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.

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.

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.

Effect of Modulation on Rayleigh-Benard Convection-II

2003 ◽  
Vol 58 (2-3) ◽  
pp. 176-182
Author(s):  
B. S. Bhadauria
Keyword(s):  
Rayleigh Number ◽  
Temperature Gradient ◽  
Fluid Layer ◽  
Critical Rayleigh Number ◽  
Periodic Part ◽  
Prandtl Numbers ◽  
Rigid Walls ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard ◽  
Horizontal Fluid Layer

The linear stability of a horizontal fluid layer, confined between two rigid walls, heated from below and cooled from above is considered. The temperature gradient between the walls consists of a steady part and a periodic part that oscillates with time. Only infinitesimal disturbances are considered. Numerical results for the critical Rayleigh number are obtained for various Prandtl numbers and for various values of the frequency. Some comparisons with known results have also been made.

Unsteady Heating of Rayleigh-Benard Convection

2004 ◽  
Vol 59 (4-5) ◽  
pp. 266-274
Author(s):  
B. S. Bhadauria
Keyword(s):  
Fluid Layer ◽  
Critical Rayleigh Number ◽  
Dependent Part ◽  
Prandtl Numbers ◽  
Periodic Temperature ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard ◽  
Unsteady Heating ◽  
Horizontal Fluid Layer ◽  
Time Periodic

The linear thermal instability of a horizontal fluid layer with time-periodic temperature distribution is studied with the help of the Floquet theory. The time-dependent part of the temperature has been expressed in Fourier series. Disturbances are assumed to be infinitesimal. Only even solutions are considered. Numerical results for the critical Rayleigh number are obtained at various Prandtl numbers and for various values of the frequency. It is found that the disturbances are either synchronous with the primary temperature field or have half its frequency. - 2000 Mathematics Subject Classification: 76E06, 76R10.

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.

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 Effects of modulation on Rayleigh-Benard convection. Part I

2004 ◽  
Vol 2004 (19) ◽  
pp. 991-1001 ◽  
Author(s):  
B. S. Bhadauria ◽  
Lokenath Debnath
Keyword(s):  
Rayleigh Number ◽  
Linear Stability ◽  
Fluid Layer ◽  
Critical Rayleigh Number ◽  
Horizontal Layer ◽  
Time Dependent ◽  
Steady Temperature ◽  
Prandtl Numbers ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard

The linear stability of a horizontal layer of fluid heated from below and above is considered. In addition to a steady temperature difference between the walls of the fluid layer, a time-dependent periodic perturbation is applied to the wall temperatures. Only infinitesimal disturbances are considered. Numerical results for the critical Rayleigh number are obtained at various Prandtl numbers and for various values of the frequency. Some comparisons have been made with the known results.

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.

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.

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