Aeroelastic Effects of a Very Flexible Transonic Wing:

Fluid-Structure Interaction Study

Dario Arestadario.aresta@tecnico.ulisboa.pt

Instituto Superior Tecnico, Universidade de Lisboa, Portugal

November 2017

Abstract

The challenging requirements of the transportation sector target a more efficient fuel consumption,leading to slender wing design. Because of the longer extension, the new configurations exhibitmuch higher deformations. Therefore, it is necessary to predict the influences in the flight perfor-mances, especially in transonic speeds, where instability problems such as flutter occur. This workperforms a computational aeroelastic study of a three-dimensional transonic wing model, part of theFP7-NOVEMOR project. In order to reach this goal, a new partitioned Fluid-Structure Interaction(FSI) solver have been compiled, starting from Open-source software. It has been tested with anincompressible flow benchmark and then extended to high speed flows. The final results show largedisplacements, leading to a clear motion of the shocks waves along the wing chord and great variationin the flight performances.Keywords: Fluid-Structure Interaction, Partitioned, Transonic, Aeroelasticity

Introduction

The transport sector is nowadays challenging thenew requests of the international community. Theinnovation, demanded by the European Commis-sion with the Horizon 2020 proposal, seek a moresustainable and resource efficient transport. Inorder to reduce the induced drag, hence to im-prove fuel consumption, a clear trend in aircraftdesign is the increased aspect-ratio of the wing.The new configurations usually expose a slenderand lightweight structure to airflow loads that in-duce an high deformation and fluctuations. In thiscase a flight performance analysis requires more so-phisticated methods than a linear analysis adoptedin small deformation case, yielding to a nonlin-ear aeroelastic study [2]. A common way to per-form aeroelastic studies is to couple a computa-tional structural dynamics solver with an unsteadyaerodynamic code using linearized schemes. Thisthesis focuses on the interesting and debated aeroe-lastic behavior of a general aviation wing in tran-sonic flow. In the transonic flight regime, a nonlin-ear response approach should be preferred becauseof the motion of the shock waves that play an im-portant role in the stability prediction [18]. For thescope of this work, a more comprehensive methodhave been adopted, in order to develop a code thatis versatile to different flow Mach numbers.

FSI interactions have been studied since theygained the interest of the engineering community.

The early stages of coupled systems analysis startedwith the firsts studies of deforming spatial domain.Later, researches regarding numerical methods ap-plied to fluid-structure interaction (FSI) [10, 15] in-vestigated different approaches that range from seg-regated to monolithic schemes starting the generalframework that, even nowadays, is under study anddevelopment.

Regarding the aerospace sector, fluid-structureinteraction studies have been developed to providea good investigation method that address the aeroe-lastic phenomena, such as flutter and divergence [3].In this sense, different studies in transonic wind-tunnel observed the limit cycle oscillations regimeand compared the experimental data with compu-tational analyses. The conclusion was that it is re-quired a detailed structural wing model to predicta correct motion [19].

In order to perform FSI analysis for a transonicwing, a preliminary partitioned algorithm is pre-sented and tested in the Turek’s benchmark prob-lem [21]. Then, the wing case is introduced witha geometrical description and the flight conditionis defined along with computational analyses forthe fluid and the structure model defined. Finally,aeroelastic studies are performed.

Aeroelastic Framework

The development of a new FSI software can begreatly fasten thanks to the use of preexistent soft-

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wares with similar applications. The author pre-ferred Open-source softwares because of the open-ness of their codes to new implementation andthe presence of an active community of develop-ers that share their experience. OpenFOAM (for”Open source Field Operation And Manipulation”)was selected as fluid solver, thanks to its widerange of fluid model supported, namely compress-ible/incompressible, laminar, turbulent, etc. It alsoincludes mesh motion solvers for various applica-tions, that is the basic of every FSI application. Forthe simplicity of the input files, CalculiXv2.12 wasused in this work as structural solver. This pack-age, written by Guido Dhondt [7], can execute staticand dynamic simulations. Besides this, it includesa wide range of elements and constitutive relations.

