Applications of the unsteady vortex-lattice method in aircraft aeroelasticity and flight dynamics

Abstract

The unsteady vortex-lattice method provides a medium-fidelity tool for the prediction of non-stationary aerodynamic loads in low-speed, but high-Reynolds-number, attached flow conditions. Despite a proven track record in applications where free-wake modelling is critical, other less-computationally expensive potential-flow models, such as the doublet-lattice method and strip theory, have long been favoured in fixed-wing aircraft aeroelasticity and flight dynamics. This paper presents how the unsteady vortex-lattice method can be implemented as an enhanced alternative to those techniques for diverse situations that arise in flexible-aircraft dynamics. A historical review of the methodology is included, with latest developments and practical applications. Different formulations of the aerodynamic equations are outlined, and they are integrated with a nonlinear beam model for the full description of the dynamics of a free-flying flexible vehicle. Nonlinear time-marching solutions capture large wing excursions and wake roll-up, and the linearisation of the equations lends itself to a seamless, monolithic state-space assembly, particularly convenient for stability analysis and flight control system design. The numerical studies emphasise scenarios where the unsteady vortex-lattice method can provide an advantage over other state-of-the-art approaches. Examples of this include unsteady aerodynamics in vehicles with coupled aeroelasticity and flight dynamics, and in lifting surfaces undergoing complex kinematics, large deformations, or in-plane motions. Geometric nonlinearities are shown to play an instrumental, and often counter-intuitive, role in the aircraft dynamics. The unsteady vortex-lattice method is unveiled as a remarkable tool that can successfully incorporate all those effects in the unsteady aerodynamics modelling.

Introduction

Unsteady aerodynamics based on potential-flow theory has long furnished the basic foundation in low-speed aircraft aeroelasticity and dynamic-load analysis. The doublet-lattice method [1], in particular, has long been the fundamental tool of the aeroelastic community, and has provided a robust approach for non-stationary aerodynamic prediction, although simpler approaches based on strip theory and indicial functions are still used at the conceptual level [2]. However, the applicability of the existing design tools is being stretched as novel concepts are developed and conventional vehicles see their wing stiffness substantially reduced. Aircraft with coupled aeroelasticity and flight dynamics, and lifting surfaces undergoing complex kinematic motions or large deformations are becoming customary, requiring a new paradigm for modelling, analysis, and control synthesis. The unsteady vortex-lattice method, an aerodynamic model with a long and successful track record, can be re-engineered to provide again an advantageous alternative in many of these scenarios.

The surge in the number of high-altitude long-endurance (HALE) aircraft, including extreme cases such as solar-powered platforms, attests some of the changes aviation is experiencing. These unmanned aerial vehicles (UAV) are not subject to many traditional restrictions on their overall configuration, and they usually consist of very light, slender and thus flexible structures. This may cause geometrically nonlinear deformations during normal operation and overlap of the aircraft's aeroelastic and flight dynamics natural frequencies, which, in turn, makes them particularly vulnerable to atmospheric disturbances. As a result, any successful modelling effort depends upon a multidisciplinary outlook that integrates aeroelasticity, flight dynamics and controls into a common framework [3], [4].

A substantial research effort has been carried out in recent years towards this goal, and even though other alternatives have been explored [5], [6], [7], [8], in most cases the presence of a characteristic dominant dimension in the primary flexible structures has motivated descriptions of the nonlinear structural dynamics through composite beam models and of the non-stationary aerodynamics by means of 2D strip theory [4], [9], [10], [11], [12], [13], [14], [15], [16]. This computationally inexpensive approach has led to the identification of critical phenomena in the behaviour of these vehicles, but neglects relevant 3D flow physics, such as the accurate prediction of wing-tip effects [17], and the aerodynamic interference between wakes and lifting surfaces [18]. Solution of the Navier–Stokes equations for 3D, unsteady flow-fields with large motions of the solid boundaries is still very demanding, not least because of the small meshes needed for boundary-layer resolution. Euler codes are more efficient, but they are also hindered by the requirement of creating and distorting the mesh as the geometrically nonlinear structure deforms. These methods have therefore found very limited applications so far in aircraft aeroelasticity and flight dynamics with large wing deformations [19], [20], [21].

