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The purpose of this paper is to develop a new local radial basis function collocation method (LRBFCM) for one‐domain solving of the non‐linear convection‐diffusion equation, as it appears in mixture continuum formulation of the energy transport in solid‐liquid phase change systems.

The method is structured on multiquadrics radial basis functions. The collocation is made locally over a set of overlapping domains of influence and the time stepping is performed in an explicit way. Only small systems of linear equations with the dimension of the number of nodes in the domain of influence have to be solved for each node. The method does not require polygonisation (meshing). The solution is found only on a set of nodes.

The computational effort grows roughly linearly with the number of the nodes. Results are compared with the existing steady analytical solutions for one‐dimensional convective‐diffusive problem with and without phase change. Regular and randomly displaced node arrangements have been employed. The solution is compared with the results of the classical finite volume method. Excellent agreement with analytical solution and reference numerical method has been found.

A realistic two‐dimensional non‐linear industrial test associated with direct‐chill, continuously cast aluminium alloy slab is presented.

A new meshless method is presented which is simple, efficient, accurate, and applicable in industrial convective‐diffusive solid‐liquid phase‐change problems with non‐linear material properties.

The axisymmetric steady‐state convective‐diffusive thermal field problem associated with direct‐chill, semi‐continuously cast billets has been solved using the dual reciprocity boundary element method. The solution is based on a formulation which incorporates the one‐phase physical model, Laplace equation fundamental solution weighting, and scaled augmented thin plate splines for transforming the domain integrals into a finite series of boundary integrals. Realistic non‐linear boundary conditions and temperature variation of all material properties are included. The solution is verified by comparison with the results of the classical finite volume method. Results for a 0.500[m] diameter Al 4.5 per cent Cu alloy billet at typical casting conditions are given.

The discretisation scheme presented for an unstructured triangular grid, is able to reflect the physical properties of stationary, nonlinear heat conduction even in the numerical case and leads to a stabilisation of iterative solution techniques for nonlinearities.

The generalized integral transform technique (GITT) is an hybrid numerical‐analytical method that has been successfully applied in convection‐diffusion problems, where the original potentials are replaced by eigenexpansion series, and the system of partial differential equations is transformed into a finite system of ordinary differential equations, allowing to obtain an error controlled solution without any kind of grid generation. This paper aims at the application of GITT to the transient version of the classical differentially heated square cavity problem, considering fluid properties as functions of temperature. Comparing results to some previously reported data for constant fluid properties validates the computational procedure. The solution for variable fluid properties with Boussinesq approximation is presented for several values of inclinations, at Rayleigh number of 10³ and a Prandtl number of 0.71, demonstrating GITT capability of capturing circulating cells formation and evolution at a low Rayleigh number. New correlations for leaning angle and aspect ratio are presented.

The numerical investigation into the stirring induced by an alternating magnetic field, applied in the axial direction of a closed axisymmetric container of conducting fluid, is presented. The interaction between the azimuthal current and magnetic field results in Lorentz forces in the meridional plane which induce the fluid flow. The magnetic Reynolds number is assumed to be smaller than the frequency magnetic Reynolds number. The electromagnetic equations are thus decoupled from the fluid flow equations. The electromagnetic field is first solved, and the body forces determined from this are introduced into the Navier‐Stokes equations. With the flow field known, the quality of mixing is determined by solving the tracer dispersion equation. The finite element method based on a Galerkin formulation is used for the solution of the equations. Three cases are examined: a finite length cylinder, a finite length cylinder with rounded corners and a sphere. In general, two vortices are formed, the equatorial vortex closest to the equator and the end vortex at the closed end. Results show that the introduction of the rounded corner increases the size and strength of the end vortex with the opposite effect on the equatorial vortex. Of the three frequency magnetic Reynolds numbers considered (Rw=30, 100 and 800), Rw=100 results in the best mixing for all cases. Rounding the corner of the cylinder only results in a definite improvement of mixing at Rw=800. The sphere results in even better mixing than this at Rw=800, but is worse than the first two geometries for Rw=30 and 100 when the interaction parameter is large.

