Nomenclature

b

span

$C_D$

drag coefficient

$C_L$

lift coefficient

$C_M$

pitching moment coefficient

$C_P$

pressure coefficient

c

chord

CG

center of gravity

CTW

conventional tube-and-wing

D

drag

$\mathcal{D}$

design weights

$\mathcal{J}$

objective function

L

lift

$L/D$

lift-to-drag ratio

M

mach number

MAC

mean aerodynamic chord

MFW

maximum fuel weight

MTOW

maximum takeoff weight

MZFW

maximum zero fuel weight

N

number of grid nodes

nmi

nautical miles

OEW

operating empty weight

OML

outer mold line

RANS

Reynolds-Averaged Navier Stokes

SBW

strut-braced wing

t

thickness

TSFC

thrust specific fuel consumption

u, v, w

parametric coordinates

x, y, z

Cartesian coordinates

2.0 Approach to evaluating aircraft fuel efficiency

Representative aircraft concepts are developed through the application of a conceptual MDO framework [Reference Chau and Zingg22], which consists of low-order models for aerodynamics, structures, weight and balance, propulsion and performance – providing a means for capturing most of the first-order effects that are relevant to cruise performance and block fuel. The objective of each MDO problem is to achieve a minimum block fuel design over a nominal range mission at a cruise Mach number of 0.78. This represents a typically flown mission for the Embraer E190-E2, and is separate from the design missions used to size the design weights of each aircraft, as described in Ref. [Reference Chau and Zingg22].

For the strut-braced-wing concept, design variables are focused on the sizing and optimisation of the wing, strut, horizontal and vertical tails, propulsion system, and operating conditions. For the wing and tail systems, design variables include the root and tip chord lengths and thickness-to-chord ratios of each wing, strut and tail segment. Constraints on these degrees of freedom include a maximum design wing loading, which places a lower bound on the wing area for takeoff and landing considerations based on the Embraer E190-E2; a minimum fuel volume, which maintains a minimum wing and strut outer mold line (OML) for the fuel tanks based on the maximum fuel weight (MFW); and minimum tail volume ratios based on a reference T-tail design. Other wing and strut parameters such as span, sweep and dihedral, as well as the attachment locations of the strut with the wing and fuselage, are not included since they are assumed to have a strong dependence on structures and aeroelastic phenomena such as flutter, going beyond the capabilities of the low-order models. These parameters are set based on reference concepts found in Bradley et al. [Reference Bradley, Droney and Allen8].

Another design variable for the strut-braced-wing concept is a maximum thrust parameter for resizing the propulsion system, which is constrained by a minimum thrust-to-weight ratio also based on the Embraer E190-E2, and top-of-climb excess thrust requirements. Initial cruise altitude design variables are also included for each design mission, which help determine the optimum cruise $C_L$ values for minimum block fuel over the nominal mission. These design variables are constrained by a number of disciplinary effects such as the thrust required at top-of-climb, and tradeoffs with climb fuel.

For the conventional tube-and-wing configuration, a concept is developed as-drawn, namely, the wing, horizontal and vertical tails, propulsion systems, and operating conditions are defined as closely as possible to the Embraer E190-E2. Given that the wing thickness-to-chord ratios are not publicly available, however, these are included as design variables and are driven toward a minimum block fuel solution while subject to a minimum fuel volume constraint.

Other design considerations for the strut-braced-wing concept include an assumption that the strut produces minimal lift, and hence is developed primarily as a structural member that generates as little drag as possible. The structures of the wing and strut are also constructed from composite materials, which is a technology that is readily available and provides an advantage over the aluminum wing composition of the Embraer E190-E2-like conventional tube-and-wing. Technologies such as natural laminar flow, active flow control, and advanced propulsion systems are not considered, given the objective of isolating the benefit offered by the configuration technology when assuming current technology levels. Additional details on the conceptual design of each aircraft are provided in Chau and Zingg [Reference Chau and Zingg22].

