Porous Box Research Articles

Oscillatory convection in a two-dimensional porous box with asymmetric lateral boundary conditions

The onset of convection in a two-dimensional porous cavity is investigated where the cavity is subject to asymmetric boundary conditions at the lateral walls: one vertical wall is thermally conducting and impermeable, while the other is thermally insulating and open. At the open boundary the saturating fluid flows freely in and out from a hydrostatic reservoir in contact with the porous medium. The top and bottom of the box are impermeable and perfectly conducting. It is shown that the mode for onset of convection is oscillatory in time. This corresponds to a disturbance traveling as a wave through the box from the impermeable wall to the open wall. The preferred eigensolution, its oscillation frequency, and critical Rayleigh number are calculated numerically for different aspect ratios of the porous box, and these values are confirmed by means of suitable asymptotic analyses.

Characteristics of solidification and melting in the water‐saturated porous medium cooled from the top (response of solid–liquid interface due to time‐varying cooling temperature)

AbstractThis paper describes the response of a solid–liquid interface in a water‐saturated porous box to a time‐varying cooling temperature. Spherical soda glass beads with an average diameter of 5 mm constitute a porous matrix. The lower boundary of the matrix is kept at 8°C at all times during the experiments, while the upper plate is set at a temperature, lower than the liquids 0°C. After a steady state is reached, the cooling temperature is varied periodically with a fixed amplitude of 4°C. The solid–liquid interface positions are measured and the characteristic amplitudes and the phase delays are determined for different periods ranging from one hour to ten hours at four different cooling temperatures. It has been found that the amplitude of the interface is proportional to the cooling temperature period length, and that a thicker solid layer causes larger phase delays. The proposed one‐dimensional model has been found appropriate for predicting the response of the horizontally averaged position of the solid–liquid interface. © 2004 Wiley Periodicals, Inc. Heat Trans Asian Res, 33(5): 330–341, 2004; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/htj.20015

On a freely floating porous box in shallow water waves

An analytical solution of the flow field, obtained by the interaction of a linear shallow water wave with a freely floating porous box, is derived. The relatively small drift forces obtained in this new solution indicate the advantage in future use of porous structures as floating breakwaters.

Coriolis effect on free convection in a long rotating porous box subject to uniform heat generation

The Coriolis effect on free convection in a long rotating porous box subject to uniform heat generation is investigated analytically. A three dimensional analytical solution is presented for large values of the porous media Ekman number. The convection results from internal heat generation which produces temperature gradients orthogonal to the centrifugal body force. Two types of thermal boundary conditions are considered for the top and bottom walls of the box. The first type is associated with perfectly conducting boundaries, i.e. the same temperature is imposed on both the top and bottom walls while the second type corresponds to a perfectly conducting top wall and adiabatic bottom wall. The solution to the nonlinear set of partial differential equations is obtained through an asymptotic expansion of the dependent variables in terms of two small parameters representing the reciprocal Ekman number in porous media and the aspect ratio of the domain. Secondary circulation in the form of one or two vortices is obtained in a plane orthogonal to the leading free convection plane.

Centrifugally generated free convection in a rotating porous box

An analytical method of solution to the non-linear problem of centrifugally generated free convection in a rotating porous box is presented. The free convection results from differential heating of the horizontal walls leading to temperature gradients perpendicular to the centrifugal body force. The validity of the solution was found to be restricted to small values of the centrifugal Rayleigh number, although this restriction was not explicitly imposed. The results are compared to an asymptotic solution of the corresponding problem for a long rotating porous box. A direct extraction and substitution of the dependent variables was found to be useful in this case for de-coupling the non-linear partial differential equations, resulting in a set of independent non-linear ordinary differential equations which was solved analytically. The solution results, their significance and their validity domain are discussed in their physical context.

Three-Dimensional Free Convection in a Long Rotating Porous Box: Analytical Solution

A three-dimensional analytical solution to the steady-state free convection problem in a long rotating porous box is presented for large values of the porous media Ekman number. The convection results from differential heating of the horizontal walls leading to temperature gradients orthogonal to the centrifugal body force. The solution to the nonlinear set of partial differential equations was obtained through an asymptotic expansion of the dependent variables in terms of two small parameters representing the reciprocal Ekman number in porous media and the aspect ratio of the domain. The results are focused towards the Coriolis effect on the flow. Secondary circulation was obtained in a plane orthogonal to the leading free convection plane. The results show that the Coriolis effect on free convection is controlled by a combined dimensionless group representing the ratio of the centrifugal Rayleigh number to the porous media Ekman number.

On the onset of convection in a water-saturated porous box: Effect of conducting walls

On the onset of convection in a water-saturated porous box: Effect of conducting walls

On the onset of convection in a water-saturated porous box: Effect of conducting walls

The critical Rayleigh number for the onset of convection in a water-saturated porous box, heated from below, has been determined by the Galerkin method and by an approximate analytical technique. The present problem differs from previous ones which have been published in that we consider the effect of conducting vertical boundaries on the critical number and on the cell pattern at the critical number. Moreover, we emphasize fault or fracture zone-like box geometries; that is, geometries in which one box dimension is much shorter than the other and much shorter than the height. The results indicate that for fault/fracture zone geometries, in which the long vertical walls are conducting and the short end walls are insulated, the critical Rayleigh number is roughly four orders of magnitude greater than the critical number for a horizontal porous slab. The flow at the onset of convection takes the form of a roll with its axis along the long horizontal dimension of the box. There is, however, little difference between the critical numbers for two and three dimensional cell patterns. These results indicate that convection may occur in naturally occurring faults and fracture zones in the earth's crust only if the permeability is of the order of Darcies. In natural systems, the Rayleigh number would probably not exceed the critical number greatly, and the flow may tend to be fully three dimensional.

On the thermo-hydromagnetic stability in a porous box

The critical Rayleigh number for the onset of convection in a rectangular porous box saturated with a conducting fluid in the presence of a normal magnetic field and a temperature gradient is studied. The linear and energy methods of determining the stability limit are employed and the critical Rayleigh numbers are obtained. It is observed that the critical Rayleigh number as obtained by linear theory is greater than that of the nonlinear theory and hence a region of subcritical instability is predicted which depends on the Hartmann number.