The onset of convective instability in a triply diffusive fluid layer

1989 ◽  
Vol 202 ◽  
pp. 443-465 ◽  
Author(s):  
Arne J. Pearlstein ◽  
Rodney M. Harris ◽  
Guillermo Terrones
Keyword(s):  
Rayleigh Number ◽  
Linear Stability ◽  
Fluid Layer ◽  
Neutral Curve ◽  
Interesting Consequence ◽  
Stability Criteria ◽  
Neutral Curves ◽  
The Stability ◽  
Rayleigh Numbers ◽  
Periodic Bifurcation

The onset of instability is investigated in a triply diffusive fluid layer in which the density depends on three stratifying agencies having different diffusivities. It is found that, in some cases, three critical values of the Rayleigh number are required to specify the linear stability criteria. As in the case of another problem requiring three Rayleigh numbers for the specification of linear stability criteria (the rotating doubly diffusive case studied by Pearlstein 1981), the cause is traceable to the existence of disconnected oscillatory neutral curves. The multivalued nature of the stability boundaries is considerably more interesting and complicated than in the previous case, however, owing to the existence of heart-shaped oscillatory neutral curves. An interesting consequence of the heart shape is the possibility of ‘quasi-periodic bifurcation’ to convection from the motionless state when the twin maxima of the heart-shaped oscillatory neutral curve lie below the minimum of the stationary neutral curve. In this case, there are two distinct disturbances, with (generally) incommensurable values of the frequency and wavenumber, that simultaneously become unstable at the same Rayleigh number. This work complements the earlier efforts of Griffiths (1979a), who found none of the interesting results obtained herein.

Exploration of Coriolis Force on the Linear Stability of Couple Stress Fluid Flow Induced by Double Diffusive Convection

2019 ◽  
Vol 141 (12) ◽  
Author(s):  
S. B. Naveen Kumar ◽  
I. S. Shivakumara ◽  
B. M. Shankar
Keyword(s):  
Linear Stability ◽  
Coriolis Force ◽  
Couple Stress ◽  
Fluid Layer ◽  
Linear Instability ◽  
Neutral Curve ◽  
Couple Stress Fluid ◽  
Diffusive Convection ◽  
Convective Cells ◽  
Rayleigh Numbers

Abstract In this paper, the effect of Coriolis force is explored on convective instability of a doubly diffusive incompressible couple stress fluid layer with gravity acting downward. A linear stability analysis is used to obtain the conditions for the onset of stationary and oscillatory convection in the closed form. Being a multiparameter instability problem, results for some isolated cases have been presented to illustrate interesting corners of parameter space. It is found that the neutral curve for oscillatory onset forms a closed-loop which is separate from the neutral curve for stationary onset indicating the requirement of three critical thermal Rayleigh numbers to specify the linear instability criteria instead of the usual single value. Besides, the simultaneous presence of rotation and the addition of heavy solute to the bottom of the layer exhibit an intriguing possibility of destabilizing the system under certain conditions, in contrast to their stabilizing effect when they are present in isolation. The implication of couple stresses on each of the aforementioned anomalies is clearly brought out. The spatial wavelength of convective cells at the onset is also discussed.

The Effect of Suction on the Stability of Fluid between Horizontal Plates at Differing Temperatures

1985 ◽  
Vol 199 (2) ◽  
pp. 145-151 ◽  
Author(s):  
M M Sorour ◽  
M A Hassab ◽  
F A Elewa
Keyword(s):  
Rayleigh Number ◽  
Fluid Layer ◽  
Critical Rayleigh Number ◽  
Linear Stability Theory ◽  
Stability Criteria ◽  
Thermal Boundary Conditions ◽  
Wide Range ◽  
Fluid Layers ◽  
The Stability ◽  
Horizontal Fluid Layer

The linear stability theory is applied to study the effect of suction on the stability criteria of a horizontal fluid layer confined between two thin porous surfaces heated from below. This investigation covers a wide range of Reynolds number 0 ≥ Re ≥ 30, and Prandtl number 0.72 ≥ Pr ≥ 100. The results show that the critical Rayleigh number increases with Peclet number, and is independent of Pr as far as Re < 3. However, for Re > 3 the critical Rayleigh number is function of both Pr and Pe. In addition, the analysis is extended to study the effect of suction on the stability of two special superimposed fluid layers. The results in the latter case indicate a more stabilizing effect. Furthermore, the effect of thermal boundary conditions is also investigated.

