Instabilities of convection rolls in a high Prandtl number fluid

1971 ◽  
Vol 47 (2) ◽  
pp. 305-320 ◽  
Author(s):  
F. H. Busse ◽  
J. A. Whitehead
Keyword(s):  
Prandtl Number ◽  
Three Dimensional ◽  
Wave Number ◽  
Stationary Convection ◽  
Wave Numbers ◽  
Theoretical Predictions ◽  
High Prandtl Number ◽  
The Stability ◽  
Convection Rolls ◽  
Rayleigh Numbers

An experiment on the stability of convection rolls with varying wave-number is described in extension of the earlier work by Chen & Whitehead (1968). The results agree with the theoretical predictions by Busse (1967a) and show two distinct types of instability in the form of non-oscillatory disturbances. The ‘zigzag instability’ corresponds to a bending of the original rolls; in the ‘cross-roll instability’ rolls emerge at right angles to the original rolls. At Rayleigh numbers above 23,000 rolls are unstable for all wave-numbers and are replaced by a three-dimensional form of stationary convection for which the name ‘bimodal convection’ is proposed.

Experiments on the stability of convection rolls in fluids whose viscosity depends on temperature

1978 ◽  
Vol 89 (3) ◽  
pp. 553-560 ◽  
Author(s):  
Frank M. Richter
Keyword(s):  
Rayleigh Number ◽  
Thermal Boundary Layer ◽  
Initial Conditions ◽  
Three Dimensional ◽  
Hexagonal Cell ◽  
Low Viscosity ◽  
Viscosity Increase ◽  
The Stability ◽  
Convection Rolls ◽  
Rayleigh Numbers

The stability of two-dimensional convection rolls has been studied as a function of the Rayleigh number, wavenumber and variation in viscosity. The experiments used controlled initial conditions for the wavenumber, Rayleigh numbers up to 25 000 and variations in viscosity up to a factor of about 20. The parameter range of stable rolls is bounded by a hexagonal-cell regime at small Rayleigh numbers and large variations in viscosity. Otherwise, the rolls are subject to the same transitions as have already been studied in fluids of uniform viscosity. The bimodal instability leading to a stable three-dimensional pattern occurs at smaller values of the average Rayleigh number as the variations in viscosity increase. This appears to be a consequence of the low viscosity of the warm thermal boundary layer associated with the original rolls.

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.

Finite-amplitude convection in the presence of finitely conducting boundaries

2001 ◽  
Vol 432 ◽  
pp. 351-367 ◽  
Author(s):  
M. WESTERBURG ◽  
F. H. BUSSE
Keyword(s):  
Thermal Conductivity ◽  
Prandtl Number ◽  
Fluid Layer ◽  
Finite Amplitude ◽  
Onset Of Convection ◽  
Theoretical Predictions ◽  
Square Cells ◽  
Convection Rolls ◽  
Rayleigh Numbers ◽  
Horizontal Fluid Layer

Finite-amplitude convection in the form of rolls and their stability with respect to infinitesimal disturbances is investigated in the case of boundaries of the horizontal fluid layer which exhibit a thermal conductivity comparable to that of the fluid. It is found that even when rolls represent the preferred mode at the onset of convection a transition to square cells may occur at slightly supercritical Rayleigh numbers. The phenomenon of inertial convection in low Prandtl number fluids appears to become more pronounced as the conductivity of the boundaries is reduced. Modulated convection rolls have also been found as solutions of the problem. But they appear to be unstable in general. Experimental observations have been made and are found in general agreement with the theoretical predictions.

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.

Linear stability analysis of mixed-convection flow in a vertical pipe

2000 ◽  
Vol 422 ◽  
pp. 141-166 ◽  
Author(s):  
YI-CHUNG SU ◽  
JACOB N. CHUNG
Keyword(s):  
Reynolds Number ◽  
Prandtl Number ◽  
Mixed Convection ◽  
Shear Instability ◽  
Convection Flow ◽  
Mixed Convection Flow ◽  
Vertical Pipe ◽  
Prandtl Numbers ◽  
The Stability ◽  
Rayleigh Numbers

A comprehensive numerical study on the linear stability of mixed-convection flow in a vertical pipe with constant heat flux is presented with particular emphasis on the instability mechanism and the Prandtl number effect. Three Prandtl numbers representative of different regimes in the Prandtl number spectrum are employed to simulate the stability characteristics of liquid mercury, water and oil. The results suggest that mixed-convection flow in a vertical pipe can become unstable at low Reynolds number and Rayleigh numbers irrespective of the Prandtl number, in contrast to the isothermal case. For water, the calculation predicts critical Rayleigh numbers of 80 and −120 for assisted and opposed flows, which agree very well with experimental values of Rac = 76 and −118 (Scheele & Hanratty 1962). It is found that the first azimuthal mode is always the most unstable, which also agrees with the experimental observation that the unstable pattern is a double spiral flow. Scheele & Hanratty's speculation that the instability in assisted and opposed flows can be attributed to the appearance of inflection points and separation is true only for fluids with O(1) Prandtl number. Our study on the effect of the Prandtl number discloses that it plays an active role in buoyancy-assisted flow and is an indication of the viability of kinematic or thermal disturbances. It profoundly affects the stability of assisted flow and changes the instability mechanism as well. For assisted flow with Prandtl numbers less than 0.3, the thermal–shear instability is dominant. With Prandtl numbers higher than 0.3, the assisted-thermal–buoyant instability becomes responsible. In buoyancy-opposed flow, the effect of the Prandtl number is less significant since the flow is unstably stratified. There are three distinct instability mechanisms at work independent of the Prandtl number. The Rayleigh–Taylor instability is operative when the Reynolds number is extremely low. The opposed-thermal–buoyant instability takes over when the Reynolds number becomes higher. A still higher Reynolds number eventually leads the thermal–shear instability to dominate. While the thermal–buoyant instability is present in both assisted and opposed flows, the mechanism by which it destabilizes the flow is completely different.