Arbitrary Lagrangian-Eulerian formulation andDynamic Mesh Motion

Dealing with problems that simulate a dynamicand strong deformation of the continuum requires aslightly different kinematic description of the phe-nomena, to take into account the relation betweenthe moving mesh and the deformed domain. Ina Lagrangian description, each node of the meshfollows the material particle it belongs to, duringtime. This is the algorithm that is usually adoptedin structural mechanics. On the contrary, an Eule-rian description, popular in fluid dynamics, assumesthat the nodes does not follow the particles of theflow, and remains fixed in space, during time. Whenit comes to FSI problems, the deformation of thestructure implies some changes of the shape in thefluid domain. By the way, if the nodes of the fluidfollow the deformation of the structure they will notanymore be fixed in space and will not track the dis-placement of the fluid particles as well. Thus, theywill move in an arbitrary way, depending on theproblem needing. This is an Arbitrary Lagrangian-Eulerian (ALE) description. The equations in theALE differential form for mass, momentum and en-ergy in the so-called reference system, Rζ are:

∂ρ

∂t ζ+ c · ∇ρ = −ρ∇ · v, (1a)

ρ

(∂v

∂t ζ+ (c · ∇)v

)= ∇ · σ + ρb, (1b)

ρ

(∂E

∂t ζ+ c · ∇E

)= ∇ · (σ · v) + v · ρb. (1c)

In this set of equations: ρ is the mass density, vthe material velocity, σ the Cauchy stress tensor,b the specific body force and E the specific totalenergy and c the convective velocity, that is therelative velocity between the material and the mesh[8].

Once the conservation new laws are defined, it is

necessary to implement a mesh motion solver thatupdate the mesh while keeping it as regular as pos-sible in response to the deformation of the bound-aries. In FSI problems, the motion of the boundarythat represents the interface between the structureand the fluid is not known a priori, but is solution-dependent, thus the motion of the mesh must berecalculated for each time step. OpenFOAM in-cludes a vertex based finite volume method for auto-matic mesh update that compute the motion start-ing from a Laplace equation with a diffusion co-efficient that can varies through the domain [11].The solution of the Laplace equation gives a regu-lar mesh composed by lines of equal potential, thatis a repositioning of the internal nodes in a way thatare equidistant to the connected nodes. Thus theequation:

∇ · (γ∇u) , (2)

is solved to find the new nodes positions. γ is thediffusion field, while u is the point velocity that isused to change the point position:

xt+1 = xt + u∆t. (3)

During the simulations, the Laplacian algorithmdemonstrated good performance especially using aninverse relation on the diffusivity term. This pre-vent bad mesh quality during the motion and over-lapping of cells.

FSI algorithm

Fluid-structure interaction analysis is usually clas-sified as a 3-field problem that involve the fluid,the structure and the motion of the mesh. Thepartitioned approach to FSI employs a segregatedsolution procedure where the fluid and structureare sequentially solved and some tools provide thecommunication between them. This allow to uselargely developed software rather than cast the twodomains in a single system, that would require aground-up coding of the entire software. In thisscheme the fluid field is solved first, then the pres-sure and shear forces at the interface are transferedto the solid solver and a predicted motion of thesolid is computed. Then the automatic mesh solverreceive as input the displacements and velocities ofthe structure, and give as output the new mesh.Then the process is reiterated till the end of simu-lation is reached.

On the interface between fluid and structure, thefollowing relations are set:

σfn = σsn on Γ, (4a)

vf = vs on Γ, (4b)

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to ensure the mechanical and kinematic continuity(Γ is the FSI interface), being σ the stress tensorand v the velocity. f stands for fluid, while s forsolid.

As described here, the algorithm does not checkthat the predicted dynamics correspond to the ac-tual motion of the structure. For these reason it iscalled loosely-coupled. Loosely coupled time ad-vancing methods does not have a correct energybalance at the interface, so they show the so-calledadded-mass effect [5]. To ensure the correspondencebetween the predicted and actual motion it is nec-essary to apply a strongly-coupled algorithm. Itconsists in a corrector step through sub-iterativeprocess that check residuals set by the user at eachtime step. The scheme of this procedure is illus-trated in Fig.1.

