Convection lines on a heated bottom plate at large Rayleigh number

1987 ◽  
Vol 92 (C5) ◽  
pp. 5489
Author(s):  
Nobuyuki Tamai ◽  
Takashi Asaeda
Keyword(s):  
Rayleigh Number ◽  
Bottom Plate ◽  
Large Rayleigh Number

Convection fluid dynamics in a model of a fault zone in the earth's crust

1978 ◽  
Vol 88 (4) ◽  
pp. 769-792 ◽  
Author(s):  
D. R. Kassoy ◽  
A. Zebib
Keyword(s):  
Fluid Dynamics ◽  
Rayleigh Number ◽  
Porous Material ◽  
Fault Zone ◽  
Slug Flow ◽  
Tectonic Activity ◽  
Two Dimensional ◽  
Core Flow ◽  
Number Flow ◽  
Large Rayleigh Number

Faulted regions associated with geothermal areas are assumed to be composed of rock which has been heavily fractured within the fault zone by continuous tectonic activity. The fractured zone is modelled as a vertical, slender, two-dimensional channel of saturated porous material with impermeable walls on which the temperature increases linearly with depth. The development of an isothermal slug flow entering the fault at a large depth is examined. An entry solution and the subsequent approach to the fully developed configuration are obtained for large Rayleigh number flow. The former is characterized by growing thermal boundary layers adjacent to the walls and a slightly accelerated isothermal core flow. Further downstream the development is described by a parabolic system. It is shown that a class of fully developed solutions is not spatially stable.

Convection by a horizontal thermal gradient

2007 ◽  
Vol 586 ◽  
pp. 41-57 ◽  
Author(s):  
H. J. J. GRAMBERG ◽  
P. D. HOWELL ◽  
J. R. OCKENDON
Keyword(s):  
Boundary Layer ◽  
Rayleigh Number ◽  
Thermal Gradient ◽  
Melting Process ◽  
Uniform Heating ◽  
Boundary Layer Analysis ◽  
Layer Analysis ◽  
Convection Problem ◽  
Large Rayleigh Number ◽  
Batch Melting

This paper considers a paradigm large-Prandtl-number, large-Rayleigh-number forced convection problem suggested by the batch melting process in the glass industry. Although the fluid is heated from above, non-uniform heating in the horizontal direction induces thermal boundary layers in which colder liquid is driven over hotter liquid. This leads to an interesting selection problem in the boundary layer analysis, whose resolution is suggested by a combination of analytical and numerical evidence.

Stability of the flow in a differentially heated inclined box

1971 ◽  
Vol 47 (3) ◽  
pp. 547-576 ◽  
Author(s):  
John E. Hart
Keyword(s):  
Rayleigh Number ◽  
Travelling Waves ◽  
Bottom Plate ◽  
List Type ◽  
Mean Flow ◽  
Small Diffusion ◽  
Longitudinal Modes ◽  
Vertical Slot ◽  
The Mean ◽  
Rayleigh Numbers

The effect of sloping boundaries on thermal convection is studied theoretically and in the laboratory in the context of a model in which fluid is contained in a differentially heated rectangular box of small aspect ratio (depth/length), inclined at an angle δ to the vertical. Like its two limiting cases, Bénard convection and convection in the vertical slot, a basic state which exists for low Rayleigh numbers becomes unstable as this parameter is increased. The types of instability and indeed the manner in which the motions become turbulent depend crucially on δ. In our work with water the following general picture of the primary instabilities applies: For 90° > δ > 10° with the bottom plate hotter, the instabilities are stationary longitudinal convectively driven rolls with axes oriented up the slope. Near δ = 10° there is an upper and lower Rayleigh number cut off. If the Rayleigh number is too small diffusion damps the instabilities, but if it is too large they are damped by the development of a stable upslope temperature gradient in the mean flow.For 10° > δ > −10° (negative angles imply a hotter upper plate), transverse travelling waves oriented across the slope are the first instabilities of the mean flow. They obtain their kinetic energy via the working of the upslope buoyancy force.For - 10° > δ > −85° longitudinal modes are again observed. These are rather curious in that they may exist when the stratification $-\hat{g}\cdot\nabla T $ is everywhere positive. The necessary energy for these modes comes out of the mean velocity field and out of the mean available potential energy.Agreement between the stability theory and the experiments is generally quite good over the whole range of δ, considering the approximations involved in finding a suitable basic flow solution.For Rayleigh numbers less than ∼ 106 turbulence is only possible for positive angles. For 85° > δ > 20° the development of unsteadiness involves the occurrence and the breaking of wavy longitudinal vortices in a manner reminiscent of the development of turbulence in cylindrical Couette flow.

The effect of rotation on vertical Bridgman growth at large Rayleigh number

2000 ◽  
Vol 409 ◽  
pp. 185-221 ◽  
Author(s):  
M. R. FOSTER
Keyword(s):  
Rayleigh Number ◽  
Binary Alloy ◽  
Biot Number ◽  
Vertical Axis ◽  
Taylor Number ◽  
Magnetic Forces ◽  
Radial Segregation ◽  
Vertical Bridgman ◽  
Thermal Layer ◽  
Large Rayleigh Number

Convection effects in the melt of a vertical Bridgman furnace, used for solidifying a dilute binary alloy, are known to cause significant, and undesirable, non-uniformity in the alloy. We have found previously that non-axisymmetry significantly degrades the performance of the furnace at large Rayleigh number, Ra, and small Biot number, [Bscr ]. There have been a number of studies on improvement of the alloy quality by the introduction of additional forces into the melt flow – magnetic forces or d'Alembert forces due to various sorts of acceleration of the ampoule. In this paper, we explore the effects on the radial segregation generated by rotating the ampoule about its vertical axis. We determine that the magnitude of segregation is proportional to the product of [Bscr ] and the thickness of the thermal layer on the crystal–melt interface. As the rotation, as measured by a Taylor number, [Tscr ], increases beyond O(Ra1/3), the thermal layer thickens and so the segregation increases. Finally, at [Tscr ] = O(Ra1/2), the thermal adjustment occurs on outer scales, and hence the solutal concentration increases to O([Bscr ]). Hence rotation about the vertical axis actually degrades performance!

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