The Finite Volume Method

The second derivative with respect toy tiien takes a simple form 

By contrast, the second derivative witii respect to jc is mnch more complex, 

Through this approach, we can employ a finite difference discretization on a regular grid in (f, ri) space however, the differential eqnation now involves more complex derivatives. The finite element method, described below, allows us to solve BVPs in complex geometries without performing such coordinate transformations (which are not always possible anyway). 

our focus has been on the finite difference method, which is easy to implement in domains of rather simple geometry. In complex domains, it is difficult to place a grid and keep track of neighbors when the grid points are required to he along the coordinate axes. Here, we discuss another method that is not subject to this condition. We again consider the 2-D Poisson equation but now instead of the microscopic equation 

To obtain a set of algebraic equations for the node field values, we apply the macroscopic balance to each cell, denoted as the control volmne that is centered on the nodal position (Xi,yj), 

The finite volume method has become a very popular method of deriving discretizations of partial differential equations because these schemes preserve the conservation properties of the differential equation better than the schemes based on the finite difference method. 

The main disadvantages of FVMs are low accuracy and low convergency rates. Compared to FDMs, the main disadvantage of the FVM is that methods of order higher than second are more difficult to develop in 3D because the FVM approach requires two levels of approximation considering the interpolation and integration processes. The FDM only requires approximations of the derivatives and interpolation. 

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The commercial CFD codes use the finite volume method, which was originally developed as a special finite difference formulation. The numerical algorithm consists of the following steps ... 

Versteeg, H. K. and Malalasekera, W., An Introduction to Computational Fluid Dynamics—The Finite Volume Method, Addison Wesley Longman Ltd., 1995. 

The finite volume method, a very eommon method for solving fluid flow problems (Versteeg and Malalasekera, 1995). The balanee equations are solved for eaeh grid eell using an iterative solution approaeh, as the underlying physieal phenomena are eomplex. 

Also a simulation of the flow field in the methanol-reforming reactor of Figure 2.21 by means of the finite-volume method shows that recirculation zones are formed in the flow distribution chamber (see Figure 2.22). One of the goals of the work focused on the development of a micro reformer was to design the flow manifold in such a way that the volume flows in the different reaction channels are approximately the same [113]. In spite of the recirculation zones found, for the chosen design a flow variation of about 2% between different channels was predicted from the CFD simulations. In the application under study a washcoat cata-... 

The equations for both laminar and turbulent flows, and the finite volume methods used to solve them, have been presented extensively in the literature (Patankar, 1980 Mathur and Murthy, 1997 Ranade, 2002 Fluent, 2003). The following summary focuses on aspects of particular concern for simulation of packed tubes and also those options chosen for our own work. 

The discretized equations of the finite volume method are solved through an iterative process. This can sometimes have difficulty converging, especially when the nonlinear terms play a strong role or when turbulence-related quantities such as k and s are changing rapidly, such as near a solid surface. To assist in convergence a relaxation factor can be introduced ... 

Mathematically, the PPDF method is based on the Finite Volume Method of solving full Favre averaged Navier-Stokes equations with the k-e model as a closure for the Reynolds stresses and a presumed PDF closure for the mean reaction rate. 

The finite volume methods have been used to discretised the partial differential equations of the model using the Simple method for pressure-velocity coupling and the second order upwind scheme to interpolate the variables on the surface of the control volume. The segregated solution algorithm was selected. The Reynolds stress turbulence model was used in this model due to the anisotropic nature of the turbulence in cyclones. Standard fluent wall functions were applied and high order discretisation schemes were also used. 

P 61] The numerical simulations were based on the solution of the incompressible Navier-Stokes equation and a convection-diffusion equation for a concentration field by means of the finite-volume method [152], The Einstein convention of summation over repeated indices was used. For pressure-velocity coupling, the SIMPLEC algorithm and for discretization of the species concentration equation the QUICK differencing scheme were applied. Hybrid and the central differencing schemes referred to velocities and pressure, respectively (commercial flow solvers CFX4 and CFX5). 

The computational fluid dynamics investigations listed here are all based on the so-called volume-of-fluid method (VOF) used to follow the dynamics of the disperse/ continuous phase interface. The VOF method is a technique that represents the interface between two fluids defining an F function. This function is chosen with a value of unity at any cell occupied by disperse phase and zero elsewhere. A unit value of F corresponds to a cell full of disperse phase, whereas a zero value indicates that the cell contains only continuous phase. Cells with F values between zero and one contain the liquid/liquid interface. In addition to the above continuity and Navier-Stokes equation solved by the finite-volume method, an equation governing the time dependence of the F function therefore has to be solved. A constant value of the interfacial tension is implemented in the summarized algorithm, however, the diffusion of emulsifier from continuous phase toward the droplet interface and its adsorption remains still an important issue and challenge in the computational fluid-dynamic framework. 

Raithby, G.D. Discussion of the finite-volume method for radiation, and its application using 3D unstructured meshes. Numerical Heat Transfer, Part B, 1999. 35, 389-405. 

This problem falls into a category of strongly coupled fluid-structure interaction (FSI) problems due to comparable stiffnesses of the container and its liquid content. Hence, accurate prediction of containers behaviour requires a liquid-container interaction model. Here, a two-system FSI model based on the Finite Volume Method is employed, and a good agreement is found between measured and predicted pressure and strain histories. 

C.J. Greenshields, H.G. Weller, A. Ivankovic, The Finite Volume Method for Coupled Fluid Flow and Stress Analysis, Computer Modeling and Smubtion in Engineering, 4, (1999), 213-218. 

A. Ivankovic, A. Karac, E. Dendrinos, K. Parker, Towards Early Diagnosis of Atherosclerosis The Finite Volume methods for Fluid-Structure Interaction, Biorheology 39 (2002), 401-407. 

This paper reports the mathematical modelling of electrochemical processes in the Soderberg aluminium electrolysis cell. We consider anode shape changes, variations of the potential distribution and formation of a gaseous layer under the anode surface. Evolution of the reactant concentrations is described by the system of diffusion-convection equations while the elliptic equation is solved for the Galvani potential. We compare its distribution with the C02 density and discuss the advantages of the finite volume method and the marker-and-cell approach for mathematical modelling of electrochemical reactions. 

We use the finite volume method for numerical approximation of the eqns (4)-(14). The whole area is treated as a set of rectangular blocks - see Fig.l... 

It follows from the above discussion and numerical results that even a simple convective-diffusive model of concentration behaviour mechanism gives realistic results and yields a satisfactory description of the formation of the gaseous layer under the anode surface. The model may be improved by adding the electrolyte circulation and electromagnetic forces yet we hope that it will not change the main conclusions. The finite volume method proves to be a flexible and sufficiently accurate numerical technique for solving both the equations for the Galvani potential and the reactant concentrations. The marker-and-cell approach makes it possible to outline the electrode surfaces easily. 

Following the choice of grid type, one has to select the approximations to be used in the discretization process. For the finite volume method, one has to select the methods of approximating surface and volume integrals. The choice of method of approximation influences the accuracy and computational costs. The number of nodes involved in approximation controls the memory requirements, speed of the code and difficulty in implementing the method in the computer program. More... 

Before discussing the finite volume method, it is worthwhile to examine the desired properties of the numerical solution method, which are summarized below ... 

In the finite volume method, discretized equations are obtained by integrating the governing transport equations over a finite control volume (CV). In this section, general aspects of the method are briefly discussed using a generic conservation equation for quantity, 0. Patankar (1980), Versteeg and Malalasekara (1995) and Ferziger and Peric (1995) may be referred to for a more detailed description. 

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