Grid refinement

Examples of disconnected and connected grid domains are shown in Fig. 12.a and 12.b, respectively. The assumption on connectedness for a domain with a narrow bridge will be satisfied if we make the step small enough or refine the grid in this part of the domain. Fig. 12.b illustrates the case where the connectedness of the grid is stipulated by the proper choice of its step rather than by successive grid refinements. 

As a rule, equations of gas dynamics are discontinuous. From a physical point of view it is fairly common to distinguish weak discontinuities relating to cutting waves and strong discontinuities relating to shock waves . For these reasons successive grid refinement can be made with caution when the accurate account of accuracy of numerical methods is performed. 

A comprehensive and more extensive overview of the pros and cons of LB with respect to applications can be found in Succi (2001). By the way, LB methods are continuously improved to increase speed and accuracy, particularly by introducing grid refinement techniques and advanced techniques for arbitrarily shaped boundaries (e.g., Rohde et al., 2002, 2003, 2006 Rohde, 2004). 

Substantial improvements in LB techniques have been elfected—in terms of immersed or embedded boundary methods for dealing with moving and curved boundaries (impeller blades, solid particles) and of grid refinement techniques— which have had a positive impact on the fast proliferation of dedicated CFD tools. Here, too, the details of the computational techniques do matter. 

Rohde, M., Extending the Lattice-Boltzmann method—novel techniques for local grid refinement and boundary conditions , Ph.D. Thesis, Delft University of Technology, Delft, Netherlands (2004). 

For simple flows where the mean velocity and/or turbulent diffusivity depend only weakly on the spatial location, the Eulerian PDF algorithm described above will perform adequately. However, in many flows of practical interest, there will be strong spatial gradients in turbulence statistics. In order to resolve such gradients, it will be necessary to use local grid refinement. This will result in widely varying values for the cell time scales found from (7.13). The simulation time step found from (7.15) will then be much smaller than the characteristic cell time scales for many of the cells. When the simulation time step is applied in (7.16), one will find that Ni must be made unrealistically large in order to satisfy the constraint that Nf > 1 for all k. 

In an unstructured mesh each node can have a different number of neighbours and elements have different shapes and sizes. Therefore connectivity information must be explicitly defined and stored. The unstructured grid approach, that has gained popularity with the enormous advancements of computer technology, allows handling complex geometries with a lower number of elements and a much easier realization of local and adaptive grid refinement. [77]... 

Figure 8 compares Model M with Model G in terms of their predictions of the axial profiles of voidage under various grid resolutions. For FCC particles, when using Model G, the solids were distributed uniformly across the riser height. It seems that the grid refining has little... 

From a computational viewpoint, the method does not require the inversion of large matrices, and thus computer memory requirements are small. Typical diffusion controlled reactions often produce sharp gradients in the concentration field [47]. Grid refinement to take these into account in three dimensions is difficult. The analogous problem for pairwise Brownian dynamics, which is the optimal location of the initiation points for the trajectories on the spherical initiation surface is much simpler to accomplish. Furthermore, the computations can easily be performed in parallel, since the result from each trajectory is independent of the rest. This also allows for sequential refining of... 

For a uniform Cartesian grid, this approximation is of second-order accuracy. Even for a non-uniform grid, the error reduction with respect to grid refinement is similar to that of a second-order approximation. Higher order polynomials can be used to estimate the required gradients. For example, a fourth-order approximation for the gradient at face e on the uniform Cartesian grid can be written ... 

Grid types import from different pre-processors co-located/ staggered (un) structured Automatic grid refinement tools, addition of grid elements Geometry modifications (change scale/cell type etc.) without re-meshing... 

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