows in an enclosure filled with vertically layered porous media

Natural convection in an enclosure filled with two layers of porous media is investigated numerically. Constant heat flux is imposed on the left vertical wall and the right wall is assumed to be at a low temperature. The focus of the work is on the validity of the Darcy model when various combinations of fluid Rayleigh number, Darcy number and permeability ratios are considered for fixed values of the modified Rayleigh numbers. It is found that the boundary effects (Brinkman term) have significant importance at higher modified Rayleigh numbers (Rayleigh number based on permeability, Ra_m). Calculations are performed for a modified Rayleigh number up to 10⁵. The results showed that, for the investigated range of parameters, the flow structure and heat transfer could be different than what Darcy model predicts. Two circulations are predicted for Ra_f=1×10⁸, for two different cases, Da=1×10⁻³, K_r=1000 and Da=1×10⁻⁴, K_r=100 (K_r=K₁/K₂). For K_r>1, increasing permeability ratio decreases flow penetration from layer 1 to layer 2 while reverse is true for K_r<1. For low Ra_m (Ra_m?10³) and K_r=1000, the heat transfer is conductive in the right layer, while this is true for the left layer for K_r=0.001. It is possible to obtain no-slip velocity boundary conditions both at the walls and at the interface between the porous layers even for very low permeability.     

Thermo-bioconvection in a suspension of gravitactic micro-organisms in vertical cylinders

We investigate the effect of heating or cooling from below at constant temperature and constant heat flux on the development of gravitactic bioconvection in vertical cylinders with stress free sidewalls. The governing equations are the continuity equation, the Navier–Stokes equations with the Boussinesq approximation, the diffusion equation for the motile micro-organisms and the energy equation. The control volume method is used to solve numerically the complete set of governing equations. The governing parameters are the thermal and bioconvection Rayleigh numbers, the bioconvection Peclet number, the Lewis number, the Schmidt number and the aspect ratio. We found that subcritical bifurcations of bioconvection became supercritical bifurcations when the thermal Rayleigh number Ra_T is different than zero. For Ra_T < 0, i.e. for cooling from below, we have opposing buoyancy forces, the convection is decreased and the critical thermo-bioconvection Rayleigh number is increased with respect to that of bioconvection. For Ra_T > 0, i.e. for heating from below, we have cooperating buoyancy forces, the convection is increased and the critical thermo-bioconvection Rayleigh number is decreased with respect to that of bioconvection. Heating and cooling from below at constant temperature and heat flux modify considerably the pattern formation of the gravitactic bioconvection.     

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.     

Turbulent natural convection of liquid deuterium,hydrogen and nitrogen within enclosed vessels

Quasi-steady natural convection of liquid deuterium, hydrogen, and nitrogen within a sphere, hemisphere, horizontal cylinder, and vertical cylinder has been studied experimentally for the case of a nearly uniform wall temperature. A single expression relating the Nusselt and Rayleigh numbers, Nu = 0·104Ra^0·352, fits the deuterium and nitrogen data over the range 7 × 10⁸ < Ra < 6 × 10¹¹, while the hydrogen Nusselt numbers are 8 per cent lower. The temperature field within the vessels is virtually free of horizontal temperature gradients. A single dimensionless temperature profile characterizes the vertical temperature distribution for each vessel shape, with the profiles for the sphere, hemisphere, and horizontal cylinder being nearly identical.     

Three-dimensional analyses of double diffusive convection in a two-layer system at high Rayleigh number

Three-dimensional numerical computations of double diffusive natural convection were carried out at the Prandtl number Pr=6, the Lewis number Le=100, the aspect ratio A=2, the Rayleigh number Ra=10⁷ and various buoyancy ratios N for a two-layer system which consists of a pure water (upper layer) and an aqueous solution (lower layer) with lateral heating and cooling. Salt-fingers with the termination of bulbous shapes were formed owing to the penetration of convective flow in a layer into the other layer at N=0.3 and plumes were formed by the collision of solutal fragments against the interface at N=0.6 or 1.     

Natural convection of gas/vapour mixtures in a porous medium

A two-component model of the natural convection of gas/vapour mixtures is described in which the temperature dependence of mixture properties is incorporated. Attention is focused on a twodimensional packed bed with isothermal top and bottom plates and insulated side walls ; nitrogen/water vapour and helium/water vapour mixtures are considered. When the mixture density is a monotonie function of temperature and the temperature difference is small (≈ 10°C), predicted values of the critical Rayleigh number (Ra_c), modes of convection, flow and temperature distribution agreed well (1–2 percent in Ra_c) with corresponding values derived from an analogous single-component model based on properties evaluated at the mean temperature; the increasing difference between such values as the temperature difference increases is also shown.     

Rayleigh–Bénard convection in a supercritical fluid along its critical isochore in a shallow cavity

We perform a numerical investigation of the Rayleigh–Bénard convection in supercritical nitrogen in a shallow enclosure with an aspect ratio of 4. The transient and steady-state fluid behaviors over a wide range of initial distances to the critical point along the critical isochore are obtained and analyzed in response to modest homogeneous bottom heating. On account of the fluid layer being extremely thin, density stratification is notably excluded from consideration herein, which leads to the dominating role of the Rayleigh criterion in the onset of convection. Following the Boussinesq approximation, we find the power law scaling relationships over five decades of the Rayleigh number (Ra) for various transient quantities including the exponential growth rate of the mean enstrophy in the cavity and the characteristic times of the development of convective motion. The correlation of the Nusselt number versus the Rayleigh number shows asymptotic features at the two ends of the Ra spectrum, which incidentally correspond to different convection patterns. Under the regime of high Ra, the heat transfer through the fluid cavity is enhanced by the turbulent bursts of thermal plumes from the boundary layers. On the other hand, under the regime of low Ra, it is the orderly multicellular flow that moves heat from the bottom of the layer to the top, which includes a transition from a four-cell structure to a six-cell structure with decreasing Ra.     

