The Lagrangian smoothed particle hydrodynamics (SPH) method is employed to obtain a meso-/micro-scopic pore-scale insight into the transverse flow across the randomly aligned fibrous porous media in a 2D domain. Fluid is driven by an external body force, and a square domain with periodic boundary conditions imposed at both the streamwise and transverse flow direction is assumed. The porous matrix is established by randomly embedding a certain number of fibers in the square domain. Fibers are represented by position-fixed SPH particles, which exert viscous forces upon, and contribute to the density variations of, the nearby fluid particles. An additional repulsive force, similar in form to the 12-6 Lennard-Jones potential between atoms, is introduced to consider the no-penetrating restraint prescribed by the solid pore structure. This force is initiated from the fixed solid material particle and may act on its neighboring moving fluid particles. Fluid flow is visualized by plotting the local velocity vector field; the meandering fluid flow around the porous microstructures always follow the paths of least resistance. The simulated steady-state flow field is further used to calculate the macroscopic permeability. The dimensionless permeability (normalized by the squared characteristic dimension of the fiber cross section) exhibits an exponential dependence on the porosity within the intermediate porosity range, and the derived dimensionless permeability-porosity relation is found to have only minor dependence on either the relative arrangement condition among fibers or the fiber cross section (shape or area).