Purpose
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.
Design/methodology/approach
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.
Findings
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.
Originality/value
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 main objective of the present paper is to define a new benchmark test for macrosegregation in axisymmetry and to verify a novel meshless method on it. The test case represents a solidification of Al4.5wt%Cu alloy in two different types of geometries, a solid and a hollow cylinder, cooled at the vertical boundaries. The volume averaging method is used to formulate the coupled mass, energy, momentum, and species transport equations for solid-liquid flow. The lever rule is used for determination of liquid and solid fraction. The meshless numerical approach, verified in this paper, is called the diffuse approximate method. The method is formed by using the weighted least squares approximation, where the second-order polynomial basis and Gaussians are used as trial and weight functions, respectively. The method is localised with the use of subdomains, each containing thirteen computational nodes. The explicit Euler scheme is used to perform the temporal integration. The fractional step method is used to couple the pressurevelocity fields. The stability of the method is attained by an adaptive shift of the computational node and Gaussian weight in the upstream direction. Results are presented for three geometrically different simulations. The results are compared with the classical finite volume method. All results show a very good agreement with the finite volume method. The simulations
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