Control Volume Finite Element Method Research Articles

Recent advances in finite element methods for multi-phase flow processes in porous media

In this paper, we will focus on three essentially important issues considering the modeling of multi-phase flow in heterogeneous porous media. Depending on the governing processes, the multi-phase flow equation system can show either predominant advection or diffusion/dispersion. Numerical methods have to be able to capture both cases. Furthermore, realistic porous media are always heterogeneous. The physically correct description of the interfacial behaviour is numerically very challenging, as discontinuities in primary variables may occur. In this paper, we review these issues and present the box scheme, a control volume finite element method, including an interface condition as one numerical method capable of tackling the three problems. In the end, we will give the results of a 3D simulation.

Second Law Analysis in Convective Heat and Mass Transfer

This paper reports the numerical determination of the entropy generation due to heat transfer, mass transfer and fluid friction in steady state for laminar double diffusive convection, in an inclined enclosure with heat and mass diffusive walls, by solving numerically the mass, momentum, species conservation and energy balance equations, using a Control Volume Finite-Element Method. The influences of the inclination angle, the thermal Grashof number and the buoyancy ratio on total entropy generation were investigated. The irreversibilities localization due to heat transfer, mass transfer and fluid friction is discussed for three inclination angles at a fixed thermal Grashof number.

Modeling of effective elastic properties for polymer based carbon nanotube composites

Effective elastic properties of the nanocomposites filled with carbon nanotubes (CNTs) are investigated by the asymptotic expansion homogenization (AEH) method. In order to implement the homogenization method, a control volume finite element method (CVFEM) is employed in contrast to the previous studies. It is assumed that the nanocomposites have geometric periodicity with respect to local length scale and the elastic properties of nanocomposites can be represented by those of the representative volume element (RVE). Random orientation of the CNTs embedded in the nanocomposites is considered by using the orientation tensor. The effective elasticity tensor predicted by the homogenization method is compared with analytical and experimental results. In the experiment, the CNT surface is treated by oxygen plasma to improve interfacial bonding between the CNT and the matrix and to disperse the CNTs homogeneously in epoxy resin because the perfect interfacial bonding is presumed in the homogenization method. Homogeneous CNT dispersion is experimentally identified by the field emission scanning electronic microscope (FESEM). It is found that the numerically calculated elastic modulus is in good agreement with that obtained by analytic model.

Asymptotic expansion homogenization of permeability tensor for plain woven fabrics

Homogenization method is adopted to predict the permeability tensor for glass fiber plain woven fabrics. Numerical calculation of the permeability tensor is an encouraging task because the permeability tensor is a key parameter in resin transfer molding (RTM). Based on multi-scale approach of the homogenization method, permeability of the micro-unit cell for the fiber tow is computed and compared with that obtained from flow analysis for the same micro-unit cell. In order to calculate the permeability tensor of macro-unit cell for the plain woven fabrics, the Stokes and Brinkman equations which describe inter-tow and intra-tow flows, respectively, are employed as governing equations. The effective permeabilities homogenized by considering intra-tow flow are compared with those obtained experimentally. Control volume finite element method (CVFEM) is used as a numerical method. It is shown that the asymptotic expansion homogenization method is an attractive tool to predict the effective permeability for heterogeneous media.

On the control volume finite element methods and their applications to multiphase flow

In this paper we systematically reviewthe control volume finite element (CVFE) methods for numericalsolutions of second-order partial differential equations.Their relationships to the finite differenceand standard (Galerkin) finite element methods are considered.Through their relationshipto the finite differences, upstream weightedCVFE methods and the conditions on positivetransmissibilities (positive flux linkages)are studied. Through their relationshipto the standard finite elements,error estimates forthe CVFE are obtained. These estimates arecomparable to those for the standard finite element methodsusing piecewise linear elements. Finally, anapplication to multiphase flows in porousmedia is presented.

A stable second-order mass-weighted upwind scheme for unstructured meshes

In this paper, an original second-order upwind scheme for convection terms is described and implemented in the context of a Control-Volume Finite-Element Method (CVFEM). The proposed scheme is a second-order extension of the first-order MAss-Weighted upwind (MAW) scheme proposed by Saabas and Baliga (Numer. Heat Transfer 1994; 26B:381–407). The proposed second-order scheme inherits the well-known stability characteristics of the MAW scheme, but exhibits less artificial viscosity and ensures much higher accuracy. Consequently, and in contrast with nearly all second-order upwind schemes available in the literature, the proposed second-order MAW scheme does not need limiters. Some test cases including two pure convection problems, the driven cavity and steady and unsteady flows over a circular cylinder, have been undertaken successfully to validate the new scheme. The verification tests show that the proposed scheme exhibits a low level of artificial viscosity in the pure convection problems; exhibits second-order accuracy for the driven cavity; gives accurate reattachment lengths for low-Reynolds steady flow over a circular cylinder; and gives constant-amplitude vortex shedding for the case of high-Reynolds unsteady flow over a circular cylinder. Copyright © 2005 John Wiley & Sons, Ltd.

