1 Introduction

1.1 Overview

The pursuit of climate neutrality in aircraft design, a key topic for aviation development [1,2,3], presents significant challenges that demand innovative and disruptive approaches. Conventional aircraft configurations, such as the tube-and-wing, have reached a level of maturity; incremental improvements in overall aircraft design introduce more and more reduced performance gains. To achieve ambitious emissions reduction targets, disruptive unconventional configurations are a solution worth to be explored [4, 5]. Among these, relevant investigation have been performed on blended-wing-body [6, 7], truss-braced wings [8, 9], and box-wing [10, 11]. These configurations challenge traditional design principles and explore novel possibilities to enhance aerodynamics and structural weight. The research on these unconventional configurations requires addressing complex engineering challenges, as developing specific predictive models for structural and aerodynamic behaviour and using high-fidelity physics-based approaches to properly characterize these features. Hence, the efforts in exploring these configurations are high but, at the same time, the potential benefits achievable can lead to substantial reductions in fuel consumption, emissions, and noise. The development and implementation of unconventional configurations require multidisciplinary collaboration, advanced simulation techniques and high computational capabilities.

In this context, this work proposes the detailed description of the aerodynamic design process of a specific unconventional configuration, called box-wing [12, 13]. This configuration, which theoretically is proposed as being more efficient than conventional aircraft, from a theoretical [14,15,16,17,18] and a practical point of view [10, 11, 19, 20], has been the object of interest for many years (e.g. [21,22,23]) and was finally analysed in detail from an aerodynamic point of view within the European collaborative project PARSIFAL [24], as discussed in detail in the following paragraphs.

1.2 State of the Art

Box-wing aircraft configuration has been a subject of investigation in the field of aerodynamics for several years; it consists of two wings positioned horizontally, with one wing located forward and the other aft, connected by properly designed tip-wings. The main features that characterize this particular lifting system is its theoretical capability of minimizing the induced drag among all the lifting configurations with the same span and lift ([14, 15, 25, 26]). Another relevant aerodynamic feature of the box-wing configuration is its increased lift generation capability, as the wings arrangement allows for a larger wing area and a higher lift generation compared to a conventional tube-and-wing, with the same wingspan [27,28,29]. The box-wing configuration may also exhibit improved structural efficiency; indeed, the wing system is over-constrained to the fuselage resulting into reduced structural stresses and, consequently, in the possibility of designing thinner and lighter structures [30, 31]. However, the focus of this paper is devoted to the deep investigation and understanding of the box-wing aerodynamics, with a focus on its transonic flight performance, which received limited attention in the literature. Large research efforts have been spent in the last decade at University of Pisa, by means of several research projects dedicated to the study of this particular architecture, which has been renamed PrandtlPlane in honour of Prandtl. However, the knowledge gained in numerical and experimental studies was limited to aircraft flying at low speed, far from the transonic conditions. Relevant examples of these results are the application of the box-wing concept to a seaplane, as done in the IDINTOS project, in which conceptual, numerical, and experimental advanced activities lead to the development of a full scale box-wing prototype [32,33,34]. Following research involved the conceptual investigation of commuter/aero-taxi box-wing aircraft [35], regional hybrid-electric solutions [36,37,38], up to the medium range liner investigated in the framework of the PARSIFAL project [28, 29] (see Fig. 1). Applications of this lifting system have been explored also for the new Urban Air Mobility (UAM) sector, with investigations on box-wing eVTOL (Vertical Take-Off and Landing) aircraft [39, 40].

Fig. 1
figure 1

Box-wing for different transport categories: a seaplane, b commuter, c regional, d short-medium range

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The configuration presented in this paper, from a collaborative aerodynamic design perspective, is that relating to the medium range and studied in the PARSIFAL project.

1.3 The PARSIFAL Project

The research detailed in this paper has been undertaken as part of the PARSIFAL project (‘Prandtlplane ARchitecture for the Sustainable Improvement of Future AirpLanes’) [41], an initiative funded by the Horizon 2020 program of the European Union. Aligned with the research area of Smart, green and integrated transport: Mobility for Growth, specifically within the field of Breakthrough Innovations, the project aimed to achieve the following principal objectives:

  1. i)

    Demonstrate the feasibility of implementing the PrandtlPlane configuration on aircraft with comparable dimensions to medium-range airplanes such as the Airbus A320 and Boeing 737; the scope is to enhance passenger capacity by over 50% while concurrently improving fuel efficiency and ensuring compatibility with existing airport infrastructure.

  2. ii)

    Develop advanced design tools, methodologies, and procedures to enable an extensive investigation of the applicability of the PrandtlPlane concept to other aircraft categories, extending its potential benefits beyond medium-range airplanes.

The PARSIFAL project was a collaborative effort involving six main partners; the University of Pisa served as the project coordinator, with additional contributions from Delft University of Technology (Delft, Netherlands), ONERA (Meudon, France), ENSAM (Bordeaux, France), DLR (Hamburg, Germany), and SkyBox Engineering (Pisa, Italy); other contributions came from specific collaboration of other involved external partners.

The activities planned and conducted within the PARSIFAL project involved multidisciplinary and interconnected aspects, such as the structural analysis of the box-wing system [42,43,44,45], its aeromechanical features [46,47,48], the integration of systems [49], the assessment of its environmental impact [50], and of course its aerodynamic analysis, which is the topic of this paper.

