text:
Ilias Petropoulos
email: ilias.petropoulos@onera.fr

Thomas Streit
email: th.streit@dlr.de

Thomas Kilian
email: thomas.kilian@dlr.de

Martin Kruse
email: martin.kruse@dlr.de

Numerical aerodynamic performance assessment of HLFC wing configurations using far-field drag analysis

This paper presents the aerodynamic performance analysis of realistic large passenger aircraft equipped with a Hybrid Laminar Flow Control (HLFC) system. To this purpose, the design of two HLFC aircraft geometries is first outlined, both designed by DLR based on an Airbus HLFC variant of the XRF-1 geometry. Numerical simulations were performed with the TAU RANS solver of DLR, using an extension of its LST-based automatic transition prediction capability for cases with boundary layer suction. Aerodynamic performance assessment was then performed using the ONERA ffd72 far-field drag analysis software, allowing comparisons between configurations in terms of phenomenological drag components. The overall assessment of the different configurations considers both the laminarization benefit using HLFC technology compared to a fully turbulent reference aircraft and the performance penalty when the HLFC wing is operated under fully turbulent conditions. Results of the analysis are consistent with the geometry modifications and the flow physics, highlighting an overall HLFC benefit via the reduction of friction and viscous pressure drag components. A wave drag penalty, inherent to the laminarization of the outer wing upper surface, is identified in the original HLFC geometry designed by DLR and reduced in its further improved version.

INTRODUCTION

The environmental impact of civil aviation coupled with the growth of aircraft traffic have motivated the orientation of regulations towards restrictions for more fuel-efficient aircraft. To this end, the Advisory Council for Aviation Research and innovation in Europe (ACARE) has introduced stringent requirements aiming at a reduction of the environmental footprint of the aviation sector [START_REF] Kruse | Strategic Research and Innovation Agenda (SRIA)[END_REF]. Goals of the European Vision FlightPath 2050 address, among others, the reduction of pollutant emissions and perceived noise of flying aircraft, sustainability (alternative fuels, recycling) and atmospheric impact. To achieve these ambitious objectives, improved aircraft components, and even disruptive technologies, must be developed both individually and collectively, with respect to their integration on the aircraft. The role of aerodynamic design methods can thus contribute to the reduction of pollutant emissions by the design of more fuel-efficient aircraft (i.e. reduced consumption for a given mission), in particular by reducing aerodynamic drag. The subject of aerodynamic drag is however complex. Despite the existence of numerous approaches to drag reduction at a theoretical or elementary level, it is not often the case in terms of their application at an industrial level. Civil transport aircraft in particular make the integration of disruptive technologies difficult in practice, due to the need for manufacturing adaptations, design methodology validation, as well as cost, maintenance, certification and safety concerns. A promising way to achieve an important reduction of aerodynamic drag, under a reasonable increase of configuration complexity, is to delay transition in order to maintain a laminar flow over parts of the aircraft surface. The delay of transition of the flow from laminar to turbulent can be achieved by Hybrid Laminar Flow Control (HLFC) systems [START_REF] Krishnan | Review of hybrid laminar flow control systems[END_REF]. These couple the application of airfoil geometries adapted to natural laminar flow principles [START_REF] Hue | Experimental and numerical methods for transition and drag predictions of laminar airfoils[END_REF][START_REF] Hue | Wind-tunnel and CFD investigations focused on transition and performance predictions of laminar wings[END_REF] with laminar flow control. Over the recent years, an evaluation of the impact of HLFC technology for large passenger aircraft is being carried out within the frame of the HLFC-WIN project, funded by the Clean Sky 2 Joint Undertaking under the European Union's Horizon 2020 research and innovation programme. This programme is a research collaboration between Aernnova, DLR, ONERA and SONACA group, with Airbus as an industrial partner. In order to perform this assessment, an integral approach is followed, combining aerodynamic and structural studies, manufacturing, system design and integration (WIPS, Krueger flaps, suction system), wind-tunnel testing, small-scale demonstrators and a large-scale ground-based-demonstrator (GBD). These numerical and experimental studies are performed within a complete validation and verification process. In particular, this project aims at reaching a TRL4 (technology readiness level) on integrated component level for HLFC technology on transonic wings of midto long-haul transport aircraft. The present paper focuses on the evaluation of HLFC benefits solely on aerodynamic performance. The assessment presented in this paper was performed based on numerical simulations of realistic aircraft configurations at a cruise operating point, including engines with power-on boundary conditions. In particular, the design process and analysis presented in this paper concern the designed wing geometry of the GBD of the HLFC-WIN project. The extended transition prediction capability of the TAU Code for HLFC designs was recently developed in German Aeronautical Research Program LuFo In.Fly.Tec/ Applaus. The paper is structured as follows. At a first step, the aircraft configuration under investigation is presented, followed by a description of the methods used and the HLFC wing design process. A numerical aerodynamic performance assessment is then presented for the HLFC configurations designed by DLR within the HLFC-WIN project. In particular, comparisons are performed between an initial geometry and a further improved one, aiming at improving performance based on results of the far-field drag analysis of the initial geometry. In addition to free transition computations, a refined assessment is presented, accounting for turbulent wedges between the suction panel segments of the final HLFC system. Finally, an outlook on further improvement in terms of physical modelling for HLFC configurations is given, followed by conclusions.

DESCRIPTION OF THE AIRCRAFT CONFIGURATION & SCOPE OF GEOMETRY MODIFICATIONS

The studies carried out in this work use the Airbus XRF-1 research large passenger aircraft as a reference geometry [START_REF] Kroll | DLR project Digital-X: towards virtual aircraft design and flight testing based on high-fidelity methods[END_REF][START_REF] Carini | Towards industrial aero-structural aircraft optimization via coupled-adjoint derivatives[END_REF]. This geometry contains the fuselage, wing, nacelle, pylon, flap-track fairings and vertical tail plane. A perspective view of the aircraft is shown in Figure 1. CFD computations and far-field analyses consider only half of the aircraft geometry. The geometry corresponds to a target 1g flight shape. For the far-field drag analysis presented in this paper, the priority was put on the accurate aerodynamic performance analysis at cruise conditions. For this reason, with the exception of fully turbulent computations of the HLFC geometries, offdesign aerodynamic performance was not considered. Geometry modifications were restricted to the outer part of the wing, outboard of the leading edge kink and close to the engine position (colour-shaded region in Figure 1), i.e. for span position η ≥ 0.319. A more detailed description of the design requirements is given in section 4.1. At a first step, a HLFC variant of the XRF-1 aircraft was designed by Airbus. The aerodynamic design studies of DLR were then performed using this geometry as a starting point. The HLFC wing surface geometry selected for the HLFC-WIN ground base demonstrator is denoted GBD-DLR-2. A further design is the GBD-DLR-3 geometry, which improves the aerodynamic performance of the GBD-DLR-2 geometry. The initial geometry used in the design is the HLFC variant of the XRF-1 aircraft designed by Airbus, denoted as the Airbus 1g HLFC geometry. For a consistent far-field performance analysis among the different configurations, the Airbus 1g HLFC case will not be considered, since it has a smaller spanwise thickness distribution than the reference geometry and the new DLR designs.

DESIGN AND SIMULATION METHODS

The numerical simulations carried out in the course of this work can be classified into HLFC wing design and analysis tasks. The following sections describe the methodology and the numerical methods used in each case.

Design method

The DLR inverse transonic 3D design method was used to design the HLFC outer wing [START_REF] Streit | DLR transonic inverse design code, extensions and modifications to increase versatility and robustness[END_REF]. This allows the design of adapted wing sections using user-specified target pressure distributions. The inverse design method is an integrated module of the DLR FLOWer code [START_REF] Kroll | MEGAFLOW -Numerical Flow Simulation for Aircraft Design[END_REF] for block structured meshes. The used target pressure distributions are generated with the aim to increase laminar extent and/or reduce shock strength. In the case of HLFC target pressure distributions, also the shape of the pressure distribution in the nose suction region is important. A HLFC pressure distribution presents a suction peak within the suction region. The height and extension of this peak has to be chosen carefully, considering its influence on transition position and shock strength. Furthermore, the change of the suction peak with off-design conditions has to be considered. The CFD meshes used for the HLFC wing design use a reduced level of geometrical representation without flap track fairings and engine pylon. They comprise separate wing, body and engine meshes and CFD solutions are obtained using the chimera (overset mesh) technique.

Together with these simplifications, the inverse design method has the advantage that it is a robust, efficient and accurate CFD method, ideal for a large number of design iterations and extensive off-design analysis.

CFD analysis for complete configurations

For the final analysis of the designed HLFC configurations, the full configuration including flaptrack fairings and engine pylon is considered. For this purpose, the wing sections designed with the inverse design process are transferred into a CAD model of the complete aircraft. The CAD model forms the basis for the analysis CFD mesh, which was created using the mesh generator SOLAR [10]. With a resolution of about 30E6 mesh points, this mesh is much finer than the design meshes.

For the CFD simulations, DLR's flow solver TAU [7] was used. TAU solves the Reynolds-averaged Navier-Stokes equation (RANS) on a cell vertex, finite volume formulation. For the spatial discretization, a central Jameson scheme with matrix dissipation is applied. Time integration is performed with an implicit backward Euler scheme, using LU-SGS. Turbulence modelling was performed with the Spalart-Allmaras SA-negative model. TAU's iterative automatic transition prediction module [START_REF] Krimmelbein | Industria-lization of the Automatic Transition Prediction in the DLR TAU Code[END_REF] is employed to evaluate the local laminar/turbulent transition position for a user-defined number of spanwise stations. At each station, pressure distributions and geometrical data are extracted from the current CFD solution and passed to a differential boundary layer solver. Employing the local conical wing assumption, the solver calculates the local boundary layer profiles.

The transition location is predicted by means of local linear stability analysis and a 2-N factor transition criterion, calibrated for HLFC application. The new transition locations for all stations investigated form a transition polyline. The transition polyline is then passed back to the CFD solver to distinguish laminar and turbulent parts of the flow. The process of CFD calculation and transition prediction is repeated iteratively. Convergence is usually reached within six iterations. To allow for the treatment of HLFC wing sections, the basic transition prediction module was enhanced to read-in user specified suction distributions. Consideration of the suction distributions in the boundary layer solver then allows for transition prediction for HLFC configurations. Details on methods and codes involved in the transition prediction module are given in the following section.