These concepts are used to create 3D models of each aircraft that include the wing (and strut), fuselage, and horizontal tail. This allows for a multipoint optimisation of each wing and tail system that captures the dominant aerodynamic effects, including the lift and interference drag from the presence of the fuselage. The optimisation of each aircraft is performed for a range of lift coefficients and Mach numbers that include and surround the nominal cruise point. To enable a more exploratory approach to design optimisation, a more general formulation of the constraints is employed to maintain the feasibility of each aircraft. For example, minimum wing (and strut) volume and minimum maximum thickness-to-chord ratio constraints are defined, which together act as surrogates for maintaining a minimum structural depth of the wing (and strut) that is independent of the internal structural topology. These constraints also ensure that there is sufficient volume for containing the fuel tanks required to store the MFW.

A construction of the full aircraft performance is then accomplished through a synthesis of the high-fidelity aerodynamic estimates with low-fidelity approximations from the components not modeled in the RANS simulations. The updated cruise fuel approximation can then be combined with the approximations for warmup, taxi, takeoff, climb, descent and landing to provide an estimate of the mission block fuel for each aircraft.

3.0 Aircraft characteristics

Aircraft concepts representing the strut-braced-wing and conventional tube-and-wing configurations in the regional jet class, herein referred to as the SBW100 and CTW100, respectively, are developed based on the design missions and top level aircraft requirements of the Embraer E190-E2. This includes a design range and payload of 3,100nmi and 104 passengers, respectively. As described in Section 2, however, evaluations of cruise performance and block fuel consider a nominal mission with a range of 500nmi, the design payload and a Mach number of 0.78. Planform views of the aircraft concepts are shown in Fig. 1.

Figure 1.

Aircraft concepts developed through conceptual MDO [Reference Chau and Zingg22].

Table 1 provides an overview of the aircraft characteristics and their nominal performance based on the low-order models. Compared to the CTW100, the SBW100 has the advantage of a 60% larger wing aspect ratio, which trades efficiently against penalties in added wing wetted area. In order to realise the full potential of this high aspect ratio wing, however, the SBW100 must operate at a much higher cruise $C_L$ . For a given wing loading, this results in a significantly elevated cruise altitude. It is important to note, however, that achieving the optimum lift coefficient for maximising the lift-to-drag ratio must trade with increased climb fuel and potentially higher fuselage weights as a result of increased cabin pressurisation loads. These factors are accounted for in the conceptual design process, with the design $C_L$ of the SBW100 representing the optimum $C_L$ for minimum block fuel.

The SBW100 also has the advantage of a 24.5% lower wing weight compared to the CTW100 owing to the structural efficiency of the strut-bracing and the benefit of composite structures. This is offset, however, by an 11.9% heavier fuselage, which comes from weight penalties for fuselage-mounted landing gears and a larger horizontal tail moment arm, which translates to larger fuselage bending loads in the empirical equations. Together, the aerodynamic and structural advantages combine to provide an overall reduction in the design weights of the SBW100. For more details on the aircraft concepts developed through the low-order MDO framework, see Chau and Zingg [Reference Chau and Zingg22].

4.0 High-fidelity aerodynamic shape optimisation framework

The high-fidelity aerodynamic design of each aircraft is performed through the application of an aerodynamic shape optimisation framework called Jetstream that has been developed at the University of Toronto Institute for Aerospace Studies. It consists of an integrated geometry parameterisation and mesh-deformation scheme [Reference Hicken and Zingg23, Reference Osusky, Buckley, Reist and Zingg24], a free-form and axial deformation geometry control system [Reference Gagnon and Zingg25], a structured multiblock Newton-Krylov-Schur flow solver for the RANS equations fully coupled with the Spalart-Allmaras turbulent model [Reference Osusky and Zingg26], the discrete-adjoint method for flow- and mesh-dependent gradient evaluation [Reference Pironneau17, Reference Jameson18], and SNOPT [Reference Gill, Murray and Saunders27] for gradient-based optimisation. In the following sections, a brief description of each component is provided.

Table 1.

Aircraft characteristics from conceptual MDO.

^a All operating conditions and cruise parameters are in reference to the start of cruise for the nominal mission.

5.0 Optimisation problem setup

In the following sections, the problem setup for each high-fidelity aerodynamic shape optimisation is presented. This includes a description of the baseline geometries, which not only serve as the initial designs for each gradient-based optimisation problem, but also determine some of the design variable bounds as described in Section 4.2. An overview of the computational grids, which have been developed to balance grid resolution and computational cost, is also provided. Sections 5.3 and 5.4 then present the multipoint optimisation formulations, and the design variables and constraints, respectively.

