Finite Volume Methods

Biodiesel, as an alternative fuel, is gradually receiving more popularity for use in internal combustion engines. However questions continue to arise with regard to its compatibility with elastomeric materials. The present work aims to investigate the comparative degradation of physical properties for different elastomers [e.g. ethylene propylene diene monomer (EPDM), silicone rubber (SR), polychloroprene (CR), polytetrafluroethylene (PTFE) and nitrile rubber (NBR)] upon exposure to diesel and palm biodiesel. Static immersion tests in B0(diesel), B10 (10% biodiesel in diesel), B20, B50 and B100(biodiesel) were carried out at room temperature (25 °C) for 1000 h. Different physical properties like, changes in weight and volume, hardness and tensile strength were measured at every 250 h of immersion time. Compositional changes in biodiesel due to exposure of different elastomers were investigated by Gas chromatography mass spectroscopy (GCMS). The overall sequence of compatible elastomers in palm biodiesel is found to be PTFE > SR > NBR > EPDM > CR.

http://www.sciencedirect.com/science/article/pii/S0360544210007139

Abstract
Dam-break flows usually propagate along rivers and floodplains, where the processes of fluid flow, sediment transport and bed evolution are closely linked. However, the majority of existing two-dimensional (2D) models used to simulate dam-break flows are only applicable to fixed beds. Details are given in this paper of the development of a 2D morphodynamic model for predicting dam-break flows over mobile beds. In this model, the common 2D shallow water equations are modified, so that the effects of sediment concentrations and bed evolution on the flood wave propagation can be considered. These equations are used together with the non-equilibrium transport equations for graded sediments and the equation of bed evolution. The governing equations are solved using a matrix method, thus the hydrodynamic, sediment transport and morphological processes can be jointly solved. The model employs an unstructured finite volume algorithm, with an approximate Riemann solver, based on the Roe-MUSCL scheme. A predictor–corrector scheme is used in time stepping, leading to a second-order accurate solution in both time and space. In addition, the model considers the adjustment process of bed material composition during the morphological evolution process. The model was first verified against results from existing numerical models and laboratory experiments. It was then used to simulate dam-break flows over a fixed bed and a mobile bed to examine the differences in the predicted flood wave speed and depth. The effects of bed material size distributions on the flood flow and bed evolution were also investigated. The results indicate that there is a great difference between the dam-break flow predictions made over a fixed bed and a mobile bed. At the initial stage of a dam-break flow, the rate of bed evolution could be comparable to that of water depth change. Therefore, it is often necessary to employ the turbid water governing equations using a coupled approach for simulating dam-break flows.

Abstract
An efficient numerical scheme is outlined for solving the SWEs (shallow water equations) in environmental flow; this scheme includes the addition of a five-point symmetric total variation diminishing (TVD) term to the corrector step of the standard MacCormack scheme. The paper shows that the discretization of the conservative and non-conservative forms of the SWEs leads to the same finite difference scheme when the source term is discretized in a certain way. The non-conservative form is used in the solution outlined herein, since this formulation is simpler and more efficient. The time step is determined adaptively, based on the maximum instantaneous Courant number across the domain. The bed friction is included either explicitly or implicitly in the computational algorithm according to the local water depth. The wetting and drying process is simulated in a manner which complements the use of operator-splitting and two-stage numerical schemes. The numerical model was then applied to a hypothetical dam-break scenario, an experimental dam-break case and an extreme flooding event over the Toce River valley physical model. The predicted results are free of spurious oscillations for both sub- and super-critical flows, and the predictions compare favourably with the experimental measurements

