Turbulence modulation by settling inertial aerosols in Eulerian-Eulerian and Eulerian-Lagrangian simulations of homogeneously sheared turbulence

Settling inertial aerosols dispersed at high enough concentration modulate the structure of the suspending turbulent flow. The present study addresses this modulation in a canonical flow, which is homogeneously sheared turbulence (HST). Eulerian-Eulerian (EE) and Eulerian-Lagrangian (EL) simulations are conducted in the semidilute regime, i.e., such that the particulate phase is sufficiently dilute to neglect particle-particle interaction but sufficiently concentrated to be strongly coupled with the carrier flow. Four cases are considered for which the Stokes number based on the Kolmogorov timescale is ${\mathrm{St}}_{\eta }=0.06$ or 0.19 and mass loading is $M=0.125$ or 0.5. Turbulence is initialized with a Taylor microscale Reynolds number ${\mathrm{Re}}_{{\lambda }_{,}0}=29$ and is strongly sheared, resulting in a shear number $S_0^{*}=27$. Simulations of the suspension with ${\mathrm{St}}_{\eta }=0.06$ and $M=0.125$ show no turbulence modification compared with single-phase HST. However, when the mass loading is increased to $M=0.5$ the aerosols cluster, despite their small inertia, and measurably modify the carrier flow. These aerosols enhance turbulence in the carrier phase, causing an increase of the growth rates of the turbulent kinetic energy (TKE) and dissipation rate among other effects. The opposite behavior emerges when the Stokes number is ${\mathrm{St}}_{\eta }=0.19$. These aerosols cause the attenuation of turbulence by reducing the growth rates of the TKE and dissipation rate in HST. This attenuation is weaker for the larger mass loading case $M=0.5$ due to stronger forcing by the particles on the gas in the gravity direction. Turbulence modulation in the EE simulation is verified to hold excellent agreement with EL simulations. In the former approach, simulating a few integral lengths of turbulence leads to a large number of Lagrangian particles, up to $1.7\times {10}^9$ particles in the present work. Eulerian-Eulerian simulations are performed with a positivity-preserving kinetic-based formulation that assumes an anisotropic-Maxwellian distribution of the subgrid particle velocity probability distribution function. The excellent agreement with EL simulations shows that this EE simulation strategy is a robust and predictive method in the semidilute regime.

Figure 1

Illustration of the subgrid particle velocities with the anisotropic-Maxwellian distribution as closure. Arrows represent the particle fluctuating velocity $\mathbf{c}-{\mathbf{u}}_p$, where $\mathbf{c}$ is the particle velocity and ${\mathbf{u}}_p$ is the cell averaged velocity. (a) In general, the anisotropic-Maxwellian distribution allows particle-trajectory crossing (${\mathbf{P}}_p\ne 0$). (b) In regions of the flow where the particle pressure ${\mathbf{P}}_p$ vanishes, the velocity fluctuations vanish (${\mathbf{P}}_p\to 0$) and the anisotropic-Maxwellian collapses onto the monokinetic distribution. (c) In regions where ${\mathbf{P}}_p$ becomes isotropic, the anisotropic-Maxwellian collapses onto a Maxwellian distribution (${\mathbf{P}}_p\to \mathrm{\Theta }{\delta }_{ij}$).

Figure 2

Sketch of the geometrical configuration. A background base shear $\mathrm{\Gamma }{x}_2{\mathbf{e}}_1$ drives a turbulent gas loaded with particles. Gravitational acceleration aligned with the shear direction, i.e., $\mathbf{g}=-g{\mathbf{e}}_2$, causes the particles to settle.

Figure 3

Evolution of (a) the product of the largest wave number resolved by the grid and the Kolmogorov scale and (b) the normalized integral length scale during the integration time $\mathrm{\Gamma }t\le 4$.

Figure 4

Snapshots of isocontours of vorticity magnitude in the $x_1-x_2$ plane at $\mathrm{\Gamma }t=4$ from a single-phase flow simulation of homogeneously sheared turbulence.

Figure 5

Isocontours of vorticity magnitude in the $x_1-x_2$ plane at $\mathrm{\Gamma }t=4$ from EL and EE simulations in Table 1.

Figure 6

Turbulence statistics from EL (open symbols), EE (closed symbols), and single-phase HST (lines) simulations for the two cases with ${\text{St}}_{\eta }=0.06$ particles: (a) normalized dissipation rate, (b) normalized turbulent kinetic energy, (c) ratio of shear production to dissipation rates, and (d) normalized streamwise Taylor microscale. Symbols $\square$ and $\blacksquare$ refer to case d50L having a mass loading $M=0.125$. Symbols $◊$ and $⧫$ refer to case d90H having a mass loading $M=0.5$.

Figure 7

Diagonal components of the anisotropic Reynolds stress tensor from EL (open symbols), EE (closed symbols), and single-phase HST (lines) simulations for the two cases with ${\text{St}}_{\eta }=0.06$ particles: (a) $M=0.125$ and (b) $M=0.5$.

Figure 8

Turbulence statistics from EL (open symbols), EE (closed symbols), and single-phase HST (lines) simulations for the two cases with ${\text{St}}_{\eta }=0.19$ particles: (a) normalized dissipation ratio, (b) normalized turbulent kinetic energy, (c) ratio of shear production to dissipation rates, and (d) normalized streamwise Taylor microscale. Symbols $○$ and $•$ refer to case d90L having a mass loading $M=0.125$. Symbols $▵$ and $▴$ refer to case d90H having a mass loading $M=0.5$.

Figure 9

Diagonal components of the anisotropic Reynolds stress tensor from EL (open symbols), EE (closed symbols), and single-phase HST (lines) simulations for the two cases with ${\text{St}}_{\eta }=0.19$ particles: (a) $M=0.125$ and (b) $M=0.5$.

Figure 10

Isocontours of volume fraction in the $x_1-x_2$ plane at $\mathrm{\Gamma }t=4$ from EL and EE simulations in Table 1.

Figure 11

Volume fraction fluctuations in EL (open symbols) and EE (closed symbols) simulations for (a) ${\text{St}}_{\eta }=0.06$ and (b) ${\text{St}}_{\eta }=0.19$. Symbols are the same as in Figs. 6 and 8.