5.2 Computational grids

For performing aerodynamic analyses, structured multiblock grids are created for each aircraft model, which are characterised by O-H blocking topologies, i.e. O- and H-grid blocking topologies in the near- and far-fields, respectively. For the CTW100, the optimisation-level grid consists of 14.41 million nodes distributed across 558 blocks, while for the SBW100, there are 26.51 million nodes over 1,355 blocks. The difference in the number of blocks, and by extension the number of grid nodes for a similar maximum number of nodes per block, is due to how the number of blocks scales with geometric complexity. The SBW100 grid also includes a local O-O blocking topology surrounding the strut, where the second O-grid layer projects onto the lower surface of the wing. This provides the necessary boundary of surface patches used to decouple the deformations of the wing and strut where they intersect with one another, as described in Section 5.4.

These optimisation level grids have been generated following the medium mesh gridding guidelines of the Fourth Drag Prediction Workshop (DPW), which are representative of current drag prediction standards [Reference Vassberg, Tinoco, Mani, Rider, Zickuhr, Levy, Brodersen, Eisfeld, Crippa, Wahls, Morrison, Mavriplis and Murayama39]. In general, this resolution offers a reasonable tradeoff between computational cost and accuracy, while including sufficient grid resolution for capturing the relevant flow features. In order to obtain mesh-independent aerodynamic functionals for the initial and optimised designs, Richardson extrapolation is performed with the addition of two finer grid levels, L1 and L2, which have two and four times as many grid nodes as the optimisation-level grid L0, respectively. Table 2 provides an overview of the grid characteristics, with off-wall spacings and $y^+$ values in reference to the initial designs.

Table 2.

Grid information. Optimisation is performed on the L0 grid levels, while the L0-L1-L2 grid families are used to perform grid convergence studies.

^a Off-wall spacings are in units of mean aerodynamic chord.

^b Maximum average values across each set of design points.

5.3 Multipoint objective

A common approach to multipoint optimisation for the aerodynamic design of an aircraft is to formulate the objective function as a weighted integral over a range of operating conditions in order to improve robustness at off-design conditions across a range of operating parameters [Reference Buckley and Zingg40]. For example, a weighted integral over the Mach number and $C_L$ design space can be defined as

(1) \begin{equation} \mathcal{J} = \int\limits_{C_{L_1}}^{C_{L_2}}\int\limits_{M_1}^{M_2} \mathcal{D}(M,C_L)\ C_D(M,C_L)\ \mathrm{d}M\mathrm{d}C_L\end{equation}

where $\mathcal{D}(M,C_L)$ are user-defined design weights that can be used to place more priority on nominal operating points that are expected to be encountered more frequently over the operating envelope. This integral can be approximated through a quadrature method given by [Reference Li, Huyse and Padula41]

(2) \begin{equation} \mathcal{J} = \sum\limits_{i=1}^{N_{C_L}}\sum\limits_{j=1}^{N_M} w_{i,j}\ \mathcal{D}(M_i,C_{L_j})\ C_D(M_i,C_{L_j}) \approx \int\limits_{C_{L_1}}^{C_{L_2}}\int\limits_{M_1}^{M_2} \mathcal{D}(M,C_L)\ C_D(M,C_L)\ \mathrm{d}M\mathrm{d}C_L\end{equation}

where $w_{i,j}$ are the weights of the quadrature method. This approach is useful when aerodynamic shape optimisation is used in the context of designing a real and very robust aircraft, especially when the nominal design point is complemented by other design points that provide constraints related to low-speed aerodynamics, high-speed buffet and flutter. In the present paper, however, the objective is to assess the relative performance of the transonic strut-braced-wing configuration while accounting for a range of cruise conditions, especially at higher Mach numbers with respect to the nominal value of 0.78, and while ignoring higher-order effects. For this purpose, it is sufficient and less costly to consider a small and discrete set of operating conditions that includes multiple $C_L$ values and higher Mach numbers. Although point-optimisation at each of the design points can lead to excessively optimistic estimates of the overall performance of a given aircraft, several studies have suggested that robust designs can be achieved with a modest number of discrete operating conditions in a multipoint optimisation problem formulation [Reference Nemec, Zingg and Pulliam42, Reference Kenway and Martins43].