The shallow water system plays an important role in the numerical simulation of oceanic models, coastal flows and dam-break floods. Several kinds of source terms can be taken into account in this model, such as the influence of bottom topography, Manning friction effects and Coriolis force. For large scale oceanic phenomena, the Coriolis force due to the Earth's rotation plays a central role since the atmospheric or oceanic circulations are frequently observed around the so-called geostrophic equilibrium which corresponds to the balance between the pressure gradient and the Coriolis source term. The ability of numerical schemes to well capture the lake at rest, has been widely studied. However, the geostrophic equilibrium issue, including the divergence free constraint on the velocity, is much more complex and only few works have been devoted to its preservation. In this manuscript, we design finite volume schemes that preserve the discrete geostrophic equilibrium in order to improve significantly the accuracy of numerical simulations of perturbations around this equilibrium. We first develop collocated and staggered schemes on rectangular and triangular meshes for a linearized model of the original shallow water system. The crucial common point of the various methods is to adapt and combine several strategies known as the Apparent Topography, the Low Mach and the Divergence Penalisation methods, in order to handle correctly the numerical diffusions involved in the schemes on different cell geometries, so that they do not destroy geostrophic equilibria. Finally, we extend these strategies to the non-linear case and show convincing numerical results.

In this paper, a numerical investigation of the two modes of heat transfer, natural convection and surface thermal radiation, in a tilted slender cavity such a collector is presented. The 2-D conservation of mass, momentum and energy are coupled with a radiative model through the boundaries and solved by the finite volume method. The studied parameters are: aspect ratios (8≤A≤16), inclination angles (15°≤≤35°) and Rayleigh numbers (104≤Ra≤106). The results indicated that the radiative surface radiation coupled with the natural convection modifies the flow patterns and the average heat transfer in the slender cavity between the absorber plate and the glass in the collector. The convective heat transfer coefficient and the radiative heat transfer coefficient as a function of the aspect ratio and the inclination angles are shown. It was found that the radiative heat transfer contributes more than 40% of the total heat transfer. A comparison between the present Nusselt numbers against the ones used for the design of solar collectors reported in the literature is presented.

Chapter 1:  Introduction: Definition of CFD, Application of CFD Technique, Main elements of a CFD code: 1-Pre-processor, 2-Solver, 3-Post-processor, Types of Fluids, Types of Fluid Motion, General Transport Equation of Fluid Flow, Examples.

Chapter 2: Steady State Diffusion: Introduction, 1D steady state diffusion (Cartesian Coordinates): 1-Grid generation , 2- Discretisation: Diffusion Coefficient, Gradient (Flux), Source term, 3-Solution of Discretized Equations – TDMA, Solved Examples: 1D steady state diffusion: Examples 1, 2 & 3, 2D steady state diffusion (Cartesian Coordinates), 3D steady state diffusion (Cartesian Coordinates), Solved Example 4, 1D steady state diffusion (Polar Coordinates), Home Work, 1D steady state diffusion (Spherical Coordinates), Projects.

Chapter 3: Transient Diffusion: 1D unsteady Heat Conduction (Cartesian Coordinates),: 1- Fully Explicit, 2- Crank-Nickalson, 3- Fully Implicit, Solved Example 1, Fully implicit time scheme for 2D and 3D unsteady Heat Conduction (Cartesian Coordinates), 1D Unsteady Heat Conduction (Polar Coordinates), Home Work 1, 1D Unsteady Heat Conduction (spherical Coordinates), Home Work 2, Projects.

Chapter 4: Steady State Convection-Diffusion: Introduction, 1D steady state convection-diffusion, the central differencing scheme, Example 1, Properties of discretisation schemes: 1-Conservativeness, 2-Boundedness, 3-Transportiveness, Assessment of the central differencing scheme, the upwind differencing scheme, Example 2, Assessment of the upwind differencing scheme, the hybrid differencing scheme, Example 3, Assessment of the hybrid differencing scheme, Hybrid differencing scheme for multi-dimensional convection-diffusion, The power-law scheme, Higher order differencing schemes for convection-diffusion problems: 1-Quadratic upwind differencing scheme (QUICK scheme), Example 4, Assessment of the QUICK scheme, stability of the QUICK scheme, Projects.

