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Purpose The purpose of this paper is to highlight the advantages of a simplified algorithm to solve the problem of heat and mass transfer in porous medium by reducing the number of partial differential equations from four to three. Design/methodology/approach The approach of the present paper is to develop a simplified algorithm to reduce the number of equations involved in conjugate heat transfer in porous medium. Findings Developed algorithm/method has many advantages over conventional method of solution for conjugate heat transfer in porous medium. Research limitations/implications The current work is applicable to conjugate heat transfer problem. Practical implications The developed algorithm is useful in reducing the number of equations to be solved, thus reducing the computational resources required. Originality/value Development of simplified algorithm and comparison with conventional method.

Purpose – The purpose of this paper is to propose a computational model for thin layers, for problems of linear time-dependent heat conduction. The thin layer is replaced by a zero-thickness interface. The advantage of the new model is that it saves the need to construct and use a fine mesh inside the layer and in regions adjacent to it, and thus leads to a reduction in the computational effort associated with implicit or explicit finite element schemes. Design/methodology/approach – Special asymptotic models have been proposed for linear heat transfer and linear elasticity, to handle thin layers. In these models the thin layer is replaced by an interface with zero thickness, and specific jump conditions are imposed on this interface in order to represent the special effect of the layer. One such asymptotic interface model is the first-order Bövik-Benveniste model. In a paper by Sussmann et al., this model was incorporated in a FE formulation for linear steady-state heat conduction problems, and was shown to yield an accurate and efficient computational scheme. Here, this work is extended to the time-dependent case. Findings – As shown here, and demonstrated by numerical examples, the new model offers a cost-effective way of handling thin layers in linear time-dependent heat conduction problems. The hybrid asymptotic-FE scheme can be used with either implicit or explicit time stepping. Since the formulation can easily be symmetrized by one of several techniques, the lack of self-adjointness of the original formulation does not hinder an accurate and efficient solution. Originality/value – Most of the literature on asymptotic models for thin layers, replacing the layer by an interface, is analytic in nature. The proposed model is presented in a computational context, fitting naturally into a finite element framework, with both implicit and explicit time stepping, while saving the need for expensive mesh construction inside the layer and in its vicinity.

Purpose – The purpose of this paper is to propose a parallel partition of unity method to solve the time-dependent convection-diffusion equations. Design/methodology/approach – This paper opted for the time-dependent convection-diffusion equations using the finite element method and the partition of unity method. Findings – This paper provides one efficient parallel algorithm which reaches the same accuracy as the standard Galerkin method (SGM) but saves a lot of computational time. Originality/value – In this paper, a parallel partition of unity method is proposed for the time-dependent convection-diffusion equations. At each time step, the authors only need to solve a series of independent local sub-problems in parallel instead of one global problem.

PurposeThe authors' objective in this paper is to find the numerical solutions of obstacle, unilateral and contact second‐order boundary‐value problems.Design/methodology/approachTo achieve this, the authors formulate a spatially adaptive grid refinement scheme following Galerkin's finite element method based on a weighted‐residual. A residual based a‐posteriori error estimation scheme has been utilized for checking the approximate solutions for various finite element grids. The local element balance has been considered as an error assessment criterion. The approach utilizes piece‐wise linear approximations utilizing linear Langrange polynomials. Numerical experiments indicate that local errors are large in regions where the gradients are large.FindingsA comparison of the spatially adaptive grid refinement with that of uniform meshing for second order obstacle boundary value problems confirms the superiority of the scheme without increasing the number of unknown coefficients.Originality/valueThe authors believe the work has merit not only in terms of the approach but also of the problem solved in the paper.

