Convection in a rotating medium above a thermally inhomogeneous horizontal surface

The linear stationary problem of convection in a medium rotating about a vertical axis above a thermally inhomogeneous horizontal surface is theoretically investigated. Attention is mainly focused on the case of a homogeneous medium, but certain stratification effects and especially the convection characteristics in binary mixtures (for example, in saline sea water) are also considered. When the rotation is rapid (large Taylor numbers) the convective cells are strongly elongated in the vertical direction, though they also contain a thin Ekman boundary layer. The importance of the boundary conditions on the horizontal surface (in parallel with the no-slip conditions, more general conditions that may follow from the quadratic turbulent friction model are considered) is shown. In the case of binary mixtures, the differential diffusion and rotation effects may together result in the appearance of “induced salt fingers”, the deep penetration of convection into an arbitrarily stably stratified medium. The convective motions may then have a considerable effect on the background vertical temperature and admixture distributions. Attention is drawn to an original manifestation of the analogy between the rotation and stratification effects: in a non-rotating, stably stratified medium, near a thermally inhomogeneous vertical surface, the convection also penetrates deep into the medium, but in the horizontal direction, so that, when the coordinate system is rotated through 90°, the solution coincides with the case of a rotating non-stratified fluid considered here.     

Natural convection flow from a continuously moving vertical surface immersed in a thermally stratified medium

 Natural convection boundary layer flow over a continuously moving isothermal vertical surface immersed in a thermally stratified medium has been investigated here. The non-linear coupled partial differential equations governing the non-similar flow have been solved numerically using an implicit finite difference scheme. For small values of the streamwise distance the partial differential equations are solved by using a perturbation expansion procedure and also using the Shanks transformation. The results indicate that the thermal stratification significantly affects both the surface shear stress and the surface heat transfer. The buoyancy parameter and the Prandtl number increase significantly, both the surface shear stress and heat transfer. Also the buoyancy force gives rise to an overshoot in the velocity profile. Received on 1 February 2000     

An integral method for free convection from a vertical heated surface in a thermally stratified porous medium

The Kárman-Pohlhausen integral procedure was exploited to solve the problem of free convection from a vertical heated surface imbedded in a thermally stratified porous medium. Upon considering the Darcy law, the energy integral relation, and the auxiliary relation associated with the curvature of the temperature profile at the wall, a closed-form expression for the Nusselt number was reduced for the case where the distributions of the wall temperature and ambient temperature vary according to power-functions of distance. Results on an isothermal plate are found to be in close agreement with those from the exact solution. Effects of the thermal stratification on local heat transfer rates are also discussed.

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Freie Konvektion von einer beheizten, vertikalen Fläche in der thermischen Schichtung eines porösen Mediums

Zusammenfassung Die Kárman-Pohlhausen-Integration wurde angewandt, um das Problem der freien Konvektion von einer vertikalen Fläche, die in ein thermisch geschichtetes poröses Medium eingebettet ist, zu lösen. Unter Hinzuziehung des Darcyschen Gesetzes der integralen Energiebeziehung und einer Hilfsbeziehung, die mit der Krümmung des Temperaturprofiles an der Wand zusammenhing, wurde eine geschlossene Form für die Nusselt-Zahl gefunden, jeweils bezogen auf den Fall, wenn die Verteilung der Wandtemperatur und der Umgebungstemperatur entsprechend eines Exponentialansatzes mit dem Abstand abnimmt. Ergebnisse, die für eine isotherme Platte mit dieser Methode berechnet wurden, sind in guter Übereinstimmung mit denen der exakten Lösung. Es werden auch Einflüsse der thermischen Schichtung auf den örtlichen Wärmeübergang diskutiert.

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Nomenclature A, B, C, D shape factors - f profile function - g acceleration due to gravity - K permeability - m parameter associated with thermal stratification - n power-law exponent of the wall temperature - Nux local Nusselt number - Pr Prandtl number - Rax local Rayleigh number - T temperature - u, v velocity components inx andy directions - x, y boundary layer coordinates Greek symbols shape parameter - \ coefficient of thermal expansion - boundary layer thickness - similarity variable     

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     

Mixing in a stably stratified medium by horizontal shear near vertical walls

Stratified environmental flows near boundaries can have a horizontal mean shear component, orthogonal to the vertical mean density gradient. Vertical transport, against the stabilizing force of gravity, is possible in such situations if three-dimensional turbulence is sustained by the mean shear. A model problem, water with a constant mean density gradient flowing in a channel between parallel vertical walls, is examined here using the technique of large eddy simulation (LES). It is found that, although the mean shear is horizontal, the fluctuating velocity field has significant vertical shear and horizontal vorticity, thereby causing small-scale vertical mixing of the density field. The vertical stirring is especially effective near the boundaries where the mean shear is large and, consequently, the gradient Richardson number is small. The mean stratification is systematically increased between cases in our study and, as expected, the buoyancy flux correspondingly decreases. Even so, horizontal mean shear is found to be more effective than the well-studied case of mean vertical shear in inducing vertical buoyancy transport as indicated by generally larger values of vertical eddy diffusivity and mixing efficiency.     

