Numerical study of natural convection flow in a vertical slot

The stability of convective motion, generated by a lateral temperature difference across a vertical slot, is studied numerically over a range ofGr=5000 to 1.5 × 10⁵,Pr=0.01 to 10, andA=8,16 and 20. Various cellular flow structures and temperature patterns are illustrated. Several branches of solutions characterized by different numbers of the cells in the flow patterns as well as by both steady and unsteady multicellular patterns are found for low-Prandtl-number fluid in the vertical slot. Meanwhile, the behaviors of the temperature variation and heat transfer are also discussed. The project supported by the National Natural Science Foundation of China (59776011) and by the Returnee from Abroad Funding of Academia Sinica.     

Natural convection from a vertical cylinder in a thermally stratified porous medium

The problem of non-Darcy natural convection adjacent to a vertical cylinder embedded in a thermally stratified porous medium has been analyzed. Nonsimilarity solutions are obtained for the case that the ambient temperature increases linearly with height of the cylinder. A generalized flow model was used in the present study to include the effects of the macroscopic viscous term and the microscopic inertial force. Also, the thermal dispersion effect is considered in the energy equation. Thus, the main aim of this work is to examine the effects of thermal stratification and non-Darcy flow phenomena on the free convection flow and heat transfer characteristics. It was found that the present problem depends on six parameters, namely, the local thermal stratification parameter ξ, the boundary effect parameter Bp, the modified Grashof number Gr^*, wall temperature exponent m, the curvature parameter ω, and the modified Rayleigh number based on pore diameter Ra _d . The impacts of these governing parameters on the local heat transfer parameter are discussed in great detail. Also, representative velocity and temperature profiles are presented at selected values of the thermal stratification parameter. In general, the local heat transfer parameter is increased with increasing the values of m, ω, and Ra _d ; while it is decreased with increasing the values of ξ, Bp, and Gr^*. Received on 19 May 1998     

Effects of thermal dispersion and stratification on combined convection on a vertical surface embedded in a porous medium

The effects of thermal dispersion and thermal stratification on mixed convection about a vertical surface in a porous medium are studied. The conservation equations that govern the problem are reduced to a system of nonlinear ordinary differential equations. The resulting equations are solved on the basis of the local similarity approximation. The results indicate that both dispersion and stratification effects have considerable influence on the heat transfer rate.     

Non-isothermal Poiseuille flow and its stability in a vertical annulus filled with porous medium

In this article, the non-isothermal Poiseuille flow and its stability in a vertical annulus filled with porous medium are investigated. The flow is induced by external pressure gradient and buoyancy force due to linearly varying inner wall temperature. The non-Darcy model along with Boussinesq approximation has been used. The Chebyshev spectral-collocation method has been adopted to solve the governing equations related to basic flow as well as its stability. Special attention is given to understand the effect of curvature parameter of the annular geometry on the flow, heat transfer rate and stability of the stably stratified flow. A comprehensive numerical experiment indicates that reducing gap between two concentric cylinders decreases the heat transfer rate as well as the maximum magnitude of the flow velocity. It stabilizes the flow which has been shown through stability analysis. Furthermore, appropriateness of the Forchheimer term in the momentum equation has been examined by investigating the flow regime as well as its stability in the presence and absence of Forchheimer term. Finally, it has been found from the energy analysis at critical point that the thermal-buoyant instability is the only mode of instability for the considered range of different parameters.     

Free convection from a vertical plate with nonuniform surface heat flux and embedded in a porous medium

An analysis is presented for the calculation of heat transfer due to free convective flow along a vertical plate embedded in a porous medium with an arbitrarily varying surface heat flux. By applying the appropriate coordinate transformations and the Merk series, the governing energy equation is expressed as a set of ordinary differential equations. Numerical solutions are presented for these equations which represent universal functions and several computational examples are provided.     

Effects of anisotropy on the free convection from a vertical plate embedded in a porous medium

The effects of anisotropy on the steady laminar boundary-layer free convection over a vertical impermeable surface are analysed by using the method of integral relations. If the permeability in the direction orthogonal to the plate is greater than the permeability along the plate, then there is an increase in the temperature field.     