Figure 1: Strongly coupled FSI algorithm.

The relaxation step in a fixed-point iteration isgiven by:

ui = ui−1 + ωi (ui − ui−1) = ωiui + (1− ωi)ui−1,(5)

where i refers to the iteration number, u is thedisplacement applied, while u is the displacementcomputed by the structural solver. The relaxationfactor ωi can be constant during the sub-iterativeprocess or can varies dynamically. Dynamic valueallow for faster convergence. The Aitken scheme[14] was adopted in this work, that gives a relax-ation factor equal to:

ωi+1 =ui−2 − ui−1

ui−2 − ui−1 − ui−1 + ui=

= ωiui−2 − ui−1

ui−2 − ui−1 − ui−1 + ui=

= −ωiri−1

ri − ri−1=

= −ωi(ri−1)T (ri − ri−1)

|ri − ri−1|2.

(6)

This parameter demonstrated a good time con-vergence, it is easy to implement and cheap to com-pute. In Fig. 2 the speed of the fixed-relaxationmethod is compared to the Aitken’s relaxation

method. The results refer to the benchmark testpresented later. The clear drop of number of itera-tions leads to a great reduction of the CPU-time inthe fixed-point relaxation process.

Figure 2: Comparison of the fixed-point itera-tion speed between fixed relaxation (blue line) andAitken’s relaxation (red line).

At the end end of each iteration, a check is ex-ecuted to verify whether the inner-loop has con-verged or not. In this work the residuals are evalu-ated on the displacements:

ui − ui−1 < ε, (7)

where ε is a tolerance set by the user.The algorithm described above has been coded in

an OpenFOAM ’s new application. The code can besummarized in the following list:

1. The software starts with a pre-processing phasethat finds the correspondence between thenodes and elements at the interface. This isrequired because, even if a conforming meshesis considered, when the elements are exportedseparately for fluid and structure mesh, thenumbering of nodes and faces at the interfaceis not the same. At the beginning, the meshcorresponding to the fluid and the one corre-sponding to the structure are red, saving thenodes’s coordinates and elements’s matrix con-nection in different lists. Then the nodes’ coor-dinates are compared to find a correspondenceand the numbering is reordered in a way thatit is the same for the different lists. Then thefaces in the interface are compared to find thecorrespondence of the elements, to transfer thestresses.

2. Afterwards the simulation starts with the ex-ternal loop, that is active till the end of theruntime. At the beginning of each time stepthe structure is solved. Here the interface loadsare calculated as:

ps = pw, (8a)

τs = ρν ∇v, (8b)

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being pw the pressure on the generic interfaceface and ps the pressure applied on the solidinterface. The shear stress applied on the solidinterface, τs, is calculated through the shearstress law for Newtonian fluids. ∇v is the ve-locity gradient, ρ the fluid density, and ν thekinematic viscosity. Then the external struc-tural solver is called through a system call andthe predicted displacements and velocities arestored into variables.

3. After that, the sub-iterative process starts witha relaxation factor ω1 = 1. Afterwards thevelocities are transferred to the mesh motionsolver that computes the new positions of thenodes and the convective velocity field.

4. Now the flow field is solved.

5. At the end of the iteration a new predictionof the structure motion is computed and com-pared with the previous through a check on theresiduals. If the iteration did not converge, thenew displacements and velocities are recalcu-lated using the Aitken’s relaxation technique.The same scheme is repeated at each time step.

BenchmarkIn order to verify the accuracy of the algorithmdeveloped, a quantitative comparison with othersolvers is required. Among all the literature’sbenchmarks, the Turek’s proposal [21] was preferredbecause of its popularity in FSI studies. It studies aflow around an elastic body result in significant self-induced structural oscillations. The original studywas conducted with a fully implicit monolithic ap-proach. The domain consists in a simple 2D flowaround a cylinder with an elastic bar attached. Re-ferring to Fig.3 the geometry, at the start time, isfully described by the following quantities:

• The domain bounding box is L = 2.5 and H =0.41;

• Considering the left bottom corner as the ori-gin, the center of the cylinder is at C =(0.2, 0.2) with radius r = 0.05. Thus the posi-tion of C gives to the geometry an asymmetri-cal configuration, the channel on the top of thebar is larger than the other;

• The bar is defined by the length and height,respectively l = 0.35 and h = 0.02. The rightbottom corner is at (0.6, 0.19);

• A control point, called A, is located at(0.6, 0.2).