Methods and tools employed for stability analysis and transition prediction

Despite the emergence of more sophisticated stability analysis methods, LST remains the preferred technical tool for stability analysis in computationally intensive design and analysis activities outlined here. Both the design process and the final analysis use the STABTOOL program suite [START_REF] Schrauf | LILO 2.1 -Users Guide and Tutorial[END_REF][START_REF] Schrauf | COCO -A Program to Compute Velocity and Temperature Profiles for Local and Nonlocal Stability Analysis of Compressible, Conical Boundary Layers with Suction[END_REF] for transition prediction based on local linear stability theory (LST).

For the stability analysis of 3D wing boundary layers, the assumption of spanwise locally conical flow conditions is introduced at first. This assumption allows a numerically efficient calculation of the laminar boundary layer profiles per wing section. The boundary layer code coco [START_REF] Schrauf | COCO -A Program to Compute Velocity and Temperature Profiles for Local and Nonlocal Stability Analysis of Compressible, Conical Boundary Layers with Suction[END_REF] was used to calculate compressible, conical flow boundary layer profiles.

Coco supports calculations with prescribed boundary layer suction, requiring a special pre-processor provided by G. Schrauf.

To calculate the growth rates of Tollmien-Schlichting (TS) and cross flow (CF) instability modes the local linear stability solver lilo was employed. To solve the LST eigenvalue problem, a fixed frequency and fixed propagation direction approach is used for TS waves, while a fixed frequency/fixed wavenumber approach is chosen for CF waves. TS mode evaluation is restricted to a propagation direction of i.e. along the group velocity direction. For cross flow, only stationary modes (f=0Hz) are considered. Separate N-factors for TS and CF modes are obtained by integrating the respective growth rates along the group velocity direction.

To predict the transition location, a 2-N factor transition criterion is used. The N-factor limit curve was adapted for HLFC design purposes. In addition, a ReΘ criterion is evaluated to check for possible attachment line transition.

DESIGN OF HLFC WING CONFIGURATIONS

In this section, the aerodynamic design of the wing surface of the ground-based demonstrator will be described. All results shown here correspond to the complete aircraft configuration. They are restricted to the cruise design point. The aerodynamic performance results for the designed geometries will be described separately in Chapter 6.

Design requirements

Aerodynamic requirements: Design flow conditions are: the cruise design point given in chapter 2. Off-design is considered in the range: 0.45 ≤CL ≤0.55, 0.81 ≤M∞≤ 0.85. For all flow conditions the active engine boundary conditions (fan intake and primary/secondary stream exhausts) provided by Airbus for the cruise point were used. For the cruise point, the design should not significantly alter the circulation distribution of the Airbus HLFC geometry. This reduces the influence of induced drag changes to overall aerodynamic performance. Geometry requirements: A detailed HLFC design of the outer wing ( ≥ 0.319) is performed. Thickness requirements are: maintain or increase front spar airfoil thickness and maximum airfoil thickness of the Airbus-HLFC geometry.

Transition position: For HLFC-WIN the laminar region is restricted to the upper side. For the shape design, a generic suction distribution was used. It was defined based on the suction distribution used by Airbus for its HLFC design. For consistency with the Airbus results, the transition position was obtained by DLR using incompressible stability analysis and flight incompressible critical NTS/NCF values [START_REF] Schrauf | Large Scale Laminar Flow Test Evaluated with Linear Stability Theory[END_REF]. However, it is important to mention that for this Mach number: a) a free flight calibration of critical N-factors is not available b) compressible and incompressible stability analysis may lead to designs with different geometry and performance potential [START_REF] Streit | NLF Potential of Laminar Transonic Long Range Aircraft[END_REF].

Design methodology and results

In the early stage of the design, CFD results were obtained for wing-body meshes and wing-body-engine meshes. Comparison with CFD results obtained for the analysis mesh, showed that: a) for the inverse design it was sufficient and necessary to include the engine in the design process, b) pylon and FTF's could be neglected since their influence was reduced to a local span region. The geometry of the GBD-DLR-2 design was obtained in 2 steps since in comparison to the XRF-1 geometry, the Airbus HLFC 1g geometry has a decreased thickness distribution in the outer wing, the first design step was to modify the thickness distribution. For that purpose, the lower side was changed in order to regain thickness. As shown in Figure 2 this leads to a GBD-DLR-2 pressure distribution with a decreased minimum pressure for the lower side. This shape modification was done without altering the upper wing surface pressure distribution and the local lift value. In comparison to the initial Airbus HLFC design, the maximum airfoil thickness was increased by t/c=1% in the outer wing for dimensionless span  > 0.4. In the second design step, the upper side was considered for the free transition case and target pressure distributions were generated in order to increase the laminar extent of the Airbus HLFC geometry. For that purpose, target pressure distributions were used in the design which did not change the shock strength, but changed the nose suction peak and the pressure gradient. Figure 2 shows the pressure distribution of the finally designed GBD-DLR-2 geometry using these target pressure distributions. In Figure 2 this is compared to results obtained for the Airbus HLFC 1g (free transition) and the XRF-1 (turbulent) cases. The leading edge region of the HLFC design airfoils generally exhibits lower leading edge radii and reduced curvature. As a result, this leads to a reduced pressure gradient in this region on the pressure side. Overall, both HLFC geometries exhibit a more regular acceleration of the flow up to roughly 30%-40% of the chord on the pressure side than the reference wing, a favourable effect for the preservation of a laminar region. On the suction side and at the outboard part of the wing, the pressure coefficient evolutions of HLFC wings are quite different from that on the reference configuration. The acceleration following the most forward weak recompression (which is overall weaker for the HLFC wings) is favourable for the delay of transition. In addition, the upstream recompression can limit the intensity of shockwaves despite the acceleration over an important portion of the chord. The reference turbulent wing overall shows an arrangement of multiple shock waves. HLFC wings on the other hand aim at extending the laminar region by presenting a more progressive acceleration, which leads to a single shock wave around 60% of the chord. This is clearly visible closer to the wing tip, at =0.87, where the reference geometry results in a first shock closer the mid-chord position, followed by a decompression and a second shock further downstream. HLFC geometries however result in a more progressive acceleration up to a single shock formed in an intermediate position with respect to the two shocks of the turbulent wing. Concerning the comparison between fully turbulent and laminar/turbulent computations on the HLFC wings, an important difference is observed at the shockwave location. Differences on the pressure coefficient are already observed at =0.40. Τhe shockwave position is moved further downstream in the laminar/turbulent cases, compared to the fully turbulent computation of each corresponding HLFC configuration. The GBD-DLR-2 design is also influenced by the flow properties of the fixed shape inner wing. As discussed above, the inner wing shows complex shock system. Unfortunately, the leading edge part of the -shock (see Figure 4) that originates from the inner turbulent wing (wing root section) partially restricts the extension of laminar boundary layer in the outer wing part (HLFC wing part). This is seen in the pressure distribution for sections =0.40 and =0.44 (see Figure 2), where the more upwind placed shock will lead to a transition. In addition to the -shock, Figure 3 and Figure 4 show an inner wing shock originating at the inboard pylon wing intersection. The transition position for the DLR-HLFC designs is given in Figure 4. Note that the GBD-DLR-2 geometry has extended the laminar extent of the Airbus HLFC geometry case by an additional 1.7% of the wing surface.

As will be shown in the far-field drag analysis in chapter 6, the GBD-DLR-2 configuration showed an increase of wave drag in comparison to the reference and the Airbus HLFC 1g geometries. This wave drag increase occurred around a region of span position n=0.6. This occurs despite the GBD-DLR-2 and the Airbus HLFC configuration showing no significant difference in shock strength in the pressure distributions. Therefore, an additional geometry was designed denoted GBD-DLR-3. Target pressure distributions were used for the GBD-DLR-3 case with a slightly larger recompression between the transition position and the shock. As shown in Figure 5 for the pressure distribution of wing section =0.65, the finally designed GBD-DLR-3 shows this property. Furthermore, the GBD-DLR-3 geometry was designed in such a way that the angle of attack for which the cruise design point lift value is achieved is closer to the angle of attack at which this occurs for the Airbus HLFC 1g geometry. Thus, differences to the latter case in induced drag and pressure distribution for inner wing and body should be reduced. In comparison to the GBD-DLR-2 geometry, the GBD-DLR-3 shows an improved aerodynamic performance (see chapter 6) and a slightly increased laminar boundary layer extent (see Figure 4). Results show that the circulation distributions for the designed geometries are very similar to the circulation distribution of the Airbus HLFC 1g configuration (results not shown here). The latter differs from the one of the XRF-1 geometry. Besides the HLFC designs described here, further designs which consider the inner wing and the tip region of the HLFC-WIN configuration were performed. They will be described in a future contribution which considers the design and analysis of the HLFC-WIN configurations in more detail.

FAR-FIELD DRAG ANALYSIS METHOD

Near-field drag coefficients are computed by performing an integration of forces on the aircraft skin. On the other hand, a far-field evaluation of drag components is performed through integrals in the flow field. This allows a phenomenological breakdown of drag into physical components, namely viscous, wave and induced drag. Furthermore, a far-field evaluation of drag is more accurate than its near-field counterpart, thus also allowing for a quantification of spurious drag production. Far-field drag analyses are performed using the ffd72 post-processing software of ONERA. The formulation and employed methodology are described in [START_REF] Van Der Vooren | Drag/thrust analysis of jet-propelled transonic transport aircraft; definition of physical drag components[END_REF][START_REF] Destarac | Far-Field / Near-Field Drag Balance and Applications of Drag Extraction in CFD, In: CFD-based Aircraft Drag Prediction and Reduction[END_REF]. The far-field method relies on an accurate physics-based definition of control volumes for the integration of viscous pressure, wave and induced drag. The aforementioned software and methodology have been extensively applied in the research and industry over the past decades. Near-field drag is defined as the sum of pressure and friction drag:

CD f CD p CD nf   (1)
Far-field drag on the other hand is defined as the sum of the viscous, wave and induced drag components:

CD i CD w CD v CD ff    (2)
where:

CD f CD vp CD v   (3)
Spurious drag production is inherent to numerical computations, and is a product of numerical approximation (e.g. mesh discretization, artificial dissipation). A quantification of spurious drag is made possible through the difference of the two drag calculations:

CD ff CD nf CD sp   (4)
Finally, the far-field drag extraction method provides the drag balance:

CD sp CD i CD w CD v CD f CD p      ( 5
)
Additional variables and performance measures are computed in the case of motorized configurations (see e.g. [START_REF] Van Der Vooren | Drag/thrust analysis of jet-propelled transonic transport aircraft; definition of physical drag components[END_REF]), in particular specific drag/thrust bookkeeping methods. A view of integration volumes is shown in Figure 6.

