Improving prediction of preferential concentration in particle-laden turbulence using the neural-network interpolation

We develop a neural-network interpolation (NNI) to improve the prediction of preferential concentration in simulations of particle-laden turbulence. The NNI uses the particle position and velocity on neighboring grid points to estimate the fluid velocity at the particle position via fully connected neural networks. By avoiding the requirement for superresolution of the entire field and additional interpolations, the NNI offers computational efficiency and simplifies implementation. To evaluate the effectiveness of NNI and compare it with other interpolation methods, we conduct simulations on two-dimensional homogeneous isotropic turbulence subjected to high-wave-number forcing. This specific turbulent flow has a long inertial range and rich small-scale structures, posing a challenge for velocity interpolations and subsequently accurate prediction of preferential concentration. We compare the results on flow fields, energy spectra, and preferential concentration against the reference data obtained from direct numerical simulations at a range of the Stokes number from 0.1 to 5.0. The comparison demonstrates that the NNI can recover the effect of small-scale motion on particle distribution, so it improves the prediction accuracy of the preferential concentration from a priori test results of the large-eddy simulation on coarse grids.

Figure 1

(a) Schematic of the input and output of neural-network interpolation (NNI) and Lagrangian interpolation (LGI). (b) Flowchart of NNI.

Figure 2

Summary of different approaches of tracking particles.

Figure 3

Instantaneous velocity contours of the (a) high-resolution (HR) field with resolution ${1024}^2$ by direct numerical simulation (DNS), (b) low-resolution (LR) field with resolution ${128}^2$, and reconstructed fields with resolution ${1024}^2$ by (c) Lagrangian interpolation (LGI), (d) convolutional neural network (CNN), and (e) neural-network interpolation (NNI).

Figure 4

$L_1$ error of the velocity field in Lagrangian interpolation (LGI), convolutional neural network (CNN), and neural-network interpolation (NNI) cases, with resolutions ${128}^2, {64}^2$, and ${32}^2$ of the low-resolution (LR) field.

Figure 5

Energy spectra of the high-resolution (HR) field by direct numerical simulation (DNS), the low-resolution (LR) field by average pool, and the reconstructed fields by Lagrangian interpolation (LGI), convolutional neural network (CNN), and neural-network interpolation (NNI). The purple dotted line marks the cutoff wave number, and the black dotted line marks the wave number where the forcing is injected.

Figure 6

Instantaneous vorticity contours and particle distributions (marked in the left half domain using yellow dots) obtained from the (a) high-resolution (HR) field (${1024}^2$) by direct numerical simulation (DNS), (b) low-resolution (LR) field (${128}^2$), and reconstructed fields (${1024}^2$) by (c) Lagrangian interpolation (LGI), (d) convolutional neural network (CNN), and (e) neural-network interpolation (NNI).

Figure 7

Particle distributions obtained via (a) direct numerical simulation (DNS) and simulations based on the low-resolution (LR) field with (b) Lagrangian interpolation (LGI), (c) convolutional neural network (CNN), and (d) neural-network interpolation (NNI) at $t=15$ for the case with $St=1$. The particle is color-coded by $\left|\omega \right|$. The arrows in zoom-in subplots denote $\mathbf{u}\left({\mathbf{x}}_p\right)$ for each particle.

Figure 8

Direct numerical simulation (DNS) results on (a) the probability density function (PDF) of the particle number in each cell at $t=0$ and the average over $t=10$–15, when the particle distribution reaches a stationary state, along with the Poisson distribution. The gray zone denotes the variance of the PDF at $t=10–15$. (b) Evolution of $D_C$ in DNS cases with $St=0.3$, 1.0, and 4.0.

Figure 9

Comparison of $\langle D_C\rangle$ in direct numerical simulation (DNS), Lagrangian interpolation (LGI), convolutional neural network (CNN), and neural-network interpolation (NNI) based on the low-resolution (LR) field (${128}^2$) for different $St$. The error bars denote the range of $D_C$.

Figure 10

(a) Comparison of $\langle D_C\rangle$ in direct numerical simulation (DNS), Lagrangian interpolation (LGI), convolutional neural network (CNN), and neural-network interpolation (NNI) based on low-resolution (LR) fields with different grid resolutions ${128}^2$ (solid lines), ${64}^2$ (dashed lines), and ${32}^2$ (dash-dotted lines) for different $St$. (b) $L_1$ error of $\langle D_C\rangle$ of the LGI, CNN, and NNI results with different grid resolutions.

Figure 11

Computational costs of different methods for simulating particle-laden two-dimensional (2D) homogeneous isotropic turbulence (HIT).

Figure 12

(a) Energy spectra of the high-resolution (HR) field in direct numerical simulation (DNS), low-resolution (LR) field in large-eddy simulation (LES), and reconstructed LES fields with Lagrangian interpolation (LGI), convolutional neural network (CNN), and neural-network interpolation (NNI). The purple dotted line marks the cutoff wave number $k_c$. (b) Comparison of $\langle D_C\rangle$ obtained via LGI, CNN, and NNI based on the LR fields by the average pool (solid lines) in the a priori test and LES (dashed lines) in the a posteriori test for different $St$, along with the DNS results (black solid line).