To this end, a five-point multipoint optimisation is considered that consists of the nominal operating point, two operating points subjected to a $\pm 10\%$ change in $C_L$ from the nominal point, and two high-speed conditions with a $+0.03$ change in Mach number from the nominal point at $-10\%$ and $-20\%$ $C_L$ . These design points cover a range of expected operating conditions with an emphasis on those most likely to lead to significant wave drag penalties. The objective function is then given by

(3) \begin{equation} \mathcal{J} = \sum^5_{i=1}\mathcal{D}(M_i,C_{L_i})\ C_D(M_i,C_{L_i})\end{equation}

where the design weights are selected such that the nominal operating condition has a two-fold priority over all other design points. The design points considered are listed in Table 3.

Table 3.

Single-point [Reference Chau and Zingg22] and multipoint optimisation design weights and operating conditions.

5.4 Design variables and constraints

Design variables include the angle-of-attack, as well as geometric degrees of freedom provided by the free-form and axial deformation geometry control systems shown in Fig. 2. For the CTW100, the parameterised wing and horizontal tail are each embedded within FFD volumes with 12 and 4 FFD-volume cross-sections, respectively, each consisting of 11 FFD-volume control point pairs. For the wing, these provide twist and section shape design variables, while for the horizontal tail, section shape design variables are omitted to maintain a symmetric design, and the twist design variables are linked across the FFD volume to provide incidence angle control. Axial curves are positioned at the quarter chord of each FFD volume, and hence each lifting surface, which define the local coordinate system for the twist and section shape design variables.

Figure 2.

Geometry control systems with FFD volume entities in blue, and axial curve entities in fuschia.

A similar setup is used for the SBW100. The FFD volume of the wing consists of 19 FFD-volume cross-sections for the same number of cross-sections per unit span, with additional cross-sections introduced near the wing-strut junction to provide sufficient geometric control for addressing the transonic interference effects. Additional FFD volumes are also included to embed the main and vertical strut surfaces, as well as the transition strut, which connects the end of the main strut to the start of the vertical strut. These strut segments are shown in the inset of Fig. 2(b). As with the CTW100, each FFD-volume cross-section consists of 11 control point pairs, which provide twist and section shape design variables for the wing and strut. The incidence angle of the horizontal tail is also included as a design variable, and axial curves are positioned at the quarter chord of each lifting surface.

A novel junction deformation scheme is also included to enable the optimisation of the wing-strut junction where the geometry control system of the wing and strut intersect [Reference Chau and Zingg22]. Similar junction deformation schemes are also included at each fuselage intersection, where changes in the geometry over the embedded aerodynamic surfaces are propagated across the fuselage patches of each aircraft [Reference Osusky, Buckley, Reist and Zingg24]. This allows for changes to the overall incidence angle of each lifting surface, and is specifically used to define the incidence angle degree of freedom for each horizontal tail. Table 4 provides a summary of the design variables involved in each problem, as well as their bounds.

For simulating steady, level flight, each multipoint optimisation problem includes nonlinear constraints for maintaining a constant lift and zero pitching moment. The pitching moment constraint, which would be considered at all operating conditions if elevators were modeled, is included only at the nominal design point for each aircraft. As mentioned in Section 2, minimum volume and minimum $(t/c)_\mathrm{max}$ constraints are also included, which take the form of nonlinear constraints.

A number of linear geometric constraints are also included to further constrain the design space. For example, the minimum distance between each chordwise pair of FFD-volume control points is limited to 50% of its baseline value, while fixed leading- and trailing-edge constraints are included to prevent shear twist and the translation of each FFD-volume cross-section. Other linear geometric constraints are included to help simplify the design of the strut near the wing-strut junction. These include a constant twist constraint for the vertical strut FFD-volume cross-sections, and a linear interpolation twist constraint across the FFD-volume cross-sections of the transition strut. A linear interpolation twist constraint is also applied across the three wing FFD-volume cross-sections across the wing-strut junction to prevent the optimiser from introducing a very sudden wash-in to compensate for the local loss of sectional lift caused by the presence of the strut. A summary of the linear and nonlinear constraints is provided in Table 5.

