Angular multiscale statistics of turbulence in a porous bed

Direct numerical simulation of pore-scale turbulence is performed in a unit cell of a face-centered cubic lattice at three different pore Reynolds numbers (300, 500, and 1000). The pore-geometry gives rise to very low porosity resulting in rapid acceleration and deceleration of the flow in different regions. Eulerian statistics of mean velocity and turbulent kinetic energy are first described to provide a good understanding of the overall flow topology. In addition, the spatial variances and probability density functions for the instantaneous and temporal fluctuation velocities are computed to assess the higher-order statistics. The flow field is then analyzed using angular Lagrangian multiscale statistics of fluid particles to study their directional change at different timescales. Two power laws are observed for the evolution of the mean absolute angle as a function of time lag, demarking the early time and an intermediate, inertial range. The effect of the geometric confinement on the asymptotic behavior of the angular statistics is examined in detail. An asymptotic limit different than $\pi /2$ (corresponding to equidistribution of the mean angle, observed for three-dimensional isotropic turbulence with periodic boundaries) or $\frac{2}{3}\pi$ (observed for two-dimensional wall-bounded turbulence) is obtained, representing the strong effect of geometric confinement on turbulence. Besides, the probability density functions (PDFs) for the instantaneous curvature angles are computed and the normalized PDFs are fit to a Fisher distribution. Furthermore, a Monte Carlo-based stochastic model is developed to predict the asymptotic curvature angle.

Figure 1

(a) Schematic of a face-centered porous unit cubic cell; (b) same as (a) except some beads are removed and a particle trajectory illustrates the definition of the curvature angle $\mathrm{\Theta }$ between subsequent Lagrangian particle increments.

Figure 2

Illustration of the two power laws and the asymptotic limit of the mean curvature angle $\theta$ evolution over time lags in log-log scale, where ${\tau }_K$ is the Kolmogorov timescale and $T^L$ the integral timescale (reproduced according to Bos et al. [46]).

Figure 3

Visualization of mean velocity in streamwise direction (${\overline{U}}_x$) and TKE distributions at three Reynolds numbers. Top, ${\text{Re}}_p=300$; center, ${\text{Re}}_p=500$; bottom, ${\text{Re}}_p=1000$. The left side column shows velocity contour, and the right-hand side TKE.

Figure 4

PDFs of the instantaneous velocity component in $x$ direction (a), and (b) showing the same data as (a) but in the lin-log scale; (c) and (d) are showing the velocity component in $y$ direction, where the solid line represents the Gaussian distribution with zero mean and unit standard deviation.

Figure 5

The PDFs for fluctuating velocity components in $x$ direction (a), and (b) showing the same data as (a) but in the lin-log scale; (c) and (d) are showing the velocity component in $y$ direction, where the solid line represents the Gaussian distribution with zero mean and unit standard deviation.

Figure 6

Lagrangian auto-correlation with $x$ axis normalized by the corresponding integral timescale $T_{11}^L$, (a) ${\text{Re}}_p=300$, (b) ${\text{Re}}_p=500,$ and (c) ${\text{Re}}_p=1000$, where is ${\rho }_{11}^L$, ${\rho }_{22}^L$, and ${\rho }_{33}^L$; (d) energy spectrum of ${\rho }_{11}^L$ for all three Reynolds numbers.

Figure 7

(a) Evolution of the mean absolute angle $\theta$ for ${\text{Re}}_p=300$ , ${\text{Re}}_p=500$ , and ${\text{Re}}_p=1000$ ; probability density functions for $\cos \left(\mathrm{\Theta }\right)$, (b) ${\text{Re}}_p=300$, (c) ${\text{Re}}_p=500,$ and (d) ${\text{Re}}_p=1000$; the red dashed line denotes the value of uniform distribution, 0.5.

Figure 8

PDFs of the normalized instantaneous curvature angle, short times in solid colored lines, long times in dashed color lines, and the bold black solid line represents the $F_{2,3}$ distribution.

Figure 9

(a) Schematic of sample blocks in a mask function, (b) illustration of how the blue mask functions are placed inside of the porous bed, (c) application of the stochastic model to a two-dimensional periodic channel and corresponding two-dimensional mask functions denoted by circles of different radii.

Figure 10

Two-dimensional periodic channel: ensemble averaged curvature angle $\langle {\theta }_S\rangle$ as a function of mask function diameter $D_S$ normalized by the channel width $L_y$.

Figure 11

Three-dimensional, triple periodic porous unit cell: ensemble averaged curvature angle ${\theta }_S$ as a function of mask function radius $D_S$ normalized by the hydraulic diameter $D_H$.