Fine structure of a stratified flow near a flat-plate surface

The pattern of disturbances arising during the motion of a strip along a horizontal surface in a continuously stratified fluid with identified upstream and attached internal waves, boundary layers, and edge singularities is calculated in the liner approximation. The flow pattern behind a flat plate moving with a constant velocity in a continuously stratified fluid is studied with the use of the optical schlieren technique; transformation of waves and finely structured elements of the flow with increasing plate velocity is analyzed. The calculated and experimentally observed patterns of internal waves at low velocities are demonstrated to be in good agreement. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 6, pp. 77–91, November–December, 2007.     

Travelling waves near a critical point of a binary fluid mixture

Travelling waves of densities of binary fluid mixtures are investigated near a critical point. The free energy is considered in a non-local form taking account of the density gradients. The equations of motions are applied to a universal form of the free energy near critical conditions and can be integrated by a rescaling process where the binary mixture is similar to a single fluid. Nevertheless, density solution profiles obtained are not necessarily monotonic. As indicated in Appendix, the results might be extended to other topics like in finance or biology.     

Non-Darcy free convection in a thermally stratified porous medium along a vertical plate with variable heat flux

Free convection along an impermeable vertical plate embedded in a thermally stratified, fluid-saturated porous medium is analyzed. The wall heat flux is varied in a power-law form. The non-Darcian effects, such as solid-boundary viscous resistances, high-flow-rate inertia forces, near wall nonuniform porosity distribution and thermal dispersion, have been considered in the present study. Due to the variation of porosity in the near wall region, the stagnant thermal conductivity also varies accordingly. The nonsimilar system of transformed equations is solved with Keller's Box method. It is shown that the thermal stratification effect and the higher value of the exponent m can increase the local Nusselt number. Also the non-Darcian and thermal dispersion effects significantly influence the velocity and temperature profiles and local Nusselt number.     

Convection of a binary mixture under conditions of thermal diffusion and variable temperature gradient

Instability of a plane horizontal layer of an incompressible binary gas mixture stratified in the gravity field under the action of a transverse temperature gradient modulated in time is studied. The case of solid impermeable boundaries of the layer, where the flux of matter vanishes, is considered. The analysis is based on the Floquet method applied to linearized equations of convection in the Boussinesq approximation. It is shown that there are regions of parametric instability at finite frequencies. In addition to the synchronous or subharmonic response to an external action, the instability may be related to quasiperiodic disturbances. Depending on the amplitude and frequency, modulation can stabilize the unstable basic state and also destabilize the equilibrium of the fluid. The threshold values of convection for modulations of temperature and translational vertical vibrations are compared.     

Free convection on a vertical plate with uniform and constant heat flux in a thermally stratified micropolar fluid

This paper analyzes flow and heat transfer characteristics of the free convection on a vertical plate with uniform and constant heat flux in a thermally stratified micropolar fluid. The dimensionless forms of boundary layer equations and their associated boundary conditions have been derived and the numerical results have been obtained using the method of cubic spline collocation with a finite difference scheme. The effects of the micropolar and stratification parameters on the dimensionless wall temperature, skin friction parameter and wall couple stress are discussed.     

Concentration jump on the plane surface in a binary gas mixture

From the solution of linearized Boltzmann equations with the collision operators in the Gross and Krook form the authors have obtained an expression for the concentration jump of a vapor evaporating from a plane liquid surface into a stationary gas, for which the liquid surface is impermeable. Langmuir [1] was the first to point out that at the surface of an evaporating liquid the vapor pressure differs from saturated vapor pressure, i.e., a jump occurs in vapor concentration. An expression was obtained by Brock [2] for the concentration jump at the plane surface in the binary gas mixture with identical molecular masses of components by gas kinetic analysis. No solution has yet been obtained to the problem at different molecular masses of components.Moscow. Translated from Izvestiya Akademii Nauk SSSR. Mekhanika Zhidkosti i Gaza, No. 2, pp. 119–122, March–April, 1972.     