The model of the fluid considered is Newtonianand incompressible. Fluid properties are defined

Figure 3: Geometry of the Turek’s benchmark [21].

through the density ρf and the viscosity νf . Astarting point for the coding of the FSI solverwas pimpleDyMFoam, distributed with the officialOpenFOAM ’s release. This is a version of pimple-Foam that supports dynamic mesh. It is based onthe incompressible governing equations. The equa-tions are solved through a PIMPLE (merged PISO-SIMPLE) algorighm. It consists in a pressure-velocity coupling scheme and can be seen as a varia-tion of the PISO (Pressure Implicit Split Operator).algorithm, with an external corrector that loops afixed number of times defined by the user. It doessupport the dynamic change of the time step in or-der to match the desired condition on the Courantnumber [17].

The structure is elastic and compressible. Thecorresponding properties are declared as Youngmodulus E, Poisson ratio νs and the shear modulusµs. The equations are solved in a unconditionallystable implicit scheme in CalculiX.

The initial condition is a flow with velocity equalto zero. The boundary condition on the inlet is aparabolic velocity profile with a smooth increase intime, starting from a null velocity. The parabolicprofile has a mean velocity U . The upper and lowerboundaries, as well as the cylinder and bar patches,have a no-slip condition.

Three different benchmarks have been proposedby Turek. The first results in a steady state solu-tion, while the others show a periodic behavior intime. For the purpose of this comparison, the FSI3case was reproduced. The values for the fluid andstructure properties and boundary condition are re-called in Tab.1.

For this test, the Reynolds number is:

Re =2rU

νf= 200.

Table 1: Benchmarks’s constantsStructure Fluid

ρs[103 kgm3 ] 1 ρf [103 kgm3 ] 1

E[106 Nm2 ] 5.6 νf [10−3m2

s ] 1νs 0.4 U [ms ] 2

Many computational studies investigated the

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Figure 5: Mesh, velocity and pressure fields whenA has maximum vertical positive displacement. Ve-locity and pressure in SI base unit. The pressure isreferred as the incompressible value used in Open-FOAM, p/ρ.

transition of the wake of the flow past a circularcylinder [4]. It is well known that, even if theboundary conditions are static, the flow is not andshows periodic oscillations. To capture this flowmotion, it was necessary to increase the mesh res-olution. The domain have been discretized witha fully structure mesh for both structure and fluid.Respectively they have 35600 and 400 hexahedrons.

The results obtained with the FSI algorithm testare shown in Fig.4. The amplitude registered is0.0681m and the frequency of 5.55Hz. A compari-son with the reference values is in Tab.2.

Figure 4: Vertical displacement of A in FSI test.

Table 2: Comparison with the benchmark’s results.Calculated Turek Error

∆z[m] 0.0681 0.0684 0.44%Freq.[Hz] 5.55 5.47 1.4%

A graphical representation of the pressure andvelocity fields, when A has its maximum verticalpositive displacement is displayed in Fig.5.

Wing model

The wing considered in this work is part of the FP7-NOVEMOR project (Novel Air Vehicle Configura-tions: From Fluttering Wings to Morphing Flight)[16]. The Reference Aircraft is a regional jet with113 PAX in a single economic class and provide op-erational flexibility to fly different missions at thetransonic regime. The wing geometry was definedusing three different sections, distributed span-wise.The first is at the root, then one is located at thetrailing-edge discontinuity and finally one at the tip.The positions of the sections, as well as the respec-tive chords, are presented in Tab.3.

Table 3: Geometrical data of the wing.Station Chord [m] Y [m]Root 7.57 0Break 3.99 6.18Tip 1.73 15.73

Here, Y is the span direction. A graphical repre-sentation is provided in Fig.6.