At low subsonic speeds, 3D unsteady potential-flow methods provide excellent tools for aerodynamic analysis: without incurring excessive computational costs, they incorporate 3D effects, interference and wake modelling. These methods, however, are not appropriate when the wing enters stalled conditions, to predict viscous drag, or at low Reynolds number. Neither are they adequate at very high altitudes, potentially within the range of operations of HALE UAVs, due to dominance of rarefied flow effects. Note however that the critical load conditions will occur during climb and descent operations in the lower atmosphere, which is the scope of aeroelastic analysis.

The equation governing low-speed potential flows is Laplace's equation for the velocity potential. One of the key features of this linear differential equation is that a 3D flow-field problem can be converted to a reduced-dimension equivalent one by distributing singularity (elementary) solutions over the surface where the flow potential must be found (boundary-value problem). As a result, the numerical solution in terms of singularities is faster compared to field methods where the unknown quantities are distributed in the entire volume surrounding the body—although it must be also acknowledged that whereas the former requires the inversion of a fully populated matrix, the latter results in a sparsely banded one. The reduction of the 3D computational domain to a surface problem has led to the rapid development of computer codes for the implementation of potential-flow methods, and thanks to their flexibility and relative economy they continue to be widely used despite the availability of more exact approaches.

The foundations of potential-flow vortex-lattice methods (VLM) can be traced back to Helmholtz's seminal work on vortex flows and Joukowski's contributions on circulation [22], but the earliest formulations appeared in the 1930s. Rosenhead [23] studied a 2D vortex-layer by replacing it with a system of vortex filaments. He showed that the vortex sheet rolled up with time, and the rationale behind it was given by Lavrent'ev [24]. The term “vortex lattice” was coined by Falkner in 1943 [25]. The concept was simple but it relied on a numerical solution, so it was not until digital computers became available that practical implementations became widespread. In the meantime, other lifting-surface methods were also pursued, such as the kernel-function method [26], but its development was hindered by the fact that it required a priori knowledge of appropriate pressure modes, and was therefore configuration dependent [27], [28], [29], [30], [31].

Hedman [32] established the now classical (steady) VLM in 1965: he idealised the mean aerodynamic surface into small trapezoidal lifting elements each containing a horseshoe vortex with its bound spanwise element along the swept quarter-chord of the element, and locating the collocation points for the non-penetration boundary condition at the three-quarter chord. The downwash at each collocation point was computed through the Biot–Savart law, and compressibility was accounted for by means of the Prandtl–Glauert transformation. The method continued to be widely used in the following decades, as exemplified by the workshop held at NASA Langley in 1976 [33], and alongside different implementations of the code [34], [35], [36] various relevant numerical issues were also addressed: increasing convergence and accuracy [37], [38], [39], [40], [41], accounting for wake roll-up instead of assuming a flat wake [42], and modelling leading-edge separation [43], [44], [45], [46], [47]. The steady VLM has since been employed in a number of applications, such as modelling yacht sails [48], in the computation of stability derivatives and flight dynamics analysis [49], [50], [51], in aerodynamic interference of aircraft [52], [53], [54], or in multidisciplinary optimisation [55], among many others.

As the VLM was initially limited to steady load calculations it was natural to develop an unsteady equivalent. Albano and Rodden [1] extended the VLM to harmonically oscillating surfaces for an assumed flat wake. Replacing the vortex sheet by one of (equivalent) oscillating doublets, the doublet-lattice method (DLM) was obtained. Indeed, it reduces to the original VLM at zero reduced frequency [56]. The DLM underwent further refinements conducted mainly by Rodden and different collaborators [56], [57], [58], [59], to become broadly used for unsteady load computations and the prevalent tool in subsonic aircraft aeroelasticity [60].