AERODYNAMIC PERFORMANCE ANALYSIS

This section presents the aerodynamic performance analysis of the aircraft configurations designed by DLR and described in section 4. The geometries considered in this study are the original HLFC geometry designed by DLR (denoted as GBD-DLR-2) and its improved counterpart (denoted as GBD-DLR-3) which was designed by DLR taking into account results of the farfield drag analysis of the original GBD-DLR-2 version.

Analysis based on a free transition approach

In laminar/turbulent computations, suction is taken into account in transition prediction but this only influences the laminar region extent, as the boundary conditions employed do not include active suction across the surface. Still, the laminarization of the boundary layer naturally has a significant effect on skin friction. Figure 7 shows a skin friction distribution at a 60% wingspan position for the five different cases. The reduction of skin friction for the laminar region of the flow is clearly visible, and extends across an important part of the chord at this spanwise position. As expected, this advantage no longer exists in fully turbulent flow over the same geometries. Between the two HLFC wings, the GBD-DLR-3 shows a slightly lower skin friction at the end of the laminar region. Global lift and near-field/far-field drag coefficient increments of the HLFC configurations are presented in Table 1. Increments are computed with respect to the corresponding coefficient value of the reference XRF-1 aircraft. Note also that computations were performed at approximately the same lift coefficient, which results in a different angle of attack. At a first step, the two HLFC geometries are compared against the reference XRF-1 in fully turbulent conditions. This corresponds to the worst-case scenario in terms of laminar area extent, but remains important for the overall aircraft performance assessment as such conditions may occur in a number of flight events and conditions. Concerning the fully turbulent computations, friction drag is similar for the three configurations (difference less than 0.5%), the total near-field drag difference being thus mainly attributed to the skin pressure drag component. The difference of pressure drag with respect to the reference aircraft case is important for the two HLFC geometries at fully turbulent conditions. A far-field drag decomposition of the total drag force highlights that the main contribution to this is a notable increase of wave drag. This pressure drag penalty in turbulent conditions is reduced on the GBD-DLR-3, because of geometry modifications aiming at reducing the wave drag identified in the far-field drag analysis of the GBD-DLR-2 geometry. Still, relative variations are generally increased, as absolute wave drag levels are low on such a reference transonic wing. An increase of induced drag is also observed, albeit at a lower level compared to that in terms of wave drag. It is interesting to note that, even in fully turbulent conditions, the HLFC geometries yield a reduced viscous pressure drag with respect to the reference XRF-1 aircraft. The overall drag of the aircraft evaluated by the far-field analysis is however higher than the reference. On the other hand, the laminar/turbulent computations with a free transition approach indicate that both geometries exhibit an improved overall performance with respect to the XRF-1 aircraft, taking into account the laminarization introduced by the geometry modifications and the HLFC system. In terms of nearfield values, drag reduction is obtained both in terms of pressure and friction drag. Whereas the laminarization benefit is often attributed to the friction drag component only, these results highlight that an important benefit is also obtained in terms of viscous pressure drag. The absolute increments for these two components are of the same order of magnitude. This benefit is found to be even more important in the improved GBD-DLR-3 configuration. The wave drag increase compared to the reference aircraft remains present in laminar/turbulent computations, but at a lower level compared to fully turbulent conditions. Furthermore, an additional benefit is obtained for both HLFC geometries in terms of induced drag. The improved aerodynamic performance of the GBD-DLR-3 is overall apparent, as its total farfield drag benefit with respect to the reference is notably higher than that of the GBD-DLR-2. It is also important to note that the modifications introduced for the reduction of the wave drag penalty do not degrade aerodynamic performance in terms of any individual drag component with respect to the GBD-DLR-2.

The computed drag coefficient increments are consistent with the modifications introduced in each of the HLFC geometries. Finally, the absolute value of spurious drag was found to remain at a similar level between the different cases. This was found to be low, which is the reason for the large proportional differences in the tabulated data. The above results highlight the importance of an accurate far-field drag assessment. First, an appropriate far-field drag analysis can provide a considerably increased precision compared to a near-field evaluation of drag, and thus a more accurate prediction of the drag benefit due to laminarization. On the other hand, the phenomenological decomposition is essential to the improvement of the aerodynamic design and the accurate understanding of the physical mechanisms involved. Looking only into near-field drag components one could indeed consider that the benefit of laminarization is mainly obtained by a reduction of friction drag. As expected, a far-field breakdown shows that the benefit of laminarization is in fact distributed between the friction and viscous pressure drag components. In other words, when working solely based on the pressure drag value, the viscous pressure drag benefit can be masked by a wave drag or induced drag contribution. The far-field drag analysis thus indicates the possibility of separately addressing the issue of the wave drag penalty in this case. This importance is demonstrated by the improved performance of the GBD-DLR-3 geometry, compared to the GBD-DLR-2. An improvement is observed both in terms of a larger drag benefit due to laminarization and in a reduction of the drag penalty if the wing is operated in fully turbulent conditions. The identification of areas responsible for performance gains, or losses, is thus important in order to efficiently guide aerodynamic design. Such an indication can be given by the approximate spanwise distribution of aerodynamic coefficients, presented in Figure 8. The lift coefficient distribution shows a slight transfer of load towards the inboard part of the wing for the two HLFC designs. This is more apparent on fully turbulent conditions and is mitigated on the laminar/turbulent computations. Minor modifications are also noted on the friction and viscous pressure drag components near the engine position. Furthermore, in both distributions, the benefit of laminarity is clear on the outboard part of the wing. Finally, the wave drag increase identified in Table 1 is clearly apparent in the spanwise distribution of wave drag, and is particularly localized on the outboard part of the wing. Trends are overall similar between the GBD-DLR-2 and GBD-DLR-3 configurations, whereas differences are consistent with the global drag coefficient increments presented in Table 1.

Adapted transition prediction taking into account technology effects

The following step towards a more reliable evaluation of the HLFC aerodynamic performance benefit is a more accurate prediction of the laminar region. For this reason, additional computations were performed using a refined definition of the laminar region, accounting for technology effects consistent with the HLFC systems designed in the HLFC-WIN project. Regarding laminar/turbulent transition, the suction magnitude was reduced without compromising the laminar region extent and the suction area was adjusted to the spanwise variation in the front spar location. Turbulent wedges were introduced at segment boundaries, where a 15° opening angle assumption was made. Finally, the transition line from stability analysis results in nine sections across all four HLFC segments was superimposed with the aforementioned turbulent wedges and lead to a more realistic laminar area extent. It should also be noted that this limits the inboard extent of the laminar region with respect to the previous free transition computations. This is because free-transition computations were performed before the exact definition of the panel extent in the finalized HLFC system.

The laminar region modification in this updated transition computation can be seen in Figure 9, which shows a comparison of skin friction and the transition line for the two types of laminar/turbulent computations. The same figure also shows a comparison of the transition line on the wing upper surface for all the laminar/turbulent computations presented above. This comparison gives a clearer view of the laminar region reduction when considering a limited spanwise extent of the laminar region, as well as introducing turbulent wedges between segment boundaries. As shown earlier (see Figure 4), the geometry modifications introduced in the GBD-DLR-3 geometry are also confirmed to have a minimal but beneficial effect on the extent of the laminar region, compared to the DLR-2. The influence of the adapted transition prediction on the aerodynamic performance is summarized in Table 2. Most near-field and far-field coefficients are penalized by the reduced laminar region due to increased viscous drag. Induced drag also shows a minor increase, whereas wave drag is slightly reduced with respect to the free transition computation. The overall penalty when considering a more realistic laminar region compared to the computation with free transition is shown to be reduction of the drag benefit by 0.29% with respect to the total far-field drag of the reference XRF-1 aircraft. In terms of the spanwise distribution of drag components, modifications due to the adaptation of the transition prediction were found to be small, but consistent with the overall values presented in Table 2.

Further improvement to physical modelling

The presented simulation methodology for HLFC configurations considers the effect of boundary layer suction only in the context of transition prediction. In the RANS simulation, however, only laminar and turbulent flow regions are distinguished and the effect of suction on the boundary layer development is not accounted for. Since the area of the suction surfaces is relatively small in relation to the total wetted area, it is assumed that the influence of this simplification on the total resistance is negligible in a first approximation. The inclusion of boundary layer suction into RANS simulations and a quantification on HLFC performance is the subject of ongoing work at DLR. The results will be presented in an upcoming paper.

CONCLUSIONS

This paper presents an aerodynamic performance assessment of realistic HLFC wing configurations, designed by DLR within the Clean Sky 2 HLFC-WIN project. Aerodynamic performance has been assessed using the ffd72 far-field drag analysis software of ONERA, based on fully turbulent and laminar/turbulent CFD computations at a cruise operating point including engines with power-on boundary conditions. Two HLFC wing geometries have been analysed, both designed by DLR based on a HLFC variant of the XRF-1 research geometry, designed by Airbus. The HLFC region is restricted to the outer wing. The first corresponds to the original geometry for the large-scale ground-based demonstrator of the HLFC-WIN project. The second is a further improved version of this geometry aiming at improving its aerodynamic performance, considering results of the far-field drag analysis of the original version. Far-field drag analysis shows that HLFC designs introduce an important drag benefit, both in terms of the friction and viscous pressure drag components. A wave drag penalty, inherent to the laminarization of the outer wing upper surface, is also identified. The estimated benefit of the improved HLFC geometry with respect to the original XRF-1 aircraft is estimated to be in the order of 3.71% at cruise conditions. This benefit is however slightly reduced (reduction in the order of 0.29% for the original HLFC geometry) when considering an adapted transition approach, which takes into account technology integration effects (exact suction panel placement, and turbulent wedges between segments). Both approaches are in good agreement, but the latter is essential at a second design step, where the drag prediction difference between the two approaches lies beyond the error margin for a realiable technology evaluation.

Geometry

These observations highlight that the physical insight and quantification of the influence of the different mechanisms behind drag reduction, as well as the use of numerical methods of adequate fidelity, are essential steps towards the design and implementation of HLFC technology in future passenger aircraft.

Figure 1 .

