Finite volume formulation

Fluent is a commercially available CFD code which utilises the finite volume formulation to carry out coupled or segregated calculations (with reference to the conservation of mass, momentum and energy equations). It is ideally suited for incompressible to mildly compressible flows. The conservation of mass, momentum and energy in fluid flows are expressed in terms of non-linear partial differential equations which defy solution by analytical means. The solution of these equations has been made possible by the advent of powerful workstations, opening avenues towards the calculation of complicated flow fields with relative ease. 

In these equations fi is the coluirm mass of dry air, V is the velocity (u, v, w), and (jf) is a scalar mixing ratio. These equations are discretized in a finite volume formulation, and as a result the model exactly (to machine roundoff) conserves mass and scalar mass. The discrete model transport is also consistent (the discrete scalar conservation equation collapses to the mass conservation equation when = 1) and preserves tracer correlations (c.f. Lin and Rood (1996)). The ARW model uses a spatially 5th order evaluation of the horizontal flux divergence (advection) in the scalar conservation equation and a 3rd order evaluation of the vertical flux divergence coupled with the 3rd order Runge-Kutta time integration scheme. The time integration scheme and the advection scheme is described in Wicker and Skamarock (2002). Skamarock et al. (2005) also modified the advection to allow for positive definite transport. 

The finite volume formulation for diffusion-type problems can be extended for solving the... 

To illustrate the discretization of a typical transport equation using the finite-volume formulation (Patankar, 1980 Versteeg and Malalasekera, 1995), a generalized scalar equation can be used with the rectangular control volume shown in Figure 5-6a. The scalar equation has the form... 

The numerical procedure for solving the governing equations was based on the Inter-Phase Slip Algorithm (IPSA) [15]. This is an iterative proc ure, operating on a Finite-Volume formulation of the conservation equations for mass and momentum for the two phases. CHAM LTD, UK, incorporates the three-dimensional numerical solver and grid generator that were employed in this work in the PHOENICS software. The numerical models and their solution are described by Spalding [15],... 

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 ... 

The boundedness of (<5r)n is to be understood in the sense that it remains an infinitesimal of the same order. We underscore the fact that, in the formulation of the frozenness theorem for vm = 0, the field or H/p change proportionally to the infinitesimal vector distance between infinitely close fluid particles. Therefore, a rapid dynamo is possible even in the case when the motion occurs in a finite volume. 

The boundary conditions are defined in the same way as with the flow analysis network. The nodes whose control volumes are empty or partially filled are assigned a zero pressure, and the gate nodes are either assigned an injection pressure or an injection volume flow rate. Just as is the case with flow analysis network, a mass balance about each nodal control volume will lead to a linear set of algebraic equations, identical to the set finite element formulation of Poisson s or Laplace s equation. The mass balance (volume balance for incompressible fluids) is given by... 

Based on the control volume approach and using the three-dimensional finite element formulations for heat conduction with convection and momentum balance for non-Newtonian fluids presented earlier, Turng and Kim [10] and [17] developed a three-dimensional mold filling simulation using 4-noded tetrahedral elements. The nodal control volumes are defined by surfaces that connect element centroids and sides as schematically depicted in Fig. 9.33. 

Proceeding from an Ogden-type material formulation, which is extended towards an inelastic porous media application, volumetric extension terms are developed which describe the finite volume change including the concept of a volumetric compaction point. Thus, the equilibrium part of the mechanical... 

The radiative source term is a discretized formulation of the net radiant absorption for each volume zone which may be incorporated as a source term into numerical approximations for the generalized energy equation. As such, it permits formulation of energy balances on each zone that may include conductive and convective heat transfer. For K—> 0, GS —> 0, and GG —> 0 leading to S —> On. When K 0 and S = 0N, the gas is said to be in a state of radiative equilibrium. In the notation usually associated with the discrete ordinate (DO) and finite volume (FV) methods, see Modest (op. cit., Chap. 16), one would write S /V, = K[G - 4- g] = Here H. = G/4 is the average flux... 

Note that the boundary conditions have no effect on the finite difference formulation of interior nodes of the medium. This is not surprising since the control volume used in the development of the formulation does not involve any part of the boundary. You may recall that the boundary conditions had no effect on the differential equation of heat conduction in the medium either. 

The assumed direction of heat transfer at surfaces of a volume element has no effect on the finite difference formulation. 

Node 2 is a boundaiy node subjected to convection, and the finite difference formulation at that node is obtained by wriling an energy balance on the volume element of thickness Ax/2 at that boundary by assuming heat transfer to be into the medium at all sides ... 

The development of finite difference formulation of boundary nodes in two- (or three-) dimensional problems is similar to the development in the one-dimensional case discussed earlier. Again, the region is partitioned between the nodes by forming volume elements around the nodes, and an energy balance is written for each boundary node. Various boundary conditions can be handled as discussed for a plane wall, except that the volume elements ill the two-dimensional case involve heat transfer in the y-direction as well as the x-direction. Insulated surfaces can still be viewed as mirrors, and the... 

The exterior surface of the Trombe v/ail is subjected to convection as well as to heat flux. The explicit finite difference formulation at that boundary is obtained hy writing an energy balance on the volume element represented by node 5,... 

The finite difference formulation of transieiii heat conduction problems is based on an energy balance that also accounts for tire variation of the energy content of the volume element during a time interval At. The heat transfer and heat generation terms are expressed at the previous time. step fin the explicit method, and at the new time step i I 1 in the implicit method. For a general node III, the finite difference formulations are expressed as... 

C Define these terms used in the finite difference formulation node, nodal network, volume element, nodal spacing, and difference equation. 

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