Nicolò Fabbiane

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Laminar boundary-layer flows can transition from the laminar to the turbulent state in different ways depending on the disturbance level in the free-stream. The typical path in low-disturbance environments sees the onset of primary linear instabilities that, once reached a critical amplitude, brake down to turbulence via a secondary instability mechanism [1].

In incompressible boundary-layer flows, the most unstable linear intstablities are the two-dimensional Tollmien-Schlichting waves (TSWs). These are convective instabilities, since they grow exponentially while being convected by the flow [2].

Tollmien-Schlichting waves in a laminar boundary-layer flow over a flat plate. Vertical perturbation velocity (colormap) and streamlines (black lines).

Laminar-to-turbulence transition can be delayed by attenuating the amplitude of these instabilities. Several works can be found in the literature on the reactive control of TSWs by means of sensors and actuators placed at the wall [e.g. 3, 4].

In this work, we investigate the use of a compliant wall in order to achieve passively a similar control. Since we consider an incompressible material – Poisson ratio ν = 0.5 – the resulting base position of the wall is flat, as the reference case over a rigid plate.

A flexible patch (orange) is inserted in the wall to attenuate the growth of the Tollmien-Schlichting waves.

Before assessing the performance of the control strategy, we should make sure not to trigger any additional fluid-structural instability. The linear global stability analysis of the coupled system around the base-state highlights the presence of unstable modes associated to travelling-wave-flutter (TWF) [5], when considering a purely elastic material. These unstable modes are directly related to the free vibration modes of the patch that are pushed in the unstable region of the spectrum by the interaction with the fluid.

One way to stabilise the system consists in choosing a material with a viscoelastic behaviour, i.e. for which viscosity – and, hence, dissipation – is present in the constitutive law of the material.

Eigenvalues from the global stability analysis. Rigid-wall (blue squares), elastic patch (white circles), and viscoelastic patch (black circles).

Now that we have a globally stable system, the spatial amplification of the convectively unstable TSW can be investigated by means of the analysis of the resolvent operator [6]. This operator describes the linear response q of the system for a given forcing field f oscillating at a given frequency ω.

Sipp & Marquet [6] have shown that TSWa are associated to the linear optimal harmonic response of the flow or, in other words, the linear response of the flow to the worst possible harmonic forcing. This forcing can be computed via an optimisation that maximise the energy gain σ², defined as the ratio between the energy of the flow perturbation |q|²_E and of the forcing |f|²_E.

Energy gain associated to the response to the optimal harmonic forced. Rigid wall (blue line) and viscoelastic patch (black line).

When comparing the curve for the fluid-only resolvent over a rigid plate to the one for the fluid-structural one over the viscoelastic patch, a reduction in the response amplitude is observed in the range of frequencies associated to TSW. 

However, an additional peak appears in the coupled case: this is associated to the TWF modes that, even if stable, are still reactive to external forcing. 

Linear response to the optimal harmonic forcing in the Tollmien-Schlichting wave (TSW) range. Vertical perturbation velocity in the fluid (blue colormap) and in the solid (orange colormap). Rigid wall (top) and viscoelastic patch (bottom).

Most interestingly, the TSW control by the compliant wall follows a mechanism of wave cancellation similar to the one exploited to reactive control techniques, such as the ones in [3] and [4].

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References

[1] W.S. Saric, H.L. Reed & E.J. Kerschen (2002). Boundary-layer receptivity to freestream disturbances. Ann. Rev. Fluid Mech. 34(1): 291–319. doi: 10.1146/annurev.fluid.34.082701.161921

[2] P.J. Schmid & D.S. Henningson (2012). Stability and Transition in Shear Flows. "Applied Mathematical Sciences" Series, Springer. doi: 10.1007/978-1-4613-0185-1

[3] M. Kotsonis, R. Giepman, S. Hulshoff & L. Veldhuis (2013). Numerical study of the control of Tollmien–Schlichting waves using plasma actuators. AIAA J. 51(10). doi: 10.2514/1.J051766

[4] N. Fabbiane, S. Bagheri & D.S. Henningson (2017). Energy efficiency and performance limitations of linear adaptive control for transition delay. J. Fluid Mech. 810: 60–81. doi: 10.1017/jfm.2016.707

[5] A.D. Lucey & P.W. Carpenter (1995). Boundary layer instability over compliant walls: comparison between theory and experiment. Phys. Fluids 7 (10): 2355–2363. doi: 10.1063/1.868748

[6] D. Sipp & O. Marquet (2013). Characterization of noise amplifiers with global singular modes: the case of the leading-edge flat-plate boundary layer. Theor. Comput. Fluid Dyn. 27 (5): 617–635. doi: 10.1007/s00162-012-0265-y

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In its operational life, a wing encounters flight conditions that are different to the nominal cruise flight, because of change of course needs (maneuvers) or external velocity fluctuations (gusts). The behaviour of the wing in these condition is decided by its aeroelastic properties. If not designed correctly, they could lead to catastrophic consequences; however, if mastered, they could improve safety margins, handling performances, and even flight comfort.

