Turbulent Properties of Stationary Flows in Porous Media

In this study, we investigate the flow dynamics in a fixed bed of hydrogel beads using particle tracking velocimetry to compute the velocity field in the middle of the bed for moderate Reynolds numbers ($\mathrm{Re}=[124,169,203,211]$). We discover that even though the flow is stationary at the larger scales, it exhibits complex multiscale spatial dynamics reminiscent of those observed in classical turbulence. We find evidence of the presence of an inertial range and a direct energy cascade, and are able to obtain a value for a “porous” Kolmogorov constant of $C_2=3.1\pm 0.3$. This analogy with turbulence opens up new possibilities for understanding mixing and global transport properties in porous media.

Figure 1

Long-time exposure of a generic experiment. The tracer particles are shown in grays, and the hydrogel beads are the empty spaces in the middle. The Lagrangian dynamics is not trivial and is observed at the local scale. It is reflective of the complex nature of the system.

Figure 2

Schematic representation of the experimental setup.

Figure 3

Left: Cumulative integral of the correlation function $\mathcal{R}$. The plateau (horizontal dashed lines) allows us to compute $L_y$. Right: Integral correlation lengths calculated for the streamwise ($y$) and transversal ($x$) directions.

Figure 4

Second-order structure functions for each velocity component. A two-thirds scaling (compared in dashed lines) is evident at the small scales $dr<0.2d$.

Figure 5

Top: The acceleration-velocity structure function, which is supposed to remain constant in a homogeneous and isotropic turbulent flow, and its negative sign indicates the direction of the energy cascade. Bottom: Averaged energy dissipation rate as a function of ${\sigma }_u^3$. The dashed red line shows Eq. (2).