Finite-volume algorithm

Fig. 4.5 Nondimensional temperature profiles produced by viscous dissipation in the annular region between a moving rod and a stationary guide. The fluid is characterized by Pr = 4000, which is typical of lubricating oils. The rod and guide geometry is characterized by r lhr — 10, meaning that the rod radius is 10 gap thicknesses. The temperature profiles are parameterized by P = -j jr—. These solutions were generated by a 12-node finite-volume algorithm implemented in Excel.

The turbulent gas/liquid flow in baffled tanks with turbine stirrer can be predicted. A mathematical model has been developed for turbulent, dispersed G/L flow. The time-averaged two phase momentum equations were solved by using a finite volume algorithm. The turbulent stresses were simulated with a K-fi-model. The distribution of gas around the stirrer blades is predicted for the first time. This model also enables an a priori prediction of the drop in the power dissipated by the stirrer in the presence of gas. Predicted flow characteristics for the gas/liquid dispersion show good agreement with the experimental data. 

The alternative interpretation is that modeling and numerical issues should deliberately be combined. The governing equations are then solved in physical space often using a second order accurate finite difference or finite volume algorithm. 

Basic Finite Volume Algorithms Used in Computational Fluid Dynamics... 

As alternatives to the MINI element, Rajupalem et al. (1997) and Talwar et al. (1998) used an equal-order velocity-pressure finite element formulation, as proposed by Rice et al. (1986), in their 3D injection molding simulation, and Chang and Yang (2001) used a 3D finite-volume approach based on the SIMPLE finite volume algorithm of Patankar (1980). Since the finite volume method does not... 

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 ... 

In summary, DQMOM is a numerical method for solving the Eulerian joint PDF transport equation using standard numerical algorithms (e.g., finite-difference or finite-volume codes). The method works by forcing the lower-order moments to agree with the corresponding transport equations. For unbounded joint PDFs, DQMOM can be applied... 

When the boundary-layer approximations are applicable, the characteristics of the steady-state governing equations change from elliptic to parabolic. This is a huge simplification, leading to efficient computational algorithms. After finite-difference or finite-volume discretization, the resulting problem may be solved numerically by the method of lines as a differential-algebraic system. 

Figure E.l represents a highly simplified view of an ideal structure for an application program. The boxes with the rounded borders represent those functions that are problem specific, while the square-comer boxes represent those functions that can be relegated to problem-independent software. This structure is well-suited to problems that are mathematically systems of nonlinear algebraic equations, ordinary differential equation initiator boundary-value problems, or parabolic partial differential equations. In these cases the problem-independent mathematical software is usually written in the form of a subroutine that in turn calls a user-supplied subroutine to define the system of equations. Of course, the analyst must write the subroutine that describes the particular system of equations. Moreover, for most numerical-solution algorithms, the system of equations must be written in a discrete form (e.g., a finite-volume representation). However, the equation-defining sub-...

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. 

Coupled methods (transport model coupled with hydrogeochemical code) For coupled models solving the transport equation can be done by means of the finite-difference method (and finite volumes) and of the finite-elements method. Algorithms based on the principle of particle tracking (or random walk), as for instance the method of characteristics (MOC), have the advantage of not being prone to numerical dispersion (see 1.3.3.4.1). 

A reactor engineer frequently encounters turbulent, multiphase and reactive flows, which are more complex than those discussed in the previous chapter. In this chapter, modifications or special techniques/algorithms required to extend the finite volume method to handle such complexities are discussed. In addition, some of the practical issues involved in carrying out numerical simulations of complex flow models are also discussed. 

In this approach, the finite volume methods discussed in the previous chapter can be applied to simulate the continuous fluid (in a Eulerian framework). Various algorithms for treating pressure-velocity coupling, and the discussion on other numerical issues like discretization schemes are applicable. The usual interpolation practices (discussed in the previous chapter) can be used. When solving equations of motion for a continuous fluid in the presence of the dispersed phase, the major differences will be (1) consideration of phase volume fraction in calculation of convective and diffusive terms, and (2) calculation of additional source terms due to the presence of dispersed phase particles. For the calculation of phase volume fraction and additional source terms due to dispersed phase particles, it is necessary to calculate trajectories of the dispersed phase particles, in addition to solving the equations of motion of the continuous phase. 

All computations are performed using the code OpenFOAM [17], an open source computational fluid dynamics (CFD) toolbox, utilizing a cell-center-based finite volume method on a fixed unstructured numerical grid and employing the solution procedure based on the pressure implicit with splitting of operators (PISO) algorithm for coupling between pressure and velocity in transient flows [18]. 

Prefer numerical algorithms that satisfy conservation laws (such as finite volume methods) over algorithms that do not force the numerical solution to satisfy conservation laws (such as finite difference methods). 

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... 

Most of the present finite volume procedures that are used to solve the continuity and momentum equations are derived from the approximate implicit interphase slip algorithm (IPSA) framework proposed by Spalding [176, 177, 178, 180]. Rather than representing a particular algorithm and... 

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