PurposeThe purpose of this paper is to investigate a large‐eddy simulation, using low order numerical discretization and upwinding schemes on unstructured grids, for a turbulent free jet at Mach number 0.95. The accuracy and stability performance is discussed for the finite element/volume upwinding numerical code used.Design/methodology/approachThis code is equipped with a self‐adaptive upwinding method which has been previously developed to reduce the numerical dissipation of applied low order flux calculation on unstructured elements using Roe's scheme. Herein, this method is used to numerically investigate a high Reynolds, compressible turbulent free jet and compare the results with a recently published set of experimental data. The effect of grid size is also investigated. A reasonable good agreement with the experimental measurements is obtained.FindingsBased on the results, it is concluded that the developed self‐adaptive upwinding scheme provides a considerably better emulation of the flow regime in comparison to the full‐upwinding scheme. Different case studies have been carried out to assess the performance of self‐adaptive upwinding method and the effect of the subgrid model.Originality/valueThis paper presents an original research on self‐adaptive upwinding scheme and the effect of the subgrid model on a compressible turbulent free jet.

PurposeThis paper sets out to implement bounded high‐order (HO) resolution schemes in a hybrid finite volume/finite element method for the solution of the radiative transfer equation.Design/methodology/approachThe hybrid finite volume/finite element method had formerly been developed using the step scheme, which is only first‐order accurate, for the spatial discretization. Here, several bounded HO resolution schemes, namely the MINMOD, CLAM, MUSCL and SMART schemes, formulated using the normalized variable diagram, were implemented using the deferred correction procedure.FindingsThe results obtained reveal an interaction between spatial and angular discretization errors, and show that the HO resolution schemes yield improved accuracy over the step scheme if the angular discretization error is small.Research limitations/implicationsAlthough the HO resolution schemes reduce the spatial discretization error, they do not influence the angular discretization error. Therefore, the global error is only reduced if the angular discretization error is also small.Practical implicationsThe use of HO resolution schemes is only effective if the angular refinement yields low‐angular discretization errors. Moreover, spatial and angular refinement should be carried out simultaneously.Originality/valueThe paper extends a methodology formerly developed in computational fluid dynamics, and aimed at the improvement of the solution accuracy, to the hybrid finite volume/finite element method for the solution of the radiative transfer equation.

PurposeThe aim of this paper is to determine a parallel computational methodology for simultaneously predicting the macro/micro scale phenomena occurring during solidification processing with external electromagnetic stirring.Design/methodology/approachMacro and micro phenomena occurring in an electromagnetically‐stirred solidifying melt are simulated using a numerical model that integrates the finite element methodology for transport phenomena and the Monte‐Carlo cellular‐automata method for microstructure formation. Parallel algorithm is introduced to enhance the computational efficiency.FindingsComputed results show that parallel algorithm can be effective in enhancing the computational efficiency of a combined macro/micro model if it is applied appropriately. Also, electromagnetically induced stirring can have a strong effect on the nucleation and grain growth and hence the final solidification microstructure.Originality/valueThis paper fulfils a need for developing an efficient numerical methodology to simulate complex electromagnetically‐assisted transport phenomena and microstructure formation during solidification processing systems.

PurposeThis paper seeks to introduce an optimization method for maximizing the effectiveness of the micro compact heat exchanger (MHE) under various geometrical parameters.Design/methodology/approachOptimization is realized using the neuro‐genetic methodology which combines the application of artificial neural network (ANN) together with genetic algorithms (GA). The analyses are divided into two main sections; the first being the modeling and prediction using finite element method, the second being the neuro‐genetic optimization. Initial results obtained from the finite element modeling are utilized for training in ANN. Subsequently, optimization is done using GA, once a well trained ANN is achieved.FindingsANN accurately predicts the effectiveness of the MHE and compares well with those obtained from the finite element simulation. Optimization shows a significant improvement in the maximum effectiveness of the MHE achievable for the given range of input parameters. Additionally, computational effort has been minimized and simulation time has been drastically reduced.Research limitations/implicationsThis analysis is valid for constant fluid properties and for steady‐state conditions. Additionally, optimization is limited to the range of the trained input parameters.Practical implicationsThis paper is very useful for practical design of various types of heat exchangers.Originality/valueThis paper will be useful for the design of the MHE where its performance can be analyzed for a given range of geometries with minimal effort. This methodology will also be applicable for other types of heat exchangers.

