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Local thermal non-equilibrium analysis of the thermoconvective instability in an inclined porous layer
Get PDFThe two-temperature model of local thermal non-equilibrium (LTNE) is employed to investigate the onset of secondary convective flow in a fluid-saturated porous layer inclined to the horizontal and heated from below. The layer is assumed to be bounded by impermeable plane parallel walls with uniform and unequal temperatures. The linear instability of the stationary pure-conduction single-cell basic flow is studied by employing a normal mode decomposition of the disturbances. A Squire-like transformation is adopted to map all the oblique roll modes onto equivalent transverse roll modes. It is shown that the longitudinal rolls are the most unstable modes at the onset of the instability. The neutral stability condition for the longitudinal modes corresponds to that for a horizontal layer, by scaling the Darcy-Rayleigh number with cosine of the inclination angle to the horizontal. This scaling law, coincident with that well-known for the local thermal equilibrium (LTE) regime, implies a monotonic increment in the stability of the basic flow as the inclination to the horizontal increases.</p
FORM-DRAG EFFECTS ON UNSTABLE BUOYANT FLOWIN A VERTICAL POROUS LAYER WITH OPEN BOUNDARIES
No full textThe onset of convective instability in a vertical porous layer with open boundaries is analysed. The base flow is partly buoyancy-induced and partly caused by an imposed vertical pressure gradient. The linearised perturbation equations expressing the local mass and energy balances, as well as the momentum balance modelled according to Darcy-Forchheimer’s law, are solved numerically in terms of normal modes. The neutral stability curves and the critical Darcy-Rayleigh number are obtained versus the form-drag parameter
On the onset of convection in a highly permeable vertical porous layer with open boundaries
Get PDFThe unstable nature of buoyant flow in a vertical porous slab with a pure conduction temperature distribution is investigated. The permeable and isothermal boundaries are subject to a temperature difference, which is responsible for the basic stationary and parallel vertical flow in the slab. The momentum transfer is modeled by adopting the Darcy-Forchheimer law, thus including the quadratic form-drag contribution. The instability to small-amplitude perturbations is tested by parameterizing the basic stationary flow through the Darcy-Rayleigh number and the form-drag number. The modal analysis is carried out numerically with a pressure-temperature formulation of the governing equations for the perturbations. The neutral stability curves and the critical values of the wave number and of the Darcy-Rayleigh number are obtained for different prescribed values of the form-drag number
Unstable Mixed Convection Flow in a Horizontal Porous Channel with Uniform Wall Heat Flux
Get PDFBuoyancy-induced instability of the horizontal flow in a plane-parallel porous channel is analysed. A model of momentum transfer is adopted where a quadratic form-drag contribution is taken into account. The basic fluid flow is parallel and stationary. Due to the uniform wall heating and the effect of the buoyancy, the velocity and the vertical temperature gradient depend on the vertical coordinate. The dynamics of small-amplitude perturbations on the basic mixed convection flow is studied numerically. Transverse, longitudinal and general oblique rolls are investigated. It is proved that the longitudinal rolls are the normal modes triggering the instability at the lowest Darcy–Rayleigh numbers. The condition of neutral stability is studied for different values of the form-drag parameter and of the Péclet number
When does the onset of convection in an inclined porous layer become subcritical?
Get PDFWe consider the onset of convective instability in an inclined porous layer heated from below. Linearised stability theory tells us that there always exists a band of wavenumbers within which small-amplitude disturbances will grow, but this is true only when the inclination of the layer is less than 31.49032∘. At higher inclinations such disturbances always decay. However, it is also widely known that nonlinear convection may be computed for larger inclinations. This paper provides an initial explanation for how these two facts may be reconciled. It is generally assumed that the onset of convection in an inclined layer is supercritical, and, while this is certainly true when the layer is horizontal, there is no reason to assume that it remains so for other inclinations. The present paper, then, is a combined weakly-nonlinear and numerical investigation of the effect of inclination on the manner of onset. The weakly nonlinear analysis shows that the transition from a supercritical onset to a subcritical one takes place when the inclination is 24.247627∘, and this is confirmed using a detailed and focussed set of nonlinear numerical simulations
Three-dimensional convective planforms for inclined Darcy-Bénard convection.
Get PDFWe 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
On the Use and Misuse of the Oberbeck–Boussinesq Approximation
Get PDFThe Oberbeck–Boussinesq approximation is the most commonly employed theoretical scheme for the study of natural or mixed convection flows. However, the misunderstanding of this approximated framework is a possibility that may cause the emergence of paradoxes or, at least, incorrect conclusions. In this paper, the basic features of the Oberbeck–Boussinesq approximation are briefly recalled and three simple examples where this theoretical scheme may be misused are provided. Such misuses of the approximation lead to erroneous conclusions that, in the examples presented in this note, entail violations of the principle of mass conservation. A discussion about the Oberbeck–Boussinesq approximation as an asymptotic theory obtained by letting the product of the thermal expansion coefficient and the reference temperature difference tend to zero is also presented
Local Thermal Non-equilibrium Analysis of the Instability in a Vertical Porous Slab with Permeable Sidewalls
No full textBuoyant 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
Viscous heating and instability of the adiabatic buoyant flows in a horizontal channel
Get PDFThe stability of buoyant flows occurring in the mixed convection regime for a viscous fluid in a horizontal plane-parallel channel with adiabatic walls is investigated. The basic flow features a parallel velocity field under stationary state conditions. There exists a duality of flows, for every prescribed value of the mass flow rate across the channel cross-section, caused by the combined actions of viscous dissipation and the buoyancy force. As pointed out in a previous study, only the primary branch of the dual solutions is compatible with the Oberbeck-Boussinesq approximation. Thus, the stability analysis will be focused on the stability of such flows. The onset of the thermal instability with small-amplitude perturbations of the basic flow is investigated by assuming a very large Prandtl number, which is equivalent to a creeping flow regime. The neutral stability curves and the critical parametric conditions for the onset of instability are determined numerically
Thermal instability of the buoyant flow in a vertical cylindrical porous layer with a uniform internal heat source
Get PDFThe buoyancy-induced parallel flow in a vertical cylindrical porous layer is
analysed. A radial thermal gradient caused by a uniformly distributed heat
source is assumed to induce the buoyant flow. The layer boundaries are modelled
as isothermal and permeable to an external fluid reservoir. The onset of the
convective instability is analysed by linearising the governing equations for
the perturbations. The governing parameters driving the instability are the
heat-source Rayleigh number and the ratio between the internal radius and the
external radius. Neutral stability curves and the critical values of the
Rayleigh number, the perturbation wave number and the angular frequency are
computed numerically. It is shown that axisymmetric modes form the most
dangerous mode of instability.Comment: 12 pages, 4 figure
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