Solution of the Momentum Equation

In this section, particular pressure-based methods designed to solve the momentum equation are outlined. The numerical methods for solving the momentum component equations differ considerably from those designed to solve the generic scalar transport 

The survey of the numerical methods for single phase flow given here is, to a considerable extent, based on standard textbooks on CFD [141, 49, 202]. The elementary theory is included in this chapter to form a sound basis for the extended algorithms employed solving multiphase problems. The multiphase methods are presented in a sect 12.11. 

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Solution of the momentum equation requires specification of an initial condition and boundary conditions. Initially the value of w must be specified for all r. The most straightforward boundary-condition specification would call for specifying values of w at both walls for all time. In general, however, the boundary conditions can be more complex functions, related to the values and spatial derivatives of w at the walls. 

It is easy to see that changing the value of Ar produces a different V profile through the solution of the momentum equation. Then, in turn, different V profiles produce different u profiles through the solution of the continuity equation. It is possible to find a value of Ar such that the continuity equation is satisfied at the top boundary and the known axial-velocity boundary conditions is satisfied, that is, uj = f/iniet However, it is inefficient to carry out such an iteration explicitly. Rather, it it is more efficient to implement an implicit boundary condition. 

For pressure-based techniques, the lack of an independent equation for the pressure complicates the solution of the momentum equation. Furthermore, the continuity equation does not have a transient term in incompressible flows because the fluid transport properties are constant. The continuity reduces to a kinematic constraint on the velocity held. One possible approach is to construct the pressure field so as to guarantee satisfaction of the continuity equation. In this case, the momentum equation still determines the respective velocity components. A frequently used method to obtain an equation for the pressure is based on combining the two equations. This means that the continuity equation, which does not contain the pressure, is employed to determine the pressure. If we take the divergence of the momentum equation, the continuity equation can be used to simplify the resulting equation. 

Solutions of the momentum equation (Eq. 6.117) [45] yield velocity distributions generally similar to those of Fig. 6.19, and the skin friction parameter/" shown by the line labeled 1 in Fig. 6.21. The skin friction coefficient is given by... 

The solution of Eq. 10.55 describing the conservation of energy requires the solution of the momentum equation for a specified constitutive relationship. The previous section provides this information for a power-law fluid. This section will treat the fully developed heat transfer... 

Several algorithms do exist in the literature for numerical computation of fluid flow problems on the basis of primitive variables, in a finite volume framework. One of the most commonly used algorithms of this kind is the SIMPLE (semi-implicit method for pressure-linked equations) algorithm [2]. With reference to a generic staggered control volume for solution of the momentum equation for u (see Fig. 3) and with similar considerations for Ihe other velocity components, major steps of the SIMPLE algorithm can be summarized as follows ... 

It is of course possible to redo the simple extruder analysis with a pressure-dependent viscosity. Equation 4.7b defines the die characteristic equation. Solution of the momentum equation with a moving wall then leads directly to Equation 3.15b for the wall velocity as a function of the flow rate and system geometry, even when the viscosity is pressure dependent. 

For a complex, or even a simple flow simulation, there will be one equation of this form for each variable solved, in each cell in the domain. Furthermore, the equations are coupled, since for example, the solution of the momentum equations will affect the transport of every other scalar quantity. It is the job... 

S-3.3.5 Numerical Diffusion. Numerical diffusion is a source of error that is always present in finite volume CFD, owing to the fact that approximations are made during the process of discretization of the equations. It is so named because it presents itself as equivalent to an increase in the diffusion coefficient. Thus, in the solution of the momentum equation, the fluid will appear more viscous in the solution of the energy equation, the solution will appear to have a higher conductivity in the solution of the species equation, it will appear that the species diffusion coefficient is larger than in actual fact. These errors are most noticeable when diffusion is small in the actual problem definition. 

A work is presently made in order to obtain more realistic modelling of the relocation process. A solution of the momentum equation of the molten materials flowing along vertical structures is being developed. Capillarity, viscosity and gravity effects will be considered. 

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