Purpose – The purpose of this paper is to present outcomes of efforts made over the last 20 years to extend the applicability of the Richardson extrapolation procedure to numerical predictions of multidimensional, steady and unsteady, fluid flow and heat transfer phenomena in regular and irregular calculation domains. Design/methodology/approach – Pattern-preserving grid-refinement strategies are proposed for mathematically rigorous generalizations of the Richardson extrapolation procedure for numerical predictions of steady fluid flow and heat transfer, using finite volume methods and structured multidimensional Cartesian grids; and control-volume finite element methods and unstructured two-dimensional planar grids, consisting of three-node triangular elements. Mathematically sound extrapolation procedures are also proposed for numerical solutions of unsteady and boundary-layer-type problems. The applicability of such procedures to numerical solutions of problems with curved boundaries and internal interfaces, and also those based on unstructured grids of general quadrilateral, tetrahedral, or hexahedral elements, is discussed. Findings – Applications to three demonstration problems, with discretizations in the asymptotic regime, showed the following: the apparent orders of accuracy were the same as those of the numerical methods used; and the extrapolated results, measures of error, and a grid convergence index, could be obtained in a smooth and non-oscillatory manner. Originality/value – Strict or approximate pattern-preserving grid-refinement strategies are used to propose generalized Richardson extrapolation procedures for estimating grid-independent numerical solutions. Such extrapolation procedures play an indispensable role in the verification and validation techniques that are employed to assess the accuracy of numerical predictions which are used for designing, optimizing, virtual prototyping, and certification of thermofluid systems.

Purpose – This paper aims to present a computationally efficient finite element model for the simulation of isothermal immiscible two-phase flow in a rigid porous media with a particular application to CO2 sequestration in underground formations. Focus is placed on developing a numerical procedure, which is effectively mesh-independent and suitable to problems at regional scales. Design/methodology/approach – The averaging theory is utilized to describe the governing equations of the involved unsaturated multiphase flow. The level-set (LS) method and the extended finite element method (XFEM) are utilized to simulate flow of the CO2 plume. The LS is employed to trace the plume front. A streamline upwind Petrov-Galerkin method is adopted to stabilize possible occurrence of spurious oscillations due to advection. The XFEM is utilized to model the high gradient in the saturation field front, where the LS function is used for enhancing the weighting and the shape functions. Findings – The capability of the proposed model and its features are evaluated by numerical examples, demonstrating its accuracy, stability and convergence, as well as its advantages over standard and upwind techniques. The study showed that a good combination between a mathematical model and a numerical model enables the simulation of complicated processes occurring in complicated and large geometry using minimal computational efforts. Originality/value – A new computational model for two-phase flow in porous media is introduced with basic requirements for accuracy, stability, and convergence, which are met using relatively coarse meshes.

Purpose – The purpose of this paper is to present an efficient algorithm based on a multi‐level adaptive mesh refinement strategy for the solution of ill‐posed inverse heat conduction problems arising in pool boiling using few temperature observations.Design/methodology/approach – The stable solution of the inverse problem is obtained by applying the conjugate gradient method for the normal equation method together with a discrepancy stopping rule. The resulting three‐dimensional direct, adjoin and sensitivity problems are solved numerically by a space‐time finite element method. A multi‐level computational approach, which uses an a posteriori error estimator to adaptively refine the meshes on different levels, is proposed to speed up the entire inverse solution procedure.Findings – This systematic approach can efficiently solve the large‐scale inverse problem considered without losing necessary detail in the estimated quantities. It is shown that the choice of different termination parameters in the disc...

