Thermo-Gravitational Convection in a Near-Critical Fluid in a Side-Heated Enclosed Cavity

Convection near a thermodynamic critical point in a square cavity with lateral heating is investigated numerically on the basis of the Navier-Stokes equations for a compressible gas with a Van-der-Waals equation of state. Comparison of a near-critical fluid and a perfect gas with parameters equal to those of the real fluid near the critical point shows that, with the development of convection, the dynamics of these two media are qualitatively different; however, a certain similarity is observed for the steady-state regime. The dependence of the steady-state flow and heat transfer characteristics on the nondimensional governing parameters is investigated.     

Rayleigh-Benard Convection in a Near-Critical Fluid in the Neighborhood of the Stability Threshold

Steady-state Rayleigh-Benard convection in a medium with parameters close to the thermodynamic critical point is simulated within the framework of the complete Navier-Stokes equations with a two-scale representation of the pressure and the Van-der-Waals equation of state. A calibration relation is obtained for a realistic Rayleigh number in a compressible stratified medium. The parameters of the numerical simulation are determined from experimental data for near-critical helium on the basis of the calibration relation. The threshold Rayleigh numbers are found without and with allowance for stratification and a comparison with the experimental and theoretical data is carried out. The effect of compressibility of the near-critical fluid on steady-state convection flows is investigated beyond the stability threshold and the effect of adiabatic compression of the medium is analyzed.__________Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, 2005, pp. 48–61.Original Russian Text Copyright © 2005 by Polezhaev and Soboleva.     

Effects of Viscous Dissipation on Unsteady Free Convection in a Fluid Past a Vertical Plate Immersed in a Porous Medium

The effects of viscous dissipation on unsteady free convection from an isothermal vertical flat plate in a fluid saturated porous medium are examined numerically. The Darcy–Brinkman–Forchheimer model is employed to describe the flow field. A new model of viscous dissipation is used for the Darcy–Brinkman–Forchheimer model of porous media. The simultaneous development of the momentum and thermal boundary layers are obtained by using a finite difference method. Boundary layer and Boussinesq approximation have been incorporated. Numerical calculations are carried out for various parameters entering into the problem. Velocity and temperature profiles as well as local friction factor and local Nusselt number are shown graphically. It is found that as time approaches infinity, the values of friction factor and heat transfer coefficient approach steady state.     

Unsteady Free Convection Boundary Layer Flows of a Bingham Fluid in Cylindrical Porous Cavities

Transport in Porous Media - We consider two unsteady free convection flows of a Bingham fluid when it saturates a porous medium contained within a vertical circular cylinder. The cylinder is...     

Throughflow Effects on Penetrative Convection in Superposed Fluid and Porous Layers

The effect of vertical throughflow on the onset of penetrative convection simulated via internal heating in a two-layer system in which a layer of fluid overlies and saturates a layer of porous medium is studied. Flow in the porous medium is governed by Forchheimer-extended Darcy equation, and Beavers?CJoseph slip condition is applied at the interface between the fluid and the porous layers. The boundaries are considered to be rigid, however permeable, and insulated to temperature perturbations. The eigenvalue problem is solved using a regular perturbation technique with wave number as a perturbation parameter. The ratio of fluid layer thickness to porous layer thickness, ??, the direction of throughflow, and the presence of volumetric internal heat source in fluid and/or porous layer play a decisive role on the stability characteristics of the system. In addition, the influence of Prandtl number arising due to throughflow is also emphasized on the stability of the system. It is observed that both stabilizing and destabilizing factors can be enhanced because of the simultaneous presence of a volumetric heat source and vertical throughflow so that a more precise control (suppress or augment) of thermal convective instability in a layer of fluid or porous medium is possible.     

Unsteady Convection in a Binary Mixture in the Neighborhood of a Plane Vertical Surface

The problem of the development of convection in a binary mixture in the neighborhood of an infinite vertical plate, on which a constant (after initial switch-on) heat flux and zero admixture flow are given, is solved. In particular, the cases of neutral and stable density stratification of the medium are considered. It is found that heat transfer to the medium can lead not to an increase but to a decrease in its temperature. This can be interpreted as the effective negative heat capacity of the stratified binary mixture.     

Free Fluid Convection in a Closed Contour in a Heat-Conducting Half-Space

The steady-state convection of a fluid in a thin porous vertical ring located in a heat-conducting half-plane is considered. For this problem approximate equations are derived. For a circular ring an analytic solution is obtained. For an elliptic ring a numerical-analytic solution is found. The Nusselt number and the fluid flow rate as functions of the Rayleigh number, the aspect ratio, and the contour depth are investigated.Many studies have been devoted to fluid convection in a porous ring [1–3]. In [1] two-dimensional convection with an isothermal internal boundary was considered when a temperature stratification is given on the outer boundary. A feature of this problem is the fact that the ring is located inside an impermeable heat-conducting medium in which a thermal gradient directed vertically downward is specified at a large distance from the ring. In [2, 3] two-dimensional convection in an annular ring occupied by a porous medium was investigated. From the results obtained in these studies it follows that in the formulation considered the hydraulic approximation can be used with satisfactory accuracy. In the present study this question is discussed more concretely and the necessary estimates are found. The results obtained could be useful for investigating hydrothermal convection in the Earth's crust, which has important geophysical applications [4–6].     

