Journal Publications

A numerical lifting-line method (implemented in an open-source software package) is presented that can accurately estimate the aerodynamics of wings with arbitrary sweep, dihedral, and twist. Previous numerical lifting-line methods have suffered from grid convergence challenges and limitations in accurately modeling the effects of sweep, or have relied on empirical relations for swept-wing parameters and have been limited in their application to typical wing geometries. This work presents novel improvements in accuracy, flexibility, and speed for complex geometries over previous methods. In the current work, thin-airfoil theory is used to correct section lift coefficients for sweep, providing a more general closure to the lifting-line problem. A linearized solution is presented, which can be used as a rapid approximation for the full solution, or as an initial guess for the nonlinear system of equations to speed convergence. Sensitivities to model parameters are investigated, and appropriate recommendations for these parameters are given. Agreement with Prandtl’s classical lifting-line method is excellent in the case of straight wings. Comparison with experimental data shows that this method can reasonably predict lift, drag, and lift distribution for a range of wing configurations. The speed and accuracy of this method make it well-suited for preliminary design and optimization.

The impressive maneuverability demonstrated by birds has so far eluded comparably sized uncrewed aerial vehicles (UAVs). Modern studies have shown that birds’ ability to change the shape of their wings and tail in flight, known as morphing, allows birds to actively control their longitudinal and lateral flight characteristics. These advances in our understanding of avian flight paired with advances in UAV manufacturing capabilities and applications has, in part, led to a growing field of researchers studying and developing avian-inspired morphing aircraft. Because avian-inspired morphing bridges at least two distinct fields (biology and engineering), it becomes challenging to compare and contrast the current state of knowledge. Here, we have compiled and reviewed the literature on flight control and stability of avian-inspired morphing UAVs and birds to incorporate both an engineering and a biological perspective. We focused our survey on the longitudinal and lateral control provided by wing morphing (sweep, dihedral, twist, and camber) and tail morphing (incidence, spread, and rotation). In this work, we discussed each degree of freedom individually while highlighting some potential implications of coupled morphing designs. Our survey revealed that wing morphing can be used to tailor lift distributions through morphing mechanisms such as sweep, twist, and camber, and produce lateral control through asymmetric morphing mechanisms. Tail morphing contributes to pitching moment generation through tail spread and incidence, with tail rotation allowing for lateral moment control. The coupled effects of wing–tail morphing represent an emerging area of study that shows promise in maximizing the control of its morphing components. By contrasting the existing studies, we identified multiple novel avian flight control methodologies that engineering studies could validate and incorporate to enhance maneuverability. In addition, we discussed specific situations where avian-inspired UAVs can provide new insights to researchers studying bird flight. Collectively, our results serve a dual purpose: to provide testable hypotheses of flight control mechanisms that birds may use in flight as well as to support the design of highly maneuverable and multi-functional UAV designs.

The locus of aerodynamic centres of a finite wing is the collection of all spanwise section aerodynamic centres, and depends on aspect ratio, wing sweep and planform shape. This locus is of great importance in the positioning of vortex elements in lifting-line theory. Traditionally, these vortex elements are placed along the quarter-chord of a wing, leading to inaccurate predictions of aerodynamic coefficients for swept wings due to the discontinuity in the line of vorticity at the wing root. An analytical solution was presented by Küchemann in 1956 to determine the locus of aerodynamic centres as a function of sweep. While experimental studies have been performed to visualise this locus, no large amount of data is available to fully evaluate the accuracy of Küchemann’s analytical solution. In the present study, a numerical approach is taken using a high-order panel method for inviscid, incompressible flow to calculate the locus of aerodynamic centres for elliptic wings over a wide range of sweep angles, aspect ratios and profile thicknesses. An inviscid panel method is chosen over full CFD solutions because of their ability to isolate the inviscid phenomena. Küchemann’s prediction is compared to this numerical data. The root mean square error is calculated for each wing in a broad design space to determine the accuracy of Küchemann’s theory. It is shown to be remarkably accurate over the range of cases studied, with the root mean square error staying below 4% for all wings with aft sweep and aspect ratios higher than  𝑅𝐴=5  . The actual difference between Küchemann’s prediction and numerical data is lower than that for the majority of the span for many of the wing designs considered, with the RMS error being skewed by the results at the tip. Results demonstrate that Küchemann’s analytical equations can be used as an accurate approximation for the locus of aerodynamic centres and could be used in modern numerical lifting-line algorithms

