Regimes of haline convection during the evaporation of groundwater containing a dissolved admixture

The problem of formation of salt concentration profile in high-permeability soil duringwater evaporation and solution upflow is considered. The numerical experiments performed showed that the salt concentration profile may be either stable or unstable. As instability develops, there arises natural haline convection whose different regimes are described and analyzed. If the evaporation intensity is moderate, in soil the curvilinear upward or circulatory flow that fills the entire layer is established. The intense evaporation leads to the formation of a small-scale structure of salt “fingers”. Boundaries between regimes are determined.     

Convective heat loss of water in a circular pipe cooled at constant rate

A numerical study is made of the heat loss by natural convection of water within a horizontal circular cylinder with wall temperature decreasing at a constant rate. The particular situation of water with maximum density at 4 °C is formulated in dimensionless relations based on a linear relationship between the water thermal expansion coefficient and the temperature. Such an approach leads to an exhaustive solution in terms of a non linear Rayleigh number. The link is also established with the standard situation where the hypothesis of a linear relationship between density and temperature is applicable. In particular it is shown that the quasi steady state results obtained for a standard situation become equilibrium curves to which the system tends with increasing difference between the temperature of the boundary and 4 °C. A complete numerical solution is obtained for non linear Rayleigh numbers ranging between 0 and 10⁷. Previous numerical and experimental results on the horizontal circular cylinder are also discussed and recast in terms of the present dimensional approach.     

Instability of an Axisymmetric Vortex in a Stably Stratified, Rotating Environment

We investigate the stability of a barotropic vorticity monopole whose stream function is a Gaussian function of the radial coordinate. The model is based on the inviscid Boussinesq equations. The vortex is assumed to exist on an $f$-plane, in an environment with constant, stable density stratification. In the unstratified, nonrotating case, we find growth rates that increase monotonically with increasing vertical wave number, the so-called “ultraviolet catastrophe” characteristic of symmetric instability. This type of instability leads to rapid turbulent collapse of the vortex, possibly accompanied by wave radiation. In the limit of strong background stratification and rotation, the vortex exhibits a scale-selective instability which leads to the formation of stable lenses. The transition between these two regimes is sharp, and coincides approximately with the centrifugal stability boundary. Received 6 December 1996 and accepted 1 November 1997     

Laboratory Flow Experiments for Visualizing Carbon Dioxide-Induced,Density-Driven Brine Convection

Injection of carbon dioxide (CO₂) into saline aquifers confined by low- permeability cap rock will result in a layer of CO₂ overlying the brine. Dissolution of CO₂ into the brine increases the brine density, resulting in an unstable situation in which more-dense brine overlies less-dense brine. This gravitational instability could give rise to density-driven convection of the fluid, which is a favorable process of practical interest for CO₂ storage security because it accelerates the transfer of buoyant CO₂ into the aqueous phase, where it is no longer subject to an upward buoyant drive. Laboratory flow visualization tests in transparent Hele-Shaw cells have been performed to elucidate the processes and rates of this CO₂ solute-driven convection (CSC). Upon introduction of CO₂ into the system, a layer of CO₂-laden brine forms at the CO₂-water interface. Subsequently, small convective fingers form, which coalesce, broaden, and penetrate into the test cell. Images and time-series data of finger lengths and wavelengths are presented. Observed CO₂ uptake of the convection system indicates that the CO₂ dissolution rate is approximately constant for each test and is far greater than expected for a diffusion-only scenario. Numerical simulations of our system show good agreement with the experiments for onset time of convection and advancement of convective fingers. There are differences as well, the most prominent being the absence of cell-scale convection in the numerical simulations. This cell-scale convection observed in the experiments may be an artifact of a small temperature gradient induced by the cell illumination.     

Experimental study of free convection in an elliptical annular enclosure in blunt and slender orientations

In this paper, free convection heat transfer between two elliptical cylinders having different elliptical ratios in blunt and slender situations was studied experimentally. Three pairs of elliptical cylinders having the same radius ratio of 2.0, the same surface area and different elliptical ratios of 0.662, 0.866 and 0.968 were cut using the computed numerically controlled wire-cut machining. The tests were carried out by keeping a constant heat flux on the inner cylinder while cooling the outer one to be isothermal. The effects of vertical eccentricity, lateral eccentricity, angle of attack of the inner cylinder on natural convection for both blunt and slender situations of each pair were investigated. Empirical correlation was deduced within an acceptable uncertainty for the experimental results. Compatible and satisfactory to the conscience agreement was found in the comparison among the results of present and previous works. In the vision of the comparison, it was found that; the vertical eccentricity can enhance free convection by about 15% than the concentric case, the horizontal eccentricity can enhance natural convection by about 10% than concentric case and slender situation offers about 40% enhancement in free convection than the blunt situation for the same elliptical ratio.     