1.4 Aim of the Work

The purpose of this article is to present in detail the design development of the box-wing aircraft from an aerodynamic perspective, focusing on the relevance of the collaborative approach and the application of multi-fidelity strategies employed to perform the design and analysis of such a configuration. The description of the aerodynamic design challenges, the way these have been faced, and the overall performance analysis of the box-wing are deeply detailed in the paper. The contents and analysis proposed in this paper are intended to provide a comprehensive overview on the aerodynamic design concerning the development of a transonic airliner with box-wing lifting architecture. In the literature, extensive space is given to the aerodynamic analysis of such a configuration considering the incompressible flow, while few information is available on transonic aerodynamics, mostly of a theoretical nature, whereas this work is part of a more comprehensive analysis of the design problem and consequent engineering integration. Transonic aerodynamic characterisation to support the specific design for the box-wing aircraft is a key aspect presented in this research. Furthermore, the implementation of a collaborative multi-fidelity approach, provided a broad and detailed aerodynamic analysis framework to widely characterise the aerodynamic design of such a non-conventional lifting system. Specifically, the article is structured as follows: in Sect. 2 the methodology employed throughout the entire project is detailed, focusing on the collaborative and multi-fidelity approach used, together with the description of design tasks and related tools; Sect. 3 provides a wide description of the high-speed aerodynamic performance assessment, starting from the explorative and conceptual phase, up to the results obtained by means of high-fidelity numerical simulations. In Sect. 4 the focus is given to the transonic aerodynamic performance penalties related to the design of a V-shaped vertical tailplane, together with the optimization strategy employed to face this issue. Finally, in Sect. 5 the conclusions are given.

2 Methodology

2.1 A Collaborative Approach

The adoption of a collaborative approach among diverse institutions for the assessment of aerodynamic performance in the aircraft design process confers multiple advantages [51, 52]. One of the key advantages is the involvement of expertise from various research institutions, which brings together a broad range of know-how, experience, methodologies, tools and practices, enabling a comprehensive analysis of aircraft aerodynamic performance. The involvement of multiple institutions allows for cross-validation and verification of the research outcomes, leading to more reliable and accurate results. In a field that typically requires huge computational resources, as the detailed aerodynamic assessment, collaborative efforts provide access to a broader range of resources, including advanced numerical models, testing facilities, and validated datasets. Nevertheless, a collaborative approach needs the alignment of formats and models (as efficiently done in the CPACS project [53]), and the definition common schemas, to minimize difficulties related to the handling of workflows involving heterogeneous simulation environments. Despite these challenges, the collaborative approach promotes and enhances interdisciplinary collaboration [54].

This was the case of the PARSIFAL project, in which such a collaborative approach has been planned and then implemented into the project activities to properly develop the aerodynamic design and performance assessment of the box-wing aircraft. Hence, in a multi-level and multi-fidelity scheme, the activities have been divided and interconnected into a specific plan, by assigning to each involved institution specific tasks; specifically, the piece of research described in this paper presented the following shared workplan:

  • University of Pisa (UniPi) was in charge of managing the overall collaborative structure and to provide technical support for the early and conceptual design; specifically, UniPi provided the initial expertise on the box-wing configuration and preformed the conceptual aircraft design and refinement, up to the preliminary stage, supporting the activities by means of low-to-medium fidelity methodologies.

  • ONERA provided tools, methods and expertise related to the high-fidelity assessment of box-wing performance in any flight conditions; ONERA activities focused on the aerodynamic performance evaluation by means of RANS simulations, and lifting system shape optimization, up to the final detailed stages of the design process.

  • Cubit was involved as an external contributor to the PARSIFAL project to provide its computational capabilities and expertise in assessing complex flows, outside the multi-level scheme of the aircraft development; Cubit indeed provided support in the burn-in phases of the design, performing high-fidelity explorative studies on box-wing transonic issues, and at the end of the design process, performing the vertical tailplane shape optimization.

2.2 Multi-fidelity Approach

In the aircraft design process, the assessment of aerodynamic performance plays a crucial role in determining efficiency (i.e. the lift-to-drag ratio), stability, and overall performance. This assessment may typically be conducted using high-fidelity computational fluid dynamics (CFD) simulations, which provide accurate predictions of the flow behaviour around the aircraft [55]. However, these simulations are computationally expensive and time-consuming, limiting their applicability for extensive design exploration. To overcome this challenge, a multi-fidelity approach [56,57,58] is a typical methodology to efficiently integrate aerodynamic assessment within the overall aircraft design process. This approach combines high-fidelity simulations with lower-fidelity methods, such as potential solver, databases extrapolations or empirical correlations, to achieve a trade-off between accuracy and simulation cost suitable for each design stage. The application of these hybrid approaches enables the exploration of a wide range of design variations and the assessment of the related aerodynamic performance. Hence, a range of tools and methods are available to analyse and predict the aerodynamic performance, even of complex configurations. These tools include established textbook methods [59,60,61], empirical databases [62,63,64], potential methods [66,67,68], computational fluid dynamics solvers based on inviscid (Euler solvers [69]) and viscous flow (RANS solvers [70,71,72,73,74]) models, and advanced techniques such as Large Eddy Simulation (LES [75]) and Detached Eddy Simulation (DES [76]). Each tool offers specific capabilities towards different aspects of aerodynamic assessment, hence the proper selection of the adequate methodology to be used in a specific design stage is crucial for the effectiveness of the overall design process. An overall summary of the available methods and tools for aerodynamic assessments is provided in Table 1.

Table 1 Methods, models and tools for aerodynamic investigation
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The integration of a multi-fidelity approach in the aerodynamic design allows the exploration of several design possibilities by considering a wider range of parameters and configurations, and the assessment of trade-offs between conflicting design goals (i.e. maximizing efficiency while ensuring stability) since the early stage of the design process. Overall, the proper application of multi-fidelity methods enables the aerodynamic design process to be efficiently orchestrated, starting from design requirements and progressing towards the final assessment of high-fidelity performance.

To summarise, Table 2 reports the collaborative multi-fidelity scheme used for the aerodynamic development of the box-wing within the PARSIFAL project.

