Convection in a rotating cylinder. Part 1 Linear theory for moderate Prandtl numbers

1993 ◽  
Vol 248 ◽  
pp. 583-604 ◽  
Author(s):  
H. F. Goldstein ◽  
E. Knobloch ◽  
I. Mercader ◽  
M. Net
Keyword(s):  
Rayleigh Number ◽  
Aspect Ratio ◽  
Prandtl Number ◽  
Boundary Conditions ◽  
Rotation Rate ◽  
Critical Rayleigh Number ◽  
Onset Of Convection ◽  
Aspect Ratios ◽  
Prandtl Numbers ◽  
Rayleigh Numbers

The onset of convection in a uniformly rotating vertical cylinder of height h and radius d heated from below is studied. For non-zero azimuthal wavenumber the instability is a Hopf bifurcation regardless of the Prandtl number of the fluid, and leads to precessing spiral patterns. The patterns typically precess counter to the rotation direction. Two types of modes are distinguished: the fast modes with relatively high precession velocity whose amplitude peaks near the sidewall, and the slow modes whose amplitude peaks near the centre. For aspect ratios τ ≡ d/h of order one or less the fast modes always set in first as the Rayleigh number increases; for larger aspect ratios the slow modes are preferred provided that the rotation rate is sufficiently slow. The precession velocity of the slow modes vanishes as τ → ∞. Thus it is these modes which provide the connection between the results for a finite-aspect-ratio System and the unbounded layer in which the instability is a steady-state one, except in low Prandtl number fluids.The linear stability problem is solved for several different sets of boundary conditions, and the results compared with recent experiments. Results are presented for Prandtl numbers σ in the range 6.7 ≤ σ ≤ 7.0 as a function of both the rotation rate and the aspect ratio. The results for rigid walls, thermally conducting top and bottom and an insulating sidewall agree well with the measured critical Rayleigh numbers and precession frequencies for water in a τ = 1 cylinder. A conducting sidewall raises the critical Rayleigh number, while free-slip boundary conditions lower it. The difference between the critical Rayleigh numbers with no-slip and free-slip boundaries becomes small for dimensionless rotation rates Ωh2/v ≥ 200, where v is the kinematic viscosity.

Heat transport by parallel-roll convection in a rectangular container

1987 ◽  
Vol 185 ◽  
pp. 205-234 ◽  
Author(s):  
R. W. Walden ◽  
Paul Kolodner ◽  
A. Passner ◽  
C. M. Surko
Keyword(s):  
Rayleigh Number ◽  
Heat Transport ◽  
Critical Rayleigh Number ◽  
Equation Model ◽  
Onset Of Convection ◽  
Infinite Layer ◽  
Lateral Extent ◽  
Prandtl Numbers ◽  
Rayleigh Numbers ◽  
Good Agreement

Heat-transport measurements are reported for thermal convection in a rectangular box of aspect’ ratio 10 x 5. Results are presented for Rayleigh numbers up to 35Rc, Prandtl numbers between 2 and 20, and wavenumbers between 0.6 and 1.0kc, where Rc and kc are the critical Rayleigh number and wavenumber for the onset of convection in a layer of infinite lateral extent. The measurements are in good agreement with a phenomenological model which combines the calculations of Nusselt number, as a function of Rayleigh number and roll wavenumber for two-dimensional convection in an infinite layer, with a nonlinear amplitude-equation model developed to account for sidewell attenuation. The appearance of bimodal convection increases the heat transport above that expected for simple parallel-roll convection.

Correlations for Natural Convection Between Heated Vertical Plates

1991 ◽  
Vol 113 (1) ◽  
pp. 97-107 ◽  
Author(s):  
S. Ramanathan ◽  
R. Kumar
Keyword(s):  
Rayleigh Number ◽  
Aspect Ratio ◽  
Prandtl Number ◽  
Vertical Direction ◽  
Numerical Results ◽  
Maximum Temperature ◽  
Parallel Plates ◽  
Plate Temperature ◽  
Aspect Ratios ◽  
Prandtl Numbers