Figure 6: Top view of the transonic wing.

The geometrical model, drawn in a CAD soft-ware, has a reference area Sw = 63.8m2. This willbe later used to calculate the force coefficients. Inexternal flow analysis is required to specify the ge-ometrical boundaries that limit the domain exten-sion. In this case a bounding box have been chosen.Supposing that the root of the airfoil is attached toone face of the box, and its center is in the origin,the bounding box extends from (−40, 0, −40) mto (100, 30, 40) m, where the first direction is theflow direction and the second the span-wise direc-tion. These length have been designed in a way thatthe boundaries’s gradient are so small that there isno interaction with the flow around the wing.

Fluid model

The flight condition is transonic flight with theMach number M = 0.78 and an altitude h =38000ft. The standard atmosphere have been con-sidered to evaluate the density, temperature andstatic pressure at this altitude [1]. The values arerecalled in Tab.4.

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Table 4: Flight conditions.Altitude [ft] 38000Mach 0.78Pressure [Pa] 20646Temperature [K] 217Speed of sound [m/s] 295Density [kg/m3] 0.332

OpenFOAM ’s suite includes different solvers de-signed for high speed flows that support mesh mo-tion. The comparison between the pressure-basedsonicFoam and the density-based rhoCentralFoam,in a set of different cases that includes a flow overa wedge, a diamond airfoil and a two dimensionalblunt body reveled that rhoCentralFoam has betterperformance [9]. In fact, to reproduce results of thesame quality, sonicFoam requires three times morecells than its competitor. In compressible flows theproperties are transported not only by the flow, butalso by the propagation of waves. Thus the compu-tation of the fluxes need to consider the transport ineach direction. In rhoCentralFoam the Kurganov’smethod is applied [13].

Coming to the mesh definition, two chances areoffered to the user: a unstructured or structuredmesh. The first can be defined also as a ”free”mesh, because it consists of an irregular patternthat cover the whole domain, through elements ofdifferent topology. On the other hand, a structuredmesh is characterized by a ”mapping” of the domainwith a repeatable pattern. Even if the unstruc-tured method allows an easy meshing of complexgeometries with less interaction of the user, it usu-ally leads to inaccuracy and non-convergence of thesolution [12]. The author noticed that, in Open-FOAM, the higher skewness of the unstructuredmeshes can cause severe oscillation and crashes ofthe code, especially when a grading of the cells sizeis required to capture wall interaction. Thus, astructured mesh was preferred in this study.

In order to reduce the resources and computa-tional time of the simulations, different meshes havebeen compared, to find a good trade-off betweenaccuracy and computational cost. The number ofcells and the CPU-time is presented in Tab.5. Thecomputational time is referred to a simulation withthe same boundary conditions and integration time,with an angle of attack equal to zero.

Table 5: Mesh comparison.Mesh Cells CPU Time [h]Fine 260694 115Coarse 108894 27

The lift and drag coefficient, coming from the

different simulations have been compared to refer-ence values [20], in Tab.6. During the simulationsa Shear Stress Transport (SST in short) k − ω tur-bulence model have been used.

Table 6: Force coefficient comparison, experimentaldata from [20].

Data CL CDExp. 1.30 0.0123Fine 1.32 (1.5%) 0.0095 (22.8%)Coarse 1.34 (3.1%) 0.0141 (14.6%)

It is clear that the run time does not exactlyscale with the number of cells, it also depends onthe overall quality of the mesh. A comparison ofthe mesh quality can be done evaluating the skew-ness and non orthogonality of the cells. The firstmeasures how much, the line connecting two cellcenters across the merged patches is far from thecenter of the face, while the other refers to the non-orthogonality between this line and the face. Tab.7shows that the ”mid” mesh is the worst in terms ofskewness and number of non-orthogonal cells. Eventhough the finer mesh, as expected, leads to betterresults, the CPU-time required have been consid-ered not compatible with a FSI analysis. For thisreason the coarser mesh have been adopted for therest of the work.

