Stochastic modeling of fluid acceleration on residual scales and dynamics of suspended inertial particles in turbulence

In numerical simulations with highly turbulent flows, the smallest scales are filtered; thereby, the effects of intermittency on those scales are neglected. When the flow is loaded by heavy small particles, the decimation of rapid changes in the velocity may lead to wrong results. This paper provides an approach to take account of subfiltered events of strong velocity jumps on the motion of heavy small particles. The idea is to force the filtered Navier-Stokes equations by a stochastic acceleration term with statistical properties identified by experiments and DNS. To this end, the stochastic model for supplemented acceleration contains the lognormal stochastic process for its norm (with long-range correlations) and the new stochastic model for the acceleration direction (with short-range correlations). The latter represents the Ornstein-Uhlenbeck process in Cartesian coordinates with relaxation to the local direction of the resolved vorticity; thereby, the geometry of highly stretched vortical structures is introduced in the designed model. Both stochastic processes depend on the local Reynolds number. The proposed flow model is applied for simulation of the background turbulence in which heavy particles are released and tracked. The assessment of single and two-time statistics of the particle acceleration and velocity clearly illustrates the advantage of the proposed flow model.

Figure 1

(a) Results from integration of Eq. (6): PDF evolution in time of the unit vector component (shown at each 10 time steps) and the autocorrelation function; $T_{\text{Diff}}=0.01$; number of released stochastic particles is 10 000; number of time steps 500; the final time is equal to one. (b) Results from integration of Eq. (8): the PDF evolution in time (shown at each 10 time steps) and the autocorrelation function for the unit vector component $e_1;h_1=-0.5;T_{\text{Diff}}=0.1;T_{\text{rel}}=0.01$; number of released stochastic particles is 10 000; number of time steps 500; the final time is equal to one.

Figure 2

DNS of box turbulence, ${\text{Re}}_{\lambda }=140$; the correlation coefficient between block directions of acceleration of fluid particles and vorticities seen by those particles in the block; $l/\eta$ indicates different sizes of blocks. ${\theta }_a$ is the direction angle of the particle acceleration vector and ${\theta }_{\omega }$ is the direction angle of the vorticity vector seen by the particle.

Figure 3

Fluid tracer velocity increments at different time lags in the box turbulence: DNS (—), standard LES $(\square )$; LES-SSAMRW [16] $(*)$; LES-SSAM revised with Eq. (8) $(•)$; $\left(▴\right)$ Gaussian fit. PDFs are shifted toward the lower part with increasing of time lag.

Figure 4

The autocorrelation function of the fluid particle acceleration in the box turbulence given by DNS (—), LES-SSAMRW [16] $(*)$, LES-SSAM revised $(•)$; ${\text{Re}}_{\lambda }=140$. For comparison, the autocorrelation function of the Lagrangian time derivative of the filtered velocity from LES (- - -) is also presented.

Figure 5

Standardized PDF's of the particle acceleration, $\text{St}=0.05,0.3,1.0$, from DNS (—), standard LES (- - -), LES-SSAMRW $(*)$, and LES-SSAM revised $(•)$. (a) ${\text{Re}}_{\lambda }=140$. (b) ${\text{Re}}_{\lambda }=70$.

Figure 6

Acceleration variance as a function of the Stokes number in the dimensionless form $A_0=\langle a_p^2\rangle \langle {\epsilon }^{-3/2}\rangle {\nu }^{1/2}$; ${\text{Re}}_{\lambda }=70,140$. Results from DNS, standard LES, LES-SSAMRW, and LES-SSAM revised.

Figure 7

(a–c) Distributions of the heavy particle velocity increment at different time lags and different Stokes numbers: $\text{St}=0.05,0.3,1.0$, from DNS (—), standard LES $(\square )$, LES-SSAMRW $(*)$, and revised LES-SSAM $(•)$. The background turbulence: ${\text{Re}}_{\lambda }=140$; PDFs are shifted toward the lower part with increasing the time lag.

Figure 8

(a, b) The autocorrelation function of the heavy particle acceleration at the moderate Stokes number, $\text{St}=0.3,1.0$, in the box turbulence given by DNS (—), standard LES (- - -), LES-SSAMRW $(*)$, and LES-SSAM revised $(•)$; ${\text{Re}}_{\lambda }=140$. (c) $\text{St}=3.0$; ${\text{Re}}_{\lambda }=70$.

Figure 9

Standardized PDFs of the acceleration norm of the particle at $\text{St}=0.05,0.3,1.0$ from DNS (—), standard LES (- - -), and LES-SSAM revised $(•)$. ${\text{Re}}_{\lambda }=140$.

Figure 10

(a, b) The autocorrelation function of the acceleration norm of the heavy particle at the moderate Stokes number, $\text{St}=0.3,1.0$, from DNS (—), standard LES (- - -), LES-SSAMRW $(*)$, and LES-SSAM revised $(•)$; ${\text{Re}}_{\lambda }=140$. (c) $\text{St}=3.0$; ${\text{Re}}_{\lambda }=70$.

Figure 11

The particle velocity probability distribution function at $\text{St}=0.3,1.0;{\text{Re}}_{\lambda }=140$. Comparison from DNS (—), standard LES (- - -), LES-SSAMRW $(*)$, and LES-SSAM revised $(•)$.

Figure 12

The Lagrangian velocity autocorrelation function of particle at the moderate Stokes number $\text{St}=0.3,1.0;{\text{Re}}_{\lambda }=140$. Comparison from DNS (—), standard LES (- - -), LES-SSAMRW $(*)$, and LES-SSAM revised $(•)$.

Figure 13

The mean-square dispersions of particles at $\text{St}=0.3,1.0;{\text{Re}}_{\lambda }=140$. Comparison from DNS (—), standard LES $\left(-··-\right)$, LES-SSAMRW $(*)$, and LES-SSAM revised $(•)$. The dashed lines show the ballistic $\left(\sim {\tau }^2\right)$ and diffusive $\left(\sim \tau \right)$ regimes.

Figure 14

PDF of the vorticity seen in the flow by the particle at $\text{St}=1.0$; flow is simulated by DNS (—), standard LES $\left(-··-\right)$, LES-SSAMRW $(*)$, and LES-SSAM revised $(•)$. ${\omega }_p$ is the vorticity seen by the particle and ${\omega }_f$ is the fluid vorticity.

Figure 15

Spatial distributions of the number fraction of particles located in zones of low vorticity relatively to a considered particle being located in the same vorticity interval; $\text{St}=1.0$. (a) DNS. (b) Standard LES. (c) LES-SSAM. (d) LES-SSAMRW.