A Nonlinear Elimination Preconditioner for Fully Coupled Space-Time Solution Algorithm with Applications to High-Rayleigh Number Thermal Convective Flow Problems

Natural-convection flow in the presence of a magnetic field in an enclosure heated from bottom and cooled from top is considered. The fluid (molten sodium) properties are function of temperature. To solve the governing non-linear differential equations (mass, momentum and energy) a finite volume code based on PATANKAR’s SIMPLER method is utilized. The results for different Rayleigh and Hartmann numbers show that the strength of the magnetic field has significant effects on the flow and temperature fields. The convection becomes stronger as the Rayleigh number increases while the magnetic field suppresses the convective flow and the heat transfer rate. When the magnetic field is weak and the Rayleigh number is high, the convection is dominant.

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Natural convection in a mushy layer

Journal of Fluid Mechanics ◽

10.1017/s0022112091001787 ◽

1991 ◽

Vol 224 ◽

pp. 335-359 ◽

Cited By ~ 128

Author(s):

M. Grae Worster

Keyword(s):

Boundary Layer ◽

Natural Convection ◽

Rayleigh Number ◽

Solid Fraction ◽

Convective Flow ◽

Mushy Layer ◽

Governing Equations ◽

Downward Flow ◽

Asymptotic Regime

Governing equations for a mushy layer are analysed in the asymptotic regime Rm [Gt ] 1, where Rm is an appropriately defined Rayleigh number. A model is proposed in which there is downward flow everywhere in the mushy layer except in and near localized chimneys, which are characterized by having zero solid fraction. Upward, convective flow within the chimneys is driven by compositional buoyancy. The radius of each chimney is determined locally by thermal balances within a boundary layer that surrounds it. Simple solutions are derived to determine the structure of the mushy layer away from the immediate vicinity of chimneys in order to demonstrate the gross effects of convection upon the solidification within the layer.

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A two-mass expanding exact space-time solution

General Relativity and Gravitation ◽

10.1007/s10714-010-1081-6 ◽

2010 ◽

Vol 43 (1) ◽

pp. 191-205 ◽

Cited By ~ 26

Author(s):

Jean-Philippe Uzan ◽

George F. R. Ellis ◽

Julien Larena

Keyword(s):

Space Time ◽

Time Solution

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Time-Dependent Double-Diffusive Instability in a Density-Stratified Fluid along a Heated Inclined Wall

Journal of Heat Transfer ◽

10.1115/1.3450872 ◽

1978 ◽

Vol 100 (4) ◽

pp. 653-658 ◽

Cited By ~ 4

Author(s):

C. F. Chen

Keyword(s):

Rayleigh Number ◽

Convective Flow ◽

Stratified Fluid ◽

Horizontal Layer ◽

Critical Values ◽

Time Dependent ◽

Free Layer ◽

Heating Wall ◽

Vertical Walls ◽

Double Diffusive

We consider a stably stratified fluid contained between two parallel sloping plates. At t = 0, the lower plate is given a step increase in temperature; a time-dependent convective flow is generated. Stability of such a flow with respect to double-diffusive mechanism is analyzed. The method is the same one used by Chen [6] in treating a similar problem with vertical walls. The predicted critical values of Rayleigh number and wavelength compare favorably with those observed experimentally. No overstability is encountered up to 75 deg of inclination of the heating wall to the vertical. For a horizontal layer, however, instability starts in an overstable mode. The frequency of the overstable mode compares favorably with that predicted by Veronis [7] for a free-free layer.

