Possibility for survival of macroscopic turbulence in porous media with high porosity

Abstract

The direct numerical simulation (DNS) study by Jin et al. (J. Fluid Mech, vol. 766, 2015, pp. 76–103) shows that the turbulent structures are generally restricted in size to the pore scale, leading to the pore-scale prevalence hypothesis (PSPH). Although the PSPH has been validated under most conditions, it might become invalid as the porosity approaches unity. In order to investigate the valid domain of the PSPH, we have studied the turbulent flows in porous matrices which have one or two length scales using DNS and macroscopic simulation methods. The large porous elements are made of staggered arrays of square cylinders, which might stimulate strong macroscopic (large-scale) turbulence. The small porous elements are made of aligned arrays of spheres or cubes, which suppress the macroscopic turbulence. The analyses are performed for various values of the Reynolds number, Darcy number, pore-scale ratio and porosity. Turbulent two-point correlations, integral length scales and premultiplied energy spectra are calculated from the DNS and macroscopic simulation results to determine the length scale of the turbulent structures. Our numerical results show that the flow becomes turbulent when the Reynolds number is sufficiently large. However, the length scale of turbulence is not considerably affected by the Reynolds number, Darcy number and pore-scale geometry. The PSPH is valid when the porosity has small or medium values. At a sufficiently large Reynolds number, large-scale turbulence survives if the porosity is larger than a critical value. Our DNS and macroscopic simulation results show that this critical value is in the range 0.93–0.98 for porous matrices with large Darcy numbers (0.3–1.26 using the definition in this study). The dependence of the critical porosity on the pore-scale geometry still needs to be further investigated.