Table 2 Multi-fidelity approach used in the PARSIFAL project for the collaborative aerodynamic design of box-wing
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3 High-Speed Aerodynamic Assessment

3.1 Explorative CFD Transonic Analyses

The aircraft addressed by the PARSIFAL project is a medium-range transport box-wing configuration capable of transporting about 300 passengers, with a wingspan limited to 36 metres, in compliance with the ICAO Aerodrome Code C standard [88]; the purpose of adopting such a lifting configuration is to operate from a wider range of even smaller airports, to increase the point to point air transport, and at the same time to obtain a significant reduction of the fuel consumption per passenger-kilometre and, thus, a reduction in specific greenhouse gas emissions: this objective passes through the design of a aerodynamic configuration with the highest lift-to-drag ratio. According to the TLARs, the cruise flight is performed at Mach 0.79, a transonic flight condition. Aerodynamic performance in such a flight regime is highly dependent on the local geometry of the lifting system, the interactions with the flow are very complex and are strongly influenced by three-dimensional effects; these factors make practically impossible to use simplified models to preliminarily assess the aerodynamic performance of the box-wing. In addition, the lack of literature on the aerodynamics of the box-wing in transonic flight has required detailed aerodynamic analyses in the preliminary stage of the aerodynamic design process [89]. The aerodynamic analyses were conducted with the STAR CCM+ software, employing steady compressible RANS on the box-wing semi-model (fuselage and lifting system) and imposing the symmetry condition on longitudinal plane, to gather information on the aircraft’s overall transonic behaviour. A mesh sensitivity analysis allowed to determine reliable, time-effective grid characteristics. In particular, grids from 10 to 110 million cells led to selection of a mesh with around 30 million trimmed volume cells, with a surface mesh from 5 mm to 25 mm elements; the wake refinement extended up to 50 m with a maximum size of 500 mm. Additionally, a prism layer consisting of 25 layers was implemented, with a height of 100 mm and a growth rate of 1.1; the resulting y+ values varied between 30 and 150; Fig. 2 reports some details of the selected grid.

Fig. 2
figure 2

Details of the selected grid

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Compressibility effects were accounted for by treating the air as “ideal gas” and enabling the energy equation. The turbulence model adopted is the “k-ε Realizable”; this choice aimed to reduce numerical instability and computational time. The airplane surfaces were assigned as “wall” boundary condition, while the external sides of the computational domain were assigned as “free-stream” condition and specific values of Mach and altitude. The logics with which the initial numerical analysis campaign was set up followed two paths:

  1. i)

    Identification of a admissible design space with respect to the drag rise, by leveraging the main macro design variables towards the transonic design, as wing loading or sweep angles;

  2. ii)

    Identification of local issues, e.g. critical zones for the occurrence of strong shock waves.

The outcomes of the investigations have been employed for two primary purposes. First, they have been utilized to fine-tune the low-fidelity design procedures, involving the adjustment of boundaries associated with design variables and optimization constraints within the design tool illustrated in Sect. 2.2; this calibration process enhances the accuracy and reliability of the design procedure. Secondly, the obtained results contributed to the advancement of the existing limited understanding regarding the transonic aerodynamic characteristics exhibited by the box-wing airframe.

For the exploratory study on wing loading, four different configurations were selected with front wing loading values of 800, 700, 600 and 500 kg/m2, respectively. All the lifting systems are provided with the same supercritical airfoil (NASA SC2041), with constant percentage thickness along the span. The gradual improvement of the flow around the lifting system is evident as the value of the wing loading decreases (see Fig. 3); this is also inferred from the increase in cruise lift-to-drag ratio, as summarised in Table 3.

Fig. 3
figure 3

Mach contours on box-wing front and tip wings varying wing loading

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Table 3 Performance comparison varying W/S
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Similar analyses were carried out to evaluate the effects of varying front and rear sweep angles, and airfoil thickness. The information extracted from these analyses provided significant indications on the design choices to be made in the initial stages of the following aerodynamic design, as detailed in Sect. 3.2. An interesting aerodynamic behaviour has been identified at fillet between the front wing and the vertical wing; in this area, localised flow accelerations can cause the onset of shockwaves that can propagate over the entire vertical wing, causing massive increases in wave drag. Different analyses were performed in which the local geometry of the fillet and of the vertical wing were modified to obtain improvements in the flow acceleration field; for example, modifications in chords and twists of the fillet, and in the leading edge sweep angle of the vertical wing were evaluated (Fig. 4). Detailed description of this study can be found in [89].

Fig. 4
figure 4

Mach contour pressure on box-wing varying tip-wing geometry

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3.2 Conceptual Aerodynamic Design

After the high-fidelity exploratory phase, the aerodynamic design was initialised by low/medium-fidelity aerodynamic solvers, such as the vortex lattice method (VLM), to analyse a large number of configurations with computational time and cost typical for conceptual design phase. To do this, we used the AEROSTATE (AERodynamic Optimization with STAtic stability and Trim Evaluator) code. This software integrates the aerodynamic design of the box-wing lifting system within a constrained optimisation procedure; the implementation aspects and application of this code are described in detail in [90,91,92,93,94,95]. The aerodynamic solver integrated in AEROSTATE is AVL [66], a VLM code used to compute induced drag, lift coefficient, and aerodynamic derivatives related to the longitudinal stability and trim. Instead, parasitic drag of the lifting surfaces is evaluated by Eq. (1):

$$D_{p} = \rho V^{2} \mathop \int \limits_{0}^{b/2} C_{d} (y)c(y){\text{d}}y$$
(1)

