scholarly journals On nonlinear overstable convection rolls in a rotating system

1984 ◽  
Vol 25 (3) ◽  
pp. 406-418 ◽  
Author(s):  
N. Riahi
Keyword(s):  
Prandtl Number ◽  
Travelling Waves ◽  
Vertical Axis ◽  
Horizontal Layer ◽  
Finite Amplitude ◽  
Oscillatory Convection ◽  
Nonlinear Analyses ◽  
Rotating System ◽  
The Stability ◽  
Convection Rolls

AbstractFinite amplitude oscillatory convection rolls in the form of travelling waves are studied for a horizontal layer of a low Prandtl number fluid heated from below and rotating rapidly about a vertical axis. The results of the stability and nonlinear analyses indicate that there is no subcritical instability and that the oscillatory rolls are unstable for the ranges of the Prandtl number and the rotation rate considered in this paper.

Thermal convection in gas–droplet mixtures with phase transition

1975 ◽  
Vol 70 (1) ◽  
pp. 89-112
Author(s):  
T. Kambe ◽  
R. Takaki
Keyword(s):  
Water Vapour ◽  
Density Gradient ◽  
Thermal Convection ◽  
Fluid Layer ◽  
Travelling Waves ◽  
Horizontal Layer ◽  
Phase Changes ◽  
Droplet Growth ◽  
The Stability ◽  
Droplet Density

Thermal convection in a three-component fluid consisting of an inert carrier gas, a condensable vapour and small liquid droplets dispersed throughout the gaseous components has been investigated both theoretically and experimentally. The theoretical study is concerned with the stability of a horizontal fluid layer subject to gradients of both temperature and droplet density. The stability is characterized by four parameters: two material constants, that is, a modified Prandtl number P and a constant Q proportional to Dm − κ (Dm is the mutual mass diffusivity of the two gaseous constituents, κ the thermometric conductivity of the gas phase), a modified Rayleigh number R and a parameter S defined as the ratio of the droplet density gradient to the gas density gradient. It is shown for positive R that, irrespective of the value of R, the system is stable for S > S∞ (S∞ is a constant dependent on P and Q) and unstable for S < Q (Q is normally less than S∞) and that for the intermediate range Q < S < S∞ a transition from stability to instability occurs via an oscillatory state as R is increased through a critical value depending on S. It is shown that the stability is governed largely by both vapour diffusion through the inert gas and droplet growth or decay due to phase changes.In the experiments, thermal convection in a three-component fluid consisting of air, water vapour and water droplets was investigated. The cloud of droplets was mainly formed by injecting cigarette smoke into a horizontal layer of air saturated with water vapour. After the injection several phases of motion were observed successively. Among them there were travelling waves and steady cellular convection. Measurements were made of the critical Rayleigh numbers for the onset of the phases, the scale of the steady convection cells and the speed of the travelling waves. It is found that all the qualitative features of the experiment are explained by the theory.

Finite Amplitude Longitudal Convection Rolls in an Inclined Layer

1973 ◽  
Vol 95 (3) ◽  
pp. 407-408 ◽  
Author(s):  
R. M. Clever
Keyword(s):  
Fluid Layer ◽  
Horizontal Layer ◽  
Finite Amplitude ◽  
Two Dimensional ◽  
Governing Equations ◽  
Inclined Layer ◽  
Experimental Heat ◽  
Inclined Fluid Layer ◽  
Longitudinal Coordinate ◽  
Convection Rolls

For the case of a large Prandtl number, buoyancy driven flow in an inclined fluid layer, it is shown that all longitudinal-coordinate-independent solutions of the governing equations are obtainable from a knowledge of the existing results for two-dimensional convection in a horizontal layer, heated from below. The rescaling here yields results which compare favorably with those of existing experimental heat transport values.

