Throughflow effects on convective instability in superposed fluid and porous layers

1991 ◽  
Vol 231 ◽  
pp. 113-133 ◽  
Author(s):  
Falin Chen
Keyword(s):  
Porous Layer ◽  
Convective Instability ◽  
Fluid Layer ◽  
Critical Rayleigh Number ◽  
Vertical Direction ◽  
Single Layer ◽  
Layer Depth ◽  
Onset Of Convection ◽  
Precise Control ◽  
Porous Layers

We implement a linear stability analysis of the convective instability in superposed horizontal fluid and porous layers with throughflow in the vertical direction. It is found that in such a physical configuration both stabilizing and destabilizing factors due to vertical throughflow can be enhanced so that a more precise control of the buoyantly driven instability in either a fluid or a porous layer is possible. For ζ = 0.1 (ζ, the depth ratio, defined as the ratio of the fluid-layer depth to the porous-layer depth), the onset of convection occurs in both fluid and porous layers, the relation between the critical Rayleigh number Rcm and the throughflow strength γm is linear and the Prandtl-number (Prm) effect is insignificant. For ζ ≥ 0.2, the onset of convection is largely confined to the fluid layer, and the relation becomes Rcm ∼ γ2m for most of the cases considered except for Prm = 0.1 with large positive γm where the relation Rcm ∼ γ3m holds. The destabilizing mechanisms proposed by Nield (1987 a, b) due to throughflow are confirmed by the numerical results if considered from the viewpoint of the whole system. Nevertheless, from the viewpoint of each single layer, a different explanation can be obtained.

Experimental investigation of convective stability in a superposed fluid and porous layer when heated from below

1989 ◽  
Vol 207 ◽  
pp. 311-321 ◽  
Author(s):  
Falin Chen ◽  
C. F. Chen
Keyword(s):  
Porous Layer ◽  
Fluid Layer ◽  
Critical Rayleigh Number ◽  
Linear Stability Theory ◽  
Convective Stability ◽  
Onset Of Convection ◽  
Temperature Distributions ◽  
Critical Wavelength ◽  
Crystal Film ◽  
Flux Curve

Experiments have been carried out in a horizontal superposed fluid and porous layer contained in a test box 24 cm × 12 cm × 4 cm high. The porous layer consisted of 3 mm diameter glass beads, and the fluids used were water, 60% and 90% glycerin-water solutions, and 100% glycerin. The depth ratio ď, which is the ratio of the thickness of the fluid layer to that of the porous layer, varied from 0 to 1.0. Fluids of increasingly higher viscosity were used for cases with larger ď in order to keep the temperature difference across the tank within reasonable limits. The top and bottom walls were kept at different constant temperatures. Onset of convection was detected by a change of slope in the heat flux curve. The size of the convection cells was inferred from temperature measurements made with embedded thermocouples and from temperature distributions at the top of the layer by use of liquid crystal film. The experimental results showed (i) a precipitous decrease in the critical Rayleigh number as the depth of the fluid layer was increased from zero, and (ii) an eightfold decrease in the critical wavelength between ď = 0.1 and 0.2. Both of these results were predicted by the linear stability theory reported earlier (Chen & Chen 1988).

Thermal Stability of Horizontally Superposed Porous and Fluid Layers

1989 ◽  
Vol 111 (2) ◽  
pp. 357-362 ◽  
Author(s):  
M. E. Taslim ◽  
U. Narusawa
Keyword(s):  
Porous Layer ◽  
Fluid Layer ◽  
Critical Rayleigh Number ◽  
Convective Motion ◽  
Darcy Number ◽  
Wave Number ◽  
Thermal Conductivity Ratio ◽  
Porous Layers ◽  
Fluid Layers ◽  
Comprehensive Picture

The results of stability analyses for the onset of convective motion are reported for the following three horizontally superposed systems of porous and fluid layers: (a) a porous layer sandwiched between two fluid layers with rigid top and bottom boundaries, (b) a fluid layer overlying a layer of porous medium, and (c) a fluid layer sandwiched between two porous layers. By changing the depth ratio dˆ from zero to infinity, a set of stability criteria (i.e., the critical Rayleigh number Rac and the critical wave number ac) is obtained, ranging from the case of a fluid layer between two rigid boundaries to the case of a porous layer between two impermeable boundaries. The effects of k/km (the thermal conductivity ratio), δ (the square root of the Darcy number), and α (the nondimensional proportionality constant in the slip condition) on Rac and ac are also examined in detail. The results in this paper, combined with those reported previously for Case (a) (Pillatsis et al., 1987), will provide a comprehensive picture of the interaction between a porous and a fluid layer.