By previously calculating the airfoil drag coefficient by means of XFOIL [65], and then integrating it along the wingspan (specifically, y is a spanwise station, c is the chord, V is the free stream speed, and ρ is the air density). Fuselage and nacelles drag contribution is computed by the simplified models proposed in [59]. The AEROSTATE optimization procedure is set as follows:

$$\left\{ {\begin{array}{*{20}c} {\min \left( { - f\left( x \right)} \right)} \\ {W_{{{\text{des}}}} - \varepsilon_{L} \le L\left( x \right) \le W_{{{\text{des}}}} + \varepsilon_{L} } \\ {\varepsilon_{M} \le C_{M} \left( x \right) \le \varepsilon_{M} } \\ {SM_{\min } \le SM\left( x \right) \le SM_{\max } } \\ {\max \left( {c_{l} \left( y \right)} \right) \le c_{l\max } } \\ {\left( {L/S} \right)_{\min } \le \left( {L/S(x)} \right)_{{{\text{wing}}}} \le \left( {L/S} \right)_{\max } } \\ {lb < x < ub} \\ \end{array} } \right.$$
(2)

where x is the vector of the design variables, limited by the lower and upper boundaries, lb and ub, respectively. With reference to Fig. 5, the design variables \({\text{x}}_{{\text{i}}}\) considered for each lifting surface are: chords and twists for wings root, kink and tip reference sections, sweep and dihedral angles for the corresponding wing-bay, and longitudinal positions of the two lifting surfaces; the wingspan b is fixed equal to 36 m. The objective function f(x) to be maximised is fixed equal to the lift-to-drag ratio L/D.

Fig. 5
figure 5

Geometric design variables of box-wing lifting system [92]

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The set of constraints, related to aerodynamics and aerodynamics features, are:

  1. i)

    Vertical equilibrium: hence the total lift L is equal to the design weight Wdes;

  2. ii)

    Pitch equilibrium: hence the pitching moment coefficient \({\text{C}}_{\text{M}}\) is equal to zero in the reference flight condition without any elevator deflection;

  3. iii)

    Longitudinal static stability: thus the static margin SM must be positive and constrained within a prescribed interval;

  4. iv)

    Limitation on the local lift coefficient \({\text{c}}_{\text{l}}\left({\text{y}}\right)\) in all the spanwise sections;

  5. v)

    Limitation on the wing loading L/S of each lifting surface.

The information obtained in the high-fidelity exploratory phase described in Sect. 3.1 served specifically to properly calibrate the constraints iv and v, as well as the values of the boundaries of the design variables lb and ub.

AEROSTATE was extensively used in the conceptual phase of the PARSIFAL project, to evaluate thousands of configurations, with a wide variety of characteristics, and to explore the entire available design space in the broadest possible way. This exploratory investigation and the related definition of trends between performance and design variables is extensively described in [94]; in this paper, for the sake of brevity, only the final stages of this conceptual exploration are reported. In particular, Fig. 6 shows the lift-to-drag ratio trends vs the design wing loading; as described in [94], there is an increase in lift-to-drag ratio as the wing loading increases. However, it has been necessary to find a trade-off between the performance of the box-wing in incompressible flow and those in transonic.

Fig. 6
figure 6

Lift-to-drag ratio versus front wing loading and reference surface

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For this reason, the initial reference configuration has been searched in a region of the design space that could ensure this trade-off; the feasible region is qualitatively represented by the red square in Fig. 6. The configuration chosen does not represent a final design, but is the starting point for the subsequent high fidelity refinement; therefore a conservative and improvable configuration has been selected, with good performances (but not the optimum achievable) and far from macro-critical issues. The selected AEROSTATE output is reported in Fig. 7; this starting point configuration has been named MS1 (MileStone 1 of the PARSIFAL project).

Fig. 7
figure 7

AEROSTATE output for the starting point box-wing (MS1 configuration)

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3.3 Preliminary CFD Assessment of the Baseline Configuration

The preliminary CFD-based aerodynamic and aeromechanical analyses were carried out by UniPi to identify the macroscopic critical issues before proceeding with the detailed aerodynamic assessment of the aircraft; the steady CFD analyses have been performed with Ansys Fluent software [77]. The computational grid is made as follows: the prism layer has 20 layers, 100 mm high, a geometric progression with a factor of 1.1, while the remaining volume grid is made by tetrahedra with a growth rate of 1.2; the total number of volume cells for the semi-model, is about 18 million (Fig. 8). For the analysis of the lateral-directional stability where the symmetry condition cannot be used, the resolution has been reduced to reduce in computational time accordingly; the solver is the steady RANS with the k-ε realizable turbulence model.

Fig. 8
figure 8

Mach contours comparison in cruise condition at front wing tip (left) and rear wing root (right) of MS1 and MS1.2

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In the case of the longitudinal and directional stability, the reference operating conditions are an altitude of 3000 m and a freestream speed of V = 131 m/s, consistent with the hypothesis of incompressible flow. In the case of cruise, the compressibility effects were simulated, and the flight condition is M = 0.79 and h = 11000 m (Fig. 9). Only specific analyses have been performed, as the detailed higher-fidelity assessment was carried out after this design phase. This preliminary CFD-based evaluations turned out to be necessary to identify possible critical issues and introduce specific modifications to the aerodynamic shape before starting the high fidelity assessment.

Fig. 9
figure 9

Contour of Cp for β = 1° (left) and β = 5° (right) of MS1.2

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These preliminary analyses were useful to tune the twist distribution along the span on view to reduce drag, to assess the static stability and to design the vertical tailplane (VTP); the results lead to the modification of the longitudinal position of the rear wing (Fig. 10) and, to the preliminary sizing and integration of the vertical V-shaped tailplane [96]. The design of the fins has been carried out following previous studies that highlighted the effectiveness of this layout towards the solution of aeroelastic issues for the rear wing [97, 98].

Fig. 10
figure 10

Left: Updated configuration (MS1.2-grey) compared with the baseline (MS1-blue). Right: MS1.2 with VTP

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The updated box-wing configuration has been renamed MS1.2, and a detailed description of these updates is proposed in [28, 94].