Weakly nonlinear oscillatory convection in a rotating fluid

1992 ◽  
Vol 436 (1896) ◽  
pp. 33-54 ◽  
Keyword(s):  
Standing Wave ◽  
Perturbation Technique ◽  
Vertical Axis ◽  
Travelling Wave ◽  
Base Flow ◽  
Two Dimensional ◽  
Oscillatory Convection ◽  
Weakly Nonlinear ◽  
Wave Disturbances ◽  
The Stability

The problem of weakly nonlinear two- and three-dimensional oscillatory convection in the form of standing waves is studied for a horizontal layer of fluid heated from below and rotating about a vertical axis. The solutions to the nonlinear problem are determined by a perturbation technique and the stability of all the base flow solutions is investigated with respect to both standing wave and travelling wave disturbances. The results of the stability and the nonlinear analyses for various values of the rotation parameter τ and the Prandtl number P (0 ≼ P < 0.677) indicate that there is no subcritical instability and that all the base flow solutions are unstable. Disturbances with highest growth rates are found to be some particular disturbances superimposed on two-dimensional base flow. Particular standing wave disturbances parallel to two-dimensional base flow are the most unstable ones either for sufficiently small P or for intermediate values of P with τ below some critical value τ *. Travelling wave disturbances inclined at an angle of about 45° to the wave vector of two-dimensional base flow are the most unstable disturbances either for P sufficiently close to its upper limit or for intermediate values of P with τ ≽ τ *. The dependence on P and τ of the nonlinear effect on the frequency and of the heat flux are also discussed.

Oscillatory instabilities of convection rolls at intermediate Prandtl numbers

1986 ◽  
Vol 164 ◽  
pp. 469-485 ◽  
Author(s):  
E. W. Bolton ◽  
F. H. Busse ◽  
R. M. Clever
Keyword(s):  
Prandtl Number ◽  
Fluid Layer ◽  
Stability Boundary ◽  
Experimental Studies ◽  
Interesting Property ◽  
Number Range ◽  
Close Relationship ◽  
Prandtl Numbers ◽  
The Stability ◽  
Convection Rolls

The analysis of the instabilities of convection rolls in a fluid layer heated from below with no-slip boundaries exhibits a close competition between various oscillatory modes in the range 2 [lsim ] P [lsim ] 12 of the Prandtl number P. In addition to the even-oscillatory instability known from earlier work two new instabilities have been found, each of which is responsible for a small section of the stability boundary of steady rolls. The most interesting property of the new instabilities is their close relationship to the hot-blob oscillations known from experimental studies of convection. In the lower half of the Prandtl-number range considered the B02-mode dominates, which is characterized by two blobs each of slightly hotter and colder fluid circulating around in the convection roll in a spatially and time-periodic fashion. At higher Prandtl numbers the BE 1-mode dominates, which possesses one hot blob (and one cold blob) circulating with the convection velocity. Just outside the stability boundary there exist other growing modes exhibiting three or four blobs which may be observable in experiments.

Three-dimensional thermal cellular convection in rectangular boxes

1988 ◽  
Vol 192 ◽  
pp. 249-286 ◽  
Author(s):  
K. R. Kirchartz ◽  
H. Oertel Jr
Keyword(s):  
Three Dimensional ◽  
Nonlinear Effects ◽  
Horizontal Layer ◽  
Stationary Solutions ◽  
Time Dependent ◽  
Linear Stability Theory ◽  
Experimental Investigations ◽  
Oscillatory Convection ◽  
Aspect Ratios ◽  
Convection Rolls

The extension of the classic Rayleigh–Bénard problem of a horizontal layer heated from below to the three-dimensional convection in rectangular boxes is dealt with in this paper both numerically and experimentally. Also discussed is the influence of shear flows in tilted boxes and the transition to time-dependent oscillatory convection. Three-dimensional numerical simulations allow the calculation of stationary solutions and the direct simulation of oscillatory instabilities. We limited ourselves to laminar and transcritical flows. For studying the particular characteristics of three-dimensional convection in horizontal containers, we carefully selected two container geometries with aspect ratios of 10:4:1 and 4:2:1. The onset of steady cellular convection in tilted boxes is calculated by an iterative application of a combined finite-difference method and a Galerkin method. The appearance of longitudinal and transverse convection rolls is determined by means of inter-ferometrical measuring techniques and is compared with the results of the linear stability theory. The spatial flow structure and the transition to oscillatory convection is calculated for selected examples in the range of supercritical Rayleigh numbers. Experimental investigations concerning the stability behaviour of the steady solutions with regard to time-dependent disturbances show a distinct influence of the Prandtl number and confirm the importance of nonlinear effects.

