Instability of Vertical Throughflows in Bidisperse Porous Media

In this paper, the instability of a vertical fluid motion, or throughflow, is investigated in a horizontal bidisperse porous layer that is uniformly heated from below. By means of the order-1 Galerkin approximation method, the critical Darcy–Rayleigh number for the onset of steady instability is determined in closed form. The coincidence between the linear instability threshold and the global nonlinear stability threshold, in the energy norm, is shown.

Finite Darcy–Prandtl Number and Maximum Density Effects on Gill's Stability Problem

The simultaneous effect of a time-dependent velocity term in the momentum equation and a maximum density property on the stability of natural convection in a vertical layer of Darcy porous medium is investigated. The density is assumed to vary quadratically with temperature and as a result, the basic velocity distribution becomes asymmetric. The problem has been analyzed separately with (case 1) and without (case 2) time-dependent velocity term. It is established that Gill's proof of linear stability effective for case 2 but found to be ineffective for case 1. Due to the lack of Gill's proof for case1, the stability eigenvalue problem is solved numerically and observed that the instability sets in always via traveling-wave mode when the Darcy–Prandtl number is not larger than 7.08. The neutral stability curves and isolines are presented for different governing parameters. The critical values of Darcy–Rayleigh number corresponding to quadratic density variation with respect to temperature, critical wave number, and the critical wave speed are computed for different values of governing parameters. It is found that the system becomes more stable with increasing Darcy–Rayleigh number corresponding to linear density variation with respect to temperature and the Darcy–Prandtl number.

Three-Dimensional Convective Planforms for Inclined Darcy-Bénard Convection

We investigate the onset of convection in an inclined Darcy-Bénard layer. When such a layer is unbounded in the spanwise direction it is generally known that longitudinal rolls comprise the most unstable planform. On the other hand, when a layer has a sufficiently small spanwise width, then transverse rolls form the most unstable planform. However, the layer remains stable to transverse roll disturbances when the inclination is above roughly 31 degrees from the horizontal. This paper considers the transition between these two extreme cases where the spanwise width takes moderate values and where rectangular cells are considered. It is found that the most unstable planform is quite strongly sensitive to the magnitude of the spanwise width and that there are large regions of parameter space within which three-dimensional convection patterns have the smallest critical Darcy-Rayleigh number.

The Horton–Rogers–Lapwood problem for an inclined porous layer with permeable boundaries

A formulation of the Horton–Rogers–Lapwood problem for a porous layer inclined with respect to the horizontal and characterized by permeable (isobaric) boundary conditions is presented. This formulation allows one to recover the results reported in the literature for the limiting cases of horizontal and vertical layer. It is shown that a threshold inclination angle exists which yields an upper bound to a parametric domain where the critical wavenumber is zero. Within this domain, the critical Darcy–Rayleigh number can be determined analytically. The stability analysis is performed for linear perturbations. The solution is found numerically, for the inclination angles above the threshold, by employing a Runge–Kutta method coupled with the shooting method.

The convection of a Bingham fluid in a differentially-heated porous cavity

Purpose – The purpose of this paper is to determine the manner in which a yield stress fluid begins convecting when it saturates a porous medium. A sidewall-heated rectangular cavity is selected as the testbed for this pioneering work. Design/methodology/approach – Steady solutions are obtained using a second order accurate finite difference method, line relaxation based on the Gauss-Seidel smoother, a Full Approximation Scheme multigrid algorithm with V-cycling and a regularization of the Darcy-Bingham model to smooth the piecewise linear relation between the Darcy flux and the applied body forces. Findings – While Newtonian fluids always convect whenever the Darcy-Rayleigh number is nonzero, Bingham fluids are found to convect only when the Darcy-Rayleigh number exceeds a value which is linearly dependent on both the Rees-Bingham number and the overall perimeter of the rectangular cavity. Stagnation is always found in the centre of the cavity and in regions close to the four corners. Care must be taken over the selection of the regularization constant. Research limitations/implications – The Darcy-Rayleigh number is restricted to values which are at or below 200. Originality/value – This is the first investigation of the effect of yield stress on nonlinear convection in porous media.