3.4 High-Fidelity Aerodynamic Performance Assessment

This Section presents the assessment of the aerodynamic performance of the MS1.2 configuration in cruise conditions by means of high-fidelity CFD computations and of refined drag estimation through the so-called far-field drag analysis [83, 99]. To perform high-fidelity CFD analyses, two models of the updated box-wing have been considered to specifically assess aerodynamic performance, namely: a lifting-system/fuselage assembly (Fig. 10-left) [100] and a configuration with the integration of the VTP (Fig. 10-right).

3.4.1 Computational Approach

RANS computations have been carried out by means of the in-house ONERA finite-volume solver elsA [81, 82] (ONERA-Airbus-Safran property) using an overset-grid approach. Body-fitted structured grids have been generated around the box-wing lifting-system, the fuselage, the vertical tail and around each corresponding intersection zone (collar grids). A Cartesian-octree background mesh has been then created automatically using the ONERA Cassiopée software library [101], with an extension of 400 m (nearly 100 mean aerodynamic chords) away from the aircraft surface along the three Cartesian directions. The generated body-fitted grids for the MS1.2 configuration are shown Fig. 11, while detailed views of the inherent overlapping regions, where interpolations take place, are given in Fig. 12. The normal wall spacing is kept almost uniform everywhere with a size of ~5 μm, corresponding to a maximum value of y+~ 0.8 for the considered cruise conditions (Mach number M = 0.79 and altitude = 11.0 km). The total number of cells for each box-wing half-model is 35.7×106 without considering the vertical tail, and 43.7×106 when integrating the VTP.

Fig. 11
figure 11

Overview of the body-fitted structured meshes generated for the CFD analysis of the MS1.2 configuration

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Fig. 12
figure 12

Overset mesh assemblage for the MS1.2 configuration. Orange coloured regions denote the interpolation fringes among the different body-fitted grids

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The steady RANS equations are supplemented by wall viscous boundary conditions on the solid surfaces, by appropriate far-field conditions on the external boundary of the background grid and by symmetry plane conditions at the longitudinal plane. The Spalart-Allmaras turbulence model with the QCR correction [102] is adopted and the Jameson scheme is used for the inviscid flux discretisation. The resulting discretised equations are solved using pseudo-time iterations with dual time-stepping and multigrid acceleration to converge towards the desired mean flow solution at the best accuracy in terms of aerodynamic coefficients.

3.4.2 Performance Assessment in Cruise Conditions

The lift (CL(AoA)) and polar (CL(CD)) curves based on the CFD computations around the MS1.2 in cruise conditions are illustrated in Fig. 13, where results obtained on the MS1 (without vertical tails) are also included for the sake of comparison. Note that all the non-dimensional aerodynamic coefficients are referred to the same reference wing surface of 266.7 m2. With reference to the lift curve in Fig. 13 (left), the MS1.2 configuration features an increased value of the zero lift AoA with a corresponding downshift of the lift curve compared to the MS1 configuration. On the contrary, the comparison of the drag polar in Fig. 13 (right) shows a small difference especially in the neighbourhood of the design condition (CL = 0.45–0.5), with a slight improvement for higher CL values and a performance reduction for lower ones. The results in Fig. 13 (right) also clearly show the drag penalty associated with the integration of the vertical tails on the MS1.2 configuration, which is higher at low CL values.

Fig. 13
figure 13

Comparison of the aerodynamic performance of the MS1 and MS1.2 at cruise conditions: Lift curve (left). Polar curve (Right). MS1.2 configuration with vertical tails is denoted as “MS1.2 VTP”

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The far-field drag decomposition [83] has been carried out for a fine performance assessment by evaluating the different drag contributions. Thanks to this analysis the polar curves of Fig. 13 (right) are decomposed according to the classical drag breakdown into the different physical drag sources: friction drag (CDf), viscous pressure drag (CDvp), i.e. the pressure drag contribution due to viscous phenomena, wave drag (CDw) and induced drag (CDi). The far-field drag formulation implemented in the ffd code developed at ONERA is based on the work of Van der Vooren and Destarac in [83]. Further details of the ONERA’s so-called one vector variant can be found in [103]. The results are illustrated in Fig. 14 (left). As in the previous figures, the results corresponding to the MS1 configuration are also included for comparison. In Fig. 14, it can be noticed that the friction drag is almost unchanged between both configurations: only when integrating the vertical tail planes, a shift of nearly 9.0 counts occurs, consistently with the associated increase of the wet surface. On the contrary the MS1.2 design (without vertical tails), features an improvement of the wave drag in the range of high lift coefficient values, by delaying the associated drag rise. In the same lift coefficient range, a small improvement of the viscous drag component is also noticed. These results can be associated with the amelioration of the box-wing design at the tip of the front wing (carried out by University of Pisa, see Sect. 3.3) to take care of compressibility effects which are responsible for performance reduction. This is illustrated by comparing the local isentropic Mach number distributions in Fig. 15. However, the onset of the separation is only slightly delayed. A small reduction in terms of the induced drag performance is observed for the MS1.2 compared to the MS1 configuration which is better highlighted in Fig. 14 (right) by comparing the Oswald efficiency \(e\): for the MS1.2 PrP the \(e\) factor is particularly reduced for CL values lower than 0.4. In the same figure, the lift-to-drag ratio (L/D) is also illustrated: the configurations have similar aerodynamic performance, only differing by a small shift of the MS1.2 curve toward slightly higher lift values and slightly higher L/D values.