Unsteady MHD Flow on a Rotating Cone in a Rotating Fluid

Unsteady flow over an infinite permeable rotating cone in a rotating fluid in the presence of an applied magnetic field has been investigated. The unsteadiness is induced by the time-dependent angular velocity of the body, as well as that of the fluid. The partial differential equations governing the flow have been solved numerically by using an implicit finite-difference scheme in combination with the quasi-linearization technique. For large values of the magnetic parameter, analytical solutions have also been obtained for the steady-state case. It is observed that the magnetic field, surface velocity, and suction and injection strongly affect the local skin friction coefficients in the tangential and azimuthal directions. The local skin friction coefficients increase when the angular velocity of the fluid or body increases with time, but these decrease with decreasing angular velocity. The skin friction coefficients in the tangential and azimuthal directions vanish when the angular velocities of fluid and the body are equal but this does not imply separation. When the angular velocity of the fluid is greater than that of the body, the velocity profiles reach their asymptotic values at the edge of the boundary layer in an oscillatory manner, but the magnetic field or suction reduces or suppresses these oscillations.     

Unsteady Natural Convection Within a Porous Enclosure of Sinusoidal Corrugated Side Walls

Numerically investigation of free convection within a porous cavity with differential heating has been performed using modified corrugated side walls. Sinusoidal hot left and cold right walls are assumed to receive sudden differentially heating where top and bottom walls are insulated. Air is considered as working fluid and is quiescent, initially. Numerical experiments reveal 3 distinct stages of developing pattern including initial stage, oscillatory intermediate, and finally steady-state condition. Implicit Finite Volume Method with TDMA solver is used to solve the governing equations. This study has been performed for the Rayleigh numbers ranging from 100 to 10,000. Outcomes have been reported in terms of isotherms, streamline, velocity and temperature plots and average Nusselt number for various Ra, corrugation frequency, and corrugation amplitude (CA). The effects of sudden differential heating and its resultant transient behavior on fluid flow and heat transfer characteristics have been shown for the range of governing parameters. The present results show that the transient phenomena are enormously influenced by the variation of the Rayleigh Number with CA and frequency.     

Convection in a Horizontal Fluid Layer with a Uniform Internal Heat Source

On the basis of a numerical simulation of convection in a horizontal fluid layer with a uniform heat source it is concluded that the convective heat flux is constant over the entire convection layer not only in the case of steady-state external conditions but also in the case of heating (cooling) of the fluid layer at a constant rate. The convective heat flux is mainly determined by the Rayleigh number and depends only slightly on the layer heating (cooling) rate.     

Numerical Modeling of Unsteady Two-Dimensional Convection in a Finite-Length Horizontal Layer Heated from Below

The development of two-dimensional thermo-gravitational convection in an elongated horizontal layer bounded by solid surfaces with the bottom instantaneously heated is investigated. The characteristics of the transition from the heat conduction regime to the convective regime are considered. The flow pattern and the heat transfer properties are described from the initial instant, which corresponds to the isothermal fluid at rest, up to the attainment of the steady-state roll-convection regime. A criterial dependence between the Rayleigh number and the nondimensional time of onset of the influence of thermo-gravitational convection on heat transfer is obtained.     

Fully Developed Mixed Convection Flow in a Vertical Channel Containing Porous and Fluid Layer with Isothermal or Isoflux Boundaries

An analysis of fully developed combined free and forced convective flow in a fluid saturated porous medium channel bounded by two vertical parallel plates is presented. The flow is modeled using Brinkman equation model. The viscous and Darcy dissipation terms are also included in the energy equation. Three types of thermal boundary conditions such as isothermal–isothermal, isoflux–isothermal, and isothermal–isoflux for the left–right walls of the channel are considered. Analytical solutions for the governing ordinary differential equations are obtained by perturbation series method. In addition, closed form expressions for the Nusselt number at both the left and right channel walls are derived. Results have been presented for a wide range of governing parameters such as porous parameter, ratio of Grashof number and Reynolds number, viscosity ratio, width ratio, and conductivity ratio on velocity, and temperature fields. It is found that the presence of porous matrix in one of the region reduces the velocity and temperature.     

Unsteady Self-Similar Flow of a Viscous Incompressible Fluid in a Plane Divergent Channel

The problem of two-dimensional unsteady flow of a viscous incompressible fluid in a sector-like domain is considered. Initially a strictly radial flow is imposed, which makes it possible to seek solutions within the class of self-similar flows. A numerical method based on mixed finite-difference and spectral spatial discretization is developed, making it possible to find the self-similar solution efficiently. The process of development and establishment of the steady Hamel-Jeffery and Moffatt flows is modeled mathematically.     

