Boundary conditions for shear flow past a permeable interface modeled as an array of cylinders

The boundary condition relating the macroscopic jump in the tangential velocity across a permeable interface consisting of a particulate lattice to the shear rate prevailing on either side of the interface is discussed. The computation of the velocity jump hinges on the realization that shear flow on one side of the interface induces a slip velocity on that side and a streaming drift velocity on the other side. The direction and magnitude of the slip and drift velocities depend on the interface constitution, solid fraction, and Reynolds number. Numerical computations are performed for a model two-dimensional interface consisting of a periodic array of cylinders. In the case of longitudinal unidirectional flow, the boundary conditions are defined in terms of previously computed drift and slip velocity coefficients for any ratio of the shear rates above and below the interface and any Reynolds number. To study the behavior in the complementary case of transverse flow, the Navier-Stokes equation is solved numerically using a finite-difference method on an orthogonal grid generated by conformal mapping, using the stream function/vorticity formulation. The results reveal that inertial effects promote the magnitude of the slip and drift velocity, and illustrate the streamline pattern near the interface.