On the stability of two-dimensional convection rolls in an infinite Prandtl number fluid with stress-free boundaries

1989 ◽  
Vol 40 (2) ◽  
pp. 153-162 ◽  
Author(s):  
M. Schnaubelt ◽  
F. H. Busse
Keyword(s):  
Prandtl Number ◽  
Two Dimensional ◽  
Free Boundaries ◽  
The Stability ◽  
Convection Rolls

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.

The oscillatory instability of convection rolls in a low Prandtl number fluid

1972 ◽  
Vol 52 (1) ◽  
pp. 97-112 ◽  
Author(s):  
F. H. Busse
Keyword(s):  
Prandtl Number ◽  
Reasonable Agreement ◽  
Fluid Layer ◽  
Three Dimensional ◽  
Convective Motion ◽  
Critical Value ◽  
Two Dimensional ◽  
Free Boundaries ◽  
Vertical Vorticity ◽  
Convection Rolls

The instability of convection rolls in a fluid layer heated from below is investigated for stress-free boundaries in the limit of small Prandtl number. It is shown that the two-dimensional rolls become unstable to oscillatory three-dimensional disturbances when the amplitude of the convective motion exceeds a finite critical value. The instability corresponds to the generation of vertical vorticity, a mechanism which is likely to operate in the case of a variety of roll-like motions. In all aspects in which the theory can be related to experiments, reasonable agreement with the observations is found.

Stability of convection rolls in a layer with stress-free boundaries

1985 ◽  
Vol 150 ◽  
pp. 487-498 ◽  
Author(s):  
E. W. Bolton ◽  
F. H. Busse
Keyword(s):  
Three Dimensional ◽  
Finite Amplitude ◽  
Two Dimensional ◽  
Free Boundaries ◽  
Stability Boundaries ◽  
Prandtl Numbers ◽  
Steady Solutions ◽  
The Stability ◽  
Convection Rolls ◽  
Steady Convection

Steady finite-amplitude solutions for two-dimensional convection in a layer heated from below with stress-free boundaries are obtained numerically by a Galerkin method. The stability of the steady convection rolls with respect to arbitrary three-dimensional infinitesimal disturbances is investigated. Stability is found only in a small fraction of the Rayleigh-number-wavenumber space where steady solutions exist. The cross-roll instability and the oscillatory and monotonic skewed varicose instabilities are most important in limiting the stability of steady convection rolls. The Prandtlnumbers P = 0.71, 7, 104 areemphasized, but the stability boundaries are sufficiently smoothly dependent on the parameters of the problem to permit qualitative extrapolations to other Prandtl numbers.

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.

Inertial Bénard–Marangoni convection

1997 ◽  
Vol 350 ◽  
pp. 149-175 ◽  
Author(s):  
THOMAS BOECK ◽  
ANDRÉ THESS
Keyword(s):  
Prandtl Number ◽  
Marangoni Number ◽  
Marangoni Convection ◽  
Pseudospectral Method ◽  
Critical Value ◽  
Two Dimensional ◽  
Conductive Temperature ◽  
Reasonable Approximation ◽  
Convection Rolls ◽  
Dimensional Surface

Two-dimensional surface-tension-driven Bénard convection in a layer with a free-slip bottom is investigated in the limit of small Prandtl number using accurate numerical simulations with a pseudospectral method complemented by linear stability analysis and a perturbation method. It is found that the system attains a steady state consisting of counter-rotating convection rolls. Upon increasing the Marangoni number Ma the system experiences a transition between two typical convective regimes. The first one is the regime of weak convection characterized by only slight deviations of the isotherms from the linear conductive temperature profile. In contrast, the second regime, called inertial convection, shows significantly deformed isotherms. The transition between the two regimes becomes increasingly sharp as the Prandtl number is reduced. For sufficiently small Prandtl number the transition from weak to inertial convection proceeds via a subcritical bifurcation involving weak hysteresis. In the viscous zero-Prandtl-number limit the transition manifests itself in an unbounded growth of the flow amplitude for Marangoni numbers beyond a critical value Mai. For Ma<Mai the zero-Prandtl-number equations provide a reasonable approximation for weak convection at small but finite Prandtl number. The possibility of experimental verification of inertial Bénard–Marangoni convection is briefly 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.

scholarly journals Finite amplitude convection between stress-free boundaries; Ginzburg–Landau equations and modulation theory

