Experimental laminar Rayleigh-Bénard convection in a cubical cavity at moderate Rayleigh and Prandtl numbers

Rayleigh-Bénard convection in a cubical cavity with adiabatic or conductive sidewalls is experimentally analyzed at moderate Rayleigh numbers (Ra ≤ 8 × 10⁴) using silicone oil (Pr=130) as the convecting fluid. Under these conditions the flow is steady and laminar. Three single-roll-type structures and an unstable toroidal roll have been observed inside the cavity with nearly adiabatic sidewalls. The sequence from the conductive state consists of a toroidal roll that evolves to a diagonally oriented single roll with increasing Rayleigh number. This diagonal roll, which is stabilized by the effect of the small but finite conductivity of the walls, shifts its axis of rotation towards to two opposite walls, and back to the diagonal orientation to allow for the increase in circulation that occurs as the Rayleigh number is further increased. Conduction at the sidewalls modifies this sequence in the sense that the two initial single rolls finally evolve into a four-roll structure. Once formed, this four-roll structure remains stable when decreasing the Rayleigh number until the initial single diagonally oriented roll is again recovered. The topology and the velocity fields of all structures, characterized with visualization and particle image velocimetry, respectively, are in good agreement with numerical results reported previously for the cavity with adiabatic walls, as well as with the numerical predictions obtained in the present study for perfectly conducting lateral walls. Received: 10 August 1998/Accepted: 1 August 2000     

Buoyancy and cross flow effects on heat transfer of multiple impinging slot air jets cooling a flat plate at different orientations

The present article reports on heat transfer characteristics associated with multiple laminar impinging air jet cooling a hot flat plat at different orientations. The work aims to study the interactions of the effects of cross flow, buoyancy induced flow, orientation of the hot surface with respect to gravity, Reynolds numbers and Rayleigh numbers on heat transfer characteristics. Experiments have been carried out for different values of jet Reynolds number, Rayleigh number and cross flow strength and at different orientations of the air jet with respect to the target hot plate. In general, the effective cooling of the plate has been observed to be increased with increasing Reynolds number and Rayleigh number. The results concluded that the hot surface orientation is important for optimum performance in practical applications. It was found that for Re ≥ 400 and Ra ≥ 10,000 (these ranges give 0.0142 ≤ Ri ≤ 1.59 the Nusselt number is independent on the hot surface orientation. However, for Re ≤ 300 and Ra ≥ 100,000 (these ranges give 1.59 ≤ Ri ≤ 42.85): (i) the Nusselt number for horizontal orientation with hot surface facing down is less that that of vertical orientation and that of horizontal orientation with hot surface facing up, and (ii) the Nusselt number of vertical orientation is approximately the same as that of horizontal orientation with hot surface facing up. For all surfaces orientations and for the entire ranges of Re and Ra, it was found that increasing the cross flow strength decreases the effective cooling of the surface.     

Experimental and numerical study on laminar natural convection in a cavity heated from bottom due to an inclined fin

Natural convection heat transfer in an inclined fin attached square enclosure is studied both experimentally and numerically. Bottom wall of enclosure has higher temperature than that of top wall while vertical walls are adiabatic. Inclined fin has also adiabatic boundary conditions. Numerical solutions have been done by writing a computer code in Fortran platform and results are compared with Fluent commercial code and experimental method. Governing parameters are Rayleigh numbers (8.10⁵ ≤ Ra ≤ 4 × 10⁶) and inclination angle (30° ≤ and ≤ 120°). The temperature measurements are done by using thermocouples distributed uniformly at the wall of the enclosure. Remarkably good agreement is obtained between the predicted results and experimental data. A correlation is also developed including all effective parameters on heat transfer and fluid flow. It was observed that heat transfer can be controlled by attaching an inclined fin onto wall.     