 1 Figure 1. Perspective view of the XRF-1 aircraft highlighting the wing region considered for geometry modifications and HLFC application. The nominal flight conditions at cruise are the following: 83 . 0   M , 5 . 0  C L , 6 10 47 Re   and a flight altitude of 36,000 ft.For the far-field drag analysis presented in this paper, the priority was put on the accurate aerodynamic performance analysis at cruise conditions. For this reason, with the exception of fully turbulent computations of the HLFC geometries, offdesign aerodynamic performance was not considered. Geometry modifications were restricted to the outer part of the wing, outboard of the leading edge kink and close to the engine position (colour-shaded region in Figure1), i.e. for span position η ≥ 0.319. A more detailed description of the design requirements is given in section 4.1. At a first step, a HLFC variant of the XRF-1 aircraft was designed by Airbus. The aerodynamic design studies of DLR were then performed using this geometry as a starting point. The HLFC wing surface geometry selected for the HLFC-WIN ground base demonstrator is denoted GBD-DLR-2. A further design is the GBD-DLR-3 geometry, which improves the aerodynamic performance of the GBD-DLR-2 geometry. The initial geometry used in the design is the HLFC variant of the XRF-1 aircraft designed by Airbus, denoted as the Airbus 1g HLFC geometry. For a consistent far-field performance analysis among the different configurations, the Airbus 1g HLFC case will not be considered, since it has a smaller spanwise thickness distribution than the reference geometry and the new DLR designs.

Figure 2 .

 2 Figure 2. Pressure distribution for outer wing sections. Results for geometries with free transition for Airbus HLFC 1g (black), GBD-DLR-2 (green) and GBD-DLR-3 (red) and for reference XRF-1 geometry (blue) with turbulent boundary layer.

Figure 3 .

 3 Figure 3. Upper surface pressure distribution contours for the GBD-DLR-2 geometry.

Figure 4 .

 4 Figure 4. Free transition position for HLFC outer wing and shock position.

Figure 5 .

 5 Figure 5. Comparison of pressure distribution in the shock region for =0.65. Solutions for turbulent boundary layer with analysis mesh

Figure 6 .

 6 Figure 6. Perspective view of the integration volumes for viscous pressure (grey) and wave drag (black).

Figure 7 .

 7 Figure 7. Friction coefficient across the upper wing surface at η=0.60 for the reference aircraft and its HLFC variants in fully turbulent and laminar / turbulent (free transition) conditions.

Figure 8 :

 8 Figure 8: Approximate spanwise distribution of near-field and far-field coefficient increments with respect to the reference XRF-1 aircraft, computed with the ffd72 software.

Figure 9 .

 9 Figure 9. Magnitude of skin friction coefficient for the HLFC GBD-DLR-2 variant with free or adapted transition (top/middle) and transition line comparison for the laminar/turbulent computations (bottom).

Table 2 :

 2 Influence of a refined transition prediction on near-field and far-field drag coefficient increments (w.r.t. the reference XRF-1 aircraft) computed with the ffd72 software for the two HLFC geometries at cruise conditions. The laminar surface percentage is computed with respect to the wing surface, excluding the pylon and FTFs. HLFC GBD-DLR-2 (free transition) HLFC GBD-DLR-2 (adapted transition)

Table 1 : Near-field and far-field drag coefficient increments (w.r.t. the reference XRF-1 aircraft) computed with the ffd72 software for the two HLFC geometries in fully turbulent and laminar/turbulent conditions at cruise. The laminar surface percentage is computed with respect to the wing surface, excluding the pylon and FTFs.

 1

											Laminar
	Geometry	Transition type AoA(°) ΔCDp	ΔCDf	ΔCDvp	ΔCDw	ΔCDi	ΔCDsp	ΔCDff	surface
											[%]
	XRF-1	Turbulent	2.634	-	-	-	-	-	-	-	-
	GBD-DLR-2	Turbulent	2.781	+2.05% -0.15%	-1.79% +99.77% +0.85% -20.90% +2.27%	-
		Free transition	2.682	-1.13% -4.69% -16.92% +85.29% -0.20% -10.17% -2.86%	14.26 %
	GBD-DLR-3	Turbulent	2.745	+1.55% -0.35%	-1.64% +83.91% +0.33% -17.51% +1.59%	-
		Free transition	2.637	-1.85% -5.01% -19.19% +70.11% -0.52% -19.21% -3.71%	14.42 %

ACKNOWLEDGEMENTS

This work was funded by the European Union within the frame of the HLFC-WIN project. This project is part of the Clean Sky 2 Joint Undertaking under the European Union's Horizon 2020 research and innovation programme (Grant agreements CS2-GAM-2018-LPA-AMD-807097-38 and CS2-GAM-2020-LPA-AMD-945583-11). The HLFC code development was funded by German Federal Ministry for Economic Affairs and Energy within the German Aeronautical Research Program LuFo V-3/2018-2022 In.Fly.Tec/Aplaus under grant agreement no 20A1705C.


domain: spi phys

text:
Quentin Bennehard
email: quentin.bennehard@onera.fr

AERODYNAMIC SHAPE OPTIMIZATION OF A SHORT MEDIUM RANGE BLENDED WING BODY AIRCRAFT

This work presents the aerodynamic shape design and optimization of a short medium range blended wing body configuration. It relies on previous activities performed at ONERA that led to a fuel efficient planform obtained from a dedicated Overall Aircraft Design process. Starting from this planform, the airfoil shape and the twist law of the glider configuration are optimized. A new optimization chain allows automatizing unstructured mesh generation on a parametric geometry. Using this new chain based on a parametric geometry and unstructured Euler simulations, gradient-based optimizations are done using a finite difference approach. RANS evaluations are then conducted on the optimized configuration to assess the aerodynamic performance benefits in terms of lift over drag ratio (one count improvement, while satisfying the trim condition). Finally, the first design iterations to integrate modern turbo-fan engines using boundary layer ingestion are presented.

INTRODUCTION 1.Context

While air traffic is expected to continue increasing after the end of the Covid crisis, massive gains are necessary for the aeronautics industry to meet the targets set to limit climate change. In this light, optimizing existing aircraft configurations could not be sufficient anymore, and exploring new configurations along with new propulsion technologies are now a necessity. The Blended Wing Body (BWB) concept is the focus of the current paper. This aircraft layout has already been widely studied because it offers great aerodynamic benefits compared to the traditional tube and wing configuration. While new engine technologies are explored (hydrogen or hybrid electric propulsion), the greater interior volume offered by the BWB geometry is also a key advantage. The present work focuses on the conception and optimization of a BWB aircraft suitable for short medium range mission, segment serviced today mainly by the A320 and the B737.

The literature is extensive on the BWB geometry for long-range mission, as it is on this type of mission that the better aerodynamic performance during cruise translates in a greater fuel economy. Liebeck [START_REF] Liebeck | Design of the blended wing body subsonic transport[END_REF] conducted a comprehensive design review of a 450 pax BWB. He demonstrates that the aerodynamic gain mainly comes from a 33% reduction of the wetted surface compared to the traditional tube and wing configuration. His work also deducts a fuel burn per seat reduction of 32%, coming from the great synergy offered by the BWB configuration. Indeed, one wing body ensures payload housing, control, structural and aerodynamic functions. Liebeck also highlights that the main aerodynamic challenge of this configuration lies in the central body. In this part, the geometry is supposed to generate lift at a moderate angle of attack to keep the passengers deck levelled. These requirements, suggest using airfoils having a positive aft camber. However, a positive aft camber airfoil generates a pitch down moment. Without a tail stabilizer and trailing edge surface deflections, it is then challenging to trim the configuration during cruise. Therefore, considering these constraints and the great number of potential geometric variables, the conception of a BWB shape is a good candidate to apply aerodynamic shape optimization. Great progresses in this field have been made in the last decades. Lyu and Martins tackle such a problem in [START_REF] Lyu | Aerodynamic design optimization studies of a blended-wing-body aircraft[END_REF], on a large BWB. They use adjoint RANS based optimization and they achieve to obtain a configuration with a lift over drag ratio (L/D) of 21.9. Meheut and al. [START_REF] Meheut | Aerodynamic shape optimizations of a blended wing body configuration for several wing planforms[END_REF] used the same kind of setups to optimize a BWB subjects to low speed rotational take off constraint.

In his work, Liebeck also suggests that the BWB configurations could be scaled laterally to derive smaller configuration from a large one. Short-medium range trip represents the majority of the air traffic, and are responsible for 40% of its CO2 emissions [START_REF] Wilkerson | Analysis of emission data from global commercial aviation: 2004 and 2006[END_REF], which points out the need for a technological shift on this segment as well. On this type of aircraft, considering a shorter chord length and the fact that the mean human being height is constant, the relative thickness of the central airfoils needs to be increased. In theory, this deteriorates the aerodynamic performance of the aircraft. It is then difficult for a thick airfoil to achieve a significant lift at low drag while keeping a low angle of attack. This fact has been highlighted in the work of Kanazaki et al [START_REF] Kanazaki | Challenge of design exploration for small blended wing body using unstructured flow solver[END_REF].

Previous work at ONERA

The current work is based on a multi-disciplinary optimization (MDO) of a short-medium range BWB carried out at ONERA in the frame of the European project NACOR. This MDO uses an Overall Aircraft Design (OAD) process developed since 2015 at ONERA [START_REF] Gauvrit-Ledogar | Multidisciplinary overall aircraft design process dedicated to blended wing body configurations[END_REF]. It is constituted by six disciplinary modules: Aircraft Geometry, Propulsion, Aerodynamics, Structure & Weight, Mission & Performance and Handling Qualities. A description of the low level Aerodynamics Module is given in [START_REF] Moens | A L0 aerodynamic model for aircraft multidisciplinary design and optimization process[END_REF].

The top-level aircraft requirements (TLAR) considered for this MDO are the following: -Payload of 150 pax with 90 kg per pax -Range of 5100 km -Mach 0.78

The process allows exploring a great variety of BWB planforms. A MDO problem is defined with objective to minimize the mission fuel weight, the main constraints being the take-off distance, the climb duration and the handling qualities. The results of the MDO can be seen Figure 1 and Table 1. The resulting configuration has been named SMILE (Small -Medium range Integrated Light and Efficient aircraft). Its planform constitutes the starting point of the work presented here after. The OAD process also gives the size and arrangement of the pressurized zone, as seen Figure 2. The objective of this work is hence to obtain an efficient 3D aerodynamic shape, starting from the planform obtained by the MDO. Engine integration is first not taken into account, as the objective is first to get an efficient glider shape.

This article is organized as follows. First the optimization chain and the baseline geometry are presented. In section 3, the different optimization problems treated are presented. Section 4 discusses the optimization results. Section 5 briefly tackles the challenges observed in engine integration with boundary layer ingestion, followed by the conclusions.

