Solutal Rayleigh Number Research Articles

The onset of penetrative double-diffusive convection

The onset of double-diffusive convection in a horizontal fluid layer is studied. The density is assumed to depend quadratically on the temperature and linearly on the solute concentration. Under the Boussinesq approximation, the linear stability of the conduction state is investigated with respect to the oscillatory and steady convection modes. For steady onset, the critical thermal Rayleigh number is found to be a double-valued function of the solutal Rayleigh number as long as the relative maximum of the density profile exists within the fluid layer. Driving mechanisms of the steady convections are discussed.

Numerical simulation for thermohaline multiple equilibrant system in non-rectangular domains

In this paper, incompressible, double-diffusive convection is simulated using finite-difference schemes. The Navier-Stokes equations are expressed in terms of stream function and vorticity. Because of the existence of large velocity, temperature and salinity gradients in boundary layers, a boundary-fitted coordinate system is used to concentrate the grid points near the wall and fit complex boundaries. The finite-difference methods used include the high-order accurate upwind difference scheme. It is shown that the scheme is a good candidate for direct simulations of double-diffusive convection flows. The proposed method is first applied to symmetry breaking and overturning states in thermohaline-driven flows in trapezoid basins. The basic phenomena agree well with those by Dijkstra and Molemaker (1997 J. Fluid Mech. 331 169) and Quon and Ghil (1992 J. Fluid Mech. 245 449), but symmetry breaking and overturning states can occur in an asymmetric geometrical region without perturbations. Then the method is applied to double-diffusive convections in a cavity with opposing horizontal temperature and concentration gradients at large thermal (Rt), solutal (Rs) Rayleigh numbers and Lewis number. There are three straight sides and a sine curve side in the cavity. Basically, numerical results are in agreement with those of Lee and Hyun (1990 Int. J. Heat Mass Transfer 33 1619) qualitatively, but eddies mixing in the top left-hand corner near the curved wall affects the layered structure.

Fast-onset Soret-driven convection in a colloidal suspension heated from above

Convection in a colloidal suspension with a large negative separation ratio is studied experimentally by heating the sample from above. Shadowgraph observations at large solutal Rayleigh numbers [Rtilde] are reported as a function of time. Stable spoke-pattern planforms with up and down columnar flows are observed at [Rtilde] ,10 6 . The convective flow starts in a time much smaller than that needed for the concentration gradient to reach steady state. Fast relaxation oscillations are reported for the rms value of the shadowgraph intensity. The oscillation period is in good agreement with the predictions of a recent work on self-oscillation behaviour induced by the Soret effect. However, in contrast with the prediction of the theory, the oscillations are damped and the rms attains a steady state value. Moreover, at very high stresses, the oscillations are overdamped.

Soret driven convection in a colloidal solution heated from above at very large solutal Rayleigh number.

Convection in a colloidal suspension with a large negative separation ratio psi is studied experimentally by heating from above. Shadowgraph observations at very large solutal Rayleigh numbers Rtilde; are reported as a function of time. Fast relaxation oscillations are reported for the root mean square value of the shadowgraph intensity. While pure fluids exhibit a transition to turbulent convection for Rayleigh number R approximately 10(6), stable spoke-pattern planform with up and down columnar flows are observed up to Rtilde; approximately 1.9 x 10(9). It is suggested that the surprising stability of the planform against turbulence is due to nonlinear focusing arising from the concentration dependence of the diffusion coefficient.

THE EFFECTS OF STABLE THERMAL STRATIFICATION ON THE ONSET OF DOUBLE-DIFFUSIVE CONVECTION

The time of the onset of double-diffusive convection in time-dependent, nonlinear concentration fields is investigated theoretically. When the initially quiescent horizontal fluid layer with a uniform temperature gradient experiences a sudden concentration change from below, the concentration field could invoke a convective motion. The related stability analysis, with considering Soret effect, is conducted on the basis of the propagation theory. Under the linear stability theory the concentration penetration depth is used as a length scaling factor and the linearized perturbation equations of similarity transform are solved numerically. The resulting critical time to mark the onset of regular cells is obtained as a function of the thermal Rayleigh number, the solute Rayleigh number, and the Soret effect coefficient

COMPUTATIONAL MODEL FOR SOLUTAL CONVECTION DURING DIRECTIONAL SOLIDIFICATION

Computational investigation is conducted during time-dependent behaviors induced by solutal convection in the melt for directional solidification of a binary alloy. Effects of solutal Rayleigh number on flow, heat, and mass transfer are studied for upward Bridgman configuration restricted to fluid phase. The results compare well with spectral ones exhibiting various hydrodynamic solutal transitions (symmetric, asymmetric, time-dependent). The finite-volumes method is tested for various uniform and refined meshes near the walls to obtain spectral accuracy. Calculations are extended to configurations including melt/solid interactions by the use of a homogeneous formulation and a linearized equilibrium phase diagram.

