The stability of two-dimensional linear flows of an Oldroyd-type fluid

A linear stability analysis is made for an Oldroyd-type fluid undergoing steady two-dimensional flows in which the velocity field is a linear function of position throughout an unbounded region. This class of basic flows is characterized by a parameter λ which ranges from λ = 0 for simple shear flow to λ = 1 for pure extensional flow. The time derivatives in the constitutive equation can be varied continuously from co-rotational to co-deformational as a parameter β varies from 0 to 1. The linearized disturbance equations are analyzed to determine the asymptotic behavior as time t → ∞ of a spatially periodic initial disturbance. It is found that unbounded flows in the range 0 < λ ? 1 are unconditionally unstable with respect to periodic initial disturbances which have lines of constant phase parallel to the inlet streamline in the plane of the basic flow. When the Weissenberg number is sufficiently small, only disturbances with sufficiently small wavenumber α₃ in the direction normal to the basic flow plane are unstable. However, for certain values of β, critical Weissenberg numbers are found above which flows are unstable for all values of the wavenumber α₃.     

Flow of a liquid with an anisotropic viscosity tensor: some axisymmetric flows

Of a class of idealized anisotropic liquids presented earlier [1,2], two particular cases, referred to as liquids D and F, are now analysed in some axially symmetric flows generated by relative motion of the boundaries. The liquids are locally transversely isotropic at each point at some initial instant, and the different responses associated with some different initial directions of orientation are considered, in torsional flow, in Couette flow, and in longitudinal flow between concentric circular cylinders.As in [1,2], it is found that only in special circumstances can the liquids behave in a Newtonian fashion, without change of orientation pattern. In general, even when the motion of boundaries is steady, the flow is unsteady, stresses are time-dependent, and initial transverse isotropy does not persist.     

Boundary layer structure of oscillatory open-channel shallow flows over smooth and rough beds

The boundary layer structure of oscillatory shallow open channel flows has been studied in a wide flume. Fluorescence solution was released at a porous rough bed through a diffuser covered by gravel of 0.5 cm grain size. A planar laser-induced fluorescence (PLIF) system was used to visualise the dye plumes in both vertical and horizontal planes for a qualitative understanding of the roles of large-scale flow structures in mass transport. A variety of tests were conducted for a range of oscillatory periods (30–240 s), water depths (3–16 cm) and velocity amplitudes (0.027–0.325 m/s), which cover a wide range of oscillatory flows with Reynolds numbers Re _a varied from 0.3 × 10⁴ (laminar) to 2.1 × 10⁶ (fully turbulent). For quantitative investigation, a novel technique, namely combined laser-induced fluorescence (LIF) and 2D laser Doppler velocimetry (LDV) (LIF/LDV), was developed and used to measure the velocity and solute concentration simultaneously in a vertical plane over 50 cycles. From the dye plumes revealed by the PLIF in transitional flows, there are different patterns of flow structure and solute transport with three representative stages of acceleration, deceleration and flow reversal. In the acceleration stage, turbulence was suppressed with dye layers adhering to the surface with little vertical mass transport. In the deceleration stage, flame-like turbulent structures occurred when turbulence generation was prominent. This was investigated quantitatively by recording the percentage occurrence of the adhered smooth layers per cycle. For those smooth bed cases with Re _a < 1.8 × 10⁵, the adhered smooth dye layers type of boundary layer occupied 100% of the oscillation period. Over a sufficiently high Re _a , a rough bed can generate fully turbulent oscillatory flows without the appearance of adhering dye layers. Between these two extremes, a transitional flow regime occurs in a wide range of flow conditions: Re _a > 2.7 × 10⁴ over the rough bed and Re _a > 8.3 × 10⁶ over a smooth bed.     