自然对流边界层中湍流的发生

自然对流边界层中从层流到湍流的转捩经历了浮力振型、无摩擦振型和黏性振型的三重流动不稳定性相继产生的前转捩过程,以及近壁迅速出现强湍流源,随之平缓地向自模拟的湍流边界层过渡的热转捩过程.浮力振型在修正Grashof数G>40时开始失稳并成为主要振型,在振幅分布中3种振型的临界层位置处出现3个峰值;在G>100时浮力振型消失,无摩擦振型失稳并成为主要振型,振幅分布中在近壁区还出现黏性振型的峰值;在G>170时无摩擦振型经非线性演化在外层形成较弱的湍流,但内层黏性应力仍远高于湍流应力,振幅分布中仅有与黏性振型相应的峰值,在频谱中黏性振型的基频、第一、第二、第三阶亚谐频随G的增加相继出现,此时黏性不稳定波的高频成分已转化为湍流,但低频成分仍按线性规律增长,直至湍流惯性子区开始形成;至G>800时黏性振型消失,并在G=850附近时近壁区出现强湍流源,湍流应力、湍能产生项和近壁湍流热流率剧增.在热转捩后期,湍流应力和湍能产生项明显下降,流动在内外层趋于平衡.     

Instabilities in two-liquid layers subject to a horizontal temperature gradient

We study theoretically the stability of two superposed fluid layers heated laterally. The fluids are supposed to be immiscible, the interface undeformable and of infinite horizontal extension. Combined thermocapillary and buoyancy forces give rise to a basic flow when a temperature difference is applied. The calculations are performed for a melt of GaAs under a layer of molten B₂O₃, a configuration of considerable technological importance. Four different flow patterns and five temperature configurations are found for the basic state in this system. A linear stability analysis shows that the basic state may be destabilized by oscillatory motions leading to the so-called hydrothermal waves. Depending on the relative height of the two layers these hydrothermal waves propagate parallel or perpendicular to the temperature gradient. This analysis reveals that these perturbations can alter significantly the liquid flow in the liquid-encapsulated crystal growth techniques. PACS 47.20.Dz, 47.20.Bp, 47.54.+r, 47.27.Te, 44.25.+f, 47.20.Ma     

Numerical investigation of variable viscosities and thermal stratification effects on MHD mixed convective heat and mass transfer past a porous wedge in the presence of a chemical reaction

An analysis is presented to investigate the effects of variable viscosities and thermal stratification on the MHD mixed convective heat and mass transfer of a viscous, incompressible, and electrically conducting fluid past a porous wedge in the presence of a chemical reaction. The wall of the wedge is embedded in a uniform nonDarcian porous medium in order to allow for possible fluid wall suction or injection. The governing boundary layer equations are written into a dimensionless form by similarity transformations. The transformed coupled nonlinear ordinary differential equations are solved numerically with finite difference methods. Numerical calculations up to the thirdorder level of truncation are carried out for different values of dimensionless parameters. The results are presented graphically, and show that the flow field and other quantities of physical interest are significantly influenced by these parameters. The results are compared with those available in literature, and show excellent agreement.     

On natural convection from a vertical plate with a prescribed surface heat flux in porous media

This paper presents a theoretical and numerical investigation of the natural convection boundary-layer along a vertical surface, which is embedded in a porous medium, when the surface heat flux varies as (1 +x ²)), where is a constant andx is the distance along the surface. It is shown that for > -1/2 the solution develops from a similarity solution which is valid for small values ofx to one which is valid for large values ofx. However, when -1/2 no similarity solutions exist for large values ofx and it is found that there are two cases to consider, namely < -1/2 and = -1/2. The wall temperature and the velocity at large distances along the plate are determined for a range of values of .Notation g Gravitational acceleration - k Thermal conductivity of the saturated porous medium - K Permeability of the porous medium - l Typical streamwise length - q _w Uniform heat flux on the wall - Ra Rayleigh number, =gK(q _w /k)l/(v) - T Temperature - Too Temperature far from the plate - u, v Components of seepage velocity in the x and y directions - x, y Cartesian coordinates - Thermal diffusivity of the fluid saturated porous medium - The coefficient of thermal expansion - An undetermined constant - Porosity of the porous medium - Similarity variable, =y(1+x ) ^/3/x ^1/3 - A preassigned constant - Kinematic viscosity - Nondimensional temperature, =(T – T )Ra^1/3 k/qw - Similarity variable, = =y(log_e x)^1/3/x ^2/3 - Similarity variable, =y/x ^2/3 - Stream function     

Combined forced and free convective flow of water at 4°C past a semi-infinite vertical plate