This paper presents the numerical results of natural convective flows between two vertical, parallel plates within a large enclosure. A parametric study has been conducted for various Prandtl numbers and channel aspect ratios. The results are in good agreement with the reported results in the literature for air for large aspect ratios. However, for small aspect ratios, the present numerical results do not agree with the correlations given in the literature. The discrepancy is due to the fact that the published results were obtained for channels where the diffusion of thermal energy in the vertical direction is negligible. The results obtained in this paper indicate that vertical conduction should be considered for channel aspect ratios less than 10 for Pr = 0.7. Correlations are presented to predict the maximum temperature and the average Nusselt number on the plate as explicit functions of the channel Rayleigh number and the channel aspect ratio for air. The plate temperature is a weak function of Prandtl number for Prandtl numbers greater than 0.7, if the channel Rayleigh number is chosen as the correlating parameter. For Prandtl numbers less than 0.1, the plate temperature is a function of the channel Rayleigh number and the Prandtl number. A correlation for maximum temperature on the plate is presented to include the Prandtl number effect for large aspect ratio channels.

Three-dimensional instability of axisymmetric buoyant convection in cylinders heated from below

1996 ◽  
Vol 326 ◽  
pp. 399-415 ◽  
Author(s):  
M. Wanschura ◽  
H. C. Kuhlmann ◽  
H. J. Rath
Keyword(s):  
Rayleigh Number ◽  
Linear Stability ◽  
Critical Rayleigh Number ◽  
Three Dimensional ◽  
Base Flow ◽  
Onset Of Convection ◽  
Buoyant Convection ◽  
Thermal Boundary Conditions ◽  
Aspect Ratios ◽  
Prandtl Numbers

The stability of steady axisymmetric convection in cylinders heated from below and insulated laterally is investigated numerically using a mixed finite-difference/Chebyshev collocation method to solve the base flow and the linear stability equations. Linear stability boundaries are given for radius to height ratios γ from 0.9 to 1.56 and for Prandtl numbers Pr = 0.02 and Pr = 1. Depending on γ and Pr, the azimuthal wavenumber of the critical mode may be m = 1, 2, 3, or 4. The dependence of the critical Rayleigh number on the aspect ratio and the instability mechanisms are explained by analysing the energy transfer to the critical modes for selected cases. In addition to these results the onset of buoyant convection in liquid bridges with stress-free conditions on the cylindrical surface is considered. For insulating thermal boundary conditions, the onset of convection is never axisymmetric and the critical azimuthal wavenumber increases monotonically with γ. The critical Rayleigh number is less then 1708 for most aspect ratios.

scholarly journals Convective Instability in Slip Flow in a Vertical Circular Porous Microchannel

2021 ◽  
Author(s):  
A. A. Avramenko ◽  
I. V. Shevchuk ◽  
A. I. Tyrinov
Keyword(s):  
Rayleigh Number ◽  
Prandtl Number ◽  
Convective Instability ◽  
Critical Rayleigh Number ◽  
Slip Flow ◽  
Onset Of Convection ◽  
Slippage Effect ◽  
Prandtl Numbers ◽  
The Galerkin Method ◽  
Thermal Conductivities

AbstractThe paper represents an analysis of convective instability in a vertical cylindrical porous microchannel performed using the Galerkin method. The dependence of the critical Rayleigh number on the Darcy, Knudsen, and Prandtl numbers, as well as on the ratio of the thermal conductivities of the fluid and the wall, was obtained. It was shown that a decrease in permeability of the porous medium (in other words, increase in its porosity) causes an increase in flow stability. This effect is substantially nonlinear. Under the condition Da > 0.1, the effect of the porosity on the critical Rayleigh number practically vanishes. Strengthening of the slippage effects leads to an increase in the instability of the entire system. The slippage effect on the critical Rayleigh number is nonlinear. The level of nonlinearity depends on the Prandtl number. With an increase in the Prandtl number, the effect of slippage on the onset of convection weakens. With an increase in the ratio of the thermal conductivities of the fluid and the wall, the influence of the Prandtl number decreases. At high values of the Prandtl numbers (Pr > 10), its influence practically vanishes.

scholarly journals Linear biglobal analysis of Rayleigh–Bénard instabilities in binary fluids with and without throughflow

