Observed flow patterns at the initiation of convection in a horizontal liquid layer heated from below

1970 ◽  
Vol 42 (4) ◽  
pp. 755-768 ◽  
Author(s):  
E. F. C. Somerscales ◽  
T. S. Dougherty
Keyword(s):  
Rayleigh Number ◽  
Critical Rayleigh Number ◽  
Test Chamber ◽  
Lower Surface ◽  
Flow Patterns ◽  
Large Temperature ◽  
Small Temperature ◽  
Initial Flow ◽  
High Prandtl Number ◽  
Rigid Surfaces

An experimental investigation has been made of the flow patterns at the initiation of convection in a layer of a high Prandtl number liquid confined between rigid, horizontal surfaces and heated from below. The experiment was designed to overcome the limitations of earlier experiments and to correspond closely to the conditions of the theory. In particular, the upper and lower rigid surfaces which enclosed the layer were made of copper which has a high thermal conductivity. To aid in the analysis of the experimental results some supplementary observations of the flow patterns were made using a glass upper plate. For small fluid depths and large temperature differences between the upper and lower surface the initial flow was in the form of hexagonal cells as predicted theoretically. With increasing Rayleigh number the cellular flow appeared to transform into rolls as predicted. For large fluid depths and small temperature differences only circular plan-form rolls were observed. This is in agreement with the results of other experimenters. It is tentatively proposed that this non-appearance of an initial cellular flow is due to the shape of the test chamber having a dominating influence on the flow pattern when the temperature gradient in the fluid is small. Measurements were also made of the development time for the flow patterns and the critical Rayleigh number.

Stability and Convection in Impulsively Heated Porous Layers

2008 ◽  
Vol 130 (11) ◽  
Author(s):  
M. J. Kohl ◽  
M. Kristoffersen ◽  
F. A. Kulacki
Keyword(s):  
Porous Medium ◽  
Rayleigh Number ◽  
Heated Surface ◽  
Critical Rayleigh Number ◽  
Axisymmetric Flow ◽  
Temperature Fluctuations ◽  
Lower Surface ◽  
Onset Of Convection ◽  
Upward Moving ◽  
The Stability

Experiments are reported on initial instability, turbulence, and overall heat transfer in a porous medium heated from below. The porous medium comprises either water or a water-glycerin solution and randomly stacked glass spheres in an insulated cylinder of height:diameter ratio of 1.9. Heating is with a constant flux lower surface and a constant temperature upper surface, and the stability criterion is determined for a step heat input. The critical Rayleigh number for the onset of convection is obtained in terms of a length scale normalized to the thermal penetration depth as Rac=83/(1.08η−0.08η2) for 0.02<η<0.18. Steady convection in terms of the Nusselt and Rayleigh numbers is Nu=0.047Ra0.91Pr0.11(μ/μ0)0.72 for 100<Ra<5000. Time-averaged temperatures suggest the existence of a unicellular axisymmetric flow dominated by upflow over the central region of the heated surface. When turbulence is present, the magnitude and frequency of temperature fluctuations increase weakly with increasing Rayleigh number. Analysis of temperature fluctuations in the fluid provides an estimate of the speed of the upward moving thermals, which decreases with distance from the heated surface.

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.

Experimental Study on the Stabilization of the No-Motion State in the Rayleigh-Benard Convection Problem

2006 ◽  
Author(s):  
Marcel C. Remillieux
Keyword(s):  
Rayleigh Number ◽  
Cooling System ◽  
Critical Rayleigh Number ◽  
Flow Patterns ◽  
Test Cell ◽  
Experimental Investigations ◽  
Pd Controller ◽  
Onset Of Convection ◽  
Motion State ◽  
Proportional Controller