The purpose of this paper is to explore the application of the mesh‐free local radial basis function collocation method (RBFCM) in solution of coupled heat transfer and fluid‐flow problems.

The involved temperature, velocity and pressure fields are represented on overlapping five nodded sub‐domains through collocation by using multiquadrics radial basis functions (RBF). The involved first and second derivatives of the fields are calculated from the respective derivatives of the RBFs. The energy and momentum equations are solved through explicit time stepping.

The performance of the method is assessed on the classical two dimensional de Vahl Davis steady natural convection benchmark for Rayleigh numbers from 10³ to 10⁸ and Prandtl number 0.71. The results show good agreement with other methods at a given range.

The pressure‐velocity coupling is calculated iteratively, with pressure correction, predicted from the local mass continuity equation violation. This formulation does not require solution of pressure Poisson or pressure correction Poisson equations and thus much simplifies the previous attempts in the field.

This paper presents an application of the dual reciprocity boundary element method (DRBEM) to transient advection‐diffusion problems. Radial basis functions and augmented thin plate splines (TPS) have been used as coordinate functions in DRBEM approximation in addition to the ones previously used in the literature. Linear multistep methods have been used for time integration of differential algebraic boundary element system. Numerical results are presented for the standard test problem of advection‐diffusion of a sharp front. Use of TPS yields the most accurate results. Further, considerable damping is seen in the results with one step backward difference method, whereas higher order methods produce perceptible numerical dispersion for advection‐dominated problems.

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 apply the variational multi-scale framework to the finite element approximation of the compressible Navier-Stokes equations written in conservation form. Even though this formulation is relatively well known, some particular features that have been applied with great success in other flow problems are incorporated.

The orthogonal subgrid scales, the non-linear tracking of these subscales, and their time evolution are applied. Moreover, a systematic way to design the matrix of algorithmic parameters from the perspective of a Fourier analysis is given, and the adjoint of the non-linear operator including the volumetric part of the convective term is defined. Because the subgrid stabilization method works in the streamline direction, an anisotropic shock capturing method that keeps the diffusion unaltered in the direction of the streamlines, but modifies the crosswind diffusion is implemented. The artificial shock capturing diffusivity is calculated by using the orthogonal projection onto the finite element space of the gradient of the solution, instead of the common residual definition. Temporal derivatives are integrated in an explicit fashion.

Subsonic and supersonic numerical experiments show that including the orthogonal, dynamic, and the non-linear subscales improve the accuracy of the compressible formulation. The non-linearity introduced by the anisotropic shock capturing method has less effect in the convergence behavior to the steady state.

A complete investigation of the stabilized formulation of the compressible problem is addressed.

This paper seeks to analyze transient convection‐diffusion by employing the generalized integral transform technique (GITT) combined with an arbitrary transient filtering solution, aimed at enhancing the convergence behavior of the associated eigenfunction expansions. The idea is to consider analytical approximations of the original problem as filtering solutions, defined within specific ranges of the time variable, which act diminishing the importance of the source terms in the original formulation and yielding a filtered problem for which the integral transformation procedure results in faster converging eigenfunction expansions. An analytical local instantaneous filtering is then more closely considered to offer a hybrid numerical‐analytical solution scheme for linear or nonlinear convection‐diffusion problems.

The approach is illustrated for a test‐case related to transient laminar convection within a parallel‐plates channel with axial diffusion effects.

The developing thermal problem is solved for the fully developed flow situation and a step change in inlet temperature. An analysis is performed on the variation of Peclet number, so as to investigate the importance of the axial heat or mass diffusion on convergence rates.

This paper succeeds in analyzing transient convection‐diffusion via GITT, combined with an arbitrary transient filtering solution, aimed at enhancing the convergence behaviour of the associated eigenfunction expansions.