The Effect of Vertical Throughflow on the Onset of Convection Induced by Internal Heating in a Layered Porous Medium

We present an analytical investigation of the effect of vertical throughflow on the onset of convection, induced by internal heating, in a composite porous medium consisting of two horizontal layers. If convection is induced by internal heating, the bulk of the convection occurs in the upper half of the layer where the buoyancy force is destabilizing. For the case of heterogeneous porous medium, if the permeability increases in the upward direction, or if the thermal conductivity decreases in the upward direction, instability is increased. It is also found that upward throughflow is stabilizing but a modest amount of downward throughflow is destabilizing.     

Double-diffusive Finger Convection in a Hele–Shaw Cell: An Experiment Exploring the Evolution of Concentration Fields,Length Scales and Mass Transfer

We present the results of an experiment conducted to explore the temporal and spatial development of double-diffusive finger convection in a Hele–Shaw cell. Two solutions each containing a different density affecting component were layered in a density stable configuration (sucrose solution over a more dense salt solution) with a nearly perturbation-free interface between. The mismatch of diffusive time scales for the two components leads to local density instabilities that generate upward and downward convecting fingers. Throughout the course of the experiment, a full-field quantitative light transmission technique was used to measure concentration fields of a dye tracer dissolved in the salt solution. Analysis of these fields yielded the temporal evolution of length scales associated with the vertical and horizontal finger structure as well as mass transfer. Distinct developmental stages are identified with strong correlation between all measures. These data provide a baseline that can be used to develop and evaluate both process-level models that simulate the full complexity of the evolving flow field and large-scale effective models that integrate over small-scale behavior.     

Stability of Solutal Convection in a Gravity Modulated Mushy Layer During the Solidification of Binary Alloys

The linear stability theory is used to investigate analytically the effects of gravity modulation on solutal convection in the mushy layer of solidifying binary alloys. The gravitational field consists of a constant part and a sinusoidally varying part, which is synonymous to a vertically oscillating mushy layer subjected to constant gravity. The linear stability results are presented for both the synchronous and subharmonic solutions. It is demonstrated that up to the transition point between the synchronous and subharmonic regions, increasing the frequency of vibration rapidly stabilizes the solutal convection. Beyond the transition point, further increases in the frequency tend to destabilize the solutal convection, but gradually. It is also demonstrated that the effect of increasing the ratio of the Stefan number and the solid composition (₀) is to destabilize the solutal convection.     

3D Numerical study of the effect of aspect ratio on mixed convection air flow in upward solar air heater

A 3D Numerical study of mixed convection air flow in upward solar air heater with large spanwise aspect ratio (A = 10 to 40) was performed using CFD commercial code Fluent 14.5 (ANSYS). The main objective of this study is to investigate the channel height's effect (aspect ratio) on flow pattern and heat transfer in upward solar air heater in the particular case of low Re and high aspect ratio. The bottom plate (absorber) was submitted to Constant Heat Flux (CHF) in the range of 200 to 1000 W/m² and Reynolds number was varied from 50 to 1000. Our results are in concordance with most of authors conclusions about Poiseuille–Rayleigh–Benard flows. In mixed convection, increasing heat flux enhances heat transfer unlike forced convection flows. Simulation results of flow visualizations and Nusselt number calculations have shown that depending on Ri*, the velocity and temperature distributions in SAH vary greatly with the channel's height. The obtained results were different from previous studies. Indeed, our investigation of channel's height was achieved for the same heat flux but different Grashof numbers. For low channel's heights (high aspect ratio), increasing heat flux has not a significant effect but for higher channel's heights, an augmentation of heat flux enhances buoyancy effects in the flow and causes high turbulence. Also, increasing Reynolds number in low channel's heights (high A), can enhance substantially heat transfer. For higher channel's heights (low A), increasing Reynolds number decreases Ri* and thus buoyancy forces. Heat transfer is reduced and so Nusselt number. The obtained results may be very useful for engineers in designing and testing solar collectors.     

A priori tests on numerical errors in large eddy simulation using finite differences and explicit filtering

When low‐order finite‐difference methods are applied in large eddy simulation (LES), the magnitude of the numerical error may be larger than that of the subgrid‐scale (SGS) term. In this paper, the effect of explicit filtering on the numerical error related to the spatial discretization of the convection term and the exact SGS term is studied a priori in the turbulent fully developed channel flow. As the filter width is increased the grid resolution is kept constant. Also filtering in the inhomogeneous wall‐normal direction is discussed. The main conclusions are related to two approaches to explicit filtering. In the traditional approach, the whole velocity field is filtered explicitly while in the alternative approach, only the non‐linear convection term of the Navier–Stokes equations is filtered explicitly. Based on the results presented in the paper it seems that the first approach leads to an unphysical situation. However, the later approach works in the desired way, and the numerical error becomes clearly smaller than the SGS term. The main difference between the two approaches seems to be the interpretation of the resolved non‐linear term in the filtered Navier–Stokes equations. Copyright © 2005 John Wiley & Sons, Ltd.     