As contemporary aerostructural research for aircraft design trends toward high-fidelity computational methods, aerostructural solutions based on theory are often neglected or forgotten. In fact, in many modern aerostructural wing optimization studies, the elliptic lift distribution is used as a reference in place of theoretical aerostructural solutions with more appropriate constraints. In this paper, the authors review several theoretical aerostructural solutions that could be used as reference cases for wing design studies, and these are compared to high-fidelity solutions with similar constraints. Solutions are presented for studies with 1) constraints related to the wing integrated bending moment, 2) constraints related to the wing root bending moment, and 3) structural constraints combined with operational constraints related to either wing stall or wing loading. It is shown that, under appropriate design constraints, theoretical solutions for the optimum lift distribution may capture aerostructural coupling sufficiently to serve as appropriate reference cases for higher-fidelity solvers. A comparison of theoretical and high-fidelity solutions for the optimum wingspan and corresponding drag reveals important insights into the effects of certain aerodynamic and structural parameters and constraints on the aerodynamic and structural coupling involved in aerostructural wing design and optimization.

U.S. national research priorities include reducing the disruption caused by sonic booms, which hindered the economic viability of supersonic aircraft in decades past. However, shock waves are unavoidable during supersonic flight; and exposure of people, animals, and structures to these shock waves on the ground cannot be eliminated entirely. It is herein shown that changing atmospheric conditions, gusts, and heights above ground can result in any supersonic aircraft that is acceptably quiet at specific atmospheric conditions becoming uncomfortably loud in other flight conditions. These performance variations are noticeable for periods ranging from 12 h to one year across the continental United States of America. Therefore, strategies to actively mitigate sonic booms may become necessary as researchers work toward resumption of overland commercial supersonic flight.

The main wings of most aircraft produce adverse yaw during roll. In order to control the lateral direction of the aircraft during the roll, the rudder is often mixed with the aileron. Lifting-line theory is used here to develop spanwise lift distributions that require the minimum number of terms in the Fourier-series solution for controlling the yawing moment during pure rolling motion using only the main wing. It is shown that the yawing moment can be controlled for arbitrary rolling moments and/or rolling rates by adding symmetric twist in the main wing. The induced-drag penalty for using this method to control the yawing moment is significant and discussed in detail. For example, it is shown that if zero yawing moment is prescribed, the induced drag can increase by 108% for a prescribed rolling moment or by 300% during a steady rolling rate relative to the induced drag in steady level flight. Because this is the minimum-series solution, it does not represent the solution for yaw control with minimum induced drag, since more terms could be used in the Fourier series describing the lift distribution to control yaw with less induced drag. However, the solutions presented here can be useful for aircraft with continuous trailing-edge technologies that are limited in spanwise deflection gradients, and provide a theoretical upper bound on the minimum induced-drag that can be expected if the main wing is used for yawing-moment control during pure roll.

Birds dynamically adapt to disparate flight behaviours and unpredictable environments by actively manipulating their skeletal joints to change their wing shape. This in-flight adaptability has inspired many unmanned aerial vehicle (UAV) wings, which predominately morph within a single geometric plane. By contrast, avian joint-driven wing morphing produces a diverse set of non-planar wing shapes. Here, we investigated if joint- driven wing morphing is desirable for UAVs by quantifying the longitudinal aerodynamic characteristics of gull-inspired wing-body configurations. We used a numerical lifting-line algorithm (MachUpX) to determine the aerodynamic loads across the range of motion of the elbow and wrist, which was validated with wind tunnel tests using three-dimensional printed wing-body models. We found that joint-driven wing morphing effectively controls lift, pitching moment and static margin, but other mechanisms are required to trim. Within the range of wing extension capability, specific paths of joint motion (trajectories) permit distinct longitudinal flight control strategies. We identified two unique trajectories that decoupled stability from lift and pitching moment generation. Further, extension along the trajectory inherent to the musculoskeletal linkage system produced the largest changes to the investigated aerodynamic properties. Collectively, our results show that gull-inspired joint-driven wing morphing allows adaptive longitudinal flight control and could promote multifunctional UAV designs.

