In this article the principles of the field operation and manipulation (FOAM) C++ class library for continuum mechanics are outlined. Our intention is to make it as easy as possible to develop reliable and efficient computational continuum-mechanics codes: this is achieved by making the top-level syntax of the code as close as possible to conventional mathematical notation for tensors and partial differential equations. Object-orientation techniques enable the creation of data types that closely mimic those of continuum mechanics, and the operator overloading possible in C++ allows normal mathematical symbols to be used for the basic operations. As an example, the implementation of various types of turbulence modeling in a FOAM computational-fluid-dynamics code is discussed, and calculations performed on a standard test case, that of flow around a square prism, are presented. To demonstrate the flexibility of the FOAM library, codes for solving structures and magnetohydrodynamics are also presented with appropriate test case results given. © 1998 American Institute of Physics.

A tensorial approach to computational continuum mechanics using object-oriented techniques

We present a multi–purpose CFD–DEM framework to simulate coupled fluid–granular systems. The motion of the particles is resolved by means of the Discrete Element Method (DEM), and the Computational Fluid Dynamics (CFD) method is used to calculate the interstitial fluid flow. We first give a short overview over the DEM and CFD–DEM codes and implementations, followed by elaborating on the numerical schemes and implementation of the CFD–DEM coupling approach, which comprises two fundamentally different approaches, the unresolved CFD–DEM and the resolved CFD–DEM using an Immersed Boundary (IB) method. Both the DEM and the CFD–DEM approach are successfully tested against analytics as well as experimental data.

Models, algorithms and validation for opensource DEM and CFD-DEM

SUMMARY
The open-source CFD library OpenFoam® contains a method for solving free surface Newtonian flows using the Reynolds averaged Navier–Stokes equations coupled with a volume of fluid method. In this paper, it is demonstrated how this has been extended with a generic wave generation and absorption method termed ‘wave relaxation zones’, on which a detailed account is given. The ability to use OpenFoam for the modelling of waves is demonstrated using two benchmark test cases, which show the ability to model wave propagation and wave breaking. Furthermore, the reflection coefficient from outlet relaxation zones is considered for a range of parameters. The toolbox is implemented in C++, and the flexibility in deriving new relaxation methods and implementing new wave theories along with other shapes of the relaxation zone is outlined. Subsequent to the publication of this paper, the toolbox has been made freely available through the OpenFoam-Extend Community. Copyright © 2011 John Wiley & Sons, Ltd.

A wave generation toolbox for the open‐source CFD library: OpenFoam®

The constrained interpolation profile method for multiphase analysis

The performance of the open source multiphase flow solver, interFoam, is evaluated in this work. The solver is based on a modified volume of fluid (VoF) approach, which incorporates an interfacial compression flux term to mitigate the effects of numerical smearing of the interface. It forms a part of the C + + libraries and utilities of OpenFOAM and is gaining popularity in the multiphase flow research community. However, to the best of our knowledge, the evaluation of this solver is confined to the validation tests of specific interest to the users of the code and the extent of its applicability to a wide range of multiphase flow situations remains to be explored. In this work, we have performed a thorough investigation of the solver performance using a variety of verification and validation test cases, which include (i) verification tests for pure advection (kinematics), (ii) dynamics in the high Weber number limit and (iii) dynamics of surface tension-dominated flows. With respect to (i), the kinematics tests show that the performance of interFoam is generally comparable with the recent algebraic VoF algorithms; however, it is noticeably worse than the geometric reconstruction schemes. For (ii), the simulations of inertia-dominated flows with large density ratios yielded excellent agreement with analytical and experimental results. In regime (iii), where surface tension is important, consistency of pressure–surface tension formulation and accuracy of curvature are important, as established by Francois et al (2006 J. Comput. Phys. 213 141–73). Several verification tests were performed along these lines and the main findings are: (a) the algorithm of interFoam ensures a consistent formulation of pressure and surface tension; (b) the curvatures computed by the solver converge to a value slightly (10%) different from the analytical value and a scope for improvement exists in this respect. To reduce the disruptive effects of spurious currents, we followed the analysis of Galusinski and Vigneaux (2008 J. Comput. Phys. 227 6140–64) and arrived at the following criterion for stable capillary simulations for interFoam: where . Finally, some capillary flows relevant to atomization were simulated, resulting in good agreement with the results from the literature.