Non-linear convection in a porous medium with internal heat sources

The paper studies non-linear thermal convection in a horizontal porous layer of fluid with nearly insulating boundaries and in the presence of internal heat sources. Square and hexagonal cells are found to be the only possible stable convection cells. Finite amplitude instability could exist for some particular forms of an internal heat source Q. For a uniform Q, the preferred flow pattern is that of hexagons for amplitude ε smaller than some critical value ε_c, while both squares and hexagonal cells are stable for ε ? ε_c. The convective motion is downward at the hexagonal cell's centers. For a non-uniform Q, the qualitative features of thermal convection depend on the actual form Q. In particular, a non-uniform Q can increase or decrease the cell's size and the critical Rayleigh number at the onset of convection, and stabilize or destabilize the convective motion in the form of hexagonal cells with either upward or downward motion at the cell's centers.     

Mixed convection from an upward facing horizontal flat plate: effect of conduction and radiation

Experiments have been conducted in a low speed horizontal wind tunnel to study the interaction of radiation and conduction on mixed convective heat transfer from an upward facing horizontal flat plate in air. Differential interferometer has been used to measure local convective heat fluxes. It has been observed that interaction between surface radiation and convection is significant for a low thermal conductivity plate material. On the basis of the previous and current studies, it can be stated that the multi-mode interaction problem is an outcome of the nature of convective boundary layer. The interaction between different modes of heat transfer would remain similar irrespective of the nature of convection (free/mixed or forced).     

The issues of the uniqueness and the stability of the homogeneous response in uniaxial tests with gradient damage models

We consider a wide class of gradient damage models which are characterized by two constitutive functions after a normalization of the scalar damage parameter. The evolution problem is formulated following a variational approach based on the principles of irreversibility, stability and energy balance. Applied to a monotonically increasing traction test of a one-dimensional bar, we consider the homogeneous response where both the strain and the damage fields are uniform in space. In the case of a softening behavior, we show that the homogeneous state of the bar at a given time is stable provided that the length of the bar is less than a state dependent critical value and unstable otherwise. However, we also show that bifurcations can appear even if the homogeneous state is stable. All these results are obtained in a closed form. Finally, we propose a practical method to identify the two constitutive functions. This method is based on the measure of the homogeneous response in a situation where this response is stable without possibility of bifurcation, and on a procedure which gives the opportunity to detect its loss of stability. All the theoretical analyses are illustrated by examples.     

NUMERICAL INVESTIGATION ON FLOW STABILITY OF RAYLEIGH-B?NARD CONVECTION OF COLD WATER IN A RECTANGULAR CAVITY COOLED AND HEATED SYMMETRICALLY RELATIVE TO THE TEMPERATURE OF DENSITY MAXIMUM

为了解具有密度极值流体瑞利-贝纳德对流特有现象和规律,利用有限容积法对长方体腔内关于密度极值温度对称加热-冷却时冷水瑞利-贝纳德对流的分岔特性进行了三维数值模拟,得到了不同条件下的对流结构型态及其分岔序列,分析了密度极值特性、瑞利数、热边界条件以及宽深比对瑞利-贝纳德对流的影响. 结果表明:具有密度极值冷水瑞利-贝纳德对流系统较常规流体更加稳定,且流动型态及其分岔序列更加复杂;相同瑞利数下多种流型可以稳定共存,各流型在相互转变中存在滞后现象;随着宽深比的增加,流动更易失稳,对流传热能力增强;系统在导热侧壁时比绝热侧壁更加稳定,对流传热能力有所减弱;基于计算结果,采用线性回归方法,得到了热壁传热关联式.     

Mixed Convection Adjacent to a Suddenly Heated Horizontal Circular Cylinder Embedded in a Porous Medium

The mixed convection caused when a horizontal circular cylinder is suddenly heated is investigated in the situation when the initial flow past the cylinder is uniform and its direction either upwards or downwards. An analytical series solution, which is valid at small times, is obtained using the matched asymptotic expansions technique. A numerical solution, which is valid at all times and for any values of the Rayleigh and Péclet numbers, is also obtained using a fully implicit finite-difference method. Three different regimes, when either the free or forced convection is dominant or when they have the same order of magnitude, are considered. In the free convection dominated regime, two vortices develop near the sides of the cylinder in both situations of an upward or downward external flow. Comparisons between the analytical and numerical results at small times, as well as a detailed discussion of the evolution of the numerical solution are presented. The numerical results obtained for large Rayleigh, Ra, and Péclet Pe, numbers show that a thermal boundary-layer forms adjacent to the cylinder for any value of the ratio Ra/e. The steady state boundary-layer analysis, similar to that performed by Cheng and Merkin, is analysed in comparison to the numerical solution obtained for large values of Ra and Pe at very large times.     