DESIGN METHODOLOGY

Optimization chain

The optimization chain used in this work is an adaptation of the Computational Aircraft Prototype Syntheses (CAPS) chain developed by the MIT [START_REF] Haimes | Multi-fidelity geometrycentric multi-disciplinary analysis for design[END_REF]. Its components are described Figure 3. Usage of this chain is new at ONERA, especially in an optimization context.

First, the chain includes a fully parametric Computer Aided Design (CAD) module: Engineering Sketch Pad (ESP) [START_REF] Haimes | The engineering sketch pad: A solid-modeling, featurebased, web-enabled system for building parametric geometry[END_REF]. In this work, ESP was retained because it creates a completely parametric geometry. The parameters used to define the shape can then be directly used as design variables of an optimization problem. Furthermore, as it possesses a CAD engine based on Open Cascade, ESP can easily compute the volumes and intersections of complex forms, allowing to easily handle the internal volume constraint of the BWB.

Second, an inviscid or viscous unstructured mesh can be generated automatically on the geometry, leveraging Pointwise scripts [START_REF] Karman | Automatic unstructured mesh generation with geometry attribution[END_REF]. Viscous layers are generated using the T-Rex capability of Pointwise.

On the obtained mesh, the open source flow solver SU2 [START_REF] Economon | SU2: An open-source suite for multiphysics simulation and design[END_REF] is then used to solve the aerodynamic field around the shape. SU2 can solve the Euler or RANS equations.

For the last ones, the Spalart-Allmaras turbulence model is used. The chain has also be made compatible with the elsA code (ONERA-Airbus-Safran property) [START_REF] Cambier | The Onera elsA CFD Software: Input from Research and Feedback from Industry[END_REF], using its unstructured capabilities.

Lastly, the in-house ONERA post process tool Far Field Drag (FFD) [START_REF] Destarac | Drag extraction from numerical solutions to the equations of fluid dynamics: the far-field philosophy[END_REF] is employed to extract and decompose the aerodynamic coefficients of the shape. These four steps are run automatically thanks to Python scripts, adapted from the CAPS chain.

An optimizer is coupled to this automatic process, in order to be able to solve the optimization problem. The choice was to use the Dakota Toolbox and the gradient based optimizer DOT MMFD (Modified Method of Feasible Directions) [START_REF] Vanderplaats | DOT user's manual[END_REF]. A gradient-free optimizer is more capable of finding the best minimum in a case of a problem with multiple local minima. However, the high number of functions calls made the use of this type of optimizer infeasible in this case where expensive CFD codes and large number of design variables are used. Rapid convergence of gradient based methods are the main motivation behind their usage, especially when they are combined with heavy CFD codes.

Even if SU2 and elsA have an ajdoint capability, the adjoint information is not fully propagated yet through the mesh and CAD processes. In this case, gradient values were then obtained via finite difference computations. The optimization chain can either be used by solving the Euler flow equations or the RANS ones, at the only cost of computer loads increase.

Figure 3. Overview of the optimization chain From the given planform presented in Figure 1, a tridimensional model was created. The geometry is defined by six sections, named p0, p01, p1, p2, p21 and p3, as seen on Figure 4. On the three central sections, modified four digit NACA airfoils were used. These airfoils were modified "by-hand", to accommodate the internal pressurized volume. On the outer wing, supercritical airfoils were used, re-using airfoils designed for the ONERA NOVA concept [START_REF] Wiart | Development of NOVA aircraft configurations for large engine integration studies[END_REF]. The central part of the wing is then defined by a cubic interpolation between the sections p0 to p2, preserving a C2 continuity at the intermediate sections. The rounded nose was obtained by imposing a normal tangent at the symmetry plane. The outer wing is defined by a linear interpolation between the last three sections p2 to p3. A two degrees dihedral angle is added on the outer wing to improve dynamic stability. In addition, winglets are added at the wing tip. They are quite large with a height of 2.5 m, as they contain rudders (not modelled in the CFD simulations) helping the lateral control of the aircraft.

Baseline geometry

The main geometric parameters are given in Table 2.

Since no planform modification will be done, they remain constant for the optimized designs. The centre of gravity is referenced by its position along the longitudinal axis, the origin being at the aircraft nose. The twist law of the baseline was determined iteratively to obtain a configuration achieving the lift needed without a high wave drag penalty. However, in this first manual design loop, it was not possible to find a configuration that achieves the trim condition at the cruise point. The optimization problem considered here after will mainly have to improve this point.

Mesh convergence study

A mesh convergence study has been performed to select the best parameters used in the automatic mesh generation process, and to validate the results given by the flow solver. The study has been made on the baseline BWB shape, SU2 solving the RANS equations. The aerodynamic conditions are presented Table 3. They are representative of the cruise segment, as computed by the OAD process. All the meshes are generated with a first cell height so that y+≈1 on the BWB surface. Figure 6 shows an overview of a mesh having 37 millions of elements. This is the mesh that has been retained for the RANS computations presented in Section 4.

OPTIMIZATION PROBLEM

Geometric parametrization

In order to obtain an efficient form, the BWB geometry has been parameterized, to allow its modification and its complete definition by a finite set of parameters. A large part of these parameters are fixed by the planform. The planform was optimized by the OAD process; it is then considered as optimal, explaining why it is unchanged in this work. As seen in Figure 4, the BWB is defined using six sections. The choice was made to parameterize the airfoils using the Class Shape Transform (CST) parameterization [START_REF] Kulfan | Universal parametric geometry representation method[END_REF]. This method uses Bernstein polynomials weighted by a shape function to define completely the upper and lower side of the airfoil.

Because of the use of finite differentiation and to limit the computational cost, each airfoil is parameterized using only eight CST coefficients (four on each side). The other parameters not fixed by the planform are the twist angles on each section. Finally, the winglet geometry is kept fixed, its shape being defined by lofting the last airfoil "p3" (see Figure 4). The angle of attack is also left free to respect the lift coefficient constraint.

In total, the optimization problem can use 55 design variables to modify the aircraft shape: -6*8 = 48 for the airfoils shapes -6 twist angles -1 angle of attack

The main geometric constraint is represented in Figure 4 by the internal pressurized volume in red. The dimensions of this volume has been fixed by the OAD process. Some margins have been added to account for the aircraft structure, allowing using this volume directly as a geometric constraint. ESP allows computing the intersectional volume between the exterior shape and the internal one. This value in then extracted to constrain the optimization process. On the outer wing section, only a minimum relative thickness constraint is used, on the three sections considered.

Optimizations

The optimization aiming at improving the aerodynamic efficiency, the objective is naturally to minimize the aircraft drag coefficient (CD), while maintaining a sufficient lift level. In our case, as the OAD process gives the aircraft mass balance and reference surface, it fixes the cruise lift coefficient (CL) at a value of 0.273. The same process also fixes the cruise aerodynamic conditions (see Table 3).

In addition to the lift constraint, as a BWB aircraft does not have any tail stabilizer, a trim constraint is thus added. To do so, at the lift value considered, the longitudinal moment coefficient computed at the centre of gravity (𝐶𝑚 𝑐𝑜𝑔 ) should be zero. This ensures that the centre of pressure coincides with the centre of gravity, implying that the aircraft is trimmed during cruise without the need to deflect any control surface, which would cause a drag penalty. The position of the centre of gravity is considered constant, as no modification of the planform is made during the optimization. For the optimization process, the position of the centre of gravity is a mean positon during cruise, as computed by the OAD process. This process also provides the centre of gravity for various aircraft load, notably zero fuel weight (ZFW) and with max fuel weight and max payload weight (MFW). The trim conditions were re-computed a posteriori for these two other centre of gravity positions.

Static margin has been computed and found positive around 2% of the mean aerodynamic chord for the baseline configuration. Since no modification of the planform is done during the optimization process, no constraint is applied on this criterion. A posteriori evaluation is made, to check the stability of the optimized design.

As highlighted in previous works [START_REF] Liebeck | Design of the blended wing body subsonic transport[END_REF][2], the challenge posed by the BWB configuration manly lies in the central body. The need for a moderate angle of attack is implemented in the optimization problem by bounding the angle of attack variable.

Only the cruise conditions were considered (single point optimization). A low-speed evaluation of the current design should be added to the presented work, notably to check the take-off rotation criteria, as mentioned in [START_REF] Meheut | Aerodynamic shape optimizations of a blended wing body configuration for several wing planforms[END_REF].

As the number of design variables is quite important and since finite differences are used to evaluation the sensitivities, three optimizations have been run successively, optimizing only subsets of the design variables at each step. The choice of the parameters and constraints on each optimization has been made after analysing carefully each resulting design, and identifying which areas could be improved.

The three optimization problems are described below:

GLIDER OPTIMIZATION

The results of the three optimizations described above are presented in Figure 8. Only the angles of attack were modified before starting the next optimization step, to start from a point where the lift constraint was satisfied.

It can be seen that the first optimization, where the trim constraint was not active, allowed to improve the L/D ratio by almost one count. Then, the next two optimizations achieve the reduction of the moment coefficient. Figure 10 presents the polar comparison of the two configurations, evaluated with RANS computations. The bottom panel of Figure 10 shows the moment coefficient computed for three different positions of the centre of gravity. Slopes of these curves remain negative, meaning a positive static margin, which ensures a stable aircraft. It can be seen that the trim condition is not totally satisfied for the mean position of the centre of gravity (solid line), the moment coefficient having a slight negative value. Further optimizations have been performed to improve this point, but they have only be realized at the price of worst L/D ratio or higher cruise angle of attack. Since the trim condition varies quite a lot with the variation of the aircraft weight, which induces a displacement of the centre of gravity by approximatively one meter, the obtained configuration was dubbed satisfying.

On the top panel of Figure 10, it can be seen that the wave drag has been reduced by two drag counts at the cruise lift coefficient. This can be also seen in Figure 7, where high-pressure zones are reduced on the outer wing of the optimized design.