2012 ◽  
Vol 713 ◽  
pp. 216-242 ◽  
Author(s):  
Jun Hu ◽  
Daniel Henry ◽  
Xie-Yuan Yin ◽  
Hamda BenHadid
Keyword(s):  
Reynolds Number ◽  
Rayleigh Number ◽  
Aspect Ratio ◽  
Critical Rayleigh Number ◽  
Two Dimensional ◽  
Binary Fluids ◽  
Transverse Modes ◽  
Aspect Ratios ◽  
Rayleigh Benard ◽  
Mode A

AbstractThree-dimensional Rayleigh–Bénard instabilities in binary fluids with Soret effect are studied by linear biglobal stability analysis. The fluid is confined transversally in a duct and a longitudinal throughflow may exist or not. A negative separation factor $\psi = \ensuremath{-} 0. 01$, giving rise to oscillatory transitions, has been considered. The numerical dispersion relation associated with this stability problem is obtained with a two-dimensional Chebyshev collocation method. Symmetry considerations are used in the analysis of the results, which allow the classification of the perturbation modes as ${S}_{l} $ modes (those which keep the left–right symmetry) or ${R}_{x} $ modes (those which keep the symmetry of rotation of $\lrm{\pi} $ about the longitudinal mid-axis). Without throughflow, four dominant pairs of travelling transverse modes with finite wavenumbers $k$ have been found. Each pair corresponds to two symmetry degenerate left and right travelling modes which have the same critical Rayleigh number ${\mathit{Ra}}_{c} $. With the increase of the duct aspect ratio $A$, the critical Rayleigh numbers for these four pairs of modes decrease and closely approach the critical value ${\mathit{Ra}}_{c} = 1743. 894$ obtained in a two-dimensional situation, one of the mode (a ${S}_{l} $ mode called mode A) always remaining the dominant mode. Oscillatory longitudinal instabilities ($k\approx 0$) corresponding to either ${S}_{l} $ or ${R}_{x} $ modes have also been found. Their critical curves, globally decreasing, present oscillatory variations when the duct aspect ratio $A$ is increased, associated with an increasing number of longitudinal rolls. When a throughflow is applied, the symmetry degeneracy of the pairs of travelling transverse modes is broken, giving distinct upstream and downstream modes. For small and moderate aspect ratios $A$, the overall critical Rayleigh number in the small Reynolds number range studied is only determined by the upstream transverse mode A. In contrast, for larger aspect ratios as $A= 7$, different modes are successively dominant as the Reynolds number is increased, involving both upstream and downstream transverse modes A and even the longitudinal mode.

Effect of a stabilizing gradient of solute on thermal convection

1968 ◽  
Vol 34 (2) ◽  
pp. 315-336 ◽  
Author(s):  
George Veronis
Keyword(s):  
Rayleigh Number ◽  
Temperature Gradient ◽  
Prandtl Number ◽  
Critical Rayleigh Number ◽  
Effective Diffusion ◽  
Stratified Fluid ◽  
Finite Amplitude ◽  
Oscillatory Motion ◽  
Linear Stability Theory ◽  
Onset Of Convection

A stabilizing gradient of solute inhibits the onset of convection in a fluid which is subjected to an adverse temperature gradient. Furthermore, the onset of instability may occur as an oscillatory motion because of the stabilizing effect of the solute. These results are obtained from linear stability theory which is reviewed briefly in the following paper before finite-amplitude results for two-dimensional flows are considered. It is found that a finite-amplitude instability may occur first for fluids with a Prandtl number somewhat smaller than unity. When the Prandtl number is equal to unity or greater, instability first sets in as an oscillatory motion which subsequently becomes unstable to disturbances which lead to steady, convecting cellular motions with larger heat flux. A solute Rayleigh number, Rs, is defined with the stabilizing solute gradient replacing the destabilizing temperature gradient in the thermal Rayleigh number. When Rs is large compared with the critical Rayleigh number of ordinary Bénard convection, the value of the Rayleigh number at which instability to finite-amplitude steady modes can set in approaches the value of Rs. Hence, asymptotically this type of instability is established when the fluid is marginally stratified. Also, as Rs → ∞ an effective diffusion coefficient, Kρ, is defined as the ratio of the vertical density flux to the density gradient evaluated at the boundary and it is found that κρ = √(κκs) where κ, κs are the diffusion coefficients for temperature and solute respectively. A study is made of the oscillatory behaviour of the fluid when the oscillations have finite amplitudes; the periods of the oscillations are found to increase with amplitude. The horizontally averaged density gradients change sign with height in the oscillating flows. Stably stratified fluid exists near the boundaries and unstably stratified fluid occupies the mid-regions for most of the oscillatory cycle. Thus the step-like behaviour of the density field which has been observed experimentally for time-dependent flows is encountered here numerically.

