Coarse-grid simulation

To meet the industrial demand for both large-scale computation and good predictability, the reasonable way out is not to simulate from the beginning of the micro-scale, but to use coarse-grid simulation with meso-scale modeling for the effects of structure. This kind of approach can be termed the "multi-scale CFD." It is entitled "multi-scale," not because the problem it solves is multi-scale, but because its meso-scale model contains multi-scale structure parameters. 

Derivative-Free Methods Derivative-free methods can simply call the simulator and use the results. A simple technique is simulated annealing which was investigated in [126] and [41]. Using a fast simulator such as a streamline method (fast by virtue of the IMPES approximation and the one-dimensional approximation along the streamlines) or a coarse grid simulator, this might be practical. 

When a coarse grid is used, wall functions are used for imposing boundary conditions near the walls (Section 11.2.3.3). The nondimensional wall distance should be 30 < y < ]Q0, where y = u,y/p. We cannot compute the friction velocity u. before doing the CFD simulation, because the friction velocity is dependent on the flow. However, we would like to have an estimation of y" to be able to locate the first grid node near the wall at 30 < y < 100. If we can estimate the maximum velocity in the boundary layer, the friction velocity can be estimated as n, — 0.04rj, . . After the computation has been carried out, we can verify that 30 nodes adjacent to the walls. 

First of all, the increased computer power makes it possible to switch to transient simulations and to increase spatial resolution. One no longer has to be content with steady flow simulations on relatively coarse grids comprising 104-105 nodes. Full-scale Large Eddy Simulations (LES) on fine grids of 106—107 nodes currently belong to the possibilities and deliver realistic reproductions of transient flow and transport phenomena. Comparisons with quantitative experimental data have increased the confidence in LES. The present review stresses that this does not only apply to the hydrodynamics but relates to the physical operations and chemical processes carried out in stirred vessels as well. Examples of LES-based simulations of such operations and processes are due to Flollander et al. (2001a,b, 2003), Venneker et al. (2002), Van Vliet et al. (2005, 2006), and Flartmann et al. (2006). 

Limited to computing capability, the following analysis confines the DNS to the fine-grid TFM simulation, which offers meso-scale closures for the correlative, coarse-grid TFM simulations. For comparison, the variational type of multi-scale CFD takes the EMMS-based models to close TFM simulations. 

Figure 5 Effect of grid resolution (A.) on the time-averaged dimensionless slip velocity (us/uT). Geldart group A particles are used. The ordinate is scaled with the terminal velocity of single particles (uT 21.84 cm/s) and the abscissa is scaled with the particle diameter dp. The domain size is 1.5 x 6 cm2, comparable to the coarse-grid used in normal simulations.

Fig. 6. Schematic of multigrid-type hybrid simulation with two grids. At the coarse grid a macroscopic model is advanced over large length and time scales. Information is passed to the macroscopic grid/coarse model from a microscopic simulation executed on a fine grid over short length and time scales. The coarse model is advanced over macroscopic length and time scales and provides to the microscopic simulation a field for constraint fine scale simulation.

The computational advantages of such multigrid methods arise from two key factors. First, microscopic simulations are carried out over microscopic length scales instead of the entire domain. For example, if the size of fine grid is 1% of the coarse grid in each dimension, the computational cost of the hybrid scheme is reduced by 10 2rf, compared with a microscopic simulation over the entire domain, where d is the dimensionality of the problem. Second, since relaxation of the microscopic model is very fast, QSS can be applied at the microscopic grid while the entire system evolves over macroscopic time scales. In other words, one needs to perform a microscopic simulation at each macroscopic node for a much shorter time than the macroscopic time increment, as was the case for the onion-type hybrid models as well. 

The discussion above focused on onion-type hybrid multiscale simulation. Finally, even though there are a limited number of examples published, I expect that the multigrid-type hybrid simulations share the same problems with onion-type hybrid multiscale models. In addition, appropriate boundary conditions for the microscopic grid model need to be developed to increase the accuracy and robustness of the hybrid scheme. Furthermore, the inverse problem of mapping coarse-grid information into a microscopic grid is ill posed. Thus, it is... 

Fig. 19.1 The model areas used for the coarse grid run yielding the forcing data (a) and for the nested simulations for the case low pressure system over Europe (b). The positioning of the nested area is shown by the frame in (a)...

The simulations nested in the coarse grid METRAS results agree in a similar way with the reference simulation in the second half of the simulation time (after about 21 h i.e. 18 CET of 29 August 2003). At this time the performance of simulation 5 is somewhat closer to the other nested simulations than before. This might be a hint that the nesting becomes less relevant and the situation is more locally driven. In the first 21 h of the simulation the two simulations with constant update intervals (3 h, 6 h) are closest to the reference case, while the adaptive update simulations (3, 4) show a high variability in performance. This is a hint that the acceleration is probably not a reliable measure to determine update intervals. The best performance is received in the present case study for a nesting every 3 h. 

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