Stability of Heat Pipes in Vapor-Dominated Systems

1999 ◽  
Author(s):  
Pouya Amili ◽  
Yanis C. Yortsos
Keyword(s):  
Rayleigh Number ◽  
Linear Stability ◽  
Oil Recovery ◽  
Critical Rayleigh Number ◽  
Stability Limit ◽  
Wave Number ◽  
Nuclear Waste Repository ◽  
Geothermal Systems ◽  
Dimensionless Numbers ◽  
The Stability

Abstract We study the linear stability of a two-phase heat pipe zone (vapor-liquid counterflow) in a porous medium, overlying a superheated vapor zone. The competing effects of gravity, condensation and heat transfer on the stability of a planar base state are analyzed in the linear stability limit. The rate of growth of unstable disturbances is expressed in terms of the wave number of the disturbance, and dimensionless numbers, such as the Rayleigh number, a dimensionless heat flux and other parameters. A critical Rayleigh number is identified and shown to be different than in natural convection under single phase conditions. The results find applications to geothermal systems, to enhanced oil recovery using steam injection, as well as to the conditions of the proposed Yucca Mountain nuclear waste repository. This study complements recent work of the stability of boiling by Ramesh and Torrance (1993).

Natural Convection in Horizontal Thin-Walled Honeycomb Panels

1973 ◽  
Vol 95 (4) ◽  
pp. 439-444 ◽  
Author(s):  
K. G. T. Hollands
Keyword(s):  
Heat Transfer ◽  
Natural Convection ◽  
Fluid Layer ◽  
Convection Heat Transfer ◽  
Correlation Equation ◽  
Wave Number ◽  
Equivalent Wave ◽  
Side Walls ◽  
The Stability ◽  
Rayleigh Numbers

This paper presents an experimental study of the stability of and natural convection heat transfer through a horizontal fluid layer heated from below and constrained internally by a honeycomb. Examination of the types of boundary conditions exacted on the fluid at the cell side-walls has shown that there are three limiting cases: (1) perfectly conducting side-walls; (2) perfectly adiabatic side-walls; and (3) side-walls having zero thickness. Experiments described in this paper approach the latter category. The fluid used is air and the honeycomb used is square-celled. Measured critical Rayleigh numbers are found to be intermediate between those applying to cases (1) and (2), and consistent with an “equivalent wave number” of approximately 0.95 times that for case (1). The measured natural convective heat transfer after instability is found to be significantly less than that predicted by the Malkus-Veronis power integral technique. However, it is found to approach asymptotically the heat transfer which would take place through a similar fluid layer unconstrained by a honeycomb. A general correlation equation for the heat transfer is given.

Rayleigh–Bénard instability generated by a diffusion flame

2013 ◽  
Vol 736 ◽  
pp. 464-494 ◽  
Author(s):  
P. Pearce ◽  
J. Daou
Keyword(s):  
Stability Analysis ◽  
Rayleigh Number ◽  
Linear Stability ◽  
Linear Stability Analysis ◽  
Diffusion Flame ◽  
Damkohler Number ◽  
Benard Convection ◽  
The Stability ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard

