Finite volume fraction effect on self-induced velocity in two-way coupled Euler-Lagrange simulations

In a two-way coupled Euler-Lagrange (EL) simulation, a particle of size comparable to the local grid spacing results in self-induced perturbation, which must be subtracted to obtain the undisturbed velocity of the particle. Several approaches have been advanced to estimate the self-induced velocity in the limit of an isolated particle. The present work addresses the effect of nonzero volume fraction in predicting the self-induced velocity of particles in an EL simulation. In addition to performing EL simulations of flow over a random distribution of stationary particles, we also perform several hundred companion simulations covering a range of Reynolds number and volume fraction. In each companion simulation we have removed one particle from the random distribution whose undisturbed flow and self-induced velocities are thereby precisely computed. By analyzing the self-induced velocity obtained from these simulations, a number of key conclusions are drawn. The most significant of them is the finding that the self-induced correction procedure of an isolated particle can be applied even at a finite volume fraction, with a simple volume fraction dependent modification, in order to accurately capture the average behavior.

Figure 1

Particle distribution within the triply periodic box and contours of particle volume fraction field at $\langle \varphi \rangle =0.0015$ (top row) and $\langle \varphi \rangle =0.1$ (bottom row). It should be noted that for visualization purposes, the particles are not drawn to scale.

Figure 2

Histogram of volume fraction field evaluated at the particle centers. The results are shown for four different values of average volume fraction. Gaussian fits are plotted in red. The mean and one standard deviation on either side are shown.

Figure 3

Contours of fluid velocity computed in the EL simulation with feedback force applied at all the particles. The results are shown for two different mean Reynolds numbers and volume fraction.

Figure 4

Histogram of deviation from unity in the streamwise component of fluid velocity measured at the particle centers obtained for all 12 cases considered. The Gaussian fits are shown in red. Again the mean and one standard deviation are shown, along with the mean computed without the self-induced velocity perturbation, which is denoted as “mean-SIV.”

Figure 5

Contours of fluid velocity divergence for the four different volume fraction distributions at ${\text{Re}}_{\sigma }=1$. The results are qualitatively similar at other Reynolds numbers.

Figure 6

Self-induced velocity from all 360 $(N-1)$-particle simulations.

Figure 7

(a) Self-induced velocity as a function of ${\text{Re}}_{@l}^{\text{un}}$. (b) A plot of $u_{@l}^{\text{si},S}-u^{\text{si},A}$ vs ${\text{Re}}_{@l}^{\text{un}}-{\text{Re}}_{\sigma }$.

Figure 8

A plot of predicted self-induced velocity using (12) vs actual values obtained from the EL simulations.