Chapter 5 : Transient Convection-Diffusion: Discretisation of Transient Convection-Diffusion problems, Solved Example of Transient Convection-Diffusion using QUICK scheme: Examples 1, Projects.

Chapter 6: Pressure-Velocity Coupling: Introduction, SIMPLE Algorithm for Pressure-Velocity Coupling in Steady State Flow, Flow chart for Steady State SIMPLE, Example 1, SIMPLE Algorithm for Pressure-Velocity Coupling in Transient flow, Flow chart for Transient SIMPLE. Projects.

APPENDIX A: Accuracy of a Flow Simulation.

Filling of an empty cryogenic vessel initially at atmospheric temperature with a cryogenic liquid at its saturation temperature will initiate a high temperature difference between the terminals of the annular space containing the thermal insulating material. This high temperature difference is behind the dependence of thermal properties of the thermal insulation material on its temperature. Based on the authors’ own research and classroom experience, this book contains two parts, in Part (I): the 1D cylindrical coordinates, non-linear partial differential equation of transient heat conduction through a temperature dependent thermal conductivity of a thermal insulation material is solved analytically using Kirchhoff’s transformation. In Part (II): the 2D body-fitted coordinates, non-linear partial differential equation of transient heat conduction through a temperature dependent thermal properties of a thermal insulation material that is subjected to a natural convection heat transfer boundary condition associated with a periodic change in ambient temperature and heat flux of solar radiation is simulated numerically using the fully implicit Finite Volume Method

Numerical simulations of two-dimensional laminar flow past a triangular cylinder placed in free-stream at low Reynolds number (10⩽Re⩽250) are performed. A finite volume method, second-order accurate in space and time, employing non-staggered arrangement of the variables with momentum interpolation for the pressure–velocity coupling is developed. Global mode analysis predicts Recr=39.9 which confirms the results of earlier studies. Vortex shedding phenomena is found to be similar to the square cylinder with no second bifurcation in the range of Re studied. A discussion on the time-averaged drag coefficient, rms of lift coefficient and Strouhal number is presented. Particle tracking and the instantaneous streaklines provide an excellent means of visualizing the von Kármán vortex street. Copyright © 2006 John Wiley & Sons, Ltd.

Turbulence and sediment transport models are incorporated into a three-dimensional hydrodynamics model to
investigate the mechanisms of morphologic evolution of scour holes within the Indian River Inlet, Delaware, USA.
The inlet bed had eroded slightly since stabilizing the channelwalls in late 1930s through 1976. The mean rate of
bed erosion roughly doubled as a response to anthropogenic activitieswithin the inlet such as the removal of ~80
piles remaining from an old bridge. Severe erosion near the in-water bridge supports and cofferdams for the
replacement bridge necessitated channel bed stabilization that alongwith the remained debris fromthe removal
of old bridge piles enhanced the growth of two deep scour holes on the bayside and oceanside of the bridge
cofferdams. Scour hole and channel bed evolution has decreased drastically since 1994. The present investigation
suggests that, initially, flowconcentration through the cofferdams of the replacement bridgewas themain reason
for scour hole development. Bed shear stress over the forward-facing slope of the scour hole entrains sediment
from the bed and extends the scour hole along the inlet and in the vertical direction. Flow separation within
the developed scour holes after channel bed stabilization enhances turbulence and appears to be the dominant
mechanism for further scour hole development.

In this article, mathematical modeling and numerical simulation of phase change materials (PCMs) used as latent heat thermal energy storage in a solar cooker have been conducted. The PCM to store thermal energy is packed in many small hollow cylinders and placed in a larger cylinder tank. Heat transfer fluid (HTF) which flows parallel to the PCM cylinder is used to distribute heat from the solar collector to the PCM storage unit and vice versa. A mathematical model describing the behaviour of temperature in the PCM and HTF  is used. Numerical solutions are obtained by transforming heat conduction equations of the PCM and HTF into enthalpy equation and solving it by using the Godunov method. Thermal performance during charging and discharging process of several selected PCMs is investigated. The simulation results showed that magnesium nitrate hexahydrate has the highest capacity to store solar thermal energy whereas erythritol can achieve the highest temperature history during charging time and at the first 54 minutes of discharging time.  The results provide an important information to design a solar cooker prototype equipped with thermal energy storage that has a good thermal performance.