Localised dynamics of laminar pulsatile flow in a rectangular channel

The exploitation of flow pulsation in low-Reynolds number micro/minichannel flows is a potentially useful technique for enhancing cooling of high power photonics and electronics devices. Although the mechanical and thermal problems are inextricably linked, decoupling of the local instantaneous parameters provides insight into underlying mechanisms. The current study performs complementary experimental and analytical analyses to verify novel representations of the pulsating channel flow solutions, which conveniently decompose hydrodynamic parameters into amplitude and phase values relative to a prescribed flow rate, for sinusoidally-pulsating flows of Womersley numbers 1.4 ≤ Wo ≤ 7.0 and a fixed ratio of oscillating flow rate amplitude to steady flow rate equal to 0.9. To the best of the authors’ knowledge, the velocity measurements – taken using particle image velocimetry – constitute the first experimental verification of theory over two dimensions of a rectangular channel. Furthermore, the wall shear stress measurements add to the very limited number of studies that exist for any vessel geometry. The amplification of the modulation component of wall shear stress relative to a steady flow (with flow rate equal to the amplitude of the oscillating flow rate) is an important thermal indicator that may be coupled with future heat transfer measurements. The positive half-cycle time- and space-averaged value is found to increase with frequency owing to growing phase delays and higher amplitudes in the near-wall region of the velocity profiles. Furthermore, the local time-dependent amplification varies depending on the regime of unsteadiness: (i) For quasi-steady flows, the local values are similar during acceleration and deceleration though amplification is greater near the corners over the interval 0 – 0.5π. (ii) At intermediate frequencies, local behaviour begins to differ during accelerating and decelerating periods and the interval of greater wall shear stress near the corners lengthens. (iii) Plug-like flows experience universally high amplifications, with wall shear stress greater near the corners for the majority of the positive half-cycle. The overall fluid mechanical performance of pulsating flow, measured by the ratio of bulk mean wall shear stress and pressure gradient amplifications, is found to reduce from an initial value of 0.97 at Wo = 1.4 to 0.28 at Wo = 7.0, demonstrating the increasing work input required to overcome inertia.     

Mechanism of flow instability and transition to turbulence

In this paper, a new mechanism of flow instability and turbulence transition is proposed for wall bounded shear flows. It is stated that the total energy gradient in the transverse direction and that in the streamwise direction of the main flow dominate the disturbance amplification or decay. Thus, they determine the critical condition of instability initiation and flow transition under given initial disturbance. A new dimensionless parameter K for characterizing flow instability is proposed which is expressed as the ratio of the energy gradients in the two directions for the flow without energy input or output. It is suggested that flow instability should first occur at the position of K_max which may be the most dangerous position. This speculation is confirmed by Nishioka et al.'s experimental data. Comparison with experimental data for plane Poiseuille flow and pipe Poiseuille flow indicates that the proposed idea is really valid. It is found that the turbulence transition takes place at a critical value of K_max of about 385 for both plane Poiseuille flow and pipe Poiseuille flow, below which no turbulence will occur regardless the disturbance. More studies show that the theory is also valid for plane Couette flows which holds a critical value of K_max of about 370.     

Binary diffusion and heat transfer in laminar mixed convection channel flows with uniform wall heat flux: extremely thin film thickness

A detailed numerical study has been performed to investigate the combined heat and mass transfer in laminar mixed convection channel flows with uniform wall heat flux. In an initial effort the liquid film on the channel wall is assumed to be extremely thin in thickness. Major dimensionless groups governing the present problem areGr _T,Gr _Mx,Pr,Sc, φ andRe. Results are specifically presented for an air-water system under various conditions. The effects of wall heating flux, the Reynolds number and the relative humidity of the moist air in the ambient on the momentum, heat and mass transfer in the flow are investigated in great detail.     

Coherent Structures in Oscillating Turbulent Boundary Layers Over a Fixed Rippled Bed

Coherent structures generated by oscillating turbulent boundary layers with or without a unidirectional current over a fixed, rippled bed are presented. The effect of ripple height and current intensity on the characteristics of these structures was investigated using a series of large-eddy simulations performed at Re _α ?=?23,163. These flows are typical in coastal regions where complex wave-current interactions occur. A cartesian flow solver was used with the rippled bed represented using the immersed boundary (IMB) method. Results are presented for three ripple steepness values and two current magnitudes. Three different types of coherent structures were identified with their size, shape and evolution largely depending on ripple steepness, while, their potential effect on sediment transport is discussed.     

