M. R. Chiarelli, M. Ciabattari, M. Cagnoni and G. Lombardi
Aerospace Unit, Department of Civil and Industrial Engineering, University of Pisa, Italy
Keywords: curved wing, drag polar curves, FSI analyses, stall of wings
the sheared wing method has been adopted to draw the curved wing), and the same airfoils. In the present case, swept and curved wings have the aspect ratio equal to AR=7.53. The two wing’s models have not twisted profiles along the half span.
Fig. 1 Geometry of the Wings (Swept and Curved).
Previous campaigns of numerical analyses has shown that, in the case of rigid wing-body models the first with a traditional swept wing and the second with a curved planform wing geometry, a CD (drag coefficient) reduction of 4% can be reached for altitude equal to 10,000 m, Mach number equal to 0.85 and a CL (lift coefficient) equal to 0.4, i.e. a condition representing a transonic cruise flight [2].
The shape of the leading edge of the curved wing have been defined by fixing the values of the slope at root and tip sections and by assuming a second order law in the span direction (in the Figure 1 the mathematical expression of the curve which defines the shape of the l.e. has been inserted).
Both in the swept and in the curved wing configurations the geometry of the airfoil, used to create the CAD models, have been introduced in the X-Z plane according to the adopted reference system (Figure 1): in this way, the selected supercritical airfoil lies on planes parallel to the longitudinal plane of the wing half models. The origin of the reference system coincides with the leading edge of the root section profile.
It is well known that the drag estimation for real configurations of aircrafts, especially in the non-linear transonic conditions or at high angles of attack, represents a challenge for the
aerospace engineers and aerodynamic scientists ([7], [8], [9]).
Even thought at the first step the models used in the present research work were not so refined, due to the limitation in the available computational resources, the numerical results obtained allow to assess, with a satisfactory level of reliability, the feasibility of the novel curved wing concept.
2 Results of the Co-Simulation Analyses The models of the swept wing and of the curved wing has been constructed assigning the properties to the structural components (skin, stringers, ribs, spars) in Abaqus® 6.11. As example a sketch of the structural model for the curved wing is shown in Figure 2. Both structural models of wings have the same dimensions for all their characteristic components. The non-linear large displacement option has been activated to carry out the coupled FE analyses.
Fig. 2 View of the Curved Wing FE Model.
The results shown in the present work (relevant only to static aeroelastic analyses) do not take into account the distributions of structural weight and non structural weight.
From a practical point of view it has been demonstrated that the structural weight has a very little influence on the deformed shape of the wing models at the examined flight conditions.
To carry out the co-simulation analysis is necessary that the CFD and the structural model are perfectly complementary in the areas where take place the exchange of the nodal forces and
Fluid-Structure Interaction Analyses of Wings with Curved Planform: Preliminary Aeroelastic Results
CEAS 2013 The International Conference of the European Aerospace Societies the nodal displacements. For this reason, in
Catia® V5 R20 the complete aerodynamic field that surround the wings has been constructed in such a way that the surface describing the upper and lower skin of the wings are correctly defined and coupled between aerodynamic and structural models.
The object of the present FSI (fluid structure interaction) analyses is a comparative study between two elastic models of wings. The drag polar curves, the CD-Mach curves, the distributions of pressure coefficient along the wings chords, the comparison of both shape and dimensions of the sonic zone around the wings (the so called bubble zone), and the stress values by Von Mises rules have been computed and drawn under the following hypothesis:
Standard Air Altitude = 10,000 m
CL = 0.4 (to draw the CD-Mach curves)
To execute the comparison the following Mach numbers have been chosen: 0.80, 0.85, 0.875. The drag polar curves have been drawn interpolating the calculated data and an estimation of the drag coefficient per cent reduction has been obtained. In this case an implicit unsteady analysis technique and the k-
turbulence model have been selected to activate the CFD code; they have been necessary about 500 hours of CPU simulation to obtain the results.
The drag polar curves (H=10,000 m, Mach=0.85) and the CD-Mach curves (H=10,000 m, CL=0.4), represented respectively in the Figure 3 and Figure 4, have been obtained for the two compared wing configurations: swept elastic wing and curved elastic wing.