Convection of a Freezing Fluid in a Square Cell After Loss of Hydrostatic Stability

The behavior of a freezing (defrosting) fluid in a square cell is studied for three different initial conditions. For a fixed Grashof number Gr, the existence of four steady-state solutions is demonstrated. On a certain range of Gr, an increased solid-phase fraction in the cell, as compared with the pure heat conduction case, is obtained. The critical values of Gr corresponding to passage from one type of solution to another are found     

Onset of Buoyancy-Driven Convection in Porous Media Saturated with Cold Water Cooled from Above

When porous media saturated with initially stagnant cold water around the density maximum temperature are cooled from above, convection may be induced in an unstable lower layer. In this study, the onset of buoyancy-driven convection during time-dependent cooling is investigated using the propagation theory, which transforms disturbance equations similarly, and also considering the density inversion effect. The critical Darcy–Rayleigh number Ra _D,c is found as a function of the dimensionless density maximum temperature θ _max. For Ra _D > Ra _D,c the dimensionless critical time τ _c to mark the onset of instability is presented as a function of Ra _D and θ _max. These critical conditions are compared with previous theoretical results.     

Centrifugal Acceleration Induced Convection in a Magnetic Fluid Saturated Anisotropic Rotating Porous Medium

A theoretical investigation is made to study the influence of magnetic field on the onset of convection induced by centrifugal acceleration in a magnetic fluid filled porous medium. The layer is assumed to exhibit anisotropy in mechanical as well as thermal sense. Numerical solutions are obtained using the Galerkin method for the eigenvalue problem arising from the linear stability theory. It is found that the magnetic field has a destabilizing effect and can be suitably adjusted depending on the anisotropy parameters to enhance convection. The effect of anisotropies of magnetic fluid filled porous media is shown to be qualitatively different from that of ordinary fluid filled porous media. This phenomenon may be helpful to increase the efficiency of suitable heat transfer devices.     

Unsteady Free Convection along an Infinite Vertical Flat Plate Embedded in a Stably Stratified Fluid-Saturated Porous Medium

The problem of unsteady free convection heat transfer from a one-dimensional (parallel) flow along an infinite vertical flat plate embedded in a thermally stratified fluid-saturated porous medium is considered. Flows are induced by a sudden change in the arbitrary temporal plate temperature. By a formal reduction of the corresponding boundary value problems to well-known Fourier heat conduction problems, analytical solutions of the Darcy and energy equations are obtained. Several special cases are discussed in detail.     

Unsteady Flow of an Oscillating Porous Disk and a Fluid at Infinity

An exact analytic solution of the unsteady Navier–Stokes equations is obtained for the flow caused by the non-coaxial rotations of a porous disk and a fluid at infinity. The porous disk is executing oscillations in its own plane with superimposed injection or suction. An increasing or decreasing velocity amplitude of the oscillating porous disk is also discussed. Further, it is shown that a combination of suction/injection and decreasing/increasing velocity amplitude is possible as well. In addition, the flow due to porous oscillating disk and a fluid at infinity rotating about an axis parallel to the z-axis is attempted as a second problem. Sommario. Si studia il flusso non stazionario prodotto dall'oscillazione di un disco poroso in un fluido e si fornisce una soluzione analitica delle equazioni di Navier–Stokes. Si discute l'effetto di una suzione/iniezione e di una variazione sull'ampiezza della velocità' di oscillazione. Infine si studia il flusso dovuto alle oscillazioni non coassiali di un disco poroso e di un fluido all'infinito.     

Vibrational Convection in an Inclined Fluid Layer Heated from Below

The stability of mechanical equilibrium of an inclined fluid layer with respect to three-dimensional perturbations under the action of high-frequency vibration is studied. It is shown that under heating from below the spiral perturbations are always the most dangerous for vibration transverse to the layer. For vertical vibration the stability limit is determined by three-dimensional perturbations whose shape depends in a complicated way on the angle of inclination of the layer and the vibrational Rayleigh number. In the limiting case of a thin vertical layer supercritical vibrational-convective motions are calculated numerically and analytically and scenarios of transition from quasi-equilibrium to irregular motions are studied.     

Unsteady Natural Convection Flow in a Square Cavity Filled with a Porous Medium Due to Impulsive Change in Wall Temperature

Unsteady natural convection flow in a two-dimensional square cavity filled with a porous material has been studied. The flow is initially steady where the left-hand vertical wall has temperature T _h and the right-hand vertical wall is maintained at temperature T _c (T _h > T _c) and the horizontal walls are insulated. At time t > 0, the left-hand vertical wall temperature is suddenly raised to which introduces unsteadiness in the flow field. The partial differential equations governing the unsteady natural convection flow have been solved numerically using a finite control volume method. The computation has been carried out until the final steady state is reached. It is found that the average Nusselt number attains a minimum during the transient period and that the time required to reach the final steady state is longer for low Rayleigh number and shorter for high Rayleigh number.