Elements of the Finite Volume Method for Flow Simulations

Elements of the Finite Volume Method for Flow Simulations... 

In this example, one periodic element (a cross-over) of the laboratory scale version of Katapak -S was selected for the detailed CFD simulation with CFX-5. This solver uses the finite volume discretization method in combination with hybrid unstructured grids. Around 1,100 spherical particles of 1 mm diameter were included in the computational domain. As the liquid flows through the catalyst-filled channels at operating conditions below the load point (cf. Moritz and Hasse, 1999), permeability of the channel walls made of the wire mesh is not taken into account by this particular model. The catalyst-filled channels are considered fully wetted by the liquid creeping down, whereas the empty channels are completely occupied by the counter-current gas. It means that the bypass flow... 

Computational fluid dynamics (CFD) based on the continuum Navier-Stokes equations Eq. 2 has long been successfully used in fundamental research and engineering design in different fluid related areas. Namrally, it becomes the first choice for the simulation of microfluidic phenomena in Lab-on-a-Chip devices and is still the most popular simulation model to date. Due to the nonlinearity arising from the convention term, Eq. 2 must be solved numerically by different discretization schemes, such as finite element method, finite difference method, finite volume method, or boundary element method. Besides, there are a variety of commercially available CFD packages that can be less or more adapted to model microfluidic processes (e.g., COMSOL (http //www.femlab.com), CFD-ACE+ (http // www.cfdrc.com), Coventor (http //www. coventor.com), Fluent (http //www.fluent.com), and Ansys CFX (http //www.ansys.com). For majority of the microfluidic flows, Re number is... 

Finite element analysis (FEM) has become a popular method for numerical simulation of flow through dies. One of the benefits of EEM is that it can handle non-linear fluids well. A newer numerical technique gaining popularity is boundary element analysis (BEM) Three-dimensional flow analysis with BEM can handle complex flow geometries well however, BEM at this point is not as good as FEM in handling nonlinear fluids. Less detailed analyses often use control volume analysis to reduce the computational effort. The different numerical techniques will be discussed in more detail in Chapter 12. 

Both models can be solved by different numerical schemes. The boundary element method has the advantage that the velocity gradient can be obtained more accurately than the finite-element method, which is important for predicting the fiber orientation (Barone and Caulk, 1986 Osswald and Thcker, 1988). However, it is restricted to single charge, flat parts and cases of uniform thickness. Hence, many simulation efforts have converged to the use of the finite-element/control-volume method (Erwin and Thcker, 1995 Osswald and Tucker, 1990) coupled with the volume of fluid (VOF) method to track the position of flow front (Hirt and Nichols, 1981). 

Numerical Methods for Fluid Flow Simulation The main classes of traditional numerical methods are finite difference methods, finite element methods, finite volume methods, spectral methods and pseudo-spectral methods. When applied to the saturation equation in the pressure-saturation formulation, the important device of a moving coordinate system has sometimes been tried. 

In the context of viscoelastic fluid flows, numerical analysis has been performed for differential models only, and for the following types of approximations finite element methods for steady flows, finite differences in time and finite element methods in space for unsteady flows. Finite element methods are the most popular ones in numerical simulations, but some other methods like finite differences, finite volume approximations, or spectral methods are also used. 

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