ENGINE INTEGRATION WITH BLI

The parametric modeller ESP allows exploring a great variety of design, when dealing with engine integration. The concept considered here is to integrate a high by pass ratio turbo-fan on the top of the BWB aircraft. In this position, the engines are supposed to ingest a part of the boundary layer that develops on the aircraft upper side. Boundary layer ingestion (BLI) has the capacity to provide additional fuel burn reduction compared to conventional engine installation. This was demonstrated experimentally and by Carrier et al. in [START_REF] Carrier | Numerical and experimental aerodynamic investigations of boundary layer ingestion for improving propulsion efficiency of future air transport[END_REF]. In a BWB context Kawai et al. [START_REF] Kawai | Blended Wing Body (BWB) Boundary Layer Ingestion (BLI) Inlet Configuration and System Studies[END_REF] have shown a potential fuel burn reduction between 5 to 10 percent. Literature on BLI installation on a BWB is quite scarce, the main works having been done by NASA and Boeing in the early 2000s, including wind tunnel testing [START_REF] Carter | Designing and testing a blended wing body with boundary-layer ingestion nacelles[END_REF]. BLI allows re-energizing the boundary layers developed on the fuselage, improving the overall energy balance of the aircraft. Additional benefits lay in the reduction of the nacelle wet surface and a weight reduction due to the absence of pylons compared to a podded integration. In the current case, the fact that a small BWB aircraft is considered, geometric constrains (mainly the internal pressurized volume) are even more pregnant than on a long range BWB.

To obtain the results presented here after, the same computational chain presented part 2.1 was used. Engines are modelled by using inlet boundary conditions in SU2, specifying density and velocity vector at the inlet and outlet of the engine. These values are obtained by using a simple thermodynamic model of a double flux turbo-fan, considering a fan pressure ratio of 1.3, representative of a modern Ultra High ByPass Ratio engine. Thrust is estimated to equilibrate the drag of a complete aircraft with an estimated L/D of 20.5. With this value, the thermodynamic model gives us the fan diameter, the pressures and mass flows, with the hypothesis of an adapted nozzle. RANS computations are again conducted, with the Spallart-Allmaras turbulence model. The boundary conditions used do not allow to account for BLI benefits (the boundary layer is lost inside the engine block). In future work, actuator disk or body force methodologies will be used to model the engine, in order to precisely quantify the effect of the BLI on the configuration. However, the current setup allows quickly spotting detached zones mainly at the inlet, the nacelle sides, and the trailing edge junction. Concerning the inlet, an S-duct shape has been employed to realize the junction between the BWB upper side and the fan face.

On the trailing edge junction, a large separation bubble can appear if the engine nozzle is significantly higher than the BWB trailing edge. This was particularly highlighted by Kim and Liou in [START_REF] Kim | Optimal Inlet Shape Design of N2B Hybrid Wing Body Configuration[END_REF] (Figure 11). This detached zone increases the overall pressure drag, and should be avoided.

Figure 11. Separation bubble on the trailing edge region, from [START_REF] Kim | Optimal Inlet Shape Design of N2B Hybrid Wing Body Configuration[END_REF] Figure 12 presents four different designs considered. On designs 1 and 2, the trailing edge issue has been taken into account, and the engine is positioned so that the nozzle coincides with the BWB trailing edge. Due to the simplified capabilities of ESP, Designs 1, 2 and 3 are modelled by making an union between a circular nacelle and the BWB body. An S-shape tube is then subtracted from the geometry to realize the S-duct entrance. Design 4 is a try to use a non-circular nacelle, which could prevent detaching zones appearing on the acute side angles formed between the nacelle and the upper side of the BWB.

Figure 13 shows the Mach field in a spanwise slice containing the fan. A detached area in the S-duct appears in Design 1. Design 2 solves this issue, with a much longer S-duct. It must be stressed that the S-duct of these two designs intersects the internal pressurized volume considered in Figure 4, which violates the geometry constraint. Complying with this constraint, imply to reduce the S-duct length. Then, there is only two solutions to prevent from an abrupt slope: either move the engine upward (which would create the trailing edge issue of Figure 11) or move the engine backward. The last solution is retained in Design 3. Because Design 3 moves the engines to the rear, the length of the acute angle is reduced, creating only a small separated bubble, even if the geometry has no fillet. Thus, Design 3 presents several advantages compared to the two others, regarding the aerodynamic performance. However, it must be stressed that the structural integration of this design appears unfavourable (see Figure 13), compared to Designs 1 and 2. A fourth different configuration (Design 4) is currently under conception and evaluation.

CONCLUSION

This study demonstrates the viability of the Blend Wing Body concept for a short medium range aircraft. An aerodynamic glider with a lift over drag ratio of 22 was designed, which places it at the forefront of aerodynamic efficiency on this aircraft segment. This study also demonstrated the capabilities of the new ONERA design and optimization chain, based on the geometric modeller ESP. The chain demonstrated good performance and robustness on this particular case, although no complete adjoint capability are currently available. This restricts gradient computation to finite difference and thus to Euler computation given the computing resources limitation. A complementary study with an adjoint capability could allow for RANS based optimization with a higher number of design variables at different flight points, further improving the configuration presented in this paper. The first design iterations of engine integration with BLI highlight the different challenges posed by this configuration, notably at the S-duct entrance, on the nacelle sides and at the trailing edge junction. In the near future, the design of the configuration using BLI engines is going to be finalized. Then, on this configuration, the BLI gains are going to be evaluated and quantified. Ultimately, these gains will be integrated in the OAD process, to be able to account for the BLI effects at the design stage.

Figure 2 .

 2 Figure 2. Cabin layout of the BWB configuration. Cargo holds are situated on both sides of the passenger cabin.

Figure 4 .

 4 Figure 4. BWB half geometry (in beige) and half geometry of the pressurized volume (in red)

Figure 5 .Figure 6 .

 56 Figure 5. Mesh convergence of the drag coefficient on the baseline configuration at cruise conditions

Optimization 1 (

 1 outer wing and twist law): 20 design variables: -16 CST parameters to describe airfoil shapes p2 and p21 -3 twist angles (sections p2, p21 and p3) ∈ [-6°, 6°] 3 -1 angle of attack < 3.5° Objective : Minimize inviscid drag coefficient CD Constraints: -𝐶𝐿 ≥ 𝐶𝐿 𝑐𝑟𝑢𝑖𝑠𝑒 -(𝑡/𝑐) 𝑝2 ≥ 0.12 -(𝑡/𝑐) 𝑝21 ≥ 0.10 Optimization 2 (central body and twist law): 28 design variables: -24 CST parameters to describe airfoil shapes p0, p01 and p1 -3 twist angles (sections p2, p21 and p3) ∈ [-6°, 6°] 3 -1 angle of attack < 3.5° Objective : Minimize inviscid drag coefficient CD Constraints: -𝐶𝐿 ≥ 𝐶𝐿 𝑐𝑟𝑢𝑖𝑠𝑒 -𝐶𝑚 𝑐𝑜𝑔 ≥ 0.0 -𝐸𝑥𝑐𝑒𝑒𝑑𝑖𝑛𝑔 𝑉𝑜𝑙𝑢𝑚𝑒 < 0.0 Optimization 3 (central body): 24 design variables: -24 CST parameters to describe airfoil shapes p0, p01 and p1 Objective : Minimize inviscid drag coefficient CD Constraints: -𝐶𝐿 ≥ 𝐶𝐿 𝑐𝑟𝑢𝑖𝑠𝑒 -𝐶𝑚 𝑐𝑜𝑔 ≥ 0.0 -𝐸𝑥𝑐𝑒𝑒𝑑𝑖𝑛𝑔 𝑉𝑜𝑙𝑢𝑚𝑒 < 0.0

  Figure 9 also shows a reduced pressure peak on the outer wing (y/b = 0.8), compared to the baseline geometry.

Figure 7 .Figure 8 .Figure 9 .

 789 Figure 7. Pressure coefficient comparison on the baseline and optimized configurations, at cruise condition

Figure 10 .

 10 Figure 10. Comparison of the baseline and optimized geometries, using RANS computations

Figure 12 .Figure 13 .

 1213 Figure 12. Overview of four different engine integration designs

Figure 14 .

 14 Figure 14. Pressure coefficient and friction vector on the surface of three designs considered Figure 14 highlights that great care should be taken when designing the side junction between the nacelle and the wing. It is particularly visible that big corner separation flows are developing in these regions. Design 2 tries to solve this issue by adding fillet junction compared to Design 1. It effectively suppresses the detached zone on the external side.Because Design 3 moves the engines to the rear, the length of the acute angle is reduced, creating only a small separated bubble, even if the geometry has no fillet. Thus, Design 3 presents several advantages compared to the two others, regarding the aerodynamic performance. However, it must be stressed that the structural integration of this design appears unfavourable (see Figure13), compared to Designs 1 and 2. A fourth different configuration (Design 4) is currently under conception and evaluation.

Table 1 .

 1 SMILE configuration mass breakdown

	Item		Weight (kg)	% MTOW
	OEW		36042	58.8 %
	PW		13500	22.0 %
	FW	SMR Mission	9091	14.8 %
		Reserve	2669	4.4 %
	MTOW		61303	100 %

Table 2 .

 2 Main geometrical characteristics of the BWB SMILE

	Mean Aerodynamic chord	12.109 m
	Reference surface	268.602 m²
	Span	37.73 m
	Centre of gravity	12.29 m

Table 3 .

 3 Aerodynamic cruise conditions

	Mach	0.78
	Altitude	12636 m
	Reynolds per meter	4.7779*10⁶

ACKNOWLEDGEMENTS

The project leading to this application has received funding from the Clean Sky 2 Joint Undertaking under the European Union's Horizon 2020 research and innovation program under grant agreement N°CS2-AIR-GAM-2022-2023-01. The author would like to thank the project partners from DLR, Dassault Aviation and AIRBUS for their interest in this topic and their guidance.


text:
Vincent Liguori
email: vincent.liguori@onera.fr

Frederic Moens
email: frederic.moens@onera.fr

Jacques Peter
email: jacques.peter@onera.fr

Eric Laurendeau
email: eric.laurendeau@polymtl.ca

Validation of a nonlinear Vortex Lattice Method for multiple wing sweep angle configurations

This paper presents the application of the non-linear vortex lattice method implemented in the CHAMPS solver developed at Polytechnique Montréal toward the aerodynamic analysis of a wing at various sweep angles. The non-linear viscous correction is applied through a local effective angle correction for spanwise sections based on 2.5D sectional viscous data from a RANS database. The results obtained with the method on the test case of a wind tunnel model designed by ONERA are compared with numerical data from 3D RANS computations and experimental data from the wind tunnel test. The wind tunnel model is a wing-body geometry which is considered for three different wing sweep configurations. The first one is an unswept wing, the second one is a conventional swept wing and the third is the unconventional configuration of a wing with a negative sweep angle. The computational results showcase the capabilities of the method with respect to predictions from a higher fidelity method and experimental data for the various configurations with a low computational cost.

