Prediction of natural convection flow using network model and numerical simulations inside enclosure with distributed solid blocks

Steady state natural convection of a fluid with Pr ≈ 1 within a square enclosure containing uniformly distributed, conducting square solid blocks is investigated. The side walls are subjected to differential heating, while the top and bottom ones are kept adiabatic. The natural convection flow is predicted employing the nondimensional volumetric flow rate (Q_max^* Q_{\max }^{*} ) by using a network model and also using numerical simulations. For identical solid and fluid thermal conductivities (i.e. k _s  = k _f ), a parametric study of the effect of number of blocks (N ²), gap size (δ) and enclosure Rayleigh number (Ra) on Q_max^* Q_{\max }^{*} is performed using the two approaches. Network model predictions are observed to agree well with that from the simulations until Raδ³ ~ 12. Considering the enclosure with blocks as a porous medium, for a fixed enclosure Ra number, increasing the number of blocks for a fixed volumetric porosity leads to a decrease in enclosure permeability, which in turn reduces the flow rate. When the number of blocks is fixed, and for a given Ra number, the flow rate increases as the porosity increases by widening the gap between the blocks.     

Unsteady natural convection in an enclosure with vertical wavy walls

In this paper, unsteady heat transfer and fluid flow characteristics in an enclosure are investigated. The enclosure consists of two vertical wavy and two horizontal straight walls. The top and the bottom walls are considered adiabatic. Two wavy walls are kept isothermal and their boundaries are approximated by a cosine function. Governing equations including continuity, momentum and energy were discretized using the finite-volume method and solved by SIMPLE method in curvilinear coordinate. Simulation was carried out for a range of Grashof number Gr = 10³–10⁶, Prandtl number Pr = 0.5–4.0, wave ratio A (defined by amplitude/wavelength) 0.0–0.35 and aspect ratio W (defined by average width/wavelength) 0.5–1.0. Streamlines and isothermal lines are presented to corresponding flow and thermal fields. Local and average Nusselt number distributions are presented. The obtained results are in good agreement with available numerical and experimental data.     

Interaction of turbulent natural convection and surface thermal radiation in inclined square enclosures

Two-dimensional numerical studies of flow and temperature fields for turbulent natural convection and surface radiation in inclined differentially heated enclosures are performed. Investigations are carried out over a wide range of Rayleigh numbers from 10⁸ to 10¹², with the angle of inclination varying between 0° and 90°. Turbulence is modeled with a novel variant of the k–ε closure model. The predicted results are validated against experimental and numerical results reported in literature. The effect of the inclination of the enclosure on pure turbulent natural convection and the latter’s interaction with surface radiation are brought out. Profiles of turbulent kinetic energy and effective viscosity are studied to observe the net effect on the intensity of turbulence caused by the interaction of natural convection and surface radiation. The variations of local Nusselt number and average Nusselt number are presented for various inclination angles. Marked change in the convective Nusselt number is found with the orientation of enclosure. Also analyzed is the influence of change in emissivity on the flow and heat transfer. A correlation relevant to practical applications in the form of average Nusselt number, as a function of Rayleigh number, Ra, radiation convection parameter, N _RC and inclination angle of the enclosure, φ is proposed.     