Minimum structural fire resistance imposed by prescriptive design methods is usually of lesser value than the actual performance of an individual building under fire. The behaviour of the entire structure as a whole is not considered. Real fire conditions are not taken into account. Full-scale fire tests are also proven not to be economically feasible as standard procedure.
The work presented in this dissertation follows a performance-based approach, with the purpose of arriving at a numerical model capable of simulating the structural response of a building under fire conditions.
A field model for a compartment fire was created with computational fluid dynamics software Fire Dynamics Simulator. A ten-storey previously designed steel-concrete composite building was modelled using commercially available finite element analysis software ABAQUS. The effects of fire from the field model were simulated with a thermal-stress analysis.

The classical Chapman-Enskog expansion is performed for the recently proposed finite-volume formulation of lattice Boltzmann equation (LBE) method [D.V. Patil, K.N. Lakshmisha, Finite volume TVD formulation of lattice Boltzmann simulation on unstructured mesh, J. Comput. Phys. 228 (2009) 5262–5279]. First, a modified partial differential equation is derived from a numerical approximation of the discrete Boltzmann equation. Then, the multi-scale, small parameter expansion is followed to recover the continuity and the Navier–Stokes (NS) equations with additional error terms. The expression for apparent value of the kinematic viscosity is derived for finite-volume formulation under certain assumptions. The attenuation of a shear wave, Taylor–Green vortex flow and driven channel flow are studied to analyze the apparent viscosity relation.

In this paper, the volume of fluid (VOF) method in the OpenFOAM open-source computational fluid dynamics (CFD) package is used to investigate the coupled heat and mass transfer by mixed convection during the evaporation of water-thin film. &e liquid film is falling down on the left wall of a vertical channel and is subjected to a uniform heat flux density, whereas the right wall is assumed to be insulated and dry. &e gas mixture consists of air and water vapor. &e governing equations in the liquid and in the gas areas with the boundary conditions are solved by using the finite volume method. &e results which include temperature, velocity, and vapor mass fraction are presented. &e effect of heat flux density, liquid inlet temperature, and mass flow rate on the heat and mass transfer are also analyzed. Better liquid film evaporation is noted for the system with a higher heat flux density and inlet liquid temperature or a lower mass flow rate. &erefore, the VOF method describes well th...

Un elemento finito lineal con sección transversal constante puede adoptar cualquier orientación en el plano y sus extremos o nodos lo ligan al resto de los elementos. La energ a cinética (T ) y potencial (V ) de un elemento elástico dinámico son el basamento en la implementación del principio de Hamilton para la definición de un elemento finito. La definición de la energía cinética y potencial es el primer paso para la formulación variacional preliminar a la enunciación por elementos finitos que se utiliza para resolver, dígase, los problemas de mecanismos que se mueven en el plano utilizando la ecuación de Hamilton. El objetivo general consistió en definir la ecuación del movimiento de un elemento finito lineal plano elástico dinámico utilizando la ecuación de Hamilton, a partir de la lagrangiana (T –V ) obtenida con el uso de un polinomio de quinto y uno de primer grados, con ocho grados de libertad, cuatro en cada nodo, que representaron las deformaciones: axial (u(x)), transversal (w(x)), pendiente ((dw(x)/dx)) y curvatura ((d2w(x)/dx2)). La deformación debido al cizalleo transversal, insignificante comparado con la deformación flexional y la axial, la inercia rotatoria y las fuerzas friccionales en las uniones,fueron desestimadas con el fin de producir un elemento amigo. Los objetivos específicos fueron producir: (a) la matriz de masa de traslación [MD], (b) la matriz giroscópica de traslación [AD], (c) la matriz de rigidez total de traslación [KD], y (d) el vector de deformación (S). Como resultado se forjó la ecuación del movimiento de un elemento finito lineal plano elástico dinámico [MD]( ¨ S) − 2¨_[AD]( ˙S ) + {[K] − _˙2[MD] − ¨_[AD]}(S) = (Q) . Se concluyó que la ecuación obtenida variacionalmente con la aplicación del principio de Hamilton es un modelo cuyo procedimiento puede ser utilizado cuando se requiera aumentar el número de grados de libertad del modelo.