In the Figure 5 and Figure 6 the distribution of the pressure coefficient for the two wing models, at section planes having the same y coordinate along the half span direction (y = 60% of the half span B and @ y =80% of B), are shown. Looking at the gradient of the pressure coefficient along the chord it can be said, as discussed in [2] and [4], that the intensity of the shock wave for the curved wing
is much lower than the intensity of the shock wave for the swept wing: so the wave drag for the curved wing model reduces respect to the swept wing model. From a physical point of view, the reduction of the adverse pressure gradient across the shock wave on the wing surface (in the X-Z plane) should provide also a retardation of the boundary layer separation, and then an increase of the whole aerodynamic efficiency of the wing. This phenomena, which depends on the interaction between the boundary layer and the shock waves, need to be investigated within a suitable and much expensive experimental tests campaign.
Fig. 3 Drag Polar Curves of the Two Elastic Wings.
Fig. 4 CD-Mach Curves of the Two Elastic Wings.
In the Table 1 the numerical values of the drag components for each part of the wings’
surfaces and the drag coefficient per cent reduction are shown (CL=0.4 and Mach=0.85).
A comparison of the in plant shape and position of the wave front (the shock wave) on the upper surfaces of the two wings proves that,
towards the tip of the wings, the area of subsonic flow downstream of the wave front is greater for the curved wing. This phenomena is highlighted in the Figure 7 and Figure 8.
Fig. 5 Comparison of the CP Distribution on the two Wings (Section @ 60% of the Half Span).
Fig. 6 Comparison of the CP Distribution on the Two Wings (Section @ 80% of the Half Span).
Table 1 CFD Results – CD Data for the Elastic Models (Drag Components Reduction: CL=0.4 & Mach=0.85).
In these figures the sonic boundary, the red surfaces which enclose the supersonic region around the wing, has a different shape and different dimensions comparing the swept wing with the curved wing, especially towards the tip of the two wings.
Fig. 7 Swept Wing – Front View of theSupersonic Zone (H=10,000 m, Mach=0.85, CL=0.4).
Fig. 8 Curved Wing – Front View of theSupersonic Zone (H=10,000 m, Mach=0.85, CL=0.4).
This fact demonstrates that the perturbation induced in the aerodynamic field by the curved wing is much less intense: that is a lower quantity of the overall asymptotic energy is dissipated in the flow around the curved wing (for a fixed transonic flight condition).
3 Comparison of Results Between Rigid and Elastic models
As an example in the Figure 9 has been represented the comparison between the CL-CD curves computed for the rigid and elastic swept wings. As can be seen the two curves overlap perfectly for low values of CL [4], [5], [6].
Fluid-Structure Interaction Analyses of Wings with Curved Planform: Preliminary Aeroelastic Results
CEAS 2013 The International Conference of the European Aerospace Societies A similar situation occurs also for the curved
wing models.
On the other hand it is to be observed that the CD0 value, that is the value of the drag coefficient corresponding to zero value of the lift, as expected, is not influenced by the deformation effects, due to the reduction of the aerodynamic loads along the span of the two wings (looking at the Figure 9 the two couples of drag polar curves overlap perfectly as CL tends to zero). This result is a proof of the goodness of the FSI analysis.
Fig. 9 CL-CD Curves (Rigid vs. Elastic).
Finally observing the results provided by the structural FE models of wings, at the clamped section for the curved wing model the stresses are reduced respect the swept wing model (a reduction of about 5% has been obtained). This fact depends on the reduction of both the bending moment and the torque at the root of the curved wing as a consequence of the in plant shape of this wing [4]. In fact, the resultant lift force for the curved wing tends to move inwards and this causes, as a consequence, the reduction of the bending moment characteristic at the root of the wing.
In the Figure 10 the distributions of the equivalent stress on the upper surface of the two FE models have been shown (reference condition: h=10,000 m, Mach =0.85, CL=0.4).
As said these results do not take into account the gravity effects but only the aerodynamic loads acting on the two wings. These results have been obtained adopting a similar structural layout for the two wing models according to works [4], [5], [6]. For these reasons, it can be
asserted that the results obtained depend only on the in plant shape of the wings.
Fig. 10 Comparison of stress on the upper skin of the wings (h=10,000 m, Mach=0.85, CL=0.4).
4 Comparison of the Stall Characteristics for a Swept and a Curved wing
To compare the aerodynamic behavior of a curved and a swept wing at high angles of attack a novel campaign of suitable numerical analyses has been set up. For these analyses new highly refined grids have been constructed: two parametric three-dimensional grids of about 9 million cells. A steady state physical model has been used to carry out preliminary analyses and the k- turbulence model, available in the commercial code Star-CCM+, has been adopted.