When referring to the vortex-lattice method, it is a common misconception (particularly within the aeroelastic community) to assume that it is limited to the steady equivalent of the doublet-lattice method. While this is true for the steady version, it is also known [61] that a panel with a piecewise constant doublet distribution is equivalent to a vortex ring around its periphery. Hence, the VLM can be directly extended to non-stationary situations, giving rise to the time-domain unsteady vortex-lattice method (UVLM). The UVLM is mentioned in many textbooks on aerodynamics, but the most comprehensive description is possibly given by Katz and Plotkin [62, Section 13.12].

Interestingly, the extension of the VLM into the unsteady aerodynamic regime was mainly driven by a viscous phenomenon, namely the need to model the leading-edge separation on delta wings. The pioneering works in the development of the UVLM were carried out by Belotserkovskii [63], Rehbach [64], and researchers at Virginia Tech [65], [66] and Technion [67], [68].

As opposed to the DLM, which is written in the frequency domain on a fixed geometry, the UVLM is formulated in the time domain and allows the shape of the force-free wake to be obtained as part of the solution procedure. The DLM offers a faster way of computing unsteady aerodynamic loads, but it is a linear method restricted to small out-of-plane harmonic motions with a flat wake. Hence, while the DLM has dominated in fixed-wing aircraft aeroelasticity, the UVLM has been gaining ground in situations where free-wake methods become a necessity because of geometric complexity, such as flapping-wing kinematics [69], [70], [71], rotorcraft [72], [73], or wind turbines [74], [75], [76], [77], [78], [79], [80]. With the advent of novel vehicle configurations and increased structural flexibility for which the underlying assumptions of the DLM no longer hold, the UVLM constitutes an attractive solution for aircraft dynamics problems and has been recently exercised in problems such as unsteady interference [53], [81], computation of stability derivatives [82], flutter suppression [83], gust response [8], [84], optimisation [85], morphing vehicles [86], and coupled aeroelasticity and flight dynamics [18].

The UVLM is suitable for quick-turnaround single-processor simulations, and as the computation of each induced velocity is independent of the rest, parallelisation of the code is straightforward. However, the method can still be constrained by computational power for large enough problems: if N is the number of bound and wake elements, the total number of operations per time step goes as $\mathcal{O}(N^2)$. Several procedures have been devised to speed up the solution process. A first possibility is linearisation of the problem. This was carried out by Hall [87], who transformed the UVLM equations into discrete state-space form. Its dominant eigenvalues would define a reduced-order model to, for instance, obtain dynamic aeroelastic stability characteristics. These ideas were explored further in Refs. [88], [89], [90], [91]. Another approach is to target the number of evaluations of the Biot–Savart law, trying to reduce it while retaining the accuracy and fidelity of the flow field. In general, the underlying principle consists of separating vortex elements into “near field” and “far field”, and making use of the diminishing influence as the distance between the point of evaluation and the vortex elements increases. The simplest alternative is to define a threshold beyond which the calculations are not performed, while retaining the influence over neighbouring elements (this could also include a wake truncation, but in this case the source of error introduced by ignoring the effect of far-field elements over their own near-field propagates and this should be done with care). A more systematic approach is to treat the effects of grouped far-field elements together, for which different alternatives exist, such as sub-grid schemes [92] or fast multipole expansions [70], [93]. The latter is derived from the general solution to the n-body problem and reduces the number of operations per time step to $\mathcal{O}(N\log N)$. Other ideas have also been investigated to decrease computational costs, e.g., in helicopter applications. For example, Bagai and Leishman [94] tackled the problem through adaptive refinement of the wake grid, with interpolation of known information onto intermediate points. For a brief introduction about other computation acceleration methods proposed in the rotorcraft community, see Ref. [95, p. 617].