Solidification of a binary mixture in a square cavity with a free surface

Solidification of an NH₄Cl-H₂O solution in a square cavity with a free surface has been studied numerically and experimentally. Solidification is characterized by the coexistence of solid, mushy (solid plus liquid), and liquid regions, and flow within the liquid and mushy regions is driven by surface tension gradients, as well as by opposing thermal/solutal buoyancy forces. Theoretical and experimental results indicate that (i) surface tension forces spawn a cellular flow at the top of the cavity and enhance its development with increasing time, (ii) water enrichment of the top cell induces remelting of adjoining solid, and (iii) final steady-state conditions are characterized by a liquidus front whose thickness increases with increasing depth. However, predicted and observed results differ with respect to the chronological pattern of multicellular flow development and the time required to achieve steady-state conditions. Differences are attributed to selected model assumptions and to uncertainties concerning mushy region structure.     

Similarity solutions for buoyancy-induced flows over a non-isothermal curved surface in a thermally stratified porous medium

A generalized similarity transformation procedure was proposed for the analysis of buoyancy-induced flows over a curved heated surface embedded in a thermally stratified porous medium. The analysis considers two-dimensional and axisymmetric non-isothermal smooth bodies of arbitrary geometrical configuration. A generalized similarity variable which adjusts its vertical scaling according to the geometry as well as the surface temperature variation was introduced to show that, for any two-dimensional or axisymmetric smooth body shape, there exists a certain class of the surface and ambient temperature distributions which admit similarity solutions. Subsequently, the set of the governing partial differential equations were transformed into a single ordinary differential equation, which was, then, solved by a standard shooting procedure based on the Runge-Kutta method, for numerous sets of parameters. The results presented here may readily be translated for the problem of free convection over any particular two-dimensional or axisymmetric smooth body within a porous medium. The effects of the surface temperature and thermal stratification on the temperature profile and isotherms were also discussed in connection with the local surface heat flux.     

Longwave instability of a binary mixture flow in a vertical channel in the presence of vibration

The longwave instability of a hybrid (thermogravitational and thermovibrational) flow of a binary incompressible liquid mixture occurring in a plane vertical channel, whose boundaries are maintained at constant but different temperatures, is studied. The investigation is carried out with account for the Soret thermal-diffusion effect on the ranges of normal and anomalous values of the mixture separation coefficient. It is shown that, owing to the properties of the system, the subharmonic response to an external action is absent. The ranges on which secondary flows arise are analytically determined using the asymptotic expansion method, both under weightlessness conditions and in the presence of the gravity effect. The parameter ranges on which longwave disturbances present the greatest danger for the main flow stability are determined.     

Pressure field around a well in a stratified inhomogeneous bed

The steady axisymmetric flow of an incompressible fluid into a vertical well hydrodynamically perfectly drilled into a stratified inhomogeneous half-space consisting of three layers with different permeabilities is considered. The boundaries of the layers are assumed to be horizontal planes and the roof of the upper layer is assumed to be impermeable. The flow obeys a linear Darcy’s law. The pressure distribution on the well is assumed to be given, which is the main obstacle to finding an exact solution of the problem. Beginning with the classical studies of Muskat and Charnyi [1, 2], approximate solutions of such problems have been constructed as a superposition of flows generated by point sources with given intensities, distributed along the well axis in accordance with a fairly simple law. In the present study, this approach is used to obtain an integral equation for the source density distribution, which is then solved numerically. Comparison with the known exact solution for a thin elongated ellipsoid (“needle”) shows that this approach makes it possible to ensure an accuracy which at any rate is sufficient for applications.     

Instability of a vertical columnar vortex in a stratified fluid

In this article, we study experimentally the evolution of a vertical columnar vortex in a stratified fluid. Three different measurement techniques are used. Particle image velocimetry allows us to monitor the time evolution of the characteristics of the vortex (Froude and Reynolds numbers). Dye visualizations reveal the existence of an instability for Froude numbers smaller than one, which creates an undulation of the vortex centerline. Synthetic schlieren visualization shows that the density structure of the unstable mode is very similar to the structure found recently numerically for the radiative instability of a Lamb–Oseen vortex (Riedinger et al. in J Fluid Mech, 2010). The experimental stability diagram and unstable wavelengths are compared with these numerical results. A secondary instability associated with the presence of critical layers is also observed for Froude numbers larger than one.