Effects of buoyancy forces on forced and free convective flow of water at 4°C past a semi-infinite vertical plate at constant temperature are studied. Flow is assumed to be vertically upwards. Similarity solutions are derived and the resulting equations are solved numerically on a computer. Velocity and temperature profiles are shown graphically and numerical values of the skin friction and the rate of heat transfer are entered in tables. It is observed that the skin friction and the Nusselt number increase with increasing Gr/Re², where Gr is the Grashof number and Re is the Reynolds number     

Magnetohydrodynamics convective flow in a vertical slot through a porous medium

An analysis is presented with magnetohydrodynamics natural convective flow of a viscous Newtonian fluid saturated porous medium in a vertical slot. The flow in the porous media has been modeled using the Brinkman model. The fully-developed two-dimensional flow from capped to open ends is considered for which a continuum of solutions is obtained. The influence of pertinent parameters on the flow is delineated and appropriate conclusions are drawn. The asymptotic behaviour and the volume flux are analyzed and incorporated graphically for the three-parameter family of solution.     

Laminar mixed convection in the entrance region of horizontal rectangular ducts

A generally applicable finite element procedure for the prediction of laminar mixed convection in horizontal straight ducts of arbitrary cross-section is presented. The procedure, based on the parabolized simplification of the complete Navier-Stokes equations and on the Boussinesq approximation of the buoyancy terms, is validated through comparisons of computed results with the available literature data for mixed convection in the entrance region of a rectangular duct of aspect ratio a=2. Uniform heating at different sides is considered as the thermal boundary condition, although the proposed formulation allows specification of most thermal boundary conditions of practical interest.     

Numerical heat transfer in a cavity with a solar control coating deposited to a vertical semitransparent wall

A transient two‐dimensional computational model of combined natural convection, conduction, and radiation in a cavity with an aspect ratio of one, containing air as a laminar and non‐participating fluid, is presented. The cavity has two opaque adiabatic horizontal walls, one opaque isothermal vertical wall, and an opposite semitransparent wall, which consists of a 6‐mm glass sheet with a solar control coating of SnS–Cu_xS facing the cavity. The semitransparent wall also exchanges heat by convection and radiation from its external surface to the surroundings and allows solar radiation pass through into the interior of the cavity. The momentum and energy equations in the transient state were solved by finite differences using the alternating direction implicit (ADI) technique. The transient conduction equation and the radiative energy flux boundary conditions are coupled to these equations. The results in this paper are limited to the following conditions: 10⁴≤Gr≤10⁶, an isothermal vertical cold wall of 21°C, outside air temperatures in the range 30°C≤T₀≤40°C and incident solar radiation of AM2 (750 W m⁻²) normal to the semitransparent wall. The model allows calculation of the redistribution of the absorbed component of solar radiation to the inside and outside of the cavity. The influences of the time step and mesh size were considered. Using arguments of energy balance in the cavity, it was found that the percentage difference was less than 4 per cent, showing a possible total numerical error less than this number. For Gr=10⁶ a wave appeared in the upper side of the cavity, suggesting the influence of the boundary walls over the air flow inside the cavity. A Nusselt number correlation as a function of the Rayleigh number is presented. Copyright © 2000 John Wiley & Sons, Ltd.     

Effect of density extremum on the solidification of water on a vertical wall of a rectangular cavity

Experiments were performed in a two-dimensional rectangular cavity to study the transient flow in an initially isothermal and motionless fluid due to a step decrease in temperature on one of the two vertical end walls. In the experiments water was used as the phase-change medium, with the cold-wall temperature maintained below the freezing temperature. The opposite vertical wall was kept at the initial temperature, greater than the temperature where the density extremum occurs. The growth of ice and the transient flow in the cavity were visualized with the aid of a tracer technique to examine the effect of density inversion. The temperature field was continuously recorded by an array of thermocouples. It was found that the density inversion of water strongly influences both the growth of ice and the convective flow in the liquid region of the test cavity.     

Non-linear dynamics and pattern formation in a vertical fluid layer heated from the side

We study both experimentally and numerically the convective flow in a tall vertical slot with differently heated walls. The flow is investigated for the fluid with the Prandtl number Pr=26, which is large enough to ensure the traveling waves as primary instability and small enough to prevent boundary layer convection. The flow evolution is determined on the base of the visual observations, power spectra and amplitude analysis. In the numerical simulations of two- and three-dimensional flows, we accept an assumption of an infinite fluid layer. The satisfactory agreement with experiment is observed, and the sequence of convection states is discovered. It starts with a plane-parallel flow as primary solution, which becomes unstable to two counter-propagating waves. It is followed by a tertiary three-dimensional flow in the form of wavy traveling waves. As the Grashof number is increased even further, a chaotically oscillating cellular pattern consisting of the pieces of broken waves arises. The formation of a structure in the form of the vertical rolls chaotically modulated along axes concludes this complicated picture.     

