The steady aerodynamics of aerofoils with porosity gradients

Abstract

This theoretical study determines the aerodynamic loads on an aerofoil with a prescribed porosity distribution in a steady incompressible flow. A Darcy porosity condition on the aerofoil surface furnishes a Fredholm integral equation for the pressure distribution, which is solved exactly and generally as a Riemann-Hilbert problem provided that the porosity distribution is Hölder-continuous. The Hölder condition includes as a subset any continuously differentiable porosity distributions that may be of practical interest. This formal restriction on the analysis is examined by a class of differentiable porosity distributions that approach a piecewise, discontinuous function in a certain parametric limit. The Hölder-continuous solution is verified in this limit against analytical results for partially porous aerofoils in the literature. Finally, a comparison made between the new theoretical predictions and experimental measurements of SD7003 aerofoils presented in the literature. Results from this analysis may be integrated into a theoretical framework to optimize turbulence noise suppression with minimal impact to aerodynamic performance.

Noncirculatory Fluid Forces on Panels with Porosity Gradients

“…decreases to O(10 −3 ) when δ = 0.1. Note that expected values of the porosity parameter in low-speed applications are δ = O(10 −2 ), as measured experimentally by Geyer et al[18] and analyzed by Hajian and Jaworski[17]. The pressure distributions for δ values of this magnitude do not show appreciable differences when compared to the nonporous case.…”

“…For a panel with a Darcy-type porosity distribution under conditions that permit omission of nonlinear flow and acoustic effects (cf. [24][25][26]), the local flow rate is linearly proportional to the porosity and the pressure distribution across the porous medium [17,27]:…”

Potential flow through a cascade of aerofoils: direct and inverse problems

“…This can be performed for any number of cascades. Finally, a useful extension to the model in aeroacoustic applications would be to allow for porous aerofoils, where a similar approach to [25] may be applied.…”

Reducing aerofoil–turbulence interaction noise through chordwise-varying porosity

“…However, the benefit of a partially porous aerofoil is in its aerodynamics rather than its acoustics; fully porous aerofoils, whilst acoustically beneficial, have significant aerodynamic penalties (Geyer et al 2010); on the other hand, partially porous aerofoils have lessened aerodynamic penalties but produce more noise than fully porous aerofoils (Iosilevskii 2011;Geyer & Sarradj 2019). The steady aerodynamics of partially porous aerofoils have previously been predicted theoretically by Iosilevskii (2011), which has been extended to aerofoils with porosity gradients by Hajian & Jaworski (2017).…”

“…This paper, therefore, investigates the effect of porosity gradients on the noise generated by aerofoil-turbulence interaction. We also implement the Riemann-Hilbert solution of Hajian & Jaworski (2017) to determine the lift coefficient, and thus a measure of the aerodynamic performance of the plates. In these models we allow an arbitrary variation in porosity along a finite perforated flat plate, modelling a thin permeable aerofoil.…”