Effect of Conduction in Bottom Wall on Darcy–Bénard Convection in a Porous Enclosure

Conjugate natural convection-conduction heat transfer in a square porous enclosure with a finite-wall thickness is studied numerically in this article. The bottom wall is heated and the upper wall is cooled while the verticals walls are kept adiabatic. The Darcy model is used in the mathematical formulation for the porous layer and the COMSOL Multiphysics software is applied to solve the dimensionless governing equations. The governing parameters considered are the Rayleigh number (100 ≤ Ra ≤ 1000), the wall to porous thermal conductivity ratio (0.44 ≤ K _r ≤ 9.90) and the ratio of wall thickness to its height (0.02 ≤ D ≤ 0.4). The results are presented to show the effect of these parameters on the heat transfer and fluid flow characteristics. It is found that the number of contrarotative cells and the strength circulation of each cell can be controlled by the thickness of the bottom wall, the thermal conductivity ratio and the Rayleigh number. It is also observed that increasing either the Rayleigh number or the thermal conductivity ratio or both, and decreasing the thickness of the bounded wall can increase the average Nusselt number for the porous enclosure.     

Effect of the surface thermal radiation on turbulent natural convection in tall cavities of façade elements

The effect of the surface thermal radiation in tall cavities with turbulent natural convection regime was analyzed and quantified numerically. The parameters considered were: the Rayleigh number 10⁹–10¹², the aspect ratio 20, 40 and 80 and the emmisivity 0.0–1.0. The percentage contribution of the radiative surface to the total heat transfer has a maximum value of  15.19% (Ra = 10⁹, A = 20) with emissivity equal to 1.0 and a minimum of 0.5% (Ra = 10¹², A = 80) with ε* = 0.2. The average radiative Nusselt number for a fixed emissivity is independent of the Rayleigh number, but for a fixed Rayleigh number diminishes with the increase of the aspect ratio. The results indicate that the surface thermal radiation does not modify significantly the flow pattern in the cavity, just negligible effects in the bottom and top of the cavity were observed. Two different temperature patterns were observed a conductive regime Ra = 10⁹ and a boundary layer regime Ra = 10¹².     

Numerical simulation of fluid flow and heat transfer characteristics in channel with V corrugated plates

A detailed numerical study is carried out to investigate fluid flow and heat transfer characteristics in a channel with heated V corrugated upper and lower plates. The parameters studied include the Reynolds number (Re = 2,000–5,500), angles of V corrugated plates (θ = 20°, 40°, 60°), and constant heat fluxs (q″ = 580, 830, 1,090 W/m²). Numerical results have been validated using the experimented data reported by Naphon, and a good agreement has been found. The angles of V corrugated plates (θ) and the Reynolds number are demonstrated to significantly affect the fluid flow and the heat transfer rate. Increasing the angles of V corrugated plates can make the heat transfer performance become better. The increasing Reynolds number leads to a more complex fluid flow and heat transfer rate. The numerical calculations with a non-equilibrium wall function have a better accuracy than with a standard wall function for solving high Reynolds numbers or complex flow problems.     

Influence of inclination angle on buoyancy-driven convection in triangular enclosure filled with a fluid-saturated porous medium

The present investigation deals with the numerical analysis of steady-state laminar buoyancy-driven convection in an inclined triangular enclosure filled with fluid saturated porous media using the Darcy law equation. One wall of the enclosure is isothermally heated and the other is cooled, while the remaining wall is adiabatic. The effect of inclination angle on natural convection is investigated by varying the angle of inclination (φ) between 0° and 360°. The governing transformed equations are solved numerically using a finite-difference method. Obtained results are shown in the form of streamlines, isotherms, mean Nusselt numbers and dimensionless stream function for different values of the Rayleigh number Ra in the range 100 ≤ Ra ≤ 1,000. It is found that the values of the maximum and minimum mean Nusselt number are reached for φ = 330° and φ = 210° , respectively. However, the lowest flow strength is formed at φ = 240° for all values of Ra.     