Transition to time-dependent convection

1974 ◽  
Vol 65 (4) ◽  
pp. 625-645 ◽  
Author(s):  
R. M. Clever ◽  
F. H. Busse
Keyword(s):  
Equations Of Motion ◽  
Three Dimensional ◽  
Horizontal Layer ◽  
Two Dimensional ◽  
Free Boundaries ◽  
Prandtl Numbers ◽  
Steady Solutions ◽  
The Stability ◽  
Convection Rolls ◽  
Good Agreement

Steady solutions in the form of two-dimensional rolls are obtained for convection in a horizontal layer of fluid heated from below as a function of the Rayleigh and Prandtl numbers. Rigid boundaries of infinite heat conductivity are assumed. The stability of the two-dimensional rolls with respect to three-dimensional disturbances is analysed. It is found that convection rolls are unstable for Prandtl numbers less than about 5 with respect to an oscillatory instability investigated earlier by Busse (1972) for the case of free boundaries. Since the instability is caused by the momentum advection terms in the equations of motion the Rayleigh number for the onset of instability increases strongly with Prandtl number. Good agreement with various experimental observations is found.

Nonlinear thermal convection with finite conducting boundaries

1985 ◽  
Vol 152 ◽  
pp. 113-123 ◽  
Author(s):  
N. Riahi
Keyword(s):  
Stability Analysis ◽  
Prandtl Number ◽  
Thermal Convection ◽  
Perturbation Technique ◽  
Three Dimensional ◽  
Horizontal Layer ◽  
Finite Amplitude ◽  
Two Dimensional ◽  
Convection Pattern ◽  
Square Cells

Finite-amplitude thermal convection in a horizontal layer with finite conducting boundaries is investigated. The nonlinear steady problem is solved by a perturbation technique, and the preferred mode of convection is determined by a stability analysis. Square cells are found to be the preferred form of convection in a semi-infinite three-dimensional region Ω in the (γb,γt, P)-space (γb and γt are the ratios of the thermal conductivities of the lower and upper boundaries to that of the fluid and P is the Prandtl number). Two-dimensional rolls are found to be the preferred convection pattern outside Ω. The dependence on γb, γt and P of the heat transported by convection is computed for the various solutions analysed in the paper.

Finite Amplitude Convection in a Rotating Porous Medium

1987 ◽  
Vol 42 (1) ◽  
pp. 13-20
Author(s):  
B. S. Dandapat
Keyword(s):  
Porous Medium ◽  
Rayleigh Number ◽  
Interstitial Fluid ◽  
Saturated Porous Medium ◽  
Critical Rayleigh Number ◽  
Vertical Axis ◽  
Horizontal Layer ◽  
Finite Amplitude ◽  
Linear Stability Theory ◽  
Onset Of Convection

The onset of convection in a horizontal layer of a saturated porous medium heated from below and rotating about a vertical axis with uniform angular velocity is investigated. It is shown that when S ∈ σ >1, overstability cannot occur, where ε is the porosity, σ the Prandtl number and S is related to the heat capacities of the solid and the interstitial fluid. It is also shown that for small values of the rotation parameter T1, finite amplitude motion with subcritical values of Rayleigh number R (i.e. R < Re, where Re is the critical Rayleigh number according to linear stability theory) is possible. For large values of T1, overstability is the preferred mode.

scholarly journals Exact energy stability of Bénard–Marangoni convection at infinite Prandtl number

2017 ◽  
Vol 822 ◽  
Author(s):  
Giovanni Fantuzzi ◽  
Andrew Wynn
Keyword(s):  
Prandtl Number ◽  
Energy Method ◽  
Marangoni Convection ◽  
Fluid Layer ◽  
Finite Amplitude ◽  
Linear Instability ◽  
Energy Stability ◽  
Thermal Boundary Conditions ◽  
Set Up ◽  
The Stability

Using the energy method we investigate the stability of pure conduction in Pearson’s model for Bénard–Marangoni convection in a layer of fluid at infinite Prandtl number. Upon extending the space of admissible perturbations to the conductive state, we find an exact solution to the energy stability variational problem for a range of thermal boundary conditions describing perfectly conducting, imperfectly conducting, and insulating boundaries. Our analysis extends and improves previous results, and shows that with the energy method global stability can be proven up to the linear instability threshold only when the top and bottom boundaries of the fluid layer are insulating. Contrary to the well-known Rayleigh–Bénard convection set-up, therefore, energy stability theory does not exclude the possibility of subcritical instabilities against finite-amplitude perturbations.

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