Unsteady Lifting-line Theory Research Articles

Dynamic hydroelasticity of composite appendages with reverse-mode algorithmic differentiation

Composite materials enable the tailoring of load-dependent, passive, shape adaptation because of the directional stiffness of the fibers. However, excessive flow-induced vibrations and dynamic instabilities represent a design challenge for composite marine appendages. We develop DCFoil, a dynamic composite foil solver that uses one-dimensional models—composite beam elements and unsteady lifting line theory—to perform static and dynamic frequency domain analysis. The program is differentiated to provide derivatives of an aggregated flutter function with respect to design variables. The flutter function characterizes dynamic hydroelastic stability. We apply the solver to investigate the hydroelastic performance of a composite fin bulb keel based on the IMOCA 60 class of racing yachts, which were reported to have flutter problems. Based on the derivatives computed for this bulb keel, we found that the foil thickness has the most significant impact on flutter speed. This work shows both the development of and utility of a program with flutter derivatives for flexible composite marine appendage design.

The effect of elastic couplings and material uncertainties on the flutter of composite high aspect ratio wings

In this paper, the stochastic aeroelastic stability analysis of a composite high aspect ratio wing with various elastic couplings is investigated. The wing consists of a rectangular spar-box made from composite materials. Due to its high aspect ratio, the wing structure is modelled using the exact beam formulation, and the aerodynamic loads acting on the wing are simulated using an unsteady lifting line theory. A composite spar-box cross section is considered, and the effect of various effective parameters on the aeroelastic stability of the wing is studied. First, a baseline composite wing is designed and the effect of three different elastic couplings on the stability of the wing is determined. These three elastic couplings that affect the aeroelastic design of the composite wing are the out-of-plane (flap)-twist, the in-plane (lag)-twist and the extension-twist. Furthermore, the effect of uniformly distributed load on the aeroelastic stability of the wing is assessed for various layups. Finally, the effect of material uncertainties on the aeroelastic stability of the composite wing with various couplings is studied. It is observed that the type of elastic coupling has a significant effect on the sensitivity of the flutter speed and frequency of the composite wing due to the randomness of the material properties. In particular, the aeroelastic behaviour of the wing with lag-twist coupling is more sensitive to the material uncertainties than other two layups.

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

Frequency-domain unsteady lifting-line theory is better developed than its time-domain counterpart. To take advantage of this, this paper proposes a method to transform time-domain kinematics to the frequency domain, perform a convolution, and then return the results back to the time domain. This paper demonstrates how well-developed frequency-domain methods can be easily applied to time-domain problems, enabling prediction of forces and moments on finite wings undergoing arbitrary kinematics. Results are presented for rectangular wings of various aspect ratios, undergoing pitch and heave kinematics. Computational fluid dynamics is used to test the effectiveness of the method in the Euler and low-Reynolds-number () regimes. Overall, the proposed method provides fast and reasonably accurate predictions of lift and moment coefficients, particularly in comparison to strip theory, which is commonly used for problems involving arbitrary kinematics.

Usefulness of Inviscid Linear Unsteady Lifting-Line Theory for Viscous Large-Amplitude Problems

Unsteady lifting-line theory (ULLT) is a low-order method capable of modeling interacting unsteady and finite wing effects at low computational cost. Most formulations of the method assume inviscid flow and small amplitudes. Although these assumptions might be suitable for small-amplitude aeroelastic problems at high Reynolds numbers, modern engineering applications increasingly involve lower Reynolds numbers, large-amplitude kinematics, and vortex structures that lead to aerodynamic nonlinearities. This paper establishes that ULLT still provides a useful solution for low-Reynolds-number, large-amplitude kinematics problems, by comparing ULLT results against those of experimentally validated computational fluid dynamics simulations at . Three-dimensional effects stabilize leading-edge vortex (LEV) structures, resulting in a good prediction of whole wing force coefficients by ULLT. Although the inviscid spanwise force distributions are accurate for small-amplitude kinematics, the ULLT cannot model three-dimensional vortical structures, and thus it cannot correctly predict the force distribution due to the LEV. It can, however, predict the shedding of LEVs to a limited extent via the leading-edge suction parameter criterion. This can then be used as an indicator of the usefulness of the force distribution results.

