Nonlinear analysis of buoyant convection in binary solidification with application to channel formation

Amberg, Gustav; Homsy, G. M.

doi:10.1017/s0022112093003672

Cited by 129 publications

(244 citation statements)

References 12 publications

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“…Equation (1d) is based on the limit of sufficiently large value of the Lewis number k/k s , where k s is the solute diffusivity. Following earlier work (Amberg & Homsy 1993;Anderson & Worster 1995), we assume the following rescaling in the limit of sufficiently small δ:…”

Section: Formulationmentioning

confidence: 99%

Nonlinear oscillatory convection in rotating mushy layers

Riahi

2006

J. Fluid Mech.

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We consider the problem of nonlinear oscillatory convection in a horizontal mushy layer rotating about a vertical axis. Under a near-eutectic approximation and the limit of large far-field temperature, we determine the stable and unstable oscillatory solutions of the weakly nonlinear problem by using perturbation and stability analyses. It was found that depending on the values of the parameters, supercritical simple travelling modes of convection in the form of hexagons, squares, rectangles or rolls can become stable and preferred, provided the value of the rotation parameter τ is not too small and is below some value, which can depend on the other parameter values. Each supercritical form of the oscillatory convection becomes subcritical as τ increases beyond some value, and each subcritical form of the oscillatory convection is unstable. In contrast to the non-rotating case, qualitative properties of the lefttravelling modes of convection are different from those of the right-travelling modes, and such qualitative difference is found to be due to the interactions between the local solid fraction and the Coriolis term in the momentum-Darcy equation. IntroductionRiahi (2003) extended the steady problem of convection in rotating mushy layers treated by Guba (2001) by following Anderson & Worster (1995) in assuming a much wider range ε δ for the amplitude of convection, taking into account the interactions between the local solid fraction and the convection associated with the Coriolis term in the momentum-Darcy equation and carrying out stability analysis of the finiteamplitude steady solutions. It was found, in particular that over most of the range of the parameter values, subcritical down-hexagons with down-flow at the cell centres and up-flow at the cell boundaries can be preferred over up-hexagons, where flow is upward at the cell centres and downward at the cell boundaries.Riahi (2002) extended the linear oscillatory problem of convection in mushy layers and in the absence of rotation due to Anderson & Worster (1996) by employing weakly nonlinear and stability analyses to determine the stable finite-amplitude oscillatory solutions. He found, in particular, that depending on the values of the parameters, only supercritical simple travelling modes of convection in the form of either righttravelling rolls (where the phase velocity of the rolls is in the direction of the component of the position vector along the wavenumber vector) or left-travelling rolls, (where the phase velocity of the rolls is in the direction opposite to that of the component of the position vector along the wavenumber vector) or supercritical standing rolls can be stable. The weakly nonlinear and stability properties of the righttravelling mode were found to be the same as those of the left-travelling mode. The author is grateful to the editor and two referees for pointing out an error due to the

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Section: Formulationmentioning

confidence: 99%

Nonlinear oscillatory convection in rotating mushy layers

Riahi

2006

J. Fluid Mech.

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“…Amberg and Homsy (1993) restricted their analysis to the case where the non-scaled original Stefan number S t =L/(C L ∆T) is an order one quantity and assumed that δ is of the same order as ε, and they computed stationary solutions in the form of hexagons and rolls only. Their hexagonal solution was determined to order ε since R 10 was non-zero to this order.…”

Section: Conclusion and Remarksmentioning

confidence: 99%

“…We consider the solidification system in a moving frame of reference ox y z , whose origin lies on the solidification front and translating at the speed V 0 with the solidification front in the positive z -direction. The reader is referred to part 1 of the present problem (Riahi2002) for the motivation and justification in using the Amberg and Homsy (1993) type of model for the present study.…”

Section: Introduction and Governing Systemmentioning

confidence: 99%

On nonlinear convection in mushy layers. Part 2. Mixed oscillatory and stationary modes of convection

Riahi

2004

J. Fluid Mech.

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“…Much work has been devoted to clarify the mechanisms underlying such convective instabilities. Fundamental investigations on convective instabilities during directional solidification of binary alloys [1][2][3][4][5] are complemented by composition dependent models [6], numerical approaches [7], experiments with forced convection/high gravity [8][9][10] and by numerous observations and results for superalloys and steels in view of their technical importance [11][12][13][14][15][16][17][18][19][20].…”

Section: Introductionmentioning

confidence: 99%

Development and application of a new freckle criterion for technical remelting processes

Böttger

Schmitz

Wahlers

et al. 2014

MATEC Web of Conferences

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Abstract. In technical solidification processes like Electro Slag (ESR) or Vacuum Arc Remelting (VAR), freckles present a serious type of defects which limit the maximum ingot size for many grades of steels and superalloys. Therefore, modelling of freckle formation is an important task for optimizing industrial remelting processes. Practically all present freckle models are based on a critical Rayleigh number. They are inspired by the "classical" assumption that freckles are caused by an inversion of the liquid density in the semisolid region. Plumes of the lighter segregated liquid evolve through perturbation of the metastable layering of the melt in the mushy zone, a mechanism which motivates its description via Rayleigh criteria. But these models are not suitable for materials like Alloy 718 which do not show a liquid density inversion, but nevertheless are prone to freckle formation in technical remelting processes. In this paper, a criterion is developed which -instead of using a Rayleigh number -is based on the evaluation of the non-isothermal component of an instantaneous down-hill flow of heavy segregated melt in a melt pool with axial symmetry. With knowledge of the exact pool geometry and the shape and properties of the mushy zone, the occurrence of freckles can be predicted. The model is applied to a technical ESR casting for which temperature fields and microstructural parameters have been obtained using CFD and 3D phase-field modelling, respectively. Furthermore, the implications are discussed which the new model offers for the understanding of freckles in technical remelting processes.

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Nonlinear analysis of buoyant convection in binary solidification with application to channel formation

Cited by 129 publications

References 12 publications

Nonlinear oscillatory convection in rotating mushy layers

Nonlinear oscillatory convection in rotating mushy layers

On nonlinear convection in mushy layers. Part 2. Mixed oscillatory and stationary modes of convection

Development and application of a new freckle criterion for technical remelting processes

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