Table 4.

Design variable information.

^a SBW100 wing root and vertical strut twist bounds are limited to $\pm3.5$ deg.

Table 5.

Linear and nonlinear constraint information.

6.0 Optimisation results

The SNOPT optimisation histories for each multipoint optimisation problem are shown in Fig. 3. The parameters are Optimality – a measure of the objective function and constraint gradients, Feasibility – a measure of the constraint violations, and the Merit Function, which represents the objective function when constraint violations are minimised. Each optimisation is considered converged when the Merit Function changes by less than a drag count over 10 or more function evaluations, Optimality has been reduced by approximately two or more orders of magnitude, and Feasibility has been satisfied to a tolerance of at least 10$^{-4}$ . For both cases, these requirements are met following 55–60 major iterations or 60–70 function evaluations, with more than 95% of the drag reductions achieved within the first 25–30 major iterations.

In order to obtain accurate predictions of the cruise drag at each design point of each multipoint optimised design, grid-converged $C_D$ values are estimated through Richardson extrapolation, using the L0, L1, and L2 grid levels described in Section 5.2. The data points are shown in Fig. 4, where the convergence of $C_D$ is monotonic for all cases. This is a necessary condition for the application of Richardson extrapolation. Although the errors in $C_D$ associated with the L0 grid level are seen to be significant, $C_D$ converges consistently across each grid level, and the relative performance is maintained as the grids are refined. As such, the multipoint optimised designs obtained on the L0 grid level can be expected to be closely comparable to those optimised on a finer grid level. For comparisons, Fig. 4 also includes grid convergence studies for the single-point optimised designs, evaluated at each design point. The estimates of the grid-converged $C_D$ values provide objective function values of $\mathcal{J} = 217$ counts and $\mathcal{J} = 208$ counts for the single-point and multipoint optimised CTW100 designs, respectively, and $\mathcal{J} = 289$ counts and $\mathcal{J} = 277$ counts for the single-point and multipoint optimised SBW100 designs, respectively.

Table 6 includes the aerodynamic performance of each multipoint optimised aircraft, with results from the single-point optimisations of Chau and Zingg [Reference Chau and Zingg22] also included for comparisons. Based on the high-fidelity models of each aircraft, the results indicate that the multipoint optimised SBW100 provides a 9.8% improvement in cruise $L/D$ and a 13.2% reduction in cruise drag over the multipoint optimised CTW100. Compared to the single-point optimised designs, the multipoint optimums show a similar relative benefit with a 0.3% improvement in both cruise $L/D$ and drag at the nominal design point, despite an increase in $C_D$ by 2 counts for both aircraft.

Figure 3.

SNOPT optimisation histories.

Figure 4.

Grid convergence studies for the single-point and multipoint optimised designs at constant lift, evaluated at each of the five design points. Drag coefficients at $N^{-2/3} = 0$ are obtained from Richardson extrapolation.

Table 6.

Optimised aircraft performance at the nominal design point.

^a Includes only the wing (and strut), fuselage, and horizontal tail contributions.

^b Includes a 5% excrescence drag markup, and profile drag contributions from the vertical tail, nacelles and pylons.

In order to obtain estimates for the full aircraft performance, low-order approximations for the drag of the aircraft components not included in the high-fidelity models are introduced, namely, for the vertical tail, nacelles and pylons of each aircraft. Skin friction drag mark-ups of 5% are also applied to approximate contributions from airframe excrescences. These results are also presented in Table 6 and indicate a 13.1% improvement in cruise $L/D$ and a 15.7% reduction in cruise drag at the nominal design point for the multipoint optimised SBW100, relative to the similarly optimised CTW100. Compared to the single-point optimised aircraft, this represents a 0.2% improvement in both relative aircraft cruise $L/D$ and drag. These results suggest that a low drag strut-braced wing can be achieved over a range of cruise conditions, especially since the performance benefits appear to be robust to off-design conditions at higher Mach numbers and lift coefficients.

