Modelling of the Meso-Segregations Under the Influence of Forced Convection: A Statistical Phase-Averaged Approach

Fautrelle, Yves; Lehmann, Peter; Quillet, G.; Medina, Mabel; Durand, Fabien; Terrail, Y. Du

doi:10.1007/978-94-015-9807-1_11

Cited by 3 publications

(6 citation statements)

References 6 publications

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“…The conservation equations for mass, momentum, solute and energy constitute the core of a macrosegregation model. These equations can be derived from three main approaches: volumetric averaging techniques [4], continuum model of mixture theory [5], and stochastic averaging [6]. This last method has been used for this work.…”

Section: Conservation Equations and Discretizationmentioning

confidence: 99%

“…The SUPG-PSPG formulation, presented for CFD finite element calculations by Tezduyar et al [10], is based on three stabilization coefficients: a τ SUPG coefficient (Streamline-Upwind/Petrov-Galerkin), a τ PSPG coefficient (Pressure-Stabilizing/Petrov-Galerkin) and a τ LSIC coefficient (least-squares on incompressibility constraint) as presented in Eq. (6).…”

Section: Conservation Of Mass and Momentummentioning

confidence: 99%

“…This phenomenon is particularly important in the solidification of large ingots. However, most three-dimensional simulations are currently made on small ingots or pieces of about 10 centimeters high [1,2]. This is due to the fact that the flow field is more complex and unstable in larger casting, and that a good accuracy in the prediction of macrosegregation requires a fine mesh in the mushy zone.…”

Section: Introductionmentioning

confidence: 99%

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3D macrosegregation simulation with anisotropic remeshing

Gouttebroze¹,

Bellet²,

Combeau³

2007

Comptes Rendus. Mécanique

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The article presents a three-dimensional coupled numerical solution of momentum, mass, energy and solute conservation equations, for binary alloy solidification. The interdendritic flow in the mushy zone is assumed to obey the Darcy's law. Microsegregation is governed by the lever rule, assuming local equilibrium at phase interfaces. The resulting energy and solute advection-diffusion equations are solved using the Streamline-Upwind/Petrov-Galerkin (SUPG) finite element method. A SUPG-PSPG velocity-pressure formulation is applied for the momentum equation. The full algorithm was implemented in the 3D code THERCAST, together with an anisotropic remeshing method. Two applications have been considered: a small ingot of Pb-48wt%Sn alloy and a large steel ingot. The numerical results of these two cases are presented with the evolution of temperature, liquid velocity, and solute concentration fields during solidification.

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Section: Conservation Equations and Discretizationmentioning

confidence: 99%

Section: Conservation Of Mass and Momentummentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

3D macrosegregation simulation with anisotropic remeshing

Gouttebroze¹,

Bellet²,

Combeau³

2007

Comptes Rendus. Mécanique

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“…Such an approach was suggested first by Furmanski [10]. A first preliminary set of equations has been proposed by Fautrelle et al [11] and Ciobanas et al [12] using the statistical phase averaging approach. In the present paper, we extend the idea proposed by Fautrelle et al [11] to propose a complete set of equations which describe both the columnar and the equiaxed solidification.…”

Section: Introductionmentioning

confidence: 99%

“…A first preliminary set of equations has been proposed by Fautrelle et al [11] and Ciobanas et al [12] using the statistical phase averaging approach. In the present paper, we extend the idea proposed by Fautrelle et al [11] to propose a complete set of equations which describe both the columnar and the equiaxed solidification. Moreover, combined with the so-called cell model [9] the averaging procedure can take into account the effects of inhomogeneous nature of the solute and temperature fields.…”

Section: Introductionmentioning

confidence: 99%

Ensemble averaged multiphase Eulerian model for columnar/equiaxed solidification of a binary alloy: I. The mathematical model

Ciobanas¹,

Fautrelle²

2007

J. Phys. D: Appl. Phys.

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A new multiphase Eulerian model for columnar and equiaxed dendritic solidification has been developed. The mean conservation equations are derived by means of a statistical phase averaging technique, and the mathematical formulation of the model can be used for both columnar and equiaxed solidification. The model uses three different phases, respectively, the columnar, the equiaxed solid and the liquid. The new set of equations enables us to simulate the columnar-to-equiaxed transition (CET) during the directional solidification of a binary alloy. Owing to the statistical nature of the model, we are able to treat rigorously the coexistence of equiaxed and columnar structures and consequently the CET phenomena. The averaged equations are closed by means of the cell model approximation. This technique can be successfully used to model the various interactions between the liquid and the solid. It may also incorporate the effects of the inhomogeneities of the various scalar fields, e.g. the solute and temperature gradients. An envelope model is used to parametrize the small scales, i.e. the dendrite scale. This leads us to distinguish two types of liquid, namely, the extra-dendritic and the inter-dendritic liquids. In part I the equations are rigorously derived in the purely diffusive case, whilst in part II we will present one-dimensional simulations of Sn–Pb and Al–Cu directional solidification experiments involving CET phenomena. Quasi-steady state CET maps are also computed.

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Ensemble Averaged Two-Phase Eulerian Model for Columnar/Equiaxed Solidification of a Binary Alloy

Ciobanas

Baltaretu

Fautrelle

2006

MSF

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A new two-phase eulerian model for the columnar and equiaxed dendritic solidification has been developed. The mean conservation equations are derived by means of a statistical phase averaging technique, and the mathematical formulation of the model can be used for both columnar and equiaxed solidification. We present a new set of equations to simulate the segregations during the columnar solidification of a binary alloy. The use of such an averaging procedure has several advantages compared with the volume averaging technique which has been widely used during the last decade. Firstly, the statistical approach does not need the definition of a representative elementary volume. Secondly, it is consistent with the random aspect of the equiaxed grain germination and motion as well as the possible turbulent behaviour of the fluid flow. The averaged equations are closed by means of the cell model approximation. This technique can be successfully used to model the various interactions between the liquid and the solid. It may also incorporate the effects of the inhomogeneities of the various scalar fields, e.g., the solute and temperature gradients. First tests of the model are presented in two asymptotic situations, namely the wholly equiaxed and the columnar solidification.

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Modelling of the Meso-Segregations Under the Influence of Forced Convection: A Statistical Phase-Averaged Approach

Cited by 3 publications

References 6 publications

3D macrosegregation simulation with anisotropic remeshing

3D macrosegregation simulation with anisotropic remeshing

Ensemble averaged multiphase Eulerian model for columnar/equiaxed solidification of a binary alloy: I. The mathematical model

Ensemble Averaged Two-Phase Eulerian Model for Columnar/Equiaxed Solidification of a Binary Alloy

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