Flow through Porous Bed of Turbulent Stream

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Abstract
An analysis of the flow through an inclined‐plane pervious substratum coupled to a turbulent, steady, uniform, and fully developed open channel flow above it is introduced. The porous material is taken to be homogeneous, isotropic, and formed by a square‐arrayed lattice made of circular cylinders with axes normal to the flow direction. Starting from the Navier‐Stokes equation for the flow through the pores, a combination of the ensemble average method and the method of multiple scales is used to derive the equations governing the macroscale flow through the bed. From them, the vertical variation of the velocity in the substratum, as well as the boundary at the pervious interface are obtained. Application of the derived boundary condition, which guarantees continuity of the total stress across the interface, to the frontier between a laminar flow and a porous material, recovers the condition proposed by Beavers and Joseph in 1967. Joint use of the obtained equation for the velocity distribution and available data, suggests that the parameter controlling its exponential decay, from the slip‐velocity at the interface to the Darcy velocity away from it, may be a function of the properties of the porous material. The analysis also indicated that the nonlinearity of the flow through the substratum originates from the curvature of the streamlines and the flow separation around the solid particles at the microscale level.

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