LARGE EDDY SIMULATION OF TURBULENT INCOMPRESSIBLE FLUID FLOWS BY A NINE-NODES CONTROL VOLUME-FINITE ELEMENT METHOD

The main purpose of this work is the numerical computation of turbulent incompressible fluid flows by a nine-node control volume finite element method (CVFEM) using the methodology of large-eddy simulation.. The domain is discretized using nine nodes finite elements and the equations are integrated into control volumes around the nodes of the finite elements. The Navier?Stokes equations are filtered for simulation of the large scales variables and the sub-grid scales stress appearing due to the filtering process are modeled through the eddy viscosity model of Smagorinsky. The two-dimensional benchmark problem of the lid-driven cavity flow is solved to validate the numerical code and preliminary results for the horizontal and vertical velocity profiles at the centerlines of the cavity and the stream functions are presented and compared with available results from the literature.

LARGE EDDY SIMULATION OF TURBULENT INCOMPRESSIBLE FLUID FLOWS BY A NINE-NODES CONTROL VOLUME-FINITE ELEMENT METHOD

The main purpose of this work is the numerical computation of turbulent incompressible fluid flows by a nine-node control volume finite element method (CVFEM) using the methodology of large-eddy simulation.. The domain is discretized using nine nodes finite elements and the equations are integrated into control volumes around the nodes of the finite elements. The Navier?Stokes equations are filtered for simulation of the large scales variables and the sub-grid scales stress appearing due to the filtering process are modeled through the eddy viscosity model of Smagorinsky. The two-dimensional benchmark problem of the lid-driven cavity flow is solved to validate the numerical code and preliminary results for the horizontal and vertical velocity profiles at the centerlines of the cavity and the stream functions are presented and compared with available results from the literature.

Modelling the migration of contaminants through variably saturated dual-porosity, dual-permeability chalk

In the Hesbaye region in Belgium, tracer tests performed in variably saturated fissured chalk rocks presented very contrasting results in terms of transit times, according to artificially controlled water recharge conditions prevailing during the experiments. Under intense recharge conditions, tracers migrated across the partially or fully saturated fissure network, at high velocity in accordance with the high hydraulic conductivity and low effective porosity (fracture porosity). At the same time, a portion of the tracer was temporarily retarded in the almost immobile water located in the matrix. Under natural infiltration conditions, the fissure network remained inactive. Tracers migrated downward through the matrix, at low velocity in relation with the low hydraulic conductivity and the large porosity of the matrix. Based on these observations, Brouyère et al. (2004a) [Brouyère, S., Dassargues, A., Hallet, V., 2004a. Migration of contaminants through the unsaturated zone overlying the Hesbaye chalky aquifer in Belgium: a field investigation, J. Contam. Hydrol., 72 (1–4), 135–164, doi: 10.1016/j.conhyd.2003.10.009] proposed a conceptual model in order to explain the migration of solutes in variably saturated, dual-porosity, dual-permeability chalk. Here, mathematical and numerical modelling of tracer and contaminant migration in variably saturated fissured chalk is presented, considering the aforementioned conceptual model. A new mathematical formulation is proposed to represent the unsaturated properties of the fissured chalk in a more dynamic and appropriate way. At the same time, the rock water content is partitioned between mobile and immobile water phases, as a function of the water saturation of the chalk rock. The groundwater flow and contaminant transport in the variably saturated chalk is solved using the control volume finite element method. Modelling the field tracer experiments performed in the variably saturated chalk shows the adequacy and usefulness of the new conceptual, mathematical and numerical model.