Spherical shell rotating convection in the presence of a toroidal magnetic field

1995 ◽  
Vol 448 (1933) ◽  
pp. 245-268 ◽  
Keyword(s):  
Magnetic Field ◽  
Prandtl Number ◽  
Spherical Shell ◽  
Lorentz Force ◽  
Thermal Convection ◽  
Toroidal Magnetic Field ◽  
Wave Number ◽  
Convective Motions ◽  
Magnetic Convection ◽  
Convection Rolls

Convective instabilities of a self-gravitating, rapidly rotating fluid spherical shell are investigated in the presence of an imposed azimuthal axisymmetric magnetic field in the form of the toroidal decay mode that satisfies electrically insulating boundary conditions and has dipole symmetry. Concentration is on two major questions: how purely thermal convection of the different forms (Zhang 1992, 1994) is affected by the Lorentz force, the strength of which is measured by the Elsasser number ∧, and in what manner purely magnetic instabilities in a spherical shell (Zhang & Fearn 1993, 1994) are associated with magnetic convection. It is found that the two-dimensionality of purely thermal convection (Busse 1970) survives under the influence of a strong Lorentz force. Convective motions always attempt to satisfy the Proudman–Taylor constraint and remain predominantly two-dimensional in the whole range of ∧, 0 ≤ ∧ ≤ ∧ c , where ∧ c ═ O (10) is the critical Elsasser number for purely magnetic instabilities. Though the optimum azimuthal wave number m of convection rolls decreases drastically, from m ~ O ( T 1/6 ) at ∧ ═ 0 to m ═ O (5) at ∧ ═ O (1). We show that there exist no optimum values of ∧ that can give rise to an overall minimum of the (modified) Rayleigh number R *; the optimum value of R * is a monotonically, smoothly decreasing function of ∧, from R * ═ O ( T 1/6 ) at ∧ < O ( T -1/6 ) to R * ═ O (–10) at ∧ ═ 20. We also show that the influence of the magnetic field on thermal convection is crucially dependent on the size of the Prandtl number. At sufficiently small Prandtl number, the Poincaré convection mode (Zhang 1994) is preferred in the region 0 ≤ ∧ < ∧ c , and is only slightly affected by the presence of the toroidal magnetic field. Analytical solutions of the magnetic convection problem are then obtained based on a perturbation analysis, showing a good agreement with the numerical solution.

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.

Stability of Mixed Convection in a Differentially Heated Vertical Channel

1998 ◽  
Vol 120 (1) ◽  
pp. 127-132 ◽  
Author(s):  
Y.-C. Chen ◽  
J. N. Chung
Keyword(s):  
Reynolds Number ◽  
Prandtl Number ◽  
Mixed Convection ◽  
Local Minimum ◽  
Wave Speed ◽  
Vertical Channel ◽  
Shear Instability ◽  
Wave Number ◽  
Wave Numbers ◽  
Prandtl Numbers

In this study, the linear stability of mixed convection in a differentially heated vertical channel is investigated for various Prandtl numbers. The results indicate that this fully developed heated flow can become unstable under appropriate conditions. It is found that both the Prandtl number and Reynolds number hold very important effects on the critical Grashof number, wave number, wave speed, and instability mechanism for higher Prandtl numbers. For low Prandtl numbers, the effects from the Prandtl number and Reynolds number are relatively small. The most significant finding is that the local minimum wave numbers can be as high as eight for Pr = 1000, which is substantially higher than those found before for other heated flows. The existence of multiple local minimum wave numbers is responsible for the sudden jumps of the critical wave number and wave speed and the sudden shift of instability type for higher Prandtl numbers. The energy budget analysis shows that the thermal-shear and shear instabilities dominate at both low and high Reynolds numbers for Pr = 0.7 and 7. It is the thermal-buoyant instability for Re < 1365 and shear instability for Re ≥ 1365 for Pr = 100. The thermal-buoyant and mixed instabilities are the possible instability types for Pr = 1000. In general, for mixed convection channel flows, the instability characteristics of differentially heated flows are found to be substantially different from those of uniformly heated flows.

The stability of pipe flow Part 1. Asymptotic analysis for small wave-numbers

1970 ◽  
Vol 43 (2) ◽  
pp. 279-290 ◽  
Author(s):  
W. P. Graebel
Keyword(s):  
Asymptotic Analysis ◽  
Poiseuille Flow ◽  
Reynolds Numbers ◽  
Transverse Component ◽  
Wave Number ◽  
Radial Velocity Component ◽  
Wave Numbers ◽  
Flow Part ◽  
The Stability ◽  
High Degree

The instability of Poiseuille flow in a pipe is considered for small disturbances. An asymptotic analysis is used which is similar to that found successful in plane Poiseuille flow. The disturbance is taken to travel in a spiral fashion, and comparison of the radial velocity component with the transverse component in the plane case shows a high degree of similarity, particularly near the critical point where the disturbance and primary flow travel with the same speed. Instability is found for azimuthal wave-numbers of 2 or greater, although the corresponding minimum Reynolds numbers are too small to compare favourably with either experiments or the initial restrictions on the magnitude of the wave-number.

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.

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