Grid size

The choice of integration technique or algorithm was unimportant as long as the grid size was fine enough to produce accurate results. 

Regarding accuracy, the finite difference approximations for the radial derivatives converge O(Ar ). The approximation for the axial derivative converges 0(Az), but the stability criterion forces Az to decrease at least as fast as Ar. Thus, the entire computation should converge O(Ar ). The proof of convergence requires that the computations be repeated for a series of successively smaller grid sizes. 

For j j =1 To 8 This outer loop varies the radial grid size to test convergence... 

Figure 8.7 shows these results for = 1 and compares them with the analytical solution. The numerical approximation is quite good, even for a coarse grid with 7=4 and 7=16. This is the exception rather than the rule. Convergence should be tested using a hner grid size. 

In a further test, increase the grid size to 100 x 100, set Pr(AB) at 1.0, and repeat the process. This test reveals the influence of concentration on the rate, since the two ingredients now have a much larger territory to roam about, making them in effect less concentrated. Compare your results with the results from the Pr(AB) =1.0 part of Study 8. lb. 

As an example (and this is a hypothetical example only), a particle is shown in Fig. 11 such as might appear on a microscope slide. This particle is gridded out in the form shown in the upper lefthand corner of Fig. 11. The number of squares in which parts of the trace of the particle is located is counted. This number is N, and the length of the grid size is g. The grid size is arbitrarily set equal to one in this example. The grid... 

Fig. 21. The product D-atom velocity-flux contour map, d

In Eq. (12), x is an artificial time that has the unit of distance. The solutions for Eq. (12) are signed distances and only those within a thickness of 3-5 grid sizes from the interface are of interest (Sussman et al., 1994, 1998 Sussman and Fatemi, 1999). Equation (12) needs to be integrated for 3-5 time steps using a time step Ax = 0.5A. 

The simulation shown in Fig. 10 is an impact of a saturated water droplet of 2.3 mm in diameter onto a surface of 400°C with an impact velocity of 65 cm/s, corresponding to a Weber number of 15. This simulation and all others presented in this study are conducted on uniform meshes (Ax — Ay — Az = A). The mesh resolution of the simulation shown in Fig. 10 was 0.08 mm in grid size, although different resolutions are also tested and the results are compared in Figs. 11 and 12. The average time-step in this case is around 5 ps. It takes 4000 iterations to simulate a real time of 20 ms of the impact process. The simulation... 

We illustrated how these equations are discretized over an appropriate numerical grid and also showed some sample results. One can readily appreciate that one must choose the grid sizes in the numerical solution of the TFM equations to be smaller than the shortest length scale at which the TFM equations afford inhomogeneities. This requirement leads to a practical difficulty when one tries to solve these microscopic TFM equations for gas-particle flows, as discussed below. 

It is readily apparent that finer and finer structures get resolved as the number of spatial grids is increased. Statistical quantities, such as average slip velocity between the gas and particle phases, obtained by averaging over the whole domain, were found to depend on the grid resolution employed in the simulations and they became nearly grid-size independent only when grid sizes of the order of a few ( 10) particle diameters were used. Thus, if one sets out to solve microscopic TFM equations, grid sizes of the order of few particle... 

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