Finite Volume Methods

This work presents a three-dimensional CFD study of a two-phase flow field in a Gas-Liquid Cylindrical Cyclone (GLCC) using CFX4.3 TM , a commercial code based on the finite volume method. The numerical analysis was made for air-water mixtures at near atmospheric conditions, while both liquid and gas flow rates were changed. The two-phase flow behavior is modeled using an Eulerian-Eulerian approach, considering both phases as an interpenetrating continuum. This method computes the inter-phase phenomena by including a source term in the momentum equation to consider the drag between the liquid and gas phases. The gas phase is modeled as a bimodal bubble size distribution to allow for the presence of free-and entrapment gas, simultaneously. The results (interface vortex shape and liquid angular velocity) show a reasonable match with experimental data. The CFD technique here proposed, demonstrates to satisfactorily reproduce angular velocities of the phases and their spatial distribution inside the GLCC. The experiments showed gas volume fractions smaller than 150 ppm along the liquid exit, however it was not possible to reproduce numerically these small volume fractions, since they had approximately the same order of magnitude of the error in the numerical method.

The purpose of the present work is to investigate the effect of magnetic field on laminar natural-convection heat transfer inside a partitioned enclosure. In our formulation of non linear governing equations, mass, momentum, and energy are applied to the partitioned enclosure. A finite volume code based on PATANKAR's SIMPLER method is utilized. Results are obtained for different Rayleigh numbers (Ra), Hartmann numbers (Ha) and baffle positions while the Prandtl number is constant (Pr = 0 733). Results show that as Ra increases the maximum absolute value of the stream function increases and causes the flow to move faster. Also, as Hartman number increases (magnetic field intensifies) the natural convection suppresses and therefore the maximum absolute value of the stream function and Nusselt number reduces. Furthermore the partition causes the flow to separate and to form two vortexes inside the enclosure.

Nanofluids are liquid/solid suspensions with higher thermal conductivity, compared to common working fluids. In recent years, the application of these fluids in electronic cooling systems seems prospective. In the present study, the laminar mixed convection heat transfer of different water-copper nanofluids through an inclined ribbed microchannel-as a common electronic cooling system in industry-was investigated numerically, using a finite volume method. The middle section of microchannel's right wall was ribbed, and at a higher temperature compared to entrance fluid. The modeling was carried out for Reynolds number of 50, Richardson numbers from 0.1 to 10, inclination angles ranging from 0° to 90°, and nanoparticles' volume fractions of 0.0-0.04. The influences of nanoparticle volume concentration, inclination angle, buoyancy and shear forces, and rib's shape on the hydraulics and thermal behavior of nanofluid flow were studied. The results were portrayed in terms of pressure, temperature, coefficient of friction, and Nusselt number profiles as well as streamlines and isotherm contours. The model validation was found to be in excellent accords with experimental and numerical results from other previous studies. The results indicated that at low Reynolds' flows, the gravity has effects on the heat transfer and fluid phenomena considerably; similarly, with inclination angle and nanoparticle volume fraction, the heat transfer is enhanced by increasing the Richardson number, but resulting in a less value of friction coefficient. The results also represented that for specific Reynolds (Re) and Richardson (Ri) numbers, heat transfer and pressure drop augmented by increasing the inclination angle or volume fraction of nanoparticles. With regard to the coefficient of friction, its value decreased by adding less nanoparticles to the fluid or by increasing the inclination angle of the microchannel.

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

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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.

This is to certify that the work in the thesis entitled "Study of Fluid Flow and Heat Transfer in Branched Pipes" submitted by Swarnendu Ganguly (Roll Number: 116ME0442) in partial fulfillment of the requirements for the degree of Bachelor of Technology in Mechanical Engineering at the National Institute of Technology Rourkela is an authentic record of research work done by him in this department under my guidance and supervision.

This paper addresses turbulent natural convection of air confined in an isosceles triangular enclosure representing conventional attic spaces of houses and buildings with pitched roofs and horizontally suspended ceilings. The values to be considered are H = 0.86 m and 2.73 m for the height while the values 1.72 m and 5.46 m were assigned to the width, W, such as the aspect ratio H/W remains 0.5. The third dimension of the cavity is considered long enough for the flow to be considered 2D. The base wall is heated at 20°C and the inclined walls are cooled at 0°C. This combination of factors leads to large Rayleigh numbers equal to 1.58 · 10 9 and 5 · 10 10 . Turbulence is modeled by a low-Reynolds-number k-e model. The system of governing equations, subject to the proper boundary conditions is solved with the finite volume method. Second-order-accurate QUICK and SIMPLE schemes were used for the discretization of the convective terms and the pressure-velocity coupling, respectively. The velocity and temperature distributions were calculated at different locations in the cavity and their mean quantities are presented. The local and average Nusselt numbers and the wall shear stresses are also presented. Since to the knowledge of the authors, no previous results on turbulent thermal convection in this geometry exist, the validation of the numerical code was performed by comparing velocity and temperature profiles against recent experimental measurements, obtained for a square cavity. Satisfactory agreement was observed.