During early phases of wing design, analytic and low-fidelity methods are often used to identify promising design concepts. In many cases, solutions obtained using these methods provide intuition about the design space that is not easily obtained using higher-fidelity methods. This is especially true for aerostructural design. However, many analytic and low- fidelity aerostructural solutions are limited in application to wings with specific planforms and weight distributions. Here, a numerical method for minimising induced drag with structural constraints is presented that uses approximations that apply to unswept planar wings with arbitrary planforms and weight distributions. The method is applied to the National Aeronautics and Space Administration (NASA) Ikhana airframe to show how it can be used for rapid aerostructural optimisation and design-space exploration. The design space around the optimum solution is visualised, and the sensitivity of the optimum solution to changes in weight distribution, structural properties, wing loading and taper ratio is shown. The optimum lift distribution and wing-structure weight for the Ikhana airframe are shown to be in good agreement with analytic solutions. Whereas most modern high-fidelity solvers obtain solutions in a matter of hours, all of the solutions shown here can be obtained in a matter of seconds.

Most modern aircraft employ discrete ailerons for roll control. The induced drag, rolling moment, and yawing moment for an aircraft depend in part on the location and size of the ailerons. In the present study, lifting-line theory is used to formulate theoretical relationships between aileron design and the resulting forces and moments. The theory predicts that the optimum aileron geometry is independent of prescribed lift and rolling moment. A numerical potential flow algorithm is used to evaluate the optimum size and location of ailerons for a wide range of planforms with varying aspect ratio and taper ratio. Results show that the optimum aileron design to minimise induced drag always extends to the wing tip. Impacts to induced drag and yawing moment are also considered, and results can be used to inform initial design and placement of ailerons on future aircraft. Results of this optimisation study are also compared to theoretical optimum results that could be obtained from morphing-wing technology. Results of this comparison can be used to evaluate the potential benefits of using morphing-wing technology rather than traditional discrete ailerons.

Implementations of lifting-line theory predict the lift of a finite wing using a sheet of semi-infinite vortices extending from a vortex filament placed along the locus of aerodynamic centers of the wing. Prandtl’s classical implementation is restricted to straight wings in flows without side slip. In this Paper, it is shown that lifting-line theory can be extended to swept wings if, at the control points where induced velocity is calculated, the second derivative of the locus of aerodynamic centers is zero and the trailing vortices are perpendicular to the locus of aerodynamic centers. Therefore, a general implementation of lifting-line theory is presented that conditionally forces the second derivative of the locus of aerodynamic centers to zero at each control point and joints each trailing vortex such that there is a finite segment of the trailing vortex that lies perpendicular to the locus of aerodynamic centers. Consideration is given to modeling the locus of aerodynamic centers and section aerodynamic properties of swept wings. The resulting general formulation is analyzed to determine sensitivity to closure parameters, accuracy, and numerical convergence. The general implementation demonstrates approximately second-order convergence when the control points are cosine clustered and produces results that closely match those of a higher-order panel method and experimental data, though requiring only a fraction of the computational cost.

For a wing in steady level flight, the lift distribution that minimizes induced drag depends on a tradeoff between wingspan and wing-structure weight. In 1933, Prandtl suggested that tapered wings have an advantage over rectangular wings due to this tradeoff. However, Prandtl’s solutions were obtained using assumptions that correspond to rectangular wings. Therefore, his claim was not analytically proven by his 1933 publication. Here, an approach similar to Prandtl’s is taken with more general approximations that apply to wings of arbitrary planform. This more general development is used to study Prandtl’s claim about tapered wings. Closed-form solutions for the optimum wingspan and corresponding induced drag are presented for wings having elliptic and linearly-tapered planforms with constraints of fixed wing loading and maximum stress. It is shown that induced drag is minimized with a triangular planform, which gives a reduction in induced drag of up to 24.44% over the rectangular planform and up to 11.71% over the elliptic planform. Numerical solutions for the lift distributions that minimize induced drag for each planform are also presented. It is shown that the optimum lift distribution produces up to 5.94% less induced drag than the elliptic lift distribution when the triangular planform is used.