Linear stability and dynamics of viscoelastic flows using time-dependent numerical simulations

Ultimately, numerical simulation of viscoelastic flows will prove most useful if the calculations can predict the details of steady-state processing conditions as well as the linear stability and non-linear dynamics of these states. We use finite element spatial discretization coupled with a semi-implicit θ-method for time integration to explore the linear and non-linear dynamics of two, two-dimensional viscoelastic flows: plane Couette flow and pressure-driven flow past a linear, periodic array of cylinders in a channel. For the upper convected Maxwell (UCM) fluid, the linear stability analysis for the plane Couette flow can be performed in closed form and the two most dangerous, although always stable, eigenvalues and eigenfunctions are known in closed form. The eigenfunctions are non-orthogonal in the usual inner product and hence, the linear dynamics are expected to exhibit non-normal (non-exponential) behavior at intermediate times. This is demonstrated by numerical integration and by the definition of a suitable growth function based on the eigenvalues and the eigenvectors. Transient growth of the disturbances at intermediate times is predicted by the analysis for the UCM fluid and is demonstrated in linear dynamical simulations for the Oldroyd-B model. Simulations for the fully non-linear equations show the amplification of this transient growth that is caused by non-linear coupling between the non-orthogonal eigenvectors. The finite element analysis of linear stability to two-dimensional disturbances is extended to the two-dimensional flow past a linear, periodic array of cylinders in a channel, where the steady-state motion itself is known only from numerical calculations. For a single cylinder or widely separated cylinders, the flow is stable for the range of Deborah number (De) accessible in the calculations. Moreover, the dependence of the most dangerous eigenvalue on De≡λV/R resembles its behavior in simple shear flow, as does the spatial structure of the associated eigenfunction. However, for closely spaced cylinders, an instability is predicted with the critical Deborah number De_c scaling linearly with the dimensionless separation distance L between the cylinders, that is, the critical Deborah number De_Lc≡λV/L is shown to be an O(1) constant. The unstable eigenfunction appears as a family of two-dimensional vortices close to the channel wall which travel downstream. This instability is possibly caused by the interaction between a shear mode which approaches neutral stability for De ≫ 1 and the periodic modulation caused by the presence of the cylinders. Nonlinear time-dependent simulations show that this secondary flow eventually evolves into a stable limit cycle, indicative of a supercritical Hopf bifurcation from the steady base state.     

Generalized relative permeability coefficients during steady-state two-phase flow in porous media,and correlation with the flow mechanisms

A parametric experimental investigation of the coupling effects during steady-state two-phase flow in porous media was carried out using a large model pore network of the chamber-and-throat type, etched in glass. The wetting phase saturation,S ₁, the capillary number,Ca, and the viscosity ratio,k, were changed systematically, whereas the wettability (contact angleθ _e ), the coalescence factorCo, and the geometrical and topological parameters were kept constant. The fluid flow rate and the pressure drop were measured independently for each fluid. During each experiment, the pore-scale flow mechanisms were observed and videorecorded, and the mean water saturation was determined with image analysis. Conventional relative permeability, as well as generalized relative permeability coefficients (with the viscous coupling terms taken explicitly into account) were determined with a new method that is based on a B-spline functional representation combined with standard constrained optimization techniques. A simple relationship between the conventional relative permeabilities and the generalized relative permeability coefficients is established based on several experimental sets. The viscous coupling (off-diagonal) coefficients are found to be comparable in magnitude to the direct (diagonal) coefficients over board ranges of the flow parameter values. The off-diagonal coefficients (k _rij /Μ _j ) are found to be unequal, and this is explained by the fact that, in the class of flows under consideration, microscopic reversibility does not hold and thus the Onsager-Casimir reciprocal relation does not apply. Thecoupling indices are introduced here; they are defined so that the magnitude of each coupling index is the measure of the contribution of the coupling effects to the flow rate of the corresponding fluid. A correlation of the coupling indices with the underlying flow mechanisms and the pertinent flow parameters is established.     

Numerical solutions of the shallow water equations with discontinuous bed topography

A simple scheme is developed for treatment of vertical bed topography in shallow water flows. The effect of the vertical step on flows is modelled with the shallow water equations including local energy loss terms. The bed elevation is denoted with z_b^‐ for the left and z_b⁺ for the right values at each grid point, hence exactly representing a discontinuity in the bed topography. The surface gradient method (SGM) is generalized to reconstruct water depths at cell interfaces involving a vertical step so that the fluxes at the cell interfaces can accurately be calculated with a Riemann solver. The scheme is verified by predicting a surge crossing a step, a tidal flow over a step and dam‐break flows on wet/dry beds. The results have shown good agreements compared with analytical solutions and available experimental data. The scheme is efficient, robust, and may be used for practical flow calculations. Copyright © 2002 John Wiley & Sons, Ltd.     

