# 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 of1 0

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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.