Conjugate heat transfer in enclosures with openings for ventilation

Conjugate heat transfer by natural convection and conduction in enclosures with openings has been studied by a numerical method. The enclosure contained a chimney consisting of a vertical solid wall, which was insulated on one side and a constant heat flux applied on the other. Vertical boundaries with openings were isothermal and horizontal boundaries adiabatic. These problems are encountered in heat transfer in buildings and heat management in electronic equipment. Two dimensional equations of conservation of mass, momentum and energy, with the Boussinesq approximation are solved using the Simpler method. Various geometrical parameters were: aspect ratio, A from 0.5 to 2.0, openings' heights, h ₁ and h ₂ from 0.10 to 0.30, orifice height, h ₃ from 0.05 to 0.15, insulation thickness, w ₁ from 0 to 0.10, wall thickness, w ₂ from 0.05 to 0.15 and chimney width, w ₃ from 0.05 to 0.15. Rayleigh number, Ra was varied from 10 ⁸ to 10 ¹² and the conductivity ratio, k _r was from 1 to 40. The results are reduced in terms of the normalized Nusselt number, Nu and volume flow rate, V? as a function of Ra number, and other non dimensional geometrical parameters. The isotherms and streamlines are produced for various Ra numbers and geometrical conditions. It is found that Nu and V? are both an increasing function of Ra, h ₁ at high Ra numbers, h ₃, and k _r. They are a decreasing function of h ₁ at low Ra numbers, h ₂, and w ₂. Nu and V? have optima with respect to w ₁, w ₃ and A.     

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.     

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.     

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.     

An experimental investigation on the airside performance of fin-and-tube heat exchangers having sinusoidal wave fins

The heat transfer and friction characteristics of the heat exchangers having sinusoidal wave fins were experimentally investigated. Twenty-nine samples having different waffle heights (1.5 and 2.0 mm), fin pitches (1.3–1.7 mm) and tube rows (1–3) were tested. Focus was given to the effect of waffle configuration (herringbone or sinusoidal) on the heat transfer and friction characteristics. Results show that the sinusoidal wave geometry provides higher heat transfer coefficients and friction factors than the herringbone wave geometry, and the difference increases as the number of row increases. The j/f ratios of the herringbone wave geometry, however, are larger than those of the sinusoidal wave geometry. Compared with the herringbone wave geometry, the sinusoidal wave geometry yielded a weak row effect, which suggests a superior heat transfer performance at the fully developed flow region for the sinusoidal wave geometry. Possible reasoning is provided considering the flow characteristics in wavy channels. Within the present geometric variations, the effect of waffle height on the heat transfer coefficient was not prominent. The effect of fin pitch was also negligible. Existing correlations highly overpredicted both the heat transfer coefficients and friction factors. A new correlation was developed based on the present data.     

Taguchi design of three dimensional simulations for optimization of turbulent mixed convection in a cavity

This study discusses the application of Taguchi method in assessing maximum heat transfer rate for the turbulent mixed convection in an enclosure embedded with rotating isothermal cylinder. The simulations were planned based on Taguchi’s L16 orthogonal array with each trial performed under different conditions of position of the cylinder, Reynolds number (Re) and Rayleigh number (Ra). The thermal lattice Boltzmann based on D3Q19 methods without any turbulent submodels was purposed to simulate the flow and thermal fields. A relaxation time method with the stability constants is introduced to solve turbulent natural convection problems. Signal-to-noise ratios (S/N) analysis were carried out in order to determine the effects of process parameters and optimal factor settings. Finally, confirmation tests verified that Taguchi method achieved optimization of heat transfer rate with sufficient accuracy.     

Experimental investigation of natural convection heat transfer in horizontal and inclined annular fluid layers

In the present study, an experimental investigation of heat transfer and fluid flow characteristics of buoyancy-driven flow in horizontal and inclined annuli bounded by concentric tubes has been carried out. The annulus inner surface is maintained at high temperature by applying heat flux to the inner tube while the annulus outer surface is maintained at low temperature by circulating cooling water at high mass flow rate around the outer tube. The experiments were carried out at a wide range of Rayleigh number (5 × 10⁴ < Ra < 5 × 10⁵) for different annulus gap widths (L/D _o = 0.23, 0.3, and 0.37) and different inclination of the annulus (α = 0°, 30° and 60°). The results showed that: (1) increasing the annulus gap width strongly increases the heat transfer rate, (2) the heat transfer rate slightly decreases with increasing the inclination of the annulus from the horizontal, and (3) increasing Ra increases the heat transfer rate for any L/D _o and at any inclination. Correlations of the heat transfer enhancement due to buoyancy driven flow in an annulus has been developed in terms of Ra, L/D _o and α. The prediction of the correlation has been compared with the present and previous data and fair agreement was found.     