https://iopscience.iop.org/article/10.1088/1749-4699/5/1/014016/pdf

Evaluating the performance of the two-phase flow solver interFoam

Solution of the implicitly discretised reacting flow equations by operator-splitting

The present paper presents, numerical computations for flow, heat transfer and chemical reactions in an axisymmetric inert porous burner. The porous media re-radiate the heat absorbed from the gaseous combustion products by convection and conduction. In the present work, the porous burner species mass fraction source terms are computed from an ‘extended’ reaction mechanism, controlled by chemical kinetics of elementary reactions. The porous burner has mingled zones of porous/nonporous reacting flow, i.e. the porosity is not uniform over the entire domain. Therefore, it has to be included inside the partial derivatives of the transport governing equations. Finite-difference equations are obtained by formal integration over control volumes surrounding each grid node. Up-wind differencing is used to insure that the influence coefficients are always positive to reflect the real effect of neighboring nodes on a typical central node. Finite-difference equations are solved, iteratively, for U, V, p’( pressure correction), enthalpy and species mass fractions, utilizing a grid of (60X40) nodes. The sixty grid nodes in the axial direction are needed to resolve the detailed structure of the thin reaction zone inside the porous media. The porous burner uses a premixed CH4-air mixture, while its radiating characteristics are computed numerically, using a four-flux radiation model. Sixteen species are included, namely CH4, CH3, CH2, CH, CH2O, CHO, CO, CO2, O2, O, OH, H2, H, H2O, HO2, H2O2, involving 49 chemical reaction equations. It was found that 900 iterations are sufficient for complete conversion of the computed results with errors less than 0.1%. The computed temperature profiles of the gas and the solid show that, heat is conducted from downstream to the upstream of the reaction zone. Most stable species, such as H2O, CO2, H2, keep increasing inside the reaction zone staying appreciable in the combustion products. However, unstable products, such as HO2, H2O2 and CH3, first increase in the preheating region of the reaction zone, they are then consumed fast in the postreaction zone of the porous burner. Therefore, it appears that their important function is only to help the chemical reactions continue to their inevitable completion of the more stable combustion products..

Numerical computation of reacting flow in porous burners with an extended ch4-air reaction mechanism

The structure of premixed methane-air flames is analyzed using the “laminar flamelet concept”. A new model based on one-dimensional set of transformed coupled second order differential conservation equations describing the species concentrations of CO2, CO, O2, CH4, H2O, H2 and N2 and the sensible enthalpy are presented in the present work. The equations are rigorously derived and solved numerically. In these equations, a reaction progress variable (c) is taken as the independent variable that varies from zero to one. A three-step chemical kinetic mechanism is adopted. This was deduced in a systematic way from a detailed chemical kinetic mechanism. The rates for the three steps are related to the rates of the elementary reactions of the full reaction mechanism. Calculations are made for different fixed values of the scalar dissipation rate (x) until the flamelet eventually reaches the extinction limit at different levels of pressure. Moreover, simultaneous and instantaneous l-D measurements of CO...

A Flamelet Model for Premixed Methane-Air Flames

The proton-exchange membrane (PEM) fuel cell works under low temperatures and hence is suitable for the automotive industry. The produced water vapor in the vicinity of the membrane may condense into liquid water, if the water mass fraction is higher than the saturation value corresponding to the local temperature. In this case the flowing fluid inside the layers of the PEMFC is a 2-phase flow. The locally homogeneous flow (LHF) model has been previously used for modeling the 2-phase flow in PEM fuel cells, with limited success. This model could not predict the blocking effect of the liquid phase, since both phases flow locally with the same velocity, according to the LHF model. In contrast to complete coupling of the two phases, assumed by the LHF model, a blocking model was used by some investigators where the liquid is totally uncoupled to the gas phase. This assumption causes the liquid to become essentially stationary inside the pores of the GDL and catalyst layers and hence only the gas phase equations need to be considered. Both of these extreme models were only successful to a limited extent. The present work considers a two-fluid mathematical model for the gas-liquid flow in PEM fuel cells. One fluid represents the continuous gas phase flow through the layers of the fuel cell. For this fluid, the governing equations of momentum, energy, mass continuity and species mass fractions, are considered with additional inter-fluid exchange source terms. The second fluid represents the dispersed liquid phase that is formed from the condensed water vapor inside the layers of the PEM fuel cell. For this fluid only the momentum, energy and mass continuity equations need to be included, as no electrochemical reactions are essentially possible. The dispersed fluid is made up of small droplets in the gas channel. However, in the porous layers of the fuel cell, the flowing layers of water represent the dispersed fluid over the solid matrix. The thickness of the creeping water layers is controlled by the wetability of the solid matrix of the porous layers of the PEM fuel cell. Numerical computations are carried out for a typical proton exchange membrane fuel cell that has experimental data. In order to obtain complete performance results, the computations are repeated for increasing fuel cell electric current densities until the limiting current is reached. The obtained two-fluid and single-phase simulations are compared with the corresponding experimental and numerical data available in the literature. The 2-fluid model shows that the blocking effect of the liquid phase starts to dominate, for cell voltage less than 0.65 V; in this case, the flowing 2-phase flow produces faster drop in cell voltage as the loading electric current increases. This phenomenon was partially hindered by the LHF model but essentially completely bypassed by the single-phase simulations. The 2-fluid simulations show that most of the liquid dispersed phase is concentrated in the cathode, reaching maximum value near the cathode catalyst layer- membrane interface. This behavior results from the lack of mobility of the liquid water inside the pores.Copyright © 2005 by ASME

