4.1.1. NLR 7301 Baseline Airfoil
The shape design optimisation of the NLR 7301 baseline airfoil is undertaken in order to generate a morphing airfoil that reaches, or even surpasses, the aerodynamic performance of the flapped configuration in terms of maximum lift, whilst avoiding the drag being significantly penalised. As a result, the morphing airfoil will be capable of operating at take-off and landing performance, which requires a substantial increase in lift. The previous step in the NLR 7301 shape design optimisation is to obtain its aerodynamic capabilities at the same angle of attack and flow conditions as the conventional configuration, exhibited in
Table 2, for
ISA Sea Level conditions.
The subsequent results are used to establish a reference in terms of initial aerodynamic coefficients before the shape optimisation. In addition, the computational set-up and the computed flow are utilised as the starting solution for the optimisation design loop.
The flow is computed using the previously validated symbiosis between the compressible RANS equations and the Spalart–Allmaras turbulence model, in a steady simulation. The new physical domain is meshed through a design loop that is subjected to the same quality requirements as the conventional high-lift configuration from
Section 3.1. Consequently, and proceeding analogously, the domain has been meshed using a C-grid typology composed of quadrilateral elements with a total of
nodes, depicted in
Figure 8. The NLR 7301 surface has been discretised by 351 nodes. The domain’s boundaries are located 14 chord lengths downstream and 10 chord lengths upstream, in the upwards and downwards directions. A wall grid spacing of
ensures sufficient grid resolution to fulfil the Spalart–Allmaras condition (
).
Again, the convergence criteria are set to a density residual reduction of four orders of magnitude, which has been achieved by a total of
iterations and a Courant–Friedrichs–Levy number of
, as depicted in
Figure 9. The resulting surface pressure distribution from
Figure 10 clearly illustrates the difference in aerodynamics between a single element and a multi-element configuration, if compared to the surface pressure distribution from the NLR 7301 plus trailing-edge flap (
Figure 5). If one focuses on each of the main elements’ upper surfaces, it can easily be seen how the leading-edge pressures from the multi-element configuration are greatly increased as a result of the flap’s gap influence in comparison with the single-element’s, achieving their peaks close to
and
, respectively. Smith [
44] identified this effect as the circulation effect: the downstream element causes the trailing edge of the adjacent upstream element to be in a region of high velocity that causes a flow inclination on its rear, which effectively increases its angle of attack. Such an effect induces considerably greater circulation on the forward element, and hence greater lift.
The resulting aerodynamic capabilities in terms of lift, drag, and moment coefficients are summarised in
Table 3, where the subscript
b refers to the baseline airfoil.
At this point, the main objective of the shape design optimisation of the NLR 7301 baseline airfoil arises by itself. Considering the lift coefficient of the conventional high-lift configuration from
Section 3.1 (
), the outcome of the optimisation design loop (
) will be achieving or surpassing a positive lift increment of
with respect to its initial value, thus demanding a considerable aerodynamic lift enhancement.
4.1.2. NLR 7301 Shape Design Optimisation
This objective, however, is not allowed to be reached at any cost since realistic designs are sought, thereby suggesting the need to introduce several geometrical constraints to induce controlled geometry deformations. This way, the resulting optimised morphing airfoil will be as close as possible to a structurally feasible design from a short-term future point of view.
In order to address the structural limitations that ought to be imposed in the subsequent designs, one must consider the structural layout of commercial jets along their wingspan, essentially composed of a wing-box structure, ribs, and secondary structures such as the mechanisms to actuate the control surfaces and the high-lift devices, all housed by the wing skin. Nowadays, the wing box represents the main structural element and has been, over time, carefully designed, with significant design changes. Considering the sensitivity of this matter, it becomes a necessity to define a virtual wing box that must not be subjected to severe deformations during the shape optimisation procedure.
This prevents the present work from exploring the design space to the full extent where unconstrained designs might, eventually, reach the targeted lift coefficient () parallel to more competitive lift-to-drag ratios. On the contrary, these designs might also imply unreachable geometries from a structural point of view, thus not fulfilling the present work’s design requirements.