Free convection in an open thermosyphon, with special reference to turbulent flow

1955 ◽  
Vol 230 (1183) ◽  
pp. 502-530 ◽  
Keyword(s):  
Heat Transfer ◽  
Boundary Layer ◽  
Rayleigh Number ◽  
Prandtl Number ◽  
Boundary Layer Flow ◽  
Radius Ratio ◽  
Tube Length ◽  
Prandtl Numbers ◽  
Layer Flow ◽  
Rayleigh Numbers

This paper describes an experimental investigation of heat transfer by free convection of a fluid in a heated vertical tube, sealed at its lower end. Heated fluid adjacent to the wall is discharged from the open end into a suitably cooled large reservoir, while a central core of cool fluid is continuously drawn into the tube by way of replacement. The system constitutes an unusual case of natural convection because the two streams of fluid, moving in opposite directions, are compelled to create their own internal boundary. Such an arrangement forms a static simulation of the Schmidt system (1951) for cooling high-temperature gas turbine blades, where sealed radial passages in the blades communicate with a reservoir in the rotor drum, and large centrifugal accelerations replace that due to gravity in the static system. The use of a scaled-up static tube in large measure compensates for the relatively small gravitational acceleration, when determining the working range of Rayleigh numbers, in this case from 10 7 to 10 13 . These are based on tube length, the fluid property values being referred to tube-wall temperature. Separate assessments are made of the effect of fluid Prandtl number (covering values from 7600 to 0·69) and tube length radius ratio (ranging from 7·5 to 47·5). In laminar flow the former is not found to be significant, but the quotient of the Rayleigh number (based on radius) and tube length-radius ratio determines the ranges of three laminar flow régimes. High values of the quotient correspond to 'boundary-layer flow’ and greatest heat transfer. This is followed first by ‘impeded non-similarity flow’ and then by ‘impeded similarity flow’ as the quotient becomes smaller, where the two streams of fluid mingle. These findings are in close agreement with theoretical prediction (Lighthill 1953). Turbulence arises in two ways. For Prandtl numbers near unity, transition occurs during the laminar impeded-flow régimes, resulting in a mixing effect and reduced heat transfer. This is predicted by Lighthill, but his discussion of turbulent flow is restricted to a Prandtl number of unity. For larger Prandtl numbers, transition takes place during laminar boundary-layer flow, yielding a conventional turbulent boundary-layer régime with increased heat transfer. The mean transitional Grashof numbers (based on radius) are in the range 10 4.4 to 10 4.6 ; they compare favourably with a pre­dicted range of from 10 4.0 to 10 4.3 . The tendency for the cool entering fluid to become turbulent renders turbulent boundary-layer flow potentially unstable. Both modes of transition eventually lead to a stable ‘fully mixed' régime where the two turbulent streams mix. This causes reduced circulation and heat transfer, the extent of the reduction varying directly with length-radius ratio and inversely with Prandtl number. The régime was predicted by Lighthill, but there are considerable dis­crepancies between estimated and experimental heat-transfer rates, and in the duration of the régime. In practice it appears to persist indefinitely, whereas Lighthill forecasts its replace­ment at high Rayleigh numbers by a stable boundary-layer flow. Empirical correlations show that fully mixed flow yields optimum heat transfer at a length-radius ratio, which is determined by the Rayleigh number. The suitability of the Schmidt system for blade cooling is briefly discussed in the light of the investigation.

