Rauno Cavallaro

Invariant Formulation for the Minimum Induced Drag Conditions of Nonplanar Wing Systems

Under the hypotheses of linear potential flow and rigid wake aligned with the freestream, a configuration-invariant analytical formulation for the induced drag minimization of single-wing nonplanar systems is presented. Following a variational approach, the resulting Euler–Lagrange integral equation in the unknown circulation distribution is obtained. The kernel presents a singularity of the first order, and an efficient computational method, ideal for the early conceptual phases of the design, is proposed. Munk’s theorem on the normalwash and its relation with the geometry of the wing under optimal conditions is naturally obtained with the present method. Moreover, Munk’s constant of proportionality, not provided in his original work, is demonstrated to be the ratio between the freestream velocity and the optimal aerodynamic efficiency. The augmented Munk’s minimum induced drag theorem is then formulated. Additional induced drag theorems are demonstrated following the derivations of this invariant proced...

Minimum Induced Drag Theorems for Joined Wings, Closed Systems, and Generic Biwings: Theory

An analytical formulation for the induced drag minimization of closed wing systems is presented. The method is based on a variational approach, which leads to the Euler–Lagrange integral equation in the unknown circulation distribution. It is shown for the first time that the augmented Munk’s minimum induced drag theorem, formulated in the past for open single-wing systems, is also applicable to closed systems, joined wings and generic biwings. The quasi-closed C-wing minimum induced drag conjecture discussed in the literature is addressed. Using the variational procedure presented in this work, it is also shown that in a general biwing, under optimal conditions, the aerodynamic efficiency of each wing is equal to the aerodynamic efficiency of the entire wing system (biwing’s minimum induced drag theorem). This theorem holds even if the two wings are not identical and present different shapes and wingspans; an interesting direct consequence of the theorem is discussed. It is then verified (but yet not demonstrated) that in a closed path, the minimum induced drag of the biwing is identical to the optimal induced drag of the corresponding closed system (closed system’s biwing limit theorem). Finally, the nonuniqueness of the optimal circulation for a closed wing system is rigorously addressed, and direct implications in the design of joined wings are discussed.

Postcritical Analysis of PrandtlPlane Joined-Wing Configurations

The postbuckling behavior of joined-wing configurations has not been fully addressed in the past. This topic is extensively discussed in this work. Starting from a baseline configuration, geometrical parameters as well as wings’ load repartition are varied to assess their influence on buckling occurrence. The snap-buckling phenomenon and postcritical pattern are investigated with the adoption of the arc-length technique. The complex load transferring through the joint induces a deformation shape that can no longer carry additional load after a critical point is reached. An abrupt snap to a configuration that is not continuously adjacent to the previous one is then observed. Comparison with the instability state obtained via eigenvalue analysis demonstrates that buckling prediction through linear-buckling analysis is inadequate, and often, the actual critical load is overestimated. This study shows that, for PrandtlPlane joined-wing configurations, increasing the height-to-wingspan ratio is beneficial as f...

Nonlinear Analysis of PrandtlPlane Joined Wings: Effects of Anisotropy

Structural geometrical nonlinearities strongly affect the response of joined wings: it has been shown that buckling evaluations using linear methods are unreliable, and only a fully nonlinear stability analysis can safely identify the unstable state. This work focuses on the understanding of the main physical mechanisms driving the wing system’s response and the snap-buckling instability. Several counterintuitive effects typical of unconventional nonplanar wing systems are discussed and explained. In particular, an appropriate design of the joint-to-wing connection may reduce the amount of bending moment transferred, and this is shown to eventually improve the stability properties, although at price of a reduced stiffness. It is also demonstrated that the lower-to-upper-wing stiffness ratio and the torsional–bending coupling, due to both the geometrical layout and anisotropy of the composite laminates, present a major impact on the nonlinear response. The findings of this work could provide useful indicat...

Minimum Induced Drag Theorems for Joined Wings, Closed Systems, and Generic Biwings: Applications

An invariant procedure for the minimization of induced drag of generic biwings and closed systems (Joined Wings) was presented in the companion paper (minimum induced drag theorems for Joined Wings, closed systems, and generic biwings: theory) and is now adopted to study several theoretical open questions regarding these configurations. It is numerically verified that a quasi-closed C-wing presents the same optimal induced drag and circulation of the corresponding closed system. It is also verified that when the two wings of a biwing are brought close to each other so that the lifting lines identify a closed path, the minimum induced drag of the biwing is identical to the optimal induced drag of the corresponding closed system. The optimal circulation of this case differs from the quasi-closed C-wing one by an additive constant. The non-uniqueness of the optimal circulation for a closed wing system is also addressed, and it is shown that there are an infinite number of equivalent solutions obtained by adding an arbitrary constant to a reference optimal circulation. This property has direct positive impact in the design of Joined Wings as far as the wing load repartition is concerned: The percentage of aerodynamic lift supported by each wing can be modified to satisfy other design constraints, and without induced drag penalty. Finally, the theoretical open question regarding the asymptotic induced drag behavior of Joined Wings, when the vertical aspect ratio approaches infinity, has been resolved. It has been shown that for equally loaded wings indefinitely distant from each other, the boxwing minimum induced drag tends to zero. In that condition, the upper and lower wings present a constant aerodynamic load. Prandtl’s approximated formula for the minimum induced drag of a boxwing (Best Wing System) cannot be used to describe the asymptotic behavior. This work also shows that the optimal distribution over the equally loaded horizontal wings of a boxwing is not the superposition of a constant and an elliptical functions. This is an acceptable approximation only for small vertical aspect ratios (of aeronautical interest).

