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!

Nonlinear Bénard convection with rotation

1973 ◽  
Vol 57 (3) ◽  
pp. 433-458 ◽  
Author(s):  
J. C. Morgan
Keyword(s):  
Rayleigh Number ◽  
Prandtl Number ◽  
Vertical Direction ◽  
Ekman Layer ◽  
Taylor Number ◽  
Eighth Order ◽  
Bénard Convection ◽  
Benard Convection ◽  
Thermal Layer ◽  
Rotational Problem

The equations for nonlinear Bénard convection with rotation for a layer of fluid, thickness d, are derived using the Glansdorff & Prigogine (1964) evolutionary criterion as used by Roberts (1966) in his paper on non-rotational Bénard convection. The parameters of the problem in this case are the Rayleigh number R = αgΔθd/vK, the Taylor number T = 4d4Ω23/v2 and the Prandtl number Pr = v/K, where α is the coefficient of volume expansion, g the acceleration due to gravity, Δθ the temperature difference between the horizontal surfaces, v the kinematic viscosity, K the thermal diffusivity and Ω3 the rotation rate about the vertical direction. The asymptotic solution for two-dimensional cells (rolls) is investigated for large Rayleigh numbers and large Taylor numbers. For rolls the convection equations are found t o be independent of the Prandtl number. However, the solutions depend upon the Prandtl number for another reason. The rotational problem differs from the non-rotational one in that the Rayleigh number and the horizontal wavenumber a of the convection are now functions of the Taylor number. These are taken to be R ∼ ρTα′ and α ∼ ATβ, where α′ and β are positive numbers. Thermal layers develop as R becomes large with ρ or T becoming large. The order in which ρ and T are allowed to increase is important since the horizontal wavenumber a also increases with T and the convection equations can be reduced in this case. A liquid of large Prandtl number such as water has v [Gt ] K. Since R ∼ O (1/vK) and T ∼ O(1/v2), ρ will be greater than T for a given (large) Δθ and Ω3. Similarly, for a liquid of small Prandtl number such as mercury v [Lt ] K, and T is greater than ρ for a given Δθ and Ω3. For rigid-rigid horizontal boundaries with ρ large and then T large the ρ thermal layer has the same structure as for the non-rotating problem. As T → ∞ three types of thermal layers are possible: a linear Ekman layer, a nonlinear Ekman layer and a Blasius-type thermal layer. When the horizontal boundaries are both free the ρ thermal layer is again of the same structure as for non-rotating BBnard convection. As T → ∞ a nonlinear Ekman layer and a Blasius-type thermal layer are possible.When T is large and then ρ made large the differential equations governing the convection are reduced from eighth order to sixth order owing to a becoming large as T → ∞. There are Ekman layers as T → ∞, when the horizontal boundaries are both rigid. The ρ thermal layers now have a different structure from the non-rotating problem for both rigid-rigid and free-free horizontal boundaries. The equation for small amplitude convection near to the marginal case is derived and the solution for free-free horizontal boundaries is obtained.

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.

Stability of Gravity Driven Convection in a Cylindrical Porous Layer Subjected to Vibration

2006 ◽  
Author(s):  
Saneshan Govender
Keyword(s):  
Porous Medium ◽  
Rayleigh Number ◽  
Linear Stability ◽  
Density Gradient ◽  
Porous Layer ◽  
Binary Alloy ◽  
Critical Rayleigh Number ◽  
Linear Stability Theory ◽  
Direct Result ◽  
Porous Cylinder

In both pure fluids and porous media, the density gradient becomes unstable and fluid motion (convection) occurs when the critical Rayleigh number is exceeded. The classical stability analysis no longer applies if the Rayleigh number is time dependant, as found in systems where the density gradient is subjected to vibration. The influence of vibrations on thermal convection depends on the orientation of the time dependant acceleration with respect to the thermal stratification. The problem of a vibrating porous cylinder has numerous important engineering applications, the most important one being in the field of binary alloy solidification. In particular we may extend the above results to understanding the dynamics in the mushy layer (essentially a reactive porous medium) that is sandwiched between the underlying solid and overlying melt regions. Alloyed components are widely used in demanding and critical applications, such as turbine blades, and a consistent internal structure is paramount to the performance and integrity of the component. Alloys are susceptible to the formation of vertical channels which are a direct result of the presence convection, so any technique that suppresses convection/the formation of channels would be welcomed by the plant metallurgical engineer. In the current study, the linear stability theory is used to investigate analytically the effects of gravity modulation on convection in a homogeneous cylindrical porous layer heated from below. The linear stability results show that increasing the frequency of vibration stabilizes the convection. In addition the aspect ratio of the porous cylinder is shown to influence the stability of convection for all frequencies analysed. It was also observed that only synchronous solutions are possible in cylindrical porous layers, with no transition to sub harmonic solutions as was the case in Govender (2005a) for rectangular layers or cavities. The results of the current analysis will be used in the formulation of a model for binary alloy systems that includes the reactive porous medium model.

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.

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