1994 ◽  
Vol 5 (3) ◽  
pp. 267-282 ◽  
Author(s):  
Andrew J. Bernoff
Keyword(s):  
Stability Theory ◽  
Finite Amplitude ◽  
Two Dimensional ◽  
Free Boundaries ◽  
Mean Flow ◽  
Onset Of Convection ◽  
Ginzburg Landau ◽  
Ginzburg Landau Equations ◽  
Modulation Theory ◽  
The Stability

The stability theory for rolls in stress-free convection at finite Prandtl number is affected by coupling with low wavenumber two-dimensional mean-flow modes. In this work, a set of modified Ginzburg–Landau equations describing the onset of convection is derived which accounts for these additional modes. These equations can be used to extend the modulation equations of Zippelius & Siggia describing the breakup of rolls, bringing their stability theory into agreement with the results of Busse & Bolton.

Instabilities of convection rolls in a fluid of moderate Prandtl number

1979 ◽  
Vol 91 (2) ◽  
pp. 319-335 ◽  
Author(s):  
F. H. Busse ◽  
R. M. Clever
Keyword(s):  
Prandtl Number ◽  
Fluid Layer ◽  
Experimental Methods ◽  
Two Dimensional ◽  
Theoretical Predictions ◽  
Convection Rolls ◽  
Rayleigh Numbers ◽  
Horizontal Fluid Layer

The instabilities of two-dimensional convection rolls in a horizontal fluid layer heated from below are investigated in the case when the Prandtl number is seven or lower. Two new mechanisms of instability are described theoretically as well as experimentally. The knot instability causes the transition to spoke-pattern convection at higher Rayleigh numbers while the skewed varicose instability accomplishes a change to larger horizontal wavelengths of the convection rolls. Both instabilities disappear in the limits of small and large Prandtl number. Although the experimental methods fail in realizing closely the infinitely conducting boundaries assumed in the theory, the observations agree in all qualitative aspects with the theoretical predictions.

Interferometric study of two-dimensional Bénard convection cells

1974 ◽  
Vol 66 (4) ◽  
pp. 739-752 ◽  
Author(s):  
R. Farhadieh ◽  
R. S. Tankin
Keyword(s):  
Rayleigh Number ◽  
Sea Water ◽  
Critical Rayleigh Number ◽  
Width Ratio ◽  
Two Dimensional ◽  
Free Boundaries ◽  
Cell Height ◽  
Bénard Convection ◽  
Benard Convection ◽  
Convection Rolls

A Mach-Zehnder interferometer was used to stud two-dimensional Bénard convection cells. The experiments were performed with distilled water and sea water in the region where density is a linear function of temperature. Two-dimensional convection rolls were formed with Rayleigh numbers as great as 23400. Reversal in the temperature profile was obtained for R/Rc ≥ 3·8, and an overshoot of about 6% was observed at R/Rc = 9·2 and 13·8. This agrees with the values predicted theoretically by Veronis (1966) for stress-free boundaries and Royal (1969) for rigid boundaries. This disagrees with the experimental results of Gille (1967), who reports an overshoot of only 1 ½% at R/Rc = 16. Many of the other results agree with those of other experimenters, such as the relation between the cell height-to-width ratio and Rayleigh number, the relation between the Nusselt number and Rayleigh number, and the value of the critical Rayleigh number.

Instabilities of convection rolls with stress-free boundaries near threshold

1984 ◽  
Vol 146 ◽  
pp. 115-125 ◽  
Author(s):  
F. H. Busse ◽  
E. W. Bolton
Keyword(s):  
Rayleigh Number ◽  
Fluid Layer ◽  
Critical Rayleigh Number ◽  
Limited Range ◽  
Finite Domain ◽  
Free Boundaries ◽  
Prandtl Numbers ◽  
The Stability ◽  
Convection Rolls ◽  
Horizontal Fluid Layer

The stability properties of steady two-dimensional solutions describing convection in a horizontal fluid layer heated from below with stress-free boundaries are investigated in the neighbourhood of the critical Rayleigh number. The region of stable convection rolls as a function of the wavenumber α and the Rayleigh number R is bounded towards higher α by the monotonic skewed varicose instability, while towards low wavenumbers stability is limited by the zigzag instability or by the oscillatory skewed varicose instability. Only for a limited range of Prandtl numbers, 0·543 < P < ∞, does a finite domain of stability exist. In particular, convection rolls with the critical wavenumber αc are always unstable.

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