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

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

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

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.

Unsteady Lifting Line Theory Using the Wagner Function for the Aerodynamic and Aeroelastic Modeling of 3D Wings

A method is presented to model the incompressible, attached, unsteady lift and pitching moment acting on a thin three-dimensional wing in the time domain. The model is based on the combination of Wagner theory and lifting line theory through the unsteady Kutta–Joukowski theorem. The results are a set of closed-form linear ordinary differential equations that can be solved analytically or using a Runge–Kutta–Fehlberg algorithm. The method is validated against numerical predictions from an unsteady vortex lattice method for rectangular and tapered wings undergoing step or oscillatory changes in plunge or pitch. Further validation is demonstrated on an aeroelastic test case of a rigid rectangular finite wing with pitch and plunge degrees of freedom.

State-Space Adaptation of Unsteady Lifting Line Theory: Twisting/Flapping Wings of Finite Span

In this paper, a low-order state-space adaptation of the unsteady lifting line model has been analytically derived for a wing of finite aspect ratio, suitable for use in real-time control of wake-dependent forces. Each discretization along the span has between 1–6 states to represent the local unsteady wake effects, rather than remembering the entire wake history which unnecessarily complicates controller design. Sinusoidal perturbations to each system degree of freedom are also avoided. Instead, a state-space model is fit to individual indicial functions for each blade element, allowing the downwash and lift distributions over the span to be arbitrary. The wake geometry is assumed to be quasi steady (no roll up) but with fully unsteady vorticity. The model supports time-varying surge (a nonlinear effect), dihedral, heave, sweep, and twist along the span. Cross-coupling terms are explicitly derived. This state-space model is then validated through comparison with an analytic solution for elliptic wings, an unsteady vortex lattice method, and experiments from the literature.

Robust Aeroelastic Stability Analysis Considering Frequency-Domain Aerodynamic Uncertainty

The problem of modeling frequency-domain aerodynamic uncertainty for a slender wing structure is investigated. Based on an unsteady lifting-line theory used for the generalized aerodynamic forces, a quite versatile uncertainty description with a clear physical interpretation is proposed. The uncertainty description is easily put in a form suitable for application of the mu framework in robust linear control. Because only frequency response matrices are required for the mu computations, the proposed uncertainty description can be used for robust stability and performance analysis without rational function approximations of the aerodynamic transfer function matrices. The usefulness of the uncertainty description and the methods available for robust aeroelastic stability analysis is demonstrated by performing aeroelastic wind-tunnel experiments.

A unified unsteady lifting-line theory

A lifting-line theory is developed for wings of large aspect ratio oscillating in an inviscid fluid. The theory is unified in the sense that the wing may be curved or inclined to the flow, and the asymptotic expansion is uniformly valid with respect to the frequency. The method is based on the integral equation formulation of the problem. The technique, pioneered by Kida & Miyai (1978), consists of asymptotically solving the Fredholm equation of the first kind which links the unknown pressure jump and the normal velocity imposed on the wing. Use of the finite-part integral theory introduced by Hadamard (1932) and of a technique developed in Guermond (1987, 1988, 1990) yields an asymptotic expansion of the surface integral in terms of the inverse of the aspect ratio. At each approximation order, the problem reduces to a classical two-dimensional integral equation, whose unknown is the pressure jump, and whose right-hand side depends only on the previous approximation orders of the solution. The first finite-span correction is explicitly calculated. An extensive numerical study is carried out, and comparisons with published results are made.

Unsteady lifting-line theory by the method of matched asymptotic expansions

An unsteady lifting-line theory is presented for a general motion of a wing of high aspect ratio. Our matched-asymptotic-expansions analysis parallels that of Van Dyke (1964) in his solution for the steady lifting line, but is complicated by the shedding of transverse vortices associated with variation of circulation with time. The principal result is an expression for the downwash due to three-dimensional effects. Numerical calculations are presented for a wing of elliptic planform following a curved path.