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.

In the present work, two-dimensional mixed convection fluid flow and heat transfer of water-(Cu, Ag, Al 2 O 3 and TiO 2 ) nanofluids in a two-sided facing lid-driven cavity partially heated from below have been investigated numerically. Two discrete heat sources are located on the bottom wall of the enclosure; however, the vertical moving walls and the ceiling are cooled at constant temperature. The remaining boundary parts of the bottom wall are kept insulated. The flow is driven by the moving two facing vertical walls in the same direction and the buoyancy force. The governing equations are solved using a second order accurate finite volume approach. The effects of the monitoring parameters in given ranges such as Reynolds ð1 6 Re 6 100Þ and Richardson numbers ð1 6 Ri 6 20Þ, solid volume fraction ð0 6 u 6 0:2Þ, the nanoparticles materials as well as the two heat sources positions are investigated. The conducted benchmark study leads to excellent accordance with previous findings. The present study analyzes and discusses the flow patterns (streamlines structures and isotherms distributions) set up by the competition between the forced flow driven by the moving walls and the buoyancy force effects, and the heat transfer rate quantified by the averaged Nusselt number along the heat source. It was found that significant heat transfer enhancement can be obtained: (i) increasing Ri at high Reynolds number (Re = 100) results in up-to 20% augmentation of heat transfer rate for all Cu volume fractions; (ii) increasing the volume fraction u, a maximum heat transfer rate increase of 47.010% is reached with Cu suspensions for u = 0.2 and Ri = 1, while a minimum increase of 7.059% is observed for TiO 2 -water nanofluid at Ri = 10 and u = 0.05; (iii) a highest heat transfer enhancement occurs when heat sources move toward the two vertical moving walls, while a lower heat transfer is obtained for heat sources located at bottom wall center.

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.

A second-order accurate scheme for the Cartesian cut-cell method developed previously by the authors [Ji H, Lien F-S, Yee E. Comput. Methods Appl. Mech. Eng. 198 (2008), 432] is generalized for application to both two-and three-dimensional inviscid compressible flow problems. A cell-merging approach is used to address the so-called ''small cell" problem that has plagued Cartesian cut-cell methods. In the present cell-merging approach, the conservative variables are stored at the cut-cell centroid (including the nonmerged and merged cut-cells) rather than at the Cartesian cell center. Although this approach results in a more complicated search algorithm for the determination of the neighbor cells (required for the computation of the spatial gradients of the conservative variables), this approach enables the straightforward formulation of a higher than first-order accurate discretization scheme in the vicinity of the (complex and irregular) internal boundaries of the flow domain. Six test cases (including detonation problems) are used to demonstrate the accuracy and capability of the adaptive cut-cell method, for which both mesh refinement and derefinement techniques are employed in the case of an unsteady shock diffraction problem.

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 the present work, two-dimensional mixed convection fluid flow and heat transfer of water-(Cu, Ag, Al 2 O 3 and TiO 2 ) nanofluids in a two-sided facing lid-driven cavity partially heated from below have been investigated numerically. Two discrete heat sources are located on the bottom wall of the enclosure; however, the vertical moving walls and the ceiling are cooled at constant temperature. The remaining boundary parts of the bottom wall are kept insulated. The flow is driven by the moving two facing vertical walls in the same direction and the buoyancy force. The governing equations are solved using a second order accurate finite volume approach. The effects of the monitoring parameters in given ranges such as Reynolds ð1 6 Re 6 100Þ and Richardson numbers ð1 6 Ri 6 20Þ, solid volume fraction ð0 6 u 6 0:2Þ, the nanoparticles materials as well as the two heat sources positions are investigated. The conducted benchmark study leads to excellent accordance with previous findings. The present study analyzes and discusses the flow patterns (streamlines structures and isotherms distributions) set up by the competition between the forced flow driven by the moving walls and the buoyancy force effects, and the heat transfer rate quantified by the averaged Nusselt number along the heat source. It was found that significant heat transfer enhancement can be obtained: (i) increasing Ri at high Reynolds number (Re = 100) results in up-to 20% augmentation of heat transfer rate for all Cu volume fractions; (ii) increasing the volume fraction u, a maximum heat transfer rate increase of 47.010% is reached with Cu suspensions for u = 0.2 and Ri = 1, while a minimum increase of 7.059% is observed for TiO 2 -water nanofluid at Ri = 10 and u = 0.05; (iii) a highest heat transfer enhancement occurs when heat sources move toward the two vertical moving walls, while a lower heat transfer is obtained for heat sources located at bottom wall center.

