Abstract:The numerical study of natural convection in a square porous cavity saturated by a Newtonian fluid is presented in this study. The vertical walls are subjected to temperatures varying sinusoidally in time and phase opposition while the upper and lower horizontal walls are thermally adiabatic. Darcy model is used, it is also assumed the fluid studied is incompressible and obeys the Boussinesq approximation. The focus is on the effect of the modulation frequency (10 ≤ ω ≤ 100) and the Rayleigh number (10 ≤ Ra ≤ 1000) on the structure of the flow and transfer thermal. The results show that the extremal stream functions (ψ max and ψ min ), the average Nusselt number at the hot (T c ) and cold (T f ) walls respectively Nucmoy and Nufmoy are periodic and periods equal to that excitatory temperatures to the range of parameters considered in this study. The results show also that oscillatory heating causes the appearance of secondary flow, whose amplification depends on the frequency of modulation of the imposed temperature. The results are shown in terms of streamlines and isotherms during a flow cycle.
Abstract:We analyze the thermal convection thresholds and linear characteristics of the primary and secondary instabilities for viscoelastic fluids saturating a porous horizontal layer heated from below by a constant flux. The Galerkin method is used to solve the eigenvalue problem by taking into account the elasticity of the fluid, the ratio between the viscosity of the solvent and the total viscosity of the fluid and the lateral confinement of the medium. For the primary instability, we found out that depending on the rheological parameters, two types of convective structures may appear when the basic conductive solution loses its stability: stationary long wavelength instability as for Newtonian fluids and oscillatory convection. The effect of the lateral confinement of the porous medium by adiabatic walls is to stabilize the oblique and longitudinal rolls and therefore selects transverse rolls at the onset of convection. In the range of the rheological parameters where stationary long wave instability develops first, we use a parallel flow approximation to determine analytically the velocity and temperature fields associated with the monocellular convective flow. The linear stability analysis of the monocellular flow is performed, and the critical conditions above which the flow becomes unstable are determined. The combined influence of the viscoelastic parameters and the lateral confinement on the characteristics of the secondary instability is quantified. The major new findings concerning the secondary instabilities may be summarized as follows: (i) For concentrated viscoelastic fluids, computations showed that the most amplified mode of convection corresponds to oscillatory transverse rolls, which appears via a Hopf bifurcation. This pattern selection is independent of both the fluid elasticity and the lateral confinement of the porous medium; (ii) For diluted viscoelastic fluids, the preferred mode of convection is found to be oscillatory transverse rolls for a very laterally-confined medium. Otherwise, stationary or oscillatory longitudinal rolls may develop depending on the fluid elasticity. Results also showed the destabilizing effect of the relaxation fluid elasticity and the stabilizing effect of the viscosity ratio for the onset of both primary and secondary instabilities.
As part of the new French law on energy transition, the Demosthene research project is studying the possibility of reusing old abandoned mines to store thermal energy in the Picardy region. The aim is to store the heat required for a small collective unit, which corresponds to a volume of water of 2,000 to 8,000 m 3 , depending on the temperature (from 15°C to 70°C). An inventory shows around 3,700 theoretically available sites in this region. These are mostly shallow dry mines, or mines that are partially flooded with around 1 m of water depth. Based on this water depth and an extraction ratio of 75%, the required mine area is approximately 10,000 m². From the forty sites that have a sufficient surface area, only one is naturally flooded, although statistically many others will exist that are currently not known. In order for this experimental site to be reproducible, the decision was made to select dry mines but with a sufficient area to achieve an artificial flooding device. Theoretically, this represents more than a thousand sites in Picardy. The most interesting one is the old limestone mine of Saint-Maximin, where a sealed basin can be built. Before installing an experimental underground thermal energy storage basin in this site, the thermomechanical and hydrothermal behavior was modeled. The aim was to optimize the position of the various sensors that will be used to monitor the basin, and to predict the future deformations induced on the walls by the thermal variations. A 100 m 3 basin, sealed with a liner, was built and fitted with eighteen sensors to measure temperature, humidity and strain. These sensors allow the stored water, the rock walls and the surrounding atmosphere to be monitored. This device must now operate for six months, i.e. a complete heating-cooling cycle, and its results will be analyzed.
We analyze the thermal convection thresholds and linear characteristics of the primary and secondary instabilities for viscoelastic fluids saturating a porous horizontal layer heated from below by a constant flux. Galerkin method is used to solve the eigenvalue problem by taking into account the elasticity of the fluid, the ratio between the viscosity of the solvent and the total viscosity of the fluid and the lateral confinement of the medium. For the primary instability, we found out that depending on the rheological parameters, two types of convective structures may appear when the basic conductive solution loses its stability: stationary long wavelength instability as for Newtonian fluids and oscillatory convection. The effect of the lateral confinement of the porous medium by adiabatic walls is to stabilize the oblique and longitudinal rolls and therefore selects transverse rolls at the onset of convection. In the range of the rheological parameters where stationary long wave instability develops first, we use a parallel flow approximation to determine analytically the velocity and temperature fields associated to the monocellular convective flow. The linear stability analysis of the monocellular flow is performed, and the critical conditions above which the flow becomes unstable are determined. The combined influence of the viscoelastic parameters and the lateral confinement on the characteristics of the secondary instability is quantified. The major new findings concerning the secondary instabilities may be summarized as follows: (i) For concentrated viscoelastic fluids, computations showed that the most amplified mode of convection corresponds to oscillatory transverse rolls which appears via a Hopf bifurcation. This pattern selection is independent of both the fluid elasticity and the lateral confinement of the porous medium; (ii) For diluted viscoelastic fluids, the preferred mode of convection is found to be oscillatory transverse rolls for a very laterally confined medium. Otherwise stationary or oscillatory longitudinal rolls may develop depending on the fluid elasticity. Results also showed the destabilizing effect of the relaxation fluid elasticity and the stabilizing effect of the viscosity ratio for the onset of both primary and secondary instabilities.
The work we present in this paper is a continuation of a series of studies on the numerical study of natural convection in a square porous cavity saturated by a Newtonian fluid. The left vertical wall is subjected to a temperature varying sinusoidally in time while the right vertical wall is either at a constant temperature, or varying sinusoidally in time. The upper and lower horizontal walls are thermally adiabatic. Darcy model is used, it is also assumed the fluid studied is incompressible and obeys the Boussinesq approximation. The focus is on the effect of the modulation frequency ( 10 100 ω ≤ ≤ ) on the structure of the flow and transfer thermal. The results show that the extremal stream functions ( max ψ et min ψ ), the average Nusselt number at the hot ( h T ) and cold ( c T ) walls respectively Nuh and Nuc are periodic in the range of parameters considered in this study. In comparison with the constant heating conditions, it is found that the variable heating causes the appearance of secondary flow, whose amplification depends on the frequency of modulation of the imposed temperature but also of the heating mode. The results are shown in terms of streamlines and isotherms during a flow cycle.
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