Continuity waves in resolved-particle simulations of fluidized beds

The results of a fully resolved simulation of up to 2000 spheres suspended in a vertical liquid stream are analyzed by a method based on a truncated Fourier series expansion. It is shown that, in this way, it is possible to identify continuity (or kinematic) waves and to determine their velocity, which is found to closely agree with the theory of one-dimensional continuity waves based on the Richardson-Zaki drag correlation.

Figure 1

Comparison of the power-law fit coefficients $n$ and $\kappa$ in Eq. (4). Black dots are from this study, white circles are from Ref. [26], and the solid and dashed lines are relations for $n$ from Refs. [24, 27], respectively.

Figure 2

Comparison of the present simulation results for the mean fluid-particle relative velocity (symbols) with the Richardson-Zaki curve (4) (lines). The dotted line and stars are for ${\rho }_p/\rho =2$; the dash-dotted line and green squares for ${\rho }_p/\rho =3.3$; the solid line and red circles for ${\rho }_p/\rho =4$; and the dashed line and light blue triangles for ${\rho }_p/\rho =5$. Insets: top right: close-up of $\varphi \approx 0.175$; bottom left: close-up of $\varphi \approx 0.087$.

Figure 3

The thin dashed line is a snapshot of the “raw” number density, i.e., the number density calculated by counting the particle centers contained in each horizontal layer of cells, for ${\rho }_p/\rho$ = 3.3 with 1000. The solid line and the thick dashed line are the number density reconstructed by a Fourier series with 15 and 30 coefficients, respectively.

Figure 4

Autocorrelation function (averaged over horizontal planes and $z$ and $t$) of the “raw” particle number density, i.e., the number density calculated by counting the particle centers contained in each horizontal layer of cells, for ${\rho }_p/\rho$ = 3.3 with 1000 particles.

Figure 5

The dashed line is the “raw” particle volume fraction, i.e., the volume fraction calculated as the fraction of volume of each horizontal cell layer falling inside particles for ${\rho }_p/\rho$ = 3.3 with 1000 particles. The solid line is the particle volume fraction reconstructed by a Fourier series with 15 coefficients.

Figure 6

Autocorrelation of the “raw” particle volume fraction $\varphi$ (i.e., the particle volume fraction obtained by counting the fraction of volume of each horizontal cell layer falling inside particles) for ${\rho }_p/\rho$ = 3.3 with 1000 particles; the quantity shown is averaged over $z$ and $t$.

Figure 7

Space-time distribution of the “raw” particle volume fraction (i.e., the particle volume fraction obtained by counting the fraction of volume of each horizontal cell layer falling inside particles) for ${\rho }_p/\rho$ = 3.3 with 2000 particles.

Figure 8

Fifteen-terms Fourier reconstruction of the volume fraction for a representative case with $\varphi =0.349$ ($N=2000$) and ${\rho }^{*}=3.3$; white corresponds to the mean volume fraction over the entire computational domain. This figure should be compared with Fig. 7 in which the “raw,” rather than the Fourier-processed, volume fraction is shown.

Figure 9

Autocorrelation of the particle volume fraction reconstructed with 15 Fourier terms for a representative case with $\varphi =0.349$ ($N=2000$) and ${\rho }^{*}=3.3$. The dashed line is obtained by a least-squares fit of the position of the maxima for each $\mathrm{\Delta }z$. This figure may be compared with Figs. 6 and 4 which show the autocorrelation of the “raw” volume fraction and number density, respectively.

Figure 10

Continuity wave speed calculated from the present simulations (symbols) compared to the relationship given in Eq. (7) shown by the dashed lines.

Figure 11

Instantaneous fluid vertical velocity vs. $z$ as calculated from the Fourier reconstruction with $N=15$ coefficients (solid line) and by averaging over horizontal cell layers for ${\rho }_p/\rho$ = 3.3 with 1000 particles.

Figure 12

Autocorrelation function of the “raw” fluid vertical velocity, i.e., the fluid vertical velocity averaged over each horizontal cell layer; the quantity shown is averaged over $z$ and $t$ for ${\rho }_p/\rho$ = 3.3 and 1000 particles.

Figure 13

Normalized cross-correlation of volume fraction and fluid vertical velocity, each reconstructed with 15 Fourier modes, for the case ${\rho }_p/\rho =3.3$ with 1000 particles. The slope of the dashed line is the speed of kinematic waves calculated from the reconstructed volume fraction as explained in the text.

Figure 14

Spatial power spectra of the Fourier-reconstructed volume fraction; $\lambda$ denotes the wavelength of the Fourier components. From left to right, the columns correspond to $\varphi$ = 0.087, 0.175, 0.262, 0.349; from top to bottom the rows are for ${\rho }_p/\rho$ = 2, 3.3, 4, and 5.

Figure 15

Power spectra of the particle volume fraction (solid line) and of the fluid vertical velocity, both reconstructed with 15 Fourier coefficients, vs the normalized wave number for ${\rho }_p/\rho$ = 3.3 with 1000 particles.

Figure 16

Power spectra of the particle volume fraction (solid line) and of the fluid vertical velocity, both reconstructed with 15 Fourier coefficients, vs the normalized frequency, for ${\rho }_p/\rho$ = 3.3 with 1000 particles.