Grid, computational

In addition to the grid density, the quality of a grid depends on a number of other factors such as the aspect ratio of the cells and the relative size of neighboring cells. A comprehensive listing of the quality criteria for computational grids is given elsewhere [83]. Some of the important points to be remembered are  

Highly skewed cells should be avoided. The angles between the grid lines of a hexahedral mesh should be 90°. Angles 40° or 140° often imply a reduced accuracy or numerical instabilities. 

The aspect ratio of hexahedral cells should be not too high, typically below 20-100. If high-aspect ratio cells are used, the accuracy and possible convergence problems depend greatly on the flow direction. 

The mesh expansion ratio, i.e. the size ratio of neighboring cells, should be kept small. Particularly in regions of large gradients, mesh size discontinuities should be avoided. 

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To resolve the problems associated with structured and unstructured grids, these fundamentally different approaches may be combined to generate mesh types which partially posses the properties of both categories. This gives rise to block-structured , overset and hybrid mesh types which under certain conditions may lead to more efficient simulations than the either class of purely structured or unstructured grids. Detailed discussions related to the properties of these classes of computational grid.s can be found in specialized textbooks (e.g, see Liseikin, 1999) and only brief definitions are given here. 

It should be emphasized at this point that the basic requirements of compatibility and consistency of finite elements used in the discretization of the domain in a field problem cannot be arbitrarily violated. Therefore, application of the previously described classes of computational grids requires systematic data transfomiation procedures across interfaces involving discontinuity or overlapping. For example, by the use of specially designed mortar elements necessary communication between incompatible sections of a finite element grid can be established (Maday et ah, 1989). 

Variational methods - theoretically the variational approach offers the most powerful procedure for the generation of a computational grid subject to a multiplicity of constraints such as smoothness, uniformity, adaptivity, etc. which cannot be achieved using the simpler algebraic or differential techniques. However, the development of practical variational mesh generation techniques is complicated and a universally applicable procedure is not yet available. 

Delaunay method - in this method the computational grid is essentially constructed by connecting a specified set of points in the problem domain. The connection of these points should, however, be based on specific rules to avoid unacceptable discreti2ations. To avoid breakthrough of the domain boundary it may be necessary to adjust (e.g. add) boundary points (Liseikin, 1999). 

In the majority of practical finite element simulations the mesh generation is conducted in conjunction with an interactive graphics tool to allow feedback and continuous monitoring of the computational grid. 

With this approach, when an element becomes severly distorted, it is eliminated from the computational grid and becomes a free mass point. Clearly, care must be taken to avoid eliminating elements that could potentially influence the problem at some later time. An example of a three-dimensional Lagrangian calculation that uses the eroding element scheme is presented in the next section. 

Simulations of multiphase flow are, in general, very poor, with a few exceptions. Basically, there are three different kinds of multiphase models Euler-Lagrange, Euler-Euler, and volume of fluid (VOF) or level-set methods. The Euler-Lagrange and Euler-Euler models require that the particles (solid or fluid) are smaller than the computational grid and a finer resolution below that limit will not give a... 

Figure 2. Computed grids for four different shapes of a vertical, axlsymmetrlc MOCVD reactor.

Absorption of wave packet amplitude that approaches the edges of the computational grid can be accomplished by periodically damping out the wave packet in small regions at the grid edges [10]. Suppose A represents this (real) operation. Instead of Eq. (2), one has... 

Figure 2.10 Multilamination flow oriented along the computational grid (left) and forming a tilted angle with the grid cells (right).

Figure 2.12 Computational grids with colocated (left) and staggered (right) arrangement of the velocity and pressure nodes.

The basic idea of the IBM is that the presence of the solid boundary (fixed or moving) in a fluid can be represented by a virtual body force field Fp applied on the computational grid at the vicinity of solid-flow interface. Thus, the Navier-Stokes equation for this flow system in the Eulerian frame can be given by... 

Since the computational grids are generally not coincident with the location of the particle surface, a velocity interpolation procedure needs to be carried out in order to calculate the boundary force and apply this force to the control volumes close to the immersed particle surface (Fadlun et al., 2000). 

In the IBM, the presence of the solid boundary (fixed or moving) in the fluid can be represented by a virtual body force field -rp( ) applied on the computational grid at the vicinity of solid-flow interface. Considering the stability and efficiency in a 3-D simulation, the direct forcing scheme is adopted in this model. Details of this scheme are introduced in Section II.B. In this study, a new velocity interpolation method is developed based on the particle level-set function (p), which is shown in Fig. 20. At each time step of the simulation, the fluid-particle boundary condition (no-slip or free-slip) is imposed on the computational cells located in a small band across the particle surface. The thickness of this band can be chosen to be equal to 3A, where A is the mesh size (assuming a uniform mesh is used). If a grid point (like p and q in Fig. 20), where the velocity components of the control volume are defined, falls into this band, that is... 

Note that the Eqs. (1), (2), and (8) are really and essentially different due to the absence or presence of different turbulent transport terms. Only by incorporating dedicated formulations for the SGS eddy viscosity can one attain that LES yield the same flow field as DNS. RANS-based simulations with their turbulent viscosity coefficient, however, essentially deliver steady flow fields and as such are never capable of delivering the same velocity fields as the inherently transient LES or DNS, irrespectively of the refinement of the computational grid ... 

All these findings of disappointing quantitative agreement with experimental data stem from the inherent drawback of the RANS-approach that there is no clear distinction between the turbulent fluctuations modeled by the Reynolds stresses and (mesoscale) fluctuations. In LES, however, the distinction between resolved and unresolved turbulence is clear and relates to the cell size of the computational grid chosen. 

Note that these closures describe SGS phenomena and hence are essentially local in space (i.e., interior to a computational grid cell). For this reason, it is... 

Fig. 15. The KATAPAK-S structure and its computational grid. Reprinted from Chemical Engineering Science, Vol. 56, van Baten et al., Radial and Axial Dispersion of the Liquid Phase within a KATAPAK-S Structure Experiments vs. CFD Simulations, pp. 813-821, Copyright (2001), with permission from Elsevier.

Figure 5.3. Atomizer geometry (top),computational grid mesh (left) and calculated Mach number contour (right) near a close-coupled atomizer (Atomization gas Ar, Ma = 1 at nozzle exit). (Reprinted with permission from Ref. 325.)...

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