It is hence possible to design the stiffness of a structure in order to pilot its aeroelastic behaviour. When composite materials are employed, this strategy takes the name of aeroelastic tailoring [1]. 

Aeroelastic response of a flexible wing to different flight conditions. Cruise load (green), high load (blue), harmonic gust (red), and resting shape (black).

Composite materials are made by two components with different roles. The fibres (e.g. carbon or glass) give rigidity to the material, while the matrix (e.g. epoxy resin) keeps together the assembly.

A typical configuration for composite materials are composite panels, where layers (plies) of aligned fibres are stacked one over the other. By changing the relative angle between the plies, it is thus possible to pilot the anisotropy of the rigidity of the panel and hence fine tune the rigidity of the overall structure [2].

Examples of composite panels and anisotropy of their membrane rigidity. From left to right: unidirectional 0°, unidirectional 30°, balanced ±30°, and balanced ±45°.

A wind-tunnel demonstrator

This activity is part of the Common Research Project (CRP) FIGURE between ONERA and DLR on the gust response of a flexible wing. The project saw the design of two composite wings – one by ONERA (here-presented) and one by DLR – and their characterisation by means of extensive wind-tunnel tests at S3Ch at the centre ONERA in Meudon in 2021. A simple structural configuration is chosen, i.e. an external glass-fibre composite shell filled with a polymer foam. The shell is diveded in 10 design regions, where the composite properties are being optimised.

Wing geometry and design regions.

The design is based on a compromise between the minimisation of two objectives respectively on the static and dynamic aeroelastic response of the wing:

• the bending moment at the root section of the wing when it is providing 2 times the nominal lift;
• the amplitude of the gust-induced oscillations of the wing-tip.

These two objectives are antagonistic, i.e. one objective must deteriorate in order to get an improvement of the other one. This means that multiple solutions equally optimal to the design problem are possible and a compromise must be taken in order to define the design point. The locus in the design space of all these possible solutions is called Pareto front.

Pareto front. Mass penalty (colormap) with respect to minimal-mass design (white square).

By analysing the Pareto front, it can be noted that a less massive and more flexible wing minimises the static root bending moment, while a stiffer and more massive wing oscillates less when forced by a gust. The final design point for the wing has been selected among the identified ones in the Pareto front as the one with the flattest resting shape, in order to facilitate its fabrication. The stacking sequences for each panel are hence retrieved by means of a specialised genetic algorithm [3].

Composite panels (thickness and rigidity) at the design point.

The article that presents ONERA's design was awarded the EREA Best Paper Award 2023.

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References

[1] C.V. Jutte & B.K. Stanford (2014). Aeroelastic tailoring of transport aircraft wings: State-of-the-art and potential enabling technologies. Technical Report NASA/TM-2014-218252, NASA.

[2] J. Dillinger (2014). Static aeroelastic optimization of composite wings with variable stiffness laminates. PhD thesis, Delft University of Technology. doi: 10.4233/uuid:20484651-fd5d-49f2-9c56-355bc680f2b7

[3] F.-X. Irisarri, A. Lasseigne, F.-H. Leroy & R. Le Riche. (2014). Optimal design of laminated composite structures with ply drops using stacking sequence tables. Composite Structures 107: 559–569. doi: 10.1016/j.compstruct.2013.08.030

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 0000-0002-0760-3547 | Google Scholar | Personal page

Even if our paths crossed earlier at KTH in Sweden, Marie and I started our collaboration at ONERA. Both interested in the control of fluid flows via fluid-structure interaction (and fluid-structure interaction in general), we joined our forces to tackle both experimentally and numerically this interesting topic, within internal research projects and doctoral theses.

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 0000-0001-7284-6361 | Google Scholar | ERC AEROFLEX

My collaboration with Olivier traces back to 2017, as he acted as advisor during my time as postdoctoral researcher at ONERA. Since then, we work together on the control of fluid flows via fluid-structure interaction, the stability of fluid-structural systems, and the data-assimilation of aeroelastic datasets.