AbstractWe investigate the Rayleigh–Bénard convection problem within the context of a diffusion flame formed in a horizontal channel where the fuel and oxidizer concentrations are prescribed at the porous walls. This problem seems to have received no attention in the literature. When formulated in the low-Mach-number approximation the model depends on two main non-dimensional parameters, the Rayleigh number and the Damköhler number, which govern gravitational strength and reaction speed respectively. In the steady state the system admits a planar diffusion flame solution; the aim is to find the critical Rayleigh number at which this solution becomes unstable to infinitesimal perturbations. In the Boussinesq approximation, a linear stability analysis reduces the system to a matrix equation with a solution comparable to that of the well-studied non-reactive case of Rayleigh–Bénard convection with a hot lower boundary. The planar Burke–Schumann diffusion flame, which has been previously considered unconditionally stable in studies disregarding gravity, is shown to become unstable when the Rayleigh number exceeds a critical value. A numerical treatment is performed to test the effects of compressibility and finite chemistry on the stability of the system. For weak values of the thermal expansion coefficient $\alpha $, the numerical results show strong agreement with those of the linear stability analysis. It is found that as $\alpha $ increases to a more realistic value the system becomes considerably more stable, and also exhibits hysteresis at the onset of instability. Finally, a reduction in the Damköhler number is found to decrease the stability of the system.

The effect of heating rate on the stability of stationary fluids

1967 ◽  
Vol 29 (2) ◽  
pp. 337-347 ◽  
Author(s):  
I. G. Currie
Keyword(s):  
Rayleigh Number ◽  
Surface Temperature ◽  
Fluid Layer ◽  
Critical Rayleigh Number ◽  
Lower Surface ◽  
Published Data ◽  
Heating Rates ◽  
Lower Surface Temperature ◽  
The Stability ◽  
Horizontal Fluid Layer

A horizontal fluid layer whose lower surface temperature is made to vary with time is considered. The stability analysis for this situation shows that the criterion for the onset of instability in a fluid layer which is being heated from below, depends on both the method and the rate of heating. For a fluid layer with two rigid boundaries, the minimum Rayleigh number corresponding to the onset of instability is found to be 1340. For slower heating rates the critical Rayleigh number increases to a maximum value of 1707·8, while for faster heating rates the critical Rayleigh number increases without limit.Two specific types of heating are investigated in detail, constant flux heating and linearly varying surface temperature. These cases correspond closely to situations for which published data exist. The results are in good qualitative agreement.

Stability of plane Poiseuille–Couette flow in a fluid layer overlying a porous layer

2017 ◽  
Vol 826 ◽  
pp. 376-395 ◽  
Author(s):  
Ting-Yueh Chang ◽  
Falin Chen ◽  
Min-Hsing Chang
Keyword(s):  
Couette Flow ◽  
Porous Layer ◽  
Poiseuille Flow ◽  
Fluid Layer ◽  
Maximum Velocity ◽  
Stability Characteristic ◽  
Layer System ◽  
Neutral Curves ◽  
Layer Mode ◽  
The Stability

This paper performs a linear stability analysis to investigate the stability of plane Poiseuille–Couette flow in a fluid layer overlying a porous medium saturated with the same fluid. The effect of superimposed Couette flow on the associated Poiseuille flow in such a two-layer system is explored carefully. The result shows that the presence of Couette flow may destabilize the Poiseuille flow at small depth ratio $\hat{d}$, defined by the ratio of the depth of the fluid layer to the depth of the porous layer, and induce a tri-modal structure to the neutral curves. At moderate $\hat{d}$, the Couette component generally produces a stabilization effect on the flow. When the velocity of the upper moving wall is large enough, a bi-modal behaviour of the neutral curves appears and a shift of instability mode occurs from the long-wave fluid-layer mode to the porous-layer mode with higher wavenumber. These stability characteristics are remarkably different from those of the plane Poiseuille–Couette flow in a single fluid layer in that the flow becomes absolutely stable when the wall velocity is over 70 % of the maximum velocity of the Poiseuille component of flow. The stability of pure Couette flow in such a two-layer system is also studied. It is found that the flow is still absolutely stable with respect to infinitesimal disturbances, which is the same as the stability characteristic of a single-layer plane Couette flow.

