Michele Celli

Darcy–Forchheimer Flow With Viscous Dissipation in a Horizontal Porous Layer: Onset of Convective Instabilities

Parallel Darcy―Forchheimer flow in a horizontal porous layer with an isothermal top boundary and a bottom boundary, which is subject to a third kind boundary condition, is discussed by taking into account the effect of viscous dissipation. This effect causes a nonlinear temperature profile within the layer. The linear stability of this nonisothermal base flow is then investigated with respect to the onset of convective rolls. The third kind boundary condition on the bottom boundary plane may imply adiabatic/isothermal conditions on this plane when the Biot number is either zero (adiabatic) or infinite (isothermal). The solution of the linear equations for the perturbation waves is determined by using a fourth order Runge―Kutta scheme in conjunction with a shooting technique. The neutral stability curve and the critical value of the governing parameter R=GePe 2 are obtained, where Ge is the Gebhart number and Pe is the Peclet number. Different values of the orientation angle between the direction of the basic flow and the propagation axis of the disturbances are also considered.

Thermal Instability of a Power-Law Fluid Flowing in a Horizontal Porous Layer with an Open Boundary: A Two-Dimensional Analysis

A two-dimensional analysis of the onset of thermal convective instability in a horizontal porous layer with open upper boundary is carried out. The saturating fluid is non-Newtonian of power-law behaviour, and its flow is represented through a suitable extension of Darcy’s law. A model of temperature-dependent viscosity is employed where the consistency index is considered as variable, while the power-law index is assumed to be constant. Numerical data for the neutral stability and for the critical values of a modified Darcy–Rayleigh number have been obtained. The feasibility of an experimental validation of the theoretical results predicted by the stability analysis is discussed in detail. An experimental set-up based on a Hele-Shaw cell is described, and preliminary results relative to the onset of convective cells are described. Observed hysteretic effects and deviations from the rheological model are identified as potential sources of uncertainty.

Nonlinear stability analysis of Darcy’s flow with viscous heating

The nonlinear stability of a rectangular porous channel saturated by a fluid is here investigated. The aspect ratio of the channel is assumed to be variable. The channel walls are considered impermeable and adiabatic except for the horizontal top which is assumed to be isothermal. The viscous dissipation is acting inside the channel as internal heat generator. A basic throughflow is imposed, and the nonlinear convective stability is investigated by means of the generalized integral transform technique. The neutral stability curve is compared with the one obtained by the linear stability analysis already present in the literature. The growth rate analysis of different unstable modes is performed. The Nusselt number is investigated for several supercritical configurations in order to better understand how the system behaves when conditions far away from neutral stability are considered. The patterns of the neutrally stable convective cells are also reported. Nonlinear simulations support the results obtained by means of the linear stability analysis, confirming that viscous dissipation alone is indeed capable of inducing mixed convection. Low Gebhart or high Péclet numbers lead to a transient overheating of the originally motionless fluid before it settles in its convective steady state.

The Effects of Double Diffusion and Local Thermal Non-equilibrium on the Onset of Convection in a Layered Porous Medium: Non-oscillatory Instability

The effect of local thermal non-equilibrium on the onset of double-diffusive convection in a porous medium consisting of two horizontal layers is studied analytically. Linear stability theory is applied. Variations of permeability, fluid conductivity, solutal diffusivity, solid conductivity, interphase heat transfer coefficient, and porosity are considered. It is found that with the introduction of double diffusion, the heterogeneity of porosity now has a major effect, comparable to the effects of heterogeneity of permeability and fluid conductivity. The general results are obtained by using a one-term Galerkin approximation. We validate this approximation by comparing these results with those obtained by using a highly accurate numerical solver. We thus established the accuracy of a one-term Galerkin approximation for stability analysis of a complicated convection problem.