Flows induced by inclined rotation in directional solidification of variable-viscosity solutions

As the unidirectional-solidifying melt rotates with respect to an inclined axis, a shear flow is induced to move parallel to the melt/solid interface while it changes its direction along the axis perpendicular to the interface, like an Ekman spiral flow. This induced flow will in turn alter the morphology of the solidifying melt/solid interface and eventually influence the quality of the final casting. In the present paper, we investigate this induced flow in the system rotating in a general way while physical conditions vary: different rotating conditions, different thermal and solutal gradients, and the non-Boussinesq effect due to variable viscosity of the melt. Results show that, in brief, the induced flow is driven by the gravity due to inclination while modified by rotation. As each of the following parameters is higher: the effective Taylor number ( T e), the thermal Rayleigh number ( R t), the solutal Rayleigh number ( R c), or the Lewis number (Le), the induced flow is of larger velocity. The variable-viscosity effect is more complex, depending on the value of other physical parameters.

Numerical study on the vertical bridgman crystal growth with thermosolutal convection

A numerical analysis has been carried out to investigate the influences of thermosolutal convection on the heat and mass transfer and solute segregation in crystals grown by the vertical Bridgman technique. The governing equations are solved by a finite-volume method using the power law scheme and the SIMPLE algorithm in which body-fitted coordinate system has been used. A primary convective cell driven by thermal gradients forms in the bulk of the domain, while a secondary convective cell driven by solutal gradients forms near interface. As the solutal Rayleigh number increases, secondary cell becomes to be stronger and has a great influence on the radial concentration along the interface.

Multiple solutions for double diffusive convection in a shallow porous cavity with vertical fluxes of heat and mass

The Darcy model with the Boussinesq approximation is used to study double-diffusive natural convection in a shallow porous cavity. The horizontal walls are subject to uniform fluxes of heat and mass, while the side vertical walls are exposed to a constant heat flux of intensity aq ′, where a is a real number. Results are presented for −20⩽R T ⩽50, −20⩽R S ⩽20, 5⩽Le⩽10, 4⩽A⩽8 and −0.7⩽ a⩽0.7, where R T, R S, Le and A correspond to thermal Rayleigh number, solutal Rayleigh number, Lewis number and aspect ratio of the enclosure, respectively. In the limit of a shallow enclosure ( A≫1) an asymptotic analytical solution for the stream function and temperature and concentration fields is obtained by using a parallel flow assumption in the core region of the cavity and an integral form of the energy and the constituent equations. In the absence of side heating ( a=0), the solution takes the form of a standard Bénard bifurcation. The asymmetry brought by the side heating ( a≠0) to the bifurcation is investigated. For high enough Rayleigh numbers, multiple steady states near the threshold of convection are found. These states represent flows in opposite directions. In the range of the governing parameters considered in the present study, a good agreement is observed between the analytical predictions and the numerical simulations of the full governing equations.

Effects of centrifugal and coriolis forces on a hydromagnetic chimney convection in a mushy layer

Nonlinear buoyancy driven convection in the melt and in cylindrical chimneys in a mushy layer during alloy solidification and subjected to an externally imposed strong magnetic field is investigated under a high-gravity environment, where the rotation axis is inclined at an angle γ to the high-gravity vector. The convection in the chimneys produces freckles in the final form of the solidified material, which are imperfections that reduce the quality of the solidified material. Asymptotic and scaling analyses, in the limit of sufficiently large solutal Rayleigh number, are applied to a nonaxisymmetric nonlinear convection in a finite Prandtl number melt and in the chimneys. Ranges in the parameter space are found where chimney convection can be intensified or weakened with either increasing Coriolis parameter T or centrifugal acceleration parameter A. Chimney convection generally decreases with increasing magnetic parameter Q (Chandrasekhar number). Within certain parameter ranges, chimney convection can have a much lower amplitude and lower rate of change with respect to A or Q if T is large and if γ<180°, as compared to the corresponding amplitude and the rate of change in the case for γ>180° keeping all the other parameter values the same as in the γ<180° case.

