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The purpose of this paper is to simulate two macrosegregation benchmarks with a newly developed stabilized finite element algorithm based on a semi-implicit pressure correction scheme.

A streamline-upwind/Petrov–Galerkin (SUPG) stabilized finite element algorithm is developed for the coupled conservation equations of mass, momentum, energy and species. A semi-implicit pressure correction method combined with SUPG stabilization technique is proposed to solve the convection flow during solidification. An analytically derived enthalpy method is adopted to solve the energy conservation equation. The nonlinearities of the energy and species equations are tackled by Newton–Raphson method. Two macrosegregation benchmarks considering the solidification of an Al-4.5 per cent Cu alloy and a Sn-10 per cent Pb alloy are simulated.

A very good agreement is achieved by comparison with the classical finite volume method and a novel meshless method for the Al-4.5 per cent Cu alloy solidification benchmark. Moreover, a unique reference numerical solution has been obtained. Besides, it is demonstrated that the stabilized finite element algorithm can capture the flow instability and channel segregation during solidification of the Sn-10 per cent Pb alloy.

A semi-implicit pressure correction method combined with SUPG stabilization technique is adopted to develop robust stabilized finite element algorithm for the macrosegregation model. A new enthalpy formulation for heat transfer problems with phase change is derived analytically.

The purpose of this paper is to simulate a macrosegregation solidification benchmark by a meshless diffuse approximate method. The benchmark represents solidification of Al 4.5 wt per cent Cu alloy in a 2D rectangular cavity, cooled at vertical boundaries.

A coupled set of mass, momentum, energy and species equations for columnar solidification is considered. The phase fractions are determined from the lever solidification rule. The meshless diffuse approximate method is structured by weighted least squares method with the second-order monomials for trial functions and Gaussian weight functions. The spatial localization is made by overlapping 13-point subdomains. The time-stepping is performed in an explicit way. The pressure-velocity coupling is performed by the fractional step method. The convection stability is achieved by upstream displacement of the weight function and the evaluation point of the convective operators.

The results show a very good agreement with the classical finite volume method and the meshless local radial basis function collocation method. The simulations are performed on uniform and non-uniform node arrangements and it is shown that the effect of non-uniformity of the node distribution on the final segregation pattern is almost negligible.

The application of the meshless diffuse approximate method to simulation of macrosegregation is performed for the first time. An adaptive upwind scheme is successfully applied to the diffuse approximate method for the first time.

The purpose of this paper is to reveal the solute segregation behavior in the molten and solidified regions during direct chill (DC) casting of a large-size magnesium alloy slab under no magnetic field (NMF), harmonic magnetic field (HMF), pulsed magnetic field (PMF) and two types of out-of-phase pulsed magnetic field (OPMF).

A 3-D multiphysical coupling mathematical model is used to evaluate the corresponding physical fields. The coupling issue is addressed using the method of separating step and result inheritance.

The results suggest that the solute deficiency tends to occur in the central part, while the solute-enriched area appears near the fillet in the molten and solidified regions. Applying magnetic field could greatly homogenize the solute field in the two-phase region. The variance of relative segregation level in the solidified cross-section under NMF is 38.1%, while it is 21.9%, 18.6%, 16.4% and 12.4% under OPMF2 (the current phase in the upper coil is ahead of the lower coil), HMF, PMF and OPMF1 (the current phase in the upper coil lags behind the lower coil), respectively, indicating that OPMF1 is more effective to reduce the macrosegregation level.

There are few reports on the solute segregation degree in rectangle slab under magnetic field, especially for magnesium alloy slab. This paper can act a reference to make clear the solute transport behavior and help reduce the macrosegregation level during DC casting.

When a multi component alloy solidifies the redistribution of solute components leads to the formation of macrosegregation patterns. Blending ideas from a number of recent publications the purpose of this paper is to provide a “best practice” on how grid convergence of a given macrosegregation simulation can be measured and determined.

The best practice is arrived at by considering a benchmark problem consisting of a 2D-casting simulation of an idealized Al-4.5%Cu alloy in a side cooled square (76×76 mm) cavity. The model for this simulation is based on a mixture treatment of the relevant heat and mass transfer equations. Simulations are made using three increasingly refined grid sizes.