Local Thermal Non-equilibrium Analysis of the Instability in a Vertical Porous Slab with Permeable Sidewalls

Buoyant flow in a fluid-saturated porous vertical slab with isothermal and permeable boundaries is performed. Two reservoirs, maintained at different uniform temperatures, confine the slab. The permeable plane boundaries of the slab are modelled by imposing a condition of hydrostatic pressure. Darcy’s law and the Oberbeck–Boussinesq approximation are employed. The hypothesis of local thermal equilibrium between the fluid and the solid phase is relaxed. A two-temperature model is adopted, so that two local energy balance equations govern the heat transfer in the porous slab. The basic stationary buoyant flow consists of a single convective cell of infinite height. The time evolution of normal mode perturbations superposed onto the basic state is investigated in order to determine the onset conditions for thermal instability. A pressure–temperature formulation is employed. Major asymptotic cases are investigated. It is shown that departure from local thermal equilibrium implies in general a destabilisation of the basic stationary flow.

Convective Instability in a Darcy Flow Heated from Below with Internal Heat Generation

The linear stability of a Darcy flow through an infinitely wide horizontal channel is here investigated. In particular, the paper is focused on the onset of thermal convection through a convective instability. The Oberbeck–Boussinesq approximation is employed to model the buoyancy term inside Darcy’s law. The channel is impermeable and heated from below by an isoflux condition. A uniform internal heat source is imposed. A steady solution is found and used as basic state for the stability analysis. This basic state is composed of a pressure gradient term and a buoyancy force term. The rotation symmetry around the vertical axis allows an important simplification: the two-dimensional case is treated here instead of the full three-dimensional case without any loss of generality. The normal modes method is employed to perform the linear stability analysis. The resulting eigenvalue problem is solved analytically for vanishing wavenumbers and numerically otherwise. Analytical solutions are used to validate the numerical procedure. Critical Rayleigh numbers for the onset of thermal convection prove that the most unstable modes are longitudinal. Temporal growth rates for longitudinal modes are calculated under slightly supercritical conditions to identify the fundamental characteristics of the most dominant mode which will be observed in experimental or numerical simulations under the same parametric conditions.

Instability of Combined Forced and Free Flow in an Inclined Porous Channel

The aim of this paper is to analyze the onset of convective instability in a plane porous channel inclined to the horizontal. A net upslope or downslope flow is considered, so that mixed convection takes place as caused by the uniform and symmetric heat fluxes prescribed on the impermeable bounding walls. The thermoconvective instability of the basic flow is studied versus small-amplitude wavelike perturbations. The hybrid analytical–numerical technique adopted in this paper, in order to track and illustrate the parametric changes of neutral stability curves, is Galerkin’s method of weighted residuals. Numerical values at significant points on the neutral stability curves are obtained by employing an accurate Runge–Kutta solver combined with the shooting method.

Unstable Forced Convection in a Plane Porous Channel With Variable-Viscosity Dissipation

Fully developed and stationary forced convection in a plane-parallel porous channel is analyzed. The boundary walls are modeled as impermeable and subject to external heat transfer. Different Biot numbers are defined at the two boundary planes. It is shown that the combined effects of temperature-dependent viscosity and viscous heating may induce flow instability. The instability takes place at the lowest parametric singularity of the basic flow solution. The linear stability analysis is carried out analytically for the longitudinal modes and numerically for general oblique modes. It is shown that longitudinal modes with vanishingly small wave number are selected at the onset of instability.

The Onset of Convection in a Sloping Layered Porous Medium: Effects of Local Thermal Non-equilibrium and Heterogeneity

We have investigated the onset of convection instability in a heterogeneous inclined porous layer. Our equations also account for local thermal non-equilibrium. We modelled the effect of heterogeneity by assuming that the layer is composed of two porous sub-layers with different properties, such as permeability, fluid conductivity, solid conductivity, interphase heat transfer coefficient and porosity. We identified which of these factors have major and which have minor effect on the instability. We also characterized the accuracy of one-term Galerkin approximation for this problem. In order to do this, we compared the results obtained by Galerkin approximation with the results obtained by a highly accurate numerical solver.

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.