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 finite volume method (FVM) was studied in Li and Wang with the error analysis in H 1 norm, and is combined with other methods, such as the finite element method (FEM) and the collocation Trefftz method (CTM) . In this paper, the renovated FVM is developed for the anisotropic diffusion problem, where there exist the mixed derivatives ∂ 2 u ∂x∂y . The computational algorithms are provided, and new partitions are proposed for existing obtuse triangles. The new FVM in this paper is as flexible as FEM, but it has two remarkable advantages over FEM: (1) The discrete algebraic equations can be formulated simple and straightforward; (2) The conservative law of flux is always obeyed for the anisotropic diffusion problem. Hence, the renovated FVM may find wide application in engineering problems.

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.

A numerical simulation of the flow past a circular cylinder which is able to oscillate transversely to the incident stream is presented in this paper for a fixed Reynolds number equal to 100. The 2D Navier-Stokes equations are solved by a finite volume method with an industrial CFD code in which a coupling procedure has been implemented in order to obtain the cylinder displacement. A preliminary work is first conducted for a fixed cylinder to check the wake characteristics for Reynolds numbers smaller than 150 in the laminar regime. The Strouhal frequency f S and the aerodynamic coefficients are thus controlled among other parameters. Simulations are then performed with forced oscillations characterized by the frequency ratio F = f 0 /f S , where f 0 is the forced oscillation frequency, and by the adimensional amplitude A. The wake characteristics are analyzed using the ti me series of the fluctuating aerodynamic coefficients and their power spectral densities (PSD). The frequency content is then linked to the shape of the phase portraits and to the vortex shedding mode. By choosing interesting couples (A, F), different vortex shedding modes have been observed, which are similar to those of the Williamson-Roshko map. A second batch of simulations involving free vibrations (so-called vortex-induced vibrations or VIV) is finally carried out. Oscillations of the cylinder are now directly induced by the vortex shedding process in the wake and therefore, the time integration of the motion is realized by an explicit staggered algorithm which provides the cylinder displacement according to the aerodynamic charges exerted on the cylinder wall. Amplitude and frequency response of the cylinder are thus investigated over a wide range of reduced velocities to observe the different phenomena at stake. In particular, the vortex shedding modes have also been related to the frequency response observed and our results at Re = 100 show a very good agreement with other studies using different numerical approaches.

Numerical techniques development for the modeling and simulation of free surface flows has generated great interests over the last decades. In hydraulic engineering, the objectives include the predictions of dam break waves' propagation, fluvial floods and other catastrophic flooding phenomenon, the modeling of estuarine and coastal circulations, and the design and optimization of hydraulic structures. Most of the flooding events involve wetting and drying lands which are critical for the numerical modeling, especially when dealing with complex topographies. Extreme slopes and abrupt changes in irregular geometries, have often led to significant numerical errors and stability difficulties, and these are more critical for propagations over complex dry beds. This paper presents a simple and efficient numerical model for the wetting and drying effects over complex bathymetries. An overview of the key methods that have been suggested since the pioneering studies is first presented. A 2-D cell-centered finite-volume scheme is then proposed for solving the shallow-water equations using both structured and unstructured fixed meshes. Steady state C-property and global mass conservation properties are satisfied using appropriate numerical fluxes and wet/ dry interfaces treatments. The resulting numerical model proved stable and robust and was validated through some benchmarks tests, including comparisons with exact solutions and experimental data, and a real case of wetting-drying simulation in a portion of the river "Rivière des Prairies" in a suburb of Laval, Quebec.

Finite volume discretizations on unstructured grids that are more than second-order-accurate have not yet gained wide acceptance. This is due to the high computational cost and memory requirements of the proposed schemes and the technical difficulties in achieving a high-order-accurate solution. It is especially true when considering flows in complex geometries. In this paper, a third-order cell-centered finite volume discretization is detailed that is based on quadratic data recovery, which is computationally efficient and has optimal memory requirements. The densitybased flow-solver discretization and solution algorithm are described briefly, followed by a detailed presentation of the least-squares-based multistep quadratic data-reconstruction methodology proposed here. The paper is concluded with numerical examples to verify the accuracy and numerical efficiency of this high-order finite volume discretization.