The best practice to determine grid resolution involves two steps: first, a visual evaluation of predicted segregation images leading to the evaluation of solute profiles along selected transects; and second, the construction of a cumulative distribution function (CDF) of the predicted segregation field. On application to the benchmark problem, it is concluded that current computer resources are insufficient to grid resolve macrosegregation patterns but that the CDF provides a useful signal of the nature of macrosegregation in a given system.

The benchmark is chosen to be representative. Exact convergence behavior, however, may depend on the system chosen.

In addition to establishing a best practice for measuring grid resolution of macrosegregation simulations the work also highlights, even in the absence of complete grid convergence, how the recently proposed CDF treatment can inform solidification modeling and process understanding.

The purpose of this paper is to present the solution of a highly nonlinear fluid dynamics in a low Prandtl number regime, typical for metal‐like materials, as defined in the call for contributions to a numerical benchmark problem for 2D columnar solidification of binary alloys. The solution of such a numerical situation represents the first step towards understanding the instabilities in a more complex case of macrosegregation.

The involved temperature, velocity and pressure fields are represented through the local approximation functions which are used to evaluate the partial differential operators. The temporal discretization is performed through explicit time stepping.

The performance of the method is assessed on the natural convection in a closed rectangular cavity filled with a low Prandtl fluid. Two cases are considered, one with steady state and another with oscillatory solution. It is shown that the proposed solution procedure, despite its simplicity, provides stable and convergent results with excellent computational performance. The results show good agreement with the results of the classical finite volume method and spectral finite element method.

The solution procedure is formulated completely through local computational operations. Besides local numerical method, the pressure velocity is performed locally also, retaining the correct temporal transient.

A single-domain multi-phase model is developed for macrosegregation and shrinkage pipe formation in castings, as functions of buoyancy- and shrinkage-induced flow. The paper aims to discuss these issues.

Using a volume of fluid (VOF) method, both the air/liquid and air/solid interfaces are tracked during shrinkage pipe formation. A set of mixture advection-diffusion equations are derived and solved for velocity, temperature, composition, and phase field evolution. The fluid mechanics of the model are verified using a transient ditch drainage problem.

Results showing the interaction of macrosegregation and pipe formation are presented for two alloys under faster and slower cooling conditions.

This model provides a comprehensive tool to investigate relationships between the developing composition distribution and shrinkage pipe formation.

This paper aims to point out the critical problems in numerical verification of solidification simulation codes and the complexity of the verification and to propose and apply a procedure of generalized verification for macrosegregation simulation.

A partial verification of a finite‐volume computational model of macrosegregation in direct chill (DC) casting of binary aluminum alloys, including the coupled transport phenomena of heat transfer, fluid flow and species transport, is performed. The verification procedure is conducted on numerical test problems, defined as subproblems with respect to the complexity of the physical model, geometry, and boundary conditions. The studied cases are thermal convection with solidification in DC casting, thermal natural convection of a low‐Prandtl‐number liquid metal in a rectangular cavity and 1D directional solidification of a binary Al‐Cu alloy. Grid‐convergence studies, code comparison with an alternative Chebyshev‐collocation method, and comparison with a reference similarity solution are used for verification.

An excellent ability of the model to accurately resolve the thermal convection in the pertinent range of Prandtl and Rayleigh numbers is shown. Concerns regarding the solution of species transport in the mushy zone remain.

The proposed verification procedure is not completed in its entirety. Further verification of the solutal and thermosolutal convection problems is required.

This paper proposes verification techniques for complex coupled solidification problems involving significant convection in the melt.

The purpose of this paper is to investigate the feasibility of solving turbulent flows based on smoothed finite element method (S-FEM). Then, the differences between S-FEM and finite element method (FEM) in dealing with turbulent flows are compared.

The stabilization scheme, the streamline-upwind/Petrov-Galerkin stabilization is coupled with stabilized pressure gradient projection in the fractional step framework. The Reynolds-averaged Navier-Stokes equations with standard k-epsilon model are selected to solve turbulent flows based on S-FEM and FEM. Standard wall functions are applied to predict boundary layer profiles.