INTRODUCTION

Potential methods such as lifting-line theory (LLT) and vortex-lattice method (VLM) are useful tools to predict aerodynamic performances of three dimensional configurations thanks to their low computational cost compared with higher fidelity methods such as threedimensional Reynolds-Averaged Navier-Stokes (RANS) equations solver [START_REF] Taylor | Low-fidelity method for rapid aerostructural optimisation and design-space exploration of planar wings[END_REF]. These potential methods have been extensively used in the literature, including for optimization due to their low computational cost. Their imple-mentation is detailed by many authors [START_REF] Drela | Flight Vehicle Aerodynamics[END_REF] [START_REF] Katz | Low-Speed Aerodynamics[END_REF].

However the downsides of these methods are the limitations related to their underlying assumptions, namely inviscid, incompressible and irrotational flows, thin lifting surfaces and flight condition in the range of small angles of attack. These assumptions are not by definition used by a RANS equations solver. This limits the usefulness of the potential methods despite their lower computational cost compared to higher fidelity options. Some authors have as such used additional methodologies to compute viscous and thickness effect on aerodynamic coefficients in optimization frameworks [START_REF] Mariens | Quasi-Three-Dimensional Aerodynamic Solver for Multidisciplinary Design Optimization of Lifting Surfaces[END_REF] [START_REF] Leifsson | Fast Low-fidelity Wing Aerodynamics Model for Surrogate-based Shape Optimization[END_REF]. It is possible to consider the effects of many phenomena that are neglected because of the previous assumptions, while still retaining the upside of a lower computational requirement than a threedimensional RANS method with the use of corrections to the potential method.

One of these correction methods is the Prandtl-Glauert correction to consider compressibility effects [START_REF] Gothert | Plane and Three-Dimensional Flow at High Subsonic Speeds[END_REF]. Another correction method that has been presented in the literature is a non-linear coupling at selected wing sections with higher-fidelity two-dimensional viscous results [START_REF] Van Dam | On the aerodynamic design of multielement high-lift systems[END_REF][8] [START_REF] Gallay | Preliminary-Design Aerodynamic Model for Complex Configurations Using Lifting-Line Coupling Algorithm[END_REF]. Solvers of the VLM type using this last correction have been presented in the literature using the name non-linear vortex-lattice method (NL-VLM).

NL-VLM has displayed good results as a method with low computational cost compared to three-dimensional RANS solvers in low-speed to transonic conditions while being able to represent non-linear physical phenomena [START_REF] ¸ugar Gabor | Fast and accurate quasi-3D aerodynamic methods for aircraft conceptual design studies[END_REF]. For this reason, the method is also a good candidate as an aerodynamic tool to be used in an aerodynamic optimization methodology when considering a large design space.

The aim of this paper is to apply the capabilities of the NL-VLM solver implemented in the CHAMPS framework developed at École Polytechnique de Montréal to predict the aerodynamic coefficients of a wing-body type configuration for multiple wing sweep angles, for which numerical RANS simulation and experimental data are available.

The paper first presents the numerical methods used. The test case and the results obtained are then presented to finally conclude on these results.

NUMERICAL METHOD

This section presents the numerical methods used in the scope of this work. The NL-VLM method implemented in the CHAMPS framework is first presented along with the method used to compute aerodynamic coefficients. Information are also provided about the 3D RANS equations solver used to validate the results of the method and the RANS equation solver used to compute the 2.5D viscous database.

Non-Linear Vortex-Lattice Method

The NL-VLM method implemented in the CHAMPS framework is based on previous works from Polytechnique Montréal's team and as such is very similar in its implementation.

Vortex-Lattice and Panel Method

The three-dimensional potential method uses a hybrid implementation using two types of potential elements. The first type is a vortex-ring element and the second type is a potential source and doublet panel method.

A vortex-lattice sheet is used over the wing planform of the aircraft while considering twist but not camber between the leading and trailing edges to compute the three-dimensional effect of the wing-induced lift. The vortex-lattice plane is a structured mesh of quadrilateral elements. The vortex rings are placed a quarter-chord of element behind the leading edge of the element and the collocation point is defined at 3/4 of the chord at the spanwise middle point for each element, in accordance with recommendations from the literature [START_REF] Deyoung | Historical evolution of vortex-lattice methods[END_REF] [START_REF] Pistolesi | Considerations on the Mutual Interference of Aerofoil Systems[END_REF].

The panel elements are used to consider the surface of thick non-lifting bodies. In this case, this refers to the main body of the considered geometry but is not by definition limited to fuselages. The implementation uses an unstructured surface mesh for the element. In this case, the elements used are triangles. A constant source and doublet distributions are applied on every panel. To ensure a unique solution, the source value is set to be equal to the normal component of the flow against the panel.

Both types of elements are assembled in a global system of equations to solve simultaneously the vortex and doublet values, the potential values being computed simultaneously.

Viscous Coupling

To include non-linear effects, an iterative coupling is done on every chordwise strip of vortex panels along the span of the wing. Each strip is corrected with a 2D or 2.5D (infinite swept wing) viscous database computed with CHAMPS, a RANS equation solver. The viscous database is computed once prior to the NL-VLM calculation. More details on the solver and the geometry considered are presented in section 2.2.2.

The coupling methodology used is van Dam's αcoupling method that was originally defined for a liftingline formulation [START_REF] Van Dam | On the aerodynamic design of multielement high-lift systems[END_REF], was extended to a VLM formulation by Mukherjee et al. [START_REF] Mukherjee | Poststall Prediction of Multiple-Lifting-Surface Configurations Using a Decambering Approach[END_REF] and for which the specific algorithm used is the same as the one presented in Gallay and Laurendeau's implementation for a viscous-corrected Weissinger method [START_REF] Gallay | Preliminary-Design Aerodynamic Model for Complex Configurations Using Lifting-Line Coupling Algorithm[END_REF].

When the iterative viscous coupling is converged for the lift coefficient value of all strips, a subsequent viscous coupling step is applied to match the local moment coefficient of each VLM chordwise section with the moment coefficient of the viscous database at the corresponding effective angle of attack. To accomplish this the gamma values of the panels on the section are multiplied by scaling factors. The scaling factors are computed by solving a least square problem corresponding to the underdetermined system. This viscous coupling in two steps to match both the local lift and moment coefficient of the viscous database is detailed in Parenteau and Laurendeau's work on unsteady NL-VLM [START_REF] Parenteau | A transonic, viscous nonlinear frequency domain Vortex Lattice Method for aeroelastic analyses[END_REF].

Force Coefficients Computation

The force coefficient computation is done using both the VLM solution and the viscous section data, taking advantage of the availability of both information. While force computation is also possible on surface modeled with source and doublet singularities, in this work only the force and moments produced by lifting surfaces, which are modeled with vortex-lattices, are considered. For the geometry considered this means that the efforts produced by the fuselage are not considered.

The lift coefficient is computed directly from the VLM mesh using the Kutta-Joukowski theorem [START_REF] Katz | Low-Speed Aerodynamics[END_REF] on the vortex filaments, which is defined as

F = 4MN ∑ i ρ ∞ Γ i U ∞ l i (n U ∞ × n l i ) (1)
where ρ ∞ and U ∞ are respectively the freestream density and velocity magnitude, Γ i and l i are respectively the strength and length of a vortex filament i among the 4MN filaments contained in a grid of M by N panels and n U ∞ and n l i are respectively the normalized vector with the orientation of the freestream velocity and the vortex filament. The amount of filament is known since the VLM mesh is a grid of M by N quadrilateral elements.

Since the vortex intensities on the wings are corrected to match the viscous local lift and moment coefficient, the moment created by the wing relative to a reference point is computed directly using the Kutta-Joukowski theorem similarly to the lift coefficient, using the equation

M tot = 4MN ∑ i ∆p × (ρ ∞ Γ i U ∞ l i (n U ∞ × n l i )) (2)
where ∆p is a vector representing the difference in position between the reference point for the moment calculation and the force acting point of the panel containing the vortex filament, which is defined as being at the 3/4 of the panel chordwise and at the midpoint of the panel spanwise. The moment component parallel to the y-axis is used to compute the aerodynamic moment coefficient.

The wing drag is evaluated in two steps. The first term D v including the pressure and viscous components of the wing drag are first evaluated by integrating the local drag coefficients of the viscous database over each of the N wing strips, as

D v = N ∑ i ( 1 2 ρ ∞ U 2 ∞ b/2 -b/2 C D v cdy) (3)
where C D v is the local drag coefficient of the viscous database at the corresponding effective angle, c is the local chord of the section and b is the spanwise width of the section.

The second term D i including the induced drag is evaluated using a Trefftz plane, which is a plane located far downstream of the wing perpendicular to the freestream [START_REF] Katz | Low-Speed Aerodynamics[END_REF]. The induced drag of the wing is computed with a spanwise line integral of the vortex intensity multiplied by the induced vertical velocity by the wake itself along the intersection of the wake and the Trefftz plane. The resulting equation

D i = N ∑ i (- ρ ∞ 2 b/2 -b/2 Γ i wdy) (4)
is mathematically equivalent to integrating the kinetic energy over the Trefftz plane.

The total drag of the wing then corresponds to the sum of both induced and viscous drag terms

D total = D v + D i .
(5)

RANS Solvers

Two different RANS solvers were used in this work. While both software programs offer various functionalities, only the ones used in this work are mentioned in this paper.

The ONERA elsA CFD software [START_REF] Cambier | The Onera elsA CFD software: input from research and feedback from industry[END_REF] (ONERA-Airbus-Safran property) was used to compute a threedimensional RANS solution for validation purposes against the NL-VLM method.

The Polytechnique Montréal CHAMPS CFD software was used to compute the 2D and 2.5D RANS solutions from which the viscous correction database of the NL-VLM method was assembled.

elsA Solver

The elsA CFD software is a RANS equations solver able to consider 3D geometries. The system of equations is solved by a cell-centered finite-volume method. More detailed information about the software and its capabilities is available in the literature [START_REF] Cambier | The Onera elsA CFD software: input from research and feedback from industry[END_REF]. In this case, the solver is used on a multiblock structured mesh.

While an exhaustive list of every turbulence model available within the solver is outside the scope of this work, multiple models were tested to identify a model which would be able to show a good agreement with experimental data. The model identified as having the best match is k-ω Menter SST model [START_REF] Menter | Two-equation eddy-viscosity turbulence models for engineering applications[END_REF] with Quadratic Constitutive Relation (QCR) [START_REF] Spalart | Strategies for turbulence modelling and simulations[END_REF].