A proof that convection in a porous vertical slab may be unstable

The stability of the natural convection parallel flow in a vertical porous slab is reconsidered, by reformulating the problem originally solved in Gill’s classical paper of 1969 (J. Fluid Mech., vol. 35, pp. 545–547). A comparison is made between the set of boundary conditions where the slab boundaries are assumed to be isothermal and impermeable (Model A), and the set of boundary conditions where the boundaries are modelled as isothermal and permeable (Model B). It is shown that Gill’s proof of linear stability for Model A cannot be extended to Model B. The question about the stability/instability of the basic flow is examined by carrying out a numerical solution of the stability eigenvalue problem. It is shown that, with Model B, the natural convection parallel flow in the basic state becomes unstable when the Darcy–Rayleigh number is larger than 197.081. The normal modes selected at onset of instability are transverse rolls. Direct numerical simulations of the nonlinear regime of instability are carried out.

Convective Instability of the Darcy Flow in a Horizontal Layer With Symmetric Wall Heat Fluxes and Local Thermal Nonequilibrium

The linear stability of the parallel Darcy throughflow in a horizontal plane porous layer with impermeable boundaries subject to a symmetric net heating or cooling is investigated. The onset conditions for the secondary thermoconvective flow are expressed through a neutral stability bound for the Darcy–Rayleigh number associated with the uniform heat flux supplied or removed from the walls. The study is performed by taking into account a condition of local thermal nonequilibrium between the solid phase and the fluid phase. The linear stability analysis is carried out according to the normal modes' decomposition of the perturbations to the basic state. The governing equations for the disturbances are solved numerically as an eigenvalue problem leading to the neutral stability condition. If compared with the asymptotic condition of local thermal equilibrium, the regime of local nonequilibrium manifests an enhanced instability. This behavior is displayed by lower critical values of the Darcy–Rayleigh number, eventually tending to zero when the thermal conductivity of the solid phase is much larger than the conductivity of the fluid phase. In this special limit, which can be invoked as an approximate model of a gas-saturated metallic foam, the basic throughflow is always unstable to external disturbances of arbitrarily small amplitude.

Plume flows in porous media driven by horizontal differential heating

In this paper we describe flow through a porous medium in a two-dimensional rectangular cavity driven by differential heating of the impermeable lower surface. The upper surface is held at constant pressure and at a constant temperature equal to the minimum temperature of the lower surface, while the sidewalls are impermeable and thermally insulated. Numerical results for general values of the Darcy–Rayleigh number $R$ and the cavity aspect ratio $A$ are compared with theoretical predictions for the small Darcy–Rayleigh number limit $(R\ensuremath{\rightarrow} 0)$ where the temperature field is conduction-dominated, and with a boundary-layer theory for the large Darcy–Rayleigh number limit $(R\ensuremath{\rightarrow} \infty )$ where convection is significant. In the latter case a horizontal boundary layer near the lower surface conveys fluid to the hot end of the cavity where it rises to the upper surface in a narrow plume. Predictions are made of the vertical heat transfer through the cavity.

Symbolic Calculation for Free Convection in a Circular Cavity With Porous Material

We consider the two-dimensional problem of steady natural convection in a circular cavity filled with porous material due to a cosine temperature variation on the boundary. We use Darcy’s law for this cavity filled with porous material. The solution is governed by dimensionless parameter Darcy-Rayleigh number. The solution is expanded for low Darcy-Rayleigh number and extended to 18 terms by computer. Analysis of these expansions allows the exact computation for arbitrarily accuracy up to 50000 figures. Although the range of the radius of convergence is small but Pade approximation leads our result to be good even for higher value of the similarity parameter.

On the boundary layer structure of differentially heated cavity flow in a stably stratified porous medium

This paper considers two-dimensional flow generated in a stably stratified porous medium by monotonic differential heating of the upper surface. For a rectangular cavity with thermally insulated sides and a constant-temperature base, the flow near the upper surface in the high-Darcy–Rayleigh-number limit is shown to consist of a double horizontal boundary layer structure with descending motion confined to the vicinity of the colder sidewall. Here there is a vertical boundary layer structure that terminates at a finite depth on the scale of the outer horizontal layer. Below the horizontal boundary layers the motion consists of a series of weak, uniformly stratified counter-rotating convection cells. Asymptotic results are compared with numerical solutions for the cavity flow at finite values of the Darcy–Rayleigh number.