Natural convection in an inclined porous enclosure with an off-center diathermal partition

A numerical study has been made of natural convection in an inclinded porous enclosure with an off-center diathermal partition. A temperature difference is imposed between the two isothermal end walls and the other two walls are assumed to be adiabatic. Numerical results are obtained for Rayleigh numbers (Ra) in the range of 10 to 500, the dimensionless partition location ( \(\bar S\) ) ranging from 0.125 to 0.875, the aspect ratios (A) of the enclosure ranging from 0.5 to 5, and the inclination angles (φ) of ?60, ?30, 0, 30, 60 degrees. It is found that the partition location has strong influence at lowRa and relatively weaker influence at highRa. The average Nusselt number reaches the minimum value when the partition is in the middle of the vertical enclosure, and the maximum Nusselt number occurs around φ = 30 degrees.     

Transient natural convection between concentric and vertically eccentric spheres with mixed boundary conditions

Transient analysis has been investigated numerically to determine heat transfer by natural convection between concentric and vertically eccentric spheres with constant heat flux on the inner wall and a specified isothermal temperature on the outer wall. The governing equations, in terms of vorticity, stream function and temperature are expressed in a spherical polar coordinate system. The alternating direction implicit method and the successive over-relaxation techniques are applied to solve the finite difference form of governing equations. A physical model is introduced which accounts for the effects of fluid buoyancy as well as eccentricity of the outer sphere. Transient solutions of the entire flow field are obtained for a range of modified Rayleigh number (10³<Ra?<5×10⁵), for a Prandtl number of 0.7 and a radius ratio of 2.0, with the outer sphere near the top and bottom of the inner sphere (ε=±0.625). Results of the parametric study conducted further reveal that the heat and flow fields are primarily dependent on the modified Rayleigh number and the eccentricity of the spherical annulus. The results of average Nusselt numbers are also compared with the results obtained for flow between two isothermal spheres.     

Numerical study of effects of pulsatile amplitude for transitional turbulent pulsatile flow in pipes with ring‐type constrictions

The effects of pulsatile amplitude on sinusoidal transitional turbulent flows through a rigid pipe in the vicinity of a sharp‐edged mechanical ring‐type constriction have been studied numerically. Pulsatile flows were studied for transitional turbulent flow with Reynolds number (Re) of the order of 10⁴, Womersley number (Nw) of the order of 50 with a corresponding Strouhal number (St) of the order of 0.04. The pulsatile flow considered is a sinusoidal flow with dimensionless amplitudes varying from 0.0 to 1.0. Transitional laminar and turbulent flow characteristics in an alternative manner within the pulsatile flow fields were observed and studied numerically. The flow characteristics were studied through the pulsatile contours of streamlines, vorticity, shear stress and isobars. It was observed that fluid accelerations tend to suppress the development of flow disturbances. All the instantaneous maximum values of turbulent kinetic energy, turbulent viscosity, turbulent shear stress are smaller during the acceleration phase when compared with those during deceleration period. Various parametric equations within a pulsatile cycle have also been formulated through numerical experimentations with different pulsatile amplitudes. In the vicinity of constrictions, the empirical relationships were obtained for the instantaneous flow rate (Q), the pressure gradient (dp/dz), the pressure loss (P_loss), the maximum velocity (V_max), the maximum vorticity (ζ_max), the maximum wall vorticity (ζ_w,max), the maximum shear stress (τ_max) and the maximum wall shear stress (τ_w,max). Elliptic relation was observed between flow rate and pressure gradient. Quadratic relations were observed between flow rate and the pressure loss, the maximum values of shear stress, wall shear stress, turbulent kinematic energy and the turbulent viscosity. Linear relationships exist between the instantaneous flow rate and the maximum values of vorticity, wall vorticity and velocity. The time‐average axial pressure gradient and the time average pressure loss across the constriction were observed to increase linearly with the pulsatile amplitude. Copyright © 1999 John Wiley & Sons, Ltd.     