The induced downwash and lift on a wing of high aspect ratio in unsteady motion

SummaryAn unsteady lifting line theory for a general motion of a wing of high aspect ratio is presented. Our analysis parallels that of Van Dyke (1964) in his solution for the steady lifting line by the method of matched asymptotic expansions but is complicated by the shedding of transverse vortices associated with variation of circulation with time. We find expressions for the downwash due to three-dimensional (finite span) effects and the lift on the wing. Calculations are presented for a wing of elliptic planform following a curved path.

Energetics and optimum motion of oscillating lifting surfaces of finite span

The energetics of an unswept wing of finite span oscillating harmonically in combined pitch and heave in in viscid incompressible flow are determined in closed form. The calculations are based on a recently developed low-frequency unsteady lifting-line theory. The energetic calculations for the wing consist of sectional and total values of thrust, leading-edge suction force, power required to maintain the wing oscillations, and energy-loss rate due to vortex shedding in the wake, where the latter quantity is only defined for the entire wing. These results are used to analyse the optimum motion of a wing oscillating harmonically: optimum motion minimizes the power input for fixed average total thrust. The optimum solution is found to be unique (at least for low reduced frequencies), in contrast to the two-dimensional optimum, which is non-unique. Numerical results are presented for the energetics and optimum motion of an elliptic wing.To understand better the structure of the known solution for the optimum motion of an oscillating two-dimensional airfoil, the solution is recast in terms of the normal modes of the energy-loss-rate matrix. It is found that one of the modes, termed here the ‘invisible mode’, plays a central role in the optimum solution and is responsible for its non-uniqueness. The three-dimensional optimum, which is unique, does not have an invisible mode.

Unsteady lifting-line theory as a singular perturbation problem

Unsteady lifting-line theory is developed for a wing of large aspect ratio oscillating at low frequency in inviscid incompressible flow. The wing is assumed to have a rigid chord but a flexible span. Use of the method of matched asymptotic expansions reduces the problem from a singular integral equation to quadrature. The pressure field and airloads, for a prescribed wing shape and motion, are obtained in closed form as expansions in inverse aspect ratio. A rigorous definition of unsteady induced downwash is also obtained. Numerical calculations are presented for an elliptic wing in pitch and heave; compared with numerical lifting-surface theory, computation time is reduced significantly. The present work also identifies and resolves errors in the unsteady lifting-line theory of James (1975), and points out a limitation in that of Van Holten (1975, 1976, 1977).

An unsteady lifting line theory of flapping wings with application to the forward flight of birds

A lifting line theory of flapping wings in steady forward flight is presented in which the unsteady features of the flow are modelled. A detailed three-dimensional model of the vortex wake is used to evaluate the unsteadiness to first order. The method gives satisfactory agreement with well-known limiting cases. Relationships between the geometric and the kinematic parameters, and the forces and the power are predicted which are compatible with the limited experimental evidence. The theory is applied to the calculation of the power curve of specific birds. Important similarities and differences are observed between the present results and those of Pennycuick (1975) and Rayner (1979c).

Some notes on unsteady lifting-line theory

A higher-order lifting-line theory for a wing in unsteady motion is discussed. Apart from the addition of higher-order terms, it also differs from the theory derived by James (1975) in its emphasis on ‘physical’ interpretations. This emphasis has made it possible to shed some new light on Prandtl's classical lifting-line theory, as well as on Weissinger's 3/4-chord theory.

Approximate Evaluation of Certain Functions Arising in Unsteady Airfoil Theory

appears in the kernel of an integral equation shown by Lawrence and Gerber [1] to govern the aerodynamics of low-aspect-ratio airfoils in unsteady motion. The Cicala function [2], F(p) = M*(oo, 3)/3, plays a similar role in Reissner's unsteady lifting line theory [3]. Although M*(a, () has been partially tabulated [1], a simple approximate expression valid over the entire domain of interest (0 ? a oo. On the other hand, the bracketed portion of the integrand in (1) is a well-behaved function of the single parameter t x/l. We therefore seek an approximation of