Numerical simulation of three-dimensional flow and analysis of filling process in compression resin transfer moulding

In compression resin transfer moulding (CRTM), resin flows inside a mould filled with fibrous reinforcement as a result of resin injection and mould compression. This process facilitates the resin flow through the fibrous reinforcement particularly for high fiber content parts. This paper presents the numerical simulation of filling process of CRTM in full three-dimensional domain. A mathematical model of three-dimensional resin flow inside the CRTM mould is presented. An effective elastic modulus is introduced for multi-layer preforms connecting the mould closing speed to deformation rate of individual layers. Control volume/finite element method (CV/FEM) is used and the numerical algorithm developed to progress the flow front is based on a quasi-steady state approach, previously employed for resin transfer moulding (RTM), which is here modified for a deformable medium. Element characteristic height (ECH) and element characteristic surface area (ECSA) are introduced to take into account irregular elements in the simulation which are often observed in complex mould cavity. The numerical algorithm presented is coded and the resulting computer code is used to predict the necessary filling parameters such as flow progression, pressure distribution and mould clamping force. The validity of the numerical algorithm is verified by comparison with analytical solutions for simple moulds. Computer code developed is also used to analyse the CRTM process with different filling modes by providing various numerical examples. The results show the effectiveness of the numerical simulation in predicting the CRTM process.

Natural convection effects in solar stills

This paper deals with a numerical simulation of natural convection flows in a triangular cavity submitted to a uniform heat flux using the Control Volume Finite Element Method. The aim of the study is to examine the thermal exchange by natural convection and effects of buoyancy forces on flow structure. The study provides useful informations on the flow structure sensitivity to the governing parameters, the Rayleigh number and the tilt angle, on the thermal exchange. In a basin still receiving a uniform heat flux, the results show that the bottom is not isotherm and the flow structure is sensitive to the cover tilt angle. Many recirculation zones can occur in the core of the cavity and the heat transfer is dependent on the flow structure. The results of this study can provide informations for the enhancement of the design of the energy systems such as solar water distillers and air conditioning systems. The physical problem concerns a two-dimensional incompressible fluid flow generated inside a triangular enclosure with isotherm upper walls and heated bottom. The hydrodynamic and thermal fields, the local Nusselt number, the temperature profile at the bottom and at the center of the cavity are investigated for a large range of Rayleigh number. The effect of the upper sides inclination is examined for the particular value of Ra=10 5.

Unstructured Control-Volume Finite-Element Method for Radiative Heat Transfer in a Complex 2-D Geometry

This article describes a new approach based on the control-volume finite-element method (CVFEM) for computing radiative heat transfer in a complex two-dimensional geometry using a general unstructured grid. To examine its accuracy and computational efficiency, five test cases are investigated, and the results obtained agree very well with other published works. In addition, the study presented in this article shows that not only this method is flexible in treating radiative heat transfer in complex geometry, but also that the computer procedure based on this numerical method needs an accurate computer process unit (CPU) time and can be combined easily with developed codes for fluid dynamics.

Temperature–Velocity Coupling of the Convective Heat Transfer Sensitivity Analysis

ABSTRACT This article deals with sensitivity studies of the temperature–velocity coupling of convective heat transfer problems with laminar fluid flow. Sensitivity coefficients contain information about the chances of a successful inverse procedure, and they can be used in a number of inverse schemes, i.e., the Levenberg-Marquardt method. Special emphasis is given in this article to the numerical solution of the direct problem, which was obtained by the control-volume finite-element method formulation. This numerical tool was used for the sensitivity coefficients evaluation and can be used for simulation of real measurements in the full inverse algorithm. The algorithm for the sensitivity coefficients evaluation and example distributions of the sensitivity coefficients are given. A proposed full inverse procedure, allowing determination of inflow velocity based on internal temperature measurements, is also included. The inverse method employs the Levenberg-Marquardt method for stabilization purposes.

Numerical analyses of radiative heat transfer in any arbitrarily-shaped axisymmetric enclosures

A numerical approach for the treatment of radiative heat transfer in any irregularly-shaped axisymmetric enclosure filled with absorbing, emitting and scattering gray media is developed. Radiative transfer equation (RTE) is formulated for a general axisymmetric geometrical configurations, and the discretized equation is conducted using an unstructured meshes, generated by an appropriate computer algorithm, and the control volume finite element method which frequently adopted in CFD problems. A computer procedure has been done to solve the discretized RTE and to examine the accuracy and the computational efficiency of the proposed numerical approach. By using this computer algorithm, five test cases, a cylindrical enclosure with absorbing and emitting medium, a diffuser shaped axisymmetric enclosure, a finite axisymmetric cylindrical enclosure with a curved wall, a furnace with axially varying medium temperature and a rocket nozzle, are treated and the obtained results agree very well with other published works. Furthermore, the developed computer procedure has an accurate CPU time and it can be coupled easily with CFD codes.