For approximating the block fuel for each nominal range mission, estimates of weight and TSFC are calculated through the low-order models, along with the fuel contributions from warmup, taxi, takeoff, climb, descent and landing, which are assumed to remain constant. This provides a block fuel savings of 7.8% for the multipoint optimised SBW100 when compared to the Embraer E190-E2-like CTW100. Compared to the aircraft designs obtained from single-point optimisation, this represents a further relative reduction of 0.2% since the drag increase experienced by the CTW100 when designed for multiple cruise conditions is marginally higher than that of the SBW100.

Comparisons of the drag performance between the single-point and multipoint optimised designs over the five design points of each aircraft are shown in Fig. 5. Here, it can be seen that the nominal performance degradation of 2 drag counts for both aircraft leads to an improvement in aerodynamic performance at the off-design conditions. For design points 2 and 3, the performance improvements are on the order of 1% and 2.5%, respectively, for each aircraft, while improvements of approximately 10–12% are achieved at design points 4 and 5. This illustrates that for both aircraft, a low drag can be maintained at the off-design cruise conditions, with significant improvements specifically at the high-speed design points, without the introduction of significant compromises to on-design performance.

Figure 5.

Cruise drag performance at each design point of the five-point operating envelope for each optimised design.

Table 7 presents approximations for the full aircraft performance of each aircraft at each of the design points. These results are obtained by adjusting the fixed weight of each aircraft at each operating condition such that the start of cruise $C_L$ matches the $C_L$ of each design point. Changes to the weight of the aircraft due to changes in the fuel required at warmup, taxi, takeoff, climb, descent and landing, as well as to the fuel reserves, are accounted for. Overall, the improvements in cruise $L/D$ and drag at the off-design conditions range from 14–16% and 11–13.5%, respectively. From Fig. 6, the savings in block fuel obtained for the missions associated with each design point remain consistent at around 7–8%.

Table 7.

Multipoint optimised aircraft performance at the on- and off-design operating conditions.

Figure 6.

Multipoint optimised block fuel burn comparisons at on- and off-design operating conditions.

Figure 7 shows the multipoint optimised spanwise lift distributions for each aircraft operating at the nominal design point, with those of the single-point optimised designs included for comparisons. For all cases, the optimised net spanwise lift distributions are elliptical in form but shifted inboard due to the presence of the trim constraint and the presumed avoidance of high outboard sectional lift coefficients that can lead to increased wave drag penalties. Moreover, the multipoint optimised spanwise lift distributions feature further inboard shifts, which are compensated by modest increases in the negative lift over each horizontal tail. Of interest, however, is that for the multipoint optimised SBW100, the inboard shift is the result of a moderate increase in positive lift over the strut, which allows the main wing to reduce its loading. This feature likely enables an improvement in performance over the high-speed design points where the cruise $C_L$ values are much lower, while aiding in reducing the overall wing loading for the high $C_L$ design point. Such a benefit suggests that lifting struts are aerodynamically favourable for transonic strut-braced wings, and may explain in part why Boeing has opted to incorporate and emphasie such a feature when transitioning from their Mach 0.70 design [Reference Bradley, Droney and Allen8] to their Mach 0.745 [Reference Droney, Sclafani, Harrison, Grasch and Beyar12] and Mach 0.80 variants [Reference Harrison, Beyar, Dickey, Hoffman, Gatlin and Viken13]. It should be noted, however, that lifting struts can lead to structural design challenges associated with the impact of non-axial loads on buckling.

Figure 7.

Optimised spanwise lift distributions computed on the L0 grid level.

Figure 8.

CTW100: Optimised design and flow features computed on the L0 grid level at the nominal design point ( $M = 0.78$ and $C_L = 0.468$ ).

Figure 9.

CTW100: Optimised aerofoil profiles and pressure distributions at two off-design cruise conditions computed on the L0 grid level.

Figure 10.

SBW100: Optimised design and flow features calculated on the L0 grid level at the nominal design point ( $M = 0.78$ and $C_L = 0.682$ ).

Figure 11.

SBW100: Optimised aerofoil profiles and pressure distributions at two off-design cruise conditions computed on the L0 grid level.

Figure 12.

SBW100: Surface pressure contours with inner ((a),left) and outer ((a),right) views of the strut, and aerofoil profiles and pressure distributions at different stations along the wing and strut. Results are for the nominal design point on the L0 grid level.