Its repeated evaluation is not the only issue concerning the Biot–Savart law. The use of discrete vortex-segments to account for the vorticity in the shed wake originates numerical difficulties due to the singularity at the vortex filament: The induced velocities at the segment itself and in its neighbourhood can be unphysically large, thereby leading to unrealistic circulation values. This may occur, for example, when a wake encounters a lifting surface or when the wake undergoes an intense roll-up causing crossings of vortex segments, making it difficult to distinguish between the known physical wake instabilities and those of numerical origin [96]. These cases require regularisation (desingularisation), which is extensively described, for instance, in Refs. [92], [97], [98], [99]. Techniques such as introducing vortex-core models [100], [101], [102], [103], using the sub-vortex technique [104], implementing distributed vorticity elements [105], or discarding wake connectivity (the “vortex-blob” method) [106], [107], [108], [109] are some of the solutions.

Finally, it is worth pointing out the need to enforce the Kutta–Joukowski condition (often shortened as the Kutta condition), necessary to guarantee the uniqueness of the solution in unsteady potential flow-fields, in addition to flow-tangency on impermeable surfaces. The Kutta–Joukowski condition is based on physical considerations, and states that the flow leaves the trailing edge of the aerofoil or wing smoothly, defining the shedding location for an infinitesimally thin wake in which all vorticity of the otherwise irrotational fluid is concentrated. The pressure difference across this sheet is zero. While in potential flows viscosity is neglected in the momentum conservation equations, the Kutta–Joukowski condition permits some aspects of viscous effects to be incorporated. It is well established for steady, attached flow situations, but its validity for time-dependent cases is controversial. Despite being applied in most numerical studies of unsteady wing-theory, in reality it may not hold for highly unsteady flows. Experimental results differ, and there is no definitive conclusion on the reduced-frequency threshold for its validity. It is however safe to assume that for reduced frequencies of $k<0.6$ the Kutta–Joukowski condition is satisfied. For further discussion, see Refs. [97], [98], [110], [111], [112], [113], [114], [115], [116].

While Ref. [62] constitutes the fundamental reference for both the theory and numerical implementation of the UVLM, this paper will focus on the use of the UVLM in aeroelasticity and flight dynamics modelling of flexible aircraft. As a result, the paper will try to give some insight into current industrial practice, identifying alternative solution methods, and accentuating situations where the UVLM is particularly advantageous. The paper is therefore organised as follows:

Potential-flow methods for unsteady aerodynamic modelling are reviewed first in Section 2, including classification and limitations, as well as brief descriptions of the most relevant models for unsteady aerodynamic load calculations. The strengths and weaknesses of each option are highlighted. Section 3 compares 3D unsteady aerodynamic models in the context of full aircraft analysis. The conventional methods in aeroelasticity and flight dynamics are surveyed, and a state-space formulation of the UVLM is presented. The benefits and shortcomings of these two methodologies are underlined. Section 4 presents a quick overview of the modelling techniques for flexible-body dynamics. Conventional linear and decoupled analyses are compared to a nonlinear geometrically exact beam model, in which structural and rigid-body degrees of freedom are included. Once the necessary tools for flexible-aircraft dynamic analysis have been outlined, Section 5 details the different possibilities for integration of the separate disciplines into a unified framework for coupled aeroelasticity, flight dynamics, and control. Weakly (loosely) or strongly (tightly) coupling the UVLM with the flexible-body equations permits fully nonlinear time-marching simulations, whereas the linearisation of the equations leads to a very powerful monolithic discrete-time state-space formulation. The latter can be used for stability analysis by solving an eigenvalue problem, in control synthesis for gust alleviation, or can be marched in time. In Section 6, both linear and nonlinear implementations will be exercised in diverse situations in order to illustrate some of the problems this methodology can deal with, and the paper will be concluded in Section 7 by summarising the main merits of the UVLM, and outlining recommendations for the unsteady-aerodynamic modelling of flexible-aircraft dynamics.