Lifting-line theory has formed the basis of wing-design understanding for nearly a century. This theory has previously been used to develop a class of optimal lift distributions that minimize induced drag for finite wings under specific structural and aerodynamic constraints. This class of lift distributions was first introduced by Prandtl in 1933 and includes the elliptic lift distribution and the bell-shaped lift distribution, among others, as special cases. Here, it is shown that, for all lift distributions in this class except the elliptic lift distribution, an antisymmetric twist distribution can be added to the wing to control the adverse yawing moment during roll. The requisite amount of this additional twist distribution depends on the prescribed yawing-moment sign and magnitude, and it can be used to produce zero adverse yaw or even proverse yaw during roll. Of course, employing this methodology to control the roll–yaw coupling produces additional induced drag relative to the case without roll–yaw coupling control. This induced-drag penalty is studied in detail and shows that a symmetric lift distribution within this class can be selected to minimize induced drag for a desired roll–yaw coupling. Roll initiation and steady rolling rate are both considered. Results can be applied to the design of morphing aircraft to minimize the use of a vertical control surface for yaw control.

Because the wing-structure weight required to support the critical wing section bending moments is a function of wingspan, net weight, weight distribution, and lift distribution, there exists an optimum wingspan and wing-structure weight for any fixed net weight, weight distribution, and lift distribution, which minimises the induced drag in steady level flight. Analytic solutions for the optimum wingspan and wing-structure weight are presented for rectangular wings with four different sets of design constraints. These design constraints are fixed lift distribution and net weight combined with 1) fixed maximum stress and wing loading, 2) fixed maximum deflection and wing loading, 3) fixed maximum stress and stall speed, and 4) fixed maximum deflection and stall speed. For each of these analytic solutions, the optimum wing-structure weight is found to depend only on the net weight, independent of the arbitrary fixed lift distribution. Analytic solutions for optimum weight and lift distributions are also presented for the same four sets of design constraints. Depending on the design constraints, the optimum lift distribution can differ significantly from the elliptic lift distribution. Solutions for two example wing designs are presented, which demonstrate how the induced drag varies with lift distribution, wingspan, and wing-structure weight in the design space near the optimum solution. Although the analytic solutions presented here are restricted to rectangular wings, these solutions provide excellent test cases for verifying numerical algorithms used for more general multidisciplinary analysis and optimisation.

Geometric and/or aerodynamic wing twist can be used to produce a lift distribution that results in a rolling moment. A decomposed Fourier-series solution to Prandtl’s lifting-line theory is used to develop analytic spanwise antisymmetric twist distributions for roll control that minimize induced drag on wings of arbitrary planform in pure rolling motion. Roll initiation, steady rolling rate, and the transition between the two are each considered. It is shown that if these antisymmetric twist distributions are used, the induced drag is proportional to the square of the rolling moment, and the induced drag during a steady rolling rate is equal to that on the wing at the same lift coefficient with no rolling rate or antisymmetric twist distribution. Results also show that if these antisymmetric twist distributions are used on straight, tapered wings without symmetric twist, any rolling maneuver for which the rolling rate and rolling moment have the same sign will always produce a yawing moment in the opposite direction. Computational results are also included, which were obtained using a gradient-based optimization algorithm in combination with a modern numerical lifting-line algorithm to find the optimum twist solutions. The resulting twist, induced drag, and yawing moment solutions compare favorably with the analytic solutions developed in the text. The solutions presented here can be used to inform the design of morphing aircraft.