Turbulent forced convective heat transfer in two-phase evaporating droplet flow through a vertical pipe

A physical model was developed to study heat transfer in turbulent dispersed flow at very high vapor quality in a vertical pipe by numerically solving the coupling governing differential equations for both phases. Major heat transfer mechanisms included in the model were the thermal nonequilibrium effects, droplet vaporization, droplet deposition on the duct wall and thermal radiative transfer. The predicted results indicated that vapor superheating is dominant for the cases with high wall superheat, otherwise droplet vaporization dominates the energy transport processes. Heat transfer during the droplet-wall interaction only exists at low wall superheat but in small amounts.     

Turbulent natural convection scaling in a vertical channel

Using direct numerical simulation (DNS) data, this study appraises existing scaling laws in literature for turbulent natural convection of air in a differentially heated vertical channel. The present data is validated using past DNS studies, and covers a range of Rayleigh number, Ra between 5.4 × 10⁵ and 2.0 × 10⁷. We then appraise and compare the various scaling laws proposed by Versteegh and Nieuwstadt, 1999, Hölling and Herwig, 2005, Shiri and George, 2008, George and Capp, 1979 with the profiles of the mean temperature defect, mean streamwise velocity, normal velocity fluctuations, temperature fluctuations and Reynolds shear stress. Based on the arguments of an inner (near-wall) and outer (channel centre) region, the data is found to support a minus one-third power law for the mean temperature in an overlap region. Using the inner and outer temperature profiles, an implicit heat transfer equation is obtained and we show that a correction term is non-negligible for the present Ra range when compared with explicit equations found in literature. In addition, we determined that the mean streamwise velocity and normal velocity fluctuations collapse in the inner region when using the outer velocity scale. We also find that the temperature fluctuations scale in inner coordinates, in contrast to the outer scaling behaviour reported in the past. Lastly, we show evidence of an incipient proportional relationship between friction velocity, u_τ, and the outer velocity scale, u_o, with increasing Ra.     

Sharp entry and transition effects for laminar combined convection of water in vertical tubes

In previous work the authors showed a sharp entry to have a definite effect on laminar forced convection to water in a vertical circular tube. The study has been extended to situations of strong aiding natural convection, and new experimental data and numerical predictions are reported. The expected entry effect is confirmed, but it is found to be less marked in a strong combined convection field than in the previous forced convection study. The experimental data also include evidence of transition from laminar flow, and a possible criterion of transition is investigated, based on the axial location of minimum local Nusselt number. This is experimental and predictive data for this criterion are compared with the experimental correlation of Lawrence and Chato based on a criterion of temperature fluctuations     

A linear stability analysis of thermal convection in spherical shells with variable radial gravity based on the Tau-Chebyshev method

The onset of thermal convection in a non-rotating spherical shell is investigated using linear theory. The Tau-Chebyshev spectral method is used to integrate the linearized equations. We investigate the onset of thermal convection by considering two cases of the radial gravitational field (i) a local acceleration, acting radially inward, that is proportional to the distance from the center r, and (ii) a radial gravitational central force that is proportional to r⁻ⁿ. The former case has been widely analyzed in the literature, because it constitutes a simplified model that is usually used, in astrophysics and geophysics, and is studied here to validate the numerical method. The latter case was analyzed since the case n = 5 has been experimentally realized (by means of the dielectrophoretic effect) under microgravity condition, in the experimental container called GeoFlow, inside the International Space Station. Our study is aimed to clarify the role of (i) a radially inward central force (either proportional to r or to r⁻ⁿ), (ii) a base conductive temperature distribution provided by either a uniform heat source or an imposed temperature difference between outer and inner spheres, and (iii) the aspect ratio η (ratio of the radii of the inner and outer spheres), on the critical Rayleigh number. In all cases the surface of the spheres has been assumed to be rigid. The results obtained with the linear theory based on the Tau-Chebyshev spectral method are compared with those of the integration of the full non-linear equations solved by using the spectral element method. By using the Tau-Chebyshev method, we were able to explore new cases that have not been previously reported in the literature.