This paper explores a completely new application of S-FEM on turbulent flows. The adopted stabilization scheme presents a good performance on stabilizing the flows, especially for very high Reynolds numbers flows. An advantage of S-FEM is found in applying wall functions comparing with FEM. The differences between S-FEM and FEM have been investigated.

The research in this work is limited to the two-dimensional incompressible turbulent flow.

The verification and validation of a new combination are conducted by several numerical examples. The new combination could be used to deal with more complicated turbulent flows.

The applications of the new combination to study basic and complex turbulent flow are also presented, which demonstrates its potential to solve more turbulent flows in nature and engineering.

This work carries out a great extension of S-FEM in simulations of fluid dynamics. The new combination is verified to be very effective in handling turbulent flows. The performances of S-FEM and FEM on turbulent flows were analyzed by several numerical examples. Superior results were found compared with existing results and experiments. Meanwhile, S-FEM has an advantage of accuracy in predicting boundary layer profile.

This paper aims to tackle the problem of thermo‐solutal convection and macrosegregation during ingot solidification of metal alloys. Complex flow structures associated with the development of channels segregate and sharp gradients in the solutal field call for the implementation of accurate methods for numerical modeling of alloy solidification. In particular, the solute transport equation is convection dominated and requires special non‐oscillarity type high‐order schemes to handle the regions of channels segregates.

In the present study, a time‐splitting approach has been adopted to separately handle solute advection and diffusion. This splitting technique allows the application of accurate total variation dimensioning (TVD) schemes for solution of solute advection. Applications of second‐order Lax‐Wendroff TVD SUPERBEE and fifth‐order weighted essentially non‐oscillatory (WENO) schemes are described in the present article. Classical numerical solution of solute transport using hybrid and central‐difference schemes are also employed for the purpose of comparisons. Numerical simulations for solidification of Pb‐18%Sn in a two‐dimensional rectangular cavity have been carried out using different numerical schemes.

Numerical results show the difficulty of obtaining grid‐independent solutions with respect to local details in the region of channels. Grid convergence patterns and numerical uncertainty are found to be dependent on the applied scheme. In general, the first‐order hybrid scheme is diffusive and under predicts the formation of channels. The second‐order central‐difference scheme brings about oscillations with possible non‐physical extremes of solute composition in the region of channel segregates due to sharp gradients in the solutal field. The results obtained using TVD and WENO schemes contain no oscillations and show an excellent capture of channels formation and resolution of the interface between solute‐rich and depleted bands. Different stages of channels formation are followed by analyzing thermo‐solutal convection and macrosegregation at different times during solidification.

Accurate prediction of local variation in the solutal and flow fields in the channels regions requires grid refinement up to scales in the order of microscopic dendrite arm spacing. This imposes limitations in terms of large computational time and applicability of available macroscopic models based on classical volume‐averaging techniques.

The present study is very useful for numerical simulation of macrosegregation during ingot casting of metal alloys.

The paper provides the methodology and application of TVD schemes to predict channel segregates during columnar solidification of metal alloys. It also demonstrates the limitations of classical schemes for simulation of alloy solidification.

This study aims to simulate the dendritic growth in Stokes flow by iteratively coupling a domain and boundary type meshless method.

A preconditioned phase-field model for dendritic solidification of a pure supercooled melt is solved by the strong-form space-time adaptive approach based on dynamic quadtree domain decomposition. The domain-type space discretisation relies on monomial augmented polyharmonic splines interpolation. The forward Euler scheme is used for time evolution. The boundary-type meshless method solves the Stokes flow around the dendrite based on the collocation of the moving and fixed flow boundaries with the regularised Stokes flow fundamental solution. Both approaches are iteratively coupled at the moving solid–liquid interface. The solution procedure ensures computationally efficient and accurate calculations. The novel approach is numerically implemented for a 2D case.

The solution procedure reflects the advantages of both meshless methods. Domain one is not sensitive to the dendrite orientation and boundary one reduces the dimensionality of the flow field solution. The procedure results agree well with the reference results obtained by the classical numerical methods. Directions for selecting the appropriate free parameters which yield the highest accuracy and computational efficiency are presented.

A combination of boundary- and domain-type meshless methods is used to simulate dendritic solidification with the influence of fluid flow efficiently.