Transient growth and bypass transition in stenotic flow with a physiological waveform

A direct analysis method is applied to compute optimal transient growth initial conditions for physiologically relevant pulsatile flows in a smooth axisymmetric stenosis with 75% occlusion. The flow waveform employed represents phase-average measurements obtained in the human common carotid artery. Floquet analysis shows that the periodic flow is stable to infinitesimal eigenmodal-type perturbations that would grow from one cycle to the next at the Reynolds numbers considered. However, the same flows display explosive transient growth of optimal disturbances, with our analysis predicting disturbance energy growths of order 10²⁵ within half a pulse period at a mean bulk flow Reynolds number Re = 300, which is significantly lower than the physiological value of Re = 450 at this location. Direct numerical simulation at Re = 300 shows that when the base flow is perturbed a small amount with the optimal growth initial condition, the disturbance grows rapidly in time in agreement with the linear analysis, and saturates to provide a locally turbulent state within half a pulse period. This transition resulting from non-normal growth mechanisms shows the flow exhibits bypass transition to turbulence. Our analysis suggests that this route to localized turbulent states could be relatively common in human arterial flows.     

Axisymmetric-conical and locally conical flows without swirling

Axisymmetric steady conical and locally conical non-swirled flows of an ideal (inviscid and non-heat-conducting) gas are considered. Like two-dimensional conical flows, the examined onedimensional (axisymmetric) flows can be conically subsonic and supersonic. If the uniform flow is not considered as a conical flow, then the type of one-dimensional conical flows can change only on the shock wave, except for the junction of two one-dimensional conical flows of different types on the C ⁺ characteristic. C ^± characteristics and streamlines are constructed for a number of locally conical flows and some already known and new conical flows.     

Mixed convection adjacent to inclined flat surfaces embedded in a porous medium

An analysis is performed to study the flow and heat transfer characteristics of laminar mixed convection boundary layer flows from inclined (including horizontal and vertical) surfaces embedded in a saturated porous medium with constant aiding external flows and uniform surface temperature. Both the streamwise and normal components of the buoyancy forces are retained in the momentum equations. Nondimensionalization of the boundary layer equations results in the following three governing parameter: (1)Gr/Re, the ratio of the Grashof number to the Reynolds number; (2)Pe _x =Re _x Pr, the Peclet number; (3) φ, the angle of inclination from the horizontal. The resulting nonsimilar equations are solved by an efficient implicit finite-difference scheme. Numerical results are presented for flows with different values ofGr/Re in the range of 0 to 50, over a wide range of the Peclet numbersPe _x, and various values of φ ranging from 0 to 90 degrees. It is found that the local surface heat transfer rate increases with increasing the local Peclet number. In addition, as the plate is tilted from the horizontal to the vertical orientation, the local Nusselt number increases for a given Peclet number and the effect of the buoyancy force on the surface heat transfer rate increases.     

Multi-layer channel flows with yield stress fluids

We present results of a computational study of visco-plastically lubricated plane channel multi-layer flows, in which the yield stress fluid layers are unyielded at the interface. We demonstrate that symmetric 3-layer flows may be established for wide ranges of viscosity ratio (m), Bingham number (B) and interface position (y_i), for Reynolds numbers Re ≤ 100. Here an inner Newtonian layer is sandwiched between 2 layers of Bingham fluid. Results are presented illustrating the variation of development length with the main dimensionless parameters and for different inlet sizes. We also show that these flows may be initiated by injecting either fluid into a steady flow of the other fluid. The flows are established quicker when the core fluid is injected into a channel already full of the outer fluid. In situations where the inner fluid flow rate is dominant we observed inertial symmetry breaking in the symmetric start-up flows as Re was increased. Asymmetry is also observed in studying temporal nonlinear stability of these flows, which appear stable up to moderate Re and significant amplitudes. In general the flows destabilize at lower Re and perturbation amplitudes than do the analogous core-annular pipe flows, but 1–1 comparison is hard. When the flow is stable the decay characteristics are very similar to those of the pipe flows. In the final part of the paper we explore more exotic flow effects. We show how flow control could be used to position layers asymmetrically within the flow, and how this effect might be varied transiently. We demonstrate that more complex layered flows can be stably achieved, e.g. a 7-layered flow is established. We also show how a varying inlet position can be used to “write” in the yield stress fluid: complex structures that are advected with the flow and encapsulated within the unyielded fluid.     