Fig. 14
figure 14

Comparison of the aerodynamic performance of the MS1 and MS1.2 at cruise conditions: Polar curve breakdown according to the far-field drag analysis (left). Oswald efficiency and L/D variation with CL (right)

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Fig. 15
figure 15

Isentropic Mach number distribution at the tip of the front wing in cruise. MS1 (left), MS1.2 (right)

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On the contrary a huge drop in L/D is observed for the MS1.2 configuration, when the vertical tails are included. By inspecting the corresponding drag breakdown in Fig. 14-left this behaviour can be essentially ascribed to a huge increase of the viscous pressure drag contribution, with smaller additional contributions from the wave and induced (decrease of the Oswald efficiency) drag components at high CL. The analysis of the flow field shows that, due to compressibility effects, important flow separations occur at the junction between the rear wing and the vertical tail, as illustrated in Fig. 16 by the skin distribution of the isentropic Mach number and in Fig. 17 by the corresponding skin friction lines. However such phenomena are likely to be greatly reduced by an ad-hoc refined design of the local shape, allowing the recovery of almost the same level of performance obtained without vertical tails, by substantially limiting the drag penalty to the friction contribution only; this piece of detailed aerodynamic design work is described in Sect. 4.

Fig. 16
figure 16

Isentropic Mach number distribution at the junction between the vertical tail and the rear wing in cruise conditions. Left view (left). Right view (right)

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Fig. 17
figure 17

Skin friction lines superposed to contour levels of the x friction vector component (Pa) at the junction between the vertical tail and the rear wing in cruise conditions. Left view (left). Right view (right)

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3.5 Euler-Based Box-Wing Shape Optimization

After presenting the high-fidelity aerodynamic assessment of the updated box-wing configuration of the PARSIFAL project, the next step in the aerodynamic development was to set CFD-based shape optimization to search for performance improvements in transonic flight. Hence, this section summarises the aerodynamic optimization of the isolated box-wing geometry to determine the maximum achievable gain in terms of L/D in cruise conditions. Euler-based optimization studies have been performed on the isolated box-wing configuration using local airfoil section parameters; full RANS analyses and the complete model for the fuselage-wings assembly are not suitable for this study, both for the computational time/cost, and for the meshing automatization; hence, the RANS approach described in Sect. 3.4 has been used only to verify the performance of the optimised configurations. For these studies the OpenVSP software [84] has been employed for the geometrical modelling and the surface mesh; TetGen [85] for the volume tetrahedral mesh generation and the SU2 solver [86], for the Euler flow simulations. For the post processing of the results, far-field drag decomposition has been carried out by using the in-house ONERA code ffd, as well as empirical viscous drag prediction, to get an estimation of the total drag coefficient CD. The total far field drag is thus approximated as the sum of the wave drag, the induced drag and the estimated viscous drag. The goal of the optimization study is to minimize the total drag coefficient sum at 3 different target lift conditions in cruise flight. For this purpose, the modified method of feasible direction implemented in the DOT module [104] of the Dakota library [105] has been employed exploiting the Dakota built-in finite-difference gradient approximation. The target CL values are imposed as constraints for the optimization process with the following values: CL=0.45, 0.50, 0.55. The local airfoil section parameters employed in the optimization study are defined at the eight reference sections of the box-wing geometry, illustrated in Fig. 18. More precisely optimization studies have been carried out based on:

Fig. 18
figure 18

Snapshot of the OpenVSP box-wing model used for the optimisation studies

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Twist optimization

Airfoil camber with airfoil camber variation being described by 1 or 3 parameters;

The concurrent optimisation of both twist and camber.

In addition to the above design variables, the design space is augmented by including the value of the angle of attack corresponding to each considered lift condition, for a total of three additional design variables.

All optimisation studies have achieved a consistent reduction of the total drag coefficient for the three design points, while satisfying the corresponding lift constraints. The best performance is obtained by including both twist and camber variables in the optimization process. For MS1.2 configuration, the corresponding optimal values of twist angles are reported in Table 4 while the comparison between the original and the optimized airfoil shapes is illustrated in Fig. 19. It is worthwhile to observe that the most important modifications occur at Sect. 1, i.e. at the root section of the front wing, where the camber is greatly reduced; on the other hand, an inverse effect can be observed at Sect. 7, i.e. At the tip section of the rear wing where the camber of both airfoils is increased. Relevant modifications are also observed for the optimized design at Sect. 3 and at Sect. 6.

Table 4 Initial and optimised values of the twist angles (degrees) for isolated box-wing configurations. The optimal values refer to the optimisation study performed by also including camber design parameters
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Fig. 19
figure 19

Comparison of baseline and optimised airfoils

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A more detailed comparison in terms of wave drag, induced drag, lift-to-drag ratio (without the contribution of viscous effects) and Oswald efficiency, among the baseline and the optimized box-wing is presented in Table 5. The optimization process successfully achieved an important improvement of the overall aerodynamic performance at each design point, up to a total of 25.6 drag counts (d.c.) reduction at CL=0.55, and in terms of L/D we have a total improvement up to 2.8 counts. The optimal design achieves the best performance by reducing both the induced drag, with very high values of Oswald efficiency e~1.43, as well as the wave drag component by almost a factor 3. Based on these Euler results it seems that the configuration achieves the desired overall aerodynamic performance improvement, but high-fidelity RANS CFD runs are needed to provide a refined assessment of these optimization results.

Table 5 Comparison of wave drag (CDw), induced drag (CDi), total efficiency (L/D)Euler and Oswald efficiency (e) for baseline and optimized MS1.2 configurations according to Euler computations
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The RANS-based comparison of the different MS1.2 designs, performed according the same computational approach described in Sect. 3.4.1, is presented in Fig. 20; in terms of the lift coefficient, Fig. 20-left, the optimized configuration features a slightly increased zero-lift AoA and a better overall performance in terms of total drag, as shown in Fig. 20-right.

