Abstract
Enhanced approximate expressions for the incipient indicial lift of thin wings in subsonic potential flow are presented in this study, featuring explicit analytical corrections for the unsteady downwash. Lifting-line and acoustic-wave theories form the basis of the method, within an effective synthesis of the governing physics, which grants a consistent generalised framework and unifies previous works. The unsteady flow perturbation consists of a step-change in angle of attack or a vertical sharp-edged gust. The proposed model is successfully evaluated against numerical results in the literature for the initial airload development of elliptical and rectangular wings with a symmetric aerofoil, considering several aspect ratios and Mach numbers. While nonlinear downwash and compressibility terms demonstrate marginal (especially for the case of a travelling gust), both linear and nonlinear geometrical effects from a significant taper ratio, sweep angle or curved leading-edge are found to be more important than linear downwash corrections (which are crucial for the circulation growth at later times instead, along with linear compressibility corrections). The present formulae may then be used as a rigorous reduced-order model for validating higher-fidelity tools and complex simulations in industrial practice, as well as for estimating parametric sensitivities of unsteady aerodynamic loads within the preliminary design of aircraft wings in the subsonic regime.
Highlights
Within aeroplane multidisciplinary design and optimisation (MDO) [1,2], unsteady airloads from pilot manoeuvres or atmospheric turbulence may effectively be calculated by adopting aerodynamic indicial-admittance functions [3,4] as reduced-order models (ROMs) [5,6], for sensitivity and uncertainty evaluation purposes [7,8,9]
This study introduces explicit analytical corrections for the unsteady downwash and enhances approximate expressions for the indicial lift of thin aircraft wings in the subsonic regime, within an effective synthesis of the governing physics, which grants a consistent generalised framework and unifies previous works
The approximate theoretical results are supported by available high-fidelity data from nonlinear Euler-based Computational fluid dynamics (CFD) simulations, which underwent rigorous convergence studies for both spatial and temporal resolutions; all details of the numerical models and computations can be found in the original works directly [65,66], from geometrical features and grid arrangement to boundary conditions and integration scheme as well as flow perturbation treatment
Summary
Within aeroplane multidisciplinary design and optimisation (MDO) [1,2], unsteady airloads from pilot manoeuvres or atmospheric turbulence may effectively be calculated by adopting aerodynamic indicial-admittance functions [3,4] as reduced-order models (ROMs) [5,6], for sensitivity and uncertainty evaluation purposes [7,8,9]. Exploiting Prandtl–Glauert’s transformation [18,19], exact results for incompressible potential flow can be generalised for low-speed compressible flow in the subsonic regime [20]; the accuracy of such an approximation depends on the rigorous applicability of the underlying fluid mechanics similitude and deteriorates with increasing Mach number towards the transonic regime [21,22] Based on these premises [23,24], a few analytical formulations have been published for the lift development of thin aerofoils Cl(τ) [25,26,27,28,29,30,31,32] and finite wings CL(τ) [33,34,35,36] in p = ρRT, subsonic potential flow [37,38,39,40,41], linearly superposing circulatory Cl(τ), CL(τ) and noncirculatory Cl(τ), CL(τ) contributions from a step in the angle of attack (AOA) or a vertical sharp-edged gust (SEG) in reduced time τ [42]. Considering AOA and SEG perturbations, theoretical results are obtained and critically compared with numerical ones in previous publications for thin airfoil at different Mach numbers [64], elliptical wings with different aspect ratio [65], and rectangular wings with different sweep angle [66]; all comparisons are explained and clarified in light of the proposed analytical derivation, within a comprehensive assessment
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