Heat transfer enhancement in grooved channels due to flow bifurcations

The flow bifurcation scenario and heat transfer characteristics in grooved channels, are investigated by direct numerical simulations of the mass, momentum and energy equations, using the spectral element methods for increasing Reynolds numbers in the laminar and transitional regimes. The Eulerian flow characteristics show a transition scenario of two Hopf bifurcations when the flow evolves from a laminar to a time-dependent periodic and then to a quasi-periodic flow. The first Hopf bifurcation occurs to a critical Reynolds number Rec₁ that is significantly lower than the critical Reynolds number for a plane-channel flow. The periodic and quasi-periodic flows are characterized by fundamental frequencies ω₁ and m· ω₁+n·ω₂, respectively, with m and n integers. Friction factor and pumping power evaluations demonstrate that these parameters are much higher than the plane channel values. The time-average mean Nusselt number remains mostly constant in the laminar regime and continuously increases in the transitional regime. The rate of increase of this Nusselt number is higher for a quasi-periodic than for a periodic flow regime. This higher rate originates because better flow mixing develops in quasi-periodic flow regimes. The flow bifurcation scenario occurring in grooved channels is similar to the Ruelle-Takens-Newhouse transition scenario of Eulerian chaos, observed in symmetric and asymmetric wavy channels.     

An improved anisotropy-resolving subgrid-scale model for flows in laminar–turbulent transition region

Some types of mixed subgrid-scale (SGS) models combining an isotropic eddy-viscosity model and a scale-similarity model can be used to effectively improve the accuracy of large eddy simulation (LES) in predicting wall turbulence. Abe (2013) has recently proposed a stabilized mixed model that maintains its computational stability through a unique procedure that prevents the energy transfer between the grid-scale (GS) and SGS components induced by the scale-similarity term. At the same time, since this model can successfully predict the anisotropy of the SGS stress, the predictive performance, particularly at coarse grid resolutions, is remarkably improved in comparison with other mixed models. However, since the stabilized anisotropy-resolving SGS model includes a transport equation of the SGS turbulence energy, k_SGS, containing a production term proportional to the square root of k_SGS, its applicability to flows with both laminar and turbulent regions is not so high. This is because such a production term causes k_SGS to self-reproduce. Consequently, the laminar–turbulent transition region predicted by this model depends on the inflow or initial condition of k_SGS. To resolve these issues, in the present study, the mixed-timescale (MTS) SGS model proposed by Inagaki et al. (2005) is introduced into the stabilized mixed model as the isotropic eddy-viscosity part and the production term in the k_SGS transport equation. In the MTS model, the SGS turbulence energy, k_es, estimated by filtering the instantaneous flow field is used. Since the k_es approaches zero by itself in the laminar flow region, the self-reproduction property brought about by using the conventional k_SGS transport equation model is eliminated in this modified model. Therefore, this modification is expected to enhance the applicability of the model to flows with both laminar and turbulent regions. The model performance is tested in plane channel flows with different Reynolds numbers and in a backward-facing step flow. The results demonstrate that the proposed model successfully predicts a parabolic velocity profile under laminar flow conditions and reduces the dependence on the grid resolution to the same degree as the unmodified model by Abe (2013) for turbulent flow conditions. Moreover, it is shown that the present model is effective at transitional Reynolds numbers. Furthermore, the present model successfully provides accurate results for the backward-facing step flow with various grid resolutions. Thus, the proposed model is considered to be a refined anisotropy-resolving SGS model applicable to laminar, transitional, and turbulent flows.     

Convective-Absolute Transition of Non-Isothermal Compressible Shear Layer

The linear stability of inviscid compressible shear layers is studied. When the layer develops at the vicinity of a wall, the two parallel flows can have velocity of the same sign or of opposite sign. This situation is examined in order to obtain first hints on the stability of separated flows in the compressible regime. The shear layer is described by an hyperbolic tangent profile for the velocity component and the Crocco relation for the temperature profile. The gravity effects and the superficial tension are neglected. By examining the temporal growth rate at the saddle point in the wave number space, the flow is characterized as being either absolutely unstable or convectively unstable. This study principally shows the non-isothermal effect on the absolute-convective transition in compressible shear flow. Results are presented, showing the amount of the backflow necessary to have this type of transition for a range of primary flow Mach number M ₁ up to 3.0. The boundary of the absolute-convective transition is defined as a function of the velocity ratio, the temperature ratio and the Machnumber.     