Measurements of heat transfer coefficients between turbulent liquid jets and a radially finned heated disk placed at the bottom of an open vertical circular cavity

 Experiments have been performed to assess the impact of an extended surface on the heat transfer enhancement for axisymmetric, turbulent liquid jet impingement on a heated round disk. The disk, with an array of integral radial fins mounted on its surface, is placed at the bottom of an open vertical circular cavity. Hydrodynamic and heat transfer data were obtained for a dielectric fluorocarbon liquid FC-77. For a fixed circular heater of diameter D=22.23 mm, several geometric parameters were tested: the nozzle diameter (4.42≤d≤9.27 mm), the confining wall diameter of the vertical cavity (22.23≤D _c≤30.16 mm), and the nozzle-to-heater spacing (0.5≤S/d≤5.0). The FC-77 flow rates varied from =0.2 to 11.0 l/min producing Reynolds numbers in the wide interval 700≤Re _d ≤44,000. For d=4.42 mm, the heat transfer response to the separation distance S/d was small but increased gradually with increasing nozzle diameter up to d=9.27 mm. The thermal resistance R _th increased with the confining wall diameter D _c and also with the nozzle diameter d. A minimum value of the thermal resistance of R _th,min=0.4 cm² K/W was attained for a combination of d=4.42 mm, D _c=22.23 mm, S/d=1, and =7.5 l/min. Based on a simplified heat transfer model, reasonable agreement was obtained between measured values of the thermal resistance and the R _th-predictions. The total fin effectiveness ɛ_f was shown to increase with increasing nozzle diameter, but was invariant with the flow rate (or the jet exit velocity). More than a three-fold heat transfer enhancement was realized through the addition of the array of integral radial fins on the heated round disk. Received on 30 August 2000 / Published online: 29 November 2001     

Transient oscillatory conjugate natural convection in a tall water cavity

A two-dimensional numerical simulation was conducted to investigate the effects of the heat capacity of the heated wall on the transient oscillatory convection in a tall cavity. Results were particularly obtained for water (Pr=6) in a tall cavity (A=6) with constant-heat-flux heating on one side and isothermal cooling on the opposing side. Significant wall heat capacity effects were found. Specifically, a heated wall of finite heat capacity can slow down the flow evolution process to steady state at low Rayleigh number and damp the flow oscillation during the initial developing period at high Rayleigh number. In periodic flow atRa* _H =4.4×10¹⁰ the wall heat capacity significantly reduces the amplitude of the temperature oscillation. A quasi-periodic flow atRa* _H =4.8×10¹⁰ was found to become periodic when the wall heat capacity was included. The wall heat capacity does not shown noticeable effect when the flow is chaotic forRa* _H =6.0×10¹⁰.In einer zweidimensionalen numerischen Simulationsrechnung wurde der Einfluß der Wärmekapazität einer beheizten Wand auf die nichtstationäre oszillatorische Konvektion in einem großen Behälter untersucht. Die Resultate beziehen sich speziell auf Wasser (Pr=6), einen tiefen Behälter (H/W=6) und konstanten Wärmezufluß auf der einen und isotherme Kühlung auf der anderen Wandseite. Es traten signifikante Effekte in Abhängigkeit von der Wandwärmekapazität auf. Insbesondere vermag eine beheizte Wand begrenzter Wärmekapazität bei niedrigen Rayleigh-Zahlen die Entwicklung des Strömungsprozesses bis zum Stationärzustand abzubremsen und bei hohen Rayleigh-Zahlen die Strömungsoszillationen während der Einschwingphase zu dämpfen. Bei periodischem Strömungszustand undRa* _H =4,4·10¹⁰ verringert die Wandkapazität die Amplitude der Temperaturschwingung beträchtlich. Es zeigte sich ferner, daß ein quasiperiodischer Strömungszustand beiRa* _H =4,8·10¹⁰ periodisch wird, wenn die Wandkapazität Einfluß nimmt. Letztere hatte keine nachweisbaren Wirkungen bei chaotischem Strömungszustand mitRa* _H =6,0·10¹⁰.     