As introduced in
Section 2.3, FFD is the preferred parametrisation method to be used, allowing us to define an FFD box with a set of 22 design variables, as illustrated in
Figure 11. It has been found that for the present work’s cases, increasing the number of design variables adds complexity to achieving controllable and smooth deformations, as well as increasing the computational time cost. A wing box defined within 20% and 60% of the chord is chosen for this work to test the morphing capabilities. Chordwise wing-box positions are usually defined during the structural design process of a three-dimensional wing and typical values are within 12% and 71% [
45,
46,
47]. It is at this point where the potential of FFD arises, where a virtual wing box is built by freezing the design variables that are red-coloured, leaving the rest (blue-coloured) in charge of the shape design optimisation.
Nevertheless, and in spite of the opposite nature of fixed and non-fixed design variables, the FFD formulation is built to ensure smooth surface continuity between their domains; thus, design irregularities are avoided. A last remark on how the FFD box is built addresses the margins between its boundaries and the NLR 7301 baseline airfoil, which are narrowed so that the differential between each of the control points’ (or design variables) locations and the airfoil nodes is minimised, and consequently the relative coordination is increased.
Observing the non-fixed design variables from
Figure 11, it is clear that the airfoil’s leading and trailing edges have the responsibility of pursuing the positive increment of lift via proper geometry deformations. In conventional three-element high-lift configurations, this is achieved by deflecting leading-edge slats and trailing-edge flaps, which directly contribute to increasing the wing’s chord (and sectional area) and camber.
Due to the direct proportionality between the sectional area and lift, adding slats and flaps to the main element results in a positive increment of lift. At the same time, an increase in camber will inherently result in augmenting the surface pressure difference between the suction and pressure sides of all elements, also leading to an increase in lift. The deflection of trailing-edge flaps causes the
curve to move upwards and towards the left as the deflection increases, which is translated into a higher maximum lift coefficient (
), as well as higher
values for the same angle of attack. Regarding the deflection of leading-edge slats, their effect upon the
curve is materialised by increasing
and the operational angles of attack by delaying stall, for an unchanged slope of the curve. Knowing this, it may seem obvious to address these lift-increasing, stall-retardant features by proper deformations of the design variables. However, morphing an airfoil alone may imply that these aerodynamic benefits are not fully obtained, essentially due to the lack of carefully designed slat and flap lifting surfaces, as well as the lack of the slots between the elements. Consequently, it becomes imperative for the present work to understand Smith’s conclusions [
44] on the aerodynamic influence between the elements of a three-element configuration, in order to give some reasoning on which aerodynamic divergences to expect between the ultimate morphed airfoil and a multi-element configuration. The first one is the slat effect, in which the pressure peak of the downstream element is considerably reduced by the action of the circulation on the upstream element, consequently reducing the adverse gradients and allowing the boundary layer to much better negotiate the modified pressure distribution. In turn, the second effect (previously observed in
Section 4.1.1), the
circulation effect, defines the lift increase of the upstream element not by a higher angle of attack, but by a greater circulation induced by the downstream element. The trailing edge of the first is at a region of high velocity that effectively increases its angle of attack. As a result, more favourable pressure distributions are obtained, leading to an increase in lift and, if compared to the very same airfoil attempting to provide the same lift, with a lower nose pressure peak—again, alleviating the pressure gradients on the boundary layer for a more competitive performance. It is precisely this high-velocity region that provides a stall-retardant effect, called the
dumping effect. Here, the boundary layer of the upstream element discharges into a flow with a higher velocity than the free-stream velocity, partially relieving it from the pressure rise imposed as it travels along the surface. As a consequence, separation problems are delayed, and higher lifts are permitted. In addition, as the boundary layer from upstream elements is dumped at velocities higher than the free-stream, the final deceleration of the wake to free-stream occurs out of contact with a wall, the so-called
off-the-surface pressure recovery. In other words, the pressure rise at the wake of upstream elements is diminished, hence transporting the greatest pressure recovery to free-stream conditions towards the trailing edge of the whole set, and out of the domains of the elements’ surfaces, helping to avoid early separation on the surface. This is a more efficient way than a deceleration in contact with a wall due to the off-the-surface deceleration being able to endure pressure rises in much shorter distances, compared to boundary layers. Lastly, it must be noted that properly designed multi-element airfoils develop their own boundary layer without any merging between adjacent elements, the
fresh-boundary-layer effect, also being the present work’s case. This works as a safety measure against separation, as breaking up the surface into several elements favours the development of thin boundary layers, which can withstand greater pressure gradients than thicker ones.