Investigating the diameter of solid particles effects on a laminar nanofluid flow in a curved tube using a two phase approach

In this paper, we report the results of our numerical studies on laminar mixed convection heat transfer in a circular Curved tube with a nanofluid consisting of water and 1 vol.% Al₂O₃. Three dimensional elliptic governing equations have been used. Two phase mixture model and control volume technique have been implemented to study flow field. Effects of the diameter of particles on the hydrodynamic and thermal parameters are investigated and discussed. Increasing the solid particles diameter decreases the Nusselt number and secondary flow, while the axial velocity augments. When the particles are in order of nano meter, increasing the diameter of particles, do not change the flow behaviors. The distribution of solid nanoparticles is uniform and constant in curved tube.     

Laminar swirling jet with allowance for buoyancy forces

There have been many investigations into swirling immersed viscous jets, and fairly detailed analyses of these studies have been given in monographs [1–3]. Nevertheless, swirling jets with free convection have not been studied. In the present paper, exact and approximate self-similar solutions are found for the problem of a swirling jet produced by a source of viscous fluid propagating vertically upward (or downward) under the influence of Archimedean forces and an initial thrust.     

Effect of Surface Corrugation on Convection in a Three-Dimensional Finite Box of Fluid Saturated Porous Material

The problem of finite amplitude thermal convection in a three-dimensional finite box of fluid saturated porous material is investigated, when the lower boundary of the fluid is corrugated. The nonlinear problem of three-dimensional convection in the box for the values of the Rayleigh number close to the classical critical value and for small values of the amplitude of the corrugations is solved by a perturbation technique. The preferred mode of convection is determined by stability analysis. In the absence of corrugation three-dimensional modes of convection can be either stable or unstable depending on the values of the aspect ratios of the box, while two-dimensional rolls are always stable, provided that the box aspect ratios allow the existence of such modes of convection. In the presence of boundary corrugation with the appropriate form, different three-dimensional or two-dimensional modes of corrugation can be stable or unstable. For a rough boundary with local roughness sites, the location, size, and number of the roughness elements plus the wave numbers of the convection modes and the box aspect ratios can all play a role leading to either stable or unstable particular three- or two-dimensional flow patterns. For a wavy boundary, resonant wave-vector excitation can lead to the preference of stable two- or three-dimensional flow patterns whose wave vectors are in a subset of those due to the wavy boundary, while nonresonant wave-vector excitation can lead to the preference of stable flow patterns whose wave vectors are not generally in a subset of those due to the wavy boundary. Heat transported by convection can either be enhanced or be reduced by certain proper forms of the corrugations and by appropriate values of the box aspect ratios. Due to the surface corrugation highly subcritical modes of convection are stable, while highly supercritical modes of convection are unstable. Received 24 July 1998 and accepted 11 April 1999     

Effects of sub-millimeter-bubble injection on transition to turbulence in natural convection boundary layer along a vertical plate in water

Temperature and velocity measurements are performed to clarify the effects of sub-millimeter-bubble injection on the transition to turbulence in the natural convection boundary layer along a vertical plate in water. In particular, we focus on the relationship between the bubble injection position L and the transition to turbulence in the natural convection boundary layer. The bubble injection positions used in our experiments are L = 1.6 and 3.6 mm. Bubble injection at L = 1.6 mm delays the transition to turbulence in the natural convection boundary layer, while that at L = 3.6 mm accelerates the transition to turbulence in the boundary layer. In the case of L = 1.6 mm, the appearance region of the liquid velocity fluctuation in the bubble-induced upward flow in the upstream unheated section is restricted to near the wall, although the peak of the liquid velocity fluctuation is high. In contrast, in the case of L = 3.6 mm, the relatively large liquid velocity fluctuation is distributed widely over the laminar boundary layer width. These results suggest that the effect of the liquid velocity fluctuation on the laminar boundary layer is quite different between L = 1.6 and 3.6 mm. It is therefore expected that the transition to turbulence in the natural convection boundary layer for the case with bubble injection is dependent on the magnitude and appearance region of the liquid velocity fluctuation in the bubble-induced upward flow in the upstream unheated section.