Compressible unsteady aerodynamics of finite-chord porous aerofoils

Abstract

A unified linear theory is developed for the unsteady loading of finite-chord aerofoils, rigid or porous, in compressible subsonic flow. The formulation combines Possio's integral operator with a convective permeable boundary condition, allowing chordwise-varying admittance while retaining the wake and unsteady Kutta condition. The loading exponents at aerofoil edges and admittance discontinuities keep their incompressible form in terms of the local permeability parameter, so weighted-Jacobi collocation carries over. The solution is verified against published incompressible results and an independent compressible formulation. Gust and heave responses and their indicial counterparts reveal that permeability controls the sensitivity of unsteady loading to compressibility. As permeability increases, the material impedance rather than the surrounding flow sets the pressure jump, but only gradually: at Mach number 0.7 the gust load on a weakly permeable surface changes by a third to a half, and closed-form steady and high-frequency limits for resistive surfaces show that it vanishes only when the permeability parameter greatly exceeds the Mach number. The porous-to-impermeable load ratio, by which a treatment is judged, cannot be obtained by combining incompressible porous and compressible rigid theories, which misjudge it by up to a factor of two. Incompressible theory overestimates this ratio for an acoustically compact chord and underestimates it for a non-compact one, by factors of 1.4 to 4.3 at Mach numbers 0.5 to 0.7 and reduced frequencies 5 to 20, overstating the load reduction. Since compactness depends on Mach number times reduced frequency, the error persists at low speed, exceeding 10 per cent at Mach number 0.05 for reduced frequencies above 10.