The discontinuous control-volume/finite-element method is applied to the one-dimensional advection-diffusion equation. The aforementioned methodology is relatively novel and has been mainly applied for the solution of pure-advection problems. This work focuses on the main features of an accurate representation of the diffusion operator, which are investigated both by Fourier analysis and numerical experiments. A mixed formulation is followed, where the constitutive equation for the diffusive flux is not substituted into the conservation equation for the transported scalar. The Fourier analysis of a linear, diffusion problem shows that the resolution error is both dispersive and dissipative, in contrast with the purely dissipative error of the traditional continuous Galerkin approximation.

A coupled solver was developed to solve the species conservation equations on an unstructured mesh with implicit spatial as well as species-to-species coupling. First, the computational domain was decomposed into sub-domains comprised of geometrically contiguous cells-a process similar to additive Schwarz decomposition. This was done using the binary spatial partitioning algorithm. Following this step, for each sub-domain, the discretized equations were developed using the finite-volume method, and solved using an iterative solver based on Krylov sub-space iterations, that is, the pre-conditioned generalized minimum residual solver. Overall (outer) iterations were then performed to treat explicitness at sub-domain interfaces and nonlinearities in the governing equations. The solver is demonstrated for both two-dimensional and three-dimensional geometries for laminar methane-air flame calculations with 6 species and 2 reaction steps, and for catalytic methane-air combustion with 19 species and 24 reaction steps. It was found that the best performance is manifested for sub-domain size of 2000 cells or more, the exact number depending on the problem at hand. The overall gain in computational efficiency was found to be a factor of 2-5 over the block (coupled) Gauss-Seidel procedure. All calculations were performed on a single processor machine. The largest calculations were performed for about 355 000 cells (4.6 million unknowns) and required 900 MB of peak runtime memory and 19 h of CPU on a single processor.

In this paper, we extend our previous work [A. Alharbi and S. Naire, An adaptive moving mesh method for thin film flow equations with surface tension, J. Computational and Applied Mathematics, 319 (2017), pp. 365-384.] on a one-dimensional r-adaptive moving mesh technique based on a mesh density function and moving mesh partial differential equations (MMPDEs) to two dimensions. As a test problem, we consider the gravitydriven thin film flow down an inclined and pre-wetted plane including surface tension and a moving contact line. This technique accurately captures and resolves the moving contact line and associated fingering instability. Moreover, the computational effort is hugely reduced in comparison to a fixed uniform mesh. Highlights (for review)

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.

In Lagrangian-Eulerian (LE) simulations of two-way coupled particle-laden flows, the dispersed phase is represented either by real particles or by computational particles. In traditional LE (TLE) simulations, each computational particle is assigned a constant statistical weight, which is defined as the expected number of real particles represented by a computational particle. If the spatial distribution of particles becomes highly non-uniform due to particle-fluid or particle-particle interactions, then TLE simulations fail to yield numerically converged solutions due to high statistical error in regions with few particles. In this work, a particle-laden lid-driven cavity flow is solved on progressively refined grids to demonstrate the inability of TLE simulations to yield numerically converged estimates for the mean interphase momentum transfer term. We propose an improved LE simulation (ILE) method that remedies the above limitation of TLE simulations. In the ILE method, the statistical weights are evolved such that the same physical problem is simulated, but the number density of computational particles is maintained nearuniform throughout the simulation, resulting in statistical error that remains nearly constant with grid refinement. The evolution of statistical weights is rigorously justified by deriving the consistency conditions arising from the requirement that the resulting computational ensemble correspond to a statistical description of the same physical problem with real particles. The same particle-laden lid-driven cavity flow is solved on progressively refined grids to demonstrate the ability of ILE simulation to achieve numerically converged estimates for the mean interphase momentum transfer term. The accuracy of the ILE method is quantified using a test problem that admits an analytical solution for the mean interphase momentum transfer term. In order to improve the accuracy of numerical estimates of the mean interphase momentum transfer term, an improved estimator is proposed to replace the conventional estimator. The improved estimator results in more accurate estimates that converge faster than those obtained using the conventional estimator. The ILE simulation method along with the improved estimator is recommended for accurate and numerically convergent LE simulations. (S. Subramaniam). 1 By particle we mean any dispersed-phase element, including solid particles, droplets and bubbles. 2 We use the term 'estimate' in the statistical sense, just as the sample mean