On the Spatio-Temporal Flow Phenomena in Low Prandtl Number Melts Part 2: Shapes of the Interfaces and the Formation of Striations

By marking the solid-liquid interfaces during a stepwise decrease of the height of a stationary molten zone of Bi 0.5 Sb 15 Te 3 reasonable and changing curvatures of the interfaces are determined. In opposite to the lower interface the upper interface changes its shape from convex to concave for aspect ratios ar ≥ 2. The differences regarding the shape of both interfaces are indirectly due to the different flow intensities in both half zones. Furthermore, in addition to experiments described in Part 1 of this paper the molten zone was moved dynamically through the sample (zone melting). By estimating solutal and thermal RAYLEIGH numbers the solutal contribution to buoyancy convection is considered as negligible compared to the thermal part for the investigated zone heights. Time-dependent convection during zone melting results in pronounced striations. With suitable models for the correlation between the oscillations of temperature, growth rate and microscopical concentration it was possible to estimate the shape and quantity of oscillating compositional inhomogeneities (striations).

Ionic mass transport correlation of double-diffusive convection in horizontal fluid layers

Here is investigated the characteristics of double-diffusive convection in thermally-stable stratified horizontal fluid layer. By employing an electrochemical technique, and adopting aqueous CuSO4-H2SO4 solution as electrolyte, experiments on ionic mass transfer have been conducted systematically. And, also a new mass transfer correlation in double-diffusive situations has been derived by extending the model of micro-scales of turbulence, which was proposed by Arpaci. The resulting correlation of the Sherwood number as a function of the thermal Rayleigh number and the solutal Rayleigh number was in good agreement with the present experimental results. The present study provides plausible understanding in controlling both mass and heat transfer rates for practical situations including double-diffusive convection.

Transient Double Diffusive Convection in a Vertical Enclosure With Asymmetrical Boundary Conditions

This study deals with the numerical analysis of transient heat and species transfer by natural convection in a binary fluid vertical layer. The cavity is differentially heated and a solutal buoyancy force is created by imposing a concentration step at one vertical wall: This refers to the experimental situation where the composition gradient inducing the solutal buoyancy force is created by melting of pure ice in a salty solution. The constitution of the flow structure and the time evolution of the heat and mass transfer characteristics are studied for opposing body forces over a range of thermal and solutal Rayleigh numbers. The numerical results allow to provide a better insight into the mechanisms driving the heat and species transfer at high Lewis number thermohaline convection. [S0022-1481(00)00303-0]

Natural convection due to heat and mass transfer in a composite system

A numerical study is performed to analyse heat and mass transfer phenomena due to natural convection in a composite cavity containing a fluid layer overlying a porous layer saturated with the same fluid. The flow in the porous region is modelled using Brinkman–Forchheimer-extended Darcy model that includes both the effect of macroscopic shear (Brinkman effect) and flow inertia (Forchheimer effect). The vertical walls of the two-dimensional enclosure are isothermal whilst the horizontal walls are adiabatic. The two regions are coupled by equating the velocity and stress components at the interface. The resulting coupled equations in non-dimensional form are solved by an alternating direction implicit method by transforming them into parabolic form by the addition of false transient terms. The numerical results show that the amount of fluid penetration into the porous layer depends strongly upon the Darcy, thermal and solutal Rayleigh numbers. Average Nusselt number decreases while average Sherwood number increases with an increase of the Lewis number. The transfer of heat and mass on the heated wall near the interface depends strongly on the Darcy number.

Displacement of Salty Water by Fresh Water in the Presence of Marangoni Convection

This paper numerically investigates the interaction between the Marangoni convection, the double diffusive convection and the intrusion force a multi-cavity system. The model consists of a two-cavity rectangular system in which the smaller cavity is located at the top left corner of the larger one. The larger cavity is filled with hot salty fluid while the smaller one contains cold fresh fluid. This problem is solved for the case of the fresh water intruding into the salty water. The interaction between the Marangoni (thermal and solutal) and the double diffusive regimes is investigated, for different solutal Rayleigh number, in detail. Finite element modelling results indicate that salinity induces stronger convection than the thermal ones and the thermal and solutal Marangoni convection further enhances the strength of the cellular flow. Numerical results indicate that surface force should be taken into consideration during the analysis and that the intruding force played an important role in the flow behaviour.

Effect of contaminant properties and temperature gradients on the efficiency of transient gaseous contaminant removal from an enclosure : a numerical study

This paper reports the results of a numerical study on the transient removal of a contaminant from a two-dimensional enclosure with one inlet and one outlet. The influence of buoyancy forces due to thermal and concentration gradients, contaminant diffusivity, inlet velocity and outlet disposition over the cleaning-time are studied. The governing equations of the laminar flow (continuity, momentum, energy and contaminant concentration) are solved by means of the SIMPLEC algorithm. Several simulations of the same test case have been made, using two numerical schemes : PLDS and SMART, in order to determinate which one provides better performances. For isothermal situations, the time required to remove the contaminant is studied parametrically as a function of the outlet position and the governing dimensionless numbers (Reynolds number, Schmidt number and solutal Rayleigh number) , while the effect of horizontal temperature differences is studied in several situations in which it plays an important role. Buoyancy forces are found to have a strong influence over the flows and, in consequence, over the cleaning times.