Convective Instability in a Melt Layer Heated From Below

1976 ◽  
Vol 98 (1) ◽  
pp. 88-94 ◽  
Author(s):  
E. M. Sparrow ◽  
L. Lee ◽  
N. Shamsundar
Keyword(s):  
Rayleigh Number ◽  
Phase Change ◽  
Wall Temperature ◽  
A Priori ◽  
Linear Stability Theory ◽  
Convective Heating ◽  
Fluid Medium ◽  
Lower Bounding ◽  
The Stability ◽  
Rayleigh Numbers

Consideration is given to the onset of convective motions in a horizontal melt layer created by solid-to-liquid phase change. The melt layer is heated at its lower bounding surface either due to convective transfer from an adjacent fluid medium or to a step change in wall temperature. The analysis is carried out for liquid melts whose densities decrease with increasing temperature. Linear stability theory is employed to determine the conditions marking the onset of motion. The results of the analysis are expressed in terms of two Rayleigh numbers. One of these, the internal Rayleigh number, is based on the instantaneous thickness and instantaneous temperature difference across the layer. The other, the external Rayleigh number, is more convenient to use in applications problems since it contains quantities which are constant and a priori prescribable. For a melting problem where the external Rayleigh number is large, instability occurs soon after the start of heating. At smaller external Rayleigh numbers, the duration time of the regime of no motion increases markedly. At large times, the stability results for convective heating coincide with those for stepped wall temperature. In addition to the results for the stability problem, results for conduction phase change (in the absence of motion) are also presented for the surface convection boundary condition.

scholarly journals Concordancing in ESP Class Rooms

2014 ◽  
Vol 4 (3) ◽  
pp. 434-439
Author(s):  
Sameh Benna ◽  
Olfa Bayoudh
Keyword(s):  
Rayleigh Number ◽  
Heat Transport ◽  
Fluid Layer ◽  
Critical Rayleigh Number ◽  
Linear Analysis ◽  
Wave Number ◽  
Gravity Modulation ◽  
Non Linear ◽  
The Stability ◽  
Time Periodic

The effect of time periodic body force (or g-jitter or gravity modulation) on the onset of Rayleigh-Bnard electro-convention in a micropolar fluid layer is investigated by making linear and non-linear stability analysis. The stability of the horizontal fluid layer heated from below is examined by assuming time periodic body acceleration. This normally occurs in satellites and in vehicles connected with micro gravity simulation studies. A linear and non-linear analysis is performed to show that gravity modulation can significantly affect the stability limits of the system. The linear theory is based on normal mode analysis and perturbation method. Small amplitude of modulation is used to compute the critical Rayleigh number and wave number. The shift in the critical Rayleigh number is calculated as a function of frequency of modulation. The non-linear analysis is based on the truncated Fourier series representation. The resulting non-autonomous Lorenz model is solved numerically to quantify the heat transport. It is observed that the gravity modulation leads to delayed convection and reduced heat transport.

scholarly journals Natural convection with mixed insulating and conducting boundary conditions: low- and high-Rayleigh-number regimes

2014 ◽  
Vol 742 ◽  
pp. 636-663 ◽  
Author(s):  
P. Ripesi ◽  
L. Biferale ◽  
M. Sbragaglia ◽  
A. Wirth
Keyword(s):  
Natural Convection ◽  
Rayleigh Number ◽  
Boundary Conditions ◽  
Two Dimensions ◽  
Turbulent Regime ◽  
Thermal Insulating ◽  
Pattern Length ◽  
The Stability ◽  
The Impact ◽  
Rayleigh Numbers

AbstractWe investigate the stability and dynamics of natural convection in two dimensions, subject to inhomogeneous boundary conditions. In particular, we consider a Rayleigh–Bénard (RB) cell, where the horizontal top boundary contains a periodic sequence of alternating thermal insulating and conducting patches, and we study the effects of the heterogeneous pattern on the global heat exchange, at both low and high Rayleigh numbers. At low Rayleigh numbers, we determine numerically the transition from a regime characterized by the presence of small convective cells localized at the inhomogeneous boundary to the onset of ‘bulk’ convective rolls spanning the entire domain. Such a transition is also controlled analytically in the limit when the boundary pattern length is small compared with the cell vertical size. At higher Rayleigh number, we use numerical simulations based on a lattice Boltzmann method to assess the impact of boundary inhomogeneities on the fully turbulent regime up to $\mathit{Ra} \sim 10^{10}$.

Sign in / Sign up

Export Citation Format

Share Document