Nusselt-Rayleigh correlations for free convection in 2D air-filled parallelogrammic enclosures with isothermal active walls

In the present study Nu-Ra-α correlations are proposed to calculate the steady-state natural convection heat transfer taking place in 2D air-filled cavities of parallelogrammic section. The thermal conditions and the dimensions of the enclosures permit to cover a large range of Rayleigh numbers, 1.7 × 10³  ≤ Ra ≤ 3.0 × 10⁹, suitable for diverse engineering applications. The two active walls of the cavities are kept vertical and isothermal at hot and cold temperatures T _h and T _c respectively. Separated by a horizontal distance H, they have the same height H and are connected by a closed adiabatic channel whose upper and lower walls can be inclined at an angle α with respect to the horizontal, varying between −60° to +60°. That gives rise to a conducting or insulating cavity, in the convective sense of the term (diode cavity). A computational model based on the finite volume method is used to solve the governing equations. The large number of treated configurations led to propose Nu-Ra-α correlations for large ranges of Ra and α which can be applied to many engineering areas. The results of this numerical study have been successfully compared with calculated and measured available data.     

Non-Darcy buoyancy driven flows in a fluid saturated porous medium: the use of asymptotic computational fluid dynamics (ACFD) approach

Natural convection in a fluid saturated porous medium has been numerically investigated using a generalized non-Darcy approach. The governing equations are solved by using Finite Volume approach. First order upwind scheme is employed for convective formulation and SIMPLE algorithm for pressure velocity coupling. Numerical results are presented to study the influence of parameters such as Rayleigh number (10⁶ ≤Ra ≤10⁸), Darcy number (10⁻⁵ ≤ Da ≤ 10⁻²), porosity (0.4 ≤ ɛ ≤ 0.9) and Prandtl number (0.01 ≤ Pr ≤ 10) on the flow behavior and heat transfer. By combining the method of matched asymptotic expansions with computational fluid dynamics (CFD), so called asymptotic computational fluid dynamics (ACFD) technique has been employed to generate correlation for average Nusselt number. The technique is found to be an attractive option for generating correlation and also in the analysis of natural convection in porous medium over a fairly wide range of parameters with fewer simulations for numerical solutions.     

Numerical simulation of laminar natural convection in a laterally heated vertical cylindrical enclosure: application to crystal growth

Laminar natural convection has been studied in a laterally heated vertical cylindrical enclosure with a free insulated surface and a centrally located constant temperature wall at the top. These conditions are a simplification of the conditions existing in a Czochralski crystal pulling system. The laminar, axisymmetric flow of a Newtonian, constant physical properties fluid under Boussinesq’s approximation has been considered. Governing equations in primitive variable form are solved numerically by control volume method. SIMPLE algorithm due to Patankar has been used for the numerical simulation. The effects of the constant wall heat flux boundary condition at the side wall have been investigated whereas the bottom wall is considered to be insulated. Streamlines and isotherms are presented for various Rayleigh numbers and Prandtl numbers. Heat flux vectors through the melt are plotted for selected cases. The axial velocity and temperature variations at different horizontal sections of the crucible have been presented graphically to explain the transport processes inside the crucible. It has been observed that in case of low Pr and high Ra, flow separation occurs at the vertical wall of the crucible which leads to an oscillatory flow as Ra increases. The investigation has been extended to the oscillatory regime of flow in the zone of supercritical Rayleigh numbers and some unsteady results are also presented. Finally a heat transfer correlation has been developed for steady-state case.     