Numerical Prediction of Permeability Tensor for Three Dimensional Circular Braided Preform by considering Intra-tow Flow

Resin transfer moulding is characterized by the permeability tensor, which is a measure of the resistance to resin flow through the preform. Complete prediction of the second order permeability tensor for three dimensional circular braided preforms is critical to an understanding of the resin transfer moulding process. The permeability can be predicted by considering resin flow through the multi-axial fibre structure. In this study, the permeability tensor for a 3-D circular braided preform was calculated by solving a boundary problem of a periodic unit cell. The flow field through the unit cell was obtained by using a 3-D control volume finite element method (CVFEM) and Darcy's law was utilized to obtain the permeability tensor. The flow analyses were carried out for two cases, one in which the fibre tow was regarded as a permeable porous medium, and one in which it was regarded as an impermeable solid. It was found that the flow within the intra-tow region of the braided preform was negligible if the inter-tow porosity was relatively high, but flow through the tow, especially flow in the thickness direction must be considered when the porosity is low. The permeability of the braided preform was measured by a radial flow experiment and compared with the predicted permeability.

Numerical simulation of mold filling in resin transfer molding using isoparametric method

The mold filling of RTM was simulated based on the control volume finite element method (CV/FEM). The formulation using isoparametric transformation was discussed in detail and a computational code based on isoparametric technique was developed. The simulation results were compared with experimental data. Different isoparametric elements, quadrilateral and triangular, were compared in the simulation. It demonstrates that the use of bilinear quadrilateral isoparametric elements in simulating the process can produce a higher precision and cost a less time than the use of triangular ones.

Comparison of Galerkin and control volume finite element for advection-diffusion problems

The control volume finite element method (CVFEM) was developed to combine the local numerical conservation property of control volume methods with the unstructured grid and generality of finite element methods (FEMs). Most implementations of CVFEM include mass-lumping and upwinding techniques typical of control volume schemes. In this work we compare, via numerical error analysis, CVFEM and FEM utilizing consistent and lumped mass implementations, and stabilized Petrov–Galerkin streamline upwind schemes in the context of advection–diffusion processes. For this type of problem, we find no apparent advantage to the local numerical conservation aspect of CVFEM as compared to FEM. The stabilized schemes improve accuracy and degree of positivity on coarse grids, and also reduce iteration counts for advection-dominated problems. Published in 2005 by John Wiley & Sons, Ltd.

Numerical study of interaction between the fluid structure and the moving interface during the melting from below in a rectangular closed enclosure

In this paper a control volume finite element method is used for solving numerically the melting of a pure material in a rectangular closed enclosure. The aim of this work is, firstly, to demonstrate the capabilities of this numerical methodology for handling such problems and, secondly, to analyse the evolution of the flow structure induced by natural convection during the melting process when the cavity is heated from below. Finally, a series of numerical experiments were undertaken to assess the effects of the aspect ratio, the cooling temperature on the top surface and the heating condition on the melting rate and the flow structure. The modelling of the melting problem is tackled with the enthalpy-porosity formulation coupled to convective heat transfer in the liquid phase.

Optimization of the Micro-Injection Molding Process using Grey Relational Analysis and MoldFlow Analysis

Microsystem technology is a compound technology including optic, mechanism, electricity, material, control and chemistry, etc. This technology can miniaturize products and increase their function, quality, trustworthiness and addendum. Micro-injection molding is a branch of the micro-electro mechanical system. The size of the product is in nanometers. It is important to produce the product for micro-injection molding technology today. The research of micro-injection molding is still in the early stages. In this study, numerical simulations of three-dimensional micro-injection moldings were performed. The governing differential equations were described by using control volume finite element method. The analysis with different polymers (such as PP, PC, PS, POM) and process parameters (injection time, mold temperature, injection temperature, injection pressure) are used to simulate the microgear for example. In order to obtain optimum results, the simulation introduces the Taguchi method and grey relational analysis to discuss the influence of each parameter in microinjection molding. The results show that the PS material is the most suitable material used in this study. The results for Mold Flow analysis and grey relational analysis are the same on this study.

AN INVESTIGATION OF THE ACCURACY OF THE CONTROL-VOLUME FINITE-ELEMENT METHOD BASED ON TRIANGULAR PRISMATIC ELEMENTS FOR SIMULATING DIFFUSION IN ANISOTROPIC MEDIA

The accuracy of the control-volume finite-element method for simulating strongly anisotropic diffusion problems on rectangular parallelepiped domains meshed with triangular prismatic elements is investigated. It is shown that a novel Gauss–Green gradient reconstruction technique provides significant improvements in accuracy over the classical finite-element shape function technique. Numerical investigations carried out on relatively coarse meshes highlight that the new Gauss–Green method offers a reduction of 80 % in solution error over the shape function method. In fact, a mesh with 27 times more elements would be required if the shape function method was to achieve the same accuracy as the new method, resulting in a substantial increase in computational overheads.