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Based on solving the Navier–Stokes equations in the two-dimensional percolation porous media for 500 different configurations, the scaling relations for the fluid permeability k and the inertial parameter β in the Forchheimer equation are studied. In the vicinity of the critical threshold pc, the fluid permeability k and the inertial parameter β will crossover from the fractal behaviors: k∼L−μ1, β∼Lμ2, where μ1≈1.0, μ2≈2.0 for the small size L, to the constants: k∼(p−pc)α1, β∼(p−pc)−α2, where α1≈4/3, α2≈8/3, α2≈8/3. Compared to the viscous flow, the resistance to flow will have a larger critical exponent for the finite Reynolds number flows. [Copyright &y& Elsevier]

We study the statistical properties of fluid flows in percolation porous media at very low Reynolds number by numerical simulations for 20,000 different configurations. It is shown that there are some important differences between fluid flows in macroscopically homogeneous and fractal porous media. For fluid flows in macroscopically homogeneous media, the pressure is definite, but the velocity is random and depends on the structural details of porous media. The permeability k for fluid flows in fractal changes with the size L as k∼L−α where α≈1.0, not approaching the constant expected from Darcy''s law. The statistical distribution of pressure in fractal is independent of the size L and can be approximated by a function of the triangular shape. [Copyright &y& Elsevier]

By analyzing the experimental data, the statistical properties of 2D turbulent wake of a heated cylinder are studied. The coherent structures may have the stronger effect upon the transverse velocity and temperature than the longitudinal velocity. In the near wake, the transverse velocity v and the temperature T have the bimodal distribution and the χ2 distribution, respectively. But the distribution of the longitudinal velocity u is near the Gaussian even in the near wake. The statistical distributions for the velocity differences and the temperature differences are studied. In the near wake, the pdfs of the transverse velocity differences are concave functions only for the time separations nΔt: nc1. Otherwise, they will be the convex functions. The pdf of the temperature difference will change from the nearly exponential distribution to the Gaussian distribution as the time separation nΔt increasing in the near wake. Using the detrended fluctuation analysis, it is found out that the longitudinal velocity and temperature have the long-range correlation with the exponent α≈0.7 in the near and intermediate wake and the transverse velocity has the weakly anti-correlation with the exponent α≈0.44 in the intermediate wake. [Copyright &y& Elsevier]

Abstract: In this article, the statistical properties for the renormalized permeability obtained from the renormalization of the correlated permeability field are investigated. In contrast to the uncorrelated porous media, the scaling of the variance of the renormalized permeability field exhibits a crossover behavior. When the correlation lengths are larger compared with the domain scale covered by the renormalization procedure, the variance of the renormalized permeability will decrease slowly and the scaling exponent will be close to zero. As the renormalization number increases, the covered domain scale will eventually become larger than the correlation lengths, and then the scaling property will transit to the uncorrelated case. The convergent values of the renormalized permeability for isotropic and anisotropic correlated media are also investigated. Both the theoretical analysis and the simulation results show that larger correlation length in one direction will lead to a larger convergent value in the corresponding direction. For the log-normal permeability field, numerical simulations show that the crossover scaling and also the convergent value for the renormalized permeability can be fitted very well by simple mathematical functions. [Copyright &y& Elsevier]