Table 7: Mesh quality comparison, OpenFOAM ’sstandard checkMesh utility units.

Mesh Max Skew. Non orth. cellsFine 1.21 1232Coarse 1.35 227

A plot of the Mach number in the plane of thewing root is in Fig.7. It clearly displays a first shockwave, right after the leading edge, that representsthe start of the supersonic region. This ends withthe second shock, allowing the flow to slow downto subsonic speeds at the trailing edge. The smallnumber of cells adopted in the coarser mesh doesnot allow the strong discontinuity across the shockwave, that appears to be more like a soft variation.

Figure 7: Mach number plot in the wing root plane

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Structural model

Modern wing structures generally include spars thatextend span-wise, ribs in the chord direction andstiffeners connected to the skin, plus a set of oth-ers equipments like fuel maneuvering systems, fueltanks, engines, etc. This thesis does not aim toa comprehensive description of this complex con-figuration, thus a simplified discretization was pre-ferred.

A reduced structural representation of the wingthrough a beam model have been investigated inother works, leading to unsatisfactory results [19].Even though the wing geometry presented fulfillthe slenderness hypothesis, the preservation of thecross-section is not. For this reason, a more com-plex structural model have been used.

Starting from the 2D mesh, that represents thewing skin in OpenFOAM, new elements have beenbuilt to increase the stiffness of the structure,and reproduce a physical bending-torsion behav-ior. Two-dimensional shell elements have been dis-tributed along the chord and the span, to discretizethe spars and ribs. The configuration is presentedin Fig.8.

Figure 8: Wing structure model formed by 2D ele-ments (without skin elements).

In order to predict the preferred shapes that thewing will assume during the loading, a modal anal-ysis is performed. It finds the eigenmodes, oscillat-ing homogeneous solutions of the linearized govern-ing equation, and the corresponding eigenfrequen-cies. Fig.9 exhibits the first bending mode, andFig.10 the first torsional mode. The second showsa strong deformation of the cross-section, that ob-viously does not respect the hypothesis of a beammodel, where the preservation of the cross-sectionshould be guaranteed.

Figure 9: First bending mode, total displacements.

Figure 10: First torsional mode, total displace-ments.

Results

Even though, loosely coupled time advancing meth-ods introduce the added-mass effect, the inaccuracyof this method arise in the case of internal fluid anal-ysis of incompressible flow with comparable den-sity value for structure and fluid [6]. Obviously,none of these hypotheses is met in the FSI tran-sonic wing study, thus a loosely-coupled approachwas preferred for the transonic wing FSI study.

Regarding the initial conditions of this analysis,the fields coming from the CFD simulations of themesh convergence study have been transferred tothis simulation. The structure was supposed to beunstressed at initial time, so the displacements arenull.

In order to simulate different cases, two config-uration have been considered. In particular, thethickness of the ribs and spars elements have beenchanged to analyze the response of wing structureswith different stiffnesses. Both of them are en-tirely made of a material with properties similarto an aluminum alloy. The relevant mechanicalproperties are the elastic modulus, E = 73GPa,the density, ρs = 2.7 103 kgm3 and the Poisson ratio,νs = 0.33. The two configurations are presentedin Tab.8. Clearly, it is expected to have larger de-formation in the second configuration, since the de-creased thickness of the spars leads to a drop in thestructural stiffness in both bending and torsion.

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Table 8: Thicknesses of the 2D elements in thestructural model.

Data Conf. 1 Conf. 2Skin [mm] 2 2Rib [mm] 40 20Spar [mm] 40 20

The displacements recorded in four points dis-tributed along the wing surface have been storedinto results to compare motion of points in the twostructural configurations. Two of them are on lead-ing edge, the other on the trailing edge. For eachgroup one is located at the tip of the edge, the otherin the middle between the tip and the wing break.For the sake of clarity, the positions are highlightedin Fig.11.

Figure 11: Positions of the four control points dis-tributed along the wing surface.

Fig.12, Fig.13, Fig.14 and Fig.15 compare the dis-placements of the first configuration (in blue) withthe second (in red).

