Božidar Šarler

Stefan's work on solid-liquid phase changes

This paper presents a detailed review of Jo~ef Stefans research on solid-liquid phase changes published in six treatises between the years 1889 and 1891. His achievements on this subject are related to the broader context of his interest in transport phenomena, particularly liquid-gas phase changes and chemical reactions in the years 1873 and 1889 respectively. Stefans eponymous work is placed in perspective between the present and the first experimental and analytical attempts to describe the solid-liquid phase change by the pioneers Blake in the 17th, and Lamt, Clapeyron and Neumann in the 18th century. In honour of Stefans work involving moving and free boundaries, the concepts of the Stefan problem and the Stefan number are widely used nowadays. The primary intention of this paper is to attempt to complete and unify the information on the historical roots which led to these two terms.

Dual reciprocity boundary element method for convective-diffusive solid-liquid phase change problems, Part 1. Formulation

Abstract A new dual reciprocity boundary element method for one-domain solving of the nonlinear convective-diffusive equation, as appears in one-phase continuum formulation of the energy transport in solid-liquid phase change systems, is described. Laplace equation fundamental solution weighting, straight line geometry and constant field shape functions on the boundary, finite-difference time discretization and scaled augmented thin plate spline global interpolation functions for transforming the domain integrals into a finite series of boundary integrals are employed in two dimensions and in axisymmetry. Iterations over the timestep are based on the Voller-Swaminathan scheme, upgraded to cope with the convective term. The technique could be applied to a wide range of solid-liquid phase change problems where finite volume or finite element solvers have been almost exclusively used in the past.

Dual reciprocity boundary element method for convective-diffusive solid-liquid phase change problems, Part 2. Numerical examples

Abstract A newly developed dual reciprocity boundary element method for solid-liquid phase change problems has been verified in this paper. Tests include numerical convergence studies for the transient Dirichlet jump problem with and without phase change, and the steady-state convective-diffusive problem with different material properties of the phases and phase change. Results are compared with the existing analytical solutions and with the results obtained by the finite volume and finite element methods. Excellent agreement has been obtained. The method turns out to be robust over the whole broad spectrum of conducted numerical tests. The axisymmeeryryc problem of direct chill continuous casting, where nonuniform meshes have been applied, serves as a ‘real’ problem example exhibiting nonlinear material properties, solidification over a temperature interval, and nonlinear boundary conditions.

Primitive variable dual reciprocity boundary element method solution of incompressible Navier-Stokes equations

Abstract This paper describes the solution to transient incompressible two-dimensional Navier–Stokes equations in primitive variables by the dual reciprocity boundary element method. The coupled set of mass and momentum equations is structured by the fundamental solution of the Laplace equation. The dual reciprocity method is based on the augmented thin plate splines. All derivatives involved are calculated through integral representation formulas. Numerical example include convergence studies with different mesh size for the classical lid-driven cavity problem at Re=100 and comparison with the results obtained through calculation of the derivatives from global interpolation formulas. The accuracy of the solution is assessed by comparison with the Ghia–Ghia–Shin finite difference solution as a reference.

Natural convection in porous media—dual reciprocity boundary element method solution of the Darcy model

This paper describes the solution of a steady state natural convection problem in porous media by the dual reciprocity boundary element method (DRBEM). The boundary element method (BEM) for the coupled set of mass, momentum, and energy equations in two dimensions is structured by the fundamental solution of the Laplace equation. The dual reciprocity method is based on augmented scaled thin plate splines. Numerical examples include convergence studies with different mesh size, uniform and non-uniform mesh arrangement, and constant and linear boundary field discretizations for differentially heated rectangular cavity problems at filtration with Rayleigh numbers of Ra*=25, 50, and 100 and aspect ratios of A=1/2, 1, and 2. The solution is assessed by comparison with reference results of the fine mesh finite volume method (FVM). Copyright

Axisymmetric augmented thin plate splines

This paper reviews the previous axisymmetric global interpolation functions used in the context of the dual reciprocity boundary element method and the axisymmetric Laplace operator. It upgrades the previous heuristic attempts with the axisymmetric form of the augmented thin plate splines. This new approach, based on the theory of radial basis functions, gives more formal mathematical support to this class of problems. The basic equations are accompanied by a set of related expressions that permit straightforward use of the developed global interpolation functions in a broad spectrum of dual reciprocity boundary element methods like discrete approximative procedures.

Analytical integration of elliptic 2D fundamental solution and its derivatives for straight-line elements with constant interpolation

Abstract This article describes the analytical integration of the elliptic 2D fundamental solution and its 1st, 2nd and 3rd derivatives with the constant function interpolation for straight-line boundary elements. As a result of the character of the integrated function, the integrals are characterized with regard to position of the source point. If it lies on the boundary Γ, then the integrals are: weak-(log r ), strong-(1/ r ) and hyper-(1/ r 2 ) singular. Otherwise, the integrals are regular. The 3rd derivatives of the fundamental solution are needed for the calculation of ∂ 2 u / ∂x 2 and ∂ 2 u / ∂y 2 of harmonic function u in the domain Ω. A comparison of the analytical and the numerical integrations is made.

Solid-Liquid Phase Change in Porous Media: Solution by Boundary Integral Method

This paper develops the dual reciprocity boundary integral method for solving coupled mass, momentum, and energy transport problems governed by the one-phase physical model of the solid-liquid phase change of a substance saturating a nonhomogenous porous matrix.

Mixture continuum formulation of convection-conduction energy transport in multiconstituent solid-liquid phase change systems for BEM solution techniques

Abstract This paper describes the two nonlinear boundary-domain integral equations for Fourier heat conduction and convection governed energy transport. The equations are compatible with the mixture continuum formulation of an incompressible multiconstituent solid-liquid phase change system. The equations assume the boundary conditions to be functions of thermal field, and thermal conductivity and specific heat to be functions of temperature and species concentrations. The constitutive enthalpy-temperature relation is assumed to be a function of the species concentrations. The integral equations are derived on the basis of time-domain weighting with the fundamental solutions of the Laplace and Fourier equations and are suitable for boundary element discrete approximative method solution techniques. The nonlinearity that appears in thermal conductivity is treated by the Kirchhoff transform and the nonlinearities of specific heat and specific latent heat phase change are both transformed into the nonlinearity of the source term. The presented equations, in connection with a similar integral description for mass, momentum and species conservation, will be used as a basis for the boundary element method computation of macroscopic transport phenomena characteristics for melting and solidification.

Simulation of a macrosegregation benchmark with a meshless diffuse approximate method

Purpose The purpose of this paper is to simulate a macrosegregation solidification benchmark by a meshless diffuse approximate method. The benchmark represents solidification of Al4.5wt%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 thirteen 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...