The aerodynamic center of an airfoil is commonly estimated to lie at the quarter-chord. This traditional estimate is based on thin airfoil theory, which neglects aerodynamic and geometric nonlinearities. Even below stall, these nonlinearities can have a significant effect on the location of the aerodynamic center. Here, a method is presented for accurately predicting the aerodynamic center of any airfoil from known lift, drag, and pitching-moment data as a function of angle of attack. The method accounts for aerodynamic and geometric nonlinearities, and it does not include small-angle, small-camber, and thin-airfoil approximations. It is shown that the aerodynamic center of an airfoil with arbitrary amounts of thickness and camber in an inviscid flow is a single, deterministic point, independent of angle of attack, and lies at the quarter-chord only in the limit as the airfoil thickness and camber approach zero. Furthermore, it is shown that, once viscous effects are included, the aerodynamic center is not a single point but is a function of angle of attack. Differences between this general solution and that predicted by the thin airfoil theory can be on the order of 3%, which is significant when predicting flutter speeds. Additionally, the results have implications for predicting the neutral point of a complete aircraft.

A parabolic trailing-edge flap is defined as a parabolic deflection of the airfoil geometry aft of a hinge point. Whereas a traditional flap deflection causes a discontinuous camber-line slope at the hinge point, a parabolic deflection produces a camber line that is first-order continuous at the hinge point. The geometry manipulation of a parabolic flap is mathematically defined such that it can be applied to any airfoil with a known camber line and thickness distribution. Small-angle and small-camber approximations are used to find analytical predictions for the ideal section flap effectiveness as well as the section pitching moment in comparison to a traditional flap. Results of the parabolic flap are compared to those of a traditional flap producing equivalent lift using thin airfoil theory and the vortex-panel method. It is shown that the ideal section flap effectiveness for a parabolic flap can be 33-50% higher than that of a traditional flap, depending on the flap-chord fraction. Additionally, a parabolic flap will produce a change in pitching moment 5-50% greater than that of a traditional flap for a given change in lift. Results may be applied in the design of modern morphing wings, for which complex flap deflections can be produced.

It is shown that the smooth-wall boundary conditions specified for commonly used dissipation- based turbulence models are mathematically incorrect. It is demonstrated that when these traditional wall boundary conditions are used, the resulting formulations allow either an infinite number of solutions or no solution. Furthermore, these solutions do not enforce energy conservation and they do not properly enforce the no-slip condition at a smooth surface. This is true for all dissipation-based turbulence models, including the k-ɛ, k-ω, and k-ζ models. Physically correct wall boundary conditions must force both k and its gradient to zero at a smooth wall. Enforcing these two boundary conditions on k is sufficient to determine a unique solution to the coupled system of differential transport equations. There is no need to impose any wall boundary condition on ɛ, ω, or ζ at a smooth surface and it is incorrect to do so. The behavior of ɛ, ω, or ζ approaching a smooth surface is that required to satisfy the differential equations and force both k and its gradient to zero at the wall.

Prandtl’s classical lifting-line equation can be used to relate the chord-length distribution, twist distribution, and section-lift distribution for an unswept finite wing. Although others have obtained specific solutions for various special cases, here a general analytic solution has been developed such that, given a specified section-lift distribution and either the chord-length distribution or the twist distribution, the third of these distributions can be obtained. If the section-lift distribution and wing planform are specified, the required twist distribution can be found from Eqs. (18) and (19). On the other hand, if the section-lift distribution and wing twist distribution are specified, the required planform can be found from Eq. (24). Example solutions are also presented for a case of special interest when designing wings for minimum induced drag. The analytic relations presented in this paper can be used for general wing design and may be particularly useful in the design of morphing wings that have the ability to alter the wing twist distribution and/or planform during flight to obtain optimum lift distributions for various flight phases.

Minimum induced drag for fixed gross weight and wingspan is obtained from the elliptic lift distribution. However, minimum induced drag for steady level flight is not obtained by imposing the constraints of fixed gross weight and wingspan. Because required wing-structure weight is a function of wingspan and lift distribution, there exist an optimum lift distribution and wingspan for a given weight and weight distribution that minimizes the induced drag in steady level flight. This optimum lift distribution can vary significantly from the elliptic lift distribution, depending on the design constraints. Analytic solutions for three such optimum lift distributions are presented for rectangular wings with varying sets of design constraints. These design constraints are 1) weight, maximum stress, and chord length fixed; 2) weight, maximum stress, and wing loading fixed; and 3) weight, maximum deflection, and wing loading fixed. It is shown that each of these optimum lift distributions results in lower induced drag than a fixed elliptic lift distribution for the same design constraints.