As expected, the second configuration has largerdisplacements. In this short simulation time, a tor-sion deformation drives the wing motion. This isclear observing that the points on the trailing edgehave positive displacements, while the ones on theleading edge have negative displacements.

Figure 12: Displacements in the z direction for thecontrol point on the trailing edge, at tip. Configu-ration 2 in red, Configuration 1 in blue.

Figure 13: Displacements in the z direction for thecontrol point on the leading edge, at tip. Configu-ration 2 in red, Configuration 1 in blue.

Figure 14: Displacements in the z direction for thecontrol point on the trailing edge, between wingbreak and tip. Configuration 2 in red, Configura-tion 1 in blue.

Figure 15: Displacements in the z direction for thecontrol point on the leading edge, between wingbreak and tip. Configuration 2 in red, Configura-tion 1 in blue.

The lift and drag values during time are plottedin Fig.16 and Fig.17, respectively. The red refersto the first configuration, the ticker, while the bluelines refer to the second configuration. As expected,the larger displacements of the second configuratonlead to greater changes in both drag and lift. Thetorsion motion of the wing leads to a drop of 15%in the lift force during the simulation. Even thoughthe drag force varies during time, the percentualchange is much less than the one registered in thelift force. The change of the drag force, in percent-age, is much lower, around 1%.

In order to describe the effects of elasticity inbending and torsion in the aerodynamics, the dy-namics of the second configuration are here pre-sented. A section at 12m from the wing root along

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Figure 16: Plot of the lift force during the simula-tion.

Figure 17: Plot of the drag force during the simu-lation.

the span direction have been selected because it ischaracterized by greater displacements, thus greaterchanges in the flow fields. The pressure field at thebeginning of the simulation is presented in Fig.18and in Fig.19 at the end of the simulation. It canbe noticed that the torsional motion of this sectionalignes the airfoil with the upstream flow velocity,leading to a lower acceleration of the fluid along thewing surface. For this reason the shock that is inFig.18 vanishes and become a more gentle variationof the pressure field in Fig.19.

Figure 18: Plot of the pressure field at the beginningof the simulation in the selected section.

Figure 19: Plot of the pressure field at the end ofthe simulation in the selected section.

ConclusionsIn this work different problems have been ad-dressed, starting from the implementation of a newFSI algorithm into an open-source software envi-ronment. Even though the algorithm can be eas-ily described through a simple flowchart, the imple-mentation was, for some aspects, cumbersome. Inthis case a big help was provided by all the activedevelopers of the open-source application employed.

Even if the analysis performed on the incompress-ible test has no relations with the main goal of thiswork, it was necessary to test if the algorithm hasbeen implemented correctly. The good performancedemonstrated on the benchmark confirmed the cor-rectness of the whole system built.

Then, the swept wing analyzed exhibited some in-stabilities in the calculations because of its complexgeometry. Trying different solutions, it was possibleto find a meshing method, in particular the Y-blockat the tip, that allowed to have stable solutions andrun the transonic simulations with a relatively smallnumber of cells.

The work is concluded with the presentation ofthe FSI results of the transonic wing. Even thoughthe final results presented could be considered notsatisfactory because of the short run time, the ex-pected motion of a very flexible wing is captured.The torsional motions, especially for the secondconfiguration leading to a drop in the lift force thatis very relevant. Regarding these simulations, thevery long computational time was a bottleneck forthe fast advancing of the work.

The author considers that this work can be en-hanced, spending more time on the following as-pects:

• The support to non conformed meshes shouldbe taken into account. In fact, the tran-sient simulations of complex structures, like thetransonic wing here discretized, can be quickenwith the use of a coarser mesh at the interface;

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• Thanks to the use of an external and extendedstructural solver, like CalculiX, different mate-rial models can be tested in FSI analysis, com-posite materials for example;

• More simulations can be executed, changingthe geometry, turbulence model and flight con-ditions;

• The coding of an internal structural solvershould be considered, calling the structuralsolver with a system call, as the author did inthis work, does not allow to run it in parallel,thus leading to a longer computational time.

• The simulations related to the results should becontinued in order to conclude this work withmore details.

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