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 0009-0008-3204-4362 | LinkedIn

Roger holds a MSc in Mechanical Engineering from Universitat Politècnica de Catalunya, complemented by a diploma in Aeronautical Engineering from Ecole Centrale Nantes. Engaging in teaching alongside his studies in Spain, he served as a lecturer in continuum mechanics and rigid body dynamics.

His doctoral project delves into the aeroelastic, reliable optimization of aeronautical composite structures. Building upon Ludovic Coelho's prior work, Roger aims to address more complex structural configurations and incorporate different levels of simulation fidelity within the optimization loop.

Supervisor: D. Lucor (UPS/LISN-CNRS).
Tutors: N. Fabbiane (ONERA), C. Julien (ONERA) & C. Fagiano (ONERA).
Funding: ONERA. | Doctoral school: SMEMAG.

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LinkedIn

Carmen has a highly international profile, holding a BSc in Mechanical Engineering from Universidade de Vigo (with a final project at the Technical University of Munich), and a MSc in Aerospace Engineering from ISAE-SUPAERO. Eclectic by nature, she also earned a diploma in piano from Conservatorio Profesional de Música de Vigo. She fluently speaks several languages, among which Spanish, English, Italian, and French.

Her thesis deals with the experimental investigation of the interaction of a normal-shock wave with a compliant wall. This involves a comprehensive approach encompassing fluid and fluid-structural experimental campaigns conducted in the transonic wind tunnels at ONERA's center in Meudon, along with the dynamical characterisation and modelling of (visco-)elastic materials. She successfully defended her PhD from Institut Polytechnique de Paris (IPP) in January 2024.

Supervisors: R. Bur (IPP/ONERA) & O. Marquet (ONERA).
Tutors: M. Couliou (ONERA) & N. Fabbiane (ONERA).
Funding: ONERA, AID. | Doctoral school: IPP. | Thesis defence: 29 January 2024.

Publications linked to the thesis

C. Riveiro Moreno, M. Couliou, N. Fabbiane, R. Bur & O. Marquet (under review). Synchronized shock wave and compliant wall interactions: experimental characterisation and aeroelastic modeling. J. Fluids Struct.

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 0000-0002-2656-1434 | LinkedIn | Current position: Research engineer at ONERA (DAAA/MSAE).

Pierre-Emmanuel joined ONERA as a postdoctoral researcher in 2022, to later continue as a research engineer in computational aeroelasticity. He earned his MSc in Applied Mathematics from INSA and Université Paul Sabatier in Toulouse, and went on to achieve his PhD in Mechanical Engineering with distinction from Technische Universität Wien.

In his postdoctoral project, Pierre-Emmanuel leverages partial experimental data obtained on transonic flexible wings to reconstruct complete pressure and deformation fields at the wall. To achieve this, he exploits knowledge from physics (aeroelasticity) and modern optimization techniques (machine learning, variational data assimilation).

Advisors: N. Fabbiane (ONERA), O. Marquet (ONERA) & F. Nicolas (ONERA).
Funding: ONERA.

Publications linked to the post-doc

P.-E. des Boscs, N. Fabbiane, F. Nicolas & O. Marquet (to be submitted). Variational Data assimilation of heterogeneous observations on a deformable transsonic wing.

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 0000-0003-2276-5189 | LinkedIn | Current position: Development engineer at ONERA (DAAA/MSAE).

Ludovic studied at Sorbonne Université, where he earned his MSc in Computational Mechanics in 2019. He firstly joined ONERA for an internship on aeroelastic coupling, to continue as a PhD student and, later, as a development engineer.

His thesis marks the first steps at ONERA on the development of optimisation strategies under uncertainty, specifically targeted to the aeroelastic behaviour of composite structures. He successfully defended his PhD in Mechanical Engineering from Université Paris-Saclay in March 2023.

Supervisor: D. Lucor (UPS/LISN-CNRS).
Tutors: C. Julien (ONERA), N. Fabbiane (ONERA) & C. Fagiano (ONERA).
Funding: ONERA. | Doctoral school: SMEMAG. | Thesis defence: 9 March 2023.

Publications linked to the thesis

L. Coelho, D. Lucor, N. Fabbiane, C. Fagiano & C. Julien (2023). Multi-scale approach for reliability-based design optimization with metamodel upscaling. Struct. Multidiscip. Optim. 66: 205. doi: 10.1007/s00158-023-03643-4

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