Mesoscale simulations of particulate flows with parallel distributed Lagrange multiplier technique Page: 16 of 47
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we consider configuration that consists of the monodisperse spheres of radius R = 0.002m
arranged in the cubic packing. We consider two flow regimes with Reynolds numbers Re = 18
p D a2
and Re = 900, where Reynolds number is defined as Re = . 'p A constant inflow velocity
v(1 - E)
field of u = 0.5 rn/s was prescribed at the left boundary and the open boundary conditions
are set up at the right boundary. Other boundaries are chosen to be periodic. The porosity is
E = 47.64%. Figure 2 shows values of friction coefficient A in different data sets that include
mentioned earlier Ergun (1952); Franzen (1979); Hovekamp (2002); Martin et al. (1951); Vortwek
and Brunn (1994). For small Re numbers, difference in the data sets for both random and cubic
packings is small (Figure 2a). For high Re numbers, there is a spread of more than half an
order magnitude for the friction coefficient in existed empirical and experimental data (Figure
2b). Figure 3 shows flow patterns at different resolutions for Re = 19. We use Richardson
extrapolation f(h) = f,,,,t + Ch to estimate the convergence order using computed values of
f (h). Here f is calculated parameter (pressure drop in our case), h is some measure of grid
spacing, C is a constant, and p is the order of convergence. Based on the results of simulations
we found that the convergence rate is close to the second order: 1.86 for Re = 18 and 1.8 for
Re = 900 (Figure 4 and Figure 6, respectively). However the low Reynolds number flow requires
less resolution to get converged solution then the high Reynolds number flow. About 48 cells
per particle are needed to describe flow behavior for Re = 18. In the case of Re = 900, the
resolution needs to be twice higher to get the same order of error. This is mainly because of
the flow separation and turbulent boundary effects that require finer grid resolutions to describe
them adequately (Figure 5). In these simulations we do not use any turbulent model. However
in future studies we may need to incorporate turbulent effects and additional drag terms to be
able to simulate high Re number flow regimes with relatively modest resolution. Another option
to improve the convergence rate is to revise the interpolation scheme for the velocity field in
the mixed cells in (15). The grid resolution needed for converged solution depends on particular
configuration and porosity. For densely packed beds, the flow characteristics are constrained by
the maximum resolution available in the void space between particles. Therefore fine-resolution
simulations are required to describe flow effects through these small void spaces for high Re
numbers. The overall agreement with available data is very good (Figure 2).
The next example is flow in a periodic domain of randomly packed spheres of radius R =
0.11283m. Re number in this configuration is about 900 and the solid packing ratio is 60%. A
superficial velocity is specified as a = 0.008 m/s. The periodic random distribution of spheres
is generated using Donev et al. (2005) algorithm. The pressure drop through the packed bed
is illustrated in Figure 7. We perform a numerical analysis to estimate the convergence rate of
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Kanarska, Y; Lomov, I & Antoun, T. Mesoscale simulations of particulate flows with parallel distributed Lagrange multiplier technique, article, September 10, 2010; Livermore, California. (digital.library.unt.edu/ark:/67531/metadc868095/m1/16/: accessed February 19, 2019), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.