Turbulent Energy Scale-Budget Equations for nearly Homogeneous Sheared Turbulence

For moderate Reynolds numbers, the isotropic relation between second-order and third-order moments for velocity increments (Kolmogorov's equation) is not respected, reflecting a non-negligible correlation between the scales responsible for the injection, transfer and dissipation of the turbulent energy. For (shearless) grid turbulence, there is only one dominant large-scale phenomenon, which is the non-stationarity of statistical moments resulting from the decay of energy downstream of the grid. In this case, the extension of Kolmogorov's analysis, as carried out by Danaila, Anselmet, Zhou and Antonia, J. Fluid Mech. 391, 1999 359-369) is quite straightforward. For shear flows, several large-scale phenomena generally coexist with similar amplitudes. This is particularly the case for wall-bounded flows, where turbulent diffusion and shear effects can present comparable amplitudes. The objective of this work is to quantify, in a fully developed turbulent channel flow and far from the wall, the influence of these two effects on the scale-by-scale energy budget equation. A generalized Kolmogorov equation is derived. Relatively good agreement between the new equation and hot-wire measurements is obtained in the outer region (40 < x ⁺ ₃ < 150) of the channel flow, for which the turbulent Reynolds number is R _λ≈ 36.     

Temporal and spatial elements of thermocapillary convection in floating zones

We investigated experimentally spatio-temporal convective flow phenomena in cylindrical liquid bridges [floating-(half-)zones] of liquids with different Prandtl-numbers (NaNO₃?Pr=7; C₂₄H₅₀?Pr=49; C₃₆H₇₄?Pr=65). The convective flow is driven by thermocapillary forces (TC-forces) and buoyancy forces. The zones were heated from above (ΔT, Ma>0) or from below (ΔT, Ma<0) to couple both effects in different ways. Optical evaluations (view from above and view from the front) in connection with thermocouple (tc) measurements (tc-tips distributed over one half of the free surface) made it possible to get very new ideas of spatio-temporal flow structures in the considered convective system. In this article we deal with some transitionary temporal phenomena accompanying the system’s way to chaotic behaviour. We present results supplementary to well-known transitions to chaos (i.e. quasi-periodic and period-doubled flow states) and introduce some very special events. Here all considerations are based on a primarily “temporal way of thinking”. We then try to illuminate several flow situations primarily from a more “spatial point of view”. Possible spatio-temporal convective flow structures are discussed by accompanying the system from a laminar flow state up to the onset of chaotic motion. Starting with former ideas of spatio-temporal flow situations we recognize 2D- and 3D-stationary flows, “pulsating” and “rotating” modes m=1 and 2, different spatial reasons for quasi-periodic and period-doubled temporal behaviour and different spatial mechanisms that cause spatio-temporal chaotic structures in the system considered. One should realize the ambiguity of a certain time-signal with respect to various spatial structures. Additionally we find out that a revision of the interpretation of very complicated Ma/Ma _c (A)-state maps now becomes necessary. These state maps show the present flow state (e.g. a time-dependent, quasi-periodic or chaotic one) depending on the geometrical parameter aspect ratio A (i.e. the zone length) and the TC-force (i.e. the Marangoni-number).     

Analytical and numerical study of double diffusive convection in a vertical enclosure

This paper presents an analytical and numerical study of natural convection of a double-diffusive fluid contained in a rectangular slot subject to uniform heat and mass fluxes along the vertical sides. Governing parameters of the problem under study are the thermal Rayleigh number, Ra _T ; buoyancy ratio, N; Lewis number, Le; Prandtl number, Pr and aspect ratio of the cavity, A. In the first part of the analytical study a scale analysis is applied to the two extreme cases of heat-transfer and mass-transfer-driven flows. In the second part, an analytical solution, based on the parallel flow approximation, is reported for tall enclosures (A?1). Solutions for the flow fields, temperature and concentration distributions and Nusselt and Sherwood numbers are obtained in terms of the governing parameters of the problem. In the limits of heat-driven and solute-driven flows a good agreement is obtained between the prediction of the scale analysis and those of the analytical solution. The numerical solutions are based on the complete governing equations for two-dimensional flows, and cover the range 1≤Ra _T ≤10⁷, 0≤N≤10⁵, 10^-3≤Le≤10³, 1≤A≤20 and Pr=7. A good agreement is found between the analytical predictions and the numerical simulation.