Linear Stability Analysis of Wall-Bounded Flows Across Anisotropic Porous Media

A simplified model to study the physical mechanisms leading to tonal noise in noise-reducing trailing edges comprises communicating channels through a porous medium. The impact of material properties and flow conditions on the onset of communication is investigated through a linear stability analysis based on a coupled Orr–Sommerfeld equation, modeling the porous medium using the Forchheimer-corrected Darcy law. The solution of the generalized eigenvalue problem results in the identification of three different instabilities: fluid, porous grazing, and communicating. The communication regime is the most unstable for thin porous media ([Formula: see text], where t is the thickness and [Formula: see text] is the half-channel height) and at a low-bulk-velocity Reynolds number ([Formula: see text]). It is characterized by an increase in the correlation and instantaneous flow between the channels. Moreover, the growth rate of the instability shows a dependence on the anisotropy of the porous medium properties, with wall-normal properties modulating the onset of communication. A velocity differential between the channels reduces the instantaneous flow through the porous medium and the correlation between the channels, explaining some of the reduction in communication in porous trailing edges under lifting conditions. Design criteria for porous trailing edges are formulated to prevent the emergence of tonal noise at high permeabilities. The porous trailing edge should be thin ([Formula: see text]), shorter than [Formula: see text], and the permeabilities should be large enough ([Formula: see text] and [Formula: see text], where [Formula: see text] and [Formula: see text] are the traces of the linear and nonlinear permeabilities, respectively, normalized with respect to the porous thickness).