It may seem reasonable to categorise these lift-empowering, stall-retardant effects as boundary layer control effects. In a way, this is precisely what the aerodynamics of a multi-element airfoil provide thanks to the gaps, which widen the aircraft’s operational range and lead its performance to be feasible at higher angles of attack without boundary layer separation.
This gives the present work a crucial hint of what is to be expected from a morphed single element. If a substantial lift increase is to be achieved by means of the non-fixed design variables (refer to
Figure 11), there exists no other way than a slat and flap-like deformation, aiming to increase the curvature as well as the chord and sectional area. However, the lack of slots and the aforementioned favourable aerodynamics will, most certainly, have a negative impact concerning stall. In this case, the absence of any external aid will force the boundary layer to develop under harsher conditions, whilst still having to fulfil our challenging demands.
Conclusively, considering the high probability of a partially separated region, it will be this work’s duty to minimise its impact on the ultimate morphed airfoil’s lift-to-drag ratio.
4.1.3. Optimisation Geometrical Constraints
Near-future structural feasibility will not only be achieved by the imposition of a virtual wing box but also by geometrical constraints, which will orchestrate the behaviour of the design variables through the entire shape design optimisation of the NLR 7301 baseline airfoil. The user’s direct influence upon the design path (or line of search) increases its accuracy towards the desired outcome.
One of the most delicate features to have under control during the whole process is the airfoil’s minimum thickness (
), which is set to be greater than
of the baseline airfoil’s maximum thickness (
):
due to manufacturability reasons. This is directly related to the chord increase that the baseline airfoil will be forced to undertake via a chord constraint. If only surface stretching was imposed, the probability of large areas below the minimum manufacturable thickness would become substantial, especially at the trailing edge, where thickness is already of low values. As a consequence, high aerodynamic loads over these thin areas could lead to a structural failure. In addition, an area constraint is implemented to help support this concept, with which the sectional area will also be forced to increase in order to avoid excessively large thin areas. All these constraints together are intended to emulate the geometrical implications of deflecting conventional slats and flaps.
In order for these concepts to be materialised as constraints, the following crude approach was used: in the conventional high-lift configuration, the flap represents 32% of the NLR 7301’s chord, and an added sectional area of 30%. Analogously to the morphed airfoil, the present work will limit its chord’s positive increment to a maximum value of 32%, while its sectional area will be forced to increase at least half of the flap’s area:
The intentions behind these constraints become now visible. The sectional area is given more freedom to explore the design space compared to the chord, as no upper limitations are set. The expected consequence is a thickness compensation during the surface stretching caused by the chord increase. Again, despite the fact that these are only rough approximations, they allow us to predefine the outer layer of the shape design optimisation’s DNA—the governing global behaviour of the deformations.
4.1.4. NLR 7301 Morphing Design
The present work’s target lies amongst the infinite solutions within. If the line of search is to be fully and accurately parametrised, the core (or inner layer) of the deformations’ behaviour must be set and this implies coding the non-fixed design variables.
Each of the design variables depends upon its own associated bi-dimensional vector of scale factors in the horizontal and vertical directions, respectively. Their value acts directly upon the amplitude and direction of the surface deformation to be applied at each optimisation step, where a higher absolute value of a scale factor comes with greater achievable deformations.
The definitive set of scale factors was obtained as a result of the reasoning presented in
Section 4.1.2, where the line of search of the shape optimisation was to be set so as to emulate a slat
and flap-like
deployment for curvature increase, ultimately having defined them for each design variable, as presented in
Table 4, where
corresponds to their indexed position along the
x axis, from leading to trailing edge, while
along the y axis or, in other words, corresponding to the suction and pressure sides, respectively.
The partition between leading and trailing edge becomes clearly noticeable with the gap spanning from , which corresponds to the virtual wing box. The leading edge has been given a higher absolute value of scale factors for both directions if compared to the trailing edge, which is translated into more freedom of movement. The reason is a higher concentration of area in the leading edge that can be exploited, and hence a lower probability of minimum thickness constraint violation or large low-thickness areas due to surface stretching. In line with this reasoning, the trailing edge design variables have gradually been given a decreasing scale factor amplitude as they approach the airfoil’s low-thickness trailing edge, thus implementing a more restrictive local movement if compared to the leading edge.