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The method of space-time conservation element and solution element - Applications to one-dimensional and two-dimensional time-marching flow problems

10.2514/6.1995-1754 ◽

1995 ◽

Cited By ~ 20

Author(s):

Sin-Chung Chang ◽

Xiao-Yen Wang ◽

Chuen-Yen Chow

Keyword(s):

Space Time ◽

Two Dimensional ◽

One Dimensional ◽

Flow Problems ◽

Time Marching ◽

Solution Element

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Natural Convective Flow in a Three Enclosure System Involving Two High Aspect Ratio Side Enclosures Joined to a Large Square Enclosure With Interfacial Heat Generation

Heat Transfer: Volume 1 ◽

10.1115/ht2003-47115 ◽

2003 ◽

Cited By ~ 1

Author(s):

Patrick H. Oosthuizen

Keyword(s):

Heat Transfer ◽

Nusselt Number ◽

Rayleigh Number ◽

Aspect Ratio ◽

Heat Generation ◽

High Aspect Ratio ◽

Convective Flow ◽

Square Enclosure ◽

Number Variation ◽

Resistance To Heat

A numerical study of free convective flow in a vertical joined three enclosure arrangement has been undertaken. In this arrangement, a vertical heated wall kept at a uniform high temperature is contained in a high aspect ratio rectangular side enclosure. This enclosure is joined to a second high aspect ratio rectangular side enclosure which has the same height as the first side enclosure, the two enclosures being separated by a vertical impermeable dividing wall which offers no resistance to heat transfer. The second side enclosure is joined to a larger square enclosure, the vertical dividing wall between these two enclosures also being impermeable and offering no resistance to heat transfer. The vertical wall of the square main flow enclosure opposite to the dividing wall is maintained at a uniform lower temperature. There is a uniform rate of heat generation in the dividing wall between the inner side enclosure and the main enclosure. The situation considered is an approximate model of a double-paned window exposed to a hot outside environment and covered by a plane blind which in turn is exposed to cooled room. In some such cases there can be significant heat generation in the blind due to the absorbtion of solar energy, this being modeled by the heat generation in the one dividing wall. The flow has been assumed to be laminar and two-dimensional and results have been obtained for a Prandtl number of 0.7. The effects of Rayleigh number, dimensionless width of the side enclosures and dimensionless heat generation rate in the blind on the Nusselt number have been investigated. The results show that for a fixed Rayleigh number and for a given dimensionless first (i.e., outer) side enclosure width, there is a minimum in the Nusselt number variation with the dimensionless width of the second side enclosure. An approximate solution for the Nusselt number variation with the dimensionless width of the second side enclosure for small values of this dimensionless width has also been derived.

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Free Convective Flow in a Two Enclosure Arrangement With a High Aspect Ratio Side Enclosure Joined to a Large Square Enclosure

10.1115/imece2000-1411 ◽

2000 ◽

Author(s):

Patrick H. Oosthuizen

Keyword(s):

Nusselt Number ◽

Rayleigh Number ◽

Aspect Ratio ◽

High Aspect Ratio ◽

Convective Flow ◽

Numerical Study ◽

Vertical Wall ◽

Heated Wall ◽

Free Convective Flow ◽

Square Enclosure

Abstract A numerical study of free convective flow in a vertical joined two-enclosure arrangement has been undertaken. In this arrangement, a vertical heated wall kept at a uniform high temperature is contained in a high aspect ratio rectangular side enclosure. This enclosure is separated from a larger square enclosure by a vertical dividing wall which is impermeable but offers no resistance to heat transfer. The vertical wall of the main flow enclosure opposite to the dividing wall is maintained at a uniform lower temperature. All remaining walls in both enclosures are adiabatic. The situation considered is an approximate model of a window exposed to a hot outside environment and covered by a plane blind which in turn is exposed to cooled room. The flow has been assumed to be laminar and two-dimensional and results have been obtained for a Prandtl number of 0.7. The effects of Rayleigh number and the dimensionless width of the side enclosure on the Nusselt number have been investigated. The results show that there is a minimum in the Nusselt number variation with side enclosure width for a fixed Rayleigh number. The effect of Rayleigh number on the conditions under which this minimum occurs and on the value of the minimum Nusselt number has been investigated.

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