Basic Finite Volume Algorithms Used in Computational Fluid Dynamics

6 Basic Finite Volume Algorithms Used in Computational Fluid Dynamics 

The purpose of this section is to outline the design of the basic finite volume solution algorithms used in computational fluid d3mamics. Other methods like finite difference, finite elements and spectral methods have been in widespread use in computational fluid d3mamics for years. However, only finite volume 

The pressure-based method was introduced by Harlow and Welch [67] and Chorin [30] for the calculation of unsteady incompressible viscous flows (parabolic equations). In Chorines fractional step method, an incomplete form of the momentum equations is integrated at each time step to 3ueld an approximate velocity field, which will in general not be divergence free, then a correction is applied to that velocity field to produce a divergence free velocity field. The correction to the velocity field is an orthogonal projection in the sense that it projects the initial velocity field into the divergence free 

Incompressible steady flows are commonly solved by pressure-based methods and methods based on the concept of artificial compressibility [183, 45]. The extension of pressure correction methods to steady flows, generally elliptic equations, has been performed by Patankar and Spalding [140] and Patankar [141]. The artificial compressibility method for calculating steady incompressible flows was proposed by Chorin [29]. In this method, an artificial compressibility term is introduced in the continuity equation, and the unsteady terms in the momentum equations are retained. Hence, the system of equations becomes hj perbolic and many of the methods developed for h3rperbolic systems can be applied. 

Generally, the design of modern solution algorithms for fluid flow problems is associated with the choice of primitive variables, the grid arrangement, and the solution approach [133]. In the class of pressure-based solution algorithms, both fully coupled and segregated approaches have been proposed. In the coupled approach, the discretized forms of the momentum and continuity 

7 Elements of the Finite Volume Method for Flow Simulations 

In this section, a survey of the basic elements of the flnite volume method, as applied to single phase flows, is provided [56, 164, 196, 248, 249]. The numerical issues considered are the approximations of surface and volume integrals, time discretizations, and space discretization of diffusive and convective (or advective) terms. 

Consider the generic transport equation for the property and assume that the velocity field and the fluid properties are known. The starting point for the FVM is the integral form of the balance equation  

The finite volume integration of (12.56) over a grid volume is essentially the same for both steady and dynamic systems to treat convection, diffusion and source terms. The first observation is that for steady problems the transient term vanishes, and the finite volume integration consists of the space dimensions only. It makes sense to consider the common features first. 

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