In this work, we examine the flux correction method for three-dimensional transonic turbulent flows on strand grids. Building upon previous work, we treat flux derivatives along strands with high-order summation-by-parts operators and penalty-based boundary conditions. A finite-volume like limiting strategy is implemented in the flux correction algorithm in order to sharply capture shocks. To achieve turbulence closure in the Reynolds-Averaged Navier–Stokes equations, a robust version of the Spalart–Allmaras turbulence model is employed that accommodates negative values of the turbulence working variable. Validation studies are considered which demonstrate the flux correction method achieves a high degree of accuracy for turbulent shock interaction flows.

It is shown that the time-dependent aerodynamic forces acting on a flapping airfoil in forward flight are functions of both axial and normal reduced frequencies. The axial reduced frequency is based on the chord length, and the normal reduced frequency is based on the plunging amplitude. Furthermore, the time-dependent aerodynamic forces are related to two Fourier coefficients, which are evaluated here from computational results. Correlation equations for these Fourier coefficients are obtained from a large number of grid- and time-step-resolved inviscid computational- fluid-dynamics solutions, conducted over a range of both axial and normal reduced frequencies. The correlation results can be used to predict the thrust, required power, and propulsive efficiency for airfoils in forward flight with sinusoidal pitching and plunging motion. Within the range of parameters typically encountered in the efficient forward flight of birds, results obtained from the correlation equations match the computational-fluid-dynamics results more closely than do those obtained from the classical Theodorsen model.

Closed-form relations are presented for estimating ratios of the induced-drag and lift coefficients acting on a wing in ground effect to those acting on the same wing outside the influence of ground effect. The closed-form relations for these ground-effect influence ratios were developed by correlating results obtained from numerical solutions to Prandtl’s lifting-line theory. Results show that these influence ratios are not unique functions of the ratio of wing height to wingspan, as is sometimes suggested in the literature. These ground-effect influence ratios also depend on the wing planform, aspect ratio, and lift coefficient.

Based on a more direct analogy between turbulent and molecular transport, a foundation is presented for an energy–vorticity turbulence model. Whereas traditional k-ε, k-ω, and k-ζ models relate the eddy viscosity to a dissipation length scale associated with the smaller eddies having the highest strain rates, the proposed model relates the eddy viscosity to a mean vortex wavelength associated with the larger eddies primarily responsible for turbulent transport. A rigorous development of the turbulent-energy-transport equation from the Navier–Stokes equations includes exact relations for the viscous dissipation and molecular transport of turbulent kinetic energy. Application of Boussinesq’s analogy between turbulent and molecular transport leads to a transport equation, which shows neither molecular nor turbulent transport of turbulent energy to be simple gradient diffusion. The new turbulent-energy- transport equation contains two closure coefficients: a viscous-dissipation coefficient and a turbulent-transport coefficient. To help evaluate closure coefficients and provide insight into the energy–vorticity turbulence variables, fully rough pipe flow is considered. For this fully developed flow, excellent agreement with experimental data for velocity profiles and friction factors is attained over a wide range of closure coefficients, provided that a given relation between the coefficients is maintained.

An improved method is presented for estimating the subsonic location of the semispan aerodynamic center of a swept wing and the aerodynamic moment components about that aerodynamic center. The method applies to wings with constant linear taper and constant quarter-chord sweep. The results of a computational fluid dynamics study for 236 wings show that the position of the semispan aerodynamic center of a wing depends primarily on aspect ratio, taper ratio, and quarter-chord sweep angle. Wing aspect ratio was varied from 4.0 to 20, taper ratios from 0.25 to 1.0 were investigated, quarter-chord sweep angles were varied from 0 to 50 deg, and linear geometric washout was varied from 4:0to 8:0 deg. All wings had airfoil sections from the NACA 4-digit airfoil series with camber varied from 0 to 4% and thickness ranging from 6 to 18%. Within the range of parameters studied, wing camber, thickness, and twist were shown to have no significant effect on the position of the semispan aerodynamic center. The results of this study provide improved resolution of the semispan aerodynamic center and moment components for conceptual design and analysis.