Throughflow effects in the Rayleigh-Bénard convective instability problem

1987 ◽  
Vol 185 ◽  
pp. 353-360 ◽  
Author(s):  
D. A. Nield
Keyword(s):  
Convective Instability ◽  
Fluid Layer ◽  
Critical Rayleigh Number ◽  
Onset Of Convection ◽  
Instability Problem ◽  
Perturbation Velocity ◽  
Different Types ◽  
Basic Temperature ◽  
Rayleigh Benard ◽  
Significant Size

The effect of vertical throughflow on the onset of convection in a fluid layer, between permeable horizontal boundaries, when heated uniformly from below, is re-examined analytically. It is shown that when the Péclet number Q is large in magnitude, the critical Rayleigh number Rc is proportional to Qn, where n = 0, 1, 2, 3 or 4, with a coefficient depending on the Prandtl number P, according to the types of boundaries. When the upper and lower boundaries are of different types, the effect of a small amount of throughflow in one direction is to decrease Rc. This is so when the throughflow is away from the more restrictive boundary. Contributions arise from the curvature of the basic temperature profile, and from the vertical transport of perturbation velocity and perturbation temperature. The decrease in Rc is small if P ∼ 1 but can be of significant size if P [Lt ] 1 or P [Gt ] 1.

Thermal Instability of a Fluid-Saturated Porous Medium Bounded by Thin Fluid Layers

1987 ◽  
Vol 109 (3) ◽  
pp. 677-682 ◽  
Author(s):  
G. Pillatsis ◽  
M. E. Taslim ◽  
U. Narusawa
Keyword(s):  
Porous Medium ◽  
Porous Layer ◽  
Thermal Instability ◽  
Numerical Solutions ◽  
Fluid Layer ◽  
Critical Rayleigh Number ◽  
Convective Motion ◽  
Wave Number ◽  
Thermal Conductivity Ratio ◽  
Fluid Layers

A linear stability analysis is performed for a horizontal Darcy porous layer of depth 2dm sandwiched between two fluid layers of depth d (each) with the top and bottom boundaries being dynamically free and kept at fixed temperatures. The Beavers–Joseph condition is employed as one of the interfacial boundary conditions between the fluid and the porous layer. The critical Rayleigh number and the horizontal wave number for the onset of convective motion depend on the following four nondimensional parameters: dˆ ( = dm/d, the depth ratio), δ ( = K/dm with K being the permeability of the porous medium), α (the proportionality constant in the Beavers–Joseph condition), and k/km (the thermal conductivity ratio). In order to analyze the effect of these parameters on the stability condition, a set of numerical solutions is obtained in terms of a convergent series for the respective layers, for the case in which the thickness of the porous layer is much greater than that of the fluid layer. A comparison of this study with the previously obtained exact solution for the case of constant heat flux boundaries is made to illustrate quantitative effects of the interfacial and the top/bottom boundaries on the thermal instability of a combined system of porous and fluid layers.

Direct numerical simulation of instabilities in a two-dimensional near-critical fluid layer heated from below

2001 ◽  
Vol 442 ◽  
pp. 119-140 ◽  
Author(s):  
S. AMIROUDINE ◽  
P. BONTOUX ◽  
P. LARROUDÉ ◽  
B. GILLY ◽  
B. ZAPPOLI
Keyword(s):  
Boundary Layer ◽  
Critical Point ◽  
Stokes Equations ◽  
Fluid Layer ◽  
Vertical Direction ◽  
Single Layer ◽  
Bulk Temperature ◽  
Two Dimensional ◽  
Piston Effect ◽  
The Stability

An analysis of the hydrodynamic stability of a fluid near its near critical point – initially at rest and in thermodynamic equilibrium – is considered in the Rayleigh–Bénard configuration, i.e. heated from below. The geometry is a two-dimensional square cavity and the top and bottom walls are maintained at constant temperatures while the sidewalls are insulated. Owing to the homogeneous thermo-acoustic heating (piston effect), the thermal field exhibits a very specific structure in the vertical direction. A very thin hot thermal boundary layer is formed at the bottom, then a homogeneously heated bulk settles in the core at a lower temperature; at the top, a cooler boundary layer forms in order to continuously match the bulk temperature with the colder temperature of the upper wall. We analyse the stability of the two boundary layers by numerically solving the Navier–Stokes equations appropriate for a van der Waals' gas slightly above its critical point. A finite-volume method is used together with an acoustic filtering procedure. The onset of the instabilities in the two different layers is discussed with respect to the results of the theoretical stability analyses available in the literature and stability diagrams are derived. By accounting for the piston effect the present results can be put within the framework of the stability analysis of Gitterman and Steinberg for a single layer subjected to a uniform, steady temperature gradient.

Onset of Convection in a Fluid Saturated Porous Layer Overlying a Solid Layer Which is Heated by Constant Flux

1999 ◽  
Vol 121 (4) ◽  
pp. 1094-1097 ◽  
Author(s):  
C. Y. Wang
Keyword(s):  
Rayleigh Number ◽  
Porous Layer ◽  
Critical Rayleigh Number ◽  
Hydrothermal Systems ◽  
Solid Layer ◽  
Onset Of Convection ◽  
Constant Flux ◽  
Saturated Porous Layer ◽  
Bottom Heating ◽  
Temperature Heating

The thermoconvective stability of a porous layer overlying a solid layer is important in seafloor hydrothermal systems and thermal insulation problems. The case for constant flux bottom heating is considered. The critical Rayleigh number for the porous layer is found to increase with the thickness of the solid layer, a result opposite to constant temperature heating.