CHAMPS Solver

The CHAMPS solver is a RANS equations solver also using a cell-centered finite-volume approach. The solver is able to consider 3D geometries and unstructured meshes. While multiple turbulence models are implemented in the solver, in this case the Spalart-Allmaras model [START_REF] Spalart | A one-equation turbulence model for aerodynamic flows[END_REF] is used. More details about the solver are available in the literature [START_REF] Parenteau | Development of Parallel CFD Applications with the Chapel Programming Language[END_REF].

NUMERICAL RESULTS

In this section the considered test case is presented alongside the methodology used to obtain the NL-VLM results. The methodologies used to obtain the RANS and experimental data are also briefly presented.

Considered Test Case

The considered test case is based on a geometry designed by ONERA to investigate the stall phenomena over a basic wing. This effort was done inside the PANDA internal project (French acronym for Prévision et ANalyse du Décrochage d'Aile). The model was tested for angles of attack ranging from approximately 0 degrees to 31 degrees, which includes a range of post-stall angles of attack. The wind tunnel conditions which are also used in the RANS and NL-VLM simulations are a Reynolds number of 0.95e10 6 and a Mach number of 0.1465. To simplify aerodynamic coefficients calculations and comparison, the same reference chord of 0.2857 m and reference area of 0.2857 sq.m are used for the results of all methods and geometries.

PANDA Geometry

The model consists of a rectangular wing with a NACA4412 airfoil connected to a body. The model is a half-geometry cut by the wind tunnel wall at its symmetry plane. The body can be compared to a fuselage but is designed to avoid having a large effect on the wing compared to a wing-at-wall case instead of realistic aircraft fuselages requirements.

The model has a variable wing sweep angle, and as such three different wing sweep angle configurations were tested, namely a 0 degree sweep angle (straight wing), a 25 degree sweep angle configuration and a -25 degree sweep angle configuration.

More details about the model geometry including dimensions and wing twist are available in a preliminary design report [START_REF] Moens | Design of a generic wing model for basic stall phenomena[END_REF] even though the final model geometry presented some differences compared to the report, which are enumerated here. First, the thickness of the body was 125,0 mm instead of the original 100,0 mm thickness mentioned. The second difference is that for the case of the positive 25 degrees wing sweep, a wingtip extension was added so that the wingtip would be parallel to the incoming flow at a distance of 1000,0 mm from the wall. A superimposed top-down view of the three geometries is shown in Fig. 1.

Meshing for Potential Calculations

The potential mesh is generated in three steps. The first is the generation of the body mesh, which is an unstructured surface mesh of triangles. While similar, the body meshes generated for each configuration are different to allow the wing and body to share coincidental nodes. The number of cells in the body meshes ranges from 2880 to 3402.

The second step is the generation of the wing mesh. The wing meshes are structured 2D grids shaped to conform to the wing planform. The leading and trailing edges of the mesh are coincidental with the geometry to model the wing twist. However, the camber of the wing is not modeled in the potential mesh since this effect is already considered by the viscous coupling. The generated wing meshes contain 441 cells divided into 9 uniformly spaced rows chordwise and 49 uniformly spaced strips spanwise for the 0 and 25 sweep angle geometries. The -25 sweep geometry wing mesh contains 297 cells divided into 9 uniformly spaced rows chordwise and 33 uniformly spaced strips spanwise. The third step is the generation of the wake mesh. A rigid wake is used in this work. The generated mesh is a structured grid of 49 spanwise strips and 137 streamwise rows. The wake is generated with a length of approximately 80 chords downstream of the wing. To consider the effect of the fuselage on the wake geometry, the contraction method of Nikolski [START_REF] Drela | Flight Vehicle Aerodynamics[END_REF] is used.

An isometric view of the potential mesh generated for the 0 sweep configuration is shown in Fig. 2. The majority of the wake is not shown for ease of view.

NACA4412 Viscous Database

As mentioned in the method definition, the airfoil geometry and viscous effects for each wing section are Since the wing sections considered in the viscous coupling are parallel to the flow, a 2.5D viscous database is used to consider the sweep effect on the wing sections of the swept wings configurations. This is done through the use of a simplified 3D mesh consisting of a 1 cell extrusion of a 2D mesh representing the geometry of an infinite swept wing.

The sign of the sweep affects only the transverse force coefficient, which is not used in the viscous coupling. For this reason, the same viscous polar is used for the positive and negative swept wing. The conditions considered for the straight wing case are a Reynolds number of 956006 and a Mach number of 0.1465. In the case of the swept wing, a Reynolds number of 866435 and a Mach number of 0.1465 are used. In both cases, the Spalart-Allmaras model is used for turbulence modeling. The resulting aerodynamic coefficients of both databases are shown in Fig. 3, Fig. 4 and Fig. 5. The sweep effect has an important effect.

Non-Linear Lattice-Vortex Method Results

The results of the NL-VLM computations are presented in this section and compared against the results from RANS computations and computational data. Since NL-VLM uses a viscous correction based on an infinite wing, a case like the PANDA geometry with a relatively low aspect ratio of 7.0 pushes the method to the boundary of its applicability and is a good test case to highlight potential limitations.

Aerodynamic Coefficients Comparison with a

High-Fidelity Numerical Method and Experimental Data

Using NL-VLM, converged solutions were obtained for the entire experimental data range of 0 to 31 degrees of angle of attack. The RANS results however are not available for every experimentally tested angle of attack. Fig. 6-14 show a comparison of the aerodynamic coefficients obtained.

A key point of interest of the method compared with a 3D RANS solver is its lower computational cost. The calculation of one angle of attack data point with the 3D RANS solver required computational power ranging from approximately 30 minutes of calculations to approximately 310 minutes while using 128 processors on a computer cluster. The calculation of one angle of attack data point with the NL-VLM solver varies between approximately 1 minute and 10 minutes on 8 processors of a workstation computer. This difference in computational requirement is significant.

The CL-alpha curves generally show a good agreement between the NL-VLM and the higher fidelity data. Despite this, some notable exceptions are present to this observation. The first is the appearance of stall at an earlier angle of 16 degrees in the NL-VLM data for the positive sweep configuration in comparison with the results obtained experimentally of 19.0 degrees and the stall angle from 3D RANS computations of 17.5.

A second one is related to the appearance of a slight lift loss before the stall angle of attack is reached for the negative sweep configuration. This lift loss is observed in the RANS results at alpha=11.0. A very slight lift loss accompanied by small amplitudes oscillations occurs at the same angle of attack in the experimental data. A lift loss before stall is also observed in the NL-VLM data but occurs at a later angle of attack of 14.0 degrees. The CL-CD curves are in the same range of values as the higher-fidelity data and show the same evolution.

However, the drag tends to be overestimated before stall and underestimated after stall in the case of the two swept geometries.

The CL-CM curves display a similar behavior as their higher-fidelity counterparts. While the 0 sweep case shows a more visible gap in Fig. 8 than the swept configurations, it should be noted that the moment coefficient has a very low variation along the different angles of attack when compared to the swept configurations. The absolute difference between the NL-VLM, RANS and experimental obtained values for the unswept wing is comparable to the swept wing cases. The lift distribution for the unswept wing is shown in Fig. 15 at the computed stall angle and the following angle in the computed stalled region. The local appearance of stall begins at the section in the root region of the wing. The appearance of local stall near the global stall angle was also observed in the root section of the wing for the 3D RANS calculations.

The lift distribution of the 25 degrees sweep wing is shown at the angle directly before and after stall in Fig. 16. Local stall in section first occurs approximately at 1/2 span of the wing. In the case of the 3D RANS calculations, the appearance of local stall was observed at approximately 2/3 of the span of the wing. It is notable that for both computational methods the appearance of local stall is not at the root or tip of the wing. 17, the first local stall of a section on the wing occurs at the root at an angle of attack of 14.0. However, the decrease in global lift from stall occurs at 18.0 when the separated zone spreads to more sections of the wing. This phenomenon of the appearance of a first separation zone inboard before stall was also observed in the RANS computation, but at an earlier angle of attack of 11.0.

CONCLUSION

The aim of this paper was the evaluation of the fidelity of NL-VLM on various wing-body geometries by comparing the results of the method with higher-fidelity computational data from a 3D RANS solver and experimental wind tunnel data. The test cases considered to achieve this objective are from a wing-body geometry tested for multiple wing sweep angles.

A summary of the NL-VLM implemented in the CHAMPS CFD software was presented, as well as the methodology used for the computation and the aerodynamic force coefficients computation. Results were obtained for the wing-body geometries with different sweep angles and were compared against data from the higherfidelity 3D RANS solver and the experimental wind tunnel data obtained in the ONERA PANDA internal research program.

The results generally show a good agreement between the method and the higher fidelity RANS method and experimental data in predicting aerodynamic force coefficients. While the method shows an expected lower accuracy in predicting 3D effects, it nevertheless captures all important features at a much lower computational cost. This makes the NL-VLM a strong candidate in the context of optimization.

Figure 1 :

 1 Figure 1: Top-down view of the three PANDA model geometries superimposed

Figure 2 :

 2 Figure 2: Isometric view of the potential mesh generated to model the 0 sweep geometry

Figure 3 :

 3 Figure 3: CL-alpha curves used to assemble the swept and unswept viscous databases

Figure 4 :

 4 Figure 4: CL-CD curves used to assemble the swept and unswept viscous databases

Figure 5 :

 5 Figure 5: CL-CM curves used to assemble the swept and unswept viscous databases

Figure 6 :

 6 Figure 6: CL-alpha curve of NL-VLM, RANS and experimental data for the 0 sweep PANDA wing-body case

Figure 7 :

 7 Figure 7: CL-CD curve of NL-VLM, RANS and experimental data for the 0 sweep PANDA wing-body case

Figure 8 :Figure 9 :

 89 Figure 8: CL-CM curve of NL-VLM, RANS and experimental data for the 0 sweep PANDA wing-body case

Figure 10 :

 10 Figure 10: CL-CD curve of NL-VLM, RANS and experimental data for the 25 degrees sweep PANDA wing-body case

Figure 13 :Figure 14 :

 1314 Figure 13: CL-CD curve of NL-VLM, RANS and experimental data for the -25 degrees sweep PANDA wingbody case

Figure 15 :Figure 16 :Figure 17 :

 151617 Figure 15: Spanwise lift coefficient along the unswept wing