Nonlinear Analysis of PrandtlPlane Joined Wings - Part II: Effects of Anisotropy

Structural geometrical nonlinearities strongly affect the response of PrandtlPlane Joined Wings: it has been shown that linear buckling evaluations are unreliable and only a fully nonlinear stability analysis can safely identify the unstable state. This work focuses on the understanding of the main physical mechanisms driving the wing system’s response and the snap-buckling instability. Several counterintuitive effects typical of unconventional non-planar wing systems are discussed and explained. In particular, an appropriate design of the joint-to-wing connection may reduce the amount of bending moment transferred, and this is shown to dramatically improve the stability properties. It is also demonstrated that the lower-to-upper-wing stiffness ratio and the torsional-bending coupling, due to both the geometrical layout and anisotropy of the composite laminates, present a major impact on the nonlinear response. How the material anisotropy modifies the Snap Buckling Region and the response is also discussed. The findings of this work could provide useful indications to develop effective aeroelastic reduced order models tailored for airplanes experiencing important geometric nonlinearities such as PrandtlPlane aircraft, Truss-braced and Strut-Braced wings and sensorcrafts.

A Computational Method for Structurally Nonlinear Joined Wings Based on Modal Derivatives

Past studies showed that the overconstrained nature of Joined Wings and the strong structural geometric nonlinearities make difficult the use of standard packages of aeroelastic solvers (usually modally reduced and frequency domain based) which have been effectively adopted by the industry for decades. We present here a study on the reduction of the computational cost in presence structural nonlinear effects that cannot be neglected in Joined Wings, even at small angles of attack and attached flow. In particular, a reduced order model is achieved with a basis constituted by vibration modes augmented with the corresponding modal derivatives. The results can be considered excellent when compared to the full order reference solution. However, a convergence test showed that the required number of vectors is relatively high and the basis needs to be often updated to achieve the best performance. More investigations will be necessary for an effective use in the industry and complicate dynamic problems involving the unsteadiness of the aerodynamics.

P ost-Critical Analysis of Joined Wings: the Concept of Snap-Divergence as a Characterization of the Instability

The concept of snap divergence and post-critical states are theoretically formulated for Joined Wings with the arc length technique. The true critical condition is compared with the divergence speed evaluated by solving an eigenvalue problem about a steady state equilibrium, showing how in some cases this last approach is not reliable and even nonconservative. The work assesses the difference of the nonlinear responses relative to mechanical loads (both conservative and follower ones) used to mimic the real loading condition and the aerodynamic forces. Two joined-wing configurations, characterized by a different location of the joint, are investigated. It is demonstrated that the lift/displacement response may hide the physical snap divergence occurrence, leading to non-physical interpretation of the stability properties of the system. Thus, as a consequence, use mechanical loading to mimic aerodynamic effects should be meditated since they may not give a reliable picture. Aeroelastic stiffening and softening effects are observed for the different cases, and it is discussed how practical instability situation may not be encompassed by the formal mathematical criterion (singularity of the system tangent matrix). Finally, physical interpretation of the static aeroelastic deformation is provided with particular emphasis on the conditions that lead to the snap divergence. The bending/torsion coupling at geometric (sweep angle of the wings) and material (composite materials) level for each wing can not be thought as an isolate property, since, due to the overconstrained nature of the system, the actions are transferred between different parts of the system. In other words, an intuitive approach that tries to fine tune the design of a part as an isolate entity may not lead to meaningful results.

Risks of Linear Design of Joined Wings: a Nonlinear Dynamic Perspective in the Presence of Follower Forces

Past work on Joined Wings pointed out the importance of including structural geometric nonlinearities since the early stages of the design. However, the attention was mainly focused on conservative loadings and several open questions needed an answer. In a effort to simulate aerodynamic-pressure-like loads, this effort focuses on non-conservative follower forces. Several numerical evaluations demonstrated that follower loadings exacerbate the risks of snap-instability in Joined Wings. Dynamic response analysis of vanishing perturbations showed that for certain combination of geometry and stiffness distributions, the phenomenon of branch-jumping is possible. This is directly linked to the existence of a bi-stable region. The presence of follower loads increases these risks with respect to conservative forces. For some configurations, a small change of a parameter (e.g. the lower-to-upper-wing bending stiffness ratio) can produce a sudden appearance of a bi-stable region and the risk of branch jumping becomes serious. This paper demonstrates that even if the design of a joined-wing system is well below the critical point (snap-buckling state) and the response appears to be quasi linear, there is a potential risk that a dynamic disturbance (perturbation) may move the system to a relatively far equilibrium state on a post-critical branch. Thus, a post-critical analysis need to be included even if a linear design point for the Joined Wing is sought. Results of this effort suggest that preliminary multidisciplinary optimizations technique can not confidently rely on linear analyses augmented with eigenvalue instability investigations. On the contrary, a post-critical study is strongly recommended for a safe design of innovative wing configurations based on the joined-wing concept.

Minimum Induced Drag Theorems for Multi-Wing Systems

Under the assumption of a rigid wake aligned with the freestream velocity, a computationally efficient induced drag minimization procedure, tailored for the preliminary design phases of generic mul...