A two-fluid mathematical model for two-phase flow in PEM fuel

The produced water vapor in the vicinity of the membrane, of a proton exchange membrane fuel cell (PEMFC), may condense into liquid water, if the water mass fraction is higher than the saturation value corresponding to the local temperature. In this case the flowing fluid inside the layers of the PEMFC is a 2-phase flow. The present mathematical model is based on the locally homogeneous flow model (LHFM) where the slip velocity between the two phases is assumed negligibly small. Therefore, the governing gas-phase and liquid-phase, for each dependent variable, can be economically added together. The resulting equations contain a ‘mixture density’, which is a function of the void fraction, species mass fractions, pressure and temperature. The resulting governing equations for u, v, T and species mass fractions together with the electric potential and mass continuity equations are solved iteratively using the SIMPLE algorithm. One solution domain is superimposed over all the layers of the PEMFC with appropriate boundary conditions applied at inlet, exit and sidewalls of the fuel cell. Special care is devoted to the electric potential, ‘Poisson-type’, equation boundary condition to prevent any escape of protons through the two diffuser layers and simultaneously insuring a non-singular matrix of finite-difference coefficients. This is because Poisson equation is notoriously known for having problems with zero gradient boundary condition. Numerical computations are carried out for a typical proton exchange membrane fuel cell that has experimental data. In order to obtain complete performance results, the computations are repeated for increasing fuel cell electric current densities until the voltage vanishes. The obtained 2-phase and 1-phase simulations are compared with the corresponding experimental and numerical data available in the literature. Systematically, the 1-phase current density is under predicted especially for values of the cell potential less than 0.8 V. On the other hand, the two-phase simulation current density, of the parallel geometry FC, is in very good agreement with the corresponding experiment data. The 2-phase flow simulations show that most of the liquid phase is concentrated in the cathode, reaching maximum value near the cathode catalyst layer-membrane interface. A new design of the serpentine PEMFC is suggested and is tested numerically. The new design involves blocking the outlet sections, either partially or fully, of the anode and/or the cathode gas channels to force the flowing fluids to diffuse into the catalyst layers at rates higher than a typical parallel geometry PEMFC. The new serpentine PEMFC design is expected to increase the concentrations of the hydrogen fuel and the oxidant in the catalyst layers and hence increase the transfer current densities. In order to obtain full simulation of the enhanced geometry of the fuel cell, the end boundary condition of the gas channels is adjusted using zero porosity to prevent any flow through the blocked area which automatically reduces the local velocity to zero value. The two-phase flow numerical results, for the modified serpentine PEMFC, indicate that the performance of the fuel cell could be enhanced appreciably.Copyright © 2004 by ASME

Karam R. Beshay

Papers

Solution of the implicitly discretised reacting flow equations by operator-splitting

Numerical computation of reacting flow in porous burners with an extended ch4-air reaction mechanism

A Flamelet Model for Premixed Methane-Air Flames

A two-fluid mathematical model for two-phase flow in PEM fuel

A Two-Phase Flow Mathematical Model for Proton Exchange Membrane Fuel Cells With Enhanced Geometry