Heat and mass transfer in a multi-porous cavity

In this paper, two-dimensional non-linear double diffusive convection in a multi-porous cavity is considered. The Darcy equation, including Brinkman term to account for the viscous effects, is used as the momentum equation. The model consists of two rectangular cavities filled with glass beads having a diameter d 1 of either 5.25 mm (Case 1) or 3.25 mm (Case 2). The smaller cavity is located at the top left corner of the larger one. The larger cavity is filled initially with hot salty fluid while the smaller one contains initially cold fresh fluid. At the initial time, the obstacle between the two cavities was released and the double diffusive phenomena were studied in details. The momentum, solutal, energy and continuity equations are solved numerically using the finite element technique. This transient problem is solved for two different Darcy numbers. For each Darcy number, the influence of the solutal Rayleigh number on double diffusive convection was studied in details. The permeability in the horizontal and vertical direction was assumed identical. A comparison of the intruding force between this case and the open flow case studied by Saghir et al [Int. J. Heat. Mass Transfer, in press] showed that it is inversely proportional to the Darcy number. Finite element modelling results indicate that salinity induces stronger convection than the thermal ones.

Linear and weakly nonlinear analysis of doubly diffusive vertical slot convection

We consider vertical slot convection in the doubly diffusive case over the full range of thermal and solute Rayleigh numbers. We quantitatively categorize the various instability regimes in the linear analyses. In the weakly nonlinear analyses, we derive the amplitude equations using the method of normal forms and study the stability of the finite-amplitude solutions. We discuss in some detail the physics of the various instabilities found in our analyses.

Modelling density-dependent flow and solute transport at the Lake Tutchewop saline disposal complex, Victoria

Intercepted saline groundwaters and drainage effluent from irrigation are commonly stored in both natural and artificial saline disposal basins throughout the Murray-Darling Basin of Australia. Their continued use as wastewater evaporation sites requires an understanding of existing groundwater dynamics. The useful of individual basins, their sustainability and possible environmental impacts remain largely unknown. In this work, the movement of salt to the underlying groundwater system from Lake Tutchewop, a saline disposal complex in north-central Victoria, was modelled in cross-section. Due to the salinity contrast between the hypersaline basin waters and the regional groundwater, it was necessary to simulate density-dependent flow behaviour. Under certain conditions, these density-stratified systems may become unstable leading to the onset of convective behaviour, which greatly increases the movement of salt from the basin to the groundwater system. Modelled concentration profiles in the aquifer system and calculated seepage rates from the basin show that Lake Tutchewop is stable under its present operating regime. The downward movement of salt is mainly controlled by diffusion and dispersion. The calibrated model was used to assess the impact of several management scenarios using time-dependent boundary conditions for lake salinity and water levels. The influence of heterogeneous basin linings on ensuing salt flux rates is examined, and results show that increased solute transport will occur under such conditions. A sensitivity analysis performed on governing variables showed that salt fluxes were most sensitive to lake salinity levels. A solute Rayleigh number defined in terms of basin salinity and hydrogeologic parameters is seen to be an effective tool for predicting the long term behaviour of such saline disposal basins. The models and concepts developed in this work may find application in the design and management of saline disposal complexes.

Effects of centrifugal and Coriolis forces on chimney convection during alloy solidification

Nonlinear natural convection in cylindrical chimneys within a mushy layer during solidification of a binary alloy is investigated under a high-gravity environment, where the rotation axis is inclined at an angle γ to the high-gravity vector. These chimneys produce freckles in the final form of the solidified materials. Freckles are imperfections that reduce the quality of the solidified materials. Asymptotic and scaling analyses are applied to weakly nonaxisymmetric convection in the chimneys. Three important nondimensional parameters are the solutal Rayleigh number R, the centrifugal acceleration parameter A and the Coriolis parameter T. It is found that, for some moderate values of the rotation rate and sinγ ≠ 0, axial convection in the chimneys decreases rapidly with increasing A above some azimuthally dependent axial level. This level can also depend on the radii of the chimneys and on γ and T. The axial convection also decreases with increasing T in certain range which depends on γ, T and R. The Coriolis effect is destabilizing for T outside this range of values, but it is less destabilizing for γ = γ0 > 180° as compared with those cases with inclined angle (γ0 − 180°).