Fig. 20
figure 20

RANS assessment of Euler optimised MS1.2 configurations w.r.t. to corresponding baselines. Lift (left) and far-field drag polar curve (right)

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Such improvement could be better investigated by inspecting and comparing the different drag contributions. From the drag breakdown it is evident that the leading improvement comes from the induced drag reduction associated with the relevant gain in terms of Oswald efficiency, as shown in Fig. 21-left, which is not only increased up to its value of 1.4, but also extended over a wider range of \({\text{C}}_{\text{L}}\), i.e. 0.4 ≤ \({\text{C}}_{\text{L}}\)≤ 0.7. In addition, improvement comes from the reduction of the wave drag component (see Table 5), and thus from the associated favourable effect in terms of viscous pressure drag, with the delay of shock-induced separation phenomena. In Table 6, a quantitative comparison of the different drag contributions from Euler and RANS computations for \({\text{C}}_{\text{L}}\)=0.5 is reported.

Fig. 21
figure 21

RANS assessment of Euler optimised MS1.2 configurations w.r.t. to corresponding baselines: (left) Oswald efficiency (right) Aerodynamic efficiency

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Table 6 Comparison between Euler and RANS based prediction of the aerodynamic performance at CL = 0.5 for optimised MS1.2 designs
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4 Vertical Tailplane Aerodynamics

4.1 Investigation on Transonic Performance of V-Tail

As shown by the results of the high-fidelity analyses presented in Sect. 3.4.2, the integration of the vertical tails negatively affect aerodynamic performance (i.e. L/D); the ffd analysis and the accurate post-processing performed by ONERA, have highlighted that this increase in drag is not only associated to the increment in wetted surface due to the addition of the V-tail, but mainly to a complex aerodynamic phenomenon of three-dimensional transonic interaction that involves the generation of severe shock waves with consequent massive boundary layer separations within the channel delimited by the fins, the rear wing and the fuselage. This complex phenomenon definitely needs to be mitigated to avoid degradation in aerodynamic performance of the box-wing in cruise, and given the extremely complex, three-dimensional, viscous and compressible nature of the flow, it must be investigated and addressed with high-fidelity simulations, such as CFD RANS analyses as described in Sect. 3.1.

Following the considerations proposed in Sect. 3.4.2, an additional part of work has been planned to:

  1. i)

    Preliminarily evaluate the nature of the aerodynamic problem, the expected critical points, and the design parameters that can be considered to improve the aerodynamic field in the V-tail area;

  2. ii)

    Set up an optimisation of the V-tail assembly based on high-fidelity compressible RANS solvers, with the aim of designing the vertical tailplane free from the transonic issues highlighted in the previous investigations.

Phase i) was conducted by UniPi on the reference box-wing with the Euler solver of Ansys Fluent; these analyses do not include the possibility of simulating viscous effects. Therefore, these are inadequate for identifying flow separations induced by strong shock waves. These models were used only for preliminary purposes to extract useful information for the local design of the fins group in transonic conditions, reducing computational cost. As an example, some Mach contours for the reference configuration (flight condition: Mach equal to 0.79, altitude 11,000 m) are shown in Fig. 22. Notwithstanding the different CFD modelling, qualitative correspondences can still be noted with the results proposed in Sect. 3.4.2. These analyses identified a relevant criticism of the rear wing-fins-fuselage assembly (hereafter referred to as the V-box) in transonic, both with regard to the need to refine the design of the fillets between the different components, and to the significant aerodynamic interference between the different surfaces, but also to the prominent “channel effect” (see Fig. 22-centre). In particular, it is observed that inside the V-box there is a longitudinal station where the flow strongly accelerates and therefore it may reach sonic conditions in the whole channel. These combined aspects cause a drastic increase in drag in cruise conditions. To limit these effects airfoils shapes, chords, twist, dihedral and sweep of the fins, together with an accurate design of the fuselage and rear wing fillets, have been considered.

Fig. 22
figure 22

Mach contours in the V-box area (Euler solver)

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4.2 CFD-Based Optimization

The next step was to carry out an optimisation of the V-box with the aim of minimising aerodynamic drag in cruise conditions [106]; this activity was conducted by means of the collaboration with Cubit. Since the simulative models to be used in this phase have very long computational times, and since an optimisation procedure must evaluate hundreds of different combinations of design variables, it was decided to use a simplified aircraft geometry, to focus on the best grid resolution in the V-box area, while eliminating areas of limited interest in the aerodynamic problem investigated. Specifically, the simplified box-wing aircraft geometry does not have a front wing or vertical tip-wings. An initial simulation was performed on this model to characterise the starting point of the following optimisation. A grid of approximately 40 million cells was used for the compressible steady RANS simulation set up according to the same specifications described in Sect. 3.1. The post-processing of the reference simulation showed the formation of the sonic barrier within the V-box (Fig. 23) and confirmed the importance and the need for a more in-depth study of the channel, specifically to seek solutions to limit the accelerations within the channel, with the aim of mitigating (or eliminating) the shock waves inside the V-box.

Fig. 23
figure 23

Mach (left) and \({\text{c}}_{\text{p}}\) (right) for the baseline V-box

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The next step was then to set up the optimisation problem; specifically, the design variables were defined as: the twist angle at the fin root \({\uptheta }_{{{\text{root}}}}^{{{\text{fin}}}}\) and tip \({\uptheta }_{{{\text{tip}}}}^{{{\text{fin}}}}\), the fin dihedral angle \({\Gamma }^{{{\text{fin}}}}\), the fillet radius between the fin and the rear wing \({\Phi }^{{{\text{rw}}}}\), and the fin and the fuselage \({\Phi }^{{{\text{fus}}}}\), the fin taper ratio \({\uplambda }^{{{\text{fin}}}}\), the lateral position of the leading edge of the fin root section \(\Delta Y_{{{\text{root}}}}^{{{\text{LE}}}}\), and its longitudinal position \(\Delta X_{{{\text{root}}}}^{{{\text{LE}}}}\); the design variables bounds are reported in Table 7.