PurposeThe purpose of this paper is to describe the development and employment of an hp‐adaptive finite element method (FEM) algorithm for solving heat transfer problems in partitioned enclosures, which has attracted the attention of both experimental and theoretical researchers in recent years.Design/methodology/approachIn the hp‐adaptive FEM algorithm presented here, both the element size and the shape function order are dynamically controlled by an a posteriori error estimator based on the L2 norm; a three‐step adaptation strategy is used with a projection algorithm for the flow solver.FindingsSimulation results are obtained for 2D and 3D natural convection within partitioned enclosures. Results show refined and enriched elements that develop near the partition edges and side walls of the enclosure, as expected. The heat transfer between the heated and cooled side walls is reduced in the presence of a partial partition.Research limitations/implicationsThe Rayleigh numbers were set to 105 in the 2D case and 103 in the 3D case. Efforts are underway to apply the hp‐adaptive algorithm to partitioned enclosures at much higher Rayleigh numbers, including comparison with available experimental data.Practical implicationsHeat transfer within partitioned enclosures occurs in many engineering situations: heat transfer across thermo pane windows, solar collectors, fire spread and energy transfer in rooms and buildings, cooling of nuclear reactors and heat exchanger design.Originality/valueThe hp‐adaptive FEM algorithm is one of the best mesh‐based algorithms for improving solution quality, whilst maintaining computational efficiency. The method shows considerable promise in solving a wide range of heat transfer problems including fluid flow.

Purpose – The purpose of this paper is to evaluate the thermal properties of carbon nanotube composites via meshless element free Galerkin (EFG) method.Design/methodology/approach – The EFG method is based on moving least square approximation, which is constructed by three components: a weight function associated with each node, a basis function and a set of non‐constant coefficients. In principle, EFG method is almost identical to finite element method. The EFG method does not require elements for the interpolation (or approximation) of field variable, but only requires a set of nodes for the construction of approximation function.Findings – The equivalent thermal conductivity of the composite has been calculated, and plotted against nanotube length, nanotube radius, RVE length, and RVE radius. Temperature distribution has been obtained and plotted with RVE length. An approximate numerical formula is proposed to calculate the equivalent thermal conductivity of CNT‐composites. Present computations show th...

PurposeThe purpose of this paper is to propose an efficient/robust numerical algorithm for solving the two‐dimensional laminar mixed‐convection in a lid‐driven cavity using the mixed finite element (FE) technique.Design/methodology/approachA numerical algorithm was based on the so‐called consistent splitting scheme, which improved the numerical accuracy of the primitive variables. In order to obtain a stable solution, two choices of mixed FEs, the Taylor‐Hood and Crouzeix‐Raviart types, were used. Two mesh layouts were considered; uniform and non‐uniform.FindingsTo verify that the proposed scheme had a second‐order accuracy, some numerical results are presented and compared with the known solution. The answer was confirmative. Numerically accurate solutions were obtained for a fixed Prandtl number, Pr=0.71, for a range of the Reynolds number, Re from 100 to 3,000, and for a range of the Richardson number, Ri from 0.001 to 100. The results from these calculations, using the mixed FE consistent splitting scheme, agreed with the existing ones.Research limitations/implicationsFurther extensions of this work could include the influence of various choices of Reynolds numbers, Prandtl numbers and Richardson numbers, and the effect of aspect ratio.Originality/valueThe present work was the first to apply a mixed FE in association with the consistent splitting scheme to the mixed convection problem.

This paper presents a two‐dimensional finite element model for convection‐dominated melting and freezing problems. The enthalpy‐porosity approach is utilized to account for the physics of the evolution of the melt velocity at the solid/liquid phase change interface. Penalty formulation is employed to treat the incompressibility constraint in the momentum equations. The streamline upwind/Petrov Galerkin method is used in combination with primitive variables to solve the governing equations. Simulations are carried out for melting of a pure phase change material in a rectangular cavity heated from below. Sample results are presented and discussed.

Methods used to track the free surface in numerical flow simulations, typically during the casting of metals into moulds are examined in this paper. The algorithm employed makes use of a mixed interpolation formulation to approximate the discretised governing equations for elimination on a Langrangian type moving mesh. Significant savings in CPU time are realised by virtue of the the air domain not being considered in the finite element analysis. The advantages and drawbacks of commonly used methods are discussed, and a method proposed for the numerical simulation of flow where the free surface boundary conditions are important. The effectiveness of the proposed algorithm in solving typical industrial flow problems is demonstrated using numerical examples. The results obtained are compared with analytical and numerical data with a view to validating the algorithm.