Experiments on Convective Instability of Large Prandtl Number Fluids in a Vertical Slot

1994 ◽  
Vol 116 (1) ◽  
pp. 120-126 ◽  
Author(s):  
S. Wakitani
Keyword(s):  
Rayleigh Number ◽  
Aspect Ratio ◽  
Prandtl Number ◽  
Traveling Wave ◽  
Critical Rayleigh Number ◽  
Transition Process ◽  
Wave Mode ◽  
Vertical Slot ◽  
Increasing Trend ◽  
The Stability

The stability of thermal convection of large Prandtl number fluids in a vertical slot (aspect ratio = 10, 15, or 20) was studied experimentally. Secondary cells began to appear from the center of the slot and then prevailed. The critical Rayleigh number showed an increasing trend as the aspect ratio was increased, and became higher than that obtained from a linear stability analysis owing to a possible traveling wave mode of instability. As the Rayleigh number was increased, either the onset of the tertiary cells or the traveling waves occurred depending on the Prandtl number. In the transition process from laminar to turbulent flow, the secondary cells around the end regions split into smaller cells and an unsteady flow structure appeared with an original secondary cell in the center region and some smaller cells along the sidewalls.

Convective Heat Transfer in a Rectangular Porous Cavity—Effect of Aspect Ratio on Flow Structure and Heat Transfer

1984 ◽  
Vol 106 (1) ◽  
pp. 158-165 ◽  
Author(s):  
V. Prasad ◽  
F. A. Kulacki
Keyword(s):  
Heat Transfer ◽  
Rayleigh Number ◽  
Aspect Ratio ◽  
Aspect Ratios ◽  
Family Of Curves ◽  
Different Temperatures ◽  
Vertical Walls ◽  
Rayleigh Numbers ◽  
Cavity Effect ◽  
Cavity Width

Two-dimensional steady natural convection in a porous rectangular cavity bounded by isothermal vertical walls at different temperatures and adiabatic horizontal walls has been studied numerically for aspect ratios less than unity and Rayleigh numbers up to 104. Results indicate the presence of multicellular flow. Also, the average Nusselt number based on cavity width is observed to be a maximum in a restricted range of aspect ratio, depending on the Rayleigh number. Effects of aspect ratio are summarized by a family of curves for constant Rayleigh number, based on cavity height, for aspect ratios from 0.05 to 100. For a cavity with fixed height, the heat transfer rate always increases as the aspect ratio is increased, except when the flow exhibits boundary layers on the vertical walls. Criteria in terms of aspect ratio and Rayleigh number have been established for the existence of different flow regimes.

The onset of convection and turbulence in rectangular layers of normal liquid 4He

1988 ◽  
Vol 189 ◽  
pp. 263-286 ◽  
Author(s):  
R. W. Motsay ◽  
K. E. Anderson ◽  
R. P. Behringer
Keyword(s):  
Nusselt Number ◽  
Rayleigh Number ◽  
Quantitative Agreement ◽  
Precision Measurements ◽  
Onset Of Convection ◽  
Normal Liquid ◽  
Intermediate Size ◽  
Prandtl Numbers ◽  
Convection Rolls ◽  
Rayleigh Numbers

We have carried out high-precision measurements of the heat transport in intermediate-size rectangular layers of convecting normal liquid 4He with Prandtl numbers of 0.52 and 0.70. The containers used for these experiments had horizontal dimensions, in units of the height d, of 13.4 × 5.95 (cell I) and 18.2 × 8.12 (cell II). The slopes N1 of the Nusselt curves were 0.56 and 0.70 respectively for cell I and cell II. These values are significantly lower than predictions for N1 for horizontally unbound layers, but comparable with results obtained in cylindrical containers of liquid helium with roughly the same number of convection rolls. For the two containers, the onset of the first instability after the onset of convection occurred at Rayleigh numbers R1 that were in reasonable quantitative agreement with the predictions of Busse and Clever for the skewed-varicose instability. For both containers, the transition at R1 was characterized by long transients ranging from ∼ 102 to ∼ 103 vertical-thermal-diffusion times. A decrease in the Nusselt number was also observed. As the Rayleigh number was increased above R1, a new steady state evolved and then additional transitions were observed. These transitions occurred at Rayleigh numbers labelled R2, R3,…, with a total of five transitions seen in cell I and a total of three transitions seen for cell II. The transition for each cell at R2 can be related quantitatively to the skewed-varicose instability, and the transition at R3 is associated with an oscillatory instability. For cell II, the time-dependence beginning at R3 persisted to the highest Rayleigh number studied, R = 11.7Rc. However, for container I, two more regimes of time-independent flow were observed; the last of these was at an unexpectedly high Rayleigh number of 6.7Rc. This work extends to lower Prandtl number recent studies made on moderate-size rectangular layers of convecting water and alcohol.

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