Hypersonic interference heating in the vicinity of surface protuberances

The understanding of the behaviour of the flow around surface protuberances in hypersonic vehicles is developed and an engineering approach to predict the location and magnitude of the highest heat transfer rates in their vicinity is presented. To this end, an experimental investigation was performed in a hypersonic facility at freestream Mach numbers of 8.2 and 12.3 and Reynolds numbers ranging from Re _∞/m = 3.35 × 10⁶ to Re _∞/m = 9.35 × 10⁶. The effects of protuberance geometry, boundary layer state, freestream Reynolds number and freestream Mach numbers were assessed based on thin-film heat transfer measurements. Further understanding of the flowfield was obtained through oil-dot visualizations and high-speed schlieren videos. The local interference interaction was shown to be strongly 3-D and to be dominated by the incipient separation angle induced by the protuberance. In interactions in which the incoming boundary layer remains unseparated upstream of the protuberance, the highest heating occurs adjacent to the device. In interactions in which the incoming boundary layer is fully separated ahead of the protuberance, the highest heating generally occurs on the surface just upstream of it except for low-deflection protuberances under low Reynolds freestream flow conditions in which case the heat flux to the side is greater.     

Non-Darcian and Anisotropic Effects on Free Convection in a Porous Enclosure

The free convective flow and heat transfer, within the framework of Boussinesq approximation, in an anisotropic fluid filled porous rectangular enclosure subjected to end-to-end temperature difference have been investigated using Brinkman extended non-Darcy flow model. The studies involve simultaneous consideration of hydrodynamic and thermal anisotropy. The flow and temperature fields in general are governed by, Ra, the Rayleigh number, AR, the aspect ratio of the slab, K*, the permeability ratio and k*, the thermal conductivity ratio, and Da, Darcy number. Numerical solutions employing the successive accelerated replacement (SAR) scheme have been obtained for 100 ≤ Ra ≤ 1000, 0.5 ≤ AR ≤ 5, 0.5 ≤ K* ≤ 5, 0.5 ≤ k* ≤ 5, and 0 ≤ Da ≤ 0.1. It has been found that [`(Nu)]{\overline {Nu}}, average Nusselt number increases with increase in K* and decreases as k* increases. However, the magnitude of the change in [`(Nu)]{\overline {Nu}} depends on the parameter Da, characterizing the Brinkman extended non-Darcy flow.     

Secondary flow patterns and mixing in laminar pulsating flow through a curved pipe

Mixing by secondary flow is studied by particle image velocimetry (PIV) in a developing laminar pulsating flow through a circular curved pipe. The pipe curvature ratio is η = r ₀/r _c  = 0.09, and the curvature angle is 90°. Different secondary flow patterns are formed during an oscillation period due to competition among the centrifugal, inertial, and viscous forces. These different secondary-flow structures lead to different transverse-mixing schemes in the flow. Here, transverse mixing enhancement is investigated by imposing different pulsating conditions (Dean number, velocity ratio, and frequency parameter); favorable pulsating conditions for mixing are introduced. To obviate light-refraction effects during PIV measurements, a T-shaped structure is installed downstream of the curved pipe. Experiments are carried out for the Reynolds numbers range 420 ≤ Re_st ≤ 1,000 (Dean numbers 126.6 ≤ Dn ≤ 301.5) corresponding to non-oscillating flow, velocity component ratios 1 ≤ (β = U _max,osc/U _m,st) ≤ 4 (the ratio of velocity amplitude of oscillations to the mean velocity without oscillations), and frequency parameters 8.37 < (α = r ₀(ω/ν)^0.5) < 24.5, where α² is the ratio of viscous diffusion time over the pipe radius to the characteristic oscillation time. The variations in cross-sectional average values of absolute axial vorticity (|ζ|) and transverse strain rate (|ε|) are analyzed in order to quantify mixing. The effects of each parameter (Re_st, β, and α) on transverse mixing are discussed by comparing the dimensionless vorticities (|ζ _P |/|ζ _S |) and dimensionless transverse strain rates (|ε _P |/|ε _S |) during a complete oscillation period.     

