Permeability and Turbulence Over Perforated Plates

Abstract

We perform direct numerical simulations of turbulent flow at friction Reynolds number $$Re_\tau \approx 500{-}2000$$ R e τ ≈ 500 - 2000 grazing over perforates plates with moderate viscous-scaled orifice diameter $$d^+\approx 40-160$$ d + ≈ 40 - 160 and analyse the relation between permeability and added drag. Unlike previous studies of turbulent flows over permeable surfaces, we find that the flow inside the orifices is dominated by inertial effects, and that the relevant permeability is the Forchheimer and not the Darcy one. We find evidence of a fully rough regime where the relevant length scale is the inverse of the Forchheimer coefficient, which can be regarded as the resistance experienced by the wall-normal flow. Moreover, we show that, for low porosities, the Forchheimer coefficient can be estimated with good accuracy using a simple analytical relation.

Turbulence and added drag over acoustic liners

“…Different mass flow rates have been simulated, which allows us to estimate the permeability coefficients of the Darcy–Forchheimer law. More details and an extensive discussion of Darcy and Forchheimer drag is available in our recent work (Shahzad, Hickel & Modesti 2022), where we calculate the Darcy permeability and the Forchheimer coefficient of perforated plates that match the present DNS dataset and compare the results with available engineering formulas.…”

“…As discussed in § 1, the friction Reynolds number over acoustic liners is , and the viscous-scaled inverse of the Forchhemeir coefficient for a geometry with , and is (Shahzad et al. 2022). Assuming that a fully rough regime exists, then (3.3) returns , which can be converted into drag variation (Modesti et al.…”

“…Fortunately, the existence of a fully rough regime simplifies the modelling of acoustic liners and makes it easy to estimate the drag increase they induce in operating conditions. As discussed in § 1, the friction Reynolds number over acoustic liners is Re τ ≈ 5400, and the viscous-scaled inverse of the Forchhemeir coefficient for a geometry with d/δ ≈ 0.08, t/d ≈ 1 and σ ≈ 0.3 is 1/α + y ≈ 55 (Shahzad et al 2022). Assuming that a fully rough regime exists, then (3.3) returns U + ≈ 6.5, which can be converted into drag variation (Modesti et al 2021):…”

“…Different mass flow rates have been simulated, which allows us to estimate the permeability coefficients of the Darcy-Forchheimer law. More details and an extensive discussion of Darcy and Forchheimer drag is available in our recent work (Shahzad, Hickel & Modesti 2022), where we calculate the Darcy permeability and the Forchheimer coefficient of perforated plates that match the present DNS dataset and compare the results with available engineering formulas. If we regard acoustic liners as porous surfaces, two relevant length scales for the flow are the square root of the Darcy permeability K y and the inverse of the Forchheimer coefficient 1/α y , besides the orifice diameter.…”