Neutral Instability and Optimum Convective Mode in a Fluid Layer with PCM Particles

2005 ◽  
Vol 127 (12) ◽  
pp. 1289-1295 ◽  
Author(s):  
Chuanshan Dai ◽  
Hideo Inaba
Keyword(s):  
Rayleigh Number ◽  
Fluid Layer ◽  
Critical Rayleigh Number ◽  
Gaussian Function ◽  
Critical Conditions ◽  
Onset Of Convection ◽  
Apparent Specific Heat ◽  
Change Material ◽  
Raphson Iteration ◽  
Convective Mode

Linear stability analysis is performed to determine the critical Rayleigh number for the onset of convection in a fluid layer with phase-change-material particles. Sine and Gaussian functions are used for describing the large variation of apparent specific heat in a narrow phase changing temperature range. The critical conditions are numerically obtained using the fourth order Runge-Kutta-Gill finite difference method with Newton-Raphson iteration. The critical eigenfunctions of temperature and velocity perturbations are obtained. The results show that the critical Rayleigh number decreases monotonically with the amplitude of Sine or Gaussian function. There is a minimum critical Rayleigh number while the phase angle is between π∕2 and π, which corresponds to the optimum experimental convective mode.

Onset of Finger Convection in a Horizontal Porous Layer Underlying a Fluid Layer

1988 ◽  
Vol 110 (2) ◽  
pp. 403-409 ◽  
Author(s):  
F. Chen ◽  
C. F. Chen
Keyword(s):  
Rayleigh Number ◽  
Porous Layer ◽  
Fluid Layer ◽  
Short Wave ◽  
Marginal Stability ◽  
Simultaneous Occurrence ◽  
Convection Pattern ◽  
Layer Depth ◽  
Unusual Behavior ◽  
Square Cells

In the directional solidification of concentrated alloys, the frozen solid region is separated from the melt region by a mushy zone consisting of dendrites immersed in the melt. Simultaneous occurrence of temperature and solute gradients through the melt and mushy zones may be conducive to the occurrence of salt-finger convection, which may in turn cause adverse effects such as channel segregation. We have considered the problem of the onset of finger convection in a porous layer underlying a fluid layer using linear stability analysis. The eigenvalue problem is solved by a shooting method. As a check on the method of solution and the associated computer program, we first consider the thermal convection problem. In this process, it is discovered that at low depth ratios dˆ (the ratio of the fluid layer depth to the porous layer depth), the marginal stability curve is bimodal. At small dˆ, the long-wave branch is the most unstable and the convection is dominated by the porous layer. At large dˆ, the short-wave branch is the most unstable and the convection is dominated by the fluid layer, with a convection pattern consisting of square cells in the fluid layer. In the salt-finger case with a given thermal Rayleigh number Ram = 50, as the depth ratio dˆ is increased from zero, the critical salt Rayleigh number Rasm first decreases, reaches a minimum, and then increases. The system is more stable at dˆ > 0.2 than at dˆ = 0. This rather unusual behavior is again due to the fact that at small dˆ, convection is dominated by the porous layer and, at large dˆ, convection is dominated by the fluid layer. However, in the latter case, the convection pattern in the fluid layer consists of a number of high aspect ratio cells.

Subcritical convective instability Part 1. Fluid layers

1966 ◽  
Vol 26 (4) ◽  
pp. 753-768 ◽  
Author(s):  
Daniel D. Joseph ◽  
C. C. Shir
Keyword(s):  
Nusselt Number ◽  
Rayleigh Number ◽  
Heat Source ◽  
Linear Theory ◽  
Convective Instability ◽  
Fluid Layer ◽  
Critical Rayleigh Number ◽  
Heat Sources ◽  
Fluid Layers ◽  
Rayleigh Numbers

This paper elaborates on the assertion that energy methods provide an always mathematically rigorous and a sometimes physically precise theory of sub-critical convective instability. The general theory, without explicit solutions, is used to deduce that the critical Rayleigh number is a monotonically increasing function of the Nusselt number, that this increase is very slow if the Nusselt number is large, and that a fluid layer heated from below and internally is definitely stable when $RA < \widetilde{RA}(N_s) > 1708/(N_s + 1)$ where Ns is a heat source parameter and $\widetilde{RA}$ is a critical Rayleigh number. This last problem is also solved numerically and the result compared with linear theory. The critical Rayleigh numbers given by energy theory are slightly less than those given by linear theory, this difference increasing from zero with the magnitude of the heat-source intensity. To previous results proving the non-existence of subcritical instabilities in the absence of heat sources is appended this result giving a narrow band of Rayleigh numbers as possibilities for subcritical instabilities.

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