Unsteady aerodynamics of porous aerofoils
Unsteady aerodynamics of porous aerofoils
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DOI：
10.1017/jfm.2020.1031
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发表时间：
2019-11
影响因子：
3.7
通讯作者：
Peter J. Baddoo;Rozhin Hajian;J. Jaworski
中科院分区：
文献类型：
--
作者：
Peter J. Baddoo;Rozhin Hajian;J. Jaworski
通过将多孔介质的材料特性嵌入到线性化边界条件中，我们将非定常薄翼型理论推广到具有广义弦向孔隙率分布的翼型。将Plemelj公式应用于所得到的边值问题，得到了一个奇异的FredholmVolterra积分方程，它不允许有解析解。我们用适当的基函数展开边界涡度分布，发展了一种数值解方案。在前沿和后缘的渐近分析表明，合适的基函数是加权的雅可比多项式，其参数与孔隙度分布有关。与切比雪夫基函数的标准选择不同，雅可比多项式基能够构造准确和快速的数值格式，而切比雪夫基函数被证明不适合于多孔翼型。给出了数值解格式在不连续孔隙度剖面、准静态问题以及环流和非环流贡献分离中的应用。进一步对奇异FredholmVolterra积分方程解的渐近分析证实了数值格式的正确性，并以标度律的形式阐明了非定常解在小的或大的约化频率下的行为。在低频下，多孔阻力占主导地位，而在高频下，在尾缘附近形成一个渐近的内区，多孔介质的有效质量占主导地位。对经典的Theodorsen和Sears函数进行了数值计算，并对这些频域函数进行傅里叶变换逆变换，分别对Wagner函数和Küssner函数进行了多孔性扩展，用于翼型的瞬变运动或阵风相遇。目前的分析结果及其支持的数值框架旨在利用孔隙率对设计策略进行非定常气动评估，这对非定常阵风抑制、降噪翼型设计和生物启发飞行具有重要意义。
Abstract We extend unsteady thin aerofoil theory to aerofoils with generalised chordwise porosity distributions by embedding the material characteristics of the porous medium into the linearised boundary condition. Application of the Plemelj formulae to the resulting boundary value problem yields a singular Fredholm–Volterra integral equation which does not admit an analytical solution. We develop a numerical solution scheme by expanding the bound vorticity distribution in terms of appropriate basis functions. Asymptotic analysis at the leading and trailing edges reveals that the appropriate basis functions are weighted Jacobi polynomials whose parameters are related to the porosity distribution. The Jacobi polynomial basis enables the construction of a numerical scheme that is accurate and rapid, in contrast to the standard choice of Chebyshev basis functions that are shown to be unsuitable for porous aerofoils. Applications of the numerical solution scheme to discontinuous porosity profiles, quasi-static problems and the separation of circulatory and non-circulatory contributions are presented. Further asymptotic analysis of the singular Fredholm–Volterra integral equation corroborates the numerical scheme and elucidates the behaviour of the unsteady solution for small or large reduced frequency in the form of scaling laws. At low frequencies, the porous resistance dominates, whereas at high frequencies, an asymptotic inner region develops near the trailing edge and the effective mass of the porous medium dominates. Analogues to the classical Theodorsen and Sears functions are computed numerically, and Fourier transform inversion of these frequency-domain functions produces porous extensions to the Wagner and Küssner functions for transient aerofoil motions or gust encounters, respectively. Results from the present analysis and its underpinning numerical framework aim to enable the unsteady aerodynamic assessment of design strategies using porosity, with implications for unsteady gust rejection, noise-reducing aerofoil design and biologically inspired flight.