Numerical investigation of transient laminar natural convection of air in a tall cavity

Transient laminar natural convection of air in a tall cavity has been studied numerically. The Navier-Stokes and Energy equations were solved by the accurate projection method (PmIII), in which the derived Poisson equation for pressure potential was solved by the approximate factorization one method (AF1). The aspect ratio of the tall cavity is 16, and the Prandtl number of air filled in the tall cavity is 0.71. To obtain the numerical results of heat transfer by natural convection of air in the tall cavity, the second order schemes for the space and time discretizations were utilized. The availability of the numerical algorithm was also assessed by considering the natural convection of air in a square cavity which is differentially heated from side walls. It was found that the overall Nusselt numbers for the Rayleigh numbers covering the range from 1000 to 100000 reveal a good agreement with measured data. When Ra takes the value in the range from 100000 to 600000, the overall Nusselt number have a relative deviation less than 18% from the experimental data. For the suddenly heating mode, the multicellular flow pattern occurs when Rayleigh number belongs to the range of Ra from 7000 to 20000. or greater than 115000. At the critical number of cats' eye instability, the cell distance is just twice of the cavity width. This is rather similar to the observed result for Bénard problem. When Ra is over 115000, a further increase of heat flux across the tall cavity causes serious cell-breaking. There are 8 cells when Ra = 600000.     

Rayleigh–Bénard convection driven by a long wavelength heating

Natural convection in a two-dimensional horizontal layer has been investigated. The layer is confined between two parallel horizontal plates. The upper plate is kept isothermal, while the lower plate has an externally imposed, long wavelength, spatially sinusoidal heating with the amplitude expressed in terms of the Rayleigh number Ra and the wavelength characterized by the wave number α. Only steady-state flow structures and their bifurcations have been considered. The detailed analysis has been carried out for two Prandtl numbers, i.e. Pr = 0.7 and Pr = 7, and only small differences in the bifurcation diagrams have been observed. When Ra < Ra _cr = 427, convection has a simple topology consisting of one pair of counter-rotating rolls per heating period. Secondary motion in the form of rolls aligned in the direction of the primary rolls and concentrated around the hot spots occurs for Ra > 427. When 427 < Ra < ～470 and α < ～0.14, the secondary motion is described by the supercritical pitchfork bifurcation. One of the branches of this bifurcation is associated with an odd number of secondary rolls per half wavelength, with rolls above the hot spots rotating in the direction opposite to the primary rolls. The other branch is associated with an even number of secondary rolls per half wavelength, with the rolls above the hot spots co-rotating with the primary rolls. The new rolls are pinched off in pairs when α decreases. When Ra > ～470 and α > ~0.14, bifurcation assumes the form of “bifurcation from infinity”. The main branch is associated with one pair of rolls per heating period for α > 0.25. Decrease in α along this branch results in the formation of secondary rolls, with the rolls at the hot spot co-rotating with the primary rolls. The lower part of the other branch is associated with one pair of rolls per heating period in the limit α → 0. Increase in α results in pinching off a single roll which counter-rotates with respect to the primary roll at the hot spot.     

Large-eddy simulation of buoyancy-driven natural ventilation in an enclosure with a point heat source

