Unsteady lifting-line theory and the influence of wake vorticity on aerodynamic loads

Abstract

Frequency-domain unsteady lifting-line theory (ULLT) provides a means by which the aerodynamics of oscillating wings may be studied at low computational cost without neglecting the interacting effects of aspect ratio and oscillation frequency. Renewed interest in the method has drawn attention to several uncertainties however. Firstly, to what extent is ULLT practically useful for rectangular wings, despite theoretical limitations? And secondly, to what extent is a complicated wake model needed in the outer solution for good accuracy? This paper aims to answer these questions by presenting a complete ULLT based on the work of Sclavounos, along with a novel ULLT that considers only the streamwise vorticity and a Prandtl-like pseudosteady ULLT. These are compared to Euler CFD for cases of rectangular wings at multiple aspect ratios and oscillation frequencies. The results of this work establish ULLT as a low computational cost model capable of accounting for interacting finite-wing and oscillation frequency effects and identify the aspect ratio and frequency regimes where the three ULLTs are most accurate. This research paves the way towards the construction of time-domain or numerical ULLTs which may be augmented to account for nonlinearities such as flow separation.

Applying Frequency-Domain Unsteady Lifting-Line Theory to Time-Domain Problems

“…Widnall [18] and Sclavounos [19] improved on the wake model by which the 2D inner problem was corrected by the simplified 3D outer problem. The importance of this wake model was demonstrated by Bird and Ramesh [20].…”

“…More recently, Boutet and Dimitriadis [34] combined a Prandtl-like wake with Wagner's theory, and Berci obtained a method for swept wings with wake vorticity modeled using a single lumped vortex ring [35]. However, as was found by Bird and Ramesh [20], Prandtl-like wake models obtain different solutions in comparison with more complete wake models such as those used by Guermond and Sellier [21].…”

“…This time marching theory required straight wings and uniform 3D induced downwash, in theory limiting its applicability to only low-frequency kinematics. However, Bird and Ramesh [20] found that this theoretical limitation had limited practical relevance for rectangular wing planforms. Crucially though, it modeled spanwise vorticity in the wake of the inner 2D domain and the correction for the change of the wake's spanwise vorticity with respect to span in the outer 3D domain.…”

“…Prandtl-like wakes neglect this. Consequently, their accuracy is limited [20]. More complicated wake models can be constructed using numerical methods, but numerical methods typically entail increased computational costs and convergence issues.…”

On the Incipient Indicial Lift of Thin Wings in Subsonic Flow: Acoustic Wave Theory with Unsteady Three-Dimensional Effects

“…It is worth stressing that a non-circulatory contribution ĈL (τ) exists from a step-change in the angle of attack of an incompressible flow [96], in the form of a Dirac-delta singularity [94] that generates apparent inertia at the start of the airload development [4]; however, this well-known effect is irrelevant for the scope of the present work on compressible flow. Alongside other formulations in the time domain [97][98][99][100][101][102][103][104][105][106], note that alternative approaches have been proposed that lead to a frequency-dependent generalisation of lifting line theory, but their formulation is less convenient for deriving explicit expressions of the incipient indicial airload, as they specialise in harmonic motion and their inherent complexity is often not practical [128]. Nevertheless, they assess when two-and three-dimensional effects may actually be decoupled as well as show that for slender wings the downwash from the tip vortices can roughly be assumed as constant along the chord of each section (unlike the wake's inflow) and affects the circulatory airload only [129,130].…”