Experimental investigation of natural convection in a vertical rib-roughened channel with asymmetric heating

An experimental study in an open-ended vertical channel is carried out in order to describe the fluid dynamics and heat transfer of transient free convection inside a vertical rib-roughened channel asymmetrically heated at various uniform heat fluxes (650, 700, and 780 W/m²) corresponding to various modified Rayleigh numbers (3.65 × 10⁶, 3.93 × 10⁶ and 4.4 × 10⁶). Two ribs are symmetrically located on each wall. The investigations focused more specifically on the influence of the ribs positions inside the channel and the modified Rayleigh number (Ra^*) both in steady-state regime and during the transitional phase occurring just after the start of the heating on the flow structure and the heat transfer performance. The results showed the appearance of large-scale flow instabilities which will develop and propagate until the development of the pocket-like vortex (reversed flow). Also, the formation and breakup of recirculation eddies, vortex banishment, besides that a separation and shifting of the boundary layer from one wall to another are identified. The best position of the ribs for heat extraction depends on the magnitude of the Rayleigh number. In that case, the top position is the optimal position for the small and the moderate modified Rayleigh numbers.     

A numerical study of the longitudinal thermoconvective rolls in a mixed convection flow in a horizontal channel with a free surface

This paper presents a numerical study of three-dimensional laminar mixed convection within a liquid flowing on a horizontal channel heated uniformly from below. The upper surface is free and assumed to be flat. The coupled Navier–Stokes and energy equations are solved numerically by the finite volume method taking into account the thermocapillary effects (Marangoni effect). When the strength of the buoyancy, thermocapillary effects and forced convective currents are comparable (Ri ⋍ O(1) and Bd = Ra/Ma ⋍ O(1)), the results show that the development of instabilities in the form of steady longitudinal convective rolls is similar to those encountered in the Poiseuille–Rayleigh–Bénard flow. The number and spatial distribution of these rolls along the channel depend on the flow conditions. The objective of this work is to study the influence of parameters, such as the Reynolds, Rayleigh and Biot numbers, on the flow patterns and heat transfer characteristics. The effects of variations in the surface tension with temperature gradients (Marangoni effect) are also considered.     

Inertial-microfluidic radial migration in solid/liquid two-phase flow through a microcapillary: Particle equilibrium position

Although equilibrium of spherical particles under radial migration has been extensively investigated, mostly in macroscale flows with characteristic lengths on the order of centimeters, it is not fully characterized at relatively small Reynolds numbers, 1 ≤ Re ≤ 100. This paper experimentally studies “inertial microfluidic” radial migration of spherical particles in circular Poiseuille flow through a microcapillary. Microparticle tracking experiments are performed to obtain the spatial distribution of the particles by adopting a depth-resolved measurement technique. Through the analysis of the radial distribution of particles, inertial microfluidic circular Poiseuille flow is shown to induce a strong radial migration of particles at substantially small Re, which is quite in contrast to the pipe flows at large Re previously reported. This particle migration phenomenon is so prominent that particle equilibrium positions are formed even at small Re. However, it turns out that there exists a certain critical Re below which particle equilibrium position is almost fixed, but above which it seems to drift toward the channel wall.     

Convection in a Lid-Driven Heat-Generating Porous Cavity with Alternative Thermal Boundary Conditions

Mixed convection flow in a two-sided lid-driven cavity filled with heat-generating porous medium is numerically investigated. The top and bottom walls are moving in opposite directions at different temperatures, while the side vertical walls are considered adiabatic. The governing equations are solved using the finite-volume method with the SIMPLE algorithm. The numerical procedure adopted in this study yields a consistent performance over a wide range of parameters that were 10⁻⁴ ≤ Da ≤ 10⁻¹ and 0 ≤ Ra _I ≤ 10⁴. The effects of the parameters involved on the heat transfer characteristics are studied in detail. It is found that the variation of the average Nusselt number is non-linear for increasing values of the Darcy number with uniform or non-uniform heating condition.