The steady aerodynamics of aerofoils with porosity gradients - PubMed (original) (raw)

The steady aerodynamics of aerofoils with porosity gradients

Rozhin Hajian et al. Proc Math Phys Eng Sci. 2017 Sep.

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.

Keywords: Riemann–Hilbert problem; silent owl flight; trailing-edge noise.

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Conflict of interest statement

We declare we have no competing interests.

Figures

Figure 1.

Figure 1.

Normalized pressure distribution of a uniformly porous flat aerofoil for different porosity parameters δ: (a) pressure jump normalized by angle of attack, −p(x)/α; (b) pressure jump normalized by the high porosity limit, −p(x)/(2_α_/δ). (Online version in colour.)

Figure 2.

Figure 2.

Porosity and pressure distributions of a thin aerofoil with a prescribed differentiable porosity distribution given by (4.8) with _a_=−0.5: (a) porosity distributions for _r_=10 and r→∞; (b) pressure distributions for _r_=10 and the singular limit as r→∞ for the flat aerofoil. The dashed line indicates Iosilevskii’s result, eqn (13) in [18]. (Online version in colour.)

Figure 3.

Figure 3.

Comparison of the predicted and measured lift coefficients of a porous SD7003 aerofoil at zero angle of attack for various porosity constants δ. (Online version in colour.)

Figure 4.

Figure 4.

Pressure distribution of a porous SD7003 aerofoil at zero angle of attack for various porosity constants δ, based on the theoretical model. (Online version in colour.)

Figure 5.

Figure 5.

Normalized pressure distribution, −p(x)/β, of a uniformly porous cambered aerofoil at zero angle of attack (_α_=0) for different porosity constants δ. (Online version in colour.)

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