We demonstrate experimentally that through the use of proportional-differential control, it is possible to stabilize the no-motion state of a fluid layer heated from below, cooled from above, and confined in an upright, circular cylinder (the Rayleigh-Be´nard problem). An array of 24 independently controlled heaters (thermal actuators), microfabricated on a silicon wafer, constitutes the bottom boundary of the test cell. A cooling system maintains the top boundary at a constant temperature. Silicon diodes located at the mid-height of the cell, above the actuators, measure the fluid's temperature. The multi-input, multi-output controller adjusts the heaters' power in proportion to the deviation of the fluid's temperatures, as recorded by the diodes, from preset values associated with the no-motion, conductive state. First, a set of experiments was conducted in the absence of a controller to determine the uncontrolled, reference state. Advantage is taken of the linear dependence of the mid-height temperature on the power input in the no-motion state. The preset temperatures are determined by extrapolating the mid-height temperatures to the desired input power values. A proportional controller is then engaged. We show that as the controller's gain increases so does the critical Rayleigh number for the onset of convection. The proportional controller allows us to increase the critical Rayleigh number by as much as a factor of 1.4. When the controller's gain is larger than a critical value, the system becomes time-wise oscillatory (Hopf bifurcation) and the controller's performance deteriorates. The oscillatory convection can be significantly damped out by engaging a proportional-differential (PD) controller. The PD controller allows us to further increase the critical Rayleigh number for the onset of convection to as much as a factor or 1.7 compared to the uncontrolled case. Further increases in the critical Rayleigh number were not possible due to the actuators' saturation. We also compared the supercritical flow patterns at the mid-height of the test cell in the presence of the controller with the flow patterns in the absence of a controller. The proportional controller modified the flow pattern from a single convective cell with ascending fluid in one half of the cell and descending in the other half, to fluid ascending at the center of the cell and descending at near the lateral wall. Our work represents an improvement over previous experimental investigations on the stabilization of Rayleigh-Be´nard convection in which the critical Rayleigh number was increased by only a factor of 1.2. Almost uniform temperature distribution at the mid-height is obtained through the combined action of proportional and derivative controllers. The Rayleigh-Be´nard convection is suppressed under conditions when, in the absence of a controller, flow would persist.

Instability of Convection and Heat Transfer of High Prandtl Number Fluids in a Vertical Slot

1996 ◽  
Vol 118 (2) ◽  
pp. 359-365 ◽  
Author(s):  
Y. Y. Jin ◽  
C. F. Chen
Keyword(s):  
Heat Transfer ◽  
Prandtl Number ◽  
Critical Rayleigh Number ◽  
Convective Motion ◽  
Flow Patterns ◽  
Convection And Heat Transfer ◽  
Number Range ◽  
Vertical Slot ◽  
High Prandtl Number ◽  
Good Agreement

The stability of convective motion of high-Prandtl-number fluids, generated by a lateral temperature difference across a vertical slot with aspect ratio 15, is studied numerically. The Prandtl number range studied is from 50 to 2000. The nonlinear governing equations are solved by a finite difference method. The predicted flow patterns and critical values are in good agreement with the recent experimental results of Wakitani (1994). It is found that the vorticity distribution along the vertical centerline of the slot is a very sensitive indicator of the onset of multicellular flow. The critical Grashof number varies almost inversely with the Prandtl number; consequently, the critical Rayleigh number is essentially independent of the Prandtl number. Heat transfer results show good agreement with the experimentally correlated values, and they are independent of the Prandtl numbers and the flow patterns.

Linear stability of rotating convection in an imposed shear flow

1997 ◽  
Vol 350 ◽  
pp. 271-293 ◽  
Author(s):  
PAUL MATTHEWS ◽  
STEPHEN COX
Keyword(s):  
Rayleigh Number ◽  
Shear Flow ◽  
Eigenvalue Problem ◽  
Linear Stability ◽  
Thermal Convection ◽  
Critical Rayleigh Number ◽  
Rotation Vector ◽  
Linear Stability Theory ◽  
Fixed Temperature ◽  
Differential Motion

In many geophysical and astrophysical contexts, thermal convection is influenced by both rotation and an underlying shear flow. The linear theory for thermal convection is presented, with attention restricted to a layer of fluid rotating about a horizontal axis, and plane Couette flow driven by differential motion of the horizontal boundaries.The eigenvalue problem to determine the critical Rayleigh number is solved numerically assuming rigid, fixed-temperature boundaries. The preferred orientation of the convection rolls is found, for different orientations of the rotation vector with respect to the shear flow. For moderate rates of shear and rotation, the preferred roll orientation depends only on their ratio, the Rossby number.It is well known that rotation alone acts to favour rolls aligned with the rotation vector, and to suppress rolls of other orientations. Similarly, in a shear flow, rolls parallel to the shear flow are preferred. However, it is found that when the rotation vector and shear flow are parallel, the two effects lead counter-intuitively (as in other, analogous convection problems) to a preference for oblique rolls, and a critical Rayleigh number below that for Rayleigh–Bénard convection.When the boundaries are poorly conducting, the eigenvalue problem is solved analytically by means of an asymptotic expansion in the aspect ratio of the rolls. The behaviour of the stability problem is found to be qualitatively similar to that for fixed-temperature boundaries.Fully nonlinear numerical simulations of the convection are also carried out. These are generally consistent with the linear stability theory, showing convection in the form of rolls near the onset of motion, with the appropriate orientation. More complicated states are found further from critical.