Table 7 V-box optimization design space
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Optimising the shape of the airfoil also has a fundamental impact on the transonic problem under investigation; however, introducing a number of parameters related to the shape of the airfoil would have resulted in a large number of configurations to be analysed that were incompatible with the computational times; thus, it was decided to optimise the airfoil a priori, simulating the two-dimensional conditions of the transonic flow around the V-box. The optimised profile was then applied to the vertical tailplane subject to optimisation. The optimisation procedure requires the parameterisation of the geometry, and the automatic handling of the geometry modifications throughout the process; this was done using the software ModeFrontier [87], which is capable of autonomously handling the design variables and generating the different configurations during optimisation. Due to the enormous number of high-fidelity simulations to be performed, the mesh used is more coarse than in the initial case, with its size having been reduced from 40 million cells to approximately 8 million. The objective function to be minimised is the drag coefficient of the fin \({\text{C}}_{\text{D}}^{\text{fin}}\), and genetic algorithms are employed for such purpose. Table 8 shows the values of the optimal design variables for the V-box found at the end of the procedure.

Table 8 Optimal design variables
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The analysis of the results of the optimal design variables shows that the solution that minimises the \(C_{{\text{D}}}^{{{\text{fin}}}}\) has a V-box that tends to widen the internal channel; the solution agrees with the analogy of a one-dimensional flow in a convergent-divergent duct, in which the opening of the duct leads to lower accelerations. This is mainly achieved by the fin dihedral angle \({\Gamma }^{{{\text{fin}}}}\), which reaches its maximum limit value, and which cannot be increased for reasons of geometric compatibility; furthermore, excessively high values of \({\Gamma }^{{{\text{fin}}}}\) would lead to reductions in yaw stiffness at low speeds. On the other hand, the lateral position of the leading edge of the fin root \({\Delta Y}_{{{\text{root}}}}^{{{\text{LE}}}}\), instead of leading to a widening of the duct, leads to a positioning of the fin root in a zone of reduced curvature of the fuselage; indeed, when the leading edge of the fin is laterally positioned farther from the centre-line of the fuselage, it results in an intersection occurring in a more curved area. This, leads to a significant increase in accelerations within that particular region. The longitudinal position of the leading edge of the fin \(\Delta X_{{{\text{root}}}}^{{{\text{LE}}}}\) tends to maximise the geometric sweep angle of the lifting surface. The improvement in terms of \(C_{{\text{D}}}^{{{\text{fin}}}}\) with respect to the baseline configuration is equal to 53%; Fig. 24 shows the Mach and \(c_{p}\) maps for the optimised configuration; the new configuration shows a great mitigation of shock waves inside the channel and the elimination of the sonic wall that affected the previous configuration. These evaluations were performed by doing a high-fidelity verification analysis on the optimised configuration, with the most refined grid (about 40 million cells), to have a robust comparison with the baseline.

Fig. 24
figure 24

Mach (left) and \({\text{c}}_{\text{p}}\) (right) for the optimized V-box

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Figure 25 shows the Mach number contours on the symmetry plane, with the maximum value of the visualization scale equal to 1, to have a more effective reading of the flow conditions in the baseline and optimised configuration. Note the absence of the shockwave that was present in the baseline configuration (and which appeared as a marked line between the rear wing and the fuselage); the result is very significant, and implies not only a drastic reduction in aerodynamic drag, but also an improvement in the effectiveness of the rudder controls, which are typically placed on the fins, and the elevator, which is positioned at the trailing edge of the rear wing.

Fig. 25
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Mach contours in the longitudinal plane: baseline (left) and optimized (right)

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5 Conclusions

This article describes the design stages of the aerodynamic development of the unconventional box-wing lifting configuration. This work aimed to underline the significance of the collaborative approach between different institutions and expertise when dealing with complex and challenging problems, emphasising the project organisation that took place within the European project PARSIFAL. The combined competences and computational resources between the University of Pisa, ONERA, and Cubit company made it possible to set up a detailed and exhaustive aerodynamic analysis and design study, with the aim of investigating in detail the aerodynamic peculiarities of the box-wing configuration. The scheduling and implementation of aerodynamic evaluations on a multi-level, multi-fidelity scheme allowed an exhaustive analysis of the aerodynamic characteristics of this unconventional architecture, starting from the conceptual exploration of the influence of the design variations on the aerodynamic and aeromechanical properties, up to the high-fidelity analysis of the performance in the transonic regime. This multi-disciplinary approach made it possible to evaluate possible critical aspects regarding flight stability, to general performance in terms of lift-to-drag ratio in all operating conditions, and to possible local critical issues in transonic flight relating to the specific design of the box-wing configuration. The design and assessment techniques employed in this context have made it possible to identify some important general findings regarding the design of this lifting system; concerns regarding the longitudinal and lateral static stability of this aircraft, which properly designed allows aeromechanical constraints to be satisfied in all flight conditions, have been resolved; advanced shape optimisation techniques have been applied to extract the maximum performance potential of this configuration, reaching span efficiency values close to those estimated by theory; finally, local problems related to transonic flight have been isolated and analysed, and specific solutions have been consistently identified to remove these critical issues. The general results proposed by the aerodynamic design and analysis described in this paper were then used as input in the higher context of the PARSIFAL project, to allow an overall impact analysis of the introduction of this aircraft in commercial air transport sector. The overall outputs of the PARSIFAL project showed that the operation of the optimised box-wing aircraft enables a significant reduction in fuel consumption per passenger-kilometre with respect to conventional competitors and, consequently a considerable decrease in its climate impact. The increased lifting capabilities of this configuration, together with its high aerodynamic efficiency, allow to meet the expected demand growth for air traffic in the coming decades, and to reduce the related share of greenhouse emissions, thus presenting itself as an excellent candidate for aviation of the near future.