Abstract Second-order accurate projection methods for simulating time-dependent incompressible flows on cell-centred grids substantially belong to the class either of exact or approximate projections. In the exact method, the continuity constraint can be satisfied to machine-accuracy but the divergence and Laplacian operators show a four-dimension nullspace therefore spurious oscillating solutions can be introduced. In the approximate method, the continuity constraint is relaxed, the continuity equation being satisfied up to ...

In several papers of Bouchut, Bourdarias, Perthame and Coquel, Le Floch ((13), (7)...), a general methodology has been developed to construct second order flnite volume schemes for hyperbolic systems of conservation laws satisfying the entire family of entropy inequalities. This approach is mainly based on the construction of an entropy diminishing projection. Unfor- tunately, the explicit computation of this projection

This study presents the extension of the volume-of-fluid solver, interFoam, for improved accuracy and efficiency when modelling dynamic liquid-gas systems. Examples of these include the transportation of liquids, such as in the case of fuel carried onboard air- and spacecraft or liquid natural gas on tankers. As part of the development three extensions are considered: Firstly, a revised surface capturing formulation is proposed; secondly, a new weakly compressible volume-of-fluid formulation is presented; and lastly, a piecewise-linear interpolation of the pressure derivative is implemented. For the evaluation of this solver, a number of test cases with dynamic liquid-gas flows are considered where the results are compared with experimental measurements as well as numerical predictions from the current interFoam solver.

The purpose of this study is to investigate the influence of the aspect ratio and Grashof number on the convection diffusion phenomena associated with solid liquid phase transition processes during PCM solidification within a rectangular cavity. The applied two-dimensional physical model is based on the enthalpy method approach. The numerical solutions are obtained with finite volume method and validated with existing numerical and experimental studies of natural convection in phase change material. Numerical analysis are performed using water as phase change material, for an aspect ratio range 0.5 to 4. The left and right walls of the cavity are maintained at hot and cold temperatures, respectively, while the upper and lower surfaces are considered adiabatic. From this study, it is found that PCM initially liquid evolves to thermal stationary regime where solid and liquid regions are present. The solidification phase change process depends considerably on the thermal and geometrical parameters of the system. Additionally, it is demonstrated that the physical approach based on a purely conductive model remains valid to cases of low values of Grashof number and aspect ratio. The findings of this research may contribute to the elaboration of an efficient modeling of PCM heat storage systems.

This paper describes a method for simulation of viscous flows with a free surface around realistic hull forms with a transom, which has been developed based on a FINFLO RANS solver with a moving mesh. A dry-transom model is proposed and implemented for the treatment of flows off the transom. The bulk RANS flow with the artificial compressibility is solved by a cell-centred finite volume multigrid scheme and the free surface deformed by wave motions is tracked by satisfying the kinematic and dynamic free-surface boundary conditions on the actual location of the surface. The effects of turbulence on flows are evaluated with the Baldwin–Lomax turbulence model without a wall function. A test case is modern container ship model with a transom, the Hamburg Test Case. The calculated results are validated and they agree well with the measured results in terms of the free-surface waves and the total resistance coefficient. Furthermore, the numerical solutions successfully captured many important features of the complicated interaction of the free surface with viscous flows around transom stern ships. In addition, the convergence performance and the grid refinement studies are also investigated. Copyright © 2001 John Wiley & Sons, Ltd.