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.

Convection in an Enclosure-Source and Sink Located Along a Single Horizontal Boundary

1978 ◽  
Vol 100 (2) ◽  
pp. 205-211 ◽  
Author(s):  
L. A. Clomburg
Keyword(s):  
Rayleigh Number ◽  
Aspect Ratio ◽  
Numerical Data ◽  
Uniform Heat Flux ◽  
Horizontal Boundary ◽  
Order Of Magnitude ◽  
High Prandtl Number ◽  
Laminar Natural Convection ◽  
Rayleigh Numbers ◽  
Source And Sink

Laminar natural convection in a two-dimensional enclosure with both source (uniform heat flux density) and sink (temperature specified) located on the top horizontal boundary is investigated numerically. Temperature and velocity profiles are presented for a high Prandtl number fluid for length Rayleigh numbers in the range 107 to 109 for length to depth ratios of 1:1 to 4:1. To generalize the results, an order of magnitude analysis is used to determine the dependence of temperature, velocity, and boundary-layer thickness scales on aspect ratio and Rayleigh number. The numerical data are well correlated using these suggested scales. The analysis shows the Nusselt number and the maximum horizontal velocity to depend on the 1/6 and 1/3 powers of the Rayleigh number, independent of aspect ratio.

Multiple Flow Patterns and Heat Transfer in Confined Jet Impingement

2002 ◽  
Author(s):  
X. Li ◽  
J. L. Gaddis ◽  
T. Wang
Keyword(s):  
Heat Transfer ◽  
Flow Field ◽  
Boundary Conditions ◽  
Jet Impingement ◽  
Flow Patterns ◽  
Computational Results ◽  
Initial Flow ◽  
Flow Parameters ◽  
Visualization Experiment ◽  
Exit Boundary

The flow field of a 2-D laminar confined impinging slot jet is investigated. Numerical results indicate that there exist two different solutions in some range of geometric and flow parameters. The two steady flow patterns are obtained under identical boundary conditions but only with different initial flow fields. Three different exit boundary conditions are investigated to eliminate artificial effects. The different flow patterns are observed to significantly affect the heat transfer. A flow visualization experiment is carried out to verify the computational results and both flow patterns are observed. The bifurcation mechanism is interpreted and discussed.

scholarly journals Nonlinear feedback in a six-dimensional Lorenz Model: impact of an additional heating term

2015 ◽  
Vol 2 (2) ◽  
pp. 475-512
Author(s):  
B.-W. Shen
Keyword(s):  
Rayleigh Number ◽  
Numerical Solutions ◽  
Critical Rayleigh Number ◽  
Critical Value ◽  
Small Scale ◽  
Solution Stability ◽  
Nonlinear Feedback ◽  
Lorenz Model ◽  
Additional Heating ◽  
The Impact

Abstract. In this study, a six-dimensional Lorenz model (6DLM) is derived, based on a recent study using a five-dimensional (5-D) Lorenz model (LM), in order to examine the impact of an additional mode and its accompanying heating term on solution stability. The new mode added to improve the representation of the steamfunction is referred to as a secondary streamfunction mode, while the two additional modes, that appear in both the 6DLM and 5DLM but not in the original LM, are referred to as secondary temperature modes. Two energy conservation relationships of the 6DLM are first derived in the dissipationless limit. The impact of three additional modes on solution stability is examined by comparing numerical solutions and ensemble Lyapunov exponents of the 6DLM and 5DLM as well as the original LM. For the onset of chaos, the critical value of the normalized Rayleigh number (rc) is determined to be 41.1. The critical value is larger than that in the 3DLM (rc ~ 24.74), but slightly smaller than the one in the 5DLM (rc ~ 42.9). A stability analysis and numerical experiments obtained using generalized LMs, with or without simplifications, suggest the following: (1) negative nonlinear feedback in association with the secondary temperature modes, as first identified using the 5DLM, plays a dominant role in providing feedback for improving the solution's stability of the 6DLM, (2) the additional heating term in association with the secondary streamfunction mode may destabilize the solution, and (3) overall feedback due to the secondary streamfunction mode is much smaller than the feedback due to the secondary temperature modes; therefore, the critical Rayleigh number of the 6DLM is comparable to that of the 5DLM. The 5DLM and 6DLM collectively suggest different roles for small-scale processes (i.e., stabilization vs. destabilization), consistent with the following statement by Lorenz (1972): If the flap of a butterfly's wings can be instrumental in generating a tornado, it can equally well be instrumental in preventing a tornado. The implications of this and previous work, as well as future work, are also discussed.

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