Mixed convection between horizontal plates—II. Fully developed flow

Fully developed velocity profiles of longitudinal convection rolls in mixed convection between horizontal plates were measured in nitrogen by laser Doppler anemometry for a range 2472 < Ra < 8300 and 15 < Re < 150. It is shown analytically and experimentally that the transverse velocities of the longitudinal convection rolls are independent of the forced flow. The experimentally and numerically obtained w-profiles (Pr = 0.71) are in good agreement with theoretical predictions (Pr → ∞) and other experimental results (Pr = 11.1 and 930) for Rayleigh-Benard convection. A detailed study of the longitudinal velocity modulation Δu[w_max(Ra), Re] is presented. Also, asymmetric roll patterns were found in spite of the small temperature differences used between the horizontal plates.     

Numerical simulation of free convection based on experimental measured conductivity in a square cavity using Water/SiO2 nanofluid

Heat transfer enhancement has been investigated in a square cavity subject to different side wall temperatures using water/SiO₂ nanofluid. An experimental setup has been used to extract the conductivity value of nanofluid. This study has been carried out for the pertinent parameters in the following ranges: the Rayleigh number of base fluid, Ra_f = 10⁵–10⁷ and the volumetric fraction of nanoparticle between 0 and 4%. The comparisons show that the mean Nusselt number increases with volume fraction for the whole range of Rayleigh numbers. Although by using the theoretical formulations for conductivity no enhancement has been observed.     

A study of laminar natural convection in a non-uniformly heated annular fluid layer

A study is made of free convection in an annular fluid layer confined between two horizontal cylinders. Results are presented for the case of an adiabatic inner boundary with a sinusoidal temperature distribution on the outer boundary. The problem is formulated in terms of the Boussinesq approximations and solved using perturbation and finite-difference techniques. It is found that the results obtained from both methods are in good mutual agreement for weak convection. The solutions reveal the existence of various flow regimes, depending on Ra and the angular position of the temperature maximum on the outer boundary, which are similar in character to earlier results for the porous medium. It is found that incipient convection may decrease Nu in one case. In particular, when heating is from below, three distinct subregimes may be obtained within the range of parameters considered (R = 2, 0 < Ra < 40,000) namely, the steady quadri-cellular flow for Ra < 1120, the circulating flow with or without secondary cells for 1120 < Ra < 40,000, and finally unsteady circulating flow for Ra > 40,000. Secondary cells begin to appear near the outer boundary at Ra = 11,000. For an arbitrary heating angle, there is always a net circulating flow around the cavity, unless heating is from the top, which leads to an enhancement of heat transfer with the surroundings. The maximum flow and heat transfer rates are obtained for a heating angle below the horizontal whose value depends on Ra.     

Heat generation effects in natural convection inside a porous annulus

The interplay between internal heat generation and externally driven natural convection inside a porous medium annulus is studied in detail using numerical methods. The axisymmetric domain is bounded with adiabatic top and bottom walls and differentially heated side walls sustaining steady natural convection of a fluid with Prandtl number, Pr = 5, through a porous matrix of volumetric porosity, ? = 0.4. The generalized momentum equation with Brinkman–Darcy–Forchheimer terms and the local thermal non-equilibrium based two-energy equation model are solved to determine the flow and the temperature distribution. Beyond a critical heat generation value defined using an internal Rayleigh number, Ra_I,cr^?, the convection transits from unicellular to bicellular mode, as the annulus T_max becomes higher than the fixed hot-wall temperature. The Ra_I,cr^? increases proportionately when the permeability based external Rayleigh number Ra_E^? and the solid–fluid thermal conductivity ratio γ are independently increased. A correlation is proposed to predict the overall annulus Nu as a function of Ra_E^?, Ra_I^?, Da and γ. It predicts the results within ± 20% accuracy.     

Natural convection slip flow in a vertical microchannel heated at uniform heat flux

Numerical solutions for steady state developing natural convection flow in air, in vertical parallel-plate microchannels are accomplished. An asymmetric heating is considered and the walls are assumed to be at uniform heat flux. A first-order model is used for slip and jump boundary conditions and an analytical solution for the fully developed flow is also given. Results are performed for air, for the heat flux ratio in the 0.0–1.0 range, for Rayleigh, Ra, and Knudsen, Kn, numbers from 10^?1 to 8 × 10³ and from 0.0 to 0.10, respectively. The maximum mass flow rate is always obtained for the highest considered Kn value, whereas the average Nusselt number, Nu, increases for lower Ra (<10) and decreases for Ra > 100. Wall temperature profiles have the lowest values for highest considered Kn value at lower Ra, whereas for the developing flow, they present opposite trends. For developing flow, velocity profiles for asymmetric and symmetric heating are completely different. In developing flow velocity profiles along the wall present the highest increases for asymmetric heating and the highest considered Kn value.