The definition of the geometrical constraints plus the design variables (
) sets the foundations that lead to this work’s targeted designs. Thus, together with the lift coefficient
being set as the objective function to be maximised, the mathematical definition of the SDO can now be displayed:
As previously introduced, the starting solution for the shape design optimisation is the NLR7301 steady RANS case from
Section 4.1.1. The same computational approach has been used for the intermediate stages during the optimisation process, a coupling between the compressible RANS and Spalart–Allmaras turbulence model in a steady-state simulation. The morphing evolution from the baseline airfoil up to the definitive morphing airfoil is depicted in
Figure 12.
A comparison between the geometrical parameters from the baseline airfoil NLR7301 and the definitive morphing airfoil is shown in
Table 5.
The relative chord and area increase of the morphed airfoil are and , meaning that the geometrical constraints as well as the targeted morphing behaviour have been respected: a comparable overall chord increase to from the flap deployment, and an area increase that has surpassed the set as the minimum in the geometrical constraints.
From a general point of view, smooth deflections have been achieved towards the desired directions thanks to the scale factors’ amplitudes and their sign convention, fairly emulating the deflection of a slat and flap devices for curvature increase. Concerning the leading edge, applying less restrictive scale factors has not led to problematic low-thickness areas, while the trailing edge, even subjected to a more restrictive margin of movement and not having surpassed the minimum thickness constraint, will certainly be the most challenging one from a structural point of view due to a larger low-thickness area, as it will have to withstand the aerodynamic loads without a structural collapse.
Regarding the aerodynamics,
Figure 13 alongside
Table 6 show how the lift and drag coefficients have evolved during the shape design optimisation from start to end and at the different intermediate stages of the process. The values shown are the last values of each of the iterative processes for every design, the convergence criteria of which correspond to a logarithmic reduction in the density equation residual of four orders of magnitude.
On the other hand, considering that unsteady features were highly likely to arise and not properly captured by a steady formulation, a maximum of iterations per design was set in case the convergence criteria were not fulfilled.
The trend in the lift curve shows the desired behaviour, boosting its climb rate from the second design up to a maximum value of
, which is considered valid for comparison with the conventional high-lift configuration and its numerically obtained Cl value (
). This substantial lift increase is accompanied by a similar trend in the drag curve, which starts increasing its rate of climb from the third to the fourth stage of the SDO, achieving a maximum value of
, a dramatic increase if compared to the NLR7301 baseline airfoil, and an indicator of an undesired unsteady feature due to the high curvature of the last designs. Both aerodynamic coefficients are plotted in
Figure 14.
Clear oscillations on both coefficients are observed, meaning that the morphed airfoil hides unsteady features that cannot be described by the steady formulation or to make the solution converge: the mean value from each of the oscillating aerodynamic coefficients is
and
for the lift and drag, respectively. For this reason, the natural step that followed these results was to address the unsteadiness via an unsteady simulation to properly describe it. As previously introduced, a dual-time stepping strategy was used. The sampling frequency was set at
Hz, equivalent to implementing a time-step of
ms. The solution reached a stable behaviour after a time-span of
s and outputted slightly different mean lift and drag coefficients, displayed in
Table 7.
The unsteady simulation revealed an almost unchanged lift coefficient and a more penalising drag coefficient.
Figure 15 shows a colour map of the Mach distribution around the morphed airfoil, as well as the instantaneous flow streamline pattern at the last time-step.
The unsteady RANS and Spalart–Allmaras turbulence model coupling predicts the presence of a pair of oscillating mean separation bubbles at the morphed airfoil’s trailing edge. Hence, it becomes now evident that this unsteady feature is the cause of a dramatic drag increase. The hypothesis from
Section 4.1.2, where a locally stalled feature on this area was predicted, materialises. It is imperative to mention that no vortex shedding occurs and the pair of recirculating mean bubbles remain oscillating but always on the trailing edge. This behaviour might show the limitations of the Spalart–Allmaras turbulence model against large unsteady features such as these.