Finite Volume Methods
As in the previous chapter, we consider only the generic conservation equation for a quantity ϕ and assume that the velocity field and all fluid properties are known. The finite volume method uses the integral form of the conservation equation as the star
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1.1 Introduction Fluids are substances whose molecular structure offers no resistance to external shear forces: even the smallest force causes deformation of a fluid particle. Although a significant distinction exists between liquids and gases, both types of fluids obey the same laws of motion. In most cases of interest, a fluid can be regarded as continuum, i.e. a continuous substance. Fluid flow is caused by the action of externally applied forces. Common driving forces include pressure differences, gravity, shear, rotation, and surface tension. They can be classified as surface forces (e.g. the shear force due to wind blowing above the ocean or pressure and shear forces created by a movement of a rigid wall relative to the fluid) and body forces (e.g. gravity and forces induced by rotation). While all fluids behave similarly under action of forces, their macroscopic properties differ considerably. These properties must be known if one is to study fluid motion; the most important properties of simple fluids are the density and viscosity. Others, such as Prandtl number, specific heat, and surface tension affect fluid flows only under certain conditions, e.g. when there are large temperature differences. Fluid properties are functions of other thermodynamic variables (e.g. temperature and pressure); although it is possible to estimate some of them from statistical mechanics or kinetic theory, they are usually obtained by laboratory measurement. Fluid mechanics is a very broad field. A small library of books would be required to cover all of the topics that could be included in it. In this book we shall be interested mainly in flows of interest to mechanical engineers but even that is a very broad area so we shall try to classify the types of problems that may be encountered. A more mathematical, but less complete, version of this scheme will be found in Sect. 1.8. The speed of a flow affects its properties in a number of ways. At low enough speeds, the inertia of the fluid may be ignored and we have creeping flow. This regime is of importance in flows containing small particles (suspensions), in flows through porous media or in narrow passages (coating techniques, micro-devices). As the speed is increased, inertia becomes important but each fluid particle follows a smooth trajectory; the flow is then said to be laminar. Further increases in speed may lead to instability that J. H. Ferziger et al., Computational Methods for Fluid Dynamics © Springer-Verlag Berlin Heidelberg 2002
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4. Finite Volume Methods
The advantage of the first approach is that the nodal value represents the mean over the CV volume to higher accuracy (second order) than in the second approach, since the node is located at the centroid of the CV. The advantage of the second approach is that CDS approximations of derivatives at CV faces are more accurate when the face is midway between two nodes. The first variant is used more often and will be adopted in this book. There are several other specialized variants of FV-type methods (cellvertex s
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