In Figure 11 a sketch of a section of the grid has been shown.
The geometry of the two wings is similar to that used in the previous analyses (Figure 1).
The grids have been constructed following a procedure that is not available in the CFD code.
Only hexagonal regular cells have been used to construct the grids around the models of the wings. For the sake of simplicity, the CFD analyses have been carried out without the elasticity effects (rigid wing models).
A low subsonic flight condition has been assumed, that is:
Standard Air Mach = 0.2
Altitude = 0,0 m (sea level)
As made in the previous works, the drag polar curves of the curved and swept wings have been constructed: these curves are shown
in the Figure 12. The data refer to mean values of the converged solutions for both CL and CD.
Fig. 11 View of the Plane Grid Around a Section of Wing Models (Structured Grid).
Fig. 12 Drag Polar Curves of the Two Rigid Wing Models (Mach=0.2, Sea Level).
As a first result it can be observed that at low subsonic flight conditions the planform shape does not affect the polar drag of the wings: that is a swept and a curved wing (having the same aspect ratio, same profiles and same value of the angle of sweep of the leading edge at the root section) have same value of efficiency for each physical value of the lift coefficient.
In other words, the curved planform does not modify the aerodynamic performances of a swept wing, having the same aspect ratio, at low subsonic flight conditions. But, as above demonstrated, in the transonic regime the curved planform improves markedly the aerodynamic efficiency of a wing.
A technical aspect that must be investigated origin of x and y axes in Figure 1) following the approximated formulas (that does not take into
The quantity X in the integrals represents the distance of a generic point on the wing with respect to the leading edge of the root section, measured in the chord direction as shown in Figure 13. In the same manner the quantity Z represents the vertical coordinate of the generic point on the surface of wings.
In the equation (3) are shown also the effects of horizontal components of vectors P·n and S·t: the quantity X0-2 contains these effects.
As can be demonstrated this last term can be neglected also for angle of attack of about 20 degree. For this reason the following numerical computation does not take into account this contribution and it is assumed X0 ≈ X0-1.
In the Figure 14 have been represented the approximated values of X0 for the two wing models as a function of the angle of attack. The centers of pressure tend to move forward from their initial positions corresponding to the angle of attack equal to zero (the angle of attack is measured respect to the x-y plane defined in the
Fluid-Structure Interaction Analyses of Wings with Curved Planform: Preliminary Aeroelastic Results
CEAS 2013 The International Conference of the European Aerospace Societies Figure 1: this plane that does not contain the
zero lift axis of the wing). From the Figure 14 it can be seen that for both the wings the minimum value of X0 is equal about 10 m.
Computing the percentage variation of X0
respect to its initial value, the graphs of Figure 15 has been obtained.
Fig. 13 Notations Used for the Computation of X0.
Fig. 14 Values of X0 for the two wings.
Fig. 15 Percentage Variation of X0 for the Two Wings.
The numeric data reported in the Figure 14 indicate that for the curved planform wing the
percentage variation of the position of the center of pressure reaches a maximum difference respect to the swept wing that is less than 2,5%
(curved 29,34% vs. swept 27,04%).
These preliminary results indicate that for low speed and high angles of attack the aeromechanical behavior of a curved wing is similar to the aeromechanical behavior of a swept wing having similar design parameters (aspect ratio, aerodynamic profiles, twist distribution, etc …). In other words, according to these last results, the curved planform wing does not modify in a negative manner the mechanics of flight of an aircraft and, in particular, the stall phenomena will be quite similar to those of a traditional swept wing configuration. On these basis, the well known deep stall phenomena, which typically concerns wings with high sweep angles, can be easily avoided adopting standard methods in the preliminary design phase of a curved wing aircraft configuration.
5 Conclusion
In the paper it has been shown that also taking into account the elasticity effects of the wing-box structure the curved planform of a wing favourably influences the distributions of both pressure coefficient and Mach along the wing span: the results discussed in the present paper, available also in [4], [5] and [6] with much more details, agree with previous results published by the authors and confirm that the feasibility of the examined novel wing configuration can be reached also adopting standard design technology for the wing-box structure.