Rising buoyant plumes from a point heat source in a naturally ventilated enclosure have been investigated using large-eddy simulation (LES). The aim of the work is to assess the performance and the accuracy of LES for modelling buoyancy-driven displacement ventilation of an enclosure and to shed more light on the transitional behaviour of the plume and the coherent structures involved. The Smagorinsky sub-grid scale model is used for the unresolved small-scale turbulence. The Rayleigh number, Ra is chosen to be in the range where spatial transition from laminar to turbulent flow takes place (Ra = 1.5 × 10⁹). The plume properties (source strength and rate of spread) as well as the ventilation properties (stratification height and temperature of stratified layer) estimated using the theory of Linden et al. are found to agree reasonably well with the LES results. The variation of the plume width with height indicates a linear variation of the entrainment coefficient rather than a constant value used by Linden et al. for a fully turbulent thermal plume. Flow visualisation revealed the nature of the large-scale coherent structures involved in the transition to turbulence in the plume. The most excited modes observed in the velocity, pressure and temperature fields spectra correspond to Strouhal number in the range 0.3 ≤ St ≤ 0.55 which is in agreement with those observed by Zhou et al. for a turbulent forced plume. Excited modes less than thisvalue (St = 0.2) were observed and may be due to low-frequency motions felt throughout the flow.     

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.     

Numerical study of steady magneto-convection around an adiabatic body inside a square enclosure in low Prandtl numbers

In this paper, the effects of Prandtl number on the steady magneto-convection around a centrally located adiabatic body inside a square enclosure are numerically investigated. Two-dimensional nonlinear governing equations are discretized using the control volume method and hybrid scheme. The equations are solved using SIMPLER algorithm. The results are displayed in the form of streamlines and isotherms when the Rayleigh number varies between 10³ and 10⁶, the Hartmann number changes between 0 and 100 and the Prandtl number ranges between 0.005 and 0.1. The ratio of the buoyancy force to the Lorentz force (Ra/Ha ²) is introduced as an index to compare the contribution of natural convection and magnetic field strength on heat transfer. The results obtained from numerical modeling show that the Prandtl number has not considerable effect on heat transfer at low Rayleigh numbers. The effect of magnetic field strength on convection is increased by increasing Prandtl number. The effect of Prandtl number on the average Nusselt number in the presence of a magnetic field is less than the case without a magnetic field.     

Prediction of unobstructed flow for co‐rotating multi‐disk drive in an enclosure

The flow structure and isotherms of hard disk drives (HDD) are investigated using a finite element method (FEM). The governing equations are based on the three‐dimensional axisymmetric Navier–Stokes partial differential equations (PDEs) with Galerkin FE formulation. Co‐rotating models are selected that include the non‐ventilated configuration within an enclosure. With various operating conditions for the disk system, the following important parameters are considered: disk number (n), rotational speed (Ro), and wall temperature. The flow structure changes rapidly when the rotational Reynolds number (Re_ϕ) is increased. The flow has a greater tendency to flow radially outwards and the swirling velocity tends to be more vertically orientated, especially for high Re_ϕ values. The isotherms only have small varying regions near the rotating axis, forming outward arcs near the wall and inward arcs near the end gap of the disk. Different from the case without the enclosure, the vorticities exist along the outer disk ends. Both the swirling velocity and isotherms indicate nearly symmetrical characteristics, as expected. A higher temperature gradient occurs near the right outer disk ends, which implies the characteristic of higher heat flux. A commercial computational fluid dynamic (CFD) code, CFX‐5, was chosen to validate the results. Copyright © 2001 John Wiley & Sons, Ltd.     

Effects of inclination angle on natural convection in enclosures filled with Cu–water nanofluid

Effects of inclination angle on natural convection heat transfer and fluid flow in a two-dimensional enclosure filled with Cu-nanofluid has been analyzed numerically. The performance of nanofluids is tested inside an enclosure by taking into account the solid particle dispersion. The angle of inclination is used as a control parameter for flow and heat transfer. It was varied from  = 0° to  = 120°. The governing equations are solved with finite-volume technique for the range of Rayleigh numbers as 10³  Ra  10⁵. It is found that the effect of nanoparticles concentration on Nusselt number is more pronounced at low volume fraction than at high volume fraction. Inclination angle can be a control parameter for nanofluid filled enclosure. Percentage of heat transfer enhancement using nanoparticles decreases for higher Rayleigh numbers.