Additional aerodynamic design features are presented next, first for the CTW100 and then for the SBW100. For the CTW100, an overview of the multipoint optimised design features and flow characteristics at the nominal design point is shown in Fig. 8, with the single-point optimisation results included for comparisons. From Fig. 8(a), it can be seen that the optimiser has allowed some shock surfaces to return over the upper surface of the wing, although these shocks are relatively weak with upstream Mach numbers less than Mach 1.1. This is consistent with the relatively small drag increase of 2 counts when optimising for multiple design points rather than one. Figure 8(c) shows optimised aerofoil profiles and pressure distributions at different spanwise stations across the wing. At the majority of the spanwise stations, the leading-edge suction peaks are reduced to favour more aft loading. At stations $2y/b = 30\%$ , 45%, and 60%, a noticeable decrease in mid-chord loading can also be seen, which may be tied to the more favourable aerodynamic performance at the off-design cruise conditions.

Figure 9 shows the single-point and multipoint optimised aerofoil profiles and pressure distributions at the same spanwise locations, but for design points 2 and 4, which are two of the more challenging off-design cruise conditions. For the $+10\%\;C_L$ design point, the pressure distributions are similar in form to those of the nominal design point, except here, it can also be seen that their features contribute to significant reductions in the strength of the shocks present over the single-point optimised design. For the pressure distributions at the Mach 0.81 design point, it can be seen that the design features help in reducing the suction peak while also reducing the favourable pressure gradient that begins near 50% chord and terminates around 65% chord. This also results in a shock strength reduction.

An overview of the design features and flow characteristics for the multipoint optimised SBW100 is provided in Fig. 10, with the single-point optimised design and flow features included for comparisons. As with the CTW100 optimisations, the multipoint optimised design experiences an increase in wave drag due to the return of shocks over the wing upper surface. Once again, however, the upstream Mach numbers associated with these shock surfaces are all less than Mach 1.1, indicating that they are weak and have a small effect on drag. Shock surfaces present within the junction of the baseline wing and strut presented in Chau and Zingg [Reference Chau and Zingg22] also remain eliminated for the multipoint optimised SBW100, even when accommodations are made to preserve off-design aerodynamic performance. Based on the pressure distributions, it can be seen that the weak shocks over the wing are positioned close to 50% chord. For the strut, the pressure distribution at 15% semispan illustrates an increase in lift, which is achieved through a minor increase in geometric twist.

Figure 11 shows the pressure distributions for the high $C_L$ design point, as well as those of the first high-speed cruise condition. Here, the same trends as those of the CTW100 can be observed, but with the pressure distributions adjusted to compensate for the higher overall lift requirements. In fact, the multipoint optimised pressure distributions suggest a higher wave drag penalty based on the magnitude of the change in pressure coefficient over the steep pressure recovery regions when compared to those of the CTW100. However, the total drag savings offered by the SBW100 suggests that this penalty is outweighed by savings in induced and viscous drag. At the high-speed design point, the pressure distributions are also quite similar to those of the CTW100. As expected, the lift carried by the strut also appears to scale with $C_L$ .

Inner and outer views of the wing-strut junction designs from single-point and multipoint optimisation are shown in Fig. 12. These include surface pressure distributions, as well as aerofoil profiles and pressure distributions at four stations along the wing, and four stations along the transition and vertical strut segments. For clarity, these aerofoil profiles have not been rotated with respect to the angle-of-attack, as has been done in the previous figures. Here, the surface pressures at the nominal cruise condition are generally smooth and free of adverse effects for both designs. This is consistent with the absence of shocks within the junction region shown in Fig. 10. The overall section designs are also quite similar between the two optimisations. For the wing, exceptions include small differences in aerofoil shape toward the leading edge as seen in sections C and D, and an overall reduction in sectional lift, corresponding to the reduction in wing loading observed in the spanwise lift distributions. For the strut, a minor reduction in outwards force distribution can be seen for the multipoint optimum. The overall similarities between the single-point and multipoint optimised designs indicate that both the novel aerofoil shapes, and the outwards force distribution are not necessarily point-designed features, and are likely to be key contributors to low drag transonic strut-braced-wing designs that are more robust to changes in operating conditions.