More in particular, analyzing the co-simulation results it can be said that for the same flight condition (h=10,000 m, CL=0.4 and Mach=0.85) the drag coefficient of the curved elastic model is less of 7% than the same of the swept elastic model (see Figure 4 and Table 1).
Moreover the bending moment and torsion moment computed for the curved elastic model are less of about 5% than the computed moments on the swept elastic model.
Consequently also the stresses in the skin of the
curved wing are less of about 5% than the same of the swept wing (Figure 10).
Analysing the graphs of the distribution of the pressure coefficient at some control sections, see Figure 5 and Figure 6, it can be concluded that the pressure rise across the shock wave is less intense and smoother in the curved wing, and consequently, the adverse pressure gradient towards the trailing edge is reduced. It can be expected that the separation of the boundary layer of the curved wing is delayed with respect to the swept wing in the transonic regime, with consequent beneficial effects on the drag (the aerodynamic efficiency tends to increase).
The plant shape of the shock wave front and the shape and dimensions of the supersonic zone around the wing (the bubble zone) are strongly influenced by the shape of the wings: from a physical point of view it means that the perturbation induced by a curved wing is less intense and then the energy dissipated in the transonic phenomena around it reduces with respect a swept wing flying at the same values comparative study the results obtained confirm that, also adopting a fluid structure interaction procedure, the effects of the curved planform configuration of a wing is not negligible from the aerodynamic point of view in the transonic regime.
Further research will be carried out in the future to draw, by the use of FSI procedure, the flutter boundary of a curved wing configuration and to study the vibrations of the tip of wings that can be induced by the turbulent unsteady flow that develops around the wings at high angles of attack.
Preliminary analyses about the aerodynamics at low speed and high angles of attack have demonstrated that for the curved wing configuration the aeromechanical properties are quite similar to that of a traditional swept wing.
In fact both the polar drag curves and the computed position of the centers of pressure are quite identical for the two type of wings.
References
[1] Chiarelli M.R., Cagnoni M., Ciabattari M., De Biasio M., Massai A., “Preliminary analysis of a high aspect ratio wing with curved planform”, 27th Congress of the International Council of the Aeronautical Sciences, Nice, France, 2010, CD-Rom ISBN 978-0-9565333-0-2.
[3] Chiarelli M.R., Lombardi G., Nibio A., “A straight wing and a forward swept wing compared with a curved planform wing in the transonic regime”, International Conference of the European Aerospace Societies, Venice, Italy, 2011, CD-Rom ISBN 978-88-96427-18-7.
[4] Ciabattari M. and Cagnoni M., “Effetti della forma in pianta sul comportamento aeroelastico di ali di elevato allungamento: confronto fra un'ala a freccia e un'ala con forma in pianta curva”, Master Degree in Aerospace Engineering, Department of Civil and Industrial engineering, University of Pisa, Italy, 2012. Ref.
http://etd.adm.unipi.it/t/etd-09132012-114137/
[5] Chiarelli M.R., Ciabattari M., Cagnoni M. and Lombardi G., “The effects of the planform shape on drag polar curves of wings: fluid-structure interaction analyses results”. STAR Global Conference 2013, Orlando, Usa, 2013. Ref.
http://www.cd-adapco.com/presentation.
[6] Chiarelli M.R., Cagnoni M., Ciabattari M., Lombardi G., “A comparison of the drag polar curves of wings using the fluid-structure interaction analyses”, XXII Congresso AIDAA, Naples, Italy, 9th – 12th September 2013.
[7] Cole J.D. and Malmuth N.D., “Wave drag due to lift for transonic airplanes”, Procedings of The Royal Society, A 2005, 461, pp. 541-560, doi:
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[8] Antunes A.P., da Silva R.G. and Azevedo J.L.F.,
“A Study of Different Mesh Generation Approaches to Capture Aerodynamic Coefficients for High-Lift Configurations”, 27th Congress of the International Council of the Aeronautical Sciences, Nice, France, 2010, CD-Rom ISBN 978-0-9565333-0-2.
[9] Gur O., W.H. Mason W.H. and Schetz J.A.,
“Full-Configuration Drag Estimation”, Journal of Aircraft, Vol. 47, No. 4, 2010, pp. 1356-1367.
Acknowledgements
The authors wish to thank the Research Coordination Department of the Tuscany Region for the granted financial support used to carry out the last part of the research work.